Turbulent transport and mixing in transitional Rayleigh-Taylor unstable flow: A priori assessment of gradient-diffusion and similarity modeling

Data from a $1152\times 760\times 1280$ direct numerical simulation [N. J. Mueschke and O. Schilling, Phys. Fluids 21, 014106 (2009)] of a Rayleigh-Taylor mixing layer modeled after a small-Atwood-number water-channel experiment is used to investigate the validity of gradient diffusion and similarity closures a priori. The budgets of the mean flow, turbulent kinetic energy, turbulent kinetic energy dissipation rate, heavy-fluid mass fraction variance, and heavy-fluid mass fraction variance dissipation rate transport equations across the mixing layer were previously analyzed [O. Schilling and N. J. Mueschke, Phys. Fluids 22, 105102 (2010)] at different evolution times to identify the most important transport and mixing mechanisms. Here a methodology is introduced to systematically estimate model coefficients as a function of time in the closures of the dynamically significant terms in the transport equations by minimizing the $L_2$ norm of the difference between the model and correlations constructed using the simulation data. It is shown that gradient-diffusion and similarity closures used for the turbulent kinetic energy $K$, turbulent kinetic energy dissipation rate $\epsilon$, heavy-fluid mass fraction variance $S$, and heavy-fluid mass fraction variance dissipation rate $\chi$ equations capture the shape of the exact, unclosed profiles well over the nonlinear and turbulent evolution regimes. Using order-of-magnitude estimates [O. Schilling and N. J. Mueschke, Phys. Fluids 22, 105102 (2010)] for the terms in the exact transport equations and their closure models, it is shown that several of the standard closures for the turbulent production and dissipation (destruction) must be modified to include Reynolds-number scalings appropriate for Rayleigh-Taylor flow at small to intermediate Reynolds numbers. The late-time, large Reynolds number coefficients are determined to be different from those used in shear flow applications and largely adopted in two-equation Reynolds-averaged Navier-Stokes (RANS) models of Rayleigh-Taylor turbulent mixing. In addition, it is shown that the predictions of the Boussinesq model for the Reynolds stress agree better with the data when additional buoyancy-related terms are included. It is shown that an unsteady RANS paradigm is needed to predict the transitional flow dynamics from early evolution times, analogous to the small Reynolds number modifications in RANS models of wall-bounded flows in which the production-to-dissipation ratio is far from equilibrium. Although the present study is specific to one particular flow and one set of initial conditions, the methodology could be applied to calibrations of other Rayleigh-Taylor flows with different initial conditions (which may give different results during the early-time, transitional flow stages, and perhaps asymptotic stage). The implications of these findings for developing high-fidelity eddy viscosity-based turbulent transport and mixing models of Rayleigh-Taylor turbulence are discussed.

Figure 1

Profiles of the turbulent viscosity ${\nu }_t$ (2) normalized by ${\nu }_c={\ell }_c^2/t_c$ at various dimensionless times.

Figure 2

Profiles of the velocity variance $\widetilde{u^{\prime \prime 2}}$ normalized by $u_c^2$ and its Boussinesq and generalized Boussinesq gradient-diffusion closures (1) (Model 1) and (15) (Model 2) at (a) $t/t_c=5.13$, (b) 12.6, (c) 25.1, and (d) 37.9.

Figure 3

Profiles of the velocity variance $\widetilde{v^{\prime \prime 2}}$ normalized by $u_c^2$ and its Boussinesq and generalized Boussinesq gradient-diffusion closures (1) (Model 1) and (15) (Model 2) at (a) $t/t_c=5.13$, (b) 12.6, (c) 25.1, and (d) 37.9.

Figure 4

Profiles of the velocity variance $\widetilde{w^{\prime \prime 2}}$ normalized by $u_c^2$ and its Boussinesq and generalized Boussinesq gradient-diffusion closures (1) (Model 1) and (15) (Model 2) at (a) $t/t_c=5.13$, (b) 12.6, (c) 25.1, and (d) 37.9.

Figure 5

Evolution of optimal coefficients $C_A^{\left(iiii\right)}$ corresponding to the three diagonal Reynolds stress components ${\tau }_{ii}$.

Figure 6

Profiles of the averaged vertical Favre fluctuating velocity $\overline{w^{″}}$ normalized by $u_c$ and its gradient-diffusion closure (23) at (a) $t/t_c=5.13$, (b) 12.6, (c) 25.1, and (d) 37.9.

Figure 7

Profiles of the vertical turbulent heavy-fluid mass fraction flux $\overline{\rho }\widetilde{m_1^{″}{w}^{″}}$ normalized by $F_c={\rho }_c{u}_c$ and its gradient-diffusion closure (25) at (a) $t/t_c=5.13$, (b) 12.6, (c) 25.1, and (d) 37.9.

