An Adaptive Multilayer Model for Density-layered Shallow Water Flows Von der Fakult¨atf¨urMathematik, Informatik und Naturwissenschaften der RWTH Aachen University zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigte Dissertation vorgelegt von Diplom-Mathematiker J¨ornThies Frings aus Aachen Berichter: Professor Dr. Sebastian Noelle Professor Dr. Manuel J. Castro D´ıaz Tag der m¨undlichen Pr¨ufung:15. Februar 2012 Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verf¨ugbar. Bibliografische Information der Deutschen Nationalbibliothek Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://dnb.d-nb.de abrufbar. ISBN 978-3-8439-0497-1 D 82 (Diss. RWTH Aachen University, 2012) © Verlag Dr. Hut, München 2012 Sternstr. 18, 80538 München Tel.: 089/66060798 www.dr.hut-verlag.de Die Informationen in diesem Buch wurden mit großer Sorgfalt erarbeitet. Dennoch können Fehler nicht vollständig ausgeschlossen werden. Verlag, Autoren und ggf. Übersetzer übernehmen keine juristische Verantwortung oder irgendeine Haftung für eventuell verbliebene fehlerhafte Angaben und deren Folgen. Alle Rechte, auch die des auszugsweisen Nachdrucks, der Vervielfältigung und Verbreitung in besonderen Verfahren wie fotomechanischer Nachdruck, Fotokopie, Mikrokopie, elektronische Datenaufzeichnung einschließlich Speicherung und Übertragung auf weitere Datenträger sowie Übersetzung in andere Sprachen, behält sich der Autor vor. 1. Auflage 2012 Acknowledgment While working on this thesis I thankfully had a lot of people supporting me|with math- ematical expertise, by keeping up my moral and sometimes even by simply taking over everyday chores, for which I had "no mind" during the final couple of months. First, I have to thank my advisor Prof. Dr. Sebastian Noelle, who is a splendid example of someone who always tries to improve himself, sometimes in the most creative ways, while staying perfectly human all along. His awareness of the people around him, may it be colleagues, co-workers or employees, is remarkable and leads to a very productive working atmosphere. Then I wish to thank Prof. Manuel Castro, who reviewed this thesis, and Prof. Gabriella Puppo, who both helped a lot in the derivation of the main ideas by advise and discussion. Moreover, it was always nice getting together at conferences and discuss general topics. Also, the colleagues at the IGPM were always helpful in a lot of ways, e.g., by balancing the various teaching obligations among all of us or with answers to questions arising in my research. Some of them became good friends during the course of the last couple of years. Friendships, which I hope to maintain in the future. Of course I also have to thank my family for supporting me and letting me follow my interests from early childhood on|without neglecting to slightly nudge me whenever I became too comfortable. Finally, I have to thank my wonderful wife Sujeela, who supported me with all her strength. I am glad that I found such a characterful and reliable partner. Financial support from the Deutsche Forschungsgemeinschaft (German Research Foun- dation) through grant GSC 111 is gratefully acknowledged. Aachen, June 4th 2012 3 4 Contents 1. Introduction 9 I. Shallow Water with uniform density 13 2. Deriving Classic Shallow Water Models 15 2.1. The Navier-Stokes equations . 15 2.1.1. Equations with dimensions . 15 2.1.2. Obtaining dimensionless Navier-Stokes equations . 18 2.2. Incompressible hydrostatic equations . 24 2.2.1. Incompressible equations . 24 2.2.2. Classic hydrostatic model . 25 2.3. Depth averaged equations . 27 2.3.1. Viscous Shallow Water model . 27 2.3.2. Balance law and quasi-linear form . 29 3. Velocity Profiles and Viscous Terms 31 3.1. The model of Gerbeau and Perthame . 31 3.1.1. Preliminaries . 31 3.1.2. Approximations to the Navier-Stokes equations . 32 3.1.3. Viscous Shallow Water . 34 3.2. Alternative models . 36 3.2.1. Variant of Kloss . 36 3.2.2. Variant of Amat . 38 3.2.3. Boussinesq coefficients . 40 4. Multilayer Shallow Water Model for Uniform Density 41 4.1. Preliminaries . 41 4.2. Multilayer Shallow Water equations . 42 5 Contents 5. Numerical Methods for Viscous Shallow Water 49 5.1. Finite Volume schemes . 49 5.1.1. Discretization and conservation laws . 49 5.1.2. Source terms . 52 5.1.3. 2nd and higher order schemes . 53 5.2. Profile-based Models . 53 5.2.1. The scheme of Gerbeau and Perthame . 53 5.2.2. The scheme of Kloss . 54 5.2.3. The scheme of Amat . 55 5.3. Multilayer Model . 57 5.3.1. The discretized system . 