Handbook on Navier-Stokes Equations Theory and Applied Anylysis 9781536102925

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PHYSICS RESEARCH AND TECHNOLOGY

HANDBOOK ON NAVIER-STOKES EQUATIONS THEORY AND APPLIED ANALYSIS

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PHYSICS RESEARCH AND TECHNOLOGY

HANDBOOK ON NAVIER-STOKES EQUATIONS THEORY AND APPLIED ANALYSIS

DENISE CAMPOS EDITOR

New York

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Copyright © 2017 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. We have partnered with Copyright Clearance Center to make it easy for you to obtain permissions to reuse content from this publication. Simply navigate to this publication’s page on Nova’s website and locate the “Get Permission” button below the title description. This button is linked directly to the title’s permission page on copyright.com. Alternatively, you can visit copyright.com and search by title, ISBN, or ISSN. For further questions about using the service on copyright.com, please contact: Copyright Clearance Center Phone: +1-(978) 750-8400 Fax: +1-(978) 750-4470 E-mail: [email protected].

NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book.

Library of Congress Cataloging-in-Publication Data Names: Campos, Denise (Editor), editor. Title: Handbook on Navier-Stokes equations: theory and applied analysis / Denise Campos, editor. Other titles: Physics research and technology. Description: Hauppauge, New York: Nova Science Publisher's, Inc., [2016] | Series: Physics research and technology | Includes bibliographical references and index. Identifiers: LCCN 2016044251 (print) | LCCN 2016047529 (ebook) | ISBN 9781536102925 (hardcover) | ISBN 153610292X (hardcover) | ISBN 9781536103083 (ebook) | ISBN 153610308X (ebook) | ISBN 9781536103083 Subjects: LCSH: Navier-Stokes equations. | Fluid dynamics--Mathematics. | Mathematical analysis. Classification: LCC QA374 .H295 2016 (print) | LCC QA374 (ebook) | DDC 518/.64--dc23 LC record available at https://lccn.loc.gov/2016044251

Published by Nova Science Publishers, Inc. † New York

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CONTENTS Preface

ix

Chapter 1

Generation of Meshes in Cardiovascular Systems I: Resolution of the Navier-Stokes Equations for the Blood Flow in Abdominal Aortic Aneurysms Alejandro Acevedo-Malavé

Chapter 2

Generation of Meshes in Cardiovascular Systems II: The Blood Flow in Abdominal Aortic Aneurysms with Exovascular Stent Devices Alejandro Acevedo-Malavé

11

Chapter 3

A Computational Fluid Dynamics (CFD) Study of the Blood Flow in Abdominal Aortic Aneurysms for Real Geometries in Specific Patients Alejandro Acevedo-Malavé, Ricardo Fontes-Carvalho and Nelson Loaiza

21

Chapter 4

Numerical Resolution of the Navier-Stokes Equations for the Blood Flow in Intracranial Aneurysms: A 3D Approach Using the Finite Volume Method Alejandro Acevedo-Malavé

31

Chapter 5

Numerical Simulation of the Turbulent Flow around a Savonius Wind Rotor Using the Navier-Stokes Equations Sobhi Frikha, Zied Driss, Hedi Kchaou and Mohamed Salah Abid

45

Chapter 6

Numerical Prediction of the Effect of the Diameter Outlet on the Mixer Flow of the Diesel with the Biodiesel Mariem Lajnef, Zied Driss, Mohamed Chtourou, Dorra Driss and Hedi Kchaou

57

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vi

Contents

Chapter 7

Computer Simulation of the Turbulent Flow Around a Six-Blade Rushton Turbine Zied Driss, Abdelkader Salah, Abdessalem Hichri, Sarhan Karray and Mohamed Salah Abid

81

Chapter 8

Study of the Meshing Choice of a Negatively Buoyant Jet Injected in a Miscible Liquid Oumaima Eleuch, Noureddine Latrache, Sobhi Frikha and Zied Driss

99

Chapter 9

Study of the Wedging Angle Effect of a NACA2415 Airfoil Wind Turbine Zied Driss, Walid Barhoumi, Tarek Chelbi and Mohamed Salah Abid

121

Chapter 10

Study of the Meshing Effect on the Flow Characteristics inside a SCPP Ahmed Ayadi, Abdallah Bouabidi, Zied Driss and Mohamed Salah Abid

143

Chapter 11

Study of the Natural Ventilation in a Residential Living Room Opening with Two No-Opposed Positions Slah Driss, Zied Driss and Imen Kallel Kammoun

159

Chapter 12

Existence, Uniqueness and Smoothness of a Solution for 3D Navier-Stokes Equations with Any Smooth Initial Velocity. A Priori Estimate of This Solution Arkadiy Tsionskiy and Mikhail Tsionskiy

177

Chapter 13

Fuzzy Solutions of 2D Navier-Stokes Equations Yung-Yue Chen

209

Chapter 14

Effective Wall-Laws for Stokes Equations over Curved Rough Boundaries Myong-Hwan Ri

229

Chapter 15

Singularities of the Navier-Stokes Equations in Differential Form at the Interface Between Air and Water Xianyun Wen

265

Chapter 16

Self-Similar Analysis of Various Navier-Stokes Equations in Two or Three Dimensions I. F. Barna

275

Chapter 17

Asymptotic Solutions for the Navier-Stokes Equations, Describing Systems of Vortices with Different Spatial Structures Victor P. Maslov and Andrei I. Shafarevich

305

Chapter 18

Analytic Solutions of Incompressible Navier-Stokes Equations by Green's Function Method Algirdas Maknickas and Algis Džiugys

325

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Contents

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Chapter 19

Analysis of the Time Step Size Effect for the Study of the Liquid Sloshing inside a Container Abdallah Bouabidi, Zied Driss and Mohamed Salah Abid

349

Chapter 20

Numerical Analysis of Navier-Stokes Equations on Unstructured Meshes K. Volkov

365

Chapter 21

Integrals of Motion of an Incompressible Medium Flow. From Classic to Modern Alexander V. Koptev

443

Chapter 22

Local Exact Controllability of the Boussinesq Equations with Boundary Conditions on the Pressure Tujin Kim and Daomin Cao

461

Index

487

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PREFACE Navier–Stokes equations describe the motion of fluids; they arise from applying Newton’s second law of motion to a continuous function that represents fluid flow. If we apply the assumption that stress in the fluid is the sum of a pressure term and a diffusing viscous term, which is proportional to the gradient of velocity, we arrive at a set of equations that describe viscous flow. This handbook provides new research on the theories and applied analysis of Navier-Stokes Equations. In Chapter 1, is described a computational fluid dynamic (CFD) approach with the aim to simulate the flow of blood through the Aorta artery with aneurysm disease. In order to resolve the Navier-Stokes equations for the blood flow, in this CFD approach a 3D Volumetric mesh is generated from a set of Computed Tomography (CT) images, which serve as input data for the computational flow solver Ansys CFX® that is used in Chapter 1 to solve numerically the Navier-Stokes equations. For the segmentation of the CT images of the specific patient studied in Chapter 1, it is used a software developed in the laboratory. This software allows to generate meshes with high resolution and enables to perform a high precision calculations in cardiovascular fluid dynamics. As part of the outcomes showed in Chapter 1, a snapshot of the CFD variables is examined with the aim that the reader can see the streamlines of the blood flow inside the Aorta artery. Other variables are reported such as the pressure field on the Aorta wall and the Wall Shear Stress (WSS) distribution. This allows to conclude the risk of the patient under consideration. In Chapter 2 it is described a computational fluid dynamics (CFD) methodology in order to analyze the features of the blood flow in patients with exovascular Stent Devices in abdominal aortic aneurysms. The Navier-Stokes equations are solved numerically for a real geometry in 3D. This geometry is obtained through a segmentation process with a software developed in the laboratory. This computational tool takes the Computed Tomography (CT) images and reconstructs in 3D the geometry of the artery of interest, in this case the Aorta artery. Once the artery is generated in 3D the surface mesh is obtained and with a process of analysis of this mesh the 3D Volumetric mesh is performed. This mesh is the input data for the computational flow solver Ansys CFX® that was employed here to resolve numerically the Navier-Stokes equations. The flow field inside the Aorta with the exovascular stent device is shown and the streamlines do not exhibit the spiral flux that is presented in the Aorta without the stent device. The pressure field and the Wall Shear Stress (WSS) distribution are shown in Chapter 2. This allows to conclude the effects of the exovascular stent placed in the Aorta artery.

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In Chapter 3 a computational fluid dynamics (CFD) study of the blood flow is proposed with the aim to show the conditions and parameters that affect the possible rupture of an abdominal aortic aneurysm for one specific patient. A segmentation procedure of CT images was made in order to construct the 3D volumetric meshes to perform the CFD calculations with the computational flow solver Ansys CFX® that is used to resolve numerically the Navier-Stokes equations. This segmentation was made with a software developed in the authors’ laboratory and allows to build the 3D model. The patient studied in Chapter3 has an aneurysm disease in their Aorta artery and the results of the CFD calculations are shown. As part of the outcomes showed in Chapter 3, the authors report the streamlines of the blood flow that exhibit this patient in the Aorta. Finally, the pressure field and the Wall Shear Stress (WSS) are shown and has a magnitude that allow to conclude that the aneurysm disease represent a considerable risk for the patient studied in Chapter 3. In Chapter 4 the Finite Volume method is employed to simulate the blood flow in an intracranial aneurysm. For that purpose, the Navier-Stokes equations are solved in the computational domain of real aneurysms in a specific patient. This computational domain is obtained from MR images with a software designed in our laboratory. This computer program transforms the MR images (in DICOM format) to the volumetric mesh where the NavierStokes equations are going to be solved. The Ansys CFX® flow solver was used here to model the blood flow through the veins with the aneurysm deformation. The streamlines were computed inside the aneurysm with an inlet flow of 0.6 m/s. It can be seen a circular trajectory in the streamlines with the maximum values at the veins adjacent to the deformation. The wall shear stress was computed and exhibits its intermediate value at the aneurysm zone. Finally, the total pressure on the aneurysm wall is presented. It can be seen that the maximum value of the pressure field is located at the aneurysm place. This represents a considerable risk for the patient studied in Chapter 4. Chapter 5 lies within the scope of the research which takes place at the Laboratory of Electro-Mechanic Systems in the field of the wind turbine. Chapter 5 aims to investigate the effect of several parameters including the computational domain and the turbulence model on the aerodynamic characteristics of the flow around a Savonius wind rotor. For thus, the authors have developed a numerical simulation using a commercial CFD code. The considered numerical model is based on the resolution of the Navier-Stokes equations in conjunction with the k-ε turbulence model. These equations were solved by a finite volume discretization method. The authors are particularly interested in visualizing the velocity field, the mean velocity and the static pressure. The authors’ numerical results were compared to those obtained by other authors. The comparison shows a good agreement. In Chapter 6, computer simulations were developed to predict the effect of the outlet diameter on the mixer flow of the diesel with the biodiesel. The Navier-Stokes equations in conjunction with the standard k-ε turbulence model were considered and solved numerically. The software “SolidWorks Flow Simulation” which uses a finite volume scheme were used to present the local characteristics of the flow. The purpose of Chapter 6 is to select the right choice giving the best performance of the considered mixer. In Chapter 7, the authors are interested on the study of the turbulent flow around a Rushton turbine. Using the CFD code Ansys-FLUENT, the finite volume method was employed to solve the Navier-Stokes equations. Chapter 7 was made using the standard k-ε turbulence model. The relative motion between the rotating impeller and the stationary baffle was considered by the multiple reference frames (MRF). The comparison between the authors

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Preface

xi

numerical results and the experimental results found from the literature shows a good agreement. In Chapter 8, the meshing effect was carried out to study the penetration of a laminar liquid jet with negatively buoyant condition in a miscible surrounding liquid. The numerical resolution of the model, using the computational fluid dynamics (CFD) module, is based on the resolution of the Navier-Stokes equations and the volume of fluid model (VOF model). These equations were solved by a finite volume discretization method using Open Source code provided in Open Foam 2.3.0. Specially, it has been used two Liquid Mixing Foam solver for the case of two miscible liquids. The numerical results consist in presentation of the volume fraction, the magnitude velocity and the dynamic pressure. Particularly, the authors are interested in the study of the profile of the volume fraction to see the elimination of the numerical diffusion. According to these results, the best model should have the biggest number of cells to have only the physical diffusion of the jet inside the miscible liquid. Predictions of the numerical results have been compared with literature data and a satisfactory agreement has been found. In Chapter 9, numerical simulations and experimental validation were carried out to study the wedging angle effect of a NACA2415 airfoil type wind turbine to evaluate its performance. The authors consider the Navier-Stokes equations in conjunction with the standard k-ε turbulence model. These equations are solved numerically to determine the local characteristics of the flow. The models tested are implemented in the open source “SolidWorks Flow Simulation” which uses a finite volume scheme. Experiments have been also conducted on an open wind tunnel to validate the numerical results. In Chapter 10, a numerical simulation was carried out to study the fluid flow inside a Solar Chimney Power Plant (SCPP). The commercial computational fluid dynamics (CFD) software ANSYS-Fluent 17.0 has been used to develop a two-dimensional (2D) steady model with standard k-ε turbulence model. The local flow characteristics in the solar chimney such as the velocity, the temperature, the dynamic pressure and the turbulence characteristics were presented and discussed. Particularly, the authors are interested on the study of the meshing effect. Four meshes were considered in this study. The numerical results were compared with experimental data. This comparison shows that the generated mesh has a direct effect on the numerical results. The appropriate mesh was chosen to have the maximum number of cells with a moderate run time. Chapter 11 presents computer simulation and experimental validation of natural ventilation of residential living room opening on two non-opposed positions. The numerical method is based on the resolution of the Navier-Stokes equations in conjunction with the standard k-ε turbulence model. These equations are solved by a finite-volume discretization method. The good comparison between the numerical results and the authors’ experimental results developed using a wind tunnel confirms the validity of the numerical method. In Chapter 12, solutions of the Navier-Stokes and Euler equations with initial conditions for 2D and 3D cases were obtained in the form of converging series, by an analytical iterative method using Fourier and Laplace transforms. There the solutions are infinitely differentiable functions, and for several combinations of parameters numerical results are presented. Chapter 12 provides a detailed proof of the existence, uniqueness and smoothness of the solution of the Cauchy problem for the 3D Navier-Stokes equations with any smooth initial velocity. When the viscosity tends to zero, this proof applies also to the Euler equations. A priori estimate of this solution is presented.

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A new method for finding solutions of “Two-dimensional Navier-Stokes equations” (2D NSE) is discussed in Chapter 13 with using an adaptive fuzzy algorithm in this investigation, and the design target of this proposed method is to construct fuzzy solutions to satisfy the 2D NSE precisely and simultaneously. For achieving this design target, two rough fuzzy solutions are formulated as regressive forms with adjustable parameters firstly. Based on these two rough fuzzy solutions, an error cost function which is the square summation of approximation errors of boundary conditions, continuum equation and NavierStokes equations is minimized by a set of adaptive laws which can optimally tune the adjustable parameters of the proposed fuzzy solutions. Furthermore, approximated error bounds between the exact solutions and the proposed fuzzy solutions with respect to the number of fuzzy rules and solution errors have also been proven mathematically. Finally, the error equations in mesh points can be proven to converge to zero for the solution finding problem of 2D NSE. The authors of Chapter 14 derive effective wall-laws for Stokes equations in three dimensional bounded domains with curved rough boundaries. No-slip boundary condition is given on the locally periodic rough boundary parts with characteristic roughness size  and nonzero boundary condition is given on the nonoscillatory smooth boundary. Based on the analysis of a boundary layer cell problem depending on the geometry of the rough domain, boundary layer approximations are constructed using orthogonal tangential



 in L -norm and O    in energy norm. Then, a Navier wall-law with error estimates of order O    vectors and normal vector on the fictitious boundary, which is of order O 

3/2

2

3/2

1,1 2 in L -norm and O    in W -norm is obtained.

The properties of the Navier-Stokes equations are extremely important for theoretical and numerical studies of fluid flow. Chapter 15 investigates the properties of the differential and integral forms of the Navier-Stoke equations for immiscible air-water flow. The analysis reveals that unlike other fluids the immiscible air-water flow is so special that the Navier-Stokes equations in the differential form have singularities at the interface due to the continuous movement of interface and discontinuous density. In contrast to the differential form, the Navier-Stokes equations in integral form hold well everywhere including at the interface, indicating the integral form can be widely used in the computation of air-water flow. In Chapter 16 the authors will shortly introduce the self-similar Ansatz as a powerful tool to attack various non-linear partial differential equations and find - physically relevant dispersive solutions. Later, they classify the Navier-Stokes (NS) equations into four subsets, like Newtonian, non-Newtonian, compressible and incompressible. This classification is arbitrary, however helps the authors to get an overview about the structure of various viscous fluid equations. They present the analytic solutions for three of these classes. The relevance of the solutions are emphasized. Lastly, the authors present an interesting system the OberbeckBoussinesq system where the two dimensional NS equation is coupled to heat conduction. This system pioneered the way to chaos studies about half a century ago. Of course, the selfsimilar solution is presented which helps the authors to enlighten the formation of the Rayleigh-Bénard convection cells. In Chapter 17, the authors present a collection of their results concerning rapidly varying asymptotic solutions of the Navier - Stokes equations. They start with the results by one of the authors, describing periodic structures in incompressible fluids and focus on the effect of

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Preface

xiii

asymptotic instability and on the special structure of equations, governing the evolution of a periodic structure. Then the authors describe the connection with the asymptotic theory for the Navier - Stokes equations and topological invariants of divergence-free vector fields and Liouville foliations. They obtain the descriptions of different types of vortex structures via equations of graphs - topological invariants of corresponding steady Euler fields. The authors study different properties of these equations and corresponding vortex structures (Reynolds stresses, turbulent viscosity, Prandl-type equations, conservation laws, etc.) They finish with their new results in this area. The Navier-Stokes equations describe the motion of fluids; they arise from applying Newton's second law of motion to a continuous function that represents fluid flow. In Chapter 18, if the authors apply the assumption that stress in the fluid is the sum of a pressure term and a diffusing viscous term, which is proportional to the gradient of velocity, they arrive at a set of equations that describe viscous flow. The Navier-Stokes equations can be transformed into a set of full-partial differential equations that are inhomogeneous and parabolic. The incompressible Navier-Stokes equations are invariant under the Galilean transform. Extension of the Galilean transform into a single integral transform allows the authors to eliminate nonlinear terms and reduce the full differential equation, with respect to time, into a partial differential equation of a single variable. Solutions in 2D Lagrangian coordinates, for a defined boundary, are then given in terms of a terms of a vorticity-velocity stream function of   . Solutions in 3D Lagrangian coordinates, for a defined boundary, are then given in terms of a vorticity-vector potential function of   A . Applying an inverse to the new proposed integral transform allows the authors to rewrite solution in Eulerian coordinates. Finally, analytical solutions were obtained for these 2D and 3D incompressible Navier-Stokes equations by applying a Green's function method. The phenomenon of liquid sloshing is a two phase problem. The numerical simulation of the unsteady flow for this phenomenon depends on several numerical parameters. Particularly, the choice of the optimum time step is essential to perform a numerical simulation. Chapter 19 focuses on the study of the time step size effect in the numerical results for the liquid sloshing application. Four time steps were tested in order to choose the optimum value. The choice of this value was based on the comparison of the authors numerical results with the experimental results founded from the literature. The local flow characteristics inside the container, such as the free surface evolution, the velocity fields, the magnitude velocity and the static pressure, were presented and discussed over time. The results show that the time step size affects significantly the numerical results performances. As explained in Chapter 20, progress in the numerical solution of the Navier-Stokes equations is associated with the development of high-resolution schemes for flux computations that produce both accurate and monotone solutions in the presence of weak and strong gas discontinuities. The development of computational fluid dynamics (CFD) and computer technology makes it possible to design and implement methods for computing unsteady three-dimensional viscous compressible flows in regions of complex geometry. Flow solution is provided using cell-centered finite volume method on unstructured meshes. The non-linear CFD solver works in an explicit time marching fashion, based on a five-step Runge-Kutta stepping procedure and piecewise parabolic method. The code uses an edgebased data structure to give the flexibility to run on meshes composed of a variety of cell types. The edge weights are pre-computed and take account of the geometry of the cell. The

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governing equations are solved with MUSCL type scheme for inviscid fluxes, and central difference scheme for viscous fluxes. The gradient and the pseudo-Laplacian are calculated at the midpoint of a control volume edge using relations adapted to the computations on a strongly stretched meshes in the boundary layer. The gradient at mesh nodes is computed using Green's identity, and a technique is proposed that ensures the conservation property of the difference scheme as applied to two-dimensional flows. Convergence to a steady state is accelerated by the use of multigrid technique, and by the application of block-Jacobi preconditioning for high-speed flows, with a separate low-Mach number preconditioning method for use with low-speed flows. The sequence of meshes is created using an edgecollapsing algorithm. A numerical analysis of the internal and external flows is performed to improve the current understanding and modeling capabilities of the complex flow characteristics encountered in engineering applications. In Chapter 21 the authors present derivation of general integral for motion of an incompressible medium flow. The proposed procedure of constructing the integral is based on known statements in the theory of differential equations and equally true for both the case of Navier - Stokes equations and for the case of Euler ones. Known integrals of Lagrange Cauchy, Bernoulli and Euler - Bernoulli are special cases of constructed new integral. In Chapter 22 the authors are concerned with the local exact controllability of the Boussinesq equations with the pressure condition on parts of boundary by controls acting on subdomains or subboundaries. Starting point of these results is to get a Carleman inequality for the system adjoint to the linearized Boussinesq system with the nonstandard boundary conditions. This is obtained by combination of our previous results for the Stokes system with the nonstandard boundary conditions and a linear parabolic equation with mixed boundary conditions. Using the Carleman inequality, the authors get an observability inequality for a system adjoint to a linearized Boussinesq system. Next, following other's way and using the observability inequality, they also prove null controllability of the linearized system by controls acting on an arbitrarily given subdomain. Finally, by using these results under some assumptions the authors prove local exact controllability of the Boussinesq system by internal or boundary controls.

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In: Handbook on Navier-Stokes Equations Editor: Denise Campos

ISBN: 978-1-53610-292-5 © 2017 Nova Science Publishers, Inc.

Chapter 1

GENERATION OF MESHES IN CARDIOVASCULAR SYSTEMS I: RESOLUTION OF THE NAVIER-STOKES EQUATIONS FOR THE BLOOD FLOW IN ABDOMINAL AORTIC ANEURYSMS Alejandro Acevedo-Malavé Multidisciplinary Center of Sciences, Venezuelan Institute for Scientific Research (IVIC), Mérida, Venezuela

Abstract In this paper, is described a computational fluid dynamic (CFD) approach with the aim to simulate the flow of blood through the Aorta artery with aneurysm disease. In order to resolve the Navier-Stokes equations for the blood flow, in this CFD approach a 3D Volumetric mesh is generated from a set of Computed Tomography (CT) images, which serve as input data for the computational flow solver Ansys CFX® that is used here to solve numerically the NavierStokes equations. For the segmentation of the CT images of the specific patient studied here, it is used a software developed in the laboratory. This software allows to generate meshes with high resolution and enables to perform a high precision calculations in cardiovascular fluid dynamics. As part of the outcomes showed here, a snapshot of the CFD variables is examined with the aim that the reader can see the streamlines of the blood flow inside the Aorta artery. Other variables are reported such as the pressure field on the Aorta wall and the Wall Shear Stress (WSS) distribution. This allows to conclude the risk of the patient under consideration.

1. Introduction Abdominal aortic aneurysms (AAA) are relatively common and are potentially lifethreatening. Aneurysms are defined as a focal dilatation in an artery, with at least a 50% increase over the vessel’s normal diameter. Thus, enlargement of the diameter of the abdominal aorta to 3 cm or more fits the definition. AAA usually results from degeneration in the media of the arterial wall, leading to a slow and continuous dilatation of the lumen of the vessel. Uncommon causes include infection, cystic medial necrosis, arthritis, trauma,

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inherited connective-tissue disorders, and anastomotic disruption. AAAs generally affect elderly white men. Smoking appears to be the risk factor most strongly associated with AAA. In addition to increasing age and male sex, other factors include increased height, weight, body mass index, and body surface area. A familiar clustering has been noted in 15-25% of patients undergoing surgical repair of AAA. Female sex, African American race, and the presence of diabetes mellitus are negatively associated with AAA. Most of the treatments for abdominal aneurysms consist of a surgical intervention to collocate a STENT in the Aorta artery [1-8]. Due to the advances in computer hardware and numerical algorithms, computational fluid dynamics (CFD) have emerged to reveal underlying physics of blood flow in human cardiac vasculature [9-22]. Patient-specific computation based on magnetic resonance imaging (MRI) and advanced computed tomography (CT) [11] is of considerable interest due to the strong anatomical, functional, and the hemodynamic interdependency of various cardiovascular structures. A ruptured aneurysm can cause severe internal bleeding, which can lead to shock or even death. Abdominal aortic aneurysms are usually asymptomatic until they expand or rupture. An expanding of AAA causes sudden, severe, and constant low back, flank, abdominal, or groin pain. Syncope may be the chief complaint, however, with pain less prominent. Most clinically significant AAAs are palpable upon routine physical examination. The presence of a pulsatile abdominal mass is virtually diagnostic but is found in fewer than half of all cases. Patients with a ruptured AAA may present in frank shock, as evidenced by cyanosis, mottling, altered mental status, tachycardia, and hypotension. Whereas the abrupt onset of pain due to rupture of an AAA may be quite dramatic, associated physical findings may be very subtle. Patients may have normal vital signs in the presence of a ruptured AAA as a consequence of retroperitoneal containment of hematoma. In this paper a methodology is proposed for the study of specific real geometries of the Aorta artery and the possible risk of a real case based on some physical parameters that arise in our CFD calculations.

2. Governing Equations The governing equations can be given by the continuity Eq. (1) and the Navier-Stokes Eq. (2):

v  0

(1)

(2) where v is the velocity, p is the pressure, η is the dynamic viscosity,  is the density. CFX® (ANSYS® 15.0, ANSYS®, Inc. Southpointe 2600 ANSYS Drive Canonsburg, PA 15317 US), a flow solver based on the finite volume method, was used to solve Eqs. (1) and (2). A 3D tetrahedral mesh was used for this calculation with 2.203.464 elements. Inside the CFX module was defined the following constants: (Blood) = 1060 Kg/m3, η (Blood) = 3.5x10-3 Pa.s and the initial value of the velocity at inlet surfaces was 0.6 m/s [23]. A

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Generation of Meshes in Cardiovascular Systems I

3

software designed in the laboratory was used to transform the CT images to the 3D volumetric mesh.

3. Segmentation of CT Images The usage of computational fluid dynamics has a considerable impact to resolve a broad quantity of problems in Science and Engineering. The precision of the computations in CFD schemes has a close relationship with the resolution of the mesh that represent the computational domain where the Navier-Stokes equation going to be solved. Is very important in cardiovascular fluid dynamics to design a methodology for the generation of meshes in a very efficient way. Due to the irregular geometries of the cardiovascular systems is necessary the development of a computational tool to make this labour in a friendly and graphical manner. In the segmentation methodology was used a software designed in the laboratory. A Computed Tomography images are loaded from a graphical interface that allow to reconstruct the 3D structure of the Aorta geometry (see Figure 1). This application uses an algorithm that marks all regions of interest and from this mask as well as a smoothing procedure can be obtained a good resolution of the Aorta in the 3D space (see Figure 2 and 3). After this step, a surface mesh is generated and an analysis of this object is performed. Through this analysis can be removed bad triangles, smooth the mesh and edit some parameters to generate the 3D volumetric mesh. Once that this surface mesh is constructed it proceeds to compute the 3D Volumetric mesh with a good resolution to perform the CFD calculation with high precision variables. As part of this methodology the 3D volumetric mesh is exported to the Ansys CFX® flow solver.

Figure 1. A Computed Tomography (CT) with the zones of the Aorta, highlighted in green color.

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Figure 2. A 3D structure of the Aorta artery is performed inside the Computed Tomography image.

Figure 3. The isolated structure in 3D of the Aorta artery with the specific geometry of the patient under consideration in this study.

4. The Numerical Resolution of Navier-Stokes Equations for the Blood Flow The blood flow is modeled as an incompressible Newtonian fluid. The governing equations are the incompressible Navier-Stokes and continuity equations (1), (2). Due to a

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Generation of Meshes in Cardiovascular Systems I

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lack of information regarding the distribution of vessel wall elasticity and thickness, the vein walls were assumed to be rigid. Non-Newtonian effects were neglected. In these CFD calculations it can be seen a flux mixed with various features, for both zones of the artery. In Figure 4 it can be seen that in the zone after inlet condition, there is a flux that is uniform and the streamlines are approximately parallel between them. When the flux of blood is penetrating around the zone adjacent to the aneurysm the flux take the form of a spiral and is decelerated with a velocity around 0.20 m/s. After that, the blood is circulating in the region where the aneurysm is located and in the neck of the deformation the flux is accelerated and reaches a velocity of 0.40 m/s. When the flux of blood enters in the aneurysm region, it can be seen again a deceleration of the blood flow and this reaches a value of 0.20 m/s in spiral form. According to the nature of the blood flow inside the aneurysm zone can be seen that this flow contribute to the mechanical stretching of the artery as well as with the evolution of the dynamics this region begin to more large in dimensions which is a risk for the patient studied here. In the pressure field reported here (see Figures 5 and 6) for the patient under consideration, it can be seen that there is a set of maximum values at the zone after the inlet condition. In this zone the flux of blood is accelerated and reaches a value around 570.8 Pa. At the neck of the aneurysm this flux is accelerated, but the pressure field diminished its value and when the flux is again decelerated at the aneurysm region the pressure has an increment in its magnitude. In Figures 7 and 8 can be seen a distribution of the Wall Shear Stress in the artery Aorta for the patient under consideration. It can be seen that the maximum value is reached at the aneurysm zone with a value around 0.200 Pa which decrease its magnitude at the other zones out of the aneurysm region. This maximum value represents a considerable risk for the patient studied here.

Figure 4. The streamlines of the blood flow in the Aorta artery.

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Figure 5. The pressure field in the Aorta artery for the specific model of the patient studied here (front view).

Figure 6. The pressure field in the Aorta artery for the specific model of the patient studied here (rear view).

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Figure 7. The Wall Shear Stress distribution in the Aorta geometry for the specific model of the patient under study (front view).

Figure 8. The Wall Shear Stress distribution in the Aorta geometry for the specific model of the patient under study (rear view).

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Alejandro Acevedo-Malavé

Conclusion In this study was proposed a CFD methodology for the modeling of blood flows in the abdominal Aorta Aneurysm for a real geometry. With a high resolution of the 3D Volumetric mesh some CFD parameters are calculated to allow to study the case presented here. The 3D Volumetric mesh was the input data to make the CFD calculations with the computational flow solver Ansys CFX®. This computational tool was employed to resolve numerically the Navier-Stokes equations for the blood flow. Some snapshots of the CFD variables were presented here. The streamlines show two regions of the blood flow that basically has a parallel lines of the field of velocity and a circular flux inside the aneurysm deformation. In the pressure fields can be seen a distribution that has a maximum value located at the neck of the aneurysm region. The WSS distribution shows a maximum value located in the aneurysm zone, this represents a considerable risk for the patient studied here.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

S.C.M. Yu, Int. J. of Heat and Fluid Flow 21 (2000) 74-83. L. Morris, P. Delassus, M. Walsh, T. McGloughlin, J. Biomech. 37 (2004) 1087-1095. Z. Li, C. Kleinstreuer, Medical Engineering and Physics 27 (2005) 369-382. L. Morris, P. Delassus, P. Grace, F. Wallis, M. Walsh, T. McGloughlin, Medical Engineering and Physics 28 (2006) 19-26. L. Morris, F. Stefanov, T. McGloughlin, J. Biomech. 46 (2013) 383-395. P. Zhang, A. Sun, F. Zhan, J. Luan, X. Deng, J. Biomech. 47 (2014) 3524-3530. H. Kandail, M. Hamady, X.Y. Xu, J. Biomech. 47 (2014) 3546-3554. A. Polanczyk, M. Podyma, L. Stefanczyk, W. Szubert, I. Zbicinski, J. Biomech. 48 (2015) 425-431. J. A. Ekaterinaris, C.V. Ioannou, A.N. Katsamouris, H. Greece, Annals of Vascular Surgery 20 (2006) 351-359. C.M. Scotti, E.A. Finol, Computers and Structures 85 (2007) 1097-1113. D.C. Barber, E. Oubel, A.F. Frangi, D.R. Hose, Medical Image Analysis 11 (2007) 648–662. F.P.P. Tan, A. Borghi, R.H. Mohiaddin, N.B. Wooda, S. Thom, X.Y. Xu, Computers and Structures 87 (2009) 680-690. X. Wangn, X. Li, Computers in Biology and Medicine 41 (2011) 812-821. A. Sheidaei, S.C. Hunley, S. Zeinali-Davarani, L.G. Raguin, S. Baek, Medical Engineering and Physics 33 (2011) 80-88. M. Malvè, A. García, J. Ohayon, M.A. Martínez, International Communications in Heat and Mass Transfer 39 (2012) 745-751. M. Piccinelli, C. Vergara, L. Antiga, L. Forzenigo, P. Biondetti, M. Domanin, Biomech. Model Mechanobiol. 12 (2013) 1263-1276. K. Sughimoto, Y. Takahara, K. Mogi, K. Yamazaki, K. Tsubota, F. Liang, H. Liu, Heart Vessels 29 (2014) 404-412. T.I. Józsa, G. Paál, International Journal of Heat and Fluid Flow 50 (2014) 342-351. A. Javadzadegan, B. Fakhim, M. Behnia, M. Behnia, European Journal of Mechanics B/Fluids 46 (2014) 109-117.

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[20] S. Karimi, M. Dabagh, P. Vasava, M. Dadvar, B. Dabir, P. Jalali, Journal of NonNewtonian Fluid Mechanics Volume 207 (2014) 42-52. [21] E. Soudaha, G. Vilaltab, M. Bordonea, F. Nietod, J.A. Vilaltacy, C. Vaquero, Rev. int. métodos numér. cálc. diseño ing. 31 (2015) 106-112. [22] R. Nagy, C. Csobay-Novák, A. Lovas, P. Sótonyi, I. Bojtár, J. Biomech. 48 (2015) 1876-1886. [23] K.H. Fraser, S. Meagher, J.R. Blake, W.J. Easson, P.R. Hoskins, Ultrasound in Med. and Biol. 34 (2008) 73-80.

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ISBN: 978-1-53610-292-5 © 2017 Nova Science Publishers, Inc.

Chapter 2

GENERATION OF MESHES IN CARDIOVASCULAR SYSTEMS II: THE BLOOD FLOW IN ABDOMINAL AORTIC ANEURYSMS WITH EXOVASCULAR STENT DEVICES Alejandro Acevedo-Malavé Multidisciplinary Center of Sciences, Venezuelan Institute for Scientific Research (IVIC), Mérida, Venezuela

Abstract In this work it is described a computational fluid dynamics (CFD) methodology in order to analyze the features of the blood flow in patients with exovascular Stent Devices in abdominal aortic aneurysms. The Navier-Stokes equations are solved numerically for a real geometry in 3D. This geometry is obtained through a segmentation process with a software developed in the laboratory. This computational tool takes the Computed Tomography (CT) images and reconstructs in 3D the geometry of the artery of interest, in this case the Aorta artery. Once the artery is generated in 3D the surface mesh is obtained and with a process of analysis of this mesh the 3D Volumetric mesh is performed. This mesh is the input data for the computational flow solver Ansys CFX® that was employed here to resolve numerically the Navier-Stokes equations. The flow field inside the Aorta with the exovascular stent device is shown and the streamlines do not exhibit the spiral flux that is presented in the Aorta without the stent device. The pressure field and the Wall Shear Stress (WSS) distribution are shown in this work. This allows to conclude the effects of the exovascular stent placed in the Aorta artery.

1. Introduction Abdominal aortic aneurysm (AAA) is a localized enlargement of the abdominal aorta such that the diameter is greater than 3 cm or more than 50% larger than normal. They usually cause no symptoms except when ruptured. Large aneurysms can sometimes be felt by pushing on the abdomen. AAA occurs most commonly in those over 50 years old, in men, and among those with a family history. Additional risk factors include smoking, high blood pressure, and

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other heart or blood vessel diseases. Genetic conditions with an increased risk include Marfan syndrome and Ehlers-Danlos syndrome. An abdominal aortic aneurysm is usually diagnosed by physical exam, ultrasound, or Compute Tomography. Plain abdominal radiographs may show the outline of an aneurysm when its walls are calcified. However, this is the case in less than half of all aneurysms. Ultrasonography is used to screen for aneurysms and to determine the size of any present. Additionally, free peritoneal fluid can be detected. It is noninvasive and sensitive, but the presence of bowel gas or obesity may limit its usefulness. CT scan has a nearly 100% sensitivity for aneurysm and is also useful in preoperative planning, detailing the anatomy and the possibility for endovascular repair. In the case of suspected rupture, it can also reliably detect retroperitoneal fluid. Alternative less often used methods for visualization of the aneurysm include MRI and angiography. An aneurysm ruptures if the mechanical stress exceeds the local wall strength; consequently, peak wall stress and peak wall rupture risk have been found to be more reliable parameters than the diameter to assess AAA rupture risk. Yu [1] studied the Steady and pulsatile flow in AAA models using Particle Image Velocimetry. In this case the author describes an experimental procedure to determine the flow characteristics in Abdominal Aortic Aneurysm for Reynolds numbers (from 400 to 1400) and Womersley numbers (from 17 to 22). The steady and pulsatile experiments were carried out. For pulsatile flow conditions is observed a recirculating vortex that is located in the zone of the aneurysm. In this study two important effects in this zone were observed as the bulk flow decelerated. First, the strength of the recirculation was strengthened. Second, the flow recirculation reduced its magnitude in this zone, but was enlarged in the transverse direction. Most of the treatments for abdominal aneurysms consist of a surgical intervention to collocate a STENT in the Aorta artery [2-8]. This stents modify the characteristics of the flux through the aneurysm zone and prevent a possible rupture. A CFD calculations of the blood flow has been carried out in these last years [9-22] and shows an analysis of the possible causes of the aneurysm rupture. In this work it is shown a CFD methodology for the study of the features of the blood flow in abdominal aortic aneurysm for a 3D geometry of a real case with an exovascular stent device.

2. Governing Equations The governing equations can be given by the continuity Eq. (1) and the Navier-Stokes Eq. (2):

v  0

(1)

(2) where v is the velocity, p is the pressure, η is the dynamic viscosity,  is the density. CFX® (ANSYS® 15.0, ANSYS®, Inc. Southpointe 2600 ANSYS Drive Canonsburg, PA 15317 USA), a flow solver based on the finite volume method, was used to solve Eqs. (1) and (2). A 3D tetrahedral mesh was used for this calculation with 1.777.290 elements. Inside the

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CFX module was defined the following constants: (Blood)=1060 Kg/m3, η (Blood)=3.5x10-3 Pa.s and the initial value of the velocity at inlet surfaces was 0.6 m/s [23]. A software designed in the laboratory was used to transform the CT images to the 3D volumetric mesh.

Figure 1. A Computed Tomography (CT) of the Aorta artery with exovascular stent device.

a

b

Figure 2. a) A 3D structure of the Aorta artery is performed inside the Computed Tomography image without the exovascular stent device. b) The 3D structure of the Aorta artery with the exovascular stent device (Blue color).

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Figure 3. The isolated structure in 3D of the Aorta artery with the specific geometry for the patient under consideration in this study with the exovascular stent device (Blue color).

3. Segmentation of CT IMAGES All CFD calculations need a meshing of the solution and geometry domain [11]. The computation of a good resolution mesh that represent a structure of interest can be very labourintensive and can be the major complication in the application of cardiovascular fluid dynamic. In the segmentation methodology we use a software designed in our laboratory. The first step is to load the CT images from a graphical interface (see figure 1). After this, the application uses an algorithm that marks all regions of interest and from this mask a 3D image of the arterial zone is performed. As the resolution of the resulting 3D image is Low in the first instance a smoothing procedure is carried out with the aim of obtaining a very soft surface without discontinuities and jumps, and by this way is constructed an image of the artery with good resolution (see figure 2). The next step is removing all regions that are out of the zone of interest and only the 3D structure of the artery is produced (see figure 3). This kind of methodology has the advantage that allows to construct a 3D realistic model of the patient under study.

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Figure 4. The streamlines of the blood flow in the Aorta artery.

Figure 5. The pressure field in the Aorta artery for the specific model of the patient studied here (front view).

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Figure 6. The pressure field in the Aorta artery for the specific model of the patient studied here (rear view).

4. The Numerical Resolution of Navier-Stokes Equations for the Blood Flow The Navier Stokes and continuity equations (1), (2) were solved numerically using the computational flow solver Ansys CFX® and the blood flow was modeled as an incompressible Newtonian Fluid where the Non-Newtonian effects were neglected. The walls of the Aorta artery were assumed to be rigid due to the nonexistence of information about thickness and elasticity of the veins. In these CFD calculations, it can be seen a flux uniform with parallel streamlines through all structure of the Aorta artery. In another study [24] can be observed that exist a zone located at the aneurysm place that the streamline are circular or in spiral form. The effect of the surgical intervention of collocate an exovascular stent device in the flux of blood is to produce a uniform flow at these deformed places in the artery wall. It can be seen in figure 4 that the blood flow accelerates at the neck region that is connected with the aneurysm zone.

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Figure 7. The Wall Shear Stress distribution in the Aorta geometry for the specific model of the patient under study (front view).

Figure 8. The Wall Shear Stress distribution in the Aorta geometry for the specific model of the patient under study (rear view).

In the pressure field can be seen (see figure 5 and 6) that the patient under consideration has a maximum value located in the aneurysm zone but due to the presence of the exovascular stent device this fact no represent a risk for the patient studied here. The WSS distribution has the same tendency and has a maximum value of 0.334 Pa located in the aneurysm zone.

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Conclusion In this work was proposed a Computational Fluid Dynamics methodology to simulate the features of the blood flow through an aneurysm disease. A real 3D geometry was constructed from Computed Tomography images of the patient under consideration. The segmentation procedure of these CT images was made with a software developed in the laboratory that allows to build a 3D Volumetric mesh with high resolution, which is the crucial importance in the computational fluid dynamics calculations with irregular geometries. This mesh was the input data for the computational flow solver Ansys CFX® that was used to resolve numerically the Navier-Stokes equations. In comparison with other work done without the stent device [24] it can be observed that the streamlines inside the Aorta artery are uniform and parallel during the whole trajectory of the blood flow. Now with the exovascular stent device the pressure field and WSS distribution do not represent a risk for the patient under consideration here.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

S. C. M. Yu, Int. J. of Heat and Fluid Flow 21 (2000) 74-83. L. Morris, P. Delassus, M. Walsh, T. McGloughlin, J. Biomech. 37 (2004) 1087–1095. Z. Li, C. Kleinstreuer, Medical Engineering & Physics 27 (2005) 369–382. L. Morris, P. Delassus, P. Grace, F. Wallis, M. Walsh, T. McGloughlin, Medical Engineering & Physics 28 (2006) 19–26. L. Morris, F. Stefanov, T. McGloughlin, J. Biomech. 46 (2013) 383–395. P. Zhang, A. Sun, F. Zhan, J. Luan, X. Deng, J. Biomech. 47 (2014) 3524–3530. H. Kandail, M. Hamady, X.Y. Xu, J. Biomech. 47 (2014) 3546-3554. A. Polanczyk, M. Podyma, L. Stefanczyk, W. Szubert, I. Zbicinski, J. Biomech. 48 (2015) 425-431. J. A. Ekaterinaris, C.V. Ioannou, A.N. Katsamouris, H. Greece, Annals of Vascular Surgery 20 (2006) 351-359. C. M. Scotti, E.A. Finol, Computers and Structures 85 (2007) 1097–1113. D.C. Barber, E. Oubel, A.F. Frangi, D.R. Hose, Medical Image Analysis 11 (2007) 648–662. F. P. P. Tan, A. Borghi, R.H. Mohiaddin, N.B. Wooda, S. Thom, X.Y. Xu, Computers and Structures 87 (2009) 680–690. X. Wangn, X. Li, Computers in Biology and Medicine 41 (2011) 812–821. A. Sheidaei, S.C. Hunley, S. Zeinali-Davarani, L.G. Raguin, S. Baek, Medical Engineering & Physics 33 (2011) 80–88. M. Malvè, A. García, J. Ohayon, M.A. Martínez, International Communications in Heat and Mass Transfer 39 (2012) 745–751. M. Piccinelli, C. Vergara, L. Antiga, L. Forzenigo, P. Biondetti, M. Domanin, Biomech. Model Mechanobiol. 12 (2013) 1263–1276. K. Sughimoto, Y. Takahara, K. Mogi, K. Yamazaki, K. Tsubota, F. Liang, H. Liu, Heart Vessels 29 (2014) 404–412. T. I. Józsa, G. Paál, International Journal of Heat and Fluid Flow 50 (2014) 342–351. A. Javadzadegan, B. Fakhim, M. Behnia, M. Behnia, European Journal of Mechanics B/Fluids 46 (2014) 109–117. S. Karimi, M. Dabagh, P. Vasava, M. Dadvar, B. Dabir, P. Jalali, Journal of NonNewtonian Fluid Mechanics Volume 207 (2014) 42–52.

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[21] E. Soudaha, G. Vilaltab, M. Bordonea, F. Nietod, J.A. Vilaltacy, C. Vaquero, Rev. int. métodos numér. cálc. diseño ing. 31 (2015) 106–112. [22] R. Nagy, C. Csobay-Novák, A. Lovas, P. Sótonyi, I. Bojtár, J. Biomech. 48 (2015) 1876–1886. [23] K. H. Fraser, S. Meagher, J.R. Blake, W.J. Easson, P.R. Hoskins, Ultrasound in Med. & Biol. 34 (2008) 73–80. [24] A. Acevedo, European Journal of Mechanics - B/Fluids (2016). Submitted.

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ISBN: 978-1-53610-292-5 © 2017 Nova Science Publishers, Inc.

Chapter 3

A COMPUTATIONAL FLUID DYNAMICS (CFD) STUDY OF THE BLOOD FLOW IN ABDOMINAL AORTIC ANEURYSMS FOR REAL GEOMETRIES IN SPECIFIC PATIENTS Alejandro Acevedo-Malavé1, Ricardo Fontes-Carvalho2 and Nelson Loaiza3 1

Multidisciplinary Center of Sciences, Venezuelan Institute for Scientific Research (IVIC), Mérida, Venezuela 2 Cardiology Department, Vila Nova de Gaia/Espinho Hospital Center, Vila Nova de Gaia, Portugal 3 National University of Colombia, Faculty of Mines, Medellin, Colombia

Abstract In this work a computational fluid dynamics (CFD) study of the blood flow is proposed with the aim to show the conditions and parameters that affect the possible rupture of an abdominal aortic aneurysm for one specific patient. A segmentation procedure of CT images was made in order to construct the 3D volumetric meshes to perform the CFD calculations with the computational flow solver Ansys CFX® that is used to resolve numerically the Navier-Stokes equations. This segmentation was made with a software developed in our laboratory and allows to build the 3D model. The patient studied here has an aneurysm disease in their Aorta artery and the results of the CFD calculations are shown. As part of the outcomes showed here we report the streamlines of the blood flow that exhibit this patient in the Aorta. Finally, the pressure field and the Wall Shear Stress (WSS) are shown and has a magnitude that allow to conclude that the aneurysm disease represent a considerable risk for the patient studied here.

1. Introduction Abdominal aortic aneurysms (AAA) are usually asymptomatic until they expand or rupture. An expanding of AAA causes sudden, severe, and constant low back, flank,

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abdominal, or groin pain. Syncope may be the chief complaint, however, with pain less prominent. Most clinically significant AAAs are palpable upon routine physical examination. The presence of a pulsatile abdominal mass is virtually diagnostic but is found in fewer than half of all cases. Patients with a ruptured AAA may present in frank shock, as evidenced by cyanosis, mottling, altered mental status, tachycardia, and hypotension. Whereas the abrupt onset of pain due to rupture of an AAA may be quite dramatic, associated physical findings may be very subtle. Patients may have normal vital signs in the presence of a ruptured AAA as a consequence of retroperitoneal containment of hematoma. At least 65% of patients with a ruptured AAA die of sudden cardiovascular collapse before arriving at a hospital. Yu [1] studied the Steady and pulsatile flow in AAA models using Particle Image Velocimetry. In this case the author describes an experimental procedure to determine the flow characteristics in Abdominal Aortic Aneurysm for Reynolds numbers (from 400 to 1400) and Womersley numbers (from 17 to 22). The steady and pulsatile experiments were carried out. For pulsatile flow conditions is observed a recirculating vortex that is located in the zone of the aneurysm. In this study two important effects in this zone were observed as the bulk flow decelerated. First, the strength of the recirculation was strengthened. Second, the flow recirculation reduced its magnitude in this zone, but was enlarged in the transverse direction. Most of the treatments for abdominal aneurysms consist of a surgical intervention to collocate a STENT in the Aorta artery [2-8]. This stents modify the characteristics of the flux through the aneurysm zone and prevent a possible rupture. A CFD calculations of the blood flow has been carried out in these last years [9-22] and shows an analysis of the possible causes of the aneurysm rupture. However, in this paper is proposed a methodology for the study of specific real geometries of the Aorta artery and the possible risk for each patient based on some physical parameters that arise in our CFD calculations.

2. Governing Equations The governing equations can be given by the continuity Eq. (1) and the Navier-Stokes Eq. (2):

 v  0

(1)

(2) where v is the velocity, p is the pressure, η is the dynamic viscosity,  is the density. CFX® (ANSYS® 15.0, ANSYS®, Inc. Southpointe 2600 ANSYS Drive Canonsburg, PA 15317 USA), a flow solver based on the finite volume method, was used to solve Eqs. (1) and (2). A 3D tetrahedral mesh was used for this calculation with 1.089.234 elements. Inside the CFX module was defined the following constants: (Blood) = 1060 Kg/m3, η (Blood) = 3.5x10-3Pa.s and the initial value of the velocity at inlet surfaces was 0.6 m/s [23]. A software designed in our laboratory was used to transform the CT images to the 3D volumetric mesh.

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A Computational Fluid Dynamics (CFD) Study of the Blood Flow …

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3. Segmentation of CT Images All CFD calculations need a meshing of the solution and geometry domain [11]. The computation of a good resolution mesh that represent a structure of interest can be very labour-intensive and can be the major complication in the application of cardiovascular fluid dynamic. In the segmentation methodology we use a software designed in our laboratory. The first step is to load the CT images from a graphical interface (see Figure 1). After this, the application uses an algorithm that marks all regions of interest and from this mask a 3D image of the arterial zone is performed. As the resolution of the resulting 3D image is Low in the first instance a smoothing procedure is carried out with the aim of obtaining a very soft surface without discontinuities and jumps, and by this way is constructed an image of the artery with good resolution (see Figure 2). The next step is removing all regions that are out of the zone of interest and only the 3D structure of the artery is produced (see Figure 3). This kind of methodology has the advantage that allows to construct a 3D realistic model of the patient under study. Finally, the 3D structure is imported to other graphical interface that allow to compute the surface mesh and with an analysis of the mesh obtained we can remove bad triangles, perform the smoothing of some zones of the surface mesh and after that we can compute the 3D Volumetric mesh that is exported to Ansys CFX® flow solver.

Figure 1. A Computed Tomography (CT) with the zones of the Aorta highlighted in green color.

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Alejandro Acevedo-Malavé, Ricardo Fontes-Carvalho and Nelson Loaiza

Figure 2. A 3D structure of the Aorta artery is performed inside the Computed Tomography image.

Figure 3.The isolated structure in 3D of the Aorta artery with the specific geometry of the patient under consideration in this study.

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Figure 4. The streamlines of the blood flow in the Aorta artery.

4. Computational Fluid Dynamics Simulations of the Blood Flow through the Aorta with the Aneurysm Disease The blood flow is modeled as an incompressible Newtonian fluid. The governing equations are the incompressible Navier-Stokes and continuity equations (1), (2). Due to a lack of information regarding the distribution of vessel wall elasticity and thickness, the vein walls were assumed to be rigid. Non-Newtonian effects were neglected. In the calculations reported here for the patient under study, there is a flux mixed with various features, for both zones of the artery. In Figure 4 it can be seen that at the zone after inlet condition, there is a flux that is uniform and the streamlines are approximately parallel between them. When the flux of blood is penetrating around the zone adjacent to the aneurysm the flux take the form of a spiral and is decelerated with a velocity around 0.35 m/s. After that, the blood is circulating in the region where the aneurysm is located and in the neck of the deformation the flux is accelerated and reaches a velocity of 0.80 m/s. When the flux of blood enters in the aneurysm region, we can see again a deceleration of the blood flow and this reaches a value of 0.20 m/s in spiral form. According to the nature of the blood flow inside the aneurysm zone, we can see that this flow contribute to the mechanical stretching of the artery as well as with the evolution of the dynamics this region begin to more large in dimensions which is a risk for the patient studied here. In the pressure field reported here for the patient under consideration, it can be seen that there is a set of maximum values at the zone after the inlet condition. In this zone the flux of blood is accelerated and reaches a value around 1200.0 Pa. At the neck of the aneurysm this flux is accelerated, but the pressure field diminished its value and when the flux is again decelerated at the aneurysm region the pressure has an increment in its magnitude.

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Alejandro Acevedo-Malavé, Ricardo Fontes-Carvalho and Nelson Loaiza

Figure 5. The pressure field in the Aorta artery for the specific model of the patient studied here (front view).

Figure 6. The pressure field in the Aorta artery for the specific model of the patient studied here (rear view).

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Figure 7.The Wall Shear Stress distribution in the Aorta geometry for the specific model of the patient under study (front view).

Figure 8.The Wall Shear Stress distribution in the Aorta geometry for the specific model of the patient under study (rear view).

In Figures 7 and 8 can be seen a distribution of the Wall Shear Stress in the artery Aorta for the patient under consideration. It can be seen that the maximum value is reached at the aneurysm zone with a value around 0.400 Pa which decrease its magnitude at the other zones

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out of the aneurysm region. This maximum value represents a considerable risk for the patient studied here. At the bifurcation zone, it can be seen that there is not an increment of the WSS value and this tendency is the same around this region, until the motion of the blood flow reaches the zone adjacent to the neck of the aneurysm (in this region the flux reaches a maximum value of the WSS distribution too).

Conclusion Here is proposed a methodology for the study of the characteristics of the blood flow through the abdominal aneurysm. The 3D model of the Aorta artery is performed from a CT image and a procedure of segmentation was carried out. From this procedure of segmentation the 3D Volumetric mesh was constructed that is the input data to the Ansys CFX® flow solver, which allows to resolve numerically the Navier-Stokes equations. A snap shot of the streamlines inside the Aorta was obtained. In this result can be seen that a spiral flux is present at the zone where the aneurysm is placed. In other regions initially the flux has some variations in the magnitude of the velocity. Around inlet condition the flux is accelerated until the next region around the neck of the aneurysm zone. In the pressure field can be seen a distribution that has a maximum value located around the inlet zone and there is a transition to the lowest values after the neck of the aneurysm region. The WSS distribution shows a maximum value located in the aneurysm zone this represents a considerable risk for the patient studied here.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

S.C.M. Yu, Int. J. of Heat and Fluid Flow 21 (2000) 74-83. L. Morris, P. Delassus, M. Walsh, T. McGloughlin, J. Biomech. 37 (2004) 1087–1095. Z. Li, C. Kleinstreuer, Medical Engineering & Physics 27 (2005) 369–382. L. Morris, P. Delassus, P. Grace, F. Wallis, M. Walsh, T. McGloughlin, Medical Engineering & Physics 28 (2006) 19–26. L. Morris, F. Stefanov, T. McGloughlin, J. Biomech. 46 (2013) 383–395. P. Zhang, A. Sun, F. Zhan, J. Luan, X. Deng, J. Biomech. 47 (2014) 3524–3530. H. Kandail, M. Hamady, X.Y. Xu, J. Biomech. 47 (2014) 3546-3554. A. Polanczyk, M. Podyma, L. Stefanczyk, W. Szubert, I. Zbicinski, J. Biomech. 48 (2015) 425-431. J. A. Ekaterinaris, C.V. Ioannou, A.N. Katsamouris, H. Greece, Annals of Vascular Surgery 20 (2006) 351-359. C.M. Scotti, E.A. Finol, Computers and Structures 85 (2007) 1097–1113. D.C. Barber, E. Oubel, A.F. Frangi, D.R. Hose, Medical Image Analysis 11 (2007) 648–662. F.P.P. Tan, A. Borghi, R.H. Mohiaddin, N.B. Wooda, S. Thom, X.Y. Xu, Computers and Structures 87 (2009) 680–690. X. Wangn, X. Li, Computers in Biology and Medicine 41 (2011) 812–821. A. Sheidaei, S.C. Hunley, S. Zeinali-Davarani, L.G. Raguin, S. Baek, Medical Engineering & Physics 33 (2011) 80–88.

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[15] M. Malvè, A. García, J. Ohayon, M.A. Martínez, International Communications in Heat and Mass Transfer 39 (2012) 745–751. [16] M. Piccinelli, C. Vergara, L. Antiga, L. Forzenigo, P. Biondetti, M. Domanin, Biomech. Model Mechanobiol. 12 (2013) 1263–1276. [17] K. Sughimoto, Y. Takahara, K. Mogi, K. Yamazaki, K. Tsubota, F. Liang, H. Liu, Heart Vessels 29 (2014) 404–412. [18] T.I. Józsa, G. Paál, International Journal of Heat and Fluid Flow 50 (2014) 342–351. [19] A. Javadzadegan, B. Fakhim, M. Behnia, M. Behnia, European Journal of Mechanics B/Fluids 46 (2014) 109–117. [20] S. Karimi, M. Dabagh, P. Vasava, M. Dadvar, B. Dabir, P. Jalali, Journal of NonNewtonian Fluid Mechanics, Volume 207 (2014) 42–52. [21] E. Soudaha,G. Vilaltab, M. Bordonea, F. Nietod, J.A. Vilaltacy, C. Vaquero, Rev. int. métodos numér. cálc. diseño ing. 31 (2015) 106–112. [22] R. Nagy, C. Csobay-Novák, A. Lovas, P. Sótonyi, I. Bojtár, J. Biomech.48 (2015) 1876–1886. [23] K.H. Fraser, S. Meagher, J.R. Blake, W.J. Easson, P.R. Hoskins, Ultrasound in Med. & Biol. 34 (2008) 73–80.

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Chapter 4

NUMERICAL RESOLUTION OF THE NAVIER-STOKES EQUATIONS FOR THE BLOOD FLOW IN INTRACRANIAL ANEURYSMS: A 3D APPROACH USING THE FINITE VOLUME METHOD Alejandro Acevedo-Malavé Centro Multidisciplinario de Ciencias, Instituto Venezolano de Investigaciones Científicas (IVIC), Mérida, Venezuela

Abstract In this study the Finite Volume method is employed to simulate the blood flow in an intracranial aneurysm. For that purpose, the Navier-Stokes equations are solved in the computational domain of real aneurysms in a specific patient. This computational domain is obtained from MR images with a software designed in our laboratory. This computer program transforms the MR images (in DICOM format) to the volumetric mesh where the NavierStokes equations are going to be solved. The Ansys CFX ® flow solver was used here to model the blood flow through the veins with the aneurysm deformation. The streamlines were computed inside the aneurysm with an inlet flow of 0.6 m/s. It can be seen a circular trajectory in the streamlines with the maximum values at the veins adjacent to the deformation. The wall shear stress was computed and exhibits its intermediate value at the aneurysm zone. Finally, the total pressure on the aneurysm wall is presented. It can be seen that the maximum value of the pressure field is located at the aneurysm place. This represents a considerable risk for the patient studied here.

1. Introduction Cerebral aneurysms are pathologic dilations of the vein's wall that frequently occur near arterial bifurcations in the circle of Willis. The most serious consequence is their rupture and intracranial hemorrhage into the subarachnoid zone, with a high mortality rate. Greater availability and improvement of neuroradiologic techniques have resulted in more frequent detection of cerebral aneurysms before rupture. Because the prognosis for subarachnoid

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hemorrhages is still poor, aneurysmal obliteration is a necessary therapeutic option. Planned aneurysm surgery could benefit from a better understanding of the process of aneurysm formation, progression, and rupture. Hassan et al. [1] proposed a hemodynamic analysis of an adult vein of Galen aneurysms. In this study, the authors applied computational fluid dynamic (CFD) analysis on the numerical grid of a 3D rotational digital subtraction angiogram obtained in a 22-year-old male patient, with an accidentally discovered vein of Galen malformation associated with a single feeder aneurysm, to understand the flow pattern through the two aneurysms and improve a general understanding of hemodynamic characteristics of this variety of fusiform aneurysm. That work provides an example of the application of CFD to 3D digital subtraction angiography for studying the flow pattern in patients with cerebrovascular disease. Shojima et al. [2] studied the magnitude and role of wall shear stress (WSS) on cerebral aneurysm. WSS is one of the main pathogenic factors in the development of saccular cerebral aneurysms. The magnitude and distribution of the WSS in and around human middle cerebral artery (MCA) aneurysms were analyzed using CFD. In this work mathematical models of MCA vessels with aneurysms were created by 3dimensional computed tomographic angiography. CFD calculations were performed by using an original finite-element solver with the assumption of Newtonian fluid property for blood and the rigid wall property for the vessel and the aneurysm. These results suggest that in contrast to the pathogenic effect of a high WSS in the initiating phase, a low WSS may facilitate the growing phase and may trigger the rupture of a cerebral aneurysm by causing degenerative changes in the aneurysm wall. The WSS of the aneurysm region may be of some help for the prediction of rupture. Cebral et al. [3] conducted a study of hemodynamic factors that play an important role in the initiation, growth, and rupture of cerebral aneurysms. This work describes a clinical study of the association between intra-aneurysmal hemodynamic characteristics of CFD models and the rupture of cerebral aneurysms. A simple flow characterization system was proposed, and interesting trends in the association between hemodynamic features and aneurysmal rupture were found. Simple stable patterns, large impingement regions, and jet sizes were more commonly seen with unruptured aneurysms. By contrast, ruptured aneurysms were more likely to have disturbed flow patterns, small impingement regions, and narrow jets. Jou et al. [4] proposed a correlation between lumenal geometry changes and hemodynamics in fusiform intracranial aneurysms. The authors examined the relationship between hemodynamics and growth of 2 fusiform basilar artery aneurysms in an effort to define hemodynamic variables that may be helpful in predicting aneurysmal growth. Two patients with basilar fusiform aneurysms of a similar size were followed for a 2-year period. The lumenal geometry and inflow and outflow rates were acquired by using MR angiography and velocimetry, respectively. The location of aneurysmal growth was identified by coregistering aneurysm models that were acquired at different times. Hemodynamic descriptors were calculated by using CFD simulations and compared with aneurysm growth pattern. Jialiang et al. [5] applied a CFD analysis on one patient-specific model of cerebral aneurysm and connected vessels constructed from 3D rotational angiography in order to understand the properties of blood flow around aneurysms and wall shear stress. From the patient model analyzed in this investigation, the authors find that the comparatively maximum WSS occurs near the root of the aneurysm, which leads to damage in the aneurysmal wall, and the maximum displacement is at the peak of the aneurysm. Linninger et al. [6] proposed a mathematical model of blood, cerebrospinal fluid and brain dynamics. By using first principles of fluid and solid mechanics a comprehensive model of

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human intracranial dynamics is proposed. Blood, cerebrospinal fluid (CSF) and brain parenchyma as well as the spinal canal are included. The compartmental model predicts intracranial pressure gradients, blood and CSF flows and displacements in normal and pathological conditions like communicating hydrocephalus. The system of differential equations is discretized and solved numerically. Fluid–solid interactions of the brain parenchyma with cerebral blood and CSF are calculated. The model provides the transitions from normal dynamics to the diseased state during the onset of communicating hydrocephalus. In this study predicted results were compared with physiological data from Cine phase-contrast magnetic resonance imaging to verify the dynamic model. Bolus injections into the CSF are simulated in the model and found to agree with clinical measurements. Jia-liang et al. [7] conducted a CFD analysis that is applied to one patientspecific model of the cerebral aneurysm located at the tip of the basilar artery, by which the differences of hemodynamic parameters before and after endovascular treatment are evaluated. Based on the model, the authors show that the flow behavior near the neck of the aneurysm sees great differences after endovascular treatment as compared with that before treatment, which also affects the WSS and the displacement distribution. In addition, the simulation process is based on a series of CFD codes. These results would be used to assess the outcome of endovascular treatment for the aneurysm occlusion. Sforza et al. [8] studied the effects of parent artery motion on the hemodynamics of basilar tip saccular aneurysms and its potential effect on aneurysm rupture. The aneurysm and parent artery motions in two patients were determined from cine loops of dynamic angiographies. The oscillatory motion amplitude was quantified by registering the frames. Patient-specific CFD models of both aneurysms were constructed from 3D rotational angiography images. Two CFD calculations were performed for each patient, corresponding to static and moving models. The motion estimated from the dynamic images was used to move the surface grid points in the moving model. Visualizations from the simulations were compared for WSS, velocity profiles, and streamlines. In both patients, a rigid oscillation of the aneurysm and basilar artery in the antero-posterior direction was observed and measured. The distribution of WSS was nearly identical between the models of each patient, as well as major intra-aneurysmal flow structures, inflow jets, and regions of impingement. The motion observed in pulsating intracranial vasculature does not have a major impact on intraaneurysmal hemodynamic variables. Parent artery motion is unlikely to be a risk factor for increased risk of aneurysmal rupture. Baharoglu et al. [9] conducted a study about aneurysm inflow angle (IA) as a discriminant for rupture in sidewall cerebral aneurysms. CFD analysis showed increasing IA leading to deeper migration of the flow recirculation zone into the aneurysm with higher peak flow velocities and a greater transmission of kinetic energy into the distal portion of the dome. Increasing IA resulted in higher inflow velocity and greater WSS magnitude and spatial gradients in both the inflow zone and dome. Goubergrits et al. [10] proposed a study of statistical WSS maps of ruptured and unruptured middle cerebral artery aneurysms. The goal of this study was to generate and analyze statistical WSS distributions and shapes in MCA saccular aneurysms. Unsteady flow was simulated in seven ruptured and 15 unruptured MCA aneurysms. In order to compare these results, all geometries must be brought in a uniform coordinate system. For this, aneurysms with corresponding WSS data were transformed into a uniform spherical shape; then, all geometries were uniformly aligned in three-dimensional space. Subsequently, the authors compared statistical WSS maps and surfaces of ruptured and unruptured aneurysms. No significant (p > 0.05) differences exist between ruptured and

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unruptured aneurysms regarding radius and mean WSS. In unruptured aneurysms, statistical WSS map relates regions with high (greater than 3 Pa) WSS to the neck region. In ruptured aneurysms, additional areas with high WSS contiguous to regions of low (less than 1 Pa) WSS are found in the dome region. In ruptured aneurysms, the authors found significantly lower WSS. The averaged aneurysm surface of unruptured aneurysms is round shaped, whereas the averaged surface of ruptured cases is multi-lobular. These results confirm the hypothesis of low WSS and irregular shape as the essential rupture risk parameters. Farnoush et al. [11] studied the effect of inflow on CFD simulation of cerebral Bifurcation aneurysms. This study investigated hemodynamic effects resulting from changes in the parent artery diameter of bifurcation type aneurysm. CFD analysis was performed on MCA models with various parent artery diameters. Calculations were performed with a steady flow rate (125 ± 12.5 ml/min) at the parent artery inlet. Energy loss (EL) was calculated from pressure and kinetic energy obtained from flow velocity. The results indicate that the high WSS and EL occur in the model with the smallest parent vessel compared to the other models for all three inflows. Results also showed that 10% variation of inflow results in an average of 23 ± 2.9% changes in WSS and 25.5 ± 0.5% changes in energy loss. These results demonstrated that for CFD analysis of MCA bifurcation type aneurysm, upstream parent vessel and inflow evaluation for an individual patient is essential. Miloš et al. [12] conducted a study to analyze the influence of the geometric parameters, blood density, dynamic viscosity and blood velocity on WSS distribution in the human carotid artery bifurcation and aneurysm, the computer simulations were run to generate the data pertaining to this phenomenon. In this work the authors evaluate two prediction models for modeling these relationships: neural network model and k-nearest neighbor model. The results revealed that both models have high predictive ability for this prediction task. The achieved results represent progress in assessment of stroke risk for a given patient data in real time. Li et al. [13] proposed a study of the influence of hemodynamics on recanalization of totally occluded intracranial aneurysms. The authors used CFD analysis and investigated the local hemodynamic characteristics at the aneurysm neck before and after total embolization, attempting to identify hemodynamic risk factors leading to recurrence of totally embolized aneurysms. In this study the overall preembolization blood flow patterns were nearly the same in the recanalized and stable groups, with no significant difference in either the maximum WSS (p = 0.914) or the spatially averaged WSS (p = 0.322) at peak systole at the aneurysm neck. After occlusion, the overall flow pattern changed, and the WSS distribution at the treated aneurysm neck differed between the 2 groups. In all of the 7 recanalized cases, both the maximum WSS and spatially averaged WSS at peak systole at the treated aneurysm neck were higher than those in the aneurysm neck before embolization. In contrast, both parameters were decreased by 70%– 80% of the stable cases. After embolization, both the maximum WSS (p = 0.021) and spatially averaged WSS (p = 0.041) at peak systole at the treated aneurysm neck were higher in the recanalized group than in the stable group. Babiker et al. [14] proposed a study about the influence of stent configuration on cerebral aneurysm fluid dynamics. The authors present a flow study that elucidates the influence of stent configuration on cerebral aneurysm fluid dynamics in an idealized wide-neck basilar tip aneurysm model. Aneurysmal fluid dynamics for three different stent configurations (half-Y, Y and, cross-bar) were first quantified using particle image velocimetry and then compared. CFD simulations were also conducted for selected stent configurations to facilitate validation and provide more detailed characterizations of the fluid dynamics promoted by different stent configurations. In vitro

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results showed that the Y stent configuration reduced cross-neck flow most significantly, while the cross-bar configuration significantly reduced the velocity magnitudes within the aneurysmal sac. The half-Y configuration led to increased velocity magnitudes within the aneurysmal sac at high parent-vessel flow rates. Experimental results were in strong agreement with CFD simulations. Simulated results indicated that differences in fluid dynamic performance among the different stent configurations can be attributed primarily to protruding struts within the bifurcation region.

Figure 1. Magnetic resonance (MR) image of a patient with an intracranial aneurysm.

Sun et al. [15] conducted a comprehensive validation of computational fluid dynamics simulations of in-vivo blood flow in patient-specific cerebral aneurysms. In this study, based on a recently proposed in-vitro quantitative CFD evaluation approach via virtual angiography, the CFD evaluation was extended from phantom to patient studies and patient-specific blood flow rates obtained by transcranial color coded Doppler ultrasound measurements were used to impose CFD boundary conditions. Virtual angiograms (VAs) were constructed which resemble clinically acquired angiograms (AAs). Quantitative measures were defined to thoroughly evaluate the correspondence of the detailed flow features between the AAs and the VAs, and thus, the reliability of CFD simulations. Wu et al. [16] proposed a study of CFD simulations, which were performed to examine the influence of a flow diverters (FD) on the hemodynamics of wide-necked and narrow-necked cerebral aneurysms. An FD with 70% porosity mesh was deployed across the neck of an ideal narrow-necked and wide-necked aneurysm model. The hemodynamics at the aneurysmal sac were changed markedly in both models. At the inflow portion of the aneurysm neck of the narrow-necked aneurysm, the peak velocity and WSS were reduced by 84% and 91%, respectively. By comparison, in the widenecked aneurysm model, the results were 47% and 21%, respectively. This study

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demonstrates that the FD markedly altered the hemodynamic conditions inside intracranial aneurysms, depending on aneurysm morphology. Therefore, hemodynamic modifications should be individually designed for aneurysms with different morphology. Aguilar et al. [17] conducted an evaluation of the influence of coil diameters on the hemodynamics of intracranial aneurysms by using CFD simulations. Three virtual treatments were performed varying packing density and coil surface area, by changing coil diameter. Hemodynamic parameters such as WSS, average velocity and dye concentration at the aneurysm site were qualitatively and quantitatively analyzed. Simulations with same coil area (958 mm2) but different packing density (21.0% and 41.6%) show similar hemodynamic alterations. Besides, the treated model with the highest coil area (1596 mm2) and intermediate packing density (33.2%) exhibited the highest reduction in the studied hemodynamic variability. These findings show that coil area plays an important role in the resistance (pressure and frictional drags) of blood flow through the coils. The aim of this paper is: (i) to demonstrate the feasibility of using patient-specific 3D MR image data from clinical studies to construct corresponding realistic patient-specific CFD models of cerebral aneurysms, (ii) to characterize these intra-aneurysmal flow patterns, and (iii) to explore their possible rupture analyzing the WSS and pressure fields.

2. Governing Equations The governing equations can be given by the continuity Eq. (1) and the Navier-Stokes Eq. (2):

v  0

(1)

1 v   (v   )v   p  v , t  

(2)

where v is the velocity, p is the pressure, η is the dynamic viscosity,  is the density. CFX® (ANSYS® 15.0, ANSYS®, Inc. Southpointe 2600 ANSYS Drive Canonsburg, PA 15317 USA), a flow solver based on the finite volume method, was used to solve Eqs. (1) and (2). A 3D tetrahedral mesh was used for this calculation with 855.838 elements. Inside the CFX module was defined the following constants: (Blood) = 1060 Kg/m3, η (Blood) = 3.5x10-3 Pa.s and the initial value of the velocity at inlet surfaces was 0.6 m/s. A software designed in our laboratory was used to transform the MR images (see Figure 1) to the 3D volumetric mesh (see Figures 2 and 3).

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Figure 2. Isolated structure of cerebral veins for the patient with the aneurysm disease (enclosed with a black square).

The software designed in our laboratory produce an isolated image of the cerebral veins (see Figure 2) that is transformed in a 3D volumetric mesh. This software was specially developed for medical image processing. After this stage, a 3D solid volume is generated from the volumetric mesh with Ansys CFX® post-processor.

Figure 3. A 3D volumetric mesh of the aneurysm model as the input data for the Ansys CFX ® flow solver.

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Figure 4. Solid 3D volume obtained from the volumetric mesh in the post-processor Ansys CFX® module.

3. Segmentation of MR Images CT medical imaging diagnostic tools is most commonly used in the scientific field that provide a wealth of information at an affordable cost. CT was introduced in the early 70s, and allows radiographic reconstruction of anatomical structures, whether bony structures, and in some cases soft structures or other organs. The anatomical segmentation is the process of selecting the edge in the medical imaging area which is desired to convert into a 3D solid. For this study, MR images, taken to a patient who had an intracranial aneurysm were worked in DICOM format, which is the standard format for medical image management. Using a software designed in our laboratory that allows 3d reconstruction from medical imaging, we proceed to the selection of the edges, in this case was selected the area of interest of the aneurysm and also we make a reduction in the model. Once the segmentation has been performed, the 3D volumetric mesh is generated and this file is then imported to the Ansys CFX® flow solver.

4. Modeling of the Blood Flow through the Aneurysm Deformation The blood flow is modeled as an incompressible Newtonian fluid. The governing equations are the incompressible Navier-Stokes and continuity equations (1), (2). Due to a lack of information regarding the distribution of vessel wall elasticity and thickness, the vein walls were assumed to be rigid. Non-Newtonian effects were neglected.

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Figure 5. Front view of streamlines for intra-aneurysmal flow patterns and adjacent veins.

In the calculations reported here, simple stable patterns, large impingement region, and large jet sizes can be seen in the aneurysm zone. In Figures 5 and 6 it can be seen that in the zone of aneurysm deformation there is a circular patron of the velocity field, with a value of 1.11e-3 m/s. On the other hand, at the vein zones the flow field is uniform and the streamlines are approximately parallel between them. In fact the velocity field reaches a value of around 5.32e-1 m/s in all adjacent branches to the aneurysm zone. At the intersection and bifurcation regions, there is a secondary circular flux with a value that is less than the value of the velocity field at the no deformed veins.

Figure 6. Rear view of streamlines for intra-aneurysmal flow patterns and adjacent veins.

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Figure 7. Front view of the WSS distribution at the aneurysm zone and adjacent veins.

The site where WSS reaches its maximum values are marked with red color in Figures 7 and 8. In these zones the blood flow produces a maximum friction with the wall of the veins. It can be seen that for the intersection and bifurcation places that the green color represent the smallest value for the WSS distribution. Because the nature of the blood flow in the aneurysm zone the WSS values are around 1.10e-2 Pa. This intermediate value that is obtained in this calculation is due to the flow patterns at intra-aneurysmal zone. It can be seen in this place that there are slowly differences in the magnitude of the WSS field. In fact at the zones around the central artery the WSS has the lowest value than other zones far from it.

Figure 8. Rear view of the WSS distribution at the aneurysm zone and adjacent veins.

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Figure 9. Front view of the pressure distribution at the aneurysm zone and adjacent veins.

The pressure distribution shows that at the aneurysm zone and the artery at the right hand side that is connected with this part of the system, the pressure field reaches its maximum value. This outcome agrees with the WSS distribution presented here. This result has a very important impact because the patient has the potential risk of the aneurysm rupture. In other works there are results that conclude that in some circumstances and geometries of the aneurysm zone the WSS and the pressure distribution does not represent a risk for the patient [3]. It can be seen that at the intersection and bifurcation places the pressure field has the lowest values, in this place there is a secondary circular flow of blood.

Figure 10. Rear view of the pressure distribution at the aneurysm zone and adjacent veins.

