Numerical Fluid Dynamics: Methods and Computations (Forum for Interdisciplinary Mathematics) [1st ed. 2022] 9789811696640, 9789811696657, 9811696640

Citation preview

Forum for Interdisciplinary Mathematics

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

Numerical Fluid Dynamics Methods and Computations

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

Numerical Fluid Dynamics Methods and Computations

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

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

ISSN 2364-6748 ISSN 2364-6756 (electronic) Forum for Interdisciplinary Mathematics ISBN 978-981-16-9664-0 ISBN 978-981-16-9665-7 (eBook) https://doi.org/10.1007/978-981-16-9665-7 Mathematics Subject Classification: 65M12, 65N12, 76M27, 76M12, 76M15, 65D17 © 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

Numerical modelling and simulations are an integral part of understanding fluid dynamics. Computational fluid dynamics (CFD) has been applied to a wide range of engineering problems including aerodynamics, natural science, environment and climate engineering, biological engineering, fluid flows and combustion analysis, to name a few. With such processes, a number of numerical methods and techniques based on finite difference, finite element and finite volume approaches are developed and implemented over the years. Utilizing these methods, there have been significant advancements and applications that allow engineers and scientists to investigate various types of problems with specific flow characteristics in fluid dynamics, e.g. compressible flows, incompressible flows, turbulent flows. These methods, with proper verification and validation, allow end users to be well prepared and more informed in the solution accuracy and efficiency. This book is another successful and fruitful research collaboration in numerical fluid dynamics topic among the editors and the contributing authors. The aim of this book is to create a global network of engineers and scientists working in this field to showcase novel methods and technologies that have been developed in the recent years. The reviewed chapters published in this book cover topics related to advancements in numerical methods and computations of fluid dynamics for realworld applications. In contrast, this book is motivated by the need to involve with interdisciplinary numerical fluid dynamics and obtain expertise beyond that which our own research topic covers. It is our hope that this book will enhance international research collaboration and to tackle and overcome scientific and technological challenges together. The writing of this book has been a very close collaborative effort, encouraged and supported by the geographically distant universities and colleagues working in fluid dynamics. We would like to thank all the authors for their unique contributions and to all the reviewers for their valuable inputs and generous giving of their time. We wish to express our appreciation to the German Jordanian University (Jordan), HTWK Leipzig University of Applied Sciences (Germany), ISAE-ENSMA, Institut Pprime (France) and Rensselaer Polytechnic Institute (USA) for their supports in allowing us to expand our international research collaborations with researchers all v

vi

Preface

over the world. Finally, we would like to give a special mention and appreciation to the Senior Publishing Editor at Springer Nature Shamim Ahmad for his cooperation and support along the entire process. Amman, Jordan Leipzig, Germany Chasseneuil-du-Poitou, France Troy, USA

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

Contents

Structure Functions for Numerical Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . L. G. Margolin and S. D. Ramsey

1

Generalized Probability Density Function of the Solution to the Random Burgers-Riemann Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . Juan Carlos Cortés and Marc Jornet

43

Semi-analytical and Numerical Study on Equatorial Rossby Solitary Waves Under Non-traditional Approximation . . . . . . . . . . . . . . . . Ruigang Zhang, Quansheng Liu, and Liangui Yang

69

High-Order Polynomial Recovery in Finite Element Advection Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Denise Vogel and Oswald Knoth

93

Breakdown of Morphing Continuum Approach for Flows Under Translational Nonequilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Jiamiao Sun, Mohamad Ibrahim Cheikh, Pedram Pakseresht, Mikel Aghachi, and James Chen Dynamics of Oscillatory Fluid Flow Inside an Elastic Human Airway . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Mayank Verma and Ashoke De Hyperbolic Balance Laws: Residual Distribution, Local and Global Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Rémi Abgrall and Mario Ricchiuto An Energy-Splitting High-Order Numerical Method for Multi-material Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Xin Lei and Jiequan Li

vii

viii

Contents

An ADER-LSTDG Scheme for the Numerical Simulation of a Global Climate Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Arturo Hidalgo and Lourdes Tello Efficient Experimental and Numerical Methods for Solving Vertical Distribution of Sediments in Dam-Break Flows . . . . . . . . . . . . . . . 291 Thomas Rowan and Mohammed Seaid

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. 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. Eric Goncalves Da Silva is Professor and Head of the Department of Fluid Mechanics and Aerodynamics, Aeronautical Engineering School ISAE-ENSMA, Poitiers, 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 ix

x

About the Editors

flow and cavitation. Recent work includes shock wave boundary layer interaction, shock–bubble interaction and investigation of three-dimensional effects on cavitation pocket. Lucy T. Zhang is Professor at the Department of Mechanical, Aerospace and Nuclear Engineering at Rensselaer Polytechnic Institute (RPI), U.S.A. She is Fellow of the American Society of Mechanical Engineers (ASME). 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. 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.

Structure Functions for Numerical Shocks L. G. Margolin and S. D. Ramsey

Abstract In this chapter, we will present an in-depth analysis of the structure of numerical shocks. Our analysis will feature the structure function, a simple technique to generate high-fidelity data for the structure of the numerical shock. We will describe two ODE models to analyze that data in terms of shock width, overshoot, and the dissipation that is explicit in the equations and implicit in the discretization. We will briefly summarize the methodology of shock capturing as employed in Lagrangian simulations and apply our ODE models to shocks generated by standard artificial viscosity formulations. We will review the recent application of finite scale theory to the structure of physical shocks, revealing the analogies between physical and numerical shocks, and exploiting them to design a new shock-capturing strategy. We will also consider shocks generated in nonoscillatory Eulerian simulations and compare their structure with comparable Lagrangian simulations. Keywords Numerical shocks · Structure functions · Finite scale equations

1 Introduction There is no cure for curiosity (Dorothy Parker). In this chapter, we will present an in depth analysis of the structure of numerical shocks. We will apply theoretical analysis based on self-similar solutions of the finite scale equations (FSEs) to both Lagrangian and Eulerian numerical simulations of the shock profile. We will quantify shock width and overshoot as a function of the parameters of artificial viscosity. We will offer new understanding of the unphysical oscillations that trail the shock. To help prepare the reader, we will provide necessary L. G. Margolin (B) Computational Physics Division, Los Alamos National Laboratory, Los Alamos, NM, USA e-mail: [email protected] S. D. Ramsey Applied Physics Division, Los Alamos National Laboratory, Los Alamos, NM, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 D. Zeidan et al. (eds.), Numerical Fluid Dynamics, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-16-9665-7_1

1

2

L. G. Margolin and S. D. Ramsey

background on shock-capturing techniques and on the FSEs. In addition, we will append an abbreviated history of the theory of shocks over the past 150 years. Shocks in fluids have structure. Unlike the idealization of a mathematical discontinuity, physical shocks have a width of several molecular mean free paths, (3–4) λ as measured in the laboratory. Similarly, planar shocks in engineering calculations have a width of several computational cells, (3–4) x. In each case, those respective quantities represent the only available length scales upon which the shock width can depend, However, in neither case is there a fundamental understanding of what determines the actual shock width. In the case of the physical shock, the difficulties go even deeper. Navier–Stokes theory predicts shock widths only half as wide as is measured in the laboratory [2, 54]. There is at present no widely accepted theory to supersede Navier–Stokes, though the FSEs have had some recent success in predicting both the width and the shape of physical shocks; see e.g., Fig. 2 in [31]. We can define an engineering scale for those problems in which the problem length scale is much larger than the molecular mean free path. In such problems, the effects of physical viscosity and heat conduction cannot be resolved. Instead, the model equations for numerical simulation are termed the regularized Euler equations [29], which generally consist of the Euler equations plus artificial viscosity [62], see Sect. 3. It is not relevant to ask whether the regularized Euler equations accurately predict the numerical shock profile, since there is no unique definition or characterization of that profile. In general, code users prefer numerical profiles that are both narrow and monotone, e.g., free of overshoots. We will show that these are conflicting goals. The FSEs govern the motion of finite-sized parcels of fluid. Several terms of those equations contain a finite length scale characterizing the size of those parcels as a parameter. The Navier–Stokes equations are a subset of the FSE, reached in the limit that the finite length scale goes to zero. Interestingly, the regularized Euler equations are also a subset of the FSE. Thus, the FSE represents a connection between the models of physical and of numerical shocks. We will begin the technical presentation of this chapter in Sect. 2 by exhibiting the FSE and discussing some of their relevant properties. Then in Sect. 3, we will review the two main techniques of shock capturing, namely artificial viscosity in Lagrangian codes and nonoscillatory differencing in Eulerian codes. Remark 1 In the computational fluid dynamics (CFD) community, numerical programs for simulating hydrodynamics are typically called codes or hydrocodes. We will distinguish in the text between model equations, which are partial differential equations, and discretized equations which are the discrete equations implemented in the codes. Lagrangian and Eulerian refer to the reference frame in which the flow is simulated; Lagrangian codes solve regularized Euler as their model equations. Nonoscillatory Eulerian codes solve discretized Euler equations. In Sect. 4, we describe our numerical laboratory, the one-dimensional code in which all the simulations of this chapter were performed. The code has two principal options, working in a purely Lagrangian framework or working in an Eulerian frame-

Structure Functions for Numerical Shocks

3

work in which the dynamics and the advection are split into two steps, a Lagrangian update and a total remap. In Sect. 5, we introduce the structure function, a method of incorporating an essentially arbitrary number of points into the shock profile so that high-fidelity estimates of the dependent variables and their derivatives can be made. The individual point values are independent and do not involve interpolation. The structure functions we will construct here depend on the self-similarity of plane shock waves, but can in principle be extended to any self-similar flow. Self-similarity is also the basis of the models we will use for analysis of the numerical data. In Sect. 6, we will derive a first-order ODE for the velocity in a plane shock which is then used to find an analytic model for the shock width and an approximate model for the unphysical oscillations behind the shock. In Sect. 7, we will apply our models to structure functions generated by standard models of artificial viscosity in Lagrangian simulations. We will show that the Lagrange codes accurately represent the shock profile when compared to the analytic solutions of the regularized Euler equations. However, those calculations are contaminated by overshoots and unphysical oscillations that can be minimized but not eliminated. In Sect. 8, we will take a closer look at the oscillations, considering different possible sources. We will conclude that the oscillations do not originate in numerical truncation error, but rather are inherent in the regularized Euler equations themselves, more precisely in the form of the standard artificial viscosity which lacks thermodynamic consistency. In Sect. 9, we return to finite scale theory to design and evaluate an artificial heat conduction that restores thermodynamic consistency to the regularization of the Euler equations. Structure functions generated with an artificial heat conduction become monotone with a critical value of the coefficients, but are wider than the profiles generated by a standard artificial viscosity. In Sect. 10, we show the structure functions generated by a nonoscillatory Eulerian code. Historically, Eulerian codes have been thought to be too diffusive for flows with shocks. However, modern codes employing nonoscillatory advection schemes can be made to produce monotone profiles while maintaining a relatively narrow width. We will end this chapter with a brief recounting of the history of modeling shocks, concentrating mainly on the theory and the experiments. Researchers in the CFD community have generally not been aware of nor concerned with the deficiencies of Navier–Stokes theory applied to shock structure. However, there has been a recent interest in employing entropy conditions to ensure the uniqueness of the numerical solutions [12, 16, 57]. We believe this is a mathematical approach to resolve the thermodynamic consistency issues that we have identified by means of the finite scale analysis. “In this house we obey the laws of thermodynamics”. (Homer Simpson)

4

L. G. Margolin and S. D. Ramsey

2 Finite Scale Equations In this section, we present the FSEs and describe the hierarchy of continuum models that result from various simplifications. We will then consider which properties of the equations are necessary to become an effective model for discretization and use in calculating flows with shocks. The FSE are designed to represent the motion of finite-sized parcels of fluid and have terms in the momentum and energy fluxes that contain a length scale characteristic of the parcel. When the parcel of fluid is associated with a computational cell, the FSE become a suitable tool to analyze the discretized equations that form the basis of a numerical code.

2.1 Introducing the FSE These are the FSE, written in one spatial dimension. ∂ρu ∂ρ + = 0, ∂t ∂x

(1)

 ∂  2 ∂ρu + ρu + P = 0, ∂t ∂x     ∂ 1 1 ∂ ρI + ρu 2 + ρIu + ρu 3 + Pu + Q = 0, ∂t 2 ∂x 2 P = (γ − 1)ρI − μ Q = −κ

∂u + Aρ ∂x

∂i + γ Aρ ∂x



∂u ∂x





∂u ∂x

(2)

(3)

2

 ∂I . ∂x

,

(4)

(5)

Comments #1 1. Notation: Here, ρ, u, and I have their usual meanings of density, material velocity, and specific internal energy. P is the total momentum flux and Q is the total heat flux. 2. We use internal energy rather than temperature, but note that for a perfect gas the former is proportional to the latter with the constant of proportionality being the specific heat at constant volume, cv , which is assumed to be constant. 3. Parameters: γ is the constant dimensionless parameter of the perfect gas law, which is the ratio of the specific heat at constant pressure, c p to cv . The sound speed, co is defined in terms of the internal energy co =

 γ (γ − 1) I.

Structure Functions for Numerical Shocks

5

The finite scale parameter A has the dimension of (length)2 and will be discussed below. Terms proportional to A represent the finite scale contributions to the continuum equations. 4. Navier–Stokes: When A → 0 the Navier–Stokes equations result, where μ is the bulk viscosity and κ is the coefficient of heat conductivity with respect to the gradient of internal energy. κ = cv k, where k is the usual coefficient of heat conduction with respect to the gradient of temperature. Note that both μ and κ have the units of (mass)/(length ∗ time) . As described in detail in the appendix, the Navier–Stokes equations predict shocks that are too narrow when compared to experimental measurements. Regularized Euler: When instead μ = ρc L co x, A = ρcq x 2 , and Q = 0, we recover the regularized Euler equations used in standard Lagrangian numerical simulations. Here, c L and cq are dimensionless parameters associated with the artificial viscosity; see Sect. 3 for details. We will refer to the computational cell size generically as x, although in Lagrangian simulations x varies in space and time. Euler: When in addition μ = κ = 0, we are left with the Euler equations. The Euler equations without regularization are not dissipative (do not dissipate kinetic energy into heat) and are not suitable to describe shocks. 5. The total momentum flux P in Eq. (4) contains three contributions. The first term is the pressure resulting from the ideal gas law, which is a good approximation to the equation of state for many gases. The second term is Newton’s law of viscosity and the third term represents the finite scale contribution. The heat flux Q in Eq. (5) consists of Fourier’s law plus a finite scale contribution. Note the explicit dependence of the finite scale fluxes on ρ. 6. The equations at all levels are written in flux form and are strictly conservative of mass, momentum, and total energy.

2.2 Model Equations for Numerical Simulation Flows with shocks are studied over a wide range of length scale from laboratory experiments with shock tubes to exploding supernova; the range of length scales covers at least 1021 orders of magnitude [20]. Independent of the physical length scale, an important distinction is made in numerical simulation between resolved calculations, in which the size of the computational cell size x is much smaller than the molecular mean free path λ and under-resolved calculations in which the shock width itself is much smaller than x. We will refer the computer codes used for the under-resolved calculations as “engineering” codes, though they encompass a wide range of problems in many branches of science and engineering. Which of the hierarchy of equations above should be the basis of the numerical simulations? Exact conservation of mass, momentum, and total energy are essential to

6

L. G. Margolin and S. D. Ramsey

accurately reproduce the Rankine–Hugoniot (jump) conditions across the shock. The other important degree of freedom in the various equations lies in the partition of the total energy between kinetic energy and internal energy. That partition is controlled by the dissipative terms in the momentum and the energy fluxes. As first noted by Rankine (see Appendix), shocks are not adiabatic, i.e., they dissipate kinetic energy and create entropy, so the Euler equations are not suitable for modeling shocks. In the case of resolved simulations, for example, in the simulations of the laboratory experiments of Schmidt and of Alsmeyer [2, 32, 54], the (compressible) Navier– Stokes equations would be an appropriate continuum model for discretization since the dissipative processes of Newtonian viscosity and Fourier heat conduction can be accurately represented. However, attempts to use the Navier–Stokes equations have been found to produce inaccurate predictions of both shock width and of profile shape, suggesting that it is the form of the dissipative terms that is at issue. The history of those attempts is recounted in the Appendix. The vast majority of practical simulations reside in the under-resolved category. The earliest attempts of using the Navier–Stokes equations as the model for underresolved problems produced solutions with large unphysical oscillations. It was quickly realized by the early pioneers that the issues lay in the lack of sufficient dissipation, i.e., the conversion of kinetic energy into heat, in the simulation. Although the Navier–Stokes equations contain physical viscosity and heat conduction, the mesh resolution did not allow sufficiently steep gradients of velocity and temperature for accurate approximations. Effectively, the simulations were using the inviscid Euler equations as the underlying model. The practice of computational fluid dynamics is often thought to be bereft of rigorous theory. Indeed, the finite scale theory was developed partially with the goal to fill that gap. However, we will take note of two early developments that play an important role in this chapter. In [13], Friedrichs and Lax show that the analytic solutions of the Euler equations may not be unique, but the physically relevant solution can be identified by adding a “vanishing viscosity” to the equations. This process, termed regularization [29], can be implemented in numerical simulation where, however, a finite amount of viscosity is required. Entropy-stable schemes for computing the Euler equations have become a major direction of research [12, 16, 57]. The second advance also involved Peter Lax. In an early unpublished Los Alamos report [23], Lax described what is now known as the finite-volume method, whose basic elements are (1) the use of integral forms of the governing equations and (2) the application of Gauss’ theorem to convert divergence terms to surface integrals. When two computational cells share a common surface (edge in 2D), the flux into one cell is exactly the flux out of the other. This idea, termed detailed balance, allows exact conservation of mass, momentum, and energy to the level of the roundoff error. By contrast, finite difference methods only conserve those quantities to the level of the truncation error. Remark 2 Note that finite-volume methods do not address the accuracy of the exchange flux, but only ensure exact conservation. Finite-volume methods led directly to the finite scale theory as we shall next describe.

Structure Functions for Numerical Shocks

7

2.3 Finite Scale Theory Finite-volume schemes differ from finite difference schemes in their interpretation of the discrete solution. For example, the density in a finite-volume code represents the density averaged over the computational cell, whereas in a finite difference code, it would represent a value at some point in the cell. This immediately leads to the question: “if every fluid point of a given volume is governed the Navier–Stokes equations, what equations govern the volume averages of density, energy and momentum?” The governing equations, the FSE, were derived rigorously in [33] using a combination of renormalization theory and inductive logic. More recently, the FSE have also been derived directly from the Boltzmann equations by generalizing the Chapman– Enskog expansion [27]. The original intent of the FSE was to provide a rationale for implicit large eddy simulation, (ILES), a numerical technique for modeling turbulent flows [33]. Properties that distinguish the FSE from Navier–Stokes include enslavement and inviscid dissipation and are described in [25, 26]. The ILES technique was first proposed by Boris [43] in the late 1980s, but remained controversial and resisted by the leaders of the turbulence community. However, the combination of the practical successes of the technique, e.g., as documented in [15], and a rationale [33] has overcome the initial objections of that community; ILES is now widely used. There are important similarities between turbulent flows and shock flows, encapsulated in the fact that both are high Reynolds number phenomena. In both cases, the length scales of dissipation, e.g., the smallest eddies and the shock width, are much smaller than the scales of advection. The first suggestions that the FSE might be applicable to physical shocks were made in [19, 37] where traveling wave solutions of the adiabatic FSE were derived. Here, adiabatic means without heat conduction; i.e., Q is set to zero in Eq. (3). That choice corresponds to standard models of artificial viscosity as described in Sect. 3. However, in physical shocks, heat conduction is as important as viscosity. The full FSEs provide a template for an artificial heat conduction that we will exploit in Sect. 9.

3 Essentials of Shock Capturing The failure of Navier–Stokes equations to accurately predict shock structure has been of little concern to the CFD community. Physical shocks may have a width of fractions of a micron. For engineering scale problems with length scales of meters, it is not possible to resolve shock structure on an affordable computational mesh. Fortunately, it is not necessary to resolve shocks in many problems. In this section, we will describe the origins of the technique of shock capturing, which is commonly used in under-resolved calculations. Shock capturing refers to both the explicit artificial

8

L. G. Margolin and S. D. Ramsey

viscosity used in Lagrangian codes and to the implicit dissipation of nonoscillatory schemes used in Eulerian codes. Remark 3 The earliest computer simulations were performed by Los Alamos staff with minimal computer capability. Those early calculations reflected the primary interests of John von Neumann, namely nuclear weapons and weather forecasting. In the former case, most of the early research was documented in classified reports that have only recently been declassified for public dissemination.

3.1 Lagrangian Codes (Artificial Viscosity) Artificial viscosity is one of the oldest artifices in CFD. It is first openly described in the well-cited 1950 paper of von Neumann and Richtmyer [62]. The very first shock calculations had employed a laborious process of fitting shocks by hand in between computational cycles and modifying the computed results appropriately. In [62], several important ideas were made public, including shock capturing, the quadratic form of artificial viscosity, the staggered mesh, and linearized stability analysis. Recently declassified Los Alamos reports help to assign proper credit for those fundamental concepts. Shock capturing is based on the recognition that some important properties of shocks, namely the jump conditions and the shock velocity do not depend on the viscosity nor on heat conduction. That fact was made explicit in the 1870 paper of Rankine [53], but its application to computing was apparently first suggested by Peierls in a 1944 letter written to von Neumann [44]. The staggered mesh refers to the typical data structure of Lagrangian codes in which position and velocity are located at vertices of the computational mesh, while thermodynamic variables, e.g., density, temperature, etc., are located at cell centers. This concept appears in [61] co-authored by von Neumann and Richtmyer, where it is attributed to unpublished lectures of von Neumann at Los Alamos. The linearized stability analysis based on Fourier analysis also appears in [61]. The artificial viscosity introduced in [62] depends on the square of the velocity gradient. In one spatial dimension,  q = cq ρ x

2

du dx

2 .

(6)

Here, x is the width of the computational cell, u is fluid velocity and ρ is the fluid density. The constant cq is dimensionless and is simply said to be of order unity. In practice, it is a user-adjustable parameter. The artificial viscosity q is added to the physical pressure. The quadratic form appears in [62] without justification. However, that form is derived in an 1948 report [47], solely authored by Richtmyer. The derivation is based on the simple idea that all shocks captured on the computational mesh should have

Structure Functions for Numerical Shocks

9

the same width in units of x independent of shock strength. The quadratic form of the artificial viscosity also appears in the theoretical discussions of the FSEs in Sect. 2 and is a central issue in this chapter. The report [47] is mainly concerned with the derivation and the stability constraints of the artificial viscosity. A second report [48], also solely authored by Richtmyer, documents the effectiveness of the artificial viscosity in simulations. Additional discussion of Richtmyer’s reports can be found in [38]. Many improvements have been proposed to artificial viscosity, e.g., to steepen the numerical profile [10], to introduce a tensor character [64], and to preserve the flow symmetry [7], etc. However, the basic quadratic form of the artificial viscosity persists today with only one important modification. In 1955 in another unpublished Los Alamos report [22], Landshoff notes that the quadratic viscosity treats the oscillations following the shock, but does not treat a small overshoot that occurs at the top of the shock. He derives a linear viscosity to add to the quadratic viscosity. In fact, Landshoff’s derivation is incorrect; he integrates a characteristic across the shock, effectively making the same mistake as Stokes (see Appendix). However, the linear viscosity does mostly eliminate overshoot at the expense of farther widening the shock by virtue of adding a first-order dissipation. The proposed form of artificial viscosity becomes       du  du  du  , q = ρ c L co x   − cq x 2   dx dx dx

(7)

where co is the sound speed and c L is another dimensionless parameter. In this form, the viscosity is turned off in expansion, i.e., when ddux > 0; see [8] for more implementation details. There is an arbitrariness to the choice of the coefficients cq and c L . In principle, the quadratic term is meant to regularize the shock, i.e., to provide the dissipative processes not resolved on the mesh, and so has a physical origin. The linear term is meant to erode the overshoot, which is believed to have an origin in truncation terms, and so has a numerical origin. Nevertheless, computational studies to optimize the choice of coefficients [1, 52] show a valley of solutions, all essentially equivalent. There is a second proposed change to the form of artificial viscosity that has (so far) received little traction in the CFD community, but which is germane to this chapter. Artificial heat conduction was proposed by Bill Noh in a well-cited paper [42] as a numerical solution to the troublesome problem of wall heating [49]. The lack of common implementation of an artificial heat conduction can be ascribed to the difficulty of finding an effective numerical formulation. Note that the FSE contain a finite scale contribution to the heat flux in Eq. (5) which we will use in Sect. 9.

10

L. G. Margolin and S. D. Ramsey

3.2 Eulerian Codes; Nonoscillatory Differencing Early Eulerian methods were less well adapted for shock calculations, especially as computers became capable of running simulations in two spatial dimensions. Pure Eulerian methods have difficulty not diffusing unresolved flow structures. Innovations like volume of fluid (VOF) methods [17] and adaptive mesh refinement (AMR) [4] have been instrumental in improving the capabilities of Eulerian methods for under-resolved flows with material boundaries and detailed geometry. However, for resolving shocks, the major innovation was the introduction of nonoscillatory differencing of the advective terms. The story of nonoscillatory differencing begins with an important theorem proved by Sergei Godunov in his Ph.D. thesis [14, 63]. The theorem states: “Linear numerical schemes for solving partial differential equations having the property of not generating new extrema (monotonicity preserving schemes), can be at most first-order accurate”. Preservation of monotonicity and high-order (or at least second-order) accuracy are both desirable properties. However, first-order schemes are too diffusive for most practical applications and for 15 or so years after Godunov, numerical methods focused on accuracy at the expense of allowing unphysical oscillations. It was Jay Boris who first realized how to overcome the barrier of Godunov’s theorem—to give up linearity. Here, linearity means that the numerical approximations are the same for every grid point. In the flux-corrected transport (FCT) algorithm [6], Boris and Book describe a method that mixes first- and second-order methods locally in such a way to preserve monotonicity. The FCT algorithm by construction uses flow dependent coefficients to approximate the transport terms and so is not linear in the sense above. It is also not quite second-order accurate by standard measures, but can be made very high order in smooth regions of the flow. The strategy for avoiding Godunov’s barrier spread quickly in the CFD community. Van Leer [60] developed a methodology based on Godunov’s method (another important contribution by Godunov). Soon there was an alphabet soup of methods and today nonoscillatory algorithms combined with finite-volume approximations, i.e., NFV schemes, are the mainstream for simulating high Reynolds number fluid dynamics. Some important properties shared by all NFV schemes are nonlinear stability (under appropriate time step conditions), exact conservation of mass, momentum, and energy, and absence of nonphysical oscillations. Further, NFV schemes are parameter free and relatively easy to implement. Preserving monotonicity provides effective control over the generation of unphysical oscillations. Monotonicity is a mathematical property of the solutions of some evolutionary equations, but the preservation of monotonicity is not a physical principle. Further, monotonicity is essentially a one-dimensional idea. In a relatively unknown paper [40], Merriam established the connection of monotonicity with a fundamental principle of physics, the second law of thermodynamics. Entropy stable schemes for the Euler equations have become a major direction of research [12, 16, 57].

Structure Functions for Numerical Shocks

11

Summary 1. The use of artificial viscosity to regularize the discrete Euler equations in a Lagrangian code introduces two new parameters, cq and c L . The choice of those parameters is a user option. 2. The nonoscillatory algorithms that regularize the discrete Euler equations in an Eulerian code are parameter free. However, there are many different formulations of nonoscillatory advection, each producing slightly different results. 3. Thus, the Eulerian codes are more automatic, while the Lagrangian codes offer more flexibility.

4 The Numerical Laboratory In the following sections, we will present and analyze numerical data generated by a one-dimensional computer code. Here, we describe what are mostly standard finitevolume methods for explicit Lagrangian and Eulerian hydrodynamics. The Eulerian capability is based on a special case of ALE (arbitrary Lagrangian–Eulerian) in which the solution is advanced in a Lagrangian calculation, followed each computational cycle by a rezone back to the original mesh and a remap of the Lagrangian solution onto the Eulerian mesh.

4.1 Lagrangian Calculation The calculation proceeds on a staggered mesh in which coordinates and velocities are stored at the cell vertices, while the thermodynamic variables, density, pressure, and internal energy, are stored at the cell centers. Mass is primarily cell-centered, but a vertex mass is required to integrate the momentum equations and is formed by averaging the masses of the adjacent cells. The code integrates an internal energy equation; however, the use of mimetic differencing [34] insures conservation of total energy to the level of roundoff error. Similarly, the finite-volume integration is based on detailed balance, so ensuring that mass and momentum are also conserved to the level of roundoff error. Exact conservation is important as the jump (Rankine–Hugoniot) conditions for a shock are determined by the conservation laws. The Lagrangian method uses a standard linear-plus-quadratic artificial viscosity as described in Sect. 3. The time integration is based on a predictor–corrector technique and is approximately second-order accurate in time. Cells in the Lagrangian simulations are not equally sized, which in principle reduces the spatial accuracy. However, we will see that the results of Lagrangian calculations are remarkably accurate, at least for the plane shock profile problem.

12

L. G. Margolin and S. D. Ramsey

4.2 Eulerian Calculation Nonoscillatory Eulerian methods can be further divided into two types, a Lagrangeplus-remap in which the dynamics and the advection are split into separate calculations and direct Eulerian in which both phases are calculated at the same time. In our Eulerian calculations in Sect. 10, we use the Lagrange-plus-remap; the Lagrangian calculation described above is followed by a rezone in which the cells are restored to their original positions followed by a remap of the Lagrange solution back onto the original grid. We use a remapper based on nonoscillatory techniques as described in Sect. 3. Our Eulerian simulations that are based on Lagrange-plus-remap also use the staggered grid, which introduces some complexity into remapping the velocity. In Sect. 10, we will use a scheme termed RRR (for Remapping, Recovery, and Repair) [35] formulated to minimize the dissipation of kinetic energy during the remap phase of the calculation. The RRR routine has an additional capability to repair small violations of monotonicity that may occur; we have not used the repair step in our calculations as its effects are not amenable to our analysis tools. Modern nonoscillatory finite-volume (NFV) methods have mostly supplanted artificial viscosity methods for Eulerian simulations. In Sect. 10, we will compare the structure functions for Eulerian simulations with those for Lagrangian simulations and show that NFV codes are as capable as the Lagrangian codes.

