Sample Results
This small sample of results demonstrate some of the
advantages moment closures have over traditional methods. The
extended solution vector leads to a very innate and natural
treatment for nonequilibrium effects. Also, the firstorder
balancelaw form of the resulting systems lead to a relative
insensitivity to grid irregularities that are commonly
encountered in realworld engineering scenarios. This is
especially true when adaptive mesh refinement is used or in
situations with moving boundaries.
NACA0012 MicroAirfoil


Figure 1 
Figure 2 


Figure 3 
Figure 4 


Figure 5 
Figure 6 
This set of figures shows both experimental and computational
results for airflow past a NACA0012 microairfoil; density
contours are pictured. The freestream values of the flow
Mach number, temperature, and density are 0.8, 257 K, and
1.161 × 10^{−4} Kg/m^{3},
respectively, and the chord length of the airfoil is
0.04 m. These conditions correspond to a Knudsen number
of 0.017 and a Reynolds number of 73. The relatively high
Knudsen number means this flow exists in the socalled
transition regime between continuum fluid dynamics and
freemolecular flow.
Experimental results measured by Allegre et al. [1] are shown
in Figure 1. Figure 2 shows a NavierStokes
solution computed by Suzuki for the same situation [2]. It
can be seen that the agreement is not good. In particular,
the density is vastly underpredicted along the trailing edge.
This is because the NavierStokes equations are a continuum
model and are not valid in this rarefied situation.
Figures 3 and 4 show DSMCbased solutions computed by Sun
and Boyd [3]. The first figure shows a standard DSMC solution
for this situation. The stochastic nature of this method is
immediately recognized by the noise in the solution.
Figure 4 shows the solution obtained using the
InformationPreserving scheme (a modified DSMC scheme for
these "lowspeed" flows). In this case, the solution is
smooth and seems quite good. However, this is still a
stochastic method and all of the benefits of a PDEbased
approach are lost. Computation time for these particlebased
solutions is also high.
Figure 5 shows a solution I computed using a 10moment
maximumentropy moment closure. This closure yields a natural
treatment for anisotropic temperatures and shear pressures,
however it has no treatment for heat transfer. It can be seen
that the solution is in good agreement with experiment and
DSMCbased results at the leading edge, however,similar to the
NavierStokes result, density is underpredicted along the
trailing edge. Figure 6 shows the solution to a novel
moment closure that I derived which provides a diffusion
correction to the base 10moment model. This correction leads
to a nonequilibrium treatment for thermal diffusion with an
anisotropic coefficient of heat transfer. It can be seen that
this result is in excellent agreement with the more expensive
DSMCbased solutions while remaining significantly cheaper to
compute.
References

Allegre, J., Raffin, M., and Lengrand, J. C., “Experimental
Flowfields Around NACA0012 Airfoils Located in Subsonic and
Supersonic Rarefied Air Streams,” Numerical Simulation of
Compressible NavierStokes Flows, edited by M. O. Bris
teau, R. Glowinski, J. Periaux, and H. Viviand, Vol. 18 of
Notes on Numerical Fluid Dynamics, Fried. Vieweg and
Sohn, Braunschweig, Germany, 1987, pp. 59–68.

Suzuki, Y., Discontinuous Galerkin Methods for Extended
Hydrodynamics, Ph.D. thesis, University of Michigan,
2008.

Sun, Q. and Boyd, I. D., “A Direct Simulation Method for
Subsonic, Mircoscale Gas Flow,” Journal of Computational
Physics, Vol. 179, 2002, pp. 400–425.

McDonald, J., and Groth, C.P.T. "Extended FluidDynamic Model
for MicronScale Flow Based on Gaussian Moment Closure".
46th AIAA Aerospace Sciences Meeting and Exhibit,
AIAA Paper 2008691, January 710, 2008, Reno, Nevada.
Vortex Shedding with AMR
This short movie shows how moment closures can be used to
compute traditional continuum flows just as easily as they
can nonequilibrium flows. The movie shows von
Kármán vortex shedding for a Reynolds number of 100.
The ability to compute solutions to such viscous situations
with a purely firstorder hyperbolic method means solutions
are less sensitive to grid quality and sharp changes in grid
resolutions do not have a large effect on numerical results.
Click the image to view the movie. It is intended to be looped.
Pitching Airfoil with Embedded Boundary
This video demonstrates the combination of a momentclosure
method with adaptive mesh refinement and an embedded moving
boundary treatment. This moving boundary treatment makes
local alterations to an initial mesh in order to track an
evolving boundary. The locality of the modification makes the
technique computationally efficient. Nevertheless, the grid
quality at the moving boundary can be significantly degraded.
Moment closures offer the ability to simulate viscous
heatconducting flows without the need to evaluate second
derivatives, and are thus much less sensitive to grid quality
issues.
The situation shown is one studied experimentally by
Landon [1]. A NACA0012 airfoil undergoes a prescribed
oscillation in a flow with a Mach number of 0.775. The
Reynolds number for this situation is
5.5 x 10^{6}. It can be seen that computed
results are in good agreement with experiment. Again, click
the image to see the movie.
References

McDonald, J., Sachdev, J.S., and Groth, C.P.T. "Use of the
Gaussian Moment Closure for the Modelling of Continuum and
MicronScale Flows with Moving Boundaries". 4th International
Conference on Computational Fluid Dynamics, ICCFD4, Ghent,
Belgium, July 1014, 2006. Edited by H. Deconinck, E. Dick,
SpringerVerlag, Heidelberg, pp. 783788, 2009.

Landon, R. H. "Compendium of unsteady aerodynamic
measurements". Advisory Report 702, NATO AGARD, August 1982.