[Q.zhang S.sohn]-Quantitative Theory of Richtmyer-Meshkov Instability in Three Dimensions(1996)
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Transcript of [Q.zhang S.sohn]-Quantitative Theory of Richtmyer-Meshkov Instability in Three Dimensions(1996)
QUANTITATIVE THEORY OF RICHTMYER-MESHKOVINSTABILITY IN THREE DIMENSIONS
Qiang Zhang and Sung-Ik Sohn
Department of Applied Mathematics and StatisticsSUNY at Stony Brook
Stony Brook, NY 11794-3600
ABSTRACT
A material interface between two fluids of different density accelerated by a
shock wave is unstable. This instability is known as Richtmyer-Meshkov (RM)
instability. Previous theoretical and numerical studies primarily focused on
fluids in two dimensions. In this paper, we present the studies of Richtmyer-
Meshkov instability in three dimensions in rectangular coordinates. There are
three main results: (1) The analysis of the linear theory of the Richtmyer-
Meshkov instability for both reflected shock and reflected rarefaction wave cases.
(2) Derivations of nonlinear perturbative solutions for the incompressible RM
instability (evaluated explicitly for the impulsive model through the third order).
(3) A quantitative nonlinear theory of the compressible Richtmyer-Meshkov ins-
tability from early to later times. Our nonlinear theory contains no free parame-
ter and provides analytical predictions for the overall growth rate, as well as the
growth rates of bubble and spike, of Richtmyer-Meshkov unstable interfaces.
1. Introduction
Richtmyer-Meshkov instability is a fingering instability which occurs at a material interface
accelerated by a shock wave. It plays an important role in studies of supernova and inertial
confinement fusion (ICF). The occurrence of this interfacial instability was first predicted theoret-
ically by Richtmyer [21] in 1960, and ten years later, confirmed experimentally by Meshkov [15].
Since then more experiments on Richtmyer-Meshkov instability have been conducted [1,4] and
several numerical simulations on the nonlinear growth rate of the RM unstable interfaces have
been performed [2,6,8,11,16-18,25]. Several theories have been developed by different
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approaches [7-10,22-24,26,27]. For long time, theories failed to give a quantitatively correct
prediction for the growth rate of RM unstable interface. Recently, the authors have developed a
quantitative nonlinear theory in two dimensions [26,27]. It provides analytic predictions for the
growth rate and amplitude of RM unstable interfaces for the case of a reflected shock. The
theoretical predictions are in excellent agreement with the results of full numerical simulations
and the experimental data [26,27]. In this paper, we present a quantitative nonlinear theory of the
growth rate of RM unstable interface in three dimensions for the case of a reflected shock.
To the author’s knowledge, little is known about the RM instability in three dimensions.
Cloutman and Wehner performed numerical simulations in two and three dimensions [6]. Only
the growth rate averaged over time was presented in [6] and the growth rate in two dimensions is
about two times larger than the experimental result. As we will see in this paper, it is important to
determine the growth rate as a function of time. Youngs has performed numerical studies of RM
turbulent mixing in three dimensions [25]. In reality, the RM instability often occurs in three
dimensions. Experimentally, it is difficult to setup a single mode disturbance at the interface in
three dimensions. Computationally, it is considerablely more expensive to perform full numeri-
cal simulations in three dimensions than that in two dimensions. These facts make the theoretical
study of the RM instability in three dimensions more important.
The development of RM unstable interfaces follows four stages. The first stage is a wave
bifurcation stage. In this stage, The incident shock wave collides with a perturbed material inter-
face and bifurcates into a transmitted shock and a reflected wave. Depending on material proper-
ties of the fluids on both sides of the interface and the incident shock strength, the reflected wave
can be either a shock or a rarefaction wave. The criterion for which reflected wave type to occur
is given in [24]. For most of real gases, the criterion is that if ρ1c 1 > ρ2c 2 the reflected wave is
a shock. Otherwise it is a rarefaction wave. Here ρi is the density and ci the speed of sound. The
subscript labels the material. The shock incidents from material 2 to material 1. If a disturbance is
presented at the material interface at the arrival of the incident shock, the transmitted shock and
the reflected wave will pick up such disturbance and move away from the material interface. The
duration of the wave bifurcation stage is very short.
The second stage is a linear stage. Accelerated by the shock, the material interface becomes
unstable and the disturbance at the interface starts to grow. Bubbles and spikes are formed at the
unstable material interfaces. A bubble is a portion of the light fluid penetrating into heavy fluid
and a spike is a portion of the heavy fluid penetrating into light fluid. Bubbles and spikes move
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in opposite directions. In the linear stage, the amplitudes of spike and bubble, relative to the wave
length of the perturbation, are small. Therefore, one can linearize Euler equations in terms of the
amplitude of the disturbance. Richtmyer developed the linear theory for the case of reflected
shock. The linear theory for the case of a reflected rarefaction wave was developed by one of the
authors and his coworkers [24]. The linear analysis of RM instability so far is confined in two
dimensions. We carry out the linear analysis of RM instability in three dimensions in this paper
(see Section 3).
Richtmyer’s impulsive model is a widely used theoretical model for the growth rate of RM
unstable interfaces [21]. The impulsive model is an approximation for the asymptotic growth rate
of the interface in the linear theory. The model approximates the incident shock as an impulse (a
delta function) and the post-shocked fluid as incompressible. The impulse occurs at the time at
which the incident shock hits the material interface. The strength of the impulse depends on the
strength of the incident shock and the material properties of the fluids. As the impulse (the
incident shock) passes through the material interface, it sets the linear growth rate of the distur-
bance and the linear growth rate remains same afterwards. The linear growth rate of the RM
unstable interface predicted by the impulsive model is given by
v imp = − ∆uAka 0 ,
where ∆u is the difference between the shocked and unshocked mean interface velocities,
A = (ρ′ − ρ)/(ρ′ + ρ) is the Atwood number. Both ρ and ρ′ are the post-shocked fluid densities.
The incident shock propagates from fluid of density ρ to fluid of density ρ′. a 0 is the post-
shocked perturbation amplitude at the interface. In his path breaking work, Richtmyer showed
three examples in which the predictions of the impulsive model agree quite well with the results
of the linear theory. A more extensive comparison between the results of the impulsive model
and the results of the linear theory over a large parameter space showed the domains of agree-
ment and disagreement [24]. Even when the prediction of the impulsive model agrees with the
result of the linear theory, it agrees in the regime where the nonlinearity is important and the
linear theory is no longer valid.
At the linear stage, the amplitude of spike and that of bubble are approximately equal. The
shapes of the spike and bubble are similar. The interface remains approximately sinusoidal. The
duration of the linear stage is longer than the bifurcation stage, but much shorter than the third
stage, the nonlinear stage.
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In the nonlinear stage, the spike grows faster than the bubble. The interface is no longer
sinusoidal and wave modes which do not present in the initial perturbation appear. Very often,
the shape of the spike becomes mushroom like. The linear theory predicts that both spike and
bubble have a constant asymptotic growth rate. However, the magnitudes of the growth rates of
the spike and bubble decay with time in the nonlinear stage. Most theoretical studies in the past
were focused on the linear stage rather than the nonlinear stage which is considerablely more
important. The duration of the nonlinear stage is much longer than that of linear stage.
Recently, a quantitative nonlinear theory of the RM instability for compressible fluids in
two dimensions has been developed by the authors [26,27] for the case of reflected shock. The
predictions from the nonlinear theory are in remarkable agreements with the results of full non-
linear numerical simulations and experimental data from linear stage to nonlinear stage. The
theory shows that decay of the growth rate of the unstable interface is due to nonlinearity rather
than compressibility. The theory also shows that the regime where the growth rate reaches a
peak is a transition regime. In that regime, the system changes from a compressible, approxi-
mately linear one to a non-linear, approximately incompressible one.
The fourth stage is a turbulent stage. In this stage, spike may pinch off to form droplets.
Secondary instability at the interface becomes pronounced. The three dimensional effects become
important. The results from numerical simulations for fluids in two dimensions are not reliable
because in reality the fluid in the three dimensions is no longer homogeneous. The physics of the
fluid in this stage is much more complicated than other stages.
The theory which we present in this paper is for the development of the RM instability
from linear to nonlinear stages in three dimension. We present three main results:
(I) The analysis of linear theory of Richtmyer-Meshkov instability in three dimensions for
both reflected shock and reflected rarefaction wave cases.
(II) Derivations of nonlinear perturbative solutions for incompressible RM instability in
three dimensions (evaluated explicitly for the impulsive model through the third order).
(III) A quantitative nonlinear theory of compressible Richtmyer-Meshkov instability in
three dimensions from early to later times.
The results of (I) and (II) are important. However, the goal of our theoretical study is (III)
which is more important. The results of (I) and (II) play essential roles in (III). We explain in
next section how the results of (I) and (II) are related to (III).
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2. Physical Picture, Mathematical Theory and Final Results
Here we describe the physical picture and the mathematical theory on which our theoretical
work is based and present the final results of our quantitative nonlinear theory of the RM instabil-
ity in three dimensions.
We adopt the following physical picture for the dynamics of the RM unstable interfaces.
The dominant effects of the compressibility occur near the shocks. This influences the material
interface at the bifurcation and linear stages. At early times, the transmitted shock and reflect
wave are in the vicinity of the material interface and magnitude of the disturbance is small.
Therefore, the compressibility is important and the nonlinearity is less important. As time
evolves, the magnitude of the disturbance at the material interface increases significantly and the
transmitted shock and reflected wave move away from the interface. The effects of compressibil-
ity are reduced and the nonlinearity starts to play a dominant role in the interfacial dynamics.
From this physical picture, we see that at early times the dynamics of the system are mainly
governed by the linearized Euler equations for compressible fluids, while at later times the
dynamics are mainly governed by the nonlinear equations for incompressible fluids. The RM
unstable system goes through a transition from a compressible and linear one at early times to a
nonlinear and incompressible one at later times. Finally we match the solution of compressible
linear theory and the solution of the incompressible nonlinear theory to obtain an analytical
expression which changes gradually from one to the other. The matched solution is the final
result of our nonlinear theory which predicts the growth rate of unstable material interface
between compressible fluids from early to later times.
For fluids in the two dimensions, the solution of the linear theory can be found in [21] and
[24]. The solution of linear theory in three dimensions will be derived in this paper.
The mathematical tool which we use to construct an approximate nonlinear solution for
incompressible fluids is Pade approximation. The analysis contains two steps. The first step is
the derivation of the generating series and the second step is the construction of Pade approxi-
mants. To determine the generating series, we expand all physical quantities in terms of the
powers of a 0 , the amplitude of the initial perturbation. We derive general formulae for the solu-
tions of n −th order quantities. Our solution procedure is recursive: the n −th order solutions are
expressed in terms of the solutions of orders less than n. We carried the calculation explicitly
through third order. The result can be regrouped to form a polynomial of the temporal variable t.
We then apply Pade approximation to the finite Taylor’s series to extend its range of validity
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beyond that of Taylor’s series itself.
The approach we outlined above for the RM instability has been remarkably successful for
fluids in two dimensions [26,27]. We demonstrate the results of our theory in two dimensions in
Figure 1. In Figure 1, a shock wave of Mach 1.2 incidents from air to SF6 . Here the reflected
wave is a shock. The initial amplitude of the perturbation is 2.4 mm, the wave length is 37.5 mm
and the pressure ahead of the shock is 0.8 bar. The post-shock Atwood number is A = 0.701.
These parameters corresponds to Benjamin’s experiments [4] on air-SF6. The solution of the
linear theory for compressible fluids, the approximate nonlinear solution for incompressible fluids
and the solution of the nonlinear theory for compressible fluids (obtained from matching) are
shown in Figure 1. The result from full nonlinear numerical simulation is also shown. At early
times the growth rate increases fast and reaches the highest peak. After the peak, the nonlinearity
starts to play a dominant role and causes the growth rate to decay with time. We define tp to be
the time associated with the highest peak and divide the time evolution of the growth rate into
two stages separated at tp . In figure 1, tp is about 150µs. The dominant dynamical behavior of
these two stages is different. For t < tp , the system is compressible and approximately linear.
For tp < t, the system is nonlinear and approximately incompressible. Figure 1 demonstrates
clearly, that the system changes gradually from a compressible and linear one to an incompressi-
ble and nonlinear one. Our theoretical prediction for the growth rate of RM unstable interface in
two dimensions is in excellent agreement with the result from full numerical simulation, as well
as with the experimental data. Benjamin has reported a growth rate of 9.2 m/s over the time
period 310-750 µs for air-SF6 experiments. The experimental result was obtained by a linear
regression analysis of the amplitude of the disturbance at the material interface. Applying the
same analysis, our theory predicted a growth rate of 9.3 m/s over the same time period [26].
Predictions of the growth rate from the impulsive model and from the linear theory are 15.6 m/s
and 16.0 m/s, respectively.
The physical picture and mathematical methods which we outlined above are applicable for
fluids in both two and three dimensions. In this paper we extend the nonlinear theory of the RM
instability to three dimensions. Our final results, given below, are analytical expressions for the
nonlinear growth rates of RM unstable interface of compressible fluids in three dimensions,
v =1 + a 0k 2vlinλ1t + max{0, a0
2k 2λ12 − λ2}k 2vlin
2 t 2
vlinhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh , (1)
v bb = −v +1+vlina 0k 2λ4λ3
−1t + vlin2 k 2(a0
2k 2λ42λ3
−2 + λ5λ3−1)t 2
vlin2 kλ3thhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh , (2)
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v sp = v +1+vlina 0k 2λ4λ3
−1t + vlin2 k 2(a0
2k 2λ42λ3
−2 + λ5λ3−1)t 2
vlin2 kλ3thhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh . (3)
Equation (1) is for the overall growth rate of the unstable interface defined as the growth rate of
the half of the distance between the bubble and spike. Equation (2) and (3) are the growth rates
of the bubble and spike, respectively. The quantitative predictions of (1)-(3) are presented in Sec-
tion 6. Here vlin is the growth rate of the linear theory in three dimensions and will be derived in
Section 3. a 0 is the initial post-shocked amplitude of perturbation at the material interface. k is
the magnitude of a wave vector of initial perturbation at the material interface. λ1, λ2, λ3, λ4 and
λ5 are dimensionless functions which depend on the post-shock Atwood number A and the polar
angle θ of the wave vector (kx, ky). The explicit expressions of λ1 and λ2 are derived in Section
4. The explicit expressions of λ3, λ4 and λ5 are given in Appendix B.
It is easy to see that in the early time, or small amplitude limits, v, −vbb and vsp approach to
vlin , as physically it should be. We comment that the predictions of our nonlinear theory given by
(1)-(3) contain no adjustable parameter and only applicable to the systems with no indirect phase
inversion. An indirect phase inversion refers to the situation where the fingers at the contact
interface gradually reverse their penetration directions after the shock-contact interaction. A
direct phase inversion corresponds to the situation where the penetration directions of the fingers
at the contact interface are reversed before the completion of the shock-contact interaction. For
the case of a reflected shock, as we consider in this paper, the indirect phase inversion usually
does not occur [24]. When the incident shock is strong enough and the adiabatic exponents of the
two fluids are different, the indirect phase inversion may occur. The direct phase inversion does
not occur for the case of a reflected shock. See [24] for more discussion about the phase inver-
sions.