4.3 Parameters Remark 4 all units are MKS. Our simulations represent a right-moving shock wave driven by a piston. The material is argon gas, whose equation of state is a perfect gas with γ = 5/3. The undisturbed material is characterized by ρo = 1.0 · 10−4 ; Io = 9.36 · 104 ;

po = 6.664.

The piston speed is u p = 745.95, which generates a Mach M = 3.38 shock. The initial mesh consists of 1000 cells, of initial width xo = 1.0 · 10−4 . The cell width in a Lagrangian calculation varies in time, but the cell mass is constant. The Rankine–Hugoniot (jump) condition for density is ρ f = ρo

γ +1 , γ − 1 + M2 2

(8)

where ρ f is the density behind the shock. When M = 3.38, ρ f = 3.168 · 10−4 . As mass is constant in a Lagrangian cell, we must have

Structure Functions for Numerical Shocks

13

ρ f x f = ρo xo , so that the Lagrangian cells behind the shock have width x f = 3.156 · 10−5 . Each of the problems was run for 16, 000 cycles, representing a sufficient time to allow a steady wave profile to build up. The time step t was fixed at a constant value such that the Courant–Friedrichs–Lewy parameter CFL = v

t = 0.2. x

(9)

Remark 5 We have chosen to study the Mach 3.38 shock mostly as a holdover from earlier studies of the physical shock [31]. That value was chosen to compare with many of the experimental results of Schmidt [54] and Alsmeyer [2]. In the measurements, that choice was found to produce the narrowest shocks. However, comparison with Fig. 10 of [54] shows that the width of all shocks beyond about Mach 2 is nearly constant. Indeed, that assumption (of constant width) is the basis of Richtmyer’s derivation of the quadratic viscosity, cf. Sect. 3. Interestingly, Richtmyer’s work preceded the experimental results by 20 years.

5 The Structure Function Verification and validation (V&V) are important elements of code development. Verification is the process of ensuring that the code is correctly solving the model equations. Validation is the process of testing whether the model equations accurately describe the physical phenomena of interest. Verification typically involves comparison of numerical solutions with analytic solutions, while validation involves comparison with experimental data. In this section, we briefly discuss convergence testing and argue that it is not a useful tool for verification of engineering codes in the presence of shocks. Then we will introduce the structure function as a useful alternative.

5.1 Convergence Testing Convergence testing is a strategy commonly used to verify continuum codes [51]. One begins by choosing a problem to which the exact solution, “truth” is known. The next step is to simulate that problem on a sequence of grids with increasing resolution. Comparison of the simulation results with the exact solution produces a set of errors as a function of resolution. If the error decreases as the resolution increases, one says the code is convergent for that test problem. In the case of a convergent code/problem, one can quantify the result by plotting the error as a function cell size x. If the graph

14

L. G. Margolin and S. D. Ramsey

can be fit by the model error ∼ (x) p , then one says the code is in an asymptotic regime for that problem and has an order of convergence p. In general, one would prefer calculations to converge at the second order ( p = 2) or higher. However, problems with under-resolved shocks can only converge at first order when “truth” is assumed to be the discontinuous solution. That result has been demonstrated rigorously for shocks [24], but should be expected in any simulation without a (resolved) length scale. This includes the many test problems used for shock flows including the self-similar solutions of Noh, of Sedov, of Guderley, etc.; see [11, 39]. It is mostly uninformative to apply convergence testing to problems with shocks. In the absence of physical dissipative mechanisms, the only length scale in the simulation is the cell size x. If one uses the quadratic viscosity of Eq. (5) the shock will look exactly the same at any resolution when scaled by x. A similar result will obtain if one uses the linear-plus-quadratic viscosity of Eq. (7), However, for verification of Lagrangian and ALE codes, there is an alternative. Since the artificial viscosity is added explicitly to the discrete equations, one can compare the numerical solutions to the analytic solutions of the Euler equations plus artificial viscosity. We will derive the analytic solutions in the next section. However, to compare those analytic solutions to code generated solutions, we must have high-fidelity numerical data. The typical shock profile only contains 3 or 4 data points, which is hardly sufficient for a detailed comparison of shock structure. Next, we will describe the construction of the structure function, a technique that can fill out the shock profile with essentially an arbitrary number of points, all of which are generated independently by the computer, i.e., not by interpolation,

5.2 Structure Function Construction A typical shock velocity profile is shown in Fig. 1, where the data has been generated in a Lagrangian simulation. Recalling that velocity is stored at the cell vertices on the staggered mesh, each of red diamonds represents the velocity at the position of a vertex in computational cycle 751. The 4 or 5 points that represent data within the shock profile are not sufficient to define the underlying structure of the profile. Now suppose, for the sake of this illustration, the calculation is proceeding at a constant = 0.5. Then it is simple to time step determined by a CFL condition of .5, i.e., vt x add the values of velocity of the same set of points from cycle 750 to the graph after translating each of their coordinate values by vt. Those are the blue dots in Fig. 1 which begin to fill out the profile. If one continued this process, the next set of points from cycle 749 would lie on top of those of cycle 751. That is the result of the particular CFL that was chosen for illustration. In principle, a more incommensurate CFL would allow any number of cycles to be superposed. However, there are some advantages to having the points in the structure function equally spaced. In practice, we have found that the choice CFL = 0.2 produces a sufficiently accurate representation of the shock profile.

Structure Functions for Numerical Shocks

15

Fig. 1 To illustrate the construction of the structure function, velocity data from two cycles are superposed to more carefully describe the shape of the velocity profile. This data is numerical in origin, taken from the simulation of the Mach 3.38 shock with artificial viscosity parameters c L = .2 and cq = 1

A typical structure function is shown in Fig. 2. The calculation that generated the nearly continuous dotted line was Lagrangian with the choices c L = 0.2 and cq = 1. The dotted line, labeled “analytic” is based on a high-order integration of the ODE of Eq. (22), resulting from the traveling wave assumption in the FSE. The structure function has a slight overshoot and a slightly smaller slope than the analytic solution, neither being obvious in the “viewgraph” norm, but both readily quantified from the data. This example is meant only for illustration and a more systematic analysis of Lagrangian simulations will be made in Sect. 7 using the models described in the next section.

6 Analysis Models “You have your way. I have my way. As for the right way, the correct way, and the only way, it does not exist. (Friedrich Nietzsche)” Our strategy in the next few sections is not to assess an error by comparing in detail the structure functions with the analytic solutions. In practice, the choice of the number coefficients c L and cq that define the artificial viscosity is motivated by conflicting desires; there is no unique choice that may be considered optimal for

16

L. G. Margolin and S. D. Ramsey

Fig. 2 Here, we compare the structure function of the velocity profile to the analytic solution generated by Eq. (22). This particular plot is for the Mach 3.38 shock with artificial viscosity parameters c L = .2 and cq = 1. The widths predicted analytically and measured from the structure function agree within 10%

all situations. In some cases, the user may seek the steepest profile. Alternately, the user may prefer a profile that is free of overshoots and unphysical oscillations. We will quantify the well-known maxim that the damping of oscillations comes at the expense of additional smearing of the shock.

6.1 Self-similar Analysis Here, we derive an ODE for the fluid velocity in a plane shock for the case of the FSEs with no heat flux, i.e., with Q = 0 replacing Eq. (5). With the identification μ = c L co x and A = cq x 2 , those equations are the model equations for a Lagrangian code with linear plus quadratic viscosity. The plane shock is a traveling wave problem. One might visualize the the problem as a piston moving to the right with velocity u p into an undisturbed fluid at rest. Then all the fluid variables will be a function of the similarity variable y = x − vt, where v will turn out to be the shock velocity. We set the boundary conditions: at y = +∞, I = Io , u = 0, ρ = ρo , p = po , at y = −∞, u = u p . Specific choice of the parameters was given in Sect. 4.3. Using the similarity assumption, Eqs. (1)–(4) can be rewritten:

Structure Functions for Numerical Shocks

d dy



17

d (−vρ + uρ) = 0, dy

(10)

 d  −vuρ + u 2 ρ + P = 0, dy

(11)

1 1 −vIρ − ρu 2 v + ρuI + ρu 3 + Pu 2 2

 = 0,

P = (γ − 1)ρI + ρχ (u), where



du χ (u) ≡ A dy

2



du −η dy

(12) (13)

 .

(14)

Equations (10)–(12) can be integrated immediately; applying the boundary conditions at y = +∞ yields uρ = v(ρ − ρo ), (15) uρ(v − u) = (P − Po ),

(16)

1 (v − u)Iρ + ρu 2 (v − u) − Pu = vIo ρo . 2

(17)

It follows then that ρ=

ρo v , v−u

 1 ρo v ρo uv − χ (u) . ρI = ρo Io + γ −1 v−u

and

(18)

(19)

Eliminating ρI between Eqs. (17) and (19) then yields χ (u) =

u v

  γ +1 v2 − co2 − uv , 2

(20)

 where co = γ (γ − 1)Io is the sound speed of the material in the undisturbed material ahead of the shock. Then equating (14) and (20) leads to a nonlinear ODE for fluid velocity:  A

du dy

2

 −η

du dy

 =

u v

  γ +1 v2 − co2 − uv . 2

The quadratic equation can be solved and put into the alternate form,

(21)

18

L. G. Margolin and S. D. Ramsey

η du = − dy 2A



η2 + 2(γ + 1)Au(u p − u) , 2A

(22)

where, for a right-moving shock, the minus sign in front of the square root term is correct. From Eq. (21) noting that du must vanish at y = −∞ where u = u p , the shock dy velocity must be the solution of the quadratic equation v2 −

γ +1 u p v − co2 = 0, 2

(23)

or (for the right-moving shock) γ +1 + v= 4



γ +1 4

2 u 2p + co2 .

(24)

6.2 Structure Function Metrics In the structure function analysis to be described in Sect. 7, we will be mainly interested in two features of the shock profile, the shock width and the overshoot. Here, we show that each of these can be estimated from the ODEs derived above, allowing a detailed comparison with numerical results generated from the structure function. Shock width model: First, we note that for η = 0, the shock does not have compact support, i.e., there is no unique visual definition of width. It is conventional to measure the shock width in terms of the maximum slope of the profile and the jumps across the shock. The width can be measured in any of the variables ρ, u, I and will be slightly different depending on the choice. Here, we will define      du   W ≡ u/ ,  d x in f 

(25)

  where u is the jump in material velocity across the shock and ddux in f is the velocity gradient measured at the inflection point. In our notation, η = c L co x and A = cq x 2 . The piston velocity u p is driving the shock into a stationary fluid. From 2 Eq. (22), one easily verifies that the inflection point, where dd xu2 = 0, is found at u = u p /2. Then after some manipulation, the analytic shock width is given by 

4 (γ + 1) 2 2 cq (M p ) x, W= cL + cL + (γ + 1)M p 2

(26)

Structure Functions for Numerical Shocks

19

where M p = u p /co is the piston Mach number. The width is proportional to x as it must be for dimensional reasons. In numerical experiments using the structure function, we will be able to vary the parameters c L and cq independently. The measured parameters may be different (larger) than what is input as a result of truncation error. Remark 6 An important difference between the theory and the numerical results is that in the Lagrangian simulations, x is not a constant. Shock oscillation model: The analysis for the shock overshoot and oscillatory behavior is more intricate. First, note that the shock profile as described by Eq. (22) is monotonic. If there were an overshoot such that at its maximum, the velocity u max > u p , then the slope would vanish there. However, from Eq. (22), the slope will only vanish when u = 0 or u = u p , showing the contradiction. On the other hand, nearly all of the numerical profiles that we will show in the following sections exhibit small or sometimes substantial overshoots. From this, we conclude that in addition to the artificial viscosity, the code must have additional terms of O(x) or O(x 2 ), i.e., truncation error. Now first-order terms, for example, = −v ∂∂ux by the due to space or time, can be grouped into the coefficient η since ∂u ∂t similarity of the plane shock. However, there is one additional independent term of 2 second-order that may appear as a truncation error, namely αu ddyu2 , where α is, for the moment, an unspecified coefficient proportional to x 2 . So here, we will consider the following model equation for the numerical shock profile which results from adding the “α term” to Eq. (21)  A

du dy

2

 −η

du dy

 + αu

d 2u u = 2 dy v

  γ +1 v2 − co2 − uv . 2

(27)

Next, let us simplify Eq. (27) near the overshoot where u ∼ u p . We define a new velocity variable w ≡ u − u p . Then we can define the small quantity, ,  = |w| /u p 0. Then the wave number k=

iη ± 2αu p

 4αu p C − η2 . 2αu p

(30)

It is the plus sign that obtains, in order to insure nonzero dissipation in the limit that α → 0. To exhibit oscillatory behavior, we must have 4αu p C > η2 which defines a critical value of the coefficient α. Then in the oscillatory regime, w has the form w = wo exp(by) cos(ay), 

where a=

4αu p C − η2 η ; b= . 2αu p 2αu p

(31)

(32)

The inverse relations are α=

bu p 1 (γ + 1) ; η = (γ + 1) 2 . 2 2 2 a +b a + b2

(33)

The critical value determining oscillatory behavior is αcrit =

η2 . 2(γ + 1) u 2p

(34)

As the solution of a second-order ODE, two boundary conditions must be supplied. wo is the initial overshoot, and is not determined by this model. For the second boundary condition, w → 0 as y → −∞. We have arbitrarily set y = 0 at the peak of the overshoot.

Structure Functions for Numerical Shocks

21

Remark 7 The introduction of the α-term will not affect our estimate of the shock width, Eq. (26). The width is measured where the gradient of velocity is steepest so that the second derivative vanishes. The model equation predicts the wavelength of the oscillations and the peak-topeak attenuation as a function of the parameters η and α. However, the ODE (28) is homogeneous and cannot predict the magnitude of the overshoot, which instead becomes a boundary condition. Conversely, if we have numerical data to estimate the wavelength and the attenuation, we can estimate the dimensionless coefficients of the linear viscosity and of the α-term. The structure functions generated by our hydrocode will provide that data, as we will demonstrate in the next sections.

7 The Standard Lagrangian Profiles Here, we study data from the standard Lagrangian viscosity models as described in Sect. 3, using the width model and the oscillation model described in Sect. 6. In our analysis, we have studied 45 individual cases in which all combinations of the parameters c L = 0, 0.05, .10, .15, .20, .25, .30, .35, .40 and cq = 0.8, 0.9, 1.0, 1.1, 1.2 are considered. In general, all of the cases are qualitatively similar. In this section, we present (in tables) data from a few combinations of parameters that typify our results. We summarize the general conclusions from all cases in narrative. Remark 8 Throughout the next few sections, we will indicate that the data is accurately fit, e.g., by a straight line or by a quadratic polynomial. In each case, we imply that the coefficient of determination or R 2 as calculated in EXCEL is greater than .99. Width model Table 1 shows results for the width and the overshoot for a range of values of cq while c L = 0.2 is held fixed. A companion study presented in Table 2 shows results for a range of values of c L , while cq = 1 is held fixed. The salient results are summarized below. 1. For the Mach 3.38 shock, the Mach piston number M p = u p /u o = 745.95/322.5 = 2.313. Then the width of the Mach 3.38 shock from Eq. (26) can be written

22

L. G. Margolin and S. D. Ramsey

Table 1 The width model in this parameter range, Eq. 35, predicts that the measured width of the √ numerically generated shock profiles will depend on cq . This is verified by the numerical data. In general, the measured width (M) is smaller than the analytic width (A). However, agreement is always within 10%. Note that the overshoot relative to the piston velocity u p decreases as the slope increases. All units are MKS cL cq Width (M) (L) Width (A) (L) Ratio Overshoot (L/T) 0.2 0.2 0.2 0.2 0.2

1.57 · 10−4 1.67 · 10−4 1.77 · 10−4 1.86 · 10−4 1.95 · 10−4

0.8 0.9 1.0 1.1 1.2

1.68 · 10−4 1.78 · 10−4 1.84 · 10−4 1.95 · 10−4 2.03 · 10−4

1.07 1.06 1.06 1.05 1.04

11.05 9.67 8.85 7.99 7.15

Table 2 The width model in this parameter range, Eq. 35, predicts that the measured width of the numerically generated shock profiles will depend linearly on c L . This is verified by the numerical data. In general, the measured width (M) is smaller than the analytic width (A). However, agreement is always with 10%. Note that the overshoot relative to the piston velocity u p decreases as the slope increases. All units are MKS cL cq Width (M) (L) Width (A) (L) Ratio Overshoot (L/T) 0.0 0.1 0.2 0.3 0.4

1.58 · 10−4 1.67 · 10−4 1.77 · 10−4 1.87 · 10−4 1.97 · 10−4

1.0 1.0 1.0 1.0 1.0

 W = 0.648 c L +

1.73 · 10−4 1.80 · 10−4 1.87 · 10−4 1.94 · 10−4 2.01 · 10−4

1.09 1.07 1.06 1.04 1.02

28.15 17.55 8.85 2.15 0.44



 c2L

 √  + 7.133cq x ≈ 0.648c L + 1.731 cq x, (35)

where the approximation is valid within our range of parameters. All the simulated √ cases verify that the measured width varies like cq , as exemplified in Table 1, and linearly with c L as exemplified in Table 2. 2. In general, the measured widths in Tables 1 and 2 are smaller than the analytic widths. The analytic widths are calculated with the initial cell size, whereas in the Lagrangian calculation, the cell size x decreases as the shock traverses the cell. As either coefficient of viscosity increases, the agreement of measured width with the analytic width gets better. However, all measured widths are within 10% of the analytic width. 3. The overshoot is measured as the maximum excess velocity relative to the predicted value of velocity behind the shock u p . When c L is held constant, the magnitude of the overshoot decreases quadratically with c L . Similarly, when c L is held constant, the magnitude of the overshoot deceases quadratically with cq .

Structure Functions for Numerical Shocks

23

Fig. 3 Data from Table 2 is used to plot the overshoot vs. the measured shock width. All simulations shown here have cq = 1. Increasing width corresponds to increased values of c L . This graph illustrates the general maxim that steeper shocks have larger overshoots. The trendline shows a quadratic relation between overshoot and shock width

4. In Fig. 3, data from Table 2 is used to plot the shock width versus the overshoot. All data has cq = 1, while increasing width indicates increasing values of c L . The resulting curve shows the overshoot depends quadratically on the width. More generally, the plot illustrates that increasing the slope of the simulation comes at the expense of larger overshoots. A similar inverse correlation exists when a sequence of results for increasing cq while c L is held constant. 5. We call attention to an unexpected result associated with the absence of an entry in the last position of Table 2. In general, we associate the magnitude of the overshoot with the peak of the largest oscillation; we also implicitly assume that this is the first peak of the oscillations. This is the case for all the entries in Tables 1 and 2, but was not the case for the simulation with the missing entry. We will return to this point in Sect. 8. Oscillation model Our next study will apply the oscillation model to the region behind the shock. We begin by considering the Lagrangian simulations for several values of cq with c L = 0. The data are summarized in Table 3. 1. The oscillation model (Sect. 6.2) assumes that the quadratic viscosity terms are negligible in the region behind the shock, implying that the wavelength and the attenuation of the oscillations are independent of cq ; this is verified by data in Table 1. In turn, the inverse relations, cf. Eq. (33) predict that η will be inde-

24

L. G. Margolin and S. D. Ramsey

Table 3 The oscillation model parameters η and α are inferred from numerical structure function measurements of the wavelength and the decay of the oscillations behind the shock. In this table, the linear artificial viscosity parameter is set to zero, c L = 0 and the quadratic viscosity parameter, cq is varied. The data shows that both η and α are independent of cq behind the shock, verifying the model assumption. All units are MKS cL cq η (L 2 /T ) α (L 2 ) α crit (L 2 ) λ (L) 0. 0. 0. 0. 0.

0.8 0.9 1.0 1.1 1.2

1.95 · 10−3 1.87 · 10−3 1.94 · 10−3 1.94 · 10−3 1.95 · 10−3

9.75 · 10−10 9.20 · 10−10 9.75 · 10−10 9.75 · 10−10 9.75 · 10−10

1.28 · 10−12 1.18 · 10−12 1.27 · 10−12 1.26 · 10−12 1.27 · 10−12

1.70 · 10−4 1.65 · 10−4 1.70 · 10−4 1.70 · 10−4 1.70 · 10−4

pendent of cq . The data also confirms that prediction. Further, one expects that the coefficient of the truncation term, α will be independent of cq and the data confirms this as well. 2. Behind the overshoot, the cell width is approximately constant, calculated in Sect. 4 as x f = 3.156 · 10−5 . Then the dimensionless coefficient of the truncation term is α/x 2f ∼ 1.0 and the truncation term could be written more generally (1.0) u

d 2u x 2 . dx2

(36)

3. The oscillation model predicts that there is a critical αc that determines whether oscillatory solutions exists. The data shows that critical value is nearly three orders of magnitude smaller than α and verifies that the solutions remain oscillatory for all the tested values of cq . Indeed, experiments (not shown) as high as cq = 10 and c L = 0 were found to be stable, but oscillatory. 4. The data shows that there is linear dissipation in the calculation even though c L = 0. On dimensional grounds, noting that the sound speed co does not appear in the simulation when c L = 0, η∼



αu 2p ;

η  = 0.083, αu 2p

using the data from Table 1 for cq = 1. We can compute an equivalent linear dissipation that would be present if α = 0, cL ∼

η = 0.187, co x f

again using data from the table for cq = 1.

Structure Functions for Numerical Shocks

25

Table 4 The oscillation model parameters η and α are inferred from numerical structure function measurements of the wavelength and the decay of the oscillations behind the shock. In this table, the quadratic artificial viscosity parameter cq = 1, and the linear viscosity parameter, c L is varied. The data show that η increases and α decreases as linear viscosity increases. However, the shock profiles remain oscillatory for all cases. The ∗ indicate that measurements were taken between the second and third peaks of the oscillation, All units are MKS cL cq η (L 2 /T ) α (L 2 ) α crit (L 2 ) λ (L) 0.0 0.1 0.2 0.3 0.35* 0.4*

1.0 1.0 1.0 1.0 1.0 1.0

1.94 · 10−3 2.32 · 10−3 2.96 · 10−3 3.93 · 10−3 7.45 · 10−3 7.82 · 10−3

9.75 · 10−10 1.04 · 10−9 1.03 · 10−9 7.55 · 10−10 7.41 · 10−10 6.88 · 10−10

1.27 · 10−12 1.81 · 10−12 2.95 · 10−12 5.20 · 10−12 1.42 · 10−11 2.06 · 10−11

1.70 · 10−4 1.70 · 10−4 1.65 · 10−4 1.50 · 10−4 1.50 · 10−4 1.40 · 10−4

5. The wavelength of the oscillations is found to be ∼ 5.4x f . In the regime where that we have studied, the wavelength of the α >> αc , true for all 45 simulations √ oscillations depends only on α and the equation of state parameter γ . Next, we consider the Lagrangian simulations for several values of c L with constant cq = 1. The data are summarized in Table 4. 1. The asterisks in the final two rows indicates that the data in these rows is not directly comparable to that in the first four rows for reasons discussed below. Nevertheless, the general trends show that the dissipation as measured by η increases as c L increases. Both α and λ are approximately constant for the smaller values of c L . The critical value αc systematically increases. 2. All the simulations in Tables 3 and 4 have oscillations. Indeed, all of the 45 runs with differing artificial viscosity parameters exhibit oscillations. 3. There are three distinct morphologies of the oscillations. Simulations in the table with c L ≤ 0.3 show a sequence of oscillations, the first of which has the maximum overshoot. In the simulation where c L = 0.35, the first maximum is smaller than the second maximum, while successive maxima decrease monotonically. In the simulation where c L = .4, the first maximum is missing. In the latter two cases, the listed values of η, α, and αc are measured between the second and third peaks, hence the asterisks. 4. The three regimes of oscillatory behavior are illustrated in Fig. 4 and will be discussed further in the next section. The particular values of c L are not meant to indicate sharp changes of behavior, but rather are typical values that exhibit the different behaviors.

26

L. G. Margolin and S. D. Ramsey

8 Overshoots and Oscillations It is usually surmised that the initial overshoot of the numerical shock profile is due to insufficient dissipation in the discretized equations [62]. Both Landshoff [22] and later Caramana et al. [8] noted that the quadratic viscosity controlled the overshoot, but was less effective in damping the subsequent unphysical oscillations. Neither of those papers explicitly connected the trailing oscillations with the overshoot as we have done in our oscillation model. Now, the different patterns of oscillations noted in the previous section raises some questions about the oscillation model. For nomenclature, we designate the peaks by position. The first maximum at the top of the shock is the overshoot, followed by the second peak, etc. The overshoot is not necessarily the maximum value of velocity. Referring to Fig. 4, it appears that the case c L = .4 has no overshoot. What then is the source of the second peak? In the same figure, the overshoot of the case c L = .35 is smaller than velocity of the second peak. In terms of the oscillation model, this would imply the parameter b in eq. (31) is negative, yet the velocity of the third peak is smaller, implying that b is positive. Altogether, there is something incorrect about the oscillation model for those cases. In constructing the oscillation model, we have accounted for all possible truncation terms up to O(x 2 ). It seems likely that a higher order dispersive truncation term would be controlled by the strong first-order dissipation. Further, Table 4 and Fig. 4 both document that in the simulations, the magnitude of all peaks decreases as c L increases.

Fig. 4 This figures illustrates the three regimes of overshoots. The extreme blowup of resolution near the top of the shock exposes the size of the individual computational cells. All three profiles exhibit oscillations with different patterns as described in the text

Structure Functions for Numerical Shocks

27

Fig. 5 The dimensionless ratio of linear to quadratic viscosity is plotted for the cases c L = 0.25, 0.35, 0.40 and all with cq = 1.0. Note that all curves have peaks and minima about the same location and are qualitatively similar. The peak of the overshoot is located approximately at y = −2.0 · 10−4 . The data is numerical in origin, taken from the respective structure functions. The magnitude of the ratio is roughly 1% in the region behind the overshoot, validating the assumption of the oscillation model that the effects of quadratic viscosity can be neglected behind the overshoot

In Figs. 5 and 6, we provide additional insights into the qualitative behavior of the linear viscosity and the α-terms in the simulations. Both figures compare results for the three cases c L = .25, c L = .35 and c L = .40 where cq = 1 in each. 1. Figure 5 shows plots of the ratio of the linear viscous term to the quadratic viscous term for all three cases. The curves show the quadratic viscosity is largest near the inflection point of the profiles where the shock width is measured. However, in the region of the overshoot and behind, the ratio is less than 1% for all three cases, vindicating the assumption that quadratic viscosity can be ignored in the model equations. 2 2. Figure 6 shows plots of the magnitude of the term ddyu2 x 2 . The three curves are again qualitatively similar. Note that this term vanishes at the inflection point as expected. 3. In addition to the qualitative similarity of the three curves, note the quantitative similarity of the cases c L = .35 and cq = .40 in both figures. There is no discordant feature in any of the curves that foretells the absence of an overshoot in the latter case.

28

L. G. Margolin and S. D. Ramsey

2

Fig. 6 The term ddyu2 x 2 is plotted for the cases c L = 0.25, 0.35, 0.40 and all with cq = 1.0 Note that all curves have peaks and minima about the same location and are qualitatively similar. The data is numerical in origin taken from the respective structure functions. The peak of the overshoot is located approximately at y = −2.0 · 10−4

If the observed sensitivity to c L is not the result of our approximations, it is likely inherent in our regularization of the Euler equations, i.e., in the form of the artificial viscosity. In particular, we will focus on the possibility that it is the lack of an artificial heat conduction (see Sect. 3) that is the issue. We can approach that possibility from both a physical and a numerical point of view. Physical heat conduction plays an important role in the shape of natural shock profiles. We allude to that importance in the Appendix where Taylor’s and Becker’s ODEs based on Navier–Stokes theory are discussed; the effect of heat conduction on calculated Navier–Stokes shock profile is quantified in Figs. 3 and 4 of [32] where it is shown to more than double the calculated shock width and to change the profile shape. One might expect a similar result when comparing the Lagrangian structure functions with and without artificial heat conduction. (SPOILER: see Fig. 7 of this chapter.) In his 1987 paper [42], Noh formulated a test problem to illustrate the phenomenon termed wall heating and then proceeded to demonstrate that an artificial heat conduction was effective in eliminating the wall heating. The Noh problem can be posed in 1, 2, and 3 dimensions, in plane, cylindrical, and spherical coordinates; it has become a standard test problem for compressible hydrocodes. Noh stated what is (in our opinion) the most important idea in his paper in the following simple words:

Structure Functions for Numerical Shocks

29

Fig. 7 Comparing the velocity profiles of a Lagrangian simulation with and without artificial heat conduction. The standard Lagrangian simulation used c L = 0.2 and cq = 1. The finite scale simulation used c L = 0.374 and cq = 0.045 in both the artificial viscosity and the artificial heat conduction

“We want to demonstrate that the wall heating error is unavoidable and is already an error in the solution of the differential equations with Q. W. Noh [42]” Finite scale theory extends the Navier–Stokes theory to describe the evolution of finite-sized volumes of fluid, including but not limited to computational cells. Additional terms beyond those of the Navier–Stokes equations appear in the momentum flux, Eq. (4) and in the energy flux, Eq. (5). The finite scale contribution to the momentum flux has exactly the form of the quadratic artificial viscosity, and the finite scale energy flux is essentially the artificial heat conduction postulated by Noh [42]. In [28], it is asserted that those finite scale fluxes have a physical origin. When considering finite-sized volumes, it is argued that the concept of a Lagrangian volume must be generalized to allow the exchange of fluid particles between neighbors. That exchange is constrained to preserve the mass of the Lagrangian volume. However, there may be a net exchange of momentum and of energy between the neighbors, which are represented by the finite scale fluxes of momentum and energy, respectively. Now consider the numerical situation when there is artificial viscosity but no artificial heat conduction. The increase of momentum in the volume implies an increase in the kinetic energy, while the lack of energy flux implies that the total

30

L. G. Margolin and S. D. Ramsey

energy remains constant. Then effectively we have converted heat into motion in violation of the second law of thermodynamics. In the next section, we will briefly summarize the application of the finite scale theory to physical shock structure and demonstrate that it can effectively eliminate overshoots and oscillations.

