Advances in Fluid Mechanics: Modelling and Simulations (Forum for Interdisciplinary Mathematics) 9789811914379, 9789811914386, 9811914370

Citation preview

Forum for Interdisciplinary Mathematics

Dia Zeidan Lucy T. Zhang Eric Goncalves Da Silva Jochen Merker   Editors

Advances in Fluid Mechanics Modelling and Simulations

Forum for Interdisciplinary Mathematics Editors-in-Chief Viswanath Ramakrishna, University of Texas, Richardson, USA Zhonghai Ding, University of Nevada, Las Vegas, USA Editorial Board Ravindra B. Bapat, Indian Statistical Institute, New Delhi, India Balasubramaniam Jayaram, Indian Institute of Technology Hyderabad, Hyderabad, India Ashis Sengupta, Indian Statistical Institute, Kolkata, India P.V. Subrahmanyam, Indian Institute of Technology Madras, Chennai, India

The Forum for Interdisciplinary Mathematics is a Scopus-indexed book series. It publishes high-quality textbooks, monographs, contributed volumes and lecture notes in mathematics and interdisciplinary areas where mathematics plays a fundamental role, such as statistics, operations research, computer science, financial mathematics, industrial mathematics, and bio-mathematics. It reflects the increasing demand of researchers working at the interface between mathematics and other scientific disciplines.

More information about this series at https://link.springer.com/bookseries/13386

Dia Zeidan · Lucy T. Zhang · Eric Goncalves Da Silva · Jochen Merker Editors

Advances in Fluid Mechanics Modelling and Simulations

Editors Dia Zeidan School of Basic Sciences and Humanities German Jordanian University Amman, Jordan Eric Goncalves Da Silva ISAE-ENSMA Institut Pprime Chasseneuil-du-Poitou, France

Lucy T. Zhang Department of Mechanical, Aerospace & Nuclear Engineering Rensselaer Polytechnic Institute Troy, USA Jochen Merker MNZ Leipzig University of Applied Sciences Leipzig, Germany

ISSN 2364-6748 ISSN 2364-6756 (electronic) Forum for Interdisciplinary Mathematics ISBN 978-981-19-1437-9 ISBN 978-981-19-1438-6 (eBook) https://doi.org/10.1007/978-981-19-1438-6 Mathematics Subject Classification: 76M55, 82B40, 74F10, 76B03, 76N10, 76N30 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Fluid mechanics modelling and simulations are important branches of computational mechanics and applied mathematics. It continues to play an important role in a variety of research topics and in real-world problems. While fluid mechanics has been regarded as an established field, computational fluid dynamics (CFD) is a subject that continues to thrive and in high demand over the past few decades due to the continuing advancements in computing capabilities. This book is the outcome of international collaboration among authors from different parts of the world working in modelling and simulations of fluid mechanics. Modelling and simulation of fluid dynamics are regarded as efficient and reliable tools in analysing flow characteristics for many applications. It is, therefore, timely to provide the international community with the latest advances in CFD developments. As a result, editors of this book invited expert researchers working in modelling and simulations of fluid mechanics to submit their recent results for possible publication in the book. The aim is to share the best and the latest developments and findings at the international level through joint networking. The themes covered in this book are selected to provide topics of fundamental and applied methods that are of current interest in the field. The contributing authors are well-known experts in the field and many of them have dedicated several years of their career to the topics covered in their chapters. The Editors are extremely thankful to the Senior Publishing Editor at Springer Nature Shamim Ahmad for supporting the idea of this book. We also would like to express our most sincere thanks and great appreciation to all the authors for their exceptional contributions to this book and to the many reviewers to spend their valuable time in reviewing the chapters and providing their comments and suggestions. We are also gratefully acknowledging the German Jordanian University, Jordan, Rensselaer Polytechnic Institute, USA, ISAE-ENSMA, Institut Pprime, France, and HTWK Leipzig University of Applied Sciences, Germany, for their kind assistance

v

vi

Preface

in supporting us to publish this book and promoting modelling and simulations of fluid mechanics research. Amman, Jordan Troy, USA Chasseneuil-du-Poitou, France Leipzig, Germany

Dia Zeidan Lucy T. Zhang Eric Goncalves Da Silva Jochen Merker

Contents

Computational Fluid Dynamics Applications in Cardiovascular Medicine—from Medical Image-Based Modeling to Simulation: Numerical Analysis of Blood Flow in Abdominal Aorta . . . . . . . . . . . . . . . Alin-Florin Totorean, Sandor Ianos Bernad, Tiberiu Ciocan, Iuliana-Claudia Totorean, and Elena Silvia Bernad

1

An Improved Density-Based Compressible Flow Solver in OpenFOAM for Unsteady Flow Calculations . . . . . . . . . . . . . . . . . . . . . . Gaurav Kumar and Ashoke De

43

Finite Volume Models and Efficient Simulation Tools (EST) for Shallow Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Martínez-Aranda, J. Fernández-Pato, I. Echeverribar, A. Navas-Montilla, M. Morales-Hernández, P. Brufau, J. Murillo, and P. García-Navarro

67

Overview of Outfall Discharge Modeling with a Focus on Turbulence Modeling Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Mostafa Taherian, Seyed Ahmad Reza Saeidi Hosseini, and Abdolmajid Mohammadian A Unified Algorithm for Interfacial Flows with Incompressible and Compressible Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Fabian Denner and Berend van Wachem Stabilized Finite Element Formulation and High-Performance Solver for Slightly Compressible Navier–Stokes Equations . . . . . . . . . . . . 209 Feimi Yu and Lucy T. Zhang

vii

About the Editors

Dia Zeidan is Associate Professor of Applied and Computational Mathematics at the German Jordanian University, Amman, Jordan, and Elected Fellow of the European Society of Computational Methods in Sciences and Engineering. An active researcher in developing mathematical and numerical tools of multiphase fluid flow problems for several years, he is recognized for research contributions in applied and computational mathematics with multiphase flows including his creative approaches to teaching and research. His work has been highly interdisciplinary, involving international collaborations with applied and computational researchers. Among various institutional obligations, he has been visitor of several important international research groups bridging with national research infrastructure gaps in Jordan. He serves on several expert review panels, as Technical Editor and Reviewer for several peer-reviewed journals and as Member of several program committees of technical conferences around the world. Lucy T. Zhang is Professor at the Department of Mechanical, Aerospace and Nuclear Engineering at Rensselaer Polytechnic Institute (RPI), U.S.A. She is an elected Fellow of American Society of Mechanical Engineers. Her research interests focus on building advanced and robust computational tools and software for accurate and efficient multiphysics and multiscale simulations that can be used for engineering applications in biomechanics, micro- and nano-mechanics, medicine and defense projects involving impacts. She is a recipient of the Young Investigator Award at the International Conference for Computational Methods. Her pioneer work in developing the immersed finite element method had been and is still being widely used in academic engineering and scientific communities. Professor Zhang is now developing open-source tools and technology, OpenIFEM, that can conveniently and efficiently couple any existing solvers for multiphysics and multiscale simulations and analysis. She is Associate Editor of the Journal of Fluid and Structures, ASME Journal of Fluids Engineering and Computer Methods and Engineering Sciences. Eric Goncalves Da Silva is Professor and Head of the Department Fluid Mechanics and Aerodynamics, Aeronautical Engineering School ISAE-ENSMA, Poitiers, ix

x

About the Editors

France. His research interests are related to the modelling and the simulation of flows for which the density is variable such as compressible flow, two-phase flow and cavitation. Recent work includes shock wave boundary layer interaction, shock–bubble interaction and investigation of three-dimensional effects on cavitation pocket. Jochen Merker is Professor of Analysis and Optimization at the Leipzig University of Applied Sciences (HTWK Leipzig), Germany, since 2015. He received his Ph.D. in Mathematics from the University of Hamburg, Germany, in 2005. Afterwards, he worked as Postdoc in Applied Analysis at the University of Rostock, Germany, received his Habilitation in 2012 and became Professor at the Applied University of Stralsund, in 2013, before he became Full Professor at HTWK Leipzig. His research focuses on partial differential equations (PDEs) and functional analysis, particularly functional analytic settings for linear and nonlinear stationary and evolution equations, especially degenerate and singular parabolic PDEs with non-local terms, and on (contact) Hamiltonian systems on manifolds. Regarding this topic, in 2018, he received an honorable mention in the international Ian Snook Prize 2017 for his contribution in the field.

Computational Fluid Dynamics Applications in Cardiovascular Medicine—from Medical Image-Based Modeling to Simulation: Numerical Analysis of Blood Flow in Abdominal Aorta Alin-Florin Totorean, Sandor Ianos Bernad, Tiberiu Ciocan, Iuliana-Claudia Totorean, and Elena Silvia Bernad

1 Introduction Cardiovascular diseases (CVDs) incidence is continuously increasing worldwide. It is well known that most of the circulatory pathologies are strongly related to an altered blood flow environment (e.g. atherosclerosis is associated to endothelial dysfunction as a result of altered Wall Shear Stress). Nowadays, the medical imaging techniques provide information regarding the anatomical characteristics but there are few techniques that could also evaluate the physiological parameters associated with the blood flow, such as pressure and velocity. Excepting Doppler echography and Magnetic Resonance Imaging which could assess non-invasively physiological A.-F. Totorean (B) Department of Mechanics and Strength of Materials, Politehnica University of Timisoara, No. 1 Mihai Viteazu Boulevard, 300222 Timisoara, Romania e-mail: [email protected] S. I. Bernad Romanian Academy—Timisoara Branch, No. 24 Mihai Viteazu Boulevard, 300223 Timisoara, Romania T. Ciocan Research Center for Engineering of Systems With Complex Fluids, Politehnica University of Timisoara, No. 1 Mihai Viteazu Boulevard, 300222 Timisoara, Romania Continental Automotive Romania, No. 1 Siemens Street, 300704 Timisoara, Romania I.-C. Totorean · E. S. Bernad University of Medicine and Pharmacy Victor Babes Timisoara, No. 2, Eftimie Murgu Square, 300041 Timisoara, Romania I.-C. Totorean Institute for Cardiovascular Diseases, Cardiology Department, No. 13A Gheorghe Adam Street, 300310 Timisoara, Romania © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 D. Zeidan et al. (eds.), Advances in Fluid Mechanics, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-19-1438-6_1

1

2

A.-F. Totorean et al.

parameters (velocity), most of the techniques are invasive and need special tools for measuring the hemodynamic parameters (“guidewire-based technology with high fidelity sensors” [2]). Nevertheless, these techniques are usually implemented for diagnosis and patient evaluation, and they are not especially dedicated for therapeutically preplanning process. Moreover, the imaging techniques are not primarily used for medical devices design and optimization. To accomplish all these goals, and to facilitate the translation process from the clinic to laboratory and back to the bedside [5], engineering specialties contribute with tools capable for: (i) patient-specific image-based model reconstruction [1]; (ii) perform non-invasively in silico simulations, using Computational Fluid Dynamics tools that could characterize the blood flow particularities within the circulatory system, in physiological or pathological condition or to assess the efficiency of cardiovascular medical devices [1, 2] or (iii) rapid prototyping technology to create physical models useful for surgical preplanning strategy, medical device development and testing such as stents or circulatory assisted devices and to experimentally validate the numerical simulations [2]. Experimental test rigs were presented in literature to analyze the blood flow under physiological or pathological conditions especially for numerical results validation. The challenge in laboratory is to mimic the patient-specific real flow conditions together with a suitable experimental model (artificial or human vascular geometry taking into consideration the real vascular properties). Polanczyk et al. [3, 4] described in their studies a new artificial circulatory model for analysis of human and artificial vessels, using an ex vivo capable of mimicking the vascular flow conditions, in which the frequency of pulsation and ejection volume can be varied independently. A comparison between real iliac arteries and silicon tubes was performed on the test rig and the changes of diameter and wall displacements were experimentally assessed [3]. Moreover, the analysis of the pulsatile blood flow in patients’ specific abdominal aorta aneurysms (AAA) was reported, putting in evidence experimentally the hemodynamic characteristics in AAA in the presence of thrombus or stent-graft [4]. The results were validated with medical data and support the idea of using this kind of experimental rig for ex vivo analysis to describe the vascular grafts mechanical behavior. The fundamental mechanics of the cardiovascular system can be represented through modeling, a powerful tool designed to improve the understanding of cardiovascular dynamics in normal as well as pathological scenarios and to provide answers to a series of questions related to diagnosis and prognosis of various pathologies within the cardiovascular medicine. Computational Fluid Dynamics tools are continuously improving in order to provide support for physics, industry or medicine. Its efficiency has been tested over the years, and it was stated to be a useful support for medical activity, as well as for medical devices development and authorization [6]. Recently, the first technology based on CFD and simulated coronary maxima hyperaemia that computes the Fractional Flow Reserve (FFR) by computed tomographic angiography (FFRCT ) was developed and introduced by HeartFlow, Inc. (Redwood City, CA) and authorized by the Food and Drug Administration (FDA) [7, 8]. Fractional Flow Reserve is a quantitative parameter that evaluates the coronary stenosis severity [7]. Furthermore,

Computational Fluid Dynamics Applications in Cardiovascular …

3

FDA has issued in the last years a guidance for computational modeling studies in medical device submission, establishing the conditions required by computational analysis used in the case of devices submission for authorization [9]. A Computational Fluid Dynamics study, and particularly the numerical analysis in cardiovascular biomechanics, includes several steps (Fig. 1): (i) pre-processing (including the medical image and patient’s physiological data acquisition together with 3D model segmentation), (ii) the simulation (using commercial or open source software packages) and (iii) post-processing (data analysis, discussions and conclusions) [13–16]. To obtain numerical results appropriate for the real physiological case there are some aspects that should be taken into consideration when setting a CFD analysis: the geometry should be similar to the patient’s vascular anatomy and the boundary conditions should respect the patient-specific conditions. The first models used for CFD studies were idealized and simplified, but over the years the complexity of the geometries increased. Since the 1990s the tendency was to use patient-specific image-based geometries especially for surgical planning [10, 11] or hemodynamics studies [12]. Over the years there have been developed commercial, open source or in-house software packages to support the research, industry of medical devices and medical practice, particularly in the field of medical image-based segmentation and numerical analysis. This chapter presents a brief review regarding the Computational Fluid Dynamics applications in cardiovascular medicine, from designing the patient-specific model to numerical analysis and post-processing. A real patient geometry of abdominal and its branches reconstructed in ITK-Snap is used for CFD analysis of blood flow using two different Solver packages: a commercial software for general use in physics, industry or medicine (Ansys Fluent) and an open source dedicated package for cardiovascular biomechanics (SimVascular). A comparative numerical analysis in steady-state condition is performed, using the pipeline specific for each package. The results are shown comparatively to assess the efficiency of putting in evidence the critical regions with altered hemodynamics parametric when using two different softwares, both based on the finite element method: a commercial with general use software versus an open source dedicated for cardiovascular simulation environment. Wall Shear Stress, pressure field, velocity field and streamlines, with medical signification are the hemodynamic parameters analyzed.

Fig. 1 Pipeline of a computational fluid dynamics analysis in cardiovascular biomechanics

4

A.-F. Totorean et al.

2 The Cardiovascular System The human cardiovascular system consists of the heart pump that provides the force required for circulation blood in the body and its intricate conduits (arteries, veins and microcirculation). The role of cardiovascular system is to develop and maintain an average blood pressure adapted to body requirements for the following purposes: tissue nutrition and removal of catabolism products—the most important function, transport of substances from one organ to another, transport of hormones regulating various functions of organs, transport of the components of the immune system and maintaining the water-electrolyte balance. The cardiovascular system is organized into two sub-systems: pulmonary circulation and systemic circulation. The blood is carried through the vascular system by arteries, which allow blood distribution to tissues, the microcirculation (including capillaries) that provides exchanges between blood and tissues and veins, which serve as tanks and collect blood to return it to the heart. The pulmonary circulation’s role is to collect oxygenated blood from the lungs and transport it to the left atrium via the pulmonary veins, as well as discharge carbon dioxide from the blood via right ventricle—pulmonary artery—lung alveoli. The system circulation provides the distribution of oxygenated blood to tissues via left ventricle—aorta—arteries and returns deoxygenated blood via two large veins (the inferior and superior venae cavae) to the right atrium [17] (Fig. 2). Blood vessels are integrated with a closed-loop circuit continuously exposed to hemodynamic forces generated by the cardiac cycle, the structural properties of the

Fig. 2 General view of the cardiovascular system circulation (with red is represented the segments that transport blood rich in oxygen and the blue color is associated with the segments transporting blood rich in carbon dioxide)

Computational Fluid Dynamics Applications in Cardiovascular …

5

vessel (length, diameter, stiffness, and tortuosity), local geometry, and associated flow characteristics in different segments of the vascular bed [2]. Hemodynamics plays an important role in the initiation, progression, and treatment of cardiovascular disease. Vascular endothelial cells (ECs) are exposed to hemodynamic forces, which regulate vascular biology and influence ECs functions. Relevant quantities such as pressure, velocity, tensile stress, and Wall Shear Stress (WSS) are key mechanical forces that regulate vascular homeostasis and function. Tensile stress is the force distributed around the vessel wall in a circumferential manner and depends on blood pressure, vessel wall thickness, and vessel radius. It regulates both paracrine and autocrine signaling and modulates angiogenesis, thus maintaining a normal vascular phenotype. Wall Shear Stress is a function of blood viscosity and velocity gradient at the vessel wall surface and is a quantitative representation of local flow disturbance [2, 18]. Laminar blood flow associated with low WSS is a protective factor for the endothelium through increased production of nitric oxide and prostacyclin and increased expression of protective transcription factors. Abnormal flow patterns and alteration of hemodynamic forces are thought to induce endothelial cell dysfunction, inflammation, flow stagnation, platelet activation and cellular proliferation (neointimal thickening), thus promoting thrombus formation and atherosclerosis. As a result, these phenomena are strongly correlated with several diseases including coronary artery disease, bypass graft failure, stent thrombosis, restenosis and aneurysm growth and rupture [2, 18]. The first studies of hemodynamics investigated flow at bifurcations. Flow patterns in bifurcations are inherently complex, including the creation of recirculation zones associated with low and oscillating Wall Shear Stress, conditions that favourize atherosclerotic plaque formation. Nowadays, the understanding of blood flow characteristics in physiological and pathological conditions has significantly improved. The development and translation of computational fluid dynamics (CFD) into cardiovascular medicine has revolutionized research and development of devices such as stents, valve prostheses and ventricular assist devices. Patient-specific and multi-scale modeling using computational tools enables individualized prediction of risk, outcomes of surgical interventions, treatment planning and optimization of cardiovascular devices [19].

3 Medical Image Segmentation Medical image-based models are very useful both for medical diagnosis and numerical or experimental analysis. The development in the field of image processing permits nowadays the possibility to create realistic patient-specific vascular models based on the medical imaging acquisition. A large number of techniques have been studied and implemented including region-based methods (tresholding and regiongrowing), pattern recognition and active contour models [1, 39, 40]. The accuracy of the model is strongly related to the imaging techniques, which are continuously

6

A.-F. Totorean et al.

Table 1 Software packages used frequently for medical image segmentation Software

Accessibility

References

3D Slicer

Open source

[20], https://www.slicer.org

Avizo

Commercial

[21], https://www.fei.com/amira-avizo/

CRIMSON

Partial free

[22], http://www.crimson.software/

InVesalius

Open source

[23], https://invesalius.github.io/

ITK-Snap

Open source

[24], http://www.itksnap.org

MeVisLab

Partial free

[25], https://www.mevislab.de/

Mimics

Commercial

[26], https://www.materialise.com/en/medical/mimics-inn ovation-suite/mimics

MITK

Open source

[27], https://www.mitk.org

Real3d VolViCon

Commercial

[28], https://real3d.pk/volvicon/

Seg3D

Open source

[29], https://www.sci.utah.edu/cibc-software/ seg3d.html

Simpleware ScanIP

Commercial

[30] https://www.synopsys.com/simpleware/software/ scanip.html

SimVascular

Open source

[31], http://simvascular.github.io/

Vitrea

Commercial

[32], https://www.vitalimages.com/

improving in terms of time and spatial resolution. The most used imaging techniques that provide images with good quality for segmentation include Computed Tomography Angiography, Magnetic Resonance Imaging, X-Ray Angiography (3D rotational angiography) and ultrasound techniques (echography and Intravascular Ultrasound technique—IVUS) [1, 2, 41]. Over the last years there have been developed software packages for imagebased segmentation useful both for medical practice and research, especially with applications and use in the cardiovascular direction, as shown in Table 1. In general, the segmentation softwares permit the export of the 3D model as a.stl file. There is a challenge regarding the 3D volume characteristics and there are few software packages that permit directly to generate the model segmentation and export a solid model. Before using the .stl or solid models for rapid prototyping or CFD analysis, there are cases when they need surface corrections and mesh repairs. The most common software packages used for these operations are shown in Table 2. Depending on the study’s purpose, the model can be also imported into Computer Aided Design (CAD) environments for other specific operations. There are situations, especially in the cardiovascular image-based modeling and simulation, when the geometry smoothed and remeshed has inlet and outlet sections that need more preparation for CFD analysis, by clipping the inlet and outlet sections. This operation is needed in order to get inlet and outlet faces characterized by their normal vector to be parallel with the blood flow direction. A possible alternative to perform the clipping operation is using the open source software package ParaView (https://www.paraview.org/) [42].

Computational Fluid Dynamics Applications in Cardiovascular …

7

Table 2 Software packages used for mesh repairs and surface corrections Software package

Accessibility References

Autodesk Meshmixer Open source

[33], https://www.meshmixer.com/

Autodesk Netfabb

Commercial

[34], https://www.autodesk.com/

Blender

Open source

[35], https://www.blender.org/

Materialize Magics

Commercial

[36], https://www.materialise.com/en/software/magics

Meshlab

Open source

[37], https://www.meshlab.net/

Open Flipper

Open source

[38], https://www.graphics.rwth-aachen.de/software/ope nflipper/

In addition, there have been developed applications for DICOM files access and mainly for medical visualization purposes which are not dedicated directly for segmentation. Furthermore, they permit 3D volume rendering and export the volume rendered as .stl file. Two common software packages having these characteristics which are used in medical practice and research are Osirix (commercial software, https://www.osirix-viewer.com/) and RadiAnt (partial free, https://www. radiantviewer.com/). Among the most common cardiovascular image-based segmentation performed for rapid prototyping, surgical planning or CFD analysis, there should be mentioned geometries associated with the aorta and its branches, the cerebral arteries—circle of Willis, the carotid arteries or the coronary arteries [26, 43–47]. Using image-based models there have been performed and frequently reported personalized patientspecific CFD studies regarding the blood flow in (i) physiological or (ii) pathological conditions, such as aneurysms, arterial stenosis, aorta dissection, stenosed cardiac valves and (iii) cardiovascular medical devices, such as stents (used in the treatment of arterial stenosis), stent-grafts and flow diverters (used mainly in the aneurysm treatment), artificial shunts (for arterio-venous shunt in case of patients following hemodialysis program), artificial grafts (used in the surgical revascularization treatment of arterial stenosis) or artificial and biological cardiac valves [43, 44, 48–52, 78].

4 Computational Fluid Dynamics and Cardiovascular Medicine A CFD analysis is performed using a geometry discretized into finite elements or volumes, where the physical constitutive laws are applied to each element/volume. The fluid flow, and particularly the blood flow in the cardiovascular system can be described using the conservation laws (conservation mass, momentum and energy). When referring to the cardiovascular system, and taking into consideration the blood as being an incompressible fluid, the mathematical characterization of the blood flow is realized by the nonlinear Navier–Stokes equations [53, 90].

8

A.-F. Totorean et al.

The Navier–Stokes equations can be described in the Eulerian formulation of motion as following [53, 90]:  ∂v + v∇v = −∇ p + μ∇ 2 v, ρ ∂t 

(1)

where ρ is fluid density,v is velocity, t is time, p is pressure, μ is fluid viscosity coefficient and ∇ is derivative operator. Equation (1) describes the change of the velocity in a specific point ( ∂v ) and flow ∂t convection (v∇v ) under the pressure gradient (∇ p) and viscous forces (μ∇ 2 v)[53]. The continuity equation is mathematically formulated as following [53, 90]: ∇v = 0.

(2)

As blood is a liquid with suspensions, there is a continuous debate regarding the blood viscosity modeling. Blood is considered to be non-Newtonian in capillaries, whereas in the larger vessels is assumed to be Newtonian. [6] For a better characterization of blood flow numerical analysis in idealized, simplified or patient-specific geometries, especially in the vicinity of walls or regions of interests such as bifurcations, constrictions-stenosis, there have been developed and implemented rheological models. It is known that the use of Newtonian fluid model may provide accurate numerical results in the case of blood flow when shear rates are higher than 100 s−1 and could underestimate the hemodynamic parameters, especially the Wall Shear Stress in the case of flows characterized by low shear stress. Moreover, for high and midrange velocities around 0, 2 ms−1 both Newtonian and non-Newtonian models provide similar numerical results, but for low shear rates it is recommended to use models describing blood viscosity. Amond the rheological models used for hemodynamic analysis, as presented in literature, there should be mentioned: the Carreau model, the Power Law model, the Generalized Power Law model, the Quemada model and the Casson model [54–57]. However, the mathematical discussions regarding blood rheology will not be assessed in this chapter, taking into consideration that (i) the paper presents a comparative numerical study associated with blood flow in a patient-specific abdominal aorta and its branches, using two different Solvers, Ansys Fluent (commercial) and SimVascular (open source), (ii) the numerical analysis is performed in steady-state condition with the prescribed inlet velocity of 0.25 ms−1 as shown in Sect. 5, (iii) SimVascular has only the Newtonian blood viscosity characterization and no other rheological model is implemented at this time, and (iv) therefore, the simulations will be performed assuming blood as a Newtonian fluid. Nowadays, there have been developed integrated software packages dedicated to Computational Fluid Dynamics analysis that have incorporated tools interconnected in a project’s workflow, for design, mesh, solver and post-processing, such as Ansys Workbench, CRIMSON or SimVascular. Depending on the methods used, a CFD analysis is mainly finite element method or finite volume method-based. However,

Computational Fluid Dynamics Applications in Cardiovascular …

9

Computational Fluid Dynamics methods’ efficiency is strongly dependent on the computing performance. Moreover, for a better understanding of the effect fluid flow has on the walls that limit the flow domain, there have been developed fluid structure interactions coupling tools, which permit the wall stress analysis under the flow dynamics. In the cardiovascular medicine FSI analysis is of great importance especially in the direction of aneurysm rupture prediction or stent and cardiac valve prosthesis design and optimization [58–60]. Computational Fluid Dynamics tools are based on Navier–Stokes or lattice Boltzmann equations, but most of the CFD software packages are solving the Navier– Stokes fluid flow governing equations, as presented in Table 3. Moreover, there have been developed software packages dedicated for cardiovascular biomechanics (e.g. CRIMSON, FEBio, HemeLB, SimVascular). On one hand the CFD’s main advantages of being used in cardiovascular medicine are to non-invasively in silico assess hemodynamic parameters useful for patient’s evaluation, diagnosis and therapy planning or medical device design and optimization. On the other hand, performing CFD analysis needs high computing resources and is time consuming, a disadvantage that sometimes is inappropriate for medical use based on translation from clinic to laboratory and back to bedside. The main Table 3 Software packages frequently used for computational fluid dynamics analysis with applications in cardiovascular medicine Software

Governing equations

Characteristics

Accessibility

References/sources

Abaqus/CFD

Navier–Stokes

Finite element method

Commercial

[61, 62]

Ansys Fluent

Navier–Stokes

Finite element method

Commercial

[63, 64]

Ansys-CFX

Navier–Stokes

Finite volume method

Commercial

[65, 66]

COMSOL

Navier–Stokes

Finite element method

Commercial

[68, 69]

CRIMSON

Navier–Stokes

Finite element method

Open source

[22, 67]

Featflow

Navier–Stokes

Finite element method

Open source

[70, 71]

FEBio

Navier–Stokes

Finite element method

Open source

[72, 73]

HemeLB

Lattice Boltzmann

Lattice Boltzman method

Open source

[74, 75]

OPENFOAM

Navier–Stokes

Finite volume method

Open source

[30, 76]

SimVascular

Navier–Stokes

Finite element method

Open source

[31, 77]

10

A.-F. Totorean et al.

directions of Computational Fluid Dynamics analysis with medical applications are shown in Table 4. One of the current trends in cardiovascular computational biomechanics is to integrate Artificial Intelligence (AI) algorithms together with CFD results in the analysis Table 4 Common cardiovascular problems investigated with Computational Fluid Dynamics and examples of studies reported in the scientific literature Problem investigated

Clinical applications

Examples of studies (first author) indicating the medical imaging source, software package used for image-based segmentation and Solver used for CFD analysis (if applicable)

Blood flow in aorta and its branches

CFD models derived from multi-detector computed tomography angiography (MDCTA) to analyze the aortic hemodynamics (distribution of Wall Shear Stress and flow pattern), an important tool in congenital heart disease diagnosis and evaluation

Zhu et al. [44] Image source: MDCTA Segmentation performed in Mimics Solver: ANSYS-FLUENT

Blood flow in carotid arteries

Using high-resolution CFD to predict the hemodynamic environment (abnormal flow, extreme Wall Shear Stress distribution and rapid directional fluctuations) in patient-based models of carotid stenosis, as a tool for understanding the multifactorial driving forces behind the progression of carotid disease

Schirmer et al. [45, 46] Image source: 3D rotational angiography Segmentation performed in Autodesk 3DS Max software Solver: ANSYS-FLUENT

Blood flow in cerebral arteries – circle of Willis

Using CFD analysis for complex study of flow-related pathophysiology in the circle of Willis. Investigation of hemodynamic factors (such as high Wall Shear Stress) that can help predict vascular sites where aneurysms are frequent and also anatomic variants at increased risk for aneurysm development

Alnaes et al. [47] Image source: computed tomography angiography, magnetic resonance angiography, and digital subtraction angiography Solver: FeatFlow Liu et al. [26] Image source: MRI Segmentation performed in MIMICS Solver: Comsol Multiphysics (continued)

Computational Fluid Dynamics Applications in Cardiovascular …

11

Table 4 (continued) Problem investigated

Clinical applications

Examples of studies (first author) indicating the medical imaging source, software package used for image-based segmentation and Solver used for CFD analysis (if applicable)

Aneurysm

Assessment of biomechanical factors that determine aneurysm initiation, growth and rupture; prediction of rupture sites

Lee et al. [48] Image source: 3D rotational angiography Segmentation performed in MIMICS Solver: COMSOL Multiphysics

Flow diverters

Analyzing blood flow velocity in a saccular aneurysm in the presence of a flow diverter can evaluate the endovascular therapy by the possibility of thrombosis formation in the aneurysm and reducing the blood flow

Ouared et al. [78] Image source: 3D rotational angiography Segmentation performed in Xtravision Philips workstation Solver: ANSYS-CFX

Aortic coarctation

Flow velocity, pressure and Wall Shear Stress were assessed using CFD as a non-invasive tool for diagnosing of aortic coarctation together with clinical imaging

Zhu et al. [44] Image source: MDCTA Segmentation performed in MIMICS Solver: ANSYS-Fluent

Aortic dissection

Evaluating the hemodynamic parameters distribution (Wall Shear Stress, Oscillatory Shear Index) inside a dilated aorta. Studies showed that the location of the oscillatory shear index might be related with the site of acute aortic dissection

Numata et al. [49] Image source: CT Segmentation performed in Osirix Solver: ANSYS-Fluent

Recent studies present the use of CFD tools in assessing the hemodynamic characteristics in the case of patients with acute type IIIb aortic dissection treated with thoracic endovascular aortic repair (TEVAR)

Polanczyk et al. [50] Image source: CTA Segmentation performed in 3DDoctor Solver: ANSYS-Fluent

(continued)

12

A.-F. Totorean et al.

Table 4 (continued) Problem investigated

Clinical applications

Examples of studies (first author) indicating the medical imaging source, software package used for image-based segmentation and Solver used for CFD analysis (if applicable)

Cardiac valves, including TAVR (transcatheter aortic valve replacement)

Optimizing the design of implantable prosthetic devices, for better surgical outcomes, and assessing the effect of post-TAVR geometry on flow characteristics

Singh-Gryzbon et al. [51] Image source: Computed Tomography Angiography Segmentation performed in 3D Slicer Solver: ANSYS-CFX

VADs (ventricular assist devices)

Assessment of the hemodynamic performance of VADs through patient-specific models

Ghodrati et al. [52] Image source: CT Segmentation performed in MIMICS Solver: ANSYS-Fluent

Coronary artery disease

Investigation of hemodynamic factors (Wall Shear Stress) known to modulate atherosclerotic evolution Investigation of vulnerable atherosclerotic plaques—the use of shear stress to predict plaque progression

Eslami et al. [83] Image source: Computed Tomography Angiography Segmentation performed in Medis QAngioCT and 3D Workbench Solver: SimVascular

Stent design

Investigation of Wall Shear Stress and flow characteristics in the vicinity of the stent in order to assess the hemodynamic performance and to optimize the stent’s design for improved clinical outcomes

Gundert et al. [84] Stent model design in SolidWorks In-house solver

Bypass surgery

Wall Shear Stress, Velocity, Streamlines and pressure were assessed to evaluate the efficiency of the Coronary Artery Bypass Grafting surgical procedure

Rezaeimoghaddam et al. [30] Image source: CT Segmentation performed in Simpleware ScanIP Solver: OpenFOAM (continued)

of blood flow in diagnosis and therapeutically planning. Nowadays, there have been reported studies regarding the application of AI in predicting cardiovascular diseases, computing hemodynamic parameters (Fractional Flow Reserve [79]) or evaluating the risks for producing cardiovascular events (e.g. aneurysm rupture [80]), but there are few studies describing the use of AI algorithms in assessing the hemodynamics

Computational Fluid Dynamics Applications in Cardiovascular …

13

Table 4 (continued) Problem investigated

Clinical applications

Examples of studies (first author) indicating the medical imaging source, software package used for image-based segmentation and Solver used for CFD analysis (if applicable)

Helical tubes for cardiovascular application and bypass surgery optimization

Using a helical tube configuration as graft in bypass surgery may decrease the post-surgical complication by improving the hemodynamic field, reducing the regions with low WSS in the anastomosis zone, and therefore to increase the graft patency

Totorean et al. [90] No segmentation Solver: ANSYS-Fluent Ruiz-Soler et al. [91] No segmentation Solver: ANSYS-CFX

Drug targeting

Blood flow characteristics strongly influence the drug delivery and especially magnetic drug targeting within the human body. Assessing flow field in the regions of interests (e.g. in the vicinity of a coronary stent, near a tumor) could improve drug targeting and therefore could increase therapy’s efficiency CFD studies combined with experimental analysis

Boutopoulos et al. [85] No segmentation Solver: ANSYS-Fluent Bernad et al. [86] No segmentation Solver: ANSYS-Fluent

FSI studies

Fluid Structure Interactions are performed to evaluate the deformation of the arterial wall under the blood flow conditions, to assess the risk of rupture in the case of aneurysm

Mo et al. [87] Image source: 3D rotational angiography Segmentation performed in MIMICS Solver: ANSYS-CFX coupled with ANSYS Mechanical in ANSYS Workbench

field of blood flow in patient-specific models [81, 82]. Li et al. [81] described a deep learning algorithm designed to evaluate the hemodynamic characteristics of blood flow in patient before and after CABG showing good correlations with CFD results. The main hemodynamic parameters assessed by Li et al. [81] were velocity field and pressure. Moreover, AI algorithms are supposed to become an alternative to CFD simulations offering results easier, needing less time and less computational cost than CFD. Nevertheless, AI algorithms need to be trained using training datasets that could be

14

A.-F. Totorean et al.

actually data obtained through CFD analysis, as it was described and used by Li et al. [81]. Among the Computational Fluid Dynamics software environments used for cardiovascular biomechanics blood flow analysis with good references and documentation in the scientific literature, there should be mentioned SimVascular and Ansys Fluent. SimVascular is an open source software package dedicated for cardiovascular system analysis that has integrated all the pipeline steps for a cardiovascular numerical analysis, starting from importing the patient-specific DICOM files for segmentation, 3D model reconstruction, mesh generation and simulation. The segmentation tool has incorporated machine learning algorithms for a more accurate reconstruction. In the case the geometrical model is reconstructed in other segmentation software, the geometry can be imported both as a solid or a 3D surface model and the rest of the analysis pipeline including meshing and simulation can be performed within SimVascular. The meshing tool permits to generate the mesh using boundary layers necessary for a better characterization of blood flow field in the vicinity of the walls. SimVascular does not have implemented turbulence models. The analysis is performed by default in unsteady-state conditions. The inlet prescribed velocity boundary condition can be defined as plug, parabolic or a Womersley profile, simply by selecting this characteristic. The walls can be defined as being rigid or deformable, SimVascular having incorporated the Fluid Structure Interaction tool. The numerical solution accuracy can be achieved by using a unique residual value as a control parameter for the nonlinear iteration. Moreover, in order to perform the simulation by using SimVascular, there is no condition for the model to be reconstructed using the SimVascular segmentation tool [7, 31, 88]. Ansys Fluent is a commercial software commonly used for fluid flow analysis in physics, industry and medicine. It is not dedicated particularly for cardiovascular system investigation, but its tools and characteristics permit to perform complex numerical analysis. The Fluent package has no tool for geometrical model segmentation and meshing, but it permits meshing adaption. It has incorporated turbulence models together with different rheological models. The numerical analysis can be performed either in steady-state or unsteady-state conditions. The prescribed inlet velocity boundary condition can be defined as plug or as a parabolic, Womersley profile by using User Defined Functions. Ansys Fluent permits full Fluid Structure Interaction analysis by coupling the flow solver with the structural solvers. The numerical solution accuracy can be achieved by using the convergence criterion and defining multiple residual values corresponding for continuity, velocity components and in the case of using turbulent models by defining residual values for the turbulence model’s components [88, 89]. In general, the numerical results can be post-processed using several commercial (e.g. Tecplot) or open source (e.g. ParaView) software packages. All these applications are dedicated for CFD data post-processing and, particularly for the cardiovascular system, permitting the analysis of the main hemodynamic parameters such as velocity field, pressure field, Wall Shear Stress or representation of streamlines and velocity vectors.

Computational Fluid Dynamics Applications in Cardiovascular …

15

5 Example: Numerical Analysis of Blood Flow in Abdominal Aorta and Its Branches 5.1 Medical Considerations Abdominal aorta and its branches represent a very important arterial segment of the cardiovascular system being responsible for providing blood supply to organs situated in the abdomen cavity and to the lower limbs (Fig. 3). The celiac artery and its branches are one of the three non-paired arteries in the abdominal cavity responsible for transporting blood to the stomach (through the left gastric artery), spleen (through the splenic artery) and to the liver (through the common hepatic

Fig. 3 Patient-specific geometry characteristics: CT axial view from the abdomen, showing the abdominal aorta and the renal arteries; b General view of abdominal aorta with its branches and their position in the abdomen—3D reconstruction in RadiAnt DICOM Viewer 2020.2; c Anatomical characteristics of the 3D arterial model—volume rendering in RadiAnt DICOM Viewer 2020.2; d The 3D model used for numerical analysis having the geometrical details (bifurcations, branches, curvatures) similar to the 3D model-volume rendered in RadiAnt DICOM Viewer 2020.2

16

A.-F. Totorean et al.

artery). The superior mesenteric artery is one of the main artery arising from the abdominal aorta supplying blood to the midgut. The left and right renal arteries are the paired-branches that transport blood to the kidneys which are mainly responsible for blood filtration. The inferior mesenteric artery is the arterial branch that transports blood to the hindgut. The abdominal aorta continues with the iliac arteries responsible for supplying the lower limbs [92, 93]. Among the pathologies affecting the circulatory system, and particularly the abdominal aorta and its branches there could be mentioned stenosis and aneurysm, their initiation and evolution being strongly related to the hemodynamic environment created by the blood flow. Nevertheless, a better understanding of the blood flow characteristics in this arterial region may contribute to identify the critical regions prone to pathological initiation and evolution together with the possibility of personalizing the therapeutical procedures.

5.2 Geometry The 3D model segmentation was based on Computed Tomography Angiography (CTA) reconstructed images with pixel spacing 0.703125/0.703125 and slice thickness 1.25 mm (Fig. 3a–b). The 3D model was reconstructed using the open source segmentation software package ITK-Snap 3.6.0, and the geometry was finally imported and used in Ansys Fluent and SimVascular softwares. Intraluminal regions associated to aorta and its branches were reconstructed based on the contrast-enhanced CTA image acquisition considering the anatomical arterial segments characterized by mean Hounsfield units (HU) distribution between 300 and 400 HU, as described in literature [94]. It is known that the blood pressure variation along a cardiac cycle may lead to arterial’s diameter modification. The medical image acquisition was performed for the entire abdominal aorta region taking into account that it is difficult to scan the entire aorta at a specific time-step of the cardiac cycle, due to the CT equipment limitation. Moreover, the DICOM files are not associated with a specific stage of blood pressure, but to the entire dynamic contrast-enhanced CTA acquisition procedure. The 3D model was reconstructed based on these files and it was further smoothened. The patient-specific geometry is characterized by one inlet section represented by the aorta and 16 outlet sections corresponding to the aorta’s branches as described in Fig. 3c–d. The pipeline for reconstructing the image-based model and preparation for the numerical analysis includes several open source software packages as shown in Fig. 4. First, the DICOM files were imported in ITK-Snap 3.6.0 and using both manual and automatic segmentation the 3D model was generated taking into consideration the anatomical details. Second, the model was exported as .stl file and further it was imported in Autodesk Meshmixer 3.5 for surface smoothing and remeshing.

Computational Fluid Dynamics Applications in Cardiovascular …

17

Fig. 4 Pipeline for medical image-based model reconstruction

Third, the next step was to prepare the inlet and outlet sections in ParaView 5.6.1, by clipping the arterial model in the input and output regions, using planes perpendicular to the arteries’ centrelines, in order to obtain flat inlet and outlet sections. The clipping operation was performed in order to design the inlet and outlet faces so that their normal vectors to be parallel to the blood flow direction.

18

A.-F. Totorean et al.

Fig. 5 Pipeline for numerical analysis in Ansys Fluent

5.3 ANSYS Fluent Versus SimVascular—Steady-State Analysis To better understand the biomechanical environment within the abdominal aorta and its main branches, a numerical investigation was performed in parallel using two different computational solvers Ansys Fluent and SimVascular. After reconstructing the 3D model, as presented in Sect. 5.1, the .stl file with the patient-specific geometry was further imported in the CFD environments (SimVascular and Ansys Workbench Fluent) for meshing and simulation. The specific steps followed in each of the two software environments considered are described as following. (a)

Pipelines for numerical analysis in Ansys Fluent and SimVascular

Figure 5 shows the steps used in Ansys Workbench for geometry preparation, meshing and simulation. The 3D model was imported as a closed surface .stl file in Ansys Workbench SpaceClaim and it was converted to solid volume. The mesh was generated in Ansys Workbench Mesh tool and it was further loaded in Fluent for numerical analysis. After defining the boundary conditions and the Solver parameters, the simulation ran until the solution reached the residuals set as quality control parameter. The numerical results were saved for post-processing and data analysis. The steps followed in SimVascular for geometry preparation, meshing and simulation are presented in Fig. 6. The 3D model was imported as .stl file in SimVascular, in the “Model section” and it was prepared as a closed surface. The mesh was generated in the meshing section for the 3D model-volume limited by the closed surface and it was further used for numerical analysis. The boundary condition and Solver parameters were set and the simulation ran until the solution reached the residual as a quality control parameter. The numerical results were converted and saved for post-processing and data analysis. The results obtained by numerical investigations and discussed in the next paragraph intend to give doctors critical information related to the complex fluid flow

Fig. 6 Pipeline for numerical analysis in SimVascular

Computational Fluid Dynamics Applications in Cardiovascular …

19

through the human cardiovascular system to better understanding the complex relationships between blood flow, the overall cardiovascular system and correlation with cardiovascular diseases. The next discussions also generated a more precise idea about the importance of numerical simulations to allow medical practitioners to create a more optimal treatment strategy. (b)

Model geometry and mesh details

The patient-specific vessel geometry is obtained from CTA data as presented in Sect. 5.2. To better understand the numerical results and to quickly quantify the results hemodynamic parameters, we used abbreviation for each investigated arterial segment according to Fig. 7. The geometry was meshed separately in SimVascular and Ansys Workbench Mesh tool using tetrahedral cells. Four different meshes with boundary layer mesh were investigated, the number of cells varied between 500,000 and 3,000,000 in both cases.

Fig. 7 Anatomical depiction of the investigated blood vessels model, including descending aorta, abdominal aorta, the major abdominal arteries, iliac arteries, and their relation to the abdominal aorta. Abbreviation of each investigated arterial segment and arterial branches. A patient-specific vessel geometry is obtained from CTA data as presented in the volume image in Fig. 3. Br_1 to br_3 represent the branches no 1, etc.

20 Table 5 Mesh characteristics in Ansys Workbench Mesh tool and SimVascular

A.-F. Totorean et al. Characteristics

Ansys Workbench Mesh

SimVascular

Number of cells

2.786.673

3.038.463

Mesh type

Tetrahedral

Tetrahedral

Boundary layer:

Yes

Yes

– Number of layers

10

10

– First layer height

0.01 mm

0.01 mm

– Layer growth rate

1.25

1

To achieves an accurate value of hemodynamic parameters (velocity distribution, pressure drop, and wall shear stress—WSS), a high mesh resolution near the walls was generated. Different arterial flow simulation studies presented in the literature [95] have shown that using about 600,000 grid nodes in geometry discretization can achieve grid independence in the WSS field. Our numerical analysis grid contained 3,038,463 cells (SimVascular analysis) and 2,786,673 cells (in Ansys Fluent investigations). Since the grid used for geometry discretization in both cases can be considered as a good compromise between accuracy and computational cost. The details regarding the number of cells and meshing characteristics used for meshing are shown in Table 5. To ensure that the obtained numerical solutions were independent of the grid’s size, mesh convergence testing was carried out. This testing was performed under mean flow conditions using the Newtonian blood rheological model. In the case of both solvers, the discretization consisted of an unstructured tetrahedral grid. For grid independence tests (in the case of both solvers), four different mesh configurations were set up with approximately 500,000, 1,000,000, 2,000,000 and 3,000,000 tetrahedral mesh cells. The grid independence tests were carried out by comparing the velocity profiles obtained in the section’s flow direction before the iliac artery bifurcation. The evaluation of the mean velocity showed that the mesh configurations contained a number of cells between 2,000,000 and 3,000,000, which led to almost similar results for both solvers. For both solvers, differences in velocity levels were larger than 6% between the 1 million elements mesh and a coarser mesh containing around 500,000 elements but less than 1.5% between the 2 million elements mesh and a finer mesh containing 3 million tetrahedral elements. As a result, the mesh of around 3 million tetrahedral elements was chosen for the k-ω SST simulations. The volumetric mesh generated for the arterial model is presented in Fig. 8. Both mesh generation tools create an unstructured volumetric mesh. As shown in Fig. 8, the mesh generated for SimVascular analysis suggests a more uniform size and cell distribution (mesh without local mesh refinement) compared to the mesh generated using Ansys mesh generator tools. These differences between the generated mesh can highlight the mesh quality’s importance to obtain an accurate value for investigated hemodynamic parameters.

Computational Fluid Dynamics Applications in Cardiovascular …

21

Fig. 8 The meshing procedure produces an unstructured volumetric mesh both for a SymVacular analysis and in b Ansys mesh generation tools. Detail regarding generated mesh: c for SimVascular and grid developed using Ansys tools (D)

In the vicinity of the wall regions, where the gradient of the velocity normal to the surface changes most drastically, boundary layer meshing is applied. Figure 9 shows the boundary layer generated in both mesh generation tools for different boundary layer characteristics (Table 5). (c)

Numerical setup

The computations were conducted using the commercial CFD package, Ansys v19.2, and, alternatively, the SimVascular software. In the next paragraph, we will compare hemodynamic parameters obtained using numerical tools to present the main advantages of the booth numerical packages. General considerations In both simulations, the flow has been described by the Navier–Stokes equations. The flow was assumed to be laminar. Blood was modeled as a Newtonian viscous fluid, incompressible, homogeneous, and isotropic, with a constant dynamic viscosity μ of the value of 0.00408 Pa.s and a particular mass density ρ of 1060 kg/m3 .

22

A.-F. Totorean et al.

Fig. 9 Boundary layer generated in SimVascular (column a) and Ansys mesh generation tools (column b). Detail regarding developed boundary layers on the different outlet sections (RRA -right renal artery, LIA—left iliac artery, and RIA—right iliac artery)

The results obtained using the Newtonian model qualitatively be similar to those predicted by non-Newtonian models [96–101]. Also, literature shows [100, 101] that the average difference in terms of the wall shear stress between Newtonian and non-Newtonian models is about 10%. In this work, the vessel walls were assumed to be rigid (in both solvers), shown in the literature that overestimates wall shear stress by only about 4.5% [101]. The Newtonian flow assumption greatly simplifies blood modeling, and it is an acceptable compromise in large artery flow numerical simulations.