In Section 3, we derive the linear theory of the RM instability for compressible fluids in
three dimensions, In Section 4, we derive the nonlinear perturbation solution for incompressible
fluids in three dimensions. This perturbation solution serves as a generating series for Pade
approximations. In Section 5, we apply Pade approximation and develop a nonlinear theory of
the RM instability for compressible fluid. In Section 6, we present the quantitative predictions of
the nonlinear theory in two cases: air-SF6 unstable interfaces and Kr-Xe unstable interfaces.
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3. Linear Theory for Compressible Fluids in Three Dimensions
As we have discussed in Section 2, at early times the unstable system is compressible and
approximately linear when the disturbance at the initial interface is small. The existing linear
theories for compressible fluids are in two dimensions only [21,24]. In this section we formulate
the governing equations and the boundary conditions for the linear theory of Richtmyer-Meshkov
instability for compressible fluids in three dimensions. The equations are derived by linearizing
the full Euler equations in three dimensions. Our theoretical derivation shows that the solutions of
the linear theory in three dimensions can be mapped from solutions of the linear theory in two
dimensions. In particular, for fixed total wave number k, the growth rates determined from the
linear theory in two dimensions and that in three dimensions are identical.
In the linear regime, the shape of the interface in three dimensions is given by
a (t)cos(kxx)cos(kyy), where kx and ky are wave numbers of the sinusoidal perturbation in x and y
directions, respectively. From the identity a (t)cos(kxx)cos(kyy) =21hha (t)[cos(kxx + kyy) +
cos(kxx − kyy)], one might think that one could apply the linear theory in two dimensions to each
cosine function on the right hand side separately and the sum of them determines the linear
theory in three dimensions. This argument may appeal intuitively, but is lack of mathematical
foundation, since the superposition theorem only applicable to terms of different modes of same
spacial variable, but not applicable to terms of different spacial variables. Here kxx + kyy and
kxx − kyy are two independent spacial variables. Therefore, it is necessary to derive the linear
theory for the RM instability in three dimensions.
As the incident shock hits the material interface, it bifurcates into a transmitted shock and a
reflected wave. Depending the material property of the fluids across the contact interface and the
incident shock strength, the reflected wave is either a shock or a rarefaction wave [24]. The linear
theories of the RM instability in three dimensions for these two different types of reflected waves
are given separately below.
3.A. Case of a reflected shock
We present the derivation for the case of reflected shock first. After shock-contact interac-
tion, the physical domain is divided into four regions separated by the transmitted shock, the con-
tact interface, and the reflected shock. We have labeled these regions 0 to 3, from bottom to top.
(See Figure 2(a) and 2(b).) The analysis is similar to Richtmyer’s [21]. We denote the dynamical
and the thermodynamical variables as follows: u→= (ux,uy,uz) = velocity; ρ = density; c = sound
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speed; p = pressure; s = specific entropy. A subscript on there variables denotes the region
number, while the superscript 0 or 1 denotes the zeroth order (unperturbed) or the first order
quantities, respectively. The reference frame is now chosen such that the contact interface is sta-
tionary after the shock-contact interaction, i.e. u10 = u2
0 = 0. In such a reference frame, the speeds
of the transmitted and reflected shock waves are denoted as w 1 and w 2 (both positive numbers),
respectively.
In regions 0 and 3, we have u00 > 0 and u3
0 < 0. In addition, all perturbed quantities are
zero in these two regions. In regions 1 and 2, the zeroth order quantities are the solutions of one
dimensional Riemann problem.
For the linear theory in three dimensions, velocity u→(x,y,z,t) and any other quantities
Q(x,y,z,t) in regions 1 and 2 can be expressed as
u→(x,y,z,t) = (ux1(z,t)sin(kxx)cos(kyy), uy
1(z,t)cos(kxx)sin(kyy), uz1(z,t)cos(kxx)cos(kyy)), (4)
Q(x,y,z,t) = Q0(z,t) + Q1(z,t)cos(kxx)cos(kyy). (5)
After linearization, the equation of continuity is
−ρ0(∂z
∂uz1
hhhh + kxux1 + kyuy
1) =∂t
∂ρ1hhhh =
c 2
1hhh∂t
∂p 1hhhh (6)
and the equations of motion are
−∂z
∂p 1hhhh = ρ0
∂t
∂uz1
hhhh , kxp 1 = ρ0
∂t
∂ux1
hhhh , kyp 1 = ρ0
∂t
∂uy1
hhhh . (7)
Differentiating (6) and substituting equations (7) into it, the pressure disturbance satisfies the
equation
∂t 2
∂2p 1hhhhh = c 2
IJL ∂z 2
∂2p 1hhhhh − k 2p 1
MJO
(8)
where k = (kx2 + ky
2)1/2 . Eqs. (6)-(8) hold for both regions 1 and 2. We have suppressed the sub-
script representing the region in Eqs. (6)-(8).
In addition to Eq. (8) for the perturbed pressure, we need to know the boundary conditions
at the contact interface, and the transmitted and the reflected shock waves. In the following, a 0(t),
a 1(t) and a 2(t) denote the perturbation amplitudes on the contact interface, and on the transmit-
ted and reflected shock waves, respectively.
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Let us consider the boundary conditions at the contact interface first. Continuity of the
pressure across the interface gives
p11 (0,t) = p2
1 (0,t). (9)
Furthermore, Newton’s second law applied to a fluid particle next to the interface gives
dt 2
d 2a 0(t)hhhhhhh = −ρ1
0
1hhhRJQ ∂z
∂p11
hhhhHJPz = 0−
= −ρ2
0
1hhhRJQ ∂z
∂p 21
hhhhhHJPz = 0+
. (10)
There are three boundary conditions at a shock interface. The tangential components of the
fluid velocity u→are continuous across the shock and the normal component of the fluid velocity
satisfies the Rankine-Hugoniot condition at the shock interface. To analyze these boundary con-
ditions, we define a normal vector n→ and two tangential vectors, t→1 and t→2 , at a shock front
z = f(x,y,t):
n→(x,y,t) = ( −∂x∂ fhhh , −
∂y∂ fhhh , 1), t→1(x,y,t) = e→y x n→, t→2(x,y,t) = n→x t→1 ,
where e→y is the y-axis unit vector. Here f is a first order quantity. The normalization factor of n→,t→1
and t→2 is 1, because (1 + fx2 + fy
2)−1/2 = 1 + O((a 0k)2).
Substituting f(x,y,t) = a 1(t)cos(kxx)cos(kyy) for the transmitted shock, it gives
n→= (a 1(t)kxsin(kxx)cos(kyy), a 1(t)kycos(kxx)sin(kyy), 1),
t→1 = (1, 0, − a 1(t)kxsin(kxx)cos(kyy)),
t→2 = (0, 1, − a 1(t)kycos(kxx)sin(kyy)).
Hence, up to first order of small quantities, the normal component of u→ is given by
n→.u→= uz(w 1t,t). Following Richtmyer’s derivation, we have the following equation for the
linearized Rankine-Hugoniot condition at the transmitted shock:
a.
1(t) = −2(ρ1
0 − ρ00)
1hhhhhhhhhhIKL w 1
1hhhh −K 1c1
2
w 1hhhhhMNOp1
1 (−w 1t,t), (11)
where K 1 is the dimensionless slopes of the Hugoniot in the P −V plane evaluated at the state 1,
i.e., for a given state (state 0). If we denote its Hugoniot in P −V plane as Ph(V), then we have
K 1 = −(ρ1
0c 1)2
1hhhhhhhRJQ dV
dPh(V)hhhhhhhHJPV=1/ρ1
0
.
Up to first order, the tangential component of u→projected onto t→1 is ux1(−w 1t,t)sin(kxx)cos(kyy)
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just behind the shock, and −u00a 1kxsin(kxx)cos(kyy) just ahead. Similarly, the tangential com-
ponent of u→ projected onto t→2 is uy1(−w 1t,t)cos(kxx)sin(kyy) just behind the shock, and
−u00a 1kycos(kxx)sin(kyy) just ahead. Therefore, from the continuity of tangential velocity at the
interface of the transmitted shock, we have
ux1(−w 1t,t) = − kxu0
0a 1(t), uy1(−w 1t,t) = − kyu0
0a 1(t),
which gives
(kxux1 + kyuy
1) = − k 2u00a 1 at z = −w 1t. (12)
When we combine (6), (7), (11) and (12), we obtain another boundary condition at the transmit-
ted shock:
IKLw 1 +
2w 1
c12
hhhhh +2K 1
w 1hhhhhMNO dt
dp11 (−w 1t,t)hhhhhhhhhhh = −(w1
2 − c12 )
RJQ ∂z
∂p11
hhhhHJPz=−w 1t
+ k 2c12u0
0w 1ρ10a 1(t). (13)
Boundary conditions at the interface of the reflected shock is similar to Eqs. (11) and (13).
They are given by
a.
2(t) =2(ρ2
0 − ρ30)
1hhhhhhhhhhIKL w 2
1hhhh −K 2c2
2
w 2hhhhhMNOp2
1 (w 2t,t), (14)
IKLw 2 +
2w 2
c22
hhhhh +2K 2
w 2hhhhhMNO dt
dp21 (w 2t,t)hhhhhhhhhh = (w2
2 − c22 )
RJQ ∂z
∂p21
hhhhHJPz=w 2t
+ k 2c22u3
0w 2ρ20a 2(t). (15)
The definition of K 2 is the same that for K 1 except that subscrip 1 is replaced by subscript 2 in ρ
and c.
Let us summarize the equations for the linear theory in three dimensions for the case of
reflected shock. The solutions of the linear theory in three dimensions for the case of reflected
shock are determined by (8)-(11), (13)-(15) and the initial conditions. Equation (8) is for the
solutions in regions 1 and 2. Equations (9) and (10) are the boundary conditions at the material
interface. Equations (11) and (13) are the boundary conditions at the transmitted shock. Equa-
tions (14) and (15) are the boundary conditions at the reflected shock. The linear theory in two
dimensions is a special case of the linear theory in three dimensions, name the case of kx = 0 or
ky = 0. However, only k, the magnitude of the wave vector, appears these equations. Neither kx,
nor ky appear explicitly in these equations. Therefore, the solutions of the linear theory in three
dimensions can be mapped from the solutions of the linear theory in two dimensions. This map-
ping is given at the end of this section.
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3.B. Case of a reflected rarefaction wave
Here we derive the linear theory of the RM instability in three dimensions for the case of
reflected rarefaction wave. In this case, configuration of the system after the wave bifurcation
contains five regions. The definition of regions 0,1,2 are 3 are same as the one for the case of
reflected shock. An additional region inside the rarefaction fan is denoted by the subscript r.
Obviously, the dynamics in region 1 and 2 are still governed by (8). The boundary conditions at
the material interface are still governed by (9) and (10). The boundary conditions at the interface
of the transmitted shock are still governed by (11) and (13). We only need to derive the equa-
tions which govern the dynamics inside the rarefaction fan and boundary conditions at the lead-
ing edge and the trailing edge of the rarefaction fan. Since the rest of equations in this section
deal with the solution inside the rarefaction fan and boundary conditions at the leading and trail-
ing edges of the rarefaction fan, we will suppress the subscript r in the rest of equations in this
section.
The solutions inside the rarefaction fan still have the functional form given by (4) and (5).
It is important to note that inside the rarefaction fan, the zeroth order (unperturbed) density and
pressure are not constants, but the functions of z. The zero-th order velocity is given by
u 0→= (0,0,uz
0(z)). Now we derive the dynamical equations inside the rarefaction fan and boun-
dary conditions at the leading and trailing edges of the rarefaction fan. The linearized continuity
equation is given by
∂t∂ρ1hhhh + kxux
1ρ0 + kyuy1ρ0 + uz
0
∂z∂ρ1hhhh + ρ1
∂z
∂uz0
hhhh + uz1
∂z∂ρ0hhhh + ρ0
∂z
∂uz1
hhhh = 0. (16)
The conservation of momentum gives
∂t
∂ux1
hhhh + uz0
∂z
∂ux1
hhhh −ρ0
1hhhkxp 1 = 0, (17)
∂t
∂uy1
hhhh + uz0
∂z
∂uy1
hhhh −ρ0
1hhhkyp 1 = 0 (18)
and
∂t
∂uz1
hhhh + uz0
∂z
∂uz1
hhhh + uz1
∂z
∂uz0
hhhh −(ρ0)2
1hhhhh∂z
∂p 0hhhhρ1 +
ρ0
1hhh∂z
∂p 1hhhh = 0. (19)
The linearized energy equation can be expressed as
∂t∂s 1hhhh + uz
0
∂z∂s 1hhhh = 0. (20)
- 13 -
p 1 , ρ1 and s 1 are related through the thermodynamic identity
p 1 = c 2ρ1 +IJL ∂s 0
∂p 0hhhh
MJOρ0
s 1 . (21)
These are the dynamical equations inside the rarefaction fan. Following a procedure similar to
the one given in [24] for the linear theory of the RM instability in two dimensions, we obtain
s 1 = 0 inside the rarefaction fan. Therefore, the solution to (20) is trivial.
We examine the boundary conditions at the leading and trailing edges of the rarefaction fan
next. At these edges all physical quantities are continuous. Let zl(t) and zt(t) denote the posi-
tions of the leading edge and trailing edge, respectively. Let al(t) and at(t) denote the amplitudes
of perturbation at the leading edge and trailing edge, respectively. At the leading edge, the con-
tinuity of density, normal velocity and tangential velocities give the following four equations
ρ1(zl(t),t) = −al(t) ∂z
∂ρ0(zl ,t)hhhhhhhh , (22)
uz1(zl(t),t) = −al(t) ∂z
∂uz0(zl ,t)hhhhhhhh , (23)
ux1(zl(t),t) = 0 (24)
and
uy1(zl(t),t) = 0. (25)
In (22)-(25), we have used the fact that the in region 3 zero-th order quantities are constants and
the first order quantities are zero.
At the trailing edge, the continuity of density, normal velocity and tangential velocities give
the following four equations
ρ1(zt(t),t) = ρ21(zr(t),t) − at(t) ∂z
∂ρ0(zt ,t)hhhhhhhh , (26)
uz1(zt(t),t) = uz 2
1(zt(t),t) − at(t) ∂z
∂uz0(zt ,t)hhhhhhhh , (27)
ux1(zt(t),t) = ux 2
1(zt(t),t), (28)
and
uy1(zt(t),t) = uy 2
1(zt(t),t), (29)
- 14 -
respectively. In (26)-(29), we have expressed the labeling for the quantities of region 2 explicitly.
