Using an 8-Million-Zone Grid

B. Kevin Edgar, S. Anderson, and Paul Woodward

University of
Minnesota &

Army High Performance Computing Research Center, Minneapolis, MN

Kurt Fickie

Army Research Laboratory, Aberdeen, MA

**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.