On the Relevance of the Dam Break Problem in the Context of Nonlinear Shallow Water Equations

The classical dam break problem has become the de facto standard in validating the nonlinear shallow water equations solvers. Moreover, the Nonlinear Shallow Water Equations (NSWE) are widely used for flooding simulations. While applied mathematics community is essentially focused on developing new numerical schemes, we tried to examine the validity of the mathematical model under consideration. The main purpose of this study is to check the pertinence of the NSWE for flooding processes. From the mathematical point of view, the answer is not obvious since all derivation procedures assumes the total water depth positivity. We performed a comparison between the two-fluid Navier-Stokes simulations and the NSWE solved analytically and numerically. Several conclusions are drawn out and perspectives for future research are outlined.

[1]  Yulong Xing,et al.  High order finite difference WENO schemes with the exact conservation property for the shallow water equations , 2005 .

[2]  P. Heinrich,et al.  Analytical Solution for Testing Debris Avalanche Numerical Models , 2000 .

[3]  P. Fischer,et al.  Entrainment in bottom gravity currents over complex topography from three‐dimensional nonhydrostatic simulations , 2004 .

[4]  Hrvoje Jasak,et al.  Error analysis and estimation for the finite volume method with applications to fluid flows , 1996 .

[5]  P. Liu,et al.  Viscous effects on transient long-wave propagation , 2004, Journal of Fluid Mechanics.

[6]  C. W. Hirt,et al.  Volume of fluid (VOF) method for the dynamics of free boundaries , 1981 .

[7]  Jean-Michel Ghidaglia,et al.  A two-fluid model for violent aerated flows , 2008, 0806.0757.

[8]  Costas E Synolakis,et al.  Tsunami science before and beyond Boxing Day 2004 , 2006, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[9]  F. Serre,et al.  CONTRIBUTION À L'ÉTUDE DES ÉCOULEMENTS PERMANENTS ET VARIABLES DANS LES CANAUX , 1953 .

[10]  Blaise Nsom,et al.  Bed slope effect on the dam break problem , 2000 .

[11]  Stephen Roberts,et al.  Numerical solution of the two-dimensional unsteady dam break , 2000 .

[12]  Bruce Hunt,et al.  Asymptotic Solution for Dam-Break Problem , 1982 .

[13]  Denys Dutykh,et al.  Influence of sedimentary layering on tsunami generation , 2008, 0806.2929.

[14]  Ming-Hseng Tseng.  Explicit finite volume non-oscillatory schemes for 2D transient free-surface flows , 1999 .

[15]  F. Marche.  Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom friction and capillary effects , 2007 .

[16]  Onno Bokhove,et al.  Flooding and Drying in Discontinuous Galerkin Finite-Element Discretizations of Shallow-Water Equations. Part 1: One Dimension , 2005, J. Sci. Comput..

[17]  Derek M. Causon,et al.  Numerical solutions of the shallow water equations with discontinuous bed topography , 2002 .

[18]  John K. Hunter,et al.  HAMILTONIAN FLUID MECHANICS , 2010 .

[19]  P. Fischer,et al.  Impact of boundary conditions on entrainment and transport in gravity currents , 2006, physics/0605177.

[20]  J. Hervouet.  A high resolution 2-D dam-break model using parallelization , 2000 .

[21]  Harry Yeh,et al.  Tsunami run-up and draw-down on a plane beach , 2003, Journal of Fluid Mechanics.

[22]  E. A. Kuznetsov,et al.  Hamiltonian formalism for nonlinear waves , 1997 .

[23]  I. Wood.  A lock exchange flow , 1970, Journal of Fluid Mechanics.

[24]  V. A. Dougalis,et al.  Numerical solution of KdV-KdV systems of Boussinesq equations: I. The numerical scheme and generalized solitary waves , 2007, Math. Comput. Simul..

[25]  C. K. Thornhill,et al.  Part IV. An experimental study of the collapse of liquid columns on a rigid horizontal plane , 1952, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[26]  Costas E. Synolakis,et al.  The runup of solitary waves , 1987, Journal of Fluid Mechanics.

[27]  G. Gottardi,et al.  Central scheme for two-dimensional dam-break flow simulation , 2004 .

[28]  H. Schäffer,et al.  Boussinesq-type formulations for fully nonlinear and extremely dispersive water waves: derivation and analysis , 2003, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[29]  M. Shoucri,et al.  Numerical solution of the shallow water equations , 1986 .

[30]  S. Zaleski,et al.  DIRECT NUMERICAL SIMULATION OF FREE-SURFACE AND INTERFACIAL FLOW , 1999 .

[31]  E. Cachan,et al.  Visco-potential free-surface flows and long wave modelling , 2009 .