We define
uk = e→k.v→= (kxux + kyuy)/k. (30)
Here e→k is a unit vector along the direction of k→
and k = √dddddkx2 + ky
2 is the magnitude of k→
. Then
(16) can be expressed as
∂t∂ρ1hhhh + kuk
1ρ0 + uz0
∂z∂ρ1hhhh + ρ1
∂z
∂uz0
hhhh + uz1
∂z∂ρ0hhhh + ρ0
∂z
∂uz1
hhhh = 0 (31)
and (19) and (20) can be combined into a single equation
∂t
∂uk1
hhhh + uz0
∂z
∂uk1
hhhh −ρ0
1hhhkp 1 = 0 (32)
The continuity of tangential velocities at the leading edge can be expressed as
uk1(zl(t),t) = 0 (33)
The continuity of tangential velocities at the trailing edge can be expressed as
uk1(zr(t),t) = uk 2
1(zr(t),t). (34)
Let us summarize the equations for the linear theory in three dimensions for the case of
reflected rarefaction wave. The solutions of the linear theory in three dimensions in the case of
reflected rarefaction wave are determined by (8)-(11), (13), (16),(19), (22), (23), (26), (27), (31),
(33), (34), the relation p 1 = c 2ρ1 and the initial conditions. Equation (8) is for the solutions in
regions 1 and 2. Equations (9) and (10) are the boundary conditions at the material interface.
Equations (11) and (13) are the boundary conditions at the transmitted shock. Equations (16),
(19) and (31) together with the relation p 1 = c 2ρ1 determine the solutions inside the rarefaction
fan. Equations (22), (23) and (33) are the boundary conditions at the leading edge of the rarefac-
tion fan. Equations (26), (27) and (34) are the boundary conditions at the trailing edge of the
rarefaction fan. As in the case of reflect shock, only k, the magnitude of the wave vector, appears
these equations. Neither kx, nor ky appears explicitly in these equations. Therefore, for the case
of reflected rarefaction wave, the solutions of the linear theory in three dimensions can also be
mapped from the solutions in two dimensions. One could further simplify the boundary condi-
tions at the leading and trailing edges of the rarefaction fan. For example, one can show that
dal(t)/dt = 0. Since we have proven that the solutions in the three dimensions can be mapped
from the solutions in two dimensions, such simplification is no longer necessary.
- 15 -
Now we derive the mapping from the solutions of linear theory in two dimensions to the
solutions of linear theory in three dimensions. To find the solution of linear theory in three
dimensions with initial perturbation a 0(0)cos(kxx)cos (kyy), we consider the solution of the linear
theory in two dimensions with an initial perturbation a 0(0)cos(kx). Here k = √dddddkx2 + ky
2 . The solu-
tion in two dimensions is given by
ρ(2d)lin (x,z,t) = ρ(2d)
1 (z,t)cos(kx),
p(2d)lin (x,z,t) = p(2d)
1 (z,t)cos(kx),
ux (2d)lin (x,z,t) = ux (2d)
1 (z,t)sin(kx),
uz (2d)lin (x,z,t) = uz (2d)
1 (z,t)cos(kx).
The shapes of the interfaces in two dimensions are given by zi(t) + ai(2d)(t)cos(kx) for i = 0, 1
and 2 or l, r. We have shown that the governing equations, (8)-(11), (13)-(15), (19), (22), (23),
(26), (27) and (31) in three dimensions depend neither on kx, nor on ky explicitly. One can see
from these equations and (6), (7), (12), (16)-(18), (22)-(30) that p(3d)1 ,ρ(3d)
1 ,uz (3d)1 and ai (3d) do not
explicitly depend on kx or ky, and ux (3d)1 and uy (3d)
1 are proportional to kx and ky, respectively.
Furthermore, kxux (3d)1 + kyuy (3d)
1 does not explicitly depend on kx and ky either. These properties
give the following mapping for the linear solutions in three dimensions.
ρ(3d)lin (x,y,z,t) = ρ(2d)
1 (z,t)cos(kxx)cos(kyy),
p(3d)lin (x,y,z,t) = p(2d)
1 (z,t)cos(kxx)cos(kyy),
ux (3d)lin (x,y,z,t) = kxk−1ux (2d)
1 (z,t)sin(kxx)cos(kyy),
uy (3d)lin (x,y,z,t) = kyk−1ux (2d)
1 (z,t)cos(kxx)sin(kyy),
uz (2d)lin (x,y,z,t) = uz (2d)
1 (z,t)cos(kxx)cos(kyy).
The interfaces are given by zi(t) + ai(2d)(t)cos(kxx)cos(kyy) for i = 0, 1 and 2 or l, r. This map-
ping holds for every region in both reflected shock and reflect rarefaction cases. In particular, for
fixed total wave number k, the growth rate determined from the linear theory in three dimensions
is same as the one in two dimensions.
A systematic study of linear theory in two dimensions for the case of reflected shock and
the case of reflected rarefaction wave is given in [24]. Using the mapping function given above,
it is easy to see that all properties and conclusions presented in [24] for the linear theory in two
dimensions also hold for the linear theory in three dimensions. In particular, the analysis of total
- 16 -
transmission phenomenon, the analysis of direct and indirect phase inversions, the observation
that freeze-out cannot occur when the adiabatic indices of the two materials are the same, the
observation that freeze-out and total transmission cannot occur simultaneously, and the
identification and analysis of an instability associated with the rarefaction wave, remain the same
for the linear theory in three dimensions. The phenomena of freeze-out was first identified by
Mikaelian [17]. The analytic expressions for the short time solutions for both the reflected shock
and reflected rarefaction cases in three dimensions can be easily mapped from the small time
solutions in the two dimensions given in [24]. All figures and tables presented in [24] for the
linear growth rate are remained the same for linear theory in three dimensions, as long as one
identifies k in [24] as the total wave vector. Therefore, with the mapping function given above,
we refer [24] for the solutions of linear theory in three dimensions.
Although the solutions of the linear theory in three dimension do not depend on the orienta-
tion of the wave vector, such independence is not true in the nonlinear theory.
4. Nonlinear Perturbation Solutions for Incompressible Fluids in Three Dimensions
As we have described in Section 2, in the nonlinear stage, the effects of compressibility are
less important. Therefore, we can approximate the fluids as incompressible. In this section we
derive the nonlinear perturbation solution for incompressible fluids in three dimensions. The
expansion is in terms of the disturbance at the initial interface.
The method of nonlinear perturbation expansion has been applied to the problem of the
interfacial fluid mixing by many researchers in the past. The second and third order perturbation
solutions for the Rayleigh-Taylor instability (interfacial instability driven by a gravitational
force) in two dimensions have been obtained by Ingraham [12] and Chang [5], respectively.
Jacobs and Catton derived third order solutions for the Rayleigh-Taylor instability in three
dimensions [13]. However, all these perturbation solutions have a very limited range of validity.
The method of nonlinear perturbation expansion has also been applied to the Richtmyer-
Meshkov instability. Haan derived the second order perturbation solution [9]. The high order
perturbation solutions for the case of the impulsive model with A = 1 have been obtained in [23]
The third and fourth order perturbations solutions of arbitrary A can be found in [26,27]. These
results are in two dimensions. As we have pointed out in Section 2, the derivation of the non-
linear perturbation solutions is only an intermediate step in our theoretical formulation. Our goal
is to construct of Pade approximants to extend the range of validity.
- 17 -
In Section 4.A, we derive perturbation solutions with general initial growth rate. A general
formula is obtained for n-th order solutions. In Section 4.B, we demonstrate our solution method
by evaluating the nonlinear solution of the impulsive model. The impulsive model has specific
initial conditions for the growth rate. Explicit expression for the growth rate of the unstable
material interface is given through third order. The nonlinear perturbation solutions derived in
this section serve as the generating series for the Pade approximants constructed in next section.
4.A. Governing Equations and Solution Procedure
In this section we derive the nonlinear solutions for incompressible systems with no exter-
nal forces and with general initial velocity along the interface. The governing equations for
inviscid, irrotational, incompressible fluids in three dimensions with no external forces are given
by
∇ 2φ(x,y,z,t) = 0 in material 1, ∇ 2φ′(x,y,z,t) = 0 in material 2, (35)
∂t∂ηhhh −
∂x∂φhhh
∂x∂ηhhh −
∂y∂φhhh
∂y∂ηhhh +
∂z∂φhhh = 0 at z = η, (36)
∂t∂ηhhh −
∂x∂φ′hhhh
∂x∂ηhhh −
∂y∂φ′hhhh
∂y∂ηhhh +
∂z∂φ′hhhh = 0 at z = η, (37)
− ρ′ ∂t∂φ′hhhh + ρ
∂t∂φhhh +
21hhρ′[( ∂x
∂φ′hhhh)2 + (∂y∂φ′hhhh)2 + (
∂z∂φ′hhhh)2]
−21hhρ[(
∂x∂φhhh)2 + (
∂y∂φhhh)2 + (
∂z∂φhhh)2] = 0 at z = η. (38)
Here the unprimed variables are the physical quantities in material 2 and the primed variables are
the physical quantities in material 1. z = η(x,y,t) is the position of the material interface at time
t. ρ and ρ′ are densities of material 2 and 1, respectively. φ and φ′ are the velocity potentials in
material 2 and 1, respectively. The velocity field is given by v→= −∇φ in material 2 and by
v→′ = −∇φ ′ in material 1. Two equations given in (35) come from the incompressibility condi-
tions. Equations (36) and (37) represent the kinematic boundary condition that a fluid particle
initially situated at the material interface will remain at the interface afterwards. Equation (38)
represents the dynamic boundary condition in which the pressure is continuous across the
material interface. We consider the single mode RM instability only in this paper. The initial
shape of the material interface is given by
η(x,y,t = 0) = a 0cos(kxx)cos(kyy)
- 18 -
and the initial velocity distribution along the material interface is given by η.(x,y,t = 0). Here,
and from now, a 0 is the amplitude of the initial disturbance. Therefore, a 0 is a constant. One
should not be confused this with the notation a 0(t) used in the linear theory (Section 3) where
a 0(t) is a function of time. η.(x,y, 0) is an arbitrary single valued function of x. The impulsive
model corresponds to a particular initial velocity distribution along the interface. This particular
initial velocity will be derived in next section from the assumption of an impulsive force.
We expand all quantities in terms of powers of a 0k, i. e. f = Σfn . Here fn = φn , φ′n and ηn ,
are proportional to (a 0k)n , and k is the magnitude of the wave vector. Then (35)-(38) can be
expressed as
n =1Σ∞
∇ 2φn = 0 in material 2,n =1Σ∞
∇ 2φ′n = 0 in material 1, (39)
n =1Σ∞
(∂t
∂ηnhhhh −i =1Σn
(∂x
∂φihhhh∂x
∂ηn −ihhhhhh +∂y
∂φihhhh∂y
∂ηn −ihhhhhh) +∂z
∂φnhhhh) = 0 at z =n =1Σ∞
ηn , (40)
n =1Σ∞
(∂t
∂ηnhhhh −i =1Σn
(∂x
∂φ′ ihhhh∂x
∂ηn −ihhhhhh +∂y
∂φ′ ihhhh∂y
∂ηn −ihhhhhh) +∂z
∂φ′nhhhhh) = 0 at z =n =1Σ∞
ηn , (41)
− ρ′n =1Σ∞
∂t
∂φ′nhhhhh + ρn =1Σ∞
∂t
∂φnhhhh +21hhρ′
n =1Σ∞
i =1Σn
[∂x
∂φ′ ihhhh∂x
∂φ′n −ihhhhhh +∂y
∂φ′ ihhhh∂y
∂φ′n −ihhhhhh +∂z
∂φ′ ihhhh∂z
∂φ′n −ihhhhhh]
−21hhρ
n =1Σ∞
i =1Σn
[∂x
∂φihhhh∂x
∂φn −ihhhhh +∂y
∂φihhhh∂y
∂φn −ihhhhh +∂z
∂φihhhh∂z
∂φn −ihhhhh] = 0 at z =n =1Σ∞
ηn . (42)
Since the boundary and initial conditions given in (40)-(42) hold at the position
z = η =n =1Σ∞
η(n) , they need to be further expanded at z = 0. After expanding the equations and col-
lecting all terms of order (a 0k)n , we have the following equations for the n −th order quantities.
∇ 2φ(n) = 0 in material 1, ∇ 2φ′(n) = 0 in material 2, (43)
∂t∂η(n)hhhhh +
∂z∂φ(n)hhhhh =
0≤i, j≤nΣ Sij
(n)(t)cos(ikxx)cos(jkyy) at z = 0, (44)
∂t∂η(n)hhhhh +
∂z∂φ′(n)hhhhhh =
0≤i, j≤nΣ S ′ ij
(n)(t)cos(ikxx)cos(jkyy) at z = 0, (45)
− ρ′ ∂t∂φ′(n)hhhhhh + ρ
∂t∂φ(n)hhhhh =
0≤i, j≤nΣ Tij
(n)(t)cos(ikxx)cos(jkyy) at z = 0, (46)
with the initial conditions
- 19 -
η(n)(x,y,t = 0) = a 0cos(kxx)cos(kyy)δ1n , (47)
η. (n)
(x,y,t = 0) =0≤i, j≤nΣ a
.ij(n)
(0)cos(ikxx)cos(jkyy). (48)
Here δ1n is Kronecker delta function. a.
ij(n)
(0) is determined by the Fourier mode decomposition
of the left hand side of (48). Sij(n) , S ′ ij
(n) and Tij(n) are determined by the Fourier mode decomposi-
tion of the right hand sides of the following equations:
0≤i, j≤nΣ Sij
(n)(t)cos(ikxx)cos(jkyy) = −sum1Σ p !
1hhh∂z p +1
∂p +1φ(a)hhhhhhhh
i =1Πp
η(ni) +sum2Σ p !
1hhh∂x∂z p
∂p +1φ(a)hhhhhhhh
∂x∂η(b)hhhhh
i =1Πp
η(ni)
+sum2Σ p !
1hhh∂y∂z p
∂p +1φ(a)hhhhhhhh
∂y∂η(b)hhhhh
i =1Πp
η(ni) at z = 0, (49)
0≤i, j≤nΣ S ′ ij
(n)(t)cos(ikxx)cos(jkyy) = −sum1Σ p !
1hhh∂z p +1
∂p +1φ′(a)hhhhhhhh
i =1Πp
η(ni) +sum2Σ p !
1hhh∂x∂z p
∂p +1φ′(a)hhhhhhhh
∂x∂η(b)hhhhh
i =1Πp
η(ni)
+sum2Σ p !
