Qiang Zhang and Sung-Ik Sohn- Nonlinear Theory of Unstable Fluid Mixing Driven by Shock Wave

8/3/2019 Qiang Zhang and Sung-Ik Sohn- Nonlinear Theory of Unstable Fluid Mixing Driven by Shock Wave

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NONLINER THEORY OF UNSTABLE FLUID MIXING DRIVEN

BY SHOCK WAVE

Qiang Zhang and Sung-Ik Sohn

Department of Applied Mathematics and Statistics

SUNY at Stony Brook

Stony Brook, NY 11794-3600

Corresponding author: Qiang Zhang

Telephone: 516-632-7567

Fax: 516-632-8490

Email: [email protected]

ABSTRACT

A shock driven material interface between two fluids of different density is

unstable. This instability is known as Richtmyer-Meshkov (RM) instability. In

this paper, we present a quantitative nonlinear theory of compressible

Richtmyer-Meshkov instability in two dimensions. Our nonlinear theory con-

tains no free parameter and provides analytical predictions for the overall growth

rate, as well as the growth rates of the bubble and spike, from early to later times

for fluids of all density ratios. The theory also includes a general formulation of

perturbative nonlinear solutions for incompressible fluids (evaluated explicitly

through the fourth order). Our theory shows that the RM unstable system goes

through a transition from a compressible and linear one at early times to a non-

linear and incompressible one at later times. Our theoretical predictions are in

excellent agreements with the results of full numerical simulations from linear to

nonlinear regimes.

PACS numbers: 47.20.Ma, 47.20.Ky


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1. Introduction

When an incident shock collides with an interface between two materials of different densi-

ties, the material interface becomes unstable. This phenomena is known as Richtmyer-Meshkov

instability and plays an important role in studies of supernova and inertial confinement fusion.

This interfacial instability was theoretically predicted by Richtmyer [1]. The first experiment of

this instability was performed by Meshkov [2]. Since then several experiments [3-4] and numeri-

cal simulations [5-12] on the growth of the RM unstable interfaces have been performed. Several

theories have been developed by different approaches [1,13-18]. For review of current researches

of Richtmyer-Meshkov instability, see [19]. Further references can be found in the proceedings of

PCTM Workshops [20] and references therein. Most of previous theoretical work focused on the

growth rate in linear regime. However, the growth of the RM unstable interface is nonlinear [8]

and, for a long time, theories failed to give a quantitatively correct prediction for the growth rate

of RM unstable interface in the nonlinear regime. Previous theoretical predictions were about

twice as large as the experimental data on air-SF6.

In this paper, we present nonlinear theory for Richtmyer-Meshkov instability for compressi-

ble fluids in the case when the reflected wave is a shock. The development of the unstable inter-

face is the formation and growth of fingers, called spike and bubble. A spike is a portion of

heavy fluid penetrating into light fluid, and a bubble is a portion of light fluid penetrating into

heavy fluid. The fingers are the dominant structures of the RM unstable interface. The overall

growth rate of RM unstable interface is defined as the growth rate of the half of the overall size of

the mixing zone between the two fluids. In the case of incompressible flow with infinite density

ratio, Hecht. et. al [17] followed Layzers approach [21] to develop a model and obtained an ana-

lytic expression for the asymptotic bubble growth rate. Our theory presented in this paper pro-

vides quantitative predictions for the overall growth rate of the unstable interface and the growth

rates of spike and bubble in compressible fluids of all density ratio in the case of a reflected shock

from early to later times. The agreements among our theoretical predictions, the results of full

nonlinear numerical simulations and experimental data are remarkable.

