Evolution of 2D jet flow at Mach number 0.5 between constant temperature walls using the 9-bit energy-dependent lattice: a long-time comparison of the scaling integrity of the flows for Reynolds number = 20,000. Left column: timestep = 300 K, temperature = 8.0 (—6), v0 = 2.0 (—3). Right column: timestep = 60 K, temperature = 2.0 (—4), v0 = 1.0 (—2).

George Vahala, College of William and Mary
Linda Vahala, Old Dominion University
Pavol Pavlo, Institute of Plasma Physics, Czech Academy

Research Objectives
Thermal lattice Boltzmann modeling (TLBM) is an ideal MPP computational tool to study nonlinear macroscopic systems. In the divertor regime, where the neutral collisionality roams from very collisional (fluid) to weakly collisional (Monte Carlo), TLBM can give a unified framework, and thus avoid the stiff problem of coupling UEDGE to Monte Carlo. Our current codes run at a kinetic Courant number CFL = 1, so that no numerical diffusion or dissipation is introduced.

Computational Approach
TLBM discretizes the Bhatnagar-Gross-Krook (BGK) kinetic equation to solve the system on a lattice. In its simplest form, the algorithm advances the distribution at time t to time t + 1 by (a) free-streaming (at CFL = 1) the distribution function from one spatial node to nearest lattice neighbors (this only involves the shift operation and use of MPI to handle boundary points in the domain decomposition); (b) recomputing local mean variables (simple summations, a local operation); (c) linear collisional relaxation (local operation).

Accomplishments
In order to improve the numerical stability of TLBM, we are employing an energy-dependent octagonal 2D lattice. Initial results are very promising as we investigate jet flow into highly stratified background.

As we move to modeling divertor physics, we have been investigating two-fluid equilibration. In particular, we have been examining the effects of velocity shear turbulence of a lighter species on a laminar heavier species. We have verified the Morse 1967 theory dealing with the rate of species temperature equilibration to velocity equilibration. This is dimension independent for weakly turbulent systems.

Significance
In the standard computational fluid dynamics approach to solving the nonlinear equations, one must handle the nonlinear Riemann problem and over 30% of the CPU time is spent in resolving the nonlinear convective derivative. In TLBM, one side-steps the Riemann problem altogether and can use Lagrangian streaming to handle the linear advective derivative. In essence, by embedding the nonlinear system into higher dimensional phase space (i.e., by going to a linearized kinetic description), we can choose a simplified system (e.g., a BGK collision operator) to recover the desired equations. This concept is similar to using multi-scale perturbation theory to solve singular problems in applied math/physics. TLBM codes are very well suited for parallel machines.

Publications
L. Vahala, D. Wah, G. Vahala, J. Carter, and P. Pavlo, "Thermal lattice Boltzmann simulation for multispecies equilibration," Phys. Rev. E 62, 507 (2000).

G. Vahala, J. Carter, D. Wah, L. Vahala, and P. Pavlo, "Paralleliza-tion and MPI performance of thermal lattice Boltzmann codes for fluid turbulence," in Parallel Computational Fluid Dynamics '99, edited by D. Keyes et al. (Elsevier Science, Amsterdam, 2000).

L. Vahala, G. Vahala, J. Carter, and P. Pavlo, "Velocity and temperature equilibration for multi-species gases using thermal lattice Boltzmann simulations," Czech. J. Physics (in press).

http://www.physics.wm.edu/~vahala/july00.html