Michael N. Macrossan

The equilibrium flux method for the calculation of flows with non-equilibrium chemical reactions

The Equilibrium Flux Method [1] is a kinetic theory based finite volume method for calculating the flow of a compressible ideal gas. It is shown here that, in effect, the method solves the Euler equations with added pseudo-dissipative terms and that it is a natural upwinding scheme. The method can be easily modified so that the flow of a chemically reacting gas mixture can be calculated. Results from the method for a one-dimensional non-equilibrium reacting flow are shown to agree well with a conventional continuum solution. Results are also presented for the calculation of a plane two-dimensional flow, at hypersonic speed, of a dissociating gas around a blunt-nosed body.

A Particle-Only Hybrid Method for Near-Continuum Flows

EPSM is a particle simulation method for the simulation of the Euler equations. EPSM is used here as part of a hybrid EPSM/DSMC method for the simulation of near continuum flows. It is used where the flow gradients are not large and the flow is expected to be in an equilibrium state. The gradient of local mean free path has been used to detect those regions where EPSM can be invoked. Results are presented for the unsteady flow of a gas in a shock tube with Knudsen numbers in the initial state of 0.01 and 0.002 either side of the diaphragm (based on the length of the initial low-pressure region). The results for the hybrid method are very close to those for pure DSMC. The execution speed of the hybrid code is 1.75 times that of standard DSMC.

A particle simulation method for the BGK equation

A particle simulation method, the “relaxation time” simulation method (RTSM), is described. In RTSM the collision phase in standard DSMC is replaced by a procedure whereby some of the particle velocities in each cell at each time step are selected from an equilibrium distribution, while conserving the total energy and momentum in the cell. The remaining velocities in each cell are not changed. The number of velocities to be changed is determined from the local relaxation time, which can be derived from the cell density and temperature and any desired viscosity law. The relaxation time method is a simulation method to solve the BGK equation. RTSM is efficient compared to DSMC, and becomes more so as the collision rate increases, so RTSM appears to be a natural candidate for near continuum flows.

A macroscopic chemistry method for the direct simulation of gas flows

In most chemistry methods developed for the direct simulation Monte Carlo (DSMC) technique, chemical reactions are computed as an integral part of the collision simulation routine. In the macroscopic chemistry method developed here, the simulation of collisions and the simulation of reactions are decoupled; reactions are computed independently, after the collision routine. The number of reaction events to perform in each cell is calculated using the macroscopic reaction rates k+, k- and equilibrium constant K*, calculated from the local macroscopic flow conditions. The macroscopic method is developed here for the symmetrical diatomic dissociating gas. For each dissociation event, a single diatomic simulator particle is selected with a probability based on its internal energy, and is replaced by two atomic particles. For each recombination event, two atomic particles are selected at random, and are replaced by a single diatomic particle. The dissociation energy is accounted for by adjusting the translational thermal energies of all particles in the cell. The macroscopic method gives density profiles in agreement with experimental data for the chemical relaxation region downstream of a strong shock in nitrogen. In the non-equilibrium regions within the shock, and along the stagnation streamline of a blunt cylinder in rarefied flow, the macroscopic method gives results in excellent agreement with those obtained using the most common conventional DSMC chemistry method in which reactions are calculated during the collision routine. The number of particles per computational cell has a minimal effect on the results provided by the macroscopic method. Unlike most DSMC chemistry methods, the macroscopic method is not limited to simple forms of k+, k- and K*. Any forms may be used, and these may be any function of the macroscopic conditions. This is demonstrated by using a two-temperature rate model, and a form of K* with a number density dependence. With the two-temperature model, the macroscopic method gives densities in the post-shock chemical relaxation region that also agree with the experimental data. For a form of K^* with a number density dependence, the macroscopic method can accurately reproduce chemical recombination behavior. In a primarily dissociative flow, the number density dependence of K* has very little effect on the flow. The macroscopic method requires slightly less computing time than the most common DSMC chemistry method.

Effects of direction decoupling in flux calculation in finite volume solvers

In a finite volume CFD method for unsteady flow fluxes of mass, momentum and energy are exchanged between cells over a series of small time steps. The conventional approach, which we will refer to as direction decoupling, is to estimate fluxes across interfaces in a regular array of cells by using a one-dimensional flux expression based on the component of flow velocity normal to the interface between cells. This means that fluxes cannot be exchanged between diagonally adjacent cells since they share no cell interface, even if the local flow conditions dictate that the fluxes should flow diagonally. The direction decoupling imposed by the numerical method requires that the fluxes reach a diagonally adjacent cell in two time-steps. To evaluate the effects of this direction decoupling, we examine two numerical methods which differ only in that one uses direction decoupling while the other does not. We examine a generalized form of Pullins equilibrium flux method (EFM) [D.I. Pullin, Direct simulation methods for compressible ideal gas flow, J. Comput. Phys. 34 (1980) 231-244] which we have called the true direction equilibrium flux method (TDEFM). The TDEFM fluxes, derived from kinetic theory, flow not only between cells sharing an interface, but ultimately to any cell in the grid. TDEFM is used here to simulate a blast wave and an imploding flow problem on a structured rectangular mesh and is compared with results from direction decoupled EFM. Since both EFM and TDEFM are identical in the low CFL number limit, differences between the results demonstrate the detrimental effect of direction decoupling. Differences resulting from direction decoupling are also shown in the simulation of hypersonic flow over a rectangular body. The computational cost of allowing the EFM fluxes to flow in the correct directions on the grid is minimal.

