A unified treatment of steady-state shallow water and two-dimensional navier-stokes equations — finite element penalty function approach

Abstract

Abstract The objectives of this paper are threefold: 1. (a) To introduce new generalized formulation of shallow water equations which permits taking into account the effect of return current distribution in the depth direction. 2. (b) To introduce a penalty formulation in the steady state solution of the shallow water equations. 3. (c) To test the validity of various arbitrary approximations introduced in simplified shallow water equations by different investigators. In the development of the first objective we start, as is common practice, introducing the only approximation necessary in shallow water formulations, i.e. the neglect of vertical acceleration terms. The essential variables remaining are the water surface elevation and the two horizontal velocities. The problem thus is still three dimensional albeit with two velocity components to be represented by an expansion in the depth coordinate direction. The problem is reduced to a set of two equations in two space dimensions but with a finite set k of parameters defining the vertical distributions. This permits, when k ⩾ 2, to obtain return currents which have been included in general formulation given by other authors at the expense of omitting horizontal viscosity and convective terms. The second objective, i.e. that of developing a “penalty” form for the steady state solution of the resulting equations is accomplished in the manner previously used by authors for Navier-Stokes equations and results in the elimination of the surface elevation variable from the final discrete form by imposing the approximate simultaneous satisfaction of the continuity constraint with a penalty number. In the last section of the paper the third objective is addressed by the study of several examples in which the full solution as well as its various approximations are used. One of the examples concerns lake circulation driven by wind forces and another wave generated currents near a shore obstacle.