Computational Fluid Dynamics Applications in Cardiovascular …

23

Boundary conditions The inflow boundary conditions were set in terms of velocity profiles: a flat inflow velocity profile for the axial velocity and zero transverse velocity components at the supra celiac aorta inlet section. The pressure outlet condition was used for all aortofemoral arterial branches (extends from the supra celiac aorta to distal the iliac artery bifurcation). Namely, the pressure was defined to be 0 Pa. The imposed boundary conditions in both solvers are summarized in Table 6. A constant average gauge pressure (p = 0) is imposed at all artery outlets in both solvers (SimVascular and Ansys Fluent) to allow the flow to redistribute between the artery branches. The artery walls were assumed to be rigid (the hypothesis of rigid graft walls was considered acceptable, based on previous studies from the literature [102, 105, 106]), and no-slip conditions were imposed on the wall. In most previous investigations, the distensibility of the blood vessel wall is neglected. For example, in the work of Torii et al. [95], the coupled FSI (fluid– solid interaction) analysis of the human RCA (right coronary artery) investigates the effects of wall compliance on coronary hemodynamics. In this study, comparing the numerical results between the rigid-wall model and FSI model showed an insignificant difference in time-averaged of the WSS. More, Zeng et al. [102] investigated a disease-free RCA using artery motions, physiological realistic compliance data, and in vivo flow waveform. The conclusion is that coronary compliance has little influence on hemodynamic parameter WSS. Numerical simulations The numerical setup (including working conditions and geometries) is identical in both used numerical solvers (Table 7). Also, in both cases, the governing equations are solved iteratively until convergence of all flow variables is achieved. The convergence criterion was set to 10–7 for the residuals of the continuity equation and X, Y, and Z momentum equations. Different research in the literature demonstrated [103, 104] that the transition from the laminar flow to turbulence is predominantly governed by the model geometry rather than flow in the larger arteries. On the other hand, evidence over a wide range of clinical applications (bifurcation stenosis, atherosclerosis, bypass graft, etc.) suggests that flow instabilities, like transition or turbulence, might be important from the hemodynamics characteristics evolution point of view. It is important to mention that in physiological flow (characterized by the moderate Reynolds number), it is difficult to predict whether the transition from laminar to turbulent flow will occur or not. Flow in the arteries bifurcation is characterized by the flow separation region and the core jet both in daughter and parent vessels. In this region, two flow regimes can be found: the acceleration zone in the flow core and deceleration flow in the vicinity of the artery’s branches’ inner wall. The near-wall flow is characterized by flow separation and vortex initiation. The flow velocity decreases quickly because of the transitions from the laminar to the turbulent stage, transitions coupled with large-scale lateral momentum transfer. To solve numerically this flow transitions and

24 Table 6 Geometric model and boundary condition used for numerical simulations

A.-F. Totorean et al. Geometric model details Number of inlets

Number of outlets

Number of investigated vessels

1

16

12

Boundary condition used for numerical simulations Section

Boundary condition SimVascular

Ansys Fluent

Superior Abdominal Aorta inlet

Velocity inlet, V = 0.25 m/s

Velocity inlet, V = 0.25 m/s

CHA br_1

Pressure outlet, P = Pressure outlet, P = 0 Pa 0 Pa

CHA br_2

Pressure outlet, P = Pressure outlet, P = 0 Pa 0 Pa

SMA br_1

Pressure outlet, P = Pressure outlet, P = 0 Pa 0 Pa

SMA br_2

Pressure outlet, P = Pressure outlet, P = 0 Pa 0 Pa

SMA br_3

Pressure outlet, P = Pressure outlet, P = 0 Pa 0 Pa

SMA br_4

Pressure outlet, P = Pressure outlet, P = 0 Pa 0 Pa

LGA br_1

Pressure outlet, P = Pressure outlet, P = 0 Pa 0 Pa

LGA br_2

Pressure outlet, P = Pressure outlet, P = 0 Pa 0 Pa

SA

Pressure outlet, P = Pressure outlet, P = 0 Pa 0 Pa

LRA br_1

Pressure outlet, P = Pressure outlet, P = 0 Pa 0 Pa

LRA br_2

Pressure outlet, P = Pressure outlet, P = 0 Pa 0 Pa

RRA br_1

Pressure outlet, P = Pressure outlet, P = 0 Pa 0 Pa

RRA br_2

Pressure outlet, P = Pressure outlet, P = 0 Pa 0 Pa

IMA

Pressure outlet, P = Pressure outlet, P = 0 Pa 0 Pa

RIA

Pressure outlet, P = Pressure outlet, P = 0 Pa 0 Pa

LIA

Pressure outlet, P = Pressure outlet, P = 0 Pa 0 Pa

All vessels wall * For

Rigid, no-slip

abbreviation, see Fig. 7

Rigid, no-slip

Computational Fluid Dynamics Applications in Cardiovascular …

25

Table 7 Numerical setup in Ansys Fluent and SimVascular Characteristics

Ansys Fluent

SimVascular

Flow regime

Steady-state

Steady-state

Turbulence model

k-ω SST model

None (using the software default characteristics)

Fluid density

1060 kg/m3

1060 kg/m3

Fluid viscosity

0.004 Pa.s

0.004 Pa.s

Boundary conditions: Inlet

Velocity inlet (constant velocity) Vinlet = 0.25 m/s

Outlet

Pressure outlet, Pout = 0 Pa (for all outlet sections)

Wall

Walls were considered rigid, non-deformable, no-slip

Solver—solution quality control parameter (convergence criterion)

Residuals values for continuity and A single residual value set to velocity components set to 10–7 10–7

quantify the evolution in the artery wall vicinity in this paper, we used the k-ω SST turbulent model. In this work, we did a numerical simulation for both solvers in conditions of steady-state analysis. The steady-state simulation was chosen to present flow analysis performances in an arterial structure with several branches for both used numerical solvers. Also, using the steady-state simulation, we avoid the unnecessary expense of performing an unsteady simulation. The numerical simulation aimed to determine the three-dimensional flow structures in the aorto-femoral artery model under steady flow conditions. We also focus on investigating the flow division changes resulting from flow variations in the arterial bifurcation during analysis. Ansys Fluent v19.2 solver To solve the nonlinear system of matrix equations derived from the finite volume method (FVM), the SIMPLE formulation was used for pressure–velocity coupling, and the second-order upwind discretization scheme was used for the momentum and species transport equation (Ansys Fluent, v19.2, Ansys, Inc. Canonsburg, PA, USA). The k-ω turbulent model was used in the present simulation because it can predict low Reynolds number transitional flows inside the bypass anastomosis region [105– 108]. SimVascular solver Blood flow is modeled using the incompressible Navier–Stokes equations. The flow solver inside of SimVascular was developed from the academic finite element code PHASTA (Parallel, Hierarchical, Adaptive, Stabilized, Transient Analysis) for solving the Navier–Stokes equations in an arbitrary domain with the streamlineupwind/Petrov–Galerkin (SUPG) and pressure-stabilizing/Petrov–Galerkin (PSPG) methods [31]. The SimVascular flow solver for velocity and pressure parameters used

26

A.-F. Totorean et al.

stabilization terms for momentum, pressure, and the incompressibility constraint. Also, to prevent instabilities at Neumann boundaries that may experience backflow due to flow reversal, backflow stabilization terms have been added to the SimVascular solver [31]. (d)

Numerical results

To quantify the effect of the abdominal aorta geometry and the abdominal arterial branches geometry, all numerical investigations were carried out under the same flow condition for both used numerical solvers. The flow field for steady flow (imposed flow at the inlet of the supra celiac artery was 3.015 L/min, corresponding to the Reynolds number of Re = 1060) conditions in both models. Arterial bifurcations hemodynamics Along the arterial wall of the bifurcation, the generated hemodynamic environment can be divided into three flow regions [109, 110] (Fig. 10a), namely, (R1) the impingement region, (R2) the acceleration region, and (R3) the recovery region. Each area generated around the apex has a different flow parameter. The evolution of the flow parameters in each region is correlated with initiation and evolution of different types of arterial diseases.

Fig. 10 Flow hemodynamics in the right renal artery (RRA). a Velocity magnitude and velocity vector distribution in a cross section of the RRA. b WSS distribution around the RRA apex. c Streamlines show an intricate flow pattern in the renal artery bifurcation. Figure (a) reveals three distinct flow regions: R1- impingement region, R2-acceleration region, and R3-recovery region

Computational Fluid Dynamics Applications in Cardiovascular …

27

As can see in Fig. 10, the artery bifurcation apex created a central stagnation point characterized by zero velocity, low WSS value, and flow division. The flow division induces flow acceleration both in branches (renal artery) and the main vessel (abdominal aorta) (Fig. 10a). This acceleration region causes loss of the endothelial cells, and from a medical point, o view is characterized by wall remodeling. On the other hand, the recovery region (R3) is characterized by an average level of the WSS (Fig. 10b), a value corresponding to the straight artery segment. It is essential to mention that the presence of the bifurcation alters the luminal flow rate. These alterations generated flow asymmetry and recirculation region in the distal part of the branch (RRA) (Fig. 10c). As in the acceleration region, the strong velocity gradient in the aorta wall’s vicinity can induce loss of endothelial cells. Hemodynamic parameters The human circulatory system is a complex three-dimensional arterial model that comprises curved, straight, and complex arterial trees, which generates a complex hemodynamic environment. Its hemodynamic profile characterizes each geometric structure. For example: in the healthy straight artery segment we have a dominant laminar flow pattern where the WSS range between 10 and 70 dynes/cm2 [111], but in the branched and curved region with strong three-dimensional spatial curvature, the flow form eddies, low WSS regions, and practically become disturbed [112]. This observation is essential from the medical point of view because in vivo observations revealed that the vascular bifurcations regions represent the most preferential location of the atherosclerotic lesions [113]. In this chapter, we investigated a non-planar three-dimensional arterial model. This model induces the inertial dominant flow regime characterized by flow separation, WSS regional variation, and in-plan flow mixing. Steady-state numerical simulations were performed using the 3D artery model in two different numerical solvers to quantify the luminal flow alterations in the presence of vessel bifurcations. The numerical simulation results focus on the main hemodynamic parameters evolutions in terms of pressure (Fig. 11), velocity (Figs. 12, 13 and 16), and wall shear stress field (Figs. 14 and 15). The arterial wall is under the influence of hemodynamic forces as well [114]: (i) pressure force (acting perpendicular to the wall), (ii) the tangential frictional force between the wall surface and blood flow (called shear stress), and (iii) circumferential stretch of the artery wall. The arterial wall is covered by endothelial cells. This cell is exposed to the action of the hemodynamic forces before being presented. Signaling from endothelial cells and hemodynamic force interaction is the fundamental determinants of vascular remodeling [115]. Figures 12 and 13 give an overview of the flow topology in the renal artery bifurcation’s distal part. The in-plane velocity field in different cross sections presents the essential features of the blood flow. As can see in Fig. 12, the inertial force imposes a change in the flow topology. More evident is the flow topology changes in the distal part of the iliac artery bifurcation.

28

A.-F. Totorean et al.

Fig. 11 Pressure distribution (Pa) for numerical solutions obtained using SimVascular solver (a1 and a2) and Ansys Fluent solver (b1 and b2), respectively. Anterior view (a1 and b1) and posterior view (a2, b2) of the pressure distribution in the investigated arterial geometry. Red arrows indicated regions with pressure differences between the results obtained using the different solvers

It is observed that the presence of the bifurcation imposed a change in the flow and induced a shift in the velocity distribution and, on the other hand, secondary flow development. Figures 12 and 13 present a detailed comparison of the flow field’s velocity patterns obtained for the same geometry. Comparison between velocity pattern computed by

Computational Fluid Dynamics Applications in Cardiovascular …

29

Fig. 12 Flow velocity distribution in a different section of the inferior abdominal aorta (I_AA), right renal artery (RRA), and superior mesenteric artery (SMA) obtained using SimVascular solver (a) and Ansys Fluent solver (b). Red arrows show the velocity field differences obtained by used solvers for each investigated section

SimVascular and field computed using Ansys Fluent shows a good agreement. The agreement both in the straight segment of the aorta and in the iliac arteries is good, with the minor exception that the Ansys Fluent solver predicts a slightly faster flow in the central section of the aorta than was predicted by numerical simulation using SimVascular.

30

A.-F. Totorean et al.

Fig. 13 Velocity field distribution in a different cross section in the iliac artery bifurcation. Velocity field distribution presents some differences between results obtained using SimVascular solvers (a) and Ansys Fluent solvers (b). The red arrows indicated differences in velocity distribution

Both numerical solutions indicated that abnormal flow conditions characterize flow in the investigated abdominal aorta segments. This odd (abnormal) flow field is characterized by flow separation in the vicinity of the arterial bifurcation, strong flow impact on the bifurcation apex, and considerable variation of the WSS level around de bifurcation and also in the proximal segment of each bifurcation branches.

Computational Fluid Dynamics Applications in Cardiovascular …

31

Fig. 14 Wall shear stress distribution in the investigated geometry. Comparison between WSS distribution obtained using Simascular (a) and Ansys Fluent solvers (b)

The acceleration and recovery region’s presence around the bifurcation apex increases the near-wall residence time, consequently inducing particle deposition in this arterial region. The presented conclusions are in agreement with literature conclusions where artery regions are identified at the area prone to the development of the intimal hyperplasia [113, 116] and thrombus formation [117]. • Wall shear stress (WSS) analysis In physiological conditions, the endothelial monolayer regulates the transport phenomena between blood and surrounding tissue [118, 119]. On the other hand, an incomplete endothelial monolayer effect (characterized by weak cell connectivity) generates an imperfect or incomplete endothelial function [120]. Suppose the endothelium is exposed to the physiological range of the WSS. In that case, the endothelial cells rearrange to align themselves to the flow direction, which is observable both in vivo and in vitro [121].

32

A.-F. Totorean et al.

Fig. 15 Details of the WSS distributions in the renal artery segment (a and b) and iliac artery bifurcation (c and d). Comparison between results obtained in SimVascular (a and c) and Ansys Fluent (b and d), respectively. Arrows indicated differences in the WSS distribution in the abdominal aorta segment (I_AA) and the superior mesenteric artery (SMA). The red rectangle showed differences in WSS distribution around the iliac artery bifurcation

This reshaping process requires the inter-cellular adhesions within the monolayer and more dynamic cell-to-cell junctions [122, 123]. Based on this data, the response of the endothelial cells depends on the applied WSS level. So, literature differentiated the WSS value into three distinguished regimes: low, high, and supraphysiological WSS values [122]. Under low WSS (8 Pa), the endothelial monolayer loses integrity and substrate coverage. Wall shear stress distribution obtained in the present numerical analysis shows that the investigated geometry generates all WSS ranges described in the literature (Figs. 14 and 15). Both numerical solutions indicated variation in the same arterial segments.

Computational Fluid Dynamics Applications in Cardiovascular …

33

Fig. 16 Detail regarding the complex flow patterns generated by the investigated arterial model. Figures a1 and a2 show the longitudinal section’s velocity field and evidence of the flow pattern changes around the arterial bifurcation area and its influence on the distal part’s flow field. Figures b and c show the flow in streamline b and flow particle distributions (c). As shown in figure c, artery bifurcation induces an increase in particle accumulation in the bifurcation apex’s vicinity. Results from Fig. a1 and a2 were obtained using SimVascular, and figures b and c were using Ansys Fluent solvers

The agreement between the results obtained with different solvers is good. The existing minor differences in WSS distribution are probably due to the difference in the generated mesh. For the quick visualization of the flow field, we used the flow pattern provided by the streamlines. The streamline is defined as a line drawn through the flow field so that the local velocity vector is tangent to the streamline at every point along the line at that instant. It is essential to mention that the streamline does not indicate the magnitude of the velocity. The investigated arterial branches are a complicated

34

A.-F. Totorean et al.

structure where flow division appears by the side of flow deceleration and acceleration. These changes in the flow behavior induce flow separation in the bifurcation regions and also streamlines divisions. To evidence, these phenomena Fig. 16 shows streamlines colored by flow velocities obtained using both numerical solvers used. As can see in Fig. 16, complex arterial geometry induces the complex flow both in the straight area and at the apices of arterial bifurcations. This complicated flow consequently causes a complex tissue response produced by the local hemodynamic microenvironments. Literature suggested that this microenvironment dominated by the high wall shear stress practically contributes to the abdominal aneurysm’s initialization and development [124].

6 Conclusions This chapter intends to promote the numerical analysis’s feasibility regarding the challenges that cardiovascular simulation currently faces. This effort aims to increase the cardiovascular system’s fundamental understanding and provide more accurate and critical information to doctors to choose a better treatment strategy for each patient with cardiovascular diseases. Computational Fluid Dynamics tools, used mainly for cardiovascular biomechanics, are continuously improving to provide numerical results with desired high accuracy for medical applications. The accuracy of the results depends on several factors, some of them being strongly related to patient-specific conditions. Thus, a cardiovascular CFD analysis should use medical image-based geometries and physiological boundary conditions. Over the years there have been developed plenty of commercial, open source or in-house software packages with good references and cardiovascular biomechanics results. The current trend in cardiovascular system analysis uses Artificial Intelligence algorithms, considered to be less time consuming and more cost-efficient than traditional Computational Fluid Dynamics analysis, with similar results as recently reported by literature [81]. On the other hand, a challenge in using AI algorithms is to have an efficient trained algorithm based on accurate training datasets. At this point, the numerical results obtained through traditional CFD studies could contribute as training datasets, as presented in the literature [81]. Both numerical simulations predict the presence of the complex flow patterns contained flow separation (acceleration and recovery area) that induce low and high WSS regions and enhanced particle residence time in the artery wall vicinity. The results of the numerical simulation are summarized in the follows: (i) (ii) (iii) (iv)

Existence of the flow separation due to the presence of the vessels bifurcation, Clear evidence of the change in the flow characteristics once it impacts the bifurcation apex, Both low and high WSS distribution in the vicinity of the bifurcation apex, Bifurcation angle directly influences hemodynamic parameters evolution (pressure, velocity, WSS) and particle deposition around the bifurcation wall.

Computational Fluid Dynamics Applications in Cardiovascular …

35

Discrepancies appear between the results obtained in the used solvers. Differences in obtained results regarding pressure or velocity value, range, and the wall shear stress are due to the different discretization schemes used during the mesh generation process. Differences between numerical results obtained can be explained taking into account the following aspects: – The turbulent behavior of the physiological flow can be approximated well using a suitable turbulence model. SimVascular solver currently does not include turbulence models, while the Ansys Fluent solver has implemented several turbulence models. – On the other hand, SimVascular solver compensates for the boundaries’ dominant effect by pre-conditioning the matrix, giving it substantial speed benefits over Ansys Fluent solver. – The solving strategies for the flow in the vicinity of the wall in flow detachment and vortex generation can be another source for the numerical differences observed between the solvers. This difference is more amplified in the artery bifurcation regions. Instead, the primary advantage of SimVascular is the integration of segmentation, meshing and solver capabilities into one streamlined software pipeline, which enhances the overview of a patient-specific project and decreases the learning curve of the complete simulation cycle. Practically, both solvers used in this chapter increase researchers’ effort in the cardiovascular system’s fundamental understanding.

References 1. Taylor, C.A., Figueroa, C.A.: Patient-specific modeling of the cardiovascular mechanics. Annu Rev Biome Eng. 11, 109–134 (2009). https://doi.org/10.1146/annurev.bioeng.10.061 807.160521 2. Randles, A., Frakes, D.H., Leopold, J.A.: Computational fluid dynamics and additive manufacturing to diagnose and treat cardiovascular disease. Trends Biotechnol. 35(11), 1049–1061 (2017). https://doi.org/10.1016/j.tibtech.2017.08.008 3. Polanczyk, A., Klinger, M., Nanobachvili, J., Huk, I., Neumayer, C.: Artificial circulatory model for analysis of human and artificial vessels. Appl. Sci. 8(7), 1017 (2018). https://doi. org/10.3390/app8071017 4. Polanczyk, A., Podgorski, M., Polanczyk, M., Piechota-Polanczyk, A., Neumayer, C., Stefanczyk, L.: A novel patient-specific Human Cardiovascular System Phantom (HCSP) for reconstructions of pulsatile blood hemodynamic inside abdominal aortic aneurysm. IEEE Access 6, 61896–61903 (2018). https://doi.org/10.1109/ACCESS.2018.2876377 5. Gray, R.A., Pathmanathan, P.: Patient-specific cardiovascular computational modeling: diversity of personalization and challenges. J. Cardiovasc. Transl. Res. 11, 80–88 (2018). https:// doi.org/10.1007/s12265-018-9792-2 6. Bluestein, D.: Utilizing computational fluid dynamics in cardiovascular engineering and medicine – what you need to know: its translation to the clinic/bedside. Artif. Organs. 41(2), 117–121. https://doi.org/10.1111/aor.12914

36

A.-F. Totorean et al.

7. Lan, H., Updegrove, A., Wilson, N.M., Maher, G.D., Shadden, S.C., Marsden, A.L.: A reengineered software interface and workflow for the open-source SimVasccular cardiovascular modeling package. J Biomech. Eng. 140(2), 0245011–02450111 (2018). https://doi.org/10. 1115/1.4038751 8. Douglas, P.S., Pontone, G., Hlatky, M.A., Patel, M.R., Norgaard, B.L., Byrne, R.A., Curzen, N., Purcell, I., Gutberlet, M., Rioufol, G., Hink, U., Schuchlenz, H.W., Feuchtner, G., Gilard, M., Andreini, D., Jensen, J.M., Hadamitzky, M., Chiswell, K., Cyr, D., Wilk, A., Wang, F., Rogers, C., De Bruyne, B.: Clinical outcomes of fractional flow reserve by computed tomographic angiography-guided diagnostic strategies versus usual care in patients with suspected coronary artery disease: the prospective longitudinal trial of FFRCT: outcome and resource impacts study. Eur. Heart J. 36(47), 3359–3367. https://doi.org/10.1093/eurheartj/ehv444 9. FDA.: Reporting of Computational Modeling Studies in Medical Devices Submission, Guidance for Industry and Food and Drug Administration Staff (2016) 10. Vannier, M.W., Marsh, J.L.: Three-dimensional imaging, surgical planning, and image-guided therapy. Radiol. Clin. North Am. 34(3), 545–563 (1996) 11. Spicer, S.A., Taylor, C.A.: Simulation-based medical planning for cardiovascular disease: visualization system foundations. Comput. Aided Surg. 5(2), 82–89 (2000). https://doi.org/ 10.1002/1097-0150(2000)5:2%3c82::AID-IGS2%3e3.0.CO;2-5 12. Milner, J.S., Moore, J.A., Rutt, B.K., Steinman, D.A.: Hemodynamics of human carotid artery bifurcations: computational studies with models reconstructed from magnetic resonance imaging of normal subjects. J. Vasc. Surg. 28(1), 143–156 (1998). https://doi.org/10.1016/ s0741-5214(98)70210-1 13. Doost, S.N., Ghista, D., Su, B., Zhong, L., Morsi, Y.S.: Heart blood flow simulation: a perspective review. BioMed. Eng. Online 15, 101 (2016). https://doi.org/10.1186/s12938016-0224-8 14. Mittal, R., Seo, J.H., Vedula, V., Choi, Y.J., Liu, H., Huang, H.H., Jain, S., Younes, L., Abraham, T., George, R.T.: Computational modeling of cardiac hemodynamics: current status and future outlook. J. Comput. Phys. 305, 1065–1082 (2016). https://doi.org/10.1016/j.jcp. 2015.11.022 15. Caballero, A.D., Lain, S.: A review on computational fluid dynamics modelling in human thoracic aorta. Cardiovasc. Eng. Technol. 4, 103–130 (2013). https://doi.org/10.1007/s13239013-0146-6 16. Wong, K.K.L., Wang, D., Ko, J.K.L., Mazumdar, J., Le, T.-T., Ghista, D.: Computational medical imaging and hemodynamics framework for functional analysis and assessment of cardiovascular structures. BioMed. Eng. Online 16(1), 35 (2017). https://doi.org/10.1186/s12 938-017-0326-y 17. Loscalzo, J.: Harrison’s Cardiovascular Medicine. McGraw-Hill Medical, New York (2010) 18. Koskinas, K.C., Chatzizisis, Y.S., Antoniadis, A.P., Giannoglou, G.D.: Role of endothelial shear stress in stent restenosis and thrombosis: pathophysiologic mechanisms and implications for clinical translation. J. Am. Coll. Cardiol. 59(15), 1337–1349 (2012). https://doi.org/10. 1016/j.jacc.2011.10.903 19. Morris, P.D., Narracott, A., von Tengg-Kobligk, H., Silva Soto, D.A., Hsiao, S., Lungu, A., Evans, P., Bressloff, N.W., Lawford, P.V., Hose, D.R., Gunn, J.P.: Computational fluid dynamics modelling in cardiovascular medicine. Heart 102(1), 18–28 (2016). https://doi.org/ 10.1136/heartjnl-2015-308044 20. Kikinis, R., Pieper, S.D., Vosburgh, K.G.: 3D slicer: a platform for subject-specific image analysis, visualization, and clinical support. In: Jolesz, F. (ed.) Intraoperative Imaging and Image-Guided Therapy. Springer, New York, NY (2014). https://doi.org/10.1007/978-1-46147657-3_19 21. Clark, D., Badea, A., Liu, Y., Johnson, G.A., Badea, C.T.: Registration-based segmentation of murine 4D cardiac micro-CT data using symmetric normalization. Phys. Med. Biol. 57(19), 6125–6145 (2012). https://doi.org/10.1088/0031-9155/57/19/6125 22. Khlebnikov, R., Figueroa, C.A.: CRIMSON: towards a software environmentfor patientspecific blood flow simulationfor diagnosis and treatment. In: Oyarzun Laura, C. et al. (ed.)

Computational Fluid Dynamics Applications in Cardiovascular …

23.

24.

25.

26.

27.

28. 29.

30.

31.

32.

33.

34.

35.

36.

37

Clinical Image-Based Procedures. Translational Research in Medical Imaging. CLIP 2015. Lecture Notes in Computer Science, vol. 9401. Springer, Cham. (2015). https://doi.org/10. 1007/978-3-319-31808-0_2 Amorim, P., Moraes, T., Silva, J., Pedrini, H.: InVesalius: an interactive rendering framework for health care support. In: Bebis, G. et al. (ed.) Advances in Visual Computing. ISVC 2015. Lecture Notes in Computer Science, vol. 9474, pp. 45–54. Springer, Cham (2015). https:// doi.org/10.1007/978-3-319-27857-5_5 Yushkevich, P.A., Piven, J., Hazlett, H.C., Smith, R.G., Ho, S., Gee, J.C., Gerig, G.: Userguided 3D active contour segmentation of anatomical structures: significantly improved efficiency and reliability. Neuroimage 31(3), 1116–1128 (2006). https://doi.org/10.1016/j.neuroi mage.2006.01.015 Huellebrand, M., Messroghli, D., Tautz, L., Kuehne, T., Hennemuth, A.: An extensible software platform for interdisciplinary cardiovascular imaging research. Comput. Methods Programs Biomed. 184, 105277 (2020). https://doi.org/10.1016/j.cmpb.2019.105277 Liu, X., Gao, Z., Xiong, H., Ghista, D., Ren, L., Zhang, H., Wu, W., Huang, W., Hau, W.K.: Three-dimensional hemodynamics analysis of the circle of Willis in the patient-specific nonintegral arterial structures. Biomech. Model Mechanobiol. 15, 1439–1456 (2016). https://doi. org/10.1007/s10237-016-0773-6 Stein, D., Fritzsche, K.H., Nolden, M., Meinzer, H.P., Wolf, I.: The extensible open-source rigid and affine image registration module of the Medical Imaging Interaction Toolkit (MITK). Comput. Methods Programs Biomed. 100(1), 79–86 (2010). https://doi.org/10.1016/j.cmpb. 2010.02.008 Real3d VolViCon User Manual. https://real3d.pk/volvicon/assets/files/Real3d_VolViCon_ Help.pdf. Accessed 02 Feb. 2021 Lopez-Perez, A., Sebastian, R., Izquierdo, M., Ruiz, R., Bishop, M., Ferrero, J.M.: Personalized cardiac computational models: from clinical data to simulation of infarct-related ventricular tachycardia. Front. Physiol. 10, 580 (2019). https://doi.org/10.3389/fphys.2019. 00580 Rezaeimoghaddam, M., Oguz, G.N., Ates, M.S., Bozkaya, T.A., Piskin, S., Lashkarinia, S.S., Tenekecioglu, E., Karagoz, H., Pekkan, K.: Patient-specific hemodynamics of new coronary artery bypass configurations. Cardiovasc. Eng. Technol. 11(6), 663–678 (2020). https://doi. org/10.1007/s13239-020-00493-9 Updegrove, A., Wilson, N.M., Merkow, J., Lan, H., Marsden, A.L., Shadden, S.C.: SimVascular: an open source pipeline for cardiovascular simulation. Ann. Biomed. Eng. 45(3), 525–541 (2017). https://doi.org/10.1007/s10439-016-1762-8 Sommer, K.N., Shepard, L.M., Mitsouras, D., Iyer, V., Angel, E., Wilson, M.F., Rybicki, F.J., Kumamaru, K.K., Sharma, U.C., Reddy, A., Fujimoto, S., Ionita, C.N.: Patient-specific 3D-printed coronary models based on coronary computed tomography angiography volumes to investigate flow conditions in coronary artery disease. Biomed. Phys. Eng. Express 6(4), 045007 (2020). https://doi.org/10.1088/2057-1976/a93f6e O’Hara, R.P., Chand, A., Vidiyala, S., Arechavala, S.M., Mitsouras, D., Rudin, S., Ionita, C.N.: Advanced 3D mesh manipulation in stereolithographic files and post-print processing for the manufacturing of patient-specific vascular flow phantoms. Proc. SPIE Int. Soc. Opt. Eng. 9789, 978909 (2016). https://doi.org/10.1117/12.2217036 Brewis, I., Mclaughlin, J.A.: Improved visualisation of patient-specific heart structure using three-dimensional printing coupled with image-processing techniques inspired by astrophysical methods. J. Med. Imaging Health Inform. 9(2), 267–273 (2019). https://doi.org/10.1166/ jmihi.2019.2644 Narata, A.P., de Moura, F.S., Patat, F., Marzo, A., Larrabide, I., Gregoire, J.-M., Perrault, C., Sennoga, C.A., Bouakaz, A.: A clinically aligned experimental approach for quantitative characterization of patient-specific cardiovascular models. AIP Adv. 10, 045106 (2020). https://doi.org/10.1063/1.5141350 Modi, Y.K., Sanadhya, S.: Design and additive manufacturing of patient-specific cranial and pelvic bone implants from computed tomography data. J. Braz. Soc. Mech. Sci. Eng. 40(10), 503 (2018). https://doi.org/10.1007/s40430-018-1425-9

38

A.-F. Totorean et al.

37. Cignoni, P., Callieri, M., Corsini, M., Dellepiane, M., Ganovelli, F., Ranzuglia, G.: MeshLab: an open-source mesh processing tool. In: Sixth Eurographics Italian Chapter Conference, pp. 129–136 (2008). https://doi.org/10.2312/LocalChapterEvents/ItalChap/ItalianChapConf 2008/129-136 38. Subramaniam, D.R., Stoddard, W.A., Mortensen, K.H., Ringgaard, S., Trolle, C., Gravholt, C.H., Gutmark, E.J., Mylavarapu, G., Backeljauw, P.F., Gutmark-Little, I.: Continuous measurement of aortic dimensions in Turner syndrome: a cardiovascular magnetic resonance study. J. Cardiovasc. Magn. Reson. 19(1), 20 (2017). https://doi.org/10.1186/s12968-0170336-8 39. Ajam, A., Aziz, A.A., Asirvadam, V.S., Muda, A.S., Faye, I., Gardezi, S.J.S.: A review on segmentation and modeling of cerebral vasculature for surgical planning. IEEE Access 5, 15222–15240 (2017). https://doi.org/10.1109/ACCESS.2017.2718590 40. Tian, Y., Chen, Q., Wang, W., Peng, Y., Wang, Q., Duan, F., Wu, Z.: A vessel active contour model for vascular segmentation. Biomed. Res. Int. 6, 106490 (2014). https://doi.org/10.1155/ 2014/106490 41. Steinman, D.A.: Image-based computational fluid dynamics modeling in realistic arterial geometries. Ann Biomed Eng. 30(4), 483–497 (2002). https://doi.org/10.1114/1.1467679 42. Hazer, D., Unterhinninghofen, R., Kostrzewa, M., Kauczor, H.U., Dillmann, R., Richter, G.M.: A workflow for computational fluid dynamics simulations using patient-specific aortic models. In: Conference Proceedings—24th CADFEM Users’ Meeting 2006— International Congress on FEM Technology with 2006 German ANSYS Conference (2006). https://www.researchgate.net/publication/242205379_A_Workflow_for_Computati onal_Fluid_Dynamics_Simulations_using_Patient-Specific_Aortic_Models 43. Berg, P., Voß, S., Saalfeld, S., et al.: Multiple aneurysms AnaTomy CHallenge 2018 (MATCH): phase I: segmentation. Cardiovasc. Eng. Tech. 9, 565–581 (2018). https://doi.org/10.1007/s13 239-018-00376-0 44. Zhu, Y., Chen, R., Juan, Y.-H., Li, H., Wang, J., Yu, Z., Liu, H.: Clinical validation and assessment of aortic hemodynamics using computational fluid dynamics simulations from computed tomography angiography. BioMed. Eng. OnLine 17, 53 (2018). https://doi.org/10. 1186/s12938-018-0485-5 45. Schirmer, C.M., Malek, A.M.: Estimation of wall shear stress dynamic fluctuations in intracranial atherosclerotic lesions using computational fluid dynamics. Neurosurgery 63(2), 326– 334; discussion 334–335 (2008). https://doi.org/10.1227/01.NEU.0000313119.73941.9E 46. Schirmer, C.M., Malek, A.M.: Computational fluid dynamic characterization of carotid bifurcation stenosis in patient-based geometries. Brain Behav. 2(1), 42–52 (2012). https://doi.org/ 10.1002/br89.25 47. Alnaes, M.S., Isaksen, J., Mardal, K.A., Romner, B., Morgan, M.K., Ingebrigtsen, T.: Computation of hemodynamics in the circle of Willis. Stroke 38(9), 2500–2505 (2007) 48. Lee, U.Y., Chung, G.H., Jung, J., Kwak, H.S.: Size-dependent distribution of patient-specific hemodynamic factors in unruptured cerebral aneurysms using computational fluid dynamics. Diagnostics 10(2), 64 (2020). https://doi.org/10.3390/diagnostics10020064 49. Numata, S., Itatani, K., Kanda, K., Doi, K., Yamazaki, S., Morimoto, K., Manabe, K., Ikemoto, K., Yaku, H.: Blood flow analysis of the aortic arch using computational fluid dynamics. Eur. J. Cardiothorac. Surg. 49(6), 1578–1585 (2016). https://doi.org/10.1093/ejcts/ezv459 50. Polanczyk, A., Piechota-Polanczyk, A., Domenig, C., Nanobachvili, J., Huk, I., Neumayer, C.: Computational fluid dynamic accuracy in mimicking changes in blood hemodynamics in patients with AcuteType IIIb aortic dissection treated with TEVAR. Appl. Sci. 8(8), 1309 (2018). https://doi.org/10.3390/app8081309 51. Singh-Gryzbon, S., Ncho, B., Sadri, V., Bhat, S.S., Kollapaneni, S.S., Balakumar, D., Wei, Z.A., Ruile, P., Neumann, F.-J., Blanke, P., Yoganathan, A.P.: Influence of patient-specific characteristics on transcatheter heart valve neo-sinus flow: an in silico study. Ann. Biomed. Eng. 48(10), 2400–2411 (2020). https://doi.org/10.1007/s10439-020-02532-x 52. Ghodrati, M., Maurer, A., Schloglhofer, T., Khienwad, T., Zimpfer, D., Beitzke, D., Zonta, F., Moscato, F., Schima, H., Aigner, P.: The influence of left ventricular assist device inflow

Computational Fluid Dynamics Applications in Cardiovascular …

53.

54.

55.

56.

57.

58.

59.

60.

61.

62. 63.

64. 65.

66. 67. 68.

69. 70.

71. 72.

73. 74.

39

cannula position on thrombosis risk. Artif. Organs. 44(9), 939–946 (2020). https://doi.org/10. 1111/aor.13705 Courchaine, K., Rugonyi, S.: Quantifying blood flow dynamics during cardiac development: demystifying computational methods. Phil. Trans. R. Soc. B. Biol. Sci. 373(1759), 2017033 (2018). https://doi.org/10.1098/rstb.2017.0330 Polanczyk, A., Podyma, M., Stefanczyk, L., Szubert, W., Zbicinski, I.: A 3D model of thrombus formation in a stent-graft after implantation in the abdominal aorta. J. Biomech. 48(3), 425–431 (2018). https://doi.org/10.1016/j.jbiomech.2014.12.033 Johnston, B.M., Johnston, P.R., Corney, S., Kilpatrick, D.: Non-Newtonian blood flow in human right coronary arteries: steady state simulations. J. Biomech. 37(5), 709–720 (2004). https://doi.org/10.1016/j.jbiomech.2003.09.016 Skiadopoulos, A., Neofytou, P., Housiadas, C.: Comparison of blood rheological models in patient specific cardiovascular system simulations. J. Hydrodyn. 29(2), 293–304 (2017). https://doi.org/10.1016/S1001-6058(16)60739-4 Caballero, A.D., Lain, S.: Numerical simulation of non-Newtonian blood flow dynamics in human thoracic aorta. Comput. Methods Biomech. Biomed. Eng. 18(11), 1200–1216 (2015). https://doi.org/10.1080/10255842.2014.887698 Lin, S., Han, X., Bi, Y., Ju, S., Gu, L.: Fluid-structure interaction in abdominal aortic aneurysm: effect of modeling techniques. Biomed. Res. Int. 2017, 7023078 (2017). https://doi.org/10. 1155/2017/7023078 Luraghi, G., Wu, W., De Gaetano, F., Matas, J.F.R., Moggridge, G.D., Serrani, M., Stasiak, J., Constantino, M.L., Migliavacca, F.: Evaluation of an aortic valve prosthesis: fluid-structure interaction orstructural simulation? J. Biomech. 58, 45–51 (2017). https://doi.org/10.1016/j. jbiomech.2017.04.004 Chiastra, C., Migliavacca, F., Martinez, M.A., Malve, M.: On the necessity of modelling fluid-structure interaction for stented coronary arteries. J. Mech. Behav. Biomed. Mater. 34, 217–230 (2014) Mao, W., Li, K., Sun, W.: Fluid-structure interaction study of transcatheter aortic valve dynamics using smoothed particle hydrodynamics. Cardiovasc. Eng. Technol. 7(4), 374–388 (2016). https://doi.org/10.1007/s13239-016-0285-7 https://www.3ds.com/. Accessed 29 Jan. 2021 Dennis, K.D., Kallmes, D.F., Dragomir-Daescu, D.: Cerebral aneurysm blood flow simulations are sensitive to basic solver settings. J. Biomech. 57, 46–53 (2017). https://doi.org/10.1016/ j.jbiomech.2017.03.020 https://www.ansys.com/products/fluids/ansys-fluent. Accessed 29 Jan. 2021 Antiga, L., Piccineli, M., Botti, L., Ene-Iordache, B., Remuzzi, A., Steinman, D.A.: An imagebased modeling framework for patient-specific computational hemodynamics. Med. Biol. Eng. Comput. 46(11), 1097–1112 (2008). https://doi.org/10.1007/s11517-008-0420-1 https://www.ansys.com/products/fluids/ansys-cfx. Accessed 29 Jan. 2021 http://www.crimson.software/. Accessed 29 Jan. 2021 Shin, E., Kim, J.J., Lee, S., Ko, K.S., Rhee, B.D., Han, J., Kim, N.: Hemodynamics in diabetic human aorta using computational fluid dynamics. PLoS ONE 13(8), e0202671 (2018). https:// doi.org/10.1371/journal.pone.0202671 https://www.comsol.com/. Accessed 29 Jan. 2021 Bayraktar, E., Mierka, O., Turek, S.: Benchmark computations of 3D laminar flow around a cylinder with CFX, OpenFOAM and FeatFlow. Int. J. Comput. Sci. Eng. 7(3), 253–266 (2012). https://doi.org/10.1504/IJCSE.2012.048245 http://www.featflow.de/. Accessed 29 Jan. 2021 Ateshian, G.A., Shim, J.J., Maas, S.A., Weiss, J.A.: Finite element framework for computational fluid dynamics in FEBIO. J. Biomech. Eng. 140(2), 0210011–02100117 (2018). https:// doi.org/10.1115/1.4038716 https://febio.org/. Accessed 29 Jan. 2021 Groen, D., Hetherington, J., Carver, H.B., Nash, R.W., Bernabeu, M.O., Coveney, P.V.: Analysing and modelling the performance of the HemeLB lattice-Boltzmann simulation

40

75. 76. 77. 78.

79.

80.

81.

82.

83.

84.

85.

86.

87.

88.

89. 90.

91.

92.

A.-F. Totorean et al. environment. J. Comput. Sci. 4(5), 412–422 (2013). https://doi.org/10.1016/j.jocs.2013. 03.002 http://www.2020science.net/software/hemelb.html. Accessed 29 Jan. 2021 https://www.openfoam.com/. Accessed 29 Jan. 2021 http://simvascular.github.io/. Accessed 29 Jan. 2021 Ouared, R., Larrabide, I., Brina, O., Bouillot, P., Erceg, G., Yilmaz, H., Lovblad, K.-O., Pereira, V.M.: Computational fluid dynamics analysis of flow reduction induced by flow-diverting stents in intracranial aneurysms: a patient-unspecific hemodynamics change perspective. J. Neurointerv. Surg. 8(12), 1288–1293 (2016). https://doi.org/10.1136/neurintsurg-2015012154 Tesche, C., Gray, H.N.: Machine learning and deep neural networks applications in coronary flow assessment: the case of computed tomography fractional flow reserve. J. Thorac. Imaging 35(1), S66–S71 (2020). https://doi.org/10.1097/RTI.0000000000000483 Canchi, T., Kumar, S.D., Ng, E.Y.K., Narayanan, S.: A review of computational methods to predict the risk of rupture of abdominal aortic aneurysms. Biomed. Res. Int. 2015, 861627 (2015). https://doi.org/10.1155/2015/861627 Li, G., Wang, H., Zhang, M., Tupin, S., Qiao, A., Liu, Y., Ohta, M., Anzai, H.: Prediction of 3D cardiovascular hemodynamics before and after coronary artery bypass surgery via deep leasrning. Commun. Biol. 4(1), 99 (2021). https://doi.org/10.1038/s42003-020-01638-1 Sankaran, S., Grady, L., Taylor, C.A.: Impact of geometric uncertainty on hemodynamic simulations using machine learning. Comput. Methods Appl. Mech. Eng. 297, 167–190 (2015). https://doi.org/10.1016/j.cma.2015.08.014 Eslami, P., Hartman, E.M.J., Albaghadai, M., Karady, J., Jin, Z., Thondapu, V., Cefalo, N.V., Lu, M.T., Coskun, A., Stone, P.H., Marsden, A., Hoffmann, U., Wentzel, J.J.: Validation of wall shear stress assessment in non-invasive coronary CTA versus invasive imaging: a patient-specific computational study. Ann. Biomed. Eng. (2020). https://doi.org/10.1007/s10 439-020-02631-9 Gundert, T.J., Marsden, A.L., Yang, W., LaDisa, J.F., Jr.: Optimization of cardiovascular stent design using computational fluid dynamics. J. Biomech. Eng. 134(1), 011002 (2012). https:// doi.org/10.1115/1.4005542 Boutopoulos, I.D., Lampropoulos, D.S., Miller, B.G.C., Loukopoulos, K., V.C.: Two-phase biofluid flow model for magnetic drug targeting. Symmetry 12(7), 1083 (2020). https://doi. org/10.3390/sym12071083 Bernad, S.I., Craciunescu, I., Sandhu, G.S., Dragomir-Daescu, D., Tombacz, E., Vekas, L., Turcu, R.: Fluid targeted delivery of functionalized magnetoresponsive nanocomposite particles to a ferromagnetic stent 519, 167489 (2021). https://doi.org/10.1016/j.jmmm.2020. 167489 Mo, X., Meng, Q., Yang, X., Li, H.: The impact of inflow angle on aneurysm hemodynamics: a simulation study based on patient-specific intracranial aneurysm models. Front Neurol. 11, 534096 (2020). https://doi.org/10.3389/fneur.2020.534096 Van de Velde, L.: Computational fluide dynamics: a clinician’s tool for femoral artery stenosis. Dissertation, University of Twente, Enschede, The Netherlands (2018). https://essay.utwente. nl/74902/1/Velde_MA_TNW.pdf. Accesed 29 Jan. 2021 ANSYS Fluent 19.2 User’s Guide. Totorean, A.F., Bernad, S.I., Susan-Resiga, R.F.: Fluid dynamics in helical geometries with applications for by-pass grafts. Appl. Math. Comput. 272(3), 604–613 (2016). https://doi.org/ 10.1016/j.amc.2015.05.030 Ruiz-Soler, A., Kabinejadian, F., Slevin, M.A., Bartolo, P.J., Keshmiri, A.: Optimisation of a novel spiral-inducing bypass graft using computational fluid dynamics. Sci. Rep. 7(1), 1865 (2017). https://doi.org/10.1038/s41598-017-01930-x Sakorafas, G.H., Sarr, M.G., Peros, G.: Celiac artery stenosis: an underappreciated and unpleasant surprise in patients undergoing pancreaticoduodenectomy. J. Am. Coll. Surg. 206(2), 349–356 (2008). https://doi.org/10.1016/j.jamcollsurg.2007.09.002

Computational Fluid Dynamics Applications in Cardiovascular …

41

93. Prakash, M.V., Rajini, T., Shasirekha, M.: The abdominal aorta and its branches: anatomical variations and clinical implications. Folia Morphol. (Warsz). 70(4), 282–286 (2011) 94. Ippolito, D., Talei Franzesi, C., Fior, D., Bonaffini, P.A., Minutolo, O., Sironi, S.: Low kV settings CT angiography (CTA) with low dose contrast medium volume protocol in the assessment of thoracic and abdominal aorta disease: a feasibility study. Br. J. Radiol. 88, 20140140 (2015). https://doi.org/10.1259/bjr.20140140 95. Torii, R., Wood, N.B., Hadjiloizou, N., Dowsey, A.W., Wright, A.R., Hughes, A.D., Davies, J., Francis, D.P., Mayet, J., Yang, G.-Z., Thom, G.S.AMc.G., Xu, X.Y.: Fluid–structure interaction analysis of a patient-specific right coronary artery with physiological velocity and pressure waveforms. Commun. Numer. Meth. Eng. 25, 565–580 (2009). https://doi.org/10. 1002/cnm.1231 96. Soulis, J.V., Giannoglou, G.D., Chatzizisis, Y.S., Farmakis, T.M., Giannakoulas, G.A., Parcharidis, G.E., Louridas, G.E.: Spatial and phasic oscillation of non-newtonian wall shear stress in human left coronary artery bifurcation: an insight to atherogenesis. Coron. Artery Dis. 17(4), 351–358 (2006). https://doi.org/10.1097/00019501-200606000-00005 97. Johnston, B.M., Johnston, P.R., Corney, S., Kilpatrick, D.: Non-newtonian blood flow in human right coronary arteries: transient simulations. J. Biomech. 39(6), 1116–1128 (2006). https://doi.org/10.1016/j.jbiomech.2005.01.034 98. Katritsis, D., Kaiktsis, L., Chaniotis, A., Pantos, J., Efstathopoulos, E.P., Marmarelis, V.: Wall shear stress: theoretical considerations and methods of measurement. Prog. Cardiovasc. Dis. 49(5), 307–329 (2007). https://doi.org/10.1016/j.pcad.2006.11.001 99. Soulis, J.V., Lampri, O.P., Fytanidis, D.K., Giannoglou, G.D.: Relative residence time and oscillatory shear index of non-Newtonian flow models in aorta. In: 2011 10th International Workshop on Biomedical Engineering, pp. 1–4. IEEE, New York (2011). https://doi.org/10. 1109/IWBE.2011.6079011 100. Perktold, K., Resch, M., Florian, H.: Pulsatile non-Newtonian flow characteristics in a threedimensional human carotid bifurcation model. J. Biomech. Eng. 113(4), 464–475 (1991). https://doi.org/10.1115/1.2895428 101. Anor, T., Grinberg, L., Baek, H., Madsen, J.R., Jayaraman, M.V., Karniadakis, G.E.: Modeling of blood flow in arterial trees. Wiley Interdiscip. Rev. Syst. Biol. Med. 2(5), 612–623 (2010). https://doi.org/10.1002/wsbm.90 102. Zeng, D., Boutsianis, E., Ammann, M., Boomsma, K., Wildermuth, S., Poulikakos, D.: A study of the compliance of a right coronary artery and its impact on wall shear stress. J. Biomech. Eng. 130(4), 041014–041111 (2008). https://doi.org/10.1115/1.2937744 103. Fischer, P.F., Loth, F., Lee, S.E., Lee, S.-W., Smith, D.S., Bassiouny, H.S.: Simulation of highReynolds number vascular flows. Comput. Methods Appl. Mech. Eng. 196(31), 3049–3060 (2007). https://doi.org/10.1016/j.cma.2006.10.015 104. Lee, S.E., Lee, S.-W., Fischer, P.F., Bassiouny, H.S., Loth, F.: Direct numerical simulation of transitional flow in a stenosed carotid bifurcation. J. Biomech. 41(11), 2551–2561 (2008). https://doi.org/10.1016/j.jbiomech.2008.03.038 105. Sherwin, S.J., Blackburn, H.M.: Three-dimensional instabilities and transition of steady and pulsatile axisymmetric stenotic flows. J. Fluid Mech. 533, 297–327 (2005). https://doi.org/ 10.1017/S0022112005004271 106. Liu, B.: The influences of stenosis on the downstream flow pattern in curved arteries. Med. Eng. Phys. 29, 868–876 (2007). https://doi.org/10.1016/j.medengphy.2006.09.009 107. Totorean, A.F., Hudrea, C.I., Bosioc, A.I., Bernad, S.I.: Flow field evolution in stented versus stenosed coronary artery. Proc. Rom. Acad. Ser. A-Math. Phys. 18(3), 248–255 (2017) 108. Totorean, A.F., Bosioc, A.I., Bernad, S.I., Susan-Resiga, R.: Critical flow region in the coronary bypass graft anastomosis. Proc. Romanian Acad., Ser. A 16(2), 201–208 (2015) 109. Meng, H., Wang, Z., Hoi, Y., Gao, L., Metaxa, E., Swartz, D.D., Kolega, J.: Complex Hemodynamics at the apex of an arterial bifurcation induces vascular remodeling resem-bling cerebral aneurysm initiation. Stroke 38, 1924–1931 (2007). https://doi.org/10.1161/STROKE AHA.106.481234

42

A.-F. Totorean et al.

110. Bernad, S.I., Susan-Resiga, D., Bernad, E.S.: Hemodynamic effects on particle target-ing in the arterial bifurcation for different magnet positions. Molecules 24(13), 2509 (2019). https:// doi.org/10.3390/molecules24132509 111. Sherwin, S.J., Caro, C.G., Watkins, N., Doorly, D.J., Peiro, J.: Influence of non-planar geometry on flow separation. J. Physiol. 513, 2 (1998) 112. Sherwin, S.J., Shah, O., Doorly, D.J., McLean, M., Watkins, N., Caro, C.G., Peiro, J., Tarnawski, M., Dumoulin, C.L.: Visualisation and computational study of flow at model planar and non-planar end-to-side arterial bypass grafts. J. Physiol. 504, 44 (1997) 113. Sunamura, M., Ishibashi, H., Karino, T.: Flow patterns and preferred sites of intimal thickening in diameter-mismatched vein graft interpositions. Surgery 141(6), 764–776 (2007). https:// doi.org/10.1016/j.surg.2006.12.019 114. Malek, A.M., Alper, S.L., Izumo, S.: Hemodynamic shear stress and its role in ather-osclerosis. JAMA 282(21), 2035–2042 (1999). https://doi.org/10.1001/jama.282.21.2035 115. Hsieh, H.-J., Liu, C.-A., Huang, B., Tseng, A.H.H., Wang, D.L.: Shear-induced endothelial mechanotransduction: the interplay between reactive oxygen species (ROS) and nitric oxide (NO) and the pathophysiological implications. J. Biomed. Sci. 21(1), 3 (2014). https://doi. org/10.1186/1423-0127-21-3 116. Sottiurai, V.S.: Distal anastomotic intimal hyperplasia: histocytomorphology, pathophysiology, etiology, and prevention. Int. J. Angiol. 8(1), 1–10 (1999). https://doi.org/10. 1007/BF01616834 117. Reininger, A.J., Heinzmann, U., Reininger, C.B., Friedrich, P., Wurzinger, L.J.: Flow mediated fibrin thrombus formation in an endothelium-lined model of arterial branching. Thromb. Res. 74(6), 629–641 (1994). https://doi.org/10.1016/0049-3848(94)90219-4 118. Dejana, E., Lampugnani, M.G., Martinez-Estrada, O., Bazzoni, G.: The molecular organization of endothelial junctions and their functional role in vascular morphogenesis and permeability. Int. J. Dev. Biol. 44, 743–748 (2000) 119. Robotti, F., Franco, D., Banninger, L., Wyler, J., Starck, C.T., Falk, V., Poulikakos, D., Ferrari, A.: The influence of surface micro-structure on endothelialization under supraphysiological wall shear stress. Biomaterials 35, 8479–8486 (2014). https://doi.org/10.1016/j.biomaterials. 2014.06.046 120. Dejana, E., Tournier-Lasserve, E., Weinstein, B.M.: The control of vascular integrity by endothelial cell junctions: molecular basis and pathological implications. Dev. Cell. 16(2), 209–221 (2009). https://doi.org/10.1016/j.devcel.2009.01.004 121. Chien, S.: Mechanotransduction and endothelial cell homeostasis: the wisdom of the cell. Am. J. Physiol. Heart Circ. Physiol. 292(3), H1209–H1224 (2007). https://doi.org/10.1152/ ajpheart.01047.2006 122. Orsenigo, F., Giampietro, C., Ferrari, A., Corada, M., Galaup, A., Sigismund, S., et al.: Phosphorylation of VE-cadherin is modulated by haemodynamic forces and contributes to the regulation of vascular permeability in vivo. Nat. Commun. 3, 1208 (2012). https://doi.org/ 10.1038/ncomms2199 123. Lampugnani, M.G., Dejana, E.: The control of endothelial cell functions by adherens junctions. Novartis Found. Symp. 283, 4–13 (2007). https://doi.org/10.1002/978047031941 3.ch2 124. Hoi, Y., Meng, H., Woodward, S.H., Bendok, B.R., Hanel, R.A., Guterman, L.R., Hopkins, L.N.: Effects of arterial geometry on aneurysm growth: threedimensional computational fluid dynamics study. J. Neurosurg. 101(4), 676–681 (2004). https://doi.org/10.3171/jns.2004.101. 4.0676

An Improved Density-Based Compressible Flow Solver in OpenFOAM for Unsteady Flow Calculations Gaurav Kumar and Ashoke De

1 Introduction OpenFOAM is an open-source C++-based platform that provides libraries for FVM discretization and solvers, grid motion and refinement, and various pre-processing and post-processing utilities [1]. OpenFOAM is used for simulating flows involving complex geometries due to the ability of OpenFOAM to use unstructured polyhedral grids. A large number of boundary conditions and grid motion solvers implemented in OpenFOAM allow simulation of a diverse class of moving body problems. It comes with many pressure-based solvers for stationary as well as moving boundary incompressible as well as compressible flow problems. But for compressible flows, rhoCentralFoam is the most popular density-based solver [2]. The rhoCentralFoam solver is very widely used in the research community for simulation of compressible flows, and many new solvers have been developed considering rhoCentralFoam as the primary flow solver for conservation equations [3, 4]. Consequently, all the derived flow solvers, which use rhoCentralFoam as a base solver, have inherited the stability issue of rhoCentralFoam at a high CFL number and all the solvers are restricted to run at a small CFL number resulting in higher computational time. In rhoCentralFoam, convective part of the governing equations are time-integrated explicitly and sequentially which often cause unboundedness in temperature field for not sufficiently small CFL number. In rhoCentralFoam, this instability issue is treated by constructing flux interpolation weights for energy variable from density, velocity and temperature field [2] in which authors find not to resolve the issue in all kinds of flow adequately. To remedy this issue, an improved G. Kumar (B) Department of Aerospace Engineering, Indian Institute of Technology, Kanpur, India e-mail: [email protected] A. De Department of Aerospace Engineering, Indian Institute of Technology, Kanpur, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 D. Zeidan et al. (eds.), Advances in Fluid Mechanics, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-19-1438-6_2

43

44

G. Kumar and A. De

solution method is considered in this paper, which solves all the governing equations simultaneously for the inviscid part. This method is stable at significantly higher CFL number compared to rhoCentralFoam due to more accurate time integration of the conservation equations. It is limited only by the stability conditions of the explicit time integration of the convection term. Implementation of high-order time discretization scheme in the solver adds at least second-order temporal accuracy to the already existing second-order spatial accuracy. Overall, this new implementation is very general and can be readily incorporated with all the other top-level solvers without much difficulty. In this manuscript, Sect. 2 describes the improved solution method and the background for the development of the new algorithm compared to rhoCentralFoam. Its efficacy is delineated through various test cases representing different physical scenarios in Sect. 3. A total of eight test cases is carefully chosen to validate the various aspects of the accuracy and efficiency of the improved solution method. Two problems consisting of 1-D unsteady-inviscid shock tube problems of Sod and ShuOsher are chosen to demonstrate the accuracy of the implementation and solution improvement obtained above rhoCentralFoam. Next, two problems consisting of 2D unsteady-inviscid flow over a forward-facing step and another over a wedge is simulated. Next, a 2D viscous supersonic free jet and an incompressible flow over backward facing step are simulated to verify the stability of the solver for viscous solutions in different Mach number regimes. Out of these six problems, rhoCentralFoam fails to provide any solution for second, fourth and sixth problems due to numerical instability. Next, a very high Reynolds number incompressible turbulent flow over an airfoil is simulated using RANS to establish its stability while using other libraries such as turbulence models. And finally, a pitching airfoil problem is solved with a moving grid to demonstrate the applicability of the solution algorithm for diverse kinds of flow problem. A brief conclusion is put forth in Sect. 4 for the improved solution method developed.

