Hypersonic Blunt Body Thermophysics Using a Unified Kinetic/Continuum Solver

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1 41st AIAA Thermophysics Conference June 2009, San Antonio, Texas AIAA Hypersonic Blunt Body Thermophysics Using a Unified Kinetic/Continuum Solver Andrew J. Lofthouse U.S. Air Force Institute of Technology, Wright-Patterson AFB, OH, The Unified Flow Solver (UFS) is used to simulate a Mach 10 flow of argon about a 2D cylinder at densities corresponding to Kn = 0.01 and Kn = UFS is a hybrid kinetic/continuum solver implementing automatic domain decomposition and mesh refinement. A BGK model is used for the kinetic solver, coupled with an Euler solver. Solution results are compared to similar results from DSMC. Good agreement is obtained for surface pressure and shear stress (with total drag predictions for UFS being within 2% of those for DSMC), surface heating is over-predicted as compared to the DSMC solutions (with UFS predicting peak heating about twice that predicted by DSMC). I. Introduction Hypersonic flight vehicles are a current topic of interest in both civilian and military research. NASA is currently designing a Crew Transport Vehicle and Crew Exploration Vehicle to replace the aging space shuttle; reentry vehicles are, by definition, hypersonic vehicles. Military requirements for reconnaissance and surveillance, as well as the mission of the United States Air Force to rapidly project power globally makes the design of hypersonic planes and missiles that can quickly traverse the globe very attractive. The design of hypersonic vehicles requires accurate prediction of the vehicle surface properties while in flight. These quantities are typically the heat flux, pressure and shear stress, from which the aerodynamic forces and moments can be calculated. These variables govern not only the aerodynamic performance of the vehicle, but also determine the selection and sizing of the thermal protection system (TPS), which protects the vehicle and occupants from the extreme temperatures encountered at hypersonic velocities. During it s trajectory through an atmosphere, a hypersonic vehicle will experience vastly different flow regimes due to the variation of atmospheric density with altitude. In addition, the high temperatures encountered due to the high velocities cause dissociation and ionization of the atmospheric gases. Reproduction of these varied flow conditions in ground-based laboratory facilities is both expensive and technically challenging. Hence, there is an extremely important role for computational models in the development of hypersonic vehicles. The Air Force currently desires a computational tool that can, with minimal user intervention, be used to predict flow features and surface properties of hypersonic vehicles along all points of its trajectory. The trajectory of a hypersonic vehicle typically includes the continuum, transition and rarefied regimes. The continuum regime is characterized by very low Knudsen numbers and relatively high densities, where the Knudsen number is defined as the ratio of the molecular mean free path to a characteristic length. Kn = λ L 1 ρl (1) The rarefied regime is characterized by large Knudsen numbers and low density. The transition regime is the intermediate regime between the continuum and rarefied regimes. Continuum regime gas flows can be accurately simulated using traditional Computational Fluid Dynamics (CFD) by solving either the Euler or the Navier-Stokes (NS) equations. The NS equations can be derived from kinetic theory based on the assumption of a small perturbation from an equilibrium velocity distribution function Assistant Professor, Department of Aeronautics and Astronautics, 2950 Hobson Way; Andrew.Lofthouse@afit.edu. Senior Member AIAA. The views expressed in this paper are those of the author and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the U.S. Government 1 of 19 This material is declared a work of the U.S. Government and is not American subject to Institute copyright of protection Aeronautics in the and United Astronautics States.