9 Pursuing the Analogy with Physical Shocks Finite scale theory provides a blueprint for implementing an artificial heat conduction. Here, we show the generalization of the self-similar analysis of Sect. 6 that includes a nonvanishing heat flux. The generalized ODE for velocity in the shock profile leads to small modifications of the width model and the oscillation model.

9.1 Self-similar Analysis of the Full FSE In [30], the traveling wave solution for the full FSE equations is derived under the assumption of a particular Prandtl number. This is the same assumption made by Becker [3] in his solutions of the Navier–Stokes equations, see discussion in the Appendix. In terms of our parameters, we assume κ = γ η. Then the analog of Eq. (21) is derived in [30] as 

du A dy

2



v−u −η v

where the constant ψ=



(γ + 1) 2γ

du − ψ u(u p − u) = 0, dy

⇒

(37)

ψ > 0.

Solving the quadratic equation (37) for the velocity gradient yields the analog of Eq. (22)   1 du = η (v − u) − η2 (v − u)2 + 4ψ Av2 u(u p − u) , dy 2v A

(38)

where we have chosen the minus sign in front of the square root to correspond to a shock moving to the right. Equation (38) may be conveniently rewritten   2ψv du 2 = (v − u) − (v − u) + Bu(u p − u) , dy Bη where the parameter

(39)

Structure Functions for Numerical Shocks

B≡

31

4 ψ Av2 η2

(40)

is dimensionless. Note that these solutions neglect any temperature dependence of the parameter η. Theoretical studies in [30] and numerical studies in [31] both indicated that B is a constant independent of shock strength. In particular, the value B = 11.1 was found to give the best comparisons with experimental data. Remark 9 We emphasize that neither the universality of B nor its magnitude were derived in the cited references. Rather these were hypotheses that were validated and evaluated by comparison with experimental data of Schmidt [54] and Alsmeyer [2]. To make the connection to the numerical shock profile, we will interpret the parameters η and A in the parameter B in terms of the artificial viscosity parameters, i.e., η = c L co x ; A = cq x 2 , then

 B = 11.1 = 4ψ

v co

2

cq , c2L

so the universality of B leads to the prediction cq = 0.329 c2L ,

(41)

for the Mach 3.38 shock. Note that the dimensionless coefficient 0.329 depends on the inverse of the Mach number, so will decrease with the strength of the shock; however, the dependence of cq on the c2L is valid for any strength shock.

9.2 Finite Scale Analysis Models To generalize the width model, we begin with Eq. (40). The velocity at the inflection 2 point, u in f , is determined by setting ddyu2 = 0. Let us more generally define aˆ = u in f /v ; bˆ = u p /v, where in particular for the Mach 3.38 shock, aˆ = 0.403 ; bˆ = 0.684. After some manipulation, we find the shock width for the finite scale shock

32

L. G. Margolin and S. D. Ramsey

   ˆ − a) 1 b(1 ˆ 2 2 c L + c L + CM cq x, W= 2ψM a( ˆ bˆ − a) ˆ where the C = 4ψ

a( ˆ bˆ − a) ˆ . (1 − a) ˆ 2

For the Mach 3.38 shock, C = 1.017 and   W = 2.10 c L + c2L + 11.62cq x.

(42)

(43)

(44)

Notice the similarity of form of Eqs. (44) and (35).   The oscillation model is easily modified by replacing η → η v−u wherever η v appears in Eqs. (32) and (33). For the Mach 3.38 shock, η → 0.597η. Finite scale numerical profile We will refer to structure functions generated by Lagrangian simulations including artificial heat conduction, as finite scale profiles. We use relation between viscous parameters c L and cq derived in Eq. (41). In studies (not shown), we have found that the critical value c L = .37 ensures a monotone velocity profile and the simulation shown here in Figs. 7 and 8 employs that value. The artificial heat conduction qh has the form         dI  2  du  dI   , (45) qh = ρ c L co x   − cq x   dx dx dx where the same values of c L and cq are used in the artificial viscosity and the artificial heat conduction (for thermodynamic consistency). Then qh is added to heat flux just as q is added to the momentum flux. In Fig. 7, we compare the structure function for the finite scale simulation with that of the standard viscosity simulation. Beyond the preservation of monotonicity in the finite scale calculation, the principal difference lies in the foot of the shock where the finite scale profile is considerably diffused relative to the standard profile. This widening of the profile should be recognized as a consequence of the intent to reproduce the shape of the physical shock, i.e., a deliberate result rather than a feature to be criticized. The center structure of the shock is least altered and the measured shock width of the finite scale profile is only 26% wider than the standard profile. This is a reminder that the definition of the shock width sometimes yields counterintuitive results. In Fig. 8, we compare the finite scale structure function with the scaled experimental data for the physical Mach 3.38 shock. The data was digitized from Fig. 1 of [2] and reproduced in Fig. 2 of [31]. The spatial scale of the experimental data, i.e., the abscissa of Fig. 8 was expanded by the factor xo /λ where xo is the initial mesh size and λ is the molecular mean free path, see Eq. 29 in [30]. The agreement of the two profiles qualitatively substantiates the analogy between λ and x given that the FSE accurately predict the experimental data.

Structure Functions for Numerical Shocks

33

Fig. 8 Comparing the simulated velocity profile with artificial heat conduction (continuous curve) with a scaled profile of a physical shock (discrete points). The scaling is described in the text, essentially expanding the coordinate by the ratio x/λ. The finite scale simulation used c L = 0.374 and cq = 0.045 in both the artificial viscosity and the artificial heat conduction

Remark 10 There is a subtle difference between physical viscosity and linear artificial viscosity, namely that the artificial viscosity is proportional to material density while physical viscosity is independent of material density. This leads to changes in the traveling wave ODEs, which are discussed in [19].

10 Nonoscillatory Eulerian Profiles Here, we exhibit the structure function of a nonoscillatory Eulerian simulation and compare it to the analogous structure functions generated by standard and finite scale Lagrangian simulations. The regularization of the Euler equations in nonoscillatory Eulerian algorithms is implicit in the nonlinear differencing of the fluxes. Here, nonlinear is used in the sense of Godunov’s theorem, cf. Sect. 3. The nonoscillatory equations contain non-differentiable functions, e.g., logical functions such as upwinding and minmod choice of slopes, that are easily implemented in the hydrocode, but which are not readily susceptible to analysis. We will be able to calculate the shock width and the (possibility of) an overshoot from the structure functions, but it is not feasible to modify the analysis models for comparisons. However, the comparisons with Lagrangian simulations are illuminating.

34

L. G. Margolin and S. D. Ramsey

Fig. 9 Comparing the simulated velocity profile with the standard Lagrangian artificial viscosity using c L = 0.2 and cq = 1 with a nonoscillatory Eulerian velocity profile using c L = 0 and cq = 1. The Lagrangian profile is slightly steeper and has a small overshoot. The Eulerian profile is monotone

In our Eulerian simulations, we have continued to use artificial viscosity in the Lagrange step. The nonoscillatory algorithms are constructed to identify the appearance of new maxima/minima in the solution and to modify or eliminate them by adding extra dissipation. In direct Eulerian methods, this is sufficient to eliminate unphysical oscillations. However, in the Lagrange-plus-remap formulation, the Lagrangian calculation may (will) create peaks that cannot be identified as unphysical by the remap step that follows. One might be concerned that the combination of explicit dissipation in the Lagrange step and the implicit dissipation of the remap step would be too diffusive, resulting in unnecessarily wide shocks. It was shown in [36] that the implicit dissipation is adaptive in calculations of turbulent flows, i.e., that increases in the explicit dissipation were compensated by decreases in the implicit dissipation. Here we find a similar adaptivity exists in our calculations of the numerical shock profile. In Fig. 9 we compare the structure functions generated by a standard Lagrangian simulation with that generated by a Lagrange-plus-remap Eulerian simulation. Both calculations use the artificial viscosity parameters c L = .2 and cq = 1. The Eulerian simulation has a measured width about 17% wider than the Lagrangian simulation, and is completely monotone. The Lagrangian simulation has an overshoot that is less than 1% of the total jump of velocity. In Fig. 10, we compare the structure functions generated by a Lagrangian simulation using artificial heat conduction with that generated by a Lagrange-plus-remap

Structure Functions for Numerical Shocks

35

Fig. 10 Comparing the velocity profiles of a finite scale Lagrangian simulation with a nonoscillatory Eulerian profile. The finite scale simulation used c L = 0.374 and cq = 0.045 in both the artificial viscosity and the artificial heat conduction. The Eulerian simulation used c L = 0 and cq = 1. The similarity of the top of the profiles indicates the presence of implicit artificial heat conduction in the nonoscillatory algorithm

Eulerian simulation. Both profiles are monotone and remarkably similar for the upper half of the shock. This suggests the presence of an implicit artificial heat conduction in the nonoscillatory algorithm. The foot of the profiles are quite different; we remind that the goal of the finite scale artificial heat conduction was to reproduce the shape of the physical shock, not to create the steepest profile.

11 Summary and Discussion Here, we summarize the main results of this chapter and hint at future directions. 1. Structure functions are a simple technique to accumulate high-fidelity data about numerical shock profiles. 2. The Euler equations regularized by artificial viscosity have simple traveling wave ODEs for material velocity that are readily evaluated. Further, the ODEs enable models for shock width and for oscillatory behavior that can be used to analyze the structure functions.

36

L. G. Margolin and S. D. Ramsey

3. Lagrangian methods using standard “linear plus quadratic artificial viscosity" provide accurate simulations of shock profile when compared to the analytic solutions of the regularized Euler equations. However, those calculations are contaminated by overshoots and unphysical oscillations that can be minimized but not eliminated. 4. The overshoots are the result of the lack of thermodynamic consistency of the standard artificial viscosity models. 5. Thermodynamic consistency can be restored by adding terms representing a linear and a quadratic artificial heat conduction to the discretized equations. The FSEs provide a derivation, an understanding and a template for those terms. 6. Modern Eulerian methods using nonoscillatory differencing offer an alternative path that provide steep shock profiles while eliminating the unphysical overshoots. The structure functions and associated results of this chapter point the way to several directions for future work. Although the structure functions constructed in this chapter depend on planar symmetry, i.e., on the similarity variable x − vt, the underlying idea is not so constrained. It should be possible to extend the concept of the structure functions to cylindrical and spherical symmetries and include other elementary problems such as the Noh problem, the Sedov problem and the Guderley problem [39]. It would also be interesting to extend the analogy between the regularization of Euler equations and the finite scale theory to multi-dimensions. The finite scale additions to the momentum and energy fluxes, which we have used as templates for artificial viscosity and heat conduction, will depend on the shape of the volume, e.g., the computational cell. Implementing such generalizations may improve the ability of the code to preserve physical symmetries. In this chapter, we have used the Lagrange-plus-remap formulation for Eulerian hydrodynamics. This is just a special case of a more general arbitrary Lagrangian– Eulerian (ALE) formulation in which the mesh motion is arbitrarily controlled by the user. In general, any remap will be dissipative implying that less regularization will be needed in the Lagrangian phase of the calculation. It would be useful to express the viscosity parameters c L and cq locally (in space) and dynamically (in time). An interesting start in that direction was published in [46]. Finally, we note that the structure function should be easily integrated and evaluated in most hydrocodes. This would provide a simple and well-defined methodology for intercode comparisons. Dedication William F. Noh spent a career at Livermore National Laboratory. Best known for the challenging test problem named for him, Bill made many important contributions to computational fluid mechanics. Understanding the good, the bad and the ugly aspects of artificial viscosity was one of Bill’s passions. In remembrance of his insights and in appreciation for many enlightening conversations (LGM), the authors dedicate this chapter to him.

Structure Functions for Numerical Shocks

37

Appendix: A Brief History of Shock Structure Studies The classical era of studying shock structure was hampered by a lack of understanding of thermodynamics, i.e., the conservation of energy (first law) and the lack of conservation of entropy (second law). The modern era, ushered in by the connection of continuum theory to kinetic theory, has uncovered new and as yet unresolved issues with Navier–Stokes theory. Here we present a brief history. The Classical Era The theoretical study of physical shock structure begins with the 1850 paper of Stokes [56]. Stokes understood the steepening of density waves by the nonlinearity of advection tending toward discontinuity. He also understood that the dissipative effects of viscosity would prevent the actual formation of a discontinuity. However, the principle of conservation of energy was just being formulated at the time and Stokes mistakenly assumed that entropy was conserved in the shock. These issues were only clarified in the 1870 paper of Rankine and the slightly later work of Hugoniot. These two papers also also made explicit the jump conditions, i.e., the relation between pre- and post-shock values of density, fluid velocity, temperature, and pressure. These early works are reproduced in [18] and discussed in detail in the excellent history of Salas [53]. The classical understanding of shock structure was summarized in the 1910 papers of Rayleigh [45] and of Taylor [58]. There, ODEs for shock structure as predicted by the Navier–Stokes equations were derived for two special cases, one with only viscosity and the other with only heat conduction. From these ODEs, one can extract the shock width W (defined in Eq. (25) and shown to be proportional to the coefficients of bulk viscosity μ and of heat conductivity k, respectively. The Interlude The formulation of the Chapman–Enskog expansion was a game-changer in several ways. Chapman and separately Enskog derived the continuum-level Navier– Stokes equations from the more fundamental molecular-level Boltzmann equation. The assumptions made in the derivation constrain the range of validity of Navier– Stokes theory. Perhaps of equal importance, the Chapman–Enskog theory establishes the relation of the microscopic parameter λ, i.e., the molecular mean free path, to the transport coefficients μ and k. The Chapman–Enskog approximation is a perturbation expansion whose perturbation parameter is the Knudsen number defined as a ratio of length scales K n ≡ λ/L ,

(46)

where L is length characteristic of the macroscopic scales of a problem. In the case of shock structure, we would identify L with the shock width W. Then a principal assumption of the expansion is that K n x0 ,

(60)

where U = [ρ, ρu, ρ E, ρφa , ρφa E a ] , U L and U R are constant states on both sides of certain position x = x0 . The solution of this Riemann problem, denoted as RP(U L , U R ), has self-similarity, U(x, t) =: v(ξ ),

ξ=

x − x0 , t > 0. t

(61)

Features of the Novel model We summarize that this novel reduced model has the following properties: • The system of equations for each material is symmetric, where (42b) and (44) are formally consistent with the seven-equation model. • This model reflects the energy exchange of two materials. Hence, it is favorable for the simulation of shocks near material interfaces. • Similar to the form of the six-equation model, it is helpful to take account of the energy equation of each material to keep the positivity of the volume fraction. • This model is easy to be discretized and extended to high-order accuracy and multi-dimensional cases. These properties imply that the corresponding numerical schemes for this model must be symmetric for material a or b. The energy exchange between two materials helps ensure the positivity of the internal energies in this novel model.

3 Numerical Approaches for Two-Material Flow Models with Instantaneous Pressure and Velocity Equilibrium In this section, we propose a numerical algorithm for the novel reduced model (42) to simulate the numerical phenomena of the compressible multi-material flow involving material interfaces, shocks and rarefaction waves more reasonably. Since the system

240

X. Lei and J. Li

(42a) is conservative, the conventional Godunov scheme is applied. However, the system (42b) is non-conservative, its numerical discretization is inspired by a fraction step method of the seven-equation model [34]. Let us start with the introduction of the conventional Godunov-type scheme by using the five-equation reduced model as an example [44].

3.1 Conventional Godunov-Type Schemes for the Five-Equation Reduced Model The conventional Godunov-type scheme is a finite volume method on the Eulerian grid. For the  1-D case,the computational domain [0, L] is divided into M fixed grid cells Ii = xi− 21 , xi+ 21 , i = 1, 2, . . . , M, with the grid size x = xi+ 21 − xi− 21 =   L/M, the cell interface xi+ 21 = i x, and the cell center xi = i − 21 x. The first four equations of the five-equation system (25) are updated from time tn to tn+1 = tn + t by the conservative Godunov scheme      n n − F W i− , W in+1 = W in −  F W i+ 1 1 2

(62)

2

where  = t/ x, W = U (4) = [ρ, ρu, ρ E, ρφa ], W in is the integral average of W over the cell Ii at time tn , and n (4) W i+ (ξ = 0), 1 = v 2

ξ=

x − xi+ 21 t − tn

, t > tn ,

(63)

with the superscript (4) referring to the four components of a vector. Here, v(ξ ) is the n ) that can be evaluated by the solution (61) of the Riemann problem RP(U in , U i+1 exact Riemann solver [48] or approximate Riemann solvers [44]. In early studies of a five-equation transport model [4], the evolutionary equation for the volume fraction z a is a simple transport equation ∂z a + u · ∇z a = 0, ∂t

(64)

∂z a + div(z a u) = z a div u. ∂t

(65)

which can be rewritten as

It is update by a Godunov-type scheme    n n n n n u , (z a )in+1 = (z a )in −  (z a u)i+ 1 − (z a u) 1 − (z a )i 1 − u 1 i− i+ i− 2

2

2

2

(66)

An Energy-Splitting High-Order Numerical Method for Multi-material Flows

241

which easily preserves positivity of the volume fraction. For the five-equation reduced model, the evolution equation (24) for the volume fraction ∂z a + div(z a u) = (z a + ) div u, ∂t

(67)

is updated by    n n n n n (z a )in+1 = (z a )in −  (z a u)i+ u . (68) 1 − (z a u) 1 − (z a + )i 1 − u 1 i− i+ i− 2

2

2

2

  n n makes it difficult to maintain positivity of The additional term in u i+ 1 − u 1 i− 2 2 the volume fraction. As shown in [44, Fig. 10], for a mixture Hugoniot (epoxy/spinel) problem, the numerical result of the five-equation reduced model is close to that of the sevenequation model, but may be far from that of the five-equation transport model. It shows that the additional term in the five-equation reduced model has an obvious effect, and the five-equation transport model is not suitable for simulating multimaterial flows. It is worth noting that there are many difficulties and challenges in numerical discretization of the non-conservative system. In the Eulerian framework, numerical solutions of non-conservative numerical schemes may converge to wrong solutions [28], providing incorrect partition of internal energies or shock wave velocity in the shock layer [51]. For the polytropic gases, a Lagrange-projection method in [53] is equivalent to the conventional Godunov-type scheme for the five-equation reduced model. We note that a conventional average of a non-conservative variable such as the volume fraction is not physically correct. An epoxy-spinel shock tube example in [53] illustrates that numerical solutions of the conventional Godunov-type scheme are difficult to converge to the exact solution for strong multi-material shocks. It results in an incorrect partition of shock energy between two materials [53]. In order to achieve more correct partition of the shock energy, we move on to another numerical approach, i.e. solving a non-equilibrium model with stiff relaxation.

3.2 Fractional Step Method for the Non-equilibrium Model As summarized in the above discussion on discretizing the five-equation reduced model for the presence of multi-material shocks, it is a difficult issue about the convergence of the numerical solutions and the positivity preserving of the volume fraction. To overcome these difficulties, non-equilibrium models with stiff relaxation are utilized. The non-equilibrium model, composed of the seven-equation model and the six-equation model, is a non-conservative hyperbolic model involving relaxation terms. A commonly used numerical approach of the non-equilibrium models is implemented by a fractional step technique with two steps: hyperbolic evolution

242

X. Lei and J. Li

and stiff relaxation. In what follows, we introduce the fractional step method for the seven-equation model (1). The seven-equation system (1) can be written in compact form as ∂U + div F (U) + H(U) · ∇z a = λRλ (U) + μRμ (U), ∂t

(69)

with ⎡ ⎡ ⎤ ⎤ ⎤ 0 ρφa ua ρφa ⎢ − pI I ⎥ ⎢ρφa ua ⊗ ua + z a pa I ⎥ ⎢ ρφa ua ⎥ ⎢ ⎢ ⎥ ⎥ ⎥ ⎢ ⎢− pI u I ⎥ ⎢ (ρφa E a + z a pa )ua ⎥ ⎢ρφa E a ⎥ ⎢ ⎢ ⎥ ⎥ ⎥ ⎢ ⎢ ⎥ , H(U) = ⎢ 0 ⎥ , (70a) ⎥ ρφb ub U=⎢ ⎢ ⎥ ⎥ ⎢ ρφb ⎥ , F (U) = ⎢ ⎢ pI I ⎥ ⎢ ρφb ub ⊗ ub + z b pb I ⎥ ⎢ ρφb ub ⎥ ⎢ ⎢ ⎥ ⎥ ⎥ ⎢ ⎣ pI u I ⎦ ⎣ (ρφb E b + z b pb )ub ⎦ ⎣ρφb E b ⎦ za 0 uI ⎤ ⎤ ⎡ ⎡ 0 0 ⎥ ⎢ ub − ua ⎥ ⎢ 0 ⎥ ⎥ ⎢ ⎢ ⎢uI · (ub − ua )⎥ ⎢ pI ( pb − pa )⎥ ⎥ ⎥ ⎢ ⎢ ⎥. ⎥ , Rμ (U) = ⎢ 0 0 Rλ (U) = ⎢ (70b) ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ ua − ub ⎥ ⎢ 0 ⎥ ⎥ ⎢ ⎢ ⎣uI · (ua − ub )⎦ ⎣ pI ( pa − pb )⎦ 0 pa − pb ⎡

As shown concretely in [34], the initial value (IV) problem of the seven-equation system (69) is solved by the hyperbolic evolution step ⎫ ∂U + div F (U) + H(U) · ∇z a = 0⎬ ∂t =⇒ U ∗ , ⎭ n IV: U(x, tn ) = U

(71)

the velocity relaxation step ⎫ ∂U = λRλ (U)⎬ ∂t =⇒ U ∗∗ , ∗ ⎭ IV: U(x, tn ) = U

(72)

and the pressure relaxation step ⎫ ∂U = μRμ (U)⎬ ∂t =⇒ U n+1 . ∗∗ ⎭ IV: U(x, tn ) = U

(73)

Let the superscript ∗ be the index symbol for the solution obtained after the hyperbolic evolution step and the superscript ∗∗ be the index symbol for the solution obtained

An Energy-Splitting High-Order Numerical Method for Multi-material Flows

243

by the velocity relaxation procedure. In the hyperbolic evolution step, the hyperbolic system in (71) can be numerically solved by the conventional Godunov-type scheme introduced in Sect. 3.1.

3.2.1

Stiff Velocity Relaxation Step

In the velocity relaxation step, we solve the system of ordinary differential equations (72) in the limit λ → ∞ for relaxing the velocities of two materials to an equilibrium value. According to the velocity equilibrium, we get ∗∗ ∗∗ = φa∗ ua∗ + φb∗ u∗b ua∗∗ = u∗∗ b = uI = u

(74)

and then the amount of internal energy exchange caused by the velocity equilibrium satisfies ∗∗ ∗ (75) E a∗∗ = E a∗ + u∗∗ I · (ua − ua ), i.e.,

3.2.2

1 ea∗∗ = ea∗ + (u∗∗ − ua∗ )2 . 2

(76)

Stiff Pressure Relaxation Step

In the pressure relaxation step, we solve the system of ordinary differential equations (73) in the limit μ → ∞ for relaxing the pressures of two materials to an equilibrium value. In this system, the total energy of the material a satisfies the equation ∂z a ∂ρφa E a = μpI ( pb − pa ) = − pI . ∂t ∂t

(77)

If we integrate (77) from tn to tn+1 to get ρφa



E an+1

−

E a∗



tn+1 +

pI tn

∂z a dt = 0, ∂t

(78)

and approximate pI as a constant pI ≈ ( p n+1 + p ∗∗ )/2 (cf. [34, 47]), the difference in internal energy after and before pressure relaxation is (ρφa ea )n+1 − (ρφa ea )∗ = (ρφa E a )n+1 − (ρφa E a )∗ tn+1 = −

pI tn

∂z a dt = − ∂t

n+1 za

pI dz a ≈ − z a∗

 p n+1 + p ∗  n+1 z a − z a∗ . 2

(79) (80)

244

X. Lei and J. Li

The incorrect partition of the internal energies between two materials in the shock layer also appears in the non-equilibrium model with stiff relaxation. A two-phase flow example in [44, Fig. 7] suggests that it is not easy to simulate a non-equilibrium model correctly by the fractional step method.

3.3 Finite Volume Methods for the Novel Reduced Model Since there are many difficulties in the numerical discretization of the five-equation models or non-equilibrium models with stiff relaxation, we try to design numerical methods for the novel reduced model. This model includes an equation of mass fraction and a one-material energy equation coupling with the Euler equations. As far as the strong multi-material shocks are concerned, the exchange of energy between two materials is well worth investigating. A benefit of this model is that its form is similar to that of the conservative Euler system, which allows to suit for the conventional Godunov-type scheme introduced in Sect. 3.1. A second-order accurate extension of the conventional Godunov-type is made using the space-time coupled generalized Riemann problem (GRP) solver [11, 13], which was originally proposed in [10] with extension to combustion models, e.g. [8, 9] and general hyperbolic balance laws [12]. The reason of making this choice is the inclusion of thermodynamics into the scheme [38] that is important in the simulation of compressible multi-material flow problems. An important issue in this type of numerical methods is related to the volume fraction positivity in the presence of shocks and even in the presence of strong rarefaction waves. Indeed, when dealing with liquid-gas mixtures, for example, the liquid compressibility is so weak that the pressure tends to be negative, resulting in computational failure in the gas sound speed computation. Such a situation occurs frequently in cavitation test problems. For this numerical method of the novel reduced model, we ameliorate this issue by simulating the partition of shock energy more accurately. Next, we introduce the concrete implementation process.

3.3.1

2-D Conventional Godunov-Type Scheme

We discretize the governing equations (42) with a cell-centered finite-volume scheme over a two-dimensional (2-D) computational domain divided into a set of polygonal cells {i }. The integral average of the solution vector U(x, tn ), x = (x, y) over the cell i at time tn is given by U in . Taking rectangular cells as an example, we denote U nj (i) as the integral average over the j-th adjacent cell  j (i) of i , as shown in Fig. 1. The volume fraction z a in i is expressed  the pressure equilibrium con by solving dition (45). The Riemann problem RP U in , U nj (i) is solved along the unit outward normal vector n j of the j-th boundary, and the corresponding Riemann solution is denoted by U i,n j . Detailed process of the computation for the Riemann solution can

An Energy-Splitting High-Order Numerical Method for Multi-material Flows

245

Fig. 1 Rectangular cells and the distribution of the solution

be implemented by the exact Riemann solver shown in Sect. 2.4 or an approximate Riemann solver in [44]. n+ 1 Approximating u in the governing Eq. (42b) as a constant ui 2 := (uin + uin+1 )/2 (similar to the approximation for pI in (79)), the finite-volume scheme of (42) with the Godunov fluxes is given by W in+1 = W in −

4 

  ij  F j W i,n j ,

j=1 n+ 21

(ρφa E a )in+1 = (ρφa E a )in − ui n+ 1 +ui 2



· g j U i,n j



+d



U i,n j

(81a)

    (ρφa u)in+1 − (ρφa u)in − ij  f j U i,n j

  n+ 21   ( pu) j − ui · ( p I) j ,

(81b)

where ij = t L j /|i |, L j is the length of the jth boundary of the cell i , |i | is the volume of i , f (U) = ρφa E a u,

g(U) = ρφa u ⊗ u, d(U) = z a + ,

(82)

 j = (•) · n j . and (•) We notice that the above discretization is different from the procedure of stiff velocity and pressure relaxation for internal energy in the fractional step method. In addition, for the polytropic gases, the positivity of volume fractions is guaranteed as long as the internal energy of each material is positive according to (45). Overall, we obtain a conventional Godunov-type scheme for the novel reduced model in two dimensions. The discretization of (44) for material b is actually updated in the following way (ρφb E b )in+1 n+ 21

+ui

=

(ρφb E b )in

−

n+ 1 ui 2



(ρφb u)in+1

−

(ρφb u)in



−

ij

 n+ 1  ( f b ) j U i, j 2

  n+ 1   n+ 1   n+ 21 n+ 21 n+ 1 , ( pu) j − ui 2 · ( · (g b ) j U i, j 2 + db U i, j 2 p I) j

(83)

246

X. Lei and J. Li

where f b (U) = ρφb E b u,

g b (U) = ρφb u ⊗ u, db (U) = z b − .

(84)

Combined with the discretization of total momentum in (81a), the sum of (81b) and (83) is the discretization of total energy in (81a). Thus, such a scheme is a numerical approach implemented by splitting the energy equation of two materials, which is termed as the energy-splitting Godunov scheme (ES-Godunov for short) in [37].