2 Mathematical Model The conservation equations for mass, momentum and energy for a fluid control volume d enclosed by surface ∂ is given by Eq. (1), where W = [ρ ρu ρ E]T represents mass density, linear momentum and total energy of the control volume and u can be substituted as [u v w]T , which make W a five component column vector.    ∂ Wd + (Fc − Fν )d S = Qd. (1) ∂t  ∂  Here, Fc , Fν and Q are convective flux, diffusive flux and volume source due to body forces and volumetric heating, respectively. The expressions for these terms are given in Eq. (2). nˆ represents unit normal vector to the face.

An Improved Density-Based Compressible Flow Solver in OpenFOAM …

45

⎡ ⎤ ⎤ ⎤ ⎡ 0 ρ(u.n) ˆ 0 T ⎢ ⎥ ⎦. ˆ + p nˆ ⎦ Fν = ⎣ ρf τ .nˆ Fc = ⎣ ρu(u.n) ⎦ Q=⎣ T (ρ E + p)(u.n) ˆ ρf.u + q˙h ˆ + κ(∇T ).nˆ (τ .n).u (2) E is defined as E = e + 21 |u|2 where e is the internal energy of the control volume.

T e relates to temperature T as e(T ) = e(T0 ) + T0 Cv (T )dT and variation of Cv (T ) with temperature can be estimated using JANAF table ([5]). τ is the viscous stress tensor, and κ is the thermal conductivity coefficient. In addition to Eqs. (1) and (2), a state equation is used, which is given by p = ρ RT for thermally perfect gases. In Eq. (1), the balance equations are represented as a transport equation of a fivecomponent column vector, which is a compact notation for writing the conservation equations. The conservation equation has four terms, namely, temporal term, convection term, diffusion term and source term. For the brevity of the manuscript, the source term is not discussed here. Still, this solver accounts for this term through various OpenFOAM libraries similar to all other previously implemented solvers in OpenFOAM. ⎡

2.1 Improved Solution Method For inviscid flow, the diffusion term (Fν ) is zero and Eq. (1) reduces to the Euler equation. This conservation equation, for inviscid flow, is hyperbolic and can be solved very efficiently using an explicit time integration method. For inviscid flow without source term, Eq. (1) can be written as Eq. (3) ∂ ∂t



 

Wd +

∂

Fc (W).d S = 0.

(3)

The vector of conservative variables (W) can be transformed to Wp = [ p u e]T , which contains the primitive variables (pressure (p), velocity (u) and internal energy (e)), using Eq. (4). ∂W p ∂W ∂W p ∂W = =P∗ . (4) ∗ ∂q ∂W p ∂q ∂q For a thermally perfect gas, P¯ and P¯ −1 can be given by (5) where I3×3 is a 3×3 identity matrix. ⎡ P¯ =

ρ p ⎢ ρu ⎣ p ρE p

⎤ 0T − Tρ ⎥ ρ I3×3 − ρu T ⎦ 2 ρ|u| ρuT − 2T

P¯ −1

⎡ ⎤  2 (γ − 1) |u|2 + Cv T − (e − e0 ) (γ − 1)uT (γ − 1) ⎢ 1 0 ⎥ − ρu =⎣ ⎦. ρ I3×3  2 T 1 |u| u 1 − (e − e ) − 0 ρ 2 ρ ρ

(5)

Hence, Eq. (3) can be written in terms of primitive variables as given in Eq. (6).

46

G. Kumar and A. De

∂ ∂t

 

Wp d

+

P¯ −1

 ∂

Fc (Wp ).d S = 0.

(6)

In rhoCentralFoam[2], one needs to time-integrate the conservation equations in a consecutive sequence for ρ, (ρu) and (ρ E) and obtain u and e using relations 2 u = (ρu)/ρ and e = (ρρE) − |u|2 . Due to this kind of lag between momentum and energy equation in rhoCentralFoam, which include values lagged from old times, this integration method is sometimes unstable, and temperature fields may become unbounded [2]. To avoid this problem in the current implementation, Fc (Wp ) is calculated from the values of p, u and e from the same time step (or same stage of multi-stage time integration) and all the conservation equations represented by Eq. (6) are integrated together from the same previous time step (or stage) values of Wp using explicit time integration method (at all stages for multi-stage time integration). The main reason for using the premitive variables is to avoid the integration of conservative variable that are a non-linear function of ρ, u and e and are used in discretization of convective flux. Since this solver is used for the solution of compressible flow problems, convective term discretization should account for the transport of flow properties due to both flow velocity as well as propagation of waves. In this regard, KT [6] and KNP [7] methods used in rhoCentralFoam are adopted in its present form, and for all the details about the implementation, the reader should refer to Sect. 3.1 of Ref. [2]. KT and KNP method has been shown in Ref. [2] to achieve excellent spatial accuracy and appropriately resolve spatial transport of flow properties due to flow velocity and wave propagation without accumulating much numerical dissipation. The only difference in implementation lies in the interpolation of the flow variables, transported due to the convection term. For the stability of the solution and the boundedness of temperature fields, authors in Ref. [2] have suggested the interpolation of only ρ, ρu and T fields across the cell centers to the faces. Other fields for the discretization are obtained from these interpolated values. But, we have used the interpolation of primitive variable p, u and e and all other flow variables are obtained from these variables. This has made the implementation of the convective flux discretization programmatically easier and more consistent. However, no improvement in numerical accuracy due to such choice of interpolation variable is claimed. Improvement in numerical accuracy is obtained only by avoidance of partial decoupling of momentum and energy equation during time integration as also observed Ref. [2]. In the case of viscous flow with the diffusion term, this explicit method can suffer from a severe time-step limitation enforced by the modification of eigenvalues of the system of equations due to diffusion term. This becomes increasingly worse as the diffusion term dominates. Alternatively, to solve the Navier-Stokes equation with large time steps as well as have a stable and accurate solution, one needs to discretize the complete governing equation (1) implicitly. This is not an easy programming task, and a complete implicit discretization including the non-linear convection term involves high complexity. Still, a similar accuracy can be obtained without a full implicit discretization of the governing equation. Operator splitting is one of the eas-

An Improved Density-Based Compressible Flow Solver in OpenFOAM …

47

iest implementation and most effective solution to this problem. Hence as a remedy to this problem, we have applied an operator-splitting method, which is similar to one implemented in rhoCentralFoam [2]. To obtain the solution of Eq. (1) for viscous flows, we first define τ¯¯ as Eq. (7) and use stokes hypothesis 3λ + 2μ = 0 τ¯¯ = μ[(∇u) + (∇u)T ] + λtr (∇u).

(7)

Using Eqs. (2) and (7), we can split the viscous flux into two parts such that first part (FνI m ) is linear in u and e, which can be discretized implicitly and second part (FνE x ) which contains non-linear terms and are discretized explicitely. This is given in Eq. (8). ⎤ ⎡ ⎤ ⎡ ⎤ 0 0 0 T ⎥ ⎣ μ∇u.nˆ ⎦ ⎣ ⎢ T τ .nˆ (μ(∇u) + λtr (∇u)).nˆ ⎦ . Fν = ⎣ +  ⎦= κ T (∇e. n) ˆ {μ((∇u) + (∇u)T ) + λtr (∇u)}.nˆ .u. ˆ + κ(∇T ).nˆ (τ .n).u Cv ⎡

(8) From the numerical studies, no quantifiable difference was observed in considering the (FνE x ) as a source term in viscous correction step or inviscid predictor step. Hence, this term is considered in the predictor step, and the corrector step is solved implicitly using Eq. (9). Following this, the pressure field is finally corrected using the state equation for consistency. Figure 1 shows the graphical representation of the new solver and it’s comparison with rhoCentralFoam.

 ∂(ρu) − ∇.(μ∇u) = 0, ∂t v



 ∂(ρe) κ − ∇. ∇e = 0. ∂t Cv v

(9)

Having discussed the implementational difference between rhoCentralFoam and the new solver, it is also important to understand why these changes improve the computational stability of the algorithm. Though it will be challenging to analyze it mathematically, it can be very easily understood using the flow physics represented by the governing equations. Let us consider the set of unsteady-inviscid conservation equations along with state equation for a calorically and thermally perfect gas in non-dimensional form as given in Eq. (10). Non-dimensional variables are calculated as x  = x/L, u = u/u l , t  = u l t/L, ρ  = ρ/ρl , p  = γ p/(ρl cl2 ), e = (γ ∗ (γ − 1))e/cl2 . Here, subscript ’l’ represents the reference quantities at the location (l) and γ is the ratio of specific heats. cl is the reference speed of sound defined as cl2 = γ RTl and Mach number (M) is defined as M 2 = γ u l2 /cl2 . In the final non-dimensionalized equations in Eq. = ∂(φ) + ∂(φu) (10), all the superscript ( ) are dropped for clarity. In Eq. (10), D(φ) Dt ∂t ∂x for a conservative variable φ and other variables have their usual meanings.

48

G. Kumar and A. De

Fig. 1 Major differences between solution algorithm of rhoCentralFoam and the new solver

D(ρ) = 0, Dt D(ρu) 1 + 2 ∇ p = 0, Dt M D( 21 ρ|u|2 ) 1 D(ρe) + 2M 2 + ∇.( pu) = 0, γ − 1 Dt Dt p = ρe.

(10)

One can see the presence of pressure gradient in the momentum equation, which is very important for providing correct forcing term. For compressible flow solution, this pressure is calculated from the state equation as p = ρe. So, it depends on the variation of thermal energy (ρe) in the domain. There are two ways to change the thermal energy: change in flow kinetic energy or pressure work as apparent from the energy equation. In some compressible subsonic and supersonic flows, change in kinetic energy can play an essential role in changing the thermal energy due to the scaling of the second term in the energy equation with M 2 . For low Mach number compressible flow (M < 0.3), the effect of change in kinetic energy becomes minimal. But at the same time in the momentum equation, a tiny change in pressure gradient will also affect the momentum transport significantly due to scaling of pressure

An Improved Density-Based Compressible Flow Solver in OpenFOAM …

49

gradient with the inverse of M 2 . Hence, for compressible flow solution at any Mach number requires the correct distribution of pressure for energy and state equation to force the momentum equation appropriately. Consider, for example, a situation where a fluid control mass gains or loses kinetic energy rapidly, such as flow behind a moving shock wave. Flow behind a moving shock wave observes a high Mach number and fluid mass through which shock passes, gains a significant amount of kinetic energy apart from pressure work done. In this situation, a lot of kinetic energy is acquired from thermal energy. This signifies a strong coupling between the first and the second term in energy equation in Eq. (10) resulting in a large change in (ρe). This large change in (ρe) causes a large change in p, which further acts as a source term in the momentum equation to redistribute this kinetic energy in linear momentum. So, in such a condition, all the terms in the momentum and energy equations strongly affect each other. Hence, considering the algorithm of rhoCentralFoam where mass, momentum and energy equations are time-integrated sequentially in an individual time step from t=t0 to t= t0 + t. The change in kinetic energy calculated from the solution of momentum equation causes the change in thermal energy due to its appearance in the energy equation. But no influence of this energy change is considered back in the momentum equation for the current time step resulting in unphysical values of kinetic energy at certain spots in the flow domain. This is also the cause of negative 2 temperature obtained in rhoCentralFoam when using e = (ρρE) − |u|2 relation. This problem is sometimes avoided by using a smaller CFL number where such rates of change are relatively mild. But, for flows where such rates of change are extreme, rhoCentralFoam fails to work even with tiny CFL number. In this improved implementation, the issue is remedied by transforming the inviscid flux of the complete set of governing equations using the transformation matrix and solving the governing equations in terms of mutually independent primitive variables, still preserving the strong coupling between the momentum and energy equations. And, this is the main reason for the significant improvement in the stability of the new implementation compared to rhoCentralFoam. Also, in such conditions where a strong shock is propagating through the flow domain, a choice of large CFL can result in significant dissipation due to high value of temporal derivatives and truncation error. Hence, a higher-order time integration method alleviates this issue to produce a manageable amount of numerical dissipation.

2.2 Time Integration Method Since the implementation of rhoCentralFoam in Ref. [2] is restricted to first-order Euler time discretization of the solver and mostly emphasizes on the spatial accuracy for obtaining accurate steady-state solutions; this aspect has resulted in a major shortcoming with the accuracy as well as stability of the solver for unsteady flow

50

G. Kumar and A. De

problems. In rhoCentralFoam, time derivative term for inviscid part of the solution is discretized using a simple explicit Euler scheme. This scheme is inherently unstable at large CFL number close to 1.0, and a non-oscillatory solution for a short duration can be obtained only at a tiny CFL number around 0.1–0.5 depending on the problem at hand. For flow problems at Mach numbers more than 1.0, characteristics of Eq. (1) are hyperbolic and it is only natural to use explicit time integration of the governing equation to obtain maximum computational efficiency. Since the OpenFOAM platform is based on the use of unstructured polyhedral volume mesh for spatial discretization, the spatial accuracy of this implementation is limited to second order. Still, a higher-order temporal accuracy can be achieved through higher-order time discretization scheme. Higher-order temporal discretization is much more desirable in flow simulations at high Mach number where the time step size is very small. And a fluid solver that can provide a stable and accurate solution at high CFL number can significantly reduce the computational time. Hence, a low storage Total Variation Diminishing (TVD) Runge-Kutta (RK) time-integration method is used here. Expressions for first, second and third-order time-integration methods are provided in Eqs. (11)–(13), respectively. A complete procedure for the solution of the Euler equation is graphically represented in Fig. 2.

Wp

Wp

n+1

(n,1)

Wp (n,2) Wp n+1

= Wp − n

Wp n+1/2 Wp (n,3) Wp n+1

(11)

  t ¯ −1 = Wp − Fc (Wp n ).d S P d ∂ 

 t ¯ −1 (n,1) = Wp − Fc (Wp (n,1) ).d S , P d ∂ 1 = (Wp n + Wp (n,2) ) 2

(12)

  t ¯ −1 Fc (Wp n ).d S P d ∂ 

 t ¯ −1 = Wp (n,1) − Fc (Wp (n,1) ).d S P d ∂ 3 1 . = Wp n + Wp (n,2) 4 4   t ¯ −1 = Wp n+1/2 − Fc (Wp n+1/2 ).d S P d ∂ 1 2 = Wp n + Wp (n,3) 3 3

(13)

n

Wp (n,1) = Wp n − Wp (n,2)

  t ¯ −1 Fc (Wp n ).d S, , P d ∂

An Improved Density-Based Compressible Flow Solver in OpenFOAM …

51

Fig. 2 Flowchart for the inviscid flow solution and multi-stage time integration

3 Numerical Results 3.1 Shock Tube Experiment The shock tube experiment has been considered as the first test case for the solver to validate the unsteady convection of the shock wave in a shock tube. This is a 1-D inviscid problem where the domain is divided into two regions “L” and “R”, which contains high-pressure gas in the left and low-pressure gas in the right, respectively. The domain extends between −5m ≤ x ≤ 5m with 1000 grid cells. The flow is initially at rest. The thermodynamic details of the flow are given in Table 1. This case setup is chosen based on Refs. [2] and [8]. The diaphragm at x = 0 is ruptured at t = 0 such that a shock wave propagates in the right direction, whereas an expansion wave moves towards the left direction. Figure 3 shows the density profile in the shock tube at t = 7 ms solved using the new solver which is compared with the analytical solution as well as solution obtained using rhoCentralFoam in OpenFOAM. For all the simulations, the cell center to face center interpolation of primitive flow variables

Table 1 Thermodynamic properties of gases before the diaphragm is ruptured (t ≤ 0) Parameters Left (L) Right (R) Density ρ(kg/m3 ) Pressure p(Pa) Temperature T (K)

1.0 105 348.4

0.125 104 278.7

52

G. Kumar and A. De

0.9

Analytical new Solver (1st order, CFL=0.4)

0.7

new Solver (3rd order, CFL=1)

0.5

rhoCentralFoam (1st order, CFL=0.4)

0.1

0.3

Density,ρ(kg m3)

new Solver (2nd order, CFL=1)

−5

−3

−1

1

3

5

Distance along tube x (m)

Fig. 3 Shock tube density profile obtained using new solver compared with analytical calculation and rhoCentralFoam solver after 7 ms from the diaphragm rupture

is obtained using VanLeer TVD scheme, and the new solver is run with RK-1(Euler), RK-2 and RK-3 time integration method. Solution using Euler time discretization at CFL = 0.4 and RK-2 method at CFL = 1.0 in new solver shows dissipation and wrong shock wave speeds. The new solver shows an excellent match of the density profile with the analytical solution when running at CFL number 1.0 for the RK-3 method. But, rhoCentralFoam shows an oscillatory solution even for CFL number 0.4. The new solver mitigates this problem due to strong coupling between conservation equations ensured by simultaneous time-integration of discretized convective flux. This improvement in accuracy of the solution and weak dependence of solution stability on CFL number is also attributed partly to the explicit high order Runge-Kutta time integration method.

3.2 Shu-Osher Shock Tube Problem: Entropy Wave Propagation The Shu-Osher shock-tube problem is the numerical computation of a normal shock front passing through a one-dimensional inviscid flow with artificial sinusoidal density fluctuations at rest. This is a difficult configuration to simulate as it includes resolving shock wave as well as smooth density fluctuations in close vicinity. A numerically diffusive solver with low order convection term discretization will

An Improved Density-Based Compressible Flow Solver in OpenFOAM …

53

Density,ρ(kg m3) 1.0 2 3 4

5

Fig. 4 Domain and initial flow conditions (at t = 0) for pressure(p), x-velocity(U) and density(ρ)

Taylor et al. (2007)

0

Simulation

0

0.2

0.4 0.6 0.8 Distance along tube x (m)

1.0

Fig. 5 The density profile at t = 0.178 s for Shu-Osher shock tube problem

smooth out all the density fluctuations whereas a high-order convection term discretization will produce oscillatory solution due to the presence of normal shock. In this simulation, VanLeer flux limiter is used, which provides excellent switching between first and second-order reconstruction for convection term discretization. Domain details and initial flow conditions are given in Fig. 4. rhoCentralFoam solver fails to provide any solution for this problem due to numerical instability, whereas a solution is obtained at t = 178 ms with 1000 cells using the new solver. The density field is compared with the resolved numerical solution of Ref. [9] in Fig. 5. The simulated result is in good agreement with the reference solution, albeit only a second-order spatial discretization scheme is used with Vanleer limiter and third-order TVD Runge-Kutta time integration method. This solution obtained is important to note because no solution could be obtained using the previous solver rhoCentralFoam with exactly the same numerical setup due to instability. With rhoCentralFoam, even for a tiny value of CFL number, the solution diverges after a few time steps due to excessive numerical oscillation which was still manageable in the previous shock-tube problem in Sect. 3.1. The cause of such numerical oscillation is the same as described in Sect. 2.1.

54

G. Kumar and A. De

3.3 Forward Step A forward-facing step introduced by Emery [10] as a test case for numerical schemes has been used in Ref. [2] for the validation of rhoCentralFoam solver implementation. The same test case has been adopted in this study to validate the solver for an inviscid, unsteady 2D convection problem. The same domain and mesh setup used in Ref. [2] 1 in both directions. rhoCentralFoam solver is run has been used here with grid size 80 at CFL number 0.2, whereas the new solver is run at CFL number 1.0 with 2 and 3 stage TVD Runge-Kutta time-integration method. Density contour lines are plotted in Fig. 6, which shows a very close agreement in results obtained using the two solvers except for the elimination of spurious oscillations from RK-2 and RK-3 simulations unlike ones seen in rhoCentralFoam. Also, computational time taken for simulating 4 s of physical time with the two solvers is compared in Table 2. rhoCentralFoam takes much longer time even compared to RK-3 simulation due to the necessity of a choice of small CFL number for the stability of rhoCentralFoam solver, however, a small amount of oscillation is still present in the solution with rhoCentralFoam.

3.4 Inviscid Shock-Vortex Interaction For further estimation of numerical stability and numerical dissipation of an inviscid solver, the shock-vortex interaction in Schardin’s problem studied by Chang et al. in Ref. [11] is adopted here. In this problem, a shock wave is traversing through an equilateral triangle (or wedge) at shock Mach number Ms = 1.34. The side of the triangular wedge is 20 mm and is confined within a shock tube of width 150mm while the wedge being placed at the centerline. Time t=0 is assigned at the event when the shock wave hits the leading edge of the wedge for the first time. As shown in Fig. 7, the same computational domain is adopted as in Ref. [11] with slip velocity conditions at the wedge, top, and the bottom wall and non-reflecting boundary conditions at the inlet and the outlet. Simulation is performed on a coarse grid where cell size is of the order 0.01 mm behind the wedge and 0.1 mm in the far-field. As shown in Fig. 8a, the new solver can predict the exact features of shockvortex interactions as mentioned in Ref. [11] without any oscillation or dissipation of structures using the TVD RK-3 method at CFL 1.0 unlike rhoCentralFoam. Figure 9 shows the density gradient field obtained using rhoCentralFoam at CFL = 0.2 and the new solver at CFL = 1.0, where the first solution is marred with spurious oscillations and false numerical waves even at a tiny CFL number. Hence, the new solution method implementation and use of a higher-order time discretization scheme have truly improved the performance of the solver to a greater extent. Also, an excellent match is obtained for comparison of computed tangential velocity in the primary vortex region with the published results of Ref. [11] in Fig. 8b.

An Improved Density-Based Compressible Flow Solver in OpenFOAM …

55

Fig. 6 Comparison of shock structures for forward step using density contours at t = 4 s for different time integration accuracy and solvers

56

G. Kumar and A. De

Table 2 Comparison of computational time for different time integration accuracy and solvers Solver Time (s) rhoCentralFoam New solver (RK-2) New solver (RK-3)

153 70 101

Slip wall t=0 Ms = 1.34

Non-reflecting BC

Non-reflecting BC

Wedge

SymmetrayPlane

Fig. 7 Domain and boundary conditions for shock-vortex interaction flow

3.5 Supersonic Jet In this subsection, a 2D axisymmetric supersonic free jet flow has been validated against the experiments of Ladenburg et al. [12]. A case with the tank pressure 60 lb/in 2 or 4.14 bar is chosen based on recommendations of Ref. [2] which suggests that it produces a Mach stem that is challenging to reproduce accounting for the correct amount of viscous and numerical dissipation. An exactly same case setup is used in the current simulation as followed in Ref. [2] and an excellent match of density contours with experimental data of Ref. [12] is seen without any spurious oscillations. The simulation is run using the TVD RK-3 time integration method at CFL 1.0. Figure 10 shows the density contours comparison of simulation using the new solver with the experiment (Ref. [12]), and an excellent match in the location of the Mach stem shows the small numerical dissipation for high Mach number flows.

An Improved Density-Based Compressible Flow Solver in OpenFOAM …

57

Fig. 8 Computational results at time = 175 µs. t = 0 is measured from the time shock wave first arrives at the leading edge of the wedge

3.6 Laminar Flow over a Backward Facing Step at Low Mach Number To validate the stability of the new solver for low Mach number or incompressible flows, a laminar subsonic flow past a backward facing step is considered at Mach number (M) = 0.1. This flow has been experimentally and numerically studied in Ref. [13]. This problem has also been validated in Ref. [14], which points out the failure of rhoCentralFoam in simulating such low Mach number flow. A domain of length 26H in downstream direction and H in the upstream direction is fixed from the step corner where H is the height of the step. The flow expands from a channel of height H (= 10 mm) to another channel of height 2H downstream of the

58

G. Kumar and A. De

Fig. 9 Comparison of density gradient fields at time = 40 µs between rhoCentralFoam (top) and the new solver (bottom)

Fig. 10 Comparison of density contours in Ludenburg jet. Our simulation (top) and original experimental data [12] (bottom). Experimental data reprinted with permission from ladenburg et al. [12] Copyright (1949) by the American Physical Society

An Improved Density-Based Compressible Flow Solver in OpenFOAM …

59

Fig. 11 Streamlines for the backward facing step simulation at Reh = 100 and 400 Table 3 Comparison of reattachment length location for flow over backward facing step at Reh = 100 and 400 with previously published results Reh = 100 Reh = 400 Experiment (Ref. [13]) QGDFoam (Ref. [14]) SimpleFoam (Ref. [14]) New solver

5 4.9 4.8 4.3

14.2 11.6 11.5 10.9

step corner. The flow is simulated for Reh = 100 and 400 and M = 0.1. To reduce the computational cost, a parabolic velocity profile is provided at the upstream inlet boundary with fixed total temperature and Neumann pressure boundary condition. A no-slip adiabatic wall boundary condition is provided at the upper and lower walls. The downstream outlet is provided with a fixed total pressure boundary condition with Neumann boundary conditions for velocity and temperature. For, further details about the case setup, Ref. [14] should be referred. Figure 11 shows the streamlines of the flow over the step and in the separation region obtained from the simulation at Reh = 100 and 400 using the new solver. The solution is in good agreement with the experiment [13] and previous numerical simulation of Ref. [14]. A discrepancy is observed in the prediction of reattachment length due to extra numerical dissipation at low Mach number. A Comparison of reattachment length prediction is shown in Table 3.

3.7 Turbulent Flow over a NACA0012 Airfoil To demonstrate the compatibility of the new solver with the turbulence model libraries available in OpenFOAM [1], the simulation of a turbulent flow over a NACA0012 airfoil is chosen at an angle of attack (AOA) = 15◦ . The Reynolds number (Re) is

60

G. Kumar and A. De

Fig. 12 Domain details for the flow over NACA0012 airfoil

Fig. 13 Span-wise vorticity contour and streamlines around NACA0012 airfoil at 15◦ AOA

based on free-stream speed (U∞ ), and the chord length (c) of the airfoil is Rec = 6 million and freestream Mach number (M∞ ) = 0.15. This flow condition lies in the fully turbulent subsonic flow regime where flow is steady and attached to the airfoil surface. For computation, a structured C-type grid has been generated around the airfoil with far-field boundary located at 500c from the 1/4-chord point of the airfoil in accordance with the validation study performed in Ref. [15]. The airfoil surface is divided into 600 segments and first cell height at the airfoil surface is chosen such that average y+ is 54. The total number of cells in the computational domain is 0.779 million. For turbulence modeling, a very well-known RANS model namely SpalartAllmaras [16] has been used, which is supposed to produce very accurate prediction of flows on airfoil surfaces. Since, the near wall region is not completely resolved, nutUSpaldingWallFunction has been used, which was proposed by Spalding [17] and it works up to a wide range of y+ values. No-slip adiabatic solid wall boundary condition is applied at the airfoil surface and a wave transmissive boundary condition at the far-field boundaries. The grid pattern near is airfoil is shown in Fig. 12. Figure 13 shows the span-wise vorticity contours and streamlines from the computed solution around the airfoil at 15◦ AOA. For quantitative evaluation of the accuracy of the simulation, the wall pressure coefficient has been plotted over the

61

−12

An Improved Density-Based Compressible Flow Solver in OpenFOAM …

2

0

−2

−4

Cp

−6

−8

−10

Gregory et al. Ladson et al. Simulation

0.0

0.2

0.4

0.6

0.8

1.0

X c Fig. 14 Wall pressure coefficient (C p ) over a NACA0012 airfoil (M∞ = 0.15, Rec = 6M) Table 4 Force coefficients for simulation of flow over NACA0012 airfoil at 15◦ AOA CL CD CFL3D (NASA [19]) Simulation

1.5461 1.5675

0.02124 0.02313

airfoil surface and compared with the experimental results of Gregory et al. [18] and Ladson et al. [19] in Fig. 14. They are in very good agreement with each other. As no pressure coefficient data are available on the pressure side of the airfoil from Gregory’s experiment [18], Ladson’s data [19] have been used but for the comparison on the suction-side, Gregory’s data [18] are used as it is more two dimensional and more appropriate for CFD validation of surface pressure coefficient. In Table. 4, force coefficients obtained from the simulation are also compared with the CFL3D code data from Ref. [15], which shows a very good agreement in the prediction of lift coefficient but the drag coefficient is slightly overpredicted due to mild dissipation produced by the inviscid flux splitting scheme near the stagnation point. Since, contribution of the pressure variation near the leading edge due to stagnation point is a major factor in determining overall drag coefficient, drag coefficient is overpredicted whereas a very good agreement is seen for the overall pressure coefficient distribution and lift coefficient prediction at such a low Mach number flow using a compressible flow solver.

62

G. Kumar and A. De

3.8 Flow over a Pitching NACA0012 Airfoil (M∞ = 0.3, Rec = 3M) Further to validate the solver for moving body problem with a non-stationary grid, a NACA0012 airfoil setup is considered for M∞ = 0.3, Rec = 3M where the airfoil is pitching about 15◦ mean AOA and 10◦ amplitude of pitching. This case is designed based on Ref. [20] where the flow data have been experimentally obtained for free-stream Mach number (M∞ ) = 0.3 and Reynolds number based on chord length (Rec ) = 3 million. This flow condition lies in the fully turbulent flow regime where the flow is unsteady and separated from the airfoil surface. For computation, a hybrid structured/unstructured grid has been generated around the airfoil with farfield boundary located at 50c from the 1/4-chord point of the airfoil. A schematic of the domain is shown in Fig. 15. The domain is subdivided into two parts at the sliding interface, outside which the domain is kept fixed whereas the domain inside the sliding interface can be rotated along with airfoil to obtain different effective angles of attack. Effective angle of attack is obtained from the difference between the freestream flow angle and the angle of the airfoil with the horizontal. A cyclicAMI boundary condition available in openFOAM is used, which enables the interpolation of flow fields across the interface [1]. The airfoil surface is divided into 440 segments and first cell height at the airfoil surface is chosen such that average y+ is 45. The total number of cells in the computational domain is 0.11 million. For turbulence modeling, a RANS model namely Spalart-Allmaras [16] has been used, which is supposed to produced very accurate prediction of flows on airfoil surfaces.

Fig. 15 Domain details for the flow over NACA0012 airfoil

An Improved Density-Based Compressible Flow Solver in OpenFOAM …

63

Fig. 16 Wall pressure coefficient (C p ) over a NACA0012 airfoil (M∞ = 0.3, Rec = 3M)

Fig. 17 Mach number contour and supersonic flow pocket over a NACA0012 airfoil (M∞ = 0.3, Rec = 3M)

Since, the near wall region is not completely resolved, nutUSpaldingWallFunction has been used [17]. No-slip adiabatic solid wall boundary condition is applied at the airfoil surface and a free-steam boundary condition at the far-field boundaries. The overall flow domain setup is shown in Fig. 15. For validation, another simulation at M∞ = 0.3, Rec = 3M and AOA = 15◦ is performed for a stationary case and distribution of pressure coefficient over the chord length is shown in Fig. 16. The only difference observed in Fig. 16 is higher value of C p on the suction side due to the shock wave generated at the leading edge. This shock wave is the consequence of the supersonic flow pocket formed at the leading edge of the airfoil due to very high acceleration as shown in Fig. 17. The pitching motion about the quarter-chord axis of

64

G. Kumar and A. De

Fig. 18 Span-wise vorticity contour and streamlines around NACA0012 airfoil at 25◦ AOA for pitching motion

the airfoil is given by θ = 15◦ − 10◦ cos(49.5t). Figure 18 shows a snapshot of the span-wise vorticity field and the streamlines when the airfoil is at 25◦ AOA. It can be seen that at high AOAs, flow is massively separated and undergoes deep stall. In Fig. 19, lift, drag and moment coefficients have been plotted over the cycle and compared with the experimental results from Ref. [20]. The results are in good agreement with experiment for upstroke cycle before stall and a deviation from experimental observation is seen in the post-stall region and down-stroke region, which is due to the use of a RANS model.

4 Conclusion An improved implementation of a density-based compressible flow solver is introduced in OpenFOAM using a coupled solution of governing equations and high-order TVD Runge-Kutta time discretization scheme. This has eradicated the instability issue of the previous implementation of density-based solver in OpenFOAM called rhoCentralFoam as well as it has improved accuracy for time-integration procedure. Multiple unsteady test cases such as 1D inviscid flow in a shock tube, Shu-Osher shock tube problem, 2D inviscid flow over a forward step and a wedge, 2D viscous flow in a supersonic free jet and low Mach number laminar flows over a backwardfacing step and airfoil have been simulated to establish the improvements gained through this new implementation. These test cases have invariably demonstrated the superior stability of the newly implemented solution algorithm in all Mach number regimes where conservation equations are simultaneously time-integrated for inviscid part of solution unlike rhoCentralFoam. However, the authors do not claim any increase in efficiency of the solver for obtaining a solution to low Mach number flows as no remedy is issued for high numerical condition number at low Mach numbers. Although this solution method has increased the stability of the solution method as was an issue with rhoCentralFoam solver. The solution is still expected to pro-

An Improved Density-Based Compressible Flow Solver in OpenFOAM …

Fig. 19 Force coefficients flow over a pitching airfoil (M∞ = 0.3, Rec = 3M)

65

66

G. Kumar and A. De

duce excess dissipation when used with first-order Euler time discretization, but the implementation of higher-order TVD Runge-Kutta method has further improved the solution method to be used at CFL = 1 without sacrificing low-dissipation properties of inviscid-flux discretization method at high Mach number. For low Mach number flows, numerical dissipation is still high in the absence of a low-diffusion flux splitting scheme designed for low Mach number. Overall, this improvement in solver is successful in improving the quality and speed of current long-duration unsteady compressible flow simulations, for not very low Mach number flows (M > 0.1), using OpenFOAM open-source toolbox. Acknowledgements Simulations are carried out on the computers provided by National PARAM Supercomputing Facility (CDAC) (www.cdac.in) and the manuscript preparation as well as data analysis have been carried out using the resources available at IITK. The support is gratefully acknowledged.

References 1. C.J. Greenshields, OpenFOAM Foundation Ltd, version 3(1), 47 (2015) 2. Greenshields, C.J., Weller, H.G., Gasparini, L., Reese, J.M.: Int. J. Numer. Meth. Fluids 63(1), 1 (2010) 3. Casseau, V., Espinoza, D., Scanlon, T., Brown, R.: Aerospace 3(4), 45 (2016) 4. Kraposhin, M., Bovtrikova, A., Strijhak, S.: Procedia Comput. Sci. 66, 43 (2015) 5. Chase M.W., Jr: J. Phys. Chem. Ref. Data, Monograph 9 (1998) 6. Kurganov, A., Tadmor, E.: J. Comput. Phys. 160(1), 241 (2000) 7. Kurganov, A., Noelle, S., Petrova, G.: SIAM J. Sci. Comput. 23(3), 707 (2001) 8. Sod, G.A.: J. Comput. Phys. 27(1), 1 (1978) 9. Taylor, E.M., Wu, M., Martín, M.P.: J. Comput. Phys. 223(1), 384 (2007) 10. Emery, A.F.: J. Comput. Phys. 2(3), 306 (1968) 11. Chang, S.M., Chang, K.S.: Shock Waves 10(5), 333 (2000) 12. Ladenburg, R., Van Voorhis, C., Winckler, J.: Phys. Rev. 76(5), 662 (1949) 13. Armaly, B.F., Durst, F., Pereira, J., Schönung, B.: J. Fluid Mech. 127, 473 (1983) 14. Kraposhin, M.V., Smirnova, E.V., Elizarova, T.G., Istomina, M.A.: Comput. Fluids 166, 163 (2018) 15. https://turbmodels.larc.nasa.gov/naca0012_val.html. Accessed 2020-09-19 16. Spalart, P., Allmaras, S.: In: 30th Aerospace Sciences Meeting and Exhibit (1992), p. 439 17. Spalding, D.: J. Appl. Mech. 28(3), 455 (1961) 18. Gregory, N., O’reilly, C.: Low-Speed aerodynamic characteristics of NACA 0012 aerofoil section, including the effects of upper-surface roughness simulating hoar frost. HM Stationery Office London (1973) 19. Ladson, C.L., Hill, A.S., Johnson, W.G. Jr: (1987) 20. McAlister, K.W., Carr, L.W., McCroskey, W.J.: (1978)

Finite Volume Models and Efficient Simulation Tools (EST) for Shallow Flows S. Martínez-Aranda, J. Fernández-Pato, I. Echeverribar, A. Navas-Montilla, M. Morales-Hernández, P. Brufau, J. Murillo, and P. García-Navarro

1 Introduction Shallow-type mathematical models are built in the context of free surface flows over the main hypothesis that the flow layer depth is smaller than a relevant horizontal length scale. They represent a simplified form of the Navier-Stokes equations so that only mass and momentum conservation are required. This approximation is applicable to a large number of physical processes. In particular, it is widely used for hydraulic/hydrological surface flows, both over rigid or movable beds. The numerical modeling of 2D shallow flows in complex geometries involving transient flow and movable boundaries has been a challenge for researchers in recent years [42]. There is a wide range of physical situations, such as flow in open channels S. Martínez-Aranda · J. Fernández-Pato · I. Echeverribar · A. Navas-Montilla · M. Morales-Hernández · P. Brufau (B) · J. Murillo · P. García-Navarro Fluid Mechanics, University of Zaragoza, Zaragoza, Spain e-mail: [email protected] S. Martínez-Aranda e-mail: [email protected] J. Fernández-Pato e-mail: [email protected] I. Echeverribar e-mail: [email protected] A. Navas-Montilla e-mail: [email protected] M. Morales-Hernández e-mail: [email protected] J. Murillo e-mail: [email protected] P. García-Navarro e-mail: [email protected] J. Fernández-Pato · I. Echeverribar Hydronia-Europe S.L, Madrid, Spain © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 D. Zeidan et al. (eds.), Advances in Fluid Mechanics, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-19-1438-6_3

67

68

S. Martínez-Aranda et al.

and rivers, tsunamis and flood modeling, which can be mathematically represented by first-order nonlinear systems of partial differential equations, whose derivation involves an assumption of the shallow-type [125]. Shallow flow models may include more sophisticated terms when applied to cases of not pure water, such as mud/debris flows, landslides or turbidity currents. Mud/debris flows are unsteady flow phenomena in which the flow changes rapidly, and the properties of the moving fluid mixture include stop and go mechanisms [55, 98]. The shallow-type equations have also been used to model load transport over an erodible bed in realistic situations involving transient flow and movable flow boundaries. With rare exceptions, the governing equations in shallow-type flows are hyperbolic. Finite volume methods, initially developed for solving problems in gas dynamics, have been accepted as a reliable and accurate tool for the numerical solution of the shallow water equations [112]. Numerical techniques were first derived on that basis for the inviscid Euler equations and have been considered a starting point in the numerical modeling of shallow water flows traditionally paying little attention to the presence of source terms associated with bed friction and variations in the bed slope [4, 41]. Nevertheless, when solving real problems, one is likely to encounter all sorts of situations, and in a wide range of them, flow is dominated by bed friction and bed variations. There is yet another important difference between the Euler equations and the shallow water (SW) equations. Contrary to the inviscid Euler equations, the SW equations ignore energy considerations in the dynamic formulation of the flow. Energy conservation requires quantification of the specific internal energy. For the Euler equations, specific internal energy can be defined using a caloric equation of state [115]. In the case of hydraulic jumps, where intense three-dimensional turbulence modifies the internal energy, it is not possible to provide an algebraic expression for this quantity. Its variation can only be determined once the sequence of depths and average velocities in the jump are determined by the momentum equation. Numerical experience has shown that appropriate discretization of source terms is necessary [52, 126]. The presence of the source terms has led to the notion of well-balanced schemes that have resulted in efficient explicit finite volume models of shallow water flow [2, 41]. When well-balanced numerical schemes are applied to real cases with irregular geometries and transient flow, involving large variations in the bed slope, advance over dry areas and also the drying of initially wetted regions. Those flow features impose heavier restrictions than the classical Courant-FriedrichsLewy (CFL) condition on the time step size [9, 10]. Well-balanced schemes pay great attention to the role of the bed slope term in the shallow water equations, but it must be stressed that the role of bed friction is not a minor issue for model users. This is especially true in wet/dry fronts that can interfere with the stability of the numerical solution [11, 82]. The recommended tendency to avoid numerical errors and oscillations arising from the friction terms is the use of an implicit treatment for this term. However, even though implicit discretization of the friction terms ensures stability, a detailed analysis shows that they lead to undesirable non-uniform discharge values in steady cases. Appropriate discretization of friction terms by following techniques based on the developments made in well-balanced schemes can remedy these situations, but the explicit nature

Finite Volume Models and Efficient Simulation Tools (EST) for Shallow Flows

69

of the required treatment may lead to oversized discrete friction forces, and as a consequence, strong reductions in the limit of the time step size [11]. The definition of augmented numerical schemes (augmented in the sense of considering an extra wave arising from the presence of source terms) allows to explain clearly all problems detailed before and provides the right steps to estimate correctly the different types of source terms involved in the problem [79, 83]. This means that no tuning parameter or any flux redistribution is necessary. Augmented numerical schemes allow recovering the stability region given by the basic CFL condition and no time reduction is necessary. Exact conservation is guaranteed. Furthermore, augmented numerical schemes allow to extend the properties of the numerical scheme to include energy arguments [81, 87]. Considering that shallow models are constructed using mass and momentum, being able to provide solutions where energy levels remain correctly bounded is of utmost importance. The reduction of computational time when simulating real or practical cases has been one of the most important challenges of computational fluid dynamics. In particular, when trying to reproduce environmental problems on a large spatial and temporal domain, the efficiency of the method becomes crucial to make a computational tool affordable and with practical use. For this reason, model reduction has traditionally been applied to decrease the total number of computations, which are directly related to the equation’s complexity and the number of spatial dimensions. However, a limit can be found when an specific level of accuracy is required, and the model simplifications [28, 50], reduced-order models [135] or emerging machine learning models [101] are not acceptable. In this context, high-performance computing (HPC) techniques are focused not on the number of operations to compute depending on the model and its dimensions, but on the speed to manage all those operations. The outline of this chapter is as follows: First, the governing shallow-type equations are derived from the incompressible Navier-Stokes system. The more complex version for variable-density flows is also included. Then first-order finite volume methods in the FV framework able to solve them numerically are detailed and some numerical results are included as an example. Furthermore, higher-order numerical approaches are introduced and compared. Efficient high-performance strategies for the computational implementation of the numerical schemes are analyzed and the main conclusions are drawn in the last section.

2 Shallow-Water-Type Approximations Shallow water models are built in the context of free surface flows over the main hypothesis that the water layer depth h is smaller than a relevant horizontal length scale. They represent a simplified form of the incompressible Navier-Stokes equations so that only mass and momentum conservation are required.

70

S. Martínez-Aranda et al.

∇ ·v =

∂v ∂w ∂u + + = 0, ∂x ∂y ∂z

(1)



 ∂τx y ∂u ∂(u 2 ) ∂(uv) ∂(uw) ∂ p ∂τx x ∂τx z + + + =− + + + , (2) ∂t ∂x ∂y ∂z ∂x ∂x ∂y ∂z   ∂τ yy ∂τ yz ∂ p ∂τx y ∂v ∂(uv) ∂(v2 ) ∂(vw) + + + =− + + + , (3) ρ ∂t ∂x ∂y ∂z ∂y ∂x ∂y ∂z   ∂τ yz ∂ p ∂τx z ∂τzz ∂w ∂(uw) ∂(vw) ∂(w2 ) + + + = −ρg − + + + , (4) ρ ∂t ∂x ∂y ∂z ∂z ∂x ∂y ∂z ρ

ρ being the fluid density, v = (u, v, w) the fluid velocity vector, g the gravity acceleration, p the pressure and τi j the components of the viscous stress tensor T in the 3D coordinate system (x, y, z). Additionally, proper boundary conditions are required at the movable free surface and the solid bottom surface, assuming rigid and impervious, limiting the fluid layer. Kinematic boundary conditions state that moving water particles do not cross any of those surfaces. At the solid bottom surface z = z b (x, y), the non-penetration condition states that v · nb = u b

∂z b ∂z b + vb − wb = 0, ∂x ∂y

(5)

where nb = (∂z b /∂ x, ∂z b /∂ y, −1)T is the unit vector normal to the solid surface and (u b , vb , wb ) are the components of the fluid velocity at the bed surface. At the moving free surface z = z s (x, y, t) the normal relative velocity must be null: ∂z s ∂z s ∂z s + us + vs − ws = 0, ∂t ∂x ∂y

(6)

(u s , vs , ws ) being the components of the fluid velocity at the free surface. Dynamic boundary conditions provide information on the pressure and viscous forces acting at the boundary surfaces. In the case of viscous flow over the rigid bed surface z b , the no-slip condition requires u b = vb = wb = 0,

(7)

whereas the normal stress continuity at the free surface z s , when neglecting surface tension effects, leads to ps = patm , where patm is the atmospheric pressure.

(8)

Finite Volume Models and Efficient Simulation Tools (EST) for Shallow Flows

71

Continuity of shear stress across the free surface helps to include the forcing due to the wind. This shear stress τ s = (τsx , τsy ) is tangent to the free surface. The tangential component of the stress matrix T at z s along the x-coordinate can be expressed as τsx = (T · ns )x = −τx x

∂z s ∂z s − τx y + τx z , ∂x ∂y

(9)

ns = (∂z s /∂ x, ∂z s /∂ y, −1)T being the unit vector normal to the free surface, and similarly for the y-coordinate. It is common to assume a known wind speed vector so that a semi-empirical law, such as Gill formula [125], can be used to estimate the modulus of the shear stress at the free surface |τs | as |τs | = ρCw Vw2 ,

(10)

where the coefficient Cw is a function of the wind speed and Vw is the wind velocity modulus. The shear stress τ b = (τbx , τby ) at the solid bed level must represent the mutual action between the flow and the rigid bed. The tangential component of the stress matrix T at z b along the x-coordinate can be expressed as τbx = (T · nb )x = −τx x

∂z b ∂z b − τx y + τx z , ∂x ∂y

(11)

and similarly for the y-coordinate. In general, this requires empirical or semiempirical closure formulations to estimate the boundary shear stresses. Solving the 3D incompressible flow system (1)–(4) is computationally expensive, and additionally, it involves a Poisson equation for the pressure field [125] in every time step. For many applications a simplified form, called the shallow water approximation, is preferable. It is based on a dimensional analysis of the relative importance of the different terms in the equations.

3 Dimensional Analysis of the Terms in the Equations The discussion concerning the relative size of the terms is linked to that of the relevant dimensionless numbers. It starts by assuming that the x and y distances along which the horizontal velocities change are of the order L, whereas that of the velocity variation along z is of the order of the water depth h, with U , W being the order of magnitude of the horizontal and vertical velocity components, respectively. Therefore, the incompressible flow continuity Eq. (1): ∂v ∂w ∂u + + = 0, ∂x ∂y ∂z

(12)

72

S. Martínez-Aranda et al.

leads to W U ≈ , L h

(13)

so that the vertical component of the velocity is scaled with respect to the horizontal component W ≈ U h/L. In case h 0). More simplified models neglect the density variations of the flow and the artificial momentum exchange term between the flow and the bed layer in the momentum Eqs. (88) and (89), leading to the most widespread system of equations for suspended load transport. Furthermore, usually this simplified approach assumes that the net exchange between the static bed layer and the suspended load can be estimated as Nb = D − E, D and E being the specific deposition and erosion rates, which can be expressed as a function of the depth-averaged volumetric concentration in the flow column φ and the capacity volumetric concentration φ ∗ , respectively: D = α ωs φ(1 − φ)m , E = α ωs φ ∗ ,

(92)

Finite Volume Models and Efficient Simulation Tools (EST) for Shallow Flows

87

where α is an empirical parameter representing the difference between the nearbed concentration and the depth-averaged concentration, m is an empirical exponent accounting for the hindering effect on the settling velocity due to high suspended concentrations and ωs denotes the specific settling velocity of the sediment particles in clear water. In complex mud/debris flows, solid and liquid phases are generally well mixed along the flow column, hence it is common to adopt α = 1 and the empirical exponent m takes a value of about 4. The capacity solid concentration φ ∗ for each solid phase is usually computed as φ∗ =

qs∗ , h|u|

(93)

where qs∗ accounts for the value of the solid transport throughout the flow column in capacity or equilibrium regime, which can be estimated using the multiple empirical relationships from the hydrodynamic variables [128]. When only the bed load transport is considered, (81) is integrated throughout a thin layer on the top of the static bed [z b , z b + η] and the exchange between the bed layer and the flow is usually neglected. Assuming the sediment concentration in the bed load layer is constant and equal to the static bed layer concentration leads to the bed level evolution equation for shallow water Exner models: (1 − ξ )

∂qb,y ∂qb,x ∂z b + + = 0, ∂t ∂x ∂y

(94)

where the mass solid discharge in the bed load layer, qb,x and qb,y , are assumed in equilibrium (or capacity) conditions and estimated from the local flow features using empirical relationships available in the literature. Most of these formulations for the bed load discharge assume a potential dependency with respect to the boundary shear stress:  (ρ p − ρw ) 3 gd p , (95) |qb | = c θ m 1 (θ − θc )m 2 ρw d p being the characteristic diameter of the solid particles and θ the dimensionless Shields stress at the bed boundary, defined as θ=

|τb | , (ρ p − ρw )gd p

(96)

and θc is the critical value of the Shields stress for the incipient motion threshold. The exponents m 1 and m 2 , as well as the coefficient c, have been adjusted to empirical data by many authors. Some of the most used formulations for uniform-graded sediments have been summarized in Table 1. For all the formulations in Table 1, assuming turbulent friction at the bed interface, it can be demonstrated that |qb | ∝ h −1/2 |u|3 [69] and hence a general formulation for

88

S. Martínez-Aranda et al.

Table 1 Coefficient c, m 1 , m 2 and θc for different bedload transport rate formulations. The parameter S0 in the Smart formulation is the bed slope Formulation c m1 m2 θc MPM [72] Nielsen [91] Fernández-Luque [65] Wong [127]

8 12 5.7

Smart [110]

4.2 S00.6

3.97 1/6 h√ n g

0 1/2 0

3/2 1 3/2

0.047 0.047 0.037

0

3/2

0.0495

1/2

1

0.047

the bed load transport rate based on the Grass law (|qb | = G|u|3 ) has been adopted by other authors [59, 68, 78]. This formulation relates the capacity bed load discharge with the depth-averaged flow velocity by means of a factor G(h, θ ) [T 2 L −1 ] which represents the interaction between the flow and the bed layer and which depends only on the flow characteristics.