2 and linearly varying transport properties (viscosity, heat transfer coefficient and diffusion coefficient). 1 Typical CFD methods also assume that the flow remains in thermodynamic equilibrium (the internal energy remains in equilibrium with the translational energy). In areas of the flow that include large gradients (such as the shock and boundary layers near the wall), these assumptions of equilibrium break down. In the rarefied flow regime (large Knudsen numbers) the flow can be computed using the direct simulation Monte Carlo (DSMC) method. 2 The DSMC method does not depend assumptions involving a small perturbation from equilibrium and hence is more accurate than CFD methods for non-equilibrium flows. It has been shown that DSMC solutions approach solutions to the Boltzmann equation. 3 Generally speaking, CFD methods for solving the NS equations are about an order of magnitude faster than the DSMC method. Note that in continuum regimes, locally a flow may behave like a rarefied flow if the local characteristic length scale is very small. On a blunt body, a high-density fore-body flow can create a rarefied flow in the wake of the vehicle. In principle, the DSMC method can be applied to any dilute gas flow, but becomes prohibitively expensive for Knudsen numbers less than It is, therefore, attractive to develop hybrid methods that can use a kinetic solver in regions of high nonequilibrium, and continuum methods in regions of local equilibrium. The Unified Flow Solver (UFS) was recently developed 4, 5 for the computation of hypersonic, nonequilibrium gas flows. It is a hybrid continuum/kinetic solver in that it contains both a continuum solver (based on the Euler or Navier-Stokes equations) and a kinetic solver (based on the Boltzmann equation). The UFS has been validated and demonstrated for a variety of simple gas flows in a variety of different flow regimes. 4 9 However, there has not been, to date, an in-depth study of the accuracy and efficiency of UFS for hypersonic body surface properties. The intent of this study is to build on the previous work which took an initial look at the capability of UFS in predicting heat flux on a hypersonic blunt body. 8 II. Background and Simulation Procedure This investigation considers a hypersonic flow of argon over a two-dimensional, 12-inch diameter cylinder, as shown in Figure 1. The free stream velocity corresponds to Mach 10 (U = 2624 m/s). The wall temperature is held constant at 500 K. The free stream density of the flow is varied such that two different regimes are considered; one near the continuum regime, and one near the rarefied regime, as shown in Table 1. Knudsen numbers are calculated based on freestream conditions and the cylinder diameter, using the hard-sphere model for the mean free path calculation. Surface and flow field properties for these flight conditions are computed using the UFS code and are compared with DSMC solutions. Table 1. Flow regimes considered. * Kn Mass Density Number Density kg/m 3 particles/m * Based on hard-sphere mean free path. DSMC results are provided from the MONACO code 10 for the flow conditions mentioned above, as described 11, 12 in a previous study. MONACO is a general 2D/3D, object-oriented, cell-based, parallel implementation of the DSMC method. It uses the Variable Hard Sphere (VHS) and Variable Soft Sphere collision models. 2, 13 It also includes variable vibrational 14 and rotational 15 energy exchange probability models. All MONACO solutions are generated using a fixed wall temperature at 500 K. Bird s VHS model is used. 2 The VHS parameters used in the DSMC computations are those for standard argon (the temperature exponent, ω, is with a reference diameter of m at a reference temperature of 1000 K). In general, the mesh used for the final solution for each case is adapted from previous solutions such that each cell size is on the order of a mean free path, with the exception of the Kn = 0.01 case, where the cell sizes near the surface are on the order of two mean free paths in size. In all cases the subcell method is used to select particles for collisions 2 to ensure physical accuracy. As mentioned, UFS is a hybrid continuum/kinetic solver for the computation of hypersonic flows. In regions of high nonequilibrium, UFS uses a kinetic solver in the form of a direct numerical simulation of the Boltzmann 2 of 19

3 Figure 1. Geometry definition. equation. 16 In these kinetic regions, the velocity distribution function is obtained directly, from which the moments are taken to obtain macroscopic gas properties. Various models for the collision integral are implemented, including a BGK approximation. For continuum regions, UFS uses a gas-kinetic Euler solver, based on Pullin s equilibrium flux method, 17 or, optionally, a Navier-Stokes solver similar to Xu s gas-kinetic BGK solver. 18 UFS automatically detects the continuum and kinetic regions based on continuum breakdown criteria, and invokes the appropriate solvers for the different parts of the domain. This hybrid kinetic/continuum approach should provide significant time savings compared to DSMC while providing increased accuracy of the flow solution compared to CFD, for certain transition regime flows. UFS is based on the Gerris framework, 19 and thus uses a cartesian cell approach, based on a quad/octree data structure. This enables the use of automatic mesh refinement, based on user-described functions. However, due to the cartesian implementation, sold boundaries are implemented with a cut-cell strategy. It is known that the accuracy of surface property prediction suffers when using cut-cell boundaries (see, for instance, Croirier s work 20 ). Current UFS capabilities include the use of Euler gas-kinetic scheme for the continuum solver, and the use of the BGK collision integral for the kinetic solver for 3D/2D planar, 2D axisymmetric, rotational and vibrational nonequilibrium (singe-species gases) and non-reacting mixtures. Other collision integral models are available for the simulation of simple gases. Additionally, a Navier-Stokes gas-kinetic scheme is available for 3D/2D planar cases and non-reacting mixtures. The gas-kinetic continuum schemes can be either first-order in space, or secondorder (with the addition of flux limiters). The Boltzmann Transport Equation represents the evolution of the particle velocity distribution function (VDF), f, in phase space. The BGK approximation to the Boltzmann equation can be written as f t + r ξ f = ν f m f (2) where ξ is the velocity vector, r is a position vector in physical space, f m is the Maxwellian equilibrium VDF and ν is the relaxation parameter. The relaxation parameter is a function of the molecular viscosity, µ, and the gas pressure, p as ν = p µ. (3) When comparing UFS solutions to DSMC solutions, it is necessary to relate the UFS BGK parameters to the DSMC VHS parameters. The gas viscosity for a VHS model is given in terms of power-law relation with the temperature as ω T µ = µ re f (4) T re f where µ is the coefficient of viscosity, T is the translational temperature, ω is the VHS temperature exponent. Internally, UFS represents all gas properties in a non-dimensional way, thus it is only necessary to replicate the VHS viscosity temperature exponent in the BGK time relaxation parameter, as ν = p µ = nkt T ω T 1 ω. (5) 3 of 19