1hhh∂y∂z p
∂p +1φ′(a)hhhhhhhh
∂y∂η(b)hhhhh
i =1Πp
η(ni) at z = 0, (50)
0≤i, j≤nΣ Tij
(n)(t)cos(ikxx)cos(jkyy) =sum1Σ p !
1hhh∂z p∂t
∂p +1hhhhhh(ρ′φ′(a) − ρφ(a))
i =1Πp
η(ni)
+21hh
sum3Σ p!q!
1hhhhh[ρ(∂z p∂x
∂p+1φ(a)hhhhhhhh
∂z q∂x
∂q+1φ(b)hhhhhhhh +
∂z p∂y
∂p+1φ(a)hhhhhhhh
∂z q∂y
∂q+1φ(b)hhhhhhhh +
∂z p+1
∂p +1φ(a)hhhhhhhh
∂z q+1
∂q+1φ(b)hhhhhhhh)
− ρ′( ∂z p∂x
∂p+1φ′(a)hhhhhhhh
∂z q∂x
∂q +1φ′(b)hhhhhhhh +
∂z p∂y
∂p+1φ′(a)hhhhhhhh
∂z q∂y
∂q +1φ′(b)hhhhhhhh +
∂z p +1
∂p +1φ′(a)hhhhhhhh
∂z q +1
∂p +1φ′(b)hhhhhhhh)]
i =1Πp
η(ni)
j =1Πq
η(mj)
at z = 0. (51)
where
sum1 : (0 < n 1 , n 2 , ...,np , p, a < n), (n 1 + n 2 + ... + np + a = n);
sum2 : (0 < n 1 , n 2 , ...,np , a, b < n),(0 ≤ p < n), (n 1 + n 2 + ... + np + a + b = n);
sum3 : (0 < n 1 , n 2 , ..., np , m1 , m2 , ...,mq , a, b <n),(0 ≤ p, q < n),
(n 1 + n 2 + ... + np + m1 + m2 + ... + mq + a + b = n).
Note that the quantities on the right hand side of (49)-(51) are already known for the n −th
order equations since they have orders lower than n . Equations (43)-(48) are linear equations for
the n −th order variables η(n) , φ(n) and φ′(n) which we are going to solve for.
- 20 -
The n −th order solution can be expressed as
η(n)(x,y,t) =0≤i, j≤nΣ aij
(n)(t)cos(ikxx)cos(jkyy), (52)
φ(n)(x,y,z,t) =0≤i, j≤nΣ bij
(n)(t)cos(ikxx)cos(jkyy)e−kijz , (53)
φ′(n)(x,y,z,t) =0≤i, j≤nΣ b ′ ij
(n)(t)cos(ikxx)cos(jkyy)e kijz . (54)
Here kij = √ddddddi 2 + j 2 k. After substituting (52)-(54) into (44)-(46) and using the orthogonality of
different Fourier modes, we have
dt
daij(n)(t)hhhhhhh − kijbij
(n)(t) = Sij(n)(t), (55)
dt
daij(n)(t)hhhhhhh + kijb ′ ij
(n)(t) = S ′ ij(n)(t), (56)
− ρ′ dt
db ′ ij(n)(t)hhhhhhhh + ρ
dt
dbij(n)(t)hhhhhhh = Tij
(n)(t) (57)
with the initial amplitude aij(n)(0) = a 0δ1nδ1iδ1j and the initial growth rate a
.ij(n)
(0).
The solutions for (55)-(57) with the initial conditions are given by
aij(n)(t) =
ρ′ + ρ1hhhhhh
0∫t
[kij(t − t ′)Tij(n)(t ′) + ρ′(S ′ ij
(n)(t ′) − S ′ ij(n)(0)) + ρ(Sij
(n)(t ′) − Sij(n)(0))]dt ′
+ a.
ij(n)
(0)t + a 0δ1nδ1iδ1j , (58)
bij(n)(t) =
ρ′ + ρ1hhhhhh[
0∫t
Tij(n)(t ′)dt ′ +
kij
1hhhρ′(S ′ ij(n)(t) − Sij
(n)(t))] +kij
1hhha.
ij(n)
(0), (59)
b ′ ij(n)(t) = −
ρ′ + ρ1hhhhhh[
0∫t
Tij(n)(t ′)dt ′ +
kij
1hhhρ(Sij(n)(t) − S ′ ij
(n)(t))] −kij
1hhha.
ij(n)
(0) (60)
for i + j ≠ 0. The growth rate can be determined from (58) and the result is
dt
daij(n)(t)hhhhhhh =
ρ′ + ρ1hhhhhh[
0∫t
kijTij(n)(t ′)dt ′ + ρ′(S ′ ij
(n)(t) − S ′ ij(n)(0)) + ρ(Sij
(n)(t) − Sij(n)(0))] + a
.ij(n)
(0).
For the case i = j = 0, a00(n) (t) = 0 from the condition of incompressibility. From (44) and (45), it
follows that S00(n) (t) = S ′ 00
(n)(t) = 0. From (46), b00(n) (t) and b ′ 00
(n)(t) are determined by
- 21 -
− ρ′b ′ 00(n)(t)+ ρb00
(n) (t) = − ρ′b ′ 00(n)(0)+ ρb00
(n) (0) +0∫t
T00(n) (t ′)dt ′.
Since the velocities are the gradients of the velocity potentials and all the source terms in (49)-
(50) involve differentiation with respect to x or/and z, the functional forms of b00(n) and b00
(n) are
irrelevant. Therefore, we will not evaluate them explicitly.
Let us summarize the recursive procedure for obtaining the nonlinear solution. We progress
from lower orders towards higher orders, starting from the first order. We first evaluate the
source terms, namely the right hand sides of (49)-(51). These source terms are known from the
lower order solutions. We determine Sij(n) , S ′ ij
(n) and Tij(n) from these source terms. Then the n −th
order solutions are simply given by (52)-(54) with aij(n) , bij
(n) and b ′ ij(n) given by (58), (59) and
(60), respectively. It is easy to show by induction that aij(n) , bij
(n) and b ′ ij(n) are polynomials of t.
The case of n=1 will be given explicitly in the Section 4.B. From induction hypothesis, we
assume that, for all n < m, bij(n) and b ′ ij
(n) are polynomials of t. Then the conclusion that aij(m) , bij
(m)
and b ′ ij(m) are polynomials follows from (58)-(60) and the mathematical properties that products,
summations, differentiation, integration of polynomials are still polynomials. Furthermore, from
the same induction procedure and the definition of sum1, sum2 and sum3, one can check that,
with respect to t, the degree of aij(n) is not greater than n and the degrees of bij
(n) and b ′ ij(n) are not
greater than n −1.
4.B. Nonlinear Solution of Impulsive Model
Let us apply the solution procedure to determine the nonlinear solution of the impulsive
model in three dimensions through third order. The impulsive model assumes that the fluids are
at rest initially and are driven by an impulsive acceleration
g = ∆uδ(t)
at time t = 0. Here ∆u represents the strength of the impulse. Then (38) becomes
(ρ′−ρ)gη − ρ′ ∂t∂φ′hhhh + ρ
∂t∂φhhh +
21hhρ′[( ∂x
∂φ′hhhh)2 + (∂y∂φ′hhhh)2 + (
∂z∂φ′hhhh)2]
−21hhρ[(
∂x∂φhhh)2 + (
∂y∂φhhh)2 + (
∂z∂φhhh)2] = 0 at z = η. (61)
After integrating (61) over time from 0− to 0+, we obtain the following initial condition
(ρ′−ρ)∆uη − ρ′φ′ + ρφ =0 at z = η = a 0cos(kxx)cos(kyy) and t = 0+. (62)
- 22 -
For t ≥ 0+, (61) reduces to (38). Therefore, our general solution procedure is valid for the impul-
sive model. The only thing we need to do is to determine the initial growth rate from (62). We
further expand (62) at z = 0. The result is
(ρ′−ρ)∆uη(1)δ1n − ρ′φ′(n) + ρφ(n) =0 ≤ i, j≤n
Σ Rij(n)cos(ikxx)cos(jkyy) at z = 0 and t = 0. (63)
Here Rij(n) are determined by the Fourier mode decomposition of the right hand sides of the fol-
lowing equation
0≤i, j≤nΣ Rij
(n)cos(ikxx)cos(jkyy) =m = 1Σ
n − 1
m !1hhhh
∂z m
∂mhhhh(ρ′φ′(n −m) − ρφ(n −m))[a 0cos(kxx)cos(kyy)]m (64)
at z = 0 and t = 0. From (53) and (54), (63) can be expressed as
(ρ′ − ρ)∆ua 0δ1nδ1iδ1j − ρ′b ′ ij(n)(0) + ρbij
(n)(0) = Rij(n) . (65)
Finally from (55),(56) and (65), we obtain the initial growth rate for the impulsive model
a.
ij(n)
(0) = σa 0δ1nδ1iδ1j +ρ′ + ρ
kijRij(n) + ρ′S ′ ij
(n)(0) + ρSij(n)(0)hhhhhhhhhhhhhhhhhhhhhhhhh . (66)
Here σ = −A∆uk for the impulsive model. Therefore, the nonlinear solutions to the impulsive
model in three dimensions are given by (52)-(54) and (58)-(60) with the initial growth rate (66).
The equations for the first order quantities are given by
∇ 2φ(1) = 0 in material 1, ∇ 2φ′(1) = 0 in material 2,
∂t∂η(1)hhhhh +
∂z∂φ(1)hhhhh = 0 and
∂t∂η(1)hhhhh +
∂z∂φ′(1)hhhhhh = 0 at z = 0,
(ρ′ − ρ)gη(1) − ρ′ ∂t∂φ′(1)hhhhhh + ρ
∂t∂φ(1)hhhhh = 0 at z = 0.
Obviously, Sij(1)(t) = S ′ ij
(1)(t) = Tij(1)(t) = Rij
(1) = 0. From (58)-(60) the first order solution is given
by
η(1) = (1 + σt)a 0cos(kxx)cos(kyy), (67)
φ(1) =k 11
σhhhha 0e−k 11zcos(kxx)cos(kyy), φ′(1) = −k 11
σhhhha 0e k 11zcos(kxx)cos(kyy). (68)
The equations for the second order quantities are given by
∇ 2φ(2) = 0 in material 1, ∇ 2φ′(2) = 0 in material 2, (69)
∂t∂η(2)hhhhh +
∂z∂φ(2)hhhhh = −
∂z 2
∂2φ(1)hhhhhhη(1) +
∂x∂φ(1)hhhhh
∂x∂η(1)hhhhh +
∂y∂φ(1)hhhhh
∂y∂η(1)hhhhh at z = 0, (70)
- 23 -
∂t∂η(2)hhhhh +
∂z∂φ′(2)hhhhhh = −
∂z 2
∂2φ′(1)hhhhhhη(1) +
∂x∂φ′(1)hhhhhh
∂x∂η(1)hhhhh +
∂y∂φ′(1)hhhhhh
∂y∂η(1)hhhhh at z = 0, (71)
− ρ′ ∂t∂φ′(2)hhhhhh + ρ
∂t∂φ(2)hhhhh = (ρ′ ∂t∂z
∂2φ(1)hhhhhh − ρ
∂t∂z∂2φ(1)hhhhhh)η1 −
21hhρ′[( ∂x
∂φ′(1)hhhhhh)2 + (
∂y∂φ′(1)hhhhhh)2 + (
∂z∂φ′(1)hhhhhh)2]
+21hhρ[(
∂x∂φ(1)hhhhh)2 + (
∂y∂φ(1)hhhhh)2 + (
∂z∂φ(1)hhhhh)2] at z = 0, (72)
− ρ′φ′(2) + ρφ(2) = (ρ′ ∂z∂φ(1)hhhhh − ρ
∂z∂φ(1)hhhhh)η1 at z = 0 and t = 0. (73)
The right hand side of these equations can be evaluated analytically from (67) and (68). Then
(70)-(73) can be written as
∂t∂η(2)hhhhh +
∂z∂φ(2)hhhhh = −
21hha0
2 σ(1 + σt)[k 11
kx2
hhhhcos(2kxx) +k 11
ky2
hhhhcos(2kyy) + k 11cos(2kxx)cos(2kyy)] at z = 0,
∂t∂η(2)hhhhh +
∂z∂φ′(2)hhhhhh =
21hha0
2 σ(1 + σt)[k 11
kx2
hhhhcos(2kxx) +k 11
ky2
hhhhcos(2kyy) + k 11cos(2kxx)cos(2kyy)] at z = 0,
− ρ′ ∂t∂φ′(2)hhhhhh + ρ
∂t∂φ(2)hhhhh =
41hh(ρ − ρ′)a0
2 σ2[1 +k 11
2
ky2
hhhhhcos(2kxx) +k 11
2
kx2
hhhhhcos(2kyy)] at z = 0,
− ρ′φ′(2) + ρφ(2) = −41hh(ρ + ρ′)a0
2 σ[1 + cos(2kxx) + cos(2kyy) + cos(2kxx)cos(2kyy)] at z = 0 and t = 0.
Obviously,
S20(2) = −
21hha0
2 σ(1 + σt)k 11
kx2
hhhh , S02(2) = −
21hha0
2 σ(1 + σt)k 11
ky2
hhhh , S22(2) = −
21hha0
2 σ(1 + σt)k 11,
S ′ 20(2) = −S20
(2), S ′ 02(2) = −S02
(2), S ′ 22(2) = −S22
(2),
T00(2) =
41hh(ρ − ρ′)a0
2 σ2 , T20(2) =
41hh(ρ − ρ′)a0
2 σ2(k 11
kyhhhh)2 , T02(2) =
41hh(ρ − ρ′)a0
2 σ2(k 11
kxhhhh)2 ,
R00(2) = R20
(2) = R02(2) = R22
(2) =41hh(ρ − ρ′)a0
2 σ.
Sij(2) = S ′ ij
(2) = Tij(2) = Rij
(2) = 0 for other values of i and j. It is straight forward to show that our
general formulae, (52)-(54) and (58)-(60), lead the following expressions for the second order
solution:
η(2) =41hhA(a 0σt)2[
k112
1hhhhkx(kxk 11 − ky2)cos(2kxx) +
k112
1hhhhky(kyk 11 − kx2)cos(2kyy)
+ k 11cos(2kxx)cos(2kyy)] + η. (2)
(0)t, (74)
- 24 -
φ(2) = [4k11
2
1hhhhha02 σ((kxk 11(1 + A) − Aky
2)σt + kxk 11) +2kx
1hhhha.
20(2)
(0)]cos(2kxx)e−2kxz
+ [4k11
2
1hhhhha02 σ((kyk 11(1 + A)−Akx
2)σt + kyk 11) +2ky
1hhhha.
