Startup process in the Richtmyer–Meshkov instability
M. Lombardini
a͒
and D. I. Pullin
Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, California 91125, USA
͑Received 2 October 2008; accepted 4 November 2008; published online 14 April 2009͒
An analytical model for the initial growth period of the planar Richtmyer–Meshkov instability is
presented for the case of a reflected shock, which corresponds in general to light-to-heavy
interactions. The model captures the main features of the interfacial perturbation growth before the
regime with linear growth in time is attained. The analysis provides a characteristic time scale
for
the startup phase of the instability, expressed explicitly as a function of the perturbation
wavenumber k, the algebraic transmitted and reflected shock speeds U
S
1
Ͻ0 and U
S
2
Ͼ0
͑defined in the frame of the accelerated interface͒, and the postshock Atwood number A
+
:
=͓͑1−A
+
͒ /U
S
2
+͑1+A
+
͒ /͑−U
S
1
͔͒/͑2k͒. Results are compared with computations obtained from
two-dimensional highly resolved numerical simulations over a wide range of incident shock
strengths S and preshock Atwood ratios A. An interesting observation shows that, within this model,
the amplitude of small perturbations across a light-to-heavy interface evolves quadratically in time
͑and not linearly͒ in the limit A →1
−
.©2009 American Institute of Physics.
͓DOI: 10.1063/1.3091943͔
I. INTRODUCTION
The Richtmyer–Meshkov instability ͑RMI͒ arises when
a shock wave interacts with a perturbed interface separating
two fluids of different densities. It combines different phe-
nomena such as, but not limited to, shock refraction, hydro-
dynamic stability, and both linear and nonlinear growth pe-
riods. Such instability occurs in a wide variety of
applications ranging from astrophysics to inertial confine-
ment fusion, including multiphase and reacting flows. Here
and hereinafter, we consider only the case of a reflected
shock corresponding in general to a “light →heavy” shock-
contact refraction.
There are two important contributions to the early time,
or small-amplitude linear growth of the instability, before
nonlinear development of the perturbation appears. First, the
baroclinic deposition of vorticity due to the direct interaction
of the incident shock with the interface, when the pressure
gradient at the shock is misaligned with the local density
gradient at the interface. If the initial interface is sharp, it can
therefore be viewed as a vortex sheet that leads to its own
self-induced distortion. The second contribution concerns the
influence of the transmitted and reflected shocks as they
leave density and vorticity perturbations behind them. Relax-
ation of these shock fronts both deposits bulk vorticity and
emits acoustic waves that, by reverberation, modifies the vor-
ticity distribution on the interface. In the weak shock limit,
the linear growth reduces essentially to the first contribution,
while for strong incident shocks, the produced transmitted
shock takes longer time to separate from the interface.
Richtmyer first derived the compressible perturbed equa-
tions and obtained a simple analytical expression for the
asymptotic linear growth rate,
1
assuming that transmitted and
reflected shocks have traveled sufficiently far, compared to
the wavelength of the perturbation, that the second contribu-
tion is subdominant. Other methods concentrating also on the
first contribution have attempted to correct the impulsive
growth rate to better model the behavior for strong incident
shocks or high Atwood ratio without loss of simplicity.
2
Be-
sides, numericists and experimentalists addressed the effect
of shock proximity by using empirical corrections to the im-
pulsive growth rate.
3–5
More complex, semianalytical studies
have taken into account all relevant phenomena
6,7
and
showed good agreement with numerical results obtained by
linearizing the Euler equations between the perturbed inter-
face and transmitted/reflected waves,
8
and with the linear
interaction analysis at low Atwood numbers of Griffond.
9
In what follows, by modeling the proximity of the reced-
ing transmitted and reflected shocks, the analysis in Sec. II
establishes a simple analytical expression for the growth rate
that captures some of the early features of the perturbation
evolution before it has reached the asymptotic growth linear
in time. As analyzed in Sec. III, the solution addresses the
early-time physics of the linear growth, with a characteristic
time
, while allowing for the determination of the
asymptotic, or later-time, growth rate by additional physics.
Section IV compares results to computations obtained from
two-dimensional numerical simulations of the RMI under
various initial conditions. Different realistic combinations of
Atwood ratio and specific heat ratios are tested, as well as
incident shock strength, initial perturbation amplitude, and
wavenumber. A more thorough parametric study of the char-
acteristic time
is presented in Sec. V.
II. ANALYTICAL MODEL
A. General formulation
At t = 0, in Cartesian axes ͑x-z͒, a plane shock traveling
to the left ͑negative z-direction͒ impacts a plane unperturbed
density interface, z=0, separating two fluids of different den-
sity, producing a transmitted shock, and a reflected shock.
a͒
Electronic mail: manuel@caltech.edu.
PHYSICS OF FLUIDS 21, 044104 ͑2009͒
1070-6631/2009/21͑4͒/044104/13/$25.00 © 2009 American Institute of Physics21, 044104-1