4 The DSMC solutions used here were obtained with a VHS temperature exponent, ω, of Thus, the UFS BGK time relaxation parameter, ν, is set proportional to T to match the VHS viscosity. It is known that the BGK model assumes a Prandtl number of unity, while that for most gases is less than unity. For argon, Pr = 2/3. A Prandtl number correction is implemented in UFS to correct this deficiency. As a hybrid method, a switching criteria must be used to determine which cells are in the continuum domain and which are in the kinetic domain. There are several switching criteria built into the code, the one used in this study is the Navier-Stokes-type switching criteria: 5 p S NS = Kn + 1 u 2 v 2 w 2 p U x y z The value of the switching criteria determines how much of the domain is solved using the kinetic solver versus the continuum solver. Larger values of S NS correspond to larger kinetic domains and larger computational expense. This work uses a value of S NS of 0.1. UFS contains the possibility to force the use of the kinetic solver in the two layers of cells closest to the surface of the cylinder, as shown in Figure 2, where only the first two layers of cells near the surface are forced to be kinetic cells. The current study uses the Euler solver for the continuum regime, and, therefore, the boundary layer near the cylinder surface must be computed using the kinetic solver, which, in this case, uses the BGK collision term. For the following cases, a larger region near the surface of the cylinder is forced to use the kinetic solver. (6) Figure 2. Detail of cartesian grid near cylinder surface showing two layers of kinetic cells (red), as well as cut-cells near the surface. III. Computational Results In the results that follow, the surface properties are presented in terms of non-dimensional coefficients for surface pressure, shear stress and heat transfer with the following definitions: C P = p p 1 2 ρ U 2 C F = C H = τ 1 ρ 2 U 2 q 1 ρ 2 U 3 where p is the pressure, τ is the shear stress, q is the heat transfer rate, p is the free stream pressure, ρ is the free stream density and U is the free stream velocity. The surface properties in each case are plotted as a (7) (8) (9) 4 of 19