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Conclusion In this work was proposed a numerical CFD model and method based on a specific patient. This model can be constructed from the MR images through a process of segmentation in an efficient manner that allows clinical studies of intra-aneurysmal hemodynamics. From MR images the 3D volumetric mesh was generated. This mesh was the input data for the Ansys CFX® flow solver that allows to resolve numerically the NavierStokes equations. A simple flow snap shot was obtained finding that at the aneurysm place the velocity has a circular pattern and the WSS distribution has an intermediate value. Interesting trends in the pressure field allow the association between hemodynamic features and aneurysmal rupture risk for the specific patient. In this aneurysm model, simple stable patterns (zones where the streamlines are parallel), large impingement regions, and jet sizes were seen.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

T. Hassan, E. V. Timofeev, M. Ezura, T. Saito, A. Takahashi, K. Takayama, and T. Yoshimoto, Am. J. Neuroradiol. 24, 1075 (2003). M. Shojima, M. Oshima, K. Takagi, R. Torii, M. Hayakawa, K. Katada, A. Morita, and T. Kirino, Stroke 35, 2500 (2004). J. R. Cebral, M. A. Castro, J. E. Burgess, R. S. Pergolizzi, M. J. Sheridan, and C. M. Putman, Am. J. Neuroradiol. 26, 2550 (2005). L. D. Jou, G. Wong, B. Dispensa, M. T. Lawton, R. T. Higashida, W. L. Young, and D. Saloner, Am. J. Neuroradiol. 26, 2357 (2005). C. Jialiang, W. Shengzhang, and Y. Wei, D. Guanghong, International Conference on BioMedical Engineering and Informatics (2008). A. A. Linninger, M. Xenos, B. Sweetman, S. Ponkshe, X. Guo, and R. Penn, Math. Biol. 59, 729 (2009). C. Jia-liang, W. Sheng-zhang, D. Guang-hong, J. Hydrodynamics 21, 271 (2009). D. M. Sforza, R. Lohner, C. Putman, and J. R. Cebral, Int. J. Numer. Meth. Biomed. Engng. 26, 1219 (2010). M. I. Baharoglu, C. M. Schirmer, D. A. Hoit, B. Gao, and A. M. Malek, Stroke 41, 1423 (2010). L. Goubergrits, J. Schaller, U. Kertzscher, N. van den Bruck, K. Poethkow, Ch. Petz, H.Ch. Hege, and A. Spuler, J. R. Soc. Interface 9, 677 (2012). A. Farnoush, Y. Qian, and A. Avolio, 33rd Annual International Conference of the IEEE EMBS (2011). R. Miloš, P. Dejan, and F. Nenad, 33rd Annual International Conference of the IEEE EMBS (2011). C. Li, S. Wang, J. Chen, H. Yu, Y. Zhang, F. Jiang, S. Mu, H. Li, and X. Yang, J. Neurosurg. 117, 276 (2012). M. H. Babiker, L.F. Gonzalez, J. Ryan, F. Albuquerque, D. Collins, A. Elvikis, and D. H. Frakes, J. Biomech. 45, 440 (2012).

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[15] Q. Sun, A. Groth, and T. Aach, Medical Physics 39, 742 (2012). [16] Y. Wua, P. Yang, J. Shen, Q. Huang, X. Zhang, Y. Qian, J. Liu, J. Clinical Neuroscience 19, 1520 (2012). [17] M. L. Aguilar, H. G. Morales, I. Larrabide, J. M. Macho, L. San Roman, and A. F. Frangi, 9th IEEE International Symposium on Biomedical Imaging: From Nano to Macro (2012).

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ISBN: 978-1-53610-292-5 © 2017 Nova Science Publishers, Inc.

Chapter 5

NUMERICAL SIMULATION OF THE TURBULENT FLOW AROUND A SAVONIUS WIND ROTOR USING THE NAVIER-STOKES EQUATIONS Sobhi Frikha, Zied Driss*, Hedi Kchaou and Mohamed Salah Abid Laboratory of Electro-Mechanic Systems (LASEM), National Engineering School of Sfax (ENIS), University of Sfax (US), Sfax, Tunisia

Abstract This work lies within the scope of the research which takes place at the Laboratory of ElectroMechanic Systems in the field of the wind turbine. This chapter aims to investigate the effect of several parameters including the computational domain and the turbulence model on the aerodynamic characteristics of the flow around a Savonius wind rotor. For thus, we have developed a numerical simulation using a commercial CFD code. The considered numerical model is based on the resolution of the Navier-Stokes equations in conjunction with the k-ε turbulence model. These equations were solved by a finite volume discretization method. We are particularly interested in visualizing the velocity field, the mean velocity and the static pressure. Our numerical results were compared to those obtained by other authors. The comparison shows a good agreement.

Keywords: Navier-Stokes equations, turbulent flow, CFD, Savonius rotor, overlap, turbulence model

Introduction Renewable energy is energy that is generated from natural processes that are continuously replenished. This includes sunlight, geothermal heat, wind, tides, water, and various forms of biomass. This energy cannot be exhausted and is constantly renewed. In *

E-mail address: [email protected] (Corresponding author).

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recent years, an interest in wind energy has been growing and wind turbines are developed to generate electricity from the kinetic power of the wind. Wind turbines can rotate about either a horizontal or a vertical axis. One advantage of the vertical-axis wind turbines is that the turbine does not need to be pointed into the wind to be effective. The Savonius wind rotor is a type of vertical-axis wind turbine. It is a drag-type device. Although with less efficiency compared with three-bucket wind turbine, the Savonius wind rotor, has the advantage of being compact, economical and aesthetic. This allows it to be easily integrated into buildings. Savonius wind turbines have good starting characteristics, operate at relatively low operating speeds and have the ability to accept the wind from any direction. For several years, many works have significantly improved the performance of Savonius rotors. For example, Kamoji et al. [1] investigated the performance of modified forms of conventional rotors with and without central shaft between the end plates. Menet and Bourabaa [2] tested different configurations of the Savonius rotor and found that the best value of the static torque coefficient is obtained for an incidence angle equal to θ = 45° and a relative overlap equal to e/d = 0.24. They compared their numerical results with those obtained by Blachwell et al. [3] and a good agreement was obtained. Ushiyama and Nagai [4] tested several parameters of the Savonius rotor including gap ratio, aspect ratio, number of cylindrical buckets, number of stages, endplate effects, overlap ratio, and bucket design. The highest efficiency of all configurations tested was 24% for a two-stage, two-bucket rotor. Saha and Rajkumar [5] compared the performance of a bladed metallic Savonius rotor to a conventional semi-circular blade having no twist. The twist produced good starting torque and larger rotational speeds and gives an efficiency of 0.14. The best torque was obtained with blades twisted by an angle α =12.5°. Akwa et al. [6] studied the influence of the buckets overlap ratio of a Savonius wind rotor on the averaged moment and power coefficients by changing the geometry of the rotor. They notice that the maximum device performance occurs for buckets overlap ratios with values close to 0.15. Khan et al. [7] tested different blade profiles of a Savonius rotor both in tunnel and natural wind conditions and they varied the overlap. The highest Cp of 0.375 was obtained for blade profile of S-section Savonius rotor at an optimum overlap ratio of 30%. Driss et al. [8] conducted a computational fluid dynamic study to present the local characteristics of the turbulent flow around a Savonius wind rotor. They compared their numerical results with experimental results and a good agreement was obtained. Ivan Dobrev, et al. [9] studied the flow through Savonius wind turbine type with aspect ratio having equal to almost 1. They made simulation with both 2D and 3D models.CFD analysis was carried out to find the behavior of Savonius wind turbine under flow field condition and performance evaluation. The simulation was validated by the experimental investigation in wind tunnel carried out with PIV (Particle image velocimetry) with rotor azimuthal position. Sharma et al. [10] have studied the performance of three-bucket Savonius rotor by the CFD software Fluent 6. The flow behavior around the rotor was also analyzed with the help of pressure, velocity and vorticity contours, for different overlap ratios. In this context, we are interested in studying the flow around a Savonius wind rotor. For thus, we have developed numerical simulations of the turbulent flow using a CFD code and we have investigated the effect of the computational domain and the turbulence model on the aerodynamic characteristics of the flow.

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1. Numerical Model Computational fluid dynamic (CFD) simulations were conducted using the commercial CFD code Fluent to study the turbulent flow around a Savonius wind rotor. This code is based on solving Navier-Stokes equations with a finite volume discretization method. The standard k-ε turbulence model has been used in the calculations.

1.1. Geometric Parameters and Boundary Conditions The examined Savonius rotor consists of two half-cylinder buckets of diameter d=0.3 m (Figure 1). In this work, we have studied different overlap values equal to (e-e’)/d=0, (ee’)/d=0.1, (e-e’)/d=0.24 and (e-e’)/d=0.3. For the boundary conditions, we take a value of V=9.95 m.s-1for the inlet velocity. For the pressure outlet, a value of p=101325 Pa is considered (Figure 2).

Figure 1. Savonius rotor.

Figure 2. Boundary conditions.

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1.2. Mathematical Formulation The mathematical formulation is based on the Navier-Stokes equations. The equations for the conservation of the mass and momentum for the compressible and incompressible flow positions in the numerical analysis can be written as follows in the Cartesian system. The continuity equation is:

ρ

(ρ ui )

t

xi

0

(1)

The momentum equation is: (ρ ui )

(ρ ui u j )

P

t

xj

xi

xj

μ

uj

uj

2

xj

xi

3

δij

ui

(-ρ ui' u'j )

xi

xj

Fi

(2)

where (-ρ ui' u'j ) is defined by:

ρui' u'j

μt

ui

uj

2

xj

xi

3

ρ k δij

(3)

In the present work, we have used the k-ε turbulence model. The transport equations for the turbulent kinetic energy k and the dissipation rate of the turbulent kinetic energy ε are written as follows:

(ρ k)

(ρ ui k)

t

xi

(ρ ε)

(ρ ui ε)

t

xi

xj

μ

xj

μ

μt

k

σk

xj

μt

ε

σε

xj

C1ε

Gk

ε k

Gk

ρε

C2 ε ρ

(4)

ε2 k

(5)

The turbulent viscosity is defined by:

μt

ρ Cμ

k2 ε

(6)

2. Effect of the Computational Domain For the choice of the numerical model, we have studied the effect of the computational domain. The inlet velocity is equal to V=5 m.s-1. We are interested in the study of the

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computational domain effect. Four different dimensions are considered which are defined by: 2 m x 1 m (Figure3.a), 3 m x 2 m (Figure3.b), 4.5 m x 3.5 m (Figure3.c) and 6 m x 4 m (Figure3.d). In these different cases, the number of cells is equal respectively to 5000, 15000, 40000 and 60000.

2.1. Numerical Results 2.1.1. Velocity Field Figure 4 presents the distribution of the velocity fields for different control domains. According to these results, it has been noted a similarity in the distribution of the velocity fields. In the inlet, the flow appears uniform and has a value of V = 5 m.s-1 which is imposed by the boundary conditions. At the meeting of the buckets, a flow deceleration appears at both concave and convex surfaces of the rotor. However, on the attack side, an acceleration of the flow has been observed. In the external attack side of the buckets, the velocity increases. Behind the rotor, there is a rapid deceleration of the velocity and the formation of two recirculation zones downstream of the two buckets. By increasing the computational domain, it has been noted that the maximum values of the velocity decrease. In these conditions, they are equal to V = 11.7 ms-1in the first case (Figure4.a), to V = 8.14 ms-1in the second case (Figure4.b), to V = 7.35 ms-1in the third case (Figure4.c) and to V = 7.03 ms-1in the fourth case (Figure4.d). In the internal attack side of the buckets, some asymmetry has been observed. Indeed, the velocity reaches a large value in the lower attack side compared to the upper attack side.

(a) 5000 cells

(b) 15000 cells

(c) 40000 cells

(d) 60000 cells

Figure 3. Different computational domains.

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(a) 5000 cells

(b) 15000 cells

(c) 40000 cells

(d) 60000 cells

Figure 4. Distribution of the velocity fields.

(a) 5000 cells

(b) 15000 cells

(c) 40000 cells

(d) 60000 cells

Figure 5. Distribution of the mean velocity.

2.1.2. Mean Velocity Figure 5 presents the distribution of the mean velocity for different control domains. In the inlet, the mean velocity is equal to the value imposed by the boundary conditions which is equal to V = 5 ms-1. This value decreases and achieved zero at the meeting of the two buckets. In the internal attack side of the upper bucket, an increase of the mean velocity is followed by a decrease. This fact is due to the infiltration of air between the two buckets. An

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increase of the mean velocity is also observed at the external attack side of the buckets. From these two attack side, two wakes characteristics of the maximum values of the mean velocity are developed. These two wakes extend downstream of the attack zones. Behind the rotor, it has been observed that the value of the mean velocity decreases rapidly. This decrease begins just downstream of the rotor and extends to the limits of the control volume. By comparing these results, it appears that the mean velocity decreases while increasing the computational domain. Indeed, for the four areas studied, the maximum values of the mean velocity are respectively equal to V = 11.7 ms-1in the first case (Figure5.a), to V = 7.91 ms-1in the second case (Figure5.b), to V = 6.92 ms-1in the third case (Figure5.c) and to V = 6.62 ms-1in the fourth case (Figure5.d).

2.1.3. Static Pressure Figure 6presents the distribution of the static pressure for different control domains. According to these results, a high pressure appears upstream of the Savonius rotor which is accentuated at the concave surface of the buckets and on the upper convex portion of the lower bucket of the Savonius rotor. In the concave surface of the lower bucket, it has been observed a rapid decrease in values of the static pressure. However, a depression appears significantly at the convex surface of the upper bucket. The most important areas of depression appear in the two attack zones of the upper bucket of the Savonius rotor. The depression zone extends downstream of the rotor to the output of the control volume. Indeed, it has been noted that the values the static pressure decreases with the increase of the computational domain. Infact, for the four studied domains, the pressure values are equal to p = 103 Pa in the first case (Figure6.a), to p = 40.2 Pa in the second case (Figure6.b), top = 32.2 Pa in the third case (Figure6.c) and to p = 26.6 Pa in the fourth case (Figure6.d).

(a) 5000 cells

(b) 15000 cells

(c) 40000 cells

(d) 60000 cells

Figure 6. Distribution of the static pressure.

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Sobhi Frikha, Zied Driss, Hedi Kchaou et al. Table 1. Static moment coefficients for different domains

CMs

5000 0.98

Cells numbers 15000 40000 0.45 0.34

60000 0.31

Menet and Cottier 0.33

2.2. Comparison with Anterior Results In this section, we are interested in comparing our numerical results with previous work of Menet and Cottier [11]. The values of the static torque coefficients found for different control domains are presented in Table 1 and compared with those found by Menet and Cottier [11]. According to those results, it has been observed that the third domain gives the best results. Indeed, in these conditions, the calculated value is very close to that found in the literature [11].

3. Effect of the Turbulence Model For the choice of the numerical model, we are interested in the study of the standard k-ε and the RNGk-ε turbulence models.

3.1. Numerical Results 3.1.1. Velocity Fields Figure 7 shows the distribution of the velocity fields for different turbulence models. Based on these results, it has been noted that the flow appears uniform and has a value of V = 5 m.s-1 in the inlet of the control volume. However, in the internal attack side of the upper bucket, there is a rapid increase of the velocity values which is followed by a decrease. This increase has been also observed at the external attack zones of the buckets. From both these sides, two wakes characteristics of the maximum values of the velocity are developed. These two wakes extend downstream of the attack zones. Furthermore, a rapid deceleration of the velocity values and the formation of two recirculation zones downstream of the two buckets have been observed. By comparing these two results, it has been noted that the velocity values obtained by the RNG k-ε model are higher than those obtained by the standard k-ε model. Under these conditions, the maximum values of the velocity obtained with the standard k-εmodel and the k-ε RNG model are respectively equal to V = 7.35 ms-1 (Figure 7.a) and to V = 7.7 ms-1 (Figure 7.b).

3.1.2. Mean Velocity Figure 8 shows the mean velocity for different turbulence models. Based on these results, it has been noted that the flow appears uniform and has a value of V = 5 m.s-1in the inlet of the control volume. Upstream of the rotor, there is a decrease of the mean velocity, it reaches zero at the meeting of the two buckets. However, in the internal attack side of the upper

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bucket, there is a rapid increase of the mean velocity values which is followed by a decrease. This increase of the mean velocity has been also observed at the external attack sides of the two buckets. From these two attack zones, the maximum value of the mean velocity is localized in both wake zones. Behind the rotor, it has been noted that the mean velocity value decreases quickly. This decrease begins just downstream of the rotor and extends to the limits of the domain. Comparing these results, it has been observed that the higher values of the mean velocity are obtained with the RNG k-ε model.

3.1.3. Static Pressure Figure 9 shows the distribution of the static pressure for different turbulence models. From these results, it has been observed a surpression zone in the upstream of the Savonius rotor which increases in the concave surface of the upper bucket and on the convex side of the lower bucket. At the concave surface of the lower bucket, there is a rapid decrease of the static pressure values. However, a depression zone appears on the convex surface of the upper bucket. The most important depression zones appear in the two attack zones of the upper bucket of the Savonius rotor. This depression zone extends in the downstream of the rotor to the outlet of the domain. Comparing these results, it appears that the higher values of the static pressure are obtained with the standard k-ε model. Under these conditions, the pressure values obtained with the k-ε model and the RNG k-ε model are respectively equal to p = 32.2 Pa (Figure9.a) and to p = 28 Pa (Figure9.b).

3.2. Comparison with Anterior Results The values of the static torque coefficients found for different turbulence models are presented inTable2 and compared with those obtained by Menet and Cottier [11]. According to those results, it has been observed that the standard k-ε model gives the best results. Indeed, in these conditions, the calculated value is very close to that found by these authors [11]. Table 2. Static moment coefficients for different turbulence models

CMs

Turbulence models Standard k-ε k-ε RNG 0.34 0.38

(a) Standard k-ε

Menet and Cottier 0.33

(b) RNG k-ε

Figure 7. Distribution of the velocity fields.

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Sobhi Frikha, Zied Driss, Hedi Kchaou et al.

(a) Standard k-ε

(b) RNG k-ε

Figure 8. Distribution of the mean velocity.

(a) Standard k-ε

(b) k-ε RNG

Figure 9. Distribution of the static pressure.

Conclusion In this chapter, numerical simulations have been developed to study the turbulent flow around a Savonius wind rotor. Particularly, we are interested in the presentation of the velocity fields, the mean velocity and the static pressure. According to the numerical results, it has been observed that the computational domain and the turbulence model have a direct effect on the aerodynamic characteristics. The numerical static torque coefficients were compared with those given by anterior results. A good agreement was obtained and confirmed the numerical method. This study permits to choose the best computational domain and the best turbulence model. In the future, we propose to study the effect of the incidence angle and the overlap of the buckets.

Nomenclature ε μ μt ρ σk σε Cp C1ε C2ε

dissipation rate of the turbulent kinetic energy, W.kg-1; dynamic viscosity, Pa.s; turbulent viscosity, Pa.s; density, kg.m-3; constant of the k-ε turbulence model; constant of the k-ε turbulence model; coefficient of the power, dimensionless; constant of the k-ε turbulence model, dimensionless; constant of the k-ε turbulence model, dimensionless;

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constant of the k-ε turbulence model, dimensionless; force components, N; production term of turbulence, kg.m-1.s-3; rotor diameter, m; bucket thickness, m; turbulent kinetic energy, J.kg-1; pressure, Pa; velocity components, m.s-1; fluctuating velocity components, m.s-1;

References [1]

Kamoji M. A., Kedare S. B., Prabhu S. V., Experimental investigations on single stage modified Savonius rotor, Applied Energy, Vol. 86, 1064-1073, 2009. [2] Menet J. L., Bourabaa N., Increase in the Savonius rotors efficiency via aparametric investigation. European Wind Energy Conference, London, 2004. [3] Blackwell B. F., Sheldahl R. E., Feltz L. V., Wind Tunnel performance data for two and three-bucket Savonius rotor. Journal of Energy, 2-3 160-164, 1978. [4] Ushiyama I., Nagai H., Optimum design configurations and performances of Savonius rotors. Wind Eng. 12-1, 59-75, 1988. [5] Saha U. K., Rajkumar M., 2005, On the performance analysis of Savonius rotor with twisted blades, J. Renew. Energy, pp. 960-1481. [6] Akwa J. V., Júnior G. A., Petry A. P., 2012, Discussion on the verification of the overlap ratio influence on performance coefficients of a Savonius wind rotor using computational fluid dynamics. Renewable Energy, 38, 141-149. [7] Khan N., Tariq I. M., Hinchey M., Masek V., 2009, Performance of Savonius Rotor as Water Current Turbine, Journal of Ocean Technology, 4, N. 2, pp. 27-29. [8] Driss Z., Abid M. S., Numerical Investigation of the Aerodynamic Structure Flow around Savonius Wind Rotor, Science Academy Transactions on Renewable Energy Systems Engineering and Technology, Vol. 2, No. 2, 196-204, 2012. [9] Dobrev Ivan, Massouh Fawaz, CFD and PIV investigation of unsteady flow through Savonius wind turbine, Energy Procedia 6 (2011) 711-720. [10] Sharma K.K. and Gupta R., Flow Field Around Three Bladed Savonius Rotor, International Journal of Applied Engineering Research, Volume 8, Number 15 (2013) pp. 1773-1782© Research India Publications. [11] Menet J.L., Cottier F., Étude paramétrique du comportement aérodynamique d’une éolienne lente à axe vertical de type Savonius, 16è Congrès Français de Mécanique, Nice, 2003.

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In: Handbook on Navier-Stokes Equations Editor: Denise Campos

ISBN: 978-1-53610-292-5 © 2017 Nova Science Publishers, Inc.

Chapter 6

NUMERICAL PREDICTION OF THE EFFECT OF THE DIAMETER OUTLET ON THE MIXER FLOW OF THE DIESEL WITH THE BIODIESEL Mariem Lajnef, Zied Driss*, Mohamed Chtourou, Dorra Driss and Hedi Kchaou Laboratory of Electro-Mechanic Systems (LASEM), National School of Engineers of Sfax (ENIS), University of Sfax (US), Sfax, Tunisia

Abstract In this chapter, computer simulations were developed to predict the effect of the outlet diameter on the mixer flow of the diesel with the biodiesel. The Navier-Stokes equations in conjunction with the standard k-ε turbulence model were considered and solved numerically. The software “SolidWorks Flow Simulation” which uses a finite volume scheme were used to present the local characteristics of the flow. The purpose of this chapter is to select the right choice giving the best performance of the considered mixer.

Keywords: mixer flow, diesel, biodiesel, Navier-Stokes equations, turbulent flow

1. Introduction In industrial process engineering, mixing is a unit operation that involves manipulation of heterogeneous physical system with the intent to make it more homogeneous. Mixing is performed to allow heat and mass transfer to occur between one or more steams, components or phases. Modern industrial processing almost always involves some form of mixing. With the right equipment, it is possible to mix a solid, liquid or gas into another solid, liquid or gas. The type of operation and equipment used during mixing depends on the state of materials *

E-mail address: [email protected]; Address: Laboratory of Electro-Mechanic Systems (LASEM), National School of Engineers of Sfax (ENIS), University of Sfax (US), B.P. 1173, km 3.5, Road Soukra, 3038, Sfax, Tunisia (Corresponding author).

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being mixed (liquid, semi-solid, or solid) and the miscibility of the materials being processed. In this context we distinguish: 



Single-phase blending: Mixing of liquids that are miscible or at least soluble in each other occurs frequently in process engineering (and in everyday life). The momentum of the liquid being added is sometimes enough to cause enough turbulence to mix the two, since the viscosity of both liquids is relatively low. If necessary, a spoon or paddle could be used to complete the mixing process. Multi-phase mixing: Mixing of liquids that are not miscible or soluble in each other often necessitates different equipment than is used for single-phase blending.

In this application, we consider the diesel and biodiesel mixture in specific proportions. View the approximation of the chemical properties between them, they can be considered as two miscible liquids. This function is provided by the mixer and it is in our case a singlephase blending. In this context, Zalc et al. (2002) explored laminar flow in an impeller stirred tank using CFD tools. They extended the analysis to include short and long time mixing performance as a function of the impeller speed. The simulated flow fields are validated extensively by particle image velocimetry (PIV). Also, they used planar laser induced fluorescence (PLIF) to compare the experimental and computed mixing patterns. Deglon and Meyer (2006) investigated the effect of grid resolution and discretization scheme on the CFD simulation of fluid flow in a baffled mixing tank stirred by a Rushton turbine. Stitt (2002) noted that multiphase reactor designs from larger scale and non-catalytic processes are now being considered. These include trickle beds, bubble columns and jet or loop reactors. Montante et al. (2001) studied the recirculation zone of the flows in the model of transition in a tank agitated by the technique using laser Doppler anemometry (LDA). They studied the influence of the position of the turbine compared to the bottom and the influence of the baffles on the hydrodynamics of the flows in stirred tanks with a Rushton turbine in order to define an optimal position for a maximum axial velocity. Murthy and Joshi (2008) tested five impeller designs namely disc turbine, variety of pitched blade down flow turbine impellers varying in blade angle and hydrofoil impeller. Alcamo et al. (2005) computed by large-eddy simulation (LES) the turbulent flow field generated in an unbaffled stirred tank by a Rushton turbine. Alvarez et al. (2002) studied a stirred tank system with a single Rushton impeller mounted in a central shaft. Using UV visualization techniques, they illustrated the 3D mechanism by which fluorescent dye is dispersed within the chaotic region of the tank. Also, they compared a system of three Rushton impellers with a system having three discs at the same locations. Guillard and Trägardh (2003) designed and tested a new model for estimating mixing times in aerated stirred tanks with three reactors which were equipped with two, three and four Rushton impellers. The results showed that the analogy model developed is independent of the scale, the geometry of the tank, the number of impellers used, the distance between impellers and the degree of homogeneity considered. Only the region in which the pulses were added was found to affect the results. Brucato et al. (1998) studied the turbulent flow generated by one and two Rushton turbines in different axial positions. They studied the effect of the grid scaling on the assessment of the three velocity components, the turbulent kinetic energy, the dissipation rate and the evolution of power number. Ammar et al. (2011) analyzed numerically the effect of the baffles length on the turbulent flows in stirred tanks

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equipped by a Rushton turbine. The numerical results from the application of the CFD code Fluent with the MRF model are presented in the vertical and horizontal planes in the impeller stream region. Driss et al. (2010) developed a computational study of the pitched blade turbines design effect on the stirred tank flow characteristics. Particularly, they studied the effects of different inclined angle, equal to 45°, 60° and 75°, on the local and global flow characteristics. Kchaou et al. (2008) compared the effect of the flat-blade turbine with 45° and -45° pitched blade turbines on the hydrodynamic structure of the stirred tank. Chtourou et al. (2011) interested to provide predictions of turbulent flow in a stirred vessel and to assess the ability to predict the dissipation rate of turbulent energy that constitutes a most stringent test of prediction capability due to the small scales at which dissipation takes place. The amplitude of local and overall dissipation rate is shown to be strongly dependent on the choice of turbulence models. In this chapter, we are interested on the study of the mixer characteristics and the effect of the diameter outlet on the mixer flow of the diesel with the biodiesel. Using the software “Solid works Flow Simulation,” we are interested on the study of the effect of its variation on mixing parameters such as velocity, pressure, vorticity and turbulence characteristics. The purpose of this chapter is to select the right choice giving the best performance.

2. Mixer Geometrical Model Figure 1 presents the different views of the considered mixer. The two inlet sections are characterized by the inlet diameters D1 = 6 mm and D2 = 24 mm and the length L = 86 mm. However, the outlet section is characterized by the second length l = 28 mm and by an outlet diameter; witch can change from one case to another. In the present applications, we have considered the outlet diameters equal to D = 4 mm, D = 8 mm, D = 8 mm, D = 10 mm (Figure 2).

(a) Overview

(b) Front view

(c) Top view

(d) Right view

Figure 1. Mixer views.

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Parameters L l D1 D2 D

Values 86 mm 28 mm 6 mm 24 mm Variable

Figure 2. Schematic representation of the mixer.

3. Boundary Conditions A boundary condition is required where fluid enters or exits the model and can be specified as a pressure, mass flow rate, volume flow rate or velocity. In our case, for the inlet mass flow, we have taken a value of 0.01 kg.s-1, and for the outlet pressure a value of p = 101325 Pa. That means at the opening side, the fluid exits the model to an area of atmospheric pressure. A summary of the boundary conditions is given in Figure 3.

4. Mesh Resolution Flow Simulation calculates the default minimum gap size using information about the faces where boundary conditions; as well as sources or fans; and goals are specified. Thus, it is recommended to set all conditions before starting to analyze the mesh. The Minimum gap size is a parameter governing the computational mesh, so that a certain number of cells per the specified gap should be generated. To satisfy this condition, the corresponding parameters governing the mesh are set by Flow Simulation as like as number of basic mesh cells, small solid features refinement level and narrow channel resolution. These parameters are applied to the whole computational domain, resolving all its features of the same geometric characteristics and not only to a specific gap. The basic mesh in many respects governs the generated computational mesh. The proper basic mesh is necessary for the most optimal mesh. So, we can control the basic mesh in several ways by changing the number of the basic mesh cells along the x, y and z axes, by shifting or inserting basic mesh planes and stretching

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or contracting the basic mesh cells locally by changing the relative distance between the basic mesh planes. The geometry can be resolved reasonably well. However, if we generate the mesh and zoom in a thin region, we will see that it may still un-solved. In order to resolve these regions properly, we take different minimum size of interval value and then control the time and more specifically number of cellules solved. The mesh that gives an acceptable execution time and a graphics resolution part a maximum number of model cells will be select. Table 1 and Figure 4 present the results according to each used mesh. As it is shown, the 0.00005 m minimum size of interval gives the most important number of fluid and partial cells. For thus, we confirms that it is the best mesh used as all the geometrical model of the mixture will be resolved. So, the corresponding simulation will give more accurate results. Figure 3.5 presents the selected mesh in different views.

Figure 3. Boundary conditions.

(a) N = 1756

(b) N = 3916

(c) N = 33761

(d) N = 80430

Figure 4. Mesh variation distribution.

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Mariem Lajnef, Zied Driss, Mohamed Chtourou et al. Table 1. Mesh variation results Study cases 1 2 3 4

Minimum size 0.005 m 0.0005 m 0.0001 m 0.00005 m

Fluid cells 1756 3916 33761 80430

Partial cells 2984 6710 51717 116933

(a) Overview

(b) Front view

(c) Top view

(d) Right view

Execution time 0 h : 1 mn : 30 s 0 h : 5 mn: 00 s 0 h : 20 mn: 00 s 1 h : 10 mn : 00 s

Figure 5. Selected mesh views.

5. Numerical Results As presented in Figure 6, four planes defined by X = 0 mm, Z = -12 mm, Y = 25 mm and Z = 32 mm are considered to visualize the Velocity fields, the velocity flow trajectories, the dynamic pressure, the total pressure, the vorticity, the turbulent kinetic energy, the dissipation rate of the turbulent kinetic energy and the turbulent viscosity.

5.1. Average Velocity Figures 7, 8, 9 and 10 present the distribution of the average velocity in the visualization planes defined respectively by X = 0 mm, Z = -12 mm, Y = 25 mm and Z = 32 mm. In each plane, the effect of the outlet diameter variation in the average velocity parameter has been observed. According to these results, it has been noted that the velocity is weak in the inlet of the entry. For the four selected planes, it has been noted that the average velocity decreases with the increase of the output section. Indeed, for a constant input velocity and an increasing diameter output, the velocity is inversely proportional to the output variation section. Indeed

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for a given output section, the mixture follows a well-determined path. As soon as the output section decreases, there is a discharge of fluid to the input and then an increase of the fluid contact and therefore a variation in the velocity input.

(a) X = 0 mm

(b) Z = -12 mm

(c) Y = 25 mm

(d) Z = 32 mm

Figure 6. Visualizations planes.

(a) D = 4 mm

(b) D = 6 mm

(c) D = 8 mm

(d) D = 10 mm

Figure 7. Distribution of the average velocity (m.s-1) in the plane X = 0 mm.

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(a) D = 4 mm

(b) D = 6 mm

(c) D = 8 mm

(d) D = 10 mm

Figure 8. Distribution of the average velocity (m.s-1) in the plane Z = -12 mm.

(a) D = 4 mm

(b) D = 6 mm

(c) D = 8 mm

(d) D = 10 mm

Figure 9. Distribution of the average velocity (m.s-1) in the plane Y = 25 mm.

5.2. Velocity Flow Trajectories Figures 11, 12, 13 and 14 present the distribution of the velocity flow trajectories in the visualization planes defined respectively by X = 0 mm, Z = -12 mm, Y = 25 mm and Z = 32 mm. In each plane, we see the effect of variation of the diameter of the outlet on the velocity flow trajectories. According to these results, it has been noted that the velocity flow trajectories variation follow the average velocity as it was previously explained.

5.3. Dynamic Pressure Figures 15, 16, 17 and 18 present the distribution of the dynamic pressure in the visualization planes defined respectively by X = 0 mm, Z = -12 mm, Y = 25 mm and Z = 32

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mm. In each plane, the effect of variation of the diameter of the outlet in the dynamic pressure distribution has been observed. According to these results, it has been noted that for the four planes, there is a compression at the input and output of the fluid area. But the volume occupied by the fluid in the interior of mixer is characterized by a depression zone. When the outlet section increases, the compression at the output decreases while the input one still constant, the same for the depression areas. When we talk about a compression or a depression zones, we mean a little variation that belongs between 10 Pa to 30 Pa.

5.4. Total Pressure Figures 19, 20, 21 and 22 present the distribution of the total pressure in the visualization planes defined respectively by X = 0 mm, Z = -12 mm, Y = 25 mm and Z = 32 mm. In each plane, the effect of the variation of the diameter of the outlet on the total pressure distribution has been observed. According to these results, it has been noted that there are no significant variation along the mixer. In fact, the variation of the total pressure is about 110 Pa, there is only a small compression areas located at the input. We may add, as in the case of the dynamic pressure, more the outlet section increases more the total pressure decreases.

(a) D = 4 mm

(b) D = 6 mm

(c) D = 8 mm

(d) D = 10 mm

Figure 10. Distribution of the average velocity (m.s-1) in the plane Z = 32 mm.

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(a) D = 4 mm

(b) D = 6 mm

(c) D = 8 mm

(d) D = 10 mm

Figure 11. Distribution of the velocity flow trajectories (m.s-1) in the plane X = 0 mm.

(a) D = 4 mm

(b) D = 6 mm

(c) D = 8 mm

(d) D = 10 mm

Figure 12. Distribution of the velocity flow trajectories (m.s -1) in the plane Z = -12 mm.

5.5. Vorticity Figures 23, 24, 25 and 26 present the distribution of the vorticity in the visualization planes defined respectively by X = 0 mm, Z = -12 mm, Y = 25 mm and Z = 32 mm. In each plane, the effect of variation of the diameter of the outlet on the vorticity distribution has been observed. According to these results, it has been noted that the phenomenon of vorticity appears with each

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section variation. Indeed, it is evident after the opening but came a little less at the outputs. We can add that each increase in the output section lead to a decrease in the output vorticity. Although, the change of section does not affect the vorticity distribution at the inlet.

(a) D = 4 mm

(b) D = 6 mm

(c) D = 8 mm

(d) D = 10 mm

Figure 13. Distribution of the velocity flow trajectories (m.s-1) in the plane Y = 25 mm.

(a) D = 4 mm

(b) D = 6 mm

(c) D = 8 mm

(d) D = 10 mm

Figure 14. Distribution of the velocity flow trajectories (m.s-1) in the plane Z = 32 mm.

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(a) D = 4 mm

(b) D = 6 mm

(c) D = 8 mm

(d) D = 10 mm

Figure 15. Distribution of the dynamic pressure (Pa) in the plane X = 0 mm.

(a) D = 4 mm

(b) D = 6 mm

(c) D = 8 mm

(d) D = 10 mm

Figure 16. Distribution of the dynamic pressure (Pa) in the plane Z = -12 mm.

5.6. Turbulent Kinetic Energy Figures 26, 27, 28 and 29 present the distribution of the turbulent kinetic energy in the visualization planes defined respectively by X = 0 mm, Z = -12 mm, Y = 25 mm and Z = 32 mm. In each plane, the effect of variation of the diameter of the outlet on the turbulent kinetic energy distribution has been observed. According to these results, it has been noted that the turbulent kinetic energy is clearly observed at a close distance entered. It may be explained by

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the fact that the vorticity itself appears before these zones. So it’s responsible on the energy creation. Although, the change of section does not affect the turbulence at the inlet.

(a) D = 4 mm

(b) D = 6 mm

(c) D = 8 mm

(d) D = 10 mm

Figure 17. Distribution of the dynamic pressure (Pa) in the plane Y = 25 mm.

(a) D = 4 mm

(b) D = 6 mm

(c) D = 8 mm

(d) D = 10 mm

Figure 18. Distribution of the dynamic pressure (Pa) in the plane Z = 32 mm.

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(a) D = 4 mm

(b) D = 6 mm

(c) D = 8 mm

(d) D = 10 mm

Figure 19. Distribution of the total pressure (Pa) in the plane X = 0 mm.

(a) D = 4 mm

(b) D = 6 mm

(c) D = 8 mm

(d) D = 10 mm

Figure 20. Distribution of the Total pressure (Pa) in the plane Z = -12 mm.

5.7. Dissipation Rate of the Turbulent Kinetic Energy Figures 30, 31, 32 and 33 present the distribution of the dissipation rate of the turbulent kinetic energy in the visualization planes defined respectively by X = 0 mm, Z = -12 mm, Y = 25 mm and Z = 32 mm. In each plane, the effect of variation of the diameter on the outlet of the dissipation rate of the turbulent kinetic energy. According to these results, it has been

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noted that the dissipation rate of the turbulent kinetic energy follows the turbulent kinetic energy. It is logical view that the turbulent dissipation rate is the power that derives from the turbulent kinetic energy.

(a) D = 4 mm

(b) D = 6 mm

(c) D = 8 mm

(d) D = 10 mm

Figure 21. Distribution of the total pressure (Pa) in the plane Y = 25 mm.

(a) D = 4 mm

(b) D = 6 mm

(c) D = 8 mm

(d) D = 10 mm

Figure 22. Distribution of the total pressure (Pa) in plane Z = 32 mm.

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(a) D = 4 mm

(b) D = 6 mm

(c) D = 8 mm

(d) D = 10 mm

Figure 22. Distribution of the vorticity (s-1) in the plane X = 0 mm.

(a) D = 4 mm

(b) D = 6 mm

(c) D = 8 mm

(d) D = 10 mm

Figure 23. Distribution of the vorticity (s-1) in the plane Z = -12 mm.

5.8. Turbulent Viscosity Figures 34, 35, 36 and 37 present the distribution of the turbulent viscosity in the visualization planes defined respectively by X = 0 mm, Z = -12 mm, Y = 25 mm and Z = 32 mm. In each plane, the effect of variation of the diameter of the outlet on the turbulent viscosity distribution has been observed. According to these results, it has been noted that the turbulent viscosity appears in the mixing zone. It can be explained by the fact that the

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viscosity is caused by the collision between the particles constituting the blend. We can add that the section change does not affect the viscosity.

(a) D = 4 mm

(b) D = 6 mm

(c) D = 8 mm

Figure 24. Distribution of the vorticity (s-1) in the plane Y = 25 mm.

(a) D = 4 mm

(b) D = 6 mm

(c) D = 8 mm

(d) D = 10 mm

Figure 25. Distribution of the vorticity (s-1) in the plane Z = 32 mm.

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(a) D = 4 mm

(b) D = 6 mm

(c) D = 8 mm

(d) D = 10 mm

Figure 26. Distribution of the turbulent kinetic energy (J.kg-1) in the plane X = 0 mm.

(a) D = 4 mm

(b) D = 6 mm

(c) D = 8 mm

(d) D = 10 mm

Figure 27. Distribution of the turbulent kinetic energy (J.kg-1) in the plane Z = -12 mm.

Conclusion This chapter presents the simulation results applied for the prediction of the influence of the outlet diameter on the mixing parameters. As the mixing operation is important and must be performed in good conditions, the choice of the geometric model of the mixer was demanding. According to the obtained results, we confirm that the 10 mm diameter outlet presents the best characteristics. In fact, in these conditions, the average velocity, the dynamic

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pressure and the vorticity are more inferior. These properties promote the passage of the fluid mixture from the mixer to the tank and hence to the engine.

(a) D = 4 mm

(b) D = 6 mm

(c) D = 8 mm

(d) D = 10 mm

Figure 28. Distribution of the turbulent kinetic energy (J.kg-1) in the plane Y = 25 mm.

(a) D = 4 mm

(b) D = 6 mm

(c) D = 8 mm

(d) D = 10 mm

Figure 29. Distribution of the turbulent kinetic energy (J.kg-1) in the plane Z = 32 mm.

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(a) D = 4 mm

(b) D = 6 mm

(c) D = 8 mm

(d) D = 10 mm

Figure 30. Distribution of the dissipation rate of the turbulent kinetic energy (W.kg-1) in the plane X = 0 mm.

(a) D = 4 mm

(b) D = 6 mm

(c) D = 8 mm

(d) D = 10 mm

Figure 31. Distribution of the dissipation rate of the turbulent kinetic energy (W.kg-1) in the plane Z = -12 mm.

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(b) D = 6 mm

(c) D = 8 mm

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(d) D = 10 mm

Figure 32. Distribution of the dissipation rate of the turbulent kinetic energy (W.kg-1) in the plane Y = 25 mm.

(a) D = 4 mm

(b) D = 6 mm

(c) D = 8 mm

(d) D = 10 mm

Figure 33. Distribution of the dissipation rate of the turbulent kinetic energy (W.kg-1) in the plane Z = 32 mm.

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(a) D = 4 mm

(b) D = 6 mm

(c) D = 8 mm

(d) D = 10 mm

Figure 34. Distribution of the turbulent viscosity (Pa.s) in the plane X = 0 mm.

(a) D = 4 mm

(b) D = 6 mm

(c) D = 8 mm

(d) D = 10 mm

Figure 35. Distribution of the turbulent viscosity (Pa.s) in the plane Z = -12 mm.

(a) D = 4 mm

(b) D = 6 mm

(c) D = 8 mm

(d) D = 10 mm

Figure 36. Distribution of the turbulent viscosity (Pa.s) in the plane Y = 25 mm.

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(a) D = 4 mm

(b) D = 6 mm

(c) D = 8 mm

(d) D = 10 mm

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Figure 37. Distribution of the turbulent viscosity (Pa.s) in the plane Z = 32 mm.

References Alcamo, R., G. Micale, F. Grisafi, A. Brucato, M. Ciofalo (2005), “Large-eddy simulation of turbulent flow in an unbaffled stirred tank driven by a Rushton turbine,” Chemical Engineering Science, 60, 2303-2316. Alvarez, M.M., J.M. Zalc, T. Shinbrot, P.E. Arratia, F.J. Muzzio (2002), “Mechanisms of mixing and creation of structure in laminar stirred tanks,” AIChE Journal, 48, 2135-2148. Ammar, M., Z. Driss, W. Chtourou, M.S. Abid (2011), “Study of the baffles length effect on turbulent flow generated in stirred vessels equipped by a Rushton turbine,” Central European Journal of Engineering, 1(4), 401-412. Brucato A., M. Ciofalo, F. Grisafi, G. Micale (1998), “Numerical prediction of flow fields in baffled stirred vessels: A comparison of alternative modelling approaches,” Chemical Engineering Science, 53, 3653-3684. Chtourou, W., M. Ammar, Z. Driss, M.S. Abid (2011), “Effect of the turbulent models on the flow generated with Rushton turbine in stirred tank,” Central European Journal of Engineering, 1(4), 380-389. Deglon, D.A., C.J. Meyer (2006), “CFD modeling of stirred tanks: Numerical considerations,” Minerals Engineering, 19, 1059-1068. Driss, Z., G. Bouzgarrou, W. Chtourou, H. Kchaou and M.S. Abid (2010), “Computational studies of the pitched blade turbines design effect on the stirred tank flow characteristics,” European Journal of Mechanics B/Fluids, 29, 236-245.

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Guillard, F., C. Trägardh (2003), “Mixing in industrial Rushton turbine agitated reactors under aerated conditions, Chemical Engineering and Processing,” 42, 373-386. Kchaou, H., Driss, Z., G. Bouzgarrou, W. Chtourou and M.S. Abid (2008), “Numerical investigation of internal turbulent flow generated by a flat-blade turbine and a pitchedblade turbine in a vessel tank,” International Review of Mechanical Engineering, 2, 427-434. Montante, G., K.C. Lee, A. Brucato, M. Yianneskis (2001), Experiments and predictions of the transition of the flow pattern with impeller clearance in stirred tanks, Computers and Chemical Engineering, 25, 729-735. Murthy, N.B., J.B. Joshi (2008), “Assessment of standard k-ε RSM and LES turbulent models in a baffled stirred agitated by various impeller designs,” Chemical Engineering Science, 63, 5468-5495. Stitt, E.H. (2002), “Alternative multiphase reactors for fine chemicals: A world beyond stirred tanks,” Chemical Engineering Journal, 90, 47-60. Wu, H. and G.K. Patterson (1989), “Laser doppler measurement of turbulent-flow parameters in a stirred mixer,” Chemical Engineering Science, 44 (10), 2207-2221. Zalc, J.M., E.S. Szalai, M.M. Alvarez, F.J. Muzzio (2002), “Using CFD to understand chaotic mixing in laminar stirred tanks,” AIChE Journal, 48, 2124-2134.

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In: Handbook on Navier-Stokes Equations Editor: Denise Campos

ISBN: 978-1-53610-292-5 © 2017 Nova Science Publishers, Inc.

Chapter 7

COMPUTER SIMULATION OF THE TURBULENT FLOW AROUND A SIX-BLADE RUSHTON TURBINE Zied Driss*, Abdelkader Salah, Abdessalem Hichri, Sarhan Karray and Mohamed Salah Abid Laboratory of Electro-Mechanic Systems (LASEM), National School of Engineers of Sfax (ENIS), University of Sfax (US), Sfax, Tunisia

Abstract In this chapter, we are interested on the study of the turbulent flow around a Rushton turbine. Using the CFD code Ansys-FLUENT, the finite volume method was employed to solve the Navier-Stokes equations. This study was made using the standard k-ε turbulence model. The relative motion between the rotating impeller and the stationary baffle was considered by the multiple reference frames (MRF). The comparison between our numerical results and the experimental results found from the literature shows a good agreement.

Keywords: rushton turbine, hydrodynamic structure, stirred tank, navier-stokes equations, turbulent flow, computer simulations

1. Introduction Computational fluid dynamics (CFD) is the science of determining a numerical solution to governing equations of fluid flow whilst advancing the solution through space or time to obtain a numerical description of the complete flow field of interest. The important aspect in CFD is to understand the linkage between the flow and the design objective. CFD emerges as one of the foremost important subject for designing various types of chemical, mechanical, pharmaceutical and bio-processes. The availability and advancements in commercial CFD software make the area more attractive and comfortable to the engineers, scientists etc, for *

Email: [email protected] (Corresponding author).

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understanding and designing the processes. Many researches, was focused on the optimisation of the design of the stirred tanks and impellers geometry. For example, Driss et al. (2010) developed a computational study of the pitched blade turbines design effect on the stirred tank flow characteristics. Particularly, they studied the effects of different inclined angle, equal to 45°, 60° and 75°, on the local and global flow characteristics. Kchaou et al. (2008) compared the effect of the flat-blade turbine with 45° and -45° pitched blade turbines on the hydrodynamic structure of the stirred tank. Stitt (2002) noted that multiphase reactor designs from larger scale and non-catalytic processes are now being considered. These include trickle beds, bubble columns and jet or loop reactors. Murthy and Joshi (2008) tested five impeller designs namely disc turbine, variety of pitched blade down flow turbine impellers varying in blade angle and hydrofoil impeller. Deglon and Meyer (2006) investigated the effect of grid resolution and discretization scheme on the CFD simulation of fluid flow in a baffled mixing tank stirred by a Rushton turbine. Montante et al. (2001) studied the recirculation zone of the flows in the model of transition in a tank agitated by the technique using laser Doppler anemometry (LDA). They studied the influence of the position of the turbine compared to the bottom and the influence of the baffles on the hydrodynamics of the flows in stirred tanks with a Rushton turbine in order to define an optimal position for a maximum axial velocity. Alcamo et al. (2005) computed by large-eddy simulation (LES) the turbulent flow field generated in an unbaffled stirred tank by a Rushton turbine. Zalc et al. (2002) explored laminar flow in an impeller stirred tank using CFD tools. They extended the analysis to include short and long time mixing performance as a function of the impeller speed. The simulated flow fields are validated extensively by particle image velocimetry (PIV). Also, they used planar laser induced fluorescence (PLIF) to compare the experimental and computed mixing patterns. Alvarez et al. (2002) studied a stirred tank system with a single Rushton impeller mounted in a central shaft. Using UV visualization techniques, they illustrated the 3D mechanism by which fluorescent dye is dispersed within the chaotic region of the tank. Also, they compared a system of three Rushton impellers with a system having three discs at the same locations. Guillard and Trägardh (2003) designed and tested a new model for estimating mixing times in aerated stirred tanks with three reactors which were equipped with two, three and four Rushton impellers. The results showed that the analogy model developed is independent of the scale, the geometry of the tank, the number of impellers used, the distance between impellers and the degree of homogeneity considered. Only the region in which the pulses were added was found to affect the results. Brucato et al. (1998) studied the turbulent flow generated by one and two Rushton turbines in different axial positions. They studied the effect of the grid scaling on the assessment of the three velocity components, the turbulent kinetic energy, the dissipation rate and the evolution of power number. Chtourou et al. (2011) interested to provide predictions of turbulent flow in a stirred vessel and to assess the ability to predict the dissipation rate of turbulent energy that constitutes a most stringent test of prediction capability due to the small scales at which dissipation takes place. The amplitude of local and overall dissipation rate is shown to be strongly dependent on the choice of turbulence models. Ammar et al. (2011) analyzed numerically the effect of the baffles length on the turbulent flows in stirred tanks equipped by a Rushton turbine. The numerical results from the application of the CFD code Fluent with the MRF model are presented in the vertical and horizontal planes in the impeller stream region.

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In this chapter, we are interested on the study of the meshing effect on the hydrodynamic structure of a stirred tank equipped with a Rushton turbine. The flow and the turbulence characteristics are compared with the experimental data of Wu and Patterson (1989).

2. Geometrical Arrangement The system investigated in this study consists of a cylindrical tank with a flat bottom. It had a diameter D of 0.3 m and a height of H = D. A standard six bladed Rushton turbine impeller with a diameter of d = D/3 and a disc diameter T = 3d/4 was placed in the tank. Both the impeller blade height b and the impeller hub diameter c are equal to d/4. The impeller blade width a is equal to d/4. The off-bottom clearance is h = D/3. Four baffles w = D/10 in diameter are equally placed around the tank (Figure 1). This system is chosen as it is more or less a research standard configuration for stirred tanks and is sufficiently small to investigate very fine grids. The water is used as the test fluid and its height is equal to the height of the tank.

Figure 1. Stirred tank equipped with a Rushton turbine.

3. Numerical Methodology Using the CFD code Ansys Fluent 17.0, three-dimensional simulations were carried out for a Reynolds number equal to Re = 40 103, which corresponds to a rotation speed of the impeller equal to N = 215 rpm. In these conditions, three different mesh-grids are tested, under the standard k-ε turbulence model. The discretized equations were solved iteratively using the SIMPLE algorithm for pressure-velocity coupling. The solution was considered converged when the total residuals for the continuity equation was dropped to below 10-5.

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3.1. Impeller Rotation Modeling Modeling the impeller rotation is complex as the relative motion between the rotating impeller blades and the stationary baffles causes a cyclic variation of the solution domain. Two commonly used models are the Multiple Reference Frames (MRF) and the Sliding Mesh (SM) models. In these models, the solution domain is divided into an inner region containing the rotating impeller and an outer region containing the stationary baffles. The simulations were accomplished using the steady-state Multi Reference Frame (MRF) approach since it requires a minimum cost time. In this approach, the grid is divided in two reference frames to account for the stationary and the rotating parts. In the present study, the mesh consists of two frames, one for the tank away from the impeller and one including the impeller. The latter rotates with the rotational speed of the impeller but the impeller itself remains stationary. The unsteady continuity and the momentum equations are solved inside the rotating frame while in the outside stationary frame the same equations are solved in a steady form. At the interface between the two frames, a steady transfer of information is made by the CFD code. One drawback of the MRF approach is that the interaction between the impeller and the baffles is weak.

3.2. Boundary Conditions The computational domain was split into two cylindrical zones, one of which is assumed to rotate with the impeller angular velocity N = 215 rpm, while the remaining space is modeled with a stationary reference. The outer zone was stationary relative to the tank walls. The wall of shaft was also split in two zones. The inner zone which is included in the rotating zone normally rotates with the same velocity as the impeller. The outer zone which is adjacent to the stationary zones should be also rotating with the same angular velocity as the rotating space. Figure 2 shows the sub-division of the computational domain with the MRF approach.

3.3. Discretization Scheme The effect of the discretization scheme on the accuracy of the predicted flow has been investigated elsewhere. For example, Aubin et al. (2004) investigated three discretization methods (First-order upwind, Upwind-central hybrid, QUICK) on a grid of 155000 control volumes and found that the choice of the discretization scheme had no effect on the mean velocities. Moreover, all three discretization schemes were found to under-predict the turbulent kinetic energy. The work of Deglon and Meyer (2006) showed that the general flow field and mean fluid velocity predictions are not strongly influenced by either the grid resolution or discretization scheme. Accurately, with a grid consisting of nearly 2 million control volumes, turbulent kinetic energy predictions are strongly influenced by both the grid resolution and discretization scheme. In the present simulation, we have used the first-order upwind discretization scheme.

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Computer Simulation of the Turbulent Flow around a Six-Blade Rushton Turbine

Figure 2. Sub-divisions of the computational domain.

(a) Grid 1

(b) Grid 2

(c) Grid 3 Figure 3. Meshing volumes for Grids.

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4. Comparison with Experimental Results The domain of integration was meshed with the commercial grid generator package ANSYS MESH ICEM CFD creating a hybrid three dimensional grid. The hybrid grid is actually an unstructured grid that contains different types of elements. The choice of an unstructured grid versus a structured one was made due to the fact that in a complex flow, details of the flow field everywhere in the tank and especially in the discharge area of the impeller and behind the baffles must be captured. The grid was refined to resolve the large flow gradients expected in those regions. The grid was also refined near the impeller blade and baffle surfaces. To determine the grid resolution required to get a solution that is both physically and numerically accurate, it is necessary to perform simulations on successively refined grids until no notable difference in the predicted values of the important solution variables are observed. In our study, grids sensitivity was conducted on three grids of significantly different resolution. The coarse, fine and the finest grid had respectively 279675, 491990 and 1532348 elements (Table 1). Table 1. Grids characteristics

Grid 1 Grid 2 Grid 3

Nodes number 55555 126323 368974

Elements number 279675 491990 1532348

4.1. Velocity Profiles Figures 4 and 5 show the experimental results from the work of Wu and Patterson (1989) superposed with the results from the CFD code relating to agitated tank. Particularly, we are interested on the axial profiles of the normalized mean radial and tangential velocities for a cylindrical area defined by the radial position r/D = 0.185, near the impeller and for different grids. The velocity profiles predicted with the coarsest grid (Grid1) deviate substantially from the results of the finer grids. However, the results of the two finest grids are almost identical. The results of these two grids are comparable with each other and the experimental data of Wu and Patterson (1989). The experimental data is shown as discontinuous points for illustrative purposes. The CFD data actually consists of 8 data points through which a smooth curve has been interpolated. In these conditions, it has been observed that the maximum value of the radial velocity component is located near the impeller and for the non-dimensional position z/H = 0.2 and z/H = 0.4. The negative values of the tangential component present the recirculation loop. For the normalized mean tangential velocity, it has been noted that the result is much more agreement with the third grid. The tangential velocity component shows a maximum value around the impeller stage.

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Figure 4. Axial profile of the normalized mean radial velocity.

Figure 5. Axial profile of the normalized mean tangential velocity.

4.2. Turbulent Kinetic Energy Profile Figure 6 shows the axial profile of the normalized turbulent kinetic energy in a cylindrical area defined by r/D = 0.29, near the impeller and in the impeller discharge stream for different grids. The profiles for all the grids show qualitative agreement with the experimental data but there is a dramatic improvement in the predicted turbulent kinetic energy as the grid resolution increases. Although the results of the finest grid are comparable with the experimental data near the impeller, the CFD model still significantly under-predicts the turbulent kinetic energy in the impeller discharge stream. Nonetheless, it should be noted that the first-order upwind scheme was used in the above simulations and it has been shown to significantly under-predict turbulence in the impeller discharge stream.

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5. Results and Discussion Our simulations results, such as the velocity field, magnitude velocity, static pressure, dynamic pressure, turbulent kinetic energy, dissipation rate of turbulent kinetic, offer local and global information about the impeller mounted on stirred tank. They give a more precise understanding of the hydrodynamic mechanism than those obtained by experimental studies. Since the differences between the grids resolution are not minor, the fine grid was employed for the simulations performed in this work. The numerical results are presented for the finest grid, in six planes: three r-z planes defined by the non-dimensional axial positions equals to z/D = 0.6, z/D = 0.3 and z/D = 0.15 and three r-θ planes defined by the angular positions equals to θ=-30°, θ = 0° and θ = 30° (Figure 7). The turbulent flow is defined by the Reynolds number Re = 4 104 corresponding to the impeller speed equal to N = 215 rpm.

k/Utip

Grid1 Grid2 Grid3 Wu and Patterson (1989)

2

0.1 0.075 0.05 0.025

z/H

0 0

0.25

0.5

0.75

1

Figure 6. Normalized turbulent kinetic energy in the impeller discharge stream

Figure 7. Visualizations Planes.

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5.1. Velocity Fields Figure 8 shows the distribution of the velocity field in the r-z planes defined by the angular positions equals to θ = -30°, θ = 0° and θ = 30°. This distribution shows the presence of a radial jet on the level of the turbine which changes against the walls of the tank with two axial flows thus forming two zones of recirculation on the two sides of the turbine. Also, it has been noted a slowing of the flow far from the Rushton impeller. Figure 9 presents the velocity fields in r-θ planes defined by the adimensional axial positions equal to z/D = 0.6, z/D = 0.3 and z/D = 0.15, According to these results, it is clear that the flow is strongly dominated by the tangential component generated by the impeller. Far from the region swept by the impellers, the rotating movement is no longer transmitted to the fluid, which remains quasi motionless.

Figure 8. Velocity fields in the r-z planes.

5.2. Magnitude Velocity Figures 10 and 11 present the distribution of the magnitude velocity in the r-θ and the r-z planes. Globally, it is has been observed that the wake characteristic of the maximum values is developed in the area swept by the turbine. Also, we find that the magnitude velocity decreases gradually away from the Rushton blade turbine and becomes very low at the bottom and at the top of the tank. In these conditions, the maximum value of the magnitude velocity reaches V=1.59 m.s-1.

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Figure 9. Velocity fields in the r-θ planes.

Figure 10. Distribution of the magnitude velocity in the r-z planes.

Figure 11. Distribution of the magnitude velocity in the r-θ planes.

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5.3. Static Pressure Figures 12 and 13 show the distribution of the distribution of the static pressure respectively in the r-z planes defined by the angular positions θ =-30°, θ = 0° and θ = 30° and the r-θ planes defined by the non-dimensional axial positions z/D = 0.15, z/D = 0.33 and z/D = 0.66. Overall, it has been noted that the compression zones characteristic of the maximum values are developed in the area swept by the turbine and especially around the blades. Also, we find that the static pressure decreases gradually away from the Rushton blade turbine and reaches medium values at the bottom and at the top of the tank. In these condition the maximum value of the static pressure reaches P = 400 Pa.

5.4. Dynamic Pressure Figures 14 and 15 show the dynamic pressure respectively in the r-z planes defined by the angular positions θ =-30°, θ = 0° and θ =30° and the r-θ planes defined by the nondimensional axial positions z/D = 0.15, z/D = 0.33 and z/D = 0.66. According to these results, it is clear that the compression zones characteristics of the maximum values of the dynamic pressure are localized in the vicinity of the turbine. Under these condition, the dynamic pressure decreases rapidly near the walls of the vessels. The maximum value of the dynamic pressure is equal to P = 1.37 103 Pa.

Figure 12. Distribution of the static pressure in the r-z planes.

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Figure 13. Distribution of the static pressure in the r-θ planes.

5.5. Turbulent Kinetic Energy Figures 16 and 17 present the distribution of the turbulent kinetic energy respectively in the r-z planes defined by the angular position θ =-30°, θ = 0° and θ = 30° and the r-θ planes defined by the non-dimensional axial positions z/D = 0.15, z/D = 0.33 and z/D = 0.66. Overall, the wake characteristic of the maximum value of the turbulent kinetic energy is located in the area swept by the turbine. Away from this area, the turbulent kinetic energy decreases gradually. At the level of the baffles and the baffle bottom of the tank, the turbulent kinetic energy increases slightly. The maximum value is equal to 5.96 10-2 J.kg-1.

Figure 14. Distribution of the dynamic pressure in the r-z planes.

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Computer Simulation of the Turbulent Flow around a Six-Blade Rushton Turbine

Figure 15. Distribution of the dynamic pressure in the r-θ planes.

Figure 16. Distribution of the turbulent kinetic energy in the r-z planes.

Figure 17. Distribution of the turbulent kinetic energy in the r-θ planes.

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5.6. Dissipation Rate of the Turbulent Kinetic Energy Figures 18 and 19 show the distribution of the dissipation rate of the turbulent kinetic energy respectively in the r-z planes defined by the angular positions θ = -30°, θ = 0° and θ = 30° a nd the r-θ planes defined by the non-dimensional axial positions z/D = 0.15, z/D = 0.33 and z/D = 0.66. According to these results, it has been noted that the maximums values of the dissipation rate of the turbulent kinetic energy are laid to the blades. Outside of this area, there is a quasi-uniform distribution characterized by turbulent dissipation rate values rather weak. The maximum value is equal to ε = 23.09 m2.s-3.

Figure 18. Distribution of the dissipation rate of the turbulent kinetic energy in the r-z planes.

Figure 19. Distribution of the dissipation rate of the turbulent kinetic energy in the r-θ planes.

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5.7. Turbulent Viscosity Figure 20 shows the distribution of the turbulent viscosity in the r-z planes defined by the angular positions equal to θ=-30°, θ= 0° and θ= 30°. From these results, it has been noted that the wake characteristic of the maximum value of turbulent viscosity is upon the area swept by the turbine. The maximum value is equal to μt = 0.4410 kg.m-1.s-1. Figure 21 presents the distribution of viscosity in the r-θ planes defined by the adimensional axial positions z/D = 0.15, z/D = 0.33 and z/D = 0.66. According to these results, it has been noted that the wake zones with low values appear around the blades of the turbine. Although, the medium values are localized around the wall of the tank and the maximum values are located in the upper zone of the vessel.

Figure 20. Distribution of the turbulent viscosity in the r-z planes..

Figure 21. Distribution of the turbulent viscosity in the r-θ planes.

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Conclusion In this paper, we are interested on the study of the grid resolution effect on the CFD simulation of the fluid flow in a stirred tank equipped with a Rushton turbine. Simulation was investigated under the steady-state mode and using the MRF approach for the impeller rotation model. The accuracy of the CFD model has been evaluated especially in terms of the predicted flow field, mean velocity components and the turbulent characteristics. The CFD model predictions have been compared with the experimental data. Overall, we determinate the numerical conditions in terms of meshing elements. For an accurate CFD simulation, we propose to use the standard k-ε turbulence model under a very fine grid resolution.

References Alcamo, R., G. Micale, F. Grisafi, A. Brucato, M. Ciofalo (2005), “Large-eddy simulation of turbulent flow in an unbaffled stirred tank driven by a Rushton turbine”, Chemical Engineering Science, 60, 2303-2316. Alvarez, M.M., J.M. Zalc, T. Shinbrot, P.E. Arratia, F.J. Muzzio (2002), “Mechanisms of mixing and creation of structure in laminar stirred tanks,” AIChE Journal, 48, 2135-2148. Ammar, M., Z. Driss, W. Chtourou, M.S. Abid (2011), “Study of the baffles length effect on turbulent flow generated in stirred vessels equipped by a Rushton turbine,” Central European Journal of Engineering, 1(4), 401-412. Brucato A., M. Ciofalo, F. Grisafi, G. Micale (1998), “Numerical prediction of flow fields in baffled stirred vessels: A comparison of alternative modelling approaches,” Chemical Engineering Science, 53, 3653-3684. Chtourou, W., M. Ammar, Z. Driss, M.S. Abid (2011), “Effect of the turbulent models on the flow generated with Rushton turbine in stirred tank”, Central European Journal of Engineering, 1(4), 380-389. Deglon, D.A., C.J. Meyer (2006), “CFD modeling of stirred tanks: Numerical considerations”, Minerals Engineering, 19, 1059-1068. Driss, Z., G. Bouzgarrou, W. Chtourou, H. Kchaou and M.S. Abid (2010), “Computational studies of the pitched blade turbines design effect on the stirred tank flow characteristics,” European Journal of Mechanics B/Fluids, 29, 236-245. Guillard, F., C. Trägardh (2003), “Mixing in industrial Rushton turbine agitated reactors under aerated conditions, Chemical Engineering and Processing,” 42, 373-386. Kchaou, H., Driss, Z., G. Bouzgarrou, W. Chtourou and M.S. Abid (2008), “Numerical investigation of internal turbulent flow generated by a flat-blade turbine and a pitchedblade turbine in a vessel tank,” International Review of Mechanical Engineering, 2, 427-434. Montante, G., K.C. Lee, A. Brucato, M. Yianneskis (2001), Experiments and predictions of the transition of the flow pattern with impeller clearance in stirred tanks, Computers and Chemical Engineering, 25, 729-735. Murthy, N.B., J.B. Joshi (2008), “Assessment of standard k-ε RSM and LES turbulent models in a baffled stirred agitated by various impeller designs,” Chemical Engineering Science, 63, 5468-5495.

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Stitt, E.H. (2002), “Alternative multiphase reactors for fine chemicals: A world beyond stirred tanks,” Chemical Engineering Journal, 90, 47-60. Wu, H. and G.K. Patterson (1989), “Laser doppler measurement of turbulent-flow parameters in a stirred mixer,” Chemical Engineering science, 44(10), 2207-2221. Zalc, J.M., E.S. Szalai, M.M. Alvarez, F.J. Muzzio (2002), “Using CFD to understand chaotic mixing in laminar stirred tanks,” AIChE Journal, 48, 2124-2134.

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Chapter 8

STUDY OF THE MESHING CHOICE OF A NEGATIVELY BUOYANT JET INJECTED IN A MISCIBLE LIQUID Oumaima Eleuch1,*, Noureddine Latrache2, Sobhi Frikha1 and Zied Driss1 1

Laboratory of Electro-Mechanic Systems (LASEM), National School of Engineers of Sfax (ENIS), University of Sfax (US), Sfax, Tunisia 2 University of Brest, FRE CNRS 3744 IRDL, Brest, France

Abstract In this chapter, the meshing effect was carried out to study the penetration of a laminar liquid jet with negatively buoyant condition in a miscible surrounding liquid. The numerical resolution of the model, using the computational fluid dynamics (CFD) module, is based on the resolution of the Navier-Stokes equations and the volume of fluid model (VOF model). These equations were solved by a finite volume discretization method using Open Source code provided in Open Foam 2.3.0. Specially, it has been used two Liquid Mixing Foam solver for the case of two miscible liquids. The numerical results consist in presentation of the volume fraction, the magnitude velocity and the dynamic pressure. Particularly, we are interested in the study of the profile of the volume fraction to see the elimination of the numerical diffusion. According to these results, the best model should have the biggest number of cells to have only the physical diffusion of the jet inside the miscible liquid. Predictions of the numerical results have been compared with literature data and a satisfactory agreement has been found.

Keywords: negatively buoyant jet, miscible liquid, meshing, Computational Fluid Dynamic, laminar flow

*

E-mail address: [email protected] (Corresponding author).

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1. Introduction A buoyant jet refers to the general situation of a fluid locally injected into another one (Pantzlaff and Lueptow, 1999). The case of a downward flow in a lighter fluid, or of an upward flow in a denser fluid, is called a positively buoyant jet since inertia of the flow and buoyancy act together in the same direction. Inversely, a negatively buoyant jet corresponds to the reverse case where buoyancy is opposed to the injection flow. This arises in numerous industrial processes or natural flows such as refueling compensated fuel tanks on naval vessels (Friedman and Katz, 1999; Friedman and Katz, 2000), waste disposal systems, ventilation of large buildings (Bains et al. 1990), brine discharge from desalination plants, or motion of plumes and clouds in the atmosphere (Turner, 1966). Such turbulent jets, or turbulent fountains, have been extensively studied. It has become common in the literature to characterize the dynamics of turbulent fountains by relationships between the initial or steady state heights, the Reynolds number and Froude number where Lin and Armfield (2008) investigated the fluid dynamics of transitional plane fountains with variation of Froude and Reynolds numbers. Turbulent forced plumes in both homogeneous and stably stratified ambient fluids have received considerable attention. A ventilated room can naturally form a two-layer stratification and it is of interest to know how cold air injected from below mixes in this environment. Noutsopoulos and Nanou (1986) studied the upward discharge of a buoyant plume into a two-layer stratified ambient and used a stratification parameter that depended on the density differences in the flow to analyze their results. By measuring temperatures in a heated turbulent air jet discharged downward into an air environment, Seban et al. (1978) showed that the centerline temperatures and the penetration depth can be predicted well by theories which consider the downward flow alone. Mizushina et al. (1982) made a similar study of fountains by discharging cold water upward into an environment of heated water and found that the reverse flow had an effect on the axial velocity measurements. They attributed the difference in their result from those of Seban et al. (1978) to the enclosure used by Seban et al. (1978) in their experiments. A theoretical study aimed at incorporating the reverse flow was first undertaken by McDougall (1981) who developed a set of entrainment equations quantifying the mixing that occurs in the whole fountain. The ideas developed in the model of McDougall (1981) have been built upon by Bloomfield and Kerr (2000) by considering an alternative formulation for the entrainment between the upflow and the downflow. Their results for the width of the whole fountain, the centerline velocity and temperature compared favorably with the experiments of Mizushina et al. (1982). Ansong et al. (2007) presented an experimental study of an axisymmetric turbulent fountain in a two layer stratified environment. The efficient method to improve the dilution rate of brine discharged from a desalination plant into the receiver is the study of the inclined negatively buoyant jet. Zeitoun et al. (1972) investigated an inclined jet discharge, focusing on an initial jet angle of 60° because of the relatively high dilution rates achieved for this angle. Roberts and Toms (1987) and Roberts et al. (1997) also focused on the 60° discharge configuration, where both the trajectory and dilution rate were measured. Cipollina et al. (2005) extended the work performed in previous studies on negatively buoyant jets discharged into calm ambient by investigating flows at different discharge angles, namely 30°, 45°, and 60. Kikkert et al. (2007) developed an analytical solution to predict the behavior of inclined negatively buoyant jets, and reasonable agreement was obtained with measurements for initial discharge angles

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ranging from 0° to 75°. Covering the entire range of angles from 0° to 90° were investigated by Jirka (2006). To our knowledge, only a few studies have been dedicated to the case of laminar jets. Lin and Armfield (2000) have investigated weak laminar plane fountains with both a uniform and a parabolic profile of the discharge velocity at the source. The behavior of plane laminar fountains with parabolic velocity inlet profile was studied using numerical simulation over the parametric range 0.25 3 without any nontrivial changes.

Appendix A.

Divergence Problem div u = f in a Rough Domain

Divergence problem is often regarded as fundamental in the study of Navier-Stokes equations. In many references rigorous estimates for solutions of the divergence problem is known, see e.g. [12], [19], Section III.3. In [17], for divergence problem in a kind of rough domain, existence of a solution with estimate irrespective of roughness size is obtained, but the result seems not applicable to our rough domain Ωε given by (2.2)-(2.5).

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In this appendix, we show that the divergence problem in Ωε given by (2.2)-(2.5) has a solution with an estimate independent of ε. Lemma A.1. Let a simply connected and bounded domain G of Rd , d ≥ 2, be expressed as G = G0 ∪

m [

k=1

Gk ,

G0 ∩ Gk 6= ∅,

Gk ∩ Gl = ∅(k 6= l),

k, l = 1, . . ., m,

k| where each Gk has cone-property and |G|G < K with some constant K > 0 for all 0 ∩Gk | k ∈ {0, . . ., m}. Moreover, suppose that for k = 0, . . . , m the problem

div uk = fk in Gk , uk |∂Gk = 0, (A.1) R where fk ∈ Lq (Gk ), 1 < q < ∞, Gk fk dx = 0, has a solution uk ∈ W01,q (Gk ) such that kuk kW 1,q (G

k)

0

≤ c0 kfk kLq (Gk )

with constant c0 independent ofRk. If f ∈ Lq (G), 1 < q < ∞, G f (x) dx = 0, then the divergence problem div u = f

in G,

u|∂G = 0,

(A.2)

(A.3)

has a solution u ∈ W01,q (G) satisfying kukW 1,q (G) ≤ Ckf kLq (G) 0

(A.4)

with constant C = C(c0 , q, K) > 0 independent of m and diam(Gk ), k = 1, . . ., m. Proof. Since the existence of solution u ∈ W01,q (G) for the problem (A.3) satisfying (A.4) is already well-known (e.g. [12] or [19], Theorem III.3.3), we shall show that the constant C in (A.4) is irrespective of m and diam(Gk ), k = 0, . . . , m, and depends only on c0 , q and K. For k = 1, . . ., m let us define fk on Gk by ( f (x) for x ∈ Gk \ G0 , fk (x) = f (x) − ak for x ∈ Gk ∩ G0 , R

R P G f (x) dx where ak = |Gk k ∩G0 | , and let f0 := f − m all k ∈ k=1 fk . Obviously, Gk fk dx = 0 for Pm {0, . . ., m} and, denoting the extension by 0 of fk to G again by fk , we have f = k=0 fk . Then, for k = 1, . . . , m using H¨older inequality and (a + b)q ≤ c¯(q)(aq + bq ) for a, b ≥ 0 we get that R R R q q q Gk |fk | dx = Gk \G0 |f (x)| dx + Gk ∩G0 |f (x) − ak | dx R R ≤ Gk \G0 |f (x)|q dx + c¯(q)( Gk ∩G0 |f (x)|q dx + |ak |q |Gk ∩ G0 |) R R = c¯(q)( Gk |f (x)|q dx + | Gk f (x)dx|q|Gk ∩ G0 |1−q )
R R q−1 k| ≤ c¯(q) Gk |f (x)|q dx + Gk |f (x)|q dx |G|G∩G q−1 0| k R ≤ c¯(q)(1 + K q−1 ) Gk |f (x)|q dx. (A.5)

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Using (A.5) we get that R R Pm R q q q |f (x)| dx + k=1 Gk ∩G0 |f (x) + fk | dx G0 |f0 (x)| dx = G0 \∪m G k k=1 R R P ≤ G0 \∪m Gk |f (x)|q dx + c¯(q) m ( Gk ∩G0 (|f (x)|q + |fk (x)|q ) dx) k=1 k=1 R R P q ≤ G0 \∪m Gk |f (x)|q dx + c¯(q)(1 + c¯(q) + K q−1 ) m k=1 Gk |f (x)| dx k=1 R = c¯(q)(1 + c¯(q) + K q−1 ) G |f (x)|q dx.

(A.6)

Thus, denoting by 0 of uk to G again by uk , we get by cone-property of P the extension 1,q Gk that u := m u ∈ W (Ω) and by (A.2), (A.5) and (A.6) that k=0 k 0 kukq

1,q

W0 (G)

P q + m 1,q k=1 ku0 + uk kW 1,q (Gk ) W0 (G0 \∪m G ) k k=1 P q ku0 kq 1,q + c¯(q) m m k=1 (ku0 kW 1,q (Gk ∩G0 ) W0 (G0 \∪k=1 Gk )
P q q + m c¯(q) ku0 k 1,q k=1 kuk kW 1,q (Gk ) W0 (G0 )
P q c¯(q)cq0 kf0 kqLq (G0 ) + m k=1 kfk kLq (Gk ) 2¯ c(q)cq0 (1 + c¯(q) + K q−1 )kf kqLq (G) .

= ku0 kq ≤

≤

≤

≤

+ kuk kqW 1,q (G ) ) k

Thus, (A.4) is proved with C = c0 (2¯ c(q)(1 + c¯(q) + K q−1 ))1/q .

2

Lemma A.2. Let 1 < q < ∞ and let Gk , k = 1, . . ., m, be the domains of (2.2) satisfying (2.3) and (2.5). Then, the divergence problem (A.1) for k = 1, . . ., m has a solution uk ∈ W01,q (Gk ) satisfying (A.2) with constant c0 independent of k. Proof. The proof of the lemma follows directly by [19], Lemma III.3.2. Here we note that the constant Ck in the statement of [19], Lemma III.3.2 is not defined for k = N . However, one can find, by checking the proof of it, that [19], Lemma III.3.2 holds with the Q −1 1/q−1 new constant CN = N |Di − Ωi|1−1/q ), see (III.3.26) on page 170 of i=1 (1 + |Fi | [19], and hence our assumptions (2.3), (2.5) are enough to apply [19], Lemma III.3.2. As a direct corollary of Lemma A.1 and Lemma A.2 we get the following statement: Corollary A.3. For our domain Ωε introduced in Section 2 the divergence problem div u = f in Ωε , u|∂Ωε = 0, R where f ∈ Lq (Ωε ), 1 < q < ∞, Ωε f (x) dx = 0, has a solution u ∈ W01,q (Ωε ) satisfying kukW 1,q (Ωε ) ≤ Ckf kLq (Ωε ) , 0

with constant C > 0 independent of ε.

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References [1] Achdou, Y. & Pironneau, O. (1995). Domain decomposition and wall-laws. C. R. Acad. Sci. Paris, S´er. I 320, 541-547. [2] Achdou, Y. & Pironneau, O. & Valentin, F. (1998). Effective boundary conditions for laminar flow over periodic rough boundaries. J. Commput. Phys. 147, 187-218. [3] Achdou, Y. & Le Tallec, P. & Valentin, F. & Pironneau, O. (1998). Constructing walllaws with domain decomposition or asymptotic expansion techniques. Comput. Methods Appl. Mech. Engrg. 151, 215-232. [4] Agmon, S. & Douglis, A. & Nirenberg, L. (1964). Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Comm. Pure Appl. Math. 17, 35-92. [5] Allaire, G. (1992). Homogenization and two-scale convergence. SIAM J. Math. Anal. 23, 1482-1518. [6] Allaire, G. & Amar, M. (1999). Boundary layer tails in periodic homogenization. ESAIM : Control, Optimisation and Calculus of Variations 4, 209-243. [7] Amirat, Y. & Climent, B. & Cara, E. F. & Simon, J. (2001). The Stokes equations with Fourier boundary conditions on a wall with asperities. Math. Methods Appl. Sci. 24, 255-276. [8] Amirat, Y. & Bodart, O. & De Maio, U. & Gaudiello, A. (2004). Asymptotic approximation of the solution of the Laplace equation in a domain with highly oscillating boundary. SIAM J. Math. Anal. 35, 1598-1616. [9] Amirat, Y. & Bodart, O. & De Maio, U. & Gaudiello, A. (2013). Effective boundary condition for Stokes flow over very rough boundaries. J. Diff. Eq. 254, 3395-3430. [10] Barrenechea1, R. G. & Le Tallec, P. & Valentin, F. (2002). New wall-laws for the unsteady incompressible Navier-Stokes equations on rough domains, M2AN. 36, 177203. [11] Basson, A. & Gerard-Varet, D. (2008). Wall-laws for fluid flows at a boundary with random roughness. Comm. Pure Appl. Math. 61, 941-987. [12] Bogovskii, M. E. (1980). Solution of some vector analysis problems connected with operators div and grad. Trudy Sem. S. L. Sobolev 80, 5-40. (In Russian.) [13] Bonnetier, E., & Bresch, D., & Miliˇsi´c, V. (2010). A priori convergence estimates for a rough Poisson-Dirichlet problem with natural vertical boundary conditions. In R. Rannacher, & A. Sequeira (Eds.), Advances in Mathematical Fluid Mechanics (pp. 105-134), Berlin, Heidelberg: Springer-Verlag. [14] Bresch, D. & Miliˇsi´c, V. (2010). High order multi-scale wall-laws: Part I, the periodic case. Quart. Appl. Math. 68, 229-253.