5.2 Flow Resistance Estimation in Sediment-Laden Flows So far, there is not a universal closure relation for representing the basal resistance τb in complex non-Newtonian flows. Stresses in fluid-solid multi-phase flows include distinct contributions from solid grains friction, intergranular fluid shear stress and solid-fluid interactions [58]. The basal resistance formulation must incorporate the bulk rheological behavior of the liquid-solid mixture in motion. Different kinds of stresses can determine this complex rheology. The modulus of the turbulentdispersive stress τt acts throughout the fluid phase and its contribution to the basal resistance at the bed interface is commonly expressed as a quadratic velocity relation: τt = ρghC f |u|2 ,

(97)

where C f = n 2 h −4/3 is a friction coefficient and n is the Manning’s roughness parameter. Similarly, a viscous stress modulus τμ appears also due to the shear deformation of the pore-fluid and generates a basal resistance contribution commonly expressed using a linear shear rate relation: τμ = μ

|u| , h

(98)

μ being the dynamic viscosity. Rheological measurements of muddy/debris flows also indicate the existence of a finite yield strength τ y which opposes the liquid-

Finite Volume Models and Efficient Simulation Tools (EST) for Shallow Flows

89

solid mixture deformation. Typical values for the yield strength τ y range from about 10–400 Pa [93, 96]. The frictional intergranular stress is usually modeled using a generalized Coulomb-type relation between the effective normal stress at the bed interface and the basal intergranular shear resistance τ f : τ f = (ρgh − Pb ) tan ϕb + c0 ,

(99)

where ϕb denotes the basal friction angle for the grain-bed interface, c0 is the cohesive strength component and Pb estimates the pore-fluid pressure at the bed interface [58]. For granular materials subject to large deformations, as in debris flows, cohesive forces are generally negligible. Estimation of the pore-pressure relaxation for mud/debris flows in motion is a challenging task for modeling multi-phase fluid-solid flows [54, 57, 58], although its effects on the reduction of the intergranular effective frictional stress seem to be demonstrated [5, 6, 56, 67, 71]. As a simplification, an idealized relation Pb = Eb ρw gh cos ψ has been used in the literature [31, 61, 95], cos ψ being the direction cosine of the bed normal with respect to the vertical axis and Eb a coefficient for estimating the excess of pore-fluid pressure over its hydrostatic value at the bed surface, which usually takes values from about 1.4 to 1.8 [55, 61]. All these kind of stresses act simultaneously along the mixture column and are combined in different ways to estimate the bulk rheological behavior of the liquidsolid flow. Non-Newtonian Bingham-type muddy mixtures do not flow until a threshold value of the tangential stress, the yield strength τ y , is reached. During the movement, the boundary basal shear stress τb is characterized by means of a cubic equation accounting for the plastic viscosity of the sediment-water mixture: 2|τb |3 − 3(τ y + 2τμ )|τb |2 + τ y3 = 0.

(100)

The Coulomb-viscous rheological model also estimates the basal resistance τb using a cubic relation accounting for the intergranular frictional stress and the viscous shear stress in the pore fluid: 2|τb |3 − 3(τ f + 2τμ )|τb |2 + τ 3f = 0.

(101)

The different basal resistance formulations considered in this work have been summarized in Table 2. The basal resistance term in the momentum Eqs. (74) and (75) is estimated as τb = sgn(u, v) |τb |.

(102)

90

S. Martínez-Aranda et al.

Table 2 Flow resistance formulations Formulation

Flow resistance relation |τb | = τt |τb | = τt + τ f |τb | = 1.5τ y + 3τμ 2|τb |3 − 3(τ y + 2τμ )|τb |2 + τ y3 = 0 |τb | = τ y + τt + κ/8 τμ 2|τb |3 − 3(τ f + 2τμ )|τb |2 + τ 3f = 0

Pure turbulent Turbulent and Coulomb Simplified Bingham Full Bingham Quadratic Coulomb-viscous

6 Numerical Resolution Using Riemann Solvers and the Finite Volume Framework Here, the pure hydrodynamical system of Eqs. (57)–(59), neglecting the hydrological source terms, is used to illustrate the numerical resolution of this kind of mathematical system on the finite volume (FV) framework. The 2D shallow water equations conform with a hyperbolic time-dependent, nonlinear system that can also be written in vector notation as ∂U + ∇ · E(U) = Sb (U) + Sτ (U), ∂t   the flux being E(U) = F(U), G(U) , where U = (h, qx , q y )T ,

qx = hu,

q y = hv

(103)

(104)

is the conserved variable vector, and F(U) and G(U) are the fluxes along the x and y coordinates: T  qx q y 1 q2 , F(U) = qx , x + gh 2 , h 2 h



qx q y q y2 1 , + gh 2 G(U) = q y , h h 2

T ,

(105) and the vector associated to the bed slope and frictional losses momentum source terms are T  Sb (U) = 0, gh S0x , gh S0y , T  Sτ (U) = 0, −gh S f x , −gh S f y .

(106) (107)

The Jacobian matrix J(U) of the convective fluxes in any normal pointing direction n is defined as [79, 111]

Finite Volume Models and Efficient Simulation Tools (EST) for Shallow Flows

J(U) =

∂F(U) ∂E · n ∂G(U) = nx + ny, ∂U ∂U ∂U

91

(108)

where (n x , n y ) are the components of the unit vector n and the fluxes derivatives ⎛ ⎞ 0 1 0 ∂F(U) ⎝ = gh − u 2 2u 0⎠ ∂U −uv v u

,

⎛ ⎞ 0 0 1 ∂G(U) ⎝ −uv v u ⎠, = ∂U gh − v2 0 2v

(109)

leading to ⎛

⎞ 0 nx ny ⎠, un y J(U) = ⎝−u(u · n) + c2 n x u · n + un x −v(u · n) + c2 n y vn x u · n + vn y where c is the surface wave speed c = λ1 = u · n − c,

√

(110)

gh. The eigenvalues of J(U) are given by

λ2 = u · n,

λ3 = u · n + c.

(111)

In order to solve numerically (103), the spatial domain is divided into computational cells using a mesh fixed in time and system (103) is integrated in each cell i using the Gauss theorem:     d U d + E · n dl = Sb (U) d + Sτ (U) d, (112) dt i ∂i i i E · n = F(U) n x + G(U) n y being the flux normal to the i cell boundary and n = (n x , n y ) the outward unit normal vector (Fig. 3). Assuming a piecewise uniform representation of the conserved variables U at the i cell for the time t = t n

Fig. 3 Schematic representation of cells in (left) orthogonal and (right) unstructured triangular meshes

92

S. Martínez-Aranda et al.

Fig. 4 Local coordinates at the intercell edge

Uin =

1 Ai

 i

U(x, y, t n ) d,

(113)

where Ai is the cell area, (112) can be expressed as d dt

 i

U d +

NE 

(E · n)k lk =

k=1

NE   k=1

i,k

[Sb (U) + Sτ (U)] d,

(114)

N E being the number of edges for the i cell, (E · n)k the value of the normal flux through each edge and lk the length of the edge. The integration of the source term at the i cell is expressed as the sum of the source term integral associated with each edge, with i,k being the area of the i cell corresponding to the kth edge. The theory of Riemann problems (RP) can be used to solve the intercell fluxes in 2D problems (114). For each kth cell edge, it is possible to define a local 1D RP by projecting (103) along n direction. Furthermore, shallow-water-type systems satisfy the rotation invariance property [21, 43] and hence can be expressed in the local framework (x, ˆ yˆ ), corresponding to normal and tangential directions to each cell edge, respectively (Fig. 4). Defining a rotation matrix Tk for the kth cell edge, with an inverse matrix T−1 k , as ⎛ ⎛ ⎞ ⎞ 1 0 0 1 0 0 ⎝ 0 n x −n y ⎠ , Tk = ⎝ 0 n x n y ⎠ , T−1 (115) k = 0 −n y n x k 0 n y nx k which satisfies the condition   (E · n)k = F(U) n x + G(U) n y k = T−1 k F(Tk U),

(116)

the local RP normal to the kth cell edge can be expressed in the local framework (x, ˆ yˆ ) as [80, 83]

Finite Volume Models and Efficient Simulation Tools (EST) for Shallow Flows

∂ Tk U ∂ F(Tk U) + = Tk Sb + Tk Sτ , ∂t ∂ xˆ  Tk Ui if xˆ < 0 Tk U(x, ˆ 0) = Tk U j if xˆ > 0,

93

(117)

where subscripts i and j indicate the left and right cells of the kth edge, respectively. ˆ = Tk U at the cell edge is defined Therefore, the set of local conserved variables U as  T ˆ = Tk U = h, h u, ˆ h vˆ , U

(118)

where uˆ = un x + vn y and vˆ = −un y + vn x are the flow velocities along the xˆ and ˆ = F(Tk U) can be yˆ coordinates, respectively, and the local convective fluxes F(U) expressed as ⎛ ⎞ h uˆ ˆ = F(Tk U) = ⎝ h uˆ 2 + 1 gh 2 ⎠. (119) F(U) 2 h uˆ vˆ The augmented value of the fluxes through the kth cell edge in (114) incorporates the non-conservative contribution of the momentum source terms Sb and Sτ into the ˆ x/2] ˆ convective fluxes (E · n)k . In order to integrate (117) over the space [−x/2, corresponding to the kth edge, the momentum source terms are linearized in time and involved in the Riemann solver as a singular source at the discontinuity xˆ = 0 so that 

x/2 ˆ

−x/2 ˆ



x/2 ˆ

−x/2 ˆ

 Tk Sb d xˆ = Tk

x/2 ˆ

−x/2 ˆ

 Tk Sτ d xˆ = Tk

x/2 ˆ

−x/2 ˆ

Sb d xˆ ≈ Tk S∨b , (120) Sτ d xˆ ≈

Tk S∨τ ,

where S∨b and S∨τ are suitable numerical source vectors along the normal direction to the cell edge, which can be expressed in the 2D framework (x, y) as    nx , H  n y , 0, 0 T , S∨b = 0, H k    n x , −T  n y , 0, 0 T , S∨τ = 0, −T k

(121)

k and T k being the bed slope and friction momentum contributions, respectively, H spatially integrated in the control volume corresponding to the kth cell edge. Using the rotation invariance property (116) and the solution of the local plane RP (117), it is possible to rewrite (114) as

94

S. Martínez-Aranda et al.

d dt

 i

U d = −

NE 

  ˆ − Sˆ ∨b − Sˆ ∨τ lk , F(U) T−1 k k

k=1

(122)

where Sˆ ∨b = Tk S∨b and Sˆ ∨τ = Tk S∨τ , allowing to define an augmented numerical flux ˆ ↓ for the kth cell edge which incorporates the integrated momentum source F(U) k terms into the convective numerical fluxes at the cell edge, ensuring the well-balance property for steady states [79]:   ˆ − Sˆ ∨b − Sˆ ∨τ . ˆ ↓ = F(U) (123) F(U) k k

Replacing (123) into (122), we obtain the updating formula for the conserved variables U at the cells Uin+1 = Uin −

NE t  −1 ˆ ↓ T F(U)k lk , Ai k=1 k

(124)

where t = t n+1 − t n is the time step. Hence the resolution procedure needs to ˆ ↓ at the cell edges ensuring (123). compute the numerical fluxes F(U) k

6.1 Approximate Riemann Solvers The plane Riemann problem (117) defined in the local framework (x, ˆ yˆ ) of the kth edge, separating the left i cell and the right j cell, can be approximated by using the following constant coefficient linear RP [111]: ˆ ˆ ∂U ∂U + Jk = (Sˆ∨b + Sˆ∨τ )k , ∂t  ∂ xˆ ˆ = Tk Un if xˆ < 0 U i ˆ x, U( ˆ 0) = ˆ i U j = Tk Un if xˆ > 0,

(125)

j

ˆ i, U ˆ j ) is a constant coefficient matrix which approximates the Jacowhere  Jk =  Jk (U   ˆ ˆ . Integrating (125) over the discrete U bian of the nonlinear RP as  Jk ≈ ∂F(U)/∂ k

space [xˆi , xˆ j ] leads to the following constraint involving conservation across discontinuities: ˆ k, ˆ k = Jk δ U δF(U)

(126)

ˆ j −U ˆ i is the conserved variables jump at the kth edge and  ˆk = U Jk is a where δ U 3 × 3 constant matrix defined as

Finite Volume Models and Efficient Simulation Tools (EST) for Shallow Flows

95

⎛

⎞ 0 1 0  Jk = ⎝ g h − u 2 2 u 0⎠ . − u v  v  u k 6.1.1

(127)

Augmented Roe Solver (A-Roe)

The Roe’s approach for solving the approximate RP at the intercell edges retains the properties of the shallow water system by assuming that the wave celerities are provided by the Jacobian matrix  Jk , leading to a set of three real eigenvalues given by  λ1k = ( u − c )k ,  λ2k =  uk ,  λ3k = ( u + c )k .

(128)

e1 , e2 , e3 )k and its inverse  Pk−1 , ek being the It is possible to define a matrix  Pk = (  eigenvectors of the approximate Jacobian matrix Jk , with the following property: ⎛ 1 ⎞  λ 0 0   k = ⎝ 0  λ2 0 ⎠ , Jk = ( P P−1 )k ,  (129) 0 0  λ3 k k is a diagonal matrix with approximate eigenvalues in the main diagonal. where  As a consequence of Roe’s linearization, the resulting approximate Riemann soluˆ tion is built of only discontinuities and U(x, t) is constructed as a sum of jumps or ˆ x, shocks. The approximate solution for U( ˆ t) at the kth edge is governed by the celerk and involves five regions connected by four waves (see Fig. 5): three waves ities in  (λ1 , λ2 , λ3 ) arising from the homogeneous hyperbolic system of equations and one steady contact wave, located at the edge xˆ = 0 and with null celerity, corresponding to the inclusion of the source term into the approximate solution of the local RP [79].

Fig. 5 Intermediate states for the A-Roe Riemann solver

96

S. Martínez-Aranda et al.

ˆ x, According to the Godunov method, it is enough to provide the solution for U( ˆ t) at the intercell position x = 0 in order to derive the updating numerical fluxes in the updating scheme (124). In order to recover the value of the approximate intermediate ˆ + at the left and right side of the (x, ˆ − and U ˆ t) plane solution, respectively states U i j ˆ i (x, ˆ − = lim U ˆ t > 0), U i − x→0 ˆ

ˆ j (x, ˆ + = lim U U ˆ t > 0), j + x→0 ˆ

the system in (125) is transformed by using  P−1 matrix as follows:   ˆ ˆ ∂ U ∂ U −1  + Jk Pk = Pk−1 (Sˆ ∨b + Sˆ ∨τ )k , ∂t ∂ xˆ

(130)

(131)

ˆ with V = expressing (125) in terms of the characteristic variables V =  Pk−1 U, 1 2 3 T (V , V , V ) . This transformation leads to a decoupled system that generates the following linear RP: ∂V ∂V k + = Bk , ∂t ∂ xˆ  ˆ i if xˆ < 0 Pk−1 U Vi =  V(x, ˆ 0) = −1 ˆ j if xˆ > 0, Vj =  Pk U

(132)

with Bk =  Pk−1 (Sˆ ∨b + Sˆ ∨τ )k , where each equation ∂V m ∂V m + λm = βkm , k ∂t ∂ xˆ

m = 1, 2, 3

(133)

involves the variable V m and the source term βkm . Equation (133) allows generating a set of independent equations that can be solved exactly for each characteristic variable V m . The solution is constructed departing the set of wave strengths ˆ k, Pk−1 δ U Ak = (α 1 , α 2 , α 3 )kT = δVk = 

(134)

and the set of source strengths Pk−1 (Sˆ ∨b + Sˆ ∨τ )k . Bk = (β 1 , β 2 , β 3 )kT = 

(135)

It is worth mentioning that α m wave strengths allow expressing a simple linear relations for both conserved variables and flux vector differences as follows:   m ˆk =  δFk = λα e k . (136) δU e)m (α k , The value of the characteristic variables at the left and right side of the intercell position in matrix form, Vi− and V+j , is

Finite Volume Models and Efficient Simulation Tools (EST) for Shallow Flows

 −1 −   −1 +    δV , V+j = V j −    δV ,     Vi− = Vi +  k k where k is a diagonal matrix ⎛ 1 ⎞  m  γ 0 0 β¯ m 2 ⎝ ⎠ 0 0 γ k = , γk = 1 − .  λm α m k 0 0 γ3 k

97

(137)

(138)

ˆ − and U ˆ + can be directly obtained by using  Now, the intermediate states U Pk i j − − + + ˆ   matrix. Vector solutions Ui = PVi and U j = PV j are recovered from (137) as follows:  ˆk = U ˆi + ˆ − = Ui + ( −   −1  P P−1 )k δ U e)m U (αγ k , i  λm 0

with the following property: ˆ+ −U ˆ − = ( −1 B)k =  −1 U P−1 (Sˆ ∨b + Sˆ ∨τ )k =  P P Jk−1 (Sˆ ∨b + Sˆ ∨τ )k . j i

(140)

Being the solution defined as a sum of jumps or shocks between the different intermediate states, the solution for the approximate linear flux function F(x, ˆ t) at the left and right side of the initial discontinuity at xˆ = 0, labeled as Fi− and F+j , respectively, in a general RP, with ˆ t > 0), Fi− = lim− F(x,

F+j = lim+ F(x, ˆ t > 0).

x→0 ˆ

x→0 ˆ

(141)

The relation between the intercell approximate fluxes can be analyzed using the RH (Rankine-Hugoniot) relation at xˆ = 0, which includes a steady contact wave [62] ˆ+ ˆ − and U between approximate solutions U i

j

ˆ+ −U ˆ − ) = 0, F+j − Fi− − (Sˆ ∨b + Sˆ ∨τ )k = S(U j i

(142)

which provides the following relation among fluxes and conserved variables in the inner regions ˆ+ −U ˆ − ) = (Sˆ ∨b + Sˆ ∨τ )k , Jk (U F+j − Fi− =  j i

(143)

and confirming condition (140). The telescopic properties of the linear solutions for the approximate flux function provide the definition of fluxes at xˆ = 0, Fi− and F+j ,

98

S. Martínez-Aranda et al.

Fi− = Fi +

 m  λ− αθ e k,

 λm 0

It must be said that, due to the presence of source terms, it is not longer possible to define a general intercell flux function independent of the solution side considered as in the homogeneous case. Therefore, in order to compute the updated values of the conserved variable using (124), the numerical flux at the intercell edges is taken as ˆ = F− . F(U) k i ↓

6.1.2

(145)

The HLLS Solver

In the framework of first-order Godunov’s method, Harten, Lax and van Leer introduced a novel Riemann solver [48], called HLL solver. This solver was derived for homogeneous RPs of two waves, providing an estimation of the intercell numerical flux considering a single intermediate region. Such flux is directly computed from the integral form of the governing equations. When dealing with non-homogeneous systems of PDEs, the HLL needs to be modified to properly account for source terms. One example was proposed by Murillo in [80] with the generation of a new solver, called HLLS, that considers the presence of an additional stationary wave at x = 0. The integral average of the approximate RP between the slowest and fastest signals ¯ can be derived by simply setting xˆ L = λ L t at time t, that will be referred to as U, and xˆ R = λ R t in control volume represented in Fig. 6. ¯ = U

 xˆ R xˆ L

ˆ x, U( ˆ t = t) d xˆ t (λ R − λ L )

Fig. 6 Intermediate states for the HLLS Riemann solver

=

ˆ j − λL U ˆ i − F j + Fi + S¯ λR U , (λ R − λ L )

(146)

Finite Volume Models and Efficient Simulation Tools (EST) for Shallow Flows

99

¯ in (146), where S¯ = (Sˆ ∨b + Sˆ ∨τ )k . Although it is possible to define the average value U ¯ ˆ the presence of the source term S introduces a variation in U in the solution across ˆ+ ˆ − and U xˆ = 0, leading to two new integral averages U i j 0  xˆ R ˆ x, ˆ x, U( ˆ t) d xˆ U( ˆ t) d xˆ ˆ − = xˆ L ˆ+ = 0 U and U . (147) L R −t λ L t λ R The introduction of the source term leads also to the definition of two new fluxes, ˆ − ) and F+ = F(U ˆ + ), with the following RH relations across the left and Fi− = F(U i i+1 j right waves, respectively ˆ− −U ˆ i ) and F j − F+ = λ R (U ˆ j −U ˆ + ), Fi− − Fi = λ L (U i j j

(148)

and the following RH relation for the steady shock, of shock speed S, at xˆ = 0 ˆ+ −U ˆ − ) = 0, F+j − Fi− − S¯ = S(U R L

(149)

since S = 0. Relation (149) gives information about fluxes F+j and Fi− , and is ˆ + and exploited here to provide information among conserved intermediate states U j ˆ − by expressing the difference (F+ − F− ) as U j j i ˆ+ −U ˆ − ) and J = J(U ˆ +, U ˆ − ), F+j − Fi− = J(U j i j i

(150)

ˆ − unknown, condition in ˆ + and U through a matrix J. Being the intermediate states U j j (149) is adopted from the A-Roe solver ˆ − ) = S, ¯ ˆ+ −U  J(U j i

(151)

¯ ˆ− = H ¯ = ˆ+ −U J−1 S. U j i

(152)

leading to

With the set of Eqs. (148, 149) and (152) it is now possible to define the intermediate states and fluxes. The intermediate states are given by [80] ˆ ˆ ¯ ¯ ˆ − = λ R U j − λ L Ui + Fi − F j + S − λ R H , U i λ R − λL ˆ ˆ ¯ ¯ ˆ + = λ R U j − λ L Ui + Fi − F j + S − λ L H , U j λ R − λL and the intermediate fluxes are

(153)

100

S. Martínez-Aranda et al.

¯ ˆ j −U ˆ i ) + λ L (S¯ − λ R H) λ R Fi − λ L F j + λ L λ R (U , λ R − λL ¯ ˆ j −U ˆ i ) + λ R (S¯ − λ L H) λ R Fi − λ L F j + λ L λ R (U F+j = . λ R − λL

Fi− =

(154)

The corresponding intercell flux for the approximate Godunov method in (124) is given by the function ⎧ if λ L > 0 ⎨ Fi ↓ − ˆ if λ L ≤ 0 ≤ λ R . F (155) F(U)k = ⎩ i ¯ F j − S if λ R < 0 In order to generate the intercell fluxes, it is necessary to compute the speeds λ R and λ L . Appropriate estimates for the wave speeds λ L and λ R are required when computing the numerical fluxes in the HLLS solver. Direct wave speed estimates, λ L = uˆ i − ci and λ R = uˆ j + c j , are the simplest methods providing minimum and maximum signal velocities and can be combined with the Roe averages [111]. Numerical experiments in [80] have proved that, in the presence of source terms, accurate results are obtained using ⎧ ¯ = 0, h i > 0, h j > 0, λ1 , λi1 ) if |S| ⎨ min( 1 ¯ if |S| = 0, λi1 < 0 < λ1j λ L = λi ⎩ 1 λ otherwise (156) ⎧ ¯ = 0, h i > 0, h j > 0, λ3 , λ3j ) if |S| ⎨ max( ¯ = 0, λ3 < 0 < λ3 λ R = λ3j if |S| i j ⎩ 3 λ otherwise that consider the presence of source terms and transcritical points. It must be recalled that the HLLS solver represents the eigenstructure system using two nonlinear waves plus the steady contact wave associated with the source term. When using the HLLS solver to compute the solution of a system which involves more than two waves—e.g. the 2D SW equations, which also account for the transport of shear momentum—a high amount of numerical diffusion will appear. For instance, when solving the 2D SW equations, the contact wave associated with the transport of the shear momentum component will be strongly smeared, increasing the diffusive mixing in the numerical solution. To circumvent this issue, the HLLC solver was introduced [116] and its augmented version, the HLLCS, was also developed [80].

Finite Volume Models and Efficient Simulation Tools (EST) for Shallow Flows

101

0.02 0.015

h+z (m)

0.01 0.005 0 -0.005 -0.01 -0.015 0

5

10

15

20

Time (s)

Fig. 7 Tsunami test case: (left) topography with gauge locations and (right) inlet boundary condition

6.2 Application to Water Flow In order to test the numerical scheme in transient conditions over irregular bed with wet/dry fronts, the simulation of a tsunami test case in a 1/400 laboratory scale is carried out in this section [63]. This case was also reproduced by other authors for testing Large Time Step schemes with CFL>1 [75] or implicit schemes [39]. Figure 7 shows the bed elevation map and the gauging points where experimental data is available. The coordinates for the three gauges are P1 = (4.52, 1.196 m), P2 = (4.52, 1.696 m) and P3 = (4.52, 2.196 m). The domain (5.5 m × 3.4 m) is discretized by means of an unstructured triangular mesh of 19000 elements. The simulations have been performed solving the system (103) using a first-order A-Roe scheme. A constant roughness Manning’s coefficient of n = 0.01 sm−1/3 is set for the turbulent friction model, and a uniform water level of h + z b = 0.0 is assumed as an initial condition. As in the experimental setup, all the boundaries are closed except the one corresponding to the offshore incoming wave, which is open and characterized by a temporal variation of the water level (h + z b ) (see Fig. 7–right). A 3D representation of the numerical results for water depth at t = 0, t = 5 s, t = 10 s, t = 13 s, t = 18 s, t = 25 s is shown in Fig. 8, where a proper solution of all the wet/dry interfaces is reached at any time of the simulation. In order to validate the model, the numerical results corresponding to the water level (h + z b ) are compared with the experimental measures provided by [63] at the P1, P2 and P3 probes (Fig. 9). A good agreement between numerical and experimental results is observed.

102

S. Martínez-Aranda et al.

Fig. 8 3D representation of the numerical results for water depth at several simulation times

6.3 Application to Sediment-Laden Erosive Flows Large-scale experiments consisting of dambreak debris flows over the erodible bed were carried out in the USGS large-scale debris-flow flume and reported by [56]. The USGS debris-flow flume is a rectangular concrete channel 95 m long, 2 m wide and 1.2 m deep with a vertical headgate placed 12.5 m downstream the channel beginning, which retains the static debris fluid until the experiment initial time. Figure 10 shows a schematic representation of the USGS debris-flow flume. All the longitudinal distances s are referred to the headgate position taken along the experimental

Finite Volume Models and Efficient Simulation Tools (EST) for Shallow Flows 0.05

P2 observed data P2 numerical

0.04

0.03

h+z (m)

h+z (m)

0.04

0.05

P1 observed data P1 numerical

0.02

103

0.03 0.02 0.01

0.01 0

0

−0.01

−0.01 0

5

0

20

15

10

5

0.05

10

15

20

Time (s)

Time (s)

P3 observed data P3 numerical

h+z (m)

0.04 0.03 0.02 0.01 0 0

5

10

15

20

Time (s)

Fig. 9 Numerical results at the three gauging points

Fig. 10 Sketch of the USGS flume experiments. Details of the equivalent horizontal distance, vertical elevation and channel slope are also indicated

104

S. Martínez-Aranda et al.

Table 3 Parameters for the debris mixture used in the USGS experiments Liquid density ρw 1000 kg/m3 Solid dry density ρ p 2700 kg/m3 Initial sediment 0.60 concentration φ0 Water content Cw 0.40 Initial debris density ρ 2020 kg/m3 Plastic viscosity μ 1.66 Pa · s Yield stress τ y 393 Pa Basal resistance angle 40◦ ϕb Manning roughness Concrete: 0.018 s m−1/3 coeff. n Erodible layer: 0.021 s m−1/3 Sed. size-class Fines Sand Gravel Grain diameter ds 0.016 0.4 12 (mm) Bed porosity ξ 0.42 Bed water content Cbw 0.183

flume floor. The channel has a 31◦ uniform slope until s = 74 m, where the flume begins to flatten following a catenary curve and evolving to a 4◦ slope at s = 82.5 m. There, the flume debouches onto a 15 m long, 8 m wide and 2.4◦ slope concrete runout surface. The features of the different solid phases presented in the initial debris mixture and the partially saturated bed were previously reported in [55]. The initial debris volume comprised a fully saturated mix of water and 56% gravel–37% sand–7% mud grains, called SGM mixture, where mud refers to particles smaller than 0.0625 mm. Table 3 shows the main parameter used in the simulations for characterizing the debris mixture. The equivalent volumetric bulk concentration for the initial debris fluid is φ0 = 0.6 and the initial volume stored upstream the headgate is 6 m3 . The flume floor is covered from s = 6 m to s = 53 m with a 12 cm thick layer of partially saturated SGM mixture. The porosity and water content of the erodible bed layer vary for each experiment, leading to different entrainment rates and wave-front velocities. Once the headgate was opened, the dambreak wave progressed over the erodible bed until it reached the runout surface where it stopped. For all the experiments, the wave-front position was tracked using image techniques and video frames. The entrainment from the erodible bed to the debris flow was estimated by measuring the bed layer volume before and after the passage of the dambreak wave. The flow thickness was measured at the control sections P1 s = 32 m and P2 s = 66 m using laser sensors mounted above the flume.

Finite Volume Models and Efficient Simulation Tools (EST) for Shallow Flows

105

Fig. 11 Spatial flow distribution at (top) t = 6 s and (bottom) t = 18 s after the gate opening using the Turbulent and Coulomb (TC) resistance

106

S. Martínez-Aranda et al.

The simulations are performed solving the Eqs. (73)–(79)–(80)–(83) and (86) with an A-Roe scheme [70] over an unstructured triangular mesh of around 57000 cells, with an averaged area of 100 cm2 . An analysis of the numerical results obtained using different rheological formulations (see Table 2) to estimate the debris basal stresses is included herein. Each rheological model requires a careful calibration of the involved parameters in order to approximate the empirical measurements. The lower part of Fig. 11 shows the bulk debris density in color scale and the upper part depicts the longitudinal profile of the specific volumetric concentration

Fig. 12 Temporal evolution of (left) the dambreak wave-front location and (right) debris flow volume with different rheological relationships for the debris stress

Finite Volume Models and Efficient Simulation Tools (EST) for Shallow Flows

107

Table 4 RMSE for the flow free surface level at the control sections (P1 and P2) and for the wave-front location Formulation RMSE RMSE WSL P1 WSL P2 Front loc. (m) (m) (m) Full Bingham (FB) Simplified Bingham (SB) Turbulent and Coulomb (TC) Quadratic (QD) Coulomb-viscous (CV)

4.46 · 10−2 4.95 · 10−2

3.92 · 10−2 4.10 · 10−2

5.97 4.72

3.83 · 10−2

2.56 · 10−2

2.72

5.16 · 10−2 1.81 · 10−2

2.94 · 10−2 4.05 · 10−2

4.38 3.55

for each sediment class in the mixture at t = 6 s and t = 18 s after the gate opening. As the dambreak wave progresses downstream, bed material is incorporated into the flow, increasing the solid phase volume in the debris and hence its density. It is worth mentioning that the coarse sediment fraction dominates along the front and central regions of the debris wave, whereas the finer sediment fraction shows higher concentrations along the tail zone. This sorting of the different solid phases composing the flow mixture is typical of debris flow dynamics and mainly caused by the differences in the solid phases velocity. This sorting of the sediment classes is also partially caused by the different bed entrainment rates for each sediment fraction from the erodible bed. The dambreak wave-front position is plotted against time in Fig. 12 (left) for all the friction formulations simulated. The full Bingham (FB) and simplified Bingham (SB) formulations show higher wave-front velocities than those observed during the whole experiment, whereas the Coulomb-viscous (CV) rheological relation predicts slower front velocities. The quadratic (QD) formula overestimates the wave-front velocity at the first stage but the runout distance is smaller than that measured in the experiment. Only the Turbulent and Coulomb (TC) formulation is able to correctly estimate the wave-front advance process. Figure 12 (right) shows the temporal evolution of debris volume for the resistance formulations considered. During the dambreak wave propagation, the volume of the debris flow increases as a consequence of the entrainment from the erodible bed, regardless of the basal resistance formulation considered. However, only the Turbulent and Coulomb (TC) relationship is able to properly estimate the maximum debris volume observed in the experiment. Table 4 shows the root mean square error (RMSE) for the temporal evolution of the flow free surface level at the control sections P1 (s = 32 m) and P2 (s = 66 m) with the different rheological relationships simulated. The RMSE for the wavefront location advance is also shown. The lowest RMSE is obtained with the CV formulation for the free surface level at the control section P1, whereas the TC relation shows the best results for both the free surface level at P2 and the temporal evolution of the wave-front location.

108

S. Martínez-Aranda et al.

7 Higher-Order Approaches in the Presence of Source Terms In the last decade, motivated by the recent technological development of computers, the genesis of new generations of high-resolution schemes has occurred. In the framework of finite volume schemes, the introduction of the weighted essentially non-oscillatory (WENO) reconstruction technique [64, 109] is a major step when seeking arbitrary order of accuracy in space. It must be noted that nonlinear reconstruction schemes, such as the WENO method, are required for the simulation of flows in the presence of a discontinuity in order to avoid Gibbs oscillations. On the other hand, the preservation of high order in time was usually done by means of Runge-Kutta (RK) time integrators, which require multiple temporal sub-steps and may sometimes be inefficient due to Butcher’s barrier [105]. This issue was circumvented by the ADER approach [113], which allows a fully discrete scheme of arbitrary order in space and time. While RK integrators require multiple sub-steps, where variables need to be reconstructed and fluxes computed (RP solved), ADER schemes are single-step methods, at the cost of the resolution of a high-order extension of the RP, called derivative Riemann problem (DRP) [114]. A broad variety of DRP solvers have been proposed up to date [19, 32, 45, 47, 73]. It is of particular interest the application of ADER schemes to the resolution of the shallow water (SW) Eqs. (57)–(59) [19, 20, 73, 124]. Parallel to the development of WENO-ADER schemes, a rather different approach for the construction of arbitrary order schemes in space and time was also developed. This gave rise to a new family of schemes called discontinuous Galerkin (DG) methods. Their origin is attributed to Reed and Hill [102] and the fundamentals of such schemes can be found in a series of papers by Cockburn and Shu published in the 1980s and 1990s [26, 27]. The DG scheme can be regarded as a compromise between finite elements (compactness of the stencils, easy hp-refinement and parallel scalability) and finite volumes (intrinsic conservativeness, presence of discontinuities between cells and use of numerical fluxes). The DG methods, in combination with either RK or ADER time integration, challenge the performance of their finite volume counterparts. Their properties make them very suitable for the resolution of the SW equations [3, 18, 22, 121–123, 131]. In this section, WENO-RK, WENO-ADER and DG-RK schemes based on augmented Riemann solvers are addressed. For the sake of simplicity, the schemes are presented in their 1D version. Multi-dimensional extensions are straightforward, provided a multi-dimensional reconstruction method is used. Unlike Godunov’s first-order scheme, where the information is represented by piecewise constant data inside cells, higher-order schemes usually consider piecewise polynomial data of a degree matching the prescribed accuracy. Figure 13 shows the typical discretization of a 1D domain. Subscripts L and R are defined with reference to the cell center. For the sake of clarity, the following notation is introduced:

Finite Volume Models and Efficient Simulation Tools (EST) for Shallow Flows

109

Fig. 13 Mesh discretization and common notation for higher-order schemes. Dashed line represents cell averages and continuous line represents the reconstructed data

ui R =

lim u i (x),

u (i+1)L =

− x→xi+1/2

lim u i+1 (x),

+ x→xi+1/2

(157)

where u(x) is some reconstructed variable.

7.1 The WENO Reconstruction The WENO method uses a variable set of stencils where lower-order polynomials are first constructed. Then, these lower-order polynomials are combined to create either a higher-order polynomial in smooth regions (optimal reconstruction) or a non-centered reconstruction able to capture discontinuities in non-smooth regions. The definition of a smoothness indicator allows to distinguish between those two cases. The traditional WENO reconstruction method, also referred to as WENO-JS, is briefly recalled here. The departing data for the WENO reconstruction procedure are the cell averaged values of a function u (x) defined in a" computational # grid composed of N cells, with cells and cell sizes defined by i = xi− 21 , xi+ 21 . In order to construct a WENO reconstruction of degree (2k − 1) on the cell i = [xi− 21 , xi+ 21 ] for the function u(x), k different stencils are needed. These stencils are given by Sr (i) = {i−r , ..., i+k−r −1 } (r = 0, ..., k − 1), where r represents the number of cells on the left-hand side of i . These stencils are used to generate a larger stencil T (i) = ∪rk−1 =0 Sr (i) = {i−k+1 , ..., i+k−1 }. The general procedure of the WENO reconstruction is summarized below: 1. Definition of the optimal weights There is a unique polynomial pr (x) defined in each stencil Sr , which is a kth-order approximation of the function u(x) on the stencil Sr (i) if this function is smooth inside it [109]. pr (x) is expressed as a linear combination of the cell averages in the stencil. At xi R , the approximation of u(xi R ) is given by u i(rR) = pr (xi+ 21 ) =

k−1  j=0

   k cr(k) j u i−r + j = u x i R + O x ,

(158)

110

S. Martínez-Aranda et al.

where cr(k) j are coefficients derived from the Lagrange interpolation formula. The same procedure can be used to obtain a polynomial q(x), which is a (2k − 1)th order approximation of the function u(x) on the large stencil T (i). At xi R , this approximation of u(xi R ) is given by 2k−1 

u i R = q(xi+ 21 ) =

   2k−1  (2k−1) ck−1, . j u i−k+ j = u x i R + O x

(159)

j=1

Note that T (i) is symmetric. The (2k − 1)th order approximation in (159) can also be expressed as a linear convex combination of the kth order reconstructions provided by (158) as ui R =

k−1 

    γr u i(rR) = u xi R + O x 2k−1 ,

(160)

r =0

where γr are the optimal weights that can be easily computed algebraically [109]. 2. Definition of the smoothness indicator: smoothness indicator for WENO-JS The smoothness indicator, βr , measures the smoothness of the initial data and is able to detect the presence of discontinuities. In the traditional WENO-JS method [109], this indicator reads βr =

k−1   l=1

xi− 1 2

 x 2l−1

xi+ 1

∂ l pr (x) ∂ xl

2 d x,

r = 0, ..., k − 1.

(161)

2

3. Definition of the WENO-JS weights Departing from the optimal weights, it is possible to define the WENO-JS weights, denoted by ωrJ S . First, the αrJ S coefficients are formulated and then normalized leading to the WENO ωrJ S weights αrJ S =

γr (βr + )2

αr ωrJ S = $k−1 l=0

αl

,

r = 0, ..., k − 1

(162)

with  a properly defined small parameter. The final WENO approximation of u(x) at xi+ 21 is given by ui R =

k−1  r =0

    ωrJ S u i(rR) = u xi R + O x 2k−1 .

(163)

Finite Volume Models and Efficient Simulation Tools (EST) for Shallow Flows

111

7.2 Augmented WENO-RK Schemes for the SW Equations Let us consider the 1D version of (103). When using a high order of accuracy, the semi-discrete 1D version of Eq. (124) inside i can be expressed as ∂Ui = L(Ui , Si ), ∂t

(164)

with L(Ui , Si ) = −

 t t  − + Fi+1/2 − Fi−1/2 S¯ i ,i , + x x R L

(165)

+ + − − = Fi−1/2 (U(i−1) R , Ui L ) and Fi+1/2 = Fi+1/2 (Ui R , U(i+1)L ) the cell enterwith Fi−1/2 ¯ ing and leaving numerical fluxes, respectively, and Si R ,i L a suitable discretization of the source term inside the cell:  xi R ¯Si R ,i L ≈ S(U, z) d x. (166) xi L

With the aim of computing the numerical fluxes, augmented solvers are used. The A-Roe or the HLLS solver, introduced in Sects. 6.1.1 and 6.1.2 respectively, can be used. According to the notation introduced in this section, U(i−1) R and Ui L stand for the reconstructed data on the left and right sides of the interface at xi−1/2 , provided by the WENO reconstruction. Note that (165) resembles to the first-order scheme in (124), with an additional term which accounts for the variation of the variables inside cells. When using piecewise constant data (first order of accuracy), such variation is null and this term vanishes. Analogously, the operator in Eq. (165) can be rewritten in terms of fluctuations, generally denoted by δM, leading to L(Ui ) = −

t − + [δMi+1/2 + δMi R ,i L + δMi−1/2 ], x

(167)

where − − + + δMi+1/2 = Fi+1/2 − Fi R , δMi−1/2 = Fi L − Fi−1/2 ,

δMi R ,i L = Fi R − Fi L − S¯ i R ,i L.

(168)

Well-balanced schemes are able to preserve the quiescent equilibrium with machine accuracy, which means that the spatial operator in Eq. (165) vanishes. In order to construct a well-balanced scheme, the following requirements must be met: • Under equilibrium, the reconstructed data must satisfy the equilibrium condition: this is achieved by applying the WENO reconstruction to the equilibrium vari-

112

S. Martínez-Aranda et al.

able. The variables h + z b , hu and z b are reconstructed, and h is computed as the difference of h + z b and z b . − + and δMi−1/2 must be zero. This is • Under equilibrium, wall fluctuations, δMi+1/2 automatically achieved by means of augmented solvers. It must be noted that only geometrical source terms, i.e. bed elevation, or other sources with a geometrical re-interpretation, must be accounted for at cell interfaces. • Under equilibrium, the cell-centered fluctuation, δMi R ,i L , must be null. This is achieved by means of a suitable choice for the approximation of the integral of the source term. Gaussian quadrature ensuring exact integration may be enough for preserving quiescent equilibrium but other approaches must be used when seeking moving water equilibria [88], e.g. Richardson extrapolation and Romberg integration using an energy-preserving integral of the source term. To achieve high order in time, Eq. (164) can be integrated over the time step t = [t n , t n+1 ] using a TVD Runge-Kutta (RK) method. A popular choice is the third-order strong stability preserving (SSP) RK method: Ui(1) = Uin − tL(Un , Sn ),   Ui(2) = 43 Uin + 41 Ui(1) − tL(U(1) , S(1) ) ,   Uin+1 = 13 Uin + 23 Ui(2) − tL(U(2) , S(2) ) .

(169)

7.3 Augmented WENO-ADER Schemes for the SW Equations When using the ADER approach, a fully discrete version of the system in (124) inside i can be obtained: Uin+1 = Uin −

t − t ¯ + [F [Si ,i ], − Fi−1/2 ]+ x i+1/2 x R L

(170)

− + and Fi−1/2 defined as time-integral averages of the with the numerical fluxes Fi+1/2 time-dependent fluxes evolved in time at the interfaces  t  t 1 1 − − + Fi R (τ ) dτ, Fi−1/2 = Fi+L (τ ) dτ, (171) Fi+1/2 = t 0 t 0

and S¯ i R ,i L a suitable approximation of the integral of the source term inside the cell given by  t  xi R 1 S d x dτ. (172) S¯ i R ,i L ≈ t 0 xi L

Finite Volume Models and Efficient Simulation Tools (EST) for Shallow Flows

113

As in Eq. (167), the scheme can also be written in fluctuation form, which is omitted for the sake of brevity. When using the flux-expansion approach [117], numerical fluxes are expanded using a Taylor power series in time and the following expression is obtained: K 

Fi−,(k) R

t k , (k + 1)!

K 

t k . (k + 1)! k=1 k=1 (173) The coefficients of the expansion in Eq. (173) will be computed using suitable Riemann solvers for the resolution of the DRP. The LFS solver from [88] can be used in combination with the A-Roe solver. The leading terms of the expansion in (173) are computed by solving the socalled DRP0 , which is defined the reconstructed data at cell interfaces. When using the A-Roe solver, the solution of the DRP0 for the fluxes reads m $ Nλ  − (0) m  λ α − β −,(0) i+1/2  = Fi(0) + m=1 ei+1/2 , Fi−,(0) R R (174)  $ Nλ  + (0) +,(0) (0) m +,(0) m  λ = F − α − β  e , F(i+1) m=1 (i+1) L i+1/2 i+1/2 L − Fi+1/2 = Fi−,(0) + R

+ = F+,(0) Fi+1/2 (i+1) L +

F+,(k) (i+1) L

(0) and F(i+1) the physical fluxes defined at the left and right interfaces, with Fi(0) R L

 m λ˜ ±

i+1/2



 λ ± | λ| = 2

m , i+1/2

m  ±   ±,(0) m λ (0) β β = , i+1/2  λ i+1/2

(175)

(0) α (0) the wave strengths given by the projection δUi+1/2 = Pi+1/2 Ai+1/2 , with Ai+1/2 =   (0),1 (0),Nλ T (0) , ..., α and β the source strengths associated with each wave, α i+1/2 T  (0) (0) 0 Pi+1/2 Bi+1/2 , with Bi+1/2 = β (0),1 , ..., β (0),Nλ i+1/2 . As in the given by S¯ i+1/2 =  scalar case, a suitable approximation of the integral of the source term across the (0) is required. interface S¯ i+1/2 When using the LFS derivative Riemann solver, the higher-order terms of (173) are computed using an extension of the A-Roe solver that uses time derivatives of (k) (k) and D(k) the conserved variables, Di(k) (i+1) L , as initial condition, where ∂t U = D . R The proposed solution for the approximate derivative fluxes at the interfaces is m $ Nλ  − (k) m  = Ji+1/2 Di(k) + m=1 ei+1/2 , λ α − β −,(k) i+1/2  Fi−,(k) R R (176)  $ Nλ  + (k) +,(k) (k) m +,(k) m    e , F(i+1)L = Ji+1/2 D(i+1)L − m=1 λ α − β i+1/2 i+1/2

with

114

S. Martínez-Aranda et al.

 m λ˜ ±

i+1/2



 λ ± | λ| = 2

m , i+1/2

m  ±   ±,(0) m λ (0) β β = . i+1/2  λ i+1/2

(177)

The wave strengths, α (k) , are given in this case by the projection of the jump of (k) (k) (k) D onto the Jacobian’s eigenvectors basis as δDi+1/2 = Pi+1/2 Ai+1/2 , with Ai+1/2 =   (k),1 T (k),Nλ (k) , ..., α . The source strengths, β , associated with each wave are α i+1/2 computed as proposed in the original approach [87]. Note that the time derivatives of the conserved quantities can be computed by means of the Cauchy-Kovalewski procedure, which allows to use the original system of partial differential equations to express time derivatives in terms of spatial derivatives, which can be reconstructed using the sub-cell WENO reconstruction method [24]. Regarding the conservation properties of the scheme, the same requirements of the WENO-RK scheme apply to the WENO-ADER scheme. It must be pointed out that time derivatives vanish under steady state; therefore there is no need of particular discretization methods for the high-order derivatives of the source term. (k)

7.4 Augmented DG-RK Schemes for the SW Equations Let us consider the 1D version of Eq. (103). The spatial domain, , is discretized in N cells, denoted by i = [xi−1/2 , xi+1/2 ]. The DG method can be obtained by applying the Galerkin projection method to each cell. Inside each cell, the solution is approximated by a linear combination of basis functions {φl }l=0,...,Nd as follows: U(x, t) ≈ Ui (x, t) =

Nd 

ˆ i,l (t)φl (x), U

∀x ∈ i,

(178)

l=0

ˆ i,l (t) is the lth degree of freedom of each cell and Nd + 1 is the number of where U degrees of freedom on each element. Projecting the governing equation onto each element of the basis set and integrating by parts the flux divergence term using Green’s first identity, we obtain  i

    ∂U d x + φk F− i+1/2 − φk F+ i−1/2 − φk ∂t

 i

∂φk dx = F(U) ∂x

 i

φk S(U, z)d x,

(179) + − and Fi+1/2 the cell entering and leaving numerical for k = 0, ..., Nd , with Fi−1/2 fluxes, respectively. To compute such fluxes, augmented solvers are used. The ARoe or the HLLS solver, introduced in Sects. 6.1.1 and 6.1.2 respectively, can be used. Substitution of U = Ui (x, t) in Eq. (179) yields:

Finite Volume Models and Efficient Simulation Tools (EST) for Shallow Flows 

Nd   l=0

i

φk φl d x

ˆ i,l     ∂U + φk F− i+1/2 − φk F+ i−1/2 − ∂t

 i

F(U)

∂φk dx = ∂x

115  i

φk S(U, z)d x.

(180) & % When choosing an orthogonal set of basis functions φ0 , φ1 , ..., φ Nd , Eq. (180) ˆ i,k (t): becomes a decoupled system for the unknown degrees of freedom U ˆ i,k ∂U = L(U, S), k = 0, ..., Nd, ∂t with ak =

 i

(181)

φk φk d x and

L(U, S) = −

1 ak

 

φk F −

  − φk F+ i−1/2 − i+1/2



 i

F(U)

∂φk dx − ∂x

 i

 φk S(U, z)d x .

(182)

To compute the integrals in (182), Gaussian quadrature is employed. For each cell i , the cell integrals in (182) are computed as  F(U) i

and

 NG   ∂φk ∂φk |Ti | ωq , F(U) dx ≈ ∂x ∂x q q=1

 i

φk S(U, z)d x ≈

NG 

(φk S |Ti |)q ωq ,

(183)

(184)

q=1

where N G is the number of Gaussian quadrature points, and F(Uq ), Sq and (φk )q is the evaluation of the flux, source term and basis function at the Gaussian quadrature points q, respectively; ωq are the Gaussian weights and |Ti | is the determinant of the Jacobian matrix associated with the change of coordinates from the reference element, r e f , to the real element, i . In the 1D cases presented in this work, we consider |Ti | = x/2, provided the reference interval is r e f = [−1, 1]. Equation (181) can be integrated over the time step t = [t n , t n+1 ] using a TVD Runge-Kutta (RK) method. The third-order strong stability preserving (SSP) RK method will be used: ˆ n − tL(Un , Sn ), ˆ (1) = U U i,k i,k   (2) 3 ˆ = U ˆ (1) − tL(U(1) , S(1) ) , ˆn + 1 U U i,k i,k 4 i,k 4   ˆ n+1 = 1 U ˆ (2) − tL(U(2) , S(2) ) , ˆn + 2 U U i,k i,k 3 i,k 3

(185)

ˆ n+1 . which allows to update the Nd + 1 degrees of freedom at t n+1 , denoted by U i,k

116

S. Martínez-Aranda et al.

Well-balanced schemes are able to preserve the quiescent equilibrium with machine accuracy, which means that the spatial operator in Eq. (182) vanishes. To construct a well-balanced scheme, the following requirements must be satisfied [12]: • Under equilibrium, the reconstructed data must satisfy the equilibrium condition: this is achieved by applying the DG reconstruction to the equilibrium variable. The variables h + z, hu and z are reconstructed, and h is computed as the difference of h + z and z. • Under equilibrium, the numerical flux at the interface must be equal to the physical flux: this is automatically achieved by augmented solvers. • Under equilibrium, the quadrature rule for the surface and volume integrals must be exact: this is achieved by choosing the adequate number of quadrature points, provided by the polynomial order. Under steady state, when using augmented solvers (e.g. A-Roe or HLLS solvers in Sects. 6.1.1 and 6.1.2), cell-leaving numerical fluxes in Eq. (182) are equal to the physical fluxes inside the cell and the following Rankine-Hugoniot condition at the interface is met: (186) F(i+1)L − Fi R = S¯ i+1/2 . It can be shown that if the rules above are met, Eq. (182) yields to L(U0 , S) = 0,

(187)

which guarantees the preservation of the discrete equilibrium with machine precision. The description of the DG-ADER scheme is omitted for the sake of brevity. The steps for the derivation are analogous to those of the WENO-ADER scheme. Refer to [33] for more details about DG-ADER schemes.

