Turbulent two-dimensional shallow water equations and their numerical solution

Abstract

Free surface flow of water over a shallow rough bed is characteristically turbulent due to disturbances generated by the bed resistance and diverse causes. The paper presents a derivation of the basic equations in two dimensions and their numerical solution by an extension of the method developed earlier for flow in one dimension. Starting from the three-dimensional Reynolds Averaged Navier–Stokes equations, the equations of continuity and horizontal momenta are depth averaged to derive three equations for the free surface elevation ζ and the horizontal, depth averaged velocity components (U,V). Certain closure assumptions are required for derivation of the equations. Principally, the viscous stresses are neglected, while the Reynolds stresses are assumed to depend on the vertical coordinate z only for the shearing flow over the x, y-plane representing the plane bed. Secondly, it is assumed that the instantaneous horizontal components of velocity (u,v) follow the 1/pth (p=7) power law of variation in the z-direction, in liu of the logarithmic law of the wall. For numerical solution of the three nonlinear equations of continuity and momenta, the equations are reformulated in terms of the primitive “discharge” components (Q,R) of the velocity (U,V), showing that Q and R can be functions of ζ alone. The transformed equation of continuity is treated by the Lax–Richtmyer method. The two momentum equations on the other hand, transform in to two coupled second degree equations in the derivatives of Q and R, which decouple in the important case of quasilinear straight crested waves on the water surface. The decoupled equations are numerically solved by the iterative modified Euler method, and illustrated by application to an initial elevation of a model for tsunami propagation. Both of the Lax–Richtmyer and the iterative modified Euler method are of second order. The numerical method is general and applicable in other cases such as gradual atmospheric flows over the globe.