Large-eddy simulation and Reynolds-averaged Navier-Stokes modeling of a reacting Rayleigh-Taylor mixing layer in a spherical geometry

Tenth-order compact difference code Miranda is used to perform large-eddy simulation (LES) of a hydrogen gas–plastic mixing layer in a spherical geometry. Once the mixing layer has achieved self-similar growth, it is heated to 1 keV, and the second-order arbitrary Lagrangian-Eulerian (ALE) code Ares is used to simulate mixing layer evolution as it undergoes thermonuclear (TN) burn. Both premixed (in which deuterium and tritium are initially present in the gas) and nonpremixed (in which deuterium is initially present only in the plastic) variants are considered at Atwood numbers 0.05 and 0.50. The impact of turbulent mixing on mean TN reaction rate is examined, and a four-equation $k\text{−}L\text{−}a\text{−}V$ Reynolds-averaged Navier-Stokes (RANS) model is presented. The $k\text{−}L\text{−}a\text{−}V$ model, which represents an extension of the $k\text{−}L\text{−}a$ model [Morgan and Wickett, Phys. Rev. E 91, 043002 (2015)] by the addition of a transport equation for the scalar mass fraction variance, is then applied in one-dimensional simulations of the reacting mixing layer under consideration. Excellent agreement is obtained between LES and RANS in total TN neutron production when fluctuations in reaction cross-section can be neglected.

Figure 1

Slice of $Y_H$ contours at the $y=0$ plane taken at $t=t_{\text{burn}}$ for simulation on mesh D at $A=0.05$. Contours are illustrated between $Y_H=0$ (white) and $Y_H=1$ (black).

Figure 2

Initial contours of $Y_H$ plotted with the computational mesh (every 12th zone) used for RANS simulations. Contours are illustrated between $Y_H=0$ (white) and $Y_H=1$ (gray).

Figure 3

Nonreacting mixing layer evolution as a function of generation number for four mesh resolutions at $A=0.05$ and 0.50.

Figure 4

Two measures of mixing layer development as a function of generation number on mesh D: (a) mixedness, $\mathrm{\Theta }$, and (b) growth constant, $\alpha$, at $A=0.05$ and 0.50.

Figure 5

Contours of radially averaged mass fraction as a function of time and space for simulation at $A=0.05$ on mesh D: (a) ${\tilde{Y}}_H$ vs. generation number and radial coordinate, $r/R_c$; (b) ${\tilde{Y}}_H$ vs. generation number and self-similar radial coordinate $\left(r-R_c\right)/h$.

Figure 6

Self-similarity of the heavy species mass fraction profile as a function of generation number for simulations on mesh D at $A=0.05$ and 0.50.

Figure 7

Spatial profile of photon mean free path from LES in the premixed configuration at $t_{\text{burn}}+0.0015\mu \mathrm{s}$ on mesh D.

Figure 8

Mass-weighted average temperature history of burning mixing layer LES in the premixed configuration on mesh D: (a) $A=0.05$ and (b) $A=0.50$.

Figure 9

Spatial profiles of radially averaged DT specific reaction rate from LES in the premixed configuration at $t_{\text{burn}}+0.0015\mu \mathrm{s}$ on mesh D: (a) $A=0.05$ and (b) $A=0.50$. I. Radially averaged reaction rate. II. Mean contribution to reaction rate. III. Mean contribution plus scalar variance term. IV. Mean contribution plus all second moment terms. See Eq. (65) for mathematical definitions of curves.

Figure 10

Spatial profiles of turbulent contributions to the average DT reaction rate from LES in the premixed configuration at $t_{\text{burn}}+0.0015\mu \mathrm{s}$ on mesh D: (a) $A=0.05$ and (b) $A=0.50$. V. ${\overline{\rho }}^2\widetilde{Y_D^{″}{Y}_T^{″}}$. VI. ${\tilde{Y}}_D{\tilde{Y}}_T\overline{{\rho }^{\prime }{\rho }^{\prime }}$. VII. $\overline{\rho }{\tilde{Y}}_D\overline{{\rho }^{\prime }{Y}_T^{\prime }}$. VIII. $\overline{\rho }{\tilde{Y}}_T\overline{{\rho }^{\prime }{Y}_D^{\prime }}$.

