Acoustic equations of state for simple lattice Boltzmann velocity sets

The lattice Boltzmann (LB) method typically uses an isothermal equation of state. This is not sufficient to simulate a number of acoustic phenomena where the equation of state cannot be approximated as linear and constant. However, it is possible to implement variable equations of state by altering the LB equilibrium distribution. For simple velocity sets with velocity components ${\xi }_{i\alpha }\in \{-1,0,1\}$ for all $i$, these equilibria necessarily cause error terms in the momentum equation. These error terms are shown to be either correctable or negligible at the cost of further weakening the compressibility. For the D1Q3 velocity set, such an equilibrium distribution is found and shown to be unique. Its sound propagation properties are found for both forced and free waves, with some generality beyond D1Q3. Finally, this equilibrium distribution is applied to a nonlinear acoustics simulation where both mechanisms of nonlinearity are simulated with good results. This represents an improvement on previous such simulations and proves that the compressibility of the method is still sufficiently strong even for nonlinear acoustics.

Figure 1

Comparison of the LB simulation of nonlinear sound propagation (solid line) with a corresponding Burgers equation solution (dashed line) at various nondimensionalized times $\overline{t}=t/t_{\mathrm{shock}}$. Note the spurious oscillation at the upper edge of the shock for $\overline{t}=1.5$.

Figure 2

Evolution of the amplitudes of the $n$th harmonics up to $\overline{t}=1.5$ from the fundamental $n=1$ (top, darkest line) to $n=6$ (bottom, lightest line) for the LB simulation (solid line) and the Burgers equation (dashed line).