Local wave-number model for inhomogeneous two-fluid mixing

We analyze the local wave-number (LWN) model, a two-point spectral closure model for turbulence, as applied to the Rayleigh-Taylor (RT) instability, the flow induced by the relaxation of a statically-unstable density stratification. Model outcomes are validated against data from 3D simulations of the RT instability. In the first part of the study we consider the minimal model terms required to capture inhomogeneous mixing and show that this version, with suitable model coefficients, is sufficient to capture the evolution of important mean global quantities including mix-width, turbulent mass flux velocity, and Reynolds stress, if the start time is chosen such that the earliest transitions are avoided. However, this simple model does not permit the expected finite asymptote of the density-specific-volume covariance $b$. In the second part of the study, we investigate two forms for a source term for the evolution of the spectrum of density-specific-volume covariance for the LWN model. The first includes an empirically motivated calibration of the source to achieve the final asymptotic state of constant $b$. The second form does not require calibration but, in conjunction with enhanced diffusion and drag captures the full evolution of all the dynamical quantities, namely, the mix-layer growth, turbulent mass-flux velocity, Reynolds stress, as well as the desired behavior of $b$.

Figure 1

Visualization of the density field in the MOBILE simulation run R3 at times (a) $\tau =0$; (b) $\tau =0.42$; (c) $\tau =3.164$; (d) $\tau =5.19;$ (e) $\tau =6.5$; (f) schematic of the LWN system, with a slice-through visualization from 3D simulations at $\tau =5.19$. LWN variables such as $R_{nn}(y,\tau )=\int {\widehat{R}}_{nn}(y,k,\tau )dk,a_y(y,\tau )=\int {\widehat{a}}_y(y,k,\tau )dk$ and $b(y,\tau )=\int \widehat{b}(y,k,\tau )dk$ at $\tau =5.19$ are shown in the same plot, and these plots show that the maximum value of all these variables occur at the center-plane of the RT system. The horizontal length $L_x=2\pi$ cm and the vertical length $L_y=8\pi$ cm. $R_{nn}(y,\tau )$ has units of $\mathrm{g}{\mathrm{cm}}^{-1}{\mathrm{s}}^{-2}, a_y(y,\tau )$ has units of $\mathrm{cm}{\mathrm{s}}^{-1}, g$ has units of $\mathrm{cm}{\mathrm{s}}^{-2}$ and ${\rho }_1,{\rho }_2,\overline{\rho }$ has units of $\mathrm{g}{\mathrm{cm}}^{-3}$. $b(y,\tau )$ is a dimensionless quantity

Figure 2

(a) The mix-width $W\left(\tau \right)$ (in cm); (b) mean $R_{nn}(y=0,\tau )$ (in $\mathrm{g}{\mathrm{cm}}^{-1}{\mathrm{s}}^{-2}$) for M1, M2, M3, and M4; (c) relative error of $R_{nn}(y=0,\tau )$ for M1 and M2 with respect to $R_{nn}(y=0,\tau )$ for M3. showing convergence as resolution is increased. The system coefficients are given in the first three rows of Table 2.

Figure 3

(a) Visualization of population of 2D-spectral modes in the $k_x\text{−}{k}_z$ plane at the centerline of the domain, (b) amplitude $h_0(x,y)$ (in $\mathrm{cm}$) in physical space; (c) time-evolution of mix-width $W\left(\tau \right)$ (in $\mathrm{cm}$) obtained from the MOBILE data showing the exponential and nonlinear growth stages; the black dotted line shows the start time for LWN calculations.

Figure 4

Initial spectral data for LWN taken from MOBILE R1 at $\tau =0.42$ (blue line): (a) $\widehat{b}(y=0,k)$ [cm], (b) ${\widehat{a}}_y(y=0,k)$ [${\mathrm{cm}}^2{\mathrm{s}}^{-1}$], and (c) spectra for ${\widehat{R}}_{nn}(y=0,k)$ [${\mathrm{g}\mathrm{s}}^{-1}$]. The spectral data are taken at the central line: $y=0$.

Figure 5

(a) Mix-width (in $\mathrm{cm}$) from the MOBILE data (R1, blue dashed line) and that from the LWN model calculations (T1 orange, T2 green, T3 red, T4 purple); (b) $b(y=0,\tau )$; (c) $a_y(y=0,\tau )$ (in $\mathrm{cm}{\mathrm{s}}^{-1}$); (d) $R_{nn}(y=0,\tau )$ (in $\mathrm{g}{\mathrm{cm}}^{-1}{\mathrm{s}}^{-2}$).