3.3.2

Second-Order Accurate Grp Extension

We make a second-order accurate extension of ES-Goduov by using a 2-D generalized Riemann problem (GRP) solver [10–13] (ES-GRP for short). In each cell, we project the solution vector U into the space of piecewise linear functions U in (x) = U in + σ in · (x − x i ),

(85)

of where σ in is the gradient of U inside the cell i at time t =  tn , and x i is the centroid  n n i . The quasi 1-D generalized Riemann problem GRP U i (x), U j (i) (x) is solved at the center x i, j of the j-th boundary with accuracy of second order, and then the associated Riemann solution U i,n j and the corresponding temporal derivative (∂U/∂t)i,n j are determined. Referring to the conventional Godunov-type scheme (81), the two-dimensional finite-volume GRP scheme for (42) is written as W in+1

=

W in

−

4 

 n+ 1  ij  F j W i, j 2 ,

j=1 n+ 21

(ρφa E a )in+1 = (ρφa E a )in − ui n+ 21

+ui

(86a)

    n+ 1 (ρφa u)in+1 − (ρφa u)in − ij  f j U i, j 2

  n+ 1   n+ 1   n+ 21 n+ 21 n+ 1 , ( pu) j − ui 2 · ( · g j U i, j 2 + d U i, j 2 p I) j

(86b)

n+ 1

where the mid-point value U i, j 2 is determined by n+ 1

U i, j 2 = U i,n j +

t 2



∂U ∂t

n .

(87)

i, j

Consistent with the whole computational procedure of the conventional Godunovtype scheme, a second-order GRP scheme for the novel reduced model in two dimensions is obtained. The Abgrall’s criterion of this second-order scheme was proved in [37], which requires uniform velocity and pressure to be preserved for multi-material flows. Therefore, this type of scheme has the non-oscillatory property.

An Energy-Splitting High-Order Numerical Method for Multi-material Flows

247

Case of the polytropic gases For the polytropic gases, the volume fraction of material a in i is expressed as (z a )in =

(ρφa ea )in (γa − 1) n k=a,b (ρφk ek )i (γk − 1)

(88)

through (48). The effective ratio of specific heats in i is calculated as γin

=

n k=a,b (ρφk ek )i γk (ρe)in

,

(89) n+ 21

by using (33), and the mid-point value of the ratio of specific heats γi, j interfaces is given by n+ 1  z k,i,2j 1 . = n+ 1 γi, j 2 − 1 k=a,b γk − 1

on cell

(90)

For readers’ convenience and completeness of the presentation, we put the GRP solver in Appendix.

4 Numerical Results Some numerical results of the novel reduced model by using the conventional Godunov-type schemes are presented in this section. These numerical results demonstrate the effectiveness of the schemes to guarantee the positivity of volume fractions and the correct simulation of multi-material shocks. The energy exchange is considered to guarantee approximate energy apportion of each material numerically across multi-material shocks and thus positive volume fractions. In this section, we compare the numerical results of the four-equation model (30) and our novel reduced model for 1-D examples, for which there are numerical difficulties due to non-physical oscillations generated at material interfaces when conservative schemes are applied to the four-equation model. We abbreviate Is-Godunov for the Godunov scheme of the four-equation model (30) with the isothermal hypothesis, UPV-Godunov for the Is-Godunov with the energy correction in [6] based on a UPV flow to suppress non-physical oscillations, in addition to the abbreviations: ES-Godunov and ES-GRP. The first numerical example is proposed to show the influence of the energy exchange terms on the positivity of volume fractions. The test for simulating the multi-material shock is contained in the second and third numerical examples, and these two examples compare the numerical results by the energy-splitting schemes (ES-Godunov, ES-GRP) and other common schemes (Is-Godunov, UPV-Godunov). The last three examples are about the interaction between shocks and material inter-

248

X. Lei and J. Li

faces in two dimensions. Through the comparison with the corresponding physical experimental results, the numerical results show that the current schemes for the novel reduced model perform well for two-dimensional cases with very sharp interfaces.

4.1 Test of Volume Fraction Positivity This is an inward two-fluid compression problem, for which the initial discontinuity at x = 0.12 separates air with γa = 1.4 in the left from wolfram with γb = 3.0 in the right. The initial data in the entire computational domain [0, 0.15], composed of 250 cells, are given as (ρ, u, p, φa , z a ) = ( 0.00129 , 0 , 1.01325 , 1 , 1), x < 0.12, (ρ, u, p, φb , z b ) = ( 19.237 , −200 , 1.01325 , 1 , 1), x > 0.12.

(91)

The left boundary is a solid wall and the right boundary has an inflow condition. This problem has exceedingly huge density ratio and velocity gradient. We use NO-EX to represent no energy exchange term caused by the stiff relaxation in the scheme, and list the numerical results of z a in the 199-th cell at time steps from 1 to 5 in Table 1. The numerical results show that without the process of energy exchange, the volume fraction of air at the interface becomes negative values, which immediately ruins the numerical simulation. It shows the necessity of the energy exchange terms in the current methods.

4.2 Two-Fluid Shock-Tube Problem This is a two-fluid shock-tube problem in [1]. The discontinuity initially at x = 0.3 separates air with γa = 1.4, Cv,a = 0.72 in the left from helium with γb = 1.67, Cv,b = 3.11 in the right. Then the initial data in the entire computational domain [0, 1], composed of 100 cells, are given by

Table 1 The interfacial volume fraction z a at advancing time steps solved by different schemes Scheme Step 1 Step 2 Step 3 Step 4 Step 5 ES-Godunov(NO-EX) ES-GRP(NO-EX) ES-Godunov ES-GRP

−0.09652 −0.09652 0.96143 0.96143

−0.07664 −0.07707 0.90025 0.88862

−0.05920 −0.05412 0.80975 0.71710

−0.04542 −0.03272 0.68928 0.45799

−0.03496 −0.02203 0.54981 0.19412

An Energy-Splitting High-Order Numerical Method for Multi-material Flows

(a) Density

(b) Pressure

(c) Velocity

(d) Mass fraction of material a

249

Fig. 2 Results of the two-fluid Sod problem at t = 0.008

(ρ, u, p, φa , z a ) = ( 1 , 0, 25, 1, 1, 1), x < 0.3, (ρ, u, p, φb , z b ) = ( 0.01 , 0, 20, 1, 1, 1), x > 0.3.

(92)

The exact solution of the shock-tube problem consists of a left-propagating rarefaction wave, a contact discontinuity moving at the speed of 0.83, and a right-propagating shock wave at the speed of 58.35. We compare the solutions computed by different schemes at time t = 0.008. The numerical solutions computed by Is-Godunov, UVP-Godunov, ES-Godunov and ES-GRP are shown in Fig. 2, in which the solid gray curves are the exact solution. The results solved by the current energy-splitting schemes are much closer to the exact solution than that by Is-Godunov and UPVGodunov, which shows the good performance of the current schemes in simulating multi-material shocks.

250

X. Lei and J. Li

4.3 Shock-Interface Interaction This is a shock-interface interaction problem. The interface initially at x = 0.2 separates material a with γa = 1.35, Cv,a = 2.4 in the left from material b with γb = 5.0, Cv,b = 1.5 in the right. These two materials correspond to high explosive products in the left and a confining material in the right [6]. The interface and a shock wave with the shock Mach number Ms = 1.5 initially at x = 0.16 propagate to the right at the speed of 0.5 and 1.74, respectively. Then the initial data in the computational domain [0, 1], composed of 125 cells, are given by x < 0.16, (ρ, u, p, φa , z a ) = ( 1.1201 , 0.6333 , 1.1657 , 1 , 1), 1 , 1 , 1), 0.16 < x < 0.2, (ρ, u, p, φa , z a ) = ( 1 , 0.5 , 1 , 1 , 1), x > 0.2. (ρ, u, p, φb , z b ) = ( 0.0875 , 0.5 ,

(93)

At time t = 0.0322, the interface is impacted by the shock wave. The resulting wave pattern after the interaction consists of a reflected rarefaction wave, an interface at the speed of 0.67, and a transmitted shock at the speed of 8.32. We compare the profiles of pressure and internal energy by using different methods at t = 0.07 in Fig. 3. Serious errors of the internal energy occur at the interface in Is-Godunov and UVP-Godunov solutions. In contrast, the current method can produce much better results.

(a) Pressure

(b) Specific internal energy

Fig. 3 Results of the shock-interface interaction problem at t = 0.07

An Energy-Splitting High-Order Numerical Method for Multi-material Flows

251

4.4 Shock-Bubble Interactions This example is about the interaction problem of a planar shock wave with a cylindrical gas bubble. This problem is motivated by the experiments in [26]. In the experiments, a weak shock with the shock Mach number Ms = 1.22 propagates from atmospheric air into a stationary cylindrical bubble filled with lighter helium or heavier Refrigerant 22 (R22). The computational domain [0, 2.5] × [0, 0.89] composes of 2500 × 890 square cells and the initial positions of shock and material interface are is set in Fig. 4. In this figure, L x = 2.5, L y = 0.89, L D = 0.5, L A = 0.375 and L B = 0.125. The upper and lower boundaries are solid wall boundaries, whereas the left and right boundaries are non-reflective. The air outside and the gas inside the bubble are assumed initially to be in the temperature and pressure equilibrium. For the helium bubble case, the gas in the bubble is assumed as a helium-air mixture where the mass fraction of air is 28%, which is explained in [26]. These materials are regarded as ideal gases, and the corresponding fluid state and parameters taken from [50] are presented in Table 2. Figure 5 compares the numerical shadow-graph images of the shock-helium bubble interaction problem and the shock-R22 bubble interaction problem by ESGodunov and ES-GRP, corresponding to the experiments at different times in [26]. In order to better compare the results, the initial interface (red curves) is added to the numerical shadow-graph images. Since the sound speed of helium inside the bubble is much greater than the sound speed of the air outside, the helium bubble acts as a divergent lens for the incident shock. In contrast, as the sound speed of R22

Fig. 4 Diagram of the shock-bubble interaction problem Table 2 Some parameters for the shock-bubble interaction problems in front of the shock wave Gas Air Helium + 28% Air R22 γ Cv ρ p u

1.40 0.72 1 1 0

1.648 2.44 0.182 1 0

1.249 0.365 3.169 1 0

252

X. Lei and J. Li

(a)

(a)

(b)

(b)

(c)

(c)

(d)

(d)

(e)

(e)

(f)

(f)

Fig. 5 Numerical shadow-graph images of the shock bubble interaction with Ms = 1.22. On the left is the results of helium bubble obtained by ES-Godunov (column 1) and ES-GRP (column 2) at experimental times (μs): a 32, b 62, c 72, d 102, e 427 and f 674; on the right is the results of R22 bubble obtained by ES-Godunov (column 3) and ES-GRP (column 4) at experimental times (μs): a 55, b 135, c 187, d 247, e 342 and f 1020. The corresponding experimental shadow-photographs can be found in [26, Figs. 7 and 11]

inside the bubble is much lower than that of the air outside, the R22 bubble acts as a convergent lens. The numerical shadow-graph images show a very good agreement between the second-order numerical simulations and the laboratory experiments. As ES-GRP is used, the stability issue becomes weaker along the material interface than the numerical results in [50] and much clearer discontinuity surfaces are observed than those by ES-Godunov.

An Energy-Splitting High-Order Numerical Method for Multi-material Flows

253

4.5 Two-Fluid Richtmyer-Meshkov Instability This test problem concerns the multi-material Richtmyer-Meshkov instability (RMI) phenomenon in two dimensions. The RMI phenomenon was originally theoretically predicted by Richtmyer and subsequently observed in experiments by Meshkov. Here, we simulate a simple test example of a plane shock hitting a sinusoidal perturbed interface, separating SF6 and air. A schematic diagram of the initial flow configuration is shown in Fig. 6, and the initial configuration of the perturbed interface is generated with [45] xd = 0.4 + 0.1 sin(2π(y + 0.25)), −0.5 < y < 0.5,

(94)

which separates the SF6 with γa = 1.094 in the left from the air with γb = 1.4 in the right. The initial conditions are shown in Table 3 from two cases: (i) The initial density of SF6 is ρ0 = 5.04. The computational domain is [0, 16] × [−0.5, 0.5] composed of 2048 × 128 square cells. (ii) The initial density of SF6 is ρ0 = 0.0005. The computational domain is [0, 5] × [−0.5, 0.5] composed of 640 × 128 square cells. The upper and lower boundaries satisfy a symmetric boundary condition, whereas the left and right boundaries are non-reflective boundaries. The initial conditions are put in Table 3, including a material interface with a large density difference. For these two cases, the development of instability is sensitive to the initial perturbation. The incident shock hits the material interface, resulting in a transmitted shock and a reflected shock. The accelerated interface forms a rolled-up spike shape.

Fig. 6 Diagram of the two-fluid RMI test problem Table 3 Initial data of the two-fluid RMI test problems Position Gas ρ p x < xd xd < x < 0.7 0.7 < x < 7

SF6 Air Air

ρ0 1 1.4112

1.24 1.24 0.8787

u

v

0.7143 0.7143 1.1623

0 0 0

254

X. Lei and J. Li

(a) 1-D GRP solver

(b) 2-D GRP solver

Fig. 7 Density distribution for the two-fluid RMI problem (Case (i)) by ES-GRP at t = 8.25

(a) 1-D GRP solver

(b) 2-D GRP solver

Fig. 8 Mass fraction distribution for the two-fluid RMI problem (case ii) by ES-GRP at t = 6.5. See [36]

The numerical results of Case (i) at t = 8.25 are displayed in Fig. 7. For the same problem, the numerical results based on conservative variables in [45, Fig. 25] generate non-physical oscillations at the material interface, which is not observed in the results by ES-GRP. The numerical results at t = 6.5 of Case (ii) by ES-GRP are shown in Fig. 8. It is observed in these figures that ES-GRP with the 2-D GRP solver obtains a better resolution for vortex structures compared with those by the 1-D GRP solver. The differences between the two solvers are described in Appendix, reflecting the ability of the 2-D GRP solver in capturing transversal effects.

4.6 Water-Air Shock-Interface Interaction Problems We test a shock-interface interaction example with a gas-liquid interface in [25, 46, 59]. This example illustrates the interaction of a shock in water with a

An Energy-Splitting High-Order Numerical Method for Multi-material Flows Table 4 Initial data of the water shock-air bubble interaction problem Region φair ρ (kg/m3 ) p (Pa) u (m/s) Inside bubble Outside bubble, pre-shock Outside bubble, post-shock

255

v (m/s)

1 0

1 1000

1 × 105 1 × 105

0 0

0 0

0

1323.65

1.9 × 109

−681.58

0

cylindrical air bubble. The density ratio of two materials across the bubble interface is very large. In this example, we use the so-called stiffened gas EOS ek =

pk + γk πk , ρk (γk − 1)

(95)

for each material, where πk is the constant stiffening pressure. For air and water, the constants in (95) are γair = 1.4, πair = 0Pa and γwater = 4.4, πwater = 6 × 108 Pa, respectively. For this water shock-air bubble interaction problem, the computational domain is [0, 15 mm] × [0, 12 mm] composed of 600 × 480 square cells and the initial positions of shock and material interface are set in Fig. 4 with L x = 15 mm, L y = 12 mm, L D = 6 mm, L A = 1.2 mm, L B = 1.8 mm. Boundary conditions are periodical in the vertical direction and non-reflection at the left and right boundaries. The initial conditions of this example are in Table 4. The pressure images and the density gradients of numerical results computed by ES-GRP with the acoustic 2-D GRP solver are shown in Figs. 9 and 10, respectively. These numerical results agree well with the numerical results in the earlier literature [25, 46, 59], and no obvious numerical oscillation phenomenon is observed near the bubble interface.

5 Discussion The study of compressible multi-material flows is an important topic in theory, numerics and applications, and it is carried out in various ways such as physical experiments, mathematical modelings, numerical simulations and many others. In this chapter, we focus on the design of numerical schemes with numerical demonstrations based on a reduced version of the BN-type model. Various reduced models based on different simplifying assumptions are presented in this chapter, and the corresponding numerical schemes are even diverse. Here, we show some of the typical numerical schemes for reduced models. The solutions by these numerical schemes are often deficient in capturing multi-material shocks or achieving real solutions of the complete BN-type model. To remedy this deficiency, a novel energy-splitting scheme is designed for a novel reduced model, based on

256

X. Lei and J. Li

(a) t = 0µ s

(b) t = 1.95µ s

(c) t = 3.145µ s

(d) t = 3.71µ s

Fig. 9 Interaction of water shock and air bubble: pressure images

the Godunov scheme with a second-order extension by using the GRP solver. In a sense, this scheme proposed here is compatible with the five-equation reduced model [44]. However, since the novel reduced model is similar to the conservative Euler system in form, the shortcomings of non-conservative schemes can be alleviated to some extent, especially when multi-material shocks are simulated. In addition, the positivity preserving of volume fractions is pivotal as a numerical fluid mixing rule around interfaces, for which the energy exchange is considered in the current scheme so that no pressure oscillations arise near material interfaces, even though there is a large difference in thermodynamic quantities. Several benchmark problems are tested in order to demonstrate the validity and performance of the current method. The one-dimensional problems display more accurate computation of internal energy around material interfaces. Numerical results show that the energy-splitting scheme is effective for the volume fraction positivity and the simulation of multi-material shock waves. The two-dimensional shockbubble interaction problems demonstrate the performance of ES-GRP capturing material interfaces, through the comparison with the corresponding physical experiments. It is expected that this method improves the validity of the reduced forms of the BN-type model for simulating multi-material flows.

An Energy-Splitting High-Order Numerical Method for Multi-material Flows

(a) t = 0µ s

(b) t = 1.95µ s

(c) t = 3.145µ s

(d) t = 3.71µ s

257

Fig. 10 Interaction of water shock and air bubble: density gradient images

Acknowledgements This research is supported by the Natural Science Foundation of China (11771054, 91852207,12072042), National Key Project (GJXM 92579), Foundation of LCP and the Fundamental Research Funds for the Central Universities. We appreciate Professor Matania Ben-Artzi for his many kind comments.

Appendix: The 2-D GRP Solver Since the two-dimensional case is considered, we need to solve a so-called quasi 1-D GRP of (25) by setting the adjacent interface along x = 0, W t + div F(W ) = 0, F = [ f , g], (z a )t + u · ∇za = div u, V − (x, y), x < 0, V (x, y, 0) = V + (x, y) x > 0,

(96)

258

X. Lei and J. Li

where V = [W ; z a ], V − (x, y) and V + (x, y) are two polynomials defined on the two neighboring computational cells at time t = 0, respectively. Since we just want to construct fluxes normal to cell interfaces, the tangential effect can be regarded as a source term. Therefore, we rewrite the quasi 1-D GRP (96) as W t + f (W )x = −g(W ) y , (z a )t + u(z a )x− u x = v y − v(z a ) y , V − (x, y˜ ), x < 0, V (x, y˜ , 0) = V + (x, y˜ ), x > 0,

(97)

by fixing a y-coordinate. That is, we solve the 1-D GRP at a point (0, y˜ ) on the interface, by considering the transversal effect to the interface x = 0. The value g(W ) y and v y − v(z a ) y at (0, y˜ ) takes account of the local wave propagation. The solution of this GRP is denoted as GRP (V − (x), V + (x)) and solved by a 2-D GRP solver. This appendix introduces the 2-D GRP solver used in the coding process just for completeness and readers’ convenience. The details can be found in [49]. We notice that the equation of z a is very close to the equation of mass fraction for the burnt gas in the basic “combustion model” [7], and the GRP solver for the combustion model in [7, 9] is a heuristic form of our 2-D GRP solver. The GRP solver for solving (96) has the following two versions, which are the acoustic version and the genuinely nonlinear version.

2-D Acoustic Case At any point (0, y˜ ), if V − (0 − 0, y˜ ) ≈ V + (0 + 0, y˜ ) and ∂∂Vx− (0 − 0, y˜ ) = ∂V+ (0 − 0, y˜ ) , we view it as an acoustic case. Denote V ∗ := V − (0 − 0, y˜ ) ≈ ∂x V + (0 + 0, y˜ ), and then linearize the governing equations (25) to get ∂V ∂V ∂V + A(V ) + B(V ) = 0. ∂t ∂x ∂y

(98)

We make the decomposition A(V ∗ ) = RR−1 , where  = diag{λi }, R is the (right) eigenmatrix of A(V ∗ ). Then the acoustic GRP solver takes 

   ∂V− ∂V− − R I + R−1 B(V − ) ∂ y (0−0, y˜ ) (0, y˜ ,0)  ∂ x (0−0, y˜ )  ∂ V ∂ V+ + −R− R−1 − R I − R−1 B(V + ) , ∂ x (0+0, y˜ ) ∂ y (0+0, y˜ ) (99) where + = diag{max(λi , 0)}, − = diag{min(λi , 0)}, I + = 21 diag{1 + sign(λi )}, I − = 21 diag{1 − sign(λi )}. ∂V ∂t



= −R+ R−1



An Energy-Splitting High-Order Numerical Method for Multi-material Flows

259

2-D Nonlinear Case At any point (0, y˜ ), if the difference V − (0 − 0, y˜ ) − V + (0 + 0, y˜ ) is large, we regard it as the genuinely nonlinear case and have to solve the 2-D GRP analytically. A key ingredient is how to understand g(W ) y and v y − v(z a ) y at (0, y˜ ). Here, we construct the 2-D GRP solver by two steps. (i) We solve the local planar 1-D Riemann problem W t + f (W )x = 0, t > 0, (z a )t + u(z a )x− u x = 0, V − (0 − 0, y˜ ), x < 0, w(x, y˜ , 0) = V + (0 + 0, y˜ ), x > 0,

(100)

where w = [W ; z a ], to obtain the local Riemann solution V ∗ = w(0, y˜ , 0 + 0). Just as in the acoustic case, we decompose A(V ∗ ) = RR−1 . Then we set

h(x, y) =

−g(W ) y

v y − v(z a ) y



  ⎧ ⎨ −R I + R−1 B(V − ) ∂∂Vy− , x < 0,  (0−0, y˜ ) = ∂ V − −1 ⎩ −R I R B(V + ) ∂ y+ , x > 0, (0+0, y˜ )

where I ± are defined the same as in (99). (ii) We solve the quasi 1-D GRP W t + f (W )x = −g(W ) y , t > 0, (z a )t + u(z a )x− u x = v y − v(z a ) y , V − (x, y), x < 0, w(x, y, 0) = V + (x, y), x > 0,

(101)

(102)

  to obtain ∂∂tV ∗ = ∂∂tV (0, y˜ , 0 + 0). This is done by solving the 1-D GRP for homogeneous equations W t + f (W )x = 0, t > 0, (z a )t + u(z a )x− u x = 0, V − (x, y˜ ), x < 0, w(x, y˜ , 0) = V + (x, y˜ ), x > 0.

(103)

  = ∂w (0, y˜ , 0 + 0) is obtained by solvFor the augmented Euler equations, ∂w ∂t ∗ ∂t ing a pair of algebraic equations essentially, 

   ∂u ∂p + bL = dL , ∂t  ∗  ∂t ∗ ∂u ∂p + bR = dR . aR ∂t ∗ ∂t ∗ aL

(104)

260

X. Lei and J. Li

At last, we have



∂V ∂t



 ∗

=

∂w ∂t

 ∗

+ h(x, y˜ ).

(105)

Here, if we ignore h(x, y˜ ), the solver is called 1-D GRP solver. Solvers for the GRP (102) with a general source term are presented in [12]. When specified to (97), construction of the solver can be found in [49].

References 1. Abgrall, R.: How to prevent pressure oscillations in multicomponent flow calculations: a quasi conservative approach. J. Comput. Phys. 125, 150–160 (1996) 2. Abgrall, R., Karni, S.: Computations of compressible multifluids. J. Comput. Phys. 169, 594– 623 (2001) 3. Ahn, H.T., Shashkov, M., Christon, M.A.: The moment-of-fluid method in action. Commun. Numer. Meth. Eng. 25, 1009–1018 (2009) 4. Allaire, G., Clerc, S., Kokh, S.: A five-equation model for the simulation of interfaces between compressible fluids. J. Comput. Phys. 181, 577–616 (2002) 5. Baer, M.R., Nunziato, J.W.: A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials. Int. J. Multiphase Flow 12, 861–889 (1986) 6. Banks, J.W., Schwendeman, D.W., Kapila, A.K., Henshaw, W.D.: A high-resolution Godunov method for compressible multi-material flow on overlapping grids. J. Comput. Phys. 223, 262–297 (2007) 7. Ben-Artzi, M.: The generalized Riemann problem for reactive flows. J. Comput. Phys. 81, 70–101 (1989) 8. Ben-Artzi, M., Birman, A.: Application of the generalized Riemann problem method to 1-D compressible flows with material interfaces. J. Comput. Phys. 65, 170–178 (1986) 9. Ben-Artzi, M., Birman, A.: Computation of reactive duct flows in external fields. J. Comput. Phys. 86, 225–255 (1990) 10. Ben-Artzi, M., Falcovitz, J.: A second-order Godunov-type scheme for compressible fluid dynamics. J. Comput. Phys. 55, 1–32 (1984) 11. Ben-Artzi M., Falcovitz J.: Generalized Riemann Problems in Computational Fluid Dynamics. Cambridge University Press (2003) 12. Ben-Artzi, M., Li, J.: Hyperbolic conservation laws: Riemann invariants and the generalized Riemann problem. Numer. Math. 106, 369–425 (2007) 13. Ben-Artzi, M., Li, J., Warnecke, G.: A direct Eulerian GRP scheme for compressible fluid flows. J. Comput. Phys. 218, 19–43 (2006) 14. Chang, C.H., Stagg, A.K.: A compatible Lagrangian hydrodynamic scheme for multicomponent flows with mixing. J. Comput. Phys. 231, 4279–4294 (2012) 15. Cheng, J., Shu, C.W.: Positivity-preserving Lagrangian scheme for multi-material compressible flow. J. Comput. Phys. 257, 143–168 (2014) 16. Cocchi, J.P., Saurel, R.: A Riemann problem based method for the resolution of compressible multimaterial flows. J. Comput. Phys. 137, 265–298 (1997) 17. Colella P., Glaz H.M., Ferguson, R.E: Multifluid Algorithms for Eulerian Finite Difference Methods. Preprint (1989) 18. Du, Z., Li, J.: Accelerated piston problem and high order moving boundary tracking method for compressible fluid flows. SIAM J. Sci. Comput. 42, A1558–A1581 (2020) 19. Dyadechko, V., Shashkov, M.: Moment-of-Fluid Interface Reconstruction. Technical report, Los Alamos National Laboratory (2005)

An Energy-Splitting High-Order Numerical Method for Multi-material Flows

261

20. Falcovitz, J., Alfandary, G., Hanoch, G.: A two-dimensional conservation laws scheme for compressible flows with moving boundaries. J. Comput. Phys. 138, 83–102 (1997) 21. Fedkiw, R.P., Aslam, T., Merriman, B., Osher, S.: A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comput. Phys. 152, 457–492 (1999) 22. Francois, M.M., Shashkov, M.J., Masser, T.O., Dendy, E.D.: A comparative study of multimaterial Lagrangian and Eulerian methods with pressure relaxation. Comput. Fluids 83, 126–136 (2013) 23. Galera, S., Maire, P.H., Breil, J.: A two-dimensional unstructured cell-centered multi-material ALE scheme using VOF interface reconstruction. J. Comput. Phys. 229, 5755–5787 (2010) 24. Gallouët, T., Hérard, J.M., Seguin, N.: Numerical modeling of two-phase flows using the twofluid two-pressure approach. Math. Mod. Meth. Appl. S. 14, 663–700 (2004) 25. Goncalves, E., Hoarau, Y., Zeidan, D.: Simulation of shock-induced bubble collapse using a four-equation model. Shock Waves 29, 221–234 (2019) 26. Haas, J.F., Sturtevant, B.: Interaction of weak shock waves with cylindrical and spherical gas inhomogeneities. J. Fluid Mech. 181, 41–76 (1987) 27. Hirt, C.W., Amsden, A.A., Cook, J.L.: An arbitrary Lagrangian-Eulerian computing method for all flow speeds. J. Comput. Phys. 14, 227–253 (1974) 28. Hou, T.Y., LeFloch, P.G.: Why nonconservative schemes converge to wrong solutions: error analysis. Math. Comput. 62, 497–530 (1994) 29. Jenny, P., Müller, B., Thomann, H.: Correction of conservative Euler solvers for gas mixtures. J. Comput. Phys. 132, 91–107 (1997) 30. Kamm, J.R., Shashkov, M.J.: A pressure relaxation closure model for one-dimensional, twomaterial Lagrangian hydrodynamics based on the Riemann problem. Technical report, Los Alamos National Laboratory (2009) 31. Kapila, A.K., Menikoff, R., Bdzil, J.B., Son, S.F., Stewart, D.S.: Two-phase modeling of deflagration-to-detonation transition in granular materials: reduced equations. Phys. Fluids 13, 3002–3024 (2001) 32. Karni, S.: Multicomponent flow calculations by a consistent primitive algorithm. J. Comput. Phys. 112, 31–43 (1994) 33. Karni, S.: Hybrid multifluid algorithms. SIAM J. Sci. Comput. 17, 1019–1039 (1996) 34. Lallemand, M.H., Chinnayya, A., Metayer, O.L.: Pressure relaxation procedures for multiphase compressible flows. Int. J. Numer. Methods Fluids 49, 1–56 (2005) 35. Larrouturou, B.: How to preserve the mass fractions positivity when computing compressible multi-component flows. J. Comput. Phys. 95, 59–84 (1991) 36. Lei, X., Du, Z., Li, J.: The simulation of compressible multi-fluid flows by a GRP-based energy-splitting method. Comput. Fluids 181, 416–428 (2019) 37. Lei, X., Li, J.: A non-oscillatory energy-splitting method for the computation of compressible multi-fluid flows. Phys. Fluids 30, 040906 (2018) 38. Li, J., Wang, Y.: Thermodynamical effects and high resolution methods for compressible fluid flows. J. Comput. Phys. 343, 340–354 (2017) 39. Liu, T.G., Khoo, B.C., Yeo, K.S.: Ghost fluid method for strong shock impacting on material interface. J. Comput. Phys. 190, 651–681 (2003) 40. Maire, P.H., Abgrall, R., Breil, J., Ovadia, J.: A cell-centered Lagrangian scheme for twodimensional compressible flow problems. SIAM J. Sci. Comput. 29, 1781–1824 (2007) 41. Michael, L., Nikiforakis, N.: A hybrid formulation for the numerical simulation of condensed phase explosives. J. Comput. Phys. 316, 193–217 (2016) 42. Miller, G.H., Puckett, E.G.: A high-order Godunov method for multiple condensed phases. J. Comput. Phys. 128, 134–164 (1996) 43. Mulder, W., Osher, S., Sethian, J.A.: Computing interface motion in compressible gas dynamics. J. Comput. Phys. 100, 209–228 (1992) 44. Murrone, A., Guillard, H.: A five equation reduced model for compressible two phase flow problems. J. Comput. Phys. 202, 664–698 (2005)