57 5.3.2. The implicit treatment of the viscosity terms . 58 5.3.3. Numerical costs . 58 5.4. Discussion . 59 II. Stratified Shallow Water 61 6. Two- and Multilayer Shallow Water Model 63 6.1. Derivation of multi-layer Shallow Water equations for stratified flows . 64 6.1.1. Two-layer equations . 64 6.1.2. Multi-layer equations as a balance law and a quasi-linear system . 70 6.2. Non-conservative products . 71 7. Numerical Methods for Two-layer Shallow Water 75 7.1. Roe-Scheme of Castro/Pares . 75 7.1.1. Original Roe scheme of Castro/Mac´ıas/Par´es. 76 7.1.2. Enhancements and non-conservative products . 80 7.2. Splitting Scheme of Bouchut/Morales . 80 7.2.1. The scheme . 80 7.2.2. Properties of the numerical scheme . 81 7.3. Relaxation Scheme of Abgrall/Karni . 82 7.3.1. Characteristic polynomial and eigenvalue bounds . 82 7.3.2. Relaxation model and solver . 83 6 Contents III. Two/Three-layer Adaptation 85 8. Hyperbolicity of Two- and Three-layer systems 87 8.1. Two-layer Shallow Water equations . 87 8.1.1. Approximations in O( u1 u2 ).................... 89 j − j 8.1.2. Hyperbolicity and Kelvin-Helmholtz instabilities . 92 8.1.3. Hyperbolicity indicators . 94 8.2. Three-layer Shallow Water equations . 95 8.2.1. Instabilities . 96 9. Approximation of Three-layer eigenvalues 99 9.1. Analysis of the system of equations . 100 9.1.1. Auxiliary quantities . 100 9.1.2. External eigenvalues . 101 9.1.3. Internal eigenvalues . 103 9.1.4. Eigenvalue approximations . 111 9.2. Analysis of characteristic polynomial . 112 9.2.1. Monolayer eigenvalues . 113 9.2.2. Polynomials of lower degree . 117 9.2.3. Comparison to Section 9.1 . 120 10.Two/Three-layer adaptation 123 10.1. The three-layer adaptation . 123 10.1.1. Introducing the intermediate layer - transformation formulas . 125 10.1.2. Energy estimates . 129 10.1.3. Choice of intermediate layer thickness . 133 10.1.4. Heuristic intermediate layer height . 145 10.1.5. Example: Gain and loss of hyperbolicity . 150 10.2. Layer reduction . 152 10.2.1. Simple approach . 152 10.2.2. Energy optimization . 154 IV. Viscosity and Friction 157 11.Viscosity and friction 159 11.1. Inter-Layer Friction . 160 11.2. Sublayers . 163 11.2.1. Turbulence . 165 7 Contents 11.2.2. Reviewing depth-averaged equations . 166 11.2.3. Example: Stabilizing effect of viscosity and friction on hyperbolicity 168 V. Numerical Scheme 171 12.Numerical scheme 173 12.1. Overview of the scheme . 174 12.2. Finite Volume method . 175 12.2.1. Time discretization . 177 12.2.2. Spatial discretization . 178 12.3. Viscous treatment . 182 12.3.1. Discrete velocity profiles . 183 12.3.2. Review of advection treatment . 190 12.4. Full scheme without 2/3-adaptation . 192 12.5. Two-/three-layer adaptation . 194 12.5.1. Introduction of layers . 194 12.5.2. Removal of layers . 197 12.5.3. Adapting the velocity profiles . 198 12.6. Limiters . 198 13.Numerical Results 201 13.1. Test 1 . 201 13.1.1. Initial state and zero friction case . 201 13.1.2. Friction and Viscosity . 203 13.2. Test 2 . 205 13.3. Test 3 . 209 13.3.1. Original setup . 209 13.3.2. Variation of viscosity and friction . 214 13.4. Test 4 . 214 14.Conclusion 221 14.1. Outlook . 222 Bibliography 227 A. MAPLE-Listing 237 8 1. Introduction In the study of fluid dynamics, the Navier-Stokes equations in their various forms are the fundamental model equations describing the evolution of the flow. Their complexity however, also in the case of numerical solvers, raises the desire for simplifications, at least to account for special conditions found in the application under consideration. The Shallow Water assumption is a widely used simplification, which made the derivation of a comparably simple and closed set of equations in the form of a conservation law, cf. [dSV71]. These equations are well-studied on the analytical as well as the numerical side, and various efficient numerical schemes suitable for different special cases are readily available in literature, cf.textbooks like [Vre94, Tor97, LeV02, Bou04] and research papers as [PS01, BLMR03, PC04, ABB+04, NPPN06, NXS07, GPC07], in a wider sense [GL96, GLBN97, Gos01] and, of course, all the references in the cited publications. The question of a wider applicability of these equations or at least the usage of well- tested and efficient numerical schemes in the context of more complex states of shal- low flows has been risen in recent years, and a wide array of more complex models and solvers are based on the findings for the classic Shallow Water equations. Gerbeau and.