Richtmyer proposed an impulsive model to simplify the dynamics of the RM unstable inter-

face [1]. The model approximates the incident shock as an impulse (a delta function) and the

post-shocked fluid as incompressible. The linear growth rate v imp of the fingers predicted by the

impulsive model [1] is

v imp = uAka 0 (1)


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where u is the difference between the shocked and unshocked mean interface velocities,

A = ( )/( + ) is the Atwood number and and are the post-shocked fluid densities.The incident shock propagates from fluid of density to fluid of density . a 0 is the post-

shocked perturbation amplitude at the interface. Equation (1) is a widely used theoretical modelfor RM instability. For weak shocks, the prediction of (1) agrees quite well with the asymptotic

growth rate from the linear theory for compressible fluids. A systematic comparison between (1)

and the solution of the linearized Euler equations showed significant disagreement in certain phy-

sical parameter domains [16]. 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. Most of the experimental and numerical simulations are con-

ducted from the linear regime to the nonlinear regime. This is the time range which will be con-

sidered in this paper.

We present two main results in this paper: (i) a general formulation of perturbative non-

linear solutions for interfacial mixing between incompressible fluids (evaluated explicitly for the

impulsive model through the fourth order); (ii) a quantitative nonlinear theory of compressible

RM instability from early to later times. The goal of our theoretical study is (ii), which is more

important. The results of (i) play essential roles in (ii). Now, we explain how the results of (i) is

related to (ii).

In order to develop an approximate nonlinear theory for compressible RM unstable inter-

face, we adopt the physical picture that the dominant effects of the compressibility occur near the

shocks. This influences the material interface when the shocks are in the vicinity of the material

interface, namely at early times. We assume that the initial disturbance at the interface is small.

Then, at early times 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

compressibility 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 linear and compressible one at early times to a

nonlinear and incompressible one at later times. Our approach is to qualitatively separate the

dynamics of the RM instability into two stages corresponding to early and later times. We deter-

mine an approximate solution in each stage. Then we match the early time solution and the later


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time solution to obtain an analytical expression which changes gradually from one to the other.

The matched solutions give quantitative predictions for the overall growth rate and the growth

rates of spike and bubble at compressible RM unstable interface from early to later times.

The solution of the linearized Euler equations in the case of a reflected shock can be found

in [1] and [16]. Richtmyer developed the linear theory for the case of reflected shock [1]. One of

the authors and his coworker extended the linear theory to the case of a reflected rarefaction wave

[16].

In Section 2, we derive the nonlinear perturbation solutions for incompressible fluids. These

perturbation solutions serve as a generating series for Pade approximations. In Section 3, we

apply Pade approximations and develop a nonlinear theory of RM instability for compressible

fluid. In Section 4, we present validation studies of the Pade approximations, our nonlinear

theory, and the physical picture on which our theory is based. In Section 5, we present the quan-

titative predictions of the nonlinear theory for several cases. In Section 6, we give a summary and

conclusion for our theoretical study.

2. Nonlinear Perturbation Solutions for Incompressible Fluids

As we have described in the previous section, at later times, the effects of compressibility

are less important. Therefore we can approximate the fluids as incompressible. In this section we

derive the nonlinear perturbation solutions for incompressible fluids. The expansion is in terms of

the disturbance at the initial interface.

In Section 2.A, we derive perturbation solutions with general initial growth rate. A general

formula is obtained for n-th order solutions. In Section 2.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 expressions for the growth rate of the unstable

material interface is given through fourth order. Furthermore, the symmetry properties of the

solutions of all orders have been determined analytically.

2.A. Governing Equations and Solution Procedure

Here we derive the nonlinear solutions for incompressible systems with no external forces

and with general velocity distribution along the initial interface. The governing equations for

inviscid, irrotational, incompressible fluids with no external forces are given by

2(x,z,t) = 0 in material 1, 2(x,z,t) = 0 in material 2, (2)


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t

x

x

+z

= 0 at z = , (3)

t

x

x

+z

= 0 at z = , (4)

t +

t

+2

1 [( x

)2 + (z )2]

2

1 [(

x

)2 + (z

)2] = 0 at z = . (5)

Here the unprimed variables are the physical quantities in material 1 and the primed variables are

the physical quantities in material 2. z = (x,t) is the position of the material interface at time t.

and are densities of material 1 and 2, respectively. and are the velocity potentials in

material 1 and 2, respectively. The velocity field is given by v= in material 1 and by

v = in material 2. Two equations given in (2) come from the incompressibility conditions.Equations (3) and (4) represent the kinematic boundary condition that a fluid particle initially

situated at the material interface will remain at the interface afterwards. Equation (5) 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 (x,t = 0) = a 0cos(kx) and the initial velocity distribution along the material

interface is given by .