Figure 8

Profiles of the vertical turbulent kinetic energy flux $\overline{\rho }\widetilde{K^{″}{w}^{″}}$ normalized by $F_c={\rho }_c{u}_c^3$ and its gradient-diffusion closure (25) at (a) $t/t_c=5.13$, (b) 12.6, (c) 25.1, and (d) 37.9.

Figure 9

Profiles of the vertical turbulent kinetic energy dissipation rate flux $\overline{\rho }\overline{{\epsilon }^{\prime }{w}^{\prime }}$ normalized by $F_c={\rho }_c{u}_c^3/t_c$ and its gradient-diffusion closure (26) at (a) $t/t_c=5.13$, (b) 12.6, (c) 25.1, and (d) 37.9.

Figure 10

Profiles of the vertical heavy-fluid mass fraction variance flux $\overline{\rho }\widetilde{m_1^{\prime \prime 2}{w}^{″}}$ normalized by $F_c={\rho }_c{u}_c$ and its gradient-diffusion closure (26) at (a) $t/t_c=5.13$, (b) 12.6, (c) 25.1, and (d) 37.9.

Figure 11

Profiles of the vertical heavy-fluid mass fraction variance dissipation rate flux $\overline{\rho }\widetilde{{\chi }^{″}{w}^{″}}$ normalized by $F_c={\rho }_c{u}_c/t_c$ and its gradient-diffusion closure (27) at (a) $t/t_c=5.13$, (b) 12.6, (c) 25.1, and (d) 37.9.

Figure 12

Evolution of optimal (a) mechanical ${\sigma }_{\rho }$, ${\sigma }_K$, and ${\sigma }_{\epsilon }$ and (b) scalar ${\sigma }_m$, ${\sigma }_S$, and ${\sigma }_{\chi }$ turbulent Schmidt numbers.

Figure 13

Profiles of the vertical pressure flux $\overline{p^{\prime }{w}^{″}}$ normalized by $F_c={\rho }_c{u}_c^3$ and the closure (31) at (a) $t/t_c=5.13$, (b) 12.6, (c) 25.1, and (d) 37.9.

Figure 14

Profiles of the pressure flux $F_p^{\epsilon }$ normalized by $F_c={\rho }_c{u}_c^3/t_c$ and the closure (32) at (a) $t/t_c=5.13$, (b) 12.6, (c) 25.1, and (d) 37.9.

Figure 15

Evolution of optimal pressure flux coefficients $C_{pu}$ and $C_{pu}^{\epsilon }$.

Figure 16

Profiles of the buoyancy production $P_b^{\epsilon }$ normalized by $P_c={\rho }_c{u}_c^2/t_c^2$ and its similarity closure (33) at (a) $t/t_c=5.13$, (b) 12.6, (c) 25.1, and (d) 37.9.

Figure 17

Profiles of the difference between the turbulent kinetic energy dissipation rate destruction and production $D^{\epsilon }\text{–}{P}_t^{\epsilon }$ normalized by $P_c={\rho }_c{u}_c^2/t_c^2$ and its similarity closure (34) at (a) $t/t_c=5.13$, (b) 12.6, (c) 25.1, and (d) 37.9.

Figure 18

Evolution of optimal similarity coefficients $C_{\epsilon 0}$ and $C_{\epsilon 2}$ in the $\epsilon$ transport equation.

Figure 19

Profiles of the (a) mechanical time scale ${\tau }_m$ and (b) scalar time scale ${\tau }_s$ at various dimensionless times.

Figure 20

Profile of the mechanical-to-scalar time scale ratio $R$ (38) at various dimensionless times.

Figure 21

Profiles of the heavy-fluid mass fraction variance destruction $D^S$ normalized by $D_c={\rho }_c/t_c$ and its similarity closure (39) at (a) $t/t_c=5.13$, (b) 12.6, (c) 25.1, and (d) 37.9.

Figure 22

Profiles of the heavy-fluid mass fraction variance dissipation rate mean production $P_m^{\chi }$ normalized by $P_c={\rho }_c/t_c^2$ and its similarity closure (40) at (a) $t/t_c=5.13$, (b) 12.6, (c) 25.1, and (d) 37.9.

Figure 23

Profiles of the heavy-fluid mass fraction variance dissipation rate turbulent production $P_t^{\chi }$ normalized by $P_c={\rho }_c/t_c^2$ and its similarity closure at (a) $t/t_c=5.13$, (b) 12.6, (c) 25.1, and (d) 37.9.

Figure 24

Profiles of the heavy-fluid mass fraction variance dissipation rate destruction term $D^{\chi }$ normalized by $D_c={\rho }_c/t_c^2$ and its similarity closure at (a) $t/t_c=5.13$, (b) 12.6, (c) 25.1, and (d) 37.9.

Figure 25

Evolution of optimal similarity coefficients for the $S$ and $\chi$ transport equations.