262

X. Lei and J. Li

45. Nonomura, T., Morizawa, S., Terashima, H., Obayashi, S., Fujii, K.: Numerical (error) issues on compressible multicomponent flows using a high-order differencing scheme: Weighted compact nonlinear scheme. J. Comput. Phys. 231, 3181–3210 (2012) 46. Nourgaliev, R.R., Dinh, T.N., Theofanous, T.G.: Adaptive characteristics-based matching for compressible multifluid dynamics. J. Comput. Phys. 213, 500–529 (2006) 47. Pelanti, M., Shyue, K.M.: A mixture-energy-consistent six-equation two-phase numerical model for fluids with interfaces, cavitation and evaporation waves. J. Comput. Phys. 259, 331–357 (2014) 48. Petitpas, F., Franquet, E., Saurel, R., Le Metayer, O.: A relaxation-projection method for compressible flows. Part II: Artificial heat exchanges for multiphase shocks. J. Comput. Phys. 225, 2214–2248 (2007) 49. Qi, J.: Numerical modeling of generalized riemann problem for two-dimensional euler equations and its application. Ph.D. thesis, Beijing Normal University (2017) 50. Quirk, J.J., Karni, S.: On the dynamics of a shock-bubble interaction. J. Fluid Mech. 318, 129–163 (1996) 51. Saurel, R., Petitpas, F., Berry, R.A.: Simple and efficient relaxation methods for interfaces separating compressible fluids, cavitating flows and shocks in multiphase mixtures. J. Comput. Phys. 228, 1678–1712 (2009) 52. Saurel, R., Abgrall, R.: A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150, 425–467 (1999) 53. Saurel, R., Franquet, E., Daniel, E., Le Metayer, O.: A relaxation-projection method for compressible flows. Part I: the numerical equation of state for the Euler equations. J. Comput. Phys. 223, 822–845 (2007) 54. Saurel, R., Le Métayer, O., Massoni, J., Gavrilyuk, S.: Shock jump relations for multiphase mixtures with stiff mechanical relaxation. Shock Waves 16, 209–232 (2007) 55. Shashkov, M.: Closure models for multimaterial cells in arbitrary Lagrangian-Eulerian hydrocodes. Int. J. Numer. Meth. Fluids 56, 1497–1504 (2008) 56. Shyue, K.M.: An efficient shock-capturing algorithm for compressible multicomponent problems. J. Comput. Phys. 142, 208–242 (1998) 57. Shyue, K.M.: A wave-propagation based volume tracking method for compressible multicomponent flow in two space dimensions. J. Comput. Phys. 215, 219–244 (2006) 58. Shyue, K.M.: A moving-boundary tracking algorithm for inviscid compressible flow. In: Hyperbolic Problems: Theory Numerics, Applications, pp. 989–996. Springer, Berlin, Heidelberg (2008) 59. Terashima, H., Tryggvason, G.: A front-tracking/ghost-fluid method for fluid interfaces in compressible flows. J. Comput. Phys. 228, 4012–4037 (2009) 60. Ton, V.T.: Improved shock-capturing methods for multicomponent and reacting flows. J. Comput. Phys. 128, 237–253 (1996) 61. Wood, A.B.: A Textbook of Sound. G. Bell and Sons Ltd., London (1930) 62. Xiao, F., Li, S., Chen, C.: Revisit to the THINC scheme: a simple algebraic VOF algorithm. J. Comput. Phys. 230, 7086–7092 (2011) 63. Youngs, D.L.: Modelling turbulent mixing by Rayleigh-Taylor instability. Physica D 37, 270– 287 (1989) 64. Zeidan, D., Slaouti, A., Romenski, E., Toro, E.F.: Numerical solution for hyperbolic conservative two-phase flow equations. Int. J. Comput. Methods 04, 299–333 (2007) 65. Zeidan, D., Touma, R., Slaouti, A.: Implementation of velocity and pressure non-equilibrium in gas-liquid two-phase flow computations. Int. J. Fluid Mech. Res. 41, 2547–555 (2014) 66. Zhang, Q., Fogelson, A.: MARS: An analytic framework of interface tracking via mapping and adjusting regular semialgebraic sets. SIAM J. Numer. Anal. 54, 530–560 (2016)

An ADER-LSTDG Scheme for the Numerical Simulation of a Global Climate Model Arturo Hidalgo and Lourdes Tello

Abstract In this work we obtain the numerical solution of a coupled oceanatmosphere model of interest in climate applications. The methodology followed makes use of a finite volume method with an Arbitrary Order for Derivative Riemann problem method for time integration and a Weighted Essentially Non-Oscillatory approach for spatial reconstruction. The numerical scheme used here is a recently introduced variant of the classical Arbitrary Order for Derivative Riemann problem technique, which makes use of a Local Space-Time Discontinuous Galerkin approach. The mathematical model being solved is based on that proposed by Watts and Morantine but modified with a nonlinear diffusive boundary condition in the upper boundary of the domain. Results of the simulation are shown, comparing the solution attained when there is an influence of the ocean on the surface and where this influence does not take place. Also some validation results are supplied.

1 Introduction Climate behaviour and climatic change are great concerns nowadays due to the fundamental influence of climatic events on the life on the Earth. We are getting used to many situations that can be linked to climatic effects such as floods, draughts, greenhouse, increasing heat waves, ice shrinking in glaciers and Poles, sea level increase or hurricanes becoming stronger to name just some of them. Apart from natural actors, human action is considered to be behind this increasingly problematic situation. A. Hidalgo (B) Departamento de Ingeniería Geológica y Minera. Escuela Técnica Superior de Ingenieros de Minas y Energía, Center for Computational Simulation. Universidad Politécnica de Madrid, Ríos Rosas, 21, 28003 Madrid, Spain e-mail: [email protected] L. Tello Departamento de Matemática Aplicada Escuela Técnica Superior de Arquitectura, Center for Computational Simulation, Universidad Politécnica de Madrid, Av. Juan de Herrera, 4, 28040 Madrid, Spain e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 D. Zeidan et al. (eds.), Numerical Fluid Dynamics, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-16-9665-7_9

263

264

A. Hidalgo and L. Tello

A deep and complete study of climate system is difficult to be conducted, since it incorporates a great amount of different phenomena and mechanisms that are not completely understood yet. However, during the last decades, scientists all over the world have carried out a great effort to understand this processes and develop mathematical models that are able to predict future climatic events and help making decisions to reduce the effect of climate change. Actually, mathematics have proved to be an ally to help human beings to prevent and reduce the effects of dangerous situations such as pandemics, tsunamis generation and propagation, earthquakes, landslides and many other situations that hit our world quite often. In the context of the study of global climate and its relation with climatic change, the General Circulation Models (GCMs) play a very important role to represent, in a very efficient way, the physical processes taking place in the interaction of atmosphere and ocean in order to simulate the response of global climate to the increasing greenhouse gas emission, as it was reported by the Intergovernmental Panel of Climatic Change (IPCC) in 2013. We focus our attention in energy balance models (EBMs), which were introduced independently at the end of the 1960s by Budyko and Sellers in the pioneering works [13, 54], respectively. These EBMs are the building blocks of global climate mathematical models currently used. These models have been formulated and studied by several authors such as [17, 21, 36, 40, 45, 53] amongst many others. The mathematical model considered in this work is a 2D climatic model, considering the sine(latitude) and depth as spatial coordinates, representing a coupled surface-ocean model where the unknown is the mean temperature over each parallel. The original version of this model was introduced by Watts and Morantine in [64], where the parabolic model included a dynamic and diffusive boundary condition, although in this case we resort to the model given in [23] where nonlinear diffusion and temperature-dependent coalbedo is given. The coalbedo is the fraction of the solar radiation which is absorbed by the surface of a material, such as the surface of the Earth in this case. It is the opposite effect of the albedo, which represents the fraction of solar radiation which is reflected by the material. Several numerical techniques have been proposed in the literature to solve this kind of models. The most common ones are finite element methods (FEMs), that have been applied in references such as [11, 12], or finite volume methods (FVMs) that have been used in other references such as [20, 43, 45]. In this work we resort to a finite volume strategy to solve the models of interest due to its versatility and good behaviour when applied to fluid dynamics and heat transfer problems. In addition, FVMs behave perfectly well when nonlinearities and discontinuities appear in the solution. These techniques are based on computing cell averages (also named as integral averages) of the solution at certain control volumes which constitute the mesh. Proceeding in this way, a piecewise constant function is attained, due to the obtention of cell averages of the solution. In order to calculate gradients of the solution, for instance at cell interfaces, or values of the solution in Gaussian quadrature points and also with the aim to achieve numerical schemes of order higher than one, some kind of reconstruction procedure must be followed. This reconstruction is usually based on polynomial Lagrange interpolation, using the cell averages. We

An ADER-LSTDG Scheme for the Numerical Simulation of a Global Climate Model

265

note that this reconstruction must be conservative, in the sense that the cell average of the polynomial reconstruction function must be equal to the cell average of the solution within all the control volumes conforming the stencil. In [48, 60, 63] there are full details on finite volume methods. It is relevant to note here that according to Godunov’s theorem, see [37], monotone behaviour of a numerical solution cannot be assured for linear finite-difference methods with more than first-order of accuracy, thus nonlinear reconstruction techniques must be developed in order to get monotone numerical schemes with order of accuracy higher than one. Widely used in different applications are the so-called Monotone Upwind schemes for the Conservation Laws (MUSCLE) by Van Leer (see [62]) where constant cell averages, as those appearing in Godunov’s method, are replaced by reconstructed values. This technique is usually known as MUSCL-Hancock scheme since, according to [52], it was greatly simplified in 1980 by Steve Hancock. This numerical scheme is second-order accurate, both in space and time, and it is still monotone, due to its nonlinear formulation. We refer to [60] for a very good description of MUSCLE-Hancock method. Other prominent methods that combine high order of accuracy and monotonicity are Essentially Non-Oscillatory (ENO) numerical schemes, designed by Harten, Engquist, Osher, and Chakravarthy in 1987 [39], which carry out a nonlinear reconstruction based on the smoothest divided differences to choose the interpolation stencil. Furthermore, in [38], Harten introduced the notion of subcell resolution, which is based on the idea that cell averages of a discontinuous piecewise-smooth function contain information about the exact location of the discontinuity within the cell. We must acknowledge that the contributions of Ami Harten and collaborators are of paramount importance in the field of high order numerical schemes. A variant of ENO approach consists of performing a convex combination of all possible stencils that can be considered at a particular control volume, applying a weight to each one of them. This technique is called Weighted Essentially NonOscillatory (WENO) reconstruction, first introduced in [51]. Some relevant references of this technique, and variants of it, mainly applied to hyperbolic problems, can be found in [6, 10, 29, 30, 55, 58] just to name a few of them. Following this approach, in order to achieve 2r − 1 order of accuracy, r candidate stencils are considered, each one of them with r cells, and thus r candidate polynomials of degree r + 1 are obtained. These polynomials are weighted, using particular nonlinear weights which depend on certain smoothness indicators which in turn depend on the cell averages of the unknown. The task carried out by these smoothness indicators is to detect discontinuities in the solution and apply higher values of the nonlinear weights to the more biased stencils in order to escape from the discontinuity. In the regions where the solution is smooth, the central stencils receive higher values of these weights than the more biased ones. The reason behind is that, as stated in [8], and references therein, the central stencils are the most stable above all the other stencils. This idea also gives rise to the introduction of the Central WENO (CWENO) schemes, see [3, 15, 26, 49, 50], where more compact stencils are used. Interesting references with different ways to implement WENO methodology can be found in [4, 5, 7]. We note that classical WENO approach makes use of pointwise values

266

A. Hidalgo and L. Tello

of the polynomials, whilst the WENO methodology followed here, according to [29, 30] for instance, allows to obtain full polynomials that can be evaluated wherever needed. In order to perform time integration we implement Arbitrary order for DErivative Riemann Problem (ADER) approach that was introduced to solve hyperbolic problems by E.F. Toro and collaborators ([31, 57, 59]). It is a prominent numerical approach to attain arbitrary high order of accuracy in a unique time step. This technique is based on solving the Derivative Riemann Problems (DRPs) that commonly appear at cell interfaces when dealing with finite volume schemes. ADER approach was put forward to solve hyperbolic problems, however, due to its versatility and effectiveness, its application has been extended also to reaction-diffusion problems, see for instance [1, 35, 61]. The classical ADER approach has some drawbacks, particularly when dealing with stiff source terms. Furthermore, the Cauchy-Kowalewskaya (CK) procedure used in this approach becomes rather cumbersome, especially when solving nonlinear problems and working in multi-dimensional domains. Due to this fact, a new version of ADER technique was developed by Dumbser, Enaux and Toro [27] who introduced the Local space-time discontinuous Galerkin ADER scheme (ADER-LSTDG) which treats adequately and accurately stiff source terms, and also avoids the use of CK procedure. We remark that there is a continuous version of this scheme named as Local Continuous space-time Galerkin method (ADER-CG), see for instance [9] for MHD applications. It is remarkable the use of both ADER-LSTDG and ADERCG in [24], for the 3D compressible Navier-Stokes equations. In [25] the authors introduce the PN PM schemes as a general framework which includes the FV and DG schemes as particular cases (taking M = 0 and M = N respectively). An application of ADER-LSTDG approach for reaction-diffusion problems with stiff source terms can be found in [42], where it is presented a novel nodal approach, based on the use of Lagrange basis functions, instead of the original modal technique, based on Legendre’s basis functions. In the same reference, a new predictor, based on a secondorder MUSCLE scheme, was introduced to initialize the iterative process. Some applications of ADER-LSTDG and ADER-CG schemes with PN PM formulation in the context of MHD equations are [9, 32, 33, 65]. In this work we use a LSTDG-ADER numerical method in the context of finite volume methods with WENO reconstruction in space to solve the coupled oceanatmosphere model. This is the first time in which this technique is used for this particular application. We are interested in solving the Energy Balance Model (EBM), both with and without influence of the deep ocean, that is coupling and decoupling the EBM and the Deep Ocean Model (DOM). This is achieved by activating or deactivating a term in the EBM which considers the influence of the ocean onto the surface. This term has also been useful, for instance, in models that consider land-sea distribution, in order to distinguish the regions with land or with only water. See [44] for details on this. The interest to consider these two situations (coupled and decoupled) is that both situations are of interest in climatic modelling. In the case of EBMs, without ocean effect, this model is able to predict very important effects that take place in the surface with a great influence on global climate, such

An ADER-LSTDG Scheme for the Numerical Simulation of a Global Climate Model

267

as absorption and emission of radiation, greenhouse effect, amongst many others, see [14, 21, 34, 36, 53] and references therein for details on these models and their relation with global climate. On the other hand, the consideration of the deep ocean is of great importance, due for instance to its well-known thermostatic effect which clearly affects the distribution of temperatures in the surface. The choice of ADER approach in this work is based on the fact that it is a very efficient numerical technique, especially indicated for solving very accurately Riemann problems, as those appearing at cell interfaces when working with the finite volume method, and also possess the ability of attaining an arbitrary high order of accuracy in a single time step, for sufficiently smooth solutions. Although in this work we show results for second order of accuracy, for the current problem, we refer the reader to [61] where orders of accuracy up to tenth for the nonlinear reactiondiffusion problem have been obtained, but using the classical ADER formulation. We remark that the present work makes use of a variant of classical ADER approach, namely ADER-LSTDG, which possess the ability, as aforementioned, of dealing with stiff source terms and also avoids the use of the rather cumbersome CauchyKowalewskaya procedure, by performing a weak formulation locally in time. Also relevant, for the choice of this particular numerical scheme, is the feature that in the EBM there is a discontinuous coalbedo which also justifies the use of a numerical method essentially discontinuous as the finite volume method. In the following sections we start by introducing a physical motivation after which it is described the mathematical model on which this work is based. Then we present the numerical approach, based on a finite volume scheme with WENO reconstruction in space and ADER-LSTDG for time integration. Next, a numerical assessment, by means of a manufactured solution, is carried out. The following part is devoted to some numerical results, and finally conclusions and further Research are given.

2 Physical Motivation The knowledge of the interactions between ocean and atmosphere is fundamental to understand climate behaviour and hence climatic change. There are many factors involved in global climate, however it is almost impossible to build a model containing all of them. In global climate models we focus our attention just on the most relevant ones. The evolution of the temperature in the surface of the ocean can be achieved via the energy balance models (EBMs), which estimate the energy budget on the surface of the Earth, considering the incoming solar radiation, the absorbed energy and the emitted energy. In this context it is very important the effect of the temperature-dependent coalbedo that has a feedback effect, that is, when the temperature decreases ice formation is favoured, at the same time the shiny white ice reflects almost all the incoming radiation which reduces the fraction of absorbed energy giving rise to a diminishing of the temperatures, leading to the formation of more ice. On the other hand, if the temperature increases, the mass of ice reduces, and

268

A. Hidalgo and L. Tello

so does the amount of reflected radiation, with the consequence that the temperature increases even more. The energy balance models are diagnostic models (not forecasting ones) and they are suitable to obtain qualitative aspects about the evolution of the climate system. Due to its global scale they are a very useful tool aimed to analyse past climates and also long-term phenomena, such as the Milankovitch cycles, which indicate that long lasting changes in the Earth’s climate during tens of thousands of years obey variations in the Earth’s orbit around the Sun (see [43]). The Paleoclimate studies indicate that towards the end of the last Glaciation period (around 10 thousand years ago) changes in the deep ocean took place. This suggests to couple the model of surface temperature with those including the effect of the ocean. The effect of the ocean on the surface is the smoothness of the surface temperature: the maximum value decreases and the minimum increases. This is due to the wellknown thermostatic effect of the ocean. This relevant feature is mathematically and numerically assessed both in previous references (such as in [43]), and also in the present work.

3 Mathematical Model The mathematical model under study is based on that proposed by Watts and Morantine (1990) [64] where a one-dimensional energy balance model (EBM) is coupled with a two-dimensional deep ocean model (DOM). In this work we focus our attention on a modified version proposed in [19, 20, 23, 45] where nonlinear diffusion and the action of a coalbedo depending of temperature is included. In the following subsections we describe the energy balance model and the deep ocean model.

3.1 The Energy Balance Model (EBM) The one-dimensional Energy Balance Model (EBM) is obtained from a twodimensional EBM by assuming constant temperature over each parallel. The twodimensional EBM (latitude-longitude) has as spatial domain given by a Riemannian manifold M without boundary, which represents the surface of the Earth (see [22]) 

C(x)u t − div(k(x)|∇u| p−2 ∇u) + Re (x, u) = Ra (x, u) in M, u(x, 0) = u 0 (x)

in M × (0, T ),

(1) where C(x) is the heat capacity, k(x) is the thermal conductivity, Re is the emitted energy, Ra represents the absorbed energy and the exponent p is a real number that is usually taken as p ≥ 2 (the case p = 2 is the linear one), ∇u is the gradient of the temperature u. Usually, in climate models, it is taken p = 3 as proposed in [56],

An ADER-LSTDG Scheme for the Numerical Simulation of a Global Climate Model Fig. 1 Temperature is constant over each parallel. This figure depicts a contour plot, based on a numerical result, where the colours denote the value of the temperatures in ◦ C

269

25 1

20

0.5

10

15

5 0 0 −5

−0.5

−10 −1

−15 −20

0.5 0.5

0 −0.5

0

−25

−0.5

in order to include the eddy fluxes at big scales. In this equation the subscript t denotes partial time derivative. In the one-dimensional domain the temperature is taken constantly over each parallel. Thus, the following formulation is attained ⎧ ⎨ C(x)u t − (k(x)(1 − x 2 ) p/2 |u x | p−2 u x )x + Re (x, u) = Ra (x, u) in (−1, 1) × (0, T ), (1 − x 2 ) p/2 |u x | p−2 u x = 0, x ∈ {−1, 1} , t ∈ (0, T ), ⎩ u(x, 0) = u 0 (x), x ∈ (−1, 1),

(2) The spatial coordinate in (2) stands for the sine of the latitude. The term (1 − x 2 ) p/2 has been obtained after changing to spherical coordinates in (1). The absorbed radiation is represented by Ra = Q S(x)β(u) where Q is the amount of solar radiation received by the Earth. Its value is one-fourth of the solar constant, since the area of the cross section is π R 2 , where R is the radius of the Earth, whereas 4π R 2 is the surface of the Earth. Considering that the value of the solar constant is 1360 W/m2 we obtain Q = 340 W/m2 . The function S(x) has been introduced in order to represent the distribution of the solar radiation throughout the surface of the Earth and is given in this model by S(x) = (5 − x 2 )/4 in order to represent a higher amount of heat received by the Equator and less amount of heat received by the Poles. Regarding β(u), it represents the coalbedo that is the fraction of radiation absorbed by the surface. It changes around a characteristic temperature of −10 ◦ C. The coalbedo is usually represented according to Budyko’s model or Seller’s model. Budyko’s approach is a discontinuous function, represented by a Heaviside’s function, whilst according to Seller’s model the coalbedo is represented by a continuous function. Concerning the emitted energy, it can be used either Sellers’ model, which considers the Stephan-Boltzmann’s law Re (u) = σ u 4 , or Budyko’s model where Newton’s cooling law Re (u) = Bu + A is used. In Fig. 1 we depict a numerical solution of the EBM showing that the temperature is considered constant on each parallel, which is one of the assumptions of the present model.

270

A. Hidalgo and L. Tello

Several references can be mentioned here concerning for instance the mathematical analysis of the model (2) that can be found in [18]. In [22] it is proved the existence of solutions of the problem (1) by using fixed point arguments. Multiplicity of steady states depending on the parameter Q was studied in [46]. There is a bounded interval of Q such that for every Q in that interval, problem (1) has at least three stationary solutions. In the reference [2], it is proved a S-shaped bifurcation branch for the associated stationary problem. In [45] part of this bifurcation diagram is numerically attained. There are many works dealing with the mathematical treatment of one-layer global climate models such as [21, 40, 53]. For two-layers climate model see [41].

3.2 Deep Ocean-EBM Coupled Model We are dealing with a mathematical model representing the evolution of temperature inside an ocean of depth H , which is based on the one proposed by Watts and Morantine [64] and further modified in [20]. In this model the spatial variables (x, z) are sin(latitude) and depth, respectively. The spatial domain is given by  = (−1, 1) × (−H, 0) with boundary  =  H ∪ 0 ∪ 1 ∪ −1 , where    ¯ : z = −H , 0 = (x, z) ∈  ¯ :  H = (x, z) ∈    ¯ ¯ −1 = (x, z) ∈  : x = −1 , 1 = (x, z) ∈  :

 z = 0 , x =1 .

(3)

The evolution of the temperature is governed by the equation  Ut −

KH (1 − x 2 )Ux R2

− K V Uzz + ωUz = 0

in  × (0, T ),

(4)

x

where U (x, z, t) is the temperature within the ocean, ω is the vertical velocity, K V is the vertical diffusivity, K H is the horizontal diffusivity and R represents the radius of the Earth. In the upper boundary we consider the equation Du t −

D K H0 R2



p

(1 − x 2 ) 2 |u x | p−2 u x

on 0 × (0, T ).

 x

1 + Bu + A + K V ∂U ∂n + ωxu x = ρc Q S(x)β(u)

(5) Regarding the ocean bottom, the boundary condition is given by ωxUx + K v Uz = 0,

on  H × (0, T ),

(6)

being u(x, t) the temperature on the upper boundary, ω the velocity, K V the vertical diffusivity, K H0 the horizontal diffusivity, R the radius of the Earth, D the thickness of the mixed layer (the ocean zone below the surface where the main processes of heat exchange between atmosphere and ocean take place), ρ the density, c the

An ADER-LSTDG Scheme for the Numerical Simulation of a Global Climate Model

271

specific heat coefficient, β(u) the temperature-dependent coalbedo, Q the amount of radiation received by the Earth, whose meaning has been explained in the previous section, Bu + A is the cooling term according to Newton’s law and S(x) is the insolation function. The full final system representing the coupled model: energy balance on the surface-deep ocean reads ⎧ Ut − ( KRH2 (1 − x 2 )Ux )x − K V Uzz + ωUz = 0 in  × (0, T ), ⎪ ⎪ ⎪ ⎪ ωxUx + K V U z = 0 in  H ×(0, T ), ⎪ ⎪ ⎪ 3 DKH ⎪ ⎪ + ωxu x + Bu + A ∈ ⎨ Du t − R 2 0 (1 − x 2 ) 2 |u x | u x + K V ∂U ∂n x

on 0 × (0, T ), ⎪ ⎪ ⎪ Ux = 0, on −1 × (0, T ) ∪ 1 × (0, T ) , ⎪ ⎪ ⎪ ⎪ in , U (x, z, 0) = U0 (x, z), ⎪ ⎪ ⎩ in 0 , u(x, 0) = u 0 (x),

1 Q S(x)β(u) ρc

(7) where homogeneous Neumann boundary conditions at −1 and 1 have been added. In this formulation we have used the value p = 3, as proposed in [56]. We solve this coupled model at each time step taking the solution of the EBM part as Dirichlet boundary condition to be used to solve the DOM part of this model. The reference [20] includes delay effects in the formulation, whereas in [44] land-sea distribution is incorporated to the model.

4 Numerical Scheme Concerning the numerical resolution of the coupled model, in the references [20, 43, 44] it is developed a finite volume scheme using WENO reconstruction in space and a third-order Runge-Kutta TVD (Total Variation Diminishing) technique for the evolution stage. In the following subsections we briefly describe the numerical procedure followed in this work, based on ADER-LSTDG finite volume approach with WENO procedure for space reconstruction, which is a novel approach to this model. We consider the 2D reaction-diffusion system written in Cartesian coordinates ∂ ∂ ∂ u+ f(u, ∇u) + g(u, ∇u) = S(u, x, t) ∂t ∂x ∂z

(8)

and carry out a finite volume discretization integrating over the control volume Ii j = [xi−1/2 , xi+1/2 ] × [z j−1/2 , z j+1/2 ] to get uin+1 = uinj + j where

t

t (fi+1/2, j − fi−1/2, j ) + (gi, j+1/2 − gi, j−1/2 ) + tSi j ,

xi

z j

(9)

272

uinj

A. Hidalgo and L. Tello

1 =

xi z j

fi+1/2, j =

gi, j+1/2

Si j =

x i+1/2 z j+1/2

u(x, z, t n )dzd x, xi−1/2 z j−1/2

t n+1 z j+1/2

1

t z j

1 =

t xi

1

t xi z j

f˜i+1/2, j (uh− (xi+1/2 , z, t), uh+ (xi+1/2 , z, t))dzdt,

t n z j−1/2

t n+1 x i+1/2

(10)

g˜ i, j+1/2 (uh− (x, z j+1/2 , t), uh+ (x, z j+1/2 , t))d xdt,

t n xi−1/2

t n+1 x i+1/2

tn

xi−1/2

z j+1/2 z j−1/2

S(uh (x, z, t))dzd xdt,

−/+

where uh is a space-time predictor that will be obtained according to the description followed in Sect. 4.2. The main ingredients of this process are: • High-order reconstruction of fluxes at cell interfaces, achieved using Weighted Essentially Non-Oscillatory (WENO) reconstruction. • High-order evolution in time, using ADER-LSTDG approach. In the following subsections both features are described.

4.1 Spatial WENO Reconstruction As already stated, when using a finite volume methods, a piecewise constant function is achieved at each time step, since we are working with cell averages. Thus, some type of reconstruction procedure should be used in order to be able to compute values and gradients where they are needed. Amongst the available techniques, we resort in this work to the Weighted Essentially Non-Oscillatory (WENO), which is a nonlinear reconstruction procedure since the different weights applied to the stencil of interpolation are solution-dependent. The approach followed in this work make use of entire polynomials as proposed in several references, such as [24, 27, 28], instead of generating pointwise values as in the WENO methodology introduced by Jiang and Shu [47]. This way to proceed is very useful in order to readily compute pointwise values of the reconstruction polynomial at Gaussian points both within the control volumes and in the element boundaries. The WENO technique applied in the two-dimensional case, is based upon a 2D dimension-by-dimension reconstruction process. Some references on this can be found in [29, 30, 58], but many others could be consulted. This way to proceed is computationally cheaper than the fully 2D WENO reconstruction. As WENO variants, it is interesting to mention the CWENO (Central WENO approach) as

An ADER-LSTDG Scheme for the Numerical Simulation of a Global Climate Model

273

described in [15, 16], or WENO scheme with Unconditionally Optimal Accuracy, see [5]. Let us consider the control volume Ii, j . For each Cartesian direction we consider a set of 1D stencils that are expressed as S i,s,xj

=

i+R 

Ie, j ,

e=i−L

Si,s,zj

=

j+R 

Ii,e ,

(11)

e= j−L

where L and R stands for the left and right spatial extension of the stencil. The strategy followed in this work follows the ideas given in [29, 30, 45]. According to these references, for the case of odd order schemes, three candidate stencils are taken into account whilst, for even order, four candidate stencils are considered. Thus, in the case of odd order schemes (that is, even polynomial degrees M), we have • One central stencil: s = 1, L = R = M/2, • One fully biased to the left stencil: s = 2, L = M, R = 0, • One fully biased to the right stencil: s = 3, L = 0, R = M, whereas, for even order schemes (that is, odd polynomial degrees M) • Two central stencils: s = 0, L = floor(M/2)+1, R = floor(M/2) and s = 1, L = floor(M/2), R = floor(M/2)+1, • One left-sided stencil: s = 2, L=M, R=0, • One right-sided stencil: s = 3, L = 0, R = M. We introduce the mapping [xi , xi+1 ] × [z j , z j+1 ] → [0, 1] × [0, 1] with x = xi− 21 + ξ xi , z = z j− 21 + η z j , with (ξ, η) ∈ [0, 1]. The way to perform, which is described in detail in several references, see, for instance [30], is briefly explained here. In the first stage, reconstruction is carried out in x-direction. Then, for each control volume Ii j , we have the following expressions of the reconstruction polynomials whs,x (x, t n ) =

M 

s,x ψ p (ξ )wˆ in,s j, p , ∀Si j ,

(12)

p=0

where the functions ψ p are the basis interpolation functions, usually Lagrange or Legendre type, whilst wˆ in,s j, p are the coefficients of each polynomial. Applying integral conservation on all control volumes of each stencil we have 1

xe

x e+1/2

M 

xe−1/2

ψ p (ξ(x))wˆ in,s ¯ nej , j, p d x = u

∀Iej ∈ Sis,x j .

