High-order accuracy and multigrid acceleration for two-dimensional flow computations

Computational Fluid Dynamics (CID) has come to the point of being a design tool. Hence, as a design tool, it must give physically sound and reasonably accurate results in a reasonable time frame.

It is well known that high accuracy can be achieved either by use of high order discretization or by use of finer mesh. The most commonly used convective term discretization technique, first-order upwind, creating artificial dissipation (viscosity) is also a well recognized reality. It is also clear that unlike ID and 2D cases, in 3D simulations grid independence notion does not really exist, especially for real world applications. Hence, CFD users are compelled to get predictions with ever increasing number of grid points (up to about 2 million control volumes, currently) to have better accuracy and representation of flow fields. Unfortunately, accuracy and Central Processing Unit (CPU) time are inversely related. It is also well known that the convergence rate of conventional solution techniques deteriorates very quickly with increasing mesh size. This is due to the fact that, the number of operations for a conventional Single Grid (SG) technique is proportional to the square of the number of control volumes. Hence, CPU time requirements grow non-linearly with increasing grid points for SG solvers.

In order to overcome the aforementioned problems, two new approaches, the highorder discretization of convective terms and multigrid, need to be optimized and implemented. The high-order discretization of convective terms seems to promise accurate and stable converged solutions without artificial smearing. Hence, more accurate and physically more sound predictions can be obtained. Whereas, the multigrid algorithms hold the key to order of magnitude CPU time reductions to converged solutions.

The quadratic upwind biased polynomial high order schemes, such as QUICK, mixed,UTOPIA, and multigrid algorithm FAS-FMG are tested and optimized via several benchmark cases. Results indicate that the promises of both high-order discretization and multigrid algorithm can be harvested for recirculating flow predictions.