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[15] Bresch, D. & Miliˇsi´c, V. Higher order boundary layer corrector and wall-laws derivation: a unified approach, ArXiv:math.AP/0611083v1 3 Nov 2006, [16] Bresch, D. & Miliˇsi´c, V. (2008). Towards implicit multi-scale wall laws. C. R. Acad. Sci. Paris, Ser. I 346, 833-838. ˇ (2008). Influence of wall roughness on the [17] Bucur, D. & Feireisl, E. & Neˇcasov´a, S. slip behaviour of viscous fluids, Proceedings of the Royal Society of Edinburgh 138A, 957-973. [18] Cattabriga, L. (1961). Su un problema al contorno relativo al sistema di equazioni di Stokes. Rend. Sem. Mat. Univ. Padova 31, 308-340. [19] Galdi, G. P. (2010). An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems (Second Edition), New York, Dordrecht, Heidelberg, London: Springer-Verlag. [20] Galdi, G. P. & Simader, C. G. & Sohr, H. (2005). A class of solutions to stationary Stokes and Navier-Stokes equations with boundary data in W −1/q,q . Math. Ann. 331, 41–74. [21] Friedmann, E., & Portl, J., & Richter, T., (2010). A Study of shark skin and its drag reducing mechanism. In R. Rannacher, A. Sequeira (Eds.), Advances in Mathematical Fluid Mechanics, (pp. 271-284), Berlin, Heidelberg: Springer-Verlag. [22] Gerard-Varet, D. (2009). The Navier wall-law at a boundary with random roughness. Commun. Math. Phys. 286, 81-110. [23] J¨ager, W. & Mikeli´c, A. (2000). On the interface boundary condition of Beavers, Joseph, and Saffman. SIAM J. Appl. Math. 60, 1111-1127. [24] J¨ager, W. & Mikeli´c, A. (2001). On the roughness-induced effective boundary conditions for an incompressible viscous flow, J. Diff. Eq. 170, 96-122. [25] J¨ager, W. & Mikeli´c, A. & Neuss, N. (2002). Asymptotic analysis of the laminar viscous flow over a porous bed. SIAM J. Sci. Stat. Comput. 22, 2006-2028. [26] J¨ager, W. & Mikeli´c, A. (2003). Couette flows over a rough boundary and drag reduction, Commun. Math. Phys. 232, 429-455. [27] Lions, J. L. & Magenes, E. (2002). Nonhomogeneous Boundary Value Problems and Applications. Berlin, Heidelberg, New York: Springer-Verlag. [28] Madureira, A. & Valentin, F. (2002). Analysis of curvature influence on effective boundary conditions, C. R. Math. Acad. Sci. Paris, I, 335, 499-504. [29] Madureira, A. & Valentin, F. (2006/2007). Asymptotics of the Poisson problem in domains with curved rough boundaries, SIAM J. Math. Anal. 38, 1450-1473.

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[30] Mikeli´c, A. (2009). Rough boundaries and wall laws. In E. Feireisl, & P. Kaplicky, & J. Malek (Eds.), Qualitative properties of solutions to partial differential equations: Lecture notes of Necas Center for mathematical modeling, Volume 5, (pp. 103-134), Prague: Matfyzpress, Publishing House of the Faculty of Mathematics and Physics Charles University in Prague. [31] Mohammadi, B. & Pironneau, O. & Valentin, F. (1988). Rough boundaries and walllaws. Int. J. Num. Methods Fluids, 27, 169-177. [32] Neuss, N. & Neuss-Radu, M. & Mikeli´c, A. (2006). Effective laws for the Poisson equation on domains with curved oscillating boundaries. Appl. Anal. 85, 479-502. [33] Temam, R. (1979). Navier-Stokes Equations. Amsterdam, New York, Oxford: NorthHolland.

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Chapter 15

S INGULARITIES OF THE N AVIER -S TOKES E QUATIONS IN D IFFERENTIAL F ORM AT THE I NTERFACE B ETWEEN A IR AND WATER Xianyun Wen∗ Institute for Climate and Atmospheric Science, School of Earth and Environment, University of Leeds, Leeds, England, UK

Abstract The properties of the Navier-Stokes equations are extremely important for theoretical and numerical studies of fluid flow. This chapter investigates the properties of the differential and integral forms of the Navier-Stoke equations for immiscible air-water flow. The analysis reveals that unlike other fluids the immiscible air-water flow is so special that the Navier-Stokes equations in the differential form have singularities at the interface due to the continuous movement of interface and discontinuous density. In contrast to the differential form, the Navier-Stokes equations in integral form hold well every where including at the interface, indicating the integral form can be widely used in the computation of air-water flow.

Keywords: Navier-Stokes equations, air-sea interaction, wind-wave interaction, air-water flow AMS Subject Classification: 53D, 37C, 65P

1.

Introduction

Immiscible air-water fluids exist widely and have many applications in practical problems, for example, the air-water flow in river and ocean. In an immiscible interfacial fluid the fluid domain is divided into sub-domains by interfaces, and each sub-domain is occupied by a single phase fluid with constant density and viscosity. The most distinct feature of an immiscible air-water fluid is that the thickness of the interface is zero and the fluid ∗

E-mail address: [email protected]

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Xianyun Wen

density and viscosity suddenly change from one constant to another across the interface. For example, across the interface between air and water the density suddenly jumps from around ρ = 1.0kg/m3 in the air to ρ = 1000kg/m3 in the water. Other features of viscous immiscible air-water flow are that the interface moves continuously and the velocity, pressure, and shear stress are continuous functions across the interface. The Navier-Stokes equations in differential form together with finite difference schemes have been widely used in order to investigate air-water flow in last three decades and twophase model have been developed. In the two-phase model the domains occupied by the water and air is treated as a single domain and the Navier-Stokes equations in differential form are solved in the air and water simultaneously. The two-phase model has been widely used: a comprehensive review of this approach was given by Scardovelli and Zaleski (1999). Since then a large volume of papers have been published based on the two-phase model; the publications of Tryggvason et al. (2001) and Lafrait et al. (2014) are examples. Despite intensive research in the past, seeking analytical or numerical solutions of airwater flow or interfacial (free surface) flow is still of great importance in theoretical research and practical applications. The Navier-Stokes equations are well known in fluid dynamics, but one has not paid their attention to some properties of air-water flow. It will be seen in this paper that the equations with singular derivatives are not suitable for finite difference schemes to be applied for the numerical simulation of fluid flow. The fact that the NavierStokes equations in differential form and finite difference schemes are widely used in the two-phase model indicates that the singularities of the Navier-Stokes equations in differential form should be discussed, the purpose of this paper is to reveal these properties and the paper is organised as follows: § 2 reveals the singularities at the interface of the NavierStokes equations in differential form in the two-phase model, § 3 analyses the validation of the the Navier-Stokes equations in integral form, and § 4 draws the conclusions.

2.

The Navier-Stokes Equations in Differential Form and Singularities at Interface

A two-dimensional air-water flow is shown in Figure 1. The solution domain is occupied by two immiscible incompressible, viscous fluids 1 and 2 separated by an interface. v = ui + vj is the velocity vector and p is the pressure. In the two-phase model the density and viscosity of the fluid are denoted by ρ and µ. In fluid 1 ρ = ρ1 and µ = µ1 , while in fluid 2 ρ = ρ2 and µ = µ2 . In the differential form, the incompressibility condition is expressed by ∂u ∂v + = 0, (1) ∂x ∂y the mass conservation is written as ∂ρ ∂(ρu) ∂(ρv) + + = 0, ∂t ∂x ∂y

(2)

and the momentum equations for (u, v) are given by ∂(ρu) ∂(ρuu) ∂(ρvu) ∂p ∂ ∂u ∂ ∂u + + =− + (µ ) + (µ ) ∂t ∂x ∂y ∂x ∂x ∂x ∂y ∂y

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(3)

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Fluid 2

ρ2 µ2

ρ1

Interface µ1

Fluid 1

Figure 1. The domain occupied by two immiscible fluids separated by an interface. ∂(ρv) ∂(ρuv) ∂(ρvv) ∂p ∂ ∂v ∂ ∂v + + =− + (µ ) + (µ ) − ρg ∂t ∂x ∂y ∂y ∂x ∂x ∂y ∂y where g is gravitational acceleration.

(4)

Fluid 2

ρ2

point A

µ2

ρ1

Interface at time t+∆t

µ1 Fluid 1

Interface at time t

Figure 2. Point A is in fluid 1 at time t and in fluid 2 at time t + ∆t. First, we analyse the singularities of the time-derivatives at the interface. Assume point A is a fixed point in the space shown in Figure 1, the interface is continuously moving. At time t the interface is at the location indicated by the solid line, point A is inside fluid 2, therefore ρ(t) = ρ2 , u(t) = u2 and v(t) = v2 , respectively. During time period ∆t the interface moves to the location indicated by the dashed line. Point A is inside fluid 1 at time t + ∆t, when the density and velocities at point A are ρ(t + ∆t) = ρ1 , u(t + ∆t) = u1 and v(t + ∆t) = v1 . We assume point A always stays between the two interfaces, hence when ∆t tends to zero the two interfaces will move to point A, therefore ∂ρ ρ(t + ∆t) − ρ(t) ρ1 − ρ2 = lim = lim =∞ ∆t→0 ∂t ∆t→0 ∆t ∆t

(5)

Equation (5) indicates that the time derivative of density ∂ρ ∂t is infinite at the interface because when an interface is passing an observer at point A, density ρ at the observer suddenly changes from ρ = ρ2 to ρ = ρ1 within an infinitesimal period of time. ∂(ρu) Now we calculate the time derivative ∂t in equation (3) when the interface is passing the fixed point A. Since velocities u and v are continuous functions they can be assumed to be constant during ∆t, namely u1 = u2 = u and v1 = v2 = v. Thus we have ρ(t + ∆t)u(t + ∆t) − ρ( t)u(t) ∂(ρu) = lim ∆t→0 ∂t ∆t

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Xianyun Wen ρ 1 u1 − ρ 2 u2 ρ1 − ρ2 = lim u =∞ ∆t→0 ∆t→0 ∆t ∆t

= lim

(6)

Similarly we have ∂(ρv) ρ(t + ∆t)v(t + ∆t) − ρ(t)v(t) = lim ∆t→0 ∂t ∆t ρ1 v1 − ρ2 v2 ρ1 − ρ2 = lim = lim v =∞ (7) ∆t→0 ∆t→0 ∆t ∆t Therefore, we conclude that all the time-derivatives in the Navier-Stokes equations (2)-(4) are infinite at the interface, namely the time-derivatives in the Navier-Stokes equations in differential form are singular. Second, we analyse the singularities in the advection terms on the left hand side of equations (2)-(4). As shown in Figure 1 velocities u and v are continuous functions in the solution domain including at the interface, but the density ρ has a jump across the interface. Thus ρu, ρv in equation (2), ρuu, ρvu and ρvv in equations (3) and (4) are all discontinuous functions at the interface. It is well known that the derivative at a discontinuity does not ∂(ρv) ∂(ρuu) (ρuv) ∂(ρvu) exist, hence all the x− and y− derivatives ∂(ρu) and ∂(ρvv) in ∂x , ∂y , ∂x , ∂y , ∂x ∂y equations (2)-(4) do not exist at the interface. Therefore, we have revealed that all terms on the left hand side of (2)-(4) are singular at the interface and we can see that it is the discontinuous density that causes the singularities in the time-derivatives and space-derivatives. The singularities at the interface may cause problems in numerical simulation. For example, if the time-derivative in equation (6) is approximated by a finite difference scheme then ∂(ρu) ρ(t + ∆t)u(t + ∆t) − ρ(t)u(t) ρ 1 u1 − ρ 2 u2 ≈ = (8) ∂t ∆t ∆t From equation (8) we can see that if ρ1 and ρ2 are two constants and velocity is a continuous function, then the value of the finite difference increases as the time-step ∆t decreases and the numerical derivative becomes increasingly large, consequently, the numerical solu∂(ρu) tion may become increasing inaccurate. For the advection terms, for example ∂x , since the space-derivative does not exist at the interface, it is impossible for any finite difference scheme to approximate the non-existing derivative. In fact, the spurious velocity and current near the interface appeared in the publications of Scardovelli and Zaleski (1999), Tryggvason et al. (2001) and Lafrait et al. (2014) are caused by the singularities. When using the differential form of the equations One may suggest using a moving frame. If the shape of the interface does not change in the moving frame then the time derivatives become zero, and the time-derivatives in equations (2)-(4) are valid at the interface. However, the discontinuity of the density still causes the non-existence of the space-derivatives in the moving frame. In general, the shape of the interface constantly changes even in a moving frame. For example, a breaking wave constantly changes its shape during a breaking process and the time-derivatives are always infinite at the interface even in a moving frame.

3.

The Navier-Stokes Equations in Integral Form

Equivalent to the differential form of the Navier-Stokes equations (1)-(4) are the NavierStokes equations in integral form (Hirsch 1998, Ferziger 2002 and Wen 2013). For a finite

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volume Ω in the solution domain the air and water are assumed incompressible, and then the volume of the fluid within the finite volume does not change. Therefore, the fluid volume entering the finite volume through the surface of the finite volume equals that leaving the finite volume through the surface of the finite volume. Corresponding to equation (1) the continuity equation for the volume conservation is given by Z

(n · v)dS = 0

(9)

S

where S is the surface of the finite volume, Ω is the volume of the finite volume, and n = nx i + ny j is the unit vector outward from the surface of the finite volume. When a finite volume Ω contains an interface which is continuously moving, the mass of fluid within the finite volume varies continuously with time t. The mass conservation stated that the rate of change of the total fluid mass within the finite volume is caused by the flow rate of fluid through the surface of the finite volume. Corresponding to equation (2) the continuity equation for mass conservation is then expressed as ∂ ∂t

Z

ρdΩ +

Ω

Z

(ρn · v)dS = 0.

(10)

S

Similarly the momentum equation for u is given by ∂ ∂t

Z

(ρu)dΩ +

Ω

Z

(ρn · v)udS = −

S

Z

S

nx pdS +

Z

τu dS

(11)

τv dS − mg,

(12)

S

and the momentum equation for v is given by ∂ ∂t

Z

Ω

(ρv)dΩ +

Z

(ρn · v)vdS = −

S

Z

S

ny pdS +

Z

S

R u, τ = where m = Ω ρdΩ is the total fluid mass within the finite volume, τu = µ ∂∂ n v v are the shear stress at the surface of the finite volume and ∂ u and ∂ v are directional µ ∂∂n ∂n ∂n derivatives along n direction.

ρ2

µ2

Fluid 2 Ω2

ρ1

µ1

Ω1

Fluid 1 Interface at time t

Figure 3. Interface is continuously moving within a finite volume. We are going to find out whether singularities occur in equations (10)-(12). We consider a finite volume Ω in Figure 3. A continuously moving interface separates Ω into two

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Xianyun Wen

An

ρ2

L

2

Interface

A

A

w

L

e

ρ

1

1

A

s

Figure 4. Rectangular finite volume. volumes Ω1 and Ω2 , thus Ω = Ω1 + Ω2 . The total fluid mass m within the finite volume is calculated by m=

Z

ρdΩ =

Ω

Z

ρdΩ =

Ω1 +Ω2

Z

Z

ρdΩ +

Ω1

Ω2

ρdΩ = ρ1 Ω1 + ρ2 Ω2

(13)

Clearly volumes Ω1 and Ω2 are continuous functions of time t due to the continuous movement of the interface, hence m is a continuous function. We have Z ∂ ∂m ∂Ω1 ∂Ω2 ( ρdΩ) = = ρ1 + ρ2 (14) ∂t Ω ∂t ∂t ∂t From the mean value theorem of integration we have ∂ ∂t

Z

(ρu)dΩ =

Ω

∂ (up ∂t

Z

ρdΩ) = Ω

∂ ∂ ∂ (up m) = ρ1 (up Ω1 ) + ρ2 (up Ω2 ) ∂t ∂t ∂t

(15)

where up is the mean velocity in the finite volume. Similarly we have ∂(ρv) ∂ ∂ ∂ dΩ = (vp m) = ρ1 (vp Ω1 ) + ρ2 (vpΩ2 ) ∂t ∂t ∂t ∂t

Z

Ω

(16)

Since Ω1 , Ω2 and the velocities of the viscous fluid are continuous functions, then up Ω1 , 1 up Ω2 , vp Ω1 and vp Ω2 are also continuous functions, therefore, the time derivatives ∂Ω ∂t , ∂(u Ω ) ∂(u Ω ) ∂(v Ω ) ∂(v Ω ) p 1 p 2 p 1 p 2 ∂Ω2 , ∂t , ∂t , ∂t are all valid. ∂t , ∂t We now prove that space integrations in equations (10)-(12) can be easily calculated. For a rectangular finite volume shown in Figure 4, from equation (10) we have Z

S

(ρn · v)dS =

Z

Ae

ρudS −

Z

ρudS +

Aw

Z

ρvdS −

An

Z

ρvdS = Fe − Fw + Fn − Fs

(17)

As

where Fe , Fw , Fn and Fs are the mass fluxes passing through surfaces Ae , Aw , An and As and they are given by Fe = Fn =

Z

ρudS

ZAe

An

ρvdS

Fw = Fs =

Z

Z Aw As



ρudS   

 ρvdS  

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(18)

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Now we calculate the mass fluxes or F 0 s. We use face Aw in Figure 4 as an example. Suppose L1 is exposed to fluid 1 and L2 is exposed to fluid 2, then from the mean value theorem of integration we have Z

Fw =

Aw

= uw (

Z

ρudS = uw Z

Aw

Z

ρdS +

L1

Z

ρdS = uw

ρdS L1 +L2

ρdS) = uw (ρ1 L1 + ρ2 L2 )

L2

(19)

Then equations in (18) are written as: Fe = ue (ρ1 L1 + ρ2 L2 )e



Fw = uw (ρ1 L1 + ρ2 L2 )w  

Fn = vn (ρ1 L1 + ρ2 L2 )n

Fs = vs (ρ1L1 + ρ2 L2 )s

(20)

 

where ue , uw , vn and vs are the mean velocities at surfaces Ae , Aw , An and As . Wen (2012) made a further discussion on why the mass flues given by (20) can avoid the spurious velocity near the interface. It is clear that L1 and L2 are continuous functions of time t due to the continuous movement of the interface, and then the mass fluxes F 0 s are continuous functions. Substituting equations (16) and (17) into equation (10) leads to ρ1

∂Ω1 ∂Ω2 + ρ2 = −(Fe − Fw + Fn − Fs ) ∂t ∂t

(21)

∂Ω2 0 1 Equation (21) proves that ∂Ω ∂t and ∂t are continuous functions since the mass fluxes F s are continuous functions. R From the mean value theorem of integration, the term S (ρn · v)udS in equation (11) is given by

Z

(ρn · v)udS

Z

=

S

ρuudS −

Ae

=

ue

Z

Aw

Z

ρudS − uw

Ae

=

ρuudS +

Z

Z

ρvudS − An

ρvudS

As

ρudS + un

Aw

Z

Z

ρvdS − us

An

Z

ρvdS

As

ue Fe − uw Fw + un Fn − us Fs

(22)

therefore, the momentum flux terms given by equation (22) are continuous functions since F 0 s and u are continuous functions. Similarly, we can work out the momentum flux terms in (12). The pressure term in equation (11) is given by −

Z

S

nx pdS = −

Z

pdS +

Ae

Z

pdS

(23)

Aw

For viscous fluid the pressure in Equation (23) is a continuous function everywhere in the solution domain. Equation (23) indicates that the total pressure on the interfaces of the finite volume can be calculated by the integration of the pressure. The stress term in equation (11) is given by Z

S

τu dS =

Z

Ae

τu dS +

Z

Aw

τu dS +

Z

An

τu dS +

Z

As

τu dS

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272

Xianyun Wen

h(x)

a

x

0

b

x

Figure 5. Piecewise continuous function h(x). The shear stresses in a viscous fluid are continuous functions even at the interface. Similarly, we can work out the shear stress terms in (12). From equation (20) we can see the density is represented by the constants ρ1 and ρ2 , it is no longer a variable in the integral form. The new variables are Ω1 , Ω2 , L1 and L2 which are all continuous functions due to the continuous movement of the interface. Therefore, all of (10)-(12) are valid. In order to further explain the difference between the differential and integral forms we analyse the difference between integration and differentiation. Shown in Figure 5 is dh a piecewise continuous function h(x) with a jump at point x0 . Clearly the derivative dx not exist at x0 , namely h(x) is not differentiable at point x0 , whereas the integration does Rb a h(x)dx is always integrable even if function h(x) is discontinuous within interval [a,b]. From this example we can see that the Navier-Stokes equations in the integral form are always valid because the integration well adapts to discontinuity whereas the Navier-Stokes equations in the differential form are not valid because differentiation does not tolerate discontinuity.

Conclusion A detailed investigation of the properties of the Navier-Stokes equations for an immiscible viscous multiphase fluid is performed. In the Navier-Stokes equations in differential form, the continuous movement of the interface and discontinuous density cause infinite time derivatives and and non-existing space derivatives at the interface. These singularities at the interface are important features of the Navier-Stokes equations in the differential form. All these shortcomings at the interface indicate the the immiscible multiphase fluid is a very special fluid. The main reason for the singularities to occur at the interface is that the differentiation does not tolerate the discontinuity. It is clear that the equations with singularities are difficult to use in seeking the numerical solution at the interface. Therefore, when the Navier-Stokes equations in the differential form in the two-phase model are deployed the accuracy of the numerical solution is questionable since no finite difference scheme is able to approximate the derivatives which tend to infinity or do not exist.

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The analysis in this chapter proves that the terms in the Navier-Stokes equations in integral form hold very well and are continuous in solution domain including at the interface. Therefore, the Navier-Stokes equations in integral form are recommended when deploying the two-phase model.

References [1] Ferzige, J.H.; Peric, M. Computational Methods for Fluid Dynamics; Springer: New York, 2002 [2] Hirsch, C. Numerical Computation of Internal and External Flows; John Wiley Sons Ltd: Chichester, 1998; Vol.1. [3] Lafrait, A.; Babanin, A. & Onorato, M. J. Computational Physics Vol. 271, 151-171. [4] Scardovelli, R. & Zaleski, S. Annu. Rev. Fluid Mech. Vol. 31, 567-603. [5] Tryggvason, G.; Bunner, B.; Esmaeeli, D.; Juric, D.; Al-Rawahi, N.; Tauber, W.; Han, J.; Nas, S.; & Jan, Y.-J. J. Computational Physics Vol. 169, 708-759. [6] Wen, X. In. J. Numer. Meth. Fluids Vol. 71, 316-338. [7] Wen, X. (2012). The Analytical Expression for the Mass Flux in the Wet/Dry Areas Method. ISRN Applied Mathematics. [doi:10.5402/2012/451693] 2012.

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In: Handbook on Navier-Stokes Equations Editor: Denise Campos

ISBN: 978-1-53610-292-5 c 2017 Nova Science Publishers, Inc.

Chapter 16

S ELF -S IMILAR A NALYSIS OF VARIOUS N AVIER -S TOKES E QUATIONS IN T WO OR T HREE D IMENSIONS I. F. Barna Wigner Research Center of the Hungarian Academy of Sciences, Plasma Physics Department, Budapest, Hungary

Abstract In the following chapter we will shortly introduce the self-similar Ansatz as a powerful tool to attack various non-linear partial differential equations and find - physically relevant - dispersive solutions. Later, we classify the Navier-Stokes (NS) equations into four subsets, like Newtonian, non-Newtonian, compressible and incompressible. This classification is arbitrary, however helps us to get an overview about the structure of various viscous fluid equations. We present our analytic solutions for three of these classes. The relevance of the solutions are emphasized. Lastly, we present an interesting system the Oberbeck-Boussinesq system where the two dimensional NS equation is coupled to heat conduction. This system pioneered the way to chaos studies about half a century ago. Of course, the self-similar solution is presented which helps us to enlighten the formation of the Rayleigh-B´enard convection cells.

PACS: 47.10.ad

1.

Introduction

There are no existing methods which could help us to solve non-linear partial differential equations (PDE) in general. However, two basic linear and time-dependent PDE exist which could help us. The first is the hyperbolic second order wave equation, in one dimension the well-known form is uxx − c12 utt where the subscripts mean the partial derivation, c is the wave propagation velocity (always a finite value), and u is the physical quantity which propagates. Traveling waves - in both directions - are the general solutions of this problem with the form of u(x ± ct). This is a more-or-less common knowledge in the theoretical physics community. A detailed study where traveling waves are used to solve various PDEs is [1].

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I. F. Barna

The other most important linear PDE is the diffusion or heat conduction equation. In one dimension it reads ut = auxx where a is the diffusion or heat conduction coefficient should be a positive real number. The well-known solution is the Gaussian curve which describes the decay and spreading of the initial particle or heat distribution. There is a rigorous mathematical theorem - the strong maximum principle - which states that the solution of the diffusion equation is bounded from above. The problem of this equation is the infinite signal propagation speed, which basically means that the Gaussian curve has no compact support, but that is not a relevant question now. The main point - which is the major motivation of this chapter - is that there exist a natural Ansatz (or trial function) which solves this equation. Namely, the self-similar solution, from basic textbooks the one-dimensional form is well-known [2, 3, 4] x T (x, t) = t−α f β := t−α f (η), (1) t where T (x, t) can be an arbitrary variable of a PDE and t means time and x means spatial dependence. The similarity exponents α and β are of primary physical importance since α gives the rate of decay of the magnitude T (x, t), while β is the rate of spread (or contraction if β < 0 ) of the space distribution as time goes on. The most powerful result of this Ansatz is the fundamental or Gaussian solution of the Fourier heat conduction equation (or for Fick’s diffusion equation) with α = β = 1/2. These solutions are visualized in Figure 1. for different time-points t1 < t2 . Applicability of this Ansatz is quite wide and comes up in various transport systems [2, 3, 4, 5, 6, 7]. Solutions with integer exponents are called selfsimilar solutions of the first kind (and sometimes can be obtained from dimensional analysis of the problem as well). The above given Ansatz can be generalized considering real and continuous functions a(t) and b(t) instead of tα and tβ . This transformation is based on the assumption that a self-similar solution exists, i.e., every physical parameter preserves its shape during the expansion. Self-similar solutions usually describe the asymptotic behavior of an unbounded or a far-field problem; the time t and the space coordinate x appear only in the combination of f (x/tβ ). It means that the existence of self-similar variables implies the lack of characteristic length and time scales. These solutions are usually not unique and do not take into account the initial stage of the physical expansion process. These kind of solutions describe the intermediate asymptotic of a problem: they hold when the precise initial conditions are no longer important, but before the system has reached its final steady state. For some systems it can be shown that the self-similar solution fulfills the source type (Dirac delta) initial condition, but not in every case. These Ans¨atze are much simpler than the full solutions of the PDE and so easier to understand and study in different regions of parameter space. A final reason for studying them is that they are solutions of a system of ordinary differential equations and hence do not suffer the extra inherent numerical problems of the full partial differential equations. In some cases selfsimilar solutions helps to understand diffusion-like properties or the existence of compact supports of the solution. Finally, it is important to emphasize that the self-similar Ansatz has an important but not well-known and not rigorous connection to phase transitions and critical phenomena. Namely to scaling, universality and renormalization. As far as we know even genuine pioneers of critical phenomena like Stanley [8] cannot undertake to give a rigorous clear-cut definitions for all these conceptions, but we fell that all have a common root. The starting

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point could be the generalized homogeneous function like the Gibbs potential Gs (H, ) for a spin system. Close to the critical point the scaling hypothesis can be expressed via the following mathematical rule Gs (λaH, λb) = λGs(H, ). Where H is the order parameter the magnetic field and  is the reduced temperature, a, b are the critical exponents. The same exponents mean the same universality classes. The equation gives the definition of homogeneous functions. Empirically, one finds that all systems in nature belong to one of a comparatively small number of such universality classes. The scaling hypothesis predicts that all the curves of this family M (H, ) can be ”collapse” onto a single curve provided one plots not M versus  but rather a scaled M (M divided by H to some power) vs a scaled  ( divided by H to some different power). The renormalization approach to critical phenomena leads to scaling. In renormalization the exponent is called the scaling exponent. We hope that this small turn-out helps th reader to a much better understanding of our approach. After this successful physical interpretation of this solution we may try to apply it to any kind of PDE system which has some dissipative property (all the NS equations are so) and look what kind of results we get. Unfortunately, there is no direct analytic calculation with the 3 dimensional self-similar generalization of this Ansatz in the literature. Now we show how this generalization is possible and what does it geometrically means.

Figure 1. A self-similar solution of Eq. (1) for t1 < t2 . The presented curves are Gaussians for regular heat conduction. This Ansatz can be generalized for two or three dimensions in various ways one is the following     F (x, y, z) x+y+z −α −α u(x, y, z, t) = t f := t f := t−α f (ω) (2) tβ tβ where F (x, y, z) can be understood as an implicit parameterization of a two dimensional

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I. F. Barna

surface. If the function F (x, y, z) = x + y + z = 0 which is presented in Figure 2. then it is an implicit form of a plane in three dimensions. At this point we can give a geometrical interpretation of the Ansatz. Note that the dimension of F (x, y, z) still have to be a spatial coordinate. Taking the following incompressible NS equation as an example, with this Ansatz we consider all the x coordinate of the velocity field vx = u where the sum of the spatial coordinates are on a plane on the same footing. We are not considering all the R3 velocity field but a plane of the vx coordinates as an independent variable. The NS equation - which is responsible for the dynamics - maps this kind of velocities which are on a surface to another geometry. In this sense we can investigate the dynamical properties of the NS equation truly. In principle there are more possible generalization of the Ansatz available.

Figure 2. The graph of the x + y + z = 0 plane. One is the following: u(x, y, z, t) = t−α f

p

x2 + y 2 + z 2 − a tβ

!

:= t−α f (ω)

(3)

which can be interpreted as an Euclidean vector norm or L2 norm. Now we contract all the x coordinate of the velocity field u (which are on a surface of a sphere with radius a) to a simple spatial coordinate. Unfortunately, if we consider the first and second spatial derivatives and plug them into the NS equation we cannot get a pure η dependent ordinary differential equation(ODE) system some explicit x, y, z or t dependence tenaciously remain. For a telegraph-type heat conduction equation (where is no v∇vterm) both of these Ans¨atze are useful to get solutions for the two dimensional case [7]. Lastly, in the introduction we present a table which classifies the viscous fluid equations - this alignment is arbitrary - however it gives a better and more compact view into the fascinating world of NS equations. The three fields which are marked with ’X’ will be analyzed in details in the following. The fourth group of NS equations is out of our recent

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Table 1. A classification of various NS equations type incompressible compressible

Newtonian X X

non-Newtonian X -

scope. At first we are not sure how to define it in a consistent way. Secondly, this system would be far too complex to analyse with the self-similar trial function.

2.

Incompressible Newtonian Fluids

In Cartesian coordinates and Eulerian description the NS and the continuity equations are the following: ∇v = 0, ∇p vt + (v∇)v = ν4v − +a ρ

(4)

where v, ρ, p, ν, a denote respectively the three-dimensional velocity field, density, pressure, kinematic viscosity and an external force (like gravitation). (To avoid further misunderstanding we use a for external field instead of the letter g which is reserved for a self-similar shape function.) From now on ρ, ν, a are physical parameters of the flow. For a better transparency we use the coordinate notation for the velocity v(x, y, z, t) = u(x, y, z, t), v(x, y, z, t), w(x, y, z, t) and for the scalar pressure variable p(x, y, z, t) ux + vy + wz = 0, px ut + uux + vuy + wuz = ν(uxx + uyy + uzz ) − , ρ py vt + uvx + vvy + wvz = ν(vxx + vyy + vzz ) − , ρ pz wt + uwx + vwy + wwz = ν(wxx + wyy + wzz ) − + a. ρ

(5)

The subscripts mean partial derivations. According to our best knowledge there are no analytic solution for the three dimensional most general case. However, there are numerous examination techniques available in the literature. Manwai [9] studied the N-dimensional (N ≥ 1) radial Navier-Stokes equation with different kind of viscosity and pressure dependences and presented analytical blow up solutions. His works are still 1+1 dimensional (one spatial and one time dimension) investigations. Later, in a book the stability and blow up phenomena of various isentropic Euler-Poisson, Navier-Stokes-Poisson, Navier-Stokes, Euler equations were examined [10]. Another popular and well established investigation method is based on Lie algebra there are numerous studies available. Some of them are even for the three dimensional case, for more details see [11]. Unfortunately, no explicit

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solutions are shown and analyzed there. Fushchich et al. [12] constructed a complete set of ˜ 3)-inequivalent Ans¨atze of co-dimension one for the NS system, they present nineteen G(1, different analytical solutions for one or two space dimensions. They last solution is very closed to our one but not identical, we will come back to these results later. Further two and three dimensional studies based on other group analytical method were presented by Grassi [13]. They also present solutions which look almost the same as ours, but they consider only two space dimensions. We will compare these results to our one at the end of this section. Some years ago, Hu et al. [14] presents a study where symmetry reductions and exact solutions of the (2+1)-dimensional NS were presented. Aristov and Polyanin [15] use various methods like generalized separation of variables, Crocco transformation or the method of functional separation of variables for the NS and present large number of new classes of exact solutions. Sedov in his classical work [2] (Page 120) presented analytic solutions for the tree dimensional spherical NS equation where all three velocity components and the pressure have polar angle dependence (θ) only. Even this restricted symmetry led to a non-linear coupled ordinary differential equation system with has a very nice mathematical structure. Now we concentrate on the first Ansatz (2) and search the solution of the NS PDE system in the following form:     x+y+z x+y+z −α −γ u(x, y, z, t) = t f , v(x, y, z, t) = t g , tβ tδ     x+y+z x+y+z −η w(x, y, z, t) = t− h , p(x, y, z, t) = t l . (6) tζ tθ Where all the exponents α, β, γ, δ, , ζ, η, θ are real numbers. (Solutions with integer exponents are called self-similar solutions of the first kind and non-integer exponents mean self-similar solutions of the second kind.) The functions f, g, h, l are arbitrary and will be evaluated later on. According to the NS system we need to calculate all the first time derivatives of the velocity field, all the first and second spatial derivatives of the velocity fields and the first spatial derivatives of the pressure. We skip these trivial calculations here. Note that both Eq. (5) and Eq. (6) have a large degree of exchange symmetry in the coordinates x, y and z. We want to get an ODE system for all the four functions f (ω), g(ω), h(ω), l(ω) which all have to have the same argument ω. This dictates the constraint that β = δ = ζ = θ have to be the same real number which reduces the number of free parameters, (let’s use the 0 (ω) 0 (ω) β from now on ω = x+y+z ). From this constrain follows that e.q. ux = ftα+β ≈ vy = ftγ+β tβ where prime means derivation with respect to ω. This example clearly shows the hidden symmetry of this construction which may helps us. For the better transparency we present the second equation of (5) after the substitution of the Ansatz (6) −αt−α−1 f (ω) − βt−α−1 f 0 (ω)ω + t−2α−β f (ω)f 0 (ω) + t−γ−α−β g(ω)f 0 (ω) + t−µ−β l 0 (ω) t−−α−β h(ω)f 0 (ω) = ν3t−α−2β f 00 (ω) − . ρ

(7)

To have an ODE which only depends on ω (which is now the new variable instead of time t and the radial components) all the time dependences e.g. t−α−1 have to be zero OR all

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the exponents have to have the same numerical value. After some algebraic manipulation it becomes clear that all the six exponents α − ζ included for the velocity field (the first three functions in Eq. (6)) have to be +1/2. The only exception is the term with the gradient of the pressure. There η = 1 and θ = 1/2 have to be. Now in Eq. (7) each term is multiplied by t−3/2 . Self-similar exponents with the value of +1/2 are well-known from the regular Fourier heat conduction (or for the Fick’s diffusion) equation and gives back the fundamental solution which is the usual Gaussian function. This is a fundamental knowledge, that NS is a kind of diffusion equation for the velocity field. For pressure the η = 1 exponent means, a two times quicker decay rate of the magnitude than for the velocity field.

  2 Figure 3. The KummerM − 14 , 12 , (ω+c) function for c = 1, and ν = 0.1. 6ν

  2 Figure 4. The KummerU − 14 , 12 , (ω+c) function for c = 1, and ν = 0.1. 6ν

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Figure 5. The implicit plot of the self-similar solution Eq. (15). Only the KummerU function is presented for t = 1, c1 = 1, c2 = 0, a = 0, c = 1, and ν = 0.1. The corresponding coupled ODE system is f 0 (ω) + g 0 (ω) + h0 (ω) = 0, 1 l 0 (ω) 1 − f (ω) − ωf 0 (ω) + [f (ω) + g(ω) + h(ω)]f 0 (ω) = 3νf 00 (ω) − , 2 2 ρ l 0 (ω) 1 1 − g(ω) − ωg 0 (ω) + [f (ω) + g(ω) + h(ω)]g 0(ω) = 3νg 00(ω) − , 2 2 ρ l 0 (ω) 1 1 − h(ω) − ωh0 (ω) + [f (ω) + g(ω) + h(ω)]h0 (ω) = 3νh00 (ω) − + a. (8) 2 2 ρ From the first (continuity) equation we automatically get f (ω) + g(ω) + h(ω) = c, and f 00 (ω) + g 00(ω) + h00 (ω) = 0

(9)

where c is proportional with the constant mass flow rate. Implicitly, larger c means larger velocities. After some algebraic manipulation of all the three NS equations we get the final equation 3 c 9νf 00 (ω) − 3(ω + c)f 0 (ω) + f (ω) − + a = 0. (10) 2 2 The solutions are the Kummer functions [16] f (ω) = c1 · KummerU

„

1 1 (ω + c)2 − , , 4 2 6ν

«

+ c2 · KummerM

„

1 1 (ω + c)2 − , , 4 2 6ν

«

+

c 2a − 3 3

(11)

where c1 and c2 are integration constants. The KummerM function is defined by the following series az (a)2 z 2 (a)n z n M (a, b, z) = 1 + + ... + + (12) b (b)2 2! (b)nn! where (a)n is the Pochhammer symbol (a)n = a(a + 1)(a + 2)...(a + n − 1), (a)0 = 1.

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The KummerU function can be defined from the KummerM function via the following form   π M (a, b, z) 1−b M (1 + a − b, 2 − b, z) U (a, b, z) = −z (14) sin(πb) Γ(1 + a − b)Γ(b) Γ(a)Γ(2 − b) where Γ() is the Gamma function. Exhausted mathematical properties of the Kummer function can be found in [16]. Note, that the solution depends only on two parameters where the ν is the viscosity, and c is proportional with the mass flow rate. Figure 3 and Figure 4 show the KummerM and KummerU function for c = 1 and ν = 0.1, respectively. For stability analysis we note that the power series which is applied to calculate the Kummer functions has a pure convergence and a 30 digit accuracy was needed to plot the KummerU function, otherwise spurious oscillations occurred on the figure. (Here we note, that the Malpe 12 Software was used during our analysis.) Note, that for ω = 6.5 the KummerM goes to infinity, and ω → ∞ KummerU function goes to ∞ which is physically hard to understand, which means that the velocity field goes to infinity as well. The complete self-similar solution of the x coordinate of the velocity reads u(x, y, z, t) =

t t

«– −1 1 ((x + y + z)/t1/2 + c)2 f (ω) = t c1 · KummerU + , , 4 2 6ν » „ « – 1 1 ((x + y + z)/t1/2 + c)2 c 2a −1/2 c2 · KummerM − , , + − . (15) 4 2 6ν 3 3 −1/2

−1/2

»

„

In Figure 5 an implicit plot of Eq. (15) is visualized. The KummerU function was presented only, the used parameters are the following c1 = 1, c2 = 0, t = 1, c = 1, ν = 0.1, a = 0. Note, that the initial flat surface of Figure 2 is mapped into a complicated topological surface via the NS dynamical equation. The following phenomena happened, an implicit function is presented, we already mentioned that all the x + y + z = 0 points considered to be the same. Therefore we got a multi-valued surface because for a fixed x numerical value various y+z combinations give the same argument inside the Kummer functions. Unfortunately, this effect is hard to visualize. This can be understand as a kind of fingerprint of a turbulence-like phenomena which is still remained in the equation. An initial simple singlevalued plane surface is mapped into a very complicated multivalued surface. Note, that for a larger value (now we presented KummerU() = 2 case) or for larger flow rate (c=1) the surface got even more structure. Therefore, Figure 5 presents only a principle. At this point we can also give statements about the stability of this solution, the solution the Kummer functions are fine, but for larger flow values a more precise and precise calculation of the solution surface is needed which means larger computational effort which is well known from the application of the NS equation. From the integrated continuity equation (f = c−g−f ) we automatically get an implicit formula for the other two velocity components v(x, y, z, t)+

w t

» „ «– −1 1 ((x + y + z)/t1/2 + c)2 (x, y, z, t) = −t−1/2 c1 · KummerU − , , 4 2 6ν » „ « – 1 1 ((x + y + z)/t1/2 + c)2 c 2a −1/2 c2 · KummerM − , , + − + c. (16) 4 2 6ν 3 3

For explicit formulas of the next velocity component the following ODE has to integrated  ω  g(ω) −3νg 00 (ω) + g 0 (ω) − + c − (17) + F (f 00 (ω), f 0(ω), f (ω)) = 0 2 2

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where F (f 00 (ω), f 0(ω), f (ω)) contains the combination of the first and second derivatives of the Kummer functions. This is a second order linear ODE and the solution can be obtained with the following general quadrature      Z  −c1 + R F (f 00 (ω), f 0(ω), f (ω))dω · exp −ω2 /4+cω  −3ν g(ω) = c2 + dω  × (18)   3ν   −ω 2 /4 + cω exp . (19) 3ν There are additional recursion formulas which help to express the first and second derivatives of the KummerU functions. Such technical details can be found in the original publication [17]. Unfortunately, we could not find any closed form for v(x, y, z, t) and for w(x, y, z, t). Only v the x coordinate of the velocity v field can be evaluated in a closed form in this manner. To overcome this problem, we may call symmetry considerations. The continuity equation states that the sum of all the three velocity component should give a constant. Therefore if every velocity component has the form of (11), we get a solution. As we mentioned at the beginning there are analytic solutions available in the literature which are very similar to our one. Fushchich et al. [12] present 19 different solutions for the full three dimensional NS and continuity equation. (For a better understanding we used the same notation here as well.) For the last (19.) solution they apply the following Ansatz of f (ω) g(ω) y h(ω) l(ω) u(z, t) = √ , v(y, z) = √ + , w(z, t) = √ , p(t, z) = √ (20) t t t t t √ where ω = z/ t is the invariant variable. The obtained ODE is very similar to ours (8) . h0 (ω) + 1 = 0 1 − (f (ω) + ωf 0 (ω)) + h(ω)f 0 (ω) = f 00 (ω), 2 1 (g(ω) + ωg 0 (ω)) + h(ω)g 0 (ω) = g 00(ω), 2 1 − (h(ω) + ωh0 (ω)) + h(ω)h0 (ω) + l 0 (ω) = f 00 (ω). (21) 2 The solutions are     3 1 3 1 1 1 3 −1/2 2 2 f (ω) = ( ω − c) exp − ( ω − c) w − , , ( ω − c) 2 6 2 12 4 3 2     3 1 3 5 1 1 3 −1/2 2 2 g(ω) = ( ω − c) exp − ( ω − c) w − , , ( ω − c) 2 6 2 12 4 3 2 h(ω) = −ω + c 3 l(ω) = cω − ω 2 + c1 (22) 2 where w is the Whittaker function, c and c1 are integration constants. Note that the Whittaker and the Kummer functions are strongly related to each other [16] w(κ, µ, z) = e−1/2z z 1/2+µ KummerM (1/2 + µ − κ, 1 + 2µ, z).

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More details can be found in the original work [12]. As a second comparison we show the results of [13]. They also have a modified form of (29) which is the following U1t + cU1 + U2 U1y + U3 U1z − ν(U1yy + U1zz ) = 0, U2t + U2 U2y + U3 U2z + πy − ν(U2yy + U2zz ) = 0, U3t + U2 U3y + U3 U3z + πz − ν(U3yy + U3zz ) = 0, U2y + U3z + c = 0

(24)

where Ui , i = 1..3 are the velocity components Ui (y, z, t) and π is the pressure, c stands for constants, ν is viscosity and additional subscripts mean derivations. After some transformation they get a linear PDA as follows U1t + k1 yU1y + (σ − k1 z)U1z − ν(U1yy + U1zz ) = 0

(25)

it is convenient to look the solution in the form of U1 = Y (y)T (z)Φ(t).

(26)

Note, that they also consider the full 3 dimensional problem, but the velocity filed has a restricted two dimensional(y,z) coordinate dependence. There are additional conditions but the general solution can be presented Φ = c1 exp(c2 )t     1 y2 1 3 y2 Y = c3 M −c4 , , + yc5 M − c4 , , 2 2ν 2 2 2ν     2 2 1 z 1 3 z T ≈ M c6 , , + zM − c6 , , 2 2ν 2 2 2ν

(27)

where M is the KummerM function as was presented below. The exact solution in [13] (Eq. 4.10a-4.10c) contains more constants as presented here. It is not our goal to reproduce the full calculation of [13] (which is not our work) we just want to give a guideline to their solution vigorously emphasizing that our solution is very similar to the presented one. Note that in both results the arguments of the KummerM function (11) and (27) are proportional to the square of the radial component divided by the viscosity, additionally one of the parameters is 1/2. As a last word we just would like to say, (as this example clearly shows) that the Lie algebra method is not the exhaustive method to find all the possible solutions of a PDA. It is possible that even this moderate result can give any simulating impetus to the numerical investigation of the NS equation. Our solution can be used as a test case for various numerical methods or commercial computer packages like Fluent or CFX.

3.

Compressible Newtonian Fluids

To study the dynamics of viscous compressible fluids the compressible NS together with the continuity equation have to be investigated. In Eulerian description in Cartesian coordinates these equations are the following:

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ρt + div[ρv] = 0 ρ[vt + (v∇)v] = ν1 4v +

ν2 grad div v − ∇p + a 3

(28)

where v, ρ, p, ν1,2 and a denote respectively the three-dimensional velocity field, density, pressure, kinematic viscosities and an external force (like gravitation) of the investigated fluid. As before we use a for external field instead of the letter g which is reserved for a self-similar solution. In the later we consider no external force, so a = 0. For physical completeness we need an equation of state (EOS) to close the equations. We start with the polytropic EOS p = κρn , where κ is a constant of proportionality to fix the dimension and n is a free real parameter (n is usually less than 2). In astrophysics, the Lane - Emden equation is a dimensionless form of the Poisson’s equation for the gravitational potential of a Newtonian self-gravitating, spherically symmetric, polytropic fluid. It’s solution is the polytropic EOS which we apply in the following. The question of more complex EOSs will be concerned later. Now ν1,2 , a, κ, n are parameters of the flow. To have a better overview we use the coordinate notation v(x, y, z, t) = (u(x, y, z, t), v(x, y, z, t), w(x, y, z, t)) and for the scalar density variable ρ(x, y, z, t) from now on. Having in mind the correct forms of the mentioned complicated vector operations, the PDE system reads the following: ρt + ρx u + ρy v + ρz w + ρ[ux + vy + wz ] = 0, ν2 ρ[ut + uux + vuy + wuz ] − ν1 [uxx + uyy + uzz ] − [uxx + vxy + wxz ] + κnρn−1 ρx = 0, 3 ν2 ρ[vt + uvx + vvy + wvz ] − ν1 [vxx + vyy + vzz ] − [uxy + vyy + wyz ] + κnρn−1 ρy = 0, 3 ν2 ρ[wt + uwx + vwy + wwz ] − ν1 [wxx + wyy + wzz ] − [uxz + vyz + wzz ] + κnρn−1 ρz = 0. 3

(29)

The subscripts mean partial derivations as was defined earlier. Note, that the formula for EOS is already applied. There is no final and clear-cut existence and uniqueness theorem for the most general non-compressible NS equation. However, large number of studies deal with the question of existence and uniqueness theorem related to various viscous flow problems. Without completeness we mention two works which (together with the references) give a transparent overview about this field [18, 19]. According to our best knowledge there are no analytic solutions for the most general three dimensional NS system even for non-compressible Newtonian fluids. Additional similarity reduction studies are available from various authors as well [20, 21, 22]. A full three dimensional Lie group analysis is available for the three dimensional Euler equation of gas dynamics, with polytropic EOS [23] unfortunately without any kind of viscosity. We use the above mentioned three dimensional Ansatz again. It can be easily shown than even a more general plane, like ax + by + dz + 1 = 0 makes the remaining ODE system much more complicated. (The second term in the NS equation on the right hand side (grad div v term) creates distinct a2 , b2 , c2 terms which cannot be transformed out, and a coupled system of three equations remain.)

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The Ansatz we apply is: −α

ρ(x, y, z, t) = t

f



 x+y+z = t−α f (η), u(η) = t−δ g(η), tβ v(η) = t− h(η), w(η) = t−ω i(η),

(30)

where all the exponents α, β, δ, , ω are real numbers as usual. The technical part of the calculation is analogous as was shown above. To get a final ODE system which depends only on the variable η, the following universality relations have to be hold α=β=

2 n+1

&

δ==ω=

2n − 2 , n+1

(31)

where n 6= −1. Note, that the self-similarity exponents are not fixed values thanks to the existence of the polytropic EOS exponent n. (In other systems e.g. heat conduction or noncompressible NS system, all the exponents have a fixed value, usually +1/2.) This means that our self-similar Ansatz is valid for different kind of materials with different kind of EOS. Different exponents represent different materials with different physical properties which results different final ODEs with diverse mathematical properties. At this point we have to mention that a NS equation even with a more complicated EOS like p ∼ f (ρl v m ) could have self-similar solutions. We may say in general, that EOSs obtained from Taylor series expansion taking into account many term are problematic and give contradictory relations among the exponents. The investigation of such problems will be performed in the near future but not in the recent study. Our goal is to analyze the asymptotic properties of Eq. (30) with the help of Eq. (31). According to Eq. (1) the signs of the exponents automatically dictates the asymptotic behavior of the solution at sufficiently large time. All physical velocity components should decay at large times for a viscous fluid without external energy source term. The role of α and β was explained after Eq. (1). Figure 6 shows the α(n) and δ(n) functions. There are five different regimes: • n > 1 all exponents are positive - physically fully meaningful scenario - spreading and decaying density and all speed components for large time - will be analyzed in details for general n • n = 1 spreading and decaying density in time and spreading but non-decaying velocity field in time - not completely physical but the simplest mathematical case • −1 ≤ n ≤ 1 spreading and decaying density in time and and spreading and enhancing velocity in time - not a physical scenario • n 6= −1 not allowed case • n ≤ −1 sharpening and enhancing density and sharpening and decaying velocity in time, we consider it an non-physical scene and neglect further analysis.

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Figure 6. Eq. (31) dotted line is α(n) = 2/(n + 1) and solid line is δ(n) = 2 − 4/(n + 1). The corresponding coupled ODE system is: α[f + f 0 η] = f 0 [g + h + i] + f [g 0 + h0 + i0 ], ν2 f [−δg − αηg 0 + gg 0 + hg 0 + ig 0 ] = −κnf n−1 f 0 + 3ν1 g 00 + [g 00 + h00 + i00 ], 3 ν2 00 0 0 0 0 n−1 0 00 f [−δh − αηh + gh + hh + ih ] = −κnf f + 3ν1 h + [g + h00 + i00 ], 3 ν2 00 0 0 0 0 n−1 0 00 f [−δi − αηi + gi + hi + ii ] = −κnf f + 3ν1 i + [g + h00 + i00 ], 3 (32) where prime means derivation with respect to η. The continuity equation is a total derivative if α = β, therefore we can integrate getting αf η = f [g+h+i]+c0 , where c0 is proportional to the mass flow rate. For the shake of clarity, we simplify the NS equation with introducing a single viscosity ν = ν1 = ν2 . There are still too many free parameters remain for the general investigation. We fix c0 = 0. Having in mind that the density of a fluid should be positive f 6= 0, we get αη = g + h + i. With the help of the first and second derivatives of this formula Eq. (32) can be reduced to the next non-linear first order ODE   4n − 4 n−1 0 −3κnf f + ηf = 0. (33) (n + 1)2 Note, that it is a first order equation, so there is a conserved quantity which should be a kind of general impulse in the parameter space η. Note, that this equation has no contribution from the viscous terms with ν just from the pressure and from the convective terms. The

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(34)

Note, that for {n; n ∈ Z\{−1}} exists n different solutions for n > 0 (one of them is the f (η) = 0) and n − 1 different solutions for n < 0 these are the n or (n-1)th roots of the expression. For {n : n ∈ R\{−1}} there is one real solution. It is remarkable, that for fixed κ, c1 > 0 when n and η tend to infinity, the limit of Eq. (34) tends to zero. This meets our physical intuition for a viscous flow, we get back solutions which have an asymptotic decay. (In the limiting case n = 1 (which means the δ = 0) we get back the trivial result f = const. which is irrelevant.) For the n = 2, the least radical case f (η) = η 2 /(27κ) + c1 which is a quadratic function in η however, the full density function ρ = t−2/3 [(x + y + z)2 /t4/3 ] = (x + y + z)2 /t2 has a proper time decay for large times. This is consistent with our physical picture. All the three velocity field components can be derived independently from the last three Eqs. (32). For the v = t−δ g(η) the ODE reads:   2n − 2 00 −3νg + gf − κnf n−1 f 0 = 0. (35) n+1 Unfortunately, there is no solution for general n in a closed form. However, for n = 2 the solutions can be given inserting f (η) = η 2 /(27κ) into Eq. (35). These are the Whittaker W and Whittaker M functions [16] √ 2! √ 2! c˜1 2η c ˜ 2η 2 2 √ √ √ √ g = √ M c1 √ + √ W c1 √ + η, (36) 2κ 1 2κ 1 η − 4 ν , 4 9 νκ η − 4 ν , 4 9 νκ 3 where c˜1 and c˜2 are integration constants. The M is the irregular and the W is the regular Whittaker function, respectively. These functions can be expressed with the help of the Kummer’s confluent hypergeometric functions M and U in general (for details see [16]) Mλ,µ (z) = e−z/2 z µ+1/2 M (µ − λ + 1/2, 1 + 2µ; z);

Wλ,µ(z) = e−z/2 z µ+1/2 U (µ − λ + 1/2, 1 + 2µ; z).

(37)

Is some special cases when κ = ν/2 the Whittaker functions can formally be expanded with other functions (e.g. Bessel, Err) when {c1 : c1 ∈ N\{−2, −4}}. It is easy to show with the help of asymptotic forms that the velocity field u ∼ t−1/3 [M orW (·, ·; t−4/3)] decays for sufficiently large time which is a physical property of a viscous fluid. (It is worthwhile to mention, we found additional closed solutions only for n = 1/2 and for n = 3/2 from Eq. (33,35) for the density and velocity field which contain the HeunT functions, in a confusingly complicated expression.) Now we compare our recent results to the former non-compressible ones. In the noncompressible case of the three dimensional NS equation, all the exponents have the 1/2 value - like in the regular diffusion equation - except the decaying exponent of the pressure field which is 1. For non-compressible fluids the x component of the velocity field is described with the help of the Kummer functions     1 1 (η + c)2 1 1 (η + c)2 c 2a g˜(η) = c1 U − , , + c2 M − , , + − , (38) 4 2 6ν 4 2 6ν 3 3

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where c1 and c2 are the usual integration constants. The viscosity is ν the external field is a and the c is the non-zero integration constant from the continuity equation - which can be set to zero. Additional properties of this formula was analyzed in our last study[7] in depth. Figure 7 compares the regular parts of the solutions of Eq. (36) and Eq. (38) with the same viscosity value ν = 0.1 and for c˜1 = 1, ˜c2 = 1, c1 = 0. The compressible parameters are κ = 1 and n = 2. Note that the shape function of velocity of the compressible flow has a maximum and a quick decay, the incompressible velocity shape function has no decay. However, these are the reduced one dimensional shape functions, and the total thee dimensional velocity fields have proper time decay for large time as it should be. The c1 in the Whittaker function cannot be negative because it comes from the density equation. If it is zero or any other positive number plays no difference in the form of the shape function. Figure 8 presents how the regular part of the solution Eq. (36) depends on the compressibility for a given value of viscosity. Note, the higher the compressibility the lower the maximum of the top speed of the system. As a second case study Figure 9 presents how the regular part of the solution Eq. (36) depends on the viscosity for a given value of compressibility. Higher the viscosity the higher the maximal reached speed and the range of the system. In our investigation the role of the two viscosities cannot be separated from each other therefore this effect cannot be seen more clearly. Note, that Eq. (36) is not a direct limit of Eq. (38) just a very similar one. More technical details of the calculations can be found in [24].

Figure 7. Comparison of the regular solutions for the non-compressible (solid line) Eq. (38) and the compressible case(dashed line) Eq. (36). The viscosities have the same numerical value µ = 0.1.

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Figure 8. The compressibility dependence of the regular solution of Eq. (36) for n = 2 and for ν = 0.1 viscosity. The solid line is for κ = 0.1 the dotted line is for κ = 1 and the dashed line is for κ = 2 .

Figure 9. The viscosity dependence of the regular solution of Eq. (36) for n = 2 and κ = 1. The solid line is for ν = 0.05 the dotted line is for ν = 0.1 and the dashed line is for ν = 0.5.

4.

Incompressible Non-Newtonian Fluid

Dynamical analysis of viscous fluids is a never-ending crucial problem. A large part of real fluids do not strictly follow Newtons law and are aptly called non-Newtonian flu-

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ids. In most cases, fluids can be described by more complicated governing rules, which means that the viscosity has some additional density, velocity or temperature dependence or even all of them. General introductions to the physics of non-Newtonian fluid can be found in [25, 26]. In the following, we will examine the properties of a Ladyzenskaya type non-Newtonian fluid [27, 28]. Additional temperature or density dependent viscosities will not be considered in the recent study. There are some analytical studies available for non-Newtonian flows in connections with the boundary layer theory, which shows some similarity to our recent problem [29, 30]. The heat transfer in the boundary layer of a nonNewtonian Ostwald-de Waele power law fluid was investigated with self-similar Ansatz in details by [31, 32]. The Ladyzenskaya [27] model of non-Newtonian fluid dynamics can be formulated in the general vectorial form of ρ

∂ui ∂t ∂uj ∂xj Γij

Eij (∇u)

+

ρuj

=

0

Def

=

Def

=

∂ui p ∂Γij =− + + ρai ∂xj ∂xi ∂xj

(µ0 + µ1 |E(∇u)|r)Eij (∇u)   ∂uj 1 ∂ui + 2 ∂xj ∂xi

(39)

where ρ, ui, p, ai, µ0 , µ1 , r are the density, the two dimensional velocity field, the pressure, the external force, the dynamical viscosity, the consistency index and the flow behavior index. The last one is a dimensionless parameter of the flow. The Eij is the Newtonian linear stress tensor, where x(x,y) are the Cartesian coordinates. The usual Einstein summation is used for the j subscript. In our next model, the exponent should be r > −1. This general description incorporates the next five different fluid models: Newtonian for

µ0 > 0, µ1 = 0,

Rabinowitsch for

µ0 , µ1 > 0, r = 2,

Ellis for

µ0 , µ1 > 0, r > 0,

Ostwald-de Waele for Bingham for

µ0 = 0, µ1 > 0, r > −1,

µ0 , µ1 > 0, r = −1.

(40)

For µ0 = 0 if r < 0 then it is called a pseudo-plastic fluid, if r > 0 it is a dilatant fluid [26]. In pseudoplastic or shear thinning fluid the apparent viscosity decreases with increased stress. Examples are: blood, some silicone oils, some silicone coatings, paper pulp in water, nail polish, whipped cream, ketchup, molasses, syrups, latex paint, ice . For the paper pulp the numerical Ostwald-de Waele parameters are µ1 = 0.418, r = −0.425 [26]. In shear thickening or dilatant fluid, the apparent viscosity increases with increased stress. Typical examples are sand in water or suspensions of corn starch in water (sometimes called oobleck). There are numerous videos available on most popular video sharing portal where young people having fun with a pool full of oobleck . The external force will be zero in our investigation ai = 0. In two dimensions, the

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absolute value of the stress tensor reads: |E| = [u2x + vy2 + 1/2(uy + vx )2 ]1/2 ,

(41)

where the u(u, v) coordinate notation is used from now on. (Note, that in three dimensions the absolute value of the stress tensor would be much complicated containing six terms instead of three.) Introducing the following compact notation L = µ0 + µ1 |E|r ,

(42)

our complete two dimensional NS system can be defined much shorter: ux + vy = 0, Ly L ut + uux + vuy = −px /ρ + Lx ux + Luxx + (uy + vx ) + (uyy + vxy ), 2 2 Lx L vt + uvx + vvy = −py /ρ + Ly vy + Lvyy + (uy + vx ) + (vxx + uxy ). 2 2

(43)

Every two dimensional flow problem can be reformulated with the help of the stream function Ψ via u = Ψy and v = −Ψx , which automatically fulfills the continuity equation. The system of (43) is now reduced to the following two PDEs   px L Ψyt + Ψy Ψyx − Ψx Ψyy = − + (LΨyx )x + (Ψyy − Ψxx ) ρ 2 y   py L −Ψxt − Ψy Ψxx + Ψx Ψxy = − + (Ψyy − Ψxx ) − (LΨyx )y (44) ρ 2 x

 r/2 with L = µ0 + µ1 2Ψ2xy + 21 (Ψyy − Ψxx )2 . Now, search the solution of this PDE system such as: Ψ = t−α f (η), p = t− h(η), η = x+y , where all the exponents α, β, γ are real numbers. tβ Unfortunately, the constraints, which should fix the values of the exponents become contradictory, therefore no unambiguous ODE can be formulated. This means that the new PDE system with the stream function does not have self-similar solutions. In other words the stream function has no diffusive property. This is a very instructive example of the applicability of the trial function of (1). Let’s return to the original system of (43) and use the trial function of u = t−α f (η),

v = t−δ g(η),

p = t− h(η),

η=

x+y . tβ

(45)

The next step is to determine the exponents. From the continuity equation we simple get arbitrary β and δ = α relations. The two NS dictate additional constraints. (We skip the trivial case of µ0 6= 0, µ1 = 0, which was examined in our former paper as the Newtonian fluid. [17]) Finally, we get µ0 = 0, µ1 6= 0, α = δ = (1 + r)/2, β = (1 − r)/2,  = r + 1.

(46)

Note, that r remains free, which describes various fluids with diverse physical properties, this meets our expectations. For the Newtonian NS equation, there is no such free parameter

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and in our former investigation we got fixed exponents with a value of 1/2. For a physically relevant solutions, which is spreading and decaying in time all the exponents Eq. (46) have to be positive determining the −1 < r < 1 range. This can be understood as a kind of restricted Ostwald-de Waele-type fluid. After some algebraic manipulation a second order non-autonomous non-linear ODE remains µ1 (1 + r)f 00 [2f 0 ]r/2 +

(1 − r) 0 (1 + r) ηf + f = 0, 2 2

(47)

where prime means derivation with respect to η. Note that for the numerical value r = 0 we get back the ODE of the Newtonian NS equation for two dimensions. In three dimensions the ODE reads: 3 c 9µ0 f 00 − 3(η + c)f 0 + f (η) − + a = 0. (48) 2 2 Its solutions are the Kummer functions [16]   1 1 (η + c)2 f = c1 · KummerU − , , + 4 2 6µ0   1 1 (η + c)2 c 2a + − , (49) c2 · KummerM − , , 4 2 6µ0 3 3 where c1 and c2 are integration constants, c is the mass flow rate, and a is the external field. These functions have no compact support. The corresponding velocity component however, decays for large time like v ∼ 1/t for t → ∞, which makes these results physically reasonable. A detailed analysis of (49) was presented in [17]. Unfortunately, we found no integrating factor or analytic solution for (47) at arbitrary values of r. We mention that with using the symmetry properties of the Ansatz uxx = uxy = uyy = −vxx = −vxy = −vyy the following closed form can be derived for the pressure field h = ρ(µ1 2r/2+1 f 0r+1 + f η − c˜1 f ) + c˜2 , (50) where c˜1 and c˜2 are the usual integration constants. The transition theorem states, that a second order ODE is always equivalent to a first order ODE system. Let us substitute f 0 = l, f 00 = l 0 , then f 0 = l, l0 = −



(1 − r) (1 + r) ηl + f 2 2

   / µ1 (1 + r)2r/2l r ,

(51)

where prime still means derivation with respect to η. This ODE system is still nonautonomous and there is no general theory to investigate such phase portraits. We can divide the second equation of (51) by the first one to get a new ODE, where the former independent variable η becomes a free real parameter     dl (1 − r) (1 + r) =− ηl + f / µ1 (1 + r)2r/2l r+1 . (52) df 2 2 Figure 10 shows the phase portrait diagram of (52) for water pulp the material parameters are r = −0.425 and µ1 = 0.18. We consider η = 0.03 as the ”general time variable” to

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be positive as well. With the knowledge of the exponent range −1 < r < +1, two general properties of the phase space can be understood by analyzing Eq. (52): Firstly, the derivative df /dl is zero at zero nominator values, which means l = −(1 + r)/(1 − r)f /η. This one is a straight line passing through the origin with gradient of 0 < (1 + r)/(1 − r) < ∞ for −1 < r < +1. On Figure 10, the numerical value of the gradient is −0.403/η = −13.5. Secondly, the derivative df /dl or the direction field is not defined for any negative l values because the power function l −(r+1) in the denominator is not defined for negative l arguments. It is not possible to extract a non-integer root from a negative number. The denominator is always positive. These properties dictate that there are two kinds of possible trajectories or solutions exist in the phase space. One type has a compact support, and the other has a finite range. We may consider the x axis as the velocity f ∼ v(η) and y axis as the f 0 ∼ a(η) acceleration for a fixed scaled time η = const. It also means that the possible velocity and accelerations for a general time cannot be independent from each other. The factors of the second derivative f 00 show some similarity to the porous media equation, where the diffusion coefficient has also an exponent. This is the essential original responsibility for the solution with compact support [34]. Additional numerical solutions of Eq. (47) obtained with the help of the Briot-Bouquet theorem was presented at the end of [33].

Figure 10. The phase portrait of Eq. (52) for η = 0.03, r = −0.425 and µ1 = 0.18. Two different kind of trajectories are presented: one with compact support (solid line), and the other one with compact range (dashed line). Our main result is that the velocity field of the fluid - in contrast to our former Newtonian result - has a compact support, which is the major difference. We can explain it with the following everyday example: Let’s consider two pots in the kitchen, one is filled with water

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and the other is filled with chocolate cream. Start to stir both with a wooden spoon in the middle, after a while the whole mass of the water becomes to move due to the Newtonian viscosity, however the chocolate cream far from the spoon remains stopped even after a long time.

5.

Incompressible Newtonian Fluid with Heat Conduction

In the following we analyze the dynamics of a two-dimensional viscous fluid which is coupled to heat conduction. Such systems were first investigated by Boussinesq [35] and Oberbeck [36] in the nineteenth century. Oberbeck used a finite series expansion. He developed a model to study the heat convection in fluids taking into account the flow of the fluid as a result of temperature difference. He applied the model to the normal atmosphere. More detailes of the Boussinesq approximation can be found our in original paper [37]. More than half a century later Saltzman [38] tried to solve the same model with the help of Fourier series. At the same time Lorenz [39] analyzed the solutions with computers and published the plot of a strange attractor which was a pioneering results and the advent the studies of chaotic dynamical systems. The literature of chaotic dynamics became enormous, however a modern basic introduction can be found in [40]. Later, till to the first beginning years of the millennium [39] Lorenz analyzed the final first order chaotic ordinary differential equation(ODE) system with different numerical methods. This ODE system becomes an emblematic object of chaotic systems and attracts much interest till today [41]. On the other side critical studies came to light which go beyond the simplest truncated Fourier series. Curry for example gives a transparent proof that the finite dimensional approximations have bounded solutions [42]. Musielak et al [43] in three papers analyzed large number of truncated systems with different kinds and found chaotic and periodic solutions as well. In the next we apply a completely different investigation approach, the self-similar Ansatz. We investigated one dimensional Euler equations with heat conduction as well [33] which can be understood as the precursor of the recent study. To our knowledge this kind of investigation method was not yet applied to the Oberbeck-Boussinesq (OB) system. Our major result is that the temperature field shows a strongly damped single periodic oscillation which can mimic the appearance of Rayleigh-B´enard convection cells. The question how our results connected to general chaotic and turbulence conception like intermittency, enstropy are discussed in our original study [37] as well. The start with the original system of [38] ut + uux + wuz + Px − ν (uxx + uzz ) = 0,

wt + uwx + wwz + Pz − eGT1 − ν (wxx + wzz ) = 0, 1 1 Tt1 + uTx1 + wTz1 − κ(Txx + Tzz ) = 0,

ux + wz = 0,

(53)

where u, w, denote respectively the x and z velocity coordinates, T 1 is the temperature difference relative to the average (T 1 = T − Tav ) and P is the scaled pressure over the density . The free physical parameters are ν, e, G, κ kinematic viscosity, coefficient of

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volume expansion, acceleration of gravitation and coefficient of thermal diffusivity. The first two equations are the Navier-Stokes equations, the third one is the heat conduction equation and the last one is the continuity equation all are for two spatial dimensions. The Boussinesq approximation means the way how the heat conduction is coupled to the second NS equation. Chandrasekhar [44] presented a wide-ranging discussion of the physics and mathematics of Rayleigh-Benard convection along with many historical references. Every two dimensional flow problem can be reformulated with the help of the stream function Ψ via u = Ψy and v = −Ψx which automatically fulfills the continuity equation. The subscripts mean partial derivations. After introducing dimensionless quantities the system of (53) is reduced to the next two PDEs (Ψxx + Ψzz )t + Ψx (Ψxxz + Ψzzz ) − Ψz (Ψxxx + Ψzzx ) −

σ(θx − Ψxxxx − Ψzzzz − 2Ψxxzz ) = 0,

θt + Ψx θz − Ψz θx − RΨx − (θxx + θzz ) = 0,

(54) 3

where Θ is the scaled temperature, σ = ν/κ is the Prandtl Number and R = GeHκν∆T0 is the Rayleigh number and H is the height of the fluid. A detailed derivation of (54) can be found in [38]. All the mentioned studies are investigated these two PDEs with the help of some truncated Fourier series, different kind of truncations are available which result different ODE systems. The derivation of the final non-linear ODE system from the PDE system can be found in the original papers [38, 39]. Berg´e et al. [45] contains a slightly different derivation of the Lorenz model equations, and in addition, provides more details on how the dynamics evolve as the reduced Rayleigh number changes. A detailed treatment of the Lorenz model can be found in the book of Sparrow [46]. Hilborn [47] presents the idea of the whole derivation in a transparent and clear way. Therefore, we skip this derivation. Some truncations even violates energy conservation [41] and some not. Roy and Musiliak [43] in their exhausting three papers present various energy-conserving truncations. Some of them contain higher horizontal modes, some of them contain higher vertical modes and some of them both kind of modes in the truncations. All these models show different features some of them are chaotic and some of them - in well-defined parameter regimes - show periodic orbits in the projections of the phase space. This is somehow a true indication of the complex nature of the original flow problem. It is also clear that the Fourier expansion method which is a two hundred year old routine tool for linear PDEs fails for a relevant non-linear PDE system. We may investigate both dynamical systems, the original hydrodynamical (53) or the other one (54) which is valid for the stream functions. Similar to the former studies [38, 39] try to solve the PDEs for the dimensionless stream and temperature functions in the form of Ψ = t−α f (η), θ = t− h(η), η =

x+z . tβ

(55)

Unfortunately, after some algebra it becomes clear that the constraints which should fix the values of the exponents become contradictory, therefore no unambiguous ODE can be derived. This means that the PDE of the stream function and the dimensionless temperature

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do not have self-similar solutions. In other words these functions have no such a diffusive property which could be investigated with the self-similar Ansats, which is a very instructive example of the applicability of the trial function of (55). Our experience shows that, most of the investigated PDEs have a self-similar ODE system and this is a remarkable exception. Now investigate the original hydrodynamical system with the next Ansatz u(η) = t−α f (η), w(η) = t−δ g(η), P (η) = t− h(η), T 1 (η) = t−ω l(η),

(56)

where the new variable is the usual η = (x + z)/tβ . All the five exponents α, β, δ, , ω are

Figure 11. Different shape functions of the temperature Eq. (59) as a function of η for different thermal diffusivity. The integration constants are c1 = c2 = 1 the same for all the three curves. The solid the dashed and the dotted lines are for κ = 1, 2, 5, respectively. real numbers. The f, g, h, l objects are the shape functions of the corresponding dynamical variables. After some algebraic manipulations the following constrains are fixed among the selfsimilarity exponents : α = δ = β = 1/2,  = 1 and ω = 3/2 which are called the universality relations. Note, that all exponents have a fixed numerical value which simplifies the structure of the solutions. There is no free exponential parameter in the original dynamical system, like an exponent in a EOS for the compressible NS system. These universality relations dictate the corresponding coupled ODE system which is the following f f 0η − + f f 0 + gf 0 + h0 − 2νf 00 2 2 g g 0η − − + f g 0 + gg 0 + h0 − eGl − 2νg 00 2 2 3l l 0 η − − + f l 0 + gl 0 − 2κl 00 2 2 f 0 + g0 −

= 0, = 0, = 0, = 0.

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Figure 12. Different shape functions of the pressure Eq. (62) as a function of η for different thermal diffusivity. The integration constants are taken c1 = c2 = 1 for all the three curves. We fixed the value of eG = 1 as well. The solid, the dashed and the dotted lines are for κ = 1, 2, 5 numerical values, respectively. From continuity equation we automatically get the f +g = c and f 00 +g 00 = 0 conditions which are necessary for the solutions. For the shake of simplicity we consider the c = 0 case in the following. If c 6= 0 then the Eq. (58) is slightly modified and the results are the KummerM and KummerU functions, but the shape of the functions remains the same which is crucial for the forthcoming analysis. After some algebraic manipulation the next single ODE for the shape function of the temperature distribution can be separated 2κl 00 +

l 0 η 3l + = 0. 2 2

After a quadrature the solution is " # √ !   2 √ η2 2η √ η2 − η8κ √ l = c1 4erf i e 2π κ − + 4 κη + c2 e− 8κ (4κ − η 2 ) 4 κ 4

(58)

(59)

where c1 , c2 are the usual free integration Rconstants. The erfi means the imaginary er√ x ror function defined via the integral 2/ π 0 exp(x2 )dx for more details see [16]. It is interesting, that the temperature distribution is separated from the other three dynamical variables an does not depend on the viscosity coefficients as well. We may say, that among the solution obtained from the self-similar Ansazt the temperature has the highest priority and this quantity defines the pressure and the velocity field. That is a remarkable feature. In a former study, where the one-dimensional Euler system was investigated with heat conduction [48] we found the opposite property, the density and the velocity field were much simpler than the temperature field. Figure 11 presents different shape functions of the temperature for different thermal diffusivity values. The first message is clear, the larger the thermal diffusivity the larger the shape function of the temperature distribution. A detailed analysis of Eq. (59) shows that for any reasonable κ and c values the main property of

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the function is not changing - has one global maximum and minimum with a strong decay for large ηs. A second remarkable feature is the single oscillation which is not a typical behavior for self-similar solutions. We investigated numerous non-linear PDE systems till today [17, 24, 33, 48] some of them are even not hydrodynamical [6, 49] and never found such a property. This analysis clearly shows that at least the temperature distribution in this physical system has a single-period anharmonic oscillation. For a fixed time value ”t” and a well-chosen coordinate ”z” the difference of values of η = (x + z)/tβ yields a minima and a maxima corresponds to that ∆x at which the temperature (and density) fluctuation may start the Rayleig-B´enard convection. We will see later that with the same consideration (when time and z coordinate are fixed) the velocity components u and w are different at these x coordinate points therefore the inhomogenity is present which start the rotation of the Rayleig-B´enard cell. This is the main result of the recent study. To go a step further we may calculate the Fourier transform of the shape function, Eq. (59) l(η) it we consider η as a generalized time dependence we may get the generalized spectral distribution. (An analytic expression for the Fourier transform is available, which we skip now.) The first term (which is proportion to c1 ) becomes a complex function, however the general overall shape remains the same, a single-period anharmonic oscillation with a global minimum and maximum like on Figure 11. Of course, the zero transition of the function depends on the value of κ. The second term of the Fourier transformed function which is proportional to c2 remains a Gaussian which is not interesting. For completeness we give the full two dimensional temperature field as follows T1 (x, z, t) =

c1 c2

„ » « „ « – √ √ (x+z)2 x+z 4 κ(x + z) (x + z)2 − 8κt t−3/2 4erf i + 2π κ − e + 4(κt)1/2 4t t1/2 „ « (x+z)2 (x + z)2 4κ − t−3/2 e− 8κt . (60) t

The shape function of the pressure field can be obtained from the temperature shape function via the following equation eGl h0 = (61) 2 with a similar solution to (59) " # √ ! √ η2 η2 2η √ h = c1 2κ 2πeG · erf i ηe− 8κ + c2 2eGκηe− 8κ + c3 , (62) 4 κ this can be understood that the derivative of the pressure is proportional to the temperature (Figure 12). With the known numerical value of the exponent  = 1 the scaled pressure field can be expressed as well P (x, z, t) = t−1 h([x + z]/t−1/2 ). Note, the difference between the ω and the  exponents, which are responsible for the different asymptotic decays. The temperature field has a stronger damping for large η than the pressure field. (It is worth to mention that for the three dimensional NS equation, without any heat exchange the decay exponent of the pressure term is also different to the velocity field [17]. ) At last the ODE for the shape function of the velocity component z reads 4νg 00 + g 0 η + g + eGl = 0

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Figure 13. The shape functions of the z velocity field g(η) Eq. (65) as a function of η. The solid line is the real and the dotted is the complex part. All the integration constants are taken c˜1 = c˜2 = c2 = 1. The physical constant eG = 1 as well. The κ = 0.04 and ν = 0.8. which directly depends on the temperature on l(η) and all the physical parameters ν, e, G, κ, of course. In contrast to the pressure and temperature field there is no closed solutions available for a general parameter set. The formal, most general solution is η2

η2

g = c˜2 e− 8ν + e− 8ν

Z

1 4ν

»„ „√ « « 2 – ff √ η2 η2 η 2η c˜1 − 4eGκc2 ηe− 8κ − 4eGκ 2πc1 erf i √ ηe− 8κ e 8ν dη 4 κ (64)

where c˜1 and c˜2 are the recent integration constants. Note, that the integral can be analytically evaluated if and only if ν = κ which is a great restriction to the physical system. We skip this solution now. The other way is to fix c1 = 0 and let κ and µ free. The solution has the next form of ! r η2 2 η2 η 2 4eGc2 κ2 e− 8κ − 8ν − η8ν g = c˜1 e erf − − + c˜2 e . (65) 4 ν κ−ν Note, that now the ν 6= κ condition is obtained. The c˜1 and c˜2 are the recent integration constants as above, it is interesting that if both of them are set to zero, the solution is still not trivial. For a physical system the kinematic viscosity ν > 0 is always positive, therefore in the case of c˜1 6= 0 the solution becomes complex. Figure 14 shows the shape function of the z velocity component. It is clear that the real part is a Gaussian function and the complex part is a Gaussian distorted with an error function, which is an interesting final result. In the literature we can find system which shows similarities like the work of Ernst [50] who presented a study where a the asymptotic normalized velocity autocorrelation function calculated from the linearized Navier-Stokes equation has an error function shape. To give coupling points to colleagues from other field, (like chaotic dynamics or turbulence) in the last section of the original publication [37] we calculated the enstropy of the system which is crucial quantity for two dimensional turbulence. The Fourier spectra of our

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velocity field was compared to other turbulence results as well. The question of fractional derivatives or fractal dimensions are addressed as well. Further analysis is in progress. Recently, we try to generalize the original OB system (53) with temperature dependent viscosity ν(T ) and heat conduction coefficients κ(T ). It seems that it is not possible and only the constant coefficients are available for selfsimilar analysis. To go beyond the Boussinesq approximation is also under consideration, the analysis is very exciting.