(13)

p=0

If we carry out now a data-dependent nonlinear combination of the coefficients of the polynomials for each particular stencil it is obtained

274

A. Hidalgo and L. Tello

whx (x, t n ) =

M 

ψ p (ξ )wˆ inj, p , with wˆ inj, p =

p=0

Ns 

ωs wˆ in,s j, p ,

(14)

s=1

s where the nonlinear weights are ωs = ω˜sω˜ k , with ω˜s = (σsλ+) r , with  being a small k value introduced in order to avoid division by zero. We can take, for instance,  = 10−20 and r = 3. The oscillation indicators are given by

σs =

M  M 

 pm wˆ in,s ˆ in,s j, p w j,m ,

(15)

p=1 m=1

which require the computation of the oscillation matrix M  ∂ α ψ p (ξ ) ∂ α ψm (ξ ) = · dξ. ∂ξ α ∂ξ α α=1 1

 pm

(16)

0

The second stage is conducted in order to carry out reconstruction in z-direction. The reconstruction polynomial is obtained in a similar way as the process carried out in x-direction in the previous stage. This process is performed for each particular degree of freedom wˆ inj, p . According to this, we have whs,z (x, t n ) =

M  M 

ψ p (ξ )ψq (η)wˆ in,s j, pq .

(17)

q=0 p=0

Integral conservation in z-direction is carried out for each particular degree of freedom in x-direction, for all the control volumes of the stencil in z-direction, which is Sis,z j , to yield 1

z e

z e+1/2

M 

z e−1/2

n ψq (η(z))wˆ in,s ˆ ie, p, j, pq dz = w

∀Iie ∈ Sis,z j ,

(18)

q=0

and perform the nonlinear combination whz (x, z, t n ) =

M  M 

ψ p (ξ )ψq (η)wˆ inj, pq with wˆ inj, pq =

q=0 p=0

Ns 

ωs wˆ in,s j, pq .

(19)

s=1

The final expression of the reconstruction polynomial for the control volume Ii j is wi j (ξ, η, t n ) =

M+1 M+1   k=1 l=1

wˆ ik,lj (t n )ψk (ξ )ψl (η).

(20)

An ADER-LSTDG Scheme for the Numerical Simulation of a Global Climate Model

275

Fig. 2 Sketch of the dimension-by-dimension WENO reconstruction performed. Second-order polynomials are obtained using three cells for each polynomial at each Cartesian direction

In Fig. 2 it is displayed a sketch of the dimension-by-dimension procedure carried out. Concerning the source term, we approximate the integrals appearing in (7) using appropriate Gaussian quadrature formulas. In the case of the WENO5 approach, three cells are used for each stencil (r = 3, M = 2) and the developed scheme is fifth-order accurate in space. This way to proceed is computationally less expensive than the fully two-dimensional reconstruction.

4.2 High-Order One-Step Time Discretization: ADER-LSTDG Approach ADER schemes were initially introduced for hyperbolic problems in [31, 58] and later on applied to reaction-diffusion models (see [1, 35, 42, 61]). A notable property of ADER technique is that it attains an arbitrarily high order of accuracy in a single time step. This procedure requires to solve the Riemann problems appearing at finite volumes interfaces, making use of Taylor expansions in time and the CauchyKowalewskaya (CK) procedure to express space derivatives as a function of time derivatives, making use of the differential equation itself. In this work it is applied the variant of the classical ADER technique, based on a local space-time DG predictor (ADER-LSTDG), which is of particular interest when stiff source terms are present. In the model (7), the nonlinear source term appearing in the equations is not stiff, but we use this variant of ADER technique, due to other advantages of this approach, since it allows to avoid the use of Cauchy-

276

A. Hidalgo and L. Tello

Kowalewskaya property, rather cumbersome when applied to nonlinear problems in multi-dimensions. ADER-LSTDG method was reported for the first time in [27] in the context of 1D hyperbolic problems in the finite volume framework. A remarkable application of this technique can be found in [24] where a PN PM scheme with ADERLSTDG approach was used to solve the 3D Navier-Stokes equations. In [42] ADERLSTDG was applied to systems of stiff advection-reaction-diffusion equations, with the introduction of a new initial guess for the local space-time predictor. In this work we also make use (see [42]) of a nodal formulation where the space-time basis functions are Lagrange interpolation polynomials passing through space-time Gauss-Legendre quadrature points. According to the expression (10), our goal now is to obtain the predictor uh := uh (ξ, η, τ ). In order to proceed, let us consider the following system of PDEs in reference coordinates ∂ ˆ∗ ∂ ∗ ∂ ∗ uh + f (uh , ∇uh ) + gˆ (uh , ∇uh ) = Sˆ (uh ), ∂τ ∂ξ ∂η

(21)

where

t ˆ f(uh , ∇uh ); gˆ ∗ (uh , ∇uh ) := fˆ ∗ (uh , ∇uh ) = x ∗ S (uh ) := tS(uh ),

t gˆ (uh , ∇uh )

y

(22)

and let us multiply (21) by space-time test functions θk . After integration on the reference element we get 

       ∂ ∂ ˆ∗ ∂ ∗ ∗ θk , uh + θk , f (uh , ∇uh ) + θk , gˆ (uh , ∇uh ) = θk , Sˆ (uh ) , (23) ∂τ ∂ξ ∂η

where we have used the same notation as in [24, 27, 42] that is

1 1 1

a, b :=

a(ξ, η, τ )b(ξ, η, τ )dξ dηdτ . 0

0

(24)

0

Integration by parts of the integral involving time derivative yields

= where



 ∗ ∂ θ , uh + θk , ∂ξ∂ ˆf (uh , ∇uh ) ∂τ  k  ∗ [θk , wnh ]0 + θk , Sˆ (uh ) ,

[θk , uh ]1 −





  ∂ ∗ gˆ (uh , ∇uh ) + θk , ∂η

(25)

An ADER-LSTDG Scheme for the Numerical Simulation of a Global Climate Model

1 1

1

a, b := 0

[a, b]τ :=

a(ξ, η, τ )b(ξ, η, τ )dξ dηdτ

0

0

1 1

(26) a(ξ, η, τ )b(ξ, η, τ )dξ dη

0

.

277

0

The values wnh appearing in (25) are given by Lagrange WENO (or ENO) reconstruction in space from cell averages as wnh = w(x, y, t n ) =

M+1 

ˆ ln , ˆ ln := l w l (x, y)w

(27)

l=1

where l (x, y) are reconstruction basis functions in space. We get the following iterative process ˆ l − Kξ F∗ (uˆ lm ) − Kη G∗ (uˆ lm ), K1 uˆ lm+1 = MS∗ (uˆ lm ) + F0 w

(28)

where we have introduced the matrices       ∂ θl , K1 = [θk , θl ]1 − ∂τ∂ θk , θl , Kξ = θk , ∂ξ∂ θl , Kη = θk , ∂η M = θk , θl  , F0 = [θk , ψl ]0 .

(29)

The iterative process can be written as ∗ m ∗ m ∗ m ˆ l ) + K−1 ˆ l ) − K−1 ˆ l ). ˆ l − K−1 uˆ lm+1 = K−1 1 MS (u 1 F0 w 1 Kξ f (u 1 Kη g (u

(30)

We solve this expression using a very simple fixed point method to obtain the values uˆ lm+1 to be used in uh (ξ, η, τ ) =

2 (M+1) 

l=1

θl (ξ, η, τ )uˆ l ; ∇uh (ξ, η, τ ) =

2 (M+1) 

∇θl (ξ, η, τ )uˆ l .

(31)

l=1

The iterative process is initialized here using the cell averages as initial value. However, in [42] it is proposed a different technique based on a second-order accurate MUSCL-type approach for the flux term, with a Crank-Nicolson technique for the source terms. The values and gradients of the predictor, uh , ∇uh , are then used in (21). We point out that we do not use here a locally implicit formulation for the source term, as proposed for instance in [27, 42], since our source term does not have a stiff behaviour. Nonetheless, for the interested reader, this formulation would read

278

A. Hidalgo and L. Tello

ˆ l − Kξ f ∗ (uˆ lm ) − Kη g∗ (uˆ lm ). K1 uˆ lm+1 − MS∗ (ulm+1 ) = F0 w

(32)

We note that the finite volume schemes can be considered as a subset of the so-called PN PM schemes, which were put forward in [24, 25, 28], setting N = 0. For this reason we denote so forth the schemes developed as P0 PM choosing the particular value M = 2 in our computations.

5 Assessment of the Numerical Scheme In this section we apply the technique of manufactured solution in order to validate the numerical scheme. We focus on the EBM model since it is the part of the full model where the most relevant phenomena take place. Hence, we consider the following problem Du t −

D K H0 R2

  p (1 − x 2 ) 2 |u x | p−2 u x x + Bu + A =

1 Q S(x)β(u) ρc

+ (x, t)

on (−1, 1)] × (0, T ). u(x, 0) = −100x 6 + 310x 4 − 320x 2 + 90 x ∈ (−1, 1) ∂u (−1, t) ∂x

=

∂u (1, t) ∂x

(33)

= 0 t > 0,

where (x, t) is a forcing term appearing due to the introduction of the manufactured solution. We do not write the expression of this term here, but it can be easily obtained with a computer tool. We use the data given in Table 1. The coalbedo β(u) is given by  0.24 if u ≤ −10, β(u) = (34) 0.69 if u > −10.

Table 1 Values of the physical parameters Physical parameter K H (m2 c−1 ) K H0 (m2 c−1 ) K V (m2 c−1 ) C, B Q(Wm−2 ) c(J(kg◦ C)−1 ) ρ(kgm−3 )

Value 0.049 0.555 × 10−3 0.0125 190, 2 340 1 1

An ADER-LSTDG Scheme for the Numerical Simulation of a Global Climate Model

279

Table 2 L 2 norm of the error and order of accuracy applied to the problem (33) Cells ||Err or || L 2 Order 4.57 × 10−2 6.99 × 10−3 1.04 × 10−3 2.62 × 10−4

15 30 60 120

2.71 2.75 2.00

The exact solution of (33) is given by u(x, t) =

 1  −100x 6 + 310x 4 − 320x 2 + 110 − 20. 1+t

(35)

In Table 2 we show the order of accuracy of the numerical scheme. We notice that the scheme is second-order accurate. Actually, as stated in [42] if we denote by M + 1 the number of cells in the stencil and M is even the order of accuracy is M. Thus, since in this case we are using stencils with three cells, the scheme turns out to be second-order accurate. In Fig. 3 we depict the exact solution, given by (35) of problem (33), with the numerical solution attained with two different meshes (15 cells and 60 cells), for an output time t = 0.25.

6 Numerical Results In this section we solve coupled EBM-DOM model by means of the finite volume approach with WENO reconstruction in space and ADER methodology for time integration. We focus our attention on the case in which there is an influence of the ocean onto the temperature in the surface, which give rise to the well-known thermostatic effect of the ocean. According to this particular effect, the ocean is able to keep a great amount of heat during summer period refreshing the surroundings and, in winter time, the ocean releases slowly the stored heat which makes the climate warmer. As a matter of fact, the locations which are in the vicinity of the sea have a warmer climate than those that are in the interior due to this property of big masses of water, such as the ocean. This nice property is clearly manifested in the numerical results displayed in this section. In order to solve the model we take as initial condition for the ocean interior 2 2 2 the function U (x, z, 0) = 18e−x −z + 6e6z (11e−x − 10). For the case of the upper boundary, that is, the EBM in the upper surface, the aforementioned initial condition 2 is restricted to z = 0 and then it takes the form u(x, 0) = U (x, 0, 0) = 84e−x − 60. This means that, initially, we assign larger values of the temperature to the Equator

280

A. Hidalgo and L. Tello

70

Exact solution Numerical solution (15 cells) Numerical solution (60 cells)

60

Temperature (ºC)

50 40 30 20 10 0 -10

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

x Fig. 3 Comparison of exact solution, given by (35) of the problem (33) with the numerical solution attained with two different meshes for an output time t = 0.25

and there is an exponential decrease of the temperature towards the Poles and the ocean bottom. In Fig. 4 we depict the initial condition taken for the computations. Concerning the physical data we consider the values given in Table 1. The physical coefficients used have been obtained from those proposed in [64] but performing a re-scaling to the rectangle [−1, 1] × [−1, 0]. As it was aforementioned, in the formulation of the EBM we take the function, S(x), which is non-negative, that represents how the incoming solar radiation is distributed throughout the surface, in such a way that most of the heat goes to the Equator and a little amount of heat goes to the Poles. This function is given by S(x) = 1 − 21 P2 (x) where P2 (x) = 21 (3x 2 − 1) which is the second Legendre polynomial in the interval [−1, 1]. The coalbedo β(u) is given by (34). With regard to the expression of the velocity, we introduce the formula given in (36) somehow representing water sinking from the Poles and rising towards the surface in some latitudes in the vicinity of the Equator. ω(x, z) =

10(x + 0.75)(x − 0.75) . (0.1 + 10|x + 0.75|)(0.1 + 10|x − 0.75|)

(36)

An ADER-LSTDG Scheme for the Numerical Simulation of a Global Climate Model

281

Fig. 4 Initial condition used in the computations. A highest temperature is assigned to latitudes close to the Equator whilst the lowest temperatures take place in the Poles

We take, as spatial domain, the rectangle [−1, 1] × [−1, 0], which is discretized in this examples using 40 × 20 cells. Regarding the discretization in time we have calculated the time step as 

t = min

  −1    du  α x 2 α z 2 2 , , α x K H0   , KH KV dx

where α is a diffusion parameter which controls the stability of the numerical scheme. We have used the value α = 0.35 since numerical experiments have shown that higher values of this parameter may produce an unstable numerical solution. In Fig. 5 it is displayed a comparison between the temperatures obtained in the upper boundary considering the effect of the deep ocean and also neglecting this

= 0 there is no influence of the ocean on the temperaeffect. In Eq. (5) if K V ∂U ∂n = 0 this influence is taken into consideration, ture on the surface whilst if K V ∂U ∂n hence the temperature of the ocean conditions the value of the temperature on the upper boundary. Hence, the results obtained clearly show the well-known thermostatic effect of the ocean is reproduced with the numerical results, showing that the maximum temperature is lower and the minimum higher than when this effect is not taken into account.

282

A. Hidalgo and L. Tello

30

Temperature (ºC)

20

10

0

-10 Coupled (t=1) Non-coupled (t=1) Coupled (t=3) Non-coupled (t=3) Coupled (t=6) Non-coupled (t=6)

-20

-30

-0.5

0

0.5

x Fig. 5 Solution of the EBM model for different output times t = 1, 3, 6. Comparison of the solution obtained with coupled model and the non-coupled one

Figures 6, 7 and 8 show the evolution of the temperature inside the ocean both = 0) and when it when the influence of the ocean on the EBM is considered (K V ∂U ∂n is not (K V ∂U

= 0). The results show, as in the upper boundary, that when this effect ∂n is considered the range of values of the temperatures is In Fig. 9 it is displayed the distribution of temperatures inside the ocean for several depths. In particular, the depths taken are for the coordinates: z = −0.25, z = −0.5, z = −0.75 when considering an ocean of depth equal to 1. This means that if we consider the average depth of the sea as 5000m then the depths considered are 1250 m, 2500 m, and 3750 m . The output time considered is t = 3. In the plot it is depicted the comparison inside the ocean of the coupled and non-coupled situations. We recall

= 0. As it was expected, in that coupling in this sense means that we take K V ∂U ∂n regions closer to the surface the difference between coupled and non-coupled cases are more patent than in regions far away from the surface. We can see in Fig. 9 that the difference between the coupled and the decoupled situation is more evident when time increases. The solution is smoother for the case in which the influence of the ocean on the atmosphere is considered and it is sharper when this influence is neglected. As time increases this phenomenon is more visible in the plots. Because of this, although there are intersections between both curves for each particular time, they are more patent for time t = 6 than for times t = 1 and t = 3.

An ADER-LSTDG Scheme for the Numerical Simulation of a Global Climate Model

283

Fig. 6 Solution of the coupled model for output time t = 1. Comparison of the solution obtained with coupled model (bottom) and the non-coupled one (top)

7 Conclusions This work is focused on obtaining the numerical solution of a climate model involving ocean and atmosphere. The numerical scheme is based on a finite volume technique using a dimension-by-dimension 2D WENO method for space reconstruction and an ADER-LSTDG approach for time integration. ADER procedure is a numerical technique that allows to achieve an arbitrary high order of accuracy in a single time step. The variant used in this work is useful to avoid the use of the rather cumbersome Cauchy-Kowalewskaya procedure. It is also useful when dealing with stiff source terms.

284

A. Hidalgo and L. Tello

Fig. 7 Solution of the coupled model for output time t = 3. Comparison of the solution obtained with coupled model (bottom) and the non-coupled one (top)

The mathematical model solved combines an Energy Balance Model (EBM) in the upper boundary and a two-dimensional deep ocean model (DOM). The numerical solution of the EBM is taken as Dirichlet boundary condition for the DOM. In the coupled model, more precisely in the EBM equation, there is also the term K V ∂U ∂n accounting for the influence of the ocean temperature on the surface. Results attained show that, when this coupling term appears in the model, the range of variation of the temperatures is more narrow than when the term is not considered. This result agrees with the well-known thermostatic effect of the ocean. According to the results attained, temperature variation is small in deep zones and large in the vicinity of the surface. This is due to the fact that the most relevant heat exchange processes occur in a narrow layer of the ocean, close to the surface (named mixed layer), where the influence of solar radiation is important.

An ADER-LSTDG Scheme for the Numerical Simulation of a Global Climate Model

285

Fig. 8 Solution of the coupled model for output time t=6. Comparison of the solution obtained with coupled model (bottom) and the non-coupled one (bottom)

Also, results of convergence of the numerical scheme developed are shown, using the Method of Manufactured Solution, considering an exact solution of an essentially similar problem with an additional source term. According to the results obtained, there is a good agreement between the numerical solution and the exact one.

286

A. Hidalgo and L. Tello

Depth z=0.25. Coupled Depth z=0.5. Coupled Depth z=0.75. Coupled Depth z=0.25. Non Coupled Depth z=0.5. Non Coupled Depth z=0.75. Non Coupled

16 15 14

Temperature (ºC)

13 12 11 10 9 8 7 6 5 4 -1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

x Fig. 9 Distribution of temperatures inside the ocean for several depths. Coupled ( K V ∂U ∂n = 0) and = 0) are considered. The closer we are to the surface the more evident is the non-coupled ( K V ∂U ∂n effect of the coupling

References 1. Hidalgo, A., Tello, L., Toro, E.F.: Numerical and analytical study of an atherosclerosis inflammatory disease model. J. Math. Biol. 68(7), 1785–1814 (2014). https://doi.org/10.1007/ s00285-013-0688-0 2. Arcoya, D., Diaz, J., Tello, L.: S-shaped bifurcation branch in a quasilinear multivalued model arising in climatology. J. Diff. Equ. 150(1), 215–225 (1998). https://doi.org/10.1006/jdeq.1998. 3502 3. Baeza, A., Bürger, R., Mulet, P., Zorío, D.: Central WENO schemes through a global average weight. J. Sci. Comput. 78(1), 499–530 (2019). https://doi.org/10.1007/s10915-018-0773-z 4. Baeza, A., Bürger, R., Mulet, P., Zorío, D.: On the efficient computation of smoothness indicators for a class of WENO reconstructions. J. Sci. Comput. 80(2), 1240–530 (2019). https:// doi.org/10.1007/s10915-019-00974-7 5. Baeza, A., Bürger, R., Mulet, P., Zorío, D.: An efficient third-order weno scheme with unconditionally optimal accuracy. SIAM J. Sci. Comput. 42(2), A1028–A1051 (2020). https://doi. org/10.1137/19M1260396 6. Ángel Balaguer, Conde, C.: Fourth-order nonoscillatory upwind and central schemes for hyperbolic conservation laws. SIAM J. Numer. Anal. 43(2), 455–473 (2006) 7. Balsara, D.S., Garain, S., Florinski, V., Boscheri, W.: An efficient class of weno schemes with adaptive order for unstructured meshes. J. Comput. Phys. 404, 109062 (2020). https://doi.org/ 10.1016/j.jcp.2019.109062 8. Balsara, D.S., Garain, S., Shu, C.W.: An efficient class of weno schemes with adaptive order. J. Comput. Phys. 326, 780–804 (2016). https://doi.org/10.1016/j.jcp.2016.09.009

An ADER-LSTDG Scheme for the Numerical Simulation of a Global Climate Model

287

9. Balsara, D.S., Rumpf, T., Dumbser, M., Munz, C.D.: Efficient, high accuracy ADER-WENO schemes for hydrodynamics and divergence-free magnetohydrodynamics. J. Comput. Phys. 228(7), 2480–2516 (2009). https://doi.org/10.1016/j.jcp.2008.12.003 10. Balsara, D.S., Shu, C.W.: Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J. Comput. Phys. 160(2), 405–452 (2000). https://doi.org/10.1006/jcph.2000.6443 11. Bermejo, R., Carpio, J., Díaz, J.I., Galán del Sastre, P.: A finite element algorithm of a nonlinear diffusive climate energy balance model. Pure Appl. Geophys. 165(6), 1025–1047 (2008). https://doi.org/10.1007/s00024-008-0345-5 12. Bermejo, R., Carpio, J., Diaz, J., Tello, L.: Mathematical and numerical analysis of a nonlinear diffusive climate energy balance model. Math. Comput. Model. 49(5), 1180–1210 (2009). https://doi.org/10.1016/j.mcm.2008.04.010 13. Budyko, M.I.: The effect of solar radiation variations on the climate of the earth. Tellus 21(5), 611–619 (1969). https://doi.org/10.3402/tellusa.v21i5.10109 14. Cannarsa, P., Malfitana, M., Martínez, P.: Parameter Determination for Energy Balance Models with Memory, pp. 83–130. Springer INdAM Series. Springer (2020) 15. Castro, M.J., Semplice, M.: Third and fourthorder wellbalanced schemes for the shallow water equations based on the CWENO reconstruction. Math. Comput. Model. 89(8), 304–325 (2019). https://doi.org/10.1002/fld.4700 16. Cravero, I., Puppo, G., Semplice, M., Visconti, G.: CWENO: Uniformly accurate reconstructions for balance laws. Math. Comput. 87, 1689–1719 (2018). https://doi.org/10.1090/mcom/ 3273 17. Díaz, J., Hetzer, G., Tello, L.: An energy balance climate model with hysteresis. Nonlinear Anal. Theory, Methods Appl. 64(9), 2053–2074 (2006). https://doi.org/10.1016/j.na.2005.07. 038 18. Díaz, J.I.: Mathematical analysis of some diffusive energy balance models in Climatology. In: Research Notes in Applied Mathematics no 27, pp. 28–56. Masson, Paris (1993) 19. Díaz, J.I., Tello, L.: On a climate model with a dynamic nonlinear diffusive boundary condition. Discret. Continuous Dyn. Syst. - S 1(2), 253–262 (2008). https://doi.org/10.3934/dcdss.2008. 1.253 20. Díaz, J.I., Hidalgo, A., Tello, L.: Multiple solutions and numerical analysis to the dynamic and stationary models coupling a delayed energy balance model involving latent heat and discontinuous albedo with a deep ocean. Proc. R. Soc. A. (2014). https://doi.org/10.1098/rspa. 2014.0376 21. Díaz, J.I.: On the mathematical treatment of energy balance climate models. In: Díaz, J.I. (ed.) The Mathematics of Models for Climatology and Environment, pp. 217–251. Springer, Berlin (1997) 22. Díaz, J.I., Tello, L.: A nonlinear parabolic problem on a riemannian manifold without boundary arising in climatology. Collectanea Math. 50(1), 19–51 (1999) 23. Diaz, J.I., Tello, L.: A 2D climate energy balance model coupled with a 3D deep ocean model. Electron. J. Diff. Equ. 16, 129–135 (2007) 24. Dumbser, M.: Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier-Stokes equations. Comput. Fluids 39(1), 60–76 (2010). https://doi.org/10.1016/j. compfluid.2009.07.003 25. Dumbser, M., Balsara, D.S., Toro, E.F., Munz, C.D.: A unified framework for the construction of one-step finite volume and Discontinuous Galerkin schemes on unstructured meshes. J. Comput. Phys. 227(18), 8209–8253 (2008). https://doi.org/10.1016/j.jcp.2008.05.025 26. Dumbser, M., Boscheri, W., Semplice, M., Russo, G.: Central weighted eno schemes for hyperbolic conservation laws on fixed and moving unstructured meshes. SIAM J. Sci. Comput. 39(6), A2564–A2591 (2017). https://doi.org/10.1137/17M1111036 27. 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). https://doi.org/10. 1016/j.jcp.2007.12.005

288

A. Hidalgo and L. Tello

28. Dumbser, M., Hidalgo, A., Castro, M., Parés, C., Toro, E.F.: FORCE schemes on unstructured meshes II: Non-conservative hyperbolic systems. Comput. Methods Appl. Mech. Eng. 199(9), 625–647 (2010). https://doi.org/10.1016/j.cma.2009.10.016 29. Dumbser, M., Hidalgo, A., Zanotti, O.: High order space-time adaptive ADER-WENO finite volume schemes for non-conservative hyperbolic systems. Comput. Methods Appl. Mech. Eng. 268, 359–387 (2014). https://doi.org/10.1016/j.cma.2013.09.022 30. Dumbser, M., Zanotti, O., Hidalgo, A., Balsara, D.S.: ADER-WENO finite volume schemes with space-time adaptive mesh refinement. J. Comput. Phys. 248, 257–286 (2013). https://doi. org/10.1016/j.jcp.2013.04.017 31. Toro, E.F., Millington, R.C., Nejad, L.A.M.: Towards Very High Order Godunov Schemes, pp. 907–940. Springer, US (2001) 32. Fambri, F., Dumbser, M., Köppel, S., Rezzolla, L., Zanotti, O.: ADER discontinuous Galerkin schemes for general-relativistic ideal magnetohydrodynamics. Mon. Notices R. Astron. Soc. 477(4), 4543–4564 (2018). https://doi.org/10.1093/mnras/sty734 33. Fambri, F., Dumbser, M., Zanotti, O.: Space-time adaptive ADER-DG schemes for dissipative flows: compressible Navier–Stokes and resistive MHD equations. Comput. Phys. Commun. 220, 297–318 (2017). https://doi.org/10.1016/j.cpc.2017.08.001 34. Floridia, G.: Approximate controllability for nonlinear degenerate parabolic problems with bilinear control. J. Diff. Equ. 257(9), 3382–3422 (2014). https://doi.org/10.1016/j.jde.2014. 06.016 35. Gassner, G., Loercher, F., Munz, C.D.: A contribution to the construction of diffusion fluxes for finite volume and Discontinuous Galerkin schemes. J. Comput. Phys. 224(2), 1049–1063 (2007). https://doi.org/10.1016/j.jcp.2006.11.004 36. Ghil, M., Childress, S.: Topics in geophysical fluid dynamics atmospheric dynamics, dynamo theory, and climate dynamics. In: Applied Mathematical Sciences. Springer, New York (1987). https://doi.org/10.1007/978-1-4612-1052-8 37. Godunov, S.: A difference scheme for numerical solution of discontinuous solution of hydrodynamic equations. Math. Sbornik 47, 271–306 (1959) 38. Harten, A.: Eno schemes with subcell resolution. J. Comput. Phys. 83(1), 148–184 (1989). https://doi.org/10.1016/0021-9991(89)90226-X 39. Harten, A., Engquist, B., Osher, S., Chakravarthy, S.R.: Uniformly high order accurate essentially non-oscillatory schemes, iii. J. Comput. Phys. 131(1), 3–47 (1997). https://doi.org/10. 1006/jcph.1996.5632 40. Hetzer, G.: The structure of the principal component for semilinear diffusion equations from energy balance climate models. Houston J. Math. 203–216 (1990) 41. Hetzer, G., Tello, L.: On a reaction-diffusion system arising in climatology. Dyn. Syst. Appl. 11(3), 381–401 (2002) 42. Hidalgo, A., Dumbser, M.: ADER Schemes for Nonlinear Systems of Stiff Advection– Diffusion–Reaction Equations. J. Sci. Comput. 48(1–3), 173–189 (2011). https://doi.org/10. 1007/s10915-010-9426-6 43. Hidalgo, A., Tello, L.: A Finite Volume Scheme for Simulating the Coupling between Deep Ocean and an Atmospheric Energy Balance Model, pp. 239–255. Springer, Berlin (2011) 44. Hidalgo, A., Tello, L.: On a climatological energy balance model with continents distribution. Discrete Continuous Dyn. Syst. A 35(4), 1503 (2015). https://doi.org/10.3934/dcds.2015.35. 1503 45. Hidalgo, A., Tello, L.: Numerical approach of the equilibrium solutions of a global climate model. Mathematics 8(9), 1542 (2020) 46. Díaz, J.I., Hernández, J., Tello, L.: On the multiplicity of equilibrium solutions to a nonlinear diffusion equation on a manifold arising in climatology. J. Math. Anal. Appl. 216(2), 593–613 (1997). https://doi.org/10.1006/jmaa.1997.5691 47. Jiang, G.S., Shu, C.W.: Efficient Implementation of Weighted ENO Schemes. J. Comput. Phys. 126(1), 202–228 (1996). https://doi.org/10.1006/jcph.1996.0130 48. LeVeque, R.J.: Finite-Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002)

