Introduction
The field of low-Mach external aerodynamics and aeroacoustics, have been particularly impacted by the rapid development of Lattice-Boltzmann (LB) methods in the last five to ten years. From industrial benchmarks, these methods quickly ramped up to full scale applications: full-scale cars, full-scale aircraft engines and even full-scale aircrafts, oftentimes with outstanding results.
LB methods are particularly interesting because they are easily scalable, leading to a staggering 80 to 90\% cost reduction on industrial configurations (such as drag coefficient assessment on full-scale car). This research effort aims at extending the LB framework capabilities, initially designed for low-mach athermal flows, to answer to our objectives.
We proposed in 2018 a LB model able to accurately reproduce canonical combustion problems. It is based on a hybrid method, coupling a LB solver (for mass and momentum conservation) with a finite differences solver (FD). Further details about the model and its implementation can be found in our publications.
LB part
The LB part solves the classical equation
\begin{equation}
\label{bgk}
f_i({x}_\alpha+{c}_{i\alpha}\delta t,t+\delta t)-f_i({x}_\alpha,t)=-\frac{1}{\tau}[f_i({x}_\alpha,t)-f_i^{eq}({x}_\alpha,t)],
\end{equation}
where the third order Hermite expansion of the thermal Maxwell-Boltzmann distribution projected in discrete Gauss-Hermite space is used:
\begin{equation}
\label{eq_feq}
\begin{split}
&f_i^{eq}=f_i^{(0)}=\rho w_i\big[1+\frac{c_{i\alpha}u_\alpha}{c_s^2}+\frac{A^{(2)}_{\alpha\beta}Q^{(2)}_{i\alpha\beta}}{2c_s^4}+\frac{A^{(3)}_{\alpha\beta\gamma}Q^{(3)}_{i\alpha\beta\gamma}}{6c_s^6}\big],\\
&A^{(2)}_{\alpha\beta}=u_\alpha u_\beta+(\theta-1)c_s^2\delta_{\alpha\beta},\quad Q^{(2)}_{i\alpha\beta}=c_{i\alpha}c_{i\beta}-c_s^2\delta_{\alpha\beta},\\
&A^{(3)}_{\alpha\beta}=u_\alpha u_\beta u_\gamma+(\theta-1)c_s^2[u\delta]_{\alpha\beta\gamma}, \quad Q^{(3)}_{i\alpha\beta}=c_{i\alpha}c_{i\beta}c_{i\gamma}-c_s^2[c\delta]_{\alpha\beta\gamma},
\end{split}
\end{equation}
where $[c\delta]_{\alpha\beta\gamma}=c_\alpha \delta_{\beta \gamma}+c_\beta \delta_{\alpha \gamma}+c_\gamma \delta_{\alpha \beta}$, $\delta_{\alpha\beta}$ is the Kronecker symbol and $\theta$ is the non-dimensional temperature
\begin{equation}
\label{eq_theta}
\theta=\frac{\overline{r}T}{c^2_s}=\frac{R T}{c^2_s}\sum_k\frac{Y_k}{W_k}
\end{equation}
As is classical, macroscopic quantities are recovered as
\begin{equation}
\label{m0}
\sum_{i} f^{(0)}_i=\rho,
\end{equation}
\begin{equation}
\label{m1}
\sum_{i} f^{(0)}_i c_{i\alpha}=\rho u_{\alpha},
\end{equation}
FD part
The LB solver is coupled with a classical FD solver to account for energy and species conservation as
\begin{equation}\label{eq_Econservation}
\frac{\partial e}{\partial t}+ u_\alpha\frac{\partial e}{\partial x_{\alpha}}= -\frac{1}{\rho}\frac{\partial q_{\alpha}}{\partial x_{\alpha}} + \frac{\sigma_{\alpha\beta}}{\rho}\frac{\partial u_{\alpha}}{\partial x_{\beta}}.
\end{equation}
and
\begin{equation}\label{eq_roYi}
\frac{\partial Y_k}{\partial t}+ (u_\alpha+V_{k,\alpha}) \frac{\partial}{\partial x_{\alpha}} Y_k=\frac{\dot \omega_k}{\rho}.
\end{equation}