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Research Papers: Fundamental Issues and Canonical Flows

Scaling Interface Length Increase Rates in Richtmyer–Meshkov Instabilities

[+] Author and Article Information
V. Kilchyk

Department of Mechanical Engineering,
Indiana University–Purdue University,
723 West Michigan Street, Indianapolis, IN 46202;Department of Mechanical Engineering,
Purdue University,
585 Purdue Mall, West Lafayette, IN 47907

R. Nalim

Department of Mechanical Engineering,
Indiana University–Purdue University,
723 West Michigan Street, Indianapolis, IN 46202

C. Merkle

Department of Mechanical Engineering,
Purdue University,
585 Purdue Mall, West Lafayette, IN 47907

Manuscript received February 25, 2012; final manuscript received December 7, 2012; published online February 22, 2013. Assoc. Editor: Ye Zhou.

J. Fluids Eng 135(3), 031203 (Feb 22, 2013) (7 pages) Paper No: FE-12-1091; doi: 10.1115/1.4023191 History: Received February 25, 2012; Revised December 07, 2012

The interface area increase produced by large-amplitude wave refraction through an interface that separates fluids with different densities can have important physiochemical consequences, such as a fuel consumption rate increase in the case of a shock–flame interaction. Using the results of numerical simulations along with a scaling analysis, a unified scaling law of the interface length increase was developed applicable to shock and expansion wave refractions and both types of interface orientation with the respect to the incoming wave. To avoid a common difficulty in interface length quantification in the numerical tests, a sinusoidally perturbed interface was generated using gases with different temperatures. It was found that the rate of interface increase correlates almost linearly with the circulation deposited at the interface. When combined with earlier developed models of circulation deposition in Richtmyer–Meshkov instability, the obtained scaling law predicts dependence of interface dynamics on the basic problem parameters.

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Figures

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Fig. 1

A schematic of the sinusoidal interface evolution following shock refraction. Initial interface was produced as y = aosin(2πx/λ).

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Fig. 2

Typical interface length and perturbation amplitude change in a Richtmyer–Meshkov instability (fast/slow Mach 1.7 shock, all values are normalized by the initial interface length)

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Fig. 3

Normalized residual unsteady calculations in a fast/slow M1.5 shock refraction. Every physical time step has 10 inner iterations.

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Fig. 4

The numerical setup (a) and grid (b) for the two-dimensional computations. Dimensions are scaled by a reference length of 0.5 m.

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Fig. 5

Density interface in a Mach 1.5 shock density interaction at 0.33 msec in the computation with three grid resolutions: (a) 24,090, (b) 64,561, and (c) 222,580 nodes (ao/λ = 0.5). Dimensions are scaled by a reference length of 0.5 m.

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Fig. 6

Comparison of analytically predicted and numerical results for the perturbation amplitude change in a fast/slow Mach 1.7 shock refraction over a sinusoidal interface (ao/λ = 0.25). Values are normalized by the initial interface length (interface length of a half wave 1.38 cm).

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Fig. 7

Interface evolution following Mach 1.7 shock fast/slow refraction (ao/λ = 0.25). Time is shown in milliseconds at the bottom. Dimensions are shown in 0.5 × m.

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Fig. 8

Interface evolution following Mach 1.5 shock slow/fast refraction at interface with a - 0.125 and b - 0.5 amplitude-to-wavelength ratio (ao/λ). Initial interface profile is shown with a dashed line. Time is shown in milliseconds at the bottom. Dimensions are shown in 0.5 × m.

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Fig. 9

Interface length change in fast/slow (a) and slow/fast (b) refractions with various shock strengths (ao/λ = 0.5). In slow/fast refraction cases, the interface increase was preceded by a short decrease. The decrease was caused by the interface inversion that started the interface deformation in these cases. Values are normalized by the initial interface length (half-wavelength is 1.38 cm).

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Fig. 10

Sinusoidal interface length increase (ao/λ = 0.25) following expansion wave refraction. Values are normalized by the initial interface length (half-wavelength is 1.38 cm).

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Fig. 11

Dependence of the dimensionless growth velocity on the amplitude-to-wave length ratio in Mach 1.5 shock refractions

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Fig. 12

Rate of interface length change as a function of generated circulation

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