Effects of initial condition spectral content on shock-driven turbulent mixing

The mixing of materials due to the Richtmyer-Meshkov instability and the ensuing turbulent behavior is of intense interest in a variety of physical systems including inertial confinement fusion, combustion, and the final stages of stellar evolution. Extensive numerical and laboratory studies of shock-driven mixing have demonstrated the rich behavior associated with the onset of turbulence due to the shocks. Here we report on progress in understanding shock-driven mixing at interfaces between fluids of differing densities through three-dimensional (3D) numerical simulations using the rage code in the implicit large eddy simulation context. We consider a shock-tube configuration with a band of high density gas (${\mathrm{SF}}_6$) embedded in low density gas (air). Shocks with a Mach number of 1.26 are passed through ${\mathrm{SF}}_6$ bands, resulting in transition to turbulence driven by the Richtmyer-Meshkov instability. The system is followed as a rarefaction wave and a reflected secondary shock from the back wall pass through the ${\mathrm{SF}}_6$ band. We apply a variety of initial perturbations to the interfaces between the two fluids in which the physical standard deviation, wave number range, and the spectral slope of the perturbations are held constant, but the number of modes initially present is varied. By thus decreasing the density of initial spectral modes of the interface, we find that we can achieve as much as 25% less total mixing at late times. This has potential direct implications for the treatment of initial conditions applied to material interfaces in both 3D and reduced dimensionality simulation models.

Figure 1

Schematic diagram of the IC for all simulations (see Table 1). The shock wave initially propagates to the right. Solid lines represent solid boundaries at both ends of the shock tube, while dotted lines show periodic boundaries in the transverse directions. Initial perturbations are applied to the front and back interfaces of the ${\mathrm{SF}}_6$ band, labeled $L_1$ and $L_2$, respectively.

Figure 2

Initial surface perturbations with (a) 100 modes, (b) 800 modes, and (c) 3263 modes, along with the spectral content of those surfaces shown in (b), (d), and (f) for each case, respectively. All surfaces have a standard deviation of 0.1 cm and random phasing of the modes. For (b), (d), and (f), red (medium gray) indicates modes with wave numbers that do not satisfy Eq. (9), green (light gray) indicates eligible modes that are not used in this realization of the interface surface, and blue (dark gray) shows modes which have been used to construct the associated surface. Note that in (f) all modes compliant with Eq. (9) are used.

Figure 3

A time-space diagram showing the position of the front (bottom solid line) and back (second solid line from bottom) of both mixing layers $L_1$ and $L_2$ (front is the second solid line from top, back is the top solid line) as well as the primary shock and the rarefaction wave for case S-32.

Figure 4

Snapshot 3D volume renderings of ${\mathrm{SF}}_6$ mass fraction from $x=18$ cm to $x=54$ cm for case S-32 at (a) the start of the simulation, (b) $t=1.0$ ms when the shock front is in the ${\mathrm{SF}}_6$ band, (c) $t=2.0$ ms when the shock has exited the band, and (d) $t=3.0$ ms after reshock. $Y_{\mathrm{SF}}6$ is shown with low concentrations in yellow (light gray), high concentrations in purple (dark gray), and pure air transparent. Visualizations made with vapor [44].

Figure 5

(a) Ensemble-averaged $\overline{\mathrm{Re}}$ for the S- and H-series cases, along with the $\overline{\mathrm{Re}}$ number for case V-1. The $\overline{\mathrm{Re}}$'s for all three resolutions show significant variability in time; however, there is a clear increase at late times for higher-resolution simulations. (b) Streamwise-averaged kinetic energy power spectra for cases S-1 (red/medium gray), H-1 (green/light gray), and V-1 (blue/ dark gray). All spectra are computed for wave numbers less than 16.7 ${\mathrm{cm}}^{-1}$. All cases show similar spectral features at low wave numbers, but increasing resolution leads to extended inertial ranges.

