A Modified Lattice Boltzmann Model For Shallow Water Equations

Shallow water flows,such as atmospheric flows,tidal flows,coastal flows,tsunamis,lake flows,rivers,channels and dam-break waves,is quite common in nature.These flows are usually described with the shallow water equations.So a numerical solution of the shallow water equations turns out to be a very successful tool in studying a wide range of flow problems occurring in ocean,environmental and hydraulic engineering.In this paper,a modified lattice Boltzmann(LB)model of shallow water equations is proposed to numerically solve shallow water flows,and through Chapman-Enskog analysis,the model can be accurately recovered to the two-dimensional shallow water equations.In the proposed model,the construction of the force term draws on the Guo-Zheng-Shi model,and via the addition of correction term in the force term,a tunable parameter is introduced into the fluid viscosity coefficient.The dimensionless relaxation time can be varied in a proper range through tuning the parameter,and thereby the stability of the new model can be improved.Since the calculation of the macroscopic velocity of the modified model contains external forces,so the traditional inflow and outflow boundary condition processing methods will no longer be applicable.Based on the principle of mass and momentum conservations,and the bounce-back rule for the non-equilibrium part of the distribution functions,we have been proposed an inflow and outflow boundary condition processing schemes suitable for the new model.Several shallow water flows are used to test the validity and better stability of the proposed model,including discharge-driven without external forces shallow water flow,and flow around a cylinder.The numerical results show that the proposed model can accurately simulate these two flows,and compared with the previous model,the new model can simulate the shallow water flow of larger discharge.Thus,the new model has better stability.And while parallel computing is implemented,only a small number of iteration steps are needed to achieve the same convergence criterion,which to some extent compensates for the problem of a slightly larger calculation of the new model.In summary,the new model provides a new class of options for the force term of the LB model of the shallow water equations.