1. INTRODUCTION
The Rayleigh–Taylor (RT) and Richtmyer–Meshkov
(RM) instabilities are involved in several physical phenomena,
such as supernovas, inertial confinement fusion (ICF), and
shock tube experiments. Several theoretical works have
described the linear stage of the RM instability (Richtmyer, 1960; Fraley,
1986; Mikaelian, 1994; Yang et al., 1994; Velikovich, 1996; Wouchuk &
Nishihara, 1996; Wouchuk, 2001).
Perturbation methods (Nayfeh, 1973) have been
used by several authors to describe the weakly nonlinear regime of
interfacial instabilities (Ingraham, 1954;
Holyer, 1979; Haan,
1991; Velikovich & Dimonte, 1996;
Zhang & Sohn, 1997, 1999; Berning & Rubenchik,
1998). The small parameter is the initial wave steepness,
a0 k, where a0 and
k are the initial amplitude and the wave number of the
interface, respectively.
In the previous International Workshop on the Physics of Compressible
Turbulent Mixing (IWPCTM) conference, an efficient method for the
perturbation theory in the weakly nonlinear RM instability was
presented; it was a drastic simplification of the work carried out by
Zhang and Sohn (1997). Their work follows
previous work by Holyer (1979) concerning
water waves. Our approach makes perturbation expansions for multimode
configurations easily tractable. Furthermore, the simplicity of the
model has brought out some features of the nonlinear evolution of the
RM instability. It will be shown by comparing these results with
two-dimensional numerical simulations, that these features that have
been derived within the framework of the weakly nonlinear theory hold
for the intermediate nonlinear stage.
The paper is organized as follows. In Section 2, we detail the
perturbation theory. In Sections 3 and 4, it is applied to the mode
selection process, and a class of homothetic perturbations is inferred
for the RM instability. In Section 5, we apply the theory to the RT
instability.
2. PERTURBATION METHODS
Two fluids 1 and 2 with densities ρ and ρ′,
respectively, are studied once the interface has been accelerated. The
flow is assumed to be incompressible and irrotational.
In this part, the interface is considered to be sine shaped (see
Fig. 1). The velocity potentials are Φ
and Φ′ for the fluids 1 and 2, respectively. The Atwood
number is A = (ρ′ − ρ)/(ρ′ +
ρ). Following Holyer (1979), the
phenomenon is described by the motion equation of the interface in each fluid:
Sketch of the configuration.
and the momentum equation at the interface (Bernoulli's equation)
For the RT instability, g is a constant whereas for the RM
instability, g is a Dirac function.
The system to be solved is composed of Eqs. (1)–(3). The
unknowns are the interface shape η(x, t) and the
two potentials Φ(x, z, t) and
Φ′(x, z, t). Provided that
a0 k is smaller than 1, perturbative
expansions may be used. Each unknown is expanded as (Holyer 1979)
The nth term of these series is of the order of
(a0 k)n. To solve the
problem, these terms are further expanded as
The unknowns are now
(aj(n)(t),
bj(n)(t),
b′j(n)(t)).
The expansions (5) are used in Eqs. (1)–(3), and at the order
n, the terms of the order of (a0
k)n are collected. The resulting system is
solved and leads to n + 1 systems of differential
equations.
It turns out that for the RT or RM instabilities, the small parameter
is always multiplied by either eγt or
σt, respectively; γ and σ are the linear growth
rates. It is well known that in such perturbation expansions, an
accurate approximation can usually be built with only the most secular
terms (see, e.g., Bender & Orszag, 1978).
The most secular term is defined as the term with the highest unbounded
time expression. By inspection of the results of the full perturbation
theory (Zhang & Sohn, 1997), up to the
fourth order, we put as general forms for the most secular part of
η(n), Φ(n), and
Φ′(n), for the RM instability:
For the RT instability, eγt and
γ must be used instead of σt and σ, respectively.
The validity of such method for the multimode RM instability has been
verified and reported in Vandenboomgaerde et al. (2002).
3. MODE SELECTION PROCESS IN THE RM
INSTABILITY
The perturbation theory derived in the previous section has a finite
range of validity defined as a0
kσtv = 1. This range of
validity can be increased with summation algorithms such as Padé
approximants. An important issue in nonlinear interaction studies is to
understand the mode selection process. In this section, we will see
that using the approximate theory beyond the weakly nonlinear regime
allows us to obtain qualitative results about the mode selection
process. To study the late time interaction of two modes, we follow the
same methodology as that used in Ofer et al. (1992) for the RT instability. In this paper,
two-dimensional (2D) numerical simulations provide the time evolution
of the shape of the interfaces which are then Fourier analyzed.
