### Abstract

This work investigates several key statistical measurements of turbulence induced by Rayleigh-Taylor instability using data from well resolved numerical simulations at moderate Reynolds number with the goal of determining the degree of departure of this inhomogeneous flow from that of homogeneous, isotropic turbulence. The simulations use two miscible fluids with unity Schmidt number and moderate density contrast (3/2 to 9). The results of this study should find application in subgrid-scale modeling for large-eddy simulations and Reynolds-averaged Navier-Stokes modeling used in many engineering and scientific problems.

This work was performed under the auspices of the Lawrence Livermore National Security, LLC under Contract No. DE-AC52-07NA27344.

I. INTRODUCTION

II. NUMERICAL SIMULATIONS

III. STATISTICAL MEASUREMENTS

A. Isotropy of the velocity and its derivatives

B. Length scales

C. Correlation functions

D. Structure functions

E. Component energy and vorticity spectra

1. Reynolds numbers and inertial range requirements

F. Skewness and flatness

G. Strain rate eigenvalues

IV. CONCLUSIONS

### Key Topics

- Turbulent flows
- 55.0
- Reynolds stress modeling
- 32.0
- Rayleigh Taylor instabilities
- 30.0
- Anisotropy
- 24.0
- Eigenvalues
- 17.0

## Figures

Visualization of a vertical slice of the density field in the RTI mixing layer at late time for the A = 0.5 case. 5 Heavy fluid is dark, light fluid is light. Gravity is directed downwards.

Visualization of a vertical slice of the density field in the RTI mixing layer at late time for the A = 0.5 case. 5 Heavy fluid is dark, light fluid is light. Gravity is directed downwards.

(a) Isotropy of the vorticity field is shown as a function of time, measured by the ratio of vertical to horizontal enstrophy components averaged over a vertical extent of h/3 in the middle of the mixing layer. (The gap in the A = 0.2 data resulted from data lost in a catastrophic file system failure.) (b) The growth of the product width h is shown for the different Atwood numbers.

(a) Isotropy of the vorticity field is shown as a function of time, measured by the ratio of vertical to horizontal enstrophy components averaged over a vertical extent of h/3 in the middle of the mixing layer. (The gap in the A = 0.2 data resulted from data lost in a catastrophic file system failure.) (b) The growth of the product width h is shown for the different Atwood numbers.

Vertical profiles (horizontal averages) of fluctuation intensity for (a) velocity components and (b) velocity derivative components are shown at time t = 25 in the A = 0.5 case, demonstrating statistical equivalences and the degree of anisotropy.

Vertical profiles (horizontal averages) of fluctuation intensity for (a) velocity components and (b) velocity derivative components are shown at time t = 25 in the A = 0.5 case, demonstrating statistical equivalences and the degree of anisotropy.

Vertical profiles (horizontal averages) of the fluctuation intensity of velocity derivative functions at time t = 25 in the A = 0.5 case: (a) vorticity components, (b) cross terms, (c) dissipation components, and (d) dissipation terms. The vertical position is scaled by the product width of the mixing layer h.

Vertical profiles (horizontal averages) of the fluctuation intensity of velocity derivative functions at time t = 25 in the A = 0.5 case: (a) vorticity components, (b) cross terms, (c) dissipation components, and (d) dissipation terms. The vertical position is scaled by the product width of the mixing layer h.

The correlation functions of velocity components and density for the A = 0.5 case at t = 25 using mid-slab averaging for horizontal correlations in (a) and (b), midplane averaging for vertical correlations in (a) and (c), and global vertical correlations in (d). The longitudinal horizontal correlation C uu (δx|z 0) in (a) is plotted with the separation length scaled by 1/√2. Statistically equivalent terms have been averaged together.

The correlation functions of velocity components and density for the A = 0.5 case at t = 25 using mid-slab averaging for horizontal correlations in (a) and (b), midplane averaging for vertical correlations in (a) and (c), and global vertical correlations in (d). The longitudinal horizontal correlation C uu (δx|z 0) in (a) is plotted with the separation length scaled by 1/√2. Statistically equivalent terms have been averaged together.

Midplane 2nd-order structure function for the vertical velocity, scaled by ⟨w 2⟩, in the A = 0.5 case at t = 25 for (a) transverse separations (r = δx) and (b) longitudinal separations (r = ±δz) showing r 2 behavior for small separations and approximate r 2/3 behavior for large separations. The model in (21) is shown with dashed lines using an eyeball fit for parameter B.

