Different finite element formulations for the NavierStokes equations
Abstract
Three different finite element schemes are presented for solving the two dimensional NavierStokes equations. The standard stream functionvorticity method is reviewed, including the use of solid boundaries and no slip conditions in order to obtain vorticity values on the boundaries. A further improvement for second order approximations is introduced by defining two conditions of the solid boundary followed by a Taylor expansion of the stream function around a point on the boundary. A mixed stream functionvorticity method is explored, and involves the application of the HellingerReissner principle to the steadystate NavierStokes equations. The NewtonRaphson technique is later employed to find a solution. The method is applied to the linear case to demonstrate its usefulness for low Re flows. Finally, a penalty function formulation which eliminates the pressure term is described. The use of a ninenodes quadrilateral, eighteen degree of freedom element is found to produce fast convergence in high Reynolds number flows.
 Publication:

Numerical Methods in Laminar and Turbulent Flow
 Pub Date:
 1981
 Bibcode:
 1981nmlt.proc..179D
 Keywords:

 Computational Fluid Dynamics;
 Finite Element Method;
 NavierStokes Equation;
 Steady Flow;
 Two Dimensional Flow;
 Viscous Flow;
 Boundary Value Problems;
 Convergence;
 Degrees Of Freedom;
 High Reynolds Number;
 Low Reynolds Number;
 NewtonRaphson Method;
 Stream Functions (Fluids);
 Taylor Series;
 Vorticity;
 Fluid Mechanics and Heat Transfer