(x,t = 0). Here a 0 is the amplitude of the initial disturbance. .

(x, 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 = fn . Here fn = n , n and n ,are proportional to (a 0k)

n . Then (2)-(5) can be expressed as

n =1

2(n) = 0 in material 1,n =1

2(n) = 0 in material 2, (6)

n =1

(t

(n)

i =1n

x(i)

x(n i )

+z

(n) ) = 0 at z =

n =1

(n ), (7)

n =1

(t

(n)

i =1n

x(i)

x(n i )

+ z(n)

) = 0 at z =n =1

(n) , (8)

n =1

t(n)

+ n =1

t(n)

+2

1

n =1

i =1n

[x

(i)

x(n i)

+z

(i )

z(n i)

]

2

1

n =1

i =1n

[x

(i)

x(n i)

+z

(i)

z(n i)

] = 0 at z =n =1

(n). (9)


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Since the boundary conditions given in (7)-(9) hold at the position z = =n =1

(n) , they

need to be further expanded at z = 0. After expanding the equations and collecting all terms of

order (a 0k)n , we have the following equations for the n th order quantities.

2(n) = 0 in material 1, 2(n ) = 0 in material 2, (10)

t(n)

+z

(n) =

0in Si(n)(t)cos(ikx) at z = 0, (11)

t(n )

+z

(n) =

0in Si(n)(t)cos(ikx) at z = 0, (12)

t(n)

+ t

(n) =

0in Ti(n)(t)cos(ikx) at z = 0, (13)

with the initial conditions

(n)(x,t = 0) = a 0cos(kx)1n, (14)

. (n)

(x,t = 0) =0in a

.i

(n)(0)cos(ikx). (15)

Here 1n is Kronecker delta function. a.

i

(n )(0) is determined by the Fourier mode decomposition

of the left hand side of (15). Si(n ), Si(n) and Ti(n) are determined by the Fourier mode decomposi-

tion of the right hand sides of the following equations:

0in S i(n)(t)cos(ikx) =

sum1 p !

1

zp +1p +1(a)

i =1p

(ni) +sum2 p !

1

xzpp +1(a)

x(b)

i =1p

(ni) at z = 0, (16)

0in Si(n)(t)cos(ikx) =

sum1 p !

1

zp +1p +1(a)

i =1p

(ni) +sum2 p !

1

xzpp +1(a)

x(b)

i =1p

(ni) at z = 0,

(17)

0in Ti(n)(t)cos(ikx) =

sum1 p !

1

zptp +1

((a) (a))i =1p

(ni)

+2

1

sum3 p!q!

1 [(

zpxp+1(a)

z qxq+1(b)

+zp+1

p +1(a )

z q+1q+1(b )

)

(zpx

p+1(a)

z qxq +1(b)

+zp +1

p +1(a)

z q +1q +1(b)

)]i =1p

(ni)j =1q

(mj) at z = 0

(18)

where


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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 , m 1 , m 2,

...,mq , a, b


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for i 0. For the case i = 0, a0(n) (t) = 0 from the condition of incompressibility. From (11) and

(12), it follows that S0(n) (t) = S0(n)(t) = 0. From (13), b0(n) (t) and b0(n)(t) are determined by

b0(n)(t)+ b0(n) (t) = b0(n )(0)+ b0(n ) (0) +0

t

T0(n) (t)dt.

Since the velocities are the gradients of the velocity potentials and all the source terms in (16)-

(17) involve differentiation with respect to x or/and z, the functional forms of b0(n) and b0

(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 (16)-(18). These source terms are known from the

lower order solutions. We determine Si(n)

, Si(n)

and Ti(n)

from these source terms. Then the n thorder solutions are simply given by (19)-(21) with ai

(n) , bi(n) and bi(n) given by (25), (26) and

(27), respectively. It is easy to show that a i(n) , bi

(n ) and bi(n) are polynomials of t. With respectto t, the degree of ai

(n) is not greater than n and the degrees of bi(n) and bi(n ) are not greater than

n 1.