Figure 6

(a) Turbulent kinetic energy (TKE) as a function of time for simulations S-32, S-1, H-32, H-1, and V-1. (b) Total mixedness (TMX) for the same simulations. Note that lines for all five cases in both plots are present at all times, though they are often overplotted by another case as values are very similar. For both quantities the evolution does not show significant changes as the numerical resolution is increased and the variation caused by changing the initial spectral density is considerably larger than the variation due to the change in resolution. In particular, cases H-1 and V-1 show less than 2.8% variation in TKE and less than 1.1% variation in TMX at late times.

Figure 7

(a) Turbulent kinetic energy as a function of time for simulations S-32 through S-1. (b) Relative TKE for the same simulations where the ensemble mean at each time from (a) has been subtracted for each simulation. While strong variations are seen, there is no obvious trend with $N_{IC}$.

Figure 8

(a) Time-evolution of the width of the mixing layer $L_1$ for cases S-1 through S-32. The mix widths grow slowly until the rarefaction wave hits it at about $t=1.6$ ms, after which all six cases show a rapid expansion of the layer. This expansion is reversed upon reshock, which recompresses the the layer. (b) Time-evolution of the mix width for the second interface $L_2$ for all six cases. The mix width grows slowly until reshock at about $t=1.6$ ms, where a sharp compression occurs and is followed by continued expansion. (c) Relative mix width for the front mixing layer $L_1$, where the ensemble mean of the six cases at each time has been subtracted. No clear trends with $N_{IC}$ can be seen either in the orderly evolution prior to reshock or the more chaotic behavior after reshock. (d) Relative mix width for the back interface $L_2$ where the ensemble mean of the six cases has again been subtracted at each time. While the mix widths are less chaotic than in $L_1$, there is no trend with $N_{IC}$.

Figure 9

(a) Volume-integrated mixedness TMX for cases S-32 through S-1 as a function of simulation time. Values have been normalized by the values of TMX for each case just prior to the first impact of the shock on the front interface, which we term ${\mathrm{TMX}}_0$. Two clear phases of mixing can be seen with mixing before 1.5 ms showing linear behavior followed by much more rapid, turbulent mixing. (b) Relative TMX for the same simulations with the ensemble mean subtracted at each time. (c) TMX for each interface for cases S-1, S-4, and S-32. While the initial linear phase shows little dependence on $N_{IC}$, the later turbulent phase reveals a clear trend where cases with greater $N_{IC}$ exhibit more mixing. By $t=2.5$ ms there is a monotonic relation between $N_{IC}$ and TMX for both interfaces individual and over the entire domain.

Figure 10

Volume-integrated mixedness at $t=3.0$ ms for six values of $N_{IC}$. Unique randomly generated instances of the initial perturbation spectra were used for each simulation, with five realizations of case S-1 and three each for cases S-4 and S-32. For case S-32, all three symbols are plotted here, though they are visually difficult to distinguish. While there is some evidence for significant variability due to realization noise, particularly for small values of $N_{IC}$, the larger trend of less mixedness with decreasing $N_{IC}$ is clear.

Figure 11

Time-evolution of the normalized probability distribution for $Y_{SF6}$ in mixing layer $L_1$ for (a) Case S-32 and (b) Case S-1 with light tones indicating larger fractions of the mixing layer at a given concentration and dark tones showing smaller fractions of the mixing layer volume. (c) The ratio of (b) to (a), with blue to green (darker gray) tones indicating larger values for case S-32 and red to yellow (lighter gray) tones indicating larger values for case S-1.

Figure 12

(a) Streamwise-averaged transverse power spectra of kinetic energy in mixing layer $L_1$ at $t=3.0$ ms for cases S-32 through S-1. All cases show a turbulent power spectra with a small but significant inertial range. (b) Relative power spectra of kinetic energy as a fraction of the ensemble mean for all six cases. While low $k$ power is highly variable and shows no clear dependence on $N_{IC}$, the high wave number portion of the spectra shows a clear dependence on $N_{IC}$.