02(2)
(0)]cos(2kyy)e−2kyz
+41hha0
2 σ((1 + A)σt + 1)cos(2kxx)cos(2kyy)e−2k 11z + b 00(t), (75)
φ′(2) = [4k11
2
1hhhhha02 σ((kxk 11(1 − A)+Aky
2)σt + kxk 11) −2kx
1hhhha.
20(2)
(0)]cos(2kxx)e 2kxz
+ [4k11
2
1hhhhha02 σ((kyk 11(1 − A) + Akx
2)σt + kyk 11) −2ky
1hhhha.
02(2)
(0)]cos(2kyy)e 2kyz
+41hha0
2 σ((1 − A)σt + 1)cos(2kxx)cos(2kyy)e 2k 11z + b ′00(t). (76)
and from (66)
η. (2)
(0) = a.
20(2)
(0)cos(2kxx) + a.
02(2)
(0)cos(2kyy)
=2k 11
1hhhhha02 σA[kx(kx − k 11)cos(2kxx) + ky(ky − k 11)cos(2kyy)]. (77)
The derivation for the third order quantities is given in the Appendix A. The result for η3 is
η(3) = a03 σ2t 2[( − K11
1 + K112 σt)cos(kxx)cos(kyy) + ( − K31
1 + K312 σt)cos(3kxx)cos(kyy)
+ ( − K131 + K13
2 σt)cos(kxx)cos(3kyy) + ( − K331 + K33
2 σt)cos(3kxx)cos(3kyy)]
+ η.
3(0)t (78)
where
K111 =
8k 112
1hhhhhh[ − (kx3 + ky
3)k 11 + 2kx2ky
2 + k 114 − k 11kxky(kx + ky)]A 2
+32k 11
1hhhhhh[4(kx3 + ky
3) − k 113],
K112 =
24k 113
1hhhhhhh[k 112((kx
3 + ky3) + kxky(kx + ky)) − 2k 11
5 + 2(kx3(ky
2 − kxk 11)
+ ky3(kx
2 − kyk 11)) − 2k 11kx2ky
2]A 2 −96k 11
1hhhhhh[4(kx3 + ky
3) − k 113],
K311 =
16k 112
1hhhhhhh[6kx2(ky
2 − kxk 11) − k 112(3kx
2 + ky2) + k 31(2kx(kxk 11 − ky
2) + k 113)]A 2
- 25 -
+32k 11
2
1hhhhhhh[24k 11kx3 + 3k 11
4 − k 31(kx2(9k 11 − 4kx) + ky
2k 11 + k 113)],
K312 =
48k 113
1hhhhhhh[12k 11kx2(kxk 11 − ky
2) − 2(ky2 − kxk 11)(kxk 11
2 + 2kx3) + k 11
3(8kx2 + k 11
2)
+ k 31(2(kxk 11 − ky2)(kx
2 − 2kxk 11) + k 112(kx
2 − ky2) − 2k 11
4)]A 2
−96k 11
2
1hhhhhhh[12k 11kx3 + 3k 11
4 − k 31(kx2(9k 11 − 4kx) + ky
2k 11 + k 113],
K13i = K31
i (kx → ky , ky → kx), i = 1,2,
K331 =
32
3k 112
hhhhhh ,
K332 =
32
k 112
hhhhh(4A 2 − 1)
and from (66)
η.
3(0) = a.
11(3)
(0)cos(kxx)cos(kyy) + a.
31(3)
(0)cos(3kxx)cos(kyy) + a.
13(3)
(0)cos(kxx)cos(3kyy)
= [32k 11
a03 σhhhhhh(2(−4(kx
3 + ky3) + k11
3 ) −21hha 0k 11A(a
.20(2)
(0) + a.
02(2)
(0))]cos(kxx)cos(kyy)
+ [32k 11
a03 σhhhhhh(−3(8kx
3 + k113 ) + k31
3 ) −21hha 0k 31Aa
.20(2)
(0)]cos(3kxx)cos(kyy)
+ [32k 11
a03 σhhhhhh(−3(8ky
3 + k113 ) + k13
3 ) −21hha 0k 13Aa
.02(2)
(0)]cos(kxx)cos(3kyy). (79)
The third order initial growth rate (79) comes from (66). φ(3) and φ′(3) can be easily calculated
from (53), (54), (59) and (60) and are not shown here.
Note that by setting ky = 0, (74) recovers the second order perturbation solution in two
dimensions given in [9]. By setting ky = 0 and A = 1, (79) recovers the third order solution in two
dimension obtained by Velikovich and Dimonte [23].
Velikovich and Dimonte considered the case of A =1 and incompressible fluids driven by an
impulsive force (impulsive model) in a system with infinite density ratio. The solution of the
impulsive model has its own interests. In this paper and references [26,27], we have considered
compressible fluids of arbitrary value of Atwood number driven by a shock wave. The impulsive
model is not valid at early time solution for compressible fluids driven by a shock wave. The
approximate nonlinear solutions for compressible fluids mixing driven by a shock wave will be
- 26 -
presented in the next section.
Now we discuss general properties of n-th order solutions.
Following the proofs given the Appendix C of [27], we checked that the n −th order solu-
tions have the following general properties
η(n)(A) = (−1)n +1η(n)(−A), (80)
φ′(n)(A,z) = (−1)nφ(n)(−A, −z). (81)
It has been shown in two dimensions that the source terms in the n-th order equations have cer-
tain symmetry properties. Following the proofs given in [27], we checked that these properties
hold for three dimensions also. In these symmetry properties, we do not include the possible A
dependence in σ, as we have seen in the impulsive model.
5. Pade Approximation and Nonlinear Theory for Compressible Fluids in Three Dimensions
We have systematically derived the nonlinear perturbation solutions for incompressible
fluids. In this section, we apply the Pade approximation to extend the range of validity beyond
the range of validity of the Taylor expansion itself. Then we match the linear compressible solu-
tion at early times and the nonlinear incompressible solution at later times to arrive at a nonlinear
theory for compressible fluids in three dimensions.
Let us discuss the initial conditions we are going to choose for the nonlinear solution for
incompressible fluids. As we have outlined in Section 2, we would like to develop a nonlinear
theory for compressible fluids from early to late times. This will be done by matching the solu-
tion of the linear theory for compressible fluids (valid at early time) and the solution of the non-
linear theory for incompressible fluids (valid at later time). We emphasis that our physical pic-
ture at late time solution depends on the incompressibility approximation only. The impulsive
force approximation is not made here. Since the impulsive approximation gives qualitatively
incorrect solution at early time for compressible fluid driven by a shock wave, and the purpose of
our study is to develop a quantitative theory from the compressible RM instability, the initial
conditions derived for impulsive model is not applicable here. The growth rate determined from
the linear theory for compressible fluids contains a single Fourier mode only. Therefore, to be
consistent with the solution of the linear theory, we choose the following single mode initial con-
ditions for the nonlinear solution of incompressible fluids:
η(x,y, 0) = a 0cos(kxx)cos(kyy) and η.(x,y, 0) = v 0cos(kxx)cos(kyy). (82)
- 27 -
Here we assume that a 0k is small and that v 0 is proportional to a 0 . Then from (48), we have
a.
ij(n)
(0) = v 0δ1nδ1iδ1j . (83)
v 0 will be determined later through matching. We are interested in the nonlinear solution for
incompressible fluids with the initial conditions given by (82). Since the initial condition given
by (83) is different from (66), the nonlinear solutions developed in this section are not the solu-
tion of the impulsive model. One could follow the solution procedure given in Section 4.A to
derive the nonlinear solutions. However, by comparing (82) with the initial conditions of the
impulsive model given by (66), one sees that the nonlinear solution with initial conditions given
by (82) can be obtained by setting η.
i(0) = 0 for i ≥ 2 and a 0σ = v 0 in the nonlinear solutions of
the impulsive model. Up to the third order, the solutions with the initial conditions (82) are
η(1) = (a 0 + v 0t)cos(kxx)cos(kyy), (84)
η(2) =41hhAv0
2t 2[k11
2
1hhhhkx(kxk 11 − ky2)cos(2kxx) +
k112
1hhhhky(kyk 11 − kx2)cos(2kyy)
+ k 11cos(2kxx)cos(2kyy)], (85)
η(3) = v02t 2[( − K11
1 a 0 + K112 v 0t)cos(kxx)cos(kyy) + ( − K31
1 a 0 + K312 v 0t)cos(3kxx)cos(kyy)
+ ( − K131 a 0 + K13
2 v 0t)cos(kxx)cos(3kyy) + ( − K331 a 0 + K33
2 v 0t)cos(3kxx)cos(3kyy)].
(86)
The shape of the material interface at time t is given by η(x,y,t), and the velocities at the tip
of a bubble and at the tip of a spike are given by
v bb =∂t∂ηhhh at x =
kx
πhhh , y = 0, or x = 0, y =ky
πhhh ,
v sp =∂t∂ηhhh at x = 0, y = 0, or x =
kx
πhhh , y =ky
πhhh ,
respectively. From the general properties given by (80), η can expressed as
η = ηa + ηb . (87)
Here
ηa =k =0Σ∞
η(2k +1) (88)
is an even function of A, and
- 28 -
ηb =k =1Σ∞
η(2k) (89)
is an odd function of A.
Now, Let us derive the expressions for nonlinear growth rates. Sections 5.A is for the
overall growth rate and Section 5.B is for the growth rates of bubble and spike.
5.A. Overall Growth Rate
The overall growth rate of the unstable interface is defined as a half of the difference
between the velocity of the spike and the velocity of the bubble, v =21hh(vsp − vbb). Then we have
v =∂t
∂ηahhhh =k =0Σ ∂t
∂η(2k +1)hhhhhhhh at x = y = 0.
The even order terms, namely ηb , do not contribute to the overall growth rate of the unstable
interface. This is due to the fact that for each even order, the velocities at the tip of the spike and
at the tip of the bubble are the same. From the analytical expressions of η1 and η3 given by (84)
and (86). we have the following generating series for the overall growth rate of the RM unstable
interface,
v = v 0 − v02a 0k 2λ1t + v0
3k 2λ2t 2 + O((a 0k)5), (90)
where
k 2λi =i + 1
1hhhhh(K11i + K31
i + K13i + K33
i ) , i = 1,2 (91)
and k = k 11 = √dddddkx2 + ky
2 . After expressing the wave vector in polar coordinate system, i. e.
kx = kcos(θ) and ky = ksin(θ), we have the following expressions for λ1 and λ2:
λ1 = −161hhhA 2
IKL[5 + 4cos(2θ)]1/2[ − 4 + cos(θ) − 2cos(2θ) − cos(3θ)]
+ [5 − 4cos(2θ)]1/2[ − 4 + sin(θ) + 2cos(2θ) + sin(3θ)]
+ 13[cos(θ) + sin(θ)] + 3[cos(3θ) − sin(3θ)] + 4cos(4θ)MNO
+161hhh f (θ)
and
λ2 =641hhhA 2
IKL[5 + 4cos(2θ)]1/2[ − 17 + 10cos(θ) − 4cos(2θ) − 2cos(3θ) + cos(4θ)]
- 29 -
+ [5 − 4cos(2θ)]1/2[ − 17 + 10sin(θ) + 4cos(2θ) + 2sin(3θ) + cos(4θ)]
+ 42[1 + cos(θ) + sin(θ)] + 14[cos(3θ) − sin(3θ)] + 14cos(4θ)MNO
−321hhh f(θ).
Here
f(θ) = −[5 + 4cos(2θ)]1/2[6 − 3cos(θ) + 4cos(2θ) − cos(3θ)]
− [5 − 4cos(2θ)]1/2[6 − 3sin(θ) − 4cos(2θ) + sin(3θ)]
+ 8 + 21[cos(θ) + sin(θ)] + 7[cos(3θ) − sin(3θ)].
Equation (90) is an expansion through fourth order. The range of the validity of the (90) is
quite limited. One of the standard methods to extend the range of validity beyond the range of
validity of the finite Taylor series expansion is the Pade approximation. We will follow this
approach below.
We divide the parameter space into two subspaces: a02k 2 ≥ λ2 /λ1
2 and a02k 2 < λ2 /λ1
2 . We
consider the parameter region a02k 2 ≥ λ2 /λ1
2 first. From the generating series given by (62), one
can construct three possible Pade approximants: P02 (t), P1
1 (t) and P20 (t). The full numerical
simulations in two dimensions show that the growth rate of the RM unstable interface decays in
the nonlinear regime. The nonlinear theory of the RM instability in two dimensions, developed
by the authors, showed that such decay is due to nonlinearity rather than the compressibility. The
growth rate in two dimensions is a special case of the result in three dimensions, (the case of
ky = 0 or kx = 0). Among the three possible Pade approximants, P20 (t) is the only one which has
this decaying property. Therefore, P20 (t) is the only candidate for our physical system and the
result is
v = P20 (t) =
1 + v 0a 0k 2λ1t + v02k 2(a0
2k 2λ12 − λ2)t 2
v 0hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh (92)
for a02k 2 ≥ λ2 /λ1
2 . In parameter region a02k 2 ≤ λ2 /λ1
2 we construct a lower order Pade approxi-
mant. In this case, we can construct either P01 (t) or P1
0 (t). Only P10 (t) has the property that the
growth rate decays in the nonlinear regime and the result is
v = P10 (t) =
1 + v 0a 0k 2λ1t
v 0hhhhhhhhhhhhh (93)
for a02k 2 ≤ λ2 /λ1
2 .
- 30 -
Here we comment the reason for constructing P10 (t) and P2
0 (t) Pade approximants in
parameter regions a02k 2 < λ2 /λ1
2 and a02k 2 > λ2 /λ1
2 , respectively. It is well known that singulari-
ties may occur in a Pade approximant. In our case, the P20 construction has a singularity at some
finite time in the parameter region a02k 2 < λ2 /λ1
2 . There are two conventional ways to remove a
singularity. One is to reduce the order of accuracy by taking less terms and the other one is to
take more terms. Here we choose the first approach to construct P10 Pade approximant due to the
facts that the formula is simpler and it guarantees the removal of the singularity. Furthermore, it
is the only one available for the generating series given by equation (11). Note that P20 and P1
0
constructions are continuous at the phase boundary a02k 2 = λ2 /λ1
2 . These Pade approximants
have already been validated through comparison with full numerical simulations in two dimen-
sions [26]. Since P10 approximation is based on a partial contribution of the third order term
rather than the full contribution, the theoretical prediction given by (95) will be less accurate.
However, it has been shown in Figure 3 of [26], the P10 construction still gives quite good predic-
tions.
