Finite element stream function‐vorticity solutions of the incompressible Navier‐Stokes equations

Abstract

SUMMARYThe incompressible, two-dimensional Navier-Stokes equations are solved by the finite element method (FEM) using a novel stream function/vorticity formulation. The no-slip solid walls boundary condition is applied by taking advantage of the simple implementation of natural boundary conditions in the FEM, eliminating the need for an iterative evaluation of wall vorticity formulae. In addition, with the proper choice of elements, a stable scheme is constructed allowing convergence to be achieved for all Reynolds numbers, from creeping to inviscid flow, without the traditional need for upwinding and its associated false diffusion. Solutions are presented for a variety of geometries.

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

“…These results merely reflect the fact that mesh refinement in high-gradient areas will generally improve a solution. The results may also be interpreted to indicate that the primary error introduced in "--a modelling of variable viscosity flows is due to the assumption of constant v,, rather than the omission of the last term of equation (6). This should be true of nearly all flows, because Vv, will nearly always be significant in the same vector direction as the major components of the velocity shear tensor, with which it has a cross product in the omitted term.…”

Finite element solution of the streamfunction‐vorticity equations for incompressible two‐dimensional flows

“…In the above equations i +b is the streamfunction, x and y are Cartesian co-ordinates, = a+/ay (3) and = -a+/ax (4) are the velocity components in the x-and y-directions respectively,…”

“…For two-dimensional incompressible laminar flows the streamfunction and vorticity equations can be written in dimensionless form as4*6-9 a 2 + / a X 2 + a 2 + / a y 2 = (1) and respectively. In the above equations i +b is the streamfunction, x and y are Cartesian co-ordinates, = a+/ay (3) and = -a+/ax (4) are the velocity components in the x-and y-directions respectively, -o = ao/axau/ay (5) is the vorticity, t is the time, b is the body force and Re is the Reynolds number.…”

Stream Function-Vorticity Formulation And Heat Transport using FEM For Unstructured Meshes and Complex Domains

“…In computational fluid dynamics (CFD), solving the incompressible Navier-Stokes equation is of utmost interst in various applications. The stream function-vorticity (ψ, ω z ) formulation is a way to express the Navier-Stokes equation in terms of ψ and ω z instead of the primitives pressure and velocity (Peeters et al, 1987). The advantage of using the stream function-vorticity formulation is that the pressure does not appear in the equation and, thus, we do not need to solve the pressure-velocity problem, therefore the use of linear element is suitable.…”