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Density Ratio and Entrainment Effects on Asymptotic Rayleigh–Taylor Instability

[+] Author and Article Information
Assaf Shimony

Physics Department,
NRCN,
Beer-Sheva 84190, Israel;
Physics Department,
BGU,
Beer-Sheva 84015, Israel
e-mail: shimonya@gmail.com

Guy Malamud

Physics Department,
NRCN,
Beer-Sheva 84190, Israel;
Climate and Space Sciences and
Engineering Department,
University of Michigan,
Ann Arbor, MI 48109

Dov Shvarts

Climate and Space Sciences and
Engineering Department,
University of Michigan,
Ann Arbor, MI 48109;
Physics Department,
NRCN,
Beer-Sheva 84190, Israel;
Physics Department,
BGU,
Beer-Sheva 84015, Israel

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received December 1, 2016; final manuscript received October 9, 2017; published online December 22, 2017. Assoc. Editor: Ben Thornber.

J. Fluids Eng 140(5), 050906 (Dec 22, 2017) (8 pages) Paper No: FE-16-1789; doi: 10.1115/1.4038400 History: Received December 01, 2016; Revised October 09, 2017

A comprehensive numerical study was performed in order to examine the effect of density ratio on the mixing process inside the mixing zone formed by Rayleigh–Taylor instability (RTI). This effect exhibits itself in the mixing parameters and increase of the density of the bubbles. The motivation of this work is to relate the density of the bubbles to the growth parameter for the self-similar evolution, α, we suggest an effective Atwood formulation, found to be approximately half of the original Atwood number. We also examine the sensitivity of the parameters above to the dimensionality (two-dimensional (2D)/three-dimensional (3D)) and to numerical miscibility.

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Figures

Grahic Jump Location
Fig. 4

The average generation of the bubbles as a function of the normalized time in (a) 2D simulations with IT (average over ten realizations for each Atwood number), (b) 3D simulation (one realization for each Atwood number). Solid lines—with IT, dashed lines—no IT.

Grahic Jump Location
Fig. 10

The mixing parameter, Θ as a function of the bubble merger generation, G. The 2D and the 3D results were averaged over 2 (2D) and 14 (3D) simulations.

Grahic Jump Location
Fig. 3

Density contours from identical LEEOR3D 3D simulations except for the IT numerics at the same time. The Atwood number is A=0.5: (a) with IT and (b) without IT.

Grahic Jump Location
Fig. 2

αb/s as a function of the bubble merger generation from 3D simulation using 2563 (only one simulation is shown) and 1283 cells (average over 14 simulations), presenting the convergence of the fronts of the bubbles (a) and the spikes (b) for the A = 0.5 case

Grahic Jump Location
Fig. 5

3D A = 0.7 simulation results. Bold line: averaged 3D αb/s from 14 realizations (without IT): (a) bubbles and (b) spikes. Dashed line: averaged αb/s from 14 realizations (with IT). Vertical solid line—asymptotic αb/s, horizontal dashed line present the estimated error bar.

Grahic Jump Location
Fig. 6

Θ(G) from 3D A = 0.7 simulations, dotted line: without IT, Θ is calculated from every single cell, solid line: with IT, Θ is calculated from every single cell, dashed line: with IT, Θ is averaged over blocks of 3×3×3 cells

Grahic Jump Location
Fig. 7

αb(G). The 2D and the 3D results were averaged over 2 (2D) and 14 (3D) simulations.

Grahic Jump Location
Fig. 8

The asymptotic growth factors αb and αs (the average values are plotted by solid lines and the uncertainty range are semitransparent) from 2D and 3D simulations (with and without IT) compared to the LEM experimental results [18,19] as a function of the Atwood number: (a) 2D and (b) 3D

Grahic Jump Location
Fig. 9

The ratio between the growth factors (αs/αb) (the average values are plotted by solid lines and the uncertainty range are semitransparent) from 2D and 3D simulations (with and without IT) and a logarithmic fit to the LEM experimental results as a function of the Atwood number. The dashed lines represent an estimation of the experimental uncertainty, calculated roughly by the spread of the experimental results.

Grahic Jump Location
Fig. 11

The values of (Aeff/A) (the average values are plotted by solid lines and the uncertainty range are semitransparent) from 2D and 3D simulations as a function of the Atwood number. The dashed line is the estimated effective Atwood number by Eq. (4): (a) 2D simulations and estimation and (b) 3D simulations and estimation.

Grahic Jump Location
Fig. 12

The densities of the bubbles, ρ¯b (the average values are plotted by solid lines and theuncertainty range are semitransparent) from 2D and 3D simulation and the estimation (dashed line) as a function of the Atwood number (Eq. (11)): (a) 2D simulations and estimation and (b) 3D simulations and estimation

Grahic Jump Location
Fig. 1

αb dependence on the initial amplitude of the long wavelength mode (mode k), as described in Ref. [11]

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