7.5 Application of High-Order Schemes In the next subsections two examples of application of the proposed schemes are presented. More examples can be found in [85, 89, 90].

7.5.1

Test Case 1: Comparison of WENO-ADER and DG-ADER Scheme

In this case, we compare the numerical performance of the FV WENO-ADER and DG-ADER methods for the computation of the linear version of Eqs. (57)–(59), without source terms, setting an initial Gaussian profile:

Finite Volume Models and Efficient Simulation Tools (EST) for Shallow Flows

117

Table 5 Numerical errors and convergence rates for h using L 1 error norm for the third-order optimal WENO-ADER, WENO-JS ADER and DG-ADER schemes, using CFL = 0.07. N represents the total number of cells and Nu represents the total number of unknowns solved, for each problem variable WENO-opt ADER WENO-JS ADER DG-ADER N Nu L 1 error Order Nu L 1 error Order Nu L 1 error Order 1002 2002 4002 5002

1002 2002 4002 5002

1.83E-03 3.28E-04 4.45E-05 2.29E-05

2.48 2.88 2.98

1002 2002 4002 5002

4.35E-03 1.46E-03 4.17E-04 2.54E-04

1.58 1.81 2.22

  (x − 50)2 + (y − 50)2 , h(x, y) = exp − 10

3002 6002 12002 15002

2.29E-05 2.74E-06 3.38E-07 1.73E-07

∀(x, y) ∈ ,

3.06 3.02 3.00

(188)

and u(x, y) = v(x, y) = 0 inside the domain  = [0, 100] × [0, 100]. A unique Courant-Friedrich-Lewy number, CFL= 0.07, is set for all schemes and the numerical solution is computed at t = 25 s. In Table 5, numerical errors for h are shown for four different grids. Results are computed using the WENO-ADER scheme using the optimal reconstruction, the WENO-JS ADER and the DG-ADER scheme. It can be observed that to achieve the same error magnitude, the DG-ADER scheme requires fewer cells than the WENO-ADER schemes. The observed results agree with those in [134], where WENO-FV and DG-RK methods were compared. It is observed that the WENO-JS ADER scheme requires finer meshes to provide the prescribed order of accuracy, which is not completely reached in this test case, while the optimal WENO-ADER scheme is able to converge with the theoretical order. This is due to a lack of accuracy in the recovery of the optimal weights around critical points of smooth data [8, 49]. On the other hand, the DG-ADER scheme converges at the prescribed rate, even for the coarsest grid, and involves smaller numerical errors (for the same CFL and number of cells). The main difference between WENO-ADER and DG-ADER schemes is that in the former, the stencil for the spatial reconstruction grows with the order of the scheme, while the latter is based on a local sub-cell reconstruction. This explains why the WENO-ADER scheme allows a much more relaxed CFL condition while the DG-ADER scheme is very restrictive with the time step.

7.5.2

Test Case 2: Resolution of Complex Shock Reflection and Turbulent Structures

The accurate capture of 2D shocks has been a challenging task in the framework of FV schemes for hyperbolic conservation laws. In this case, we consider the resolution of a Mach reflection structure produced by an oblique shock reflection (e.g. present in hydraulic jumps and other wave reflection phenomena in shallow water flows

118

S. Martínez-Aranda et al.

Fig. 14 Case configuration, including relevant angles and states

of engineering interest [100]). We aim to compare the performance of a first and third-order WENO-ADER scheme for the resolution of a complex wave reflection structure and analyze the benefits of a high order of accuracy for such type of flows. In this test case, we consider a supercritical flow aligned to the x-axis and confined in a straight channel with solid walls as depicted in Fig. 14. The upstream flow is deflected by a wedge of θ1 = 23.3048◦ , generating an incident attached shock, which is reflected at the top wall, producing a Mach reflection structure. The triple point of the Mach stem is the origin of a slip line that propagates downstream and separates two regions with different velocity, where Kelvin-Helmholtz vortices are likely to appear. The complexity of this flow configuration makes it a good candidate for the evaluation of the schemes. The computational domain is given by  = [0, 100] × [0, 55] m and the solid domain is defined by the points (15, 0), (80, 28) and (80, 0) m. Solid boundaries are considered on the top and bottom walls, while a supercritical boundary condition is considered at the inlet (Fr0 = 4 and h 0 = 1 m) and a transmissive boundary condition at the outlet. The solution is computed at t = 200 s using the first and third-order A-Roe WENO-ADER scheme in a 400 × 220 grid (square) and 800 × 440 grid. The Mach reflection leads to the so-called 3-shock solution. The 3-shock solution for a given incoming Froude number and deflection angle can be analytically calculated using the so-called shock polar diagram, based on the Rankine-Hugoniot conditions. Such diagram is a representation of h/ h 0 = h/ h 0 (θ ); the reader is referred to [29, 30] for further information. For this case, the solution reads β I = 37.79◦ , β R = 31.11◦ , θ2,3 = 7.92◦ and h 2,3 / h 0 = 5.175 m. The shock-polar diagram including the analytical solution and the numerical solutions is presented in Fig. 15. It is observed that the schemes accurately reproduce the post-shock solution (regions 2 and 3) and converge as the grid is refined. Note that the third-order solution is closer to the analytical solution, as expected, though the first-order scheme also provides a fairly accurate approximation of the post-shock solution. The numerical solution provided by the third-order and the first-order WENOADER scheme in a 400 × 220 grid and 800 × 440 grid is presented in Fig. 16. The analytical 3-shock solution is overlapped in a white line. It can be observed that as the order is increased and the mesh is refined, the shock angles are more accurately

Finite Volume Models and Efficient Simulation Tools (EST) for Shallow Flows

119

Fig. 15 Mach polar diagram for an incident flow with Fr0 = 4 and h 0 = 1 m, deflected by a wedge of θ1 = 23.3048o , including a detail of the region of interest. The numerical solution is depicted for the first-order scheme (square) and third-order scheme (triangle) in a 800 × 440 grid (magenta) and 400 × 220 grid (blue). The analytical solution is represented with a black dot at the intersection of the shock-polar branches

Fig. 16 Numerical solution provided by a third-order (right) and a first-order WENO-ADER scheme (left) in a 400 × 220 grid (top) and 800 × 440 grid (bottom). The analytical 3-shock solution is overlapped in a white line

120

S. Martínez-Aranda et al.

Fig. 17 Detail of the vorticity at the slip line, computed by a third-order and a first-order WENOADER scheme in the 800 × 440 grid

captured. The use of a cartesian grid prevents the alignment of the flow with the wedge, producing an artificial boundary layer and yielding large errors in the incident shock angle. This spurious effect is stronger when using the first-order scheme, where a wider boundary layer appears. It can be observed that the third-order scheme in the coarse grid gives a more accurate solution than the first-order scheme in the fine grid. Strong differences between the first and the third-order WENO-ADER schemes are observed at the slip line, as depicted in Fig. 16. It is observed that a first order of accuracy smears the contact discontinuity at the slip line, whereas the third-order scheme provides a sharper transition. Note that for the inviscid SW equations, the analytical solution for the slip line is a pure discontinuity. To better examine those differences, a detail of the region around the slip line is presented in Fig. 17. Only when using a third-order scheme, the hydrodynamic instability inherent to the slip line is triggered and Kelvin-Helmholtz vortices are reproduced. On the other hand, the first-order scheme introduces a large amount of numerical diffusion in the solution, which smears the contact discontinuity and damps the hydrodynamical instability. The results outlined above evidence the benefits of high-order schemes for the resolution of free surface shallow flows in the presence of gravity waves, quasi-2D turbulence and solid boundaries (using a cartesian grid). The application of a thirdorder WENO-ADER scheme for the resolution of resonant turbulent shallow water flows using a URANS approach can be found in [85, 86].

Finite Volume Models and Efficient Simulation Tools (EST) for Shallow Flows

121

8 HPC for Large Environmental Problems Finite-difference methods emerged as a tool for solving partial differential equations in the early 1940s at Los Alamos National Laboratory [107] with the advent of computers. Around those years, the works of numerous scientists, as Godunov [44], had given rise to what is nowadays called computational fluid dynamics (CFD). However, at the beginning CFD was strictly limited by computational resources and they provided representative calculations, more than accurate results. With time, the complexity of the computations has been increasing hand in hand with the computational power, leading to faster and more accurate calculations [76, 107]. HPC generally refers to the practice of aggregating computing power in a way that delivers much higher performance than one could get out of a typical desktop computer or workstation in order to solve large problems in science or engineering. Standard chips are no longer improving at the rate at which was predicted by Moore’s Law, so serial computers—able to compute a single operation in a clock tick—have reached a sterile point [76]. Conversely, parallel computing has emerged strongly in recent decades with systems capable of performing multiple operations in a single clock tick, adding resources in the form of cores with shared, distributed memory and more recently, with the use of graphics cards. Based on parallelization techniques that divide the computing workload into different cores that solve the scheme simultaneously, different technologies can be found. This methodology is especially suitable for CFD, as explicit numerical schemes, widely used in CFD, consist of a loop of computations that correspond to each element of the discretized domain. Therefore, there exists, among others, a loop of independent operations that can be cooperatively solved by different processors. Unfortunately, not every loop or instruction can be parallelized—efficiently—since several operations involve race conditions or collective communications between the processors. This chapter offers an overview of HPC techniques and the issues that can be found when a model is accelerated with them, as well as some results of the potential improvement that can be achieved.

8.1 HPC Techniques When a CFD algorithm is executed a computational processor handles those operations and processes them one at a time, this is, sequentially. This execution unit is called core, and is usually inside the CPU (Central Processing Unit) that could have one, two or more cores. Present-day computers usually have more than one core in the CPU so that all the computational tasks can be distributed into several executing units. High-performance computing is based on the parallelization of the workload into different processors. Nowadays, the technology has developed new executing units that are not set into a CPU or has designed new layouts more complex

122

S. Martínez-Aranda et al.

Fig. 18 Sketch of a dual-core shared memory system (a); and comparison of processes distribution between a parallelized and sequential code (b)

than a chip with two or four processors. Therefore, not only the hardware has been improving but also the programming standards or techniques that take advantage of those architectures. This chapter reviews different programming techniques and shows how they are related to different hardware layouts.

8.1.1

Shared Memory Parallel Programming

If the CPU is assumed as a compute element, compound by one or more cores, then a system that has a number of CPUs that work on the same physical address space is called a shared memory parallel computer system [46]. Although transparent to the user, too many architectures can be found regarding the real distribution of cores inside the computation device. Single- or dual-cores in chips can be found distributed in sockets sharing or not sharing paths to memory [46]. In Fig. 18a, a sketch of a shared memory with a dual-core layout can be seen. Each socket is compound by two processors, P, connected with a memory that is shared with other sockets. However, it must be pointed out that the physical layout is not directly related to the programming standard used to implement a parallel algorithm. The translation of the different numerical schemes detailed in the previous sections into efficient algorithms is not an easy task. These algorithms, especially when dealing with numerical schemes with a spatial discretization of the computational domain, are made of some sequential/dependent parts, and some others that can be run independently of each other. The last one is the so-called parallel region, for instance, the numerical flux computation in a list of cell edges. When only a single core is used, the total simulation time would be the sum of all the local times used to compute each independent process sequentially, as represented in the upper part of Fig. 18b. However, when the parallel region is distributed into all the available processors, the time reduces as represented in the lower part of Fig. 18b. To determine

Finite Volume Models and Efficient Simulation Tools (EST) for Shallow Flows

123

parallel regions, some directives must be declared at the code. These directives for the compiler depend on the used programming standard. One of the most common standards is OpenMP (Open Multi-Processing) [7], a set of directives, libraries and environment variables in the SMP (symmetric multiprocessors) model in which all threads share memory and data. It supports different API directives by which all the sequential workload can be distributed into the cores that work simultaneously. The set of OpenMP compiler directives define the parallel regions of the code allowing the rest of the code to work in serial version when necessary and are ignored if a non-OpenMP-capable compiler reads them. They are implemented for the C, C++ and Fortran languages.

8.1.2

Distributed Memory Parallel Programming

In shared memory systems every processor has access to the main and shared memory. On the opposite, in distributed memory computers each processor is linked to its local memory so that any other CPU has no direct access to it. Nowadays, single computers are not physically constructed with such a layout anymore. However, distributed memory understood as a programming model is widely extended and applied to more complex systems such as computational clusters. The workload of a CFD algorithm or model could be so large as to require a big amount of cores to distribute the operations. However, the number of available CPUs in single devices, i.e. a computer or a node, is limited by physical space. In this context, distributed memory parallel programming is used to manage clusters, i.e. different compute nodes with their own memory connected by a communication network. Each of the nodes consists of a shared memory computer that can have two or more CPUs connected to the same local memory, and the communication network connects all the nodes enlarging the number of available cores into different devices. The main feature of these programming models in comparison with shared memory standards is that a problem distributed into different cores that have no remote access to a common memory must be divided and sent back and forth between processes. Thus, the standards for these models are the message passing standards, as MPI (Message Passing Interface) [40]. This paradigm is much more tedious than shared memory programming, since processes must be mapped to the nodes or computer instances, that are connected through a connecting network as seen in Fig. 19. However, it is widely used in large-scale supercomputers, which are always addressed as distributed memory systems. In particular, as creators point out in their main guide [40], MPI is not a language or an implementation. MPI is a message-passing library interface specification. It addresses primarily the message-passing parallel programming model providing extensions to the classical message-passing model. All MPI operations are expressed as functions, subroutines or methods according to the different supported languages: C, C++ or Fortran.

124

S. Martínez-Aranda et al.

Fig. 19 Sketch of a distributed memory system

8.1.3

Graphics Processing Units

The graphic cards, or Graphics Processing Units (GPU), were developed to control and manage the graphic operations for video games and other graphical applications. With the boost of fast graphics render demand, those devices were improved by intersecting their development with a parallel technology. Nowadays, these devices are also compound by their own local memory and a number of cores that can be seen as compute elements. Thus, their development has turned these devices into a more general tool that can offer its capabilities for other purposes, such as the engineering problem acceleration. This approach is also known as GPGPU (General-purpose computing on Graphics Processing Units) and it allows the coder to implement algorithms that can run on a GPU hardware using a high-level language. These devices are continuously evolving—as the CPU does—and can contain hundreds of processing units in one single device that turns the massive parallelization techniques into affordable. The basis of the computation organization is the same as in the shared memory parallel programming but with a different hardware layout and, thus, a different programming model. The main advantage resides in the high number of available cores that can be used for workload distribution, which leads to higher speed-ups than shared memory computers without an increment of investment in large facilities (as in distributed memory parallel systems). The main drawback is that programming in GPU usually requires more effort than programming using OpenMP or other shared memory strategies. A very simplified sketch of a GPU layout can be seen in Fig. 20, where a high number of processors gathered into groups is represented, together with a global memory (L2). The complexity of the GPU architecture is beyond the scope of the chapter. However, it is important to note that a GPU works with thousands of cores and relatively small memory layers, compared to a CPU. There exist many platforms to use these devices for high-performance computing. NVIDIA developed the CUDA (Compute Unified Device Architecture) toolkit, a computing platform and programming model where the developer still programs in its familiar language (C, C++ or Fortran) and incorporates extensions that express massive amounts of parallel operations. However, the use of the CUDA toolkit is not as direct or easy as the directives that a user may use for shared memory parallel programming. For this reason, NVIDIA, in collaboration with PGI and others, is continuously developing alternatives or improvements to turn some implementing issues

Finite Volume Models and Efficient Simulation Tools (EST) for Shallow Flows

125

Fig. 20 Sketch of a GPU layout

transparent to the programmer, as the OpenACC standard [94]. Another alternative is OpenGL, a cross-platform API focused on rendering 2D and 3D applications [106]. With the development of machine learning (ML), applications using Keras [25] or scikit-learn [97], or other platforms oriented to the use of GPUs for those operations, have been developed, as TensorFlow [1]. To avoid the problem of portability and the possibility of developing codes that may run in both GPUs and CPU-based systems, some platforms have prioritized the portability to the efficiency, as Kokkos [35]: a library and programming model than can be run in different kinds of devices once it has been implemented. From the platforms listed in the former paragraph, the most extended programming platform is the CUDA toolkit [92], developed by NVIDIA for its GPUs. CUDA allows the programmer to implement the code in GPU in its familiar programming environment just by incorporating expressions for the parallel parts of the code. This technology is continuously growing and the devices are constantly improving regarding the number of cores, speed on the data transfer and increasing the efficiency of the CUDA toolkit. For instance, the necessity of data transfer (I/O) between CPU and GPU requires a computational effort that could entail a bottleneck on a simulation. Recently, NVIDIA unveiled “GPUDirect” storage, a new capability that enables its GPUs to talk directly with NVM-Express storage without the need to involve the host CPU and system memory [37]. Hence, newer cards present better performances than former ones. The GPU has its own local memory and the use of CUDA implies the use of directives that specifically indicate that in the compiler if a variable must be allocated into the CPU or the GPU memory, as well as the copy between devices, it must be indicated by the programmer by means of directives. Therefore, CUDA could be considered as a “low-level” programming platform, and NVIDIA is continuously developing alternatives to turn this transparent to the user (see Sect. 8.3).

126

8.1.4

S. Martínez-Aranda et al.

Hybrid Computing: Multi-GPU

Hybrid computation refers to the use of shared memory/GPUs and distributed memory parallel programming models to increase the level of parallelism and accelerate the computations. Unfortunately, the process of involving more resources and obtaining a proportional acceleration is not trivial and some metrics are usually considered to quantify the performance of the model: the load balance, the weak speed-up and the strong speed-up [51]. The efficiency on the usage of the computing resources is measured by means of the load balance. Due to the special characteristics of each type of problem, the workload distribution among the resources may be affected, leading to an imbalance. Some strategies such as the domain decomposition—as opposed to the static domain decomposition—are oriented to improve the load balance. The strong and weak speed-ups analyze the scalability of the model, i.e. how the model performs under different considerations. First, the strong speed-up measures the relative acceleration obtained when increasing the computing resources given a fixed problem. On the contrary, the weak speed-up accounts for the efficiency achieved when increasing the problem size and the computing resources proportionally. More details of these metrics can be found in [51]. For the particular case of environmental flows, the combination of GPU offloading and domain decomposition techniques results are very attractive to handle large spatial and temporal domains at fine scales in a reasonable runtime. Several 2D shallow water models have been reported to use multiple GPUs to perform their calculations. The novel and preliminary work by Sætra and Brodtkorb [104] analyzed the scaling (weak and strong) and synchronization using a 4-GPU model implemented in CUDA for square grids. Recently, Turchetto et al. [118, 119] have extended the BUQ (quadtree) grid scheme detailed in [120] to multiple GPUs using CUDA and MPI. This model enables high-resolution spatial discretization, at the extra cost of increasing the complexity when dealing with domain decomposition. To overcome this downside, different partitioning algorithms as well as their dynamic load balance implementation are proposed as a sharp and effective solution. Real-world configurations have also been studied out by means of multi-GPU models. Xia et al. [130] reported the simulation of a storm in a 2500 km2 catchment with HiPIMS using 8 GPUs and 100 million grid cells around 2.5 times faster than in real time. The massive flood caused by Hurricane Harvey was analyzed in [77, 108] using TRITON, the multi-architecture open source model developed at Oak Ridge National Laboratory [74]. The ten-day flood event is simulated in a spatial domain covering 6800 km2 of Harris County, Texas, USA. The results in terms of computational efficiency was evaluated using Summit supercomputer for a different number of cells (up to 272 million) and configurations (multiple CPUs and GPUs), analyzing the model performance and scalability. It is shown that a very large problem can be computed by a multi-GPU configuration in less than 30 min, unlocking the use of these models for operational purposes. The use of CUDA-Aware MPI is also stressed out as a way to bypass hard copies to the CPU, allowing GPU to GPU direct communication. However, the scalability (see Fig. 21, source data [77]) is

Finite Volume Models and Efficient Simulation Tools (EST) for Shallow Flows

127

Fig. 21 Runtime and speed-up (left) and percentage of GPU, MPI, I/O and “other” time over the total runtime (right) achieved by each configuration for Hurricane Harvey test case in [77]

compromised as a result of the MPI communication, the input/output file and the load imbalance due to the static decomposition.

8.2 Application to a Large Environmental Problem When simulating real situations, encompassing thousands of kilometers and simulating events that can last days or even months, the dimensions of the problem can increase up to unaffordable to compute with single-core devices. The Ebro River is the largest river in Spain and its middle reach causes frequently heavy damages due to flood events. This part of the basin collects all the runoff from the mountains and increases its discharge up to 2500 m3 /s, while its mean discharge is only around 200 m3 /s. 2D simulations with a shallow water model were carried out by means of a firstorder explicit finite volume method (145) implemented to run on GPU. The final computational mesh contains around 870000 elements and encompasses 477 km2 . Four historical flood events were simulated and an analysis of computational times with four different GPUs (see Table 6) was performed and can be seen in [34]. When carrying out a speed-up analysis it must be noticed that GPUs are constantly improving and although being the same technology, different architectures could present very different results. As seen in Table 7, the GPU4 provides much faster results than the others. As the HPC turns into an affordable computation of very large and detailed domains, high-computational demanding results can be obtained. As seen in Fig. 22a, the flooded area can be analyzed at each observation time in a large spatial discretization. These kind of results are pretty useful for risk evaluation in adjoining areas, and are also widely extended as a calibration tool if they can be compared with an observed flooded extension. In this case, the flooded area was measured and a comparison between computed and observed extensions was used for topography adjustment. In Fig. 22b, the polygon that defines the extension of the observed flooded area

128

S. Martínez-Aranda et al.

Table 6 Characteristics of the different GPU devices Type Processor CUDA cores GPU1 GPU2

GPU GPU

GPU3 GPU4

GPU GPU

a Including

GTX 780 GTX Titan Black Tesla K40 Tesla V100

Memory

Year

2304 2880

3072 MB 6144 MB

2013 2015

2880 5120a

12 GB 16 GB

2013 2017

the new development: 8 × 640 Tensor Cores

Table 7 Different computational times for the four events carried out in different processing devices Case

Max. discharge (m3 /s)

Flood tG PU 1 (h) duration (d)

tG PU 2 (h)

tG PU 3 (h)

tG PU 4 (h)

Case 1 Case 2 Case 3 Case 4

1797 1800 2000 2600

8.91 24.27 5.25 21.0

5.88 17.25 3.83 16.36

3.69 10.676 2.38 10.40

1.19 3.63 0.495 3.45

7.26 21.14 4.75 21.27

(a) Flooded area at t = 314 h

(b)

(c)

(d)

Fig. 22 3D representation of flooded area at t = 314 h for Case 4 (a), observed flooded area (b), computed flooded area (c) and superposition of both (d)

is shown in yellow; in Fig. 22c, the computed flooded area is represented and colored by water depth; finally, figure (d) represents the superposition of both images and helps to detect disagreement. This level of detail allows the modeler to take into account every singularity of the topography that may produce a change in the flood event evolution. As seen in Fig. 23, where time evolution of discharge can be seen at different gauging points of the river, hydrograph shapes may be perturbed during large and long flood events

Finite Volume Models and Efficient Simulation Tools (EST) for Shallow Flows

129

Fig. 23 Discharge time evolution observed and computed at the inlet, at Tudela (intermediate gauging point) and at Zaragoza (outlet) for the four different flood events analyzed

due to the presence of levees or dikes existing in the floodplain. This can be seen in Case 1, where the two initial peaks present in the inlet are not recorded in the outlet section located in Zaragoza.

8.3 HPC Challenges and Perspectives The new era of exascale computing requires the design and implementation of new smart algorithms that are able to leverage the computer power and the level of parallelism within its hardware. The access to these cutting-edge technologies, once impossible for end-users, has been democratized with the renting low-cost cloud services such as those offered by Amazon or Google. In the HPC context, the most typical situation is the adaptation of a serial or quasi-serial implementation to the next-generation architectures. Refactoring existing models with millions of lines of

130

S. Martínez-Aranda et al.

legacy code for HPC entails four major issues: reliability, scalability, programming effort and performance portability. Reliability or bit-for-bit comparison is often difficult to achieve when offloading the computation to the GPUs due to parameterization complexity and heavy legacy of code structures, as well as to the floating-point behavior of graphics hardware. Recent strategies based on statistical description of the modeled phenomena are devoted to ensure reproducibility when bit-for-bit cannot be expected [66]. Strong and weak scalability is also a question of interest when porting a code to HPC architectures, being input/output and communication times the main bottlenecks as shown in [77]. A parallel code that is able to minimize the ratio between computation and communication time is always chosen as the best option. However, the choice is not always straightforward since higher degrees of parallelism usually implies an increasing demand for communication. Two additional concepts are also linked to scalability: locality and work load balancing. Understanding the way the memory is organized and how to access data efficiently is crucial to achieve a good performance. Consequently, models dealing with structured and Cartesian grids are generally more suitable to get a greater performance—when ported to heterogeneous architectures— than those using unstructured grids, although some strategies such as mesh reordering have been demonstrated to be very convenient for this type of grid [60]. On the other hand, work load balancing refers to the use of “enough work” for each process at the same time. As long as more processors are used, synchronization overhead increases and this may result in a decreased performance. This fact is particularly noticeable when dealing with flood models and wet/dry cells as reported in [76]. It is worth noting that while CUDA is very difficult to be outperformed in terms of computation speed, the process of translating most of the code is often tedious and overwhelming. With the aim of optimizing these tasks, OpenACC was launched as a programming standard for parallel programming of heterogeneous CPU/GPU architectures. Although it is still less mature and does not include many of the explicit synchronization techniques available in OpenMP for instance, it seems to be a promising and simple strategy to obtain acceptable speed-ups with less programming effort. Additionally, not only fast and extremely parallel techniques are desired but also to manage the heterogeneity across target machines. Newly developed applications can benefit from the suite of libraries that are emerging to ensure performance portability such as Kokkos—a set of template specializations for different architectures—or Legion—a parallel programming system allowing task parallelism.

9 Final Comments Starting with the derivation of the equations governing complex shallow flows, the presence of source terms requires the construction of new numerical schemes appropriate to the nature of the equations, instead of using extensions of schemes constructed for the simple, homogeneous case. The definition of augmented numerical schemes considering an extra wave arising from the presence of source terms provides

Finite Volume Models and Efficient Simulation Tools (EST) for Shallow Flows

131

the correct steps to estimate correctly the different types of source terms involved in the problem. Augmented numerical schemes allow recovering the stability region given by the basic CFL condition and no time reduction is necessary. Exact conservation is guaranteed. It can be said that the renewed descriptions made for augmented numerical schemes are the key to explain and avoid important difficulties in real applications. These advances, in combination with the geometric flexibility inherent to unstructured grid models, provide an excellent tool in environmental engineering. The key for model developers is productivity. The implementation of the fastest and most efficient algorithms usually brings with it the use of less friendly programming environments and languages, which translates into a learning time that sometimes turns out to be essential for the accomplishment of short-term projects. However, recent advances in massive parallelization techniques for 2D hydraulic models are able to reduce computer times by orders of magnitude, making 2D applications competitive and practical for operational flood prediction in large river reaches. Moreover, high-performance code development can take advantage of generalpurpose and inexpensive graphical processing units (GPU), allowing to run 2D simulations more than 100 times faster than old generation 2D codes, in some cases. The use of parallel techniques and HPC is therefore mandatory to enable new scientific discoveries. The disadvantages are clearly offset by the potential arising from heterogeneous computing. In the particular field of geophysical flows and environmental problems, the study of large temporal and spatial domains at finer resolution is then becoming affordable, hence unlocking new numerical schemes, models and increasing the complexity of existing applications. The multi-disciplinary approach resulting from complex mathematical formulation, robust and accurate numerical schemes and efficient computational algorithms is enclosed in the “Efficient Simulation Tool” concept and is essential for the simulation of environmental surface processes with realistic temporal and spatial scales. Acknowledgements This work is part of the PGC2018-094341-B-I00 research project funded by the Ministry of Science and Innovation/FEDER. Additionally, Mario Morales-Hernández was partially supported by the U.S. Air Force Numerical Weather Modeling Program.

References 1. Abadi, M. et al.: TensorFlow: Large-scale machine learning on heterogeneous systems (2015). Software available from tensorflow.org 2. Audusse, E., Bouchut, F., Bristeau, M.O., Klein, R., Perthamen, B.: A fast and stable wellbalanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25(6), 2050–2065 (2004) 3. Beisiegel, N., Behrens, J.: Quasi-nodal third-order bernstein polynomials in a discontinuous galerkin model for flooding and drying. Environ. Earth Sci. 74(11), 7275–7284 (2015) 4. Bermudez, A., Vázquez-Cendón, M.E.: Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23, 1049–1071 (1994) 5. Berti, M., Genevois, R., Simoni, A., Tecca, P.R.: Field observations of a debris flow event in the dolomites. Geomorphology 29(3), 265–274 (1999)

132

S. Martínez-Aranda et al.

6. Berti, M., Simoni, A.: Experimental evidences and numerical modeling of debris flow initiated by channel runoff. Landslides (2005) 7. Board, O.A.R.: OpenMP application programming interface. High-Perform. Comput. Center Stuttg. (2015) 8. Borges, R., Carmona, M., Costa, B., Don, W.S.: An improved weighted essentially nonoscillatory scheme for hyperbolic conservation laws. J. Comput. Phys. 227(6), 3191–3211 (2008) 9. Brufau, P., García-Navarro, P., Vázquez-Cendón, M.: Zero mass error using unsteady wettingdrying conditions in shallow flows over dry irregular topography. Int. J. Numer. Methods Fluids 45(10), 1047–1082 (2004) 10. Brufau, P., Vázquez-Cendón, M.E., García-Navarro, P.: A numerical model for the flooding and drying of irregular domains. Int. J. Numer. Methods Fluids 39(3), 247–275 (2002) 11. Burguete, J., García-Navarro, P., Murillo, J.: Friction term discretization and limitation to preserve stability and conservation in the 1d shallow-water model: application to unsteady irrigation and river flow. Int. J. Numer. Methods Fluids 54, 403–425 (2008) 12. Caleffi, V., Valiani, A.: A well-balanced, third-order-accurate RKDG scheme for SWE on curved boundary domains. Adv. Water Res. 46, 31–45 (2012) 13. Caleffi, V., Valiani, A.: A 2D local discontinuous Galerkin method for contaminant transport in channel bends. Comput. Fluids 88, 629–642 (2013) 14. Calhoun, N.C., Clague, J.J.: Distinguishing between debris flows and hyperconcentrated flows: an example from the eastern swiss alps. Earth Surf. Process. Landf. 43(6), 1280–1294 (2018) 15. Cao, Z., Hu, P., Pender, G.: Multiple time scales of fluvial processes with bed load sediment and implications for mathematical modeling. J. Hydraul. Eng. 137(3), 267–276 (2011) 16. Cao, Z., Li, Y., Yue, Z.: Multiple time scales of alluvial rivers carrying suspended sediment and their implications for mathematical modeling. Adv. Water Res. 30(4), 715–729 (2007) 17. Cao, Z., Xia, C., Pender, G., Liu, Q.: Shallow water hydro-sediment-morphodynamic equations for fluvial processes. J. Hydraul. Eng. 143(5), 02517,001 (2017) 18. Castro, C.E., Behrens, J., Pelties, C.: Optimization of the ADER-DG method in GPU applied to linear hyperbolic PDES. Int. J. Numer. Methods Fluids 81(4), 195–219 (2016) 19. Castro, C.E., Toro, E.F.: Solvers for the high-order Riemann problem for hyperbolic balance laws. J. Comput. Phys. 227(4), 2481–2513 (2008) 20. Castro, C.E., Toro, E.F., Käser, M.: ADER scheme on unstructured meshes for shallow water: simulation of tsunami waves. Geophys. J. Int. 189(3), 1505–1520 (2012) 21. Castro, M., Fernández-Nieto, E., Ferreiro, A., García-Rodríguez, J., Parés, C.: High order extensions of ROE schemes for two-dimensional nonconservative hyperbolic systems. J. Sci. Comput. 39, 67–114 (2009) 22. Caviedes-Voullième, D., Gerhard, N., Sikstel, A., Müller, S.: Multiwavelet-based mesh adaptivity with discontinuous galerkin schemes: Exploring 2D shallow water problems. Adv. Water Res. 103559 (2020) 23. Cea, L., Puertas, J., Vázquez-Cendón, M.E.: Depth averaged modelling of turbulent shallow water flow with wet-dry fronts. Arch. Comput. Methods Eng. 14(3), 303–341 (2007) 24. Cheng, J.B., Toro, E.F., Jiang, S., Tang, W.: A sub-cell WENO reconstruction method for spatial derivatives in the ADER scheme. J. Comput. Phys. 251, 53–80 (2013) 25. Chollet, F. et al.: Keras (2015) 26. Cockburn, B., Hou, S., Shu, C.W.: The runge-kutta local projection discontinuous galerkin finite element method for conservation laws. iv. the multidimensional case. Math. Comput. 54(190), 545–581 (1990) 27. Cockburn, B., Shu, C.W.: TVB runge-kutta local projection discontinuous galerkin finite element method for conservation laws. II. General framework. Math. Comput. 52(186), 411– 435 (1989) 28. Costabile, P., Macchione, F., Matale, L., Petaccia, G.: Flood mapping using LIDAR DEM. limitations of the 1-D modeling highlighted by the 2-D approach. Nat. Hazards 77(2), 181–204 (2015)

Finite Volume Models and Efficient Simulation Tools (EST) for Shallow Flows

133

29. Defina, A., Susin, F.M., Viero, D.P.: Numerical study of the guderley and vasilev reflections in steady two-dimensional shallow water flow. Phys. Fluids 20(9), 097–102 (2008) 30. Délery, J., Dussauge, J.P.: Some physical aspects of shock wave/boundary layer interactions. Shock waves 19(6), 453 (2009) 31. Denlinger, R.P., Iverson, R.M.: Flow of variably fluidized granular masses across threedimensional terrain: 2. Numerical predictions and experimental tests. J. Geophys. Res.: Solid Earth 106(B1), 553–566 (2001) 32. Dumbser, M., Enaux, C., Toro, E.F.: Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. J. Comput. Phys. 227(8), 3971–4001 (2008) 33. Dumbser, M., Munz, C.D.: Ader discontinuous galerkin schemes for aeroacoustics. Comptes Rendus Mécanique 333(9), 683–687 (2005). Computational AeroAcoustics: from acoustic sources modeling to farfield radiated noise prediction 34. Echeverribar, I., Morales-Hernández, M., Brufau, P., García-Navarro, P.: 2D numerical simulation of unsteady flows for large scale floods prediction in real time. Adv. Water Res. 134, 103–444 (2019) 35. Edwards, H.C., Trott, C.R., Sunderland, D.: Kokkos: Enabling manycore performance portability through polymorphic memory access patterns. J. Parallel Distrib. Comput. 74(12), 3202– 3216 (2014). Domain-Specific Languages and High-Level Frameworks for High-Performance Computing 36. Egashira, S., Honda, N., Itoh, T.: Experimental study on the entrainment of bed material into debris flow. Phys. Chem. Earth Part C: Solar Terr. Planet. Sci. 26(9), 645–650 (2001) 37. Feldman, M.: NVIDIA GPU accelerators get a direct pipe to big data. The Next Platform (2019) 38. Fernández-Pato, J., García-Navarro, P.: Numerical simulation of valley flood using an implicit diffusion wave model. Ingeniería del agua 20(3), 115–126 (2016) 39. Fernández-Pato, J., Morales-Hernández, M., García-Navarro, P.: Implicit finite volume simulation of 2D shallow water flows in flexible meshes. Comput. Methods Appl. Mech. Eng. 328, 1–25 (2018) 40. Message-Passing Interface Standard Forum: MPI: A Message-Passing Interface Standard, Version 3.1 (2015) 41. García-Navarro, P., Vázquez-Cendón, M.: On numerical treatment of the source terms in the shallow water equations. Comput. Fluids 29(8), 951–979 (2000) 42. García-Navarro, P., Murillo, J., Fernández-Pato, J., Echeverribar, I., Morales-Hernández, M.: The shallow water equations and their application to realistic cases. Environ. Fluid Mech. 19, 1235–1252 (2019) 43. Godlewski, E., Raviart, P.A.: Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer, New York (1996) 44. Godunov, S.: A difference scheme for numerical solution of discontinuous solution of hydrodynamic equations. Math. Sbornik 47, 271–306 (1959) 45. Goetz, C.R., Dumbser, M.: A novel solver for the generalized Riemann problem based on a simplified lefloch-raviart expansion and a local space-time discontinuous galerkin formulation. J. Sci. Comput. 69(2), 805–840 (2016) 46. Hager, G., Wellein, G.: Introduction to High Performance Computing for Scientists and Engineers. CRC Press, Taylor & Francis Group (2011) 47. Harten, A., Engquist, B., Osher, S., Chakravarthy, S.R.: Uniformly high order accurate essentially non-oscillatory schemes, III. In: Upwind and High-Resolution Schemes, pp. 218–290. Springer (1987) 48. Harten, A., Lax, P., van Leer, B.: On upstream differencing and godunov type methods for hyperbolic conservation laws. SIAM Rev. 25, 35–61 (1983) 49. Henrick, A.K., Aslam, T.D., Powers, J.M.: Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points. J. Comput. Phys. 207(2), 542–567 (2005) 50. Horritt, M.S., Bates, P.D.: Evaluation of 1D and 2D numerical models for predicting river flood nundation. J. Hydrol. 268, 89–99 (2002)

134

S. Martínez-Aranda et al.

51. Houzeaux, G., Borrell, R., Fournier, Y., Garcia-Gasulla, M., Göbbert, J.H., Hachem, E., Mehta, V., Mesri, Y., Owen, H., Vázquez, M.: Computational Fluid Dynamics—Basic Instruments and Applications in Science. Cambridge University Press (2017) 52. Hubbard, M., García-Navarro, P.: Flux difference splitting and the balancing of source terms and flux gradients. J. Comput. Phys. 165, 89–125 (2000) 53. Kimura, I.T.H.: Fundamental properties of flows in open channels with dead zone. J. Hydraul. Eng. 123, 98–107 (1997) 54. Iverson, R.M.: The physics of debris flows. Rev. Geophys. 35(3), 245–296 (1997) 55. Iverson, R.M., Logan, M., LaHusen, R.G., Berti, M.: The perfect debris flow? aggregated results from 28 large-scale experiments. J. Geophys. Res.: Earth Surf. 115, F03–005 (2010) 56. Iverson, R.M., Reid, M.E., Logan, M., LaHusen, R.G., Godt, J.W., Griswold, J.P.: Positive feedback and momentum growth during debris-flow entrainment of wet bed sediment. Nat. Geosci. 4, 116–121 (2011) 57. Iverson, R.M., Vallance, J.W.: New views of granular mass flows. Geology 29(2), 115–118 (2001) 58. Jakob, M., Hungr, O.: Debris-flow hazards and related phenomena. Springer Praxis Books. Springer, Berlin, Heidelberg (2005) 59. Juez, C., Murillo, J., García-Navarro, P.: A 2D weakly-coupled and efficient numerical model for transient shallow flow and movable bed. Adv. Water Res. 71, 93–109 (2014) 60. Lacasta, A., Morales-Hernández, M., Murillo, J., García-Navarro, P.: An optimized GPU implementation of a 2D free surface simulation model on unstructured meshes. Adv. Eng. Softw. 78, 1–15 (2014) 61. Lancaster, S.T., Hayes, S.K., Grant, G.E.: Effects of wood on debris flow runout in small mountain watersheds. Water Res. Res. 39(6), 21 (2003) 62. Li, J., Chen, G.: The generalized Riemann problem method for the shallow water equations with bottom topography. Int J. Numer. Methods Eng. 65, 834–862 (2006) 63. Liu, P.F., Yeh, H., Synolakis, C.: Advanced numerical models for simulating tsunami waves and runup. Advances in Coastal and Ocean Engineering, vol. 10. World Scientific (2008) 64. Liu, X.D., Osher, S., Chan, T., et al.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115(1), 200–212 (1994) 65. Luque, R.F., Beek, R.V.: Erosion and transport of bed-load sediment. J. Hydraul. Res. 14(2), 127–144 (1976) 66. Mahajan, S., Evans, K.J., Kennedy, J.H., Xu, M., Norman, M.R., Branstetter, M.L.: Ongoing solution reproducibility of earth system models as they progress toward exascale computing. Int. J. High Perform. Comput. Appl. 33(5), 784–790 (2019) 67. Major, J.J., Iverson, R.M.: Debris-flow deposition: Effects of pore-fluid pressure and friction concentrated at flow margins. GSA Bull. 111(10), 1424–1434 (1999) 68. Martínez-Aranda, S., Murillo, J., García-Navarro, P.: A 1D numerical model for the simulation of unsteady and highly erosive flows in rivers. Comput. Fluids 181, 8–34 (2019) 69. Martínez-Aranda, S., Murillo, J., García-Navarro, P.: A comparative analysis of capacity and non-capacity formulations for the simulation of unsteady flows over finite-depth erodible beds. Adv. Water Res. 130, 91–112 (2019) 70. Martínez-Aranda, S., Murillo, J., García-Navarro, P.: A robust two-dimensional model for highly sediment-laden unsteady flows of variable density over movable beds. J. Hydroinformatics 22(5), 1138–1160 (2020) 71. McArdell, B.W., Bartelt, P., Kowalski, J.: Field observations of basal forces and fluid pore pressure in a debris flow. Geophys. Res. Lett. 34(7), L07–406 (2007) 72. Meyer-Peter, E., Müller, R.: Formulas for bed-load transport. In: Report on 2nd Meeting on International Association on Hydraulic Structures Research, pp. 39–64. Stockholm, Sweden (1948) 73. Montecinos, G., Castro, C.E., Dumbser, M., Toro, E.F.: Comparison of solvers for the generalized Riemann problem for hyperbolic systems with source terms. J. Comput. Phys. 231(19), 6472–6494 (2012)

Finite Volume Models and Efficient Simulation Tools (EST) for Shallow Flows

135

74. Morales-Hernandez, M., Evans, K.J., Gangrade, S., Kao, S.C., Sharif, M.B., Ghafoor, S.K., Kalyanapu, A.J., Dullo, T.T.: Triton, version 00 (2020) 75. Morales-Hernández, M., Hubbard, M., García-Navarro, P.: A 2D extension of a Large Time Step explicit scheme (CFL>1) for unsteady problems with wet/dry boundaries. J. Comput. Phys. 263, 303–327 (2014) 76. Morales-Hernández, M., Sharif, M.B., Gangrade, S., Dullo, T.T., Kao, S.C., Kalyanapu, A., Ghafoor, S.K., Evans, K.J., Madadi-Kandjani, E., Hodges, B.R.: High-performance computing in water resources hydrodynamics. J. Hydroinformatics 22(5), 1217–1235 (2020) 77. Morales-Hernández, M., Sharif, M.B., Kalyanapu, A., Ghafoor, S., Dullo, T.T., Gangrade, S., Kao, S.C., Norman, M.R., Evans, K.J.: TRITON: A multi-GPU open source 2D hydrodynamic flood model. Environ. Model. Softw. (submitted) (2020b) 78. Murillo, J., García-Navarro, P.: An exner-based coupled model for two-dimensional transient flow over erodible bed. J. Comput. Phys. 229, 8704–8732 (2010) 79. Murillo, J., García-Navarro, P.: Weak solutions for partial differential equations with source terms: application to the shallow water equations. J. Comput. Phys. 229, 4327–4368 (2010) 80. Murillo, J., García-Navarro, P.: Augmented versions of the HLL and HLLC Riemann solvers including source terms in one and two dimensions for shallow flow applications. J. Comput. Phys. 231, 6861–6906 (2012) 81. Murillo, J., García-Navarro, P.: Energy balance numerical schemes for shallow water equations with discontinuous topography. J. Comput. Phys 236, 119–142 (2012) 82. Murillo, J., García-Navarro, P., Burguete, J.: Time step restrictions for well balanced shallow water solutions in non-zero velocity steady states. Int. J. Numer. Meth. Fluids 56, 661–686 (2008) 83. Murillo, J., Navas-Montilla, A.: A comprehensive explanation and exercise of the source terms in hyperbolic systems using Roe type solutions. Application to the 1D–2D shallow water equations. Adv. Water Res. 98, 70–96 (2016) 84. Nadaoka, K., Yagi, H.: Shallow-water turbulence modeling and horizontal large-eddy computation of river flow. J. Hydraul. Eng. 124(5), 493–500 (1998) 85. Navas-Montilla, A., Juez, C., Franca, M., Murillo, J.: Depth-averaged unsteady rans simulation of resonant shallow flows in lateral cavities using augmented Weno-ADER schemes. J. Comput. Phys. 395, 511–536 (2019) 86. Navas-Montilla, A., Martínez-Aranda, S., Lozano, A., García-Palacín, I., García-Navarro, P.: 2D experiments and numerical simulation of the oscillatory shallow flow in an open channel lateral cavity. Adv. Water Res. 103836 (2020) 87. Navas-Montilla, A., Murillo, J.: Energy balanced numerical schemes with very high order. the augmented roe flux ADER scheme. Application to the shallow water equations. J. Comput. Phys. 290, 188–218 (2015) 88. Navas-Montilla, A., Murillo, J.: Asymptotically and exactly energy balanced augmented fluxADER schemes with application to hyperbolic conservation laws with geometric source terms. J. Comput. Phys. 317, 108–147 (2016) 89. Navas-Montilla, A., Murillo, J.: 2D well-balanced augmented ADER schemes for the shallow water equations with bed elevation and extension to the rotating frame. J. Comput. Phys. 372, 316–348 (2018) 90. Navas-Montilla, A., Solán-Fustero, P., Murillo, J., García-Navarro, P.: Discontinuous galerkin well-balanced schemes using augmented Riemann solvers with application to the shallow water equations. J. Hydroinformatics 22(5), 1038–1058 (2020) 91. Nielsen, P.: Coastal Bottom Boundary Layers and Sediment Transport. World Scientific, Advanced Series on Ocean Engineering (1992) 92. NVIDIA: NVIDIA CUDA Programming Guide Version 3.0 (2010). http://developer.nvidia. com/object/cuda_3_0_downloads.html 93. O’Brien, J., Julien, P.: Laboratory analysis of mud flow properties. J. Hydraul. Eng. 114, 87–887 (1988) 94. OpenACC-Standard.org: The OpenACC Application Programming Interface (Version 3.1) (2020)

136

S. Martínez-Aranda et al.

95. Ouyang, C., He, S., Tang, C.: Numerical analysis of dynamics of debris flow over erodible beds in Wenchuan earthquake-induced area. Eng. Geol. 194, 62–72 (2015) 96. Parsons, J., Whipple, K., Simioni, A.: Experimental study of the grain-flow, fluid-mud transition in debris flows. J. Geol. (2001) 97. Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., Vanderplas, J., Passos, A., Cournapeau, D., Brucher, M., Perrot, M., Duchesnay, E.: Scikit-learn: machine learning in python. J. Mach. Learn. Res. 12, 2825–2830 (2011) 98. Pierson, T.: Hyperconcentrated flow—transitional process between water flow and debris flow. Debris-flow Hazards and Related Phenomena. Springer, Berlin, Heidelberg, Germany (2005) 99. Ponce, V.: Diffusion wave modeling of catchment dynamics. J. Hydraul. Eng. 112, 716–727 (1986) 100. Pummer, E., Schüttrumpf, H.: Reflection phenomena in underground pumped storage reservoirs. Water 10(4), 504 (2018) 101. Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics informed deep learning (part II): datadriven discovery of nonlinear partial differential equations (2017). arxiv:abs/1711.10566 102. Reed, W.H., Hill, T.: Triangular mesh methods for the neutron transport equation. Technical report, Los Alamos Scientific Laboratory, N. Mex. (USA) (1973) 103. Rickenmann, D., Weber, D., Stepanov, B.: Erosion by debris flows in field and laboratory experiments. In: 3rd International Conference on Debris-Flow Hazards: Mitigation. Mechanics, Prediction and Assessment, pp. 883–894. Millpress, Rotterdam (2003) 104. Sætra, M.L., Brodtkorb, A.R.: Shallow water simulations on multiple gpus. In: Jónasson, K. (ed.) Applied Parallel and Scientific Computing, pp. 56–66. Springer, Berlin Heidelberg (2012) 105. Schwartzkopff, T., Dumbser, M., Munz, C.D.: Fast high order ADER schemes for linear hyperbolic equations. J. Comput. Phys. 197(2), 532–539 (2004) 106. Segal, M., Akeley, K.: The OpenGL Graphics System: A Specification—Version 4.0 (Core Profile). Silicon Graphics International (2010) 107. Shang, J.S.: Three decades of accomplishments in computational fluid dynamics. Progress Aerosp. Sci. 40, 173–197 (2004). www.sciencedirect.com 108. Sharif, M.B., Ghafoor, S.K., Hines, T.M., Morales-Hernändez, M., Evans, K.J., Kao, S.C., Kalyanapu, A.J., Dullo, T.T., Gangrade, S.: Performance evaluation of a two-dimensional flood model on heterogeneous high-performance computing architectures. In: Proceedings of the Platform for Advanced Scientific Computing Conference, PASC ’20. Association for Computing Machinery, New York, NY, USA (2020) 109. Shu, C.W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In: Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, pp. 325–432. Springer (1998) 110. Smart, G.: Sediment transport formula for steep channels. J. Hydraul. Eng. 3, 267–276 (1984) 111. Toro, E.: Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. Springer, Berlin, Germany (1999) 112. Toro, E.: Shock-Capturing Methods for Free-Surface Shallow Flows. Wiley (2001) 113. Toro, E., Millington, R., Nejad, L.: Primitive upwind numerical methods for hyperbolic partial differential equations. In: Sixteenth International Conference on Numerical Methods in Fluid Dynamics, pp. 421–426. Springer (1998) 114. Toro, E., Titarev, V.: Solution of the generalized riemann problem for advection–reaction equations. Proceedings of the Royal Society of London. Ser. A: Math. Phys. Eng. Sci. 458(2018), 271–281 (2002) 115. Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. Springer, Berlin, Heidelberg (2009) 116. Toro, E.F., Spruce, M., Spears, W.: Restoration of the contact surface in the HLL Riemann solver. Shock Waves 4, 25–34 (1994)

Finite Volume Models and Efficient Simulation Tools (EST) for Shallow Flows

137

117. Toro, E.F., Titarev, V.A.: TVD fluxes for the high-order ADER schemes. J. Sci. Comput. 24(3), 285–309 (2005) 118. Turchetto, M., Palù, A.D., Vacondio, R.: A general design for a scalable MPI-GPU multiresolution 2D numerical solver. IEEE Trans. Parallel Distrib. Syst. 31(5), 1036–1047 (2020) 119. Turchetto, M., Vacondio, R., Paù, A.D.: Multi-gpu implementation of 2D shallow water equation code with block uniform quad-tree grids. In: G.L. Loggia, G. Freni, V. Puleo, M.D. Marchis (eds.) HIC 2018. 13th International Conference on Hydroinformatics, EPiC Series in Engineering, vol. 3, pp. 2105–2111. EasyChair (2018) 120. Vacondio, R., Dal Palú, A., Ferrari, A., Mignosa, P., Aureli, F., Dazzi, S.: A non-uniform efficient grid type for GPU-parallel shallow water equations models. Environ. Model. Softw. 88, 119–137 (2017) 121. Vater, S., Behrens, J.: Well-balanced inundation modeling for shallow-water flows with discontinuous galerkin schemes. In: Finite Volumes for Complex Applications VII-Elliptic, Parabolic and Hyperbolic Problems, pp. 965–973. Springer (2014) 122. Vater, S., Beisiegel, N., Behrens, J.: A limiter-based well-balanced discontinuous galerkin method for shallow-water flows with wetting and drying: one-dimensional case. Adv. Water Res. 85, 1–13 (2015) 123. Vater, S., Beisiegel, N., Behrens, J.: A limiter-based well-balanced discontinuous galerkin method for shallow-water flows with wetting and drying: triangular grids. Int. J. Numer. Methods Fluids 91(8), 395–418 (2019) 124. Vignoli, G., Titarev, V.A., Toro, E.F.: ADER schemes for the shallow water equations in channel with irregular bottom elevation. J. Comput. Phys. 227(4), 2463–2480 (2008) 125. Vreugdenhil, C.: Numerical Methods for Shallow Water Flow. Kluwer Academic Publishers, Dordrecht, The Netherlands (1994) 126. Vázquez-Cendón, M.E.: Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry. J. Comput. Phys. 148(2), 497– 526 (1999) 127. Wong, M., Parker, G.: Reanalysis and correction of bed-load relation of meyer-peter and müller using their own database. J. Hydraul. Eng. 132(11), 1159–1168 (2006) 128. Wu, W.: Computational River Dynamics. Inc. CRC Press, NetLibrary (2007) 129. Wu, W., Vieira, D., Wang, S.: One-dimensional numerical model for nonuniform sediment transport under unsteady flows in channel networks. J. Hydraul. Eng. 130(9), 914–923 (2004) 130. Xia, X., Liang, Q., Ming, X.: A full-scale fluvial flood modelling framework based on a highperformance integrated hydrodynamic modelling system (HIPIMS). Adv. Water Res. 132, 103–392 (2019) 131. Xing, Y., Zhang, X., Shu, C.W.: Positivity-preserving high order well-balanced discontinuous galerkin methods for the shallow water equations. Adv. Water Resour. 33, 1476–1493 (2010) 132. Yen, B., Tsai, C.S.: On noninertia wave versus diffusion wave in flood routing. J. Hydrol. 244(1), 97–104 (2001) 133. Yulistiyanto, B., Zech, Y., Graf, W.: Flow around a cylinder: shallow-water modeling with diffusion-dispersion. J. Hydraul. Eng. 124(4), 419–429 (1998) 134. Zhou, T., Li, Y., Shu, C.W.: Numerical comparison of weno finite volume and runge-kutta discontinuous galerkin methods. J. Sci. Comput. 16(2), 145–171 (2001) 135. Zokagoa, J.M., Soulaïmani, A.: A pod-based reduced-order model for uncertainty analyses in shallow water flows. Int. J. Comput. Fluid Dyn. 32(6-7), 278–292 (2018). https://doi.org/ 10.1080/10618562.2018.1513496

Overview of Outfall Discharge Modeling with a Focus on Turbulence Modeling Approaches Mostafa Taherian, Seyed Ahmad Reza Saeidi Hosseini, and Abdolmajid Mohammadian

1 Introduction and Background 1.1 Environmental Impacts of Effluent Discharge These days, the disposal of effluents from power plants, mining operations, and coastal desalination facilities into water bodies has become a serious global concern because of the subsequent public health-related and environmental problems [1]. These effluents, which can be in the form of brine or thermal discharges, may reduce the water quality, damage near-shore recreational sites, or even alter the marine ecosystems as a consequence of increasing turbidity, changing ambient temperature, etc. In particular, regarding the brine discharges, since the effluent has a density higher than the ambient water, it sinks to the sea-bottom and creates a salty desert in the area of the outlet, which results in almost permanent adverse effects on marine habitats, such as the prevention of growth of marine species and the reduction of fish cultures [2]. Results have also shown that brine discharges from desalination plants are more likely to affect the local natural environment at the near-field of the discharge point [2, 3]. In addition, the direct discharge of thermal effluents makes the ambient flow environmentally unstable, which requires further dispersion facilities to protect the

M. Taherian (B) · S. A. R. Saeidi Hosseini · A. Mohammadian Department of Civil Engineering, University of Ottawa, 75 Laurier Ave E, Ottawa, ON K1N 6N5, Canada e-mail: [email protected] S. A. R. Saeidi Hosseini e-mail: [email protected] A. Mohammadian e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 D. Zeidan et al. (eds.), Advances in Fluid Mechanics, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-19-1438-6_4

139

140

M. Taherian et al.

Table 1 International brine discharge regulations [1, 6] Region/Authority

Salinity increment limit (ppt)

Absolute salinity limit (ppt)

Compliance point (relative to discharge) (m)

Sydney, Australia

≤1

–

50–75

Gold coast, Australia

≤2

–

120

Oman

≤2

–

300

US EPA

≤4

–

–

Okinawa, Japan

≤1

–

Mixing zone boundary

Huntington beach, CA

–

≤ 40

304.8

Carlsbad, CA

–

≤ 40

304.8

receiving water bodies. Thus, a quick dispersion and mixing of both types of effluent discharges is required to diminish the adverse marine environmental effects.

