[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

- 9 -

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)

- 11 -

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

∂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

∂uz0

hhhh + uz1


∂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


− ρ′ ∂t∂φ′(n)hhhhhh + ρ

∂t∂φ(n)hhhhh =

0≤i, j≤nΣ Tij


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


∂x∂η(b)hhhhh

i =1Πp

η(ni)

+sum2Σ p !

1hhh∂y∂z p


∂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


∂x∂η(b)hhhhh

i =1Πp

η(ni)

+sum2Σ p !

1hhh∂y∂z p


∂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


∂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


∂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


21hh

∂z 3

∂3φ′(1)hhhhhhη(1)2 +

∂z 2

∂2φ′(1)hhhhhhη(2))

+ (∂x∂z


∂x∂φ′(2)hhhhhh)

∂x∂η′(1)hhhhhh + (

∂y∂z∂2φ′(1)hhhhhhη(1) +

∂y∂φ′(2)hhhhhh)


- 38 -

+∂x

∂φ′(1)hhhhhh

∂x∂η(2)hhhhh +


∂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


∂t∂z 2

∂3φ(1)hhhhhh]η(1)2 + (ρ′ ∂t∂z


∂t∂z∂2φ(1)hhhhhh)η(2)

− ρ′( ∂x∂φ′(1)hhhhhh

∂x∂z∂2φ′(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′(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


741hhhhhhMJO

cos(5 θ) +IJL−

256 f 4

3 f 2hhhhhh +256 f 3


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)


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


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)


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

[1] A. N. Aleshin, E. G. Gamalii, S. G. Zaytsev, E. V. Lazareva, I. G. Lebo, and V. B. Roza-

nov, Sov. Tech. Phys. Lett., 14, 466, (1988); A. N. Aleshin, E. V. Lazareva, S. G. Zaytsev,

V. B. Rozanov, E. G. Gamalii and I. G. Lebo, Sov. Phys. Dokl. 35, 159, (1990).

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[3] C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and

Engineers, McGraw-Hill, (1978).

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[5] C. T. Chang, Phys. Fluids 2, 656, (1959).

[6] L. D. Cloutman, and M. F. Wehner, Phys. Fluids A 4, 1821, (1992).

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[7] G. Fraley, Phys. Fluids, 29, 376, (1986).

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(1993).

[9] S. W. Haan, Phys. Fluids B 3, 2349, (1991).

[10] J. Hecht, U. Alon and D. Shvarts, Phys. Fluids, 6, 4019, (1994).

[11] R. L. Holmes, J. W. Grove and D. H. Sharp, J. Fluid Mech. 301, 51, (1995).

[12] R. L. Ingraham, Proc. Phys. Soc. B 67, 748, (1954).

[13] J. W. Jacobs and I. Catton, J. Fluid Mech. 187, 329, (1988).

[14] P. A. Lagerstrom, Matched Asymptotic Expansions, Springer-Verlag, New-York, (1988).

[15] E. E. Meshkov, NASA Tech. Trans. F-13, 074, (1970); E. E. Meshkov, Instability of

shock-accelerated interface between two media, In Dannevik, W., Buckingham, A., and

Leith, C., editors, Advances in Compressible Turbulent Mixing, pages 473--503. National

Technical Information Service, U.S. Department of Commerce, 5285 Port Royal Rd,

Springfield, VA 22161, (1992).

[16] K. A. Meyer and P. J. Blewett, Phys. Fluids 15, 753, (1972).

[17] K. O. Mikaelian, Phys. Rev. Lett. 71, 2903, (1993).

[18] T. Pham and D. I. Meiron, Phys. Fluids A 5, 344, (1993).

[19] A. Pozzi, Applications of Pade Approximation Theory in Fluid Dynamics, World Scientific

Publishing Co., (1994).

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Heidelberg, Teubner, Leipzig, 484, (1905).

[21] R. D. Richtmyer, Comm. Pure Appl. Math. 13, 297, (1960).

[22] R. Samtaney and N. J. Zabusky, Phys. Fluids A 5, 1285, (1993).

[23] A. L. Velikovich and G. Dimonte, Phys. Rev. Lett. 76, 3112, (1996).

[24] Y. Yang, Q. Zhang and D. H. Sharp, Phys. Fluids A, 6, 1856, (1994).

[25] D. L. Youngs, Laser and Particle Beams, 14, 725, (1994).

[26] Q. Zhang and S.-I. Sohn, Phys. Lett. A, 212, 149, (1996).

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Wave, (Phys. Fluids, accepted).

- 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.

[q.zhang s.sohn]-quantitative theory of richtmyer-meshkov instability in three dimensions(1996)

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