Midplane 2nd-order structure function for the vertical velocity, scaled by ⟨w 2⟩, in the A = 0.5 case at t = 25 for (a) transverse separations (r = δx) and (b) longitudinal separations (r = ±δz) showing r 2 behavior for small separations and approximate r 2/3 behavior for large separations. The model in (21) is shown with dashed lines using an eyeball fit for parameter B.

One-dimensional spectra of (a) velocity and (b) vorticity at midplane at t = 25 in the A = 0.5 case. Vertical spectra (in k 3) for velocity components were constructed using Fourier transforms of vertical autocorrelation functions. Statistically equivalent terms have been averaged together.

One-dimensional spectra of (a) velocity and (b) vorticity at midplane at t = 25 in the A = 0.5 case. Vertical spectra (in k 3) for velocity components were constructed using Fourier transforms of vertical autocorrelation functions. Statistically equivalent terms have been averaged together.

Compensated 2D horizontal spectra for the vertical velocity component E 33(k) (thick lines) and for the sum of all velocity components E ii (k) (thin lines) calculated near the midplane at t = 25 for the A = 0.5 case. In (a) the k −5/3 Kolmogorov scaling is used, and in (b) the k −7/4 Rayleigh-Taylor scaling proposed by Zhou 21 is used.

Compensated 2D horizontal spectra for the vertical velocity component E 33(k) (thick lines) and for the sum of all velocity components E ii (k) (thin lines) calculated near the midplane at t = 25 for the A = 0.5 case. In (a) the k −5/3 Kolmogorov scaling is used, and in (b) the k −7/4 Rayleigh-Taylor scaling proposed by Zhou 21 is used.

Horizontal midplane spectra for (a) vertical and (b) horizontal kinetic energy rate terms: buoyancy production (dashed line), pressure-strain (thin solid line), and dissipation (medium solid line). The pressure-strain term is a sink (–) for the vertical kinetic energy and a source (+) for the horizontal kinetic energy. The thick curves show the corresponding vertical and horizontal velocity spectra.

Horizontal midplane spectra for (a) vertical and (b) horizontal kinetic energy rate terms: buoyancy production (dashed line), pressure-strain (thin solid line), and dissipation (medium solid line). The pressure-strain term is a sink (–) for the vertical kinetic energy and a source (+) for the horizontal kinetic energy. The thick curves show the corresponding vertical and horizontal velocity spectra.

Vertical profiles of (a) skewness and (b) flatness for velocity derivatives at t = 25 for the A = 0.5 case.

Vertical profiles of (a) skewness and (b) flatness for velocity derivatives at t = 25 for the A = 0.5 case.

Histograms of the cosine of angles between strain rate eigenvectors, vorticity, and density gradient in the middle of the mixing layer at t = 25 in the A = 0.5 case. Values near unity correspond to parallel alignment of the vectors, and values near zero correspond to orthogonal alignment.

Histograms of the cosine of angles between strain rate eigenvectors, vorticity, and density gradient in the middle of the mixing layer at t = 25 in the A = 0.5 case. Values near unity correspond to parallel alignment of the vectors, and values near zero correspond to orthogonal alignment.

## Tables

Velocity and velocity-derivative isotropy in homogeneous isotropic turbulence (theoretical) and late-time (t = 25) Rayleigh-Taylor instability flow with gravity in direction 3.

Velocity and velocity-derivative isotropy in homogeneous isotropic turbulence (theoretical) and late-time (t = 25) Rayleigh-Taylor instability flow with gravity in direction 3.

Turbulence statistics for the A = 0.5 case at t = 25 for different spatial averaging domains.

Turbulence statistics for the A = 0.5 case at t = 25 for different spatial averaging domains.

Late-time turbulence statistics for different Atwood numbers with “mid-slab” average.

Late-time turbulence statistics for different Atwood numbers with “mid-slab” average.

Mean properties of strain rate eigenvalues, vorticity, and density gradient in the middle of the mixing layer at late time in the simulations for different Atwood numbers.

Mean properties of strain rate eigenvalues, vorticity, and density gradient in the middle of the mixing layer at late time in the simulations for different Atwood numbers.

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