1.2 Environmental Regulations The consideration of environmental standards and regulations, such as US Environmental Protection Agency (EPA) [4] and EU Water Framework Directive (WFD) [5], is vital during the design, implementation, and monitoring of outfall systems, irrespective of the type of discharge. The reason behind this is that without such regulations it is impossible to check if one discharge system is properly designed and capable of efficiently providing the required mixing at various distances. The requirements for wastewater and thermal discharges are being quite well developed. For instance, discharges into rivers, lakes, and coastal waters of the US are regulated via the National Pollutant Discharge Elimination System (NPDES) [1]. Some regulations for brine discharges and their compliance points are indicated in Table 1. The international regulations are usually defined based on either the salinity increment or the absolute salinity at a specific distance from outfalls.

1.3 Outfall Discharge System as a Solution For the purpose of minimizing environmental impacts while being in agreement with regulatory demands, the effluent needs to be discharged through an outfall discharge system in the form of turbulent buoyant jets from diffuser ports or nozzles (see Fig. 1), after a proper level of land-based treatment. These facilities can be introduced as a remedy for solving the limited mixing behavior issues as the effluents enter the receiving environments. The ambient fluid is quickly mixed with the buoyant jet by turbulent entrainment resulting from the velocity gradient between the surrounding

Overview of Outfall Discharge Modeling …

141

Fig. 1 Schematic of a tunneled marine outfall system

water and the jet [1]. This buoyant jet also grows in size due to the entrainment, which results in the adequate dispersion of effluents. Prior research has shown the effectiveness of outfall systems. For instance, the impact of diffuser installation on brine plume discharge from two desalination plants located in San Pedro del Pinatar, Murcia, SE Spain (2006–2011) was examined by [7]. Field measurements indicated that the installation of the diffuser significantly improved the dispersion of the brine discharge and the brine plume was completely diluted over a shorter distance from the discharge point. It was also found that the use of the diffuser increased the dilution rate of samples taken 1 km away from the outfall by almost 17 times, which clearly shows the significant effect of a diffuser on brine dispersion.

2 Outfall Discharge Mixing Behavior and Classification 2.1 Behavior of the Discharge: Near- and Far-Field Regions The understanding of the behavior of discharges is closely related to the definitions and distinctions between different terms associated with outfall mixing. Table 2 presents some common terminologies of outfall mixing. Assume a jet discharge with an angle relative to the horizontal of θ0 , jet nozzle diameter of d 0 , and port height of h0 which releases an effluent with a density of ρ0 , concentration of C0 , and discharge velocity of U0 into the stagnant receiving environment with an ambient density of ρa (< ρ0 ), concentration of Ca , and flow depth of H a . As shown in Fig. 2, the mixing process of this type of outfall discharge is typically divided into two regions.

142

M. Taherian et al.

Table 2 Outfall mixing and mixing zone terminology [8] Term

Definition

Mixing Zone or Allocated Impact Zone (AIZ)

Refers to an area in which rapid mixing occurs and the numeric water quality criteria can go beyond what is allowed, although the intense toxic conditions have to be halted Certain water quality provisions must be followed at the end of the AIZ

Regulatory Mixing Zone or Legal Mixing Zone (LMZ)

As defined by the appropriate regulatory authority, can be a length, an area, or a volume of the receiving water bodies

Near-Field or Hydrodynamic Mixing Zone

The zone where mixing is only induced by the processes resulting from the outfall. Near-field characteristics are closely dependent on the discharge parameters, and are under the control of the designer

Far-Field

The ambient oceanic turbulence is the reason for the mixing processes in this region. Far-field mixing is not controlled by the engineer

Initial Dilution

General terminology for the quick mixing occurring in the proximity of the outfall

Zone of Initial Dilution (ZID)

Refers to a small zone in the immediate vicinity of the diffuser where dilution is mainly induced by both the momentum and buoyancy of the discharge

The first region, which is located in the proximity of the discharge point, is named the near-field. This region is often characterized by its initial mixing, which significantly depends on the discharge configuration design and the effluent and ambient properties. Normally, a brine discharge system should be designed in such a way that maximum dilution can happen in the near-field region. In this region, the dilution rate is very high and intense mixing occurs since mixing is a consequence of initial buoyancy and momentum of the high velocity discharge and their interaction with the ambient fluid. Furthermore, mixing and flow characteristics are dominated by small scales (~meters and ~minutes). According to [9], the location where selfinduced turbulence collapses under the effect of the induced density stratification is recognized as the end of the near-field, although it has not been uniquely defined. The second region, which occurs beyond the near-field and further away from the discharge point or diffuser, is called the far-field, where the effluent moves along the bottom of the ocean as a density current. In this region, self-induced turbulence almost vanishes and mixing is predominantly caused by ambient turbulence, and the dilution rate is much lower compared to the near-field. Furthermore, mixing and flow features are dominated by large scales (~kilometers and ~hours) [10]. More precisely, the different flow behaviors of a brine jet discharge can be categorized into four regions (see Fig. 2), as specified by the following. In region (1), the momentum greatly impacts the ascending trajectory of the jet because of the inclined

Overview of Outfall Discharge Modeling …

143

Fig. 2 Near- and far-field regions in a jet discharge

initial velocity. The momentum continuously decreases up to the point where the jet maximum height occurs; at this point, the momentum and buoyancy force are equal. In region (2), the buoyancy force is the predominant force, which leads the jet to descend until it reaches the seabed, where further dilution occurs as the effluent expands. Region (3) is called the transition zone and acts as a “spreading layer”. This region covers the area between the impact point and the far-field region. Finally, in region (4), the far-field, the effluent acts as a density current [10].

2.2 Outfall Discharge Principles and Classification The Froude number is an important non-dimensional parameter in fluid dynamics and is mainly related to the influence of gravity on fluid motion. It can be defined as

144

M. Taherian et al.

u Fr = √ , gl

(1)

where u is the characteristic flow velocity, g is the acceleration of the external field, and l is the characteristic length depending on the geometry of the flow channel. However, when it comes to buoyant jets, the densimetric Froude number (F 0 ) is more common, which can be expressed as follows: U0 F0 =   , gl 

g = 

ρ , ρ0

(2) (3)

where g is the modified gravitational buoyancy acceleration and ρ is the density difference between the jet flow and the ambient fluid. In order to design an ocean outfall system, it is essential to adopt the most efficient discharge form to reach rapid and efficient mixing. The classification of buoyant jets can vary depending on the different standards. Buoyant jets can be classified based on the discharge location, the jet density relative to the ambient water, or the discharge direction. Regarding the discharge location, wastewaters may be discharged either as surface or submerged jets. Surface discharges do not achieve efficient mixing and rapidly change to density currents moving at the bottom of the surrounding water. However, the dilution rates for submerged jets are much higher compared to surface discharges since the surrounding fluid entrains into the jet in all directions. Previous studies confirm the idea that efficient mixing of the highly concentrated brine effluent can be reached by submerged discharge in an upward direction as a high velocity turbulent jet [7, 9, 11]. Considering the jet density compared to the surrounding water density, jets can be categorized into two main groups: positively and negatively buoyant jets, as shown in Fig. 3. A positively buoyant jet has a density less than the ambient fluid density (ρ0 < ρa ) so that both momentum and buoyancy affect the flow dynamics. Thermal discharges from power plants into colder ambient water and sewage discharges into denser ocean water can be the most applicable examples of positively buoyant jets. The effluent rises because of buoyancy forces until it reaches the water surface and then spreads horizontally as plumes. However, the effluent may not reach the water surface in a stratified ambient. In stratified receiving water, the effluent rises to a level where its density is the same as the ambient water, then spreads horizontally. In contrast, a negatively buoyant jet has a density higher than the ambient water (ρ0 > ρa ); as a result, its trajectory is completely different. Since the jet density is higher than the receiving water, the negative buoyancy causes the dense jet to reach a terminal rise height and then fall back to the lower boundary. Once the dense jet impacts the seabed, it spreads as a density current.

Overview of Outfall Discharge Modeling …

145

Fig. 3 Schematic of flow discharges in positively and negatively buoyant jets. (U, T, S denote the velocity, temperature, and salinity parameters, respectively; the subscripts 0 and a represent the jet and ambient characteristics, respectively.)

A buoyant jet can also be discharged horizontally (horizontal jet) or upward, either vertically (vertical jet) or at some angle to the horizontal (inclined jet). The vertical dense jet in stagnant water tends to fall back on the source. This feature disturbs the dilution process and causes lower dilution, which may not be desirable in outfall discharge design [1, 9, 12]. In contrast, inclined jets can clear the effluent from the jet discharge site because of their horizontal component of momentum [13].

3 Outfall Discharge Modeling 3.1 Governing Equations The mathematical representation of fluid flow processes in outfall discharges generally involves a set of partial differential equations (PDEs) of continuity and momentum, which are known as Navier–Stokes equations. The conservation of mass is governed by the continuity equation, which can be written in the incompressible form (which means the pressure does not affect the density of fluid) as follows: ∂v ∂w ∂u + + = 0, ∂x ∂y ∂z

(4)

where u, v, and w are the mean velocity in the x, y, and z directions, respectively. The momentum equations for the three-dimensional incompressible fluids and in non-conservation form can also be expressed as

146

M. Taherian et al.

         ∂u ∂u ∂u 1 ∂p ∂ ∂ ∂ Du =− + νe f f + νe f f + νe f f , (5) Dt ρ ∂x ∂x ∂x ∂y ∂y ∂z ∂z

         1 ∂p ∂ ∂ ∂ ρ − ρ0 ∂v ∂v ∂v Dv =− + νe f f + νe f f + νe f f −g , Dt ρ ∂y ∂x ∂x ∂y ∂y ∂z ∂z ρ

(6)          Dw ∂w ∂w ∂w 1 ∂p ∂ ∂ ∂ =− + νe f f + νe f f + νe f f , Dt ρ ∂z ∂x ∂x ∂y ∂y ∂z ∂z (7) where t is the time, νe f f is the effective kinematic viscosity (νe f f = ν + νt ; νt is the turbulent kinematic viscosity), and p represents the fluid pressure. It is worthwhile mentioning that to capture the variable density impacts, the buoy0 only exists in the momentum equation in the vertical direction (yancy term g ρ−ρ ρ coordinate), and the term ρ exists in all momentum equations. For an incompressible fluid, the density can be calculated for both the surrounding fluid and the jet considering temperature (T ) and salinity (S). In this regard, normally the empirical equation proposed by Millero and Poisson is applied [14]: ρ(T, S) = ρt + aS + bS 3/2 + cS 2 ,

(8)

where empirical parameters ρt (the density of water changing with the temperature), a, b, and c can be defined as presented in Table 3. Along with the Navier–Stokes equations, the advection–diffusion equation governs the fluid concentration and temperature evolution over time in the system as follows: ∂ 2C DC ∂ 2C ∂ 2C = D( 2 + + ), Dt ∂x ∂ y2 ∂z 2  2  ∂ T DT ∂2T ∂2T νt ν with ke f f = = ke f f , + + + 2 2 2 Dt ∂x ∂y ∂z Pr t Pr

(9)

(10)

where C is the fluid concentration/salinity, D is the diffusion coefficient, T is the fluid temperature, ke f f is the heat transfer coefficient, Pr is the Prandtl number, and Pr t is the turbulent Prandtl number. Once the simplifying assumptions have been used, the governed PDEs are solved using available numerical techniques. It should be noted that the successful applications of these governing equations on outfall discharge modeling have been validated in prior research [11, 15–19].

−9.09529 × 10−3 7.6438 × 10−5

−4.0899 × 10−3 1.0227 × 10−4 –

0.824493

−5.72466 × 10−3

4.8314 × 10−4

a

b

c

–

−1.6546 × 10−6

γ

β

999.842594

ρt

α

Coefficients 6.793952 × 10−2

Empirical Parameters

Empirical equation parameters (T [°C]):α + βT + γ T 2 + δT 3 + θ T 4 + ϕT 5

Table 3 Definition of empirical equation parameters used for the determination of density [14]

–

–

−8.2467 × 10−7

1.001685 × 10−4

δ

–

–

5.3875 × 10−9

−1.120083 × 10−6

θ

–

–

–

6.536332 × 10−9

ϕ

Overview of Outfall Discharge Modeling … 147

148

M. Taherian et al.

3.2 Solution Methods and Simulation Techniques for Discharge Modeling There are three different solution approaches for the modeling of the behavior of outfall discharges, including length-scale models, jet integral models, and computational fluid dynamics (CFD) models [20]. Length-scale models, which are driven from the dimensional analysis of jet and plume mixing behavior, are a type of semiempirical approach requiring large amounts of experimental and field data in order to generate a relatively accurate performance (see Sects. 4.1.1 and 4.2.1 for details). As a result, these models are obtained from highly simplified formulas for the flow characterization of outfall discharges and can only be reliable in obtaining general knowledge of their design. CORMIX and NRFIELD are among the common simulation tools being used based on the length-scale models. Jet integral models solve the flow governing equations based on their integration over the cross-section and then these PDEs are transformed into a simple set of ordinary differential equations (ODEs) while using numerical solution methods such as the Runge Kutta formula. In these models, the velocity profile of jets is assumed to be axisymmetric and Gaussian, having no radial variations. The deficiency in comprehension of horizontal and lateral boundary effects, unsteady ambient flow situations, and unstable near-field conditions (re-entrainment of effluent into the jet as an example) may cause some limitations in the applications of jet integral models. JetLag of VISJET software, UM3 of Visual Plumes software, and CoreJet of CORMIX software are among the available simulation tools being used based on the jet integral models. The application of CFD modeling in the simulation of a large number of turbulent flows in nature, as well as engineering, has been increased recently [8]. In CFD modeling, some turbulence closure assumptions are usually adopted to be able to numerically solve the equations of continuity and momentum; to review various CFD methods see [21]. In general, two widely-used discretization methods are developed to address governing equations of the flow problems, namely finite difference method (FDM) and finite volume method (FVM). Different software packages and codes are being applied in the CFD numerical simulations such as ANSYS Fluent, OpenFOAM (OPEN Field Operation And Manipulation), and SU2 code. With the recent advent of progressive CFD tools and robust computational resources, there is a great opportunity to overcome the deficiency of length-scale and jet integral models in the simulation of outfall discharge systems. CFD techniques can provide more detailed information on the flow fields of outfall discharges without considering some of the simplified assumptions of previous approaches. However, they require more sophisticated computational procedures to close the system of equations. Since most of the flow in outfalls are turbulent, the analysis of different applicable turbulence modeling approaches is important in order to accurately determine the discharge behaviors.

Overview of Outfall Discharge Modeling …

149

3.3 Turbulence Modeling Turbulence modeling involves a large number of scale motions that need to be resolved or modeled. The three methods of Reynolds-averaged Navier–Stokes (RANS), large eddy simulation (LES), and direct numerical simulation (DNS) are the main numerical methods used to model turbulent flows. The application of DNS for prediction of mixing behavior of outfall discharges is limited since it requires very high computational cost, although it can resolve all the turbulent flow problems. In this section, the two most practical modeling approaches of RANS and LES in outfall discharge modeling are described.

3.3.1

Reynolds-Averaged Navier–Stokes (RANS)

The main focus of the RANS method is on the effects of turbulence on the mean flow field properties. In this approach, Navier–Stokes equations are usually time averaged prior to being numerically solved. More precisely, any instantaneous flow variable is decomposed into mean and fluctuating components. Afterward, they are replaced in the original equations, and then the obtained equations are time averaged [22,  23]. Figure 4 represents fluctuating (∅ (t)) and mean (∅) components of an instantaneous variable (∅(t)). Each instantaneous variable (∅(t)) can be demonstrated as in Eq. (11): 

∅(t) = ∅ + ∅ (t).

(11)

For instance, the instantaneous values of velocity (u) and pressure ( p) can be decomposed as fluctuating components of u  and p  , and mean values as u and p: 

Fig. 4 An instantaneous variable and its mean and fluctuating components

ui = ui + ui ,

(12)

p = p + p .

(13)

150

M. Taherian et al.

The spread of the fluctuation component about the mean is usually represented by the variance (∅ )2 :   2 1 ∅ = t

t  2  ∅ dt.

(14)

0

  2   2   2 The velocity variances u 1 , u 2 , and u 3 are employed to define the turbulent    kinetic energy per mass (k) as Eq. (15), where u 1 , u 2 , and u 3 are the fluctuations of the velocity components.     2   2   2 . k = 0 · 5 u1 + u2 + u3

(15)

Time averaging of Navier–Stokes equations leads to so-called RANS equations, which can be expressed as follows: ∂u i = 0, ∂ xi

   ∂u j ∂u i ∂u i ∂u i ∂p ∂ μ − ρu  i u  j + ρg, + ρu j ρ =− + + ∂t ∂x j ∂ xi ∂ xi ∂x j ∂ xi

(16)

where μ is the dynamic viscosity of the fluid. The process of time averaging results in the introduction of six extra unknowns of the Reynolds stress tensor (−ρu  i u  j ). These unknown terms must be estimated to be able to close the system of equations; otherwise, the closure problem emerges. Therefore, various turbulence models are developed to perform the estimations, which introduce additional transport equations in most cases. Turbulence models can be categorized based on the number of these extra transport equations into: • Algebraic models (zero additional equation) In these models, eddy viscosity is computed based on an algebraic equation. • Spalart–Allmaras models (one additional equation) In these models, one turbulent quantity is obtained based on a transport equation, and another turbulent quantity is acquired based on an algebraic expression. • k − ε and k − ω models (two additional equations) These models are more complicated and apply two transport equations, which describe transport of two scalars. For instance, in the case of the k − ε model, one transport equation is applied to describe transport of the turbulent kinetic energy and one for its dissipation. • Reynold stress models (seven additional equations)

Overview of Outfall Discharge Modeling …

151

These models consider six transport equations for the Reynolds tensor and one equation for the length scale of the turbulence. From another point of view, the RANS turbulence models can be classified into Eddy viscosity model and Reynolds stress model (RSM), as discussed below.

Eddy Viscosity Model This model has been proposed based on the Boussinesq assumption, which declares that Reynolds stresses are proportional to mean rates of deformation:  



−ρu i u j = τi j = μt

∂u j ∂u i + ∂x j ∂ xi



2 − ρkδi j , 3

(17)

where μt is the eddy viscosity and δi j is the Kronecker delta: δi j = 1i f i = j, δi j = 0 i f i = j.

(18)

The Kronecker delta causes the correct answers for the normal Reynold stresses, where i = j. k−ω, shear stress transport (SST) k−ω, standard k−ε, re-normalization group (RNG) k − ε, and realizable k − ε can be the best examples of the turbulence models based on the eddy viscosity model.

Reynolds Stress Model (RSM) This model applies the Reynolds stress transport equations to obtain the Reynolds stress tensor. Accordingly, the direct effects of the Reynolds stress field can be achieved. This leads RSM to perform better compared to the Eddy viscosity model under the conditions where the flow fields are sophisticated [23]. The Reynolds stress (Ri j ) and the transport equation of Ri j are presented as Eqs. (19) and (20), respectively: Ri j = −

τi j = ui u j − ui u j , ρ

∂ Ri j ∂ Ri j = Pi j + Di j − εi j + i j + i j , + uj ∂t ∂x j ∂R

∂R

(19) (20)

where ∂ti j is the rate of change of Ri j , u j ∂ xijj is the transport term by convection, Pi j is the production rate of Ri j , Di j is the transport term by diffusion, εi j is the dissipation

152

M. Taherian et al.

rate of Ri j , i j is the transport term by turbulent pressure-strain interactions, and i j is the transport term by rotation. In RSM, three terms: convection, production, and rotation maintain their exact forms, and the terms of diffusion, dissipation rate, and pressure-strain interactions need to be modeled; see [24] and [25] for the required models.

3.3.2

Large Eddy Simulation (LES)

The LES approach was first proposed by [26]. LES is a method in which the different behavior of large and small eddies is considered. The large eddies are almost anisotropic, and their behavior is affected by external forces, the geometry of the domain, and boundary conditions, while the small eddies are almost isotropic and homogeneous [23, 27]. Furthermore, the large eddies are the reason for most of the momentum transfer, and incorporate most of the turbulence energy [27]. Unlike the RANS models in which all eddies are modeled, in the LES approach, the unsteady nature of the large eddies is directly computed and just small eddies are modeled using turbulence models. LES employs the following steps to simulate the behavior of flows. The procedure of the LES approach is demonstrated in Fig. 5. At first, a spatial filter function and a filter size are determined to distinguish between the large and small eddies. Afterward, the filter is applied to the flow equations; the large eddies are resolved and the small eddies are eliminated. At the end, the small eddies are modeled using subgrid scale (SGS) models. It is worth noting that the accuracy of LES strictly depends on the adopted SGS model [28]. The small eddies can be accurately modeled since they are independent of boundary conditions and the flow type [22]. The filtering operation in this method is defined as follows:

Fig. 5 Spatial filtering in the LES approach

Overview of Outfall Discharge Modeling … ∞ ˚

153

       G x.x . ϕ x .t d x1 d x 2 d x 3 ,

ϕ(x.t) =

(21)

−∞

      where G x.x . is the filtered function,  is the filter size, and ϕ x .t is the original function. Box filter, Gaussian filter, and spectral cutoff are the most common filtering functions in LES. The filter size () should not be smaller than the cell size. The truncation error can be almost eliminated when the filter size is larger than the cell size [29]. It is usually considered to be the cube root of the cell volume [23]: =

 3

xyz,

(22)

where x, y, and z are the grid cell sizes in x, y, and z directions. The filtered Navier–Stokes equations including continuity and momentum equations can be presented for an incompressible fluid as Eqs. (23) and (24), respectively. ∂(u i ) = 0, ∂ xi

  ∂τi j ∂(u i ) ∂ u i u j 1 ∂p ∂ 2ui + =− +ν − , ∂t ∂x j ρ ∂ xi ∂x j∂x j ∂x j

(23)

(24)

where u i is the filtered velocity, p is the filtered pressure field, and τi j indicates the SGS stresses. The difference between the filtered velocity and the actual velocity is indicated in Fig. 6. The SGS stress can be defined as follows: τi j = u i u j − u i u j .

(25)

The SGS stresses in LES are somewhat like the Reynolds stresses in RANS, but they are created as a result of spatial filtering operation, not time averaging. Furthermore, like the Reynold stresses in RANS, they need to be modeled. They can be modeled either via standard or dynamic Smagorinsky methods. Fig. 6 Comparison of the filtered velocity and the actual velocity

154

3.3.3

M. Taherian et al.

Comparison of Turbulence Modeling Approaches

The relative advantages and disadvantages of applicable turbulence modeling approaches for the prediction of mixing behavior of outfall systems can be explained by considering their conceptual differences. RANS has some advantages over LES in simulating outfall discharges. The main relative advantages have to do with the computational cost and the validation. Time averaging and modeling all eddies allow RANS to have reasonable computational cost, which makes it quite favorable for industrial and full-scale simulations. When an engineer’s focus is on the steady-state flow, simulation of detailed instantaneous flow is a waste of time and money. The computation of unsteady flow equations in LES leads it to require a much larger computational expense compared to RANS. In addition to the fact that high computational time is undesirable for industrial applications, it may also lead to the unreliability of simulation results. The high computing times required for LES prevent performing multiple grid-dependency tests and sensitivity analysis in industrial studies, which may result in unreliable predictions [30]. Furthermore, the performance of RANS is validated by many studies, while there are fewer LES models, resulting in a lack of validation compared to the RANS method; therefore, the risk of applying LES models for field scale diffusers can be somewhat high. More precisely, to the authors’ knowledge, there is no validated LES model for the simulation of multiport diffusers, even at the research stage. The limited studies on single port diffusers have mentioned the weakness of SGS models in dealing with low turbulence intensity [31], the reproduction of the wall interaction processes [32], and the convective mixing due to the buoyancy-induced instability [33]. In addition, a need for considering stratified effects in SGS models has been noted [34]. Due to the mentioned advantages, RANS has been the most widely used approach in industrial applications for turbulent flow simulations in recent decades [22]. On the other hand, LES has some unique features that distinguish it from RANS. The main superiority of LES over RANS is its accuracy and the detailed information that it can provide. Directly computing and capturing the large eddies which incorporate most of the turbulence energy has made LES a more accurate approach compared to RANS, in which all eddies are modeled. The dimensions of this difference may be even more highlighted for field-scale simulations. The accuracy of RANS has always been a concern in turbulent flow problems due to time averaging [28]. Furthermore, unlike RANS, LES is capable of providing statistics of the resolved fluctuations. With the recent advances in computational resources, more attention is given to these models, and the application of LES for industrial problems is becoming more promising [22, 28]. It should be noted that taking advantage of hybrid approaches such as detached eddy simulation (DES) has shown great potential in the simulation of turbulent flows, although it has not been adopted to simulate outfall discharges yet. For comparison, Table 4 lists the differences between different turbulence modeling approaches.

Overview of Outfall Discharge Modeling …

155

Table 4 Comparison between different applicable turbulence modeling approaches for outfall discharge modeling Turbulence models Advantage

Limitations

Applications

DNS

1. Resolves NS equations with no turbulence approximation 2. Resolves all temporal and spatial turbulence scales

1. Extreme computational cost 2. Very fine meshes required to capture all the spatial scales

Simple flow or low Reynolds number flow/ less applicable to engineering problems

RANS

Most computationally Cannot be accurate inexpensive and easy to within the entire flow implement field

LES

1. High accuracy, as it directly computes large scale eddies 2. Much cheaper than DNS computationally due to modeling of only small scales

DES

Combines the benefits 1. Difficult to couple of both RANS and LES RANS-LES interface 2. Unaffordable for large computational domains

Most suitable for steady flow

1. High computational Unsteady flow with cost vortex and recirculation 2. Small-scale turbulence theory still needs development for complex geometries Massively separated flow

4 Outfall Discharge Analysis and Design 4.1 Single Port Effluent Discharges 4.1.1

Dimensional Analysis

The primary flow parameters for a single brine jet in stagnant ambient fluid are illustrated in Fig. 7. At first, due to the initial vertical momentum flux produced by the diffuser, the jet moves upward. The buoyancy forces continuously decrease this momentum flux until it almost vanishes at the point where the jet reaches its maximum. Subsequently, the jet turns downward and falls back to the sea bottom and develops as a gravity current. Typically, jet-densimetric Froude number F 0 , discharge volume flux Q 0 , discharge momentum flux M0 , and discharge buoyancy flux B0 determine the primary flow characteristics of jet. These variables in the dimensional analysis are defined as follows: Q0 =

π d02 π d02 2 ρ0 − ρa  U0 , M 0 = U = U0 Q 0 , B0 = g Q 0 = g0 Q 0 , 4 4 0 ρa

(26)

156

M. Taherian et al.

y

x

Fig. 7 Definition sketch for typical inclined jet parameters 

where g0 is the reduced gravitational buoyancy acceleration at the source. Furthermore, the flow characteristics mainly depend on some length scales derived from the volume, momentum, and buoyancy fluxes. These length scales are as below [35]: 3/4

lM =

M0

1/2 B0

and l Q =

Q0 1/2

M0

,

(27)

l Q quantifies the distance over which the volume flux of the entrained ambient fluid becomes approximately equal to the initial volume flux. Therefore, for distances from the nozzle much larger than lQ , the initial volume flux will not dynamically be of significance. lM is a measure of the distance over which the buoyancy generates momentum approximately equal to the initial momentum. At distances from the nozzle much greater than lM the effect of initial momentum flux becomes negligible and the buoyant jet has essentially become a plume [ 9, 36, 37]. It should be mentioned that since the direction of momentum and buoyancy are different in negatively buoyant jets, the initial momentum flux will always be an important parameter [9]. Considering the assumption that states the flow is fully turbulent, any dependent variable, for example the terminal rise height yt , is a function of Q 0 , M0 , and B0 only: yt = f (Q 0 , M0 , B0 ).

(28)

The maximum rise height can be expressed in terms of the two length scales:   lM yt . = f lM lQ

(29)

For l M > > l Q , the effect of the source volume flux becomes negligible, and Eq. (29) turns into:

Overview of Outfall Discharge Modeling …

yt = K, lM

157

(30)

where K is a constant. Equation (30) can be rewritten as below for a round jet. yt = c1 F0 , d0

(31)

where F0 is the jet-densimetric Froude number and c1 is a constant. Applying the same procedure, the other geometric jet parameters including the maximum centerline height yc and its relevant horizontal location xc , the thickness of the spreading layer yl , the impact point location xi , and the length of the near-field xn can be similarly obtained, as Eq. (32). Moreover, the minimum dilution at the horizontal location xc (Sc ), the impact dilution Si , and the ultimate dilution Sn are found to be constant as follows: yl xc xi xn Sc Si Sn yc , , , , = c2 , c3 , c4 , c5 , c6 ; , , = c7 , c8 , c9 . d0 F0 d0 F0 d0 F0 d0 F0 d0 F0 F0 F0 F0 (32) These constants (c1 to c9 ) are estimated by experiments and specify the trajectory of the jet and dilution rate at different points for the jet.

4.1.2

Numerical Studies

Generally, employing the proper discharge characteristics (like angle and configuration) and appropriate ambient water characteristics (like water depth and forcing currents) would be beneficial in obtaining an efficient mixing of effluent discharges into the receiving water bodies. Extensive studies have been conducted on the flow behavior of jets and plumes, taking into account the optimization of the discharge systems’ performance as the main objective. Among the conducted studies, the use of numerical simulations in comparison with the experimental works is less explored for the assessment of outfall discharge systems, although the numerical methods can be seen as an inexpensive and practical approach in this field. Literature reviews show that improvements during the last two decades in the computational resources provide an ability to extract accurate numerical results for the entire field of outfall regions. Thus, the applications of numerical methods to jet and plume-type flows still need further attention. In the following paragraphs, a summary of numerical studies on single port effluent discharges is presented. Hwang et al. [38] numerically investigated the initial mixing of a vertical buoyant jet discharge into a density stratified cross-flow using the k − ε turbulence model, which successfully shows the evolution of vortex pairs as a consequence of jet and ambient flow interactions. Blumberg et al. [39] and Zhang and Adams [40] employed

158

M. Taherian et al.

the far-field CFD circulation models to predict the near-field plume characteristics of sewage outfalls. Vafeiadou et al. [41] investigated the inclined negatively buoyant jets using the ANSYS CFX tool while applying the shear stress transport turbulence model to close the problem. This model is based on combining the k − ω and k − ε models. They compared the results of the numerical model with the experimental works of Roberts et al. [9] and Bloomfield and Kerr [42], demonstrating relatively acceptable predictions for the terminal rise heights of the negatively buoyant jets studied. However, considerable discrepancies were observed for the forecast of the return point. Furthermore, Kim and Cho [43] numerically studied the mixing characteristics of both the surface and submerged heated flow discharges issuing into ambient crossflow by applying the RNG k − ε method as the turbulence closure method. The three-dimensional model was successfully built using the commercial CFD software package: Flow-3D. The effect of various levels of water depth (shallow and deep ambient water) was also investigated on the initial mixing of buoyant flows. Applying the ANSYS CFX model, Oliver et al. [44] examined the geometrical and bulk flow parameters of inclined negatively buoyant jets. Two sets of k − ε simulations, one based on the standard two-equation k − ε turbulence approach and the other based on the calibration of turbulent Schmidt number in the tracer transport equation, were employed in this study. Although a comparison of the results from the standard and calibrated k − ε approaches was more accurate than the analytical and integral model predictions, the implementation of these approaches could not provide a precise prediction for the bulk flow parameters at the centerline maximum height. Papanicolaou et al. [45] proposed Gaussian distribution and top-hat integral models for the prediction of concentration and velocity in negatively buoyant jets. In addition, different jet entrainment coefficient values were applied to measure the geometrical parameters and dilution at the return point locations and the maximum centerline. Findings showed that the geometrical characteristics were underestimated, particularly when using the Gaussian formulation. HUAI et al. [46] applied the realizable k − ε turbulence closure model to predict the behavior of wall buoyant jets and then conducted a comparison between the numerical results (such as centerline trajectory, cling length, and dilutions) and the analytical and experimental data. A linear relationship between the dilutions of temperature and velocity and the distance from the nozzle was also suggested. Christodoulou et al. [47] proposed that the centerline trajectory and the upper boundary of inclined negatively buoyant jets with discharge angles in the range of 30° to 85° could be estimated in a non-dimensional form by a second degree polynomial. Palomar et al. [48] studied the accuracy of different commercial software tools including VISJET, Visual Plumes, and CORMIX (CorJet) to model the behavior of brine jet discharges into both stagnant and dynamic receiving water environments. Results revealed that these commercial models underestimated the geometrical characteristics such as the terminal rise height of the jets. In addition, the dilution at the impact point was predicted with a deviation of 50% to 65%. Consequently, the commercial software models should be considered highly conservative when approximating the dilution levels. Oliver et al. [49] presented a modified integral model for

Overview of Outfall Discharge Modeling …

159

the simulation of a negatively buoyant jet’s performance in the near-field region. According to the obtained results, it was claimed that the developed model could predict both the geometrical and dilution parameters with a higher accuracy than the previously introduced models by considering the influence of additional mixing in various initial conditions for submerged outfall discharges of brine effluents. Kheirkhah Gildeh et al. [16] numerically investigated the three-dimensional fields of velocity and temperature for both saline and thermal wall jets. The buoyancy modified solver was applied to simulate negatively buoyant jets in the OpenFOAM software. A modified version of PisoFoam solver was created that involves the density equation as a function of salinity and temperature. Seven turbulence models were used to evaluate the accuracy of RANS models for the simulation of jet discharges. The Launder-Reece-Rodi (LRR) turbulence models were introduced as the most reliable approach among those studied. Kheirkhah Gildeh et al. [11] employed different turbulence models for the simulation of the inclined dense jet discharges (with the two angles of 30° and 45°) into the stationary shallow waters, demonstrating that the LRR and the realizable k − ε models could predict this flow type more precisely than the other turbulence closure models examined. Kheirkhah Gildeh et al. [15] also applied the LRR and Realizable k −ε turbulence models to investigate the geometrical and flow properties of 30° and 45° inclined negatively buoyant jets with a densimetric Froude number varying from 10 to 34. Moreover, the influence of buoyancy on the turbulence model was studied by modifying the standard k − ε model using the standard Boussinesq gradient diffusion hypothesis (SGDH) and general gradient diffusion hypothesis (GGDH) approaches. Considering that the buoyancy term in the turbulence model resulted in a wider spread of dense jets on the inner half, this points out the more realistic results according to the physical experiments. In addition, Zhang et al. [33] presented a numerical simulation of a submerged negatively buoyant jet in the OpenFOAM software and applied the “twoLiquidMixingFoam” solver with both the Smagorinsky and dynamic Smagorinsky SGS models, then validated the obtained data with the experiments conducted. More recently, Ardalan and Vafaei [50] numerically modeled 45° inclined thermal-saline effluent discharges into the stationary environment using the realizable k − ε turbulence model. The experimental data were applied to calibrate the model results. The findings demonstrated an acceptable simulation performance for the prediction of geometrical characteristics in the thermal-saline jets studied. A summary of some prominent numerical studies using the RANS approaches to simulate single jets is presented in Table 5. These studies confirm the predictive capability of these models in simulating the mixing behavior of single jets. Among different RANS models, the realizable k − ε and LRR models can be considered as the most reliable models for single jets and can be utilized when designing outfall systems. Furthermore, Table 6 presents a summary of numerical studies using the LES models for the simulation of single jets. It is worthwhile mentioning that the CFD modeling is also useful for the investigation of some real-world jet and plume applications that may encounter with multiphase flows. Discharging particulate matters in dredge disposal operations as

160

M. Taherian et al.

Table 5 Prominent numerical studies using the RANS modeling approaches for the simulation of single jets and their remarks Ref. Applied models

Jet type

Outcome/Remarks

[51] Standard k − ε, 60° inclined realizable k − ε, dense jet standard k − ω, and RSM

• The realizable k − ε model provided the most accurate predictions of the mixing in the negatively buoyant plume and the spreading bed layer • The RSM models required unacceptable processing time for the level of accuracy required for the problem

[44] Standard k − ε and calibrated k−ε

Inclined dense jet

• A k − ε model based on the calibration of the turbulent Schmidt number was proposed • k − ε models led to more accurate predictions of the behavior of inclined dense jets compared to integral models and analytical solutions • Both models underestimated the flow spread and the centerline dilution at the maximum height • There was no improvement in the quality of the predictions of bulk flow parameters at the centerline maximum height when usingk − ε models compared to analytical solutions

[52] Standard k − ε, realizable k − ε, RNG k − ε, and standard k − ω

Heated and • The realizable k − ε model provided the most accurate unheated predictions of the jet characteristics confined vertical • The heated jet was a lazy plume jets • The maximum velocity was theoretically predicted

[46] Realizable k − ε Horizontal wall jet

• The realizable k − ε model could successfully predict velocity distribution and temperature dilutions • Velocity profile had a Gaussian form after the distance of 5d 0 from the nozzle • The distributions of velocity and temperature dilutions indicated a similarity along the axial direction at centerline in the near-field

[16] Standard k − ε, Thermal and realizable k − ε, saline wall jet RNG k − ε, k − ω SST, LRR, and Launder-Gibson

• The LRR and realizable k − ε models led to the most accurate solutions for saline and thermal wall jets • Stream-wise and span-wise profiles for velocity and temperature were self-similar after an initial distance from the nozzle • The stream-wise temperature profiles had a general Gaussian pattern at various distances from the nozzle • Bed slope shortened the cling length • The bed roughness effect on the temperature field was less than on the velocity field

[53] Realizable k − ε 60°, 80°, and and LRR 85° inclined dense jets

• Brine discharges could be accurately modeled using the realizable k − ε and LRR models • The terminal rise height was underestimated by about 5% using both models • The terminal rise height for 80° jets was higher than 85° jets • The LRR model captured secondary flows and buoyancy-induced forces since it considers the effects of the stress anisotropy (continued)

Overview of Outfall Discharge Modeling …

161

Table 5 (continued) Ref. Applied models

Jet type

Outcome/Remarks

[11] Realizable 30° and 45° k − ε, RNG inclined dense k − ε, nonlinear jet k − ε, LRR, and Launder-Gibson

• The LRR and realizable k − ε models resulted in more accurate solutions compared to the other models • The nonlinear k − ε model led to the least accurate results • The dilution was slightly underestimated by all models • Port inclination did not have a significant effect on the dilution at the return point • The cross-sectional velocity and concentration profiles of the outer half of the jet had an axisymmetric Gaussian pattern

[15] Realizable k − ε 30° and 45° and LRR inclined dense jet

• The LRR model led to a slightly better prediction of the jet geometry and dilution characteristics since it could capture secondary flows and buoyancy-induced forces • After the potential core region (x/d 0 > 3), the maximum velocity at the jet centerline decreased almost linearly • Considering the density-induced term in the turbulence model caused the inner part of the jet to spread more widely

[50] Realizable k − ε 45° thermal-saline jet

• Thermal-saline discharges could be accurately modeled by the realizable k − ε model • The accuracy of the predictions decreased from the jet-like regions toward the plume-like region • The outer side boundary of the flow was predicted less accurately compared to the inner side boundary • The salinity and temperature dilution ratios in thermal-saline jets were 93% and 89% for the return point and the centerline peak, respectively • The combination of saline and thermal effluent could be a proper approach to optimize the mixing of effluent

[18] RNG k − ε

• The accuracy of the RNG k − ε model in simulating inclined jets in linearly stratified fluids was evaluated • The prediction errors for the terminal rise height, the depth of the under-flow, and the spreading layer thickness were about 4%, 9%, and 9%, respectively • A new length scale that characterizes the effects of ambient stratification was proposed

45° inclined plane jet

well as sand and sediment during land reclamation can be the best examples of these applications, in which using predictive tools can shed light on their design development. In this regards, the successful use of multiphase CFD-based modeling to analyze the mixing and flow behaviors of single port sediment jet and plumes has been confirmed in previous studies [56–58].

162

M. Taherian et al.

Table 6 Prominent numerical studies using the LES modeling approach for the simulation of single jets and their remarks Ref. Jet type

Outcome/Remarks

[54] Vertical downward jet

• Buoyancy flux had a positive effect on the penetration rate • The penetrative distances driven by the initial buoyancy and momentum fluxes were independent • The total penetration distance can be treated as a linear combination of the penetrative mechanisms driven by the initial buoyancy and momentum fluxes

[31] Wall jet

• LES model, compared to standard k − ε and k − ω models, led to more accurate results of both the kinematic and scalar mixing characteristics • Standard k − ε and k − ω models were unable to predict the strong anisotropic spreading of the wall jet near the boundary • The weakness of the Smagorinsky SGS model in dealing with low turbulence intensity in regions far from the centerline led to faster reduction of spanwise turbulence intensity in those regions

[55] Horizontal jet

• The velocity decay rate was dependent on the Richardson number • The Richardson number had a significant positive effect on the vertical deflection • The anisotropy increased when increasing the Richardson number • The co-flow affected the jet trajectory and the radial half-widths, but not the turbulent energetics • The Reynolds number, Re, had no effect on the total radial spread, the jet trajectory, and the turbulent fluctuations • The ratio of scalar and velocity spreads was non-constant • Coherent vortex rings were formed on the upper side of the jet, while intermittent coherent vortices and small-scale structures were produced on the lower part

[33] 45° inclined jet

• The LES model was able to predict the geometric characteristics with a slight over-prediction of 10% and the return point dilution with an under-prediction of 20% • The LES approach outperformed the integral models • The jet spread widths were reasonably predicted in regions close to the nozzle, but not beyond the centerline peak • Both the Smagorinsky and dynamic Smagorinsky SGS models were not fully able to capture the convective turbulence under the influence of buoyancy

[32] 45° and 60° inclined jets • The LES model was able to simulate reasonably well the geometric characteristics • The return point dilution was underestimated by 20% • The LES model was able to reproduce the localized concentration build-up at the impact point (continued)

Overview of Outfall Discharge Modeling …

163

Table 6 (continued) Ref. Jet type

Outcome/Remarks

[34] 45° inclined jet

• The LES model was able to reproduce well the time-averaged first order mixing characteristics • Turbulence kinetic energy spectrum decreased with the log slope of − 5/3 at the frequency from 2 to 20 Hz near and beyond the centerline peak • SGS models needed to be improved in such a way that incorporate the stratification effects

4.2 Multiport Effluent Discharges 4.2.1

Dimensional Analysis

Although the mixing of inclined dense single jets has been widely investigated for many years, the behavior of multiport diffusers has recently gained more attention. These diffusers have been applied in some areas around the world; the relevant instances include the Australian cities of Melbourne, Perth, and Sydney [1]. Results demonstrated that discharges from a multiport diffuser can achieve a higher dilution in comparison with an equivalent single port discharge [12, 59]. Marine outfalls may be designed as a diffuser with multiple ports (multiport diffuser). There are different designs depending on the site-specific conditions. For example, nozzles can be aligned uni-directionally in the offshore direction to decrease impacts on the sensitive receivers [60]. Jirka [61] reported a summary of other types of diffuser designs. In modern outfall designs, the wastewater is typically discharged through a number of risers mounted on the outfall; each riser is fitted circumferentially with 2–8 horizontal nozzles [60]. Figure 8 shows the perspective view of these risers with a 6-nozzle configuration. With this type of nozzle arrangement, experiments have shown that the diffuser length can be minimized [62], resulting in a cost-effective construction. Considering the multiport diffuser illustrated in Fig. 9 (with discharge either from one or both sides) whose port spacing is L p , the constants on the right-hand side of L Eq. (32) then become functions of d0 Fp 0 [1]:     Lp Lp xi yt ; ; = f = f d0 F0 d0 F0 d0 F0 d0 F0       Lp Lp Lp xn Si Sn ; ; . = f = f = f d0 F0 d0 F0 F0 d0 F0 F0 d0 F0

(33)

Therefore, the effect of the port spacing is completely encapsulated in the L dimensionless parameter d0 Fp 0 [1]. L

When the jets are adequately far from each other ( d0 Fp 0  1), the jets do not have any interactions and behave as single jets, and Eq. (32) can be applied for the multiport

164

M. Taherian et al.

Rosette Riser

Jet Nozzles

Sea Bed

Riser Spacing

Outfall Diffuser

Fig. 8 Perspective view of multiport rosette diffuser on a submarine outfall (the obtained jet flows are shaped like a rose, and is called a rosette jet group) Lp

Fig. 9 Definition diagram for the multiport dense jet L

instead of Eq. (33). On the other hand, when the ports are close together d0 Fp 0  1, they are considered as if they are discharged from a line. Thus, the jet parameters are introduced per unit diffuser length rather than the individual jet parameters. The discharge volume, momentum, and buoyancy fluxes per unit length can be introduced as q0 , m0 , and b0 , respectively:

Overview of Outfall Discharge Modeling …

q0 =

QT  ; m 0 = U0 q0 ; b0 = g0 q0 , L

165

(34)

where QT is the total discharge from the diffuser and L is the diffuser length. The terminal rise height for a line source is as follows, instead of Eq. (28): yt = f (q0 , m 0 , b0 ),

(35)

which following a dimensional analysis becomes: yt b2/3 = Constant. m0

(36)

For a long diffuser, b0 and m 0 can be expressed as B0 /L p and M0 /L p , respectively. Considering the concept of the Froude number, Eq. (36) can be expressed as [1]: L p −1/3 yt = c10 ( ) . d0 F0 d0 F0

(37)

The other geometrical parameters can be presented using the same procedure, and for dilution [1]:   L p 1/3 St = c11 , F0 d0 F0

(38)

where c10 and c11 are experimental constants. Equations (37) and (38) show the negative effect of the port spacing on the terminal rise height and the positive effect of it on the dilution [1].

4.2.2

Numerical Studies

There are a number of studies using CFD to investigate the jet interaction in multiport diffuser discharges. Anderson and Spall [63] studied twin parallel plane momentum jet discharges using CFD and compared them with their experimental observations. A recirculation region was successfully simulated, which was also observed in experiments. Law et al. [64] reported the first CFD computation of rosette jet groups with 20 8-port risers, which required a run time of four days. Kuang et al. [65] carried out a numerical study to investigate the plume interaction above an alternating diffuser. The negative cavity pressure was confirmed by the computation. The merging process between neighboring jets was found. The dependency of the degree of interaction on nozzle spacing was also observed. The computation was however performed with an assumed infinite water depth, which is somewhat different from the actual situation. Xiao et al. [66] conducted a numerical study of four tandem jets in a cross-flow. The

166

M. Taherian et al.

sheltering influence observed in the experiments was well reproduced. Jet trajectories were in agreement with experimental data. Rear jets were deflected to a similar degree by the ambient cross-flow. The induced velocity and pressure field, which are difficult to determine comprehensively in an experiment, were also obtained. The flow field showed a decrease in the effective cross-flow velocity after being sheltered by the front jet. Lai and Lee [60] also presented a general semi-analytical model for the dynamic interaction of multiple buoyant jets in a stagnant environment. Results showed that jet merging and mixing can be remarkably influenced by jet interactions. Model predictions of the jet trajectories, merging height, the centerline velocity, and concentration of the buoyant jet group are corroborated by the experimental findings. Some of the prominent studies on the modeling of multiport jets are presented in Table 7. The prediction capabilities of the RANS models for rosette buoyant, multiple inclined, and multiple vertical jets have been indicated in recent studies. These studies show that multiple jets can be simulated using the RANS models with reasonable error. Regarding the previous studies focusing on the different turbulence closures, the RNG k − ε model led to more accurate results compared to the other RANS models for various multiple jet types. Thus, this model can be applied as a reliable model to predict the mixing behavior of multiport diffusers. Although the effect of port spacing on the mixing behavior of multiple jets has been well studied, the effect of port inclination on the mixing process has not been well recognized. Furthermore, to the authors’ knowledge, multiple jets have not yet been simulated using the LES models. The reason may be the high computational cost and complexity when it comes to the simulation of the complex geometry of multiport discharges.

