A Boltzmann-based finite volume algorithm for surface water flows on cells of arbitrary shapes

An explicit two–dimensional conservative finite volume model for shallow water equations is formulated and tested. The algorithm for the mass and momentum fluxes at the control surface of the finite volume is obtained from the solution of the Bhatnagar–Gross–Krook (BGK) Boltzmann equation. Unlike classical methods, BGK schemes do not require an ad–hoc splitting of advection and diffusion. The BGK scheme is second order in both time and space. The formulation of the BGK algorithm is performed for a cell of arbitrary irregular shape, but the test cases are conducted using a structured grid of quadrilateral cells. Two approximate Riemann solvers, the HLLC scheme and the two–stage Hancock–HLLC scheme, where HLL stands for Harten, Lax and van Leer and C stands for contact discontinuity, are also considered. The second–order accuracy of HLL and Hancock–HLLC schemes is obtained by MUSCL approach, where MUSCL is the acronym for Monotone Upstream–centered Schemes for Conservation Laws. The data reconstruction for all three schemes is carried out by theVan Leer limiter. The test cases involve strong shocks and expansion waves. The accuracy of the schemes are measured using an absolute error norm and a waviness error norm. The HLLC scheme is highly oscillatory for Courant number larger than 0.5, while the BGK and the Hancock–HLLC schemes are applicable for Courant numbers as high as 1.0. For a fixed value of the central processing unit (CPU) time, the absolute error of the Hancock–HLLC is slightly smaller than that of the BGK while the waviness error of the BGK is quite close to that of Hancock–HLLC. This is because (i) the Hancock–HLLC is a two–step method while the BGK is a single–step method (i.e., the Hancock–HLLC requires storage of intermediate variables, but the BGK does not), and (ii) the Hancock–HLLC schemes requires larger number of grid points than the BGK scheme for the same level of accuracy. For example, to achieve an absolute error of 0.01, the BGK requires about 600 grid points while the Hancock–HLLC requires about 800 grid points. Both the BGK and Hancock–HLLC schemes have similar convergence properties. Unlike exact or approximate Riemann solvers, BGK fluxes accounts for both waves and diffusion. The ability of the BGK scheme to model diffusion is illustrated using a viscous flow problem. Excellent agreement between the analytical and computed viscous flow solution is found. Although the BGK and Hancock–HLLC schemes perform similarly for hyperbolic problems, BGK schemes have the added advantage of being able to solve hyperbolic–parabolic problems without the need for an ad–hoc operator splitting. This is important given that the artificial splitting of advection and diffusion is known to cause artificial widening in shear layers and introduces artificial transient in regions with sharp gradients. Such problems arise when the splitting operation fails to faithfully represent the correct coupling between the physics of advection and the physics of waves.

[1]  W. Gray,et al.  A kinetic flux vector splitting scheme for shallow water flows , 2003 .

[2]  M. Hanif Chaudhry,et al.  Depth-averaged open-channel flow model , 1995 .

[3]  Mohamed Salah Ghidaoui,et al.  Low-Speed Flow Simulation by the Gas-Kinetic Scheme , 1999 .

[4]  Rolf D. Reitz,et al.  One-dimensional compressible gas dynamics calculations using the Boltzmann equation , 1981 .

[5]  Kun Xu,et al.  A multidimensional gas-kinetic BGK scheme for hypersonic viscous flow , 2005 .

[6]  J. Zhou.  Lattice Boltzmann Methods for Shallow Water Flows , 2003 .

[7]  Pilar García-Navarro,et al.  1‐D Open‐Channel Flow Simulation Using TVD‐McCormack Scheme , 1992 .

[8]  Shiyi Chen,et al.  LATTICE BOLTZMANN METHOD FOR FLUID FLOWS , 2001 .

[9]  Fotis Sotiropoulos,et al.  Numerical Simulation of Transcritical Flow in Open Channels , 1997 .

[10]  Kun Xu,et al.  A gas-kinetic BGK scheme for the Navier-Stokes equations and its connection with artificial dissipation and Godunov method , 2001 .

[11]  M. Hanif Chaudhry,et al.  Computation of flows in open-channel transitions , 1992 .

[12]  William G. Gray,et al.  A Boltzmann based model for open channel flows , 2001 .

[13]  M. Ghidaoui,et al.  Numerical modeling of island wakes in shallow waters , 2004 .

[14]  S. Orszag,et al.  Extended Boltzmann Kinetic Equation for Turbulent Flows , 2003, Science.

[15]  D. Zhao,et al.  Approximate Riemann solvers in FVM for 2D hydraulic shock wave modeling , 1996 .

[16]  D. Causon,et al.  The surface gradient method for the treatment of source terms in the shallow-water equations , 2001 .

[17]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[18]  K. Karlsen,et al.  Operator spltting methods for systems of convection-diffusion equations: Nonlinear error mechanisms and correction strategies , 2001 .

[19]  Kun Xu,et al.  Linear and nonlinear analysis of shallow wakes , 2006, Journal of Fluid Mechanics.

[20]  F. White.  Viscous Fluid Flow , 1974 .

[21]  B. P. Leonard,et al.  The ULTIMATE conservative difference scheme applied to unsteady one-dimensional advection , 1991 .

[22]  S. Orszag,et al.  Expanded analogy between Boltzmann kinetic theory of fluids and turbulence , 2004, Journal of Fluid Mechanics.

[23]  Manuel Torrilhon,et al.  Stability and consistency of kinetic upwinding for advection–diffusion equations , 2006 .

[24]  William G. Gray,et al.  A Boltzmann-based mesoscopic model for contaminant transport in flow systems , 2001 .

[25]  Stephen Roberts,et al.  Explicit schemes for dam-break simulations , 2003 .

[26]  T. Teichmann,et al.  Introduction to physical gas dynamics , 1965 .

[27]  Jiaquan Deng.  BGK Boltzmann models for free-surface water flows and contaminant transport , 2000 .

[28]  G. Mellor.  USERS GUIDE for A THREE-DIMENSIONAL, PRIMITIVE EQUATION, NUMERICAL OCEAN MODEL , 1998 .

[29]  H. Ted Davis,et al.  Application of the lattice‐Boltzmann method to study flow and dispersion in channels with and without expansion and contraction geometry , 1999 .