References [1] B.H. Gilding and R. Kersner, Traveling Waves in Nonlinear Diffusion-Convection Reactions, Progress in Nonlinear Differential Equations and Their Applications, Birkh¨auser Verlag, Basel-Boston-Berlin, 2004, ISBN 3-7643-7071-8. [2] L. Sedov, Similarity and Dimensional Methods in Mechanics CRC Press 1993. [3] G.I. Baraneblatt, Similarity, Self-Similarity, and Intermediate Asymptotics, Consultants Bureau, New York 1979. [4] Ya.B. Zel’dovich and Yu.P. Raizer, Physics of Shock Waves and High Temperature Hydrodynamic Phenomena, Academic Press, New York 1966. [5] T. Vicsek, E. Somfai and M. Vicsek, J. Phys. A: Math. Gen 25, (1992) L763. [6] I.F. Barna and R. Kersner, J. Phys. A: Math. Theor. 43, (2010) 375210. [7] I.F. Barna and R. Kersner, Adv. Studies Theor. Phys. 5, 193 (2011). [8] H.E. Stanley, Rev. Mod. Phys. 71, (1999) S358. [9] Y. Manwai, J. Math. Phys. 49 (2008) 113102. [10] Y. Manwai, Some Problems on a Class of Fluid Dynamical Systems: Euler-Poisson, Navier-Stokes-Poisson, Euler and Navier-Stokes Equations, VDM Verlag, 2009. [11] V.N. Grebenev, M. Oberlack and A.N. Grishkov, Journ. of Nonlin. Mathem. Phys 15 (2008) 227. [12] W. I. Fushchich, W. M. Shtelen and S. L. Slavutsky, J. Phys. A: Math. Gen. 24 (1990) 971. [13] V. Grassi, R.A. Leo, G. Soliani and P. Tempesta, Physica 286 (2000) 79*, ibid 293 (2000) 421. [14] X.R. Hu, Z.Z. Dong, F. Huang et al., Z. Naturforschung A 65 (2010) 504. [15] S. N. Aristov and A. D. Polyanin, Russ. J. Math. Phys. 17 (2010) 1. [16] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions Dover Publication., Inc. New York.

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[17] I.F. Barna, Commun. in Theor. Phys. 56 (2011) 745-750. [18] J.G. Heywood, Acta Mathematica 136, 61 (1976). [19] R.B. Barrar, Services Technical Information Agency Comment Service Center AD 1711 Report, work done at Harvard University under Contract N 5ori-07634, 1952. [20] J. Xia-Yu, Commun. Theor. Phys. 52, 389 (2009). [21] K. Fakhar, T. Hayat, C. Yi and T. Amin, Commun. Theor. Phys. 53, 575 (2010). [22] D.K. Ludlow, P.A. Clarkson and A.P. Bassomx, J. Phys. A: Math. Gen. 31, 7965 (1998). [23] M. Nadjafikhah, http://arxiv.org/abs/0908.3598. [24] I.F. Barna and L. M´aty´as, Fluid. Dyn. Res. 46, 055508 (2014). [25] G. Astarita and G. Marucci, Principles of non-Newtonian fluid mechanics McGrawHill 1974, ISBN -0-07-084022-9. [26] G. B¨ohme, Non-Newtonian Fluid Mechanics, North-Holland Series in Applied Mathematics and Mechanics 1987. [27] O.A. Ladyzhenskaya, Boundary Value Problems of Mathematical Physics V, Amer. Math. Soc. Providence, RI 1970. [28] H-O. Bae and H.J. Choe, Theory of non-Newtonian flow, Trends in Mathematics Inf. Cent. for Math. Sci., 1, 201 (1998). [29] A.M. Ishak, H. Merkin, R. Nazar and I. Pop, Z. angew. Math. Phys. 59, 100 (2008). [30] M. Benlahsen, M. Guedda and R. Kersner, Math. Comput. Mod. 47, 1063 (2008). [31] G. Bogn´ar, Comp. and Mathem. with Appl. 61, 2256 (2011) [32] G. Bogn´ar, Numerical Simulation, 10, 1555 (2009). [33] I.F. Barna, G. Bogn´ar and K, Hricz´o, Math. Model. and Anal. 21, 83 (2016). [34] Private communication with Prof. Dr. Robert Kersner. [35] M.J. Boussinesq, Rendus Acad. Sci (Paris), 72, 755 (1871). [36] A. Oberbeck, Annal. der Phys. und Chemie Neue Folge 7, 271 (1879). [37] I.F. Barna and L. M´aty´as, Chaos, Solitons and Fractals, 78, 249 (2015). [38] B. Saltzman, J. Atmos. Sci. 19, 329 (1962). [39] E. N. Lorenz, J. Atmos. Sci. 20, 130 (1963) ibid, 26, 636 (1969) ibid. 63, 2056 (2005). [40] T. T´el and M. Gruiz, Chaotic Dynamics Cambridge University Press, 2006.

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[41] C. Lainscsek, Chaos 22, 013126 (2012). [42] J. H. Curry, Commun. Math. Phys. 60, 193 (1978), SIAM J. Math. Anal. 10, 71 (1979). [43] D. Roy and Z. E. Musielak, 32, 1038 (2007) ibid, 31, 77 (2007), ibid, 33, 1064 (2007). [44] S. Chandrasekhar, Hydrodynamic and Hydrodynamic Stability, Chapter II, Dover, New York 1984. [45] P. Berg´e, Y. Pommeau and C. Vidal, Ordre Within Chaos, Appendix D., J. Wiley, New York 1984. [46] C. Sparrow, The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors Springer-Verlag, New York, 1982. [47] R.C. Hilborn, Chaos and Nonlinear Dynamics Appendix C, Oxford University Press 2000. [48] I.F. Barna and L. M´aty´as, Miskolc. Math. Notes. 14, 785 (2013). [49] I.F. Barna, Laser. Phys. 24, 086002 (2014). [50] M. H. Ernst, E. H. Hauge and J. M. J. van Leeuwen, Phys. Rev. Lett. 25, 1254 (1970). [51] G. Peng and Y. Jiang, Physica A, 389, (2010) 41410. [52] Y. Pomeau and P. Manneville, Commun. Math. Phys. 74, (1980) 189. [53] D.C. Wilcox, Turbulence modeling for CFD, DCW Industries 1993.

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In: Handbook on Navier-Stokes Equations Editor: Denise Campos

ISBN: 978-1-53610-292-5 c 2017 Nova Science Publishers, Inc.

Chapter 17

A SYMPTOTIC S OLUTIONS FOR THE N AVIER -S TOKES E QUATIONS , D ESCRIBING S YSTEMS OF VORTICES WITH D IFFERENT S PATIAL S TRUCTURES Victor P. Maslov and Andrei I. Shafarevich M. V. Lomonosov Moscow State University, Moscow, Russia

Abstract In the chapter, we present a collection of our results concerning rapidly varying asymptotic solutions of the Navier — Stokes equations. We start with the results by one of the authors, describing periodic structures in incompressible fluids and focus on the effect of asymptotic instability and on the special structure of equations, governing the evolution of a periodic structure. Then we describe the connection with the asymptotic theory for the Navier — Stokes equations and topological invariants of divergence-free vector fields and Liouville foliations. We obtain the descriptions of different types of vortex structures via equations of graphs - topological invariants of corresponding steady Euler fields. We study different properties of these equations and corresponding vortex structures (Reynolds stresses, turbulent viscosity, Prandl-type equations, conservation laws, etc.) We finish with our new results in this area.

1.

Introduction

The velocity field u(x, t) of an incompressible viscous fluid (a vector field depending on the time t in R3 ) satisfies the Navier–Stokes equations ∂u + (u, ∇)u + ∇P = ν∆u, ∂t

(∇, u) = 0,

(1)

where P (x, t) is the pressure and ν is the viscosity coefficient. We consider the vortex structure of the Taylor described by the asymptotic solution of Eqs. (1) of the form 

 S(x, t) u(x, t) = U , x, t + hU1 + . . . , h

  S(x, t) P =Π , x, t + hΠ1 + . . . , h

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Victor P. Maslov and Andrei I. Shafarevich

where h → 0, h2 = ν, and S(x, t) is a one-, two-, or three-dimensional vector function. The spatial properties of such a field are determined by the dependence of the function U (z, x, t) on the vector of “fast” variables z = S/h. We consider the following three situations (for other structures see [1], [2], [3], [4], [5], [6], [7], [8], [9]): 1. A periodic structure with one-dimensional function S — this situation was studied in [1], [2]. We describe the effect of “asymptotic instability” of the corresponding vortex structure. 2. A system of vortices forming a two-dimensional film: We assume that S(x, t) is a two-dimensional vector function, U (z, x, t) → V (x, t) as |z1 | → ∞, and the function U is 2π-periodic in z2 (the function Π has the same properties). The field U then describes a periodic system of vortex filaments located at a distance of the order of h from each other; moreover, their axes form a smooth two-dimensional surface (vortex film). This surface is determined by the equation S1 (x, t) = 0, and the vortex axes are orthogonal to the vectors ∇S1 and ∇S2 . We describe the connection of such structure with topological invariants of divergence-free vector fields defined on a two-dimensional cylinder. 3. A system of vortex filaments entirely filling the space: If the vortex filaments of a periodic system are not arranged as a two-dimensional surface but fill the entire volume, then the velocity field again has form (2), and S = (S1 , S2 ), but the function U is 2π-periodic in each of its fast variables z1 and z2 . This structure is connected with topological invariants of divergence-free vector fields defined on a two-dimensional torus. 4. A periodic system of localized vortices: A periodic structure consisting of vortices, each of which is localized near a point, is described by a function U depending on the three-dimensional vector of fast variables z = S/h, S = (S1 , S2 , S3). The dependence on each of the variables zj is 2π-periodic. The corresponding topological object is a Liouville foliation on a 3D torus. In what follows, we describe the asymptotic solutions of the Navier–Stokes equations corresponding to these four situations. The proofs of the formulated theorems can be found in [1], [2], [3], [4], [5], [6], [7], [8], [9].

2. 2.1.

One-Dimensional Periodic Structure Phase of a Periodic Structure

Let S be a scalar function; we suppose that U is 2π-periodic with respect th the “fast” variable z = S/h. For arbitrary smooth periodic function f (z) we denote by f¯ the average of f : Z 2π 1 ¯ f= f (z)dz, 2π 0 and by fosc the oscillatory part of f : fosc = f − f¯. For the functions f with the zero average, ∂ −1 f will denote the integral of this function, which also has zero average.

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We substitute (2) in (1), equate the coefficient of h−1 in the left-hand side to zero, and obtain the following assertion. Assertion 2.1. If Eqs. (1) admit asymptotic solutions of form (2), then the functions U , S satisfy the equations ∂S + (V, ∇)S = 0, ∂t

(a, ∇S) = 0,

Πosc = 0,

(3)

¯ and a = Uosc . where V = U Remark 2.1. The periodic structure propagates along the trajectories of the mean flow V . This situation is typical for incompressible fluid.

2.2.

Equations of the Periodic Structure. Reynolds Stresses and Asymptotic Instability

Now we equate to zero the coefficient of h0 . Theorem 2.1. This equation can be reduced to the form   ∂V + (V, ∇)V + ∇Π = − (a, ∇)a + a(∇, a) , ∂t

(∇, V ) = 0,

∂a ∂a ∇S ⊗ ∇S ∂V + (V, ∇)a + (V1 , ∇S) + (E − 2 ) a+ ∂t ∂y (∇S)2 ∂x   2 ∂a −1 ∇S ∂S 2∂ a + (a, ∇)a − ∂y (∇, a) + (a, a) = (∇S) . ∂y (∇S)2 ∂x2 ∂y 2 osc

(4)

(5)

Here ∂y−1 f denotes the the average-free integral of the average-free periodic function f , V1 = U1 . Remark 2.2. The right hand side in the equation (4) coincides with well-known Reynolds stresses. Remark 2.3. Equation (5) contains term with the correction V1 . This term, however, can be killed by introducing the additional “phase shift”. Assertion 2.2. Let a(y, x, t) = ˆa(y + c(x, t), x, t) and let c satisfy the equation ∂c + (V, ∇)c + (V1 , ∇S) = 0. ∂t Then the function ˆa satisfies (5) with V1 = 0. So Eqs. (4), (5) define the main term of asymptotic solution up to the phase shift c. The latter function depends on V1 and can be found from the next approximation — the corresponding equations define a closed system for V1 , c, a1, where a1 is defined up to the term of the form c1 (x, t)∂a/∂y. The function c1 is defined from the further approximation, etc. In each approximation we obtain the functions Vk , ak and the “phase-shift correction” ck−1 ; ak is defined up to the summand ck (x, t)∂a/∂y. Dependence of the functions ck on Vk+1 means that the asymptotic solution is “asymptotically unstable”: if one changes the initial velocity field by the quantity O(h), the solution changes as O(1) (however, the mean flow V and the profile ˆa do not change).

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3. 3.1. 3.1.1.

Victor P. Maslov and Andrei I. Shafarevich

Vortex Films Euler Equations and Topological Invariants of Vector Fields on a Two-Dimensional Cylinder Euler Equations on a Two-Dimensional Cylinder

Now we consider asymptotic solutions of form (2) describing vortex films. We let Mt denote a surface in the space R3 determined by the equation S1 (x, t) = 0. Because the function U (z, x, t) − V (x, t) decreases rapidly as |z1 | → ∞, it can be replaced (mod O(h)) with its restriction to Mt . Moreover, we can assume that S1 (x, t) is the distance to Mt along the normal (more precisely, any smooth function in R3 coinciding with this distance in a sufficiently small neighborhood of the surface Mt ), and S2 and U coincide with S2 |Mt and U |Mt near the surface. With this fact taken into account, we substitute (2) in (1), equate the coefficient of h−1 in the left-hand side to zero, and obtain the following assertion. Assertion 3.1. If Eqs. (1) admit asymptotic solutions of form (2), then the function U satisfies the equations (v, ∇z )v + ∇z Π = 0,

(∇z , v) = 0,

(v, ∇z )w = 0,

(6) (7)

where ∇z denotes differentiation with respect to the fast variables z in the Euclidean metric g ij = (∇Si, ∇Sj )|Mt on the cylinder Rz1 × S1z2 , vj = (∂Sj /∂t + dSj (U ))|Mt are the velocity field components orthogonal to the vortex film and tangent to it but orthogonal to the axes of the vortex filaments, m is the unit vector orthogonal to ∇S1 , ∇S2 (i.e., directed along the axes of the filaments), and w = (U, m)|Mt is the velocity field component directed along the axes of the vortices. Remark 3.1. It follows from the choice of the functions S1 and S2 that this Euclidean metric has the form ds2 = dz12 + (∇S2 )2 |Mt dz22 . The equations for the two-dimensional vector field v are the stationary Euler equations on the cylinder; the last equation for the function w means that it is constant on the trajectories of the vector field v. 3.1.2.

Integral Identities for Solutions of the Euler Equations

In what follows, we need solutions v of the Euler equations such that almost all their trajectories are closed. Such fields satisfy the condition that v1 → 0 as z1 → ∞; they have some special properties, which we discuss below. Assertion 3.2. Let v be a solution of Euler equations (6), and let v1 → 0 as z1 → ∞. Then we have the identities Z

2π

v1 v2 dz2 =

0

∂ ∂z1

Z

∞

v1 v2 dz1 = 0,

−∞

Z

0

2π

(v12 + Π) dz2 =

∂ ∂z2

Z

∞

−∞


(v22 − ω22 ) + (∇S2 )|2Mt (Π − Π0 ) dz1 = 0,

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Asymptotic Solutions for the Navier-Stokes Equations where ω2 = lim v2 = z1 →∞



 ∂S2 + (V, ∇)S2 , ∂t Mt

309

Π0 = lim Π. z1 →∞

Remark 3.2. The obtained relations are similar to the identities in [11], [12] satisfied by the solutions of the Euler or Navier–Stokes equations decreasing at infinity. But we note that in our case, functions of one of the variables z, not constants, are zero. 3.1.3.

Parameterization of Solutions of the Euler Equations

The next stage in the asymptotic procedure is to obtain equations for the parameters on which solutions of Eqs. (6) depend. A hypothesis according to which the parameters on which solutions of the Euler equations depend are topological invariants of vector fields with zero divergence with respect to the flow region area-preserving diffeomorphisms (volume-preserving in the three-dimensional case) was discussed in [3] – [5] (also see [13], [14], [15], [16] for the relation between the flow topology and the Euler equations). We present this hypothesis adapted to our situation. We consider a divergence-free vector field v(z) on a two-dimensional cylinder. We assume that all equilibriums of this field are nondegenerate and, in addition, almost all trajectories are closed. We let Γ denote the cylinder quotient by the trajectories of v; it is clear that Γ is a graph whose vertices correspond to equilibriums and whose edges correspond to domains smoothly foliated into closed trajectories of v. There is a natural parameterization on each edge: a periodic trajectory γ is associated with the number Z 1 I= z1 dz2 2π γ (the action variable). The parameterized graph Γ is an invariant of the field v with respect to area-preserving diffeomorphisms of the cylinder. Moreover, an (also invariant) function associating each periodic trajectory of the field v with its frequency ω(I) is defined on this graph. It is clear that this function is continuous on Γ and smooth on each edge. If the field v satisfies the Euler equations, then the frequency function and the Bernoulli integral satisfy the relation B = v 2 /2 + Π. It is convenient for our purposes to use the function B instead of ω. Conjecture 3.1. There is an open (in an appropriate sense) subset of the set of pairs Γ and B, where Γ is a parameterized graph and B is a function continuous on Γ and smooth on the edges, such that there is a smooth solution v, Π of Euler equations (6) for each pair in this open subset, the graph Γ is the set of trajectories of the field v, and B is the Bernoulli integral. Remark 3.3. Any divergence-free field v on the cylinder such that almost all its trajectories are closed can be represented as a skew gradient of the scalar function ψ (stream function). In the coordinates z, we have v1 = −

∂ψ , ∂z2

v2 =

∂ψ , ∂z1

ψ = Cz1 + ψ0 (z1 , z2 ),

where C is a constant, ψ0 is a periodic function of z2 , and ψ0 → 0 as z1 → ∞. It is clear that Γ is a Reeb graph (set of level lines) of the function ψ.

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Equation (7) means that w is a function well defined on Γ. The complete set of “parameters” on which solutions of Eqs. (6) and (7) depend hence consists of the triples Γ, B, w, where Γ is a parameterized graph and B and w are functions continuous on Γ and smooth on the edges. The functions B and w also depend on the “slow” variables x and t, x ∈ Mt . Below, we obtain equations for these functions determining the evolution of the vortex system.

3.2.

Cokernel of the Linearized Euler Equations

Equations for the parameters appear in the analysis of the next approximation in the asymptotic procedure (see, e.g., [17]). Namely, we equate the coefficients of h0 arising in the left- and right-hand sides of Eqs. (1) after solutions (2) are substituted in Eqs. (1). We obtain equations for the field U1 , i.e., linearized Eqs. (6) and (7) with a nonzero right-hand side. The sought equations follow from the solvability conditions for this problem, i.e., from the conditions that the right-hand sides are orthogonal to the cokernel of the operators obtained by linearizing the left-hand sides of (6) and (7). We first describe the cokernel of the linearized Euler operator on the two-dimensional cylinder; it consists of divergence-free vector fields ξ satisfying the equation (v, ∇z )ξ −

∂v ∗ ξ + ∇z χ = 0. ∂z

(8)

Assertion 3.3. The cokernel of the linearized Euler operator is infinite-dimensional: any divergence-free field commuting with v belongs to it. Remark 3.4. This space in the cokernel of the linearized Euler operator is generated by a variation in an arbitrary function B and can be interpreted as the space of functions on the graph Γ. Indeed, we introduce the action–angle variables I and ϕ ([13]) in an arbitrary domain of the cylinder smoothly foliated into the trajectory of v. The fields v and ξ in these coordinates have the respective forms ω(I) ∂/∂ϕ and λ(I) ∂/∂ϕ, where ω(I) and λ(I) are the frequencies of these fields on the trajectory corresponding to the parameter I. The field ξ is thus determined by the function λ(I) defined on the graph Γ. The cokernel of the operator corresponding to Eq. (7) obviously consists of functions constant on the trajectories of the field v, i.e., of functions on the graph Γ. The cokernel of the linearized operator of system (6), (7) hence contains pairs of functions defined on this graph.

3.3.

Solvability Conditions for the Equation for the Correction

The equations in the first-order approximation (i.e., the relations obtained by equating of the first-order terms arising after (2) is substituted in (1)) have the forms (v1 , ∇z )v + (v, ∇z )v1 + ∇z Π1 = −F, (∇z , v1 ) = −G, (v, ∇)w1 + (v1 , ∇)w = −H,

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where the vector function F and the scalar functions G and H are expressed in terms of v, w, Π, v1j = dSj (U1 )|Mt , and w1 = (U1 , m)|Mt . The conditions of orthogonality to the space in the cokernel of the linearized Euler operator (mentioned above) imply a relation for F , G, and H. More precisely, we have the following assertion. Theorem 3.1. If there are smooth solutions of Eqs. (9), then the functions F , G, and H satisfy the relations ∂B (F, v) dϕ + a(I) = 0, ∂I γ

Z

∂a + ∂I

Z

G dϕ = 0,

γ

Z

H dϕ + a

γ

∂w = 0, ∂I

(10)

where B = Π + v 2 /2 is the Bernoulli function, γ is an arbitrary closed trajectory of the field v, ϕ is the angular coordinate on the trajectories, and a is an auxiliary function on the graph Γ. Remark 3.5. The conditions written above follow from the conditions that the vector of right-hand sides of Eqs. (9) is orthogonal to the abovementioned infinite space in the cokernel of the linearized Euler operator. Namely, relations (10) are the orthogonality conditions written in a specially chosen “basis of the equipment of this space,” which, roughly speaking, consists of δ-type fields supported on the trajectories of the field v. Precisely such a choice of the “basis” permits reducing the problem of determining the orthogonality conditions to the problem of averaging along the trajectories.

3.4.

Equations of Motion of a Vortex Film

We consider Eqs. (10) and pass to the limit as |z1 | → ∞ in them. Because U (z, x, t) → V (x, t), we immediately obtain the following assertion. Assertion 3.4. The vector field V (x, t) satisfies the Euler equations ∂V + (V, ∇)V + ∇P0 = 0, ∂t

(∇, V ) = 0,

(11)

where P0 = lim|z1 |→∞ Π. Because almost all trajectories of the field v are closed, we see that v1 → 0 as |z1 | → ∞, which implies the equation of motion of the vortex film, i.e., of the surface Mt determined by the equation S1 (x, t) = 0: 

 ∂S1 + (V, ∇)S1 = 0. ∂t S1 =0

(12)

Remark 3.6. It follows from (12) that the surface Mt moves along the flow, i.e., the normal velocity of a point of this surface coincides with the component of the velocity field V normal to this surface.

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3.5.

Victor P. Maslov and Andrei I. Shafarevich

Evolution of a Vortex System

Expressions (10) are equations determining the evolution of the vortex structure “parameters,” i.e., of the functions B and w defined on the graph Γ and depending on a point of the moving surface Mt . These equations must be considered together with Euler equations (6), (7). The following assertion shows their structure more clearly. Theorem 3.2. Relations (10) are equivalent to the system of equations ∂B ∂B ∂ 2 ∂ + (< U >, ∇)B + a + Q(B, w) = D B, ∂t ∂I ∂I ∂I ∂a + < (∇, U ) > +E(B, w) = 0, ∂I

(13)

∂w ∂w ∂ 2 ∂ + (< U >, ∇)w + a + K(B, w) = D w. ∂t ∂I ∂I ∂I Here, the angle brackets denote averaging along the trajectories of the field v, D 2 = (1 + (∇S2 )2 )|Mt



∂z ∂ϕ

2  ,

and the functions Q, E, and K depend on B, w, and I and a point of the moving surface Mt . In particular, 

   ∂ ∂ ∂Sk −Q = + (U, ∇) Π + vk + 2(U, ∇) + vk (U, Sk00U ) + ∂t ∂t ∂t    ∂Sj ∂U ∂Π + vk z1 (∇Sk ), + (U, ∇Sj ) + ∇Sj ∂t ∂zj ∂zj Mt

(the sums are taken over repeated indices, and Sj00 denotes the matrix of second-order derivatives of the function Sj ). ∂ ∂ ∂ ∂ Remark 3.7. The terms ∂I D 2 ∂I B and ∂I D 2 ∂I w in Eqs. (13) describe the influence of the fluid viscosity on the vortex system under study. We note that the viscosity coefficient D 2 depends on the unknown functions B and w themselves. A similar phenomenon appears in the description of developed turbulence: the dynamical equations contain the so-called turbulent viscosity depending on an unknown velocity field. We note that the expression for the turbulent viscosity is unknown in advance; it is chosen from physical considerations or considerations of maximal simplicity of the model. In our case, D 2 is a completely definite function of B and w (although it is defined in a rather complicated way).

Equations (13) in the variable I are given on the graph Γ. In what follows, we discuss the conditions that must be satisfied by the functions a, B, and w at the vertices of the graph.

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Asymptotic Solutions for the Navier-Stokes Equations

3.6.

313

Kirchhoff Conditions

Obviously, the functions w and B are continuous on the graph Γ. At each vertex, the derivatives of these functions and the function a satisfy the Kirchhoff relations in the theory of electric circuits. Namely, we have the following assertion. Assertion 3.5. At each inner vertex (i.e., at a vertex of degree greater than 1) of the graph Γ, the functions a, D 2 ∂B/∂I, and D 2 ∂w/∂I satisfy the Kirchhoff conditions     2 ∂B 2 ∂B D = D , ∂I out ∂I in

aout = ain ,



D

2 ∂w

∂I



out

=



D

2 ∂w

∂I



,

(14)

in

where the subscript out denotes the sum of the limits of the corresponding function at a given vertex over the outgoing edges and the subscript in denotes the sum of its limits over the incoming edges. Remark 3.8. The function a is zero at vertices of degree 1 of the graph Γ (they correspond to the elliptic equilibriums of the field v). Remark 3.9. All inner vertices of the graph Γ in general position are vertices of degree 3. The Kirchhoff conditions then become       2 ∂B 2 ∂B 2 ∂B a1 = a2 + a3 , D = D + D , ∂I 1 ∂I 2 ∂I 3       2 ∂w 2 ∂w 2 ∂w = D + D , D ∂I 1 ∂I 2 ∂I 3 where the subscript 1 denotes the limit over an incoming edge and the subscripts 2 and 3 denote the limits over the outgoing edges.

3.7.

Reynolds Stresses

It is well known, that averaging hydrodynamic equations leads to the appearance of terms describing the influence of fluctuations on the average field (Reynolds stresses). The role of the average field in our situation is played by the integral of the velocity field over the two-dimensional cylinder of “fast” variables z. Namely, we let U0 denote the decreasing part of the velocity field, U0 = U − V , and let the bar denote averaging over the cylinder: f¯ =

Z

2π

0

dz2

Z

∞

f dz1 . −∞

Theorem 3.3. The vector field U 0 satisfies the relations ∂U 0 + (V, ∇U 0 ) + (U 0 , ∇)V + (U 0 , ∇)U 0 + (Θ, ∇)Θ + Θ(∇, Θ) + ∇P = 0, ∂t (∇, U 0 ) = 0,

where Θ = U0 − U 0 and P = Π − Π0 .

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Victor P. Maslov and Andrei I. Shafarevich

Remark 3.10. The terms κ = (Θ, ∇)Θ + Θ(∇, Θ) in (15) coincide exactly with the Reynolds stresses. They have the form κi = (∇, ΘΘi ) in coordinates. Equations (15) are the conditions that the vector of the right-hand sides of Eqs. (9) is orthogonal to a constant vector field. Of course, such a field commutes with any field, in particular, with v and therefore belongs to the cokernel of the linearized Euler operator.

4.

System of Vortex Filaments Entirely Filling the Space

The function U is periodic in both “fast” variables z in the case where a system of vortex filaments entirely fills the space. From the technical standpoint, the difference from the preceding case is that there is no surface Mt in the case of a vortex system distributed over the space, and there are hence no expansions in the distance to this surface that create additional terms in the formulas in Sec. 3. Moreover, there is no passing to the limit as z1 → ∞, and hence no vector field V . The variables z1 and z2 become equivalent.

4.1.

Euler Equations and Topological Invariants of Vector Fields on the Torus

We assume that the variables z range over a two-dimensional torus T2 . We substitute (2) in Eqs. (1), equate the coefficient of h−1 in the left-hand side to zero, and obtain the following assertion. Assertion 4.1. If Eqs. (1) admit asymptotic solutions of form (2), then the function U satisfies the Euler equations (v, ∇z )v + ∇z Π = 0, (∇z , v) = 0,

(16)

(v, ∇z )w = 0, where ∇z denotes differentiation with respect to the fast variables z in the Euclidean metric g ij = (∇Si , ∇Sj ) on the torus T2 , vj = ∂Sj /∂t + dSj (U ), w = (U, m), and m is the unit vector orthogonal to ∇S1 and ∇S2 . Remark 4.1. Constructing asymptotic solutions of form (2) requires replacing the derivatives (a, ∇) (where a is a vector field) with the derivatives (a, ∇) + h−1 dSj (a)∂/∂zj , i.e., a connection whose horizontal space is defined by the vectors ai ∂/∂xi + bj ∂/∂zj such that bj = dSj (a) arises in the Cartesian product R3x × T2z . The graph Γ is defined as in Sec. 3, and the hypothesis of parameterization of the solutions of Eqs. (16) is formulated similarly. Conjecture 4.1. There is an open subset (in an appropriate sense) in the set of triples Γ, B, w, where Γ is a parameterized graph and B and w are functions continuous on Γ and smooth on the edges, such that there is a smooth solution v, π, w of Euler equations (16) for each triple in this open subset, the graph Γ is the set of trajectories of the field v, and B is the Bernoulli function for this solution.

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Cokernel of the Linearized Euler Equations and the Solvability Conditions for the Equation of the Correction

In the considered situation, the cokernel consists of divergence-free vector fields ξ on the torus T2 that satisfy the equation curlz



∂v ∗ (v, ∇z )ξ − ξ ∂z



= 0.

The following assertion can be proved similarly to Assertion 3.3. Assertion 4.2. The cokernel of the linearized Euler operator is infinite-dimensional, i.e., it contains any divergence-free field commuting with v. The first-order approximation equations (i.e., the relations obtained by equating the first-order terms after (2) is substituted in Eqs. (1)) have form (9), where the vector function F and the scalar functions G and H are expressed in terms of v, w, Π, v1j = dSj (U1 ), and w1 = (U1 , m). The conditions of orthogonality to the space in the cokernel of the linearized Euler operator (mentioned in Assertion 4.2) imply a relation for F , G, and H. The following theorem can be proved similarly to Theorem 3.1. Theorem 4.1. If there are smooth solutions of Eqs. (9), then the functions F , G, and H satisfy (10), where B is the Bernoulli function, γ is an arbitrary closed trajectory of the field v, ϕ is the angular coordinate on the trajectories, and a is an auxiliary function on the graph Γ.

4.3.

Equations of a System of Vortex Filaments: Kirchhoff Conditions

Equations (10) can be rewritten as a system of equations for B and w (we recall that these functions are defined on the graph Γ and depend on the additional parameters x and t). The following assertion can be proved similarly to Theorem 4.1 in [6]. Theorem 4.2. Relations (10) are equivalent to system of equations (13), where the angle brackets denote averaging along the trajectories of the field v, 2

D =g



∂z ∂ϕ

2 

,

g = det (∇Si, ∇Sj ), and Q, E, and K depend on B, w, I, x, and t. The functions w and B are continuous on the graph Γ, and the functions a, D 2 ∂B/∂I, and D 2 ∂w/∂I satisfy Kirchhoff conditions (11) at each vertex of degree greater than 1. The function a is zero at the vertices of degree 1 of the graph Γ.

4.4.

Reynolds Stresses

The Reynolds equations are derived as in [6]. Namely, the following assertion can be proved by integrating Eqs. (9) over the torus T2 .

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Theorem 4.3. If Eqs. (9) have a smooth solution, then ∂U ¯ + (Θ, ∇Θ) + Θ(∇, Θ) = 0, + (U, ∇U) + ∇Π ∂t

(∇, U) = 0,

(17)

where Θ = U − U and the bar denotes averaging over the torus T2 . Remark 4.2. If the vortices are arranged in a film, then the asymptotic description of such a structure contains two vector fields that are independent of the fast variables. One of them, i.e., the limit field V (x, t), satisfies the Euler equations and determines the motion of the film. The second, i.e., the average field U0 , satisfies the Reynolds equations (containing the field V and the Reynolds stresses). There are no vortices of the limit field in the case of a distributed system, but there is an average field related to the Reynolds stresses. Remark 4.3. The vortex structure described in this section does not become a solution of the Euler equations in the limit of vanishing viscosity (h → 0). The weak limit of this structure satisfies the Reynolds equations; the “envelope” of fast oscillations (which is obviously nonzero) is a function of the slow variables x and t obeying a complex law. To calculate this law, it is necessary to find the maximum of the solution of Eqs. (13) with respect to the fast variables.

5.

Periodic Structure Consisting of Localized Vortices

Finally, in a fluid, we consider a structure consisting of vortices localized at one point and periodically located in the space. In this case, S ∈ R3 , and the function U (z, x, t) is 2π-periodic in the fast variables z = S/h, i.e., the variables z vary on the three-dimensional torus T3 .

5.1.

Three-Dimensional Euler Fields and the Fomenko Invariants of Liouville Foliations

We substitute (2) in Eq. (1), equate the coefficient of h−1 in the left-hand side to zero, and obtain the following assertion. Assertion 5.1. If Eqs. (1) admits asymptotic solutions of form (2), then the function U satisfies the Euler equations (v, ∇z )v + ∇z π = 0,

(∇z , v) = 0,

(18)

where ∇z denotes differentiation with respect to the fast variables z in the Euclidean metric g ij = (∇Si , ∇Sj ) on the torus T3 and vj = ∂Sj /∂t + dSj (U ). Remark 5.1. Constructing asymptotic solutions of form (2) requires replacing the derivatives (a, ∇) (where a is a vector field) with the derivatives (a, ∇) + h−1 dSj (a)∂/∂zj , i.e., a connection whose horizontal space is defined by the vectors ai ∂/∂xi + bj ∂/∂zj such that bj = dSj (a) arises in the Cartesian product R3x × T3z .

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The central role in the constructions in the preceding sections was played by the hypothesis of parameterization of the solutions of the Euler equations by topological invariants of divergence-free vector fields. The vector field v is three-dimensional in our situation. The description of invariants of arbitrary three-dimensional fields with zero divergence is apparently a transcendentally complicated problem. But the topology of Euler fields in general position is specific and similar to the topology of integrable Hamiltonian systems with two degrees of freedom. The invariants of such systems are described in the framework of the theory of their topological classification [18]. We determine the required invariants and formulate the corresponding hypothesis of parameterization. We recall (see [13]) that solutions of the stationary Euler equations have the following properties: 1. The vector field Ω = curlz v commutes with v (here curlz is the curl in the fast variables z). 2. The vector product v × Ω is the gradient of the Bernoulli function B = (v, v)/2 + π0. 3. A nonsingular compact connected level set of the function B is homeomorphic to the torus. This torus is invariant under the phase flows of the fields v and Ω, and motion on it is conditionally periodic by virtue of either of these fields. The topology of stationary Euler fields is therefore similar to the topology of completely integrable Hamiltonian systems with two degrees of freedom, i.e., such a field determines a foliation of a three-dimensional torus into two-dimensional tori, and this foliation is similar to the Liouville foliation of an isoenergetic surface. This fact permits introducing an invariant of such fields, which is similar to the Fomenko invariant (molecule [18]) arising in the theory of Hamiltonian systems. Namely, we let Γ denote the set of compact level surfaces of the Bernoulli integral B and consider the fields v for which Γ is a graph such that all inner points of its edges correspond to nonsingular level surfaces (i.e., to two-dimensional tori). We note that the Bernoulli integral B cannot be a Morse function, which is a consequence of the following simple assertion (cf. [18]). Assertion 5.2. Any isolated critical point of the function B is degenerate. Therefore, Γ is not a Reeb graph. The theory of topological classification of Hamiltonian systems with two degrees of freedom treats graphs that are sets of level surfaces of the so-called Morse–Bott function [18]. It is important for our purposes that Γ is a graph. It is clear that the graph Γ constructed with respect to the field v remains unchanged if this field is transformed by a volume-preserving diffeomorphism of the three-dimensional torus. Moreover, this graph bears a natural parameterization that is invariant under such diffeomorphisms. Namely, we consider an arbitrary vertex of degree 1 of the graph Γ and associate an arbitrary point of the graph edge incident to it (i.e., a torus invariant under the flows of the fields v and Ω) with a number I equal to the volume of the corresponding solid torus divided by 4π 2 . An invariant parameterization is thus defined on this edge, and we orient the edge in the direction of the increasing parameter. We further consider the second vertex incident to this edge. We assign this vertex the value of the parameter I0 equal to the

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limit of I on the considered edge and parameterize all the other edges incident to this vertex as follows. We first parameterize the edge in the direction away from the vertex through which we already passed. We further assign the point a on the edge a number I(a) such that the following two conditions are satisfied:

a. The number I(a2) − I(a1) is equal to the volume of the three-dimensional film bounded by the tori a1 and a2 if the edge is oriented from a1 to a2 . b. The limit of I(a) as the point tends to the already passed vertex is equal to I0 . As a result, we obtain a parameterization on all edges incident to this vertex. This process can be continued until we parameterize all edges. Namely, we assume that some edges are parameterized at a certain stage. We then consider an arbitrary vertex incident to one of the parameterized edges. All edges incident to this vertex can be divided into two classes: already parameterized edges (we call them incoming edges) and edges that are not yet parameterized (we call them outgoing edges). We assign the chosen vertex the value of the parameter I0 equal to the sum of the limits of I over all incoming edges. We then introduce a parameter on each outgoing edge following the procedure described above. We have thus defined a topological invariant of the field v, i.e., the parameterized graph Γ. We further note that there are additional invariants on this graph, i.e., functions of the frequencies of the field v. We consider an arbitrary edge of the graph. This edge determines a three-dimensional torus domain smoothly foliated into two-dimensional tori. We fix a basis of cycles on the invariant tori in this domain. This basis varies smoothly from torus to torus, and we let ϕ = {ϕ1 , ϕ2}, ϕj ∈ [0, 2π], denote the angular coordinates on the tori corresponding to this basis. In the coordinates I and ϕ (we further call them the action–angle variables by analogy with the theory of integrable Hamiltonian systems; see [13]), the field v has the form ∂ ∂ v = ω1 (I) + ω2 (I) , ∂ϕ1 ∂ϕ2 and the functions ωj (I) (frequencies) are defined to be invariant under diffeomorphisms. We have thus determined the set of invariants of the field v, which consist of the parameterized graph Γ and a pair of smooth functions ω1 and ω2 on its edges. Conjecture 5.1. There is an open subset (in an appropriate sense) in the set of triples Γ, ω1 , ω2 , where Γ is a parameterized graph and ωj are smooth functions on its edges, such that there is a smooth solution v, π of Euler equations (18) for each triple in this open subset, the graph Γ is the set of compact level surfaces of the Bernoulli integral, and the functions ωj are the frequencies of the field v on the corresponding tori. The equations determining the evolution of the parameters on which the vector v depends are equations for the vector of the frequencies ω and are given for one of the variables on the graph Γ.

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Cokernel of the Linearized Euler Equations

The cokernel of the linearized Euler operator consists of the divergence-free vector fields ξ on the torus T3 satisfying the equations curlz



∂v ∗ (v, ∇z )ξ − ξ ∂z



= 0.

(19)

Assertion 5.3. The cokernel of the linearized Euler operator is infinite-dimensional: any divergence-free field commuting with v and Ω = curlz v satisfies Eq. (19). Remark 5.2. The considered space in the cokernel of the linearized Euler operator is generated by a variation in the arbitrary functions ω1 and ω2 and can be interpreted as the space of pairs of functions on the graph Γ. Indeed, we introduce the above-described action–angle variables I and ϕ in an arbitrary domain of the three-dimensional torus smoothly foliated into the tori B = const. The fields v and Ω in these coordinates have the respective forms ω1 (I)∂/∂ϕ1 + ω2 (I)∂/∂ϕ2 and λ1 (I)∂/∂ϕ1 + λ2 (I)∂/∂ϕ2, where ωj (I) and λj (I) are the frequencies of these fields on the invariant torus corresponding to the parameter I. On the other hand, any divergence-free field ξ commuting with v and Ω has the form b1 (I)∂/∂ϕ1 + b2 (I)∂/∂ϕ2 in the same coordinates, where bj (I) are arbitrary smooth functions. In other words, the Liouville tori of the field v are also invariant manifolds for the field ξ, which is therefore determined by its frequency vector b1 , b2. The set of divergence-free fields commuting with v and Ω can thus be interpreted as the set of pairs b1 , b2 of functions of one variable I, which ranges over the graph Γ. This fact is an argument supporting Hypothesis 3 of parameterization of the set of solutions of the Euler equations.

5.3.

Solvability Conditions for the Correction

The equations of the first-order approximation (i.e., the relations obtained by equating the first-order terms arising after (2) is substituted in (1)) have the form (v1 , ∇z )v + (v, ∇z )v1 + ∇z π1 = −F,

(∇z , v1) = −G,

(20)

where the vector function F and the scalar function G are expressed in terms of v and j v1 = dSj (U1 ). The conditions of orthogonality to the space in the cokernel of the linearized Euler operator (mentioned above) imply a relation for F and G. More precisely, we have the following assertion. Theorem 5.1. If there are smooth solutions of Eqs. (20), then the functions F and G satisfy the relations Z Z Z ∂B ∂a F dφ + a(I) = 0, + G dφ = 0, (Ω, F ) = 0, (21) ∂I ∂I Λ Λ Λ where B is the Bernoulli function, Λ is an arbitrary nonsingular torus of the Liouville foliation, Ω = curlz v, dφ = dφ1 ∧ dφ2 , and a is an auxiliary function on the graph Γ.

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Remark 5.3. We use the flat metric gij to identify the vector field F on the torus T3 with the 1-form β. Equations (21) can be written as Z Z ∂B β(v) dφ + a = 0, β(Ω) dφ = 0. ∂I Λ Λ These relations permit calculating the restriction of the form β to the Liouville torus Λ: β|Λ = a

∂B σ, ∂I

where

λ1 dϕ2 − λ2 dϕ1 . ω1 λ2 − ω2 λ1 Here, ωi and λi are the respective frequencies of the fields v and Ω on the torus Λ. σ=

5.4.

Equations of the Coherent Structure

Equations (21) can be rewritten as a system of equations for the vector of frequencies ω of the field v (we recall that this vector is defined on the graph Γ and depends on the additional parameters x and t). Namely, we have the following assertion. Theorem 5.2. Relations (18) are equivalent to the equations for the vector ω ∂ω ∂ω ∂ 2ω ∂B ∂a + (w, ∇ω) + Q + rω = D2 2 + a , + R = 0. (22) ∂t ∂I ∂I ∂I ∂I Here, the coefficients of the 2×2 matrices Q and r, the scalar functions R and D, and the vector field w can be expressed in terms of ω and the coefficients of the flat metric on T3 calculated at points of the torus Λ. In particular, D2 =< det g >, where g is the induced Riemannian metric on Λ and the angle brackets denote averaging over Λ.

5.5. 5.5.1.

Kirchhoff Conditions for the Equations of the Coherent Structure Three-Atoms and Admissible Frequencies on the Liouville Tori

Each edge of the graph Γ corresponds to a three-dimensional toric domain smoothly foliated into two-dimensional tori. We now consider an arbitrary vertex of this graph that represents a singular level set of the Bernoulli integral B. We consider a neighborhood of this singular set. We assume that it is a three-atom in the sense of [18], i.e., it is a compact connected orientable manifold with the edge equipped with the structure of a Seifert fibration, and the latter is either trivial or has type (1, 2) (see [18]). The edge of the manifold consists of several tori; a torus is said to be outer if the parameter I increases when approaching it and inner otherwise. Remark 5.4. In the topological theory of integrable Hamiltonian systems with two degrees of freedom (see [18]), the neighborhoods of singular fibers of the Liouville foliation are three-atoms by the Morse–Bott condition on the additional integral and the requirement of topological stability. Of course, similar requirements can be imposed on the Euler field v in our situation, but to know which conditions guarantee the “atomic” structure of singular sets is not important for our purposes, and we therefore assume that such a structure is known in advance.

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We now consider the corresponding inner vertex of the graph. There are several edges incident to this vertex, some of them are incoming, and the others are outgoing (we recall that the graph is oriented in the direction of the increasing parameter I). We let η and ζ denote admissible bases of cycles on the nonsignificant tori of an atom (see [18]) and recall their definition. The first basis cycle η on the Liouville torus is a fiber of the Seifert fibration; as the torus tends to the critical circle, this cycle also tends to this circle bypassed once or twice depending on whether the Seifert fibration is trivial or not on the atom. The second cycle ζ completes the cycle η to a basis. If the Seifert fibration is trivial, then this cycle is constructed as the intersection of a Liouville torus with a two-dimensional surface that is a global cross section of the Seifert fibration. But if the foliation is nontrivial, then the second cycle is determined as the intersection of the torus with the global cross section of the “trivialized” Seifert fibration (the latter is obtained by deleting small neighborhoods of the critical circle from the atom). The cross section must then satisfy the condition that its intersection ζ 0 with the boundary of a tubular neighborhood of the critical circle is related to the fiber η of the Seifert fibration and to the vanishing cycle κ by the expression η = κ+2ζ. Remark 5.5. Of course, an admissible basis of cycles is defined nonuniquely. Namely, we can add a cycle kη to each cycle ζ, and the sum of the integers k over all boundary tori of a given atom must be zero (see [18]). The choice of a basis of cycles determines the frequencies of the field v on the nonsingular tori of the Liouville foliation in the considered atom. We let ω0 denote the frequency corresponding to the cycle η (a fiber of the Seifert fibration) and let ω 0 denote the frequency corresponding to the cycle ζ. It is clear that ω0 is a function smooth on an atom and constant on each Liouville torus (we note that ω 0 is nonsmooth, and its asymptotic form as it passes through the critical circle is given, e.g., in [18]). 5.5.2.

Kirchhoff Conditions on a Three-Atom

Theorem 5.3. At each inner vertex (i.e., at a vertex of degree greater than 1) of the graph Γ corresponding to a three-atom, the functions a, D2 ∂B/∂I, and D2 ∂ω0 /∂I satisfy the Kirchhoff conditions         2 ∂ω0 2 ∂ω0 2 ∂B 2 ∂B aout = ain , D = D , D = D , ∂I out ∂I in ∂I out ∂I in where the subscript out denotes the sum of limits of the corresponding function at a given vertex on the outgoing edges and the subscript in denotes the sum of its limits on the incoming edges. Here, D2 =< g 11 >, where g 11 is an element of the Euclidean metric in the space R3z in the action–angle coordinates, and this element corresponds to the coordinate I. Corollary 5.1. Let all vertices of the graph Γ represent three-atoms. For each edge of the graph, we choose a basis of cycles on the Liouville tori of the corresponding domain of the three-dimensional torus and thus define the frequencies ω1 and ω2 of the field v. We further choose an admissible basis of cycles η and ζ on the Liouville tori of each atom (see [18] and associate each vertex of degree m of the graph Γ with a set of integer-valued

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two-dimensional vectors A1 , . . . , Am defined as follows. We consider the jth edge of the graph incident to this vertex (the index j ranges from 1 to m, and the outgoing vertices correspond to the first k numbers). On the Liouville tori corresponding to this edge, there is a basis of cycles γ1 and γ2 generating the frequencies ω1 and ω2 and a basis of cycles η and ζ. These bases can be expressed in terms of each other, for example, as γ1 = A1j η + A2j ζ (we note that the same coefficients express the frequency ω0 in terms of the frequencies ω1 and ω2 ). The Kirchhoff conditions for ω0 at the graph vertices can now be rewritten in terms of the frequencies ω1 and ω2 . Namely, the preceding theorem implies the formulas k X j=1

j

(Aj , ω ) =

m X

(Aj , ω j ),

j=k+1

where ω j is the limit of the vector of frequencies ω = {ω1 , ω2 } on the jth edge of the graph at a given vertex. Remark 5.6. The function a is zero at the vertices of degree 1 of the graph Γ (they correspond to the case of degeneration of a torus into a circle rather than to the case of bifurcation of tori). Remark 5.7. We assume that all vertices of Γ correspond to atoms. It is natural to assume that these atoms are stable, i.e., do not decay under the action of a small deformation (indeed, in the problem of the coherent structure, the field v depends on additional parameters x and t whose variations must generally lead to the decay of unstable atoms). Such atoms can be of the following three types: A (the torus degenerates into a circle, and the atom is associated with a vertex of degree 1 of the graph G), B (two tori bifurcate into one, and the atom is associated with a vertex of degree 3 and has the structure of a trivial Seifert fibration), and A∗ (one torus bifurcates into one torus, such an atom is associated with a vertex of degree 2, and the corresponding Seifert fibration is nontrivial). In this case, the Kirchhoff conditions on atoms of types B and A∗ together with the conditions of vanishing at the vertices corresponding to atoms of type A completely determine the function a. In other words, this function can be eliminated from the equations for the coherent structure just as in the cases of classical Prandtl equations, equations of one-dimensional coherent structures [1], and equations of stretched vortices [3] — [4].

5.6.

Reynolds Stresses

The following assertion can be proved by integrating Eqs. (21) over the threedimensional torus T3 . Theorem 5.4. If Eqs. (21) have a smooth solution, then ∂V + (V, ∇V ) + ∇P + (w, ∇w) + w(∇, w) = 0, ∂t

(∇, V ) = 0,

where V = U , w = U − U , and the bar denotes averaging over the torus T3 .

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(23)

Asymptotic Solutions for the Navier-Stokes Equations

6.

323

Relation to the Theory of Fully Developed Turbulence

The asymptotic description of the vortex structures described above and the theory of turbulence have several features in common: 1. Presence of Reynolds stresses: The average velocity field satisfies equations similar to the Reynolds equations in all cases described above. As is known, Reynolds stresses play a significant role in the theory of turbulence because they can increase the energy of the average field. 2. An analogue of turbulent viscosity: The obtained equations of motion of vortex structures contain terms describing the influence of viscosity on these structures. These terms contain a coefficient similar to turbulent viscosity, which is generally a nonlinear function of the field velocities. 3. Kolmogorov scale and Kolmogorov law: The leading part of the above-described solutions of the Navier–Stokes equations satisfies the Euler equations. These solutions are therefore unstable under the action of smaller-scale perturbations. More precisely, we linearize the Navier–Stokes equations for a solution of form (2) and consider the asymptotic solutions of the linearized equations of the form u = eiσ(z,x,t)/ε (η(z, x, t) + εη1 (z, x, t) + . . . ),

ε → 0.

(24)

It turns out that such solutions increase with time. The following assertion can be proved just as in [19]. Assertion 6.1. If the linearized Navier–Stokes equations admit a solution of form (24), then the function η(z, x, t) satisfies a system of linear equations of the form η˙ + A(t)η = 0, where η˙ is the derivative of η along the trajectories of the field U (z, x, t) and A(t) is a matrix function (the explicit formulas for A can be obtained just as in [19]). The solutions of this system generally increase as t → ∞. Moreover, there are fields U that satisfy the Euler equations and increase exponentially. The instability leads to an increase in the small-scale perturbations and therefore to a variation in the vortex structure U . Moreover, √ the new arising multiphase structure again satisfies the Euler equations if only ε  h. The small-scale perturbations continue to increase under this condition, and their growth is related to the fact that the leading part of the coherent structure satisfies equations. The increase therefore continues √ the Euler 3/2 until the new scale is equal to h h = h , i.e., ν 3/4 (Kolmogorov law). In this case, the leading part of the solution satisfies the Navier–Stokes-type equations including the case with viscous terms.

Acknowledgments This work is supported by the Russian Foundation for Basic Research (Grant Nos. 1601-00378 a and 14-01-00521 a) and the Program for Supporting Leading Scientific Schools (Grant No. NSh-7962.2016.1).

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References [1] V. P. Maslov, Uspekhi Mat Nauk, 1986, 41, 1935. [2] V. P.Maslov, Asymptotic Methods for Solving Pseudodifferential Equations (Nauka, Moscow, 1987) [in Russian]. [3] A. I. Shafarevich, Differ. Uravn., 1998, 34 (6), 11191130 [Diff Eq 1998, 34 (6), 11241134]. [4] A. I. Shafarevich, Dokl. Akad. Nauk RAN, 1998, 358 (6) 752755. [5] A.I. Shafarevich, Operator Theory: Advances and Applications, 2001, v. 132, 347359. [6] Maslov V.P., Shafarevich A.I. Rapidly Oscillating Asymptotic Solutions of the NavierStokes Equations, Coherent Structures, Fomenko Invariants, Kolmogorov Spectrum, and Flicker Noise. Russian Journal of Mathematical Physics, 2006, v.13, N 4, 414425. [7] V. P.Maslov and A. I. Shafarevich, Asymptotic solutions of the NavierStokes equations describing periodic systems of localized vortices, Math. Notes 90 (5), 686700 (2011). [8] V.P. Maslov, A.I. Shafarevich, Mathematical Notes, 2012, Vol. 91, No. 2, pp. 207216. [9] V.P. Maslov, A.I. Shafarevich, Theoretical and Mathematical Physics, 2014, 180(2): 967982. [10] V.P. Maslov, A.I. Shafarevich, Fundamental and Applied Mathematics, 2015, V 20(3), 191-212. [11] S. Yu. Dobrokhotov, A. I. Shafarevich, Russ. J. Math. Phys., 1994, 2, 133–135. [12] S. Yu. Dobrokhotov, A. I. Shafarevich. Fluid Dynam., 1996, 31, 511-514. [13] V. I. Arnold and B. Khesin, Topological Methods in Hydrodynamics (Appl. Math. Sci., Vol. 125), Springer, New York (1998). [14] Moffatt H.K. J. Fluid Mech., 1985, v. 159, pp. 359-378. [15] Moffatt H.K. Revista Brasiliera de Ciencias Mecanicas IX, 1987, pp. 93-101. [16] Moffatt H.K. Fluid Dynamics Research, Holland, 1988, v.3, pp. 22-30. [17] S. Yu. Dobrokhotov and V. P. Maslov, J Soviet Math, 1981, 16, 1433–1487. [18] A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian Systems, Vols. 1, 2 (Udmurtskii Universitet, Izhevsk, 1999; Chapman and Hall/CRC, Boca Raton, 2004). [19] V. P. Maslov, A. I. Shafarevich, J Appl Math Mech, 1998, 62, 389–396.

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ISBN: 978-1-53610-292-5 c 2017 Nova Science Publishers, Inc.

Chapter 18

A NALYTIC S OLUTIONS OF I NCOMPRESSIBLE N AVIER –S TOKES E QUATIONS BY G REEN ’ S F UNCTION M ETHOD Algirdas Maknickas1,∗ and Algis Dˇziugys2,† 1 Institute of Mechanical Science, Vilnius Gediminas Technical University, Vilnius, Lithuania 2 Lithuanian Energy Institute, Kaunas, Lithuania

Abstract The Navier–Stokes equations describe the motion of fluids; they arise from applying Newton’s second law of motion to a continuous function that represents fluid flow. If we apply the assumption that stress in the fluid is the sum of a pressure term and a diffusing viscous term, which is proportional to the gradient of velocity, we arrive at a set of equations that describe viscous flow. The Navier–Stokes equations can be transformed into a set of full-partial differential equations that are inhomogeneous and parabolic. The incompressible Navier–Stokes equations are invariant under the Galilean transform. Extension of the Galilean transform into a single integral transform allows us to eliminate non-linear terms and reduce the full differential equation, with respect to time, into a partial differential equation of a single variable. Solutions in 2D Lagrangian coordinates, for a defined boundary, are then given in terms of a terms of a vorticity–velocity stream function of ω − ψ. Solutions in 3D Lagrangian coordinates, for a defined boundary, are then given in terms of a vorticity–vector potential function of ω − A. Applying an inverse to the new proposed integral transform allows us to rewrite solution in Eulerian coordinates. Finally, analytical solutions were obtained for these 2D and 3D incompressible Navier–Stokes equations by applying a Green’s function method.

PACS: 47.10.ad, 47.11.Kb, 47.32.CKeywords: Navier–Stokes equations, Spectral methods, Vortex dynamics, Existence, Uniqueness, Regularity theory, Vortex methods AMS Subject Classification: 76D03, 76D05, 76M22, 76M23 ∗ †

E-mail address: [email protected] (Corresponding Author) E-mail address: [email protected]

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1.

Algirdas Maknickas and Algis Dˇziugys

Introduction

Clay institute problem formulation. The Navier–Stokes equations may be described as Newton’s second law for fluid flow. In order to emphasize the importance of the solution and to encourage others to solve it, the Clay Institute asks for a proof of one of the following four statements (Fefferman, 2016). Two statements are described below: A. Existence and smoothness of Navier–Stokes solutions on Rn . Take the positive coefficient (the viscosity) ν > 0 and n = 3. Let the initial velocities be any smooth, divergence-free vector many times differentiable field satisfying inversely proportional in k’th power of the space coordinate and convergence to zero on to the infinity. Take the time dependent force field to be identically zero. Then there exist smooth functions of pressure and velocities, on 3D space and time, that satisfy the NavierStokes equations and the continuity conditions for incompressible fluids where the initial velocities are known. The pressure and velocities are functions of space and time and the kinetic energy of fluid is bounded for each time moment. B. Existence and smoothness of Navier–Stokes solutions on Rn /Zn . Take positive coefficient (the viscosity) ν > 0 and n = 3. Let initial velocities be any smooth, divergence-free vectors satisfying a periodic condition; we take the external forces to be identically zero. Then there exist smooth pressure and velocity functions on 3D space and time (of real numbers) that satisfy the Navier–Stokes equations and the continuity condition for incompressible fluids where the initial velocities are known. The initial velocities are periodic and the pressure and velocity functions are described for all space and time. The additional statements are the counterparts of statements A and B, respectively. The Clay Institute summarize the importance of this problem as follows: ”Fluids are important and hard to understand... Since we don’t even know whether these solutions exist, our understanding is at a very primitive level. Standard methods from PDE appear inadequate to settle the problem. Instead, we probably need some deep, new ideas”. Existence of 2D Solution. The existence of the 2D solution has been known since 1963. Ladyzhenskaya (1969) proved existence and uniqueness of the 2D Navier–Stokes equations by using a Galerkin method approximation. She expressed the generalized solution as a sum of orthogonal functions with coefficients depending only on time. As a result, the Navier– Stokes equations transformed into a nonlinear system of differential equations in terms of time–dependent coefficients. The quadratic sum of the time–dependent coefficients for each time moment is bounded. The main problem of proving the 3D version of this theorem was the non-linearity of the proposed system of differential equations for the time–dependent coefficients. Weak formulation. Multiplication of momentum equations by a test function and integration in the fluid domain yields, after some rearranging, the weak formulation of the Navier–Stokes fluid equations. Hoffman & Johnson (2004) considered the Clay Institute Prize Problem and asked them to add a mathematical analytical proof of existence, smoothness and uniqueness (or a converse) solution to the incompressible Navier–Stokes equations. They argued that the present formulation of the Prize Problem, which asks for a

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strong solution, is not reasonable in the case of turbulent flow which always occurs for higher Reynolds numbers, and they proposed to focus instead on weak solutions. Authors asserted that since weak solutions are known to exist by a basic result Leray (1934), only the uniqueness of weak solutions remains an open problem. Spectral methods in fluids. The weak formulation of fluid equations allows us to solve this set of equations numerically by replacing partial differential equations with a n-th order linear system of algebraic equations. Today, researchers mostly use two types of algorithms. First, they can be derived by dividing the fluid domain into n subdomains and applying a polynomial test function for each subdomain. This method is called the finite element method. The disadvantage of this method is the big consumption of computational resources for complicated fluid domains. Therefore, researchers applied spectral methods of orthogonal polynomials, which are very popular in quantum mechanics or classical electrodynamics, as test functions in the whole fluid domain. In this case, quick convergence can be achieved by solving a linear system of algebraic equations using smaller computational resources (see Boyd (2013)). Therefore, the non-linearity of fluid flow equations is still applicable. ALE time integration. In order to avoid the non-linearity, an arbitrary LagrangianEulerian description of fluid flow can be used. The algorithms of continuum mechanics usually make use of two classical descriptions of motion: the Lagrangian and the Eulerian description (see Malvern (1969)). Lagrangian algorithms, where each individual node of the computational mesh follows the motion of the associated particle, are mainly used in structural mechanics. In contrast, Eulerian algorithms are widely used in fluid dynamics. In this case, the computational mesh is fixed and the continuum moves with respect to the grid. Donea et al. (2004) developed the arbitrary Lagrangian–Eulerian (ALE, in short) description in an attempt to combine the advantages of the above classical kinematical descriptions, while minimizing their respective drawbacks as far as possible. They proposed that at the beginning of a simulation, the computational domain is discretized in Lagrangian form. They get individual sections of the network by changing the unfavorable geometry; the element nodes are displaced so far that a numerically unstable network is avoided. This method has the advantage over other adaptive methods since both the number of nodes and the number of elements remains unchanged. The proposed method was widely applied in the numerical investigation of fluid flow, however, Lagrangian and Eulerian descriptions can be applied not only in numerical cases but also in the analytical case of investigating fluid flow. Aims and scopes. The authors would like to present, in this chapter, an extension of the aforementioned ideas of analytical solution of the incompressible Navier–Stokes equations. In order to arrive at an analytical solution, we avoid the non-linear term in the investigated partial differential equations. We achieve this by applying an integral Galilean transform. In the section ”Time derivative”, we show that the incompressible Navier–Stokes equations is invariant under Galilean transform using a reference frame moving with constant velocity. The section ”Linearization of Navier–Stokes equations” shows the linearization of the equations after extending the Galilean transform to the integral Galilean transform. The rest of the chapter describes a method for obtaining analytical solutions of linearized incompressible Navier–Stokes equations by using a Green’ss function approach.

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We then go from a Lagrangian description to a Eulerian description in order to find a solution of the incompressible Navier–Stokes equations in Eulerian coordinates. In this case, the Green’s function approach uses basic orthogonal functions as partial solutions of the appropriate differential equations; the elliptic coordinate system in 2D and elliptic cylindrical coordinate system in 3D was chosen. Additionally, the energy estimate and uniqueness of the solution was investigated. The chapter ends by applying a technique of expansion of finite order onto the obtained solution for giving analytical expressions of 2D lid-driven cavity as an example of the ability to solve the Navier–Stokes equation in a Eulerian description.

2.

Time Derivative The incompressible fluid equations are expressed as follows:   ∂v 1 1 + (v · ∇)v − ν∆v + ∇p = f, ∂t ρ ρ ∇ · v = 0, ∂v∂Ω v∂Ω = f , =g ∂n

(1) (2) (3)

The equation of fluid motion (1) can be expressed using the convective time derivative d ∂ = + dt ∂t to obtain

(v · ∇)

dv 1 − ν∆v = (−∇p + f ), dt ρ

(4)

(5)

which is an inhomogeneous parabolic-like equation for the full time derivative where ν = µ/ρ. It is a well-known fact that the Galilean coordinate transform is invariant over the Navier–Stokes equations (Frisch, 1995). Definition 1. A group G of transformations of v(r, t) is the symmetry group of Navier– Stokes equations if and only if ∀g ∈ G, v a NS solution =⇒ gv a NS solution. Theorem 2.1. A Galilean coordinate transformation group Gal gu v(r, t) = v(r − ut, t) + u, u ∈ Rd

is symmetry group of the Navier–Stokes equations. Proof. Set ˜ (r, t) = v(r − ut, t) + u v

(6)

˜ Now we apply operator (4) onto v ˜ (r, t) = ∂t v(r − ut, t) − (u · ∇)v(r − ut, t) ∂t v

(˜ v · ∇)˜ v(r, t) = (v(r − ut, t) + u) · ∇v(r − ut, t)

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Summation of left and right parts gives ˜ (r, t) + (˜ ∂t v v · ∇)˜ v(r, t) = ∂t v(r − ut, t) + v(r − ut, t) · ∇v(r − ut, t)

(9)

On other hand, the transformations of spatial derivatives with respect to the coordinates are ∂x = ∂x x0 · ∇0 + ∂x t0 ∂t0 = ∂x0

∂x2

0

0

0

= (∂x x · ∇ + ∂x t ∂t0 )∂x0 =

(10) ∂x20

(11)

Corollary 2.2. For non-constant u, the displacement of coordinate system of each fluid Rt parcel moving with velocity u = v(r − t0 vdτ, t) transforms the full-time derivative of Navier–Stokes equations into a partial-time derivative Dv Dt

= ∂t v(r0 , t) + ∂t r 0 ∂r0 v(r0 , t) Z t Z t = ∂t v(r − vdτ, t) + ∂t (r − vdτ )∂r0 v(r0 , t) = ∂t v(r −

where the equality ∂t (r −

3.

Rt

t0

Z

t0 t

t0

vdτ, t) = ∂t v(r 0 , t)

t0

vdτ ) ≡ 0 was applied.

Linearisation of Navier-Stokes Equations Let’s start from Navier-Stokes equations (Navier, 1823; Stokes, 1845) 1 ∂v + v · ∇v = − ∇p + ν∇2 v ∂t ρ

We know that we have a full time derivative on the left-hand side Dv 1 = − ∇p + ν∇2 v Dt ρ Now we could integrate both sides by time and obtain for v(r, t) Z Z t 1 t v(r, t) = − ∇r p(r, τ )dτ + ν ∇2r v(r, τ )dτ + v(r, 0) ρ 0 0

(12)

(13)

We obtained the formal solution for velocities as follow v(r, t) = F(r, t)

(14)

Eq. (14) is valid for any r0 in the boundary where we try to solve NS equation. We could choose r0 so that Z r0 = r −

t

v(r, τ )dτ

0

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and v(r −

Z

t

0

v(r, τ )dτ, t) = F(r −

Z

t

v(r, τ )dτ, t)

(16)

0

or by taking full derivative of both sides we obtain Rt

Rt v(r, τ )dτ, t) DF(r − 0 v(r, τ )dτ, t) = = (17) Dt Dt Z t Z t −∇r0 p(r − v(r, τ )dτ, t) + ν∇2(r−R t v(r,τ )dτ,t)v(r − v(r, τ )dτ, t) (18) Dv(r −

0

0

0

0

It is easy to prove that Dv(r −

Rt

∂v(r − 0 v(r, τ )dτ, t) = Dt

Rt 0

v(r, τ )dτ, t) . ∂t

(19)

Rt

(20)

because Dv(r −

Rt

Rt

v(r, τ )dτ, t) ∂v(r − = Dt

0

0

v(r, τ )dτ, t) ∂v(r − + ∂t

v(r, τ )dτ, t) ∂r0 ∂r0 ∂t

0

and ∂(r − ∂r0 = ∂t

Rt 0

v(r, τ )dτ, t) = v − v = 0. ∂t

(21)

So, we obtain parabolic equations

∂v (r0 , t) 1 = − ∇r0 p r0 , t + ν∇2r0 v r0 , t ∂t ρ

(22)

which are analytically solvable for defined boundaries using, for example, the Green’s function method. When we obtain velocities v(r0, t) in the local coordinates. R t We could easily return to the global coordinates by inserting the new coordinates r0 + 0 v(r, τ )dτ into the obtained analytical solution which implies that ∇r0 +R t v(r,τ )dτ v(r0 0

+

Z

0

t

v(r, τ )dτ, t) = ∇2r v(r, t)

and Rt

Rt v(r, τ )dτ, t) ∂v(r0 + 0 v(r, τ )dτ, t) = + ∂t ∂t R Z t t ∂(r0 + 0 v(r, τ )dτ, t) · ∇r0 +R t v(r,τ )dτ v(r0 + v(r, τ )dτ, t) = 0 ∂t 0 ∂v(r, t) ∂r ∂v(r, t) + · ∇r v(r, t) = + v(r, t) · ∇r v(r, t). ∂t ∂t ∂t ∂v(r0 +

0

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Vorticity-Stream Formulation

4.1.

The Plane Flow

According to Peyret (2002), in a two-dimensional plane geometry the vorticity vector ω = ∇×v

(23)

has only one nonzero component ω, such that ω = ωk

(24)

where k is the unit vector normal to the plane (x, y) of the flow. The vorticity ω is expressed by ω = ∂x vy − ∂y vx (25) The equation satisfied by ω is obtained by applying the curl operator to Navier–Stokes equations Eq.(1), so that the pressure gradient term disappears. The result is ∂t ω − ν∇2 ω = F

ω∂Ω = ∇ × v∂Ω

(26) (27)

Finally, the forcing term F is F = (∇ × f ) · k

(28)

The velocity vector v is defined in terms of the stream function ψ by v = ∇ (ψk)

(29)

vx = ∂y ψ, vy = −∂x ψ

(30)

so that The equation satisfied by ψ is obtained by applying the curl operator to Eq.(29) and using the definition (23), namely, ∇2 ψ + ω = 0 (31) Let us assume that v satisfies the boundary condition (3). From the definition of the stream function (29) we have, on the boundary, ∇ψ · t = −v∂Ω · n

∇ψ · n = −v∂Ω · t

(32) (33)

where t is the unit vector tangent to r with clockwise orientation. Now, by integrating the first equation (32), (33) along the boundary r, we obtain Z s ψ= v∂Ω · nds, on ∂Ω (34) s0

where s is the curvilinear abscissa along ∂Ω and s0 corresponds to a given arbitrary fixed point of ∂Ω. Lastly, it may be useful to calculate the pressure from the knowledge of the

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velocity field. Going back to the Eulerian coordinates, one possible way is to consider the pressure as the solution of the Poisson equation obtained by taking the divergence of Navier–Stokes Eq.(1), namely, ∇2 p = ∇ · (f − (v · ∇) v) in Ω

(35)

associated with the Neumann condition deduced from the projection of NS Eq.(1) on the normal to the boundary
∂n p = f + ν∇2 v − (v · ∇) v − ∂t v · n on ∂Ω (36) In deriving Eq.(35) we have taken into account that ∇ · v = 0.

4.2.

Elliptic Coordinates

Our first step towards the solution will be to consider the transformation of Laplace’s operator from Cartesian coordinates to an elliptic system. Using the transformation equations x = aξη , ∀ξ ∧ ∀η ∈ [−1, 1]
1
1 y = a ξ2 − 1 2 1 − η2 2

The scale factors related to this transformation are "   2 # 12 1  2 ∂x 2 ∂y ξ − η2 2 hξ = + =a ∂ξ ∂ξ ξ2 − 1 "   2 # 12 1  2 ∂x 2 ∂y ξ − η2 2 hη = + =a ∂η ∂η 1 − η2

(37) (38)

(39)

(40)

The scale factors that have been previously calculated will allow us to construct the Laplacian operator, which will have the form      1 ∂ hη ∂ ∂ hξ ∂ 2 ∇ = + . (41) hξ hη ∂ξ hξ ∂ξ ∂η hη ∂η We can obtain ∇

2

= +

or ∂t ω − +

  
1 ∂
1 ∂ 1 2 2 2 2 ξ −1 ξ −1 a2 (ξ 2 − η 2 ) ∂ξ ∂ξ   1 ∂ 1 ∂

1 − η2 2 1 − η2 2 ∂η ∂η   
1 ∂
1 ∂ 1 2 2 2 2 ξ −1 ξ −1 a2 (ξ 2 − η 2 ) ∂ξ ∂ξ   1 ∂ 1 ∂

1 − η2 2 1 − η2 2 ω = 0. ∂η ∂η

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(42)

(43)

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A partial solution can be found in form
ω = exp −k2 t S(ξ)H(η).

(44)

Inserting it into Eq. (43) and dividing by S(ξ)H(η) gives 
1  
12 ∂ ξ2 − 1 2 ∂ 1 2 2  k + ξ −1 S a2 (ξ 2 − η 2 ) S ∂ξ ∂ξ 
1   2 2 1
1−η ∂ ∂ 1 − η2 2 H = 0 + H ∂η ∂η

(45)

Next, the separation can be achieved by letting each equation which depends on ξ and η to be constant m2 as follows  
21 ∂
1 ∂ ξ2 − 1 2 S(ξ) + k 2 a2 ξ 2 S(ξ) = ∂ξ ∂ξ  
1 ∂
1 ∂ 1 − η2 2 1 − η2 2 H(η) − k 2 a2 η 2 H(η) = ∂η ∂η ξ2 − 1

m2 S(ξ)

(46)

−m2 H(η)

(47)

or after simplification


∂2 ∂ S(ξ) + ξ S(ξ) + k2 a2 ξ 2 S(ξ) = m2 S(ξ) 2 ∂ξ ∂ξ 2
∂ ∂ 1 − η2 H(η) − η H(η) − k2 a2 η 2 H(η) = −m2 H(η). ∂η 2 ∂η ξ2 − 1

(48) (49)

Solution we will search in form

S(ξ) =

∞ X

si ξ i

(50)

hi η i

(51)

i=0

H(η) =

∞ X i=0

Inserting it into Eq. (48) and Eq. (49) gives recurrent relations between coefficients
1 si+2 = (i2 − m2 )si − k2 a2 si−2 (i + 1)(i + 2)
1 (i2 − m2 )hi − k2 a2 hi−2 hi+2 = (i + 1)(i + 2)

(52) (53)

Finally, we obtains partial solutions

k2 a2 k2 a2 , , ξ) 4 4 k2 a2 k2 a2 Smk = S(m2 − , , ξ) 4 4 k2 a2 k2 a2 Hmk = C(m2 − , , η) 4 4 k2 a2 k2 a2 Hmk = S(m2 − , , η) 4 4 Smk = C(m2 −

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(54) (55) (56) (57)

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Algirdas Maknickas and Algis Dˇziugys

where even function C(b, q, ξ) is Mathieu’s cosinus and odd function S(b, q, ξ) is Mathieu’s sinus functions, accordantly. As described in Arfken et al. (2005), the Green’s function is then defined by   ∂ 2 G(r1 , r2 , t!, t2 ) = δ 3 (r1 − r2 )δ(t1 − t2 ). ∇r 1 − (58) ∂t1 We define orthogonal functions ωn as solutions of the homogeneous Helmholtz differential equation in a corresponding boundary ∇2 ωn (r) − kn2 ωn (r) = 0.

(59)

Thus, the Green’s function can be expressed as follows Polyanin (2001): ∞ X ωn (r1 )ωn (r2 )

G(r1 , r2 , t, τ ) =

n=0

kωn k2

exp (−kn2 (t − τ )).