5 function of the angle around the cylinder, with the stagnation point being located at an angle of zero (see Figure 1). IV. Kn = 0.01 The first case considers a flow of argon at a global Knudsen number of This case represents a flow at the lower edge of the transitional, or slip, regime, just above the continuum regime. The meshes used for both the UFS and DSMC simulations are shown in Figure 3. Note that UFS uses a cartesian mesh that has been automatically adapted based on the solution in order to resolve areas of large gradients. The DSMC simulation uses an unstructured triangle mesh that has been adapted based on the computed mean free path. In addition to the meshes, also shown is the decomposition of the UFS domain into kinetic (dark gray) and continuum (light gray) regions. The kinetic (or BGK) solver is used in the area of the shock and the wake, where the selection of one or the other is based on the selection criteria in Equation 6. A fixed area immediately around the cylinder surface is also forced to use the kinetic solver. (a) Mesh (b) Kinetic (dark) and Continuum (light) Domains Figure 3. UFS and DSMC mesh comparison. Note that UFS uses a cartesian mesh, with an automatic algorithm and mesh adaptation capabilit, while the DSMC code uses an unstructured triangle mesh that has been adapted based on the computed mean free path. UFS automatically decomposes the domain into kinetic (dark gray) and continuum (light gray) regions. Nondimensional representations of the field properties (including temperature, density, pressure and Mach number) computed with each method are compared in Figure 4. Overall there seems to be fairly good agreement between the two methods. Slight differences in the temperature contours in the wake are shown. The UFS temperature also increases more rapidly ahead of the shock than DSMC. This is most evident in the Mach contours (which seems to show a thicker shock predicted by UFS), and will be examined in more detail below. The stagnation region also exhibits some differences in density and pressure. It is expected that at these flow conditions that the UFS solution would compare very well with the DSMC. One possible explanation for the discrepancy is that the UFS solution has not reached convergence. The stagnation region tends to take a large time to converge, and, due to time constraints, the Kn = 0.01 case was only run to 15,000 time steps. It is possible, therefore, that the UFS solution has not converged completely. At the present time, it is difficult to judge when a UFS solution has reached convergence. The best (and perhaps only) way to determine if a solution has converged is by monitoring the gas properties at one or more points in the computational domain. For the present case, gas properties at two points were monitored the first along the stagnation line just upstream of the stagnation point, and the second just after the cylinder in the wake. The properties of normalized density, velocity and pressure are plotted for these two points in Figure 5. It can bee seen in these figures that the solution is still developing. In the stagnation region, the density and pressure are still showing slight variations with time. In the wake, the density and pressure are still clearly increasing with time. Therefore, this solution is not a converged solution. 5 of 19

6 (a) Temperature (b) Density (c) Pressure (d) Mach Figure 4. Temperature field and density ratio field (normalized by freestream density) predicted by UFS (top) and DSMC (bottom) for Kn = 0.01 case of Mach 10 flow of argon. 6 of 19

7 (a) Monitor Point 1 (stagnation region) (b) Monitor Point 2 (wake) Figure 5. Gas properties at monitoring points plotted as a function of time. The surface properties of pressure, shear stress and heating are plotted in Figures 6-8. Details of the surface pressure near the stagnation region and of the shear stress near max skin friction are also shown. The surface pressure again shows differences that are not expected, that again might be attributed to a non-converged solution. At the stagnation point, the difference between UFS and DSMC is approximately 2%. The stagnation region detail illustrates the difficulty in obtaining smooth surface properties with a cartesian mesh. The regular pattern in the surface pressure is also related to the level of refinement and the resolution of the geometry. Each point in the UFS solution corresponds to one cell on the surface. Each group of points corresponds to one facet on a discretized cylinder. The cylinder geometry was defined with 256 faces; it is these faces that give the pattern in surface pressure shown. (a) Surface Pressure (b) Surface Pressure Detail Figure 6. Surface pressure and detail near stagnation region. The surface shear stress plots also show discrepancies that are not expected. These are greatest at about Φ = The detailed plot focuses on this region. Notice also the scatter in the UFS data. There is similar 7 of 19

