Two-length-scale turbulence model for self-similar buoyancy-, shock-, and shear-driven mixing

The three-equation $k\text{−}L\text{−}a$ turbulence model [B. Morgan and M. Wickett, Three-equation model for the self-similar growth of Rayleigh-Taylor and Richtmyer-Meshkov instabilities, Phys. Rev. E 91, 043002 (2015)] is extended by the addition of a second length scale equation. It is shown that the separation of turbulence transport and turbulence destruction length scales is necessary for simultaneous prediction of the growth parameter and turbulence intensity of a Kelvin-Helmholtz shear layer when model coefficients are constrained by similarity analysis. Constraints on model coefficients are derived that satisfy an ansatz of self-similarity in the low-Atwood-number limit and allow the determination of model coefficients necessary to recover expected experimental behavior. The model is then applied in one-dimensional simulations of Rayleigh-Taylor, reshocked Richtmyer-Meshkov, Kelvin-Helmholtz, and combined Rayleigh-Taylor–Kelvin-Helmholtz instability mixing layers to demonstrate that the expected growth rates are recovered numerically. Finally, it is shown in the case of combined instability that the model predicts a mixing width that is a linear combination of Rayleigh-Taylor and Kelvin-Helmholtz mixing processes.

Figure 1

Mixing layer growth in a one-dimensional RT mixing layer. (a) Mixing layer half width, turbulence kinetic energy, and turbulence length scales are plotted as a function of time squared for both the $k\text{−}L\text{−}a$ and $k\text{−}2L\text{−}a$ models. (b) The RT growth parameter, ${\alpha }_b=h/\left(Ag{t}^2\right)$, is plotted as a function of nondimensional time for both the $k\text{−}L\text{−}a$ and $k\text{−}2L\text{−}a$ models.

Figure 2

Normalized self-similar profiles in a one-dimensional RT mixing layer at $A=0.05$: (a) heavy species mass fraction $Y_H$, (b) turbulence kinetic energy $k$, (c) mass-flux velocity $a_y$, and (d) density–specific-volume covariance $b$ profiles, for the $k\text{−}L\text{−}a$ and $k\text{−}2L\text{−}a$ models compared with LES results [48].

Figure 3

Normalized self-similar profiles of turbulence length scales in a one-dimensional RT mixing layer are compared between the $k\text{−}L\text{−}a$ and $k\text{−}2L\text{−}a$ models.

Figure 4

Mixing width as a function of time for the Vetter and Sturtevant shock tube experiment [19]. Symbols indicate experimental data, while the solid line indicates simulation results with the $k\text{−}2L\text{−}a$ model.

Figure 5

Self-similar profiles and mixing layer evolution of a KH shear layer simulated with the $k\text{−}L\text{−}a$ model for several choices of ${\mathrm{\Phi }}^{-1}$: (a) self-similar velocity profile, (b) steady-state turbulence kinetic energy profile, (c) mixing layer half width as a function of time, and (d) mixing layer growth parameter as a function of time. For all $k\text{−}L\text{−}a$ results, $C_{L2t}=C_{L2d}$, where $C_{L2d}$ is determined by Eq. (44) except for the baseline KLA results, in which $C_{L2t}=C_{L2d}=0.0$. Experimental data are taken from Bell and Mehta [18].

Figure 6

Self-similar profiles and mixing layer evolution of a KH shear layer simulated with the $k\text{−}2L\text{−}a$ model for several choices of $\delta /\mathcal{A}$: (a) self-similar velocity profile, (b) steady-state turbulence kinetic energy profile, (c) mixing layer half width as a function of time, and (d) mixing layer growth parameter as a function of time. For all $k\text{−}2L\text{−}a$ results, ${\mathrm{\Phi }}^{-1}=0.035$, and $C_{L2t}$ is determined by Eq. (48). Experimental data are taken from Bell and Mehta [18].

Figure 7

KH mixing layer half width and growth parameter for several choices of $\delta /\mathcal{A}$ simulated using the $k\text{−}2L\text{−}a$ model with the stress-limiter modification described in Appendix pp2. For all simulation results, ${\mathrm{\Phi }}^{-1}=0.035$, and $C_{L2t}$ is determined by Eq. (48). Experimental growth rate is taken from Bell and Mehta [18].

Figure 8

The ratio of length scales $L_d/L_t$ across a KH shear layer is plotted for several choices of $\delta /\mathcal{A}$. For each case, ${\mathrm{\Phi }}^{-1}=0.035$ and $C_{L2t}$ is determined by Eq. (48).

Figure 9

Self-similar profiles and mixing layer evolution of variable-density KH shear layers simulated with the $k\text{−}2L\text{−}a$ model for several choices of ${\rho }_1/{\rho }_2$: (a) self-similar velocity profile, (b) steady-state turbulence kinetic energy profile, (d) mixing layer half width as a function of time, and (d) mixing layer growth parameter as a function of time. For all cases, the expected values are ${\mathrm{\Phi }}^{-1}=0.035$ and $\delta /\mathcal{A}=0.08$.

Figure 10

Richardson number as a function of time for several $k\text{−}2L\text{−}a$ simulations of a combined RT-KH mixing layer with varying levels of gravitational acceleration. Acceleration is indicated in terms of a multiplier on Earth's gravity such that $g=G{g}_e$.

Figure 11

Four measures of mixing layer and turbulence intensity evolution for $k\text{−}2L\text{−}a$ simulations of combined RT-KH instability with increasing gravitational acceleration: (a) mixing layer half width as a function of time, (b) turbulence intensity ${\mathrm{\Phi }}^{-1}$ as a function of time, (c) nondimensional growth rate $\dot{h}/\mathrm{\Delta }U$ as a function of time, and (d) nondimensional growth rate as a function of Richardson number.

Figure 12

The ratio of length scales $L_d/L_t$ at four different times across a combined RT-KH mixing layer for $G=10$.