On the finite element simulation of fluid motion with high Reynolds or Grashof number

Abstract

Numerical Simulation of free convection problems with usual Finite Element (FE) formulations run into stability problems for large Grashof numbers. This is true even for simple models. In regions where the stream velocity or the temperature field is constant along the stream lines, the stiffness matrix may become ill-conditioned for large stream velocities although its physical contribution vanishes. Besides a costly increase in the number of elements and corresponding decrease of the length unit, a change in the unit system can not improve the matrix condition. The ill-conditioning arises from the usual decomposition of the nonlinear convective terms into a velocity-dependent stiffness matrix which represents a differential operator and a solution vector on which the stiffness matrix operates. The problem should be solved by an adapted choice of the matrix representation of the nonlinear convective terms. Alternatively - and necessarily for Newton-Raphson iteration of the nonlinear equations - the ill-conditioning terms may be neglected within those elements, where their physical contribution is negligible.