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 non-equilibrium effects. Also, the first-order balance-law form of the resulting systems lead to a relative insensitivity to grid irregularities that are commonly encountered in real-world engineering scenarios. This is especially true when adaptive mesh refinement is used or in situations with moving boundaries.

NACA0012 Micro-Airfoil

naca experiment naca Navier-Stokes
Figure 1 Figure 2
naca DSMC naca IP
Figure 3 Figure 4
naca Gaussian naca Gaussian HT
Figure 5 Figure 6

This set of figures shows both experimental and computational results for airflow past a NACA0012 micro-airfoil; density contours are pictured. The free-stream values of the flow Mach number, temperature, and density are 0.8, 257 K, and 1.161 × 10−4 Kg/m3, 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 so-called transition regime between continuum fluid dynamics and free-molecular flow.

Experimental results measured by Allegre et al. [1] are shown in Figure 1. Figure 2 shows a Navier-Stokes 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 Navier-Stokes equations are a continuum model and are not valid in this rarefied situation. Figures 3 and 4 show DSMC-based 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 Information-Preserving scheme (a modified DSMC scheme for these "low-speed" 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 PDE-based approach are lost. Computation time for these particle-based solutions is also high.

Figure 5 shows a solution I computed using a 10-moment maximum-entropy 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 DSMC-based results at the leading edge, however,similar to the Navier-Stokes 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 10-moment model. This correction leads to a non-equilibrium 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 DSMC-based solutions while remaining significantly cheaper to compute.


  1. 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 Navier-Stokes 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.
  2. Suzuki, Y., Discontinuous Galerkin Methods for Extended Hydrodynamics, Ph.D. thesis, University of Michigan, 2008.
  3. 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.
  4. McDonald, J., and Groth, C.P.T. "Extended Fluid-Dynamic Model for Micron-Scale Flow Based on Gaussian Moment Closure". 46th AIAA Aerospace Sciences Meeting and Exhibit, AIAA Paper 2008-691, January 7-10, 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 non-equilibrium 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 first-order 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 moment-closure 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 heat-conducting 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 106. It can be seen that computed results are in good agreement with experiment. Again, click the image to see the movie.


  1. McDonald, J., Sachdev, J.S., and Groth, C.P.T. "Use of the Gaussian Moment Closure for the Modelling of Continuum and Micron-Scale Flows with Moving Boundaries". 4th International Conference on Computational Fluid Dynamics, ICCFD4, Ghent, Belgium, July 10-14, 2006. Edited by H. Deconinck, E. Dick, Springer-Verlag, Heidelberg, pp. 783-788, 2009.
  2. Landon, R. H. "Compendium of unsteady aerodynamic measurements". Advisory Report 702, NATO AGARD, August 1982.