Figure 11

Spatial profiles of normalized standard deviation of the form $\sqrt{\widetilde{f^{″}{f}^{″}}}/\tilde{f}$ for (a) temperature and (b) DT reaction cross-section from LES in the premixed configuration at $t_{\text{burn}}+0.0015\mu \mathrm{s}$ on mesh D.

Figure 12

Comparison of spatial profiles between LES (mesh D) and RANS in the premixed configuration at $t_{\text{burn}}$ for $A=0.05$: (a) mass fraction profiles, $\tilde{Y}$, and (b) scalar variance profile, $\widetilde{Y^{″}{Y}^{″}}$.

Figure 13

Comparison of total neutron production as a function of time for LES (mesh D) and RANS in the premixed configuration: (a) $A=0.05$ and (b) $A=0.50$.

Figure 14

Mass-weighted average temperature history of burning mixing layer LES in the nonpremixed configuration on mesh D: (a) $A=0.05$ and (b) $A=0.50$.

Figure 15

Spatial profiles of radially averaged DT specific reaction rate from LES in the nonpremixed configuration at $t_{\text{burn}}+0.0015\mu \mathrm{s}$ on mesh D: (a) $A=0.05$ and (b) $A=0.50$. I. Radially averaged reaction rate. II. Mean contribution to reaction rate. III. Mean contribution plus scalar variance term. IV. Mean contribution plus all second moment terms. See Eq. (65) for mathematical definitions of curves.

Figure 16

Spatial profiles of turbulent contributions to the average DT reaction rate from LES in the nonpremixed configuration at $t_{\text{burn}}+0.0015\mu \mathrm{s}$ on mesh D: (a) $A=0.05$ and (b) $A=0.50$. V. ${\overline{\rho }}^2\widetilde{Y_D^{″}{Y}_T^{″}}$. VI. ${\tilde{Y}}_D{\tilde{Y}}_T\overline{{\rho }^{\prime }{\rho }^{\prime }}$. VII. $\overline{\rho }{\tilde{Y}}_D\overline{{\rho }^{\prime }{Y}_T^{\prime }}$. VIII. $\overline{\rho }{\tilde{Y}}_T\overline{{\rho }^{\prime }{Y}_D^{\prime }}$.

Figure 17

Spatial profiles of normalized standard deviation of the form $\sqrt{\widetilde{f^{″}{f}^{″}}}/\tilde{f}$ for (a) temperature and (b) DT reaction cross-section from LES in the nonpremixed configuration at $t_{\text{burn}}+0.0015\mu \mathrm{s}$ on mesh D.

Figure 18

Comparison of spatial profiles between LES (mesh D) and RANS in the nonpremixed configuration at $t_{\text{burn}}$ for $A=0.05$: (a) mass fraction profiles, $\tilde{Y}$, and (b) scalar variance profile, $\widetilde{Y^{″}{Y}^{″}}$.

Figure 19

Comparison of total neutron production as a function of time for LES (mesh D) and RANS in the nonpremixed configuration: (a) $A=0.05$ and (b) $A=0.50$.

Figure 20

The $k\text{−}L\text{−}a\text{−}V$ model applied to simulation of a planar 1D RT mixing layer at $A=0.05$: (a) steady-state scalar variance profile and (b) steady-state $b$ profile compared with LES [40] and with similarity solution.

Figure 21

The $k\text{−}L\text{−}a\text{−}V$ model applied to simulation of a planar 1D RT mixing layer at $A=0.05$: mixedness as a function of generation number compared with LES [40].