(60)

So, G(ξ1 , η1 , ξ2 , η2 , t, τ ) = ∞ 2 1 X S(bnk , qk , ξ1 )S(bnk , qk , η1)S(bnk , qk , ξ2 )S(bnk , qk , η2 )e(−k (t−τ)) π2

(61)

n,k=0

We can use Green’s function method Polyanin (2001) by using third Green’s identity (Green, 1828) which satisfy specific boundaries  ZZ  1 ∂φ(r2 ) ∂G(r1 , r2 ) φ(r 1 ) = G(r1 , r2 ) (62) − φ(r 2 ) dSr2 2π ∂n ∂n ∂Ω

∂φ(r2 ) = f, ∀r 2 ∈ ∂Ω ∂n φ(r 2 ) = g, ∀r2 ∈ ∂Ω

(63) (64)

for writing solution as follow ω(r 1 , t) =

1 ν Z

0

∂ ω(r 2 , t) = ∂n ω(r 2 , t) =

Zνt ZZ 0 νt Z Z

dτ dSr2 F (r2 , τ )G(r1 , r2, νt, τ ) +

Ω

dτ dSr2 ω(r2 , 0)G(r1 , r2, 0, τ ) +

Ω νt Z

Z 1 dτ d`r2 G(r1 , r2 , νt, τ )|∂Ω (∇ × f (r2 , τ )) · k − 2π 0 ∂Ω Z νt Z 1 ∂G(r1 , r2 , νt, τ ) dτ d`r2 (∇ × g(r2 , τ )) · k 2π 0 ∂n ∂Ω ∂Ω

(65)

∇ × f (r2, t), ∀r2 ∈ ∂Ω

(66)

∇ × g(r2 , t), ∀r2 ∈ ∂Ω

(67)

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for given boundaries f and g. Here differentiation along the normal n denotes X ∂ ∂ F (r, t) = ni F (r, t) ∂n ∂ri

(68)

i

Now, we should solve Poisson equation for stream function in elliptic coordinates   
1 ∂
1 ∂ 1 2 2 2 2 ξ − 1 ξ − 1 (ξ 2 − η 2 ) ∂ξ ∂ξ   1 1
∂
∂ + 1 − η2 2 1 − η2 2 ψ = −ω ∂η ∂η

(69)

Applying separation of variables ψ(ξ, η) = T (ξ)U (η) for time independent Laplace equation   
1 ∂
1 ∂ 1 2 2 2 2 ξ − 1 ξ − 1 (ξ 2 − η 2) ∂ξ ∂ξ   1 1
∂
∂ + 1 − η2 2 1 − η2 2 T (ξ)U (η) = 0 (70) ∂η ∂η gives two Chebyshev equations for T and U functions


∂2 ∂ T (ξ) + ξ T (ξ) = m2 T (ξ) ∂ξ 2 ∂ξ
∂2 ∂ 1 − η2 U (η) − η U (η) = −m2 U (η) 2 ∂η ∂η ξ2 − 1

(71) (72)

Two last equations Eq.(71) and Eq. (72) are similar. It can be shown by multiplication of Eq. (72) with −1. So, the Green’s function for stream function is G(ξ1 , η1, ξ2 , η2 ) =

∞ 4 X 1 Tm (ξ1 )Tm(η1 )Tn (ξ2 )Tn (η2 ) 2 π (1 + δm0 )(1 + δn0 )

(73)

m,n=0

where Tm = cos (n arccos x) is Chebyshev polynomial. So, the solution for velocity stream function is expressed as follow ZZ ψ(ξ, η, t) = − ω(ξ2 , η2 , t)G(ξ, η, ξ2, η2)hξ hη dξ2 dη2 + Ω Z Z ∂G(ξ, η, ξ2, η2 ) v∂Ω · nd` d`2 − ∂n ∂Ω ∂Ω ∂Ω Z Z s2 ∂v∂Ω (ξ2 , η2) · n d` G(ξ, η, ξ2, η2)|∂Ω d`2 (74) ∂n ∂Ω s0 Finally, solution of Navier-Stokes equations in Lagrangian elliptic coordinates are as follow  2 1 ! 1 ∂(hξ ψ) 1 ∂ ξ − η2 2 vξ (ξ, η, t) = − =− ψ (75) hξ hη ∂η a(ξ 2 − η 2 ) ∂η ξ2 − 1  2 1 ! 1 ∂(hη ψ) 1 ∂ ξ − η2 2 vη (ξ, η, t) = = ψ (76) hξ hη ∂ξ a(ξ 2 − η 2 ) ∂ξ 1 − η2

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Applying Galilean integral transform ξe = ξ +

Z

t

vξ (ξ, η, τ )dτ

(77)

vη (ξ, η, τ )dτ

(78)

0

ηe = η +

Z

t

0

gives solution in Eulerian coordinates v(ξe , ηe, t) =

u(ξe , ηe, t) =

1 ∂ − 2 2 a(ξ − η ) ∂η a(ξ 2

1 ∂ 2 − η ) ∂ξ

 12

! ψ ξ=ξe ,η=ηe !  2  21 2 ξ −η ψ 1 − η2 

ξ2 − η2 ξ2 − 1

(79)

(80)

ξ=ξe ,η=ηe

The same way pressure p can be expressed in Eulerian description ZZ p(ξ, η, t) = ∇ · (f − (v · ∇) v) G(ξ, η, ξ2, η2 )hξ hη dξ2 dη2 − Ω Z
f + ν∇2 v − (v · ∇) v − ∂t v · n G(ξ, η, ξ2, η2)|∂Ω d`

(81)

∂Ω

5.

Vorticity - Vector Potential Formulation

5.1.

The Volume Flow

Similar to 2D flow, in a three-dimensional plane geometry the vorticity vector ω = ∇×v

(82)

The equation satisfied by ω is obtained by applying the curl operator to Navier-Stokes equations Eq.(1), so that the pressure gradient term disappears. The result is ∂t ω − ν∇2 ω = F ω ∂Ω = ∇ × v∂Ω

(83) (84)

Finally, the forcing term F is F = ∇×f

(85)

The velocity vector v is defined in terms of vector potential function A by v = ∇×A

(86)

The equation satisfied by A is obtained by applying the curl operator to Eq.(86) ∇ × (∇ × A) = ∇(∇ · A) − ∇2 A and using the definition (82), namely, ∇2 A + ω = 0

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for gauge fixing ∇ · A = 0. Each vector A can be decomposed by Helholtz decomposition into the curl and gradient parts A = ∇ × Ac + ∇φ

(88)

All physical observables are required to be gauge invariant. Therefore, the curl part of decomposition ∇ × Ac can be chosen. Let us assume that v satisfies the Dirichlet or Neumann boundary condition. From the definition of the vector potential function (86) we have, on the boundary, ∇ × A∂Ω = v∂Ω on ∂Ω ∂∇ × A∂Ω ∂v∂Ω = ∂n ∂n

(89) (90)

Now, solving system of equations (89) along the boundary appropriate boundaries functions Ai∂Ω should be found. Also, we can write solution as follow A∂Ω = (∇×)−1 v∂Ω ∂A∂Ω ∂ = (∇×)−1 v∂Ω ∂n ∂n

(91) (92)

where (∇×)−1 denotes inverse curl operator and can by found by using equations (Sahoo, 2010)

−

(∇×)−1 v∂Ω = ˛ eˆ1 ˛ ˛ ˛“ R hR1 ” ˛ 1 + 1 − ˛ 3 +2 (∂u1 ) ˛ ˛ 1 h2 h3 v∂Ω

“ R + 1 3

eˆ2 hR2 ” −

+ 21 (∂u2 ) 2 h1 h3 v∂Ω

“ R 1 + 3

eˆ3 hR3 ” −

+ 12 3 h1 h2 v∂Ω

˛ ˛ ˛ ˛ ˛ (∂u3 )˛˛ ˛

(93)

R R − + where + (∂u1 ) does integration w.r.t. u1 , while other two variables of the integrandRare to be treated fixed. If the integrand is the 3rd component (along eˆ3 ) of a vector, + then (∂u1 ) acts only on the part of the integrand having the variable u3 . Similarly R− (∂u1 ) acts only on the part of the integrand does not contain the variable u3 . Lastly, the pressure from the knowledge of the velocity field can be similar to the 2D flow pressure. ∇2 p = ∇ · (f − (v · ∇) v) in Ω associated with the Neumann condition deduced from the projection of NS Eq.(1) on the normal to the boundary
∂n p = f + ν∇2 v − (v · ∇) v − ∂t v · n on ∂Ω In deriving 3D analogue of Eq.(35) we have taken into account that ∇ · v = 0.

5.2.

Elliptic Cylindrical Coordinates

Our first step towards the solution will be to consider the transformation of Laplace’s operator from Cartesian coordinates to an elliptic cylindrical coordinates system. Using the

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Algirdas Maknickas and Algis Dˇziugys

transformation equations x = aξη , ∀ξ ∧ ∀η ∈ [−1, 1]
1
1 y = a ξ2 − 1 2 1 − η2 2

(94) (95)

z = z

(96)

The previously calculated scale factors (39), (40) will allow us to construct the Laplacian operator, which will have the form      1 ∂2 ∂ hη ∂ ∂ hξ ∂ 2 ∇ = + 2 (97) + hξ hη ∂ξ hξ ∂ξ ∂η hη ∂η dz we can obtain ∇

2

= +

or

  
1 ∂
1 ∂ 1 2 2 2 2 ξ −1 ξ −1 a2 (ξ 2 − η 2 ) ∂ξ ∂ξ   2 1 ∂ 1 ∂

∂ + 2 1 − η2 2 1 − η2 2 ∂η ∂η dz

(98)




1  
12 ∂ ξ2 − 1 2 ∂ 2 ∂t ω −  2 2 ξ − 1 a (ξ − η 2 ) ∂ξ ∂ξ +


1   2 1 ∂
1 − η2 2 ∂ ∂ 1 − η2 2 + 2ω = 0 a2 (ξ 2 − η 2 ) ∂η ∂η dz

A partial solution will be searched for in the form
ωi = exp −k2 t S(ξ)H(η)Z(z), ∀i, i ∈ [ξ, η, z]

(99)

(100)

After inserting the partial solution into Eq.(99) and dividing by S(ξ)H(η)Z(z) we obtain 
1  
1 ∂ ξ2 − 1 2 ∂ 2 2 2 k +  2 2 ξ − 1 S a (ξ − η 2 )S ∂ξ ∂ξ 
1   2 1 ∂
1 − η2 2 ∂ 2 2 + 1 ∂ Z = 0 (101) + 1 − η H a2 (ξ 2 − η 2 )H ∂η ∂η Z ∂z 2

Let’s denote

1 ∂ 2Z = −p2 Z ∂z 2

(102)

So, we obtain „ « ´ 21 ∂ ` 2 ` ´ ´ 12 ∂ ξ −1 ξ −1 S(ξ) + k2 − p2 a2 ξ 2 S(ξ) ∂ξ ∂ξ „ « ` ´1 ∂ ` ´1 ∂ ` ´ 1 − η2 2 1 − η2 2 H(η) − k2 + p2 a2 η2 H(η) ∂η ∂η `

2

=

m2 S(ξ)

(103)

=

−m2 H(η)

(104)

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339

or after simplification
∂2
∂ S(ξ) + ξ S(ξ) + k 2 − p2 a2 ξ 2 S(ξ) 2 ∂ξ ∂ξ 2

∂ ∂ 1 − η2 H(η) − η H(η) − k 2 + p2 a2 η 2 H(η) 2 ∂η ∂η ξ2 − 1

= m2 S(ξ)

(105)

= −m2 H(η)

(106)

We will search solution of Eq.(124) and Eq.(124) in form S(ξ) =

∞ X

si ξ i

(107)

hi η i

(108)

i=0

H(η) =

∞ X i=0

which implies recurrent equations for coefficient si and hi si+2 = hi+2 = or



1 (i2 − m2 )si − k2 − p2 a2 si−2 (i + 1)(i + 2)

1 (i2 − m2 )hi − k2 + p2 a2 hi−2 (i + 1)(i + 2)

(k2 − p2 )a2 (k2 − p2 )a2 , , ξ) 4 4 (k2 − p2 )a2 (k2 − p2 )a2 Smkp = S(m2 − , , ξ) 4 4 (k2 + p2 )a2 (k2 + p2 )a2 , , η) Hmkp = C(m2 − 4 4 (k2 + p2 )a2 (k2 + p2 )a2 Hmkp = S(m2 − , , η) 4 4 Smkp = C(m2 −

(109) (110)

(111) (112) (113) (114)

where the even function C(b, q, ξ) is Mathieu’s cosine and the odd function S(b, q, ξ) is Mathieu’s sine function. When we choose the k = p function, Smpp reduces to the Chebyshev polynomial Tm . So, the Green’s functions can now be expressed as follows: G(ξ1 , η1 , z1 , ξ2 , η2 , z2 , t, τ ) = ×

∞ 2 X p 2 a2 p 2 a2 2 T (ξ )S(n − , , η1 ) n 1 π 2 n,p=0 2 4

× Tn (ξ2 )S(n2 −

2 p 2 a2 p 2 a2 , , η2 )e−p (t−τ) 4 2

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(115)

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Algirdas Maknickas and Algis Dˇziugys

As in 2D case, we could use Green’s function method Polyanin (2001) by using a third Green’s identity (Green, 1828) for writing solution as follows:

ω(r 1 , t)

=

1 ν Z

Zνt Z Z Z 0 νt

0

1 2π 1 2π ∂ ω(r 2 , t) ∂n ω(r 2 , t)

dτ dVr2 F (r2 , τ )G(r1 , r2 , νt, τ ) +

Ω

ZZ Z

dτ dVr2 ω(r2 , 0)G(r1 , r2 , 0, τ ) + Ω ZZ νt dτ dSr2 G(r 1 , r 2 , νt, τ )|∂Ω [∇ × f (r2 , τ )] − 0 ∂Ω ˛ Z νt ZZ ∂G(r 1 , r 2 , νt, τ ) ˛˛ dτ dSr2 [∇ × g(r 2 , τ )] ˛ ∂n 0 ∂Ω ∂Ω Z

(116)

=

∇ × f (r2 , t), ∀r 2 ∈ ∂Ω

(117)

=

∇ × g(r2 , t), ∀r 2 ∈ ∂Ω

(118)

Now, we should solve Poisson equation for vector potential function in elliptic coordinates 
1  
12 ∂ ξ2 − 1 2 ∂ 2  ξ −1 a2 (ξ 2 − η 2 ) ∂ξ ∂ξ 
1   2 1 ∂
1 − η2 2 ∂ ∂ + 1 − η2 2 + 2  A = −ω (119) a2 (ξ 2 − η 2 ) ∂η ∂η ∂z

Applying separation of variables A(ξ, η, z) = T (ξ)U (η)Z(z) for time independent Laplace equation gives expression as follow   +

Let’s denote


1  
12 ∂ ξ2 − 1 2 ∂ 2 ξ −1 a2 (ξ 2 − η 2 ) ∂ξ ∂ξ


1   2 1 ∂
1 − η2 2 ∂ ∂ 1 − η2 2 + 2  T (ξ)U (η)Z(z) = 0 a2 (ξ 2 − η 2 ) ∂η ∂η ∂z

1 ∂ 2Z = −p2 Z ∂z 2

(120)

(121)

So, we obtain  
21 ∂
21 ∂ 2 ξ −1 ξ −1 S(ξ) + p2 a2 ξ 2 S(ξ) = ∂ξ ∂ξ  
1
1 2 2 ∂ 2 2 ∂ 1−η 1−η H(η) − p2 a2 η 2 H(η) = ∂η ∂η 2

m2 S(ξ)

(122)

−m2 H(η)

(123)

or after simplification


∂2
∂ S(ξ) + ξ S(ξ) + p2 a2 ξ 2 − m2 S(ξ) = 0 2 ∂ξ ∂ξ 2

∂ ∂ 2 2 2 2 1 − η2 H(η) − η H(η) − p a η − m H(η) = 0 ∂η 2 ∂η ξ2 − 1

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(124) (125)

Analytic Solutions of Incompressible Navier–Stokes Equations ...

341

which implies recurrent equations for coefficient si and hi si+2 = hi+2 =


1 (i2 − m2 )si − p2 a2 si−2 (i + 1)(i + 2)
1 (i2 − m2 )hi − p2 a2 hi−2 (i + 1)(i + 2)

(126) (127)

Finally, we obtains partial solutions

p2 a2 p2 a2 , , ξ) 4 4 p2 a2 p2 a2 Smp = S(m2 − , , ξ) 4 4 p2 a2 p2 a2 Hmp = C(m2 − , , η) 4 4 p2 a2 p2 a2 Hmp = S(m2 − , , η) 4 4 Smp = C(m2 −

(128) (129) (130) (131)

where even function C(b, q, ξ) is Mathieu’s cosinus and odd function S(b, q, ξ) is Mathieu’s sinus functions, accordantly. So, we can write Green’s function G(ξ1 , η1 , z1 , ξ1 , η1 , z1 , t, τ ) = × ×

2 π2 ∞ X

S (bmp , qp , ξ1 ) S (bmp , qp, η1 ) sin (πpz1 )

m,p=0

S (bmp , qp, ξ2 ) S (bmp , qp, η2 ) sin (πpz2 )

(132)

So, the solution for velocity potential function is expressed as follow A(ξ, η, z, t)

=

ZZ Z − ω(ξ2 , η2 , z2 , t)G(ξ, η, z, ξ2 , η2 , z2 )hξ hη dξ2 dη2 dz + Ω ˛ ZZ ∂G(ξ, η, z, ξ2 , η2 , z2 ) ˛˛ A∂Ω (ξ2 , η2 , z2 , t) ˛ d(∂Ω) − ∂n ∂Ω ∂Ω ZZ ∂A∂Ω (ξ2 , η2 , z2 , t) n· G(ξ, η, z, ξ2 , η2 , z2 )|∂Ω d(∂Ω) ∂n ∂Ω

(133)

Finally, solution of Navier-Stokes equations in Lagrangian elliptic cylindrical coordinates are as follow hξ eˆξ hη eˆη eˆz 1 ∂ ∂ ∂ v(ξ, η, z, t) = (134) ∂ξ ∂η ∂z hξ hη hξ Aξ hη Aη Az Applying Galilean integral transform

ξe = ξ +

τ

Z

0

ηe = η + ze = z +

Z

Z

0

vξ (ξ, η, z, τ )dτ

(135)

vη (ξ, η, z, τ )dτ

(136)

vz (ξ, η, z, τ )dτ

(137)

τ

τ

0

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gives solution in Eulerian coordinates hξ eˆξ hη eˆη eˆz 1 ∂ ∂ ∂ v(ξe , ηe, ze , t) = ∂ξ ∂η ∂z hξ hη hξ Aξ hη Aη Az ξ=ξ

(138) e ,η=ηe ,z=ze

The same way pressure p can be expressed as follow

−

p(ξ, η, t) = ZZ Z ∇ · (f − (v · ∇) v) G(ξ, η, z, ξ2 , η2 , z2 )hξ hη dξ2 dη2 dz ZZ Ω ` ´ f + ν∇2 v − (v · ∇) v − ∂t v · n G(ξ, η, z, ξ2 , η2 , z2 )|∂Ω d(∂Ω)

(139)

∂Ω

6.

A Priori Energy Estimate

Basic energy estimate for the Navier-Stokes equations follows from an integration of the equations. We multiply eq. (83) in Eulerian description by ωi , i ∈ [1, 2, 3], integrate over Ω, apply the divergence theorem, and use the BC that ωi = 0 on ∂Ω to obtain1 Z Z Z 1 d |∇ωi|2 dV = fi ωi dV (140) ωi2 dV + 2 dt Ω Ω Ω d where dt is full derivative. Integrating this equation with respect to time and using the initial conditions ωi (x, 0) = gi , we get Z tZ Z tZ Z Z 1 1 2 2 |∇ωi | dV dτ = fi ωi dV dτ + (141) ω dV + g 2 dV 2 Ω i 2 Ω i 0 Ω 0 Ω

for 0 ≤ t ≤ T , we have from the Cauchy inequality with ε that Z t Z 1/2 Z t Z 1/2 Z tZ 2 2 fi ωi dV dτ ≤ fi dV dτ ωi dV dτ 0 Ω 0 Ω 0 Ω Z tZ Z tZ 1 ≤ ωi2 dV dτ fi2 dV dτ + ε 4ε 0 Ω 0 Ω Z Z tZ 1 2 ≤ ωi2 dV f dV dτ + εT max 0≤t≤T Ω 4ε 0 Ω i

(142)

Thus, taking the supremum of (141) over t ∈ [0, T ] and using this inequality with εT = 1/4 in the result, we get 1 max 4 0≤t≤T

Z

Ω

ωi2 (x, t)dV +

Z

0

T

Z

Ω

|∇ωi |2 dV dt ≤ T

Z

0

T

Z

Ω

fi2 dV dt +

1 2

Z

Ω

gi2 dV

(143)

It follows that we have an a priori energy estimate for ωi . Now, we multiply (87) by Ai , integrate over Ω, apply the divergence theorem, and use the BC that Ai = 0 on ∂Ω Z Z 2 |∇Ai| dV = Ai ωi dV (144) Ω

1

Ω

If we use Lagrangian description of Navier-Stokes equations, the result is the same

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Analytic Solutions of Incompressible Navier–Stokes Equations ... we have from the Cauchy inequality with ε that Z 1/2 Z 1/2 Z 2 2 Ai ωi dV ≤ Ai dV ωi dV Ω Ω Ω Z Z 1 ≤ A2i dV dτ + ε ωi2 dV 4ε Ω Ω The same way, we have from Poincar`e inequality Z Z 2 Ai dV ≤ C |∇Ai|2 dV Ω

343

(145)

(146)

Ω

Thus, using this inequality with ε = 21 C in the result, we get Z Z 2 Ai dV ≤ C ωi2 dV Ω

(147)

Ω

It follows that we have an a priori energy estimate for Ai

7.

Uniqueness of Solutions

If vi1 , vi2 are two solutions with the same data fi1 = fi2 , gi1 = gi2 , then vi = vi1 − vi2 is solution with zero data fi = 0, gi = 0. To show uniqueness, it is therefore sufficient to show that the only weak solution with zero data is vi = 0. It can be shown by using inequalities Z Z TZ 1 2 max (ωi1 − ωi2 ) (x, t)dV + |(∇ωi1 − ∇ωi2 )|2 dV dt ≤ 4 0≤t≤T Ω 0 Ω Z TZ Z 1 2 T (fi1 − fi2 ) dV dt + (gi1 − gi2 )2 dV (148) 2 Ω 0 Ω and

Z

Ω

(Ai1 − Ai2 )2 dV ≤ C

Z

Ω

(ωi1 − ωi2 )2 dV

(149)

If we have solution with zero data fi = 0, gi = 0, the left side of inequalities are zero to, because integration function are positive in boundary Ω and for 0 ≤ t ≤ T .

8.

Expansion of Finite Order

Experience shows that a simple truncation of an infinite series, leads to poor precision and fluctuations also known as Gibbs oscillations near points where the function f (x) is not continuously differentiable (Weiße et al., 2006). f (x) ≈ =

N −1 X m=0

< f, φm > µm φm (x) < φm , φm >

1 √ π 1 − x2

µ0 + 2

N −1 X m=1

µm Tm (x)

!

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(150)

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Algirdas Maknickas and Algis Dˇziugys

where denotes < f, φm >=

Z1

ω(x)f (x)φm(x)dx

(151)

−1

√ 2 and ω(x) is orthogonality weight function, which √ equals to 1 − x for Chebyshev polynomials of second kind Um and equals to 1/ 1 − x2 for Chebyshev polynomials of the first kind, Tm. √ 2 √ Here, orthogonal polynomials φm (x) = Tm (x)/ 1 − x and weight functions ω(x) = 1 − x2 were used. A common procedure is to damp these oscillations. This relies on an appropriate modification of the expansion coefficients, µm → gm µm , which depends on the order of the approximation N, ! N −1 X 1 fKP M (x) = √ gm µm Tm (x) . (152) g0 µ0 + 2 π 1 − x2 m=1 In more abstract terms, this truncation of the infinite series to order N together with the corresponding modification of the coefficients is equivalent to the convolution of f (x) with a kernel or Green’s function truncation of the form KN (x, y) =

N −1 X m=0

gm φm (x)φm (y). < φm , φm >

(153)

Therefore, we can write Z1 p fKP M (x) = 1 − y 2 KN (x, y)f (y)dy

(154)

0

and in 3D space 1Z q ZZ

(1 − y12 )(1 − y22 )(1 − y32 )KN (x, y)f(y)dV

fKP M (x)

=

KN (x, y)

= KN (x1 , y1 )KN (x2 , y2 )KN (x3 , y3 )

(155)

0

(156)

D = The simplest kernel, which is usually attributed to Dirichlet, is obtained by setting gm 1. This leads to the Gibbs oscillations mentioned above. J Jackson (1911, 1912) showed that with a similar kernel with gm J gm

πn π (Nm + 1) cos Nπn +1 + sin N +1 cot N +1 = N +1

(157)

a continuous function f can be approximated by a polynomial of degree N − 1 such that   1 kf − fKP M k∞ ∼ O (158) N

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9. 9.1.

345

Example Lid Driven Cavity 2D

The lid-driven cavity problem has long been used as a test or validation case for new codes or new solution methods. The problem geometry is simple and two-dimensional, and the boundary conditions are also simple. The standard case is a fluid contained in a square domain with Dirichlet boundary conditions on all sides, with three stationary sides and one moving side (with a velocity tangent to the side) (Lid-driven cavity problem, retrieved 2016). This problem has been solved as both a laminar flow and a turbulent flow, and many different numerical techniques have been used to compute these solutions. Since this case has been solved many times, there is a great deal of data to compare with. A good set of data for comparison is the data of Ghia et al. (1982), since it includes tabular results for various of Reynolds numbers. These simulation results are obtained using a non-primitive variable approach. This problem is a nice one for testing for several reasons. First, as mentioned above, there is a great deal of literature to compare with. Second, the (laminar) solution is steady. Third, the boundary conditions are simple and compatible with most numerical approaches. Let’s begin by analyzing the vorticity equation Eq. (27). There are no external forces. There are no external pressure gradients. The curl of const boundary velocity equals to zero. Therefore, the solution of vorticity equation equals to zero and our analytical solution will be vorticity free. We have const velocity U on boundary. This implies that a boundary condition for the stream function is ψ(x) = U x, or in elliptic coordinates ψ(ξ, η) = U aξη. The solution by using expression Eq.(74) and the Jackson kernel is as follows: ψ(ξ, η)

amn

=

=

N −1 J J 4U X gm gn amn Tm (ξ)Tn (η) π2 m,n=0 (1 + δm0 )(1 + δn0 )

Z1

ξ

2

s

1−

(159)

1 a2 (ξ 2 − 1)

−1

×

„

nξ

∂Tm (ξ)Tn (η) hξ nη ∂Tm (ξ)Tn (η) + ∂ξ hη ∂η

«

η=ξ

r

dξ

(160)

1− 2 12 a (ξ −1)

where equations Eq.(94), Eq.(94), y = 1 were used for η finding and n = (nξ , nη ) expressions are derived in Appendix. Now, we can write expressions for the velocities u(ξ, η)

v(ξ, η)

=

1 ∂hξ ψ hξ hη ∂η

=

  N−1 J J gn amn Tm (ξ) 1 ∂Tn (η) Tn (η) ∂hξ 4U X gm + π 2 m,n=0 (1 + δm0 )(1 + δn0 ) hη ∂η hξ hη ∂η

= −

(161)

1 ∂hη ψ hξ hη ∂ξ

  N−1 J J gn amn Tn (η) 1 ∂Tm (ξ) Tm (ξ) ∂hη 4U X gm = − 2 + π m=0 (1 + δm0 )(1 + δn0 ) hξ ∂ξ hξ hη ∂ξ

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(162)

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Algirdas Maknickas and Algis Dˇziugys

Expressions (161) and (162) can be simplified by using relations dTm(x) = mUm−1 (x) dx

(163)

as follows: =

1 hη ∂ψ hη ∂η

=

« „ N −1 J J Tn (η) ∂hξ gn amn Tm (ξ) nUn−1 (η) 4U X gm + π2 m,n=0 (1 + δm0 )(1 + δn0 ) hη hξ hη ∂η

=

−

1 ∂hξ ψ hξ ∂ξ

=

−

„ « N −1 J J gn amn Tn (η) mUm−1 (ξ) 4U X gm Tm (ξ) ∂hη + π2 m=0 (1 + δm0 )(1 + δn0 ) hξ hξ hη ∂ξ

amn

=

a(1) nm

=

(2) a(1) nm + anm s 1 Z dξξ 2 1 −

u(ξ, η)

v(ξ, η)

−1

a(2) nm

Z1

=

dξξ 2

s

1−

(164)

(165) (166)

1 (nξ mTn (η)Um−1 (ξ)) r η=ξ 1− 2 12 a2 (ξ 2 − 1) a (ξ −1) 1 a2 (ξ 2 − 1)

„

nhξ nη Tm (ξ)Un−1 (η) hη

−1

«

η=ξ

r

(167)

(168) 1− 2 12 a (ξ −1)

Integration by time gives the Eulerian coordinates ξe = ξ + u(ξ, η)t

(169)

ηe = η + v(ξ, η)t

(170)

Finally, the solution of the NS equations is u(ξe , ηe ) = u(ξ + u(ξ, η)t, η + v(ξ, η)t)

(171)

v(ξe , ηe) = v(ξ + u(ξ, η)t, η + v(ξ, η)t)

(172)

Conclusion The main idea of the paper is linearization of the Navier-Stokes equations by integral Galileo transformation, which transforms Navier-Stokes equations from Euler frame reference to Lagrange frame reference. This linearization means that so linearized Navier-Stokes equations can be solved in each point of space at any time moment applied in Lagrange frame reference by usual known methods of mathematical physics and after transform into Euler frame reference, each point of the fluid becomes addicted to each own history of motion.

Acknowledgments This work was partly supported by the Project of Scientific Groups (Lithuanian Council of Science), Nr. MIP-13204.

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Appendix Normal Vector on Boundary We will find normal on boundary U = y − L by using expression as follow n=

∇U k∇U k

Normal vector equals in Cartesian coordinates to   1 ∂U ∂U n= , = (0, 1) k∇U k ∂x ∂y Normal vector equals in elliptic coordinates to   ∂U ∂U 1 , n = k∇U k ∂ξ ∂η   1 ∂U ∂x ∂U ∂y ∂U ∂x ∂U ∂y = + , + k∇U k ∂x ∂ξ ∂y ∂ξ ∂x ∂η ∂y ∂η   1 ∂y ∂y = , = k∇U k ∂ξ ∂η ! p p 1 − η2 · ξ η · ξ2 − 1 1 p ,− p = q 2−1 η 2 ·(ξ 2 −1) (1−η 2 )·ξ 2 ξ 1 − η2 + ξ 2 −1 1−η 2
1 (1 − η 2 )ξ, −η(ξ 2 − 1) = p (1 − η 2 )2 ξ 2 + η 2 (ξ 2 − 1)2

(173)

(174)

(175)

References Arfken, G., Weber, H. & Harris, F. (2005), Mathematical Methods for Physicists, 6th ed., Elsevier Academic Press, USA. Boyd, J. (2013), Chebyshev and Fourier Spectral Methods, Second Edition (Revised), Dover Publications, Inc. Mineola, New York. Donea, J., Huerta, A., Ponthot, J.-P. & Ferran, A. (2004), Arbitrary Lagrangian–Eulerian Methods, John Wiley & Sons, Ltd., pp. 1–25. Fefferman, C. L. (2016), ‘Existence and smoothness of the navier–stokes equation’. URL: http://www.claymath.org/sites/default/files/navierstokes.pdf. Frisch, U. (1995), Turbulence: The Legacy of A. N. Kolmogorov, Cambridge University Press, United Kingdom. Ghia, U., Ghia, K. N. & Shin, C. T. (1982), ‘High-Resolutions for incompressible flow using the Navier-Stokes equations and a multigrid method’, Journal of Computational Physics 48, 387–411.

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Green, G. (1828), ‘An essay on the application of mathematical anlysis to the theories of elasticity and magnetism’, Printed for the Author by Weelhouse T. Nottigham. Also,Mathematical papers of Georg Green, Chelsen Pulishing Co. 1970, p. 1-15. Hoffman, J. & Johnson, C. (2004), On the uniqueness of weak solutions of navier-stokes equations: Remarks on a clay institute prize problem. URL: http://www.csc.kth.se/ jhoffman/archive/papers/clayprize.pdf Ladyzhenskaya, O. A. (1969), The mathematical theory of viscous incompressible flow., 2nd edn, Gordon and Breach, Science Publishers, New York-London-Paris. Leray, J. (1934), ‘On the movement of a viscous liquid filling the space’, Acta Mathematica 63, 193–248 [Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Mathematica 63, 193–248]. Lid-driven cavity problem (retrieved 2016). URL: http://www.cfd-online.com/Wiki/Lid-driven cavity problem. Malvern, L. (1969), Introduction to the Mechanics of a Continuous Medium., Prentice-Hall: Englewood Cliffs. Navier, C. L. M. H. (1823), ‘Report on the motion of fluids’, Rep. Academy of Sciences Inst. Fr. 6, 389–416 [Memoire sur les Loisdu Movement des Fluides, Mem. de la Acad. R. Sci. Paris 6, 389–416]. Peyret, R. (2002), Spectral Methods for Incompressible Viscous Flow, Vol. 148 of Applied Mathematical Sciences, Springer-Verlag New York. Polyanin, A. D. (2001), Handbook of linear partial differential equations for engineers and scientists, Chapman & Hall/CRC. Sahoo, S. (2010), ‘Inverse Vector Operators’, arXiv:0804.2239v3 pp. 1–8. Stokes, G. (1845), ‘On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids’, Trans. Cambridge Phil. Soc. 8, 287–305. Weiße, A., Wellein, G., Alvermann, A. & Fehske, H. (2006), ‘The kernel polynomial method’, Rev. Mod. Phys. 78, 275–306. URL: http://link.aps.org/doi/10.1103/RevModPhys.78.275.

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Chapter 19

ANALYSIS OF THE TIME STEP SIZE EFFECT FOR THE STUDY OF THE LIQUID SLOSHING INSIDE A CONTAINER Abdallah Bouabidi, Zied Driss and Mohamed Salah Abid Laboratory of Electro-Mechanic Systems (LASEM), National Engineering School of Sfax (ENIS), University of Sfax (US), Sfax, Tunisia

Abstract The phenomenon of liquid sloshing is a two phase problem. The numerical simulation of the unsteady flow for this phenomenon depends on several numerical parameters. Particularly, the choice of the optimum time step is essential to perform a numerical simulation. This chapter focuses on the study of the time step size effect in the numerical results for the liquid sloshing application. Four time steps were tested in order to choose the optimum value. The choice of this value was based on the comparison of our numerical results with the experimental results founded from the literature. The local flow characteristics inside the container, such as the free surface evolution, the velocity fields, the magnitude velocity and the static pressure, were presented and discussed over time. The results show that the time step size affects significantly the numerical results performances.

Keywords: sloshing phenomenon, turbulent, flow, CFD, finite volume, time step effect

1. Introduction CFD technique is currently used to predict and analyze the fluid flow in many engineering sections. Particularly, the CFD technique is used to predict the fluid flow characteristics inside the partially filled container. Many works have been devoted in to investigate the liquid sloshing. For example, Bouabidi et al. [1] analyzed the effect of the vertical baffle height to suppress the sloshing violence in the partially filled tank using the 

E-mail address: [email protected] (Corresponding Author).

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CFD code. Their study confirmed that the sloshing decreases with the increase of the baffle height. Ohayon et al. [2] developed a new computational model to analyse the liquid motion in a partially filled container with elastic structure. In their study, the free surface tension effect is considered. Rafiee et al. [3] proposed a numerical method based on the Smoothed Particle Hydrodynamics (SPH) technique to predict the liquid sloshing. In addition, they carried out a series of experiments to study experimentally the liquid sloshing. The comparison between their numerical and experimental results showed a good agreement which confirmed the validity of the proposed numerical method. El-Kamali et al. [4] conducted a series of numerical simulations to investigate the sloshing phenomenon in the partially filled tanks with complex geometry using a three dimensional finite element method. Yan et al. [5] studied the liquid sloshing for the cases of lateral and longitudinal external excitations. Hou et al. [6] analyzed numerically the liquid sloshing phenomenon. They used the volume of fluid (VOF) to resolve the two phase problem and the dynamic mesh technique to impose the external excitations. They performed the numerical simulation for the case of single excitation and multiple coupled excitations. They showed that the sloshing is more violent in the case of multiple coupled excitations. Also, they confirmed that the sloshing is very violent for the case of the resonance. Singal et al. [7] developed a CFD analysis of a kerosene fuel tank to reduce liquid sloshing using the commercial finite volume package ANSYS FLUENT. They found that the Kerosene liquid interface that the sloshing in the fuel tank was significantly reduced with the introduction of baffles in the fuel tank. Bouabidi et al. [8] studied the liquid sloshing in a battery cell with mixing element partially filled with liquid using the commercial CFD code “Fluent”. They studied the pumping phenomena under pressure variation in the tank. A test bench was developed to investigate the sloshing experimentally in the battery cell. The comparison between their numerical and the experimental one showed the validity of their numerical model. Sanapala et al. [9] used the CFD technique to simulate the dynamics of liquid sloshing and its control in a storage tank for spent fuel applications. They developed a systematic numerical simulations in a container subjected to seismic excitations. To suppress the sloshing loads, they tested the efficient of two baffle types; the ring and the vertical baffle. Bouabidi et al. [10] carried out a series of experiments to study a hydrostatic pump in a partially filled container. To understand the source of the hydrostatic pump, the performed a CFD simulation to predict the static pressure distribution generated in the tank under sloshing loads. In another work, Bouabidi et al. [11] conducted a series of numerical simulations to investigate the effect of the external excitation frequency. They used the VOF (volume of fluid) model to resolve the two phase problem. Their study confirmed that the sloshing behavior depends on the frequency value. In fact, the sloshing loads increase with the frequency increase. In this chapter, we propose to examine the liquid sloshing in a partially filled container subjected to an external sinusoidal excitation. Particularly, we are interested on the study of time step size effect. Table 1. Standard k-ε model constants

C1ε

C2ε

Cμ

σk

σε

1.44

1.92

0.09

1

1.3

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Figure 1. Geometry arrangement.

2. Geometry Configuration The geometrical arrangement is similar to Panigrahy et al. [12] application. As shown in Figure 1, the considered 2d geometry is defined by the length L = 0.6 m and the height equal to H = 0.6 m. The tank is partially filled with liquid with the height h = 0.1 m.

3. Numerical Model The commercial CFD code “Fluent” is used to develop the simulation of the sloshing problem in the partially filled tank. The mathematical model is given by Navier Stokes equations defined as follows:

   (  ui )  0 t xi

(1)

u u j   p  (  ui )  (  ui u j )    ( i  )  Fi t x j xi x j x j xi

(2)

where ui (m.s-1) present the velocity components, ρ (kg.m-3) presents the density, p (Pa) presents the pressure and µ represents the viscosity. The external body force written as follows: 2

Fi

ρg j

ρ

X t2

(3)

where X presents the external sinusoidal excitation. The turbulent kinetic energy k and its dissipation rate ε are given by the following equations:

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t

t

ρk

ρε

xi

xi

ρ k ui

ρ ε ui

xj

xj

μ

μ

μt σε

μt σk

k xj

ε xj

C1ε

Gk

ε Gk k

Gb

ρ ε YM

C3ε Gb

C2ε ρ

Sk

ε2 k

Sε

(4)

(5)

where Gk and Gb are the generation of the turbulent kinetic energy (kg.m-1.s-3) respectively due to the mean velocity gradients and buoyancy. YM is the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate. The turbulent viscosity μt (Pa.s) is given by:

μt

ρC μ

k2 ε

(6)

The default values of the constants C1ε, C2ε, Cμ, σk and σε are presented in Table 1.

(a) ts = 0.2 s

(b) ts = 0.1 s

(c) ts = 0.05 s

(d) ts = 0.01 s

Figure 2. Free surface deformation at t = 1 s.

4. Numerical Results The numerical simulation is carried out for four different time step values equals to ts = 0.2 s, ts = 0.1 s, ts = 0.05 s and ts = 0.01 s. For each time step value, the free surface

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evolution, the velocity fields, the dynamic pressure are presented over time; at t = 1 s, t = 2 s, t = 3 s and t = 4 s.

4.1. Free Surface Deformation Figures 2, 3, 4 and 5 show the time evolution of the liquid motion including the free surface deformation at t = 1 s, t = 2 s, t = 3 s and t = 4 s for the different considered time steps equals to ts = 0.2 s, ts = 0.1 s, ts = 0.05 s and ts = 0.01 s. According to these results, it has been noted that the sloshing appears in the partially filled container under the external excitation effect. The results show that the free surface evolution in the container depends on the time step value. In fact, the free surface evolution for ts = 0.2 s is very different to the free surface evolution for the others considered time steps. Whereas, the results of ts = 0.05 s and ts = 0.01 s are very similar for the different instants.

(a) ts = 0.2s

(b) ts = 0.1 s

(c) ts = 0.05 s

(d) ts = 0.01 s

Figure 3. Free surface deformation at t = 2 s.

4.2. Velocity Fields Figures 6, 7, 8 and 9 present the velocity fields inside the container over time equal to t = 1 s, t = 2 s, t = 3 s and t = 4 s for the different considered time steps equals to ts = 0.2 s, ts = 0.1 s, ts = 0.05 s and ts = 0.01 s. According to these results, it has been observed that the sloshing generates the appearance of velocity fields in the whole volume of the container. In addition, the recirculation zones appear over time for the different considered cases. Indeed, it

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has been noted that the velocity fields are very similar for the two steps equals to ts = 0.05 s and ts = 0.01 s. Although, the velocity fields corresponding to ts = 0.2 s and ts = 0.1 s are very different between them and compared to the other instants.

(a) ts = 0.2 s

(b) ts=0.1s

(c) ts = 0.05s

(d) ts = 0.01s

Figure 4. Free surface deformation at t = 3 s.

4.3. Magnitude Velocity The distribution of the magnitude velocity caused by the liquid sloshing for the different time steps equal to ts = 0.2 s, ts = 0.1 s, ts = 0.05 s and ts = 0.01 s are shown in Figures 10, 11, 12 and 13 respectively for the different instants equals to t = 1 s, t = 2 s, t = 3 s and t = 4 s. According to these results, it has been noted that the sloshing generates an acceleration zones characterized by the maximum value of the magnitude velocity. For the different considered cases, the acceleration zone characteristic of the maximum values of the magnitude velocity appears in the region located between tow phase in the right and in the left of the areas close to the wall. The results indicate that the distribution of the magnitude velocity depends on the time steps values. In fact, the maximum values appear for the time step equal to ts = 0.2 s at t = 1 s, t = 2 s, t = 3 s and t = 4 s. However, the minimum value has been observed for the time step equal to ts = 0.01 s at t = 1 s, t = 2 s,t = 3 s and t = 4 s.

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(a) ts=0.2s

(b) ts=0.1s

(c) ts=0.05s

(d) ts=0.01s

Figure 5. Free surface deformation at t = 4 s.

(a) ts = 0.2 s

(b) ts = 0.1 s

(c) ts = 0.05 s

(d) ts = 0.01 s Figure 6. Velocity field at t = 1 s.

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(a) ts = 0.2 s

(b) ts = 0.1 s

(c) ts = 0.05 s

(d) ts = 0.01 s

Figure 7. Velocity field at t = 2 s.

(a) ts = 0.2 s

(b) ts = 0.1 s

(c) ts = 0.05 s

(d) ts = 0.01 s

Figure 8. Velocity field at t = 3 s.

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(a) ts = 0.2s

(b) ts = 0.1 s

(c) ts = 0.05 s

(d) ts = 0.01 s Figure 9. Velocity field at t = 4 s.

(a) ts = 0.2 s

(b) ts = 0.1 s

(c) ts = 0.05 s

(d) ts=0.01s

Figure 10. Distribution of the magnitude velocity at t = 1 s.

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(a) ts = 0.2 s

(b) ts= 0.1 s

(c) ts = 0.05 s

(d) ts = 0.01 s

Figure 11. Distribution of the magnitude velocity at t = 2 s.

(a) ts = 0.2 s

(b) ts = 0.1 s

(c) ts = 0.05 s

(d) ts = 0.01 s

Figure 12. Distribution of the magnitude velocity at t = 3 s.

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(a) ts = 0.2 s

(b) ts = 0.1 s

(c) ts = 0.05 s

(d) ts = 0.01 s

359

Figure 13. Disruption of the magnitude velocity at t = 4 s.

4.4. Static Pressure The static pressure distribution caused by the liquid sloshing for the different time step size values equal to ts = 0.2 s, ts = 0.1 s, ts = 0.05 s and ts = 0.01 s are shown in Figures 14, 15, 16 and 17 respectively for the different instants equal to t = 1 s, t = 2 s, t = 3 s and t = 4 s. The numerical results show that the sloshing generates the apparition of the compression and the depression zones in the whole volume of the container. From these results, it was found that the compression zone is located in the bottom wall of the tank. However, the depression zone is located just above the tank for all the considered time steps size. The location of the compression and the depression zones depends on the external excitation direction. Since the external excitation is a sinusoidal excitation, the location of the compression and the depression zones change every half period time. In the other hand, it has been noted that the time step size affects the static pressure distribution. The exception has been observed for the two time steps values equal to ts = 0.05 s and ts = 0.01 s. The static pressure distribution for these time steps presents the same observations.

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(a) ts = 0.2s

(b) ts = 0.1 s

(c) ts = 0.05 s

(d) ts = 0.01 s

Figure 14. Distribution of static pressure at t = s.

(a) ts = 0.2 s

(b) Time step size ts = 0.1 s

(c) ts = 0.05 s

(d) ts = 0.01 s

Figure 15. Distribution of static pressure at t = 2 s.

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(a) ts = 0.2 s

(b) ts = 0.1 s

(c) ts = 0.05 s

(d) ts = 0.01 s

Figure 16. Distribution of static pressure at t = 3 s.

(a) ts = 0.2 s

(b) ts = 0.1 s

(c) ts= 0.05 s

(d) ts = 0.01s

Figure 17. Distribution of static pressure at t = 4 s.

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Abdallah Bouabidi, Zied Driss and Mohamed Salah Abid ts = 0.2 s

ts = 0.05 s

ts = 0.1s

ts = 0.01 s

Experimental [12]

Figure 18. Static pressure profile.

5. Comparison with Experimental Results Figure 18 shows the variation of the static pressure in the point P1 defined by x = 0 m and y = 0.05 m. According to these results, it has been noted that the static pressure varies periodically with the time since the container is subject to a periodic sinusoidal excitation. Indeed, we found that the time step values equals to ts = 0.05 s and ts = 0.01 s give results very close to the experimental results reported by Panigrahy et al. [12]. The good agreement confirms the validity of our numerical results. In another hand, the time steps size equal to ts = 0.01 s is characterized by a simulation time longer than ts = 0.05 s. For thus, we have considered the time step size value equal to ts = 0.01 s for the future simulations.

Conclusion In this chapter, the phenomenon of liquid sloshing is examined in a partially filled container subjected to an external sinusoidal excitation using the numerical model. A series of numerical simulations were conducted to predict the time step size effect on the sloshing behavior. Four time step values are considered. The results show that the time step value affects significantly the sloshing behavior. The comparison with the experimental results gives that the two time steps size values equal to ts = 0.05 s and ts = 0.01 s present good agreement. In addition, the numerical results for the liquid sloshing, the velocity fields, the magnitude velocity and the static pressure distribution are very close for the two time steps size equals to ts = 0.05 s and ts = 0.01 s. In the future, we propose to study the liquid sloshing in container equipped with different baffle configurations.

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References [1]

Bouabidi A., Driss Z., Abid M.S., Vertical Baffles Height Effect On Liquid Sloshing In An Accelerating Rectangular Tank, International Journal of Mechanics and Applications, 3, 105-116, 2013. [2] Ohayon R., Soize C., Vibration of structures containing compressible liquids with surface tension and sloshing effects. Reduced-order model, Computational Mechanics (2014) DOI 10.1007/s00466-014-1091-4. [3] Rafiee A., Pistani F., Thiagarajan K., Study of liquid sloshing: numerical and experimental approach, Computational Mechanics, 47, 65-75, 2011. [4] El-Kamali M, Schotté J S, Ohayon R, Computation of the equilibrium position of a liquid with surface tension inside a tank of complex geometry and extension to sloshing dynamic cases, Computational Mechanics 46, 169-184, 2010. [5] Yan G., Rakheja S., Siddiqui K., Experimental study of liquid slosh dynamics in a partially filled tank, Journal of Fluids Engineering, 7, 131-145, 2009. [6] Hou L, Li F (2012) A Numerical Study of Liquid Sloshing in a Two-dimensional Tank under External Excitations. Journal of Marine Science and Application 11: 305-310. [7] Singal V., Bajaj J., Awalgaonkar N., Tibdewal S., CFD Analysis of a Kerosene Fuel Tank to Reduce Liquid Sloshing, Procedia Engineering, 69, 1365-1371, 2014. [8] Bouabidi A., Driss Z., Kossentini M., Abid M.S., Numerical and Experimental Investigation of the Hydrostatic Pump in a Battery Cell with Mixing Element, Arabian Journal for Science and Engineering, 41, 1595-1608, 2016. [9] Sanapala V.S., Velusamy K., Patnaik B.S.V., CFD simulations on the dynamics of liquid sloshing and its control in a storage tank for spent fuel applications, Annals of Nuclear, Energy, 94, 494-509, 2016. [10] Bouabidi A., Driss Z., Abid M.S., Study of hydrostatic pump created under liquid sloshing in a rectangular tank subjected to external excitation, International Journal of Applied Mechanics, 08, 1-15, 2016. [11] Bouabidi A., Driss Z., Cherif N., Abid M.S., Computational investigation of the external excitation frequency effect on liquid sloshing phenomenon, WSEAS Transactions on Fluid Mechanics, 11, 1-9, 2016. [12] Panigrahy, P. K., Saha, P. K., Maity, U. K., Experimental studies on sloshing behavior due to horizontal movement of liquids in baffled tanks, Ocean Engineering, 36, 213222, 2009.

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ISBN: 978-1-53610-292-5 c 2017 Nova Science Publishers, Inc.

Chapter 20

N UMERICAL A NALYSIS OF N AVIER –S TOKES E QUATIONS ON U NSTRUCTURED M ESHES K. Volkov∗ Faculty of Science, Engineering and Computing, Kingston University, London, UK ITMO University, St. Petersburg, Russia

Abstract Progress in the numerical solution of the Navier–Stokes equations is associated with the development of high-resolution schemes for flux computations that produce both accurate and monotone solutions in the presence of weak and strong gas discontinuities. The development of computational fluid dynamics (CFD) and computer technology makes it possible to design and implement methods for computing unsteady three-dimensional viscous compressible flows in regions of complex geometry. Flow solution is provided using cell-centered finite volume method on unstructured meshes. The non-linear CFD solver works in an explicit time marching fashion, based on a five-step Runge–Kutta stepping procedure and piecewise parabolic method. The code uses an edge-based data structure to give the flexibility to run on meshes composed of a variety of cell types. The edge weights are pre-computed and take account of the geometry of the cell. The governing equations are solved with MUSCL type scheme for inviscid fluxes, and central difference scheme for viscous fluxes. The gradient and the pseudo-Laplacian are calculated at the midpoint of a control volume edge using relations adapted to the computations on a strongly stretched meshes in the boundary layer. The gradient at mesh nodes is computed using Green’s identity, and a technique is proposed that ensures the conservation property of the difference scheme as applied to two-dimensional flows. Convergence to a steady state is accelerated by the use of multigrid technique, and by the application of block-Jacobi preconditioning for highspeed flows, with a separate low-Mach number preconditioning method for use with low-speed flows. The sequence of meshes is created using an edge-collapsing algorithm. A numerical analysis of the internal and external flows is performed to improve the current understanding and modeling capabilities of the complex flow characteristics encountered in engineering applications. ∗

E-mail address: [email protected]

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K. Volkov

Keywords: Navier–Stokes equations, CFD, finite volume method, unstructured mesh, high resolution scheme, multigrid, preconditioning

1.

Introduction

Finite volume method and high resolution schemes are applied to discretization of Euler and Navier–Stokes equations in the domain of complex geometry.

1.1.

Discretization

The development of computational fluid dynamics (CFD) and computer technology makes it possible to design and implement methods for computing unsteady 3D viscous compressible flows in regions of complex geometry. Traditionally, CFD simulations are performed on structured meshes (regular meshes with quadrilateral cells on a surface and hexahedral cells in a space). Regularity means that the mesh represents an sequence of data that is ordered according to certain rules. Finite difference and finite volume algorithms and modern high-order accurate monotone methods are relatively easy to implement on structured meshes. However, the range of geometric objects described by structured meshes, including block-structured meshes, is limited. A feature of unstructured meshes is that the mesh nodes are arbitrarily located over the physical domain, which means that there are neither distinct mesh directions nor mesh structure similar to structured meshes. The number of cells containing a particular node varies from node to node. Mesh nodes are combined in polygons (in two dimensions) or polyhedra (in three dimensions). As a rule, triangular and quadrilateral cells are used in a plane, and tetrahedra and prisms are used in a space. A major advantage of unstructured meshes over regular ones is that they provide more flexibility in the discretization of physical domains of complex geometry, and unstructured mesh generation can be fully automated. Local mesh refinement and mesh adaptation to the solution are rather easily to implement on unstructured meshes. The Navier–Stokes equations on unstructured meshes are discretized using finite element and finite volume techniques. The application of unstructured meshes increases the computational costs per one mesh node. A reasonable tradeoff is to use combinations of structured and unstructured meshes in various sub-domains (hybrid meshes), thus allowing one to enhance the advantages and reduce the shortcomings intrinsic to each mesh type. Hybrid meshes are widely used in CFD. Numerous applications are available concerning the design and implementation of computational algorithms on unstructured meshes [1–3] and the discretization of Euler and Navier–Stokes equations [4–13]. In contrast to detailed finite element techniques, finite volume algorithms on unstructured meshes lack unified principles for discretizing inviscid and viscous fluxes and source terms. A rather frequent situation is that discretization methods with different characteristics are combined together. Progress in the numerical solution of Navier–Stokes equations is associated with the development of high-resolution schemes (HRS) for flux computation that produce both accurate and monotone solutions in the presence of weak and strong gas discontinuities.

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The monotonic upwind scheme for conservative laws (MUSCL) [5, 6] is widely used in the design of high-order accurate monotonized difference schemes. Instead of fluxes from neighboring cells, the variables themselves are extrapolated, their derivatives are limited, and the improved values are substituted into the flux expressions. An advantage of the MUSCL is that the accuracy of a scheme can be improved by changing the order of interpolation within a cell. A finite volume discretization of Navier–Stokes equations on a hybrid mesh is considered. Mesh generation is separated from the discretization of Navier–Stokes equations, while the representation and storage of mesh node coordinates is not discussed. Advantages of the approach proposed are that it covers both structured and unstructured meshes and uses high-resolution schemes. The control volume is defined as the cell-vertex median dual control volume to discretize governing equations. The relations for determining the gradient and the pseudo-Laplacian are adapted to the computations on a strongly stretched mesh in the boundary layer.

1.2.

Preconditioning

Flow simulation at low Mach numbers (M < 0.3) is usually based on the incompressible form of Navier–Stokes equations (the velocity field is solenoidal, and ∇v = 0), and steady state problems are solved using time marching scheme. Since the incompressible continuity equation involves only the velocity components, it is not directly related to pressure, which is expressed in terms of density for compressible flows. This difficulty is overcome by using artificial compressibility (the time derivative of pressure is introduced into the continuity equation). Additionally, pressure correction methods are applied (a difference scheme is constructed for the increments of the unknowns and Poisson equation is solved for the pressure correction at each time step) or the splitting of unknowns is used (the projection method and its modifications). In many CFD problems of practical interest, the flow ranges from essentially subsonic speeds (inlet flows, re-circulation zones) to supersonic speeds (nozzle flows, local supersonic regions in airfoil flows). To avoid the separation of the computational domain into sub-domains depending on the characteristic Mach number and the application of a simplified mathematical model corresponding to Mach number in each sub-domain (viscous incompressible model for M  1, viscous compressible model for M < 1, and inviscid compressible model for M > 1), a mathematical model based on the compressible Navier– Stokes equations is used. The implementation of this approach requires some additional efforts related to the stability of the computations and the convergence acceleration of the iterative process. A characteristic feature of low-speed flow simulation based on the compressible Euler or Navier–Stokes equations is that the numerical solution becomes unstable, and the convergence rate of the iterative process decreases due to the small difference between the speeds of acoustic and convective waves [9, 13–22]. The time integration step is determined by the speed of the fastest wave, while the time required for reaching a steady state depends on the speed of the slowest wave. When explicit finite difference schemes and stretched meshes are used in the boundary layer, the time step is limited by acoustic solution modes and the mesh size in the direction normal to the wall [9, 14, 15], ∆t = O(∆y/c). This condition is

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several orders of magnitude more restrictive than the conditions required for the resolution of convective modes, ∆t = O(∆x/vx ) = O(∆x/vy ). Preconditioning makes it possible to modify the governing equations in a such way that the eigenvalues of the Jacobian (wave propagation speeds) of the modified equations have identical orders of magnitude [13, 16]. Physical preconditioning is widely used to solve the Euler equations on structured meshes, but it is difficult to implement on unstructured meshes, and numerical preconditioning is applied [13–15]. In [17–19] semi-implicit and implicit preconditioning methods were developed, since explicit methods do not ensure that low-frequency solution modes are damped. Specifically, the difference equations were solved with a multigrid method, and the mesh was coarsened in three Cartesian directions simultaneously [17]. To facilitate the multigrid method, an approach was proposed based on consecutive mesh coarsening in each coordinate direction [18, 19] (semi-coarsening algorithm). In another implementation of the approach proposed in [17] (line-implicit J-Jacobi preconditioning), the preconditioning matrix was constructed taking into account the inviscid and viscous terms in the direction normal to the flow. An advantage of the modified approach is that its computational costs remain the same for 2D and 3D problems [17] (mesh is coarsened in a single coordinate direction). The shortcomings of the approaches used in [17–19] are associated with the increased complexity and difficulties related to the parallelization of implicit algorithms. Although high-order difference schemes (second-order or higher) are widely used to discretize the Navier–Stokes equations, the modified equations are usually discretized using a first-order scheme, which leads to an underestimated time step [20]. An additional shortcoming of the approaches of [9, 13–20] is that they are designed for a narrow series of external aerodynamic problems, which imposes constraints on the choice of a limiter and a transition criterion. The original equations are multiplied by a preconditioning matrix when the Mach number inside the computational domain is lower than ηM2∞ , where M∞ is the freestream Mach number and η is a coefficient (η > 1). Despite its successful use in the case of structured meshes [15, 21, 22], global preconditioning is inapplicable when the Mach number at the inlet boundary is unknown (internal flows) or its choice is a complicated problem. A way out is to use a limiter computed using the local Mach number or the local pressure field (local preconditioning). A block preconditioning method for Navier–Stokes equations on structured and unstructured meshes is considered. In the case of an unstructured mesh, it is shown that the preconditioning matrix is different for second- and fourth-order schemes. The capabilities of the approach are demonstrated by computing the flow past an airfoil at low Mach numbers. The computational algorithms with scalar preconditioning, block preconditioning and without preconditioning are compared in terms of their efficiency.

1.3.

Multigrid Method

Given a system of difference equations, the method used for its solution has a large effect on the stability of the numerical procedure and the convergence rate of the iterative process. A universal technique for solving systems of difference equations is the multigrid

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method, which relies on a sequence of meshes and transition operators from one mesh to another. The solution process begins with the coarsest mesh. The solution obtained on the coarse mesh is interpolated to a fine mesh and is used as an initial approximation in an iterative process, which requires a relatively small number of iterations to achieve a prescribed accuracy. It is taken into account that some iterative methods (e.g., Gauss–Seidel method) converge at a high rate at the first few iteration steps (due to the fast suppression the highfrequency Fourier components of the initial error in the eigenvector expansion) but then slow down. The low-frequency harmonics converge more slowly and comprise most of the error. The multigrid method is not a fixed technique but rather a template and a build-up construction whose implementation efficiency depends on the adaptation of its components to the problem considered. The simplest implementation of classical multigrid methods is the cascade method, which consists of an interpolation procedure with an iterative smoother. A multigrid algorithm for the standard five-point discretization of Poisson equation in a unit square was formulated in [23], and the underlying ideas were developed in [24]. A considerable contribution to the extension of the multigrid algorithm to nonlinear equations, the development of a multilevel adaptation technique, and the nested iteration method was made in [25,26]. Modern numerical algorithms rely on multilevel multigrid methods with a sequence of meshes generated explicitly or implicitly. The multigrid cycle consists of the following steps: pre-smoothing, residual calculation at the current mesh level, restriction and correction of the residual on the coarse mesh, prolongation and interpolation of the error to a fine mesh, correction of the fine mesh solution using the correction interpolated from the coarse mesh (coarse mesh correction), and postsmoothing for the suppression of the high-frequency error components appearing after the interpolation to the fine mesh. The computations are terminated when a prescribed accuracy is achieved. The smoothing method (smoother) damps the high-frequency error modes and appears the element of the multigrid method that is most dependent on the type of the problem [27]. The role of the smoother is not so much to reduce the total error but rather to smooth it (suppress the high frequencies) so that the error admits a good approximation on the coarse mesh. Standard smoothers are linear iterative methods (Jacobi, Gauss–Seidel and incomplete LU -factorization methods). In many respects, the quality of the multigrid method is determined by the chosen sequence of meshes and interpolation operator. Quality criteria include the convergence factor, which shows how fast the method converges (how many iteration steps are required to achieve the prescribed level of the residual) and the complexity of the restriction and prolongation operators, which determines the number of operations and the amount of storage required for each iteration step. Two approaches can be distinguished, algebraic multigrid (Algebraic Multigrid, AMG) and geometric multigrid (Geometric Multigrid, GMG). In the algebraic approach, discrete equations on a sequence of nested meshes are formed without nested mesh generation, while, in the geometric approach, a hierarchy of meshes is constructed by merging control volumes of the fine mesh (as a result, the meshes of various levels do not need to be stored

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as separate files). Depending on the topology of the original mesh, coarse meshes have irregular cells with different numbers of faces (several control volumes of the fine mesh are joined in a control volume of the coarse mesh). The simplicity of the construction of mesh levels leads to the simplicity of the construction of restriction and prolongation operators. The geometric approach seems more suitable for nonlinear problems, since the nonlinearities of the original system are passed downward the mesh hierarchy (from the fine to coarse mesh). As applied to the Navier–Stokes equations, the approach violates the sum of the accuracy orders of the restriction and prolongation operators [26]. In CFD problems, various strategies are used to implement the multigrid approach. 1. An independent sequence of meshes is produced by a black box mesh generator [28]. The procedure for mesh generation at various mesh levels is not completely automated (meshes of various levels are constructed manually), while there is no connection between the mesh topology and the multigrid method. However, an arbitrary mesh generator can be used, which makes this approach rather flexible. 2. For complicated problems, an approach is used in which edge weights of the coarse mesh are constructed by merging control volumes of a fine mesh (agglomeration multigrid) [29]. The use of a hybrid mesh does not lead to additional difficulties, while ensuring high efficiency. 3. A sequence of nested meshes is constructed by introducing new nodes with the subsequent mesh adaptive refinement [2]. The generation of meshes begins with the coarsest one. As a result, to the quality of the fine mesh near the wall in viscous problems is difficult to control. Despite the completely automated procedure for nested mesh generation, difficulties also arise in flow simulation around bodies of complex geometry with splines used to reproduce their surfaces. 4. A procedure based on the collapse of an edge in the fine mesh is used to pass to the next mesh level. The generation of meshes begins with the finest mesh. This approach is rather flexible and efficient and provides first and second-order accuracy for the restriction and prolongation operators [2, 30]. A sequence of structured nested meshes is constructed straightforwardly (mesh nodes are removed in turn in each coordinate direction). A full coarsening approach to the construction of triangular and tetrahedral nested meshes was developed in [31]. Two mesh nodes joined by an edge are replaced with a single node, and all the cells associated with this edge are rebuilt. In the course of mesh coarsening, the number of cells divided by the number of nodes remains a constant. Additional constraints can be used to obtain an isotropic mesh (for example, near the wall). The approach of [31] is difficult to generalize to hybrid meshes, since, in the restructuring of cells, their shape (the number of corner points) has to be preserved wherever possible in order to prevent the violation of the nested mesh topology. On a triangular or tetrahedral mesh, the replacement of two nodes with a single one (the collapse of an edge) leads to the vanishing of a cell, while, on a hybrid mesh, a cell vanishes when several edges collapse. Mesh coarsening in a regular hexahedral domain leads to a halved number of nodes but does not reduce the number of edges. As a result, the approach of [31] does not preserve the regular structure of coarse meshes. In [32] a graph of mesh nodes was considered and two constraints were introduced, one of which is that the collapse of mesh edges does not produce cells with a negative volume

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and the other is that the length of any neighboring edge does not exceed a given length (in mesh coarsening, the edge length is doubled). The approach of [32] yields a sequence of high quality meshes in an isotropic case (inviscid flows). For viscous flows, the approach produces a too coarse mesh near the wall that fails to ensure the required accuracy [33]. The modified approach developed in [33] (directional coarsening) generates a mesh in a boundary layer with cells contracted in the normal direction to the wall and with the additional constraint that the shortest cell edge does not collapse. An edge collapses is there is another edge oriented in the normal direction to the wall. The modified approach preserves the structured part of the mesh near the wall (for example, around an airfoil). The loss of mesh regularity away from an airfoil has a weak effect on the quality of the numerical solution. A shortcoming of the approach is that the coarse mesh cells become more oblique. The multigrid method can also be applied together with block-Jacobi preconditioning [34]. The preconditioning damps all the convective modes, while the multigrid method used on a mesh coarsened in the normal direction to the wall (across the boundary layer) guarantees the suppression of the acoustic modes. A multigrid method is proposed for solving Navier–Stokes equations on unstructured mesh using edge collapsing algorithm. The approach involves a modified restriction operator and a nested mesh generation procedure for viscous flows (the boundary layer on the wall is taken into account). The discretization of the governing equations on a sequence of meshes is easy to implement, since the finite volume method developed has no constrains on the number of cells and the cell shape (for hybrid meshes, the cell shape in the original mesh changes in the transition to the next mesh level). The capabilities of the approach are demonstrated by computing a turbulent airfoil flow.

1.4.

Wall Boundary Conditions

Reynolds-averaged Navier–Stokes (RANS) equations are widely used for computing turbulent flows. They are closed with different levels of complexity, and turbulence models are usually classified according to the number of differential equations introduced in addition to the momentum and energy equations. The differential turbulence models that have gained considerable attention in CFD simulations are the Spalart–Allmaras (SA) model [36], k–ε model [37], and their modifications [38]. In high-Reynolds number formulations, the equations of Spalart–Allmaras and k–ε models are suitable for describing flows away from the wall [36, 37]. Near-wall flows are usually modeled using wall functions, when the viscous sub-layer and the transition zone of the boundary layer are not resolved, but are described by semi-empirical formulas (wall functions). The accuracy is improved by solving simplified boundary layer equations in a near-wall control volume [39]. To obtain a flexible description of domains of complex geometry, which are encountered in CFD applications, numerical methods on unstructured meshes are developed. In turbulent flow computations on unstructured meshes combined with the wall functions, the solution depends substantially on the mesh step size near the wall. On the wall, the no-penetration boundary condition (vn = 0) is usually applied to the normal velocity, while the no-slip boundary condition (vτ = 0) is used for the tangential

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velocity (rigid boundary conditions). Although the slip boundary condition (vτ 6= 0) on the wall (weak boundary conditions) contradicts physical reality (rarefied flows are not considered), this approach is used in CFD, but mainly in finite element discretizations of the Navier–Stokes equations [40]. The influence exerted by the wall on the flow is taken into account in the form of mesh shear stresses and additional mesh generation of turbulence caused by the difference of the tangential velocity profile from the logarithmic distribution near the wall [4]. In weak boundary conditions, no-penetration boundary condition for the normal velocity is preserved (vn = 0), while the tangential velocity is computed using shear stresses on the wall (vτ 6= 0). This study deals with various formulations and numerical implementations of weak boundary conditions on the wall in computations of turbulent flows on unstructured meshes. The advantages and shortcomings of the wall functions and the features of their implementation for Spalart–Allmaras and k–ε models are discussed. Implementations of weak boundary conditions in the framework of the finite volume discretization of RANS equations are proposed. The capabilities of the approach are demonstrated by solving benchmark CFD problems. The influence of the near-wall mesh size on the accuracy of the numerical solution is shown, and the mesh dependence of CFD solution computed with wall functions and weak boundary conditions is examined.

2.

Governing Equations

The calculations are based on the compressible flow CFD code designed with air as the working fluid. The perfect gas law is used to link the density, pressure and temperature.

2.1.

Conservative Formulation

The governing equation is
∂Q + ∇ · F I + F V = H, ∂t

(1)

where Q is the vector of conservative variables, F I is the vector of inviscid fluxes, F V is the vector of viscous fluxes, and H is the model-dependent source term. The system of equations (1) includes the mass conservation equation (continuity equation), momentum conservation equation and energy conservation equation. The conservation equations written in the form (1) are applicable to absolute and relative velocity formulations. To complete the system of equations (1), an equation of state for an ideal gas is required   q2 p = ρRT = (γ − 1)ρ e − , 2 where γ, R, T are the specific heat capacity ratio, the gas constant and the temperature, respectively. The velocity magnitude is q 2 = |v|2 − r 2 ω 2 ,

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where r is the rotation radius, and ω is the rotation speed. The viscous stress tensor and the strain rate tensor are defined as i 2 1h τ = 2µs − µ (∇ · v) I, s = (∇v) + (∇v)∗ , 3 2

where I is the unit tensor. The molecular viscosity, µ, is a function of temperature. It is modeled with Sutherland’s law  3/2 T T ∗ + S0 µ = , µ∗ T∗ T + S0

where µ∗ and T∗ are a reference viscosity and temperature, and S0 is a constant determined experimentally, so that µ∗ = 1.68 × 10−5 kg/(m·s), T∗ = 273 K, and S0 = 110.5 K for air. The heat flux is given by Fourier law   γ µ p ∇q = −λ∇T = − ∇ . (γ − 1) Pr ρ The thermal conductivity, λ, is linked to the specific heat capacity at constant pressure, cp , and the Prandtl number, Pr, so that λ = cpµ/Pr, and Pr = 0.7 for air. To model flows that include rotating boundaries, the rotating frame of reference is used. The flow may be unsteady in an inertial frame (a domain fixed in the laboratory frame), but steady relative to the rotating non-inertial frame (the domain moving with the rotating part). When the governing equations are solved in a rotating frame of reference, the acceleration of the fluid is augmented by additional terms that appear in the momentum equations. The absolute velocity (velocity in inertial reference frame), V , and the relative velocity (velocity relative to the rotating reference frame), v, are related by V = v + r × ω, where ω is the angular velocity of the rotating frame, and r is the radius vector in the rotating frame. In the rotating frame of reference, the source term includes external volume forces, such as Coriolis force and centrifugal force. The Coriolis force, F cor, and centrifugal force, F cen , are computed as F cor = −2ρ ω × v,

F cen = −ρ ω × (ω × r) .

It is assumed, the x-axis is the axis of rotation, so ω = {ω, 0, 0}, and ω = 0 in the absolute frame of reference. In a rotating frame of reference, the entrainment velocity does not contribute to the mass balance, and the continuity equation remains invariant (in contrary to the momentum equation). The RANS equations take the same form as the equations (1) if the definitions of the viscosity and thermal conductivity are modified to incorporate both molecular and turbulent contributions. The total viscosity (effective viscosity) and the total thermal conductivity (effective thermal conductivity) are given by   µ µt µe = µ + µt , λe = λ + λt = cp + , Pr Prt

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where µt and λt are the eddy viscosity and the eddy thermal conductivity, respectively, and Prt is the turbulent Prandtl number (Prt = 0.9 for air). For turbulent calculations, the RANS equations, written in the form (1), are solved using a turbulence model for closure.

2.2.

Cartesian Coordinates

In Cartesian coordinates (x, y, z), the unsteady viscous compressible flow is described by the equation ∂Q ∂Fx ∂Fy ∂Fz + + + = H. ∂t ∂x ∂y ∂z

(2)

Equation (2) is supplemented with the ideal gas equation of state  
1 2 2 2 p = (γ − 1)ρ e − v + vy + vz . 2 x

The vector of conservative variables Q and the flux vectors Fx , Fy , Fz have the form   ρ  ρvx     Q=  ρvy  ,  ρvz  ρe 

  Fx =   



  Fy =   



  Fz =   

ρvx ρvx vx + p − τxx ρvxvy − τxy ρvx vz − τxz (ρe + p)vx − vx τxx − vy τxy − vz τxz + qx ρvy ρvy vx − τyx ρvy vy + p − τyy ρvy vz − τyz (ρe + p)vy − vx τyx − vy τyy − vz τyz + qy ρvz ρvz vx − τzx ρvz vy − τzy ρvz vz + p − τzz (ρe + p)vz − vx τzx − vy τzy − vz τzz + qz



  ,  



  ,  



  .  

The components of the viscous stress tensor and the heat flux components are given by   ∂vi ∂vj 2 ∂vk ∂T τij = µe + − δij , qi = −λe . ∂xj ∂xi 3 ∂xk ∂xi

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Here, t is time, ρ is the density, vx , vy , vz are the velocity components in the x, y and z directions, p is the pressure, e is the total energy per unit mass, T is the temperature, γ is the ratio of specific heat capacities. In cartesian coordinates, the Coriolis force and the centrifugal force are found from F cor = 2ρ ω (vz j − vy k) ,

F cen = ρ ω 2 (yj + zk) ,

where j and k are the unit vectors in directions y and z, respectively. The source term has the form   0   0    H=  ρ ω(y ω + 2vz )  .  ρ ω(z ω − 2vy )  0

2.3.

Boundary Conditions

To obtain a well-posed problem, appropriate boundary conditions must be imposed on the boundary of computational domain. Boundary conditions used in calculations include inviscid wall (slip and no-penetration boundary conditions), viscous wall (no-slip and no-penetration boundary conditions), free stream, inflow, outflow, periodic and centreline boundary conditions. No parameters are required for inviscid wall (the appropriate solid wall boundary condition is zero velocity normal to the wall). For viscous wall, three velocity components (two in 2D case) or the rotation speed for an annular model can be specified if the wall is moving. A constant wall temperature may be applied or the wall can be adiabatic. For a free stream boundary, the density, pressure, Mach number and angle of attack and yaw angle must be specified. To define the inflow boundary conditions, for 2D calculations, one needs to provide a total pressure, total temperature, an inflow angle and the turbulence properties. In 3D cases, 1D profiles of these parameters can be specified, and both the tangential and radial flow angles are required. For the outflow boundary, a static pressure and reverse flow temperature are required. If the reverse flow temperature is set at zero (advised if unknown), the reverse flow temperature would be taken identical to the temperature of the flow going out of the boundary. Periodic boundary conditions are specified in circumferential directions, and symmetric boundary conditions are specified on the centreline.

3.

Turbulence Models

The closure of RANS equations written in the form of (2) is based on the Spalart– Allmaras and k–ε turbulence models.

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3.1.

K. Volkov

Spalart–Allmaras Model

In addition to equation (2), the Spalart–Allmaras model includes the transport equation for the working variable νe, which is an analogue of turbulent viscosity [36] i o ∂ρe ν 1n h + (ρv · ∇) νe = ∇ (µ + ρe ν ) ∇e ν + cb2 ρ∇e ν · ∇e ν + Sνe. ∂t σ

(3)

The source term in equation (3) takes into account the generation and dissipation of the turbulent viscosity    νe 2 eν − cw1 fw − cb1 ft2 Sνe = cb1 ρSe , κ2 d | {z } | {z } generation

dissipation

where d is the distance from the control volume center to the nearest wall. The working variable νe is related to the turbulent viscosity, νt = fv1 νe. The turbulence production term is modeled by the relation Se = Sfv3 +

νe fv2 . κ 2 d2

The source term is calculated using the vorticity S = |Ω| = (2Ωij Ωij )

1/2

,

1 Ωij = 2



∂vi ∂vj − ∂xj ∂xi



.

To ensure the correct behavior of the working variable in the logarithmic layer (e ν = κyuτ ), the damping function is introduced fv1 =

χ3 , χ3 + c3v1

The functions fv2 and fv3 have the form fv2 = 1 −

χ , 1 + χfv1

χ=

νe . ν

fv3 = 1.

Function fw is important in the outer region of the boundary layer fw = g



1 + c6w3 g 6 + c6w3

1/6

,

g = r + cw2 (r 6 − r),

r=

νe

κ 2 d2 Se

.

The function g is used as a limiter on overestimated values of fw . The parameters r and fw are set to unity in the logarithmic layer and decrease in the outer region. The constants of the model are specified as cb1 = 0.1355, cb2 = 0.622, σ = 2/3, cv1 = 7.1, cw1 = cb1 /κ 2 + (1 + cb2 )/σ, cw2 = 0.3, cw3 = 2.0, κ = 0.42.

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377

k–ε Model

In addition to equation (2), the k–ε turbulence model includes the transport equations for the turbulent kinetic energy and its dissipation rate, which have the form [37]    ∂ρk µt + (ρv · ∇) k = ∇ µ + ∇k + P − ρε; (4) ∂t σk    ∂ρε µt ε + (ρv · ∇) ε = ∇ µ + (5) ∇ε + (cε1 P − cε2 ρε) . ∂t σε k The eddy viscosity is calculated using the Kolmogorov–Prandtl formula, µt = cµ ρk2 /ε. The constants of the model are specified as cµ = 0.09, σk = 1.0, σε = 1.3, cε1 = 1.44, cε2 = 1.92. The turbulence production term in the equations (4) and (5) is determined by the relation [37]   1 ∂vi ∂vj 1/2 P = µt |S|, |S| = (2Sij Sij ) , Sij = + . 2 ∂xj ∂xi The rotation of the flow is taken into account using the Kato–Launder modification of the term describing turbulence generation [41]   1 ∂vi ∂vj 1/2 1/2 1/2 P = µt |S| |Ω| , |Ω| = (2Ωij Ωij ) , Ωij = − . 2 ∂xj ∂xi For flows with curved streamlines, the semi-empirical constants of the k–ε model are modified by multiplying them by correction functions depending on turbulent Richardson number [42].

4.

Finite Volume Discretization

An unstructured mesh discretization of the Navier–Stokes equations based on the finite volume method and high resolution difference schemes is described. The control volume is defined as the cell-vertex median dual control volume. The fluxes through the faces of internal and boundary control volumes are written identically. The gradient and the pseudoLaplacian are calculated at the midpoint of a control volume edge using relations adapted to the computations on a strongly stretched mesh in the boundary layer.

4.1.

Governing Equations

In conservative variables, the equation governing an unsteady three-dimensional viscous compressible flow (1) or (2) is written as ∂Q + ∇ · F (n, Q, ∇Q) = H(Q, ∇Q). ∂t

(6)

Here, Q(x, t) is the vector of conservative variables at the point x at time t, F (n, Q, ∇Q) is the flux through a surface whose orientation is specified by the outward unit normal n, and

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H(Q, ∇Q) is the source term. The source term takes into account the non-inertial character of the frame of reference (the effect of the Coriolis and centrifugal forces). Equation (6) is supplemented with the ideal gas equation of state. In the case of turbulent flows, turbulence model equations (with the molecular transfer coefficients replaced by their effective values) are also added to equation (6). The normal velocity on the wall satisfies the no-penetration condition (vn = 0), and the no-slip condition (vτ = 0) is applied to the tangential velocity on the wall. A variety of boundary conditions are available for the wall temperature and at the inlet and outlet the computational domain. They are easy to set and implement, and their type is not important for the discretization of the Navier–Stokes equations [1–3]. By splitting the flux into inviscid and viscous components, F (n, Q, ∇Q) = F I (n, Q) + F V (n, Q, ∇Q), | {z } | {z } | {z } full flux

inviscid flux

(7)

viscous flux

equation (6) can be rewritten as h i ∂Q + ∇ · F I (n, Q) + F V (n, Q, ∇Q) = H(Q, ∇Q). ∂t

(8)

Define the residual vector

R(Q) = ∇ · F (n, Q, ∇Q) − H(Q, ∇Q). Then equation (8) becomes ∂Q + R(Q) = 0. ∂t

(9)

Equation (6), (8) or (9) is discretized by the finite volume method on a hybrid mesh. The turbulence model equations are discretized in the same manner as the Navier– Stokes equations. The source terms in the turbulence model equations are discretized so as to guarantee that the desired functions are bounded according to their physical meaning [3, 4].

