Liouville models of particle-laden flow

Abstract

Langevin (stochastic differential) equations are routinely used to describe particle-laden flows. They predict Gaussian probability density functions (PDFs) of a particle's trajectory and velocity, even though experimentally observed dynamics might be highly non-Gaussian. Our Liouville approach overcomes this dichotomy by replacing the Wiener process in the Langevin models with a (small) set of random variables, whose distributions are tuned to match the observed statistics.

Figures (8)

• Figure 1: Particle position PDF $f_X$ at times $t=[0,\ 0.05, \ 0.8]$ given by \ref{['eq: heat_kernel_solution_general']} with $D=2.5$ and $u_\text{p}=5$ for normal and triangular distributions of the parameter $\Xi$. The uniform distribution has been omitted for clarity (see Fig. \ref{['fig: PL_fx_unif']}).
• Figure 2: Temporal evolution of $f_X(x_\text{p};t)$, PDF of the particle trajectory $X_\text{p}(t)$, predicted by the Liouville solution \ref{['eq: heat_kernel_solution_general']} for (a) normal, (b) uniform and (c) triangular distributions of the parameter $\Xi$. The deterministic parameters are set to $D=2.5$ and $u_\text{p}=5$.
• Figure 3: Temporal evolution of $f_X(x_\text{p};t)$ and $f_U(u_\text{p};t)$, PDFs of the particle trajectory $X_\text{p}(t)$ and velocity $U_\text{p}(t)$, predicted by the Liouville solutions \ref{['eq: velocity_solution_general_f_xp']} and \ref{['eq: velocity_solution_general_f_up']} for normal (left column), uniform (middle column) and triangular (right column) distributions of the parameter $\Xi$. The deterministic parameters are set to $\tau_\text{p}=10$, $v_0=1$ and $D = 1/50$.
• Figure 4: Temporal evolution of the thermal Reynolds number $Re_T(t)$ alternatively predicted with our model \ref{['eq: FHHS_mean_up_and_sigma_up']} (solid lines with square symbol), the model of Ref.lattanzi2022fluid (dotted lines with triangle symbol) and the Langevin approach of Ref.lattanzi2022stochastic (dashed lines with diamond symbol), for several values of the mean Reynolds number $\mathit{Re}_\text{m}$. The open circles indicate the PR-DNS data tenneti2016stochasticlattanzi2022stochastic, to which $\sigma_\Xi = \sigma_\Xi(\mathit{Re}_\text{m})$ and $\mathcal{C}_1 = \mathcal{C}_1(\mathit{Re}_\text{m})$ are fitted (Fig. \ref{['fig: PR_DNS_FHCS_sigma_alpha']}). The average volume fraction and density ratio are $\omega=0.1$ and $\rho_\text{p}/\rho_\text{f}=100$ respectively. For all cases $\mathcal{C}_2=1.2$ and $\tau_\text{p} = 0.14$.
• Figure 5: Functional dependencies of the model parameters $\sigma_{\Xi}$ and $\mathcal{C}_1$ on the mean Reynolds number $\mathit{Re}_\text{m}$. The squares and circles indicate the values of $\sigma_{\Xi}$ and $\mathcal{C}_1$ obtained via fitting \ref{['eq: FHHS_mean_up_and_sigma_up']} to the PR-DNS data tenneti2016stochastic for $\rho_\text{p}/\rho_\text{f}=100$ and $\omega = 0.1$. These data are represented by $\log(\sigma_{\Xi}) = 0.06258 \log(\mathit{Re}_\text{m}) + \log(0.7866)$ and $\log(\mathcal{C}_1) = 2.446 \log(\mathit{Re}_\text{m}) + \log(0.5411)$, with the coefficient of determination $R^2=0.998$.