A finite element formulation is presented for solution of problems of two-dimensional flow (vertical elevation). The equations describing the phenomena are assumed to be the two-dimensional version of the Navier Stokes equations coupled to the advection-diffusion equation for variations in density. In these nonlinear equations the conventional coefficient of viscosity is replaced by a turbulent exchange coefficient when flow is in the turbulent range. The Galerkin method of weighted residuals is described for the finite element method and the use of the Newton-Raphson iterative scheme is explained with respect to incorporation of nonlinear terms in the solution procedure. The result of this derivation is a set of nonlinear simultaneous equations, and implications and programming considerations are discussed. Examples of both constant density (homogeneous) flow and density stratified flow over a submerged weir are included. These examples are compared to experimental results and their accuracy is assessed.