5 Knowledge Gaps According to the literature review on outfall discharge modeling, there are some main gaps in the existing numerical modeling that need to be considered. Addressing these deficiencies can lead to more realistic predictions of outfall discharge behavior via modeling. The major knowledge gaps can be expressed as follows: • The effect of neighboring boundaries including the bottom layer roughness, the bed slope, and shallow water surface on the mixing behavior of jets requires further research. • With the advancements in computational resources, more attention should be given to hybrid simulation approaches. • To capture precisely the coherence structures and turbulence behavior of discharge flow, the use of instantaneous CFD analysis instead of time averaging should be further explored. • Simulation difficulties such as mesh generation deficiencies and instability of models for outfall discharge modeling, particularly for complex multiport diffuser configuration, need to be considered.

Overview of Outfall Discharge Modeling …

167

Table 7 Prominent numerical studies on the simulation of multiple jets and their remarks Ref. Method

Applied models

Multiple jet type

Outcome/Remarks

[63]

Numerical

Differential Reynolds stress (RSM) and standard k−ε

Dual, parallel planar jets

• A numerical model

[67]

Theoretical

–

Uni-directional • A theoretical model multiple jets • Individual jets asymmetrically spread during jet merging • Models that employed the summation of velocity and tracer distributions underestimated centerline dilutions

[68]

Entrainment restriction

–

Uni-directional • A theoretical model multiple jets • The merging effect became significant on the centerline velocities and concentrations after about double the distance where the nearby optical jet boundaries intersected

[69]

Numerical

Eddy viscosity

Multiple inclined jets

• A numerical model • Unlike empirical mixing zone models, the complex interplay of the ambient flow and thermal jets and jets interactions could be properly resolved via the CFD model

[60]

Semi-analytical –

Rosette buoyant jets

• A Semi-analytical model • Jet interactions significantly affected mixing behavior • Dynamic interactions between the jet groups were negatively affected by ambient cross-flow

[19]

Numerical

Rosette buoyant jets with upward and downward 30°, 45°, and 60° inclined ports

• A numerical model • The port inclination affected mixing behavior • The RNG k − ε outperformed the standard k − ε • Jets became less diluted in the cases with greater angles • The better performance of the RNG k − ε was due to the consideration of the influence of the Reynolds number on the effective turbulence transport, incorporating a new term in the ε, and determination of the inverse effective Prandtl numbers (continued)

Standard k − ε and RNG k − ε

168

M. Taherian et al.

Table 7 (continued) Ref. Method

Applied models

Multiple jet type

Outcome/Remarks

[17]

Numerical

Standard k − ε and RNG k − ε

Multiple 60° inclined jets

• A numerical model • The port spacing significantly impacted mixing behavior • The RNG k − ε model outperformed the standard k − ε • Both the standard and RNG k − ε models could provide results of terminal rise height and impact dilution with errors smaller than 15%

[70]

Artificial intelligence (AI)

Single-gene Multiple 60° genetic inclined jets programming (SGGP) and multigene genetic programming (MGGP)

• AI-based models • The MGGP models outperformed the SGGP models and the existing regression-based empirical models

[71]

AI

MGGP

Multiple vertical jets

• AI-based models • The non-dimensional vertical displacement of the merging point and concentration where the jets were well mixed became more sensitive to port spacing than to the Froude number • The non-dimensional centerline concentration was more sensitive to the Froude number than to port spacing • There was no clear trend regarding the sensitivity of the non-dimensional vertical displacement of the merging point versus the Froude number

[72]

Numerical

k − ω(SST)

Multiple 60° inclined jets

• A numerical model • Cross-flow conditions significantly affected mixing behavior • Quasi-quiescent conditions produced the largest areal extent across the salinity spectrum • Cross-flow decreased the salinity footprint • Impact distances and dilutions were about 10% lower for the logarithmic cross-flow regime compared to the equivalent uniform velocity regime (continued)

Overview of Outfall Discharge Modeling …

169

Table 7 (continued) Ref. Method

Applied models

Multiple jet type

Outcome/Remarks

[73]

Standard k − ε, RNG k − ε, standard k − ω, and k − ω (SST)

Multiple vertical jets

• A numerical model • The port spacing significantly influenced the mixing behavior • The RNG k − ε outperformed other RANS models • The port spacing positively affected the characteristic jet width in the streamwise direction • The jets with larger port spacing merged at locations farther from the nozzle • A new formula for centerline concentration profiles considering the effect of port spacing was proposed

Numerical

• One of the most important challenges in applicability of CFD methods is to define the appropriate mesh and boundary conditions in a way that provide stable results. The definition of high mesh resolutions required for the accurate and stable solution can be translated into high computational cost. Meshfree CFD methods such as smoothed particle hydrodynamics (SPH), which have been recently studied in fluid mechanics problems [74–78] with the idea of replacing the mesh with a set of arbitrarily distributed particles, can overcome the conventional CFD methods’ deficiencies. This method as the next-generation of CFD approach should be investigated further for the modeling of outfall discharges. • The applications of lattice Boltzmann method in the modeling of fluid flow within the complex outfall geometries are worthwhile to examine. • A more realistic scenario involving ambient flow and wave currents representative of a coastal situation should be studied in future research. • Dynamic of ambient stratification in CFD analysis of outfall discharge requires further investigation. • Investigating the flow discharge behavior beyond the impact point, intermediate and far-field regions, in CFD studies should be conducted in detail. • Modifications to turbulence models, such as the incorporation of buoyancy terms, have demonstrated an effective way to enhance the prediction of jet characteristics and should be further explored. • As AI techniques have been introduced as promising approaches in the prediction of mixing behavior in recent years, collecting rich data based on experiments and CFD simulations can be a valuable basis for future AI studies.

170

M. Taherian et al.

6 Concluding Remarks Outfall systems have been introduced as a solution for the adverse environmental impacts on marine life resulting from the disposal of effluents in oceans and seas. Outfall discharges protect marine habitats by providing rapid mixing, which leads to a drop in salinity and thermal levels. Modeling of outfall discharge empowers engineers to predict the mixing behavior of discharges and design proper discharges that meet water quality standards at the point of compliance. This chapter provided a comprehensive description of the basic concepts involved in outfall discharge modeling and a critical review on the current state-of-the-art in this field. Thanks to the advancements in computational resources over the past two decades, the application of CFD modeling in the prediction of jet-mixing behavior has been dramatically increased. It was shown that numerical modeling is a promising tool for simulating both outfall single and multiple discharges, and it can be predicted that, in the near future, CFD modeling will be widely applied in engineering designs to decrease experimental and field costs. Considering the pros and cons of different turbulence modeling approaches, the LES approach provides more accurate results compared to the RANS approach, although it requires higher computational cost. Therefore, as computational resources improve, more attention should be given to the application of the LES for outfall discharge simulation, compared to the RANS. Regarding different turbulence closures adopted in the RANS approach, realizable k − ε and RNG k − ε models have shown better performance in the simulation of single and multiple jets, respectively. These comparisons would provide valuable insight into the efficient modeling of outfalls for engineering applications. Finally, by the identification of knowledge gaps and research directions, this study can be addressed as a reference for future researchers conducting numerical assessments on the outfall discharges.

Nomenclature Abbreviations AI AIZ CFD DES DNS EPA FDM FVM GGDH LES

Artificial Intelligence Allocated Impact Zone Computational Fluid Dynamics Detached Eddy Simulation Direct Numerical Simulation Environmental Protection Agency Finite Difference Method Finite Volume Method General Gradient Diffusion Hypothesis Large Eddy Simulation

Overview of Outfall Discharge Modeling …

LMZ LRR MGGP NPDES ODE OpenFOAM PDE RANS RNG RSM SGDH SGGP SGS SPH SST WFD ZID

Legal Mixing Zone Launder-Reece-Rodi Multigene Genetic Programming National Pollutant Discharge Elimination System Ordinary Differential Equation OPEN Field Operation And Manipulation Partial Differential Equations Reynolds-Averaged Navier–Stokes Re-Normalization Group Reynolds Stress Model Standard Gradient Diffusion Hypothesis Single-Gene Genetic Programming Sub-Grid Scale Smoothed Particle Hydrodynamics Shear Stress Transport Water Framework Directive Zone of Initial Dilution

Mathematical Symbols a, b, and c B0 b0 C C0 Ca c1 to c11 D Di j d0 F0 g  g  g0 Ha h0 k ke f f L lM Lp lQ M0

Empirical parameters in Millero and Poisson equation Discharge buoyancy flux Discharge buoyancy flux per unit length Fluid concentration Initial jet fluid concentration Ambient fluid concentration Constants in dimensional analysis Diffusion coefficient Transport term by diffusion Jet nozzle diameter Jet-densimetric Froude number Acceleration due to gravity Modified acceleration due to gravity Initial modified acceleration due to gravity Ambient flow depth Port height Turbulent kinetic energy per mass Heat transfer coefficient Diffuser length Jet-to-plume Length Scale Port-spacing Discharge Length Scale Discharge momentum flux

171

172

m0 p p p Pi j Pr Pr t Q0 q0 Ri j Re Re0 S S0 Sc Si Sn St T T0 Ta t U0 Ua u u u, v, and w xc xi xn yc yl yt

M. Taherian et al.

Discharge momentum flux per unit length Fluid pressure Fluctuating component of pressure Mean/filtered component of pressure Production rate of Ri j Prandtl number Turbulent Prandtl number Discharge volume flux Discharge volume flux per unit length Reynolds stress Reynolds numbers Jet Reynolds Number Salinity Jet discharge dilution Centerline peak dilution Impact dilution Ultimate dilution Terminal peak dilution Temperature Jet discharge temperature Ambient fluid temperature Time Jet discharge velocity Ambient flow velocity Fluctuating component of velocity Mean/filtered component of velocity Mean velocity in the x, y, and z directions Horizontal distance to jet terminal rise height Horizontal distance to jet impact point Horizontal distance to near-field location Maximum centerline height Thickness of the spreading layer Maximum terminal rise height

Greek symbols  x, y, and z ρ δi j εi j θ0 μ

Filter size Grid cell sizes in x, y, and z directions Density difference between the jet flow and the ambient fluid Kronecker delta Dissipation rate of Ri j Jet discharge angle relative to the horizontal Dynamic viscosity of the fluid

Overview of Outfall Discharge Modeling …

μt ν νe f f νt i j ρ0 ρa ρt τi j ∅(t)  ∅ ∅ i j

173

Eddy viscosity kinematic viscosity Effective kinematic viscosity Turbulent kinematic viscosity Transport term by turbulent pressure-strain interactions Jet discharge density Ambient flow density Density of water changing with the temperature in Millero and Poisson empirical equation Sub-grid scale stresses Instantaneous variable Fluctuating component of an instantaneous variable Mean component of an instantaneous variable Transport term by rotation

References 1. Missimer, T., Jones, B., Maliva, R. (eds.): Intakes and Outfalls for Seawater Reverse-Osmosis Desalination Facilities: Innovations and Environmental Impacts. Springer, New York, NY, USA (2015) 2. Einav, R., Lokiec, F.: Environmental aspects of a desalination plant in Ashkelon. Desalination 156, 79–85 (2003). https://doi.org/10.1016/S0011-9164(03)00328-X 3. Hashim, A., Hajjaj, M.: Impact of desalination plants fluid effluents on the integrity of seawater, with the Arabian Gulf in perspective. Desalination 182, 373–393 (2005). https://doi.org/10. 1016/j.desal.2005.04.020 4. EPA, U.: Water Quality Standards Handbook, 2nd edn. DC, USA, Washington (1994) 5. Chave, P.: The EU water framework directive. IWA Publishing (2001) 6. Abessi, O.: Brine disposal and management—planning, design, and implementation. In: Sustainable Desalination Handbook: Plant Selection, Design and Implementation. Elsevier Inc., Amsterdam, pp. 259–303 (2018). https://doi.org/10.1016/B978-0-12-809240-8.00007-1 7. Loya-Fernández, Á., Ferrero-Vicente, L.M., Marco-Méndez, C., et al.: Quantifying the efficiency of a mono-port diffuser in the dispersion of brine discharges. Desalination 431, 27–34 (2018). https://doi.org/10.1016/j.desal.2017.11.014 8. Jenkins, S., Paduan, J., Roberts, P., et al.: Management of brine discharges to coastal waters: recommendations of a science advisory panel: submitted at the request of the California water resources (2012) 9. Roberts, P.J.W., Ferrier, A., Daviero, G.: Mixing in inclined dense jets. J. Hydraul. Eng. 123, 693–699 (1997). https://doi.org/10.1061/(ASCE)0733-9429(1997)123:8(693) 10. Palomar, P., Losada, I.: Impacts of brine discharge on the marine environment: modelling as a predictive tool. In: In Desalination: Trends and Technologies; Michael, S., Ed. IntechOpen, London, UK (2011) 11. Kheirkhah Gildeh, H., Mohammadian, A., Nistor, I., Qiblawey, H.: Numerical modeling of 30◦ and 45◦ inclined dense turbulent jets in stationary ambient. Environ. Fluid Mech. 15, 537–562 (2015). https://doi.org/10.1007/s10652-014-9372-1 12. Fischer, H., List, J., Koh, C., et al.: Mixing in inland and coastal waters. Elsevier Inc., San Diego, CA, USA (2013) 13. Pincince, A.B., List, E.J.: Disposal of brine into an estuary. Water Pollut. Control Fed. 45, 2335–2344 (1973)

174

M. Taherian et al.

14. Millero, F., Oceanographic, AP-DSRPA.: International one-atmosphere equation of state of seawater. J. Deep Res1. 28, 625–629 (1981). https://doi.org/10.1016/0198-0149(81)90122-9. 15. Kheirkhah Gildeh, H., Mohammadian, A., Nistor, I., et al.: CFD modeling and analysis of the behavior of 30° and 45° inclined dense jets—new numerical insights. J. Appl. Water Eng. Res. 4, 112–127 (2016). https://doi.org/10.1080/23249676.2015.1090351 16. Kheirkhah Gildeh, H., Mohammadian, A., Nistor, I., Qiblawey, H.: Numerical modeling of turbulent buoyant wall jets in stationary ambient water. J. Hydraul. Eng. 140,(2014). https:// doi.org/10.1061/(ASCE)HY.1943-7900.0000871 17. Yan, X., Mohammadian, A.: Numerical modeling of multiple inclined dense jets discharged from moderately spaced ports. Water (Switzerland) 11, 14–16 (2019). https://doi.org/10.3390/ w11102077 18. Yan, X., Mohammadian, A., Chen, X.: Numerical modeling of inclined plane jets in a linearly stratified environment. Alexandria Eng. J. 59, 1857–1867 (2020b). https://doi.org/10.1016/j. aej.2020.05.023 19. Yan, X., Mohammadian, A., Chen, X.: Three-dimensional numerical simulations of buoyant jets discharged from a rosette-type multiport diffuser. J. Mar. Sci. Eng. 7, 1–15 (2019). https:// doi.org/10.3390/jmse7110409 20. Taherian, M., Mohammadian, A.: Buoyant jets in cross-flows: review, developments, and applications. J. Mar. Sci. Eng. 9, 61 (2021). https://doi.org/10.3390/jmse9010061 21. Sotiropoulos, F.: Introduction to statistical turbulence modelling for hydraulic engineering flows. In: Computational Fluid Dynamics. Applications in Environmental Hydraulics. Wiley, Chichester, UK (2005) 22. Moukalled, F., Mangani, L., Darwish, M.: Fluid Mechanics and Its Applications The Finite Volume Method in Computational Fluid Dynamics. Springer (2016) 23. Versteeg, H.K., Malalasekera, W.: An introduction to computational fluid dynamics: the finite volume method, 2nd edn. Pearson Education (2007) 24. Launder, B., Reece, G., Rodi, W.: Progress in the development of a Reynolds-stress turbulence closure. J. Fluid Mech. 68, 537–566 (1975). https://doi.org/10.1017/S0022112075001814 25. Rodi, W.: Turbulence models and their applications in hydraulics-a state-of-the-art review. Delft, The Netherlands (1984) 26. Smagorinsky, J.: General circulation experiments with the primitive equations: I the basic experiment. Mon. Weather Rev. 91, 99–164 (1963). https://doi.org/10.1175/1520-0493(196 3)091%3c0099:GCEWTP%3e2.3.CO;2 27. Zhiyin, Y.: Large-eddy simulation: Past, present and the future. Chinese J. Aeronaut. 28, 11–24 (2015). https://doi.org/10.1016/j.cja.2014.12.007 28. Mohammadian, A., Gildeh, H.K., Nistor, I.: CFD modeling of effluent discharges: a review of past numerical studies. Water (Switzerland) 12, 856 (2020). https://doi.org/10.3390/w12 030856 29. Lund, T.: On the use of discrete filters for large eddy simulation. Annu. Res. Briefs 83–95 (1997) 30. Gant, S.: Reliability issues of LES-related approaches in an industrial context. Flow Turbul. Combust. 84, 325–335 (2010). https://doi.org/10.1007/s10494-009-9237-8 31. Zhang, S., Law, A.W.K., Zhao, B.: Large eddy simulations of turbulent circular wall jets. Int. J. Heat Mass. Trans. 80, 72–84 (2015). https://doi.org/10.1016/j.ijheatmasstransfer.2014.08.082 32. Zhang, S., Law, A.W.K., Jiang, M.: Large eddy simulations of 45° and 60° inclined dense jets with bottom impact. J. Hydro-Environ. Res. 15, 54–66 (2017). https://doi.org/10.1016/j.jher. 2017.02.001 33. Zhang, S., Jiang, B., Law, A.W.K., Zhao, B.: Large eddy simulations of 45° inclined dense jets. Environ. Fluid Mech. 16, 101–121 (2016). https://doi.org/10.1007/s10652-015-9415-2 34. Jiang, M., Law, A.W.K., Lai, A.C.H.: Turbulence characteristics of 45° inclined dense jets. Environ. Fluid Mech. 19, 27–54 (2019). https://doi.org/10.1007/s10652-018-9614-8 35. Roberts, P.J.W., Toms, G.: Ocean outfall system for dense and buoyant effluents. J. Environ. Eng. 114, 1175–1191 (1988)

Overview of Outfall Discharge Modeling …

175

36. Doneker, R.L., Jirka, G.H.: Cormix User Manual 6.0E: A Hydrodynamic Mixing Zone Model and Decision Support System for Pollutant Discharges into Surface Waters. U.S. Environmental Protection Agency, Washington DC (2007) 37. Wright, S.J.: Buoyant jets in density-stratified crossflow. J. Hydraul. Eng. 110, 643–656 (1984) 38. Hwang, R.R., Chiang, T.P., Yang, W.C.: Effect of ambient stratification on buoyant jets in cross-flow. J. Eng. Mech. 121, 865–872 (1995). https://doi.org/10.1061/(asce)0733-9399(199 5)121:8(865) 39. Blumberg, A.F., Ji, Z.-G., Ziegler, C.K.: Modeling outfall plume behavior using far field circulation model. J. Hydraul. Eng. 122, 610–616 (1996). https://doi.org/10.1061/(asce)0733-942 9(1996)122:11(610) 40. Zhang, X.-Y., Adams, E.E.: Prediction of near field plume characteristics using far field circulation model. J. Hydraul. Eng. 125, 233–241 (1999). https://doi.org/10.1061/(asce)0733-942 9(1999)125:3(233) 41. Vafeiadou, P., Papakonstantis, I., Christodoulou, G.: Numerical simulation of inclined negatively buoyant jets. In: In Proceedings of the 9th International Conference on Environmental Science and Technology. Rhodes Islands, Greece, pp. A1537–A1542 (2005) 42. Bloomfield, L., Kerr, R.: Inclined turbulent fountains. J. Fluid Mech. 451, 283–294 (2002). https://doi.org/10.1017/S002211200100652 43. Kim, D.G., Cho, H.Y.: Modeling the buoyant flow of heated water discharged from surface and submerged side outfalls in shallow and deep water with a cross flow. Environ. Fluid Mech. 6, 501–518 (2006). https://doi.org/10.1007/s10652-006-9006-3 44. Oliver, C.J., Davidson, M.J., Nokes, R.I.: k-ε predictions of the initial mixing of desalination discharges. Environ. Fluid Mech. 8, 617–625 (2008). https://doi.org/10.1007/s10652-0089108-1 45. Papanicolaou, P., Papakonstantis, I., Christodoulou, G.: On the entrainment coefficient in negatively buoyant jets. J. Fluid Mech. 614, 447–470 (2008). https://doi.org/10.1017/S00221120 0800350 46. Huai, W.X., Li, Z.W., Qian, Z.D., et al.: Numerical simulation of horizontal buoyant wall jet. J. Hydrodyn. 22, 58–65 (2010).https://doi.org/10.1016/S1001-6058(09)60028-7 47. Christodoulou, G., Hydraulics, IP-E.: Simplified estimates of trajectory of inclined negatively buoyant jets. In: Christodoulou, G,C., Stamou, A.I. (eds.) Environmental Hydraulics- Proceedings of the 6th International Symposium on Environmental Hydraulics. Taylor & Francis, Milton Park, Didcot, pp. 165–170 (2010) 48. Palomar, P., Lara, J., Losada, I.: Near field brine discharge modeling part 2: validation of commercial tools. Desalination 290, 28–42 (2012) 49. Oliver, C.J., Davidson, M.J., Nokes, R.I.: Predicting the near-field mixing of desalination discharges in a stationary environment. Desalination 309, 148–155 (2013). https://doi.org/10. 1016/j.desal.2012.09.031 50. Ardalan, H., Vafaei, F.: CFD and experimental study of 45° inclined thermal-saline reversible buoyant jets in stationary ambient. Environ. Process 6, 219–239 (2019). https://doi.org/10. 1007/s40710-019-00356-z 51. Plum B, Webb T, Young J (2008) Modelling of Desalination Plant Outfalls. Sch. Aerospace, Civ. Mech. Eng., p. 24. 52. El-Amin, M.F., Sun, S., Heidemann, W., Müller-Steinhagen, H.: Analysis of a turbulent buoyant confined jet modeled using realizable k-ε model. Heat Mass Trans. und Stoffuebertragung 46, 943–960 (2010). https://doi.org/10.1007/s00231-010-0625-3 53. Gildeh, H.K., Mohammadian, A., Nistor, I., Qiblawey, H.: Numerical modelling of brine discharges using OpenFOAM. In: Proceedings of the International Conference on New Trends in Transport Phenomena. Ottawa, ON, Canada, p. 51 (2014) 54. Wang, R.Q., Law, A.W.K., Adams, E.E., Fringer, O.B.: Large-eddy simulation of starting buoyant jets. Environ. Fluid Mech. 11, 591–609 (2011). https://doi.org/10.1007/s10652-0109201-0 55. Ghaisas, N.S., Shetty, D.A., Frankel, S.H.: Large eddy simulation of turbulent horizontal buoyant jets. J. Turbul. 16, 772–808 (2015). https://doi.org/10.1080/14685248.2015.1008007

176

M. Taherian et al.

56. Azimi, A., Zhu, D., Rajaratnam, N.: Computational investigation of vertical slurry jets in water. Int. J. Multiph. Flow 47, 94–114 (2012). https://doi.org/10.1016/j.ijmultiphaseflow.2012. 07.002 57. Chan, S.N., Lai, A.C.H., Law, A.W.K., Eric Adams, E.: Two-Phase CFD Modeling of Sediment Plumes for Dredge Disposal in Stagnant Water. In: Estuaries and Coastal Zones in Times of Global Change, pp. 409–423. Springer, Singapore (2020) 58. Chan, S.N., Lee, K.W.Y., Lee, J.H.W., et al.: Numerical modelling of horizontal sediment-laden jets. Environ. Fluid Mech. 14, 173–200 (2014). https://doi.org/10.1007/s10652-013-9287-2 59. Lee, J., Chu, V.: Turbulent Jets and Plumes: A Lagrangian Approach. Kluwer Academic Publishers, Hingham, MA, USA (2003) 60. Lai, A.C.H., Lee, J.H.W.: Dynamic interaction of multiple buoyant jets. J. Fluid Mech. 708, 539–575 (2012). https://doi.org/10.1017/jfm.2012.332 61. Jirka, G.H.: Integral model for turbulent buoyant jets in unbounded stratified flows part 2: plane jet dynamics resulting from multiport diffuser jets. Environ. Fluid Mech. 6, 43–100 (2006). https://doi.org/10.1007/s10652-005-4656-0 62. Isaacson, M.S., Koh, R.C.Y., Brooks, N.H.: Plume dilution for diffusers with multiport risers. J. Hydraul. Eng. 109, 199–220 (1983). https://doi.org/10.1061/(asce)0733-9429(1983)109: 2(199) 63. Anderson, E.A., Spall, R.E.: Experimental and numerical investigation of two-dimensional parallel jets. J. Fluids Eng. Trans. ASME 123, 401–406 (2001). https://doi.org/10.1115/1.136 3701 64. Law, A., Lee, C., Qi, Y.: CFD modeling of a multiport diffuser in an oblique current. In: Proceedings of the Marine Waste Water Discharges, MWWD. Istanbul, Turkey (2002) 65. Kuang, C., Lee, J., Liu, S., Gu, J.: Numerical study on plume interaction above an alternating diffuser in stagnant water. China Ocean Eng. 20, 289–302 (2006) 66. Xiao, Y., Lee, J., Tang, H., Yu, D.: Three-dimensional computations of multiple tandem jets in crossflow. China Ocean Eng. 20, 99–112 (2006) 67. Wang, H.J., Davidson, M.J.: Jet interaction in a still ambient fluid. J. Hydraul. Eng. 129, 349–357 (2003). https://doi.org/10.1061/(asce)0733-9429(2003)129:5(349) 68. Yannopoulos, P.C., Noutsopoulos, G.C.: Interaction of vertical round turbulent buoyant jets— Part I: entrainment restriction approach. J. Hydraul. Res. 44, 218–232 (2006). https://doi.org/ 10.1080/00221686.2006.9521677 69. Tang, H.S., Paik, J., Sotiropoulos, F., Khangaonkar, T.: Three-dimensional numerical modeling of initial mixing of thermal discharges at real-life configurations. J. Hydraul. Eng. 134, 1210– 1224 (2008). https://doi.org/10.1061/(asce)0733-9429(2008)134:9(1210) 70. Yan, X., Mohammadian, A.: Evolutionary modeling of inclined dense jets discharged from multiport diffusers. J. Coast Res. 36, 362–371 (2020a). https://doi.org/10.2112/JCOASTRESD-19-00057.1 71. Yan, X., Mohammadian, A.: Evolutionary prediction of multiple vertical buoyant jets in stationary ambient water. Desalin. Water Treat. 178, 41–52 (2020b). https://doi.org/10.5004/ dwt.2020.24938 72. Baum, M.J., Gibbes, B.: Field-scale numerical modeling of a dense multiport diffuser outfall in crossflow. J. Hydraul. Eng. 146, 05019006 (2020). https://doi.org/10.1061/(asce)hy.19437900.0001635 73. Yan, X., Ghodoosipour, B., Mohammadian, A.: Three-dimensional numerical study of multiple vertical buoyant jets in stationary ambient water. J. Hydraul. Eng. 146, 04020049 (2020a). https://doi.org/10.1061/(asce)hy.1943-7900.0001768 74. Aristodemo, F., Marrone, S., Federico, I.: SPH modeling of plane jets into water bodies through an inflow/outflow algorithm. Ocean Eng. 105, 160–175 (2015). https://doi.org/10.1016/j.oce aneng.2015.06.018 75. Hu, X.Y., Adams, N.A.: A SPH model for incompressible turbulence. Procedia IUTAM 18, 66–75 (2015). https://doi.org/10.1016/j.piutam.2015.11.007 76. Liu, S., Wang, X., Ban, X., et al.: Turbulent details simulation for SPH fluids via vorticity refinement. Comput. Graph. Forum. 40, 54–67 (2021). https://doi.org/10.1111/cgf.14095

Overview of Outfall Discharge Modeling …

177

77. Meister, M., Winkler, D., Rezavand, M., Rauch, W.: Integrating hydrodynamics and biokinetics in wastewater treatment modelling by using smoothed particle hydrodynamics. Comput. Chem. Eng. 99, 1–12 (2017). https://doi.org/10.1016/j.compchemeng.2016.12.020 78. Tran-Duc, T., Phan-Thien, N., Khoo, B.C.: A smoothed particle hydrodynamics (SPH) study of sediment dispersion on the seafloor. Phys. Fluids 29, 083302 (2017). https://doi.org/10.1063/ 1.4993474

A Unified Algorithm for Interfacial Flows with Incompressible and Compressible Fluids Fabian Denner and Berend van Wachem

1 Introduction Numerical methods and algorithms for interfacial flows have, so far, typically been developed either for the simulation of compressible fluids or the simulation of incompressible fluids, as a result of the different dominant physical mechanisms, the different mathematical characteristics of the governing equations and the different numerical challenges at different flow speeds. While in reality all fluids are compressible, with a compressibility given as β = dρ/(ρ d p), where ρ is the density and p is the pressure, a frequent assumption in modelling fluid flows is that the fluid is incompressible (dρ = 0). From a numerical viewpoint, the distinction between incompressible and compressible fluids is important and the Mach number M = U/a, with U the flow speed and a the speed of sound, is of particular importance, as it determines the mathematical nature of the governing conservation laws. For incompressible fluids, the density is constant along the fluid particle trajectories and the speed of sound is a → ∞, with M → 0, whereas for compressible fluids, the density is variable (dρ = 0), with a finite speed of sound (0 < a < ∞) and Mach number (0 < M < ∞). For large flow speeds (M > 0.1), pressure and density are strongly coupled, especially for supersonic flows, whereas this pressure–density coupling diminishes in the incompressible flow regime (M → 0), where density changes vanish (dρ → 0). This acoustic degeneration as well as the pressure–velocity coupling present the major challenges for the modelling of flows with M < 0.1, whereas the stability in different Mach number regimes and the robust resolution of discontinuities are pertinent issues for the F. Denner (B) · B. van Wachem Chair of Mechanical Process Engineering, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany e-mail: [email protected] B. van Wachem e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 D. Zeidan et al. (eds.), Advances in Fluid Mechanics, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-19-1438-6_5

179

180

F. Denner and B. van Wachem

modelling of flows with M > 0.1. These different mathematical characteristics and numerical requirements has made the development of algorithms that can simulate mixed compressible–incompressible two-phase fluids a difficult task. In this chapter, we propose a fully coupled pressure-based algorithm for interfacial flows at all speeds, including compressible and incompressible fluids, in which the discretised governing equations are solved for pressure, velocity and temperature. This algorithm features a conservative finite-volume discretisation of the governing equations that is identical for incompressible fluids (dρ = 0) and compressible fluids (dρ = 0) [22]. Through an appropriate linearisation, the discretised continuity equation serves as a transport equation for the density in the case of a compressible fluid, with density formulated as a function of pressure by an equation of state, and as a constraint on the velocity field for incompressible fluids, with pressure acting as a Lagrange multiplier. Hence, the proposed algorithm is applicable to incompressible, compressible and, crucially, mixed compressible–incompressible interfacial flows in all Mach number regimes. The bulk phases are represented and advected using a Volume-of-Fluid method (VOF), and the bulk phases are coupled at the interface using the acoustically conservative interface discretisation (ACID) method [29], which was originally proposed for fully compressible interfacial flows. Results of representative test cases, including flows with surface tension, the pressure-driven collapse of a bubble and a shock–drop interaction, are presented to validate and scrutinise the proposed algorithm for compressible-incompressible interfacial flows.

2 State of the Art As a consequence of the very different mathematical characteristics and numerical requirements of the simulation of compressible and incompressible fluids, as explained above, two different classes of algorithms have emerged: density-based algorithms and pressure-based algorithms. In the following, the main developments of both classes of algorithms related to interfacial flows are briefly reviewed in Sects. 2.1 and 2.2, respectively, and, in Sect. 2.3, currently available numerical algorithms able to simulate mixed compressible–incompressible interfacial flows are discussed.

2.1 Density-Based Algorithms Density-based algorithms, where the sought primary variables are the density, momentum and energy of the flow, are widely applied for compressible flows. With respect to the modelling of compressible interfacial flows, notable examples are the two-fluid Baer–Nunziato (BN) model [5], where each fluid is represented by its own set of governing equations, and the five-equation models [3, 52], with a separate continuity equation for each phase, and shared momentum and energy equations. In these algorithms, an exact or approximate Riemann solver is employed to evaluate the

A Unified Algorithm for Interfacial Flows …

181

fluxes at cell faces of the computational mesh [17, 36, 60]. The ghost-fluid method (GFM) [31] has established itself as an alternative to solving a Riemann problem, partly due to its conceptual simplicity. Recent developments have also seen a combination of GFM with Riemann solvers [48, 49] to improve the stability of simulations with strong shock–interface interactions and compressible gas–liquid flows. Since pressure is not directly solved for with a density-based algorithm, the pressure field has to be reconstructed based on the applied thermodynamic model via an equation of state, which may be difficult in interfacial cells where two bulk phases coexist [2, 3, 52], and which requires hydrodynamically and thermodynamically plausible fluid properties. While density-based algorithms are naturally suited for compressible flows, they are poorly suited for low-Mach number flows [43, 71], where the coupling of pressure and density vanishes. Although density-based algorithms have been applied to low-Mach number flows with some success, this requires computationally expensive pre-conditioning techniques [65]. A different approach for single-phase flows at all speeds was pursued by van der Heul et al. [67], in which the energy equation is reformulated as an equation for pressure, while the continuity equation still serves as an equation for density. Similar methods were subsequently presented by Park and Munz [53] and Cordier et al. [18]. These algorithms formally converge to the incompressible flow equations in the limit of zero Mach number (M → 0) [53, 67], with the reformulated energy equation enforcing a divergence-free velocity field. Boger et al. [11] extended the work of Park and Munz [53] to interfacial flows.

2.2 Pressure-Based Algorithms Pressure-based algorithms, in which the continuity equation is cast into a discretised equation for pressure (with pressure acting as the Lagrange multiplier that enforces ∇ · u → 0 for M → 0), while density is either constant (incompressible fluids) or evaluated using a suitable equation of state (compressible fluids), are preferably applied to simulate the flow of incompressible fluids and weakly compressible flows, and may yield significant advantages for low-Mach number compressible flows, since the acoustic degeneration at low Mach numbers does not pose a problem [39]. The success of pressure-based algorithms is facilitated by the unique role of pressure in all Mach number regimes [39], with the pressure–velocity coupling dominant at low Mach numbers and the pressure–density coupling dominant at high Mach numbers [44, 68], and the convenient fact that the fully conservative formulation of the governing conservation laws can still be satisfied accurately [22, 68]. The majority of pressure-based algorithms for incompressible interfacial flows are founded on pressure-correction methods, such as projection methods [8, 15], the SIMPLE method [54] and its subsequent derivatives, or the PISO method [42], or fully coupled algorithms [26]. Pressure-based algorithms for compressible flows are much less prominent in the literature than their density-based counterparts, partly because deriving stable and efficient numerical schemes for the transonic regime [20]

182

F. Denner and B. van Wachem

as well as formulating consistent shock-capturing schemes have proven difficult for pressure-based algorithms [9, 71]. Nonetheless, because pressure plays an important role in all Mach number regimes, pressure-based algorithms have a distinct potential for applications in all Mach number regimes or in which the Mach number varies strongly. Starting with the work of Harlow and Amsden [38], a variety of pressurebased algorithms for compressible single-phase flows has been developed [19, 20, 43, 68, 72]. Denner et al. [22] presented a fully coupled pressure-based algorithm able to predict single-phase flows of incompressible, ideal-gas and real-gas fluids at all speeds (0 ≤ M ≤ 239) accurately and robustly, using the same fully coupled pressure-based algorithm and conservative finite-volume discretisation. Despite these developments for single-phase flows, it was only recently that the first pressure-based algorithm in conservative form for compressible interfacial flows at all speeds was proposed [29]. This algorithm was proposed in conjunction with a new interface discretisation method that retains the acoustic features of the flow, without the need to employ a Riemann solver, and has been shown to be a promising alternative to traditional density-based algorithms, with the pressure-based formulation of the governing equations facilitating the definition of consistent mixture rules at the interface that apply naturally to flows in all Mach number regimes. This algorithm was subsequently extended to polytropic interfacial flows [23], i.e. treating the flow as a polytropic process.

2.3 Algorithms for Compressible–Incompressible Flows Caiden et al. [13] were the first to propose a numerical algorithm dedicated to the simulation of general compressible–incompressible flows, solving different governing equations for the compressible fluid and the incompressible fluid, coupling the bulk phases at the interface using the GFM. A similar method to describe the behaviour of compressible bubbles in an incompressible fluid was subsequently presented by Aanjaneya et al. [1]. Wadhwa et al. [70] proposed a method for computing incompressible liquid drops in a compressible gas, representing the interface as a matching moving mesh and solving the flow of the incompressible fluid using the artificial compressibility method of Chorin [15]. Others proposed algorithms for compressible– incompressible flows applying the same governing equations in non-conservative form in both fluids [10, 14, 73] and these algorithms are, in general, only applicable to incompressible and low-Mach number flows. However, the application of the governing equations in conservative form together with a conservative discretisation is a prerequisite for an accurate prediction of flows in all Mach number regimes [41], in particular shock waves, rarefaction fans and contact discontinuities. The densitybased algorithm of Boger et al. [11] is readily applicable to interfacial flows in all Mach number regimes, including the incompressible limit (M → 0), yet incompressible fluids (M = 0) have not been considered in the context of such an algorithm. In addition to the difficulties encountered when developing numerical methods for interfacial flows, such as a precise discrete force balance between surface tension

A Unified Algorithm for Interfacial Flows …

183

and the pressure gradient, Caiden et al. [13] and Billaud et al. [10] identified the transmission of waves at the interface, as well as physically realistic and compatible discrete formulations of the governing equations describing the incompressible fluid and the compressible fluid, as the main additional difficulties associated with an accurate prediction of compressible–incompressible interfacial flows. The algorithm presented in this chapter addresses and resolves these difficulties, as demonstrated by the chosen validation cases.

3 Governing Equations The conservation laws governing both incompressible and compressible fluid flow at all speeds are the conservation of mass, momentum and energy, given as ∂ρ + ∇ · (ρu) = 0, ∂t

∂(ρu) + ∇ · (ρu ⊗ u) = −∇ p + ∇ · τ + S, ∂t ∂p ∂(ρh) + ∇ · (ρuh) = − ∇ · q + ∇ · (τ · u) + S · u, ∂t ∂t

(1) (2) (3)

respectively, where t is time, u is the velocity vector, p is pressure, ρ is the density and h is the specific total enthalpy. The stress tensor τ for the considered Newtonian fluids is given as   2 τ = μ ∇u + (∇u)T − μ (∇ · u) I, (4) 3 where μ is the dynamic viscosity. The heat flux due to thermal conduction is typically described by Fourier’s law as q = −k∇T , with k the thermal conductivity and T the temperature. All external forces applied to the flow, e.g. the force due to surface tension, are gathered in the volumetric source term S. The Volume-of-Fluid (VOF) method [40] is adopted to capture the fluid interface between two immiscible bulk phases, applying an indicator function field ζ , defined as  0 if x ∈ a , ζ (x) = (5) 1 if x ∈ b , where a and b are the subdomains occupied by fluid a and fluid b, respectively, and  = a ∪ b is the computational domain. Because the interface is a material front propagating with the flow, the indicator function ζ is advected with the underlying fluid velocity by the advection equation ∂ζ + u · ∇ζ = 0. ∂t

(6)

184

F. Denner and B. van Wachem

4 Thermodynamic Closure The governing conservation laws given above require closure by an appropriate thermodynamic model, defining the thermodynamic properties of the fluids. Following the approach recently proposed by Denner et al. [22], the density ρ and the specific isobaric heat capacity c p are defined by a set of input parameters (ρ0 , c p,0 , cv,0 and 0 ) that enables the formulation of a unified thermodynamic closure for incompressible and compressible fluids. For a compressible fluid, the stiffened-gas model [37, 47] is applied, in which the density is defined as p + γ 0 0 , (7) ρ= R0 T where 0 is a material-dependent pressure constant, R0 = c p,0 − cv,0 is the specific gas constant, with the constant reference-specific isobaric heat capacity c p,0 and the constant reference-specific isochoric heat capacity cv,0 , and γ0 = c p,0 /cv,0 is the heat capacity ratio. The specific isobaric heat capacity is given as [29] c p = c p,0

p + 0 , p + γ 0 0

(8)

and the specific total enthalpy is defined as h = cp T +

u2 . 2

(9)

The speed of sound follows as  a=

γ0

p + 0 . ρ

(10)

For 0 = 0, the stiffened-gas model describes a calorically perfect ideal gas, with ρ = p/(R0 T ) and c p = c p,0 . The density of an incompressible fluid is, by definition, constant and given as ρ = ρ0 ,

(11)

with ρ0 a predefined density value. The specific total enthalpy is given by Eq. (9) and the specific isobaric heat capacity c p of an incompressible fluid is also assumed to be constant, with (12) c p = c p,0 . In order to incorporate compressible and incompressible fluids in the same numerical framework, the definitions for the density ρ and the specific isobaric heat capacity

A Unified Algorithm for Interfacial Flows …

185

c p are unified by the binary operator C, given as  1 , for compressible fluids, C= 0 , for incompressible fluids.

(13)

This binary operator is used as a coefficient for the compressible part and, analogously, 1 − C is used as a coefficient for the incompressible part of the unified closure model. The density for a given fluid is then defined, based on Eqs. (7) and (11), as ρ=C

p + γ 0 0 + (1 − C) ρ0 , R0 T

(14)

and the specific isobaric heat capacity is defined, based on Eqs. (8) and (12), as   p + 0 c p = c p,0 C + (1 − C) . p + γ 0 0

(15)

The speed of sound is given, following Eq. (10), as  a=C

γ0

p + 0 + (1 − C) a∞ , ρ

(16)

where a∞ is a very large velocity (here: a∞ = 1032 m s−1 ) that represents the infinite speed of sound of incompressible fluids and ensures a computationally meaningful definition of the speed of sound throughout the computational domain. The appropriate formulations of the density ρ, the specific isobaric heat capacity c p and the speed of sound a for incompressible fluids (C = 0), perfect-gas fluids (C = 1, 0 = 0) and stiffened-gas fluids (C = 1, 0 > 0) are readily recovered.

5 Numerical Framework The proposed numerical algorithm is based on a fully coupled pressure-based algorithm for incompressible and compressible fluids with a finite-volume discretisation of the governing equations [22]. The numerical framework is predicated on a collocated variable arrangement, meaning that the primary solution variables p, u and T , as well as all fluid properties, are stored at the centre of the mesh cells. First, the finite-volume discretisation underpinning the numerical algorithm is briefly discussed (Sect. 5.1), followed by the definition of the advecting velocity (Sect. 5.2) and the discretisation of the governing equations (Sect. 5.3). The particular form the discretised governing equations assume in the specific case of the incompressible limit is discussed in Sect. 5.4 and the solution procedure is described in Sect. 5.5.

186

F. Denner and B. van Wachem

5.1 Finite-Volume Discretisation Considering, for example, the advective–diffusive transport of a general fluid variable, φ, is given as ∂(ρφ) + ∇ · (ρuφ) = ∇ · ( φ ∇φ), (17) ∂t where t is the time, ρ is the fluid density, u is the fluid velocity and φ is the diffusion coefficient of φ. Reformulating Eq. (17) in its integral form with respect to the control volume V , given as ˚ V

∂(ρφ) dV + ∂t

˚

˚ ∇ · (ρuφ) dV = V

∇ · ( φ ∇φ) dV,

(18)

V

allows a rather straightforward finite-volume discretisation of the advective–diffusive transport of φ. In the interest of simplicity and brevity, the mesh is henceforth assumed to be Cartesian with a local mesh spacing x. Extending the presented methods to unstructured meshes is easily achieved by introducing corrections for mesh skewness and non-orthogonality, for instance, described in [22]. The transient term is discretised in the following using the second-order backward Euler scheme, which is given for cell P as ˚ V

3 (n+1) ∂ − 4 (t−t) + (t−2t) P P P dV ≈ V P + O(t 2 ), ∂t 2t

(19)

where = ρφ, t is the time-step, superscript (n + 1) denotes the implicitly sought solution, superscript (t − t) denotes the previous time-level and superscript (t − 2t) denotes the previous-previous time-level. The discrete form of the advection term is obtained by applying the divergence theorem ‹ ˚ ∇ · (ρuφ) dV = ρuφ d, (20) ∂V

V

where  is the outward-pointing surface vector on the surface ∂ V of control volume V . Since the surface of a discrete control volume is constituted by a finite number of flat faces f of area A f , as illustrated in Fig. 1, and applying the midpoint rule [34,

P

f

Q nf

Fig. 1 Schematic illustration of mesh cell P with its neighbour mesh cell Q and the shared mesh face f , where n f is the unit normal vector of face f , pointing out of cell P

A Unified Algorithm for Interfacial Flows …

187

51], the advection term follows as ‹   ρuφ d ≈ ρfϑfφf Af = m˙ f φ f . ∂V

f

(21)

f

The face value φ f is typically not readily available and has to be interpolated from the cell-centred values, which in a finite-volume sense represent the values of φ averaged over the respective cell volume, using an appropriate interpolation scheme. The advecting velocity ϑ f = u f · n f , where n f is the normal vector of face f (pointing outwards with respect to cell P, see Fig. 1), represents the velocity normal to face f and forms, together with the face density ρ f and the face area A f , the mass flux m˙ f = ρ f ϑ f A f through face f . In the context of the presented algorithm, the advecting velocity ϑ f is determined using a momentum-weighted interpolation from the cell-centred values of velocity and pressure, as further discussed in Sect. 5.2. Since the values of density and velocity reside at the cell centres, a suitable interpolation must be applied, such as upwind differencing, central differencing or a TVD scheme. Similar to the advection term, applying the divergence theorem in conjunction with the midpoint rule, the diffusion term of the transport equation (18) is discretised as ˚    ∇ · ( φ ∇φ) dV ≈

φ, f ∇φ| f · n f A f , (22) V

f

where the diffusion coefficient φ at face f is defined by a harmonic average of the cell-centred values as [32] 2

φ, f = −1 , (23) −1

φ,P + φ,Q with cell Q the neighbour cell of cell P, as illustrated in Fig. 1. The face-centred gradient of φ along the normal vector n f is approximated with second-order accuracy as φQ − φP , (24) ∇φ| f · n f ≈ x where x is the mesh spacing. Applying the divergence theorem, the spatial gradient of φ averaged over the cell volume V P is readily computed as ∇φ P ≈

1  φf nf Af. VP f

(25)

188

F. Denner and B. van Wachem

5.2 Advecting Velocity The momentum-weighted interpolation (MWI) is applied to define an advecting velocity ϑ = u f · n f at cell faces, which is used in the discretised advection terms of the governing equations. MWI provides a robust pressure–velocity coupling for flows with low Mach numbers and flows of incompressible fluids by introducing a cell-to-cell pressure coupling and by applying a low-pass filter acting on the third derivative of pressure [7, 33], thus avoiding pressure–velocity decoupling due to the collocated variable arrangement. Following the unified formulation of the MWI proposed by Bartholomew et al. [7], the definition of the advecting velocity includes modifications to the original formulation of the MWI, as introduced by Rhie and Chow [59], to account for large density ratios and source terms occurring in interfacial flows, and the transient nature of the considered problems. As demonstrated by Bartholomew et al. [7], only the driving pressure gradient ∇ P = ∇ p − S, which is the pressure gradient associated with the flow field, should be coupled to the velocity field, whereas source terms and other external contributions should not be coupled to the velocity field. Taking this into account, the advecting velocity ϑ f at face f is given as 

 ρ ∗f ∇ PP ∇ PQ ˆ · nf + ϑ f = u f · n f − d f ∇ Pf · n f − 2 ρP ρQ  ρ ∗(t−t) f ˆ ϑ (t−t) − u(t−t) · nf . + df f f t

(26)

The first term on the right-hand side of Eq. (26) is the velocity perpendicular to the face f , with u f obtained by a linear interpolation of the velocities at the adjacent cell centres. The second term, which introduces the aforementioned low-pass filter, and the third term, which ensures a time-step independent solution, on the right-hand side of Eq. (26) are the correction terms introduced by the MWI, both weighted by the coefficient [7, 26]

VP VQ + eP eQ (27) dˆ f =

, ∗ ρf VP VQ 2+ + t e P eQ where e P and e Q are the sum of the coefficients of the primary variable u arising from the advection and shear stress terms of the momentum equations [22]. Equation (26) is strictly speaking only applicable to equidistant Cartesian meshes, as considered in this study, but the extension to unstructured meshes is straightforward [7]. The driving velocity gradient normal to the face f is approximated following Eq. (24) as PQ − PP , (28) ∇ Pf · n f ≈ x

A Unified Algorithm for Interfacial Flows …

189

providing a direct coupling between neighbouring nodal values of pressure. The driving pressure gradients at the cell centres are evaluated using Eq. (25) and weighted by the local density. The face density ρ ∗f is interpolated by a harmonic average, which is necessary for a consistent definition of the coefficient of the pressure term as well as the efficacy of the density weighting [7]. This density weighting applied to the cell-centred pressure gradient in Eq. (26) has been shown to yield robust results for flows with large and abrupt changes in density [7], demonstrated for incompressible interfacial flows with a density ratio of up to 1024 [26, 27]. The MWI formulation given in Eq. (26) is independent of the applied time-step and the error in kinetic energy introduced by the MWI is proportional to x 3 [7]. Hence, the convergence of the second-order accurate finite-volume method remains unaffected. Including the transient term, i.e. the third term on the right-hand side of Eq. (26), has previously been shown to be important for a correct transient evolution of pressure waves [7, 72], which is particularly pertinent for acoustic effects in compressible flows.

5.3 Discretised Governing Conservation Laws Since the discretisation of the governing conservation laws is identical for singlephase and interfacial flows, the following presentation of the discretisation of the governing conservation laws focuses on single-phase flows. The extension of this discretisation to interfacial flows by an appropriate definition of the fluid properties and using the ACID method is described in Sect. 6. Applying the numerical schemes described in the previous sections, the discretised continuity equation (1) for cell P is given as

 ∂ρ

VP + (ρ f ϑ f )(n+1) A f = 0,

∂t P f

(29)

where superscript (n + 1) denotes implicitly sought solutions. A Newton linearisation [20, 72] is applied to the advection term of the discretised continuity equation, given as (n+1) (n) (n) (ρ f ϑ f )(n+1) ≈ ρ (n) + ρ (n+1) ϑ (n) (30) f ϑf f f − ρf ϑf , where superscript (n) denotes the latest available solution. The Newton linearisation facilitates a smooth transition from low to high-Mach number regions [20, 68] and provides an implicit pressure-velocity coupling for flows at low Mach numbers and flows of incompressible fluids [22, 72]. The discretised momentum equations (2) of cell P are given as

190

F. Denner and B. van Wachem

  (n+1) ∂ρu j

VP + (ρ f ϑ f u j, f )(n+1) A f = − pf n j, f A f

∂t P

+

⎡

f

f

(n+1)  u (n+1) j,Q − u j,P μ∗f ⎣ x f

⎤

(n)

(n)

∂u i 2 ∂u k +

n i, f −

n i, f ⎦ A f + S j,P V P , ∂x j 3 ∂ xk f

f

(31) where  f denotes linear interpolation of the values at the adjacent cell centres and (n) (n+1) (n+1) (n) + ρ (n) u j, f (ρ f ϑ f u j, f )(n+1) ≈ ρ (n) f ϑ f u j, f f ϑf (n) (n) (n) (n) + ρ (n+1) ϑ (n) f f u j, f − 2 ρ f ϑ f u j, f .

