# Low Diffusion (LD) Hybrid Particle Method

Particle-based algorithms are developed for simulating hypersonic flows, high-altitude rocket exhaust flows, and other gas flow problems involving a wide range of density values or characteristic length scales among different flowfield regions. These types of flows are most commonly simulated using a hybrid particle-continuum approach, which employs a CFD solver in continuum regions and the direct simulation Monte Carlo (DSMC) method in nonequilibrium regions where the assumptions underlying the governing Navier-Stokes or Euler equations break down. While hybrid CFD-DSMC techniques have been effectively used for a variety of challenging test cases, the inherent scatter in DSMC generally prevents such an approach from allowing strong two-way coupling between continuum and nonequilibrium regions. This limitation becomes particularly problematic when simulation of an unsteady flow is desired.

Hybrid “all-particle” algorithms have been developed to permit stronger coupling and potentially allow for unsteady flow simulation, while dramatically reducing the size and complexity of the source code relative to hybrid CFD-DSMC techniques. These alternative methods use a modified version of DSMC in continuum regions, with procedures that in effect resample particle velocities from an equilibrium distribution during each time step. Despite the benefits of an all-particle hybrid approach (in particular, strong coupling, simplicity and ease of implementation) this type of scheme has not received widespread acceptance, due mainly to the large inherent numerical diffusion in existing DSMC-based particle methods for continuum flow simulation. As these equilibrium particle methods assume free molecular fluxes between neighboring computational cells, the numerical transport coefficients tend to scale with cell size and become extremely large when (as is typically required for continuum flow simulation) the cell size is much greater than the local mean free path.

In the current project, a new extension to the DSMC method is developed for the simulation of near-equilibrium compressible gas flows. Instead of resampling particle velocities from an equilibrium distribution, here all particles are assigned velocities roughly equal to the local bulk velocity, so that particles tend to move along gas streamlines. This reproduces the path of real gas molecules over macroscopic length scales, where the influence of Brownian motion is suppressed due to the large disparity between the cell size and mean free path. In the numerical procedures, particles are moved through the grid in such a way that each particle remains fixed with respect to a Lagrangian cell over the time step interval. The Lagrangian cell in turn is moved and deformed according to gas bulk properties, following a set of approximations based on kinetic theory. As particles travel along streamlines and no free-molecular flux assumptions are used in the new method, numerical diffusion effects are greatly reduced relative to existing DSMC-based equilibrium particle methods. Statistical scatter is greatly reduced as well, due to the deterministic nature of particle trajectories and a lack of random processes in the simulation procedures. When differences in required cell sizes and the required number of sampling time steps are taken into account and comparable levels of accuracy and precision are desired, representative simulations using the new method are between one and two orders of magnitude faster than simulations performed using existing DSMC-based techniques.

The new equilibrium particle scheme, called for convenience the low diffusion (LD) particle method, has been integrated into a hybrid code with DSMC for the simulation of rarefied/continuum flows involving a wide range of characteristic length scales. A standard continuum breackdown parameter is used to dynamically assign flowfield regions to either LD or DSMC domains, and two layers of buffer cells are employed to transfer information during each time step between these domains. In a buffer cell, all particles are cloned just before movement procedures during a given time step, and the clones are assigned appropriate LD or DSMC particle properties. All clones remaining in the source cell are then destroyed immediately following particle movement routines, in such a way that strong two-way coupling of flowfield information occurs between DSMC and LD regions. Initial tests have shown excellent agreement between hybrid LD-DSMC simulation results and results from a DSMC simulation of the same flow. The figure shown here gives Mach number contours for a Mach 6 flow of N_{2} around a specularly reflecting cylinder, with hybrid simulation results at the top and DSMC results at the bottom. Purple lines indicate boundaries between DSMC regions, in the wake and around the shock, and LD regions elsewhere in the flow.