2.B. Nonlinear Solution of Impulsive Model

Let us apply the solution procedure to determine the nonlinear solution of the impulsive

model through fourth order. The impulsive model approximates that the fluids are at rest at t


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Here R i(n) are determined by the Fourier mode decomposition of the right hand sides of the fol-

lowing equation

0in Ri(n)cos(ikx) =

m = 1

n 1

m !

1

zm

m ((n m) (n m))[a 0cos(kx)]m (31)

at z = 0 and t = 0. From (20) and (21), (31) can be expressed as

( )ua 01n1i bi(n)(0) + b i(n)(0) = Ri(n) . (32)

Finally from (22),(23) and (32), we obtain the initial growth rate for the impulsive model

a.

i

(n)(0) = a 01n1i + +

ikRi(n ) + Si(n)(0) + Si(n)(0)

. (33)

Here = Auk for the impulsive model. Therefore the nonlinear solutions to the impulsive

model are given by (19)-(21) and (25)-(27) with the initial growth rate given by (33).

The explicit expressions through the fourth order are given by

(1) = (1 + t)a 0cos(kx), (34)

(2) =2

1 Aka0

2 2t2cos(2kx), (35)

(3) = 24

1 k2a0

3 [(4A 2 + 1)2t3 + 3t2 + 6t]cos(kx)

+81 k2a0

3 [(4A 2 1)2t3 3t2]cos(3kx), (36)

(4) = 12

1 k3a0

4 [4A 33t4 + 6At2 + At]cos(2kx)

+12

1 k3a0

4 [(8A 3 4A)3t4 8A2t3 + 3At2]cos(4kx). (37)

The first and second order solutions shown above are identical to the ones obtained previously by

Richtmyer [1] and Haan [14], respectively. The derivations for (34)-(37) are given in the Appen-

dices A, B and C.

Now, we examine the symmetry properties ofn th order quantities.

From (18) and (31), Ti(n) and R i

(n) can be expressed as

Ti(n) = Fi(n) Fi(n) , (38)

Ri(n) = Gi(n) Gi(n) . (39)

The definitions ofFi(n) , Fi(n) , Gi(n) and Gi(n) are obvious from (18) and (31).


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The n-th order source terms have the following symmetry

Si(n)(A) = (1)n +1Si(n)(A), (40)

Fi(n)(A) = (1)n Fi(n)(A), (41)

Gi(n)(A) = (1)n Gi(n)(A). (42)

The n-th order solutions have the following symmetry

(n)(A) = (1)n +1(n)(A), (43)

(n)(A,z) = (1)n(n)(A, z). (44)

In (40)-(44), remains fixed. In other words, we do not change the sign of A in the definition of

. The proofs for (40)-(44) are given in Appendix D. From (43) we see that (n) is an even (odd)

function ofA when n is odd (even). Therefore can be expressed as

= a + b .

Here a =k=0

(2k+1) contains all odd orders of a 0 and is an even function of A, and b =k=1

(2k)

contains all even orders of a 0 and is an odd function ofA. We mention again that, here we keep

fixed and do not include the implicit dependence ofA in .

In Fig. 1, we illustrate the development of unstable interfaces for several different values of

the Atwood number, using the perturbation solutions through fourth order given by (34)-(37) at

t = 0.3, 0.6, 0.9 and 1.2. Figures 1(a), 1(b), 1(c) and 1(d) are for A = 0.1, 0.4, 0.7, and 1.0,

respectively. In Fig. 1, as well as Figs. 2 and 3 shown later, we choose a 0 = 1/2,k = 1 and = 1.