Equations (92) and (93) can be combined together to form
v =1 + v 0a 0k 2λ1t + max{0, a0
2k 2λ12 − λ2}v0
2k 2t 2
v 0hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh . (94)
We emphasize that one should not be confused the range of validity of Taylor series expan-
sions with the range of the validity of the Pade approximants. It is well known that the Pade
approximants have a larger range of convergence than the Taylor series. We refer a recent book
[19] by Pozzi for more systematic presentation of applications of the Pade approximation in fluid
dynamics. In the preface of that book the authors wrote, "The principal advantage of Pade
approximants with respect to the generating Taylor series is that they provide an extension
beyond the interval of convergence of the series". Also in the book [3] by Bender and Orszag,
the authors wrote, "Pade approximants often work quite well, even beyond their proven range of
applicability" and "there is no compelling reason to use Pade summation because the Taylor
series already converge for all z and the improvement of convergence is not astounding. The real
power of Pade summation is illustrated by its application to divergent series". This mathematical
property is clearly demonstrated in Figure 4 in the next section for the RM unstable system.
Equation (94) is an approximate nonlinear solution for incompressible fluids. From the phy-
sical picture which we gave earlier, they are also approximate nonlinear solutions for compressi-
ble fluids at later times. At early times, the solution is given by the linear theory for compressible
- 31 -
fluid, v lin . In order to develop a nonlinear theory for compressible fluids, we need to construct
expressions which smoothly match the linear solution for compressible fluids at early times and
the nonlinear solution for incompressible fluids at later times. Furthermore, the matching should
allow us to determine v 0 .
We can adopt the techniques of asymptotic matching developed in boundary layer prob-
lems. In a boundary layer problem, the dynamics in a thin layer near the boundary, called the
inner layer, is quite different from the dynamics in the region away from the boundary, called the
outer layer. One determines the solution in the inner layer (the inner solution) and the solution at
the outer layer (the outer solution) separately, and matchs these two solutions to form matched
asymptotics. Since our system is an initial value problem, rather than a boundary value problem,
a boundary condition is replaced by the initial conditions and the spacial variable is replaced by
the temporal variable.
In our case, the inner solution is the linear compressible solution and the outer solution is
the nonlinear incompressible solution given by (94). A recipe to determine v 0 in (94) was pro-
posed by Prandtl at the beginning of this century, namely by taking the large time limit of inner
solution and small time limit of the outer solution, and setting them equal [20]. Therefore, we
have the equation v lin(t → ∞) = η.
a(0,t → 0). Here η.
a is given by (94). This equation leads to
v 0 = vlin∞ = v lin(t → ∞). Then, (94), the outer solution for the overall growth rate, becomes
vincomp =1 + vlin
∞ a 0k 2λ1t + max{0, a02k 2λ1
2 − λ2}vlin∞2
k 2t 2
vlin∞
hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh . (95)
Equation (95) is a nonlinear incompressible solution with an initial growth rate given by vlin∞ . For
weak shocks, vlin∞ in (95) can be approximated by the linear solution of the impulsive model.
Finally, following the procedure proposed by Prandtl [20] (see also chapter 2 of [14]), we add the
inner and outer solutions and subtract the common part (vlin∞ in our case) to arrive at matched
asymptotics for the overall growth rate
vmatch = v lin +1 + vlin
∞ a 0k 2t + max{0, a02k 2λ1
2 − λ2}vlin∞2
k 2t 2
vlin∞
hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh − vlin∞ . (96)
The essence of the matched asymptotic technique is to blend the inner and outer solutions
smoothly. The technique proposed by Prandtl requires to calculate the asymptotic velocity of the
linear theory. We prefer to construct a simpler matched solution which has the same order of
accuracy. The facts that (94) approaches v 0 at early times and that the growth rate of the linear
- 32 -
theory for compressible fluids approaches an asymptotic constant vlin∞ at later times show that an
alternative way of matching can be achieved by replacing v 0 with v lin in (94). Then, we have Eq.
(1). In the small amplitude or short time limits, (1) recovers the result of linear theory. There-
fore, (1) provides a quantitative prediction for the growth rate of the RM instability from linear
regime to nonlinear regime.
In Figure 3, we compare the predictions of (96) and (1) for θ = 0 and π/4. The total wave
length, defined as 2π(kx2 + ky
2)−1/2 , is fixed to 37.5 mm. All physical parameters are the same as
the ones in Figure 1. Figure 3 shows that (1) and (96) indeed have the same accuracy. In fact, we
have checked that (1) and (96) have the same accuracy for all values of θ.
If we set ky = 0 in (1), it recovers the result of the nonlinear theory in two dimensions
developed recently by the authors [26]:
v =1 + vlina 0k 2t + max{0, a0
2k 2 − A 2 +21hh}vlin
2 k 2t 2
vlinhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh . (97)
Here k = kx and A is the post-shock Atwood number.
For the symmetric case in three dimensions, i. e. the case kx = ky, we have
λ1 =81hh(2 − 5√dd2 + 4√dd5 − √ddd10 )A 2 +
81hh(4 + 7√dd2 − 6√dd5 + √ddd10 )
= 0.088866A 2 + 0.455671
and
λ2 =161hhh(7 + 7√dd2 − 9√dd5 + 3√ddd10 )A 2 −
161hhh(4 + 7√dd2 − 6√dd5 + √ddd10 )
= 0.391357A 2 − 0.227835.
Then, the growth rate for the case kx = ky is given by
v = vlinRQ1 + vlina 0k 2(0.088866A 2 + 0.455671)t
+ max{0, a02k 2(0.088866A 2 + 0.455671)2
− (0.391357A 2 − 0.227835)}vlin2 k 2t 2H
P−1
. (98)
Here k = √dddddkx2 + ky
2 = √dd2 kx = √dd2 ky.
- 33 -
The phase boundary for the two dimensional case (θ = 0 or π/2) is given by
a02k 2 = A 2 −
21hh . The phase boundary for the symmetric case (kx = ky, or equivalently θ = π/4) is
given by
a02k 2 = (0.679984A 2 − 0.227835)/(0.088866A 2 + 0.455671)2 .
5.B. Growth Rates of Bubble and Spike
From (87), the growth rates of spike and bubble can be expressed as
vsp=∂t
∂ηahhhh +∂t
∂ηbhhhh at x = y = 0,
vbb= −∂t
∂ηahhhh +∂t
∂ηbhhhh at x = y = 0.
Here we have used the facts that ηa contains odd cosine Fourier modes and that ηb contains even
cosine Fourier modes. η.
b(0,0,t) represents21hh(vsp + vbb).
Following the solutions procedure developed in Section 4, we can calculate the explicit
expression for η4 . However, due to the complexity of the expression, we only present the final
result for η.
b(0,0,t).
η.
b(0,0,t) = kλ3v02t − k 3λ4a 0v0
3t 2 − k 3λ5v04t 3 . (99)
The first term of the right hand side of (99) comes from η. (2)
. and the second and third terms of
the right hand side of (99) come from η. (4)
. Here λ3, λ4 and λ5 are the functions of post-shocked
Atwood number A and the orientation angle θ of the wave vector. Their explicit expressions are
given in Appendix B. Applying the Pade approximation to (99), we have
η.
b(0,0,t) =1+v 0a 0k 2λ4λ3
−1t + v02k 2(a0
2k 2λ42λ3
−2 + λ5λ3−1)t 2
v02kλ3thhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh . (100)
Equation (100) is an approximate nonlinear solutions for incompressible fluids. From the match-
ing determined in the overall growth rate, we obtain quantitative expressions for the growth rates
of the bubble and spike in compressible fluids in three dimensions. The results are given by (2)
and (3). We have checked that (a02k 2λ4
2λ3−2 + λ5λ3
−1) are non-negative for all values of A and θ.
Therefore, our theoretical predictions of the bubble and spike growth rates given by (2) and (3)
have no singularity.
- 34 -
6. Quantitative Predictions
Before we present the quantitative prediction of our nonlinear theory for the RM unstable
system in three dimensions, we demonstrate that the range of validity of Pade approximations is
significantly larger than that of its generating series (Taylor expansions).
In Figure 4, we compare the overall growth rate of the compressible unstable interface
between air and SF6 in two dimensions from the predictions of the perturbation solutions given
by η.
1 and η.
1 + η.
3 (with v 0 = vlin∞ ), the prediction of the Pade approximation given by (95) for
incompressible system (the outer solution), and the result of the full nonlinear numerical simula-
tion. All physical parameters are the same as the ones in Figure 1. Note that η.
2 and η.
4 do not
contribute to the overall growth rate. Figure 4 shows that the range of validity of the nonlinear
perturbation solutions are very limited and the Pade approximation has successfully extended the
range of validity.
Now, we present the quantitative predictions of our nonlinear theory for the overall growth
rate and the growth rates of the bubble and spike for commonly considered in experiments and
numerical simulations. We present two cases here. The first one is an air-SF6 unstable interface
and the second one is a Kr−Xe unstable interface. For these two cases, experiments and full non-
linear numerical simulations have been conducted in two dimensions.
In Figure 5, we compare the predictions of our analytical prediction given by (1), the linear
theory, Richtmyer’s impulsive model for the growth rate and amplitude of air-SF6 interface for
several different values of θ. The physical parameters are same as the ones in Figure 1. The total
wave length is fixed to 37.5 mm, which is the same as the one in Figure 1. Figure 5(a) is for the
growth rate and Figure 5(b) is for the amplitude. The agreements between our theoretical predic-
tion for two dimensional RM unstable interface and the data of full numerical simulations in two
dimensions are remarkable. We have shown in [26] that our theoretical prediction for two dimen-
sional RM unstable interface are also in excellent agreement with experimental data.
In Figure 6, we compare the predictions of our analytical prediction given by (1), the linear
theory and Richtmyer’s impulsive model for the growth rate and amplitude of Kr−Xe interface.
The interface is accelerated by a strong shock of Mach number 3.5 moving from Kr to Xe. The
reflected wave is also a shock. The initial amplitude of the perturbation is 5 mm, the total wave
length is 36mm, and the pressure ahead of the shock is 0.5 bar. The post-shock Atwood number
is A = 0.184. These physical parameters correspond to Zaytsev’s recent experiments in two
dimensions. The dimensionless initial perturbation amplitude, which is defined as initial
- 35 -
perturbation amplitude times total wave number, is 0.87. This amplitude is about two times as
large as than the dimensionless amplitude 0.40 given in Figure 5 for air-SF6. For comparison, the
result of full non-linear numerical simulation in two dimensions is also shown. The agreement
between our theoretical predictions and the results of full nonlinear numerical simulations is also
remarkable.
Figures 5 and 6 show that the growth rates for different orientation angle θ with fixed total
wave number k are qualitatively similar, but quantitatively different. For the physical systems
shown in Figures 5 and 6, the symmetric interface in three dimensions (kx = ky) is more unstable
than the interface of same total wave number k in two dimensions.
In Figure 7, we compare the predictions of our analytical prediction for the bubble and
spike given by (2) and (3), the linear theory and Richtmyer’s impulsive model for air-SF6 inter-
face with several different values of θ. The physical parameters are same as the ones used in Fig-
ure 5. Figure 7(a) and 7(b) are for the growth rate and the amplitude of the bubble, respectively.
Figure 7(c) and 7(d) are for the growth rate and the amplitude of the spike, respectively. In Fig-
ure 6(c) and 6(d), the solid curves are the predictions of the nonlinear theory and the values
increase as θ varies from 0 (or, π/2) toward π/4. The selected valued of θ are same as the ones in
Figure 7(a). In Figure 8, we compare the predictions of our analytical prediction for the bubble
and spike given by (2) and (3), the linear theory and Richtmyer’s impulsive model for Kr-Xe
interface with several different values of θ. The physical parameters are same as the ones used in
the Figure 6. Figure 8(a) and 8(b) are for the growth rate and the amplitude of the bubble, respec-
tively. Figure 8(c) and 8(d) are for the growth rate and the amplitude of the spike, respectively.
The solid curves are the predictions of the nonlinear theory. The selected values of θ are same as
the ones in Figure 7(a). In Figure 8(a) and 8(b), the values decrease (the magnitudes increase) as
θ varies from 0 (or, π/2) toward π/4. In Figure 8(c) and 8(d), the values increase as θ varies
from 0 (or, π/2) toward π/4.
Figures 7 and 8 show that the growth rates of the bubble and spike for different orientation
angle θ with same total wave number k are qualitatively similar, but quantitatively different. The
orientation angle θ has more influence on the growth rate of spike than that of bubble.
In Figures 9 and 10, we plot λ1 and λ2 as functions of the orientation angle θ and the
Atwood number A. Figure 10(a) is for A < 0.64 and Figure 10(b) is for A > 0.64. Figure 9
shows that, for fixed Atwood number A, λ1 decreases monotonically as θ changes from θ = 0 (or
π/2) toward π/4. From Figure 10(a) we see that, for fixed Atwood number A < 0.64, λ2 increases
- 36 -
monotonically, as θ changes from θ = 0 (or π/2) to π/4. Thus from Figures 9 and 10(a) and (1) it
follows that, for A < 0.64, the growth rate increases monotonically as the orientation angle θ
changes from θ = 0 (or π/2) to π/4. Therefore, for fixed total wave number k and fixed Atwood
number A < 0.64, the symmetric interface in three dimensions is most unstable, while the inter-
face in two dimensions is least unstable. Figure 10(b) shows that λ2 has a local maxima near
θ = 0 (or π/2), and it has a minimum at θ = π/4 for fixed large Atwood number.
Although we expect that the range of the validity of the Pade approximant is significantly
larger than that of primitive perturbation expansion, the range of the validity of the Pade approxi-
mant is still not infinity. Therefore, our theory may not applicable at asymptotic large times. In
reality, the unstable system becomes turbulent at very late times. The physics of fluid turbulence
involves much more than just the nonlinearity.
Now let us examine the phase boundary between P20 and P1
0 constructions of Pade approxi-
mant. In (1) and (95), the Pade approximation is based on P20 formula when a0
2k 2 > λ2 /λ12 , and
on P10 formula when a0
2k 2 < λ2 /λ12 . Therefore, the phase boundary is determined by
a02k 2 = λ2 /λ1
2 . We have checked that λ1 is always non-negative. In Figure 11(a), we plot λ2 /λ12
for A ≤ 0.7. Figure 11(a) shows that, for A ≤ 0.7, (1) and (95) are based on P20 approximant for
all values of a 0k and θ since λ2 /λ12 is negative. In Figure 11(b), we plot λ2 /λ1
2 for A > 0.7. Fig-
ure 11(b) shows that, for A > 0.7, (1) and (95) are based on either P20 or P1
0 , depending on
whether a02k 2 is larger or less than λ2 /λ1
2 . In Figure 12, we plot the phase boundary as a function
of a 0k and A for several fixed orientation angle θ. The phase boundary curves in Figure 12 are
determined by a 0k = [max{0, λ2 /λ12}]1/2 . As one can seen from Figure 12, the phase domain
covered by P20 construction is much bigger than the the phase domain covered by P1
0 construc-
tion.