4.2.

Finite Volume Method

Integrating equation (6) over the control volume Vi with the boundary ∂Vi, whose orientation is set by the outward unit normal n = {nx , ny , nz }, and applying the Gauss theorem, obtain Z I h Z i ∂ Q dΩ + F (n, Q, ∇Q) − (vb · n)Q dS = H(Q, ∇Q) dΩ, (10) ∂t Vi

∂Vi

Vi

where vb is the velocity of the boundary ∂Vi of control volume Vi . Assuming that the mesh value at the center of the control volume is the mean integral value of the corresponding continuously distributed value Z 1 Qi = Q dΩ, Vi Vi

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equation (10) can be reduced to ∂Qi + Ri (Q) = 0. ∂t The residual vector in equation (11) is given by the relation ( I ) Z h i 1 Ri(Q) = F (n, Q, ∇Q) − (vb · n)Q dS − H(Q, ∇Q) dΩ . Vi ∂Vi

(11)

(12)

Vi

The residual at the node i is calculated by summing the fluxes through the faces of the control volume 1 X Ri = Fij ∆sij . Vi j∈Ei

Taking into account the splitting of the flux into inviscid and viscous components (7), the residuals due to the discretization of inviscid and viscous fluxes are found from the relations   I X X 1 1 I  RIi = F I (n, Q)dS = FijI ∆sij + Fik ∆sik  , (13) Vi Vi j∈Ei

∂Vi

RVi

1 = Vi

I

F V (n, Q, ∇Q)dS =

∂Vi

k∈Bi

1 X V Fij ∆sij . Vi

(14)

j∈Ei

Residual (13) is due to the discretization of inviscid fluxes and takes into account the contribution of the internal and boundary faces of the control volume, while residual (14) is due to the discretization of viscous fluxes and takes into account only the contribution of V the internal faces (no-slip and no-penetration conditions are set on the wall, so Fik = 0 for ∀ k ∈ Bi ). In the finite volume discretization of Navier–Stokes equations, the control volume coincides with the mesh cell or is centered at the mesh node [2, 5]. The latter technique (Figure 1a) requires roughly 6 times as much storage as the former but produces more accurate results in the boundary layer because of the finer mesh size in the near-wall region. Half the control volume is used near the wall (Figure 1b). A special case of the centered control volume is the cell-vertex median dual control volume (Figure 2). The cell-vertex median dual control volume Vi associated with the mesh node i = 1, . . ., N , where N is the number of nodes, is constructed in a such way that the geometric centers of mesh cells with their vertex at the node i are joined to each other through the middles of the separating faces. The control volume edge joining mesh nodes i and j is denoted by (i, j). The edge weights (surface areas of the faces) of internal faces in the control volume are antisymmetric sij = −sji for ∀ j ∈ Ei (the faces of the control volume are traversed clockwise), and the edge weights of boundary faces are symmetric sik = ski for ∀ k ∈ Bi [2]. Moreover, the following relation holds [5, 6] X X sij + sik = 0. (15) j∈Ei

k∈Bi

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b)

j

i

i

j

Figure 1. Internal (a) and boundary (b) control volumes. mesh control volume median dual control volume

Figure 2. Median dual control volume. Here, Ei is the set of internal edges associated with the node i, Bi is the set of boundary edges associated with node i, nij is the outward unit normal specifying the orientation of the edge (i, j), sij is the surface area of the edge joining the nodes i and j; nik is the outward unit normal to the boundary edge (i, k), sik is the surface area of the boundary edge joining the nodes i and k. Condition (15) guarantees that the difference scheme is conservative when applied to 3D flows [2,6]. In the quasi 3D case (for plane flows with periodic conditions in streamwise direction), the edge weights of internal control volume edges are specified as hij sij , where hij is the height of the stream tube. If its height varies, the difference scheme becomes non-conservative, since X hij sij 6= 0. j∈Ei

In contrast to implementations [5, 6], to preserve the conservation property, the mesh nodes in the plane normal to the flow direction are treated as boundary nodes (hidden group of nodes). The edge weights of boundary edges associated with these nodes are calculated by the formula X X sik + hij sij , k∈Bi

j∈Ei

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where the first term is a standard edge weight, and it is equal to zero for interior nodes. The modification of (15) ensures the conservation of the difference scheme as applied to plane and axisymmetric flows.

4.3.

Numerical Fluxes

The integral over the boundary of the control volume is calculated by summing (over all the faces) the products of the flux at the center of a face and the surface area of the face. The integral of the flux in (12) splits into two terms associated with internal and boundary faces Z X F (n, Q, ∇Q) ds = F (nij , Q, ∇Q)|xij sij + j∈Ei

∂Vi

+

X

k∈Bi

F (nik , Q, ∇Q)|x=xik sik .

In view of (16) and the mean-value theorem, relation (12) is discretized as ! X X 1 Ri = Fij sij + Fik sik − Hi Vi . Vi j∈Ei k∈Bi | {z } | {z } | {z } internal edges

boundary edges

(16)

(17)

source

Here, Fij is the flux through the internal edge (i, j), whose orientation is specified by the outward unit normal nij , Fik is the flux through the boundary edge (i, k), whose orientation is specified by the outward unit normal nik , sij and sik are the surface areas of the internal edge (i, j) and boundary edge (i, k) in the control volume. 4.3.1.

Internal Faces

The flux through the internal face (i, j) of the control volume is calculated at the face midpoint xij =

1 (xi + xj ) 2

as the half-sum of the corresponding node values times the surface area of the face Fij =

1 (Fi + Fj ) sij . 2

Summing these relations over all the internal faces (for all ∀ j ∈ Ei ) yields F =

1 X (Fi + Fj ) sij . 2 j∈Ei

Since the edge weights of internal faces for a closed control volume are antisymmetric X sij = 0, j∈Ei

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then F =

X 1 X (Fi + Fj ) sij − Fi sij . 2 j∈Ei

j∈Ei

This relation can be rewritten as F =

1 X (Fj − Fi ) sij . 2 j∈Ei

Since sij = −sji for ∀ j ∈ Ei , the contribution of each control volume face (i, j) is represented as Fij = 4.3.2.

1 (Fj − Fi ) sij . 2

Boundary Faces

The flux through the boundary face (i, k) of the control volume is calculated at the point xik =

i Xh 1 1 + (D + 2)δjk xj , 2D + 2 j∈Gk

where D is the dimension of the problem, and Gk is the set of nodes in the boundary cell k. The contribution of a face lying on the boundary of the computational domain is written separately. Since the edge weights of internal faces are symmetric and the edge weights of internal faces are antisymmetric (15), then ! X 1 X F = (Fi + Fj ) sij + Fik sik − Fik sik + sij . 2 j∈Ei

j∈Ei

As a result, the fluxes through the faces of a boundary control volume are calculated by the same relation as for internal faces 1 X F = (Fj − Fi ) sij . 2 j∈Ei

The identical expressions used for computing the fluxes through the faces of internal and boundary control volumes simplify programming of finite volume method as compared to available implementations on unstructured meshes [9–13].

4.4.

Inviscid Fluxes

It follows from (12) and (17) that RIi (Q)

1 = Vi

I

∂Vi

1 F (n, Q) dS = Vi I

X

j∈Ei

FijI sij

+

X

I Fik sik

k∈Bi

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Inviscid fluxes are discretized using MUSCL-type scheme [1], which is a combination of centered finite differences of second-order (scheme CDS2) and fourth-order scheme. A flux smoother based on characteristic variables is used to switch between these schemes [7]. The inviscid flux through a face of the control volume is given by FijI =

i 1 1h I F (Qj ) + F I (Qi) − |A| (Qj − Qi ) , 2 2

(19)

where A = ∂F I /∂Q is the inviscid Jacobian. The Jacobian is decomposed in the form |A| = T −1 |Λ|T , where T is the matrix made up of the left eigenvectors of the Jacobian, and Λ is a diagonal matrix made up of the eigenvalues of the Jacobian (λ1 = q − c, λ2 = q + c, λ3,4,5 = q). For low-speed flows, |q|  c. This imposes constraints on the time integration step and leads to an increased contribution of the smoothing term [3], which is proportional to the flow velocity. 4.4.1.

Special Case

Consider the model equation ∂Q ∂F + = 0. ∂t ∂x

(20)

The linearization of equation (20) gives ∂Q ∂Q +A = 0, ∂t ∂x

(21)

where A = ∂F/∂Q. Discretization of the equation (21) yields Qn+1 − Qnj j ∆t

+A

Qnj − Qnj−1 = 0. ∆x

(22)

When A < 0 the scheme (22) becomes unstable. To prevent instability, the matrix A is represented as the sum of symmetric and antisymmetric parts A = A+ + A− , where A+ = max{A, 0} =

1 (A + |A|) , 2

A− = min{A, 0} =

1 (A − |A|) . 2

Substitution of these relations into equation (22) yields Qn+1 − Qnj j ∆t

+



i 1 h + n A Qj − Qnj−1 + A− Qnj+1 − Qnj = 0. ∆x | {z } | {z } A>0

(23)

A 1/ 3 or using another criterion). As ε → 0, the eigenvalues have the order of the flow speed. When ε → 1 (M → 1) the preconditioning procedure is reduced to multiplying the residual by the identity matrix. The preconditioning matrix is represented in a symmetric form [22] P = N ΓN −1 , where N is the transition matrix from symmetrized to conservative variables, and Γ = diag{ε, 1, 1, 1, 1}. This choice of Γ is equivalent to the approach used in [16], which is formulated in symmetrized variables. Taking into account the decomposition of the inviscid Jacobian and the relations between the Jacobians in conservative, primitive and symmetrized variables, it can be obtained |Ac | = M N Γ−1 |ΓAs |N −1 M −1 = M N Γ−1 Ts−1 |Λ| TsN −1 M −1 , | {z } | {z } Tc

Tp−1

where M and N are transition matrices. The indices p, c and s denote primitive, conservative and symmetrized variables, respectively. The matrices Tc and Tp are written in conservative and primitive variables. 5.3.2.

Viscous Fluxes

When the preconditioning matrix is constructed for viscous fluxes, the mixed derivatives are ignored and the gradient at the face midpoint is represented as ∇Q =

∂Q l, ∂l

∂Q Qj − Qi = . ∂l |xj − xi |

The vector l is directed along the edge (i, j) from node i to node j.

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Relation (14) becomes RVi =

1 X ∂Q Bc ∆sij , Vi ∂l j∈Ei

where Bc is the viscous Jacobian in conservative variables. The mesh derivatives of the velocity involved in the viscous Jacobian can be computed more accurately if primitive variables are used. The viscous Jacobian in primitive variables is given by Bc = Bp M −1 . At interior mesh nodes, the preconditioning matrix for viscous fluxes is written as !
1 X 1 V −1 −1 Pi = Bp M ∆sij . Vi |xj − xi | j∈Ei

The viscous Jacobian in conservative variables has the form  0 0 0 0 0   0 µ 0 0 0    0 0 µ 0 0 Bc =    0 0 0 µ 0   γ µ cT γ µc − µvx µvy µvz γ − 1 Pr ρ γ − 1 Pr ρ where µ is the viscosity and Pr is the Prandtl number. 5.3.3.



     ,    

Boundary Conditions

The inviscid and viscous Jacobians are computed at each internal mesh node. At boundary nodes, the viscous Jacobian is not calculated, since the no-slip boundary condition is set on the wall. For the Euler equations, the inviscid Jacobian at boundary mesh nodes is modified so as to satisfy the no-penetration boundary condition. At boundary mesh nodes, the preconditioning matrix for inviscid fluxes is constructed in a local coordinate system (xn , xτ 1, xτ 2 ), where xn axis is normal to the surface, while xτ 1 and xτ 2 axes are tangential to it. Taking into account the direct and inverse rotation of the coordinate system, equation (11) becomes 

(I − T −1 ST )P −1 − T −1 ST

 dQ
= I − T −1 ST R, dt

where S is a matrix such that the normal momentum component vanishes and the nopenetration boundary condition on the wall is satisfied, and I is the identity matrix. From the point of view of programming, the boundary conditions are taken into account in two steps. Only the Jacobian components corresponding to flow velocities in x, y and z directions have to be changed.

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1. Step 1 (pre-multiply), j = 1, . . .5

where

e2j = A2j − δAnx , A

e3j = A3j − δAny , A

e4j = A4j − δAnz , A

δA = A2j nx + A3j ny + A4j nz . 2. Step 2 (post-multiply), i = 1, . . .5

where

ei2 − δ An e x, Ai2 = A

ei3 − δ An e y, Ai3 = A

e=A ei2 nx + A ei3 ny + A ei4 nz , δA

Here, δij is the Kronecker’s symbol. 5.3.4.

ei4 − δ An e z, Ai4 = A

e − δi2 nx − δi3 ny − δi4 nz . δA = δ A

Flux Limiter

To obtain a stable solution to the problem, it is necessary that ε = O(M2 ). In contrast to the approaches used in [9, 14, 15], a local limiter based on the local Mach number is used  ε = min 1, ηM2max ,

where 1 6 η 6 4 (η = 3 in the computations). The maximum Mach number Mmax is determined in a loop over all the edges of the unstructured mesh. The maximum Mach number is found for two nodes associated with an edge and is preserved for both nodes. The procedure is repeated several times (4 times) to produce flow regions with a common maximum local Mach number. This approach guarantees that speeds of convective and acoustic waves have identical orders of magnitude comparable to the flow speed. The resulting estimation is local and ensures the smooth behavior of the flux limiter. However, it is inapplicable to internal flows, in which the Mach number at the inlet in the computational domain is usually unknown. The limiter is calculated at the midpoint of the edge (i, j) as the half sum of the node values ε=

1 (εi + εj ) . 2

The values εi and εj at mesh nodes are determined using an algorithm consisting of the following steps. Step 1. For each mesh edge, find parameters depending on the local velocity and pressure distributions 1 (|vi + vj | + |v i − v j |)2 ; 4c2 ( ) 2 |pi − pj | 2 µ 1 P = 2 max , . c ρ |xi − xj | ρ

M2 =

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For M2 < P , set e2 = 1 M 2



M2 M2 +P P



.

Store the resulting values at each mesh node. Step 2. For each mesh node, calculate the average εi =

Ni 1 X e 2n , M Ni n=1

where Ni is the number of edges associated with the node i. Step 3. Perform ηm = 5 iterations for smoothing. Each iteration consists of searching through all the mesh edges. For each edge, set δi = max {εi , δ} ,

δj = max {εj , −δ} ,

where δ = εj − εi . For each node, find the smoothed flux limiter 1 δi εei = εi + δi + . 2 1 + Ni

Step 4. For each mesh node, find the flux limiter
εi = a 1.2 − 0.2a5 , a = min {1, ηe εi} , where η ∼ 12.

5.4.

Numerical Example

Consider the laminar inviscid compressible steady state flow over a NACA0012 airfoil at M∞ = 0.01 and a zero angle of attack. The outlet boundary was 10L away from the trailing edge, where L is the airfoil chord length. The computations were performed on a 320 × 64 structured O-mesh (its fragment is shown in Figure 5a) and on an unstructured mesh with 15000 nodes (its fragment is shown in Figure 5b). The Navier–Stokes equations were solved with a scalar and a block preconditioning (cases 1 and 2) and without preconditioning (case 3). Figure 6 and Figure 7 show the residual of the solution against the number of iterations for preconditioning (the results obtained on a structured mesh are displayed in Figure 5a). The governing equations were solved in dimensionless variables. As a result, the residuals produced by discretization of the continuity, momentum and energy equations are combined (in the computations, the lower bound for the residual was set to R ∼ 10−13 ). The block preconditioning (lines 1) is much more superior to calculations without preconditioning (line 2 in Figure 6) or to the scalar preconditioning (line 2 in Figure 7) in terms of the number of multigrid cycles required to achieve a given residual value and in terms of total CPU time. Specifically, the number of multigrid cycles decreases from 856 for the scalar preconditioning to 98 for the block one, and the CPU time is 2.6 times less in the latter case.

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a)

b)

airfoil NACA0012

L

Figure 5. Structured (a) and unstructured (b) meshes for simulation flow around airfoil. -2

R

-6

-10

2

1

-14

0

600

1200

1800

n

Figure 6. Convergence history with block preconditioning (line 1) and without preconditioning (line 2). Figure 5b shows the results obtained on an unstructured mesh, which are overall similar to the previous results. It can be seen that, for the scalar and block preconditioning, the convergence rate of multigrid iterations is somewhat slower. The number of multigrid cycles is about 8% and 5% higher than that on the structured mesh. In addition to the residual value, the conservation of the mass (at the inlet and outlet of

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R

2 -6

1

-10

-14

0

50

100

150

n

Figure 7. Convergence history with block preconditioning (line 1) and with scalar preconditioning (line 2). the computational domain) and variations in the integral flow parameters from iteration to iteration were kept track in the computations. In contrast to the local flow characteristics, the integral ones reach their steady state values after a considerably smaller number of iterations. Figure 8 displays the numerical results in the form of contour of Mach number obtained on the structured mesh. The solution produced without preconditioning is not presented, since its steady state value was not obtained in an acceptable time. a)

b)

Figure 8. Contours of Mach number with preconditioning. 2 Figure 10 shows the airfoil surface pressure coefficient, Cp = 2∆p/ρU∞ , computed

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b)

a)

Figure 9. Contours of Mach number without preconditioning. in the inviscid problem (symbols ◦). It can be seen that the numerical results fairly well agree with the exact solution of the incompressible problem obtained with the Schwarz– Christoffel transformation (solid line). Moreover, the results obtained on the structured mesh (symbols •) are also in good agreement with those produced on the unstructured mesh (symbols ◦). 1.2

-Cp

0.4

-0.4

-1.2

0

0.55

1

x/L

Figure 10. Comparison of exact (solid line) and numerical distributions of pressure coefficient over airfoil. Symbols • correspond to structured mesh, and symbols ◦ correspond to unstructured mesh.

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Multigrid Method

A multigrid method is proposed for solving the system of difference equations obtained with the finite volume discretization of Navier–Stokes equations on an unstructured mesh. A sequence of unstructured meshes is generated with edge collapsing algorithm taking into account the features of the problem (inviscid/viscous). The capabilities of the approach are demonstrated by computing inviscid and viscous compressible flows around an airfoil on structured, unstructured and hybrid meshes.

6.1.

Full Approximation Scheme

Consider the system of difference equations N (Q) = f,

(52)

where N is a discrete operator. An iterative procedure for system (52) is written as h i n+1 n −1 n Q = Q +J f − N (Q ) for n = 1, 2, . . .,

where J is the Jacobian, R = f − N (Qn ) is the residual, and the superscript n is the iteration number. According to the multigrid approach as applied to system (52), the computations are performed on a sequence of nested meshes h1 ⊃ . . . ⊃ hM (which generate a sequence of finite dimensional spaces Vh1 ⊃ . . . ⊃ VhM ) and employ a sequence of operator equations N k (Qk ) = f k ,

Qk ∈ Vhk ,

k = 1, . . ., M.

Coarse mesh correction is applied to the solution obtained on the fine mesh. In the absence of a consistent solution, O(hpk ) relative truncation error is obtained on the mesh of level k. A good approximation is relative truncation error on the mesh of level k. A good approximation is achieved when the vector Ql−1 is close to Ql at least up to O(hpl−1 ) accuracy. The coarse mesh solution Ql is used as an initial approximation for the fine mesh solution Ql−1 . Equation (52) is solved using the full approximation scheme (FAS) [25, 26]. In contrast to Newton linearization method with an adapted or fixed number of multigrid iterations at each iteration step, FAS avoids global linearization (since linearization is performed inside a cycle on the coarsest mesh) and the computation of high dimension Jacobians, while providing the possibility of using various smoothing algorithms. The inner and outer iterations do not need to be consistent. In a discrete form on a fine mesh bh) = f h , N h (Q

(53)

b h is the exact solution of the discrete system. Using Qh as an initial approximation where Q b h − Qh , the equation (53) is rewritten in and defining the error of the solution as E h = Q the form   N h Qh + E h = f h . (54)

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Subtracting N h (Qh ) from both sides of equation (54) gives   N h Qh + E h − N h (Qh ) = f h − N h (Qh ) = Rh (Qh ). Restricting the residual and the solution to the coarse mesh

N H (E H ) = IhH Rh (Qh );   h i N H IhH Qh + E H − N H (IhH Qh ) = IhH f h − N h (Qh ) .

h is the prolongation Here, IhH is the restriction operator from a fine to a coarse mesh, and IH h H operator from a coarse to a fine mesh, so QH = IhH Rh and Qh = IH R . The error and the residual prolonged to the fine mesh are smooth functions (in contrast to the solution). On the coarse mesh, the equation is solved h i f H = IhH f h − N h (Qh ) + N H (IhH Qh ).

The system of equations N H (QH ) = f H is solved by executing nc smoothing iterations on the coarse mesh (usually, nc ∼ 10). The correction in the transition from a fine to a coarse mesh has the form ThH = N H (IhH Qh ) − IhH N h (Qh ). The correction to the coarse mesh equation is such that the solution on the coarse mesh coincides with that on the fine mesh. The multigrid method consists of the following sequence of steps. 1. Generate µ1 approximations of the solution on the mesh h by applying the Gauss– Seidel method (pre-smoothing) h i e h = Qh + (J h )−1 f h − N h (Qh ) . Q

For notational simplicity, the iteration number superscript n is omitted. 2. Project the residual Rh = f h − N h (Qh ) ∈ Vh onto the space VH , hence RH = H h Ih R . 3. Find an approximate coarse mesh solution N H (QH ) = N H (IhH Qh ) + RH by executing γ cycles and nc iterations on the coarsest mesh for smoothing. 4. Interpolate the error E H = QH − IhH Qh to the fine mesh and correct the fine mesh solution   h e h = Q h + IH Q QH − IhH Qh .

5. Perform µ2 approximations of the solution on the fine mesh to suppress the interpolation error (post-smoothing). The number of smoothing iterations at each level (usually, µ1 = µ2 ), the number of recursive calls of the method at each level, γ, and the number of iterations on the coarsest mesh, nc , are prescribed. An iteration step of the multigrid method as applied to the system of difference equations is defined by the recursive procedure Qn+1 = M GM (k, Qn), where Qn and Qn+1 are the solutions at the iteration steps n and n + 1. One iteration step of the method gives a V-cycle for γ = 1 and a W-cycle for γ = 2.

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405

Prolongation and Restriction Operators

To ensure the convergence of classical multigrid methods, it is necessary that OP + OR > OE [26], where OP and OR are the orders of the prolongation and restriction operators, which are highest degrees plus one of polynomials that are precisely interpolated by the prolongation and restriction operators, and OE is the order of the problem in question (the order of the differential equation, which is equal to two for the Navier–Stokes equations). The transition operators are independent of the problem, the iterative method or the technique used for ordering the unknowns. The prolongation and restriction procedures are defined as follows. 1. At the prolongation step from a coarse to a fine mesh, the solution is reconstructed by linear interpolation  
h H ∆Qhi = ∆QH + x − x · ∇ ∆QH j for ∀ i ∈ Kj , (55) j i j

where ∆Q is a correction of the solution. The indices h and H denote the fine and coarse meshes, respectively. A scalar on a triangular or tetrahedral mesh K is found using the relation X K−1 , qjK = φK−1 (xK j )qi i i∈CjK→K−1

K where φK i are piecewise linear basis functions centered about the node xi ∈ K. The suK→K−1 perscript K denotes the mesh level. The set Cj includes the nodes of the tetrahedron K on the mesh K − 1 that contain the nodes xj . Near a boundary, the nodes xK j may lie beyond the mesh K − 1. This leads to negative edge weights (extrapolation procedure), which are set equal to zero. 2. On the restriction step from a fine to a coarse mesh, the volume-weighted average residual is calculated .X X h h RH = V R Vih . (56) j i i i∈Kj

i∈Kj

Volume-weighted averaging assumes that VjH =

X

Vih .

i∈Kj

Near the boundaries of the computational domain (for example, near the walls), it is possible that X VjH > Vih . i∈Kj

Condition (56) is replaced by the limited volume-weighted average .   X X h h H h RH = V R max V , V j i i j i . i∈Kj

i∈Kj

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The implementation of the multigrid approach is illustrated in the Figure 11 (V-cycle). The iterations begin with the mesh of the highest resolution (level 1). The number of mesh levels is nlevel. The arrows indicate the transmission of data from one mesh level to another. The horizontal arrows show that the computations (smoothing) occurs at a single mesh level. The CFL number is estimated by executing nstart iteration steps (one iteration step is used by default). The parameter npre specifies the number of iterations used for pre-smoothing at each mesh level (in the transition from a fine to a coarse mesh). The parameter npost defines the number of iterations used for post-smoothing (in the transition from a coarse to a fine mesh). The parameter ncrs specifies the number of iterations executed on the coarsest mesh. 1 multigrid cycle

level 4

ncrs

level 3

npre

level 2 level 1

npost npost

npre nstart

npre

restriction

npost prolongation

initial approximation

residual output

Figure 11. Structure of multigrid cycle (V-cycle).

6.3.

Mesh Levels

Choosing and constructing a sequence of nested meshes is an important component of the multigrid approach that determines the quality of the computational procedure. The number of nested meshes is a parameter of the problem. At most 4–5 nested meshes are usually used in practice. A further increase in the number of mesh levels has a negligible effect on the convergence rate. The nested meshes do no share nodes or edges of control volumes. There exists a oneto-one mapping between the node and edge indices on a coarse mesh and those of the highest resolution mesh. To construct sub-meshes, one has to know only the number of nodes and the fine mesh topology. Therefore, the construction of nested meshes is independent of the problem under study and is performed by a standard subroutine. A sequence of nested meshes is constructed using edge collapsing. Two nodes i and j in an edge are replaced by a single one located in the middle between them (Figure 12a). Each edge is associated with its length, which is multiplied by the edge growth factor (for example, by 2) in the transition to the next mesh level. In a boundary layer, the cell collapses in the direction of the shortest edge (Figure 12b) unless the edge length is smaller a threshold value. This preserves the mesh topology near the wall. The resulting edges are checked against the length constraint. A list of cells is constructed, the maximum possible

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angle between two edges is fixed (for example, 135 degrees), and a geometric criterion is verified (over all the cells involving the collapsing edge). a)

b) i

j

edge collapsing

Figure 12. Collapse of a cell in the direction of the shortest edge on unstructured mesh (a) and structured mesh (b). The algorithm is recursive and rather easy to implement. A list of all the mesh cells is compiled and sorted according to their size. If a mesh cell collapses, it is eliminated from the list. The algorithm terminates, when the list of cells becomes empty. The consecutive steps in the transformation of a hexahedral mesh cell into a tetrahedron are shown in Figure 13.

Figure 13. Transformation of a hexagonal cell into a tetrahedron. The algorithm for constructing nested meshes is implemented as the following recursive sequence of steps. 1. Construct the list T of all cells associated with nodes i and j of a given edge. 2. Construct the list G of all boundary edges that occur once in T . 3. Construct the set S of new cells connecting the node xn with each edge in G (connection occurs if the resulting control volumes are positive). The coordinates of a new node depend on the locations of the edge nodes. For an internal edge, the new node is placed at the middle point of the edge. For a boundary edge, the new node is placed on the wall to preserve the shape of the wall when the mesh is coarsened. The coordinates of the node xn

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are determined according to the rule  1 (x + x ),  j   2 i xn = xi ,    xj ,

if di = dj , if di > dj , if di < dj ,

where dk is the index of the boundary surface touching the node k. 4. If all the volumes in S are positive, then the set T is replaced by S.

6.4.

Numerical Example

Consider the turbulent compressible flow over an RAE2822 airfoil at ReL = 6.5 × 106 under various freestream conditions, M∞ = 0.725 and α = 2.4◦ (version 1, AGARD Test Case 06) and M∞ = 0.73 and α = 2.8◦ (version 2, AGARD Test Case 09). Turbulence was simulated using the Spalart–Allmaras model with a fixed transition at x/L = 0.03. Freestream conditions are set at the inflow boundary, and free outflow conditions are specified at the outflow boundaries. The computations were performed on an unstructured triangular mesh consisting of 11298 nodes (versions 1a and 2a) and on a hybrid mesh consisting of 19126 nodes (versions 1b and 2b). A structured mesh was used near the airfoil. The mesh sizes satisfied the conditions (∆y/∆x)min = 6.8 × 10−4 , (∆y/∆x)max = 30, (∆y/L) = 3.6 × 10−6 . On the airfoil surface, y + ∼ 1.8 ÷ 2.2. In the computations, four mesh levels were used, and the mesh generation technique was based on edge collapsing in the direction of the shortest edge (semi-coarsening method), which makes it possible to capture the boundary layer on the airfoil. Figure 14 and Figure 15 show the meshes of levels 1, 2 and 3 generated via collapsing edges. The number of mesh nodes at each level for versions 1 and 2 is presented in Table 1. In the case of a hybrid mesh, the number of cells was 24339 in the mesh of level 1 (10692 triangles and 13647 quadrilaterals), 11484 in the mesh of level 2 (5527 triangles and 5957 quadrilaterals), 5528 in the mesh of level 3 (2795 triangles and 2733 quadrilaterals), and 2894 in the mesh of level 4 (1599 triangles and 1295 quadrilaterals). For the meshes of levels 1–4, the domain ∆y/L < 5 × 10−6 contained 50, 25, 13 and 6 cells, respectively. Table 1. A number of nodes of various meshes Variant 1a, 2a 1b, 2b

Level 1 11298 24339

Level 2 3246 11484

Level 3 1129 5528

Level 4 325 2894

The aspect ratio for mesh levels 2–4 was 1:2 away from the airfoil, while near the airfoil, it was somewhat smaller than 1:2, since some of the quadrilaterals turned into triangles when a sequence of nested meshes was generated. The maximum aspect ratio was 1 for the mesh of level 1 (1 for both triangles and quadrilaterals), 0.47 for the mesh of level 2 (0.54 for triangles and 0.44 for quadrilaterals), 0.47 for the mesh of level 3 (0.51 for triangles and 0.45 for quadrilaterals), and 0.52 for the mesh of level 4 (0.57 for triangles and 0.48 for quadrilaterals).

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a)

b)

409

c)

Figure 14. Sequence of meshes around the RAE2822 airfoil for version 1.

a)

b)

c)

Figure 15. Sequence of meshes around the RAE2822 airfoil for version 2. Figure 16 shows the solution of the problem in the form of the pressure coefficient distribution over the airfoil surface obtained on a hybrid mesh for versions 1b and 2b. The integration of the pressure distribution and wall shear stresses over the airfoil surface gives

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Cx = 0.8388 and Cy = 0.0197 for version 1b, and Cx = 0.7757 and Cy = 0.0142 for version 2b. The solutions for versions 1a and 2a are not shown, since the results obtained on unstructured and hybrid meshes differ little. 1.5

-Cp a)

1.5

0.5

0.5

-0.5

-0.5

-1.5

0

0.5

1

-1.5

-Cp b)

0

0.5

1

x/L

x/L

Figure 16. Comparison of the computed pressure coefficient distribution over the airfoil surface (solid line) with the data of [35] (symbols ◦) for version 1 (a) and version 2 (b). Table 2 demonstrates the convergence of the iterative process on the initial segment of the residual (from 100 to 10−4 ) and in its entire range (from 100 to 10−8 ) (the residual norm was normalized by its initial value). The numerator and the denominator correspond to scalar preconditioning and block-Jacobi preconditioning, respectively. The number of iterations for pre-smoothing and post-smoothing was set equal to 1. Five smoothing iterations were executed on the coarsest mesh. Table 2. Convergence of iteration process Variant 1a 2a 1b 2b

Convergence 100 Number of cycles 311/122 565/171 234/78 297/170

→ 10−4 Time, s 839/341 1510/472 1563/534 1982/1149

Convergence 100 → 10−8 Number of cycles Time, s 947/240 2522/649 1233/361 3293/1052 1788/918 11956/6152 2655/1697 17645/11377

The convergence history for version 1 on unstructured and hybrid meshes is shown in Figure 17. Line 1 and line 3 correspond to the residual obtained by discretizing the continuity, momentum and energy equations, while line 2 and line 4 correspond to the residual obtained by discretizing the turbulence model equation (the computations were performed in dimensionless variables). In all cases, the residual converges to the prescribed level (R ∼ 10−10 ). The block preconditioning ensures faster convergence than scalar preconditioning on both unstructured and hybrid meshes. For versions 1a and 1b, the acceleration factor is 2.46 and 2.93 for the initial residual segment, and 3.89 and 1.94 for the entire range of the residual. Figure 18 shows the convergence factor of the multigrid method as a function of the

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R

0

a)

411

R b)

1 -3

-3

1

2 -6

-9 0

4

200

2 -6

4

3

3

400

n

600

800

-9 0

1000

200

400

n

600

800

1000

Figure 17. Residual history on unstructured (a) and hybrid (b) meshes with scalar preconditioning (lines 1, 2) and block preconditioning (lines 3, 4). number of mesh levels for version 1a. As the number of mesh levels rises from 1 to 4, the number of multigrid cycles decreases from 800 to 220. For n = 1, the residual reaches an approximately constant level R ∼ 10−5 after 1000 multigrid cycles (a further increase in the number of iterations does not reduce the residual), while for n = 4, the residual monotonically decreases to R ∼ 10−8 beyond a small initial segment. 0

R

-2 1 -4

-6

-8 0

2

200

400

600

800

n

Figure 18. Residual history on one mesh (line 1) and four nested meshes (line 2). Figure 19 depicts contours of Mach number near the airfoil obtained for version 2 (on a hybrid mesh). As the number of multigrid cycles increases from 100 (fragment a) to 400 (fragment b), the numerical solution improves, which corresponds to the residual variation shown in Figure 18 (line 2).

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b)

Figure 19. Contours of Mach number near the airfoil after 100 (a) and 400 (b) multigrid cycles. 2 ), over the Figure 20 displays the distribution of friction coefficient, Cf = 2τw /(ρ∞ V∞ airfoil surface for version 2 on a hybrid mesh. It can be seen that the results agree fairly well with the measurements from [35] (symbols ).

0.012

Cf

0.008

0.004

0

-0.004

0

0.25

0.5

0.75

1

x/L

Figure 20. Distribution of the skin friction coefficient over the airfoil surface (solid line) as compared with the measurements of [35] (symbols ).

7.

Wall Boundary Conditions

A method is proposed for implemention of weak wall boundary conditions for a finite volume discretization of RANS equations on unstructured meshes. The effect of the

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near-wall mesh resolution on the accuracy of CFD computations is analyzed, and the mesh dependence of the solution is compared for wall function and weak boundary conditions.

7.1.

Discretization of Governing Equations

The turbulence model equations written in the form of (3) or (4), (5) are discretized in the same manner as the Navier–Stokes equations. The source terms in the turbulence model equations are discretized so as to guarantee that the unknowns are bounded according to their physical interpretation. The diffusion term in equation (3) has a nonconservative form [45]. The term (∇e ν )2 containing the gradient is represented as ∇e ν n+1 ∇e ν n . To represent the gradient term in an implicit form in numerical implementation, the identity is used ∇e ν · ∇e ν = ∇ · (e ν ∇e ν ) − νe ∇ · (∇e ν) .

Assuming that the modified turbulent viscosity at the control volume center is a constant (e νp = const), gives h i (∇e ν )2 = ∇ (e ν − νep ) ∇e ν . The diffusion term in equation (3) becomes nh i o ∇ µ + ρe ν + cb2 ρp (e ν − νep ) ∇e ν .

Because of the term cb2 ρp(e ν − νep ), the total viscosity may become negative. Consider the turbulent diffusion coefficient on the control volume edge (i, j) Γij = ρij νeij + cb2 ρp (e νi + νej ) .

Setting νeij = (e νi + νej )/2 yields 1 1 Γij = (ρij + cb2 ρp ) νei + (ρij − cb2 ρp) νej . 2 2 For a constant density, the transfer coefficient is guaranteed to be positive if |cb2 | < 1 (cb2 = 0.622 is used by default). For a variable density, the total diffusion coefficient is positive. For the numerical procedure to be stable, the source term in equation (3) is linearized. Although the production and dissipation terms in equation (3) are separated, each of them is sign changing, so the total source term is subject to linearization. The relations used to discretize inviscid and viscous fluxes and the source term in equation (3) are as follows. 1. Inviscid fluxes ( Z   X 1 (ρv · ∇) νe dS = ρ(vi · nij ) νej + νei − 2 | {z } j∈Ei ∂Vi

inviscid flux

"

#)     ∗ ∗ b (e b νi ) +ϕ ρ νej − νei − |vi · nij | (1 − ϕ) ρ L ∆sij . i νj ) − Lj (e | {z } | {z } 4th order 2nd order | {z } dissipative term

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K. Volkov 2. Viscous fluxes Z

∂Vi

i o 1n h ∇ (µ + ρe ν ) ∇e ν + cb2 ρ (∇e ν )2 dS = σ

" # X 1 1 (µ + ρe ν ) + cb2 ρ (e νj − νei ) ∇e = ν ∆sij . σ 2 j∈Ei

3. Source term Z

ρS dΩ = ρ

Vi

X

j∈Ei

"

#    νe 2 c b1 Se νe − cw1 fw − 2 ft2 Vi . κ d

b ∗ is a modified pseudo-Laplacian [3]. The paramHere, ∆sij is the area of face (i, j) and L eter ϕ determines the relative contribution of the second- and fourth-order terms. The gradients of the transported variable in the Spalart–Allmaras model are usually lower than those of the sought functions in the k–ε model. As a result, the Spalart–Allmaras model is less sensitive to numerical errors on unstructured meshes in the boundary layer.

7.2.

Near-Wall Flows

The wall functions represent a number of semi-empirical formulas that relate the desired functions in a near-wall control volume to the corresponding values on the wall. 7.2.1.

Structure of Boundary Layer

A turbulent boundary layer is usually divided into several characteristic sub-domains (Figure 21) by using the distance from the wall, y + = yuτ /ν, the flow velocity, u+ = u/uτ , and the temperature, T + = (Tw − T )/Tτ , which are expressed in near-wall units, where uτ = (τw /ρ)1/2 is the friction velocity, Tτ = qw /(ρcpuτ ) is the dynamic temperature, τw is the shear stress on the wall, and Tw is the wall temperature. The turbulence characteristics are also represented in dimensionless form, k+ = k/u2τ and ε+ = εν/u4τ . In the viscous sub-layer, the viscous stresses dominate the Reynolds ones and the velocity is a linear function of the distance to the wall, u+ = y + for 0 6 y + < 11. In the logarithmic layer, the Reynolds stresses are much higher than the viscous ones, and the velocity profile is described by the logarithmic law, u+ = (1/κ) ln Ey + for 11 6 y + < 0.2δ, where δ is the thickness of the boundary layer. For a smooth wall, E = 8.8. Here, κ is the von Karman constant (κ = 0.42). In the buffer layer, the viscous and Reynolds stresses are of the same order. By matching the velocity profiles in the viscous and logarithmic sub-layers, it is easy to obtain u+ = A ln y + + B for 5 < y + < 30, where A = 4.94 and B = −2.96. It is usually assumed that the law of the wall holds for 30 < y + < 200, and the node adjacent to the wall lies within this interval. The implementation of the wall functions depends on the turbulence model used and requires an iterative procedure for finding shear stresses on the wall.

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u+

u+ =y +

inner region

u+ =2.5 log y + +5.45

outer region y/δ~0.2

viscous sub-layer log-layer buffer layer y+ ~250 y+ ~60 y +~5

log y+

Figure 21. Structure of turbulent boundary layer. 7.2.2.

Wall Functions

The shear stress distribution is assumed to be uniform within near-wall control volume. The shear stresses on the wall are calculated using the formula τw = µe ∆uS/∆y, where ∆y is the distance from the control volume center to the wall, ∆u is the difference between the tangential velocities at the near-wall node and on the wall, S is the area of a face, and µe is the effective viscosity. Since µe = µRe/u+2 , where Re = ρ∆u∆y/µ = u+ y + , therefore τw = ρ∆u2 /u+2 . The velocity is computed using the logarithmic law of the wall (Spalding’s formulation)  

1 1 + + + + + 2 + 3 y = u + exp κu − 1 − κu − (κu ) − (κu ) × 2 6 × exp (−κB) ,

(58)

where B = 5.3. The nonlinear equation (58) is solved by applying Newton’s iteration method. The form of the underlying relation depends on the Reynolds number. For Re 6 140, the velocity distribution is described by the relation  

1 1 0 = u+ + exp κu+ − 1 − κu+ − (κu+ )2 − (κu+ )3 × 2 6 × exp (−κB) −

Re . u+

(59)

The iteration begins with u+ = Re1/2 (the linear relation u+ = y + holds in the laminar sub-layer).

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For Re > 140, the velocity is found from the distribution obtained by taking the natural logarithm of (58), hence ( 
1 1 1 + + + 2 + 3 0 = u − B − ln 1 + κu + (κu ) + (κu ) × κ 2 6 ) Re (60) × exp (−κB) + + − u+ . u The nonlinear equation (60) is solved using Newton’s method with u+ = B + (1/κ) ln Re used as an initial approximation. With equation (60), the convergence of Newton’s method (60) is faster than in the case of (58). The temperature profile near the wall is described by the relation h+ + exp [−κ(B + P )] × Prt "  +  +  2  3 # κh κh 1 κh+ 1 κh+ × exp −1− − − , Prt Prt 2 Prt 6 Prt

y+ =

(61)

where 

Pr P = 9.24 −1 Prt



Pr Prt

−1/4

.

The heat flux is found using the relations ∆h qw = ρuτ + , h

qw =



µ µt + Pr Prt



∆h . ∆y

Equating these expressions gives a formula for computing the turbulent viscosity   Re 1 µt = Prt − µ. u+ h+ Pr Instead of (61), an equation solved by Newton’s method is used h+ + exp [−κ(B + P )] × Prt "  +  +  2  3 # κh κh 1 κh+ 1 κh+ Re × exp −1− − − − +. Prt Prt 2 Prt 6 Prt u

0=

The velocity is expressed from the logarithmic distribution.

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417

y b) a)

2

∆y ∆y/2

1

0 u 0 1

wall

u2

u

Figure 22. Distribution of the skin friction coefficient over the airfoil surface (solid line) as compared with the measurements of [35] (symbols ). 7.2.3.

Weak Boundary Conditions

Weak boundary conditions are formulated using the wall tangential velocity [4], which is added to the residual caused by discretization of the inviscid fluxes through the faces of a near-wall control volume (Figure 22). This requires relatively small modifications of the code [3]. The tangential velocities on the wall are determined by the relations uτ 1 = uτ 2 − τ x

∆y , µe

vτ 1 = vτ 2 − τy

∆y , µe

w τ 1 = w τ 2 − τz

∆y , µe

where τx = τw

uτ 2 − uτ 1 , ∆q

τy = τw

vτ 2 − vτ 1 , ∆q

τz = τw

wτ 2 − wτ 1 . ∆q

The tangential velocities {uτ , vτ , wτ } in local coordinates are related to the Cartesian velocities {u, v, w} as uτ = u − vn nx , vτ = v − vn ny , wτ = w − vn nz , where vn = unx + vny + wnz is the normal velocity on a control volume face. For wall functions, uτ 1 = vτ 1 = wτ 1 = 0. In a modified approach based on averaging over a near-wall control volume, the logarithmic velocity profile is prolonged to the wall (the viscous sub-layer is not resolved, as in the wall functions). The tangential velocity on the wall is derived by averaging over the logarithmic velocity distribution in the near-wall control volume (the shadowed region in the Figure 22). The tangential velocities on the wall are determined by the relations u∗τ 1 = ub

uτ 2 − uτ 1 , ∆q

vτ∗1 = ub

vτ 2 − vτ 1 , ∆q

wτ∗ 1 = ub

wτ 2 − wτ 1 . ∆q

The resulting values are used to compute the residual caused by discretization of inviscid fluxes. The boundary layer flow velocity ub = µu+ b /ρ∆y expressed in near-wall units is

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computed using the wall law (59) and the mean value theorem + + u+ b =u y −



4 1 +2 1 + 1 u + u − y+ + κu+ , 2 κ 24κE

(63)

where y + = Re/u+ . Relation (63) is derived by integrating distribution (59) over a nearwall control volume with the subsequent use of relation (58) to simplify the result. The nonlinear equation for determining the velocity u+ is solved by applying Newton’s method.

7.3.

Numerical Examples

The solution of benchmark CFD problems is discussed, and the mesh dependence of the solution obtained with the wall functions and weak boundary conditions is analyzed. 7.3.1.

Turbulent Boundary Layer on a Flat Plate

The length and width of the plate were 100 m and 20 mm, respectively. For the boundary layer flow on a flat plate, there are reliable experimental and numerical data [46,47] (the results were obtained for an incompressible fluid at M < 0.2). In the inlet section, the total pressure, total temperature and the turbulence characteristics were specified as p0∞ = 6.67 × 105 Pa, T0∞ = 300 K, k∞ = 2 m2 /s2 , ε∞ = 200 m2 /s3 , which corresponded to U∞ = 200 m/s and M∞ = 0.5). In the outlet section, the static pressure was held fixed (p = 5.56 × 105 Pa). The surface of the plate was assumed to be adiabatic. The no-penetration flow conditions were set on the upper boundary. Periodic boundary conditions were used in streamwise direction. The computations were performed on a 35 × 32 mesh with a mesh refined toward the leading edge and the surface of the plate. Along the plate, y + ∼ 8 (except for the leading edge), which was comparable with the mesh used in [47]. The convergence of the iterative process is demonstrated in Figure 23. The solid and dashed lines depict the residuals for flow characteristics and turbulence characteristics, respectively. The residual reaches a prescribed value (R ∼ 10−16 ) after 2309 iterations with wall functions and 2138 iterations with weak boundary conditions. The velocity profile in the cross section x = 0.05 m is shown in Figure 24a in comparison with the data from [46] (symbols •). The solid line corresponds to the k–ε model with wall functions, and the symbols  show the velocity profile obtained with weak boundary conditions. The results agree well with the data from [46], except for an area lying far away from the wall (for y + > 100). 2 , over the plate is shown in The distribution of the friction coefficient, Cf = τw /ρU∞ Figure 24b. As compared with the data from [46], the computations give lower coefficient values near the leading edge of the plate and higher values away of it. 7.3.2.

TurbulentBoundary Layer with a Pressure Gradient

The computational domain was specified as a curvilinear channel (Figure 25, lengths are given in mm). The lower wall was the plate surface, while the upper wall consisted of line segments and circular arcs joined so as to reproduce the pressure gradient created in the experimental setup [48]. The last segment (for x > 120 mm) was made narrowing

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R

R

-6

a)

b)

-8

-8

-10

-10

-12

-12

-14

-14

-16

-16

-18

-18

-20

0

500

1000

n

1500

2000

2500

-20

419

0

500

1000

n

1500

2000

2500

Figure 23. Convergence history with wall functions (a) and weak boundary conditions (b). 25

u+

0.006

a)

Cf b)

20 0.004 15 10 0.002 5 1 100

101

y+

102

103

0 0

0.025

0.05

x, m

0.075

0.1

Figure 24. Velocity profile in the boundary layer (a) and friction coefficient distribution (b). to guarantee the convergence of the numerical solution. For given parameters, the flow in the outlet section was supersonic, so the flow features downstream of it had no effect on the upstream parameters. Experimental and numerical data on boundary layer flows with favorable and unfavorable pressure gradients were obtained in [48–50]. 19.00

60.00

12.10

21.77

6.00

10.22

167.22

Figure 25. Geometry of computational domain.

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In the inlet section, the total pressure (p0∞ = 3.66 × 105 Pa), total temperature (T0∞ = 380 K), turbulent viscosity (e ν∞ = 10−3 m2 /s) or turbulence kinetic energy and 2 2 its dissipation rate (k∞ = 2 m /s , ε∞ = 200 m2 /s3 ) were specified. The static pressure (p = 1.013 × 105 Pa) was set in the outlet cross section. The upper wall was assumed to be adiabatic (∂T /∂n = 0). The plate had a constant temperature (Tw = 300 K). Several meshes with different y + values were used in the computations. For 40, 60 and 80 nodes across the channel, the meshes contained 2800, 4200 and 7200 nodes, respectively. The coordinate y + varied from 62 to 74 in mesh 1, from 12 to 38 in mesh 2, and from 4 to 20 in mesh 3. The Nusselt number was computed as Nux = qw x/(λ∞ ∆T ), where qw is the heat flux and ∆T = T∞ − Tw . Figure 26 and Figure 27 show the Nusselt number distributions for wall functions and weak boundary conditions, respectively. Lines 1–3 correspond to meshes with 40, 60, and 80 nodes in the transverse direction. The physical experimental data from [48] are marked with symbols •. 80

Nu . 10 3 a)

70

80

Nu . 10 3 b)

70

1 60

60

1

2 2

50

50

3

40

40

30

30

20

20

10

10

0 20

40

60

80

x, mm

100

0 20

3

40

60

80

100

x, mm

Figure 26. Distributions of local Nusselt number over the plate for k–ε model (a) and Spalart–Allmaras model (b) with wall functions. With an increasing distance from the leading edge, the Nusselt number first decreases (with a minimum occurring at x ∼ 38 mm) and then again reaches a maximum (Figure 26). The position of the local minimum is well predicted by line 3 in Figure 26a. However, the position of the local maximum is inaccurate and worse than that produced on coarser meshes, which is explained by the low y + values near the wall. Thus, the accuracy of the results based on the Spalart–Allmaras and k–ε models with wall functions is unsatisfactory. The results obtained with weak boundary conditions (Figure 27) agree better with experimental data, and the influence of the near-wall mesh resolution is alleviated. However, in this case, the results also exhibit a rather significant scatter. 7.3.3.

Flow in a Compressor Channel

The computational domain is shown in Figure28. It is bounded by the inlet and outlet channel cross sections, the profile surface, the channel walls, and periodic boundaries (the flow is aligned with x axis).

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Nu . 10 3 a)

70

80

421

Nu . 10 3 b)

70

1 60 50

60

1

2 50

2 3

40

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30

30

20

20

10

10

0 20

40

60

x, mm

80

100

0 20

3

40

60

80

100

x, mm

Figure 27. Distributions of local Nusselt number over the plate for k–ε model (a) and Spalart–Allmaras model (b) with weak boundary conditions.

Figure 28. Geometry of computational domain. In the inlet section, radial profiles of the total pressure and temperature and the flow direction angles, as well as turbulence characteristics (e ν∞ = 1.76 × 10−4 m2 /s or k∞ = −4 2 2 −3 2 3 10 m /s , ε∞ = 10 m /s depending on the turbulence model used) were specified. The static pressure distribution on the outflow boundary was held fixed. The no-slip and no-penetration boundary conditions were set on the profile surface, while slip conditions (inviscid walls) were specified on the channel walls. The profile surface was assumed to be adiabatic. Periodic boundary conditions were used in the circumferential direction. The computations were performed on two meshes with an O-type domain near the profile with nearly the same number of cells, but with different mesh sizes near the wall. Meshes 1 and 2 contained 16302 and 15518 cells, respectively, with 7872 and 7480 cells lying on the boundary. On the inlet and outlet boundaries, there were 64 and 44 cells. Each of the periodic boundaries contained 127 cells. The channel walls contained 7872 cells in mesh 1 and 7480 cells in mesh 2. In both cases, 196 cells were placed on the profile surface. The value of y + varied from 0.2 to 2 for mesh 1 and from 0.8 to 24 for mesh 2. Mesh 1 had a roughly uniform distribution of y + values on the pressure and suction surfaces of the

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K. Volkov -8

R a)

-10

R

-8

b)

-10 2

-12 4

-14

1, 2

-14

1

3

-16

3, 4

-12

-16 1, 2

1 -18

-18 4

2

3, 4

3

-20 -22 0

500

-20

1000

1500

n

-22

0

500

1000

1500

n

Figure 29. Convergence history with Spalart–Allmaras model (a) and k–ε model (b). profile (moreover, y + ∼ 1, which is lower than the bound usually used in computations based on wall functions). Convergence (R ∼ 10−16 ) was achieved after 5000 multigrid cycles (V-cycles and four level mesh were used). The convergence of the numerical solution is demonstrated in Figure 29. Lines 1 and 2 depict the numerical results obtained on mesh 1, and curves 3 and 4 show the numerical results obtained on mesh 2 with wall function and weak boundary conditions. The solid lines depict the residual produced by discretization of the continuity equation, momentum equations and energy equation (residual R1 ), while the dashed lines show the residual caused by discretization of the turbulence model equations (residual R2 ). Numerical results obtained on meshes 1 and 2 are summarized in Table 3, where N is a number of multigrid cycles. The Spalart–Allmaras model fails to achieve the prescribed residual value after 5000 iterations on mesh 1 for both wall functions and weak boundary conditions. For k–ε model, convergence on mesh 1 is achieved after 2059 and 2073 multigrid cycles, respectively. The fastest convergence takes place when the k–ε model is used on mesh 2. Specifically, the number of multigrid cycles is 881 for wall functions and 888 for weak boundary conditions. The Spalart–Allmaras model requires a somewhat larger number of multigrid cycles, 961 and 1030. Table 3. Residuals and number of iterations R1 −15.13

SA-model R2 −18.50

N 5000

R1 −16.01

k–ε model R2 −20.52

N 2059

−12.29

−18.25

5000

−16.02

−20.53

2073

−16.00

−19.84

961

−16.01

−20.64

881

−16.00

−19.75

1030

−16.00

−20.64

888

Mesh Mesh 1 + wall functions Mesh 1 + weak boundary conditions Mesh 2 + wall functions Mesh 2 + weak boundary conditions

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In Table 4, the total pressure losses, L, and the mass flow rate, m ˙ (in kg/s), in the outlet section of the channel are given against the number of multigrid cycles for various turbulence models and various formulations of boundary conditions on the profile. Note the difference between the outlet total pressure losses based on the Spalart–Allmaras and k–ε models with wall functions. In Table 4, the total pressure losses, L, and the mass flow file surface are shown in Figure 30 and Figure 31 (the notation is the same as in Figure 29). It can be seen that the near-wall mesh size and the method for boundary condition formulation have a large effect on the numerical results near the trailing edge of the profile. Wall functions and weak boundary conditions give roughly identical results on mesh 1 for both Spalart–Allmaras and k–ε models. In the case of mesh 2 (having a larger near-wall mesh size) and wall functions, the static pressure near the trailing edge of the profile has a loop-shaped distribution. The k–ε model also produces a nonmonotone pressure distribution on the upper profile surface (arched pressure distribution depicted by line 3 in Figure 31). In the case of mesh 2 and weak boundary conditions, this effect is not completely eliminated but reduced so that a coarser mesh can be used near the wall. Table 4. Total pressure losses and mass flow rate in the outlet section Mesh Mesh 1 + wall functions Mesh 1 + weak boundary conditions Mesh 2 + wall functions Mesh 2 + weak boundary conditions

SA-model L 0.0471 0.8342

k–ε model m ˙ L 0.0971 0.7854

0.0472

0.8312

0.0972

0.7854

0.0571

0.8130

0.0396

0.8305

0.0627

0.8123

0.0432

0.8315

m ˙

The streamwise velocity, vx , the circumferential velocity, vθ , and the flow rotation angle, β = arctg(vθ /vx ), in the outlet section of the channel are given in Table 5 (the velocities are in m/s, and the angles are in degrees). The computations predict similar values of local flow characteristics on mesh 1 irrespective of the near-wall simulation method. In the case of weak boundary conditions, the computed values of the velocity and angle on mesh 2 are somewhat smaller than those based on wall functions (the difference is 0.25%). The most important is the near-wall mesh resolution, which gives a 0.2% difference in the streamwise velocity, a 1.5% difference in the circumferential velocity, and a 2.1% difference in the flow direction angle on meshes 1 and 2 with wall functions. The weak boundary conditions on meshes 1 and 2 lead to a difference of 0.2%, 1.3%, and 1.8%, respectively.

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K. Volkov 1.065

p .10 5, Pa a)

1.044

p .10 5, Pa b)

1, 2

1.04

4

1.05 1, 2

1.036 4

1.035

3 3

1.02

0

0.2

0.4

s

0.6

0.8

1.032

1.028 0.9

1

0.92

0.94

0.96

s

0.98

1

1.02

Figure 30. Pressure distributions over the profile surface (a) and near its trailing edge (b) for the Spalart–Allmaras model. 1.065

p .10 5, Pa

1.044

a)

p .10 5, Pa b)

1, 2

1.04

4

1.05

1.036 4 1, 2

1.035

3 3

1.02

0

0.2

0.4

s

0.6

0.8

1.032

1.028 0.9

1

0.92

0.94

0.96

0.98

1

1.02

s

Figure 31. Pressure distributions over the profile surface (a) and near its trailing edge (b) for the k–ε model. Table 5. Velocities and flow rotation angle in the outlet section Mesh Mesh 1 + wall functions Mesh 1 + weak boundary conditions Mesh 2 + wall functions Mesh 2 + weak boundary conditions

vx 34.81

SA-model vθ β 16.18 24.94

vx 34.53

k–ε model vθ β 18.29 27.91

34.80

16.18

24.94

34.53

18.29

27.91

35.03

17.89

27.05

35.16

17.11

25.94

34.81

17.55

26.75

34.61

16.58

25.60

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The near-wall mesh resolution near the trailing edge of the channel, the method used for matching the O-type structured mesh in the boundary layer with the mesh used in the outer region, and the method for boundary condition formulation have a large effect on the quality of the numerical results (surface pressure distribution near the trailing edge, structure of the near-wake flow, total pressure distribution in the outlet section).

Conclusion An approach to the discretization of the Navier–Stokes equations on a hybrid mesh was developed within the framework of the finite volume method and high-resolution finite difference schemes in 2D and 3D. The approach makes use of the median control volume centered at a mesh node. The gradient and the pseudo-Laplacian are calculated using modified relations adapted to the computations on a strongly stretched mesh in the boundary layer. The gradient at mesh nodes is computed using Green’s identity, and a technique is proposed that ensures the conservation property of the difference scheme in 2D. A block preconditioning of the Euler or Navier–Stokes equations for simulation of inviscid or viscous low-Mach flows on structured and unstructured meshes was developed. The use of the preconditioning changes only the form of the dissipative term in the formula for inviscid fluxes, which simplifies the modifying of CFD code. In contrast to structured meshes, the preconditioning matrix on unstructured mesh is different for secondand fourth-order schemes. Although the block preconditioning leads to roughly a 25% increase in CPU time at every iteration (the preconditioning matrix is calculated once at a time integration step), its use ensures the stability of the numerical solution, speeds up the convergence of the iterative process, and reduces the total CPU time. A multigrid method was developed for solving the Euler and Navier–Stokes equations on unstructured and hybrid meshes. The method relies on a modified form of the restriction operator and involves the generation of a sequence of meshes with edge collapsing algorithm (semi-coarsening method). The method for generating meshes of different levels accurately takes into account the features of the problem (the boundary layer on the wall), preserves the topology of the original mesh, and produces high quality meshes in the near-wall region (reasonable stretching and obliqueness of the cells). The capabilities of the approach are demonstrated by computing inviscid and viscous compressible flows around an airfoil. The multigrid method in conjunction with block-Jacobi preconditioning produces a prescribed level of the residual after considerably fewer iteration steps than in the case of scalar preconditioning. Methods were proposed for formulation of weak boundary conditions on the wall for finite volume discretization of RANS equations. The mesh dependence of the solution was studied in the case of wall functions and weak boundary conditions. It was shown that the near-wall mesh size has a large effect on the numerical results. Weak boundary conditions have a number of advantages over wall functions. The basic shortcoming of weak boundary conditions is that they are of a numerical nature and lack physical justification. It should be noted that wall functions are also frequently used in situations where the law of the wall is not applicable (for example, in boundary layer computation with a pressure gradient or in the case where y + values are below or above recommended values).

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Acknowledgments This work was financially supported by the Ministry of Education and Science of the Russian Federation (agreement No 14.578.21.0111, a unique identifier of applied scientific research RFMEFI57815X0111).

Nomenclature Latin symbols c Specific heat capacity e Specific total energy f Function k Turbulent kinetic energy m Mass n Number of iterations p Pressure q Heat flux r Radius t Time vx , vy , vz Velocity components v Velocity vector x, y, z Cartesian coordinates C Constant Cf Friction coefficient Cp Pressure coefficient F Flux H Source term L Total pressure losses M Mach number Pr Prandtl number Q Conservative variables R Residual Re Reynolds number T Temperature U Symmetrized variables V Primitive variables W Characteristic variables Greek symbols γ Ratio of specific heat capacities ε Dissipation rate λ Thermal conductivity µ Dynamic viscosity ν Kinematic viscosity ρ Density

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Numerical Analysis of Navier–Stokes Equations on Unstructured Meshes ω

Rotation speed

Subscripts b e i, j, k n t w τ 0 ∗ ∞

Boundary Effective Tensor indices Normal Turbulent Wall Tangential Total Reference Free stream

427

Superscripts n Time layer + Near wall value Abbreviations 1D One-dimensional 2D Two-dimensional 3D Three-dimensional CDS Central Difference Scheme CFD Computational Fluid Dynamics CFL Courant–Friedrichs–Lewy MG Multigrid MUSCL Monotonic Upwind Scheme for Conservative Laws RANS Reynolds-Averaged Navier–Stokes RK Runge–Kutta SA Spalart–Allmaras

A.

Discretization and Linearization

A.1.

Primitive and Conservative Variables

Consider the vectors of primitive and conservative variables     ρ ρ  vx   ρvx         V =  vy  , Q =   ρvy  ,  vz   ρvz  p ρe

where ρ is the density, vx , vy , vz are the velocity components in the x, y, z directions, p is the pressure, e is the total energy per unit mass.

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K. Volkov The equation of state has the form   1 2 p = (γ − 1) ρ e − q , 2

q 2 = vx2 + vy2 + vz2 − ω 2 r 2 ,

where γ is the ratio of specific heat capacities, q is the velocity magnitude, ω is the rotation speed, r is the radius. The transfer matrices from conservative to primitive variables and back are 0

B B ∂Q =B M= B ∂V @ 0

M

A.2.

−1

B B ∂V = =B B ∂Q @

1 −vx /ρ −vy /ρ −vz /ρ (γ − 1)q2 /2

1, vx vy vz q2 /2

0 ρ 0 0 ρvx

0 0 ρ 0 ρvy

0 1/ρ 0 0 −vx (γ − 1)

0 0 0 ρ ρvz

0 0 0 0 1/(γ − 1)

0 0 1/ρ 0 −vy (γ − 1)

1

C C C, C A

0 0 0 1/ρ −vz (γ − 1)

0 0 0 0 γ −1

1

C C C. C A

Linearization of Equations in Conservative Variables

Linearization of the Navier–Stokes equations written in conservative variables in the form of (6) gives ∂Q + A∇Q = ∇ (D∇Q) . ∂t

(64)

Projecting equation (64) onto the Cartesian axes yields „ « ∂Q ∂Q ∂Q ∂Q ∂Q ∂Q ∂Q ∂ + Ax + Ay + Az = + Dxy + Dxz Dxx + ∂t ∂x ∂y ∂z ∂x ∂x ∂y ∂z „ « „ « ∂Q ∂Q ∂Q ∂Q ∂Q ∂Q ∂ ∂ Dyx + Dyy + Dyz + Dxx + Dxy + Dxz . + ∂y ∂x ∂y ∂z ∂z ∂x ∂y ∂z

(65)

The inviscid Jacobian is given by the relation A=

∂FyI ∂FxI ∂FzI nx + ny + nz = Ax nx + Ay ny + Az nz . ∂Q ∂Q ∂Q

The matrices Ax , Ay and Az have the form 0

B B B Ax = B B @

ˆ

0 1−γ 2 2 2 q − vx −vx vy −vx vz

˜ −γe + (γ − 1)q 2 vx

1

0

0

0

(3 − γ)vx vy vz” “ γ−1 γe − 2 q 2 − (γ − 1)vx2

(1 − γ)v vx 0

(1 − γ)w 0 vx

γ−1 0 0

(1 − γ)vx vy

(1 − γ)vx vz

γvx

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1

C C C C, C A

Numerical Analysis of Navier–Stokes Equations on Unstructured Meshes 0

B B B Ay = B B @ ˆ 0

B B B Az = B B @

0 −vx vy 1−γ 2 2 2 q − vy −vy vz

˜ −γe + (γ − 1)q 2 vy

0 −vx vz −vy vz 1−γ 2 2 2 q − vz ˜ ˆ −γe + (γ − 1)q 2 vz

429

0 vy

1 vx

0 0

0 0

(1 − γ)vx 0

(3 − γ)vy vz” “ γ−1 γe − 2 q 2 − (γ − 1)vy2

(1 − γ)vz vy

γ−1 0

(1 − γ)vy vz

γvy

(1 − γ)vx vy 0 vz 0

0 0 vz

1 vx vy

0 0 0

(1 − γ)vx

(1 − γ)vy

(1 − γ)vx vz

(1 − γ)vy vz

(3 − γ)vz “ ” γ−1 γe − 2 q 2 − (γ − 1)vz2

1

C C C C, C A 1

C C C C. γ−1 C A γvz

The Jacobian is represented in the form A = R |Λ| L, where Λ is a diagonal matrix composed of the eigenvalues of the Jacobian, and R and L are the matrices made up of the right and left eigenvectors of matrix A (hence, L = R−1 ). Solving the characteristic equation det(A − λI) = 0 yields the eigenvalues of the Jacobian. The matrix Λ has the form   vn − c 0 0 0 0  0 vn + c 0 0 0     Λ= 0 0 vn 0 0  .  0 0 0 vn 0  0 0 0 0 vn The right eigenvectors form the columns of the matrix R = {R1 , R2 , R3 , R4 , R5 }, where     1 1  vx + cnx   vx − cnx        vy + cny  , , R = R1 =  v − cn y  2   y   vz + cnz   vz − cnz  h0 + cvn h0 − cvn 

  R3 =   

nx vx nx vy nx + cnz vz nx − cny q 2 nx/2 + c(vy nz − vz ny )



  R5 =   



  ,  



  R4 =   

ny vx ny − cnz vy ny vz ny + cnx q 2 ny /2 + c(vz nx − vx nz )

nz vx nz + cny vy nz − cnx vz nz q 2 nz /2 + c(vxny − vy nx )



  .  

The left eigenvectors form the rows of L = {L1 , L2 , L3 , L4 , L5 }0 , where 

 1 L1 =  2 

(γ − 1)q 2 /2 + cvn − [(γ − 1)vx + cnx ] − [(γ − 1)vy + cny ] − [(γ − 1)vz + cnz ] γ −1



  ,  



 1 L2 =  2 

(γ − 1)q 2 /2 − cvn − [(γ − 1)vx − cnx ] − [(γ − 1)vy − cny ] − [(γ − 1)vz − cnz ] γ−1

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

  ,  



  ,  

430

K. Volkov  

  L3 =   

 c2 − (γ − 1)q 2 /2 nx + c(vz ny − vy nz ) (γ − 1)vxnx (γ − 1)vy nx + cnz (γ − 1)vz nx − cny −(γ − 1)nx

 

  L4 =   

 

  L5 =   



  ,  

 c2 − (γ − 1)q 2 /2 ny + c(vx nz − vz nx ) (γ − 1)vxny − cnz (γ − 1)vy ny (γ − 1)vz ny + cnx −(γ − 1)ny



 c2 − (γ − 1)q 2 /2 nz + c(vy nx − vx ny ) (γ − 1)vxnz + cny (γ − 1)vy nz − cnx (γ − 1)vz nz −(γ − 1)nz



  ,     .  

Here, c and h are the speed of sound and enthalpy, which are related to the temperature, c2 = γRT and h = cpT , where cp is the specific heat capacity at constant pressure.

A.3.

Linearization of Equations in Primitive Variables

Transition to primitive variables in equation (64) yields M

∂V + AM ∇V = ∇ (DM ∇V ) . ∂t

Multiplying this equation by M −1 on the left gives h i
∂V + M −1 AM ∇Q = ∇ (M −1 DM )∇V . ∂t

(66)

Projection of the equation (66) onto the Cartesian axes gives

  ∂V ∂ x ∂V y ∂V z ∂V xx ∂Q xy ∂Q xz ∂Q +A +A +A = D +D +D + ∂t ∂x ∂y ∂z ∂x ∂x ∂y ∂z     ∂ ∂Q ∂Q ∂Q ∂ ∂Q ∂Q ∂Q + Dyx + Dyy + Dyz + Dxx + Dxy + Dxz . ∂y ∂x ∂y ∂z ∂z ∂x ∂y ∂z

The inviscid Jacobian is given by A=

∂FyI ∂FzI ∂FxI Sx + Sy + Sz = Ax Sx + Ay Sy + Az Sz . ∂V ∂V ∂V

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(67)

Numerical Analysis of Navier–Stokes Equations on Unstructured Meshes

431

The Jacobian in primitive variables is related in conservative variables by Ax = M −1 Ax M,

Ay = M −1 Ay M,

Az = M −1 Az M,

where 

  Ax =   



  A =   y



  A =   z

The matrix A has the form    A=  

 vx ρ 0 0 0 0 vx 0 0 1/ρ   0 0 vx 0 0  , 0 0 0 vx 0  2 0 ρc 0 0 vx vy 0 ρ 0 0 vy 0 0 0 0 vy 0 0 0 0 vy 0 0 ρc2 0 vz 0 0 0 0

0 0 vz 0 0 vz 0 0 0 0

0 0 1/ρ 0 vy

ρ 0 0 0 0 0 vz 1/ρ ρc2 vz



  ,  



  .  

 vn ρSx ρSy ρSz 0 0 vn 0 0 Sx /ρ   0 0 vn 0 Sy /ρ  . 0 0 0 vn Sz /ρ  0 ρc2 Sx ρc2 Sy ρc2 Sz vn

Here, vn = vx Sx +vy Sy +vz Sz . The surface areas of the corresponding faces of the control volume are given by Sx = Snx ,

Sy = Sny ,

Sz = Snz ,

S 2 = Sx2 + Sy2 + Sz2 ,

where nx , ny , nz are the projections of the outward unit normal to a control volume face onto the Cartesian axes. The Jacobian is represented as A = R |Λ| L, where Λ is a diagonal matrix composed of eigenvalues, and R and L are the matrices made up of the right and left eigenvectors of matrix A (hence, L = R−1 ). Solving the characteristic equation det(A − λI) = 0, the eigenvalues of the Jacobian are found. The matrix Λ has the form   vn 0 0 0 0  0 vn 0  0 0    . Λ =  0 0 vn 0 0   0 0 0 vn + c  0 0 0 0 0 vn − c

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K. Volkov

The right eigenvectors form the columns of the matrix   nx ny nz ρ ρ  0 −cnz cny cnx −cnx     R =  cnz 0 −cnx cny −cny  .  −cny cnx 0 cnz −cnz  0 0 0 ρc2 ρc2

The left eigenvectors form the rows of the matrix

L=

1 2c2

0 B B B B @

2c2 nx 2c2 ny 2c2 nz 0 0

0 −2cnz 2cny cnx −cnx

2cnz 0 −2cnx cny −cny

−2cny 2cnx 0 cnz −cnz

−2nx −2ny −2nz 1/ρ 1/ρ

1

C C C. C A

Moreover, it holds that 0

|Λ|L−1

0 0 1 B B = 2 B 2c2 nx |vn | 2c @ 2c2 n |v | y n 2c2 nz |vn |

−ρcnx |vn − c| ρcnx |vn + c| 0 −2ρcnz |vn | 2ρcny |vn |

−ρcny |vn − c| ρcny |vn + c| 2ρcnz |vn | 0 −2ρcnx |vn |

−ρcnz |vn − c| ρcnz |vn + c| −2ρcny |vn | 2ρcnx |vn | 0

|vn − c| |vn + c| −2nx |vn | −2ny |vn | −2nz |vn |

For viscous fluxes, matrices are 0

D

D

xx

xy

D yy

D

B B B =B B B @

0 0 0 γµp − Prρ2

0 2µ + λ ρ 0 0 0



  = (D ) =    yx 0



   =   

xz

0

0 0 0 0 γµp − Prρ2

0 µ ρ 0 0 0 

  = (D ) =    zx 0

0

0

0

0 µ/ρ 0 0

0 0 µ/ρ 0

0 0 0 γµ Prρ

0 0 0 0 0 0 0 λ/ρ 0 0 0 µ/ρ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2µ + λ ρ 0 0

1

C C C C, C C A



  ,  

0 0

0 0

0 µ/ρ 0

0 0 γµ Prρ



   ,   

 0 0 0 0 0 0 0 0 λ/ρ 0   0 0 0 0 0  , 0 µ/ρ 0 0 0  0 0 0 0 0

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1

C C C. A

Numerical Analysis of Navier–Stokes Equations on Unstructured Meshes   0 0 0 0 0  0 µ/ρ 0 0 0      0 0 µ/ρ 0 0 zz , D =   2µ + λ   0 0 0 0 ρ  γµp γµ  − 0 0 0 Prρ Prρ2

D

A.4.

yz



  = (D ) =    zy 0

0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 λ/ρ 0 0 µ/ρ 0 0 0 0 0 0

433



  .  

Inviscid Fluxes

The relations required for discretization of the equation (66) on unstructured mesh are provided. Introduction of the notation δV = Vi − Vj leads to the matrix   (c2 δρ − δp)nx + cnz δvy − cny δvz  (c2 δρ − δp)n − cn δv + cn δv  y z x x z    2 1  (c δρ − δp)n + cn δv − cn z y x x δvy  . L δV = 2    1 (δp + ρcδv ) c   n 2ρ   1 (δp − ρcδv ) n 2ρ

Here, δvn = δvx nx + δvy ny + δvz nz . The convective term in (66) is determined by the relation (A · ∇) V = (R |Λ| L) δV , which gives 0

− ρcδvn )|vn − c| (c2 δρ − δp)|vn| + 1 (δp + ρcδvn )|vn + c| + 1 2 (δp B c2 (δv − n δv )|v | + 2cnx (δp + ρcδv )|v + c| − cnx (δp − ρcδv )|v − c| x x n n B n n n n 2ρ 2ρ 1 B B c2 (δvy − ny δvn )|vn | + cny (δp + ρcδvn )|vn + c| − cny (δp − ρcδvn )|vn − c| (R |Λ| L) δV = 2 B 2ρ 2ρ c B B c2 (δvz − nz δvn )|vn | + cnz (δp + ρcδvn )|vn + c| − cnz (δp − ρcδvn )|vn − c| 2ρ 2ρ @ c2 (δp + ρcδv )|v + c| c2 (δp − ρcδv )|v − c| n n n n 2 2

1

C C C C C. C C A

Introducing the notation a1 =

|vn − c| (δp − ρcδvn ) , 2

a2 =

gives

1 (R |Λ| L) δV = 2 c

      

|vn + c| (δp + ρcδvn ) , 2


a3 = −|vn | δp − c2 δρ ,

a1 + a2 + a3 cnx (a − a ) + c2 (δv − n δv )|v | 2 1 x x n n ρ cny 2 (a − a ) + c (δv − n δv )|v 2 1 y y n n| ρ cnz (a − a ) + c2 (δv − n δv )|v | 2 1 z z n n ρ (a1 + a2 )c2



   =  

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K. Volkov

1 = 2 c

     

a123 (d12 nx + eδvxt )/ρ (d12 ny + eδvyt )/ρ (d12 nz + eδvzt )/ρ (a1 + a2 )c2



  .  

The relation for computing the vector of inviscid fluxes becomes 

  FI =   

a123 a123 vx + d12 nx + e3 δvxt a123vy + d12 ny + e3 δvyt a123 vz + d12nz + e3 δvzt 2 a123q /2 + d12 vn + e3 (vx δvxt + vy δvyt + vz δvzt ) + (a1 + a2 )c2 /(γ − 1)



  .  

Here, a123 = a1 + a2 + a3 , d12 = (a2 − a1 )c, e = ρc2 |vn |. The increments of the velocity are calculated by the formulas δvxt = δvx −nx δvn , δvyt = δvy −ny δvn , δvzt = δvz −nz δvn .

B.

Preconditioning Matrices

B.1.

Transition Matrices

The vectors of primitive, conservative and symmetrized variables are      ρ ρ dp/ρc  vx   ρvx   dvx       , Q =  ρvy  , dU =  v dvy V = y       vz   ρvz   dvz p ρe dp − c2 dρ



  ,  

where ρ is the density, vx , vy , vz are the velocity components in the x, y, z directions, p is the pressure, e is the total energy of a unit of mass, c is the speed of sound. The equation of state is   1 2 p = (γ − 1) ρ e − q . 2 The velocity magnitude is calculated by the formula q 2 = vx2 + vy2 + vz2 − ω 2 r 2 . Here, γ is the ratio of specific heat capacities, ω is the rotation speed, r is the radius. The transition matrix from conservative to primitive variables is   1 0 0 0 0  vx  ρ 0 0 0  ∂Q   . M= =  vy 0 ρ 0 0  ∂V  vz  0 0 ρ 0 2 q /2 ρvx ρvy ρvz 1/(γ − 1)

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Numerical Analysis of Navier–Stokes Equations on Unstructured Meshes The transition matrix from symmetrized to conservative variables is  γ 1/2 ρc 0 0 0 0   1/2 ρvx  γ ρ 0 0 0  c   1/2 ρvy ∂U  0 ρ 0 0 c N= = γ ∂Q   1/2 ρvz  γ 0 0 ρ 0 c    1/2  γ p ρv ρv ρv γ 1/2 ρe x y z c γ−1 c The transition matrix from symmetrized to primitive variables is 

      ∂U L= =  ∂V     

γ 1/2 ρc

0 0 0

0

0

1 0 0

0

0

0 1 0

0

0

0 0 1

0

ρc γ 1/2

0 0 0



γ γ−1



       .      



1/2

ρc

      .      

The inviscid Jacobian in symmetrized variables is given by As =

∂FyI ∂FzI ∂FxI nx + ny + nz = Ax nx + Ay ny + Az nz . ∂U ∂U ∂U

The matrices Ax , Ay and Az have the form 

       Ax =       

c γ 1/2

vx c

0

0

0 

γ−1 γ

1/2

γ 1/2

vx

0

0

0

0

vx

0

0

0

0

vx

0

0

0

vx

0



γ−1 γ

1/2

c

0



   c     ,      

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436

K. Volkov 

       Ay =       

vy

0

c γ 1/2

0

0

vy

0

0

c γ 1/2

0

vy

0

0

0

0

vy

0

0



vz

0

0

0

vz

0

0

0

vz

c γ 1/2

0

0

0

0

0



       Az =       

B.2.

