Research Objective: To determine, if possible, whether or not the PPM gas dynamics code converges to a meaningful limit of very high Reynolds number Navier-Stokes flow when it is used to solve the Euler equations for the time-dependent, supersonic flow of air (gamma = 1.4) about a cylinder with a rough surface.

Methodology: In this experimental computational approach, the surface of the cyclinder is represented on a uniform Cartesian grid as a series of zones, or cells, which are partially filled with impenetrable fluid. The problem is thus reduced to a two-fluid hydrodynamics problem, and in the process, the surface of the cylinder is made rough on a scale comparable to the grid spacing. A series of PPM Euler simulations were performed on progressively finer grids for the Mach 4 flow of air in 2-D about a cylinder. In each case, the flows were allowed to settle into a statistically steady condition, which occurred fairly rapidly. Then, using the finest grid, the Navier-Stokes dissipation terms were added in such measure that the boundary layer at the side of the cylinder was roughly doubled, in one case, and roughly quadrupled in another. These flows had Reynolds numbers of 130,000 and 65,000 respectively. The various results have been compared in detail, and further quantitative analysis of these results is under way.

Accomplishments: The plan of action described above was carried out using the CM-5 Connection Machine of the AHPCRC last spring. This machine gives 8 Gflops performance on these 2-D applications. The Navier-Stokes run with Reynolds number 130,000 looked very much like the PPM Euler run on the next to finest grid (effective Reynolds number 170,000). Both of these results looked, in turn, very similar to the PPM Euler run on the finest grid. Time averages of these simulated flows should increase the similarities between these results, because the fine details of individual eddies will be filtered out of the data. These results present powerful evidence for the conjecture that the PPM results converge, in a statistical sense, to a meaningful high-Reynolds-number limit of viscous flows, at least in this simple 2-D case.

Significance: The D.o.E. and the Army have a continuing need to simulate supersonic flow of air about stationary and moving obstacles of complex shape. Particularly in the highly transient flows of this type it is not clear that the detailed structure of viscous boundary layers is important to the flow dynamics (but it is clear that the presence of these boundary layers is important). In some cases where complex phenomena such as transition to turbulence in a boundary layer might play an important role, projectile designers purposely introduce grooves or other surface features in order to force this transition and hence to make the flow predictable. It is important to know when the details of boundary layers matter and when they do not, in order that the most computationally effcient simulation procedure may be employed in a given problem.

Future Plans: Quantitative comparisons of time-averaged data from these runs will be made. Experimentation in 3-D flows of this type is planned, with a very simplified adaptive mesh refinement scheme planned to bring the effective surface roughness into a meaningful regime, as in the 2-D runs discussed above.

Density distributions around circular cylinders in a Mach 4 air flow