In conclusion, we have developed analytic expressions (1)-(3) explicit in terms of the
growth rate from the linear theory vlin , to predict the overall growth rate, as well as the growth
rates of the bubble and spike, for the Richtmyer-Meshkov unstable interfaces in three dimension
for the case of reflected shock with no indirect phase inversion. The theory contains the nonlinear
theory in two dimensions, developed previously by the authors. In three dimensions the non-
linear growth rates of same total wave number k but different orientation angle θ are qualitatively
similar, but quantitatively different. In particular, for fixed total wave number k and fixed Atwood
number A < 0.64, the symmetric interface in three dimensions is most unstable, while the inter-
face in two dimensions is least unstable. Our theories in two and three dimensions are based on
- 37 -
the same physical picture and mathematical methods. In two dimensions, our theory is in
remarkable agreements with the results of full nonlinear numerical simulations and experimental
data. We expect that our theory in three dimensions also should provide quantitatively correct
predictions. However, so far no experiments are available for single mode RM unstable interface
in three dimensions.
Recently, at Los Alamos National Laboratory and Indiana-Purdue University at Indianapo-
lis, full numerical simulations of the compressible RM instability in three dimensions have been
conducted by using two different methods of numerical simulator. It has been shown by the
researchers at these institutes that the predictions of our nonlinear theory for the RM unstable
interface in three dimensions are in good agreement with their numerical solutions. The results
of these numerical studies will be presented separately by these researchers.
Acknowledgement
We would like to thank Drs. J. Glimm and D. H. Sharp for helpful comments and Dr. R.
Holmes for providing the data from his numerical simulations in two dimensions. This work was
supported in part by the U. S. Department of Energy, contract DE-FG02-90ER25084, by subcon-
tract from Oak Ridge National Laboratory (subcontract 38XSK964C) and by National Science
Foundation, contract NSF-DMS-9500568.
Appendix A: Derivation of the Third Order Quantities
In this appendix, we derive the third order quantities η3 ,φ3 and φ′3 . From (43)-(46), (49)-
(51), the third order quantities are governed by
∇ 2φ(3) = 0 in material 1, ∇ 2φ′(3) = 0 in material 2, (A1)
∂t∂η(3)hhhhh +
∂z∂φ(3)hhhhh = − (
∂z 2
∂2φ(2)hhhhhhη(1) +
21hh
∂z 3
∂3φ(1)hhhhhhη(1)2 +
∂z 2
∂2φ(1)hhhhhhη(2))
+ (∂x∂z∂2φ(1)hhhhhhη(1) +
∂x∂φ(2)hhhhh)
∂x∂η(1)hhhhh + (
∂y∂z∂2φ(1)hhhhhhη(1) +
∂y∂φ(2)hhhhh)
∂y∂η(1)hhhhh
+∂x
∂φ(1)hhhhh
∂x∂η(2)hhhhh +
∂y∂φ(1)hhhhh
∂y∂η(2)hhhhh at z = 0, (A2)
∂t∂η(3)hhhhh +
∂z∂φ′(3)hhhhhh = − (
∂z 2
∂2φ′(2)hhhhhhη(1) +
21hh
∂z 3
∂3φ′(1)hhhhhhη(1)2 +
∂z 2
∂2φ′(1)hhhhhhη(2))
+ (∂x∂z
∂2φ′(1)hhhhhhη(1) +
∂x∂φ′(2)hhhhhh)
∂x∂η′(1)hhhhhh + (
∂y∂z∂2φ′(1)hhhhhhη(1) +
∂y∂φ′(2)hhhhhh)
∂y∂φ′(1)hhhhhh
- 38 -
+∂x
∂φ′(1)hhhhhh
∂x∂η(2)hhhhh +
∂y∂φ′(1)hhhhhh
∂y∂η(2)hhhhh at z = 0, (A3)
(ρ′−ρ)gη(3) − ρ′ ∂t∂φ′(3)hhhhhh + ρ
∂t∂φ(3)hhhhh = (ρ′ ∂t∂z
∂2φ′(2)hhhhhh − ρ
∂t∂z∂2φ(2)hhhhhh)η(1)
−21hh[ρ′ ∂t∂z 2
∂3φ′(1)hhhhhh − ρ
∂t∂z 2
∂3φ(1)hhhhhh]η(1)2 + (ρ′ ∂t∂z
∂2φ′(1)hhhhhh − ρ
∂t∂z∂2φ(1)hhhhhh)η(2)
− ρ′( ∂x∂φ′(1)hhhhhh
∂x∂z∂2φ′(1)hhhhhh +
∂y∂φ′(1)hhhhhh
∂y∂z∂2φ′(1)hhhhhh +
∂z∂φ′(1)hhhhhh
∂z 2
∂2φ′(1)hhhhhh)η(1)
+ ρ(∂x
∂φ(1)hhhhh
∂x∂z∂2φ(1)hhhhhh +
∂y∂φ(1)hhhhh
∂y∂z∂2φ(1)hhhhhh +
∂z∂φ(1)hhhhh
∂z 2
∂2φ(1)hhhhhh)η(1)
− ρ′( ∂x∂φ′(1)hhhhhh
∂x∂φ′(2)hhhhhh +
∂y∂φ′(1)hhhhhh
∂y∂φ′(2)hhhhhh +
∂z∂φ′(1)hhhhhh
∂z∂φ′(2)hhhhhh)
+ ρ(∂x
∂φ(1)hhhhh
∂x∂φ(2)hhhhh +
∂y∂φ(1)hhhhh
∂y∂φ(2)hhhhh +
∂z∂φ(1)hhhhh
∂z∂φ(2)hhhhh) at z = 0. (A4)
From the first and second order solutions given by (67), (68) and (74)-(76), the right hand
side of (A2)-(A4) can be evaluated explicitly. They are
∂t∂η(3)hhhhh +
∂z∂φ(3)hhhhh = S11
(3) cos(kxx)cos(kyy) + S31(3) cos(3kxx)cos(kyy)
+ S13(3) cos(kxx)cos(3kyy) + S33
(3) cos(3kxx)cos(3kyy), (A5)
∂t∂η(3)hhhhh +
∂z∂′φ(3)hhhhhh = S ′ 11
(3)cos(kxx)cos(kyy) + S ′ 31(3)cos(3kxx)cos(kyy)
+ S ′ 13(3)cos(kxx)cos(3kyy) + S ′ 33
(3)cos(3kxx)cos(3kyy), (A6)
(ρ′−ρ)gη(3) − ρ′ ∂t∂φ′(3)hhhhhh + ρ
∂t∂φ(3)hhhhh = T11
(3) cos(kxx)cos(kyy) + T31(3) cos(3kxx)cos(kyy)
+ T13(3) cos(kxx)cos(3kyy) + T33
(3) cos(3kxx)cos(3kyy). (A7)
Here
S11(3) =
32k113
1hhhhhha03 σ[(8k 11(−(kx
3 + ky3)k 11(1 + A) + 2Akx
2ky2) + 4k 11
2kxky(kx + ky)A
− k 115(1 + 6A) − 8(kx
3(ky2 − kxk 11) + ky
3(kx2 − kyk 11))A)σ2t 2
+ 8k 11(−(kx3 + ky
3)k 11(2 + A) + 2Akx2ky
2) − 2(1 + 2A)k 115)σt
- 39 -
− (8k 112(kx
3 + ky3) + k 11
5)]
−2k 11
1hhhhha 0σ[(k 112 + kxk 11 − 2kx
2)a.
20(2)
(0) + (k 112 + kyk 11 − 2ky
2)a.
02(2)
(0)]t
−21hha 0(kxa
.20(2)
(0) + kya.
02(2)
(0)), (A8)
S31(3) =
32k113
1hhhhhha03 σ[(−24k 11kx
2(kxk 11(1 + A) − Aky2) + 4(ky
2 − kxk 11)(kxk 112 + 2kx
3)A
− 2k 113(8kx
2A + 3k 112) + (3 − 2A)k 11
5)σ2t 2 + (24k 11kx2(Aky
2 − kxk 11(2 + A))
− 4k 113A(3kx
2 + ky2) − 6k 11
5)σt − 3k 112(8kx
3 + k 113)]
−2k 11
1hhhhha 0σ(3kxk 11 + 2kx2 + k 11
2)a.
20(2)
(0)t −23hha 0kxa
.20(2)
(0), (A9)
S13(3) = S31
(3) (t,kx→ky,ky→kx,a.
20(2)
(0)→a.
02(2)
(0),A), (A10)
S33(3) = −
32
k112
hhhha03 σ[(18A + 9)σ2t 2 + (12A + 18)σt + 9], (A11)
S ′ ij(3) = Sij
(3)(t,kx,ky,a.
ij(2)
(0),A→ −A), (A12)
T11(3) =
8k113
1hhhhha03 σ2[((ρ′ − ρ)A(4k 11(kxky(kx + ky) − k 11
3)
− 2(k 11(kx3 + ky
3) − 2kx2ky
2) − 3k 114) + (ρ′ + ρ)k 11(k 11
3 + 2(kx3 + ky
3)))σt
+ (ρ′ − ρ)A(2k 11(kxky(kx + ky) − k 113) − k 11
4)
+ (ρ′ + ρ)k 11(k 113 + 2(kx
3 + ky3))]
−2k 11
1hhhhha 0σ(ρ′−ρ)[(k 11 + kx)a.
20(2)
(0) + (k 11 + ky)a.
02(2)
(0)], (A13)
T31(3) =
16k113
1hhhhhha03 σ2[((ρ′ − ρ)A(4(kxk 11 − ky
2)(kx2 − 2kxk 11) + 2k 11
2(kx2 − ky
2) − 4k 114)
+ (ρ′ + ρ)k 11(kx2(9k 11 − 4kx) + ky
2k 11 + k 113))σt
+ (ρ′ − ρ)A(4k 11kx(ky2 − kxk 11) − 2k 11
4)
- 40 -
+ (ρ′ + ρ)k 11(kx2(9k 11 − 4kx) + ky
2k 11 + k 113)]
+2k 11
1hhhhha 0σ(ρ′−ρ)( − k 11 + kx)a.
20(2)
(0), (A14)
T13(3) = T31
(3) (t,kx→ky,ky→kx,a.
20(2)
(0)→a.
02(2)
(0),A), (A15)
T33(3) =
81hha0
3 σ2k 11[( − (ρ′ − ρ)A + (ρ′ + ρ))σt − (ρ′ − ρ)A + (ρ′ + ρ)], (A16)
R11(3) =
323hhha0
3 σk 11(ρ + ρ′) −21hha 0(ρ′−ρ)(a
.20(2)
(0) + a.
02(2)
(0)), (A17)
R31(3) =
32k 11
1hhhhhha03 σk13
2 (ρ + ρ′) −21hha 0(ρ′ − ρ)a
.20(2)
(0), (A18)
R13(3) = R13
(3) (t,k 31→k 13 ,a.
20(2)
(0)→a.
02(2)
(0),A), (A19)
R33(3) =
323hhha0
3 σk 11(ρ + ρ′). (A20)
After substituting Sij(3), Tij
(3) and Rij into the general formulae (52)-(54) and (58)-(60), we
have the solutions for the third order quantities. The result for η3 , determined from (52) and (58),
is expressed explicitly in (78). φ(3) and φ′(3) can be easily obtained from our general solutions
given by (53), (54), (59) and (60).
Appendix B: Expressions for λλ3, λλ4 and λλ5
The explicit expressions of λ3, λ4 and λ5 are
λ3 = C 1A 3 + C 2A,
λ4 = −C 3A 3 − C 4A,
λ5 = −C 5A 3 − C 6A.
