Funding: Some of this material is based upon work supported by the DOE [National Nuclear Security

1. Improvements to the Discrete Velocity Method for the Boltzmann EquationPeter Clarke D. Hegermiller, A.B. Morris , P.T. Bauman, P. L. Varghese, D. B. GoldsteinUniversity of Texas at AustinDepartment of Aerospace EngineeringDSMC Workshop September 2011 Funding: Some of this material is based upon work supported by the DOE [National Nuclear Security Administration] under Award Number [DE-FC52-08NA28615] and NASA’s NSTRF Fellowship program

2. Outline Motivation The Discrete Velocity Method Previous work Variance reduction using an interpolation scheme Non-uniform grids in velocity space Application of VHS and VSS collision models Future Work Inclusion of internal energy in DVM 2

3. Motivation • Discrete velocity methods are comparable to DSMC • But discrete velocity methods have several traditional problems • High Mach number flows and other flows that require large velocity bounds. • The inclusion of physics in the model such as varying types of molecular potentials, multi-species flow, internal energy, and chemical reactions are often neglected in preliminary • DVM investigations. • We wish to solve the first problem with the eventual application of • adaptive velocity grids. • The first step towards this goal is the implementation of non • uniform grids in velocity space. 3

4. DVM Formulation We begin with the scaled Boltzmann equation: Scaling Factors: The collision integral is split into replenishing and depleting parts: 1, pseudo-maxwell 1, pseudo-maxwell To solve the Boltzmann equation using DVM, we must discretize the integro-differential equation. 4

5. DVM Formulation β We separate the convection and collision parts of the equation 4th order convection: We then approximate the collision integral with finite summations: 5 Bobylev, A.V., 1976, Soviet Phys. Dokl., 20, 822-824. Krook, M., and Wu, T.T., 1977, Phys. Fluid, 20, 1589-1595

6. DVM Formulation DSMC – “Fixed mass, variable velocity particles.” DVM – “Fixed velocity, variable mass quasi-particles.” ϕ ϕ ηj ηj ηi ηi 6

7. Variance Reduction As has been previously presented by A. Morris we use a stochastic discrete velocity model: Decomposeϕ into an equilibrium part and a deviation from equilibrium part δ 7 Baker, L.L. and N.G. Hadjiconstantinou, "Variance Reduction for Monte Carlo Solutions of the Boltzmann Equation," Physics of Fluids, 17, 2005 Morris, A.B., “Variance Reduction for a Discrete Velocity Gas” 2011

8. Variance Reduction As has been previously presented by A. Morris we use a stochastic discrete velocity model: 0 δ 8 Baker, L.L. and N.G. Hadjiconstantinou, "Variance Reduction for Monte Carlo Solutions of the Boltzmann Equation," Physics of Fluids, 17, 2005 Morris, A.B., “Variance Reduction for a Discrete Velocity Gas” 2011

9. Variance Reduction Calculate depletion mass: Select random collision partners, either two from the deviation distribution or one from the deviation and one from the equilibrium distribution. or β 9

10. Interpolation Begin with conservation equations: Mass Momentum Energy xyz iz The system of equations is solved: Δϕ ex iy c o b a ix ey ez 10 Varghese, P.L., “Arbitrary Post-Collision Velocities in a Discrete Velocity Scheme for the Boltzmann Equation.” 2007

11. Code Development • DVM has been implemented using modern software engineering principles that enhance maintainability and ease of testing to allow more thorough verification of the software and, thus, increases confidence that the implementation is correct. These practices include: • Object-oriented code style to enhance encapsulation and minimize code duplication. • Source code revision control using svn. • Build system (Autotools) for portability between computing systems (code currently tested on Linux and Mac OS X environments) • Build system also enables easy addition of unit and regresssion tests. Current suite is at 42 tests. • Full documentation of code and algorithms 11

12. Variance Reduction with Interpolation We combine the variance reduction technique with the interpolation scheme that has been developed: 4th, 6th, and 8th moments of the relaxation of the BKW distribution - Interpolation - No interpolation - analytic No Interpolation Interpolation 12

13. Variance Reduction with Interpolation Mach 2 Shock density profile: - No interpolation - Interpolation 13

14. Non-uniform grids in velocity space Additions to the Discrete Velocity Method: Due to the interpolation scheme we can relax the requirement that β be a constant number βk βj βi 14

15. Non-uniform grids in velocity space 3D homogeneous relaxation with variable grid: The optimal configuration for the velocity grid is an area of active research 15

16. VHS and VSS VHS VSS Collision probability depends on relative speed. The amount depleted during a collision is now proportional to collision probability. VSS is similar to VHS except the scattering is no longer isotropic . When picking post-collision velocities, sample from the scattering distribution. 16

17. VHS Example A Mach 2 shock with VHS: DVM DSMC 17

18. Future Work Internal Energy: A major addition to the Discrete Velocity Method that allows for more accurate physics is the inclusion of internal energy ηk ηj We assign a single internal energy to each location in velocity space. Future work will allow a distribution of energies at every velocity location ηi Erot Evib 18

19. Future Work An exchange in energy between translation and internal energy is calculated using a Landau-Teller-like equation. The exchange changes the magnitude of the post-collision relative velocity vector as well as adding or subtracting from the internal energy distributions. Interpolation allows for any post collision relative velocity vector length. 19

20. Summary • We showed: • Comparison between DSMC and DVM • Combination of Variance Reduction with Interpolation • Non-uniform velocity grids • Application of VHS and VSS collision models • Future Work: • Full implementation of internal energy including distributions of energy at • every point in velocity space. • Adaptable velocity grids. 20