An ADER-LSTDG Scheme for the Numerical Simulation of a Global Climate Model

289

49. Levy, D., Nayak, S., Shu, C., Zhang, Y.: Central WENO schemes for Hamilton-Jacobi equations on triangular meshes. SIAM J. Sci. Comput. 28(6), 2229–2247 (2006). https://doi.org/10.1137/ 040612002 50. Levy, D., Puppo, G., Russo, G.: Central WENO schemes for hyperbolic systems of conservation laws. ESAIM: Modélisation mathématique et analyse numérique 33(3), 547–571 (1999). https://doi.org/10.1137/040612002 51. Liu, X.D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115(1), 200–212 (1994). https://doi.org/10.1006/jcph.1994.1187 52. Murawski, K., Stpiczy´nski, P.: Implementation of muscl-hancock method into the c++ code for the euler equations. Bull. Polish Acad. Sci.: Tech. Sci. 60(No 1), 45–53 (2012). https://doi. org/10.2478/v10175-012-0008-7 53. North, G.R.: Multiple solutions in energy balance climate models. Palaeogeography, Palaeoclimatology, Palaeoecology 82(3), 225–235 (1990). https://doi.org/10.1016/S00310182(12)80003-4 54. Sellers, W.D.: A Global Climatic Model Based on the Energy Balance of the Earth-Atmosphere System. J. Appl. Meteorol. 8(3), 392–400 (1969). https://doi.org/10.1175/1520-0450 55. Shu, C.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic Conservation Laws. NASA/CR-97-206253 ICASE Report No. 97–65 (1997) 56. Stone, P.H.: A simplified radiative-dynamical model for the static stability of rotating atmospheres. J. Atmos. Sci. 29, 405–418 (1972). https://doi.org/10.1175/1520-0469 57. Titarev, V.A., Toro, E.F.: ADER: Arbitrary High Order Godunov Approach. J. Sci. Comput. 17(1), 609–618 (2002). https://doi.org/10.1023/A:1015126814947 58. Titarev, V.A., Toro, E.F.: Finite-volume WENO schemes for three-dimensional conservation laws. J. Comput. Phys. 201(1), 238–260 (2004). https://doi.org/10.1016/j.jcp.2004.05.015 59. Titarev, V.A., Toro, E.F.: ADER schemes for three-dimensional non-linear hyperbolic systems. J. Comput. Phys. 204(2), 715–736 (2005). https://doi.org/10.1016/j.jcp.2004.10.028 60. Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. Springer, Berlin (2009) 61. Toro, E.F., Hidalgo, A.: ADER finite volume schemes for nonlinear reaction diffusion equations. Appl. Numer. Math. 59(1), 73–100 (2009). https://doi.org/10.1016/j.apnum.2007.12. 001 62. Van Leer, B.: Towards the ultimate conservative difference scheme iii. upstream-centered finitedifference schemes for ideal compressible flow. J. Comput. Phys. 23(3), 263–275 (1977). https://doi.org/10.1016/0021-9991(77)90094-8 63. Vázquez-Cendón, M.E.: Solving Hyperbolic Equations with Finite, vol. Methods. Springer International Publishing, UNITEXT (2015) 64. Watts, R.G., Morantine, M.: Rapid climatic change and the deep ocean. Clim. Change 16(1), 83–97 (1990). https://doi.org/10.1007/BF00137347 65. Zanotti, O., Fambri, F., Dumbser, M.: Solving the relativistic magnetohydrodynamics equations with ADER discontinuous Galerkin methods, a posteriori subcell limiting and adaptive mesh refinement. Mon. Notices R. Astron. Soc. 452(3), 3010–3029 (2015). https://doi.org/10.1093/ mnras/stv1510

Efficient Experimental and Numerical Methods for Solving Vertical Distribution of Sediments in Dam-Break Flows Thomas Rowan and Mohammed Seaid

Abstract In this work, a class of efficient experimental and numerical methods for recording sediment distributions in dam-break flows is developed and assessed. A new experimental platform is developed and tested enabling accuracy and high frameper-second images to be generated and evaluated. The novelty of this study is that the collected images are broken down into smaller cellular-based images which are individually assessed for their color contents. Every frame collected is first filtered to exclude any background pixels and remove the effects of photography and lighting, then the color spectra of the frame are analyzed and sensible filtering added to remove the complex effects of fluid features including ripples and air bubbles. Once these processes are complete, a simple color comparison can be utilized to assess the sediment fraction of the flow in each frame and cell. The proposed method is used on a small-scale dam-break problem and a four-stage breakdown of the dam-break is given with measured sediment levels accounted for. Mathematical models based on multilayer shallow water equations with mass exchange terms are developed in this study. The governing equations form a system of conservation laws with source terms. As numerical methods we implement a fast, accurate and well-balanced finite volume characteristics solver. Comparisons between experimental and numerical results for the vertical distribution of sediments in a dam-break problem are also discussed.

T. Rowan Department of Civil and Environmental Engineering, Imperial College London, South Kensington Campus, London SW72AZ, UK e-mail: [email protected] M. Seaid (B) Department of Engineering, University of Durham, South Road, Durham DH1 3LE, UK e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 D. Zeidan et al. (eds.), Numerical Fluid Dynamics, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-16-9665-7_10

291

292

T. Rowan and M. Seaid

1 Introduction Transport of sediments is a crucial aspect of modeling sedimentary flows, although it is very hard to know exactly what mode of sediment transport is prevalent in any flow. In general there are five modes of sediment transport; dissolved load, wash load, suspended load, intermittently suspended (or bounding) load, and bed load. Often the former three and latter two are amalgamated to give suspended and bed load, see, for instance, [7, 10, 30, 31] and further reference are therein. Many experiments for sediment transport have subsequently been conducted during the past years on various bed forms. The one-dimensional natured experiments include the small-scale dam-break problems were detailed in [11], and the stream over a dyke was also studied in the Delft hydraulics laboratory [7]. There are also many twodimensional experimental studies including a partial dam-break reported in [36], and the groin experiment conducted in [1]. These experimental investigations have been very useful for quantitative and qualitative understandings of hydrodynamics as well as morphodynamics at a laboratory scale. However, one aspect that is rarely measured is the vertical distribution of sediments within the fluid. This is mainly due to its complex nature and the difficulties in sampling of developing flows without altering the movement of sediments or the speed of fluids. In the present work, these problems are overcame by utilizing a simple small-scale one-dimensional flow and image processing along with color content to measure both sediment transport and water flow. This method still presents a number of difficulties such as parallax error, effects of lighting, and water surface effects. These drawbacks are eliminated in part by using the color content, disregarding the background among others, and considering the images gathered in both the RGB in HSI color domains to give the best results. It should be stressed that vertical distribution of sediments is very important for a full understanding of sediment transport in water flows. The techniques implemented in our study would provide a new efficient and low cost approach for gathering sediment distribution data in dam-break problems. There exists many models for suspended sediments in water flows with different empirical formulae for each type of load under study. One of the earliest and the most recognized relation was formed from the work conducted by Shields in [34]. Indeed, Shields has created the foundation for a lot of the work conducted nowadays, but it has a series of limits due to the ranges over which the data was generated (such as grain size and Reynolds numbers). However, the line between the initiation of motion and the entrainment of particles into the flow was, as Sheilds explained, difficult to assess. Extensive work has been carried out to improve this analysis and develop consistent relations for suspended sediments, see for example [8, 25, 29]. Most natural bodies of water in which sediment transport is a major feature, such as rivers and coastal waters, may be approximated by shallow water flows obtained by depth-averaging the complex free-surface flows, see [13, 14, 23, 26, 27, 40, 41] among others. For the modeling of morphodynamics in shallow water flows, the three most popular models are the Grass model [19], the Meyer-Peter & Muller model [24] and the Van-Rijn model [29]. A large amount of work has also been performed on

Effcient Experimental and Numerical Methods for Solving Vertical Distribution …

293

the effects of water flows on sediment beds, see, for example, [5, 32, 38, 39]. The predominant approach to sediment transport problems is to rely on empirical data sets, or semi-empirical relations derived from these data sets, see [33] among others. The scale of these approaches can vary widely, from particulate tracking [15], to whole estuary simulations using sediment balance models [15]. In general, the incompressible Navier–Stokes equations have been widely used as the basis for a large number of models and solutions in this field. However, the use of a fully three-dimensional approach for these flows in sediment transport simulations presents challenging formulations and it is computationally expensive. Therefore, modeling simplifications are often preferable, see [23] and further references are therein. One notable development in this direction consists of multilayer shallow water models which offer, a more efficient and accurate flow description. Both miscible [2, 3, 18] and immiscible layered models [20] have been considered in the literature. The advantage of both models is that they avoid the computationally expensive solutions of three-dimensional flows while returning stratified velocities (in two space dimensions). The shallow water equations, can be derived from the non-stationary three-dimensional Navier–Stokes equations, see, for example, [3, 18]. In the presented work, a novel miscible multilayer model is developed, this model includes movable beds and transport of sediments. The formulation of the model consists of a set of multilayer shallow water equations coupled to a set of transport equations for the suspended sediments in each layer, and a set of semi-empirical equations for erosion and deposition are used. Intra-layer mass movement is captured through mass exchange terms in both the water flow and the sediment concentrations. Thus, each layer is able to have a varying sediment concentration and flow velocities to its neighboring layers. Other models in [21] have similar resolution for the vertical concentration, but the proposed multilayer model does not require computationally demanding vertical discretizations. Recently, a similar work has been extended to account for turbulent kinetic energy in [43]. The focus in the present work is on developing a framework within which empirical or semi-empirical relations as those reported in [43] can be easily incorporated into the numerical model as required. For the entrained sediments we use the equations proposed in [9] but the presented formulation allows for other erosion and deposition equations to be easily incorporated such as those reported in [16, 19, 22, 24]. Solving the multilayer shallow water equations numerically is a complex problem owing to their nonlinear nature as well as the source terms and the free-surface aspects, compare [2, 3] among others. The inclusion of bed load and suspended sediment equations in the multilayer model creates even more complexities to a numerical solver for this fully coupled system. The problem arises from the coupling terms in these models that encompass derivatives of the unknown physical variables, that the result in the system becoming non-conservative and even non-hyperbolic. As a result, numerical methods originally developed for multilayer shallow water equations over fixed beds will encounter instabilities if applied individually to each layer. In the current study, we implement the Finite Volume Characteristics (FVC) method developed in [6] for solving single-layer shallow water equations. As well as being second-order accurate, the FVC method circumvents the solution of Riemann

294

T. Rowan and M. Seaid

problems as it is a predictor–corrector type method. In the predictor stage, the method of characteristics is used to reconstruct the numerical fluxes while the corrector stage recovers to the conservation equations using the finite volume discretization. As shown in [2, 31], the considered FVC method has been used to solve a class of multilayer shallow water equations. As shown in these studies, the FVC method is simple, conservative, non-oscillatory, and a practical solver for multilayer shallow water flows over movable beds. In the present work, further improvements to the FVC method have been implemented including: (i) a third-order Runge–Kutta scheme is applied to the time integration; (ii) a second-order splitting operation is used for the solution of the source terms, and; (iii) a cubic Spline interpolation is utilized in the predictor stage. We present numerical results in order to verify the multilayer shallow water flows over erodible beds. Finally, we demonstrate the ability of the proposed model for calculating lateral and vertical distributions of velocities for a multilayer dam-break flow over erodible bed. This paper is organized as follows: In Sect. 2 we present a full description of the experimental setup used and discuss the difficulties to overcome and potential sources of errors. In Sect. 3 we introduce the mathematical equations governing multilayer shallow water flows over a sedimentary topography. In Sect. 4, the numerical method for solving the governing equations is formulated. Included in this section is the reconstruction of numerical fluxes by applying the method of characteristics and the discretization of source terms in the model. Experimental and numerical results are presented in Sect. 5. Finally, Sect. 6 contains some concluding remarks.

2 Experimental Setup for Attaining Vertical Distribution of Sediments in Dam-Break Flows This section overviews the experimental methods used to calculate the vertical distribution of sediments in a one-dimensional dam-break flow. It focuses on both the physical techniques that are involved in creating the dam-break, as well as overviewing the digital processing techniques required to extract the measured data. It further describes the design and tuning process that was used to create accurate and repeatable results. Commercial image processing tools were used as they provide a basic toolbox that could be quickly adapted for the purposes of this study. The sizing of the entire experiment was of great concern and was dictated by the photographic equipment available and the sediment sizes. There have been a number of smallscale one-dimensional dam-break problems which have already been investigated at various lengths from, the 1.2 m Taipei experiment in [11] to the 6 m long Hanyang University experiments in [28]. With the aim to achieve a 0.5 mm to pixel precision, a camera with 1080 p resolution and 100 fps is used in our experiment. For the presented results, the bed is formed of a well-graded soil in the range of 1–0.16 mm and with an averaged particle size d = 0.25 mm. Consequently, the domain was set to a length of 900 mm and, to enable workability and minimal edge effects, the domain

Effcient Experimental and Numerical Methods for Solving Vertical Distribution …

295

Fig. 1 Design illustration of the experimental rig showing a fixed bed used in the present work

width was fixed at 75 mm. The dam-break height was initially made at 100 mm. and following testing it was increased to 120 mm to amplify the erosion observed. Having decided upon the overall dimensions, the mechanism was designed. An electronically controlled pneumatic valve was used to minimize human error and the opening time. The target set was to allow for a dam-break to occur in less than 0.02 s. The available line pressure available was 10 bar and a small tracking cylinder initially trialed. A range of designs were tested, a directly coupled linear actuator in its simplest configuration was found to be the quickest and most repeatable with an averaged opening time of 0.011 s. A bead of sealant was placed in a rebate along the edges of the dam so as to seal the edged but not impair the opening of the dam. The sealant reduced the averaged opening time to 0.018 s which was still acceptable. One conclusion that emerged early on was that even if the bed is compacted it will still seep. A range of sediments were trialed from ABS spheres at 6 mm to fine sand at 0.25 mm, nonetheless the seepage beneath the dam undermined to an impractical extent before a differential head of 100 mm could be achieved. Consequently, a solid apron behind the dam was implemented as shown in Fig. 1. In order to maximize effects of the high intensity lighting used, transparent ends and a high-sided fixed-end tank were used. This diverged from other open-ended or re-circulating tanks that are more normally used during studies of this type. The dam is positioned at 330 mm from the left-hand side which allows 640 mm for the dam-break to run over as shown in Fig. 1. This maximizes the flow evolution before it reaches the end of the tank, giving an experimental time of 0.42 s over a fixed low slip bed and a 0.48 s over a high friction sand bed. The high quality consumer grade camera was able to control for photographic concerns like white balance among other concerns (these were also checked by eye and able to be modified if required). All other laboratory

296

T. Rowan and M. Seaid

Fig. 2 A sketch detailing the problem of parallax errors related to high-speed camera

conditions were maintained throughout the experiments as far as possible. In order to mitigate the errors produced by a single camera recording, a variety of postprocessing measures were implemented. The first addressed was parallax error as shown in Fig. 2, the greater the orthogonal distance from the camera position the more skewed the image captured would be, which would affect the sediment calculations. Three mitigations were put in place: (i) the camera was centered at 450 mm not at the dam-break; (ii) a tele-centric lens was used and a cell neighbor averaging calculation was implemented. A parallax error also affected the vertical dimension of the images, though to a lesser extent than in the horizontal dimension. The next issue was that the mapping of pixels dominated by the bed into the fluid cells would artificially increase the measured concentration. Balanced against the need to capture the bed load this became a complex issue. Two steps were used to reduce this error: first, the camera was positioned at the bed height and as erosion in these experiments was minimal this almost eradicated the problem. Second, the bed height was extracted at the light change using the human eye (though in the future it is hoped to use a deep vision approach). This was possible as the increased voids in bed load as compared to a packed bed, created a noticeable light change. Algorithms were initially trialed to discern this but proved to be less accurate and more time consuming than blowing up the image and implementing a dragging point-click interpolation method. Thus, the following procedure is developed to correct these raw images and extract the required data:

Effcient Experimental and Numerical Methods for Solving Vertical Distribution …

297

Fig. 3 A screenshot of the automated program used in the current work

1. The white balanced was checked against the standard and corrected if necessary; 2. The bed and water surfaces were mapped by a point-click method and parallax errors were computed; 3. Irrelevant background pixels were removed and any remaining hue/lighting errors were corrected by multiple HSI masks; 4. A sediment color chart was created for each experimental run using known high and low areas; 5. The image was then broken down into cells and the depth-averaged concentration by interpolated from the color chart and the cell average; 6. The results were checked against the quantity of erosion in the experiment. This step also allows for an error measure to be made. It should be stressed that, although this procedure requires some human interaction in step 2, all other steps were automated and incorporated into a graphical user interface (GUI) as shown in Fig. 3.

3 Mathematical Models for Vertical Distribution of Sediments in Dam-Break Flows Multilayer flow systems are mainly obtained using a vertical discretization of the three-dimensional Navier–Stokes equations accounting for of shallow water assumptions, compare [2–4] and further references are therein. In the present study, we consider the one-dimensional version of the model to each of the layers in the fluid and include models for sediment transport accounting for mass exchange between the layers and intra-layer forces between the erodible bed and the water flow. In a

298

T. Rowan and M. Seaid

Fig. 4 A simple illustration of multilayer shallow water flows over erodible beds. Each layer k (k = 1, 2, . . . , M) is characterized with a water height h k , a water velocity u k , and a sediment concentration ck . The initial bed is denoted by B

multilayer system of a total number of M layers, the shallow water equations for each layer k = 1, 2, . . . , M read as ∂ (h k u k ) ∂h k + = G k−1/2 − G k+1/2 , ∂t ∂x ∂ ∂ (h k u k ) + ∂t ∂x



1 h k u 2k + gh 2k 2

(1)

 = −gh k

∂B + Fk , ∂x

where u k is the depth-averaged water velocity of the kth layer, B the bottom topography, g the gravitational acceleration, and h k the water height of the kth layer defined as k = 1, . . . , M, (2) h k = lk H, where H is the total water depth and lk is the proportional height of the layer, see Fig. 4 for an illustration. In (1), Fk includes the intra-layer forces defined below and G k±1/2 are mass exchange terms between the layers including erosion and deposition in the lower layer defined as

G k−1/2

⎧ k   M  ⎪ ∂h γ u γ E k − Dk ⎪ ∂(h β u β ) ⎪ −lβ + , if k = 2, . . . , M, ⎨ ∂x ∂x 1− p γ =1 = β=1 ⎪ E − D1 ⎪ ⎪ ⎩− 1 , if k = 1, 1− p

with E k and Dk represent the entrainment and deposition terms in the upward and downward directions, respectively. Following the same procedure as in[3], we sum

Effcient Experimental and Numerical Methods for Solving Vertical Distribution …

299

the first equation in (1) for all the layers to obtain a single equation for the total water height H as M ∂H  ∂ E 1 − D1 + (h k u k ) = . ∂t ∂ x 1− p k=1 In the current study, we also consider bed load and suspended sediments within the multilayer shallow water system (1). To this end, we define the depth-averaged concentration ck for the kth layer as ck =

ρk − ρw , ρs − ρk

(3)

where p is the porosity, ρw the water density, ρs the sediment density and ρk is the density of the water-sediment mixture in the layer k. Hence, the governing equations we consider for modeling multilayer shallow water flows over erodible beds are given as E 1 − D1 ∂ H  ∂(h k u k ) + = , ∂t ∂ x 1− p k=1   ∂ (h k u k ) ∂ (ρs − ρw ) 2 ∂ck ∂B 1 + h k u 2k + gh 2k = −gh k − + Fk , gh k ∂t ∂x 2 ∂x 2ρk ∂x M

(4) ∂(h k ck ) ∂ (h k u k ck ) + = E k − Dk − ck+1/2 G k+1/2 + ck−1/2 G k−1/2 − ∂t ∂x   2 ∂ c+1/2 ∂ 2 c−1/2 ,  − ∂z 2 ∂z 2 ∂B E 1 − D1 =− , ∂t 1− p where E 1 and D1 are the erosion and deposition rates on the bottom fluid layer, respectively. In this study, the inter-layer diffusion in the concentration is handled by considering the diffusion potential between two layers c,k−1/2 and c,k+1/2 and a diffusion coefficient . Note that these terms are included in the model to handle the sediment diffusion. In (4), the external force Fk acting on the kth layer accounting for friction and momentum exchange effects is given by Fk = Fk(u) + Fk(b) + Fk(w) + Fk(μ) , with Fk(u) is related to the momentum exchange between the layers by Fk(u) = u k+1/2 G k+1/2 − u k−1/2 G k−1/2 −

1 (ρ0 − ρk )(E k − Dk )u k , lk ρk (1 − p)

(5)

300

T. Rowan and M. Seaid

where ρ0 is the density of the saturated bed related to the porosity as ρ0 = ρw p + ρs (1 − p) and the intermediate velocity u k+1/2 is reconstructed using an upwind method based on the sign of the mass exchange term as u k−1/2

u k−1 , if G k−1/2 ≥ 0, = otherwise. uk , (μ)

The vertical kinematic eddy viscosity term Fk between neighboring layers as

Fk(μ)

in (5) takes into account the friction

⎧ u k−1 − u k ⎪ ⎪ −2ν , if k = M, ⎪ ⎪ (lk−1 + lk )H ⎪ ⎨ u k+1 − u k u l−1 − u k − 2ν , if k = 2, . . . , M − 1, = 2ν (l + l )H (l ⎪ k+1 k k−1 + lk )H ⎪ ⎪ u − u ⎪ l+1 k ⎪ , if k = 1, ⎩2ν (lk+1 + lk )H

where ν is the eddy viscosity. The external bed friction term Fk(b) in (5) is given as Fk(b)

⎧ ⎨

gn 2b u |u |, if k = 1, 1/3 1 1 = ⎩ H 0, otherwise, −

(6)

where n b is the Manning roughness coefficient. The surface wind force Fk(w) in (5) is defined as ⎧ ⎨ σ 2 ρa w|w|, if k = M, − (w) (7) Fk = H ⎩ 0, otherwise, where w is the relative wind velocity at 10 m above the water surface and σ is the wind stress coefficient. Note that for the bottom layer, an equation that relates the effects of an erodible bed is included in the model (1). These equations are presented in a very general form such that appropriate empirical erosion and deposition equations can be substituted with ease. Thus, to determine the entrainment and deposition terms in (1) we assume a non-cohesive sediment and we use the empirical relations reported in [9] ws (1 − Ca )m Ca , if k = 1, Dk = (8) 0, otherwise, where ws is the deposition coefficient experimentally measured in [30, 33, 42], d the averaged diameter of the sediment particle, m an exponent indicating the effects

Effcient Experimental and Numerical Methods for Solving Vertical Distribution …

301

of hindered settling due to high sediment concentrations, Ca = βc ck is the near-bed volumetric sediment concentration. Here, βc is a coefficient larger than unity used to ensure that the near-bed concentration does not exceed (1 − p) and it is defined in [10] by   1− p . βc = min 2, ck For the entrainment of the material the following relation is used ⎧ ⎨ϕ θ − θc u d −0.2 , if θ ≥ θ and k = 1, 1 c h1 Ek = ⎩0, otherwise,

(9)

where ϕ is a coefficient to control the erosion forces, θc is a critical value of Shields parameter for the initiation of sediment motion and θ is the Shields coefficient defined by u2 θ= ∗ , sgd with s = ρρws − 1 is the submerged specific gravity of sediment and u ∗ is the friction velocity defined as

u 2∗ =

g n 2b |u 1 | . h 1/3

Note that the equations used for the entrainment and deposition have been widely used in the literature for the conventional single-layer shallow water flows over erodible beds, see, for example, [5, 30, 33, 42]. It should also be pointed out that no vertical velocities are calculated in the proposed model, but the vertical sediment diffusion is a major problem for a formulation of this type. In this study, a sediment diffusion coefficient  is introduced in the multilayer model (4). For ease of presentation, we re-arrange the governing Eq. (4) into a compact vector form as ∂W ∂F(W) + = Q(W) + R(W), (10) ∂t ∂x where W is the vector of conserved variables, F(W) is the vector of flux functions, Q(W) and R(W) are the vectors of source terms defined by

302

T. Rowan and M. Seaid

⎛

⎞

k 

lα H u α ⎟ ⎜ ⎜ ⎟ ⎜ α=1 ⎟ ⎜ ⎟ 1 ⎜ H u2 + g H 2 ⎟ 1 ⎜ ⎟ 2 ⎜ ⎟ ⎜ ⎟ H u 1 c1 ⎜ ⎟ 1 ⎜ 2⎟ 2 ⎜ H u2 + g H ⎟ ⎜ ⎟ 2 ⎟, F(W) = ⎜ H u 2 c2 ⎜ ⎟ ⎜ ⎟ .. ⎜ ⎟ . ⎜ ⎟ ⎜ ⎟ 1 ⎜ H u2 + g H 2 ⎟ ⎜ ⎟ k 2 ⎜ ⎟ 1 ⎜ ⎟ ⎜ H u 2k + g H 2 ⎟ ⎜ ⎟ 2 ⎝ ⎠ H u k ck

⎛

0 ∂B (ρs − ρw ) ∂c1 ⎜ − gl1 H 2 ⎜ −g H ⎜ ∂x 2ρ1 ∂x ⎜ 0 ⎜ ⎜ ⎜ −g H ∂ B − (ρs − ρw ) gl H 2 ∂c2 ⎜ 2 ∂x 2ρ2 ∂x ⎜ Q(W) = ⎜ 0 ⎜ ⎜ .. ⎜ . ⎜ ⎜ (ρs − ρw ) ∂ck ∂B ⎜ − gl M H 2 ⎜ −g H ⎜ ∂x 2ρk ∂x ⎝ 0 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

0

⎛ ⎞ H ⎜ H u1 ⎟ ⎟ ⎜ ⎜ H c1 ⎟ ⎟ ⎜ ⎜ H u2 ⎟ ⎟ ⎜ ⎜ H c2 ⎟ ⎟ ⎜ W = ⎜ . ⎟, ⎜ .. ⎟ ⎟ ⎜ ⎜ H uk ⎟ ⎟ ⎜ ⎜ H uk ⎟ ⎟ ⎜ ⎝ H ck ⎠ B ⎛

E 1 − D1 1− p

⎜ ⎜  1  (u) ⎜ − F1 + F1(b) + F1(μ) ⎜ ⎜ l1 ⎜ ∂ 2 c,3/2 ⎜ E 1 − D1 − G 3/2 c3/2 −  ⎜ 2 ⎜  ∂z ⎜ 1  (u) (μ) ⎜ F2 + F2 − ⎜ l2 ⎜ 2 2 ⎜ R(W) = ⎜ G 5/2 c5/2 − G 3/2 c3/2 +  ∂ c,3/2 −  ∂ c,5/2 ⎜ ∂z 2 ∂z 2 ⎜ .. ⎜ ⎜ . ⎜  1  (u) ⎜ (μ) ⎜ − Fk + Fk(w) + Fk ⎜ lk ⎜ ∂ 2 c,M−1/2 ⎜ ⎜ −G M−1/2 c M−1/2 +  ⎜ ∂z 2 ⎝ E 1 − D1 . − 1− p

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

It should be stressed that the source term Q contains the first-order differential terms with respect to the coordinate x, while the remaining forces are included in the source term R. This structure is advantageous as it allows for a time splitting operator in (10), for which the source terms Q and R are treated separately in different stages of the splitting.

Effcient Experimental and Numerical Methods for Solving Vertical Distribution …

303

4 Numerical Methods for Vertical Distribution of Sediments in Dam-Break Flows To integrate the system (4), time is divided into subintervals [tn , tn+1 ] with length t = tn+1 − tn and the notation Wn is used to denote the value of a generic function W at time tn . Here, a second-order splitting procedure, like [37] is used, and carried out in three stages as: Stage 1: Solve for W∗ ∂W∗ = R(W∗ ), ∂t W∗ (tn ) = W(tn ).

t ∈ [tn , tn+1/2 ], (11)

Stage 2: Solve for W∗∗ ∂F(W∗∗ ) ∂W∗∗ + = Q(W∗∗ ), ∂t ∂x

t ∈ [tn , tn+1 ],

W∗∗ (tn ) = W∗ (tn+1/2 ).

(12)

Stage 3: Solve for W∗∗∗ ∂W∗∗∗ = R(W∗∗∗ ), ∂t W

∗∗∗

t ∈ [tn+1/2 , tn+1 ], (13)

∗∗

(tn+1/2 ) = W (tn+1 ).