γ−1 γ

1/2

c

0



0

     1/2   γ−1 c  , γ    0    vy 0

c γ 1/2

0

0

0



      0 0  .  1/2   γ−1  vz c  γ    1/2 γ−1 c vz γ

Eigenvectors of Jacobian

The structure of the matrix T in conservative variables and the matrix T −1 in primitive variables is provided in the regular case (without preconditioning) and with preconditioning. 1. Regular case 

1

  1   Tc =  nx    ny nz

vx − cnx

vy − cny

vz − cnz

vx + cnx

vy + cny

vz + cnz

vx nx

vy nx + cnz

vz nx − cny

vx ny − cnz

vy ny

vz ny + cnx

vx nz + cny

vy nz − cnx

vz nz



Tp−1

0

  0 1   = 2  c2 n x c   2  c ny c2 n z

ρcnx /2

ρcny /2

h − cvn



    1 2 v n + c(v n − v n ) , x y z z y 2 n   1 2 2 vn ny + c(vz nx − vx nz )  h + cvn

1 2 v n 2 n z

+ c(vx ny − vy nx )

ρcnz /2

−ρcnx /2 −ρcny /2 −ρcnz /2 0

ρcnz

−ρcny

−ρcnz /2

0

ρcnx

ρcny

−ρcnx

0

1/2



 1/2    −nx  .   −ny  −nz

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2. Preconditioning case 0

s+ 2cε

B B s− B B 2cε B Tc = B nx B B B ny @ nz

vx s+ −2c2 nx ε 2cε

vy s+ −2c2 ny ε 2cε

vz s+ −2c2 nz ε 2cε

vx s− +2c2 nx ε 2cε

vy s− +2c2 ny ε 2cε

vz s− +2c2 nz ε 2cε

vx nx

vy nx + cnz

vz nx − cny

vx ny − cnz

vy ny

vz ny − cnx

vx nz + cny

vy nz − cnx

vz nz



Tp−1

0

   0 1  = 2 c2 n x c    2  c ny 2

c nz

hs+ −2c2 qε 2cε

1

C C C C C 1 2 C, v n + c(v n − v n ) x y z z y n C 2 C 1 2 C v n + c(v n − v n ) y z x x z n A 2 hs− +2c2 qε 2cε

1 2 v n 2 n z

−ρs− cnx 2τ

−ρs− cny 2τ

−ρs− cnz 2τ

ρs+ cnx 2τ

ρs+ cny 2τ

ρs+ cnz 2τ

0

ρcnz

−ρcny

−ρcnz

0

ρcnx

ρcny

−ρcnx

0

+ c(vx ny − vy nx )

c 4τ



     −nx  .   −ny  c 4τ

−nz

Here, s+ = τ + (1 − ε)vn , s− = τ − (1 − ε)vn .

B.3.

Inviscid Fluxes

The variables are calculated at the edge midpoint a = (ai + aj )/2. The velocity at the boundary of the control volume is given by the formula vn = vx Sx + vy Sy + vz Sz . The surface areas of the corresponding control volume faces are Sx = Snx ,

Sy = Sny ,

Sz = Snz ,

S 2 = Sx2 + Sy2 + Sz2 ,

where nx , ny , nz are the projections of the outward unit normal to a control volume face onto the Cartesian axes. The increments of the primitive variables are given by the relation δ1 a = aj − aj . To achieve the second-order of accuracy, the relation is used δ2 a = s1 δ1 a − s2 [Lj a − Li (a)] , where Li (a) is the pseudo-Laplacian of at the node i. The flux limiters are specified as    |Lj (p)| |Li (p)| s1 = min 1, ε2 + , s2 = ε1 (1 − s1 ), |Li (p) + 2pi | |Lj (p) + 2pj | where ε1 = 1 and ε2 = 8. At wall nodes, s1 = 1.

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K. Volkov Table 6. Regular and preconditioning cases Parameter λ1 λ2 λ3 a1 a2 a3 a123 b123 d12 e

Regular case vn − c vn + c vn 1 2 |λ1 |(δp − ρcδvn) 1 2 |λ2 |(δp + ρcδvn) −|λ3 |(δp − c2 δρ) a1 + a2 + a3 (a1 + a2 )h + 12 q 2 a3 (a2 − a1 )c ρc2 |λ3 |

Preconditioning 1 1 2 (1 + ε)vn − 2 τ 1 1 2 (1 + ε)vn + 2 τ vn |λ1 |(δp − 12 s− ρδvn ) |λ2 |(δp + 12 s+ ρδvn ) −|λ3 |(δp − c2 δρ) s+ s− 2τ ε a1 + 2τ ε a2 + a3 1 1 2 + − 2τ ε (s a1 + s a2 )h + 2 q a3 a2 − a1 ρc2 |λ3 |

The vector of inviscid fluxes is computed as follows (the smoothing term affected by preconditioning is provided)   a123  a123 vx + d12 nx + eδvxt   1 1  a123 vy + d12 ny + eδvyt  |A|(Qj − Qi ) = − 2   . 2 2c  a123 vz + d12 nz + eδvzt  b123 + d12 vn + eqδq The increments of the velocity components are calculated by the formulas δvxt = δvx − δvn nx , δvyt = δvy − δvn ny , δvzt = δvz − δvn nz , qδq = vx δvxt + vy δvyt + vz δvzt . The relations for the other characteristics and coefficients are given in Table 6.

References [1] Hirsch C. Numerical computation of internal and external flows. New York, John Wiley & Sons, 1990. [2] Barth T.J. Aspects of unstructured grids and finite-volume solvers for the Euler and Navier–Stokes equations. VKI Lecture Series 1994-05. Belgium, Von Karman Institute for Fluid Dyanmics, 1994. [3] Volkov K.N. Unstructured-grid finite volume discretization of the Navier–Stokes equations based on high-resolution difference schemes. Computational Mathematics and Mathematical Physics, 2008, 48(7), pp. 1181–1202. [4] Volkov K.N. Formulation of wall boundary conditions in turbulent flow computations on unstructured meshes. Computational Mathematics and Mathematical Physics, 2014, 54(2), pp. 353–367.

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[5] Jameson A., Mavripils D. Finite volume solution of the two-dimensional Euler equations on a regular triangular mesh. AIAA Paper, 1985, 85-0435. [6] Morgan K., Perire J., Peiro J., Hassan O. The computation of three dimensional flows using unstructured grids. Computational Methods in Applied Mechanics and Engineering, 1991, 87, pp. 335–352. [7] Roe P.L. Approximate Riemann solvers, parameter vectors, and difference schemes. Journal of Computational Physics, 1981, 43, pp. 357–372. [8] Luo H., Baum J.D., Lohner R. Egde-based finite element scheme for the Euler equations. AIAA Journal, 1994, 32(6), pp. 1183–1190. [9] Crumpton P.I., Moinier P., Giles M.B. An unstructured algorithm for high Reynolds number flows on highly stretched grids. Proceedings of the 10th International Conference on Numerical Methods for Laminar and Turbulent Flows, 21–25 July 1997, University of Wales, Swansea, United Kingdom. [10] Jameson A. Transonic aerofoil calculations using the Euler equations. Proceedings of the IMA Conference on Numerical Methods in Aeronautical Fluid Dynamics, March 1981, Reading, United Kingdom. Academic Press, 1982, pp. 289–308. [11] Crumpton P.I. A cell vertex method for 3d Navier–Stokes solutions. Technical Report of the Oxford University Computing Laboratory. Oxford, University of Oxford, 1993, NA-93/09. [12] Moinier P., Giles M.B. Stability analysis of preconditioned approximations of the Euler equations on unstructured meshes. Journal of Computational Physics, 2002, 178, pp. 498–519. [13] Moinier P., Giles M.B. Compressible Navier-Stokes equations for low Mach number applications. Proceedings of the ECCOMAS Computational Fluid Dynamics Conference, 4–7 September 2001, Swansea, United Kingdom. [14] Pierce N.A., Giles M.B. Preconditioning compressible flow calculations on stretched meshes. AIAA Paper, 1996, 96-0889. [15] Pierce N.A., Giles M.B. Preconditioned multigrid method for compressible flow calculations on stretched meshes. Journal of Computational Physics, 1997, 136(2), pp. 425–445. [16] Weiss J., Smith W. Preconditioning applied to variable and constant density flows. AIAA Journal, 1995, 33(11), pp. 2050–2062. [17] Allmaras S. Analysis of semi-implicit preconditioners for multigrid solution of the 2D compressible Navier–Stokes equations. AIAA Paper, 1995, 95-1651.

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[18] Mulder W.A. A new multigrid approach to convection problems. Journal of Computational Physics, 1989, 83(2), pp. 303–323. [19] Mulder W.A. A high-resolution Euler solver based on multigrid, semi-coarsening and deflect correction. Journal of Computational Physics, 1992, 100(1), pp. 91–104. [20] Darmofal D.L., Schmid P.J. The importance of eigenvectors for local preconditioners of the Euler equations. Journal of Computationoal Physics, 1996, 127(2), pp. 728– 756. [21] Turkel E., Vatsa V., Radespiel R. Preconditioning methods for low-speed flows. AIAA Paper, 1996, 96-2640. [22] Turkel E. Preconditioning-squared methods for multidimensional aerodynamics. AIAA Paper, 1997, 97-2025. [23] Fedorenko R.P. A relaxation method for solving elliptic difference equations. Computational Mathematics and Mathematical Physics, 1961, 1(4), pp. 922–927. [24] Bakhvalov N.S. On the convergence of a relaxation method with natural constraints on the elliptic operator. Computational Mathematics and Mathematical Physics, 1966, 6(2), pp. 101–113. [25] Brandt A. Multi-level adaptive solutions to boundary value problems. Mathematics of Computation, 1977, 31(138), pp. 46–50. [26] Hackbusch W. Multigrid method and application. Berlin, Springer Verlag, 1985. [27] Jameson A., Schmidt W., Turkel E. Numerical solutions of the Euler equations by finite volume methods using Runge–Kutta time-stepping schemes. AIAA Paper, 1981, 81-1259. [28] Peraire J., Peiro J., Morgan K. Finite element multigrid solution of Euler flows past installed aero-engines. Computational Mechanics, 1993, 11(5–6), pp. 433–451. [29] Mavriplis D.J. Multigrid strategies for viscous flow solvers on anisotropic unstructured meshes. Journal of Computational Physics, 1998, 145(1), pp. 141–165. [30] Moinier P., M˝uller J-D., Giles M.B. Edge-based multigrid and preconditioning for hybrid grids. AIAA Journal, 2002, 40(10), pp. 1954–1960. [31] Crumpton P.I., Moinier P., Giles M.B. An unstructured algorithm for high Reynolds number flows on highly stretched grids. Numerical Methods in Laminar and Turbulent Flows. Pineridge Press, 1997, pp. 561–572. [32] M˝uller J.-D., Giles M.B. Edge-based multigrid schemes for hybrid grids. Numerical Methods for Fluid Dynamics, 1998, 6, pp. 425–432. [33] Moinier P., Giles M.B. Preconditioned Euler and Navier–Stokes calculations on unstructured grids. Proceedings of the 6th ICFD Conference on Numerical Methods for Fluid Dynamics, 31 March — 3 April 1998, Oxford, United Kingdom.

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[34] Pierce N.A., Giles M.B., Jameson A., Martinelli L. Accelerating three-dimensional Navier–Stokes calculations. AIAA Paper, 1997, 97-1953. [35] Cook P.H., McDonald M.A., Firmin G.N. Aerofil RAE2822 — pressure distribution and boundary layer and wake measurements. AGARD Advisory Reports, 1979, AR138. [36] Spalart P.R., Allmaras S.R. A one equation turbulence model for aerodynamic flows. AIAA Paper, 1992, 92-0439. [37] Launder B.E., Spalding D.B. The numerical computation of turbulent flows. Computer Methods in Applied Mechanics and Engineering, 1974, 3(2), pp. 269–289. [38] Volkov K.N. Simulation of turbulent flows in rotating disc cavity systems. Turbulence: Theory, Types and Simulation. USA, Nova Science, 2011, pp. 569–686. [39] Bredberg J. On the wall boundary condition for turbulence model. Report of Chalmers University of Technology, 2000, 00/4, 56p. [40] Collis S.S. Discontinuous Galerkin methods for turbulence simulation. Stanford University, Center for Turbulence Research. Technical Report, 2002, 12 p. [41] Kato M., Launder B.E. The modelling of turbulent flow around stationary and vibrating square cylinders. Proceedings of the 9th Symposium on Turbulent Shear Flows, 16–18 August 1993, Kyoto, Japan. 1993, 9, pp. 10.4.1–10.4.6. [42] Leschziner M.A., Rodi W. Calculation of annular and twin parallel jets using various discretization schemes and turbulent-model variations. Journal of Fluids Engineering, 1981, 103, pp. 353–360. [43] Volkov K.N. Multigrid techniques as applied to gas dynamic simulation on unstructured meshes. Computational Mathematics and Mathematical Physics, 2010, 50(11), pp. 1837–1850. [44] Volkov K.N. Preconditioning of the Euler and Navier–Stokes equations in lowvelocity flow simulation on unstructured grids. Computational Mathematics and Mathematical Physics, 2009, 49(10), pp. 1789–1803. [45] Deck S., Duveau P., d’Espiney P., Guillen P. Development and application of Spalart– Allmaras one-equation turbulence model to three-dimensional supersonic complex configurations. Aerospace Science and Technology, 2002, 6(3), pp. 171–183. [46] Wieghardt K., Tillman W. On the turbulent friction layer for rising pressure. NACA Report, 1951, TM-1314, 18 p. [47] Yoder D.A., Georgiadis N.J. Implementation and validation of the Chien k–ε turbulence model in the WIND Navier–Stokes code. AIAA Paper, 1999, 99-0745.

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[48] Teekaram A.J.H., Forth C.J.P., Jones T.V. Film cooling in the presence of mainstream pressure gradients. Journal of Turbomachinery, 1991, 113, pp. 484–492. [49] Volkov K.N. The effect of pressure gradient and localized injection on turbulent heat transfer on a flat plate. High Temperature, 2006, 44(3), pp. 414–421. [50] Volkov K.N., Hills N.J., Chew J.W. Simulation of turbulent flows in turbine blade passages and disc cavities. ASME Paper, 2008, GT2008-50672.

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In: Handbook on Navier-Stokes Equations Editor: Denise Campos

ISBN: 978-1-53610-292-5 c 2017 Nova Science Publishers, Inc.

Chapter 21

I NTEGRALS OF M OTION OF AN I NCOMPRESSIBLE M EDIUM F LOW. F ROM C LASSIC TO M ODERN Alexander V. Koptev∗ Admiral Makarov State University of Maritime and Inland Shipping, Saint-Petersburg, Russia

Abstract On the paper under consideration we present derivation of general integral for motion of an incompressible medium flow. The proposed procedure of constructing the integral is based on known statements in the theory of differential equations and equally true for both the case of Navier - Stokes equations and for the case of Euler ones. Known integrals of Lagrange — Cauchy, Bernoulli and Euler — Bernoulli are special cases of constructed new integral.

Keywords: motion, incompressible medium, partial differential equations, root integral, Navier–Stokes equations, Euler equations

1.

Introduction

Model of incompressible medium is one of the simplest and the most demanded in the study of various mechanical problems. In the framework of this model density and all other physical characteristics of the medium are constant and the components of the velocity vector and pressure are considered as key variables. The Navier — Stokes equations are generally accepted to describe media motion of that type [1-3]. On dimensionless variables they can be represented as

∗

∂u ∂u ∂u ∂u ∂(p + Φ) 1 +u +v +w =− + ∆u, ∂t ∂x ∂y ∂z ∂x Re

(1)

∂v ∂v ∂v ∂v ∂(p + Φ) 1 +u +v +w =− + ∆v, ∂t ∂x ∂y ∂z ∂y Re

(2)

E-mail address: [email protected]

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Alexander V. Koptev ∂w ∂w ∂w ∂w ∂(p + Φ) 1 +u +v +w =− + ∆w, ∂t ∂x ∂y ∂z ∂z Re

(3)

∂u ∂v ∂w + + = 0. ∂x ∂y ∂z

(4)

On equations (1-4) u, v, w, p denotes the velocity components and pressure; ∆ is the 3D Laplace operator in the spartial coordinates ∆=

∂2 ∂2 ∂2 + + ; ∂x2 ∂y 2 ∂z 2

Φ is the potential of external force given initially; Re denote the Reynolds number. It was given as non-negative parameter calculated by the formula U0 L . ν In the last formula U0 and L denotes velocity and length scales respectively and ν is the coefficient of kinematic viscosity of the medium. Euler equations is a particular case of the Navier — Stokes ones when viscosity equal zero. In this case the kinematic viscosity vanishes, which formally corresponds to infinity 1 value of Reynolds number. It follows that the value of Re is zero and on the right-hand side 1 of equations (1-3) group of members proportion to Re fall out of consideration. Equations (1-3) are converted to the form of Re =

∂u ∂u ∂u ∂(p + Φ) ∂u +u +v +w =− , ∂t ∂x ∂y ∂z ∂x

(5)

∂v ∂v ∂v ∂v ∂(p + Φ) +u +v +w =− , ∂t ∂x ∂y ∂z ∂y

(6)

∂w ∂w ∂w ∂w ∂(p + Φ) +u +v +w =− . ∂t ∂x ∂y ∂z ∂z

(7)

Equations (5-7) together with the continuity one (4) describes the motion of an ideal (or perfect) incompressible medium flow. These equations were derived by Leonhard Euler and bear the name of its creator. The Navier — Stokes are of great practical importance since they allow to define the basic parameters of viscous laminar medium flow and explore the special effects associated with friction. These equations have numerous applications to practical problems. Along with the traditional fields of application in recent years there has been a tendency to use the Navier — Stokes equations in new fields such as oceanography, meteorology, tribology and cardiology. The Navier — Stokes equations and the Euler ones are of interest from a purely mathematical point of view. Since they combine both of linear and nonlinear properties. Their complex study is one of the areas of modern mathematical physics [4-5]. Today, however, many issues not fully clarified and requires a deeper study. One of the main problems is the

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Integrals of Motion of an Incompressible Medium Flow

445

lack of a constructive method for solution. As what way to build the decision of 3D variant of equations in the general case while full preserving nonlinear member? This question still remains unanswered but it urgently requires for permission.

2.

Classic Integrals

An important step towards solving the equations is the construction of integrals. The provisions of classical fluid mechanics in this regard are as follows. There are three accepted integrals convenient for practical application [1-3]. For potential unsteady fluid flow occurs the Lagrange — Cauchy integral. For an incompressible medium on dimensionless variables it can be represented as U 2 ∂ϕ + = f (t), (8) 2 ∂t √ where U is the velocity module, U = u2 + v 2 + w 2 ; ϕ is the velocity potential; f (t) is an arbitrary function of time. The Lagrange — Cauchy integral is just for the potential motion as the ideal and viscous medium. Although sometimes expressed doubt that the real potential motion of viscous fluid do exist, but nevertheless formally relationship (8) is an integral of the Navier — Stokes equations for the case under consideration. In the case of steady-state motion of an ideal fluid along the stream line holds the Bernoulli’s integral. In the case of an incompressible medium on dimensionless variables it can be written as U2 p+Φ+ = Csl , (9) 2 where Csl denotes a constant generally depending on the choice of stream line. The subscript ” sl ” comes from English ” stream line ” and is underlines this pattern. In some cases constant Csl on the right-hand side does not depend on the choice of stream line. This is the case if additional condition p+Φ+

− − → → U × Ω = 0,

(10)

→ − → − where U is the velocity vector, Ω is the swirl vector, ” × ” is the sign of the vector product. In this case integral is often called as the Euler — Bernoulli integral. It can be set by equality U2 p+Φ+ = C, (11) 2 where C denotes an absolute constant not depending on the choice of the stream line. The above mentioned integrals without any exaggeration be called one of the basic relationships of classical fluid mechanics. A lot of diverse tasks were decided with their help. But even for these well known integrals not all questions clarified through. For example it is not clear how the Bernoulli integral (9) transforms if the characteristic points is not to take along the stream line, but as any different way. The ratio of (9) will already be broken and it is not clear what will happen instead. It is also not clear what will be instead

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Alexander V. Koptev

of (8) if we abandon condition of potentiality of motion. Assumptions and range of applicability for the integrals (8), (9), (11) are different. But 2 each of them contains the same combination as p + Φ + U2 which is usually called as trinomial of Bernoulli. The question naturally arises about the existence of a general root integral which also would contain the Bernoulli’s trinomial and would cover the integrals (8), (9), (11), as special cases. Construction of such an integral would be a step forward in comparison with the provisions of classical fluid mechanics.

3.

Derivation of the Root Integral

This section provides the analitical derivation of general root integral for incompressible medium flow. The procedure is based on the representation of the equations (1-4) in the divergence form as ∂Pi ∂Qi ∂Ri ∂Si + + + = 0, (12) ∂x ∂y ∂z ∂t where Pi , Qi , Ri, Si are some combinations of unknowns u, v, w, p and first derivatives on spartial coordinates of u, v, w . The following assertion is true. Each of four equations (1 - 4) can be represented in the form of (12). Indeed, for equation (4) is obvious since it is already represented in the form (12). For this equation (i = 4) we have P4 = u, Q4 = v, R4 = w, S4 = 0 and equality (12) is performed. Each of three equations (1 - 3) can also be represented as (12). To verify this it suffices to use the following transformations of nonlinear members u

∂u ∂u ∂u ∂ u2 ∂(uv) ∂(uw) ∂v ∂w +v +w = + + −u −u , ∂x ∂y ∂z ∂x 2 ∂y ∂z ∂y ∂z

u u

∂v ∂v ∂v ∂(uv) ∂ v 2 ∂(vw) ∂u ∂w +v +w = + + −v −v , ∂x ∂y ∂z ∂x ∂y 2 ∂z ∂x ∂z

∂w ∂w ∂w ∂(uw) ∂(vw) ∂ w2 ∂u ∂v +v +w = + + −w −w . ∂x ∂y ∂z ∂x ∂y ∂z 2 ∂x ∂y

Taking into account the continuity equation (4) the last two terms on the right-hand sides are transformed as follows −u

∂v ∂w ∂u ∂ u2 −u =u = , ∂y ∂z ∂x ∂x 2

−v

∂u ∂w ∂v ∂ v2 −v =v = , ∂x ∂z ∂y ∂y 2

∂u ∂v ∂w ∂ w2 −w =w = . ∂x ∂y ∂z ∂z 2 As a result, the nonlinear terms of equations (1 - 3) get a form as sum of the derivatives with respect to spatial coordinates −w

u

∂u ∂u ∂u ∂u2 ∂uv ∂uw +v +w = + + , ∂x ∂y ∂z ∂x ∂y ∂z

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(13)

Integrals of Motion of an Incompressible Medium Flow ∂v ∂v ∂w ∂uv ∂v 2 ∂vw +v +w = + + , ∂x ∂y ∂z ∂x ∂y ∂z

(14)

∂w ∂w ∂uw ∂vw ∂w 2 ∂w +v +w = + + . ∂x ∂y ∂z ∂x ∂y ∂z

(15)

u u

447

Linear terms of equations (1 - 3) has initially presented as a sum of first derivatives with respect to x, y, z, t. With this in mind and taking into account relations (13 - 15) we obtain the divergent form of equations (1 - 3) as follows ∂ 2 1 ∂u ∂ 1 ∂u ∂ 1 ∂u ∂u (u + p + Φ − )+ (uv − )+ (uw − )+ = 0, (16) ∂x Re ∂x ∂y Re ∂y ∂z Re ∂z ∂t ∂ 1 ∂v ∂ 2 1 ∂v ∂ 1 ∂v ∂v (uv − )+ (v + p + Φ − )+ (vw − )+ = 0, ∂x Re ∂x ∂y Re ∂y ∂z Re ∂z ∂t

(17)

∂ 1 ∂w ∂ 1 ∂w ∂ 1 ∂w ∂w (uw − )+ (vw − )+ (w 2 + p + Φ − )+ = 0. (18) ∂x Re ∂x ∂y Re ∂y ∂z Re ∂z ∂t Note that a similar assertion is also true for Euler equations as well. The first three of Euler equations reduce to (16 - 18) with the only difference being that the fall out of 1 consideration of terms proportional to Re . Thus, each of (16 - 18) is an equation of the form (12). This is to be shown. The form of equations of (12) is said to be canonical. These equations are linear with respect to Pi , Qi , Ri , Si and that allows the use of some of the approaches usually used to solve on a linear problems. The main benefits of providing in the form of (12) is the next. Each equation of the form (12) allows integration in generally. Equation (12) is linear. Before to construct general solution of (12) let’s previously ∂Qi i consider it particular ones. For example, a two-term equation ∂P ∂x + ∂y = 0 has a particular ∂Ψi1 i1 solution Pi = ∂Ψ ∂y , Qi = − ∂x , Ri = 0, Si = 0, where Ψi1 is an arbitrary twice continuously differentiable function in arguments x, y, z, t. Similarly, any other two-term equation, selected from (12) also gives rise to a particular ∂Ri i solution of (12). So the equation ∂P ∂x + ∂z = 0 generates a particular solution Pi = ∂Ψi2 i2 − ∂Ψ ∂z , Qi = 0, Ri = ∂x , Si = 0, where Ψi2 is also an arbitrary twice continuously differentiable function of four arguments. In total different two-term equations are as many as the number of combinations of two to four. Since C42 = 6 so the number of different two-term equations provided from (12), also equal to six. These equations are as follows

∂Pi ∂Qi ∂Pi ∂Ri ∂Pi ∂Si + = 0, + = 0, + = 0, ∂x ∂y ∂x ∂z ∂x ∂t

∂Qi ∂Ri ∂Qi ∂Si ∂Ri ∂Si + = 0, + = 0, + = 0. ∂y ∂z ∂y ∂t ∂z ∂t

(19)

The solution of each two-term equation (19) leads to a particular solution of (12). Various partial solutions in total as many as different equations (19), then there are six.

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Recorded successively they can be represented as Pi =

∂Ψi1 ∂Ψi1 , Qi = − , , Ri = 0, Si = 0, ∂y ∂x

∂Ψi2 ∂Ψi2 , Qi = 0, Ri = , Si = 0, ∂z ∂x ∂Ψi3 ∂ψi3 Pi = , Qi = 0, Ri = 0, Si = − , ∂t ∂x ∂Ψi4 ∂Ψi4 Pi = 0, Qi = , Ri = − , Si = 0, ∂z ∂y

Pi = −

Pi = 0, Qi = −

(20)

∂Ψi5 ∂Ψi5 , Ri = 0, Si = , ∂t ∂y

Pi = 0, Qi = 0, Ri =

∂Ψi6 ∂Ψi6 , Si = − . ∂t ∂z

Here Ψi1 , Ψi2 , Ψi3 , Ψi4 , Ψi5 , Ψi6 are an arbitrary twice continuously differentiable function in arguments x, y, z, t. Each particular solution can be interpreted as a four-dimensional vector which components are a function of four arguments. We have six such vectors - solutions. These particular solutions are linearly independent. Consider a linear combination with constant coefficients Ck , where k = 1, 2, ...6.. It is also a four-dimensional vector. This vector is non-zero since no one of its components does not vanish. The last statement is true, since the functions Ψij are arbitrary and any relationship between them and their derivatives are excluded. With help of these independent particular solutions we can build a general solution. It can be represented as a linear combination of six particular solutions considered above. Since functions Ψij are an arbitrary chosen each of the coefficients of linear combination can be chosen to be one. As a result, the general solution of (12) defined by relationships as the next Pi =

∂Ψi1 ∂Ψi2 ∂Ψi3 − + , ∂y ∂z ∂t

Qi = − Ri =

∂Ψi1 ∂Ψi4 ∂Ψi5 + − , ∂x ∂z ∂t

∂Ψi2 ∂Ψi4 ∂Ψi6 − + , ∂x ∂y ∂t

Si = −

(21)

∂Ψi3 ∂Ψi5 ∂Ψi6 + − . ∂x ∂y ∂z

From the above it follows that general solution of (12) defined by a set of six twice continuously differentiable functions Ψij in arguments x, y, z, t, where j = 1, 2, ..., 6. Note that all of the above applies to the full equations of the form (12). These equations

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are each of the three ones (1 - 3). By comparison to this the equation (4), for i = 4 is incomplete. This equation is three-term, as there is no term of a derivative with respect to time. Since C32 = 3, then from (4) can be identified only three different two-term equation. We have three independent particular solutions. General solution of (4) defined by three functions Ψ4k , Ψ4(k+1), Ψ4(k+2) , according to equations as the next P4 =

∂Ψ4(k+2) ∂Ψ4(k+1) ∂Ψ4(k+2) ∂Ψ4k − , Q4 = − + , ∂y ∂z ∂x ∂z R4 =

∂Ψ4(k+1) ∂Ψ4k − . ∂x ∂y

(22)

The resulting formula for the solutions of (12) define some relationships between the quantities Pi , Qi , Ri, Si , and therefore it define an implicit connection relation between unknowns u , v, w, p. We investigate such relations in more detail. Equations (16 - 18) are of the form (12) and their solutions can be constructed according to formulas (21). So, for each equation (16 - 18), i = 1, 2, 3, get four relations for Pi , Qi , Ri , Si . Total turns we have twelve ratio of this species. All ratio generated by each individual equation contain six an arbitrary functions. Then for equation (4) we have three ratio for u, v, w containing three arbitrary functions. Thus, in total we have fifteen relationships linking the main unknown u, v, w, p and twenty-one new functions. We enumerate these functions sequentially from the j = 1 and is denoted as Ψ1,j . As a result, the relationships under consideration take the form u2 + p + Φ − uv −

∂Ψ1,1 ∂Ψ1,2 ∂Ψ1,3 1 ∂u = − + , Re ∂x ∂y ∂z ∂t

1 ∂u ∂Ψ1,1 ∂Ψ1,4 ∂Ψ1,5 =− + − , Re ∂y ∂x ∂z ∂t

uw −

1 ∂u ∂Ψ1,2 ∂Ψ1,4 ∂Ψ1,6 = − + , Re ∂z ∂x ∂y ∂t

u=−

uv −

1 ∂v ∂Ψ1,7 ∂Ψ1,10 ∂Ψ1,11 =− + − , Re ∂y ∂x ∂z ∂t

1 ∂v ∂Ψ1,8 ∂Ψ1,10 ∂Ψ1,12 = − + , Re ∂z ∂x ∂y ∂t

v=− uw −

∂Ψ1,3 ∂Ψ1,5 ∂Ψ1,6 + − . ∂x ∂y ∂z

∂Ψ1,7 ∂Ψ1,8 ∂Ψ1,9 1 ∂v = − + , Re ∂x ∂y ∂z ∂t

v2 + p + Φ − vw −

(23)

∂Ψ1,9 ∂Ψ1,11 ∂Ψ1,12 + − , ∂x ∂y ∂z

1 ∂w ∂Ψ1,13 ∂Ψ1,14 ∂Ψ1,15 = − + , Re ∂x ∂y ∂z ∂t

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(24)

450

Alexander V. Koptev vw −

1 ∂w ∂Ψ1,13 ∂Ψ1,16 ∂Ψ1,17 =− + − , Re ∂y ∂x ∂z ∂t

w2 + p + Φ −

∂Ψ1,14 ∂Ψ1,16 ∂Ψ1,18 1 ∂w = − + , Re ∂z ∂x ∂y ∂t

w=−

u=

(25)

∂Ψ1,15 ∂Ψ1,17 ∂Ψ1,18 + − , ∂x ∂y ∂z

∂Ψ1,21 ∂Ψ1,20 ∂Ψ1,21 ∂Ψ1,19 − , v=− + , ∂y ∂z ∂x ∂z w=

∂Ψ1,20 ∂Ψ1,19 − . ∂x ∂y

(26)

Functions Ψ1,j appearing in the right-hand sides of (23 - 26) is said to be stream pseudo functions of the first order with number j, where j = 1, 2, ..., 21 [6-8]. This name is justified by the fact that for the simplest case of a 2D equations function Ψ1,21 coincides with the current function Ψ(x, y) well known in fluid mechanics [1-3]. Functions Ψ1,j arise naturally in the course of solving the problem. We introduce these functions as a new associated unknowns. The significance of these new unknowns is as follows. Since different combinations of main unknowns u, v, w, p are expressed in terms of functions Ψ1,j according to (23 - 26) , then we have new implicit relations between u, v, w, p by Ψ1,j . We face the question is how to simplify (23 - 26) and to give them the form of the most convenient for practical use. The following are suggested the transformations aimed at solving this particular problem. The basic idea is to avoid in possible nonlinear and non-divergent terms. The whole chain of transformations conditionally divided into three several parts.

3.1. Let’s eliminate the unknown p. This one is present only on three equations and additively. These are the first of equations (23), the second of (24) and the third of (25). Adding these equations term by term, divided the result by three and taking into account the continuity equation (4) we obtain a new ratio for p in the form of 1 ∂(Ψ1,14 − Ψ1,7 ) ∂(Ψ1,1 − Ψ1,16 ) p = −Φ − (u2 + v 2 + w 2 ) + + + 3 3∂x 3∂y ∂(Ψ1,10 − Ψ1,2 ) ∂(Ψ1,3 − Ψ1,11 + Ψ1,18 ) + . 3∂z 3∂t

(27)

Instead of two other equations initially containing p we take two new ones received as a term by term difference between the first of (23) and the second from (24), and the second from (24) and the third of (25). These new equations are the next u2 − v 2 −

1 ∂u ∂v ( − )= Re ∂x ∂y

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∂Ψ1,7 ∂Ψ1,1 ∂(Ψ1,2 + Ψ1,10 ) ∂(Ψ1,3 + Ψ1,11 ) + − + , ∂x ∂y ∂z ∂t

(28)

v 2 − w2 − −

1 ∂v ∂w ( − )= Re ∂y ∂z

∂(Ψ1,7 + Ψ1,14 ) ∂Ψ1,16 ∂Ψ1,10 ∂(Ψ1,11 + Ψ1,18 ) + + − . ∂x ∂y ∂z ∂t

(29)

3.2. We eliminate some of the nonlinear terms of the form of uv, vw, uw. In total there are six equations containing nonlinear terms of the indicated form. This is the second and third equations of (23), the first and the third of (24), the first and the second of (25). Moreover, the nonlinear terms of each species present only in two of the six equations. So, term uv present in the second equation of (23) and in the first of (24). Similarly, vw is present in the third equation of (24) and the second of (25). A term uw is present in the third of (23) and in the first of (25). Let’s consider these pairs of equations separately and find their term by term differences. The result is three new simpler linear equations which can again be present in the divergence form of (12). Consider the difference from the second of (23) and the first of (24). We come to new equation as the next ∂ v ∂ u ∂ ( + Ψ1,1 ) + (Ψ1,7 − )+ (−Ψ1,4 − Ψ1,8 )+ ∂x Re ∂y Re ∂z ∂ (Ψ1,5 + Ψ1,9 ) = 0. (30) ∂t Similarly, the difference of a third of (23) and the first of (25) leads to new equation as follows ∂ w ∂ ∂ u ( − Ψ1,2 ) + (Ψ1,4 + Ψ1,13 ) + (− − Ψ1,14 )+ ∂x Re ∂y ∂z Re ∂ (−Ψ1,6 + Ψ1,15 ) = 0. ∂t

(31)

The difference from the third of (24) and the second of (25) gives the next ∂ ∂ w ∂ v (−Ψ1,8 − Ψ1,13 ) + ( + Ψ1,10 ) + (− + Ψ1,16 )+ ∂x ∂y Re ∂z Re ∂ (−Ψ1,12 − Ψ1,17 ) = 0. ∂t

(32)

Of the six equations initially containing nonlinear terms of the above form we leave three original ones. There are the second and the third of (23) and the third of (24). And let’s further consider the three new equation (30-32).

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3.3. We transform the equation with the non-divergent terms u, v, w in the left-hand sides. In total there are six equations of this type. This is the fourth equation of (23), (24), (25) and all three equations (26). Each of the variables u, v, w is present in the equations of the additive and is present in only two of the six equations. Consider pairs of such equations and find their term by term differences. As a result we face three new linear equations in which non-divergent terms are absent. These equations have the canonical form of (12). Let’s consider the difference from the fourth equation of (23) and the first of (26). We obtain new equation as the next ∂ ∂ ∂ (−Ψ1,3 ) + (Ψ1,5 − Ψ1,21 ) + (−Ψ1,6 + Ψ1,20 ) = 0. ∂x ∂y ∂z

(33)

The difference of the fourth of (24) and the second of (26) leads to ∂ ∂ ∂ (−Ψ1,9 + Ψ1,21 ) + (Ψ1,11 ) + (−Ψ1,12 − Ψ1,19 ) = 0. ∂x ∂y ∂z

(34)

Similarly, the difference of the fourth of (25) and third of (26) gives the next ∂ ∂ ∂ (−Ψ1,15 − Ψ1,20 ) + (Ψ1,17 + Ψ1,19 ) + (−Ψ1,18 ) = 0. ∂x ∂y ∂z

(35)

Instead of the six equations initially containing non-divergent terms of u, v, w, we leave only three original ones (26), but instead the others will be considered three new equations (33-35). Thus, the conversion produced of equations (23 - 26) leds to the following interim results. Unknown p is excluded according to (27) and we obtain two new non-linear equation (28 - 29) without p. There was also six new linear equation of the form (12). This equation (30 - 35). Moreover, the last three of them are incomplete. Solutions of new linear equations given by the ratios similar to (21), (22). Each of the three equations (30 - 32) generates four relationships, containing six new arbitrary twice continuously differentiable functions of the variables x, y, z, t. Each of three equations (33 - 35) creates three new relationships containing three arbitrary twice continuously differentiable function. In total we obtain (3 · 4 + 3 · 3 = 21) twenty-one new ratio. They will involve (3 · 6 + 3 · 3 = 27) twenty-seven arbitrary twice continuously differentiable functions. We enumerate these functions sequentially starting with one and call them as stream pseudo function of the second order of the corresponding number [6-8]. The argument in favor of such a name is that in the course of the study to the canonical equations of the form (12) has been accessed twice. For these new unknown we introduce the notation Ψ2,k , where the first subscript indicates the order of the stream pseudo function, and the second one its number. As a result, we arrive at the relationships as the next v ∂Ψ2,1 ∂Ψ2,2 ∂Ψ2,3 + Ψ1,1 = − + , Re ∂y ∂z ∂t

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Integrals of Motion of an Incompressible Medium Flow u ∂Ψ2,1 ∂Ψ2,4 ∂Ψ2,5 + Ψ1,7 = − + − , Re ∂x ∂z ∂t ∂Ψ2,2 ∂Ψ2,4 ∂Ψ2,6 −Ψ1,4 − Ψ1,8 = − + , ∂x ∂y ∂t

453

−

Ψ1,5 + Ψ1,9 = −

(36)

∂Ψ2,3 ∂Ψ2,5 ∂Ψ2,6 + − . ∂x ∂y ∂z

w ∂Ψ2,7 ∂Ψ2,8 ∂Ψ2,9 − Ψ1,2 = − + , Re ∂y ∂z ∂t ∂Ψ2,7 ∂Ψ2,10 ∂Ψ2,11 + − , ∂x ∂z ∂t ∂Ψ2,8 ∂Ψ2,10 ∂Ψ2,12 = − + , ∂x ∂y ∂t

Ψ1,4 + Ψ1,13 = − −

u − Ψ1,14 Re

−Ψ1,6 + Ψ1,15 = −

−Ψ1,8 − Ψ1,13 =

∂Ψ2,9 ∂Ψ2,11 ∂Ψ2,12 + − . ∂x ∂y ∂z

∂Ψ2,13 ∂Ψ2,14 ∂Ψ2,15 − + , ∂y ∂z ∂t

w ∂Ψ2,13 ∂Ψ2,16 ∂Ψ2,17 + Ψ1,10 = − + − , Re ∂x ∂z ∂t v ∂Ψ2,14 ∂Ψ2,16 ∂Ψ2,18 − + Ψ1,16 = − + , Re ∂x ∂y ∂t −Ψ1,12 − Ψ1,17 = −

−Ψ1,3 =

(37)

(38)

∂Ψ2,15 ∂Ψ2,17 ∂Ψ2,18 + − . ∂x ∂y ∂z

∂Ψ2,21 ∂Ψ2,20 ∂Ψ2,21 ∂Ψ2,19 − , Ψ1,5 − Ψ1,21 = − + , ∂y ∂z ∂x ∂z −Ψ1,6 + Ψ1,20 =

−Ψ1,9 + Ψ1,21 =

(39)

∂Ψ2,24 ∂Ψ2,23 ∂Ψ2,24 ∂Ψ2,22 − , Ψ1,11 = − + , ∂y ∂z ∂x ∂z

−Ψ1,12 − Ψ1,19 =

−Ψ1,15 − Ψ1,20 =

∂Ψ2,20 ∂Ψ2,19 − . ∂x ∂y

∂Ψ2,23 ∂Ψ2,22 − . ∂x ∂y

(40)

∂Ψ2,27 ∂Ψ2,26 ∂Ψ2,27 ∂Ψ2,25 − , Ψ1,17 + Ψ1,19 = − + , ∂y ∂z ∂x ∂z −Ψ1,18 =

∂Ψ2,26 ∂Ψ2,25 − . ∂x ∂y

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We assume that the stream pseudo function of the second order introduced in accordance with ratios (36 - 41). Then the equation (30 - 35) performed identically and the number of equations is reduced to six. Let’s specify which of the equation remain in the defining system. These equations are the equation for pressure (27) and two equations (28 - 29) no longer contain the pressure p. Also leave unchanged six original equations. This equation the second and third of (23) , the third of (24) and three equations (26). Thus, the defining system remains nine equations. Put to further simplify the system. For this we use (36 - 41), which can be regarded as a connection relation between the stream pseudo functions of the first and second orders. We solve these equations with respect to Ψ1,j , with a view to further exclude them. Of the twenty-one equation (36 - 38) stand nine that immediately lead to the relationships of interest to us. Considering the first and second of (36), the first and third of (37), second and third of (38) , the first of (39), second of (40) and third of (41) we obtain directly Ψ1,1 = −

v ∂Ψ2,1 ∂Ψ2,2 ∂Ψ2,3 + − + , Re ∂y ∂z ∂t

u ∂Ψ2,1 ∂Ψ2,4 ∂Ψ2,5 − + − , Re ∂x ∂z ∂t w ∂Ψ2,7 ∂Ψ2,8 ∂Ψ2,9 = − + − , Re ∂y ∂z ∂t

Ψ1,7 = Ψ1,2

Ψ1,14 = −

∂Ψ2,8 ∂Ψ2,10 ∂Ψ2,12 u − + − , Re ∂x ∂y ∂t

(42)

w ∂Ψ2,13 ∂Ψ2,16 ∂Ψ2,17 − + − , Re ∂x ∂z ∂t v ∂Ψ2,14 ∂Ψ2,16 ∂Ψ2,18 = + − + . Re ∂x ∂y ∂t

Ψ1,10 = − Ψ1,16 Ψ1,3 = −

∂Ψ2,21 ∂Ψ2,20 ∂Ψ2,24 ∂Ψ2,22 + , Ψ1,11 = − + , ∂y ∂z ∂x ∂z Ψ1,18 = −

∂Ψ2,26 ∂Ψ2,25 + . ∂x ∂y

(43)

Further we find the ratio for Ψ1,4 , Ψ1,8 , Ψ1,13 . These functions appear only in three equations of (36 - 38). These equations are the third one of (36), the second of (37) and the first (38). These three linear equations form a closed subsystem with respect to unknowns Ψ1,4 , Ψ1,8 , Ψ1,13 . This subsystem has a unique solution 1 ∂ ∂ ∂ Ψ1,4 = ( (−Ψ2,7 − Ψ2,2 ) + (Ψ2,4 + Ψ2,13 ) + (Ψ2,10 − Ψ2,14 )+ 2 ∂x ∂y ∂z

Ψ1,8

∂ (Ψ2,15 − Ψ2,11 − Ψ2,6 )), ∂t 1 ∂ ∂ ∂ = ( (Ψ2,7 − Ψ2,2 ) + (Ψ2,4 − Ψ2,13 ) + (Ψ2,14 − Ψ2,10 )+ 2 ∂x ∂y ∂z

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Integrals of Motion of an Incompressible Medium Flow ∂ (−Ψ2,6 + Ψ2,11 − Ψ2,15 )), ∂t 1 ∂ ∂ ∂ Ψ1,13 = ( (Ψ2,2 − Ψ2,7 ) + (−Ψ2,4 − Ψ2,13 ) + (Ψ2,10 + Ψ2,14 )+ 2 ∂x ∂y ∂z

455 (44)

∂ (Ψ2,6 − Ψ2,11 − Ψ2,15 )). ∂t Consider further the fourth equation of (36), the second of (39) and the first of (40). These three linear equations form a closed subsystem with respect to Ψ1,5 , Ψ1,9 , Ψ1,21 . The solution is unique, it has the form Ψ1,5 =

Ψ1,21

Ψ1,9

1 ∂ ∂ ( (−Ψ2,3 − Ψ2,21 ) + (Ψ2,5 + Ψ2,24 )+ 2 ∂x ∂y

∂ (−Ψ2,6 + Ψ2,19 − Ψ2,23 )), ∂z 1 ∂ ∂ = ( (Ψ2,21 − Ψ2,3 ) + (Ψ2,5 + Ψ2,24 )+ 2 ∂x ∂y ∂ (−Ψ2,6 − Ψ2,19 − Ψ2,23 )), ∂z 1 ∂ ∂ = ( (Ψ2,21 − Ψ2,3 ) + (Ψ2,5 − Ψ2,24 )+ 2 ∂x ∂y ∂ (−Ψ2,6 − Ψ2,19 + Ψ2,23 )). ∂z

(45)

Consider the remaining six equations of (36 - 41). This equation fourth of (37) and (38), the third of (39) and (40), the first and the second of (41). These equations can be regarded as a closed linear system with respect to six unknowns Ψ1,6 , Ψ1,12 , Ψ1,15 , Ψ1,17 , Ψ1,19 , Ψ1,20 . The solution is unique, it has the form Ψ1,6 =

1 ∂ ∂ ( (Ψ2,9 − Ψ2,20 ) + (Ψ2,19 − Ψ2,11 − Ψ2,27 )+ 2 ∂x ∂y ∂ (Ψ2,12 + Ψ2,26 )), ∂z

Ψ1,12 =

1 ∂ ∂ ( (Ψ2,15 − Ψ2,23 + Ψ2,27 ) + (Ψ2,22 − Ψ2,17 )+ 2 ∂x ∂y

∂ (Ψ2,18 − Ψ2,25 )), ∂z 1 ∂ ∂ Ψ1,15 = ( (−Ψ2,9 − Ψ2,20 ) + (Ψ2,11 + Ψ2,19 − Ψ2,27 )+ 2 ∂x ∂y ∂ (Ψ2,26 − Ψ2,12 )), ∂z 1 ∂ ∂ Ψ1,17 = ( (Ψ2,15 + Ψ2,23 − Ψ2,27 ) − (Ψ2,22 + Ψ2,17 )+ 2 ∂x ∂y

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(46)

456

Alexander V. Koptev ∂ (Ψ2,18 + Ψ2,25 )), ∂z Ψ1,19 =

Ψ1,20

1 ∂ ∂ (− (Ψ2,15 + Ψ2,23 + Ψ2,27 ) + (Ψ2,22 + Ψ2,17 )+ 2 ∂x ∂y

∂ (Ψ2,25 − Ψ2,18 )), ∂z 1 ∂ ∂ (Ψ2,11 + Ψ2,19 + Ψ2,27 )+ = ( (Ψ2,9 + Ψ2,20 ) − 2 ∂x ∂y ∂ (Ψ2,26 + Ψ2,12 )). ∂z

Thus, all the stream pseudo functions of the first order Ψ1,j with j = 1, 2, ..., 21 is uniquely expressed in terms of derivatives of the stream pseudo functions of the second order by means of relationships (42 - 46). The stream pseudo functions of the first order play a supporting role and may be eliminated at all. Enough of all equations of the basic system instead of Ψ1,j make the substitutions according to (42 - 46). This reveals a pattern that facilitates further study in many ways. In the right-hand sides of the equations obtained will be attended by the derivatives of functions Ψ2,1 , Ψ2,8 , Ψ2,16 and derivatives of a defined below combinations of the remaining twenty-four functions Ψ2,k . In the next appear twelve as such combinations Ψ2,10 − Ψ2,14 ; Ψ2,4 − Ψ2,13 ; Ψ2,7 − Ψ2,2 ; Ψ2,5 + Ψ2,24 ; Ψ2,12 + Ψ2,26 ; Ψ2,3 − Ψ2,21 ; Ψ2,18 − Ψ2,25 ; Ψ2,9 + Ψ2,20 ; Ψ2,22 + Ψ2,17 ;

(47)

Ψ2,27 − Ψ2,6 ; Ψ2,11 − Ψ2,23 ; Ψ2,19 − Ψ2,15 . We take these twelve combinations of functions Ψ2,k as the new unknown. Let’s call them as stream pseudo functions of the second order with numbers from 28 till 39 inclusive. We introduce them consecutively numbered according to (47) and denote by Ψ2,k , where k = 28, 29, ..., 39. In all equations of defining system will be present functions Ψ2,1 , Ψ2,8 , Ψ2,16 and twelve new ones. As a result of these steps, the total number of employed associated unknown reduced to fifteen. We introduce a more convenient numbering of associated unknowns and new symbols, in which the stream pseudo function is indicated only the number but does not specify the order. The new designation is entered according to the following equations Ψ1 = Ψ2,31 ; Ψ2 = Ψ2,32 ; Ψ3 = Ψ2,33 ; Ψ4 = Ψ2,34 ; Ψ5 = Ψ2,35 ; Ψ6 = Ψ2,36 ; Ψ7 = Ψ2,37 ; Ψ8 = Ψ2,38 ; Ψ9 = Ψ2,39 ; Ψ10 = Ψ2,1 ; Ψ11 = Ψ2,8 ; Ψ12 = Ψ2,16 ; Ψ13 = Ψ2,28 ; Ψ14 = Ψ2,29 ; Ψ15 = Ψ2,30 .

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(48)

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As a result all of the above, the system of nine equations in question is transformed into p = −Φ −

U2 1 ∂ ∂ ∂ − d − · ( (Ψ2 − Ψ1 ) + (Ψ4 − Ψ3 )+ 2 3 ∂t ∂x ∂y

∂ (Ψ6 − Ψ5 )), ∂z 2 ∂u ∂v ∂ 2 Ψ10 ∂ 2 Ψ10 ∂ 2 Ψ11 ∂ 2 Ψ12 · (− + )=− + − − + u2 − v 2 + Re ∂x ∂y ∂x2 ∂y 2 ∂z 2 ∂z 2   ∂ 2 Ψ14 ∂ 2 Ψ15 ∂ ∂Ψ1 ∂Ψ3 ∂(Ψ5 + Ψ6 ) + + − + + , ∂x∂z ∂y∂z ∂t ∂x ∂y ∂z 2 ∂v ∂w ∂ 2 Ψ10 ∂ 2 Ψ11 ∂ 2 Ψ12 ∂ 2 Ψ12 · (− + )= + − + − Re ∂y ∂z ∂x2 ∂x2 ∂y 2 ∂z 2   ∂ 2 Ψ13 ∂ 2 Ψ14 ∂ ∂(Ψ1 + Ψ2 ) ∂Ψ4 ∂Ψ6 − + + − , ∂x∂y ∂x∂z ∂t ∂x ∂y ∂z

(49)

(50)

v 2 − w2 +

uv −

1 ∂v ∂u ∂ 2 Ψ10 1 ∂ 2 Ψ15 ∂ 2 Ψ14 ∂ 2 Ψ13 ·( + )=− + ·( + + )− Re ∂x ∂y ∂x∂y 2 ∂x∂z ∂y∂z ∂z 2   1 ∂ ∂Ψ3 ∂Ψ1 ∂(Ψ8 + Ψ9 ) · + + , 2 ∂t ∂x ∂y ∂z

uw −

1 ∂w ∂u ∂ 2 Ψ11 1 ∂ 2 Ψ15 ∂ 2 Ψ14 ∂ 2 Ψ13 ·( + )= − ·( + + )+ Re ∂x ∂z ∂x∂z 2 ∂x∂y ∂y 2 ∂y∂z   1 ∂ ∂Ψ5 ∂(Ψ9 − Ψ7 ) ∂Ψ2 · − + + , 2 ∂t ∂x ∂y ∂z

(51)

(52)

(53)

1 ∂w ∂v ∂ 2 Ψ12 1 ∂ 2 Ψ14 ∂ 2 Ψ15 ∂ 2 Ψ13 ·( + )=− + ·( + − )+ Re ∂y ∂z ∂y∂z 2 ∂x∂y ∂x2 ∂x∂z   1 ∂ ∂(Ψ7 + Ψ8 ) ∂Ψ6 ∂Ψ4 · + + , (54) 2 ∂t ∂x ∂y ∂z   1 ∂ ∂Ψ3 ∂Ψ1 ∂Ψ7 ∂ ∂Ψ5 ∂Ψ8 ∂Ψ2 u= · (− + + )+ (− + − ) , (55) 2 ∂y ∂x ∂y ∂z ∂z ∂x ∂y ∂z   1 ∂ ∂Ψ3 ∂Ψ1 ∂Ψ7 ∂ ∂Ψ9 ∂Ψ6 ∂Ψ4 v= · ( − − )+ ( + − ) , (56) 2 ∂x ∂x ∂y ∂z ∂z ∂x ∂y ∂z   1 ∂ ∂Ψ5 ∂Ψ8 ∂Ψ2 ∂ ∂Ψ9 ∂Ψ6 ∂Ψ4 w= · ( − + )+ (− − + ) . (57) 2 ∂x ∂x ∂y ∂z ∂y ∂x ∂y ∂z In equation (49) introduced a new designation. d indicated the dissipative term, calculated according to the formula vw −

d=−

U2 1 ∂ 2 Ψ13 ∂ 2 Ψ14 ∂ 2 Ψ15 + (−∆xy Ψ10 + ∆xz Ψ11 − ∆yz Ψ12 − + − ). 6 3 ∂x∂y ∂x∂z ∂y∂z

(58)

Symbols ∆xy , ∆xz , ∆yz in the last formula indicated an incomplete Laplace operators on the coordinates ∆xy =

∂2 ∂2 ∂2 ∂2 ∂2 ∂2 + , ∆ = + , ∆ = + . xz yz ∂x2 ∂y 2 ∂x2 ∂z 2 ∂y 2 ∂z 2

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Conclusion Equation (49 - 57) represents the ratio of the link between the main unknowns u, v, w, p and associated ones Ψk , where k = 1, 2, ..., 15. The order of derivatives with respect to the main unknowns u, v, w, p one less than their order in the original equations (1 - 4). Considered in the aggregate ratios (49 - 57) represent first integral of the Navier — Stokes equations. From these ratios can be easily obtained the first integral of the Euler equations as well. 1 = 0. It’s enough to put in five equations (50 - 54) Re The foolowing assertion is true. All known integrals of motion of an incompressible fluid such as the Bernoulli integral, Euler — Bernoulli and Lagrange — Cauchy ones are derived from (49 - 57) as the special cases. A brief analysis of the relationships (49 - 57) leads to the following. We have a system of nine equations with respect to fifteen unknowns. We face an excess of the number of unknowns over the number of equations. This fact plays a positive role in solving a specific practical problems since there is a possibility of modeling the solution and to satisfy the boundary and initial conditions. Among nine ratios (49 - 57) there are four as the special ones. Namely, there are ratio of (49) and (55 - 57). It is true to state that these relationships define the general structure of solutions of the Navier — Stokes equations [9]. So, from (49) it follows that the pressure p represented as algebraic sum of the four terms of different nature. These terms are the 2 potential of external force Φ, the dynamic pressure U2 and dissipative terms as d and dt , where the last is non-stationary dissipative term defined as dt =

∂ ∂ 1 ∂ ∂ · ( (Ψ2 − Ψ1 ) + (Ψ4 − Ψ3 ) + (Ψ6 − Ψ5 )). 3 ∂t ∂x ∂y ∂z

Equation (49) should be used to find p at the last stage when all the other unknowns are already defined. From the relationships (55 - 57) it follows that velocities u, v, w represented by linear combinations of the second derivative of Ψk , where k = 1, 2, 3, ..., 9. Equations (49 - 57) pave the way for the practical construction of solutions of the Navier — Stokes equations (1 - 4) [10-14]. According to the structural formulas (49), (55 - 57) values of the main unknown defined of the values of all associated unknowns. To find the Ψk where k = 1, 2, 3, ..., 15 it is necessary to solve a system of five nonlinear equations 50 - 54) with respect to fifteen unknowns. Thus, the first integral of the Navier — Stokes equations reduced to nine relationships (49 - 57). This conclusion is valid in the general case without any restrictions to the private nature of the motion characteristics, the configuration of the flow domain, the type of initial and boundary conditions and value of Reynolds number Re.

References [1] L.I. Sedov, Mekhanika sploshnoy sredy, Part 2, Moscow:Nauka, (1970) (in Russian). [2] L.G. Loitsaynskiy, Mechanics of Fluid and Gas, Moscow:Nauka, (1987) (in Russian).

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[3] N.E. Kochin, I.A. Kibel, N.V. Rose Teoreticheskaya gydromekhanika, Part 1, Moscow:Fismatlit, (1963) (in Russian). [4] O.A. Lodizhenskaya, The Mathematical Theory of Viscous Incompressible Fluid , Gordon and Breach, New York, (1969). [5] Charls L. Fefferman, Existence and Smoothness of the Navier — Stokes Equation, Preprint, Princeton Univ., Math. Dept., Princeton, NJ, USA, (2000), P. 1-5. [6] A.V. Koptev, Integrals of the Navier — Stokes Equations, Trudy Sredne-volzhskogo Matematicheskogo Obshchestva, Saransk, 6(1), (2004), C. 215-225 (in Russian). [7] A.V. Koptev, First integral and Ways of Further Integration of Navier — Stokes Equations, Izvestia Rossiyskogo gosudarsvennogo pedagogicheskogo universiteta im. Gertsena, Saint-Petersburg, 147, (2012), C. 7-17 (in Russian). [8] A.V. Koptev, First integral of motion for an incompressible fluid flow, Proc. of 11-th All-Russian Congress on fundamental problems of theoretical and applied mechanics, Privolzhskiy Federal Univ., Kazan,(2015), C. 1957-1959 (in Russian). [9] A.V. Koptev, Structure of solution of the Navier — Stokes equations, – Bulletin of the National Research Nuclear University MEPI, Moscow, 3(6), ,(2014), – P. 656-660. [10] A.V. Koptev, Nonlinear Effects in Poiseuille Problem, Journal of Siberian Federal University, Math. and Phys., Krasnoyarsk, 6(3), (2013), P. 308-314. [11] A.V. Koptev, Generator of Solution of 2D Navier — Stokes Equations, Journal of Siberian Federal University, Math. and Phys., Krasnoyarsk, 7(3), (2014), P. 324-330. [12] A.V. Koptev, The Solution of Initial and Boundarry Value Problem for 3D Navier — Stokes Equations and its Features, Izvestia Rossiyskogo gosudarsvennogo pedagogicheskogo universiteta im. Gertsena, Saint-Petersburg, 165, (2014), C. 7-18 (in Russian). [13] A.V. Koptev, Dynamic Responce of an Underwater Pipeline on the Sea Currents, Vestnik gosudarsvennogo universiteta morskogo i rechnogo flota im. adm. S.O. Makarova, Saint-Petersburg, 4(26), (2014), C. 107-114 (in Russian). [14] A.V. Koptev, Theoretical Research of the Flow Around the Cylinder of an Ideal Incompressible Medium in the Presence of a Shielding Effect, Vestnik gosudarsvennogo universiteta morskogo i rechnogo flota im. adm. S.O. Makarova, Saint-Petersburg, 2(36), (2016), C. 127-137 (in Russian).

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In: Handbook on Navier-Stokes Equations Editor: Denise Campos

ISBN: 978-1-53610-292-5 c 2017 Nova Science Publishers, Inc.

Chapter 22

L OCAL E XACT C ONTROLLABILITY OF THE B OUSSINESQ E QUATIONS WITH B OUNDARY C ONDITIONS ON THE P RESSURE Tujin Kim1,∗ and Daomin Cao2,† 1 Institute of Mathematics, State Academy of Sciences, Pyongyang, DPR Korea 2 Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences, Beijing, P. R. China

Abstract In this paper we are concerned with the local exact controllability of the Boussinesq equations with the pressure condition on parts of boundary by controls acting on subdomains or subboundaries. Starting point of these results is to get a Carleman inequality for the system adjoint to the linearized Boussinesq system with the nonstandard boundary conditions. This is obtained by combination of our previous results for the Stokes system with the nonstandard boundary conditions and a linear parabolic equation with mixed boundary conditions. Using the Carleman inequality, we get an observability inequality for a system adjoint to a linearized Boussinesq system. Next, following other’s way and using the observability inequality, we also prove null controllability of the linearized system by controls acting on an arbitrarily given subdomain. Finally, by using these results under some assumptions we prove local exact controllability of the Boussinesq system by internal or boundary controls.

PACS: 05.45-a, 52.35.Mw, 96.50.Fm Keywords: Boussinesq equations, Controllability, Carleman inequality, Mixed boundary condition, Pressure boundary condition AMS Subject Classification: 35Q35, 35A23, 76D55, 93C20 ∗

E-mail address: [email protected]. Partially supported by TWAS, UNESCO and AMSS in Chinese Academy of Sciences. † E-mail address: [email protected]. Partially supported by Science Fund for Creative Research Groups (10721101), Chinese Academy of Sciences(KJCX3-SYW-S03) and NNSF of China(10631030).

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1.

Tujin Kim and Daomin Cao

Introduction

Local exact controllability of the Boussinesq equations by controls acting on an arbitrarily given subdomain were studied in several papers. In [13] the equations with periodical boundary condition, in [9], [10], [12], [26], [16] and [15] the equations with Dirichlet boundary conditions, and in [17] equations with Navier slip boundary condition, respectively, were studied. Local exact controllability of the Boussinesq equations was obtained by combination of null controllability of a linearized system with a kind of fixed point. To get the null controllability of the linearized system, global Carleman inequalities for the system adjoint to the linearized Boussinesq system were studied. On the other hand, in practice for the fluid flow we are concerned with nonstandard boundary conditions including pressure. Existence and regularity of solutions to the NavierStokes equations (cf. [2]-[5], [7] [22]) and Boussinesq equations (cf. [1]) with condition on the pressure on parts of the boundary where there is flow were studied. Also in [19] local exact controllability of the Navier-Stokes equations with such a nonstandard boundary condition was studied. In this paper we study local exact controllability of the Boussinesq equations with condition on the pressure on parts of the boundary by controls acting on subdomains or subboundaries. Main results of this paper are Theorem 6.3 (local exact controllability by internal controls) and Theorem 6.4 (local exact controllability by boundary controls). But starting point of these main results is to get a Carleman inequality for the system adjoint to the linearized Boussinesq system with the nonstandard boundary conditions. This is obtained by combination of our previous results. In our case the assumption for smoothness of the target is weaker than the one in [13] and [26]. This paper consists of six sections. In the rest of Section 1 we explain notations. In the Section 2 we consider our problems and linearization of the Boussinesq system. In Section 3, using an inequality (which is also a kind of Carleman inequality) for the Stokes system with the nonstandard boundary condition in [19] and a Carleman inequality for a parabolic equation with mixed boundary conditions in [20], we obtain a global Carleman inequality for the system adjoint to the linearized Boussinesq system (Theorem 3.4). In Section 4 using the Carleman inequality and following the way in [13], [18], we get an observability inequality (Theorem 4.2) for the solutions to the system adjoint to the linearized one. Using the observability inequality, in Section 5 we prove the null controllability of the linearized system with the nonstandard mixed boundary condition (Theorem 5.1). In Section 6 under some assumptions we prove local exact controllability of the Boussinesq system by internal or boundary controls (Theorem 6.3, Theorem 6.4). After converting the problems to abstract differential equations all arguments in Sections 5, 6 are same as [19], and so these sections are briefly described. Let Ω ⊂ Rl (l = 2, 3) be a connected bounded open set with boundary ∂ΩS ∈ C 2 , ∂Ω = Γ0 ∪ Γ1 = Γ2 ∪ Γ3 , Γ0 ∩ Γ1 = ∅, Γ2 ∩ Γ3 = ∅, Γ3 ⊂ Γ0 , Γ2 6= ∅, Γj = Γji , where Γji are open subsets of ∂Ω. If Γ0 = ∅, then Γ1 is connected. In the case of 2-D domains Γ1 is not concave. Let n(x) be outward normal unit vector at x in ∂Ω, Q = Ω × (0, T ) and

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0 < T < ∞. Let ω and ω0 be arbitrarily given subdomains such that ω0 ⊂ ω, Qω = ω × (0, T ) and Qω0 = ω0 × (0, T ). Let Wαk (Ω) be Sobolev spaces, H 1 (Ω) = W21 (Ω), H1 (Ω) = {H 1 (Ω)}l , L2 (Ω) = {L2 (Ω)}l , L2 = {L2 (Ω)}l+1 and HΓ1i (Ω) = {v ∈ W21 (Ω) : v |Γi = 0}. Let V = {u ∈ H1 (Ω) : div u = 0, u |Γ0 = 0, u × n |Γ1 = 0}, H be the closure of V in ¯ : div u = 0}. L2 (Ω) and D(Ω) = {C ∞ (Ω) 1 Let X = V × HΓ2 (Ω), X ∗ be a dual space of X and H = H × L2 (Ω). Define the spaces W and Z by ∂u ∂u ∈ L2 (0, T ; X ∗ )}, kukW = kukL2(0,T ; X) + k kL2 (0,T ; X ∗ ) ∂t ∂t ∂u ∂u e : Z = {u ∈ L2 (0, T ; D(A)) ∈ L2 (0, T ; H)}, kykZ = kukL2 (0,T ;D(A)) kL (0,T ; H) , e +k ∂t ∂t 2 W = {u ∈ L2 (0, T ; X) :

e is defined by (3.11) in Section 3. where D(A) When there is not confusion, inner products in Hilbert spaces are denoted by (· , ·) and duality products between Banach spaces and their dual spaces by h· , ·i. Specially, (· , ·)Γi 1 is an inner product in L2 (Γi ) and h· , ·iΓi means the duality product between H 2 (Γi ) ≡ 1

1

H02 (Γi ) (cf. Theorem 11.1, ch. 1 in [21]) and it’s dual space H− 2 (Γi ). In the case that l = 2, for convenience, y = (y1 (x1 , x2 ), y2 (x1 , x2)) is identified with y¯ = (y1 , y2 , 0), and so rot y = rot y¯.

2.

Problems and Linearization ˆ be a solution to the problem Let (ˆ v, pˆ, θ)  ∂v   − ν∆v + (v · ∇)v + ∇p + θel = fˆ1 ,   ∂t      div v = 0,     ∂θ    − ∆θ + v · ∇θ = fˆ2 , ∂t
1   v |Γ0 = 0, v × n |Γ1 = c(x, t) × n, p + |v|2 Γ = ϕ1 (x, t),   1  2   
∂θ    θ |Γ2 = a(x, t), b(x)θ + = d(x, t)  Γ3  ∂n    ˆ = θˆ0 , vˆ(0) = vˆ0 , θ(0)

(2.1)

where el is the unit vector of xl . Throughout this paper we assume that b(x) ∈ L∞ (Γ3 ) and

1 1 vˆ ∈ W∞ (0, T ; Wα2 (Ω)), θˆ ∈ W∞ (0, T ; Wα2(Ω)), α > l.

(2.2)

We are concerned with controllability by controls acting on a subdomain or subboundˆ In the case of internal control we can write the problem as ary to the given trajectory (ˆ v , θ).

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follows.  ∂v   − ν∆v + (v · ∇)v + ∇p + θel = fˆ1 + χω u1 ,    ∂t     div v = 0,     ∂θ    − ∆θ + v · ∇θ = fˆ2 + χω u2 , ∂t
1    v |Γ0 = 0, v × n |Γ1 = c(x, t) × n, p + |v|2 Γ = ϕ1 (x, t),  1  2   
∂θ    = d(x, t) θ |Γ2 = a(x, t), b(x)θ +  Γ3  ∂n    v(0) = v , θ(0) = θ , 0

(2.3)

0

where χω is the characteristic function of the set ω. Then, our interest is existence of control ˆ )). (u1 , u2 ) by which (v(T ), θ(T )) = (ˆ v(T ), θ(T Setting v = vˆ + y, p = pˆ + p1 , θ = θˆ + τ in (2.3) and using the facts that −∆v = rot rot v − grad(div v), 1 (v, ∇)v = rot v × v + grad|v|2 , 2 we have the following equation for (y, q, τ )  ∂y   + νrot rot y + rot vˆ × y + rot y × vˆ + rot y × y + ∇q + τ el = χω u1 ,   ∂t     div y = 0,       ∂τ − ∆τ + vˆ · ∇τ + y · ∇θˆ + y · ∇τ = χ u , ω 2 ∂t   y |Γ0 = 0, y × n |Γ1 = 0, q|Γ1 = 0,      ∂τ
  τ |Γ2 = 0, b(x)τ + = 0,    ∂n Γ3   y(0) = y0 , τ (0) = τ0 ,

(2.4)

where q = p1 + vˆ · y + 12 |y|2 and y0 = v0 − vˆ0 , τ0 = θ0 − θˆ0 . Neglecting the two nonlinear terms rot y × y and y · ∇τ in (2.4), we arrive to the following initial boundary value problem of the linearized Boussinesq system  ∂y   + νrot rot y + rot vˆ × y + rot y × vˆ + ∇q + τ el = χω u1 ,   ∂t     div y = 0,       ∂τ − ∆τ + vˆ · ∇τ + y · ∇θˆ = χ u , ω 2 ∂t (2.5)   y | = 0, y × n | = 0, q | = 0,  Γ Γ Γ 0 1 1    
∂τ     τ |Γ2 = 0, b(x)τ + ∂n Γ3 = 0,    y(0) = y0 , τ (0) = τ0 .

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The case of boundary control is similar, and we will consider this case in the end of this paper. On the other hand, for y, w ∈ V ∩ D(Ω), ν(rot rot y, w)Ω = ν(rot y, rot w) − ν(rot y × n, w) Γ 1 (2.6) = ν(rot y, rot w) − ν(rot y, n × w) Γ = ν(rot y, rot w) = ν(y, rot rot w) 1

and

(rot vˆ × v, w ) = −(rot vˆ × w, v )

∀v, w ∈ H1 (Ω),

(rot u × vˆ, w ) = (rot u, vˆ × w ) = (u, rot (ˆ v × w))

∀u, w ∈ V.

∂τ Also, for τ ∈ HΓ12 (Ω) ∩ C ∞ (Ω) such that (b(x)τ + ∂n ) |Γ3 = 0 Z Z ∂ρ
(∆τ, ρ) = − b(x)τ ρ − (∇τ, ∇ρ) = − τ b(x)ρ + dσ + (τ, ∆ρ) ∂n Γ3 Γ3

(2.7)

(2.8)

∀ρ ∈ HΓ12 (Ω) ∩ C ∞ (Ω),

(ˆ v · ∇τ, ρ) =

Z

vˆ · nτ ρ dσ − (div (ˆ v ρ), τ ) = −(ˆ v · ∇ρ, τ )

Γ3

∀τ, ρ ∈

(2.9)

HΓ12 (Ω) ∩ C ∞ (Ω),

where vˆ |Γ0 = 0 and Γ3 ⊂ Γ0 were used. In view of (2.6) and (2.8), we give the following
Definition 2.1. A function (y, τ ) ∈ L2 (0, T ; V ) ∩ C([0, T ]; H) × L2 (0, T ; HΓ12 ) ∩
∂τ ∗ 1 ∗ C([0, T ]; L2 ) is called a solution to (2.4) if ( ∂y ∂t , ∂t ) ∈ L2 (0, T ; V ) × L2 (0, T ; (HΓ2 ) ), (y(0), τ (0)) = (y0 , τ0 ) and

∂y  , w + ν(rot y, rot w) + (rot vˆ × y + rot y × vˆ + rot y × y + τ el , w) ∂t

∂τ  + , φ + (∇τ, ∇φ) + (ˆ v · ∇τ + y · ∇θˆ + y · ∇τ, φ) + (b(x)τ, φ)Γ3 ∂t = hχω u1 , wi + hχω u2 , φi ∀(w, φ) ∈ L2 (0, T ; V ) × L2 (0, T ; HΓ12 ).

3.

A Global Carleman Inequality

In this section we get a global Carleman inequalities for the system adjoint to the linearized Boussinesq system. By (2.6)-(2.9), we see that the problem adjoint to (2.5) is  ∂w   + νrot rot w − rot vˆ × w + rot (ˆ v × w) + ρ∇θˆ + ∇p = f1 , −   ∂t     div w = 0,       − ∂ρ − ∆ρ − vˆ · ∇ρ + w · e = f , l 2 ∂t (3.1)   w | = 0, w × n | = 0, p | = 0,  Γ Γ Γ 0 1 1    
∂ρ     ρ |Γ2 = 0, b(x)ρ + ∂n Γ3 = 0,    w(T ) = wT , ρ(T ) = ρT .

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The backward problem (3.1) is equivalent to the following initial boundary value problem  ∂w   v × w) + ρ∇θˆ + ∇p = f1 ,  ∂t + νrot rot w − rot vˆ × w + rot (ˆ      div w = 0,       ∂ρ − ∆ρ − vˆ · ∇ρ + w · e = f , l 2 ∂t (3.2)   w | = 0, w × n | = 0, p | = 0,  Γ Γ Γ 0 1 1    
∂ρ   = 0, ρ |Γ2 = 0, b(x)ρ +    ∂n Γ3   w(0) = w0 , ρ(0) = ρ0 and in this section we will consider (3.2).

3.1.

Strong Solution to the Adjoint System

By orthogonal decomposition of space L2 (Ω) = H ⊕ G0, Γ1 (Ω), G0, Γ1 (Ω) = {u = grad p : p ∈ HΓ11 (Ω)}

(3.3)

without any influence to the boundary condition on the pressure on Γ1 the case of that f1 ∈ L2 (0, T ; L2(Ω)) refer to the case f1 ∈ L2 (0, T ; H). Denote by Π the projector from L2 onto H by (3.3). When f ≡ (f1 , f2 ) ∈ W ∗ and (w0, ρ0 ) ∈ H, in view of (2.6)-(2.9), we give the following
Definition 3.1. A function (w, ρ) ∈ L2 (0, T ; V ) ∩ C([0, T ]; H) × L2 (0, T ; HΓ12 ) ∩
∂ρ ∗ 1 ∗ C([0, T ]; L2 ) is called a solution to (3.2) if ( ∂w ∂t , ∂t ) ∈ L2 (0, T ; V )×L2 (0, T ; (HΓ2 ) ), (w(0), ρ(0)) = (w0 , ρ0) and

∂w  ˆ u) , u + ν(rot w, rot u) + (−rot vˆ × w + rot (ˆ v × w), u) + (ρ∇θ, ∂t

∂ρ  + , φ + (∇ρ, ∇φ) + (−ˆ v · ∇ρ + w · el , φ) + (b(x)ρ, φ)Γ3 ∂t = hf1 , ui + hf2 , φi ∀(u, φ) ∈ L2 (0, T ; V ) × L2 (0, T ; HΓ12 ). Define operators A1 ∈ L (V → V ∗ ) and A2 ∈ L (HΓ12 → (HΓ12 )∗ ) by hA1 w, ui = ν(rot w, rot u)

∀w, u ∈ V

(3.4)

and hA2 ρ, φi = (∇ρ, ∇φ) + (b(x)ρ, φ)Γ3 + k0 (ρ, φ)

∀ρ, φ ∈ HΓ12 (Ω),

(3.5)

where k0 is a constant determined in (3.8). Then |hA1 w, ui| ≤ KkwkV kukV , |hA2 ρ, vi| ≤ KkρkH 1 kvkH 1 . Γ2

Γ2

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In the case of 3-D domain with ∂Ω ∈ C 1.1 (cf. Proposition 1.1 in [1] or Lemma 1.5 in [3] if Γ0 6= ∅, and p. 286 in [7] if Γ0 = ∅) we have hA1 w, wi ≥ akwk2V , a > 0.

(3.7)

For 2-D domains where Γ1 is not concave, (3.7) is valid (cf. Lemma 5.2 in [19]). On the other hand, there exists a constant k0 ≥ 0 such that Z 1 kb(x)kL∞(Γ3 ) ρ2 dσ ≤ k∇ρk2L2 + k0 kρk2L2 , 2 Γ3

(3.8)

(cf. Theorem 1.6.6 in [6] or (1), p. 258 in [8]) and so

hA2 ρ, ρi ≥ akρk2H 1 , a > 0.

(3.9)

Γ2

Define an operator A ∈ L (X → X ∗ ) by   A1 0 A= 0 A2

(3.10)

and let e1 : D(A e1 ) = {w ∈ V : A1 w ∈ H}, A1 w = A e1 w ∀w ∈ D(A e1 ), A e2 : D(A e2 ) = {ρ ∈ H 1 : A2 ρ ∈ L2 (Ω)}, A2 ρ = A e2 ρ ∀ρ ∈ D(A e2 ), A Γ2 ! e e = A1 0 , D(A) e = D(A e1 ) × D(A e2 ), A e2 0 A 2 2 k(w, ρ)k2D(A) e = k(w, ρ)kX + kA(w, ρ)kH.

(3.11)

(3.12)

e−1 is self-adjoint in H (cf. p. 56 in [25]). Then, the operator A

Remark 3.1. If A2 ρ ∈ L2 (Ω), then there exists f ∈ L2 (Ω) such that e2 ), φ ∈ HΓ1 (Ω). (∇ρ, ∇φ) + (b(x)ρ, φ)Γ3 + k0 (ρ, φ) = (f, φ) ∀ρ ∈ D(A 2

(3.13)

Taking any φ ∈ C0∞ (Ω), from (3.13) we get

∆ρ + k0 ρ = f ∈ L2 (Ω).

(3.14)

∂ρ Thus, ∂n ∈ H −1/2 (Γ3 ) (cf. Theorem 1.2, ch. 1 in [24]) and from (3.13), (3.14) we get ∂ρ ∂n + b(x)ρ = 0.

