The direct simulation Monte Carlo method

As an alternative to deterministic methods for the solution of the Boltzmann equa- tion, let us consider the stochastic particle-based simulation method known as the direct simulation Monte Carlo (DSMC) method [Bird, 1994]. It is widely regarded as a reliable, flexible, and efficient method (in the case of multi-dimensional flows) for the direct simulation of the Boltzmann equation, with other rarefied gas models frequently using DSMC as a benchmark for comparison. A large number of repre- sentative molecules (N) are tracked through a computational mesh as they interact with walls and each other. These representative molecules (which may be referred to as DSMC particles) represent a number of real molecules. This o↵ers a degree of control over the computational intensity of the method, but, as will be discussed in the following, also a↵ects the statistical uncertainty in sampled quantities.

The DSMC method decouples the transport and collision of particles, e↵ec- tively simulating the left- and right-hand sides of equation (1.3) in two separate steps, for a time step t. Assuming that F = 0, the particles are translated sim- ply according to x(t+ t) = x(t) +c(t) t. This step also involves generation or deletion of particles (if applicable), and particle-surface/boundary interaction. The type of surface (impermeable boundary) dictates the behaviour of a reflecting particle, and commonly we deal with either specular or di↵use reflection. The post- collision velocities of a di↵usely reflected particle are sampled from an equilibrium (or Maxwellian) distribution function based on the temperatureT of the surface.

Following the transport step, pairs of particles within the same computa- tional cell are randomly selected to collide. Therefore, the spatial discretisation of the physical domain is determined as a function of , since it would be physically inaccurate to allow particles separated by many mean free paths to collide. In gen- eral, and for the simulations presented in this thesis, the maximum cell dimension

x = /3. Similarly, the mean free time (¯⌧ = /¯cr, where ¯cr is the mean (rela-

tive) speed of collisions) limits t(_⌧⌧¯), as particles should be advected without colliding. It has been rigorously shown that as N ! 1 and the discretisation pa- rameters tend to zero ( t_!0 and x_!0), the DSMC method provides solutions to the Boltzmann equation [Wagner, 1992]. A computational mesh is also required for the statistical sampling of macroscopic quantities; once a steady-state condition is achieved, the appropriate microscopic properties are averaged over a number of time steps to obtain low-noise cell-centred values of macroscopic properties. If the problem is unsteady, an ensemble average (the number of realisations required to obtain the desired fractional error in macroscopic properties has been characterised by Hadjiconstantinou et al. [2003]) is required to improve estimates.

In dilute gases, intermolecular collisions are assumed to always be binary. The exact dynamics of a collision are dependent on the molecular model selected, with those that are more realistic varying the collision cross section T as a function

of the relative speed (cr) between colliding molecules (which includes monatomic,

diatomic, and polyatomic molecules). Figure 1.6 shows the e↵ective cylinder of volume Tcr t that is formed by sweeping the collision cross section along the

trajectory of a particle. During the interval t, a collision will occur if the centre of any other particle exists within the cylinder; the probability that a collision will occur between two molecules is P = W Tcr t/Vcell, where W is the number of

real molecules represented by a DSMC particle, andVcell is the volume of a cell.

All DSMC simulations presented in this thesis make use of the variable hard sphere (VHS) model, where the molecular diameterdis a function ofcr [Bird, 1994]:

d=dref. cr,ref./cr ! 1_/2

. (1.10)

The subscript ‘ref.’ denotes reference values, and ! is the viscosity index. This model is able to reproduce the correct temperature dependence of viscosity:

µ=µref. T /Tref. ! , (1.11)

where µ is the dynamic viscosity. For a thorough review of the various molecular models, and the DSMC method, the reader is referred to Bird [1994, 2013].

Figure 1.6: Over an interval of time which is much shorter than the mean colli- sion time, assume that the molecule of interest moves through a field of stationary molecules with speed cr. A collision will occur if the centre of any field molecule

Statistical uncertainty

As macroscopic properties are sampled by averaging microscopic properties, the DSMC method is inherently statistical. Hadjiconstantinou et al. [2003] present predictions for the statistical uncertainty due to finite sampling in the presence of thermal fluctuations, and show that relative statistical error can be expressed as

✏w = w U = 1 p Nsam.Ncell 1 Map, (1.12)

where w is the standard deviation of the streamwise velocity w, U is the charac-

teristic velocity,Nsam.is the number of number of statistically independent samples

required to obtain an estimate ofw,Ncell is the number of computational particles

per cell, and is the adiabatic index. Therefore, the computational intensity asso- ciated with estimating w with a relative uncertainty ✏w using the DSMC method

scales as (✏wMa) 2; as Ma! 0 the DSMC method becomes intractable. This has

led researchers to develop computationally efficient (variance-reduced) methods for the simulation of low-speed (or more generally, low-signal) rarefied gas flows.