8 scatter in the DSMC method that is related to the statistics of a particle method. The scatter in the UFS data is not related to statistics, but to the cut-cell boundary conditions used with the cartesian method. Better boundary conditions, such as the immersed boundary method, are expected to improve the smoothness of surface properties from UFS simulations. CFD Research Corporation (the developer of UFS) is currently working to implement these types of boundary conditions into the UFS code. (a) Surface Shear Stress (b) Surface Shear Stress Detail Figure 7. Surface shear stress and detail near Φ = 45. The most disturbing difference between UFS and DSMC is found in the surface heating, as plotted in Figure 8. The peak heating predicted by UFS is more than a factor of two greater than that predicted by DSMC. A portion of this discrepancy could be explained by the lack of convergence, but similar differences are noted in the results for Kn = 0.25, which is a converged solution (as will be shown below in Figure 13. Previous work 8 showed that it is possible to obtain good agreement between DSMC and UFS for heating rates. There, the simulations were obtained with a different DSMC code, and the intermolecular potential modeled was the hard-sphere model. It was also mentioned that it is possible to achieve good agreement with a well-designed mesh, although it is vague as to what exactly constitutes a well-designed mesh. It is mentioned that cell sizes on the order of a mean free path are necessary for good results. Past experience has demonstrated that heating rates are very dependent on the cell size near the wall. In this work, the mesh is refined at the stagnation region to a level of 13. This gives a cell size near the stagnation point on the order of the local mean free path, as shown in Figure 9. It should be noted that the DSMC results for this case are not in question, as they are shown to agree well with continuum CFD calculations (with slip boundary conditions). 12 The initial mesh away from the cylinder maintains a cell size approximately equal to the local mean free path. As the gas is compressed through the shock, the mean free path decreases. Mesh adaptation decreases the local cell sizes, with cell refinement being more than required to maintain cell sizes about equal to the mean free path. Although the cell sizes near the body are refined to a maximum level of 13 near the stagnation point, the large decrease in the mean free path (or increase in density) as the gas is cooled near the relatively cold body wall causes cell size / mean free path ratio to increase, but it still remains at or near a value of unity at the wall. The variation of the gas properties along the stagnation line are detailed in Figure 10. The flow properties for both DSMC and UFS are shown along with the value of the kinetic flag to show the kinetic and continuum regions. A kinetic flag value of 1 corresponds to a kinetic region (BGK solver) and a value of 0 corresponds to a continuum region (Euler solver). The agreement between DSMC and UFS is good overall, with some exceptions. The pre-shock states are, of course, in good agreement. However, the UFS temperature starts to rise at about X/R = -1, well before the increase in temperature predicted by DSMC. There is a corresponding decrease in the velocity. The most glaring exception is in the post-shock region as UFS transitions from the kinetic solver to the continuum solver. In all cases, there is notable disagreement. It appears that UFS is prematurely switching from the kinetic solver to the continuum solver. It is not immediately clear if this is due to an non-converged solution, or if it is due to faulty switching criteria. Further studies are warranted to answer this question. 8 of 19

9 (a) Surface Heating (linear scale) (b) Surface Heating (logarithmic scale) Figure 8. Surface heating. Figure 9. Cell size (normalized by the box length, L), the mean free path, and the ratio of cell size to mean free path along the stagnation line. 9 of 19

10 (a) Density (b) Temperature (c) Pressure (d) Velocity Figure 10. Comparison of properties along the stagnation line, through the shock. A kinetic flag value of 1 corresponds to a kinetic region, while a value of 0 corresponds to a continuum region. 10 of 19

11 (a) Velocity (b) Temperature Figure 11. Near-wall properties (velocity slip and temperature jump. The near-wall properties of velocity and temperature are plotted as a function of the position around the cylinder in Figure 11. These properties give an idea of the surface velocity slip and temperature jump. Similar profiles have been demonstrated for both DSMC and continuum CFD results (with slip boundary conditions) in previous work. 21 The agreement between UFS and DSMC shown here good qualitatively, although it seems that UFS under-predicts the actual values. Note that neither UFS nor DSMC include any explicit slip boundary conditions, but they are inherent in both models. V. Kn = 0.25 This second case considered here is at a much higher Knudsen number such that it is near the rarefied regime. The DSMC and UFS meshes are seen in Figure 12. As the density is lower, the flow gradients are more diffuse and there is less mesh refinement needed. Almost the entire domain is solved using the UFS kinetic solver. Unlike the Kn = 0.01 case, which was run for only 15,000 time steps, this case was run for 55,000 time steps. The time history of the gas properties at the two monitoring points (near the stagnation point and in the wake) are plotted in Figure 13. Although there are some very slight oscillations in some properties (particularly the density and pressure), all are nearly constant over time, indicating that this case has reached a converged solution. The normalized field properties show fairly good agreement between UFS and DSMC, as shown in Figure 14, although UFS also predicts a pre-shock temperature that begins to rise earlier than DSMC. Although the BGK model is expected to be less accurate as the Knudsen number rises, it still appears that UFS is giving fairly good results. The surface properties of pressure, shear stress and heating are plotted in Figure 15. Although this simulation is converged, there is still an unexpected, and unexplained, disagreement in surface pressure near the stagnation region. The shear stress distribution shows good agreement with DSMC, and the surface property profiles are quite smooth. The UFS, however, again overpredicts the peak heating by a factor of nearly two. The stagnation line properties seen in Figure 16 show good agreement, although UFS predicts a slightly higher peak temperature, with the temperature beginning to rise earlier than DSMC. Note that the entire stagnation line region plotted here is in the kinetic domain. The mean free path and cell size (and their ratio) is again plotted for this case in Figure 17. Note that the cell size is much less than the mean free path everywhere. It may be possible to reduce the mesh refinement without losing accuracy. As with the previous case, the velocity and temperature near the wall (that would correspond to a velocity slip and temperature jump), also show good qualitative agreement, as plotted in Figure 18, although UFS predicts 11 of 19

