On the computed pressures for Navier‐Stokes problems at increasing Reynolds numbers using the penalty finite element method

Abstract

DISCUSSION The reduced integration penalty finite element method has been frequently used to compute approximate velocity solutions to Navier- Stokes problems.’-4 The velocity approximation can be post-processed to obtain an approximate pressure solution. Numerical studies for Stokes have shown that for problems where the data (boundary conditions or the forcing function) are ‘rough’, these computed pressures may exhibit oscillations for certain elements and reduced integration schemes, whereas smoother pressures are obtained for other (‘stable’) elements. An example of such a problem is the familiar ‘driven cavity’ problem. For this problem, both in the case of Stokes flow and low Reynolds number Navier-Stokes flow, when the 4-node bilinear element with 1-point Gauss integration of the penalty term is used, the pressures computed from the velocity field are found to be oscillatory, whereas when the 9-node biquadratic element with 1point integration of the penalty term is used, smooth pressure profiles are obtained. Recently, we have been investigating the use of continuation techniques and iterative methods for computing approximate solutions at higher Reynolds numbers. On examining the computed pressures for the driven cavity we observed that as the Reynolds number increased, the local pressure oscillations for the bilinear element diminish and by Re = 2000 the pressure profiles at representative sections appear smooth. As an example, results are now given for the cavity problem and bilinear I-point element for calculations on a uniform mesh of size h = 1/32 and with penalty parameter E = The pressure profile at Re = 10 along the section y = 17/64 is given in Figure 1 (marked + ) and is seen to contain local pressure oscillations. The projected pressure obtained by averaging the four element pressures adjacent to a node is also given and is smooth (marked x ). Analogous results are also observed for Stokes flow (Re = 0). However, the computed pressure profile at J’ = 17/64 for Re = 2000 appears smooth (Figure 2).+ Similar behaviour is observed at other representative sections and is not shown here. Results with penalty parameter E = and E = Approximate solutions were also computed using the mixed method with Co biquadratic velocity and Co bilinear pressure on uniform meshes with h = 1/16 and h = 1/25. This element is known to be stable. The pressure solution is smooth as anticipated and qualitatively agrees with that obtained with the penalty method. However, quantitatively the pressure profiles agree were essentially the same.