Three-equation model for the self-similar growth of Rayleigh-Taylor and Richtmyer-Meskov instabilities

In the present work, the two-equation $k\text{−}L$ model [G. Dimonte and R. Tipton, Phys. Fluids 18, 085101 (2006)] is extended by the addition of a third equation for the mass-flux velocity. A set of model constants is derived to satisfy an ansatz of self-similarity in the low Atwood number limit. The model is then applied to the simulation of canonical Rayleigh-Taylor and Richtmyer-Meshkov test problems in one dimension and is demonstrated to reproduce analytical self-similar growth and to recover growth rates used to constrain the model.

Figure 1

Normalized profiles obtained for a one-dimensional RT mixing layer of $A_T=0.05$ at non-dimensional time $\tau =t{(A_{T}g/{\lambda }_0)}^{1/2}= 324 (\square ),648 (◯),972 (△)$, and $1296 (\diamond )$: (a) $K^{*}=k/K_0$, (b) $L^{*}=L/L_0$, (c) ${\mu }_t^{*}={\mu }_t/\left(C_{\mu }\rho L_0\sqrt{2K_0}\right)$, (d) $a^{*}=(A_T^2-1)a_y/\left(C_B{A}_T\sqrt{2K_0}\right)$, (e) $b^{*}=(1-A_T^2)b/A_T^2$, and (f) $Y^{*}\equiv$ heavy fluid mass fraction.

Figure 2

Convergence of a one-dimensional RT mixing layer at $A_T=0.05$ with grid resolution of $200 (◯),400 (\square ),800 (△)$, and $1600 (\diamond )$ zones. (a) Solutions of $h_b,K_0$, and $L_0$. Dimensions are in cm for $h_b$ and $L_0$ and ${10}^{-9}$ (cm/${\mu \mathrm{s})}^2$ for $K_0$. (b) Time history of ${\alpha }_b=h_b/A_{T}g{t}^2$.

Figure 3

Normalized profiles obtained for a one-dimensional RT mixing layer of $A_T=0.05$, 0.20, 0.40, 0.60, and 0.80 at $\tau =t{(A_{T}g/{\lambda }_0)}^{1/2}=1296$: (a) $K^{*}=k/K_0$, (b) $L^{*}=L/L_0$, and (c) $a^{*}=(A_T^2-1)a_y/\left(C_B{A}_T\sqrt{2K_0}\right)$. Arrows indicate direction of increasing Atwood number.

Figure 4

Mixing width profiles calculated for three different shock tube experiments: ($a$) Leinov et al. experiment 1570 [18] ($h_0=0.110$ cm, test section = 23.5 cm), (b) Vetter and Sturtevant experiment 85 [19] ($h_0=0.224$ cm, test section = 60.0 cm), and (c) Vetter and Sturtevant experiment 87 [19] ($h_0=0.283$ cm, test section = 49.0 cm). Symbols $(◯)$ indicate experimental data. Dashed lines are power law profiles fit to the simulation data.