Note that the time scales in Fig. 1 are from linear stage to early nonlinear stage. Figure 1 shows

that, as the Atwood number approaches to 1, the spike becomes more narrow and grows faster. In

Fig. 2, we compare the shapes of a bubble and spike for A = 1. The solid curve is a spike and the

dashed curve is a bubble. Figure 2 is obtained from Fig. 1(d) by flipping the bubble portion of

the interface over the x-axis and translating it along the x-axis, so that the symmetry axis of the

spike and bubble coincide. Figure 2 shows that, as t increases, the shape of spike becomes longer

and narrower than that of bubble. In other words, the asymmetry between shapes of spike and

bubble increases with time. Figures 2(a), 2(b), 2(c) and 2(d) are for the time = 0.3, 0.6, 0.9 and

1.2, respectively. In Fig. 3, we compare the interfaces of a spike and bubble for several different

values of the Atwood number at time = 1.2. Figures 3(a), 3(b), 3(c) and 3(d) are for

A = 0.1, 0.4, 0.7, and 1.0, respectively. Figure 3 shows that the interface becomes more and more


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symmetric, as the Atwood number decreases. We see that, for fixed value of , the spike for large

A grows much faster than the one for smaller A, while the variation of A does not have a

significant influence on the growths of bubble. However, for fixed impulse strength (or, fixed

incident shock strength), the growth rates of both spike and bubble increase with A.

3. Pade Approximation and Nonlinear Theory for Compressible Fluids

We have systematically derived the nonlinear perturbation solutions for incompressible

fluids. In this section, we construct the perturbation series for the overall growth rate and the

growth rates of spike and bubble, and apply Pade approximation to extend the range of validity

beyond the range of validity of series expansion itself. Then, we match the linear compressible

solution at early times and the nonlinear incompressible solution at later times to arrive at a non-

linear theory for compressible fluids.

Let us discuss the initial conditions we are going to choose for the nonlinear solutions for

incompressible fluids. As we have outlined earlier, we would like to develop a nonlinear theory

for compressible fluids from early to late times. This will be done by matching the solution of

the linear theory for compressible fluids (valid at early times) and the solution of the nonlinear

theory for incompressible fluids (valid at later times). 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 single mode initial conditions for the nonlinear

solution of incompressible fluids:

(x, 0) = a 0cos(kx) and .

(x, 0) = v 0cos(kx). (45)

Here we assume that a 0kis small and that v 0 is proportional to a 0 . Then from (15), we have

a.

i

(n)(0) = v 01n1i . (46)

v 0 will be determined later through matching. Applying the solution procedure given in Section

2.A, We have the following perturbation solutions for the initial conditions (45):

(1) = (a 0 + v 0t)cos(kx), (47)

(2) =2

1 Akv0

2t2cos(2kx), (48)

(3) = 24

1 k2v0

2 [(4A 2 + 1)v 0t3 + 3a 0t

2]cos(kx)

+8

1 k2v0

2 [(4A 2 1)v 0t3 3a 0t

2]cos(3kx), (49)


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(4) = 12

1 k3v0

2 [4A 3v02t4 + 3Aa0

2t2]cos(2kx)

+12

1 k3v0

2 [(8A 3 4A)v02t4 8Aa 0v 0t

3 + 3Aa02t2]cos(4kx). (50)

Note that, in (47)-(50), we did not use the initial conditions of the the impulsive model. This is

due to the fact that the initial growth rate obtained from the impulsive model is based on the

impulsive approximation to the incident shock. Such an approximation is not valid at early times

for shock driven compressible fluids. The impulsive model contains two approximations: the

incompressible approximation and the impulsive approximation. In our approach, we applied

only the incompressible approximation at the later times, but not the impulsive approximation.

The spike and bubble are located at x = 0 and x = /k, respectively. Let vsp and vbb be the

growth rates at the tips of spike and bubble, respectively. Then

vsp = .

a (0,t) + .

b (0,t), vbb = .

a (0,t) + .

b (0,t).

Here we have used the facts that a contains odd cosine Fourier modes and that b contains even

cosine Fourier modes. .

a (0,t) represents the overall growth rate defined as v =2

1 (vsp vbb).