Define an operator B(t) ∈ L (X; H) by ˆ u) (B(t)(w, ρ), (u, φ))H = (−rot vˆ(t) × w + rot (ˆ v (t) × w), u)L2 + (ρ∇θ(t), + (−ˆ v(t) · ∇ρ + w · el − k0 ρ, φ) ∀(u, φ) ∈ H, e by (2.2) where k0 is one in (3.5). For (w, ρ) ∈ D(A)

1

(3.15)

1

2 kB(t)(w, ρ)kH ≤ ck(w, ρ)kX ≤ ck(w, ρ)kX · k(w, ρ)k 2

e D(A)

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,

(3.16)

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where c is independent of t. When f ∈ W ∗ and (w0 , ρ0) ∈ H, unique existence of a solution to (3.2) in the sense of Definition 3.1 is equivalent to the property of a solution u ∈ W to the problem   ∂u + (A + B)u = f, ∂t (3.17)  u(0) = (w0 , ρ0 ). By (3.6)-(3.10) and (3.16) there exists a unique solution to (3.17) and kukW ≤ c(k(w0, ρ0 )kH + kf kL2 (0,T ; X ∗ ) )

(3.18)

(cf. ch. 6 in [14]). Remark 3.2. Since hc(t), v · niΓ1 = 0 for v ∈ V , p is determined up to R a constant with respect to x. Thus, to fix it when we consider p, we assume that p satisfies Ω p(x, t) dx = 0, R namely Q p dxdt = 0. e−1 is self-adjoint in H, in the same way as Lemma 2.3 in [19] Using that the operator A we have the following

Lemma 3.1. If (w0 , ρ0) ∈ X and f ∈ L2 (0, T ; L2 (Ω)), then there exists a solution u ∈ Z to (3.17) such that ¯ kL (0,T ; H)), kukZ ≤ c(k(w0, ρ0)kX + kΠf (3.19) 2 ¯ = (Π, I), I is the unit operator on L2 (Ω). where and future Π

3.2.

A Global Carleman Inequality

¯ be a function such that Let ψ ∈ C 2 (Ω) ψ(x) > 0

∀x ∈ Ω,

ψ |∂Ω= 0 and |∇ψ| > 0

∀x ∈ Ω \ ω 0

(cf. Lemma 1.1, ch. 1 in [12]). Using this function, let us define . ϕ(x, t) = eλψ(x) [t(T − t)]8 , . ϕ(x, ˜ t) = e−λψ(x) [t(T − t)]8 ,  . 2 ¯ α(x, t) = eλψ − eλ kψkC(Ω) [t(T − t)]8 ,  . λ2 kψkC(Ω) α(x, ˜ t) = e−λψ − e [t(T − t)]8 , 2

¯ 1 − eλ kψkC(Ω) α(x, ˆ t) = α(x, t) |∂Ω = , [t(T − t)]8 1 ϕ(t) ˆ = , [t(T − t)]8

where λ is a positive parameter which will be chosen later on.

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To get a global Carleman inequality, let us consider such inequalities for the Stokes and parabolic problems. First let us consider the Stokes problem  ∂u   − ν∆u + ∇p = f˜,   ∂t   div u = 0, (3.21)   u | = 0, u × n | = 0, p| = 0,  Γ0 Γ1 Γ1    u(0) = u0 ∈ V.

Lemma 3.2. (cf. Lemma 2.7 in [19], Carleman inequality for the Stokes problem) Assume that f˜ ∈ L2 (0, T ; L2(Ω)), div f˜ ∈ L2 (0, T ; L2(Ω)). Then, there exists a numˆ > 1 such that for an arbitrary λ > λ ˆ there is a s0 (λ) with the property that for every ber λ s ≥ s0 (λ) the solutions to (3.21) satisfy  Z  2 1 ∂u 2 3 3 2 I(s) ≡ ˆ + s ϕˆ |u| e2sa dxdt + sϕ|∇u| s ϕ ˆ ∂t Q Z Z h i 3 1 2 2sα 2sˆ α ˜ ≤C |f | e + (sϕ) ˆ 4e dxdt + |div f˜|2 (sϕ) ˆ 2 e2sα dxdt (3.22) Q Q  Z h i 11 3 3 2 2 2sα + s ϕˆ |u| + (sϕ) ˆ 4 p e dxdt ∀s ≥ s0 (λ), Qω0

where C depends on λ. Next, let us consider a parabolic problem  ∂y   − ∆y = g(x, t),    ∂t

∂y
y| = 0, k(x)y + |Γ = 0,  Γ 2  ∂n 3    y(x, 0) = y0 (x) ∈ HΓ12 (Ω),

(3.23)

where k(x) ∈ L∞ (Γ3 ), g(x, t) ∈ L2 (0, T ; L2(Ω)).

Lemma 3.3. (Carleman inequality for a parabolic problem) ˆ > 1 such that for an arbitrary λ > λ ˆ there exists a s0 (λ) such There exists a number λ that for every s ≥ s0 (λ) the solutions to (3.23) satisfy Z   Z  Z  1 ∂y 2 2 3 3 2 2sα 2 2sα +sϕ|∇y| +s ϕ y e dxdt ≤ C g e dxdt+ s3 ϕ3 y 2 e2sα dxdt , sϕ ∂t

Q

Q

Qω

where C depends on λ. Proof. The proof of Theorem 2.1 (Carleman inequality for a parabolic equation with a mixed boundary condition) in [20] is valid with ones in (3.20) instead of . ϕ(x, t) = eλψ(x) [t(T − t)], . ϕ(x, ˜ t) = e−λψ(x) [t(T − t)],  . 2 ¯ α(x, t) = eλψ − eλ kψkC(Ω) [t(T − t)],  . 2 α(x, ˜ t) = e−λψ − eλ kψkC(Ω) [t(T − t)].

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Thus applying Theorem 2.1 in [20] with the change mentioned above, we arrive to the conclusion. Theorem 3.4. (Carleman inequality) Suppose that (w0 , ρ0 ) ∈ X, f1 ∈ L2 (0, T ; L2(Ω)), ˆ >1 div f1 ∈ L2 (0, T ; L2(Ω)) and f2 ∈ L2 (0, T ; L2(Ω)). Then, there exists a number λ ˆ such that for every λ > λ there exists a s0 (λ) such that for every s ≥ s0 (λ) the solutions to (3.2) satisfy Z h i 1 ∂w 2 2 ˆ + s3 ϕˆ3 |w|2 e2sα dxdt + sϕ|∇w| ˆ ∂t Q sϕ Z h 2 i 1 ∂ρ 2 3 3 2 2sα + + sϕ|∇ρ| + s ϕ |ρ| e dxdt Q sϕ ∂t Z Z h i 1 3 ˆ 2 |div f1 |2 e2sα dxdt ≤C |f1 |2 e2sα + (sϕ) ˆ 4 e2sˆα dxdt + (sϕ) (3.24) Q Q Z Z h i 11 + |f2 |2 e2sα dxdt + s3 ϕˆ3 |w|2 + (sϕ) ˆ 4 p2 e2sα dxdt Q Qω0  Z 3 3 2 2sα + s ϕ |ρ| e dxdt ∀s ≥ s0 (λ), Qω0

where C depends on λ. Proof. Put F1 = rot vˆ × w − rot (ˆ v × w) − ρ∇θˆ + f1 . By Lemma 3.1 and (2.2), F1 ∈ ˆ + div f1 ∈ L2 (0, ; L2(Ω)), L2 (0, T ; L2(Ω)). Since div F1 = div (rot vˆ × w) − div (ρ∇θ) ˜ applying Lemma 3.2 with F1 instead of f to the problem  ∂w   + νrot rot w − rot vˆ × w + rot (ˆ v × w) + ρ∇θˆ + ∇p = f1 ,   ∂t   div w = 0,    w |Γ0 = 0, w × n |Γ1 = 0, p |Γ1 = 0,    w(0) = w0 , ˆ1 we get for a s0,1 (λ), λ ≥ λ Z h i 1 ∂w 2 2 ˆ + s3 ϕˆ3 |w|2 e2sa dxdt + sϕ|∇w| ˆ ∂t Q sϕ Z h i 3 ≤C |f1 |2 e2sα + (sϕ) ˆ 4 e2sˆα dxdt Q Z 1 ˆ 2 (sϕ) + |div (rot vˆ × w) − div (ρ∇θ)| ˆ 2 e2sα dxdt Q  Z Z h i 1 11 2 2sα 3 3 2 2 2sα + |div f1 | (sϕ) ˆ 2 e dxdt + s ϕˆ |w| + (sϕ) ˆ 4 p e dxdt Q

Qω0

∀s ≥ s0,1 (λ), where C depends on λ.

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(3.25)

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On the other hand, applying Lemma 3.3 to the problem  ∂ρ   − ∆ρ − vˆ · ∇ρ + w · el = f2 ,    ∂t ∂ρ
= 0, ρ | = 0, b(x)ρ +  Γ 2  ∂n Γ3    ρ(0) = ρ0 ,

ˆ2 we have that for a s02 (λ), λ ≥ λ Z h i 1 ∂ρ 2 + sϕ|∇ρ|2 + s3 ϕ3 |ρ|2 e2sα dxdt Q sϕ ∂t Z Z
≤C |ˆ v · ∇ρ|2 + |w · el |2 + |f2 |2 e2sα dxdt + Q

Qω0

 s3 ϕ3 |ρ|2 e2sα dxdt

(3.26)

∀s ≥ s02 (λ), where C depends on λ. Taking (2.2) into account, we have Z 1 |div (rot vˆ × w)|2(sϕ) ˆ 2 e2sα dxdt Q

≤C

Z

≤C

Z

1

|rot vˆ|2 |∇w|2 e2sα (sϕ) ˆ 2 dxdt +

Q 2 2sα

|∇w| e

1 2

(sϕ) ˆ dxdt +

Q

Z

Z

Z

0

T

1 2

Z

 1 kˆ v k2W2 · kwesα k2L6 (sϕ) ˆ 2 dt 3

sα

k∇(we

0

2 2sα

T

1 )k2L2 (sϕ) ˆ 2

 dt

T

(k∇w · esα )k2L2  1 + kwsλ|∇ψ| · eλψ(x)ϕe ˆ sα )k2L2 )(sϕ) ˆ 2 dt Z  Z 1 5 2 2sα 2 2sα ≤ C(λ) |∇w| e (sϕ) ˆ 2 dxdt + |w| e (sϕ) ˆ 2 dxdt .

≤C

|∇w| e

Q

(sϕ) ˆ dxdt +

0

Q

Q

(3.27) In the same way, we get Z 1 ˆ 2 (sϕ) |div (ρ∇θ)| ˆ 2 e2sα dxdt Q Z  Z 1 5 ≤ C(λ) |∇ρ|2 e2sα (sϕ) ˆ 2 dxdt + |ρ|2 e2sα (sϕ) ˆ 2 dxdt . Q

Q

Now, taking s0 (λ) large enough, from (3.25)-(3.28) we can obtain the conclusion. 

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(3.28)

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4.

Tujin Kim and Daomin Cao

An Observability Inequality

In this section relying on the Carleman inequality, we obtain an observability inequality for the system  ∂w   + νrot rot w − rot vˆ × w + rot (ˆ v × w) + ρ∇θˆ + ∇p = f1 , −   ∂t     div w = 0,       − ∂ρ − ∆ρ − vˆ · ∇ρ + w · e = f , l 2 ∂t (4.1)    w |Γ0 = 0, w × n |Γ1 = 0, p|Γ1 = 0,     ∂ρ
  ρ |Γ2 = 0, b(x)ρ + = 0,    ∂n Γ3   w(T ) = wT , ρ(T ) = ρT , which ia adjoint to the linearized Boussinesq system. In the same way as in Lemma 2.8 in [19] we have

Lemma 4.1. Let f = (f1 , f2 ) ∈ L2 (0, T ; H) and (wT , ρT ) ∈ X. Then solutions to (4.1) satisfy

 

∂(w, ρ)

+ k(w, ρ)k ≤ c k(w, ρ)k + kf k T 3T 3T ˜ L2(0, 2 ; D(A)) L2(0, 4 ; H) L2 (0, 4 ; H) .

∂t T L2 (0, 2 ; H)

Take a function ` ∈ C ∞ ([0, T ]) such that `(t) = t ∀t ∈ [ 43 T, T ], `(t) > 0 ∀t ∈ [0, T ] and let ˜2 ˜ ¯ eλψ − eλ kψkC(Ω) , (4.2) k(x, t) = [`(t)(T − t)]8 ˜>λ ˆ is taken such that where λ ¯ < max{k(x, t) | x ∈ Ω}

95 ¯ min{k(x, t) | x ∈ Ω} 100

∀t ∈ [0, T ]

ˆ is the number in Theorem 3.4. and λ Let ˆ = min{k(x, t) | x ∈ Ω}, ˜ = max{k(x, t) | x ∈ Ω}. ¯ ¯ k(t) k(t)

(4.3)

Then, ˜

˜2

¯ eλψ(x) − eλ kψkC(Ω) k(x, t) = αλ˜ (x, t) = ∀(x, t) ∈ Ω × [t(T − t)]8



3 T, T 4



.

Theorem 4.2. (Observability inequality) Suppose that f ≡ (f1 , f2 ) ∈ L2 (0, T ; H) and (wT , ρT ) ∈ X and (w, p, ρ) is the solution to (4.1). Then there exists sˆ > 0 such that Z
kw(·, 0)k2V + kρ(·, 0)k2W 1 + (T − t)8 |w|2 + |ρ|2 esˆk dxdt 2Γ2 Q Z Z 
9 sˆkˆ
9 ˆ 2 2 10 ≤c |f1 | + |f2 | e dxdt + |w|2 + |ρ|2 e 10 sˆk dxdt , Q

Qω

where c is independent of fi and (wT (x), ρT (x)).

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Proof. Step 1. Applying the Carleman inequality to a new system obtained from (4.1), we will get the conclusion. Thus, first we get a new system. Let Z t Z t z(x, t) = w(x, τ )dτ, g(x, t) = p(x, τ )dτ, T 2

Fi (x, t) =

Z

T 2

t T 2

fi (x, τ )dτ,

σ(x, t) =

Z

t T 2

ρ(x, τ )dτ.

Then, integrating by parts we have Z τ Z t Z t Z th i ∂ˆ v (τ ) × (rot vˆ × w)(τ )dτ = rot vˆ(t) × w(τ )dτ − rot w(r)dr dτ, T T T T ∂τ 2 2 2 2 Z τ Z t Z t h i Z t h ∂ˆ i v (τ ) × rot (ˆ v × w)(τ )dτ = rot vˆ(t) × w(τ )dτ − rot w(r)dr dτ, T T T T ∂τ 2 2 2 2 Z t ∂w T
∂z T
dτ = w(t) − w = −w , T ∂τ 2 ∂t 2 2 Z t Z th Z t i ˆ )Z τ ∂∇θ(τ ˆ )dτ = ∇θˆ ρ∇θ(τ ρ(τ )dτ − ρ(r)dr dτ, T T T T ∂τ 2 2 2 2 Z t Z t Z th Z τ i ∂ˆ v (τ ) vˆ · ∇ρ(τ )dτ = vˆ∇ ρ(τ )dτ − ∇ ρ(r)dr dτ, T T T T ∂τ 2 2 2 2 Z t ∂ρ T
∂σ T
dt = ρ(t) − ρ = −ρ . T ∂t 2 ∂t 2 2

By these and (4.1), we have  T
∂z   + νrot rot z − rot vˆ × z + rot (ˆ v × z) + σ∇θˆ + ∇g = F1 − w −   ∂t 2    Z t Z t Z t  ˆ

 ∂ˆ v ∂ˆ v ∂∇θ(τ )   × z (ξ)dξ + × z (ξ)dξ + σ(τ ) dτ, − rot rot   T T T ∂τ ∂τ ∂τ   2 2 2     div z = 0,   Z t ∂σ T
∂ˆ v(τ )  − − ∆σ − vˆ · ∇σ + z · el = F2 − ρ − ∇σ(τ )dτ,   T ∂t 2 ∂τ   2     z |Γ0 = 0, z × n |Γ1 = 0, g |Γ1 = 0,      ∂σ
  σ |Γ2 = 0, b(x)σ + = 0,   ∂n Γ3    z(T ) = zT , σ(T ) = σT . (4.4) Step 2. Unlike the Carleman inequality in the observability inequality pressure does not attend. Thus, to remove pressure by the same argument as (2.28) of [18], from (4.4) we will get an estimate of kg(t)kL2(ω0 ) by other variables. To this end, on the domain ω let us consider the following system which is obtained from first two equations in (4.4) ( ν∆ z + ∇g = F, (4.5) div z = 0,

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where T
F = w(t) + rot vˆ × z − rot (ˆ v × z) − σ∇θˆ + F1 − w 2 Z t Z t Z t ˆ )

∂ˆ v ∂ˆ v ∂∇θ(τ × z (ξ)dξ + × z (ξ)dξ + σ(τ ) dτ. rot rot − T T T ∂τ ∂τ ∂τ 2

2

2

We fined a solution to (4.5) of the form z = z1 +z2 , g = g1 +g2 , where (z1 , g1 ) is a solution to  ν∆ z1 + ∇g1 = F,     div z1 = 0, (4.6) Z    g1 dx = 0.  z1 |∂ω = 0, ω0

Then, we know

kz1 kH1 (ω) + kg1 kL2 (ω) ≤ kF kH−1 (ω).

(4.7)

To estimate the right hand side of (4.7), for any u ∈ H10 (ω) we get the following estimates. Z Z t
∂ˆ v rot × z (ξ)dξu(x) dx ω T /2 ∂τ !1 !1 Z Z t 2 2 t

∂ˆ

2 v 2

∇ ≤ ckukL6 (ω) kzk dξ dξ L2 (ω) (4.8) ∂τ L3 (ω) T /2 T /2

∂ˆ v ≤ ckukH1(ω) W1 (ω) kzkL2 (T /2,t;L2(ω)) 0 α ∂τ

∂ˆ

v ≤ ckukH1(ω) W1 (ω) kwkL2(T /2,t;L2(ω)), 0 α ∂τ Z Z t
∂ˆ v × z (ξ)dξu(x) dx rot ω T ∂τ 2 !1 Z !1 Z t 2 2 t

∂ˆ

2 v 2

∇ ≤ ck∇ukL2(ω) dξ kzk dξ L2 (ω) (4.9) ∂τ L∞ (ω) T /2 T /2

∂ˆ v ≤ ckukH10(ω) W2 (ω) kzkL2 (T /2,t;L2(ω)) α ∂τ

∂ˆ

v ≤ ckukH10(ω) W2 (ω) kwkL2 (T /2,t;L2(ω)) , α ∂τ Z Z t ˆ ∂∇ θ(τ ) σ(τ ) dτ u(x) dx ω T ∂τ 2 !1 Z !1 Z t 2 2 t

∂ˆ

2 v 2

∇ ≤ ckukL6 (ω) dξ kσk dξ L2 (ω) ∂τ L3 (ω) T /2 T /2

∂ˆ v ≤ ckukH10(ω) W2 (ω) kσkL2(T /2,t;L2(ω)) α ∂τ

∂ˆ

v ≤ ckukH10(ω) W2 (ω) kρkL2(T /2,t;L2(ω)) . α ∂τ

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(4.10)

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Therefore by (2.2), (4.7)-(4.10), we get kz1 kH1 (ω) + kg1 kL2 (ω)  ≤ c kw(· , t)kL2(ω) + kz(· , t)kL2(ω) + kσ(· , t)kL2(ω) + kwkL2 (( T , t)×ω) 2 

T + kρkL2 (( T , t)×ω) + w(· , ) L (ω) + kF1 (·, t)kL2(ω) . 2 2 2

From (4.5), (4.6) we have that on ω (

ν∆ z2 + ∇g2 = 0, div z2 = 0,

(4.11)

(4.12)

which implies ∆2 z2 = 0. Thus, (cf. Subsection 1.17, ch. 2 in [27]) kz2 kC2 (¯ω0 ) ≤ ckz2 (· , t)kL2(ω) ≤ ckz(· , t) − z1 (· , t)kL2(ω) .

(4.13)

By (4.11) and (4.13) we get  kz2 kC2 (¯ω0 ) ≤ c kw(· , t)kL2(ω) + kz(· , t)kL2(ω) + kσ(· , t)kL2(ω) + kwkL2 (( T , t)×ω) 2 


T

+ kρkL2(( T , t)×ω) + w · , + kF1 (·, t)kL2(ω) . 2 2 L2 (ω)

Using this, by (4.12) we get  k∇g2 kC(¯ω0 ) ≤ c kw(· , t)kL2(ω) + kz(· , t)kL2(ω) + kσ(· , t)kL2(ω)



T


+ kwkL2 (( T , t)×ω) + kρkL2(( T , t)×ω) + w · , + kF (·, t)k . 1 L (ω) 2 2 2 2 L2 (ω) (4.14) R R To fix one, we find g such that ω0 g dx = 0 ∀t ∈ [0, T ]. Then, by (4.6) ω0 g2 dx = 0 ∀t ∈ [0, T ]. Thus, from (4.11) and (4.14) we get  kg(t)kL2 (ω0) ≤ c kw(· , t)kL2(ω) + kz(· , t)kL2(ω) + kσ(· , t)kL2(ω) 

T


+ kwkL2 (( T , t)×ω) + kρkL2(( T , t)×ω) + w · , + kF (·, t)k 1 L2(ω) , 2 2 2 L2 (ω) (4.15) which is what we want. Step 3. To apply the Carleman inequality to (4.4) we estimate the terms in the left hand sides of (4.4). First, let us estimate Z Z t Z t 2

∂ˆ v ∂ˆ v E1 = rot × z (ξ)dξ + rot × z (ξ)dξ e2sα(x,t)dxdt. − ∂τ ∂τ Q T /2 T /2

When x is fixed, α has its maximum at t = T /2 and is increasing and decreasing on [0,T/2] and [T/2,T], respectively. Thus, Z Z t

2 v ∂ˆ v rot ∂ˆ E1 ≤ c × z + rot × z e2sα(x,ξ)dξ dxdt, ∂τ ∂τ Q T /2

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which by (2.2) implies Z Z t Z t 2

∂ˆ v ∂ˆ v rot × z (ξ)dξ + rot × z (ξ)dξ e2sα dxdt − ∂τ ∂τ Q T /2 T /2 Z ≤ c(λ) (|∇z|2 + |z|2 )e2sα dxdt.

(4.16)

Q

3

The function (sϕ) ˆ 4 e2sˆα has the same property as α, and so in the same way we have Z t Z Z t 2

3 ∂ˆ v ∂ˆ v × z (ξ)dξ + × z (ξ)dξ (sϕ) ˆ 4 e2sˆαdxdt rot rot − ∂τ ∂τ T /2 Q T /2 (4.17) Z 3 2 2 2sˆ α ≤ c(λ) (|∇z| + |z| )(sϕ) ˆ 4 e dxdt. Q

Similarly, we have Z Z Q

Z Z Q

t T 2

t T 2

Z 2 ˆ ) ∂∇θ(τ 2sα σ(τ ) dτ e dxdt ≤ c(λ) |σ(τ )|2e2sα dxdt, ∂τ Q

Z 2 ˆ ) 3 3 ∂∇θ(τ |σ(τ )|2(sϕ) ˆ 4 e2sˆα dxdt, σ(τ ) dτ (sϕ) ˆ 4 e2sˆα dxdt ≤ c(λ) ∂τ Q Z Z t Z 2 ∂ˆ v(τ ) 2sα ∇σ(τ )dτ e dxdt ≤ c |∇σ|2 e2sα dxdt. T ∂τ Q Q

(4.18)

(4.19)

(4.20)

2

Also, let us estimate Z Z E2 = div Q

t

T /2

1

rot

2
1 ∂ˆ v × z (ξ)dξ (sϕ) ˆ 2 e2sα dxdt. ∂τ

Taking the property of (sϕ) ˆ 2 e2sα into account, we have Z 2
1 v div rot ∂ˆ E2 ≤ c × z (ξ)dξ (sϕ) ˆ 2 e2sα dxdt ∂τ Q Z T  Z T

∂ˆ

∂ˆ 1 1 v v 2 2 sα 2 sα 2



4 4 ≤ c(λ) · kz(sϕ) ˆ e kL6 dt + ˆ e kL2 dt 1 k∇z(sϕ) ∂τ Wα2 ∂τ W∞ 0 Z0 1 1 ˆ 2 e2sα s2 |∇α|2 ) dxdt ≤ c(λ) (|∇z|2 (sϕ) ˆ 2 e2sα + |z|2 (sϕ) Q  Z 1 2 2sα + |∇z| (sϕ) ˆ 2 e dxdt Q Z  Z 5 1 2 2 2sα 2 2sα 2 2 ≤ c(λ) (|∇z| + |z| )(sϕ) ˆ e + |z| (sϕ). ˆ e dxdt . Q

Q

(4.21)

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477

In the same way we get estimate Z Z div

2 ˆ ) 1 ∂∇θ(τ σ(τ ) dτ (sϕ) ˆ 2 e2sα dxdt T ∂τ Q 2 Z nZ o 5 1 ≤ c(λ) (|∇σ|2 + |σ|2)e2sα (sϕ) ˆ 2 dxdt + |σ|2 e2sα (sϕ) ˆ 2 dxdt . t

Q

(4.22)

Q

Step 4. Now, applying Theorem 3.4 to (4.4) and using (4.16)-(4.22), we have Z h i 1 2 |w|2 + sϕ|∇z| ˆ + s3 ϕˆ3 |z|2 e2sα dxdt ˆ Q sϕ Z h i 1 + |ρ|2 + sϕ|∇σ|2 + s3 ϕ3 |σ|2 e2sα dxdt Q sϕ nZ 3 T
2
2sα ˆ ≤ c(λ) sϕˆ 4 |F1 |2 + w x, e dxdt 2 Q (4.23) Z

T 2 e2sα dxdt + |F2 |2 + ρ x, 2 Q Z Z h i o 11 2 2sα 3 3 2 2sα 4 + (sϕ) ˆ g e + s ϕˆ |z| e dxdt + s3 ϕ3 |σ|2e2sα dxdt Qω0

Qω0

ˆ ∀s ≥ s0 (λ), λ ≥ λ. From now using (4.23), (4.15), Lemma 4.1 and Theorem 3.4, by the same arguments as (2.83)-(2.100) in [19] we arrive to the conclusion. 

5.

Null Controllability of the Linearized Problem

Relying on the observability inequality, we prove null controllability of the linearized problem, which is a base for the main results. Let us consider the problem  ∂y   L1 (y, τ ) ≡ + νrot rot y + rot vˆ × y + rot y × vˆ + τ el = −∇q + f1 + χω u1 ,   ∂t     div y = 0,       L (y, τ ) ≡ ∂τ − ∆τ + vˆ · ∇τ + y · ∇θˆ = f + χ u , 2 2 ω 2 ∂t   y |Γ0 = 0, y × n |Γ1 = 0, q |Γ1 = 0,      ∂τ
  τ |Γ2 = 0, b(x)τ + = 0,    ∂n Γ3   y(0) = y0 , τ (0) = τ0 . (5.1)
Definition 5.1. A function (y, τ ) ∈ L2 (0, T ; V ) ∩ C([0, T ]; H) × L2 (0, T ; HΓ12 ) ∩
∂τ ∗ 1 ∗ C([0, T ]; L2 ) is called a solution to (5.1) if ( ∂y ∂t , ∂t ) ∈ L2 (0, T ; V ) × L2 (0, T ; (HΓ2 ) ),

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(y(0), τ (0)) = (y0 , τ0 ) and

∂y  , w + ν(rot y, rot w) + (rot vˆ × y + rot y × vˆ + τ e3 , w) ∂t

∂τ  ˆ φ) + (b(x)τ, φ)Γ , φ + (∇τ, ∇φ) + (ˆ v · ∇τ + y · ∇θ, + 3 ∂t 1 = hf1 + u1 , wi + hf2 + u2 , φi ∀(w, φ) ∈ L2 (0, T ; V ) × L2 (0, T ; HΓ2 ). Remark 5.1. As (3.2) in Section 2 unique existence of a solution to (5.1) in the space W and Z (for smooth given data) and estimations (3.18) and (3.19), respectively, can been obtained. Let η(x, t) = −ˆ sk(x, t), θ(x, t) = (1 − χω )

eη + χω , (T − t)8

where sˆ is in Theorem 4.2, and L2 (Q, θ) be a weighted L2 -space with weight θ(x, t). Let  F1 (Q, θ) = f1 ∈ L2 (Q) : ∃f11 ∈ L2 (Q, θ), ∃f12 ∈ L2 (0, T ; HΓ11 (Ω)), f1 = f11 + ∇f12 ,   1 2 2 2 : f1 = f11 + ∇f12 ∈ F1 , kf1 kF1 (Q, θ) = inf kf11 kL2 (Q, θ) + k∇f12 kL2 (Q) F (Q, θ) = F1 (Q, θ) × L2 (Q, θ), L(y, τ ) ≡ (L1 (y, τ ), L2(y, τ )), (y, τ ) ∈ Z, n o 2 ˆ Y (Q) = (y, τ ) ∈ Z : L(y, τ ) ∈ F (Q, θ), e− 5 sˆk (y, τ ) ∈ Z , 2

ˆ

k(y, τ )kY = kL(y, τ )kF (Q,θ) + ke− 5 sˆk (y, τ )kZ . Theorem 5.1. (Null controllability) Suppose that (y0 , τ0) ∈ X, f ≡ (f1 , f2 ) ∈ F (Q, θ). Then, there exists a solution (y, p, τ, u ≡ (u1 , u2 )) to (5.1) such that (y, τ )(T ) = 0, ((y, τ ), u) ∈ Y × L2 (Qω ),


k((y, τ ), u)kY ×L2 (Qω ) ≤ c k(y0 , τ0 )kX + kf kF (Q, θ) .

Proof. The operator B ∗ (t) ∈ L(X; H) adjoint to B(t) is expressed by
B ∗ (y, τ ), (w, ρ) H = (rot vˆ × y + rot y × vˆ + τ e3 , w)L2 + (ˆ v · ∇τ + y · ∇θˆ − k0 τ, ρ)L ∀(w, ρ) ∈ H, 2

where k0 is the one in (3.5). ¯ (see Lemma 3.1), the existence of a solution By definitions of operators A, B ∗ and Π (y, τ ) ∈ Z to (5.1) (cf. Remark 5.1) is equivalent to the existence of a solution to the problem   ∂(y, τ ) + (A + B ∗ )(y, τ ) = Π(f ¯ + u), ∂t (5.2)  (y, τ )(0) = (y0 , τ0 ) ∈ X.

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Local Exact Controllability of the Boussinesq Equations ... First, suppose that f ∈ L2 (Q, θ) and f |Qω = 0. Let us consider an extremal problem  ∂(y, τ )  ¯ + u),  + (A + B ∗ )(y, τ ) = Π(f   ∂t     (y, τ )(0) = (y , τ ) ∈ X, 0

where

0

 u ∈ L2 (0, T ; L2 (Ω)),   Z Z   1 1  2 2   Jk (y, τ, u) = σk (|y| + |τ | )dxdt + mk |u|2 dxdt → inf, 2 Q 2 Q 8 ˆ − 9ˆsk(t)(T −t) 8 10(T −t+1/k)

σk = e

,

mk (x, t) =

 

−

e

ˆ 9ˆ sk(t)(T −t)8 10(T −t+1/k)8

 k,

479

(5.3)

, x∈ω ¯,

x ∈ Ω\¯ ω.

It is easy to verify the existence of a unique solution to the problem, which is denoted by (ˆ yk , τˆk , u ˆk ) ∈ Z × L2 (Q). Define F : W × U 7→ V ≡ W ∗ × H by   ∂(y, τ ) ∗ ¯ F (y, τ ) ≡ − − (A + B )(y, τ ) + Π(f + u), (y(0), τ(0)) . ∂t Since there exists a unique solution to (5.2) for any (f1 , f2 ) ∈ W ∗ and (y0 , τ0 ) ∈ H, Im D(y,τ )F (y, τ ) = W ∗ × H, where D(y,τ ) means derivation with respect to (y, τ ). Therefore, Lagrange function to the extremal problem (5.3) is Lk (y, τ, r) = Jk (y, u)+ < F (y, τ ), r > ∀r ∈ W × H.

(5.4)

Then, by the Lagrange principle (cf. Theorem 1.5, ch. 2 in [11]) there exists a ((wk , ρk ), ak ) ∈ W × H such that hσk (ˆ yk , τˆk ), hi − Z

∂h  + (A + B ∗ )h, (wk, ρk ) + (h(0), ak ) = 0 ∀h ∈ W, ∂t

[mk u ˆk · u + (wk , ρk ) · u]dxdt ≥ 0 ∀u ∈ L2 (0, T ; L2 (Ω)).

(5.5) (5.6)

Q

From (5.5) and (5.6) we have, respectively, ∂(wk , ρk ) + (A + B)(wk , ρk ) = σk (ˆ yk , τˆk ), ∂t (wk , ρk )(T ) = 0, −

(5.7)

(in addition,(wk , ρk )(0) = ak ) and mk u ˆk + (wk , ρk ) = 0.

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(5.8)

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Tujin Kim and Daomin Cao

Now applying Theorem 4.2, in the same way as Theorem 3.2 in [19] we come to the assertion. Let us consider the case that f ≡ (f1 , f2 ) ∈ F (Q, θ), f1 = f11 + ∇f12 , f11 ∈ L2 (Q, θ), f12 ∈ L2 (0, T ; HΓ11 (Ω)) and f2 ∈ L2 (Q, θ). Let (y, p, τ, u) be the solution when f = (1 − χω )(f11 , f2 ). Then, (y, p + f12 , τ, u + χω (f11 , f2 )) is a solution corresponding to f . 

6.

Local Exact Controllability of the Nonlinear Problem

Relying on the observability inequality, in this section we study local exact controllability of the Boussinesq problem. In what follows we consider the case such that the following assumption is true. Assumption 6.1. (Assumption 4.1 in [19], cf. Remark 4.2 in [19]) A1 u ∈ H

implies

∆u ∈ L2 (Ω).

Lemma 6.1. (Lemma 4.4 in [19]) Under Assumption 6.1 the operator (y, 0) 7→ rot y × y is continuous from Y (Q) to F1 (Q, θ). Lemma 6.2. Assume that if l = 3, then Γ3 = ∅ or Γ2 ∩ Γ3 = ∅. Then, for (y, τ ) ∈ Y (Q) the operator (y, τ ) 7→ y · ∇τ is continuous from Y (Q) to L2 (Q, θ). ˆ we have Proof. Applying η ≡ −ˆ sk < −ˆ sk, Z Z h i 2 2 2 η −8 ky · ∇τ kL2 (Q, θ) ≤ c |y| |∇τ | e (T − t) dxdt + |y|2 |∇τ |2 dxdt Q\Qω Qω Z ˆ ≤c |y|2 |∇τ |2 e−ˆsk (T − t)−8 dxdt Q Z 8 ˆ 3 ˆ ≤c |y|2 |∇τ |2 e− 5 sˆk e 5 sˆk (T − t)−8 dxdt Q Z 2 ˆ 2 ˆ ≤c |e− 5 sˆk y|2 |e− 5 sˆk ∇τ |2 dxdt.

(6.1)

Q

On the other hand, from 2 ˆ e− 5 sˆk y ∈ L2 (0, T ; D(A˜1 )),

∂ − 2 sˆkˆ (e 5 y) ∈ L2 (0, T ; H), ∂t

we have (cf. Proposition 2.1 and Theorem 3.1 of ch. 1 in [21]) 2

ˆ

e− 5 sˆk y ∈ L (0, T ; V ),

− 2 sˆkˆ ∞
2 ˆ ˆ ∂ − 25 sˆk

e 5 y ≤ K ke− 5 sˆk ykL2 (0,T ; D(A˜1 )) + k ∂t e ykL2 (0,T ; H) . L∞ (0,T ; V ) 2

ˆ

Now, let l = 2. From e− 5 sˆk τ ∈ L2 (0, T ; D(A˜2 )) we have that 2

ˆ

∆e− 5 sˆk τ ∈ L2 (0, T ; L2(Ω)),

− 2 sˆkˆ − 25 sˆˆ k

∆e 5 τ ≤ Kke τ kL2 (0,T ; D(A˜2 )), L (0,T ;L (Ω)) 2

2

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(6.2)

Local Exact Controllability of the Boussinesq Equations ... 2

481

ˆ

(cf. Remark 3.1) and so e− 5 sˆk τ ∈ L2 (0, T ; H α(Ω)), where α < 3/2 (cf. [23]). Thus, 2

ˆ

e− 5 sˆk ∇τ ∈ L2 (0, T ; L3 (Ω)),

− 2 sˆkˆ

2 ˆ

e 5 ∇τ ≤ K e− 5 sˆk τ L L (0,T ; L (Ω)) 2

˜ 2 (0,T ; D(A2 ))

3

.

(6.3)

2 ˆ Let l = 3. By the assumptions, from e− 5 sˆk τ ∈ L2 (0, T ; D(A˜2 )) we have 2

ˆ

e− 5 sˆk τ ∈ L2 (0, T ; H 2 (Ω)),

− 2 sˆkˆ

2 ˆ

e 5 τ ≤ K e− 5 sˆk τ L L (0,T ; H 2 (Ω)) 2

˜

2 (0,T ; D(A2 ))

,

and so (6.3) is valid for l = 3. By (6.1)-(6.3) we have

ky · ∇τ k2L2 (Q,θ) ≤ Kk(y, τ )k2Y , which ends the proof of Lemma 6.2.  Now let us consider local exact controllability of the Boussinesq system. ˆ t)) be a Theorem 6.3. (Local exact controllability by internal controls) Let (ˆ v (x, t), θ(x, solution to (2.1) which satisfies (2.2) and Assumption 6.1 and the assumption in Lemma 6.2 be satisfied. Then, there exists ε > 0 such that for any pair (v0 , θ0 ) ∈ X satisfying k(v0 , θ0 )kX ≤ ε there exist u ∈ L2 (Q) which satisfy  ∂v   − ν∆v + (v, ∇)v + ∇p + θel = fˆ1 + χω u1 ,    ∂t     div v = 0,     ∂θ    − ∆θ + v · ∇θ = fˆ2 + χω u2 ,    ∂t
1 v |Γ0 = 0, v × n |Γ1 = c(x, t) × n, p + |v|2 Γ = ϕ1 (x, t),  1  2   
∂θ   = d(x, t)  θ |Γ2 = a(x, t), b(x)θ +   ∂n Γ3     v(0) = v0 + vˆ0 , θ(0) = θ0 + θˆ0 ,      v(T ) = vˆ(T ), θ(T ) = θ(T ˆ ).

Proof. It is enough to prove local null controllability of (2.4). The existence of a solution (y, τ ) in Z to (2.4) in the sense of Definition 2.1 is equivalent to the existence of a solution (y, τ ) ∈ Z to the problem   ∂(y, τ ) + (A + B ∗ )(y, τ ) + Π(rot ¯ ¯ ω u, y × y, y · ∇τ ) = Πχ ∂t (6.4)  (y, τ )(0) = (v0 , θ0 ) ∈ X,

¯ is the one in Lemma 3.1. Let X = Y (Q) × L2 (Qω ), Z = ΠF ¯ (Q, θ) × X, where Π ¯ (y, τ, u), (y(0), τ (0))}, where A {(y, τ ), u} = {ΠN n ∂y N (y, τ, u) = + νrot rot y + rot vˆ × y + rot y × vˆ + rot y × y + τ el − χω u1 , ∂t o ∂τ − ∆τ + vˆ · ∇τ + y · ∇θˆ + y · ∇τ − χω u2 . ∂t

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By Lemma 6.1, 6.2 and the definition of space F (Q, θ), A ∈ (X → Z ). When x0 = (0, 0) ∈ X and z0 = (0, 0) ∈ Z , A (x0 ) = z0 . Also, the operator A is continuously differentiable and n
∂y A 0 (x0 )(y, τ, u) = Π − ν∆y + rot vˆ × y + rot y × vˆ + τ el − χω u , ∂t

o ∂τ − ∆τ + vˆ · ∇τ + y · ∇θˆ − χω u2 , y(0), τ (0) . ∂t By Theorem 5.1, for any f ∈ F (Q, θ) and (v0 , θ0 ) ∈ X there exists a solution ((y, τ ), u) ∈ X to   ∂(y, τ ) + (A + B ∗ )(y, τ ) − Πχ ¯ ω u = Πf, ¯ ∂t  (y, τ )(0) = (v0 , θ0 )

(cf. (5.2)). This means that A 0 (x0 ) ∈ (X 7→ Z ) is an epimorphism. Therefore, when k(v0 , θ0 )kX ≤ ε for sufficiently small ε > 0, there exists a solution (y, τ, u) ∈ X with ¯ (y, τ, u) = 0 (cf. Theorem 4.1 in [18]), which (y(0), τ (0)) = (v0 , θ0 ) to the equation ΠN means that (y, τ, u) satisfies (6.4). Also, by the definition of Y (Q) (y(T ), τ (T )) = 0, by which we come to the conclusion.  Now, let us turn to the boundary control problem. Let Γc be an open subset of Γ0 ∩ Γ2 ¯ c ⊂ (Γ0 ∩ Γ2 ). We are concerned with the following boundary control problem such that Γ  ∂v   − ν∆v + (v, ∇)v + ∇p + θel = fˆ1 ,   ∂t      div v = 0,     ∂θ    − ∆θ + v · ∇θ = fˆ2 , ∂t (6.5) 1 2
   v |Γ0 \Γc = 0, v|Γc = u1 , v × n |Γ1 = c(x, t) × n, p + |v| Γ1 = ϕ1 (x, t),   2   
∂θ   = d(x, t)  θ |Γ2 \Γc = 0, θ|Γc = u2 , b(x)θ +   ∂n Γ3    v(0) = v0 + vˆ0 , θ(0) = θ0 + θˆ0 .

ˆ t)) be a Theorem 6.4. (Local exact controllability by boundary controls) Let (ˆ v(x, t), θ(x, ∂ vˆ ∂ θˆ solution to (2.1) with a(x, t) = 0 which satisfies (2.2), ∂n |Γ0 = 0, ∂n |Γ2 = 0, Γc 6= ∅ and Assumption 6.1 and the assumption in Lemma 6.2 be satisfied. If ε > 0 is small enough,
1 for (v0 , θ0 ) ∈ X satisfying k(v0 , θ0 )kX ≤ ε there exists a control u ∈ L2 0, T ; H 2 (Γc ) ×
1 ˆ ). L2 0, T ; H 2 (Γc ) to problem (6.5) by which (v, θ)(T ) = (ˆ v, θ)(T

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483

Proof. Setting v = vˆ + y, p = pˆ + p1 , θ = θˆ + τ in (6.5) we have the following problem  ∂y   + νrot rot y + rot vˆ × y + rot y × vˆ + rot y × y + ∇q + τ el = 0,   ∂t     div y = 0,       ∂τ − ∆τ + vˆ · ∇τ + y · ∇θˆ + y · ∇τ = 0, ∂t (6.6)   y | = 0, y | = u , y × n | = 0, q | = 0,  Γc 1 Γ1 Γ1 Γ0 \Γc    
∂τ   = 0, τ |Γ2\Γc = 0, τ |Γc = u2 , b(x)τ +   Γ3  ∂n   y(0) = v0 , τ (0) = θ0 ,

where q = p1 + vˆ · y + 12 |y|2 . To prove the assertion it suffices to prove local null ˜ ⊃ Ω such that Ω ˜ T ∂Ω ⊂ Γc controllability of (6.6). Take a connected domain Ω ˜ be null extensions of v0 , θ0 ,ˆ ˜ ∈ C 2 . Let y˜0 , τ˜0 , v˜, θ, ˜ and ∂ Ω v , θˆ respectively, on Ω\Ω. 1 ˜ Then, we can see that y˜0 ∈ {u ∈ H (Ω) : div u = 0, u |∂ Ω\Γ ˜ 1 = 0, u × n |Γ1 = 0}, 1 ˜ τ˜0 ∈ {v ∈ W (Ω) : v | ˜ = 0}(cf. Theorem 11.4 of ch. I in [21]) and v˜ ∈ 2 1 2 ˜ W∞ (0, T ; Wα(Ω)), div˜ v

∂ Ω\Γ3

1 ˜ = 0, θ˜ ∈ W∞ (0, T ; Wα2(Ω))(cf. Theorem 11.4 of ch. I in [21] ˜ and Definition 1.10 of ch. IV in [14]). Set ω = Ω\Ω. Let us consider the following control ˜ × (0, T ) problem on Ω  ∂y   + νrot rot y + rot v˜ × y + rot y × v˜ + rot y × y + ∇q + τ el = χω u1 ,    ∂t     div y = 0,       ∂τ − ∆τ + v˜ · ∇τ + y · ∇θ˜ + y · ∇τ = χ u , ω 2 ∂t (6.7)   y | = 0, y × n | = 0, q | = 0,  ˜ Γ Γ 1 1 ∂ Ω\Γ1    
∂τ   = 0,  τ | ˜ 3 = 0, b(x)τ +  Γ3  ∂ Ω\Γ ∂n    y(0) = y˜ , τ (0) = τ˜ . 0 0

By the argument in the proof of Theorem 6.3, we can see that if ε > 0 small enough, ˜ × (0, T )) for (v0 , θ0 ) ∈ X satisfying k(v0 , θ0 )kX ≤ ε there exists a control u ∈ L2 (Ω 1 in (6.7) by which y(T ) = 0, τ (T ) = 0. Then, u1 = y|Γc ×(0,T ) ∈ L2 (0, T ; H 2 (Γc )) and 1

u2 = τ |Γc ×(0,T ) ∈ L2 (0, T ; H 2 (Γc )) are the controls in the assertion. 

References [1] Alekseev, G. V. & Smishliaev, A. B. (2001). Solvability of the boundary value problems for Boussinesq equations with inhomogeneous boundary conditions. J. Math. Fluid Mech. 3, 18-39. [2] Begue, C. & Conca, C. & Murat, F. & Pironneau, O. (1987). A nouveau sur les equations de Stokes et de Navier-Stokes avec des conditions aux limites sur la pression. C. R. Acad. Sci. Paris, Serie I t. 304, 23-28.

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[3] Begue, C. & Conca, C. & Murat, F. & Pironneau, O. (1988). Les e´ quations de Stokes et de Navier-Stokes avec des condition sur la pression. In H. Brezis & J. L. Lions (Eds.), Nonlinear Partial Differential Equations and their Applications, College de France, Seminar, Vol. IX (pp. 179-264), Longman Scientific & Technical. [4] Bernard, J. M. (2002). Non-standard Stokes and Navier-Stokes problems: existence and regularity in stationary case. Mathematical Methods in the Applied Sciences 25, 627-661. [5] Bernard, J. M. (2003). Time-dependent Stokes and Navier-Stokes problems with boundary conditions involving pressure: existence and regularity. Nonlinear Analysis: Real World Applications 4, 805-839. [6] Brenner, S. C. & Scott, L. R. (2008). The mathematical theory of finite element methods, Springer. [7] Conca, C. & Murat, F. & Pironneau, O. (1994). The Stokes and Navier-Stokes equations with boundary conditions involving the pressure. Japan. J. Math. Vol. 20 (2), 279-318. [8] Evans, L. C. (1998). Partial differential equations. American Mathematical Society. [9] Fern´andez-Cara, E. & Guerrero, S. & Imanuvilov, O. Yu. & Puel, J.-P. (2005). On the controllability of N-dimensional Navier-Stokes and Boussinesq systems with N-1 scalar controls. C. R. Acad. Sci. Paris, Serie I t. 340, 275-280. [10] Fern´andez-Cara, E. & Guerrero, S. & Imanuvilov, O. Yu. & Puel, J.-P. (2006). Some controllability results for the N-dimensional Navier-Stokes and Boussinesq systems with N-1 scalar controls. SIAM J. Control Optimiz. Vol. 45 (1), 146-173. [11] Fursikov, A. V. (2000). Optimal control of distributed systems, Theory and Applications. Amer. Math. Society. [12] Fursikov, A. V. & Imanuvilov, O. Yu. (1996). Controllability of evolution equation, Lecture Notes Series No. 34. SNU. [13] Fursikov, A. V. & Imanuvilov, O. Yu. (1999). Exact controllability of the NavierStokes and Boussinesq equations. Russian Math. Surveys 54, 565-618. [14] Gajewski, H. & Gr¨oger, K. & Zacharias, K. (1974). Nichtlinear operatorgleichungen und operatordifferentialgechungen. Academie-Verlag. [15] Gonz´alez-Burgos, M. & Guerrero, S. & Puel, J.-P. (2009). Local exact controllability to the trajectories of the Boussinesq system via a fictitious control on the divergence equation. Communic. Pure and Appl. Analysis Vol. 8 (1), 311-332. [16] Guerrero, S. (2006). Local exact controllability to the trajectories of the Boussinesq system, Anales de Institut Henri Poincare-AN 23, 29-61.

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[17] Imanuvilov, O. Yu. (1998). Local exact controllability for the 2-D Boussineq equations with the Navier slip boundary conditions. ESAIM: COCV (http://www.emath.fr/cocv/.) Vol. 1, 153-170. [18] Imanuvilov, O. Yu. (2001). Remarks on exact controllability for the Navier-Stokes equations, ESAIM: COCV (http://www.emath.fr/cocv/.) Vol. 6, 39-72. [19] Kim, T. & Cao, D. (2010). Local exact controllability of the Navier-Stokes equations with the condition on the pressure on parts of the boundary. SIAM J. Control and Optim. Vol. 48 (6), 3805-3837. [20] Kim, T. & Chang, Q. & Xu, J. (2008). A global Carleman inequality and exact controllability of parabolic equations with mixed boundary conditions. Acta Mathematicae Applicatae Sinica Vol. 24 (2), 265-280. [21] Lions, J. L. & Magenes, E. (1968). Problems aux lmites non homogenes et applications Vol. 1, Dunod, Paris. [22] Pironneau, O. (1986). Conditions aus limites sur la pression pour les equations de Stokes et Navier-Stokes. C. R. Acad. Sci. Paris, Serie I t. 303, 403-406. [23] Savar´e, G. (1997). Regularity and perturbation results for mixed second order elliptic problems. Comm. Partial Differential Equations V. 22 (5-6), 869-899. [24] Teman, R. (1985). Navier-Stokes Equations. North-Holland. [25] Teman, R. (1997). Infinite-dimensional dynamical systems in mechanics and physics. Springger-Verlag. [26] Wang, L. & Wang, G. (2003). Local internal controllability of the Boussinesq system. Nonlinear Analysis 53, 637-652. [27] Yegorov, Yu. V. & Shubin, M. A. (1992). Partial differential equations, 1, Encyclopedia of Mathematical Sciences, Vol. 30. Springger-Verlag, (Itogi Nauki i Texniki, Modern problems of mathematics, Fundamental directions (Russian), Vol. 30, 1988.)

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INDEX # 3D Navier-Stokes equations, xi, 177, 178

boundary surface, 408 boundary value problem, 254, 440, 464, 466, 483 bounds, xii, 209, 213, 214, 223 Boussinesq equations, xiv, 461, 462, 483, 484 building, 159, 160, 161, 162, 164, 174, 175

A Adaptive fuzzy solutions, 209 aerodynamic, x, 45, 46, 54, 121, 122, 136, 140, 141, 159, 162, 164, 368, 441 aerodynamic structure, 121, 141, 159, 162 air quality, 159, 174 air temperature, 144 air-sea interaction, 265 air-water flow, xii, 265, 266 algorithm, xii, xiv, 3, 83, 161, 209, 365, 368, 369, 371, 398, 403, 407, 425, 439, 440 amplitude, 33, 59, 82 aneurysm, ix, x, 1, 2, 8, 11, 12, 16, 17, 18, 21, 22, 25, 28, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42 angiogram, 32 angiography, 12, 32, 33, 35 Ansys CFX®, ix, x, 1, 8, 11, 16, 18, 21, 28, 31, 37, 38, 42 ANSYS-Fluent 17.0, xi, 143 aorta, 1, 11 approximation error bounds, 209 artery, ix, x, 1, 2, 4, 5, 6, 11, 12, 14, 15, 16, 18, 21, 22, 24, 25, 28, 32, 33, 34, 40, 41 asymmetry, 49 asymptomatic, 2, 21 atmospheric pressure, 60, 104, 162 atoms, 320, 321, 322 atrium, 161, 175

B Banach spaces, 463 basilar artery, 32, 33 biodiesel, x, 57, 58, 59 blood flow, ix, x, 1, 5, 8, 11, 12, 16, 18, 21, 22, 25, 28, 31, 32, 34, 35, 36, 38, 40

C cardiovascular system, 3 Carleman inequality, xiv, 461, 462, 469, 470, 472, 473, 475, 485 case study, 161, 290 Cauchy problem, xi, 177, 178, 190, 203, 204, 208 Central Europe, 79, 96 cerebral aneurysm, 31, 32, 33, 34, 35, 36 cerebrospinal fluid, 32, 33 cerebrovascular disease, 32 CFD, ix, x, xi, xiii, 1, 2, 3, 5, 8, 11, 12, 14, 16, 21, 22, 23, 25, 27, 29, 32, 33, 34, 35, 36, 42, 45, 46, 47, 55, 58, 59, 79, 80, 81, 82, 83, 84, 86, 87, 96, 97, 99, 122, 143, 144, 146, 147, 149, 156, 159, 160, 161, 174, 304, 349, 350, 351, 363, 365, 366, 367, 370, 371, 372, 396, 412, 418, 425, 427 CFD code Ansys-FLUENT, x, 81 chemical properties, 58 chemicals, 58, 80, 81, 97 circular flow, 41 circulation, 367 classes, xii, 275, 277, 280, 318 classical electrodynamics, 327 classification, xii, 275, 279, 317 clean air, 162 climate, 161, 174 closure, 231, 374, 375, 463 clustering, 2 coatings, 292 commercial, x, xi, 45, 47, 81, 86, 143, 156, 161, 285, 350, 351 complexity, 368, 369, 371 compressibility, 290, 291, 367 compression, 65, 91, 151, 152, 168, 170, 359 computation, xii, 2, 229, 265, 366, 385, 387, 403, 425, 438, 439, 441

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computational fluid dynamic (CFD), ix, 1, 18, 25, 29, 99, 101, 162, 427, 439 computational fluid dynamics, ix, x, xi, xiii, 2, 3, 11, 18, 21, 35, 55, 99, 143, 144, 160, 174, 365, 366 Computed Tomography (CT), ix, 1, 2, 3, 11 computer, x, xi, xiii, 2, 31, 34, 57, 81, 104, 123, 140, 144, 159, 285, 365, 366 computer simulations, x, 34, 57, 81 computer technology, xiii, 365, 366 computing, xiii, 365, 366, 368, 371, 382, 392, 395, 403, 416, 425, 434 conduction, xii, 161, 206, 275, 276, 277, 278, 281, 287, 296, 299, 301 conductivity, 373, 374, 389, 426 conservation, xiii, xiv, 48, 150, 266, 269, 305, 365, 372, 380, 381, 400, 425 conserving, 297 construction, 160, 231, 251, 259, 280, 369, 370, 385, 406, 445, 458 consumption, 327 controllability, 461, 463, 465, 467, 469, 471, 473, 475, 477, 479, 480, 481, 483, 484, 485 convergence, 208, 217, 218, 262, 283, 326, 327, 367, 368, 369, 392, 393, 394, 395, 400, 405, 406, 410, 416, 418, 419, 422, 425, 440 cooling, 160, 161, 442 cost, xii, 38, 84, 209 covering, 146, 246 CPU, 399, 425 CSF, 33 cyanosis, 2 cycles, 318, 321, 322, 399, 400, 404, 410, 411, 412, 422, 423

D damping, 300, 376 data structure, xiii, 365 decay, 248, 256, 276, 281, 287, 289, 290, 299, 300, 322 decomposition, 230, 262, 337, 396, 466 deformation, x, 8, 25, 31, 39, 322, 352, 353, 354, 355 depression, 51, 53, 65, 127, 152, 359 depth, 100, 115, 118, 119, 290 derivatives, 180, 244, 247, 266, 267, 268, 269, 270, 272, 278, 280, 284, 288, 301, 312, 313, 314, 316, 329, 367, 396, 397, 446, 447, 448, 456, 458 design, xii, xiii, 3, 46, 55, 59, 79, 81, 82, 96, 140, 141, 144, 159, 161, 162, 174, 175, 209, 216, 365, 366, 367 diesel, x, 57, 58, 59 differential equations, xiv, 326, 328, 371, 443, 462, 484, 485 diffusion, xi, 99, 101, 104, 108, 109, 110, 118, 276, 281, 289, 295, 413 diffusivity, 296, 298, 299 discontinuity, 111, 268, 272

discretization, x, xi, 45, 47, 58, 82, 84, 99, 101, 159, 162, 215, 366, 367, 369, 371, 372, 377, 378, 379, 388, 392, 394, 399, 402, 412, 417, 422, 425, 433, 438, 441 displacement, 32, 33, 329 distribution, ix, 1, 7, 8, 11, 17, 18, 25, 27, 28, 32, 33, 34, 38, 40, 41, 42, 49, 50, 51, 52, 53, 61, 62, 64, 65, 66, 67, 68, 70, 72, 89, 91, 92, 94, 95, 107, 111, 112, 124, 126, 131, 134, 135, 149, 150, 151, 152, 153, 154, 155, 164, 167, 169, 170, 171, 238, 239, 241, 276, 299, 300, 350, 354, 359, 362, 372, 385, 409, 410, 411, 415, 416, 417, 418, 419, 421, 423, 424, 441 divergence, xiii, 144, 179, 231, 246, 259, 260, 261, 305, 306, 309, 310, 315, 317, 319, 326, 332, 342, 395, 446, 451, 484 DOI, 174, 363 doppler, 80, 97 duality, 234, 463 dust storms, 160, 174 dynamic viscosity, 2, 12, 34, 36, 54 dynamical properties, 278 dynamical systems, 296, 297, 485

E Ehlers-Danlos syndrome, 12 electric circuits, 313 electricity, 46, 121, 143 electromagnetism, 229 embolization, 34 emission, 160 energy, xii, 33, 34, 45, 48, 54, 55, 58, 59, 62, 68, 69, 70, 71, 74, 75, 76, 77, 82, 84, 87, 88, 92, 93, 94, 113, 121, 123, 131, 132, 140, 143, 144, 146, 150, 153, 154, 155, 159, 160, 161, 169, 170, 171, 174, 175, 178, 216, 218, 224, 229, 230, 231, 287, 297, 323, 326, 328, 342, 343, 351, 352, 371, 372, 377, 399, 410, 420, 422, 426 energy conservation, 297, 372 engineering, xiv, 57, 58, 209, 228, 349, 365 environmental conditions, 144 environments, 100, 161, 174, 175 equality, 236, 239, 242, 251, 329, 445, 446 equilibrium, 348, 363 equipment, 57, 58, 311 Euler equations, xi, 177, 178, 179, 205, 208, 279, 296, 308, 309, 311, 312, 314, 316, 317, 318, 323, 368, 394, 395, 397, 439, 440, 443, 444, 447, 458 evolution, xiii, 25, 58, 82, 101, 149, 174, 305, 310, 312, 318, 349, 353, 484 exaggeration, 445 excitation, 350, 351, 353, 359, 362, 363 existence, 177, 325, 326, 347, 459, 462

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F FAS, 403 fibers, 320, 321 field tests, 122 films, 308 finite element method, 350, 484 finite volume method, x, xiii, 2, 12, 36, 81, 161, 365, 366, 371, 378, 382, 425, 440 flexibility, xiii, 147, 365, 366 flow field, ix, 11, 39, 46, 58, 79, 81, 82, 86, 96, 122, 141, 160 flow value, 283 fluctuations, 313, 343 fluid, ix, xi, xii, xiii, 1, 3, 4, 12, 23, 25, 32, 34, 35, 38, 46, 47, 58, 60, 61, 63, 65, 75, 81, 82, 83, 84, 89, 96, 99, 100, 101, 104, 113, 114, 119, 143, 144, 149, 156, 159, 161, 162, 164, 178, 179, 205, 206, 207, 209, 210, 211, 215, 218, 229, 230, 231, 262, 265, 266, 267, 269, 270, 271, 272, 275, 278, 286, 287, 288, 289, 291, 292, 293, 295, 296, 297, 303, 305, 307, 312, 316, 325, 326, 327, 328, 329, 345, 346, 349, 350, 372, 373, 418, 445, 446, 450, 458, 459, 462 fluid flow, ix, xi, xii, xiii, 58, 81, 82, 96, 101, 114, 143, 149, 156, 161, 162, 205, 229, 230, 231, 262, 265, 266, 325, 326, 327, 349, 445, 459, 462 fluorescence, 58, 82 force, 55, 161, 162, 178, 179, 240, 279, 286, 292, 326, 351, 373, 375, 444, 458 formation, xii, 32, 49, 52, 275 formula, 181, 185, 187, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 208, 283, 286, 288, 290, 377, 380, 395, 415, 416, 425, 434, 437, 444, 449, 457 Fourier transform, 177, 178, 180, 181, 182, 183, 190, 191, 192, 193, 194, 195, 197, 198, 199, 200, 204, 300 fractal dimension, 301 FRE, 99 free fields, 319 friction, 40, 348, 411, 412, 414, 417, 419, 441, 444 functional analysis, 207 functional separation, 280 fuzzy sets, 212

G Galileo, 346 gauge invariant, 337 geometric parameters, 34, 143, 161 geometrical parameters, 144 geometry, ix, xii, xiii, 3, 8, 11, 12, 14, 18, 32, 46, 58, 61, 82, 101, 145, 229, 231, 278, 327, 331, 336, 345, 350, 351, 363, 365, 366, 370, 371 glasses, 161

graph, 232, 278, 309, 310, 311, 312, 313, 314, 315, 317, 318, 319, 320, 321, 322, 370 gravitation, 279, 286, 296 Greece, 8, 18, 28, 119 greenhouse gas, 160, 161 greenhouse gas emissions, 161 grid resolution, 58, 86, 96, 164 grids, 83, 86, 87, 88, 164, 438, 439, 440, 441 growth, 32, 323, 406 growth factor, 406

H Hamiltonian, 317, 318, 320, 324 Hausdorff dimension, 207 heat capacity, 426 heat transfer, 101, 144, 150, 160, 161, 174, 175, 292 height, 2, 83, 101, 113, 119, 144, 145, 162, 297, 349, 350, 351, 380 hematoma, 2 hemorrhage, 31 high blood pressure, 11 high resolution scheme, 366 Hilbert space, 463 homogeneity, 58, 82 homogenization, 229, 262 hybrid, 84, 86, 160, 174, 366, 367, 370, 371, 378, 403, 408, 409, 410, 411, 425, 440 hydrocephalus, 33 hydrodynamic structure, 59, 81, 82

I images, ix, x, 1, 3, 11, 13, 18, 21, 22, 31, 33, 36, 38, 42 incompressible medium, xiv, 443, 444, 445 independent variable, 278, 294 industrial processing, 57 inequality, xiv, 204, 205, 234, 235, 236, 242, 253, 255, 258, 260, 342, 343, 461, 462, 469, 470, 472, 473, 475, 477, 480, 485 inertia, 100, 175 initial state, 104 integration, 86, 191, 193, 194, 196, 198, 201, 234, 236, 241, 251, 270, 271, 272, 282, 285, 289, 290, 294, 298, 299, 301, 326, 327, 337, 342, 343, 367, 383, 391, 392, 394, 395, 409, 425, 447 interface, xii, 3, 84, 108, 118, 119, 238, 263, 265, 266, 267, 268, 269, 270, 271, 272, 273, 350 internal controls, 462, 481 intracranial aneurysm, x, 31, 32, 35, 36, 38 intracranial pressure, 33 invariants, 317, 318 inversion, 185 iteration, 369, 399, 400, 401, 403, 404, 406, 410, 415, 425

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K kerosene, 350 Korea, 229, 461

L Laboratory of Electro-Mechanic Systems, 45, 57, 81, 121, 143, 159, 162, 349 laminar, xi, 58, 79, 80, 82, 96, 97, 99, 101, 118, 119, 144, 161, 205, 218, 262, 263, 345, 399, 415, 444 laminar flow, 58, 82, 99, 218, 262, 345 Laplace transform, xi, 177, 178, 183, 190, 205 laws, xii, xiii, 209, 214, 217, 218, 223, 225, 227, 229, 230, 231, 262, 263, 264, 305, 367 Lie algebra, 279, 285 Lie group, 286 linear function, 386, 414 Liquid Mixing Foam, xi, 99 liquids, xi, 58, 99, 104, 108, 114, 363

migration, 33 miscible liquid, xi, 58, 99, 102, 119 mixer flow, x, 57, 59 mixing, 57, 58, 59, 72, 74, 79, 80, 82, 96, 97, 100, 119, 350 modelling, 79, 96, 441 models, xi, 12, 22, 32, 33, 34, 35, 36, 46, 52, 53, 59, 79, 80, 82, 84, 96, 121, 144, 149, 161, 292, 297, 371, 375, 423 modifications, 36, 367, 371, 417 modulus, 244 momentum, 48, 58, 84, 146, 266, 269, 271, 326, 371, 372, 373, 397, 399, 410, 422 morphology, 36 motion, ix, x, xiii, xiv, 28, 33, 81, 84, 100, 178, 311, 316, 317, 323, 325, 327, 328, 346, 348, 350, 353, 443, 444, 445, 446, 458, 459 motion of fluids, ix, xiii, 325, 348 multidimensional, 440 multigrid, xiv, 347, 365, 366, 368, 369, 370, 371, 399, 400, 402, 403, 404, 405, 406, 410, 411, 412, 422, 423, 425, 439, 440 multiple reference frames (MRF), x, 81

M magnetic field, 162, 175, 277 magnetic resonance imaging, 33 magnetism, 348 magnitude, x, xi, xiii, 5, 12, 21, 25, 28, 32, 33, 40, 88, 89, 90, 99, 109, 110, 111, 112, 113, 118, 149, 150, 276, 281, 349, 354, 357, 358, 359, 368, 372, 395, 398, 427, 434 manifolds, 319 manipulation, 57, 225, 281, 282, 293, 299 mapping, 232, 406 mass, 2, 48, 57, 60, 104, 144, 150, 266, 269, 270, 271, 282, 283, 288, 294, 295, 372, 373, 375, 400, 422, 423, 427, 434 materials, 57, 58, 160, 287 matrix, 178, 189, 190, 191, 193, 196, 197, 198, 201, 233, 235, 236, 237, 239, 242, 248, 256, 257, 258, 312, 323, 368, 383, 386, 388, 390, 392, 393, 394, 395, 396, 397, 425, 429, 431, 432, 433, 434, 435, 436 matrixes, 394 measurements, 33, 35, 80, 100, 122, 161, 174, 412, 417, 441 mechanical stress, 12 media, 1, 443 median, 367, 377, 379, 380, 425 Mediterranean climate, 175 mellitus, 2 meshing, xi, 83, 96, 99, 101, 102, 104, 106, 107, 108, 109, 110, 111, 112, 118, 136, 143, 144, 145, 149, 150, 152, 153, 155, 156, 164 methodology, ix, 2, 3, 8, 11, 12, 18, 28, 210, 212, 223 microclimate, 162

N Navier-Stokes equations, ix, x, xi, xii, xiii, 1, 8, 11, 18, 28, 31, 45, 47, 48, 57, 81, 99, 121, 159, 162, 177, 178, 206, 209, 210, 223, 259, 262, 263, 265, 266, 268, 272, 273, 296, 305, 306, 309, 323, 325, 326, 327, 328, 329, 335, 341, 342, 346, 347, 365, 366, 367, 368, 370, 371, 372, 377, 378, 379, 392, 394, 399, 402, 405, 413, 425, 439, 441, 443, 462, 484, 485 negatively buoyant jet, 99, 100, 101, 118, 119 negativity, 257, 258 nodes, xiv, 101, 104, 105, 147, 148, 149, 150, 151, 152, 153, 154, 155, 162, 327, 365, 366, 370, 379, 380, 381, 382, 385, 395, 397, 398, 399, 405, 406, 407, 408, 420, 425, 437 nonlinear dynamic systems, 210 null, xiv, 190, 461, 462, 477, 481, 483 numerical analysis, xiv, 48, 365

O one dimension, 224, 275, 276, 290, 296 operations, 182, 190, 286, 369 optimization, 144, 174, 216, 230 ordinary differential equations, 183, 217, 276 orthogonal functions, 326, 328, 334 orthogonality, 311, 315, 319, 344 oscillation, 33, 296, 299, 300 overlap, 45, 46, 47, 54, 55

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P parallel, 5, 8, 16, 18, 25, 39, 42, 389, 441 parallelization, 368 parameter vectors, 439 parenchyma, 33 partial differential equations, xii, xiii, 178, 182, 208, 262, 264, 275, 276, 293, 297, 325, 327, 348, 443 phase transitions, 276 physical characteristics, 443 physical features, 104 physical properties, 287, 293 physics, 2, 160, 275, 291, 296, 346, 444, 485 Poisson equation, 230, 231, 332, 335, 340, 367, 369 polycarbonate, 145 porous media, 295 preconditioning, xiv, 365, 366, 368, 371, 392, 393, 394, 395, 396, 397, 399, 400, 401, 402, 410, 411, 425, 436, 438, 440 prediction models, 34 pressure boundary condition, 461 pressure gradient, 211, 218, 219, 331, 336, 345, 419, 425, 442 programming, 382, 397 propagation, 276, 389 pulp, 292, 294 pure water, 101, 102, 104, 107, 111, 112

Q QED, 224, 226, 227 quantum mechanics, 327

R radius, 34, 144, 145, 148, 278, 373, 390, 391, 392, 394, 427, 434 rate of change, 269 real numbers, 280, 287, 293, 298, 326 reference frame, x, 81, 84, 327, 373 regularity theory, 325 residential, 161, 165, 167, 169, 171, 173, 174, 175 residuals, 83, 379, 399, 418 resolution, ix, x, xi, xiii, 1, 3, 8, 14, 18, 23, 45, 60, 61, 82, 84, 86, 88, 99, 110, 159, 365, 366, 367, 368, 377, 386, 406, 412, 420, 424, 425, 438, 440 root, 32, 122, 276, 295, 443, 446 root integral, 443, 446 rotational matrix, 250 rough boundary, xii, 229, 230, 232, 235, 263 Rushton turbine, x, 58, 59, 79, 80, 81, 82, 83, 96

S

491

Savonius wind rotor, x, 45, 46, 47, 54, 55, 122, 140, 141 scaling, 58, 82, 276, 277, 386 sensitivity, 12, 86, 164 shape, 33, 34, 108, 149, 214, 230, 231, 268, 276, 279, 290, 298, 299, 300, 301, 370, 371, 407 shear, x, 31, 32, 266, 269, 272, 292, 372, 409, 414, 415 shock, 2, 22 signs, 2, 287 simulation, x, xi, xiii, 33, 34, 45, 46, 58, 61, 74, 79, 82, 84, 96, 101, 104, 119, 122, 136, 140, 141, 143, 144, 147, 156, 159, 160, 161, 162, 164, 174, 175, 215, 223, 266, 268, 327, 345, 349, 350, 351, 352, 367, 370, 371, 389, 400, 424, 425, 441 sloshing phenomenon, 349, 350, 363 smoothing, 3, 14, 23, 369, 383, 399, 403, 404, 406, 410, 438 smoothness, xi, 177, 214, 326, 327, 347, 462 software, ix, x, xi, 1, 3, 11, 13, 18, 21, 22, 31, 36, 37, 38, 46, 57, 59, 81, 101, 122, 123, 140, 143, 144, 146, 156, 161, 390 Solar Chimney Power Plant (SCPP), xi, 143, 144, 145, 147, 149, 150, 151, 152, 153, 154, 155, 156, 157 SolidWorks Flow Simulation, x, xi, 57, 121, 122 spacetime, 178, 207 specific heat, 372, 373, 375, 426, 427, 430, 434 Spectral methods, 325, 327 spiral flux, ix, 11, 28 stability, 279, 283, 320, 367, 368, 386, 390, 425 state, xiv, 33, 57, 84, 96, 100, 118, 122, 218, 223, 276, 286, 365, 367, 372, 374, 378, 399, 401, 427, 434, 445, 458 stent, ix, 11, 12, 13, 14, 16, 18, 34, 35 stirred tank, 58, 59, 79, 80, 81, 82, 83, 88, 96, 97, 140 Stokes equations, x, xii, xiii, xiv, 1, 31, 42, 101, 203, 208, 209, 223, 229, 230, 262, 266, 268, 272, 305, 326, 328, 367, 378, 392, 443, 444, 445, 458, 459, 462 storage, 350, 363, 367, 369, 379 stratification, 100 stress, ix, x, xiii, 12, 31, 32, 266, 269, 271, 272, 292, 325, 373, 374, 414, 415 stretching, 5, 25, 60, 425 structure, xii, xiii, 3, 4, 13, 16, 23, 24, 37, 59, 79, 81, 82, 83, 96, 121, 122, 141, 159, 162, 275, 280, 283, 299, 305, 306, 307, 312, 316, 320, 322, 323, 350, 366, 370, 392, 394, 424, 436, 458 subdomains, xiv, 237, 327, 462, 463 suppression, 369, 371 surface area, 2, 36, 379, 380, 381, 387, 431, 437 surface tension, 350, 363 surgical intervention, 2, 12, 16 suspensions, 292 symmetry, 102, 280, 284, 294, 328

Savonius rotor, 45, 46, 47, 51, 53, 55

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492

Index

T tachycardia, 2, 22 tanks, 58, 79, 80, 82, 83, 96, 97, 100, 350, 363 techniques, 31, 58, 82, 161, 230, 231, 236, 262, 279, 345, 366, 441 technologies, 121, 209 temperature, xi, 100, 143, 144, 146, 149, 150, 151, 160, 161, 277, 291, 296, 297, 298, 299, 300, 301, 372, 373, 375, 414, 416, 418, 420, 421, 430 temperature dependence, 291 testing, 237, 240, 257, 345 thinning, 292 tides, 45 time increment, 197, 201 time resolution, 164 time step effect, 349 topological invariants, xiii, 305, 306, 309, 317 topology, 180, 309, 317, 370, 406, 425 torus, 306, 314, 315, 316, 317, 318, 319, 320, 321, 322 total energy, 375, 426, 427, 434 trajectory, x, 18, 31, 100, 309, 310, 311, 315, 463 transformation, 276, 280, 285, 328, 332, 337, 338, 346, 402, 407 transformations, 328, 329, 446, 450 transmission, 33, 406 transparency, 279, 280 transport, 48, 162, 174, 276, 376, 377, 384 traveling waves, 275 treatment, 33, 297 tribology, 444 turbulence, x, xi, 45, 46, 47, 48, 52, 53, 54, 55, 57, 58, 59, 69, 81, 82, 83, 88, 96, 121, 122, 134, 140, 141, 143, 157, 159, 283, 296, 301, 312, 323, 352, 371, 372, 374, 375, 376, 377, 378, 410, 413, 414, 418, 420, 421, 422, 423, 441 turbulence model, x, xi, 45, 46, 47, 48, 52, 53, 54, 55, 57, 59, 81, 82, 83, 96, 121, 143, 157, 159, 371, 374, 377, 378, 410, 413, 414, 421, 422, 423, 441 turbulent, x, xiii, 45, 46, 47, 48, 54, 55, 57, 58, 59, 62, 68, 70, 71, 72, 74, 75, 76, 77, 78, 79, 80, 81, 82, 84, 87, 88, 92, 93, 94, 95, 96, 97, 100, 118, 119, 121, 122, 123, 131, 133, 134, 141, 143, 153, 154, 155, 156, 159, 160, 169, 170, 171, 172, 175, 305, 312, 323, 327, 345, 349, 351, 352, 371, 372, 373, 374, 376, 377, 378, 388, 389, 408, 413, 414, 415, 416, 420, 438, 441, 442 turbulent flow, x, 45, 46, 47, 54, 57, 58, 59, 79, 80, 81, 82, 88, 96, 121, 122, 141, 143, 159, 175, 327, 345, 371, 372, 378, 438, 441, 442

U ultrasound, 12, 35 uniqueness, 325, 343

universality, 276, 277, 287, 299 unstructured mesh, xiii, 365, 366, 367, 368, 371, 372, 377, 382, 385, 386, 387, 392, 396, 398, 400, 402, 403, 407, 412, 414, 425, 433, 438, 439, 440, 441

V validation, xi, 34, 35, 121, 122, 136, 140, 141, 159, 160, 175, 266, 345, 441 variables, ix, 1, 3, 8, 32, 33, 86, 161, 179, 211, 212, 248, 272, 276, 280, 298, 299, 306, 308, 309, 310, 313, 314, 316, 317, 318, 319, 335, 337, 340, 367, 372, 374, 377, 383, 386, 388, 392, 393, 396, 397, 399, 410, 426, 427, 428, 430, 434, 435, 436, 437, 443, 445, 452, 473 variations, 28, 322, 400, 441 vasculature, 2, 33 vector, xii, xiii, 178, 179, 180, 189, 190, 193, 196, 201, 203, 204, 216, 217, 224, 225, 229, 231, 232, 233, 236, 238, 242, 243, 246, 247, 248, 250, 251, 262, 266, 269, 278, 286, 305, 306, 308, 309, 310, 311, 313, 314, 315, 316, 317, 318, 319, 320, 322, 325, 326, 331, 336, 337, 340, 347, 372, 373, 374, 377, 378, 379, 386, 389, 393, 394, 395, 396, 403, 426, 434, 438, 443, 445, 448, 462, 463 vein, 5, 25, 31, 32, 38, 39, 134, 136, 162 ventilation, xi, 100, 159, 160, 161, 162, 174, 175 vessels, 32, 79, 91, 96, 100 viscosity, xi, xiii, 22, 48, 54, 58, 62, 72, 73, 78, 79, 95, 102, 104, 114, 133, 134, 155, 156, 171, 172, 177, 179, 211, 218, 265, 266, 279, 283, 285, 286, 288, 290, 291, 292, 295, 296, 299, 301, 305, 312, 316, 323, 326, 351, 352, 373, 374, 376, 377, 388, 389, 397, 413, 415, 416, 420, 426, 444 visualization, 12, 58, 62, 64, 65, 66, 68, 70, 72, 82 Vortex dynamics, 325 Vortex methods, 325

W Wall Shear Stress (WSS), ix, 11, 21 wall temperature, 160, 375, 378, 414 wall-laws, xii, 229, 230, 231, 262, 263 waste disposal, 100 water, xii, 45, 83, 100, 101, 102, 104, 106, 107, 108, 110, 111, 112, 113, 118, 161, 211, 265, 266, 269, 292, 294, 295 wave propagation, 275, 368 wind speed, 122, 141, 159, 160, 175 wind tunnel, xi, 46, 121, 122, 140, 141, 159, 160, 162, 173 wind turbine(s), x, xi, 45, 46, 55, 121, 122, 124, 126, 127, 131, 134, 136, 140, 141 wind-wave interaction, 265

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