Here
C 1 =IJL 64
−1hhh −32 f 4
5 f 2hhhhh +8 f 4
61hhhh −32
9 f 2hhhhMJO
cos(θ)
+IJL−
32 f 3
69hhhhh −64 f 4
13 f 2hhhhh +64 f 3
13 f 1hhhhh −32 f 4
69hhhhh +64
f 2hhh −64
f 1hhhMJO
cos(2 θ)
+IJL 128
−59hhhh +32 f 4
3 f 2hhhhh +16 f 4
57hhhhh −32
3 f 2hhhhMJO
cos(3 θ)
- 41 -
+IJL 32
−27hhhh +8 f 4
f 2hhhh +8 f 3
f 1hhhh +64 f 4
69hhhhh −64
3 f 2hhhh −64
3 f 1hhhh +64 f 3
69hhhhhMJO
cos(4 θ)
+IJL 128
−27hhhh +16 f 4
f 2hhhhh +8 f 4
5hhhhMJO
cos(5 θ) +IJL 32 f 4
9hhhhh −32 f 3
9hhhhhMJO
cos(6 θ)
+IJL 64
−1hhh −32 f 3
5 f 1hhhhh −32
9 f 1hhhh +8 f 3
61hhhhMJO
sin(θ) −81hhsin(2 θ)
+IJL 128
59hhhh −32 f 3
3 f 1hhhhh −16 f 3
57hhhhh +32
3 f 1hhhhMJO
sin(3 θ) +IJL 128
−27hhhh +16 f 4
f 1hhhhh +8 f 4
5hhhhMJO
sin(5 θ)
+IJL
−64 f 3
99hhhhh −64 f 4
13 f 1hhhhh +32137hhhh −
64 f 4
99hhhhh −64 f 4
13 f 2hhhhh +256
5 f 1hhhh +256
5 f 2hhhhMJO,
C 2 =IJL 128
−83hhhh +32 f 4
47 f 2hhhhh −128 f 4
703hhhhhh +32
9 f 2hhhhMJO
cos(θ)
+IJL 256
7 f 1hhhh +256 f 3
85f 1hhhhhh −256 f 4
85 f 2hhhhhh +128 f 3
57hhhhhh −256
7 f 2hhhh −128 f 4
57hhhhhhMJO
cos(2 θ)
+IJL 128
−51hhhh +64 f 4
43 f 2hhhhh −128 f 4
307hhhhhh +32
3 f 2hhhhMJO
cos(3 θ) +IJL 32
25hhh −4 f 4
f 2hhhh −4 f 3
f 1hhhh −64 f 4
5hhhhh −64 f 3
5hhhhhMJO
cos(4 θ)
+IJL
−321hhh +
64 f 4
7 f 2hhhhh −32 f 4
13hhhhhMJO
cos(5 θ) +IJL−
128 f 4
3hhhhhh +128 f 3
3hhhhhhMJO
cos(6 θ)
+IJL
−12883hhhh +
32 f 3
47 f 1hhhhh +32
9 f 1hhhh −128 f 3
703hhhhhhMJO
sin(θ) −641hhhsin(2 θ)
+IJL 128
51hhhh −64 f 3
43 f 1hhhhh +128 f 3
307hhhhhh −32
3 f 1hhhhMJO
sin(3 θ) +IJL 32
−1hhh +64 f 1
7 f 1hhhhh −32 f 3
13hhhhhMJO
sin(5 θ) −641hhhsin(6 θ)
+IJL
−16 f 3
7hhhhh −128 f 3
35 f 1hhhhhh −16 f 4
7hhhhh −128 f 4
35 f 2hhhhhh +32127hhhh −
32
f 1hhh −32
f 2hhhMJO,
C 3 =IJL 256
119hhhh −64 f 4
19 f 2hhhhh +512 f 4
7279hhhhhh −2
f 2hhhMJO
cos(θ)
+IJL
−256 f 3
1325hhhhhh −1024 f 3
409 f 1hhhhhhh +1024 f 4
409 f 2hhhhhhh +256 f 4
1325hhhhhh+128
13 f 1hhhhh −128
13 f 2hhhhhMJO
cos(2 θ)
+IJL 256
−231hhhhh +128 f 4
23 f 2hhhhhh +512 f 4
3353hhhhhh −64
15 f 2hhhhhMJO
cos(3 θ)
- 42 -
+IJL 64
−77hhhh +128 f 4
27 f 2hhhhhh +128 f 3
27 f 1hhhhhh +128 f 4
319hhhhhh −8
f 2hhh −8
f 1hhh +128 f 3
319hhhhhhMJO
cos(4 θ)
+IJL−
64
f 2hhh −25697hhhh +
128 f 4
15 f 2hhhhhh +512 f 4
587hhhhhhMJO
cos(5 θ) +IJL−
256 f 4
f 2hhhhhh +256 f 3
f 1hhhhhh +256 f 4
147hhhhhh −256 f 3
147hhhhhhMJO
cos(6 θ)
+IJL 256
1hhhh +512 f 4
13hhhhhhMJO
cos(7 θ) +IJL 256
119hhhh −64 f 3
19 f 1hhhhh −2
f 1hhh +512 f 3
7279hhhhhhMJO
sin(θ) −12827hhhh sin(2 θ)
+IJL 256
231hhhh −128 f 3
23 f 1hhhhhh −512 f 3
3353hhhhhh +64
15 f 1hhhhhMJO
sin(3 θ) +IJL 256
−97hhhh +128 f 3
15 f 1hhhhhh −64
f 1hhh +512 f 3
587hhhhhhMJO
sin(5 θ)
+1281hhhhsin(6 θ) +
IJL−
512 f 3
13hhhhhh −2561hhhh
MJO
sin(7 θ)
+IJL
+128 f 4
457hhhhhh +512 f 3
201 f 1hhhhhh −64507hhhh +
128 f 3
457hhhhhh +512 f 4
201 f 2hhhhhh −128
19 f 1hhhhh −128
19 f 2hhhhhMJO,
C 4 =IJL 256
−317hhhhh +64 f 4
45 f 2hhhhh −512 f 4
8271hhhhhh +64
27 f 2hhhhhMJO
cos(θ)
+IJL
−1024 f 3
557 f 1hhhhhhh −64
7 f 2hhhh +256 f 4
969hhhhhh +1024 f 4
557 f 2hhhhhhh +64
7 f 1hhhh −256 f 3
969hhhhhhMJO
cos(2 θ)
+IJL 64
−19hhhh +128 f 4
45 f 2hhhhhh −512 f 4
3923hhhhhh +128
25 f 2hhhhhMJO
cos(3 θ)
+IJL 32
57hhh −16 f 4
7 f 2hhhhh −16 f 3
7 f 1hhhhh −64 f 4
97hhhhh +128
11 f 2hhhhh +128
11 f 1hhhhh −64 f 3
97hhhhhMJO
cos(4 θ)
+IJL 128
f 2hhhh +12813hhhh +
128 f 4
9 f 2hhhhhh −512 f 4
741hhhhhhMJO
cos(5 θ) +IJL−
256 f 4
3 f 2hhhhhh +256 f 3
3 f 1hhhhhh −256 f 4
79hhhhhh +256 f 3
79hhhhhhMJO
cos(6 θ)
+IJL 256
−1hhhh −512 f 4
25hhhhhhMJO
cos(7 θ) +IJL 256
−317hhhhh +64 f 3
45 f 1hhhhh +64
27 f 1hhhhh −512 f 3
8271hhhhhhMJO
sin(θ) +163hhh sin(2 θ)
+IJL 64
19hhh −128 f 3
45 f 1hhhhhh +512 f 3
3923hhhhhh −128
25 f 1hhhhhMJO
sin(3 θ) +IJL 128
13hhhh +128 f 3
9 f 1hhhhhh +128
f 1hhhh −512 f 3
741hhhhhhMJO
sin(5 θ)
+IJL 512 f 3
25hhhhhh +2561hhhh
MJO
sin(7 θ) +IJL
+32 f 4
15 f 2hhhhh −32237hhhh +
32 f 3
15 f 1hhhhh −512
93 f 1hhhhh −512
93 f 2hhhhh +64 f 3
181hhhhh +64 f 4
181hhhhhMJO,
C 5 =IJL 1536
781hhhhh −768 f 4
221 f 2hhhhhh +384 f 4
2687hhhhhh −12
f 2hhhMJO
cos(θ)
- 43 -
+IJL
−768 f 3
171 f 1hhhhhh +768 f 4
171 f 2hhhhhh −24 f 3
77hhhhh +24 f 4
77hhhhh −3072
233 f 2hhhhhh +3072
233 f 1hhhhhhMJO
cos(2 θ)
+IJL 1536
−475hhhhh +256 f 4
11 f 2hhhhhh +768 f 4
2503hhhhhh −96
13 f 2hhhhhMJO
cos(3 θ)
+IJL 24
−7hhh +128 f 4
13 f 2hhhhhh +128 f 3
13 f 1hhhhhh +16 f 4
23hhhhh −96
5 f 2hhhh −96
5 f 1hhhh +16 f 3
23hhhhhMJO
cos(4 θ)
+IJL−
96
f 2hhh −1536253hhhhh +
768 f 4
43 f 2hhhhhh +256 f 4
153hhhhhhMJO
cos(5 θ)
+IJL−
192 f 4
f 2hhhhhh +192 f 3
f 1hhhhhh −768
f 1hhhh +96 f 4
29hhhhh +768
f 2hhhh −96 f 3
29hhhhhMJO
cos(6 θ)
+IJL 1536
11hhhhh +768 f 4
f 2hhhhhh +48 f 4
1hhhhhMJO
cos(7 θ) +IJL 768
−1hhhh +512 f 3
1hhhhhh +512 f 4
1hhhhhhMJO
cos(8 θ)
+IJL 1536
781hhhhh −768 f 3
221 f 1hhhhhh −12
f 1hhh +384 f 3
2687hhhhhhMJO
sin(θ) −38459hhhh sin(2 θ)
+IJL 1536
475hhhhh −256 f 3
11 f 1hhhhhh −768 f 3
2503hhhhhh +96
13 f 1hhhhhMJO
sin(3 θ) +IJL 1536
−253hhhhh +768 f 3
43 f 1hhhhhh −96
f 1hhh +256 f 3
153hhhhhhMJO
sin(5 θ)
+3841hhhhsin(6 θ) +
IJL−
48 f 3
1hhhhh −153611hhhhh −
768 f 3
f 1hhhhhhMJO
sin(7 θ)
+IJL
+1536 f 3
341 f 1hhhhhhh +1536 f 4
3341hhhhhhh +1536 f 3
3341hhhhhhh +1536 f 4
341 f 2hhhhhhh −7682415hhhhh −
192
27 f 1hhhhh −192
27 f 2hhhhhMJO,
C 6 =IJL 192
−65hhhh +768 f 4
101 f 2hhhhhh −384 f 4
2563hhhhhh +96
11 f 2hhhhhMJO
cos(θ)
+IJL
−384 f 4
87 f 2hhhhhh +192
17 f 2hhhhh −192 f 4
635hhhhhh +384 f 3
87 f 1hhhhhh −192
17 f 1hhhhh +192 f 3
635hhhhhhMJO
cos(2 θ)
+IJL 384
31hhhh +256 f 4
13 f 2hhhhhh −384 f 4
1307hhhhhh +48
5 f 2hhhhMJO
cos(3 θ)
+IJL 64
47hhh −64 f 4
13 f 2hhhhh −64 f 3
13 f 1hhhhh −384 f 4
475hhhhhh +96
5 f 2hhhh +96
5 f 1hhhh −384 f 3
475hhhhhhMJO
cos(4 θ)
+IJL 96
f 2hhh +38435hhhh +
768 f 4
5 f 2hhhhhh −384 f 4
289hhhhhhMJO
cos(5 θ) +IJL−
96 f 4
f 2hhhhh +96 f 3
f 1hhhhh −192 f 4
41hhhhhh +192 f 3
41hhhhhhMJO
cos(6 θ)
- 44 -
+IJL−
768 f 4
f 2hhhhhh −384 f 4
17hhhhhhMJO
cos(7 θ) +IJL 768
1hhhh −512 f 3
1hhhhhh −512 f 4
1hhhhhhMJO
cos(8 θ)
+IJL 192
−65hhhh +768 f 3
101 f 1hhhhhh +96
11 f 1hhhhh −384 f 3
2563hhhhhhMJO
sin(θ) +38447hhhh sin(2 θ)
+IJL 384
−31hhhh −256 f 3
13 f 1hhhhhh +384 f 3
1307hhhhhh −48
5 f 1hhhhMJO
sin(3 θ) +IJL 384
35hhhh +768 f 3
5 f 1hhhhhh +96
f 1hhh −384 f 3
289hhhhhhMJO
sin(5 θ)
+1281hhhhsin(6 θ) +
IJL 384 f 3
17hhhhhh +768 f 3
f 1hhhhhhMJO
sin(7 θ)
+IJL
−1536 f 4
3633hhhhhhh +7682187hhhhh −
1536 f 3
3633hhhhhhh −78 f 3
145 f 1hhhhhh −78 f 4
145 f 2hhhhhh +384
49 f 1hhhhh +384
49 f 2hhhhhMJO.
Here
f 1 =IJL
−23hh cos(2 θ) +
25hh
MJO
1/2
,
f 2 =IJL 2
3hh cos(2 θ) +25hh
MJO
1/2
,
f 3 = IL−4 cos(2 θ) +5
MO
1/2 ,
f 4 = IL 4 cos(2 θ) +5
MO
1/2 .
References
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- 46 -
Captions
Figure 1. A demonstration of our physical picture and theoretical prediction from our
approach in two dimensions. A Mach 1.2 shock incidents from air to SF6 . The solution of
linear theory, the approximate nonlinear solution for incompressible fluids and the solution
of the nonlinear theory are shown. The result of full nonlinear numerical simulation is also
shown. The theoretical predictions are in excellent agreement with the result of full numer-
ical simulation. It demonstrates clearly, that the system changes smoothly from a compres-
sible and linear one to an incompressible and nonlinear one. The transition occurs qualita-
tively at tp , the time associated with the highest peak. Here tp is about 150µs.
Figure 2. An illustration of wave bifurcation due to shock-contact interaction: (a) before the
interaction; (b) the case of reflected shock after bifurcation; (c) the case of reflected rarefac-
tion after bifurcation. The symbols C, IS, TS and RS stand for contact discontinuity,
incident shock, transmitted shock and reflected shock, respectively. LE and TE denote the
leading edge and the trailing edge of the rarefaction wave, respectively.
Figure 3. Comparison of the results of the matched asymptotics given by (96) and the
matched nonlinear theory given by (1) for the growth rate of air-SF6 unstable interface.
The physical parameters here are identical to the ones in Fig. 1. The comparison shows that
(96) and (1) have same accuracy. The curves labeled (i) correspond to θ = 0 (or, π/2) and
the curves labeled (ii) correspond to θ = π/4.
Figure 4 Comparison of the predictions for the overall growth rates of the compressible
unstable interface between air and SF6 . A shock of Mach number 1.2 incidents from air to
SF6 . Figure 4 is the comparison of predictions of the perturbation solutions, η.
1 and
η.
1 + η.
3 , the prediction from the Pade approximation given by (95), and the result from the
full nonlinear numerical simulation.
Figure 5. Comparison of the results of the linear theory, impulsive model, and nonlinear
theory for the overall growth rate given by (1), for air-SF6 unstable interface in three
dimensions for several different values of θ, the orientation of the wave vector (kx,ky). For
comparison, the results of a full nonlinear numerical simulation in two dimensions are also
shown. The physical parameters are same as the ones used in Figure 1. (a) is for the
growth rate and (b) is for the amplitude. Both the linear theory and impulsive model do not
depend on θ, while the results of the nonlinear theory do. In two dimensions the predictions
of the nonlinear theory are in good agreements with the results of the full nonlinear
- 47 -
numerical simulation in two dimensions.
Figure 6. Comparison of the results of the linear theory, impulsive model, and non-linear
theory for the overall growth rate given by (1), for Kr-Xe unstable interface in three dimen-
sions with several different values of the θ, the orientation of the wave vector (kx,ky). A
shock of Mach number 3.5 incidents from Kr to Xe. The dimensionless preshocked inter-
face amplitude is 0.87. For comparison, the results of a full nonlinear numerical simulation
in two dimensions are also shown. (a) is for the growth rate and (b) is for the amplitude.
Both the linear theory and impulsive model do not depend on θ, while the results of non-
linear theory do. In two dimensions the predictions of the nonlinear theory are in good
agreements with the results of the full nonlinear numerical simulation in two dimensions.
Figure 7. Comparison of the results of the linear theory, impulsive model, and non-linear
theory for the bubble and spike given by (2) and (3), for air-SF6 interface with several dif-
ferent values of θ. The physical parameters are same as the ones used in Figure 5. In (c)
and (d), the solid curves are the predictions of the nonlinear theory and the values increase
as θ varies from 0 (or, π/2) toward π/4. The selected valued of θ are same as the ones in
Figure 7(a).
Figure 8. Compare of the results of the linear theory, impulsive model, and non-linear
theory for the bubble and spike given by (2) and (3), for Kr-Xe interface with several dif-
ferent values of θ. The physical parameters are same as the ones used in the Figure 6. In
each figure, the solid curves are the predictions of the nonlinear theory. The selected valued
of θ are same as the ones in Figure 7(a). In (a) and (b), the values decrease (the magnitudes
increase) as θ varies from 0 (or, π/2) toward π/4. In (c) and (d), the values increase as θ
varies from 0 (or, π/2) toward π/4.
Figure 9. Plot of λ1 as a function of the orientation angle θ and the Atwood number A.
Figure 10. Plot of λ2 as a function of the orientation angle θ and the Atwood number A.
(a) is for A < 0.64 and (b) is for A > 0.64.
Figure 11. Plot of λ2 /λ12 as a function of θ and A. (a) is for A ≤ 0.7 and (b) is for A > 0.7.
Figure 12. Plot of phase boundary between P20 and P1
0 constructions of Pade approximant
for several fixed values of θ. The phase boundaries are determined by
a 0k = (max{0,λ2 /λ12})1/2 . The growth rates from the two domains are continuous at the
phase boundary.