.

b(0,t) represents2

1 (vsp + vbb). Therefore we have

.

a (0,t) = v 0 a 0v02k2t + (A 2

2

1 )v0

3k2t2 + O ((a 0k)5), (51)

.

b (0,t) = Akv02t 2Ak3a 0v0

3t2 +3

4 k3(A 3 A)v0

4t3 + O((a 0k)6). (52)

The range of the validity of this perturbation solutions 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 series is a Pade approximation [22,23]. Applying the Pade approximation to (51), we

have [18]

.

a (0,t) = 1 + v 0a 0k2t + max{0, a0

2k2 A 2 +2

1 }v0

2k2t2

v 0

. (53)

Equation (53) is based on the P20 Pade approximant when a0

2k2 A 2 2

1 and on the P1

0 Pade

approximant when a02k2 < A 2

2

1 . The physical reason for choosing these Pade approximants is

that the overall growth rate decays at large times [8,10].


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Similarly, we construct the P21 Pade approximant for (52), and the result is

.

b (0,t) =1 + 2k2a 0v 0t + 4k

2v02 [a0

2k2 +3

1 (1 A 2)]t2

Akv02t

. (54)

Following the same proof given in Appendix D, we have also checked that, for the single

mode initial conditions given by (45), the perturbations solutions of all orders satisfy the same

symmetry properties given by (40)-(44). Note that the Pade approximants given by (53) and (54)

satisfy the symmetry properties shown in (43).

Equations (53) and (54) are approximate nonlinear solutions for incompressible fluids.

From the physical picture which we gave earlier, they are also approximate nonlinear solutions

for compressible fluids at later times. At early times, the solution is given by the linear theory for

compressible 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 match-

ing 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 next to the boundary, called inner

layer, is quite different from the dynamics in the region away from the boundary, called outer

layer. One determines the solution in the inner layer, called inner solution, and the solution at the

outer layer, called outer solution, separately, and match these two solutions to form matched

asymptotics. Both inner and outer solutions contain integration constants which are determined

by boundary conditions and matching which will be described shortly. We follow the same pro-

cedure. 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 spatial 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 (53). A recipe to determine v 0 in (53) 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 [24]. Therefore, we

have the equation v lin(t ) = .

a (0,t 0). Here .

a is given by (53). This equation leads to

v 0 = vlin = v lin(t ). Then, (53), the outer solution for the overall growth rate, becomes

vincomp =1 + vlin

a 0k2t + max{0, a0

2k2 A 2 +

2

1 }vlin

2 k2t2

vlin

. (55)


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Similarly, the outer solutions for growth rates of the spike and bubble can be obtained by setting

v 0 to vlin in

.a +

.b and

.a +

.b, respectively. Equation (55) is a nonlinear incompressible

solution with an initial growth rate given by vlin . For weak shocks, vlin

in (55) can be approxi-

mated by the linear solution of the impulsive model given by (1). Finally, following the pro-cedure proposed by Prandtl [24] (see also chapter 2 of [25]), 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 0k2t + max{0, a0

2k2 A 2 +2

1 }vlin

2 k2t2

vlin

vlin . (56)

The matched solution for the spike and bubble growth rates can be obtained in a similar way.

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 would like to construct a simpler matched solution which has the same order of

accuracy. The facts that (53) approaches v 0 at early times and that the growth rate of the linear

theory for compressible fluids approaches an asymptotic constant vlin at later times show that

another way of matching can be achieved by replacing v 0 with v lin in (53) and (54). Then we

have

v =1 + v lina 0k

2t + max{0, a02k2 A 2 +

2

1 }vlin

2 k2t2v lin (57)

for the overall growth rate,

v bb = v +1 + 2k2a 0v lint + 4k

2vlin2 [a0

2k2 +3

1 (1 A 2)]t2

Akvlin2 t

(58)

for the growth rate of bubble and

v sp = v +1 + 2k2a 0v lint + 4k

2vlin2 [a0

2k2 +3

1 (1 A 2)]t2

Akvlin2 t (59)

for the growth rate of spike. Therefore we have achieved our goal of constructing approximate

theories for the overall growth rates and the growth rates of the bubble and spike in compressible

fluids. We emphasize that the range of validity of (57)-(59) are not limited to the range of vali-

dity of generating series (47)-(50). This is well known from the theory of Pade approximation


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- 15 -

[22,23].

It is easy to see that in the early time, or small amplitude limits, (57)-(59) approach v lin.

