1Temple University, Philadelphia, Pennsylvania, U. S. A.
2University of Maryland, Maryland, U. S. A.
We will present some new developments for a class of 4th-order Essentially Compact methods(EC4), originally developed by E and Liu, for solving unsteady viscous incompressible flows in the vorticity-stream function formulation. Application of EC4 to model problems with relatively complicated physics or domains, as well as some pratical aspects of the method, will be discussed.
A straightforward patch mesh refinement technique, which is easily incorporated into the original scheme, will be outlined. It is particularily important here that we did not have to sacrifice the use of standard fast Poisson solvers when solving the Poisson-like equations which arise in the scheme. We will also present results of the use of a very effective far-field boundary condition (Chang and Sa), for the stream function, which is derived from a high order expansion in terms of moments of the vorticity. These new developments maintain the simplicity, high accuracy, and robustness of the original scheme.
As a detailed illustration of an application of the above mentioned methodoligies, we will present computations of the flow past a cylinder for Reynolds numbers up to 100,000. Even at this considerably high Reynolds number, the flow is completely resolved. Results of the method applied to the Boussinesq equation, as well as a zero Mach combustion problem, will also be discussed.