Heat transfer and transport property contrast effects on the compressible Rayleigh-Taylor instability

In extreme environments, the Rayleigh-Taylor instability (RTI) may occur under large variations in density and temperature and with fluid transport properties strongly dependent on temperature. Direct numerical simulations of the 3D fully compressible RTI are conducted, examining the idealized configuration of a hotter, less dense fluid pushing against a colder, denser fluid. Various temperature ratios and transport property configurations are explored to examine how heat conduction, large variations in transport properties, and sudden changes in transport properties can affect the evolution of the mixing layer. Nonuniform fluid expansion and contraction induced by heat transfer can significantly affect local density differences and overall instability growth, causing profile asymmetries about the initial interface for flow and mixing statistics. The departures from classical self-similar development of the instability along with misalignment between regions of mixing and regions of most intense turbulent activity caused by both heat transfer and transport property contrasts are examined. After sudden changes in fluid transport properties, which may occur as a result of rapid heating (e.g., in inertial confinement fusion), the flow quickly responds and begins to relax towards quasi-self-similar late-time evolution. For many dynamical quantities such as vorticity and dissipation, this late-time evolution resembles that of the configuration that already started with the final transport property magnitudes, suggesting that these quantities depend only on the transport properties and not on past flow history, provided that the density field distributions for the flows remain similar. On the other hand, the mixing evolution after the transport property change is unique, implying that both property magnitudes and previous history are impactful on the mixing. These simulations demonstrate how various temperature-related effects are extremely important to consider in compressible RTI flows with large temperature variations.

Figure 1

Schematic of the RT instability problem.

Figure 2

(a) Nondimensional mass fraction, density, pressure, and temperature profiles for the isothermal base case and (b) nondimensional temperature profile for the temperature ratio three cases.

Figure 3

(a) Normalized spectrum $E_{\eta }\left(k\right)$ of the initial perturbations in wavenumber space, dashed line indicates the most unstable wavenumber for the 2D version of the problem; (b) the initial perturbations in physical space.

Figure 4

Example mask function $\xi$ used for boundary filtering; red region indicates where the mask is exactly zero.

Figure 5

(a) Time history of the spike and bubble side penetration depths based on the 5% average mole fraction limits (Sec. 3b, Fig. 8). (b) Average mole fraction $\langle f_1\rangle$ profiles at various times, centered around the mixing midplane location $z_{50}$ and scaled by mixing height $h_{95}$ (Sec. 3b, Fig. 9) (c) Time history of Young's mixing parameter $\theta$ [Sec. 3d, Fig. 19, Eq. (39)]. (d) Time history of the average Favre turbulent kinetic energy and the average dissipation rate of turbulent kinetic energy $\epsilon$, volume averaged within the mixing zone [Sec. 3e, Fig. 22, Eq. (31)].

Figure 6

(a) Vertical variation of the turbulent Reynolds number at $t=10$ for all cases, with the dashed line indicating the location of the initial interface position $z_{\mathrm{int}}$; (b) time history of the maximum turbulent Reynolds number for all cases.

Figure 7

Estimates of the growth rate $\alpha$ calculated for various $t_0$ values using Eq. (37) for (a) isothermal case A, (b) $\mathrm{T}1/\mathrm{T}2=3$ constant property case B, and (c) $\mathrm{T}1/\mathrm{T}2=3$ variable property case C.

Figure 8

Time history of (a) spike penetration depths, (b) bubble penetration depths, and (c) mixing heights based on the 5% mole fraction limits.

Figure 9

Average mole fraction $\langle f_1\rangle$ profiles at various times, centered around the mixing midplane location and scaled by mixing height $h_{95}$ for (a) isothermal case A, (b) $\mathrm{T}1/\mathrm{T}2=3$ constant property case B, and (c) $\mathrm{T}1/\mathrm{T}2=3$ variable property case C.

Figure 10

Average density $\langle \rho \rangle$ profiles at various times, centered around the mixing midplane location and scaled by mixing height $h_{95}$ for (a) isothermal case A, (b) $\mathrm{T}1/\mathrm{T}2=3$ constant property case B, and (c) $\mathrm{T}1/\mathrm{T}2=3$ variable property case C.

Figure 11

Comparison of early time ($t=4$) density contours at the interface for (a) isothermal case A, (b) $\mathrm{T}1/\mathrm{T}2=3$ constant property case B, and (c) $\mathrm{T}1/\mathrm{T}2=3$ variable property case C.

Figure 12

Comparison of late time ($t=12$) density contours at the interface for (a) isothermal case A, (b) $\mathrm{T}1/\mathrm{T}2=3$ constant property case B, (c) $\mathrm{T}1/\mathrm{T}2=3$ variable property case C, and (d) $\mathrm{T}1/\mathrm{T}2=3$ transitional property case D.

Figure 13

Visualizations of temperature, dilatation and density tracking the evolution of a single spike in the variable property case C for $t=4,5,6,7,8$.

Figure 14

Visualizations of temperature, dilatation and density tracking the evolution of a single bubble in the variable property case C for $t=4,5,6,7,8$.

Figure 15

Time history comparison of the 0.95, 0.5, and 0.05 average mole fraction $\langle f\rangle$ locations (solid lines) and the boundaries of regions with the highest ${\text{Re}}_t$ (dashed lines) for (a) isothermal case A, (b) $\mathrm{T}1/\mathrm{T}2=3$ constant property case B, (c) $\mathrm{T}1/\mathrm{T}2=3$ variable property case C, and (d) $\mathrm{T}1/\mathrm{T}2=3$ transitional property case D.

Figure 16

(a) Weighted vorticity PDF at the plane location of highest average vorticity and (b) the fraction $F_A$ of the plane that is considered turbulent across the mixing layer at time $t=10$.

Figure 17

PDFs of the signed Atwood number inside the turbulent regions at the (a) $\langle f_1\rangle =0.75$, (b) $\langle f_1\rangle =0.5$, and (c) $\langle f_1\rangle =0.25$ plane locations for time $t=10$.

Figure 18

(a) Spatial variation of the Batchelor scale ${\lambda }_b$ at $t=10$ for all cases, (b) average 1D mole fraction $f_1$ spectra at the mixing midplane location $z_{50}$ at $t=10$; the two dashed lines indicate the approximate limits of the inertial range.

Figure 19

Time variation of Young's mixing parameter $\theta$.

Figure 20

Spatial variation of the density-specific volume perturbation correlation for (a) $t=6$ and (b) $t=10$.

Figure 21

PDFs of light fluid mole fraction $f_1$ at the (a) $\langle f_1\rangle =0.75$, (b) $\langle f_1\rangle =0.5$, and (c) $\langle f_1\rangle =0.25$ plane locations for time $t=10$.

Figure 22

(a) Time histories of the dissipation of turbulent kinetic energy, domain averaged over the mixing layer extents, and several moments throughout the evolution of case D are highlighted; (b) time histories of the total mixing rate TMR.

Figure 23

Average 1D compensated u velocity spectras at the peak turbulence location at the various times described in Fig. 22: (i) initial state, (ii) peak of dissipation, (iii) middle of recovery period, and (iv) late-time state.

Figure 24

PDFs of (a) vorticity-strain alignment angles and (b) mass fraction gradient-strain alignment angles at the peak turbulence location for the latest time in the simulation ($t=13$).

Figure 25

Time-step stability limit for a representative 1D RT test case and comparison to various scalings.