[32]  Denys Dutykh,et al.  Viscous potential free-surface flows in a fluid layer of finite depth , 2007, 0705.1281.

[33]  Vasily Titov,et al.  Implementation and testing of the Method of Splitting Tsunami (MOST) model , 1997 .

[34]  Vasily Titov,et al.  Numerical Modeling of Tidal Wave Runup , 1998 .

[35]  P. Raviart,et al.  Numerical Approximation of Hyperbolic Systems of Conservation Laws , 1996, Applied Mathematical Sciences.

[36]  Fayssal Benkhaldoun,et al.  A Non Homogeneous Riemann Solver for shallow water and two phase flows , 2006 .

[37]  Min Chen,et al.  Boussinesq Equations and Other Systems for Small-Amplitude Long Waves in Nonlinear Dispersive Media. I: Derivation and Linear Theory , 2002, J. Nonlinear Sci..

[38]  Eckart Meiburg,et al.  Analysis and direct numerical simulation of the flow at a gravity-current head. Part 1. Flow topology and front speed for slip and no-slip boundaries , 2000, Journal of Fluid Mechanics.

[39]  A. Kumbaro,et al.  An Approximate Linearized Riemann Solver for a Two-Fluid Model , 1996 .

[40]  P. Liu,et al.  Experimental and numerical investigation of viscous effects on solitary wave propagation in a wave tank , 2006 .

[41]  Jean-Michel Ghidaglia,et al.  On the numerical solution to two fluid models via a cell centered finite volume method , 2001 .

[42]  J. J. Stoker.  Water Waves: The Mathematical Theory with Applications , 1957 .

[43]  D. Bresch,et al.  Existence of Global Weak Solutions for a 2D Viscous Shallow Water Equations and Convergence to the Quasi-Geostrophic Model , 2003 .

[44]  W.-Y. Chang,et al.  Simulations of Dam-Break Flows Using Free Surface Capturing Method , 2008 .

[45]  Dong Xu,et al.  Numerical Simulation of Two-Dimensional Dam-Break Flows in Curved Channels , 2007 .

[46]  Denys Dutykh,et al.  Water waves generated by a moving bottom , 2007 .

[47]  William C. Webster,et al.  A derivation of the Green-Naghdi equations for irrotational flows , 2001 .

[48]  Didier Bresch,et al.  On the construction of approximate solutions for the 2D viscous shallow water model and for compressible Navier-Stokes models , 2006 .

[49]  A. J. Hermans.  Theory of Water Waves , 2011 .

[50]  Derek M. Causon,et al.  Calculation of shallow water flows using a Cartesian cut cell approach , 2000 .

[51]  Peter Stansby,et al.  The initial stages of dam-break flow , 1998, Journal of Fluid Mechanics.

[52]  A. I. Dyachenko,et al.  Theory of weakly damped free-surface flows: A new formulation based on potential flow solutions , 2007, 0704.3352.

[53]  Hervé Capart,et al.  Riemann wave description of erosional dam-break flows , 2002, Journal of Fluid Mechanics.

[54]  G. Chechkin,et al.  S. N. Kruzhkov's lectures on first-order quasilinear PDEs , 2009 .

[55]  Jean-Michel Ghidaglia,et al.  Simulation of Free Surface Compressible Flows Via a Two Fluid Model , 2008, ArXiv.

[56]  J. Bona,et al.  Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: II. The nonlinear theory , 2004 .

[57]  F. Harlow,et al.  Numerical Calculation of Time‐Dependent Viscous Incompressible Flow of Fluid with Free Surface , 1965 .

[58]  Denys Dutykh,et al.  Tsunami generation by dynamic displacement of sea bed due to dip-slip faulting , 2007, Math. Comput. Simul..

[59]  Ryan J. Lowe,et al.  The non-Boussinesq lock exchange problem , 2000 .

[60]  Jean-Michel Ghidaglia,et al.  ON BOUNDARY CONDITIONS FOR MULTIDIMENSIONAL HYPERBOLIC SYSTEMS OF CONSERVATION LAWS IN THE FINITE VOLUME FRAMEWORK , 2002 .

[61]  Stéphane Popinet,et al.  A front-tracking algorithm for accurate representation of surface tension , 1999 .

[62]  J. C. Luke.  A variational principle for a fluid with a free surface , 1967, Journal of Fluid Mechanics.

[63]  H. Rusche.  Computational fluid dynamics of dispersed two-phase flows at high phase fractions , 2003 .

[64]  Steven J. Ruuth,et al.  A New Class of Optimal High-Order Strong-Stability-Preserving Time Discretization Methods , 2002, SIAM J. Numer. Anal..

[65]  Oguz Yilmaz,et al.  The initial stage of dam-break flow , 2009 .