Figure 6

Profiles of (a) $b(y,\tau )$, (b) $a_y(y,\tau )$ (in $\mathrm{cm}{\mathrm{s}}^{-1}$), and (c) $R_{nn}(y,\tau )$ (in $\mathrm{g}{\mathrm{cm}}^{-1}{\mathrm{s}}^{-2}$) at time $\tau =5.3$ for the MOBILE data (R1, blue dashed line) and that from the LWN model calculations (T1 orange, T2 green, T3 red, T4 purple).

Figure 7

(a) Mix-width (in $\mathrm{cm}$) from the MOBILE data (R1, blue dashed line) and that from the LWN model calculations (T1 orange, T2 green, T3 red, T4 purple) (b) comparison of mix-width growth rate $\alpha \left(\tau \right)$ for the case $C_{rp1}=0.5$ in (a); (c) $b(y=0,\tau )$; (d) $a_y(y=0,\tau )$ (in $\mathrm{cm}{\mathrm{s}}^{-1}$); (e) $R_{nn}(y=0,\tau )$ (in $\mathrm{g}{\mathrm{cm}}^{-1}{\mathrm{s}}^{-2}$).

Figure 8

Results of comparison between MOBILE calculation (R1, dashed lines) and that from the LWN model calculation (T7, solid lines) at center-plane, and at distances of 6%$L_y$ and 10%$L_y$ from the center. Plots of (a) $b(y,\tau ),$ (b) $a_y(y,\tau )$ (in $\mathrm{cm}{\mathrm{s}}^{-1}$), and (c)$R_{nn}(y,\tau )$ (in $\mathrm{g}{\mathrm{cm}}^{-1}{\mathrm{s}}^{-2}$).

Figure 9

Results of comparison between MOBILE data R3 (dashed line) and LWN calculations (T7, solid lines) (a) Mix-width (in $\mathrm{cm}$) comparison; results at center-plan and at distances of 6%$L_y$ and 10%$L_y$ from the center of (b) $b(y,\tau ),$ (c) $a_y(y,\tau )$ (in $\mathrm{cm}{\mathrm{s}}^{-1}$), and (d) $R_{nn}(y,\tau )$ (in $\mathrm{g}{\mathrm{cm}}^{-1}{\mathrm{s}}^{-2}$).

Figure 10

(a) Plots of $\frac{b(y=0)}{\text{max}\left[b\right(y=0\left)\right]}$ for different Atwood numbers from MOBILE data R1, R2, R3. (b) Comparison of $b(y=0,\tau )$ among results from MOBILE data R1 (blue dashed line), modified LWN (orange line with circles), and LWN (green line). (b) Comparison of $a_y(y=0,\tau )$ (in $\mathrm{cm}{\mathrm{s}}^{-1}$) for the same runs.

Figure 11

The evolution of $b(y,\tau )$ at the the center-plane and at two distances away from the centerplane, computed using a source term as in Eq. (33), for two different start times, (a) $\tau =0.42$ (run PT1, Table 4) and (b) $\tau =0.0$ (run PT2, Table 4).

Figure 12

The evolution of mix-layer (in $\mathrm{cm}$), for two different start times, (a) $\tau =0.42$ (run PT1, Table 4) and (b) $\tau =0.0$ (run PT2, Table 4).

Figure 13

Results of comparison between MOBILE data R1 (dashed line) and LWN calculations (run PT3 in Table 4, solid lines) (a) Mix-width (in $\mathrm{cm}$) comparison; results at center-plane and at distances of 6%$L_y$ and 10%$L_y$ from the center of (b) $b(y,\tau ),$ (c) $a_y(y,\tau )$ (in $\mathrm{cm}{\mathrm{s}}^{-1}$), and (d) $R_{nn}(y,\tau )$ (in $\mathrm{g}{\mathrm{cm}}^{-1}{\mathrm{s}}^{-2}$).

Figure 14

(a) Comparison of $a_y(y=0,\tau )$ (in $\mathrm{cm}{\mathrm{s}}^{-1}$) among results from MOBILE data (blue dashed line), and LWN runs with $C_d=0.5$ (orange line), $C_d=1.0$ (green line), $C_d=1.5$ (red line), and $C_d=2.0$ (purple line). All other coefficients are same as in run T7. (b) Comparison of $R_{nn}(y=0,\tau )$ (in $\mathrm{g}{\mathrm{cm}}^{-1}{\mathrm{s}}^{-2}$) for these runs.