A fast simulation method with arbitrary viscosity law

A new approach to DSMC collision modelling, called viscosity-DSMC or μ-DSMC, is described in which the time-averaged temperature is used to set the characteristic collision cross-section in each cell such that the Chapman-Enskog viscosity is that given by any desired viscosity law μ = μ (T ), including a curve fit to experimental data. For example, a hard sphere collision model, with hard sphere collision probability, used with a different molecular size in each cell can reproduce a Sutherland viscosity law. Similarly, a variable hard sphere collision model can reproduce the viscosity given by the more complicated generalized hard collision model, by making the reference cross-section a function of the temperature. This model is used to calculate the structure of a plane 1D shock and the results agree closely with those from standard DSMC using the GHS model. A particularly simple method is to use the Maxwell VHS model, in which all collision pairs are equally likely, to produce any desired viscosity law. The time-averaged cell temperature is available in standard DSMC as part of the procedures which determine the steady state flow and the new methods are as fast as, or faster than standard DSMC. Unlike more complicated models with realistic viscosities, the new procedures are compatible with the Borgnakke-Larsen energy exchange scheme and the established chemistry models for DSMC. VISCOSITY OF REAL GASES AND THE VHS MODEL The most commonly used DSMC collision model is the variable hard sphere (VHS) which gives rises to a power law viscosity μ ∝ T ω . Although this viscosity law is reasonably accurate over a limited range of temperature for a particular gas it does not represent a realistic viscosity over all temperatures. For T < 2000 K, the viscosity of a typical gas displays a significant deviation from a simple power law as a result of the long-range attractive forces between molecules. Fig. 1 shows the viscosity of argon compared with the two extremes of the power law, ω = 2 , corresponding to hard spheres with constant cross-section, and ω = 1 corresponding to a ‘Maxwell molecule’. Also shown is the Sutherland viscosity, μ/μ1 = (T/T1) (1+Ts/T1)/(1+Ts/T ), (1) where Ts = 142 K and μ1 = μ (T1) which fits the data better than any power law. When using the VHS model in any particular DSMC calculation one can determine the range of temperatures expected in the flow and choose VHS parameters which match the viscosity reasonably well. However it would be better to have a collision model which fitted the experimental data over all temperatures. Realistic potentials, such as the Morse potential [2], the Lennard-Jones potential [3, 4] and the Maitland-Smith potential [5] have been used in DSMC as have other collision models [6, 7] which produce realistic viscosity laws, but these are not widely used. This is partly because of their relative complexity, but also because of the difficult of using the Borgnakke-Larsen energy exchange scheme [8] when the collision probability does not match that of the VHS collision model. The combination of the Borgnakke-Larsen (BL) exchange model and the VHS model is now the de facto standard of DSMC. Similarly, the DSMC procedures for chemically reacting flow [9] are built around the VHS model. Here I show how an arbitrary viscosity law μ = μ (T ) can be realized in DSMC using simple collision models which are compatible with the BL exchange scheme. This is good not only in itself, but also because it makes the construction of hybrid DSMC/Navier-Stokes solvers easier; the viscosity law in both the DSMC and Navier-Stokes code can be the best available, rather than that dictated by computational practicality. The new method runs as fast, or faster than the standard VHS collision model. The new method, called μ-DSMC, is to adjust the size of any simple collision model based on the local temperature to produce any desired viscosity at that temperature. We use the hard sphere and VHS collision models as the basis of the new method and in each case produce a viscosity law different from the usual one for that collision model. The method has been tested in Couette flow and for highly non-equilibrium flow in the interior of a shock and has been shown to produce essentially the same results as DSMC using more complicated collision models. A particular form of the general method, based on the Maxwell limit of VHS, is described and compared with standard DSMC in a zero-dimensional velocity relaxation calculation and the supersonic flow around a blunt-faced cylinder. In all cases the new method agrees well with standard DSMC; the deviations are small and confined to small regions of the flow. In all cases shown here a monatomic gas, with γ = 5/3 has been used. HARD SPHERE MODEL WITH SUTHERLAND VISCOSITY The Chapman-Enskog viscosity for hard spheres, with diameter d is μ = 5m 16 (πRT ) σ , where σ = πd2 is the the total collision cross-section and R = k/m is the ordinary gas constant. A given viscosity law μ = μ (T ) can be implemented by setting the cross-section in each cell as σ (T̄ ) = 5m 16 (πRT̄ ) μ (T̄ ) , (2) where T̄ is the time-averaged translational kinetic temperature in the cell. This method has been tested for supersonic Couette flow in Ref. [10], using the Sutherland viscosity (Eq. 1) for μ in Eq. 2. The measured shear stress τmeas = ρcxcy, was determined from the steady-state velocity distribution and compared with the theoretical shear stress τt = μsuthdux/dy, where μsuth is the Sutherland viscosity evaluated for the cell temperature. The ratio τmeas/τt in Fig. 2, is close to unity across most of the flow except in the Knudsen layer near the wall. The figure also shows that μmeas/μt is significantly different from unity when μt is calculated using the hard sphere viscosity; in other words the hard sphere and Sutherland viscosities are significantly different for these temperatures. Thus, the hard sphere collision model with a different collision cross-section in each cell displays the Sutherland law as required.