12 (a) Mesh (b) Kinetic and Continuum Domains Figure 12. UFS and DSMC mesh comparison. Kinetic and continuum domains for the UFS simulation. (a) Monitor Point 1 (stagnation region) (b) Monitor Point 2 (wake) Figure 13. Gas properties at monitoring points plotted as a function of time. 12 of 19

13 (a) Temperature (b) Density (c) Pressure (d) Mach Figure 14. Temperature field and density ratio field (normalized by freestream density) predicted by UFS (top) and DSMC (bottom) for Kn = 0.25 case of Mach 10 flow of argon. 13 of 19

14 (a) Surface Pressure (b) Surface Shear Stress (c) Surface Heating Figure 15. Surface properties. 14 of 19

15 (a) Density Ratio (b) Temperature (c) Pressure Ratio (d) Velocity Magnitude Figure 16. Stagnation line properties 15 of 19

16 Figure 17. Cell-size / mean free path on stgnation streamline. values slightly less than DSMC. (a) Velocity (b) Temperature Figure 18. Near-wall properties (velocity slip and temperature jump. 16 of 19

17 VI. Total Drag and Peak Heating The total drag and peak heating for both cases are summarized in Tables 2 and 3. Total drag predictions are based on the surface pressure and shear stress, and UFS predictions remain within 2% of DSMC for both cases considered. However, UFS predicts peak heating about twice that of DSMC. The cause of the disagreement is unknown. Table 2. Total drag a for flow of argon about a cylinder. Drag/Length [N/m] Kn DSMC UFS % Difference % % Table 3. Peak heating for a flow of argon about a cylinder. Peak Heating kw/m 2 Kn DSMC UFS % Difference % % VII. Computational Expense The computational details for the simulations presented here are summarized in Table 4. It was expected that UFS would give reasonably accuracy, as compared to DSMC, for less computational work. It is obvious that this is not the case. However, very little effort was made to reduce the computational expense. Further efficiency could be attained by reducing the level of mesh refinement (although this will have an impact on the solution accuracy), and the actual computational domain can be reduced further than it is. It is quite apparent that the UFS simulations use a much larger number of cells, especially with the Kn = 0.01 case, than do the DSMC simulations (which have also not been tuned to reduce the computational expense). It was also expected that the more rarefied case would require more computational expense since a larger percentage of the domain would require the kinetic solver. This is not the case precisely because of the level of refinement used in the lower Knudsen number case in order to resolve the flow details (such as the shock) and in an attempt to get smooth surface properties that compared well with DSMC. VIII. Conclusions and Future Work The Unified Flow Solver solutions for a flow of argon at Kn = 0.01 and Kn = 0.25 were compared with DSMC solutions. Both surface property results and flow details were compared. There is reasonable agreement between the two solvers for both cases considered, with the exception of surface heating. Although some of the disagreement can be attributed to a non-converged solution for Kn = 0.01, this is not the case for Kn = The cartesian mesh approach with automatic algorithm and mesh refinement is very attractive, as it reduces the amount of time for operator intervention in setting up each problem. However, the use of cut-cell boundary conditions presents particular problems when attempting to obtain surface properties. New boundary conditions are expected to improve this situation. Further work is necessary to resolve still open questions. These include: Ensure solution convergence. 17 of 19