To complete the time integration, the explicit third-order Runge–Kutta method [35] is used for each stage in (11)–(13). For instance, to advance the solution of (11) from time tn to time tn+1 the following is used W (1) = Wn + tR(Wn ), 3 1 W (2) = Wn + W (1) + 4 4 1 n 2 (2) n+1 W = W + W + 3 3

1 tR(W (1) ), 4 2 tR(W (2) ). 3

(14)

The asterisk is dropped off of the variables for ease in the notation. Note that the Runge–Kutta method (14) is total variation diminishing (TVD), third-order accurate in time, and stable under the usual Courant–Friedrichs–Lewy (CFL) condition involving eigenvalues of the system under study. It should also be noted that explicit expressions of the eigenvalues for the system (10) are not trivial to find and as for multilayer shallow water equation over fixed beds there may exist situations for which eigenvalues become complex. In these cases, the multilayer system (10) is not hyper-

304

T. Rowan and M. Seaid

bolic and yields to the so-called Miles-Howard instability at the water interfaces [12]. As a consequence, most finite volume methods which are based on Riemann solvers would fail to resolve the system (10) for the multilayer shallow water equations over erodible beds. In the present study, we consider the Finite Volume Characteristics (FVC) method introduced in [6] and used in [2] for the numerical solution of multilayer shallow water flows over fixed beds. In this section, we briefly describe the FVC formulation for the system (10) and further details can be found in [2, 6]. Note that the FVC method does not require the calculation of the eigenvalues for the multilayer system (4). However, the selection of time steps is carried out using the eigenvalues associated with the single-layer counterpart of the system (4) which are defined in [5] as λ1 = 0, λ2,k = u k , λ3,k = u k −



gh k , λ4,k = u k +



(15) gh k ,

k = 1, 2 . . . , M.

Note that the eigenvalues (15) are for the single-layer sediment transport system associated with (4) using the water heights h k and not the total height H . This results in a system of (3M + 1) equations for which each uncoupled layer, its associated four eigenvalues are given by (15). It should also be stressed that similar approach has been considered in [2] for multilayer shallow water flows over fixed beds for which eigenvalues of it single-layer counterpart have been used in the simulations. In the current study, the courant number is set to Cr = 0.85 in all the computations and the time stepsize t is adjusted at each step according to the CFL condition t = Cr

max

k=1,2,...,M

x       ,  |λ1 | , λ2,k  , λ3,k  , λ4,k 

where x is the spatial discretization step, λ1 , λ2,k , λ3,k and λ4,k are the approximated eigenvalues in (15).

4.1 Spatial Discretization The spatial domain is discretized into control volumes [xi−1/2 , xi+1/2 ] centered at xi with a step size x. For the space discretization of equations (12), the following notations are used x

Wi± 21 (t) = W(t, xi± 21 ),

1 Wi (t) = x

i+ 2

1

W(t, x)d x xi− 1 2

Effcient Experimental and Numerical Methods for Solving Vertical Distribution …

305

to denote the point-values and the approximate cell-average of the variable W at the gridpoint (t, xi± 21 ) and (t, xi ), respectively. Integrating Eq. (12) with respect to space over the control volume, the following semi-discrete equations are obtained Fi+1/2 − Fi−1/2 dWi + = Qi , dt x

(16)

where Fi±1/2 = F(Wi±1/2 ) are the numerical fluxes at the cell interfaces x = xi±1/2 . In Eq. (12), Qi is a consistent discretization of the source term Q in (12). In order n , the Method of Characteristics (MoC) is to reconstruct the numerical fluxes Fi∓1/2 applied to the advective version of the system (12). Without accounting for the source term R(W) we reformulate equations in (12) into the following advective form ∂H + ∂t

 M j=1

 lju j

 ∂u j ∂H =− , lj H ∂x ∂x j=1 M

∂ Qk ∂(H + B) (ρs − ρw ) ∂ Qk ∂u k ∂ck + uk = −Q k − gH − , glk H 2 ∂t ∂x ∂x ∂x 2ρk ∂x ∂ Pk ∂ Pk ∂u k + uk = −Pk , ∂t ∂x ∂x where the discharge Q k = H u k and the sediment remittance Pk = H ck . The above system can also be rearranged in a compact vector form as Dk U k = Sk (U) , Dt with

Dk Dt

k = 0, 1, 2, . . . , 2M,

(17)

is the total derivative defined by Dk ∂ ∂ = + Uk , Dt ∂t ∂x

k = 0, 1, 2, . . . , 2M,

(18)

where U = (U0 , U1 , . . . , U2M )T , S (U) = (S0 , S1 , . . . , S2M )T with ⎛

M  ∂u j − lj H ⎜ ∂x ⎜ j=1 ⎜ ⎞ ⎛ ⎜ H ∂ + B) (ρs − ρw ) ∂u ∂c1 (H 1 ⎜ −Q − gH − gl1 H 2 ⎜ ⎜ Q1 ⎟ 1 ∂ x ∂ x 2ρ ∂x ⎜ ⎟ ⎜ 1 ⎜ ⎜ P1 ⎟ ∂u 1 ⎜ ⎟ ⎜ −P1 U = ⎜ . ⎟ , S(U) = ⎜ ∂x ⎜ ⎜ .. ⎟ ⎜ ⎟ ⎜ .. ⎜ ⎝Q ⎠ . M ⎜ ⎜ PM ⎜ −H u M ∂u M − g H ∂ (H + B) − (ρs − ρw ) gl M H 2 ∂c M ⎜ ∂x ∂x 2ρ M ∂x ⎝ ∂u M −PM ∂x

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

306

T. Rowan and M. Seaid

Fig. 5 A schematic diagram showing a control volume and the main quantities used in the calculation of the departure points X i+1/2 (τ ) that will reach xi+1/2 at time tn+1 . The exact trajectory is represented by a solid line and the approximate trajectory with a dashed line

and the advection velocity Uk is defined as ⎧ M  ⎪ ⎪ ⎪ l j u j , if ⎪ ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ ⎨ Uk = u k + 1 , if ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ if u , ⎪ ⎪ ⎩ k 2

k = 0, k = 1, 3, 5, . . . ,

(19)

k = 2, 4, 6, . . . .

Note that we used k = 0 in the above equations to only formulate the compact advective form (17) for all the equations and it does not refer to any layer in the system. The principal idea of the FVC method is to use the method of characteristics to approximate the numerical fluxes in (16). Hence, the characteristic curves associated with the system (17) are solutions of the initial-value problems   d X k,i+1/2 (τ ) = Uk,i+1/2 τ, X k,i+1/2 (τ ) , τ ∈ [tn , tn+1 ] , dτ (20) X k,i+1/2 (tn ) = xi+1/2 ,

k = 0, 1, . . . , 2M,

where X k,i+1/2 (τ ) is the departure point defined at time τ of a particle that will reach xi+1/2 at time tn+1 . It should be noted that the FVC method does not follow the flow particles forward in time, as a Lagrangian method does, instead it traces backwards the position at time tn of particles that will reach the points of a fixed

Effcient Experimental and Numerical Methods for Solving Vertical Distribution …

307

mesh at time tn+1 , see Fig. 5 for an illustration. Therefore, the FVC method avoids the grid distortion difficulties that the conventional Lagrangian schemes have. Accurate approximation of the characteristic curves X k,i+1/2 (tn ) is crucial to the overall accuracy of the FVC method. Some authors approximate the solutions of (20) using a second-order explicit Runge–Kutta scheme, which is not accurate enough to maintain a particle on its curved trajectory, see for instance [17]. In [2, 6], a second-order extrapolation based on the mid-point rule is used to approximate the solution of (20), but this method involves an iterative procedure which may become computationally demanding. In the present study, we consider the third-order explicit Runge–Kutta method (14). Thus, the procedure to approximate the solution of the differential equations (20) can be achieved by (1) n = xk,i+1/2 − tUk,i+1/2 , Xk,i+1/2 3 1 (1) (2) = xk,i+1/2 + Xk,i+1/2 − Xk,i+1/2 4 4 1 2 (2) X k,i+1/2 (tn ) = xk,i+1/2 + Xk,i+1/2 − 3 3

1 n tUk,i+1/2 , 4 2 n tUk,i+1/2 . 3

(21)

(1) It should be stressed that since the departure points X k,i+1/2 (tn ) and the stages Xk,i+1/2 (2) and Xk,i+1/2 would not necessarily lie on a mesh point in the computational domain, the solution at the departure points must be obtained by interpolation from known values at the gridpoints of the element where the points X k,i+1/2 (tn ) and the stages (1) (2) Xk,i+1/2 and Xk,i+1/2 are localized. In the current work, we use the cubic Spline interpolation to approximate the solutions at the characteristics points. Other highorder interpolation procedures can also be applied. Once the characteristics curves X k,i+1/2 (tn ) in (20) are calculated, a solution at the cell interface xi+1/2 is approximated as

    n+1 k tn , X k,i+1/2 (tn ) , := Uk tn+1 , xi+1/2 = U Uk,i+1/2

(22)

  k tn , X k,i+1/2 (tn ) is the solution at the departure point X k,i+1/2 (tn ) approxwhere U imated by the cubic interpolation using the gridpoints of the control volume where it belongs, i.e.,      k tn , X k,i+1/2 (tn ) = P Uk tn , X k,i+1/2 (tn ) , U

(23)

where P is the cubic Spline interpolating operator. Notice that authors in [2, 6] used the Lagrange interpolation polynomials in (23). In what follows we use the first-order Euler scheme to illustrate the formulation of the FVC method but in all our simulations the third-order Runge–Kutta scheme (14) is used. Thus, applied to Eq. (17), the characteristic solutions are computed in the predictor stage of the FVC method as

308

T. Rowan and M. Seaid

   t n n+1 n i+1/2 Hi+1/2 =H − lk u nk,i+1 − u nk,i , Hi+1/2 x k=1    t n n Q n+1 Q k,i+1/2 u nk,i+1 − u nk,i + k,i+1/2 = Q k,i+1/2 − x   n n n i+1/2 Hi+1 + Bi+1 ) − (Hin + Bin + gH (24)    n 2  n+1/2 (ρs − ρw ) n+1/2 i+1/2 ci+1 − ci , glk H n 2 ρk,i+1/2  n  t n n+1 n k,i+1/2 Pk,i+1/2 u =P − − u nk,i , P x k,i+1/2 k,i+1    n  n n i+1/2 k,i+1/2 = Q k tn , X k,i+1/2 (tn ) and P k,i+1/2 where H = H tn , X 0,i+1/2 (tn ) , Q =   Pk tn , X k,i+1/2 (tn ) are the solutions at the departure points X k,i+1/2 (tn ) computed using  the cubic Spline interpolation. To calculate the numerical fluxes Fi±1/2 = F Wi±1/2 , the intermediate states Wi±1/2 are updated using the characteristic solutions Ui±1/2 in the predictor stage (24). Thus, using the first-order Euler scheme for illustration only, the solution in the FVC method (16) is obtained using the following corrector stage M

M  t   n n − (lk H u k )i−1/2 (lk H u k )i+1/2 , x k=1   n n  t 1 1 2 2 2 2 H uk + g H − − H uk + g H − x 2 2 i+1/2 i−1/2

H n+1 = H n − n Q n+1 k,i = Q k,i



n+1 Pk,i

 n  (ρs − ρw )  n 2  n t n n in Bi+1 i g H − Bi−1 lk H − ck,i+1 − ck,i−1 n x 2 ρk,i  t  n n n . = Pk,i − − (H u k ck )i−1/2 (H u k ck )i+1/2 x

 

(25) ,

n in and ρ For the reconstruction of the terms H k,i , we use the same concept as in [2, 6] to guarantee that the discretization of the flux gradients and source terms in (16) are well balanced. Hence,

 n  n in = 1 Hi+1 , + 2Hin + Hi−1 H 4

n ρ k,i =

 1 n n n ρk,i+1 + 2ρk,i . + ρk,i−1 4

(26)

Note that the discretization of equations (11) and (13) is straightforward and it is omitted here. It should also be mentioned that the considered FVC method is fully conservative by construction and the non-conservative system (17) is used only to compute the intermediate states for the numerical fluxes in (16).

Effcient Experimental and Numerical Methods for Solving Vertical Distribution …

309

5 Experimental and Numerical Results The aim of this section is to present both the collected data from the experimental study and compare them to the numerical results obtained using the multilayer shallow water model. This should demonstrate that the measurements are reasonable and validate the computed results. To this end the numerical model is first compared with a three-dimensional Navier–Stokes solver using icoFOAM algorithm on the OpenFOAM software for a dam-break problem over a fixed bed. Then, the experimental data is presented and compared to those results obtained using the FVC method solving the multilayer shallow water equations. We present computational results using the sediment characteristics listed in Table 1. These sediment parameters have been recommended in many experimental studies on sediment transport applications, see for instance [30, 33, 42]. The configuration of the domain along with the experimental setup are illustrated in Fig. 6. The experimental work was carried out at the hydraulics laboratory at University of Durham. In our simulations, the water density ρw = 1000 kg/m3 , the gravity acceleration g = 9.81 m/s2 and the number of layers in the multilayer shallow water model is fixed to 10 for all examples in this section. First we examine the mesh convergence in the proposed FVC method for solving dam-break problems. To this end, we consider the case of dam-break problem over a fixed bed. Hence, B = 0 and initially, H (0, x, y) =

0.10 m, if x ≤ 0.33 m, 0.016 m, if x > 0.33 m,

u(0, x, y) = 0 m/s.

We consider four meshes with x = 0.036, 0.018, 0.009 and 0.0045 m using 10 layers in the multilayer model. The obtained results for the water depth at time t = 0.42 s are presented in Fig. 7. As can be seen for the last two meshes with x = 0.009 m and x = 0.0045 m, the differences in the results obtained for the water depth in Fig. 7 are very small. It is easy to see that solutions obtained using x = 0.036 m are far from those obtained by the other meshes. Increasing the density of control volumes, the results for the x = 0.009 m and x = 0.0045 m are roughly similar. Results obtained for the water velocity and not reported here for brevity show the same trends. This ensures grid convergence of the numerical results. Hence, the mesh with x = 0.009 m is used in all our next computations. The reasons for choosing this mesh structure lie essentially on the computational cost required for each mesh configuration and also on the numerical resolution obtained.

Table 1 Sediment parameters for the bed type used in our simulations for the erosion and deposition formulae d p ϕ θc ws nb ε ρs 0.0625 mm

0.5

0.0004

0.0145

0.0002

0.011

0.005

1420 kg/m3

310

T. Rowan and M. Seaid

Fig. 6 Configuration of the scour after an apron domain (top) and a photo of the experimental setup (bottom) used in the current study

Fig. 7 Numerical results obtained for the water depth using the FVC method on four different meshes for the dam-break problem over a fixed bed

Effcient Experimental and Numerical Methods for Solving Vertical Distribution …

311

Fig. 8 Three-dimensional results obtained using OpenFOAM simulation for the dam-break problem over a fixed bed at time t = 0.42 s

Our next concern is to validate our results to those obtained using the threedimensional Navier–Stokes equations. The objective is to discern the ability of the presented model to capture sediment transport in both the horizontal and vertical dimensions. Unfortunately, for this field of research the sedFOAM program, one of the most highly advanced sediment tools in OpenFOAM, it is not able to simulate the dam-break problem presented in this study. Thus, prior to the addition of sediment to the domain, the dam-break over a fixed bed is considered and the IcoFOAM solver is implemented. In Fig. 8 we present the water free-surface obtained at time t = 0.42 s for the three-dimensional simulations. Here, a mesh with 7569 elements and 4543 nodes is used in the three-dimensional computations along with a fixed time step t = 0.01. A comparison between the cross section of the IcoFOAM results at y = 0.05 and those obtained using the FVC method for 10-layer model is depicted in Fig. 9. The experimental results obtained for this case are also included in this figure. It is clear from these results that the location and the speed of the moving water front obtained using the multilayer model agree well to those computed using the three-dimensional IcoFOAM model. Note that fluctuations with different amplitudes appear in the three-dimensional results which are completely absent in the results obtained using the multilayer model. These fluctuations are mainly caused by the two-phase flow equations and turbulent effects accounted for in the IcoFOAM model and are also in good agreement with the experimental for this class of dam-break flows. We next consider sediment transport in our experimental setup and compare the measured results to those obtained using the FVC method solving the multilayer shallow water equations. At time t = 0,

312

T. Rowan and M. Seaid

Fig. 9 Comparison between three-dimensional results obtained using OpenFOAM simulation, the proposed multilayer model and the experimental results for the dam-break problem over a fixed bed at time t = 0.42 s

B(0, x, y) =

and H (0, x, y) =

0 m, if x ≤ 0.33 m, 0.00475 m, if x > 0.33 m

0.12 m, if x ≤ 0.33 m, 0.016 m, if x > 0.33 m,

c(0, x, y) = 0,

u(0, x, y) = 0 m/s.

In Fig. 10 we present the experimental and numerical results obtained at two different instants t = 0.23 s and t = 0.46 s. Here, we display the bed profile, the water depth and the distribution of the sediments using the concentrations ck , k = 1, 2, . . . 10. As the sediment is well distributed and the erosion rates are comparable, we do not observe ripple formation or any other effect of armoring, as the experimental results demonstrate in these results. The results obtained for this dam-break problem show that using a detailed description of sediments in the multilayer shallow water model, it is possible to accurately represent the vertical distribution of sediments constituting the bed. The proposed model performs very well for this case and it captures the correct flow and sediments structures without requiring complicated techniques or three-dimensional representations for the free-surface flows over erodible beds. For a better insight, we depict in Fig. 11 the bed profile B and the averaged concentration c for the experimental and numerical simulations at time t = 0.46 s. There is a good agreement between the results obtained using the experimental setup and the multilayer shallow water model for both the bed and sediment concentration distributions. The FVC method performs well for this test example and produces highly accurate and stable numerical results using reasonably coarse meshes. In general the sediment and flow features for this dam-break problem are computed with no numerical artifacts or spurious oscillations. According to these results the stability and shock capturing abilities of the FVC method are validated.

Effcient Experimental and Numerical Methods for Solving Vertical Distribution …

313

Fig. 10 Experimental results (first column) and numerical results (second column) obtained for the dam-break problem over an erodible bed at time t = 0.23 s (first row) and t = 0.46 s (second row)

Fig. 11 Comparison between experimental and numerical results for the bed B (left) and the concentration c (right) obtained for the dam-break problem over an erodible bed at time t = 0.46 s

314

T. Rowan and M. Seaid

Fig. 12 Comparison between experimental and numerical results obtained for the dam-break problem over an erodible bed at time t = 0.23 s (first row) and t = 0.46 s (second row) Table 2 Errors between the experimental and numerical results obtained for the bed, water height and averaged concentration at four different times t = 0.12, 0.24, 0.35 and 0.46 s t [s] B H c L 2 -error L ∞ -error L 2 -error L ∞ -error L 2 -error L ∞ -error 0.12

1.34E–06

1.69E–04

2.14E–03

0.24 0.35 0.46

4.22E–06 5.76E–06 8.74E–06

7.73E–04 9.52E–04 1.58E–03

1.33E–03 7.23E–04 6.95E–04

3.592 = E–02 2.07E–02 1.36E–02 1.13E–02

4.25E–02

1.82E–01

1.09E–01 1.85E–01 4.72E–01

1.72E–01 2.12E–01 2.23E–01

In Fig. 12, we further compare the experimental and numerical results for this dambreak problem over an erodible bed. Here, we plot the profiles of the bed and water depth at two different times t = 0.23 s and t = 0.46 s. The dam collapses at t = 0 and the flow system develops a shock wave heading downstream and a rarefaction wave traveling upstream. This is shown, in line with other experiments of this type, in Fig. 12. Furthermore, no localized undershoots or overshoots were detected in either the flow velocity or the sediment concentration, even in the presence of steep gradients detected during the simulation. The results obtained for the sediment concentrations shown in Fig. 10 illustrate the sedimentary diffusion and profiles of the sediment concentration can clearly be seen diffusing up through the layers. The proposed FVC method accurately approximates the solution to this dam-break problem over the erodible bed. Our final concern with this dam-break problem is to quantify the errors on the obtained experimental and numerical results. To this end, we present in Table 2, values of the L 2 -error and L ∞ -error between the experimental and numerical results obtained for the bed B, water height H and averaged concentration c evaluated at four different times t = 0.12, 0.24, 0.35 and 0.46 s. Under the considered sediment conditions and for all times, the errors in the sediment concentration are larger than those obtained for the bed and the water height. There is relatively low level of errors in our numerical simulations which is deeply encouraging for the multilayer shallow

Effcient Experimental and Numerical Methods for Solving Vertical Distribution …

315

water model proposed in this study. To the knowledge of the authors, this is the first time that sediment concentration has been computed and compared to a simulation in both the vertical and horizontal dimensions for a dam-break problem.

6 Conclusions A new class of experimental and numerical methods is presented in this study for the vertical distribution of sediments in dam-break flows. The experimental tools employ a high-speed camera and image processing toolbox to extract the concentration data for sediment transport. This new approach allows for the vertical distribution of sediments in a flow to be accurately and efficiently assessed. These techniques have been combined with a novel experimental setup that enables a small-scale dam-break to be investigated. Fine sand has been used in the experimental study and measurements have been collected for its vertical distribution in the dam-break problem. Through the development of this model, multilayer shallow water equations and sediment transport equations (including erosion and deposition terms) have been coupled and considered. Mass exchange terms are accounted for in the inter-layered coupling for both water flow and sediment transport as well as including the vertical sediment diffusion. In order to solve the coupled system, the finite volume characteristics method was implemented along with a second-order splitting procedure to account for the source terms. The finite volume characteristics method is second-order accurate and encompasses a predictor–corrector type procedure in two stages. Firstly, the method reconstructs the numerical fluxes using the method of characteristics. This results in an upwind discretization of the characteristic variables and avoids the computation of the Riemann problem. Secondly, the solution is updated using the finite volume discretization of the conservation system. The method combines the desirable qualities of the finite volume discretization and the method of characteristics to create a simple solver for multilayer shallow water flows over erodible beds. In future work it is expected to extend this method from two-dimensional models in order to model a wider range of problems. It is also expected to include the effects of turbulence in both flow fields and sediment transport. The collected experimental data was used for comparisons to the numerical results obtained using the proposed multilayer shallow water model. The results have also used for validations with numerical results obtained using the three-dimensional Navier–Stokes solver on the OpenFOAM software for a dam-break problem over a flat bed. These comparisons revealed very encouraging results for both the experimental method and the presented multilayer shallow water model. In additions, the proposed finite volume characteristics method exhibited good shape, high accuracy and stability behavior for all sediment transport regimes considered. The experimental method is low cost and easily replicable with great promise in being adaptable to efficiently assess more complex problems like composite beds. The presented results demonstrate the capability of the multilayer models that can provide insight to complex suspended sediments and bed-load transport in dam-break flows. Further work

316

T. Rowan and M. Seaid

in this research field should include composite beds in multi-phase flows as well as particle tracking for the vertical distribution of sediments in dam-break flows. It is anticipated to use multiple cameras and composite images in the future to provide three-dimensional data and sediment capture.

References 1. Abderrezzak, K., Moran, A., Tassi, P., Ata, R., Hervouet, J.: Modelling river bank erosion using a 2D depth-averaged numerical model of flow and non-cohesive, non-uniform sediment transport. Adv. Water Resour. 93, 75–88 (2015) 2. Audusse, E., Benkhaldoun, F., Saida, S., Seaid, M., Tassi, P.: A fast finite volume solver for multi-layered shallow water flows with mass exchange. J. Comput. Phys. 272, 23–45 (2014) 3. Audusse, E., Bristeau, M., Pelanti, M., Sainte-Marie, J.: Approximation of the hydrostatic Navier-Stokes system for density stratified flows by a multilayer model: Kinetic interpretation and numerical solution. J. Comput. Phys. 230, 3453–3478 (2011) 4. Audusse, E., Bristeau, M., Perthame, B., Sainte-Marie, J.: A multilayer Saint-Venant system with mass exchanges for Shallow Water flows. Derivation and Numerical Validation. M2AN Math. Model. Numer. Anal. 45, 169–200 (2009) 5. Benkhaldoun, F., Sari, S., Seaid, M.: A flux-limiter method for dam-break flows over erodible sediment beds. Appl. Math. Model. 36, 4847–4861 (2012) 6. Benkhaldoun, F., Seaid, M.: A simple finite volume method for the Shallow Water equations. J. Comput. Applied Math. 234, 58–72 (2010) 7. Bijker, E.: Some considerations about scales for coastal models with movable bed. Delft Hydraulics Lab. 50, 1–128 (1967) 8. Brownlie, W.: Prediction of flow depth and sediment discharge in open channels. Report Number KH-R-43A. Keck Laboratory of Hydraulics and Water Resources, California Institute of Technology (1981) 9. Cao, Z., Carling, P.: Mathematical modelling of alluvial rivers: reality and myth. Part 1: General overview. Water Maritime Eng. 154, 207–220 (2002) 10. Cao, Z., Pender, G., Wallis, S., Carling, P.: Computational Dam-Break Hydraulics over Erodible Sediment Bed. J. Hydraulic Eng. 67, 149–152 (2004) 11. Capart, H., Young, D.: Formation of jump by the dam-break wave over a granular bed. J. Fluid Mech. 372, 165–187 (1998) 12. Chumakova, L., Menzaque, F., Milewski, P., Rosales, R., Tabak, E., Turner, C.: Shear instability for stratified hydrostatic flows. Commun. Pure Appl. Math. 62, 183–197 (2009) 13. Cunge, J., Holly, F., Verwey, A.: Practical Aspects of Computation River Hydraulics. Pitman Publishing Inc., Boston (2002) 14. Dawei, S., Weibing, F.: A two-dimensional numerical model of suspended sediment transport using in the radial sand ridge field. In: International Conference on Mechatronic Science, Electric Engineering and Computer, pp. 826–830 (2011) 15. Dyke, P.: Modeling Coastal and Offshore Processes (2007) 16. Einstein, H.: Formulas for the Transportation of Bed Load. Trans. Am. Soc. Civil Eng. 107, 561–573 (1949) 17. El-Amrani, M., Seaid, M.: Numerical simulation of natural and mixed convection flows by Galerkin-characteristic method. Int. J. Numer. Meth. Fluids. 53(12), 1819–1845 (2007) 18. Fernández-Nieto, E., Koné, E., Chacón, T.: A multilayer method for the hydrostatic Navier– Stokes equations: a particular weak solution. J. Sci. Comput. 57, 1–30 (2013) 19. Grass, A.: Sediment Transport by Waves and Currents. SERC Lond. Cent. Mar. Technol. (FL29) (1981)

Effcient Experimental and Numerical Methods for Solving Vertical Distribution …

317

20. Gula, J., Zeitlin, V., Bouchut, F.: Instabilities of buoyancy-driven coastal currents and their nonlinear evolution in the two-layer rotating shallow water model. Part 2. Active lower layer. J. Fluid Mech. 665, 209–237 (2010) 21. Huang, S., Sun, Z., Xu, D., Xia, S.: Vertical distribution of sediment concentration. J. Zhejiang Univ. Sci. 9, 1560– (2008) 22. Kubatko, E., Westerink, J.: Exact discontinuous solutions of Exner’s bed evolution model: Simple theory for sediment bores. J. Hydraulic Eng. 133 (2007) 23. Liu, X., Beljadid, A.: A coupled numerical model for water flow, sediment transport and bed erosion. Comput. Fluids 154, 273–284 (2017) 24. 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 (1948) 25. Paphitis, D.: Sediment movement under unidirectional flows: an assessment of empirical threshold curves. Coastal Eng. 43, 227–245 (2001) 26. Rahuel, J., Holly, F. et al.: Modeling of riverbed evolution for bedload sediment mixtures. J. Hydraulic Eng. 11, 1521–1542 (2016) 27. Rahuel, J., Holly, F., Holly Chollet, J., Yang, G.: Modeling of riverbed evolution for bedload sediment mixtures. J. Hydraulic Eng. 115 (1989) 28. Rehman, K., Cho, Y.: Bed evolution under rapidly varying flows by a new method for wave speed estimation. J. Comput. Phys. 43, 357–372 (2016) 29. van Rijn, L.: Sediment pick-up functions. J. Hydraulic Eng. 110, 1494–1502 (1984) 30. van Rijn, L.: Unified view of sediment transport by currents and waves. I : initiation of motion, bed roughness, and bed-load transport. J. Hydraulic Eng. 113, 649–667 (2007) 31. Rowan, T., Seaid, M.: Efficient computational models for shallow water flows over multilayer erodible beds. Eng. Comput. 37, 401–429 (2019) 32. Rowan, T., Seaid, M.: Two-dimensional numerical modelling of shallow water flows over multilayer movable beds. Appl. Math. Model. 88, 474–497 (2020) 33. Rubey, W.: Settling velocity of gravel, sand, and silt particles. Am. J. Sci. 148, 325–338 (1933) 34. Shields, A.: Anwendung der Ahnlichkeitsmechanik und der Turbulenzforschung auf die Geschiebebewegung. Mitteilung der preussischen Versuchsanstalt fur Wasserbau und Schiffbau 26 (1936) 35. Shu, C.: Total variation diminishing time discretizations. SIAM J. Sci. Stat. Comput. 9, 1073– 1084 (1988) 36. Soares-Frazao, S., et al.: Dam-break flows over mobile beds: experiments and benchmark tests for numerical models. J. Hydraulic Res. 50, 364–375 (2012) 37. Strang, G.: On the construction and the comparison of difference schemes. SIAM J. Numer. Anal. 5, 506–517 (1968) 38. Terzaghi, K., Peck, R., Mersi, G.: Soil Mechanics in Engineering Practice. Wiley, New York (1996) 39. Vercruysse, K., Grabowski, R., Rickson, R.: Suspended sediment transport dynamics in rivers: multi-scale drivers of temporal variation. Earth-Sci. Rev. 166, 38–52 (2017) 40. Wu, W.: Depth-averaged two-dimensional numerical modeling of unsteady flow and nonuniform sediment transport in open channels. J. Hydraulic Eng. - ACSE 130, 1013–1024 (2004) 41. Wu, W., Vieira, D.: One-dimensional channel network model CCHE1D 2.0. National Center for Computational Hydroscience and Engineering (2002) 42. Wu, W., Wang, S.: Formulas for sediment porosity and settling velocity. J. Hydraulic Eng. 132, 858–862 (2006) 43. Yang, Z., Zou, Z., Xue, W., Sun, D.: Experimental study of near-bed concentration and sediment vertical mixing parameter for vertical concentration distribution in the surf zone. Int. J. Sediment Res. 35, 27–41 (2020)