Two-point spectral model for variable-density homogeneous turbulence

We present a study of buoyancy-driven variable-density homogeneous turbulence, using a two-point spectral closure model. We compute the time-evolution of the spectral distribution in wave number $k$ of the correlation of density and specific volume $b\left(k\right)$, the velocity associated with the turbulent mass flux $\mathit{a}\left(k\right)$, and the turbulent kinetic energy $E\left(k\right)$, using a set of coupled equations. Under the modeling assumptions, each dynamical variable has two coefficients governing spectral transfer among modes. In addition, the velocity $\mathit{a}\left(k\right)$ has two coefficients governing the drag between the two fluids. Using a prescribed initial condition for $b\left(k\right)$ and starting from a quiescent flow, we first evaluate the relative importance of the different coefficients used to model this system and their impact on the statistical quantities. We next assess the accuracy of the model, relative to direct numerical simulation of the complete hydrodynamical equations, using $b,\mathit{a}$, and $E$ as metrics. We show that the model is able to capture the spectral distribution and global means of all three statistical quantities at both low and high Atwood number for a set of optimized coefficients. The optimization procedure also permits us to discern a minimal set of four coefficients which are sufficient to yield reasonable results while pointing to the mechanisms that dominate the mixing process in this problem.

Figure 1

Convergence of outcomes at increasing resolution for initial Gaussian distribution for $b\left(k\right)$, and $a_y\left(k\right),E\left(k\right)$ set to zero at fixed viscosity $\nu ={10}^{-4}$. (a) Time evolution plots of the turbulent kinetic energy $E\left(t\right)$ at different system resolutions and (b) kinetic energy spectra at different resolutions.

Figure 2

Comparison of results from codes using Crank-Nicolson (blue line with circles) and MacCormack (orange line) schemes for time integration. Plots showing time evolution of (a) the mean density-specific volume covariance $b\left(t\right)$, (b) $a\left(t\right)$, and (c) the turbulent kinetic energy $E\left(t\right)$.

Figure 3

Time evolution plots of (a) $b\left(t\right)$, (b) $a\left(t\right)$, and (c) $E\left(t\right)$ for different values of $C_{b1}$. Time evolution plots of (d) $b\left(t\right)$, (e) $a\left(t\right)$, and (f) $E\left(t\right)$ for different values of $C_{b2}$. The values of the parameters which do not vary in each case are shown in Table 1. The number of $k$ modes used is 512.

Figure 4

Time evolution plots of (a) $b\left(t\right)$, (b) $a\left(t\right)$, and (c) $E\left(t\right)$ for different values of $C_{a1}$. Time evolution plots of (d) $b\left(t\right)$, (e) $a\left(t\right)$, and (f) $E\left(t\right)$ for different values of $C_{a2}$. The values of the parameters which do not vary in each case are shown in Table 1.

Figure 5

Time evolution plots of (a) $b\left(t\right)$, (b) $a\left(t\right)$, and (c) $E\left(t\right)$ for different values of $C_{r1}=2C_{r2}$. The values of the other parameters are shown in Table 1.

Figure 6

Time evolution plots of (a) $b\left(t\right)$, (b) $a\left(t\right)$, and (c) $E\left(t\right)$ for different values of $C_{rp1}$. Time evolution plots of (d) $b\left(t\right)$, (e) $a\left(t\right)$, and (f) $E\left(t\right)$ for different values of $C_{rp2}$. The values of the parameters which do not vary in each case are shown in Table 1.

Figure 7

3D visualization of the density field for the $At=0.05$ DNS run $({1024}^3$ mesh) at (a) initial time and (b) turbulent kinetic energy peak time as described in Ref. [45].

Figure 8

Plots of ${\chi }^2$ function calculated using Eq. (26) for different values of $C_{rp1},C_{rp2},C_{b1},C_{b2},C_{a1},C_{a2}$, and $C_{r1}$. This minimum value of the ${\chi }^2$ function occurs at the optimum value of each variable.

Figure 9

Time evolution of flow with initial $At=0.05$. (a) Mean density-specific volume covariance $b\left(t\right)$, (b) mean $a\left(t\right)$, and (c) turbulent kinetic energy $E\left(t\right)$ obtained from the results of the DNS calculations (blue line with circles) and from the spectral model [R1(orange line) and R2 (green line)] with parameter values listed in Table 3. DNS runs in this case are at a resolution of ${1024}^3$ (the viscosity is ${10}^{-4})$.

Figure 10

Spectra of flows with initial $\text{At}=0.05$. Plots of $b\left(k\right)$ versus the wave number $k$ at times (a) $t=0.8$, (b) $t=3.2$, (c) $t=6.4$, and (d) $t=14$. $a_y\left(k\right)$ versus the wave number $k$ at times (e) $t=0.8$, (f) $t=3.2$, (g) $t=6.4$, and (h) $t=14$. Energy $E\left(k\right)$ versus the wave number $k$ at times (i) $t=0.8$, (j) $t=3.2$, (k) $t=6.4$, and (l) $t=14$. The parameters for these data are given in Table 3 with the corresponding DNS data resolution at ${1024}^3$.

Figure 11

Time evolution of flow with initial $\text{At}=0.75$. (a) Mean density-specific volume covariance $b\left(t\right)$, (b) mean $a\left(t\right)$, and (c) turbulent kinetic energy $E\left(t\right)$ obtained from the results of the DNS calculations (blue line with circles) and from the spectral model code (orange line) with parameter values listed in Table 3 (run R1), and the corresponding DNS resolution is ${1024}^3$ (the viscosity is ${10}^{-4})$.

Figure 12

Spectra of flows with initial $At=0.75$. Plots of $b\left(k\right)$ versus the wave number $k$ at times (a) $t=0.2$, (b) $t=1.8$, (c) $t=3.0$, and (d) $t=5.0$. $a_y\left(k\right)$ versus the wave number $k$ at times (e) $t=0.2$, (f) $t=1.8$, (g) $t=3.0$, (h) $t=5.0$. Energy $E\left(k\right)$ versus the wave number $k$ at times (i) $t=0.2$, (j) $t=1.8$, (k) $t=3.0$, (l) $t=5.0$. The parameters for these data are given in Table 3 with the corresponding DNS data resolution at ${1024}^3$.