µ-DSMC: a general viscosity method for rarefield flow

A modified DSMC method for rarefied flows is described, by which any viscosity law µ = µ(T) may be simulated. The method is simple to implement. The collision cross-section of a simple collision model, such as the hard sphere or variable hard sphere (VHS) is made to vary from cell to cell, based on the time-averaged cell temperature and the required viscosity at that temperature. The method is here demonstrated for two viscosity laws which fit experimental data better than does the hard sphere or variable hard sphere viscosity laws, but in principle the method can use the experimental data directly. The new method is tested in two different flows: high speed Couette flow and a plane 1D shock. For Couette flow, the shear stress and heat transfer, calculated from the velocity distribution, agree with the theoretical values calculated from the flow gradients and the theoretical transport coefficients. For the plane 1D shock, the new method is compared with the generalized hard sphere (GHS) model. The new method produces profiles of density and temperature within the shock which are generally indistinguishable from the GHS results except for a deviation in the Tx temperature component in a small region ahead of the shock. This deviation depends on the shock Mach number; for the worst case it is 4.6%. The deviation can be reduced by basing the imposed viscosity on the maximum component of kinetic temperature (in this case Tx) rather than the mean kinetic temperature. The new method is shown to be insensitive to the number of simulator particles used in each cell. Three translational degrees of freedom are considered here. However, because µ-DSMC is based on a hard sphere or VHS cross-section, it is compatible with the most commonly used Borgnakke-Larsen energy exchange model for translational-rotational energy exchange.

Modified Generalised Hard Sphere Collision Model for Direct Simulation Monte Carlo Calculations

The generalised hard sphere collision model (GHS) was introduced by Hassan and Hash [Physics of Fluids A, v5(3), 738-744 (1993)] and is a generalization of the Sutherland collision model suggested by Kuscer [Physica, v158, 784-800 (1989)]. Despite its superior modelling of realistic gas viscosities, compared to the Variable Hard Sphere collision model, the GHS model is rarely used because of its great computational expense compared to the VHS model. We show here how a slight modification of the GHS model makes it no more than 15% more computationally expensive than the VHS model, while retaining its superior viscosity modelling. All that is required is that the collision probability be limited for collision speeds approaching zero, rather than increase to infinity as it does for the original GHS model. A particularly simple modification is to use a Maxwell collision cross-section (equal probabilities) for collision energies less than the attractive energy of a realistic molecular model (characteristic temperature T* approximate 90 - 150 K). For temperatures above T*, the GHS viscosity is retained, while for temperatures less than T* the viscosity is slightly different from the GHS viscosity, but arguably more realistic.

Rarefied, superorbital flows in an expansion tube

This paper describes a free‐piston driven expansion tube and its instrumentation. The facility is used to generate rarefied flows at speeds of approximately 10 km/s. Although the flow in the tube itself is in the continuum regime, rarefied flow conditions are achieved by allowing the test gas to further expand as a free jet into the facilitys test section. The test flow is surveyed to provide bar‐gauge pressure measurements. Numerical simulation is then used to describe more fully the test flow properties. The flows produced are suitable for the aerodynamic testing of small models at superorbital speeds and should provide data that are suitable for the calibration of Direct Simulation Monte‐Carlo codes.

Hypersonic Flow over a Wedge with a Particle Flux Method

We have investigated the use of DSMC as a pseudo-Euler solver in the continuum limit by using a modification of Pullins Equilibrium Particle Simulation Method (EPSM). EPSM is a particle-based method which is in effect the large collision rate limit of DSMC yet requires far less computational effort. We propose a modification of EPSM, the Particle Flux Method (PFM), which is intermediate between EPSM and a conventional finite volume continuum flow solver. The total mass, momentum and energy in each cell are stored. Flux particles are created at every time step and move in free flight over a short decoupling time step, carrying mass momentum and energy between cells. The new method has been demonstrated by calculating the hypersonic flow over a wedge, for which DSMC calculations are available [Bondar, Markelov, Gimelshein and Ivanov, AIAA Paper 2004-1183 (2004)]. Because of an inherent dissipation, related to the cell size and time step, the shock was thicker than that found in the DSMC calculations, but the shock location was the same. PFM is not prohibitively expensive and has some advantages over conventional continuum based flow solvers, in terms of robustness arising from its firm basis in the physics of molecular flow.

A Computational Investigation of Inviscid Hypervelocity Flow of a Dissociating Gas Past a Cone at Incidence

Calculations have been performed for the inviscid hypervelocity flow of nitrogen past a 15 degree semi-angle sharp cone at an incidence of 30 degree, at an enthalphy sufficiently high to produuce dissociation/recombination chemistry downstream of the bow shock wave. A spatially second-order-accurate EFM (Equilibrium Flux Method) scheme for the numerical solution of the inviscid Euler equations was used, combined with the Lighthill-Freeman model of the non-equilibrium ideal dissociating gas. The computations have been sued to gain an understanding fo the interaction between the ags dynamics and the finite-rate chemistry. Inviscid flow has been considered to ensure that the only physical length scales in the flow are those associated with the chemical reactions. It was found that a chemical length scale L_s, based on the local dissociation length behind the shock on the windward plane of symmetry is an important governing parameter of the flow. However, as the flow length-scale becomes large and the flow approached the limiting case of equilibrium chemistry, L_s is not the dominant chemical length-scale, particularly in the leeward flow which contains a shock-vortex structure. A simple modelling technique has been used to determine a more appropriate length scale L_r for the leeward flow, based on the equilibrium conditions behind the leeward cross-flow shock.