A variety of gas flow problems are characterized by the presence of rarefied and continuum domains. In a rarefied domain, the mean free path of gas molecules is comparable to (or larger than) a characteristic scale of the system. The rarefied domains are best described by particle models such as Direct Simulation Monte Carlo (DSMC); or, they involve solution of the Boltzmann kinetic equation for the particle distribution function. The continuum flows are best described by Euler or Navier-Stokes equations in terms of average flow velocity, gas density, and temperature and are solved by computational fluid dynamics (CFD) codes. The development of hybrid solvers combining kinetic and continuum models has been an important area of research over the last decade. Potential applications of such solvers range from high-altitude flight to gas flow in microsystems.

Figure 1. UFS key components

The key parameter governing selection of the appropriate physical model is the Knudsen number (Kn), defined as the ratio of the local mean free path to the characteristic size of the system. For flights at high altitude, where the atmospheric density is relatively low, the Kn is essentially a representation of gas density, which is the predominant factor that mandates model selection. For gas flows in microsystems, the small dimension of the system has the most influence on the Kn and therefore dictates selection of a kinetic model.

Until recently, most attempts to create hybrid gas flow solvers have involved coupling DSMC codes with CFD codes.1 As part of a Small Business Innovation Research project, AFRL partnered with CFD Research Corporation and the Russian Academy of Sciences to develop a Unified Flow Solver (UFS) based on the direct numerical solution (DNS) of the Boltzmann transport equation combined with kinetic schemes for the CFD equations. Choosing a DNS rather than a DSMC-based approach enabled the developers to use similar numerical techniques for the rarefied and continuum domains and thus facilitate coupling of the rarefied and continuum solvers. The UFS can automatically identify kinetic and continuum domains using preestablished criteria, introduce and remove kinetic patches, and select suitable solvers to maximize the accuracy and efficiency of simulations.

Figure 1 shows the key components of the UFS. The Boltzmann kinetic solver implemented in the UFS utilizes the numerical algorithms and computational methods described by Aristov and Tcheremissine.2,3 The continuum solvers are based on kinetic schemes employing numerical algorithms similar to those of the Boltzmann solver.4 The remaining UFS components include criteria for domain decomposition into rarefied and continuum parts and coupling algorithms.

The collaborative team built the UFS on a tree-based Gerris Flow Solver framework with a dynamically adaptive grid and the support of complex solid boundaries.5 This architecture allows the shape of a boundary—a spacecraft, for example—to be specified using common geometry tools and inserted into a computational domain as a standard file. The solver automatically generates the computational grid, refining it near the boundary where necessary. In the process of simulation, the grid can automatically adapt (refine or coarsen) according to flow patterns. The steep parameter gradients of shock waves, for example, would dictate a refined grid structure. Depending on local continuum breakdown criteria, the UFS can automatically switch between the deterministic Boltzmann solver and the continuum solvers. The parallelized code can run on multiprocessor systems, solving complex three-dimensional geometries in single-component atomic gases.

Figure 2. UFS 2-D supersonic gas flow simulation around a cylinder, Mach=3, Kn=1.0, 0.5, and 0.1 (from top to bottom). Density distributions display on the left side; the computerized grid indicates the kinetic (red) and continuum (white) domains on the right.
Figure 3. UFS simulation of a nozzle with subsonic flow (Mach=0.2) at its entry. The image depicts the domain decomposition (upper) and Mach number distributions (lower) for Kn=0.01 (left) and 0.001 (right). The kinetic domain displays in red and the continuum domain, in white.

The UFS enables accurate and efficient simulation of gas flows for the entire Kn range. Figure 2 (see page 22) illustrates a two-dimensional (2-D) simulation of supersonic gas flow (at Mach=3) around a cylinder for three different Kn values. The left side of each image depicts the density profiles, and the right side shows the computational grid and separation of the kinetic (red) and continuum (white) domains. Figure 3 (see page 22) shows UFS simulation results for a nozzle with subsonic flow (Mach=0.2) at the nozzle entry for Kn values=0.01 and 0.001. The UFS employed both the Boltzmann and Euler solvers to perform these calculations. The grid adaptation is based on the parameter δ = log (ρ) + log (μ), where ρ is the gas density and μ is the gas velocity.

In conclusion, the UFS is capable of automatically switching between a deterministic Boltzmann solver and a continuum kinetic solver depending on local gradients of gas density, flow velocity, and temperature. Currently, scientists are developing extensions to the UFS code to treat multicomponent mixtures and molecular gases with internal degrees of freedom. With these extensions, the UFS will be valuable for such practical applications as the simulation of transatmospheric flights at hypersonic velocities and space exploration analysis. The UFS will also be extremely valuable for advanced material processing and semiconductor manufacturing, primarily for flows at low speed, where species transport is induced by temperature gradients in low-pressure plasma processing reactors.

The UFS team has developed a Web site containing additional technical information, including results from case studies and contact information. This content is accessible at http://info-ufs.wpafb.af.mil/UFS_index.html.

Dr. V. I. Kolobov (CFD Research Corporation) and Ms. Melissa Withrow (Azimuth Corporation), of the Air Force Research Laboratory's Air Vehicles Directorate, wrote this article. For more information, contact TECH CONNECT at (800) 203-6451 or place a request at http://www.afrl.af.mil/techconn_index.asp. Reference document VA-H-05-12.


  1. Aristov, V. V., et al. "Construction of a Unified Continuum/Kinetic Solver for Aerodynamic Problems." AIAA Journal of Spacecraft and Rockets, vol 42, no 4 (2005): 598.
  2. Aristov, V. V. "Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows." Dordrecht, Kluwer Academic Publishers, 2001.
  3. Tcheremissine, F. G. "Direct Numerical Solution of the Boltzmann Equation." RAREFIED GAS DYNAMICS: 24th International Symposium on Rarefied Gas Dynamics, AIP Conference Proceedings 762. Monopoli, Bari, Italy (Jul 04): 677.
  4. Xu, K. "A Gas-Kinetic BGK Scheme for the Navier-Stokes Equations and Its Connection With Artificial Dissipation and Godunov Method." Journal of Computational Physics, vol 171 (2001): 289- 335.
  5. Popinet, S. "Gerris: A Tree-Based Adaptive Solver for the Incompressible Euler Equations in Complex Geometries." Journal of Computational Physics, vol 190 (2003): 572.

Air Force Research Laboratory Technology Horizons Magazine

This article first appeared in the June, 2006 issue of Air Force Research Laboratory Technology Horizons Magazine.

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