Acceleration of iterative algorithms for Euler equations of gasdynamics

Abstract

of the grid size). The solution is considered converged when the maximum absolute residual is one order of magnitude smaller than the truncation error. The convergence history for the stream-function vorticity formulation is similar. The residual of the linear Poisson equation for the stream function is always of machine accuracy and the residual of the vorticity transport equation reduces quadratically; convergence is obtained after six iterations. For the biharmonic equation, convergence takes six iterations and is also quadratic. The solutions in terms of the wall skin-friction distribution are presented in Fig. 2. The results for the three formulations are in excellent agreement and correlate well with the skin-friction distribution presented by Briley. 8 Next, the model problem of a separated flow in a symmetrical diffuser, introduced by Inoue, 9 is examined. The diffuser problem is solved using the stream-function vorticity and the biharmonic formulation. The inlet and outlet boundary of the diffuser are at x = - 1.0 and x = 3.0, respectively. The shape factor of the diffuser wall A = -0.089. The centerline is located at.y = 1.0 and the Reynolds number based on this reference length and the free-stream velocity is R = 6250. The inflow conditions provide the initial conditions for the entire flowfield. Convergence is quadratic and machine zero is reached in 6-7 iterations. The solutions in terms of the wall skin-friction distribution are shown in Fig. 3. The results for the two formulations are in excellent agreement and correlate well with Inoue's results, except for the outflow conditions. In Fig. 3 the solution for the higher Reynolds number, R - 12500, with a larger separation region is also presented. The biharmonic equation is the most efficient of the three formulations in terms of CPU time and storage requirements. The biharmonic program is more than two-times faster and requires more than a factor two less memory than the streamfunction vorticity program. The primitive variable method is the slowest among the three formulations and it puts severe demands on the computer storage requirements. In this case the bandwidth is 0(67V) and this coupled with the increase in the number of variables results in more than a four-fold increase in storage and CPU time as compared to the biharmonic program. Conclusions Three formulations of the two-dimensional Navier-Stokes equations are solved numerically using Newton's method and a direct solution routine for banded matrices. The fully implicit solution techniques use second-order central differencing for all the terms and are shown to be reliable and to provide quadratic convergence. The biharmonic formulation is most efficient in terms of CPU time and memory without loss of accuracy. Finally, while it is well known that iterative methods (line overrelaxation or ADI) for biharmonic equations have very slow rates of convergence, the present study, using direct solvers, indicates that the biharmonic formulation is the most recommended.