Permanence of large eddies in Richtmyer-Meshkov turbulence with a small Atwood number

The large-scale properties of a self-similar Richtmyer-Meshkov turbulent mixing zone with a small Atwood number and in an infinite domain are investigated. The existence of large-scale invariants is predicted for the velocity spectrum. By contrast, the concentration and concentration flux spectra do not display any such invariants. Their large-scale properties are instead controlled either by the velocity spectrum or by nonlinear backscattering processes depending on the initial conditions. The existence of large-scale invariants for the velocity spectrum allows one to relate the self-similar growth rate of the turbulent mixing zone to the infrared slope of the velocity spectrum. Besides, it also implies that large scales keep their initial anisotropy so that the return to isotropy of the turbulent mixing zone is only partial. Finally, it allows one to estimate the prefactors entering the power laws governing the decay of Richtmyer-Meshkov turbulence, the growth of its length scales, and its mixing level. The different assumptions and predictions of this work are verified by performing implicit large eddy simulations of a Richtmyer-Meshkov turbulent mixing zone.

Figure 1

Evolution of (a) the mixing zone width $L$ and (b) the nondimensionalized second-order moments of velocity and concentration for the simulation with $s_0=2$.

Figure 2

Evolution of the kinetic energy spectrum for different $s_0$.

Figure 3

Velocity spectra $E_{33}$ and $E_{11}$ at initial and final times for different $s_0$.

Figure 4

Comparison between ${\mathtt{e}}_{3c}$ and its linear contribution ${\mathtt{e}}_{3c}^{\mathrm{lin}}$ at $t=6t_{\text{ref}}$ for $s_0=1$ and $s_0=2$.

Figure 5

Comparison between ${\mathtt{e}}_{cc}$ and its linear contribution ${\mathtt{e}}_{cc}^{\mathrm{lin}}$ at $t=6t_{\text{ref}}$ for $s_0=1$ and $s_0=2$.

Figure 6

Growth exponent ${\mathrm{\Theta }}_{\text{inst}}$: (a) as a function of time; (b) as a function of $s_0$ at the end of the simulation.

Figure 7

Mixing width constant ${g_2}_{\text{inst}}$: (a) as a function of time; (b) as a function of $s_0$ at the end of the simulation.

Figure 8

Anisotropy ratio $R=〈\overline{u_1^{\prime 2}}〉/〈\overline{u_3^{\prime 2}}〉$: (a) as a function of time; (b) as a function of $s_0$ at the end of the simulation.

Figure 9

Mixing parameter $\mathrm{\Xi }$: (a) as a function of time; (b) as a function of $s_0$ at the end of the simulation.

Figure 10

Growth exponent $\mathrm{\Theta }$ as a function of the mixing parameter $\mathrm{\Xi }$.

Figure 11

2D kinetic energy spectra at the center of the mixing zone for different $s_0$ at $t/t_{\text{ref}}\approx 1$.

Figure 12

Nondimensionalized nonlinear transfer terms at different times greater than $t_0$ for the simulation with $s_0=2$. (a) Transfer of the kinetic energy spectrum. (b) Transfer of the concentration flux spectrum. (c) Transfer of the concentration spectrum.

Figure 13

Linear production terms for the simulation with $s_0=2$ at different times greater than $t_0$. (a) ratio ${\mathtt{e}}_{33}^G/{\mathtt{e}}_{33}$ and (b) ratio ${\mathtt{e}}_{3c}^G/{\mathtt{e}}_{3c}$.

Figure 14

Nondimensionalized spectra of kinetic energy, concentration, and concentration flux for $s_0=2$, taken at different times $t\ge t_0$. (a) Kinetic energy spectrum. (b) Concentration spectrum. (c) Concentration flux spectrum.