A comparison of primitive variables and streamfunction–vorticity for fem depth‐averaged modelling of steady flow

Abstract

SUMMARYThe governing equations for depth-averaged turbulent flow are presented in both the primitive variable and streamfunction-vorticity forms. Finite element formulations are presented, with special emphasis on the handling of bottom stress terms and spatially varying eddy viscosity. The primitive variable formulation is found to be preferable because of its flexibility in handling spatial variation in viscosity, variability in water surface elevations, and inflow and outflow boundaries. The substantial reduction in computational effort afforded by the streamfunction-vorticity formulation is found not to be sufficient to recommend its use for general depth-averaged flows. For those flows in which the surface can be approximated as a fixed level surface, the streamfunction-vorticity form can produce results equivalent to the primitive variable form as long as turbulent viscosity can be estimated as a constant.

Progress in Surface Water Modeling

“…Despite the schemes relatively good performance in idealized linear flow computations, field applications which are not heavily damped [ Baptista et al , 1989] are still rather wiggly and indicate that other mechanisms such as geometric boundary irregularities, perturbations in elevation boundary conditions and small scale variations in bathymetry can still excite spurious modes in primitive based solutions. DeVantier [1989] presents and compares several FE solutions to the steady rigid lid two dimensional depth average flow equations with horizontal momentum transport modeled using an eddy viscosity closure model. He compares a primitive equation mixed interpolation formulation and stream function vorticity (Ψ ‐ ω) equation formulations in applications to a cavity driven flow problem as well as a flow through an asymmetric expansion.…”

A Compact Finite-Difference Scheme Based on the Projection Method for Natural-Convection Heat Transfer

“…The primitive variable method is preferable to the stream function-vorticity method because of its flexibility in handling complex boundary conditions [18]. Zhang et al [19] studied a compact finite-difference scheme based on the projection method for Navier-Stokes flows.…”

A compact finite differences exact projection method for the Navier–Stokes equations on a staggered grid with fourth-order spatial precision

“…Another advantage is that they do not suffer from pressure-decoupling problems. On the other hand, vorticity-stream function methods are often considered inefficient when dealing with 3D problems and when complex boundary conditions are present [9]. A popular alternative, based on primitive variables, is the use of projection methods [10,11].…”