For the RM instability, the range of validity of the approximate
theory, checked numerically, is tv =
(a0 kσ)−1. As σ
is proportional to k (Richtmyer,
1960), tv is proportional to
k−2. For practicality, we present in this
paper a nearby mode study in order to have near ranges of validity from
the two modes. In a first step, 2D simulations of RM instability are
performed with the code CADMÉE (Mügler et al.,
1996, 2000). In these
configurations, a 1.26 Mach number shock wave moves from helium to air. The
two wave numbers of the initial interface perturbation are
k1 = 274.889 m−1 and
k2 = 314.159 m−1. The initial amplitudes
for the two modes are either a01 = 10−3
m or a02 = 0.35 × 10−3 m. The
postshocked Atwood number is A = 0.756. The final time of the
2D simulations is 1.8 × 10−3 s. At time as early
as 4 × 10−4 s, mushroomlike features appear at
the interface. This can be seen in Figure 2,
which shows the isocurves of the concentration (c), c
= 0.25, c = 0.5, and c = 0.75 at different times for
the following two-mode configuration: {(k1,
a01),(k2,
a02)}.
One of the two-mode configurations. The two gases are helium and air.
Interface shapes for t = 0, 0.4 ms, 0.8 ms, 1.2 ms, and 1.6
ms.
A spectral analysis of such shapes is performed. Figure 3a,c,e,g shows the results obtained from
numerical simulations and compares the single-mode evolution with the
two-mode growth of k1 and k2
for various initial amplitudes. Full and dashed lines represent single
and two-mode dynamics, respectively.
Single- and two-mode configurations. Growth of the modes
k1 and k2 versus time. Full and
dashed lines represent the single-mode and two-mode dynamics,
respectively. Curves a, c, e, and g are from CADMÉE simulations.
Curves b, d, f, and h are from the approximate perturbation
theory.
Conclusions about RM mode selection process are the same as the ones
obtained in Ofer et al. (1992) about
the RT instability.
Figure 3b,d,f,h shows the growths of
k1 and k2, which are obtained
with the P21 Padé approximant of
the fifth order approximate perturbation expansion. Vertical bars on
each curve indicate the range of validity of the perturbation
expansion. It can be verified that theoretical and numerical results
are in good agreement in the range of validity of the theory. Beyond
that time, the P21 Padé
approximant exhibits an oversaturated behavior: At large time, the
growth from the extended approximate theory is roughly half the
simulation values. However, the following qualitative remarks can be
made: Figures 3a,b deals with the configuration
{(k1, a02),(k2,
a01)}. For the single-mode dynamics, the relative
evolution of the two modes is qualitatively recovered by the theory. In
the simulation of the two-mode dynamics, the mode
k1 is suppressed, whereas the mode
k2 behavior is only slightly modified by the
interaction. In the same way, theoretical results show that the growth
of the mode k1 saturates whereas the mode
k2 is not perturbed.
Such a good qualitative agreement about the mode selection process is
also found for the three other configurations. We believe that this
agreement indicates that the mode selection process is driven by early
time dynamics.
4. CLASS OF HOMOTHETIC INTERFACES IN THE RM
INSTABILITY
The amplitude a(t) of a single-mode interface is
rearranged as follows:
The right-hand side of Eq. (7) can be considered as the Taylor
expansion of a single function F [A,
a0 kσt] . So, when the
same physical parameter values for two configurations 1 and 2 are used
(same Atwood number), and as σ is proportional to k, Eq.
(7) gives
Equation (8) defines a class of homothetic configurations: For a
given set of physical parameters, the growth of one configuration
allows us to determine the growth of all the configurations of the same
class.
To confirm this conclusion, we have run two simulations with the
CADMÉE code. In these simulations, a 1.0962 Mach number shock
moves from helium to air. The postshocked Atwood number is 0.764. The
two configurations are: (a01,
k1) = (0.35 × 10−3 m,
274.855 m−1) and (a02,
k2) = (0.525 × 10−3 m,
224.418 m−1), which satisfies the equality
a01 k12 =
a02 k22. Figure 4a shows the growths of the perturbations
obtained from the two-dimensional simulations. Full and mark-dotted
lines represent configurations 1 and 2, respectively. Figure 4b shows the growths of the perturbation 1,
obtained either from the simulation (full line) or from Eq. (8).