[66]  Stuart B. Dalziel,et al.  Mixing in lock-release gravity currents , 1996 .

[67]  J. Boussinesq,et al.  Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. , 1872 .

[68]  D. Dutykh.  Group and phase velocities in the free-surface visco-potential flow: New kind of boundary layer induced instability , 2008, 0810.4697.

[69]  N. J. Zabusky,et al.  Shallow-water waves, the Korteweg-deVries equation and solitons , 1971, Journal of Fluid Mechanics.

[70]  J. Ghidaglia,et al.  A compressible two-fluid model for the finite volume simulation of violent aerated flows. Analytical properties and numerical results , 2008, 0805.2457.

[71]  H. P. Greenspan,et al.  Water waves of finite amplitude on a sloping beach , 1958, Journal of Fluid Mechanics.

[72]  S. Ooi.  High resolution numerical simulations of lock-exchange gravity-driven flows , 2006 .

[73]  Jean-Frédéric Gerbeau,et al.  Derivation of viscous Saint-Venant system for laminar shallow water , 2001 .

[74]  Jean-Michel Ghidaglia,et al.  Une méthode volumes finis à flux caractéristiques pour la résolution numérique des systèmes hyperboliques de lois de conservation , 1996 .

[75]  T. Benjamin.  Gravity currents and related phenomena , 1968, Journal of Fluid Mechanics.

[76]  Patrick J. Lynett,et al.  Analytical solutions for forced long waves on a sloping beach , 2003, Journal of Fluid Mechanics.

[77]  C. Synolakis,et al.  The run-up of N-waves on sloping beaches , 1994, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[78]  K. Anastasiou,et al.  SOLUTION OF THE 2D SHALLOW WATER EQUATIONS USING THE FINITE VOLUME METHOD ON UNSTRUCTURED TRIANGULAR MESHES , 1997 .

[79]  R. Miura.  Conservation Laws for the Fully Nonlinear Long Wave Equations , 1974 .

[80]  E. Barthélemy,et al.  Nonlinear Shallow Water Theories for Coastal Waves , 2004 .

[81]  Traian Iliescu,et al.  Large eddy simulation of stratified mixing in two-dimensional dam-break problem in a rectangular enclosed domain , 2007 .

[82]  L. R. Scott,et al.  An evaluation of a model equation for water waves , 1981, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[83]  Eckart Meiburg,et al.  The non-Boussinesq lock-exchange problem. Part 2. High-resolution simulations , 2005, Journal of Fluid Mechanics.

[84]  T. Y. Wu,et al.  Long Waves in Ocean and Coastal Waters , 1981 .

[85]  J.-M. Ghidaglia.  FLUX SCHEMES FOR SOLVING NONLINEAR SYSTEMS OF CONSERVATION LAWS , 2001 .

[86]  Patrick Bar-Avi,et al.  Water Waves: The Mathematical Theory with Application , 1995 .

[87]  Tadepalli,et al.  Model for the Leading Waves of Tsunamis. , 1996, Physical review letters.

[88]  P. Lax.  Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves , 1987 .

[89]  M. Ishii.  Thermo-fluid dynamic theory of two-phase flow , 1975 .

[90]  C. Mei.  The applied dynamics of ocean surface waves , 1983 .

[91]  Nektarios Chrysoulakis,et al.  Mathematical Modelling of Tsunami Waves , 2010, ERCIM News.

[92]  S. Shao,et al.  INCOMPRESSIBLE SPH METHOD FOR SIMULATING NEWTONIAN AND NON-NEWTONIAN FLOWS WITH A FREE SURFACE , 2003 .

[93]  Denys Dutykh,et al.  Comparison between three-dimensional linear and nonlinear tsunami generation models , 2007 .

[94]  Jean-Michel Ghidaglia,et al.  The normal flux method at the boundary for multidimensional finite volume approximations in CFD , 2005 .

[95]  Costas E. Synolakis,et al.  NOAA Technical Memorandum OAR PMEL-135 STANDARDS, CRITERIA, AND PROCEDURES FOR NOAA EVALUATION OF TSUNAMI NUMERICAL MODELS , 2007 .

[96]  Thierry Colin,et al.  Long Wave Approximations for Water Waves , 2005 .

[97]  Stuart B. Dalziel,et al.  Gravity currents produced by lock exchange , 2004, Journal of Fluid Mechanics.

[98]  M M Hafez,et al.  Innovative Methods for Numerical Solution of Partial Differential Equations , 2001 .

[99]  A. Radder.  HAMILTONIAN DYNAMICS OF WATER WAVES , 1999 .

[100]  P. M. Naghdi,et al.  A derivation of equations for wave propagation in water of variable depth , 1976, Journal of Fluid Mechanics.