Upwinding finite-element method for heat transfer problems with large convection

In this paper, the concept of optimal weighting function in Shen (1978) is briefly discussed, and a specific form is constructed for the steady two-dimensional convective heat transfer equation, to be used with triangular elements and linear shape functions. It is then applied to a test problem corresponding to the fully developed channel flow pushing a slippery piston. The problem is related to an important practical one in the industrial injection-molding process. It also bears resemblance to the controversial thermal entry problem. Even with the 'soft' condition of the first derivative of T with respect to x nearly zero at one boundary, the upwinding is shown to be indispensable at large Peclet numbers.