18 Table 4. Computational details for a flow of argon about a cylinder. Total CPU time is the wall time multiplied by the number of CPUs. DSMC Kn Cells Particles Time Steps CPUs Total CPU Time [hours] , , , , , UFS Kn Cells Time Steps CPUs Total CPU Time [hours] ,731 15, , ,053 55, ,360 Investigate reduction in computational expense while maintaining accuracy. Investigate the effect of switching parameter on solution accuracy. Determine cause of disagreement in peak heating predictions. IX. Acknowledgements This work is supported by a grant from the U.S. Air Force Office of Scientific Research, monitored by John Schmisseur. Computer resources were provided by the U.S. Air Force Research Laboratory DoD Supercomputing Resource Center at Wright-Patterson AFB, OH. The author would like to acknowledge the helpful assistance of Robert Arslanbekov and Vladimir Kolobov of CFD Research Corporation, and of Eswar Josyula of the U.S. Air Force Research Laboratory. References 1 Vincenti, W. G. and Kruger, Jr., C. H., Introduction to Physical Gas Dynamics, Krieger Publishing Company, Bird, G. A., Gas Dynamics and the Direct Simulation of Gas Flows, Oxford University Press, Oxford, Wagner, W., A convergence proof for Bird s direct simulation Monte Carlo method for the Boltzmann equation, Journal of Statistical Physics, Vol. 66, No. 3-4, Feb. 1992, pp Kolobov, V. I., Bayyuk, S. A., Arslanbekov, R. R., Aristov, V. V., Frolova, A. A., and Zabelok, S. A., Contruction of a Unified Continuum/Kinetic Solver for Aerodynamic Problems, Journal of Spacecraft and Rockets, Vol. 42, No. 4, July August 2005, pp V. I. Kolobov, V. I., Arslanbekov, R. R., Aristov, V. V., Frolova, A. A., and Zabelok, S. A., Unified Solver for Rarefied and Continuum Flows with Adaptive Mesh and Algorithm Refinement, Journal of Computational Physics, Vol. 223, 2007, pp Kolobov, V. I., Yang, H. Q., Bayyuk, S. A., Aristov, V. V., Frolova, A. A., and Zabelok, S. A., Unified Methods for Continuum and Rarefied Flows, 42nd AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, jan 2004, AIAA Paper Kolobov, V. I., Arslanbekov, R. R., Aristov, V. V., Frolova, A. A., and Zabelok, S. A., Unified Flow Solver for Aerospace Applications, 44th Aerospace Sciences Meeting and Exhibit, Reno, NV, jan 2006, AIAA Paper Arslanbekov, R. R., Kolobov, V. I., Frolova, A., Zabelok, S., and Josyula, E., Evaluation of Unified Kinetic/Continuum Solver for Computing Heat Flux in Hypersonic Blunt Body Flows, 39th AIAA Thermophysics Conference, Miami, FL, jun 2007, AIAA Paper Josyula, E., Arslanbekov, R. R., Kolobov, V. I., and Gimelshein, S. F., Evaluation of a Kinetic/Continuum Solver for Hypersonic Nozzle- Plume Flow, Journal of Spacecraft and Rockets, Vol. 45, No. 4, July August 2008, pp Dietrich, S. and Boyd, I. D., Scalar and Parallel Optimized Implementation of the Direct Simulation Monte Carlo Method, Journal of Computational Physics, Vol. 126, No. 2, 1996, pp Lofthouse, A. J., Boyd, I. D., and Wright, M. J., Effects of continuum breakdown on hypersonic aerothermodynamics, Physics of Fluids, Vol. 19, No. 2, 2007, pp Lofthouse, A. J., Scalabrin, L. C., and Boyd, I. D., Velocity Slip and Temperature Jump in Hypersonic Aerothermodynamics, Journal of Thermophysics and Heat Transfer, Vol. 22, No. 1, 2008, pp Koura, K. and Matsumoto, H., Variable Soft Sphere Molecular Model for Air Species, Physics of Fluids A, Vol. 4, No. 5, May 1992, pp Vijayakumar, P., Sun, Q., and Boyd, I. D., Vibrational-Translational Energy Exchange for the Direct Simulation Monte Carlo Method, Physics of Fluids, Vol. 11, No. 8, Aug. 1999, pp of 19

19 15 Boyd, I. D., Analysis of Rotational Nonequilibrium in Standing Shock Waves of Nitrogen, AIAA Journal, Vol. 28, No. 11, Nov. 1990, pp Aristov, V. V., Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows, Kluwer Academic Publishers, Pullin, D. I., Direct Simulation Methods for Compressible Inviscid Ideal-Gas Flow, J. Comput. Phys., Vol. 34, 1980, pp 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, pp Popinet, S., Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries, Journal of Computational Physics, Vol. 190, 2003, pp Coirier, W. J., An Adaptively-Refined, Cartesian, Cell-Based Scheme for the Euler and Navier-Stokes Equations, Ph.D. thesis, The University of Michigan, Lofthouse, A. J., Scalabrin, L. C., and Boyd, I. D., Velocity Slip and Temperature Jump in Hypersonic Aerothermodynamics, 45th AIAA Aerospace Sciences Meeting and Exhibit, 2007, AIAA Paper of 19