Rotational symmetry of the multiple-relaxation-time collision model

We point out that the minimal components of the tensorial moments of the distribution that can be independently relaxed in collision without violating rotational symmetry are its irreducible representation (irrep) of SO(3), and a generic multiple-relaxation-time collision model can be constructed by independently relaxing these components. As the simplest example the irreps of the second moment are the traceless deviatoric stress and an isotropic tensor which is conserved in monatomic gases. Applying the decomposition to the thermal lattice Boltzmann model for polyatomic gases [Phys. Rev. E 77, 035701(R) (2008)], the shear and bulk viscosities are decoupled by two independent relaxation times. The hydrodynamic equation of the model is obtained via Chapman-Enskog calculation and verified by numerical simulation.

Figure 1

Time history of the amplitude of pressure perturbation in acoustic waves with the varying ratio of bulk to shear viscosities. The solid black, dotted black, dashed black, solid red, and solid blue curves denote the cases with ${\nu }_b/\nu =100,500,1000,1500,2000$, respectively. The other parameters are ${\tau }_{21}=0.5005,\gamma =1.3,Pr=2.1$, and ${\tau }_1={\tau }_3$.

Figure 2

Relative error of the decay rate with the varying ratio of bulk viscosity to shear viscosity. Two cases are tested: ${\tau }_{21}=0.5005$ and ${\tau }_{21}=0.6$. In both cases $\gamma =1.3,\mathrm{Pr}=2.1$, and ${\tau }_1={\tau }_3$. To be seen is that the relative error is generally below 1% over a large range of ${\nu }_b/\nu$.

Figure 3

Sound speed: analytical solution $c_s=\sqrt{\gamma {\theta }_0}$ versus numerical measurement with varying specific heat ratio. Parameters are $\mathrm{Pr}=2.1,{\tau }_{21}=0.501,{\nu }_b/\nu =100$, and ${\tau }_1={\tau }_3$.