(32)

The discretised energy equation (3) of cell P is given as

  TQ(n+1) − TP(n+1) ∂ρh

∂ p

(n+1) Af V + (ρ ϑ h ) A = V + k ∗f P f f f f P

∂t P ∂t x P f f ⎡⎛

(n)

(n) ⎞

(n) ⎤

 (n+1) ∂u ∂u ∂u 2 j i k

⎦ + u i, f μ∗f ⎣⎝ n j, f A f

+

⎠−

∂ xi ∂x j 3 ∂ xk f

+

(n+1) u i,P

f

f

f

Si,P V P , (33)

with (n) (n+1) (n+1) (n) (ρ f ϑ f h f )(n+1) ≈ ρ (n) + ρ (n) hf f ϑf h f f ϑf (n) (n) (n) (n) + ρ (n+1) ϑ (n) f f h f − 2ρf ϑf h f ,

and h (n+1) = c p T (n+1) +

u(n),2 . 2

(34)

(35)

The advecting velocity ϑ f is given by Eq. (26) and is the same in all equations, ensuring a consistent transport of the conserved quantities. Following the work of Ferziger [32], the viscosity μ∗f and the thermal conductivity k ∗f at face f are given by a harmonic average of the values at the adjacent cell centres. Previous studies [20, 29] have demonstrated substantial improvements with respect to the performance and stability of the numerical solution algorithm associated with a Newton linearisation of all governing equations compared to the more frequently applied Picard linearisation, i.e. (φη)(n+1) ≈ φ (n) η(n+1) . In particular for flows with large Mach numbers and with sharp changes in Mach number, such a linearisation enables a smooth transition from low- to high-Mach number regions [20, 43, 45]. To this end, the advection terms of the governing equations as well as

A Unified Algorithm for Interfacial Flows …

191

the transient terms of the momentum and energy equations are discretised using a Newton linearisation, see Eqs. (32) and (34), as described in more detail in [20].

5.4 Incompressible Limit For flows in the incompressible limit, with M → 0, the density is constant along the fluid particle trajectories [16], with ∂ρ Dρ = + u · ∇ρ = 0. Dt ∂t

(36)

Consequently, the continuity equation is no longer effective as a transport equation for density, but becomes a constraint on the velocity field with ∇ · u → 0 [16], enforced by pressure which acts as a Lagrange multiplier [64]. Inserting Eq. (36) into Eqs. (1)–(3), the governing conservation laws for flows in the incompressible limit reduce to ∇ · u = 0,

∂u + ∇ · (u ⊗ u) = −∇ p + ∇ · τ + S, ρ ∂t

∂p ∂h ρ + ∇ · (uh) = − ∇ · q + ∇ · (τ · u) + S · u. ∂t ∂t

(37) (38) (39)

Inserting ρ = ρ0 , see Eq. (11), into the discretised governing conservation laws presented above, the discretised continuity equation (29) reduces to 

ϑ f A f = 0,

(40)

f

the discretised momentum equations (31) become ⎛

⎞

 

∂u j ρP ⎝ VP + ϑ f u j, f A f ⎠ = − p f n j, f A f

∂t P

+

 f



μf

f

f



 ∂u j

∂u i

2 ∂u k

+ − n i, f A f + S j,P V P , ∂ xi f ∂x j f 3 ∂ xk f

and the discretised energy equation (33) becomes

(41)

192

F. Denner and B. van Wachem

⎛

⎞



 

∂h ∂ T ∂ p

n i, f A f ρ P ⎝ VP + ϑf h f Af⎠ = VP + kf

∂t P ∂t ∂ x i P f f f 



  ∂u j

∂u i

2 ∂u k

+ u i, f μ f + − n j, f A f + u i,P Si,P V P .

∂ xi f ∂x j f 3 ∂ xk f f

(42)

The transient terms are discretised in the same manner as in Sect. 5.3. These are precisely the governing conservation laws of the incompressible limit, Eqs. (37)– (39), discretised with the schemes as discussed above. In addition, Eqs. (40) and (41) are identical to the discretised continuity and momentum equations of the fully coupled pressure-based algorithm for incompressible interfacial flows of Denner et al. [26].

5.5 Solution Procedure The discretised governing conservation laws are solved simultaneously in a single linear system of equations, Aφ = b. Following Denner et al. [22], the system of equations is solved for the primary solution variables χ , which are the pressure p, the velocity vector u and the temperature T . For a three-dimensional computational mesh with N cells, the linear system of governing equations is given as ⎛ Ap ⎜B p ⎜ ⎜C p ⎜ ⎝D p Ep

Au Bu Cu Du Eu

Av Bv Cv Dv Ev

Aw Bw Cw Dw Ew

⎞ ⎛ ⎞ φp 0 ⎜ ⎟ 0⎟ ⎟ ⎜ φu ⎟ ⎜ ⎟ 0⎟ ⎟ · ⎜ φ v ⎟ = b, 0 ⎠ ⎝φ w ⎠ ET φT

(43)

where the coefficient submatrix Aχ of size N × N holds the coefficients of the primary variable χ associated with the continuity equation (29), submatrices Bχ , Cχ , Dχ , all of size N × N , hold the coefficients of the primary variable χ associated with the momentum equations (31), for the x-, y- and z-axes of the Cartesian coordinate system, and submatrix Eχ , also of size N × N , contains the coefficients of the primary variable χ associated with the energy equation (33). The subvector φ χ of length N holds the solution of the primary variable χ . All contributions from previous nonlinear iterations or previous time-levels are gathered in the right-hand side vector b of length 5N . The solution procedure performs nonlinear iterations in which the linear system of governing equations (43) is solved using the Block–Jacobi preconditioner and the BiCGSTAB solver of the software library PETSc [6], as described in detail by Denner et al. [29]. No underrelaxation of the governing equations is required.

A Unified Algorithm for Interfacial Flows …

193

6 Interface Treatment The discretised governing equations are extended to interfacial flows using an interface advection method (Sect. 6.1), an appropriate definition of the fluid properties in the interface region (Sect. 6.2) and the acoustically conservative interface discretisation (ACID)1 [29] (Sect. 6.3). In order to represent the two interacting fluids discretely, the indicator function ζ translates into a colour function ψ, defined for cell P as ˆ 1 ζ dV. (44) ψP = V P VP The interface is, thus, located in every cell with 0 < ψ < 1.

6.1 Interface Advection To advect the fluid interface between two fluids, Eq. (6) is applied to the colour function, Eq. (44), and reformulated as ∂ψ + ∇ · (uψ) − ψ ∇ · u = 0. ∂t

(45)

The advection of the fluid interface is then described by Eq. (45) using an appropriate discretisation method. Two different VOF methods, an algebraic VOF method [29] and a piecewise-linear interface calculation (PLIC) method with Lagrangian advection of the interface [69], are considered for the advection of the fluid interface. In the algebraic VOF method [29], the advection equation of the colour function, Eq. (45), is discretised using the Crank–Nicolson scheme for the discretisation of the transient term and the CICSAM scheme [66] for the spatial interpolation of the colour function ψ. In the applied VOF-PLIC method, the interface is reconstructed based on the local colour function ψ and the normal vector of the interface [61]. The interface advection equation (45) is then rewritten in integral form and the reconstructed interface is advected using the Lagrangian split advection scheme of van Wachem and Schouten [69]. In both considered interface advection methods, the advection is based on the same advecting velocity ϑ f as for all advection terms of the governing equations, thus ensuring an accurate prediction of volume changes [29]. Nevertheless, the finite-volume discretisation and pressure-based algorithm presented in Sect. 5 are not limited to the employed VOF methods and other methods to represent the bulk phases and advect the interface, including level-set and fronttracking methods, may equally be applied. 1

The term “acoustically-conservative” refers to the acoustic properties of this discretisation method in the context of fully compressible flows and is not indicative of its application to incompressible fluids.

194

F. Denner and B. van Wachem

6.2 Fluid Properties The definitions of the fluid properties largely follow the principles outlined by Denner et al. [29]. The density of the fluid is defined based on the colour function ψ as ρ = (1 − ψ) ρa + ψ ρb ,

(46)

where the partial densities ρa and ρb of the bulk phases are given by Eq. (14). This linear interpolation of the density is required in order to satisfy the discrete conservation of mass, momentum and energy. In the context of compressible flows, it is equivalent to an isobaric closure assumption [3, 62], while for incompressible fluids this formulation reduces to the typically used definition of density [12]. The heat capacity ratio also follows from the isobaric closure assumption as 1−ψ ψ 1 = Ca + Cb , γ −1 γ0,a − 1 γ0,b − 1

(47)

where Ca and Cb are the binary compressibility operators defined in Eq. (13) associated with fluid a and fluid b, respectively. The specific isobaric heat capacity is defined by a mass-weighted interpolation [29], which is essential for the conservation of the total energy, given as ρ c p = (1 − ψ) ρa c p,a + ψ ρb c p,b ,

(48)

where the partial densities ρa and ρb are given by Eq. (14), density ρ is given by Eq. (46), and the partial specific isobaric heat capacities c p,a and c p,b are given by Eq. (15). The viscosity μ and thermal conductivity k are defined as μ = (1 − ψ) μa + ψ μb , k = (1 − ψ) ka + ψ kb .

(49) (50)

The Continuum Surface Force (CSF) model of Brackbill et al. [12] is applied to model the force due to surface tension, represented by the source term S = σ κ ∇ψ,

(51)

where κ is the interface curvature, which is computed using a second-order height function method [30], and σ is the surface tension coefficient. In the interest of conciseness, but without loss of generality, the surface tension coefficient is assumed to be constant.

A Unified Algorithm for Interfacial Flows …

195

6.3 Coupling of the Bulk Phases The ACID method [29] assumes that, for the purpose of discretising the governing conservation laws for a given cell, all cells in its finite-volume stencil are assigned the same colour function value, i.e. the colour function is assumed to be constant in the entire finite-volume stencil. The relevant thermodynamic properties that are discontinuous at the interface, i.e. density and enthalpy, are then evaluated based on this locally constant colour function field, as described by Denner et al. [29] in detail. This recovers the contact discontinuity associated with the interface [4, 63] and enables the application of the conservative discretisation described in Sect. 5, identical to the one applied for single-phase flows. Denner et al. [29] reported robust and accurate results for acoustic and shock waves in interfacial flows, supporting the notion that the interface discretisation indeed conserves the acoustic features of the flow and retains the conservative discretisation of the governing equations.

6.3.1

Density Treatment

Under the assumption that the colour function ψ is constant throughout the finitevolume stencil of cell P, as proposed by Denner et al. [29], the density interpolated to face f of cell P is given as ρ f = ρU +

 ξf   ρ D − ρU , 2

(52)

where ξ f is the flux limiter determined by the applied differencing scheme, e.g. a TVD differencing scheme [28]. The density ρU at the upwind cell U and ρ D at the downwind cell D are given based on the colour function value of cell P by Eq. (46) so that   (53) ρU = ρa,U + ψ P ρb,U − ρa,U , and

  ρ D = ρa,D + ψ P ρb,D − ρa,D .

(54)

The densities at previous time-levels are evaluated in a similar fashion based on the colour function value of cell P, with  (t−t) (t) (t−t) (t−t) ρ , (55) = ρ + ψ − ρ ρ (t−t) P a,P P b,P a,P and analogously for ρ (t−2t) . P

196

6.3.2

F. Denner and B. van Wachem

Enthalpy Treatment

The specific total enthalpy at face f is given, again assuming the colour function ψ is constant throughout the finite-volume stencil of cell P, as [29] ρ f h f = ρU h U +

 ξf    ρ D h D − ρU h U , 2

(56)

with ρ f given by Eq. (52). The specific total enthalpy of the upwind and downwind cells are uU2 , 2 u2 h D = cp,D TD + D , 2 h U = cp,U TU +

(57) (58)

respectively, ρU is given by Eq. (53) and ρ D is given by Eq. (54). The specific isobaric heat capacities cp,U and cp,D are defined by Eq. (48) with ψ P as

and

ρU cp,U = ρa,U c p,a,U + ψ P (ρb,U c p,b,U − ρa,U c p,a,U ),

(59)

ρ D cp,D = ρa,D c p,a,D + ψ P (ρb,D c p,b,D − ρa,D c p,a,D ).

(60)

Since the specific enthalpy is partially sought implicitly, formulated implicit with respect to the primary solution variable temperature, see Eq. (35), a deferred correction approach is applied to enforce Eq. (56) [29]. The specific total enthalpy at the previous time-levels is given as = c,(t−t) TP(t−t) + h (t−t) P p,P

u(t−t),2 P , 2

(61)

with  ,(t−t) (t−t) (t−t) (t) (t−t) (t−t) (t−t) (t−t) ρ (t−t) ρ c = ρ c + ψ c − ρ c P p,P a,P p,a,P P b,P p,b,P a,P p,a,P , (62) and analogously for h (t−2t) and c,(t−2t) . P p,P 6.3.3

Further Observations

The corrections applied by ACID to the discretised governing equations vanish in the bulk phases and are non-zero only at fluid interfaces [29]. The fluid properties are piecewise constant at the interface and the corresponding error is proportional to x, as commonly found in VOF, level-set and front-tracking methods [57]. With respect

A Unified Algorithm for Interfacial Flows …

197

to the different fluid combinations that can occur at the interface, the following can be observed: • At compressible–compressible interfaces, problems associated with a discontinuous change of fluid properties are circumvented with ACID, while retaining the information carried by compression and expansion waves, as comprehensively demonstrated by Denner et al. [29]. Hence, acoustic waves, shock waves and rarefaction fans can interact with and at the interface. The proposed algorithm then becomes an enhanced version of the algorithm for compressible interfacial flows of Denner et al. [29], including viscous stresses and surface tension. • At incompressible–incompressible interfaces, ACID retains the incompressible formulation of the discretised governing equations. In fact, the non-conservative formulation of the governing equations at the interface originally proposed by Brackbill et al. [12] is obtained. The proposed algorithm then reduces to a nonisothermal version of the fully coupled algorithm for incompressible interfacial flows of Denner and van Wachem [26], as for instance used in [27]. • At compressible–incompressible interfaces, the compressibility of the compressible fluid stays, dependent on the local colour function value, partly intact and ACID provides a transition from the compressible fluid to the incompressible fluid. Compression and expansion waves are able to interact with such a compressible– incompressible interface, but compressible effects are not transmitted into the incompressible fluid.

7 Validation As already discussed, for incompressible–incompressible interfacial flows, the proposed algorithm is equivalent to the incompressible algorithm of Denner and van Wachem [26], which has been extensively tested and validated against analytical solutions [24, 27], experiments [21, 25] and other numerical methods [24]. For compressible-compressible interfacial flows, the proposed algorithm becomes the compressible algorithm of Denner et al. [29], which has been validated thoroughly with a broad range of test cases. Thus, the validation presented below focuses on the application of the proposed algorithm to predict compressible–incompressible interfacial flows. Five representative test cases are considered to demonstrate the application of the proposed algorithm to compressible–incompressible interfacial flows: a bubble with surface tension in equilibrium (Sect. 7.1), the viscous damping of capillary waves (Sect. 7.2), the reflection of an acoustic wave at a fluid interface (Sect. 7.3), the pressure-driven collapse of a bubble (Sect. 7.4) and the shock interaction with a water drop (Sect. 7.5).

198

F. Denner and B. van Wachem

7.1 Bubble in Equilibrium A circular bubble of a compressible gas surrounded by an incompressible fluid is simulated to study the parasitic currents associated with the numerical treatment of the surface tension contribution. Similar to the test cases used in previous studies [35, 56], the considered two-dimensional bubble has a diameter of dB = 0.8 m and a surface tension coefficient of σ = 1.0 N m−1 . The ambient pressure is p0 = 1.0 Pa, the incompressible fluid has a density of ρ0,f = 1.0 kg m−3 and the compressible gas has a density of ρg = 1.0 kg m−3 at p0 . The compressible gas has a heat capacity ratio of γ0 = 1.4 and both fluids have a specific isobaric heat capacity of c p,0 = 1008 J kg−1 K−1 . The viscosity μ, which is the same for both fluids, follows from the considered Laplace number, La = ρ0,f dB σ/μ2 = 120. The bubble is placed at the centre of a domain with edge length 2 m × 2 m, represented by an equidistant Cartesian mesh with 64 × 64 mesh cells, and the applied time-step is t = 10−3 s, which satisfies the capillary time-step constraint [27]. The fluid interface is advected using the VOF-PLIC method.  Figure 2 shows the total kinetic energy E kin = NP=1 ρ P u2P /2, with N the total number of mesh cells, in the computational domain as a function of the dimensionless time τ = t/tμ , with tμ = ρ0,f dB2 /μ the viscous timescale. Since the bubble is in equilibrium, the observed kinetic energy is the result of parasitic currents only. The imbalance caused by errors in the numerical evaluation of the interface curvature leads to an initial production of parasitic currents and the associated kinetic energy. As the interface topology relaxes towards a numerical equilibrium, the production of parasitic currents diminishes and the existing parasitic currents dissipate as a result of viscous stresses. Consequently, the kinetic energy in the domain decays rapidly and the exact balance is recovered on the discrete level.

10

−2

10−4 10−6 Ekin [J]

Fig. 2 Kinetic energy E kin as a result of parasitic currents as a function of dimensionless time τ = t/tμ , with tμ = ρ0,f dB2 /μ the viscous timescale, of the bubble in equilibrium. The gas inside the bubble is compressible, whereas the fluid surrounding the bubble is incompressible

10−8 10−10 10−12 10−14 −16

10

−18

10

0

0.5

1 τ

1.5

2

A Unified Algorithm for Interfacial Flows …

199

7.2 Capillary Waves To demonstrate the accurate prediction of viscous flows and surface tension by the proposed algorithm, the oscillation of a standing capillary wave between two viscous fluids is simulated. This test case is particularly challenging because the wave amplitude is small and the temporal evolution of the wave amplitude, governed by the dispersion (due to surface tension) and attenuation (due to viscous stresses) of the capillary wave, is very sensitive to the implementation of the viscous stress terms, the surface tension model, numerical diffusion and spurious oscillations [24, 56]. The analytical solution for the initial value problem of a freely oscillating capillary wave with infinitesimal amplitude, derived by Prosperetti [58], which is valid for two-phase systems in which both fluids have the same kinematic viscosity ν = μ/ρ or in which one fluid is neglected, serves as a reference. The considered capillary wave has a wavelength of λ = 0.1 m, both fluids have a kinematic viscosity of ν = 1.6394 × 10−4 m2 s−1 and uniform initial temperature T0 = 300 K, and the surface tension coefficient is σ = 0.25 π −3 N m−1 . The density of the two fluids are ρa = 1 kg m−3 and ρb = 16 kg m−3 . The computational domain has the dimensions λ × 3λ and is represented by an equidistant Cartesian mesh with x = λ/32. A compressible–compressible flow, with γ0 = 1.4, as well as a compressible–incompressible flow (fluid ’b’ is considered to be incompressible) are considered. The fluid interface is advected using the algebraic VOF method. Figure 3 shows the dimensionlesswave amplitude A/A0 as a function of the dimensionless time tω0 , where ω0 = σ k 3 /(ρa + ρb ) is the undamped angular frequency of the capillary wave and k = 2π/λ is the wavenumber, with initial wave amplitude A0 = 0.01λ, obtained for both flow types. In both cases, the computed result is in excellent agreement with the analytical solution of Prosperetti [58], with respect to the amplitude as well as the frequency.

1

0

−0.5 −1

Simulation Theory

0.5 A/A0

0.5 A/A0

1

Simulation Theory

0

−0.5 0

5

10

tω 0

15

20

(a) Compressible-compressible flow

25

−1

0

5

10

tω0

15

20

25

(b) Compressible-incompressible flow

Fig. 3 Temporal evolution of the dimensionless amplitude A/A0 of a capillary wave with wavelength λ = 0.1 m and initial amplitude A0 = 0.01λ as a function of dimensionless time tω0 , obtained for a compressible–compressible flow and a compressible–incompressible flow. The results are compared against the analytical solution of Prosperetti [58]

200

F. Denner and B. van Wachem

7.3 Acoustic Waves The propagation of a single acoustic wave in an air-water flow, where air is treated as a compressible fluid and water is taken to be either a compressible fluid described by the stiffened-gas model or an incompressible fluid with constant density, is simulated in a one-dimensional domain with mesh spacing x = 2 × 10−3 m, with initial velocity u 0 = 1 m s−1 , initial pressure p0 = 105 Pa and initial temperature T0 = 300 K. Air has the fluid properties γ0,Air = 1.4 and 0,Air = 0 Pa, and the density at initial conditions is ρAir = 1.16 kg m−3 . Compressible water has the fluid properties γ0,Water = 4.1, 0,Water = 4.4 × 108 Pa with a density at initial conditions of ρWater = 1000 kg m−3 . Incompressible water has a constant density of ρ0,Water = 1000 kg m−3 . The acoustic wave is initiated by the inlet-velocity

⎧ ⎨u + u sin 2π f t + 3 π if t < f −1 0 0 2 u in = , ⎩ u 0 − u 0 if t ≥ f −1

(63)

with u 0 the amplitude and f = 2500 s−1 the frequency of the velocity perturbation. Two perturbation amplitudes are considered, u 0 ∈ {0.01 u 0 , 100 u 0 }, to study the linear and nonlinear acoustic regimes. The applied time-step t satisfies an acoustic Courant number of Coa = aWater t/x = 0.34, in order to adequately resolve the acoustic waves. However, contrary to explicit algorithms, Coa ≤ 1 is not a necessary requirement for the stability of the proposed implicit algorithm. In the linear acoustic regime (u a0 and ρ ρ0 ), which applies to the considered velocity perturbation amplitude of u 0 = 0.01 u 0 , the perturbation leads to a sound wave. Based on linear acoustic theory [4], the acoustic wave reflected at the air–water interface should have a pressure amplitude of reflected pAir,0 =

incident pAir,0 , 2Z Water −1 Z Water − Z Air

(64)

incident = Z Air u 0 is where Z = ρa is the characteristic acoustic impedance and pAir,0 the pressure amplitude of the incident wave. While the theoretical pressure amplireflected incident = pAir,0 if water is considered to be an tude of the reflected wave is pAir,0 incompressible fluid (ρWater = const., aWater = ∞), the reflected pressure amplireflected incident

0.999 pAir,0 if water is considered to be a compressible tude is pAir,0 fluid (ρWater = f ( p, T ), aWater = 1343 m s−1 ). Hence, the differences in the results between the incompressible and compressible treatments of water should be negligible, at least in the linear acoustic regime, apart from the transmitted wave that should only be present for the compressible treatment of water. Figure 4 shows the pressure profile of the acoustic waves, after they have interacted with the interface, for both considered perturbation amplitudes using both the compressible treatment and the incompressible treatment of the liquid phase. For the

A Unified Algorithm for Interfacial Flows …

201

Fig. 4 Pressure profile of acoustic waves with different amplitudes in a one-dimensional domain after the interaction with an air–water interface at t = 2 ms, in which air (left of the interface) is treated as a compressible fluid and water (right of the interface) is treated as an incompressible fluid (aWater = ∞) or as a compressible fluid (aWater = 1343 m s−1 ). The amplitude of the applied initial velocity perturbation is (a) u 0 = 0.01 u 0 and (b) u 0 = 100 u 0 . The pressure amplitudes of the reflected and transmitted waves according to linear acoustic theory are shown in (a) as a reference

small perturbation amplitude, u 0 = 0.01 u 0 , the amplitude of the reflected pressure wave is in excellent agreement with linear acoustic theory with both treatments of the liquid phase. When the liquid phase is treated as a compressible fluid, a pressure wave is transmitted through the interface, while no wave is transmitted when the liquid phase is treated as an incompressible fluid, as expected. For the large perturbation amplitude, u 0 = 100 u 0 , the pressure profile of the acoustic wave departs significantly from its originally sinusoidal shape as a result of nonlinear wave steepening. Nevertheless, the reflected waves predicted by the compressible treatment and the incompressible treatment of the liquid phase are in excellent agreement, suggesting that the compressible–incompressible treatment of gas-liquid flows can be applied even when large amplitude acoustic waves are present in the gas phase.

202

F. Denner and B. van Wachem

7.4 Bubble Collapse The pressure-driven growth and collapse of gas bubbles, in particular cavitation, is a widely observed phenomenon, and the ability to focus large amounts of energy by a collapsing bubble is being utilised in an increasing number of applications. The simplest analytical model for the pressure-driven collapse of a bubble in a liquid is the Rayleigh–Plesset equation [55] without considering viscous dissipation and surface tension, given as pg − p∞ 3 , (65) R R¨ + R˙ 2 = 2 ρ0, where R is the bubble radius, ρ0, is the (constant) density of the liquid, pg is the (uniform) gas pressure inside the bubble and p∞ is the ambient liquid pressure at infinite distance from the bubble. Interestingly, Eq. (65) is based on the assumption of an incompressible liquid, with ρ0, = const., and a compressible gas bubble. The collapse of a spherical gas bubble due to an overpressure in the liquid, the so-called Rayleigh collapse [46], is considered to validate the prediction of pressuredriven flows of a compressible gas in contact with an incompressible liquid by the proposed algorithm. Following the set up of Denner et al. [23], a bubble with initial radius R0 and an initial gas pressure of pg = 4000 Pa is situated in a liquid with an initial pressure of p (r ) = p∞ + ( pg − p∞ ) R0 /r , with p∞ = 105 Pa and r the radial coordinate. The gas has a heat capacity ratio of γ0 = 1.4 and the liquid has a constant density of ρ0, = 1000 kg m−3 . Viscosity, thermal conduction and surface tension are neglected. The fluid interface is advected using the algebraic VOF method. Because the liquid is incompressible and viscous dissipation and thermal conduction are neglected, the oscillations of the bubble radius should continue indefinitely with unchanged frequency and amplitude. Figure 5 shows the dimensionless  radius R/R0 as a function of the dimensionless time t/tR , where tR = 0.915 R0 ρ0, / p∞ is the Rayleigh collapse time, obtained with different spatial and temporal resolutions using the proposed algorithm. The solution of the Rayleigh–Plesset equation (65) is also shown as a reference and regarded here as the exact solution of the dynamic bubble behaviour under consideration. The results predicted by the proposed algorithm and by the Rayleigh–Plesset equation are in excellent agreement, as shown in Fig. 5, for a sufficiently large spatial and temporal resolution.

7.5 Shock–Drop Interaction Following the work of Meng and Colonius [50], the interaction of a shock wave, initially travelling in air with Mach number Ms = 1.47, with a water drop is simulated. The two-dimensional domain is represented by an equidistant Cartesian mesh with 100 cells per drop diameter. The shock wave separates the post-shock region

A Unified Algorithm for Interfacial Flows …

203

Fig. 5 Temporal evolution of the dimensionless radius R/R0 of a collapsing gas bubble in an incompressible liquid, compared against the solution of the Rayleigh–Plesset equation (65). a Results for different time-steps t obtained with a spatial resolution of x = R0 /400; b Results −4 with different mesh spacings x obtained with  a time-step of t = 10 tR . Time t is normalised by the Rayleigh collapse time, tR = 0.915 R0 ρ0, / p∞

(I) and the pre-shock region (II), which are initialised with u I = 246.24 m s−1 , pI = 2.35 × 105 Pa, TI = 450.56 K and u II = 0 m s−1 , pII = 1.00 × 105 Pa, TII = 347.22 K, respectively. The fluid properties of air are γ0,Air = 1.4 and 0,Air = 0 Pa, with ρII,Air = 1 kg m−3 and aII,Air = 374.17 m s−1 in the pre-shock region. The liquid drop is treated as an incompressible fluid, with constant density ρ0,Water = 1000 kg m−3 , or as a compressible fluid, with the fluid properties γ0,Water = 4.4 and 0,Water = 6 × 108 Pa, and ρII,Water = 1000 kg m−3 and aII,Water = 1624.94 m s−1 in the pre-shock region. The fluid interface is advected using the algebraic VOF method. Figure 6 shows the contours of the velocity magnitude |u| and the density gradient |∇ρ| obtained t = 16 μs after the first shock–drop interaction, treating water as an incompressible or a compressible fluid. While the velocity profile and the shock structures are virtually identical in the gas phase, the compressible treatment of the water drop allows density waves to propagate in it, while no density waves are propagating in the incompressible drop. With respect to the execution time of the simulations, the incompressible treatment of the drop achieves a significant speed-up of the simulation, because the stiff pressure–density coupling imposed by the stiffened-gas model for the compressible liquid, and the associated decrease in convergence rate, can be circumvented with the incompressible treatment. In addition, a larger time-step can be applied with the incompressible treatment of the liquid, because no shock waves have to be resolved in the liquid. As a result, simply switching to an incompressible treatment of the water drop accelerates the simulation by factor 1.8, and also adjusting the applied time-step brings an overall acceleration of the simulation of factor 3.9.

204

F. Denner and B. van Wachem

(a) Velocity

(b) Density gradient

Fig. 6 Contours of the velocity magnitude |u| and the density gradient |∇ρ| after a shock with Ms = 1.47 interacted with a two-dimensional water drop, simulating the water drop as an incompressible fluid (upper half) and as a compressible fluid (lower half). The density waves inside the compressible drop are clearly visible, while no density waves are transmitted to the incompressible drop

Although the compressibility of the liquid will undoubtedly have a considerable influence on the complex behaviour of the drop after the interaction with the shock wave, see e.g. [50], this test case demonstrates the robustness of the proposed algorithm.

8 Conclusions In this chapter, we have presented a new and promising route to construct numerical algorithms for the simulation of both compressible and incompressible interfacial flows, as well as mixtures thereof, based on a unified thermodynamic closure model, a finite-volume discretisation and a fully coupled pressure-based algorithm. The proposed thermodynamic closure model and treatment of the fluid properties at the interface, in conjunction with the ACID method to couple the interacting bulk phases [29] and the Newton linearisation of the governing equations [20], bridge the different numerical requirements for the simulation of incompressible and compressible fluids, and facilitate the simulation of compressible–incompressible interfacial flows. If, however, only incompressible fluids or only compressible fluids are considered, the presented algorithm is equivalent to previously proposed pressure-based algorithms for incompressible interfacial flows [26] or compressible interfacial flows [29], respectively. The proposed algorithm has been successfully validated using five representative test cases, each featuring a combination of a compressible fluid and an incompressible fluid separated by an interface: a bubble with surface tension in equilibrium, the

A Unified Algorithm for Interfacial Flows …

205

viscous damping of capillary waves, the reflection of an acoustic wave at a gas–liquid interface, the pressure-driven collapse of a bubble and the shock interaction with a water drop. In particular, the presented simulations of interactions of an acoustic wave with a compressible–incompressible interface and of a high-Mach compressible flow with an incompressible fluid (i.e. the shock–drop interaction) are unique capabilities that have not been demonstrated in the literature before. In addition, treating liquids as incompressible, while still resolving acoustic effects in the gas phase, has been shown to yield substantial performance benefits. For gas–liquid flows in general, the strength of the proposed algorithm lies in the reliable prediction of compressible effects in the gas phase, without paying the additional cost of resolving marginal physical effects in the liquid phase, if the application allows this. For instance, acoustic waves in the gas phase are known to promote interfacial instabilities [74], which however have a negligible influence on the behaviour of the liquid phase in subsonic flows; especially, the compressibility of the liquid does not influence the acoustics in the gas phase, as shown by the presented results. With the presented algorithm the simplification of an incompressible fluid cannot only be invoked for low-Mach flows with respect to the gas phase, as considered in previous studies [10, 14, 73], but practically for any flow velocity. While a compressible liquid is not a valid simplification for large liquid Mach numbers, for subsonic gas flows, for instance, in subsonic fuel injection and spray atomisation processes, the presented results suggest an incompressible liquid to be a reasonable assumption. Acknowledgements This research was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), grant numbers 420239128 and 447633787.

References 1. Aanjaneya, M., Patkar, S., Fedkiw, R.: A monolithic mass tracking formulation for bubbles in incompressible flow. J. Comput. Phys. 247, 17–61 (2013) 2. Abgrall, R., Karni, S.: Computations of compressible multifluids. J. Comput. Phys. 169, 594– 623 (2001) 3. Allaire, G., Clerc, S., Kokh, S.: A five-equation model for the simulation of interfaces between compressible fluids. J. Comput. Phys. 181(2), 577–616 (2002) 4. Anderson, J.D.: Modern Compressible Flow: With a Historical Perspective. McGraw-Hill, New York (2003) 5. Baer, M., Nunziato, J.: A two-phase mixture theory for the deflagration-to-detonation transition (ddt) in reactive granular materials. Int. J. Multiphase Flow 12(6), 861–889 (1986) 6. Balay, S., Abhyankar, S., Adams, M.F., Brown, J., Brune, P., Buschelman, K., Dalcin, L., Eijkhout, V., Kaushik, D., Knepley, M.G., May, D.A., McInnes, L.C., Gropp, W.D., Rupp, K., Sanan, P., Smith, B.F., Zampini, S., Zhang, H., Zhang, H.: PETSc Users Manual. Tech. Rep. ANL-95/11 - Revision 3.8, Argonne National Laboratory (2017) 7. Bartholomew, P., Denner, F., Abdol-Azis, M., Marquis, A., van Wachem, B.: Unified formulation of the momentum-weighted interpolation for collocated variable arrangements. J. Comput. Phys. 375, 177–208 (2018) 8. Bell, J.B., Colella, P., Glaz, H.M.: A second-order projection method for the incompressible Navier-Stokes equations. J. Comput. Phys. 85(2), 257–283 (1989)

206

F. Denner and B. van Wachem

9. Bijl, H., Wesseling, P.: A unified method for computing incompressible and compressible flows in boundary-fitted coordinates. J. Comput. Phys. 141, 153–173 (1998) 10. Billaud, M., Gallice, G., Nkonga, B.: A simple stabilized finite element method for solving two phase compressible-incompressible interface flows. Comput. Methods Appl. Mech. Eng. 200(9–12), 1272–1290 (2011) 11. Boger, M., Jaegle, F., Weigand, B., Munz, C.D.: A pressure-based treatment for the direct numerical simulation of compressible multi-phase flow using multiple pressure variables. Comput. Fluids 96, 338–349 (2014) 12. Brackbill, J., Kothe, D., Zemach, C.: Continuum method for modeling surface tension. J. Comput. Phys. 100, 335–354 (1992) 13. Caiden, R., Fedkiw, R., Anderson, C.: A numerical method for two-phase flow consisting of separate compressible and incompressible regions. J. Comput. Phys. 166, 1–27 (2001) 14. Caltagirone, J.P., Vincent, S., Caruyer, C.: A multiphase compressible model for the simulation of multiphase flows. Comput. Fluids 50(1), 24–34 (2011) 15. Chorin, A.J.: A numerical method for solving incompressible viscous flow problems. J. Comput. Phys. 2(1), 12–26 (1967) 16. Chorin, A.J., Marsden, J.E.: A Mathematical Introduction to Fluid Mechanics. Springer, Berlin (1993) 17. Coralic, V., Colonius, T.: Finite-volume WENO scheme for viscous compressible multicomponent flows. J. Comput. Phys. 274, 95–121 (2014) 18. Cordier, F., Degond, P., Kumbaro, A.: An asymptotic-preserving all-speed scheme for the Euler and Navier-Stokes equations. J. Comput. Phys. 231(17), 5685–5704 (2012) 19. Darwish, M., Moukalled, F.: A fully coupled Navier-Stokes solver for fluid flow at all speeds. Numer. Heat Tr. B-Fund. 65(5), 410–444 (2014) 20. Denner, F.: Fully-coupled pressure-based algorithm for compressible flows: linearisation and iterative solution strategies. Comp. Fluids 175, 53–65 (2018) 21. Denner, F., Charogiannis, A., Pradas, M., Markides, C.N., van Wachem, B., Kalliadasis, S.: Solitary waves on falling liquid films in the inertia-dominated regime. J. Fluid Mech. 837, 491–519 (2018) 22. Denner, F., Evrard, F., van Wachem, B.: Conservative finite-volume framework and pressurebased algorithm for flows of incompressible, ideal-gas and real-gas fluids at all speeds. J. Comput. Phys. 409, 109348 (2020) 23. Denner, F., Evrard, F., van Wachem, B.: Modeling acoustic cavitation using a pressure-based algorithm for polytropic fluids. Fluids 5(2), 69 (2020) 24. Denner, F., Paré, G., Zaleski, S.: Dispersion and viscous attenuation of capillary waves with finite amplitude. Euro. Phys. J. Spec. Top. 226, 1229–1238 (2017) 25. Denner, F., van Wachem, B.: Compressive VOF method with skewness correction to capture sharp interfaces on arbitrary meshes. J. Comput. Phys. 279, 127–144 (2014) 26. Denner, F., van Wachem, B.: Fully-coupled balanced-force VOF framework for arbitrary meshes with least-squares curvature evaluation from volume fractions. Numer. Heat Tr. BFund. 65(3), 218–255 (2014) 27. Denner, F., van Wachem, B.: Numerical time-step restrictions as a result of capillary waves. J. Comput. Phys. 285, 24–40 (2015) 28. Denner, F., van Wachem, B.: TVD differencing on three-dimensional unstructured meshes with monotonicity-preserving correction of mesh skewness. J. Comput. Phys. 298, 466–479 (2015) 29. Denner, F., Xiao, C.N., van Wachem, B.: Pressure-based algorithm for compressible interfacial flows with acoustically-conservative interface discretisation. J. Comput. Phys. 367, 192–234 (2018) 30. Evrard, F., Denner, F., van Wachem, B.: Height-function curvature estimation with arbitrary order on non-uniform Cartesian grids. J. Comput. Phys.: X 7, 100060 (2020) 31. Fedkiw, R., Aslam, T., Merriman, B., Osher, S.: A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comput. Phys. 152(2), 457–492 (1999) 32. Ferziger, J.: Interfacial transfer in Tryggvason’s method. Int. J. Numer. Methods Fluids 41, 551–560 (2003)

A Unified Algorithm for Interfacial Flows …

207

33. Ferziger, J., Peri´c, M.: Computational Methods for Fluid Dynamics, 3rd edn. Springer, Berlin (2002) 34. Ferziger, J.H., Peric, M., Street, R.L.: Computational Methods for Fluid Dynamics, 4th edn. Springer International Publishing, Berlin (2020) 35. Fuster, D., Popinet, S.: An all-Mach method for the simulation of bubble dynamics problems in the presence of surface tension. J. Comput. Phys. 374, 752–768 (2018) 36. Garrick, D.P., Owkes, M., Regele, J.D.: A finite-volume HLLC-based scheme for compressible interfacial flows with surface tension. J. Comput. Phys. 339, 46–67 (2017) 37. Harlow, F., Amsden, A.: Fluid Dynamics. Monograph LA-4700, Los Alamos National Laboratory (1971) 38. Harlow, F.H., Amsden, A.A.: A numerical fluid dynamics calculation method for all flow speeds. J. Comput. Phys. 8(2), 197–213 (1971) 39. Hauke, G., Hughes, T.J.: A comparative study of different sets of variables for solving compressible and incompressible flows. Comput. Methods Appl. Mech. Eng. 153(1–2), 1–44 (1998) 40. Hirt, C., Nichols, B.: Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys. 39(1), 201–225 (1981) 41. Hou, T.Y., Floch, P.G.L.: Why nonconservative schemes converge to wrong solutions: error analysis. Math. Comput. 62(206), 497–530 (1994) 42. Issa, R.: Solution of the implicitly discretised fluid flow equations by operator-splitting. J. Comput. Phys. 62, 40–65 (1985) 43. Karimian, S.M.H., Schneider, G.E.: Pressure-based computational method for compressible and incompressible flows. J. Thermophys. Heat Trans. 8(2), 267–274 (1994) 44. Karki, K.C., Patankar, S.V.: Pressure based calculation procedure for viscous flows at all speeds in arbitrary configurations. AIAA J. 27(9), 1167–1174 (1989) 45. Kunz, R., Cope, W., Venkateswaran, S.: Development of an implicit method for multi-fluid flow simulations. J. Comput. Phys. 152(1), 78–101 (1999) 46. Lauterborn, W., Lechner, C., Koch, M., Mettin, R.: Bubble models and real bubbles: Rayleigh and energy-deposit cases in a Tait-compressible liquid. IMA J. Appl. Math. 83(4), 556–589 (2018) 47. Le Métayer, O., Massoni, J., Saurel, R.: Élaboration des lois d’état d’un liquide et de sa vapeur pour les modèles d’écoulements diphasiques. Int. J. Therm. Sci. 43(3), 265–276 (2004) 48. Liu, C., Hu, C.: Adaptive THINC-GFM for compressible multi-medium flows. J. Comput. Phys. 342, 43–65 (2017) 49. Liu, T., Khoo, B., Yeo, K.: Ghost fluid method for strong shock impacting on material interface. J. Comput. Phys. 190(2), 651–681 (2003) 50. Meng, J.C., Colonius, T.: Numerical simulations of the early stages of high-speed droplet breakup. Shock Waves 25(4), 399–414 (2015) 51. Moukalled, F., Mangani, L., Darwish, M.: The Finite Volume Method in Computational Fluid Dynamics: An Advanced Introduction with OpenFOAM and Matlab. Springer, Berlin (2016) 52. Murrone, A., Guillard, H.: A five equation reduced model for compressible two phase flow problems. J. Comput. Phys. 202(2), 664–698 (2005) 53. Park, J.H., Munz, C.D.: Multiple pressure variables methods for fluid flow at all Mach numbers. Int. J. Numer. Methods Fluids 49(8), 905–931 (2005) 54. Patankar, S.: Numerical Heat Transfer and Fluid Flow. Hemisphere Publishing Company (1980) 55. Plesset, M.S.: The dynamics of cavitation bubbles. J. App. Mech. 16, 277–282 (1949) 56. Popinet, S.: An accurate adaptive solver for surface-tension-driven interfacial flows. J. Comput. Phys. 228(16), 5838–5866 (2009) 57. Popinet, S.: Numerical models of surface tension. Annu. Rev. Fluid Mech. 50, 49–75 (2018) 58. Prosperetti, A.: Motion of two superposed viscous fluids. Phys. Fluids 24(7), 1217–1223 (1981) 59. Rhie, C.M., Chow, W.L.: Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA J. 21(11), 1525–1532 (1983) 60. Rohde, C., Zeiler, C.: A relaxation Riemann solver for compressible two-phase flow with phase transition and surface tension. Appl. Numer. Math. 95, 267–279 (2015)

208

F. Denner and B. van Wachem

61. Scardovelli, R., Zaleski, S.: Analytical relations connecting linear interfaces and volume fractions in rectangular grids. J. Comput. Phys. 164(1), 228–237 (2000) 62. Shyue, K.M.: A volume-fraction based algorithm for hybrid barotropic and non-barotropic two-fluid flow problems. Shock Waves 15(6), 407–423 (2006) 63. Toro, E.F.: Riemann Solvers and Numerical Fluid Dynamics: A Practical Introduction, 3rd edn. Springer (2009) 64. Toutant, A.: General and exact pressure evolution equation. Phys. Lett. A 381(44), 3739–3742 (2017) 65. Turkel, E., Fiterman, A., van Leer, B.: Preconditioning and the Limit to the Incompressible Flow Equations. Tech. rep., NASA CR-191500 (1993) 66. Ubbink, O., Issa, R.: A method for capturing sharp fluid interfaces on arbitrary meshes. J. Comput. Phys. 153, 26–50 (1999) 67. van der Heul, D., Vuik, C., Wesseling, P.: A conservative pressure-correction method for flow at all speeds. Comp. Fluids 32(8), 1113–1132 (2003) 68. Van Doormaal, J., Raithby, G., McDonald, B.: The segregated approach to predicting viscous compressible fluid flows. J. Turbomach. 109(April 1987), 268–277 (1987) 69. van Wachem, B., Schouten, J.: Experimental validation of 3-D Lagrangian VOF model: Bubble shape and rise velocity. AIChE J. 48(12), 2744–2753 (2002) 70. Wadhwa, A.R., Abraham, J., Magi, V.: Hybrid compressible-incompressible numerical method for transient drop-gas flows. AIAA J. 43(9), 1974–1983 (2005) 71. Wesseling, P.: Principles of Computational Fluid Dynamics. Springer, Berlin (2001) 72. Xiao, C.N., Denner, F., van Wachem, B.: Fully-coupled pressure-based finite-volume framework for the simulation of fluid flows at all speeds in complex geometries. J. Comput. Phys. 346, 91–130 (2017) 73. Yamamoto, T., Takatani, K.: Pressure-based unified solver for gas-liquid two-phase flows where compressible and incompressible flows coexist. Int. J. Numer. Methods Fluids (2018) 74. Zhou, Z.W., Lin, S.P.: Effects of compressibility on the atomization of liquid jets. J. Propuls. Power 8(4), 736–740 (1992)

Stabilized Finite Element Formulation and High-Performance Solver for Slightly Compressible Navier–Stokes Equations Feimi Yu and Lucy T. Zhang

1 Introduction Capturing the acoustic perturbation in flow fields is difficult due to the gaps exist within the temporal and spatial scales between the physics of acoustics and fluid dynamics. This disparity is especially extreme when the Mach number is small and reaches the incompressible limit [18]. For that reason, it is difficult for direct numerical simulations (DNS) that employs fully compressible Navier–Stokes equations to properly predict the acoustic behaviors, as the small acoustic pressure fluctuations eventually converge to the thermodynamic pressure [24]. Computational aeroacoustics (CAA) is a branch of acoustics that focuses on sound production in fluid flows which considers both fluid and acoustics. Lighthill’s aeroacoustic analogy [22, 23] is considered as the first method to predict the acoustic field based on a given flow field. It approximates the acoustic effects from the turbulent fluctuations of a given free flow, which is interpreted as a quadrupole source in the wave equation to be solved. The aeroacoustic analogy concept is further developed into Ffowrcs Williams–Hawkings analogy (FW–H) [35] that additionally considers monopole and dipole sources from moving hard walls. Another approach of CAA is the splitting method originally proposed by Hardin and Pope [17]. It splits the computation into a linear superposition of the incompressible fluid dynamics and acoustics perturbations, where the solution from the incompressible fluid field is substituted into the acoustic perturbation equations to solve the acoustic field. There are many variations developed later in the splitting methods family, such as linearized perturbed compressible equations (LPCE) [27, 28], acoustic perturbation equations (APE) [14], and linearized acoustic perturbation equations [24]. These CAA methods assume an incompressible flow and the acoustic field is computed based on that without any F. Yu · L. T. Zhang (B) Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 D. Zeidan et al. (eds.), Advances in Fluid Mechanics, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-19-1438-6_6

209

210

F. Yu and L. T. Zhang

feedback to the flow field. In light of considering the non-linear coupling between the fluid and acoustic fields, we developed a slightly compressible fluid formulation in a previous work [42], which does not decompose the variables but integrates all the variables in one set of Navier–Stokes equations. Navier–Stokes equations have been known for their non-linearity. Numerically solving this set of equations has been a challenge for decades due to the numerical instability it causes. Finite difference/finite volume methods (FDM/FVM) are often the more popular choices when it comes to solving Navier–Stokes equations due to its intuitiveness and speed. In most case, there is no complex connectivity for structured grids and there is no matrix to be formed for explicit time scheme, thus they do not require any expensive iterative solver. However, it requires structured grids, which generates discretization issues when the geometry of the computational domain is more complex. Moreover, the pressure solution often requires special treatment to be coupled with the velocity in order to solve Navier–Stokes equations in a staggered manner. Finite element method (FEM) is a discretization method that makes use of the virtual work concept to construct a weak formulation of a set of original partial differential equations (PDE) [19] such as Navier–Stokes equations. Utilizing isoparametric mapping of geometric shapes of elements, it allows for various element shapes to represent complex computational domains. To obtain the weak form, test functions based on Petro–Galerkin finite element method are often used to solve for the Navier– Stokes equations, as the convection terms are add into the non-linearity. For cases with high Peclet numbers or advection-dominated flows, numerical instabilities are often observed. Additionally, simple elements such as bilinear quadrilateral and constant triangle elements generates spurious pressure modes because they do not meet the inf-sup conditions for incompressible flow, whereas inf-sup stable elements have quadratic or higher polynomial degrees for the velocity components [8]. There have been methods for the stabilization of the incompressible form of Navier–Stokes equations. Some stabilization methods are aimed at adding stabilization parameters in the finite element system matrix. The finite element system matrix is a block matrix that includes four blocks, where there is a zero block (there is no term to couple pressure and pressure test functions), two symmetric blocks (pressure gradient block and velocity divergence block), and the velocity–velocity block which includes the velocity mass, convection, and viscous terms. The stabilization parameter is specifically imposed in the velocity–velocity block, due to the instability caused by velocity convection. For example, grad-div stabilization developed by Olshanskii [25, 26] is successful in stabilizing the advection terms while keeping the symmetry of the remaining terms by adding a penalty term to the velocity–velocity block. However, there are shortcomings with grad-div method as it is not equal-order, i.e., it still needs inf-sup stable elements. Thus, additional computational load is resulted. Moreover, the stabilization was specifically designed for incompressible fluid and cannot be applied to other forms of the Navier–Stokes equations. Tezduyar et al. [29, 30] proposed the streamline-upwind Petrov–Galerkin (SUPG) and pressure-stabilizing/Petrov–Galerkin (PSPG) methods. SUPG/PSPG methods add extra stabilization terms to the test functions in the weak form. These stabilization

Stabilized Finite Element Formulation and High-Performance Solver …

211

terms are versatile and are easily applicable for advection–diffusion equations as well as general compressible and incompressible flows. Using SUPG and PSPG methods, a stabilized, equal-order formulation of the Navier–Stokes equations can be obtained. Nevertheless, problems still remain in the system block matrix: SUPG and PSPG adds extra entries to the system block matrix which make the symmetric blocks no longer symmetric and the zero block non-zero. These significantly increase the difficulty in solving the linearized system with respect to approximating the inverse of the system matrix. Washio et al. [34] introduced a robust preconditioner based on incomplete LU (ILU) factorization, which is proven to be effective on flow problems with SUPG/PSPG stabilization. However, ILU factorization is serial per se, and its parallelized counterparts sacrifice either accuracy or efficiency. In this work, a finite element solver based on the SUPG/PSPG stabilization and the ILU-based preconditioner is presented, with an aim to improve the efficiency and efficacy in modeling aeroacoustic problem using finite elements with proper stabilization. This solver is implemented for a slightly compressible fluid that is derived for aeroacoustic [42]. The solver is implemented and incorporated into OpenIFEM [9, 40, 41], an open-source fluid–structure interaction framework using deal.II finite element library [3, 4, 7]. OpenIFEM has modular components that can model fluids and solids independently, or conduct coupled fluid–structure interactions using the immersed finite element method [31–33, 39]. This chapter is organized as follows: In Sect. 2, we first introduce the governing equations for aeroacoustics using slightly compressible form of Navier–Stokes equations, along with the discretized form with SUPG/PSPG stabilization and linearization scheme. To solve the discretized form, a preconditioner and its implementation details are discussed in Sect. 3. It is followed by two verification cases for accuracy with a preconditioner performance test in Sect. 4. Finally, the conclusions are drawn in Sect. 5.

2 Slightly Compressible N–S for Aeroacoustics 2.1 Governing Equations Aeroacoustic can be described as slightly compressible flow with an isentropic process, which is a reversible adiabatic process where the entropy S is constant. In normal acoustic situations, air compression/expansion is much closer to isentropic than other processes such as isothermal process. When air is compressed by shrinking its volume V , not only the pressure P increases, the temperature T increases as well due to the increased internal energy from the compression work, which is slowly diffused to the surrounding environment. However, in acoustic vibrations, the compressed air immediately expands and temperature changes are not given time to diffuse away to thermal equilibrium. Since heat diffusion occurs much slower than acoustic vibrations, this compression/expansion process is adiabatic and reversible

212

F. Yu and L. T. Zhang

and the entropy remains constant. Therefore, temperature field does not play a major role in the governing equation. As a result, the energy equation involving T does not need to be coupled with continuity and momentum equations. The pressure and density fluctuations p and ρ from equilibrium pressure p0 and density ρ0 are p = P − p0 and ρ  = ρ − ρ0 . For slightly compressible flows with low Mach number, we can assume p and ρ  are much smaller than the total pressure P and the equilibrium density ρ0 , i.e., p/P