Homothetic single-mode dynamics. a: Growths of the perturbations 1
and 2 are plotted with full and mark-dotted lines, respectively.
Results are obtained with the CADMÉE simulation. b: The growth
of the perturbation 1. The full line is from the CADMÉE
simulations. The mark-dotted curve is obtained from configuration 2 by
using Eq. (8).
Numerical and theoretical results cannot be distinguished. The range
of validity of the approximate theory for these two configurations is
tvs = 0.72 × 10−3
s. Figure 4b shows that Eq. (8) is valid in
the intermediate nonlinear regime. Furthermore, even when mushroom
shapes appear, the homothetic transform allows us to determine one
interface shape from the other one. Figure 5a
shows the shapes of interfaces 1 and 2 at t = 2.87 ×
10−3 s (full and mark-dotted lines, respectively).
Homothetic single-mode dynamics at t = 2.87 ms. a: Shape of
perturbations 1 and 2 are plotted with full and mark-dotted lines,
respectively. b: Shape of perturbation 1. The full line is from
simulation. The mark-dotted curve is obtained from a homothetic
transform of shape 2.
We have applied z and x homotheties of scale
k2 /k1 = 0.82 to shape 2.
The result is plotted in the mark-dotted line in Figure 5b. It has to be compared with shape 1 in
full line. Once again, the agreement is excellent even for the vortex
structures of the mushroom.
5. SINGLE-MODE WEAKLY NONLINEAR RT
INSTABILITY
As explained in Section 2, simplified results can also be obtained
for the RT instability. The range of validity of the expansions is now
defined as a0 k exp(γt) = 1.
The same algorithm is used as for the RM instability.
To check the accuracy of the approximate perturbation method, its
predictions have been compared with simulations done by Menikoff and
Zemach (1983). The physical parameters are
g = 1, A = 1, k = 1, a0
= −0.01. This leads to tv = 4.6. In
Figure 6, the amplitude of the spike and the
bubble are plotted versus time. The 9th- and 11th-order approximate
perturbation theory and the numerical results are in full and dashed
lines, respectively.
Growths of the tips of the spike and the bubble. Ninth- and
11th-order perturbation results and numerical ones are plotted in full
and dashed lines, respectively.
For t slightly greater that tv,
the series diverges. However, within the range of validity, numerical
and theoretical curves cannot be distinguished.
A qualitative comparison has been made with some results of Baker
et al. (1980) concerning the
influence of the Atwood number on the growth of the spike and bubble.
For a small Atwood number, the spike and bubble have the same nonlinear
growth; on the other hand, for an Atwood number close to 1, the spike
grows faster than the bubble. The same conclusions can be found using
the approximate perturbation theory.
6. CONCLUDING REMARKS
We have developed an efficient way of deriving a perturbation method
for the Richtmyer–Meshkov instability in the nonlinear regime.
This model holds provided the flows are potential and incompressible.
Solutions have been obtained as perturbation series. Such expansions
can be drastically simplified by retaining, at each order, the terms
with the highest power in time. Nonlinear behavior of interfaces can
then be easily calculated even if high-order expansions are required.
Several conclusions can be drawn from this study:
- The range of validity of the theory, checked numerically, is
tv with a0
kσtv = 1 for the single-mode
case.
- It turns out that the selection process in the nonlinear regime can
be predicted from the weakly nonlinear theory. Padé
approximants, despite an oversaturated growth for the mushroomlike
stage, allow us to predict the wave number of the leading mode. This
has been verified with comparisons between theory and simulation for a
two-mode interface.
- Within such a theory, we are able to define homothetic interfaces.
For single-mode configurations, homothetic interfaces have the same
values of a0 k2 and A.
For two perturbations of the same class, the ratio of the
time-dependent amplitudes is the ratio of the wavelengths. Moreover, 2D
simulations in the intermediate nonlinear regime have shown that the
shapes of single-mode interfaces scale as the ratio of the
wavelengths.
The kind of solution technique that has been used for the RM
instability is appropriate to describe the weakly nonlinear regime of a
single-mode RT instability. This has been checked against numerical
simulation.
