Shock Waves

, Volume 25, Issue 1, pp 35–45 | Cite as

Testing an analytic model for Richtmyer–Meshkov turbulent mixing widths

Original Article

Abstract

We discuss a model for the evolution of the turbulent mixing width \(h(t)\) after a shock or a reshock passes through the interface between two fluids of densities \(\rho _A\) and \(\rho _B\) inducing a velocity jump \(\Delta V\). In this model, the initial growth rate is independent of the surface finish or initial mixing width \(h_0\), but its duration \(t^{*}\) is directly proportional to it: \(h(t)=h_0 +2\alpha A\Delta Vt\) for \(0\le t\le t^{*}\), and \(h(t)=h^{*}({1+(\dot{h}^{*}/\theta h^{*})(t-t^{*})})^{\theta }\) for \(t\ge t^{*}\). Here \(A\) is the Atwood number \((\rho _B -\rho _A)/(\rho _B +\rho _A), \alpha \) and \(\theta \) are dimensionless, \(A\)-dependent parameters measured in past Rayleigh–Taylor experiments, and \(\beta \) is a new dimensionless parameter we introduce via \(t^{*}=(h_0 /\Delta V)\beta \). The mixing width \(h\) and its derivative \(\dot{h}\) remain continuous at \(t=t^{*}\) since \(h^{*}=h_0 +2\alpha A\Delta Vt^{*}\) and \(\dot{h}^{*}=2\alpha A\Delta V\). We evaluate \(\beta \sim 6\) at \(A\approx 0.7\) from air/SF\(_{6}\) experiments and propose that the transition at \(t=t^{*}\) signals isotropication of turbulence. We apply this model to the recent experiments of Jacobs et al. (Shock Waves 23:407–413, 2013) on shock and reshock, and discuss briefly the third wave causing an unstable acceleration of the interface. We also consider the experiments of Weber et al. (Phys Fluids 24:074105, 2012) and argue that their smaller growth rates reflect density gradient stabilization.

Keywords

Turbulent mix Shocks Reshocks Rayleigh–Taylor Richtmyer–Meshkov National Shock-Tube Facility 

Glossary

\(\rho _A (\rho _B)\)

Density of fluid \(A\) (fluid \(B\))

\(A\)

Atwood number

\(M_s\)

Mach number

\(W_i\)

Speed of incident shock wave

\(\Delta V\)

Velocity change of the \(A/B\) interface induced by a shock or reshock

\(g\)

Acceleration of the \(A/B\) interface

\(y\)

Coordinate along which a shock moves, usually vertical. Also, coordinate of a density gradient between fluids \(A\) and \(B\)

\(x\)

Coordinate transverse to \(y\)

\(\lambda \)

Wavelength of a perturbation

\(k\)

\(2\pi /\lambda \)

\(\eta \)

Perturbation amplitude

\(\eta _0\)

Initial value of \(\eta \)

\(\dot{\eta }_0\)

Initial value of \(\mathrm{d}\eta /\mathrm{d}t\)

\(\gamma \)

Growth rate of a Rayleigh–Taylor (RT) perturbation

\(\Gamma \)

\(\gamma /\sqrt{g}\)

b

Superscript denoting the bubble part of a perturbation or turbulent mixing width into the heavier fluid

s

Superscript denoting the spike part of a perturbation or turbulent mixing width into the lighter fluid

\(h\)

Turbulent mixing width

\(h_0\)

Initial value of \(h\)

\(\dot{h}_0\)

Initial value of \(\mathrm{d}h/\mathrm{d}t\)

\(\alpha \)

Coefficient for the turbulent mixing width in a constant acceleration

\(\theta \)

Power for the evolution of \(h\) when \(g=0\)

\(t^{*}\)

Time after shock or reshock when \(h\) changes from growth linear with time (\(h\sim \alpha t)\) to power-law growth \((h\sim t^{\theta })\)

\(h^{*}\)

\(h(t=t^{*})\)

\(\dot{h}^{*}\)

\((\mathrm{d}h/\mathrm{d}t)_{t=t^{*}}\)

\(\beta \)

Nondimensional coefficient relating \(t^{*}\) to \(h_0\) and \(\Delta V\)

\(d\)

Density gradient scale length

erf

Error function

erfc

Complimentary error function \(1-\mathrm{erf}\)

\(D\)

Diffusion coefficient

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Copyright information

© Springer-Verlag Berlin Heidelberg (outside the USA) 2014

Authors and Affiliations

  1. 1.Lawrence Livermore National LaboratoryLivermoreUSA

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