Equations (58) and (59) show that the spike grows faster than the bubble. We comment that our

nonlinear theories given by (57)-(59) contain no adjustable parameter. Equations (57)-(59) are

applicable to the systems with no indirect phase inversion only. An indirect phase inversion is

defined for the situation a 0(0+)v lin(t ) < 0. (We use the notations 0 and 0+ to represent the

times just before and after shock contact interaction, respectively.) For the case of reflected

shock, the indirect phase inversion usually does not occur [16]. The comparison between results

of the matched asymptotic technique (56) proposed by Prandtl and the matching given by (57)

will be presented in the next section.

4. Validation of the the nonlinear theory and physical picture

In this section, we validate our nonlinear theory as well as our physical picture. Our theoret-

ical derivations contain two major steps: one is the Pade approximation, and the other is the

matching between the compressible linear solutions and the incompressible nonlinear solutions.

To check the validity of Pade approximation, we consider the incompressible fluids with A = 1.

For this system, Hecht et. al. [17] have developed a Layzer-type [21] potential flow model for the

bubble. The model assumes that the fluid is incompressible and irrotational, and approximates the

shape of the interface near the tip of the bubble as a parabola. An asymptotic bubble growth rate

of 2/3kthas been determined analytically. The prediction of the Layzer-type potential flow model

over finite time scale can be obtained numerically. It has been shown that the numerical solution

of the Layzer-type model for a bubble is in excellent agreement with the result of a full nonlinear

numerical simulation in the case of A = 1, (see Fig. 1 of [17]). In Fig. 4, we show the comparison

among the predictions of the Layzer-type potential flow model, the perturbation solutions given

by (47)-(50), and the Pade approximants given by (55). Here, the asymptotic bubble growth rate

of the Layzer-type model is based on the analytic expressions 2/3kt, while the finite time solution

of the Layzer-type model is determined numerically. The parameters used here are

a 0k = vlin = 1/2. It is clear from Fig. 4, that the range of the validity of the perturbation solution

is small, but the range of the validity of the Pade approximant is quite large. This is an important

feature of Pade approximation . As a text book says, "The real power of Pade summation is illus-

trated by its application to divergent series"(see page 385 of [23]).


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- 16 -

In Fig. 5(a), we compare the predictions of the perturbation solutions, .

1 and .

1 + .

3 (with

v 0 = vlin ), the prediction from the Pade approximation given by (55) for incompressible system

(the outer solution), and the result from the full nonlinear numerical simulation for the overall

growth rate of the compressible unstable interface between air and SF6 . Note that

.

2 and

.

4 do

not contribute to the overall growth rate. The compressible interface is accelerated by a weak

shock of Mach number 1.2 moving from air to SF 6 . The reflected wave is a shock. The initial

amplitude of the perturbation is a 0(0) = 2.4mm, the wave length is 37.5mm and the pressure

ahead of the shock is 0.8 bar. These physical parameters are taken from Benjamins experiments

[4]. The post-shock Atwood number is A = 0.701. Figure 5(a) shows again that the range of vali-

dity of the nonlinear perturbation solutions are very limited and the Pade approximation has suc-

cessful extended the range of validity. The prediction of the perturbation solution deviates from

the nonlinear theory around t = 250 s. At this time the dimensionless amplitude of the fingers at

the interface, k, is about 0.8. Figure 5(a) shows that the prediction of (55) is accurate even at

intermediate time scales.

In Fig. 5(b), we compare the results from the compressible linear theory, the incompressible

nonliner theory given by (55), the matched nonlinear theory for compressible fluids given by

(57), and a full nonlinear numerical simulation for the overall growth rate of an air-SF 6 unstable

interface. The physical parameters here are identical to the ones shown in Fig. 5(a). Figure 5(b)

shows that the technique of the matching is successful.

We comment that the difference between the predictions of (57) and the linear theory is due

to the nonlinear effects; while the difference between the predictions of (55) and (57) is due to the

compressible effects. Figure 5(b) shows that the dynamics of the RM unstable interface indeed

changes from an approximately linear and compressible dominant one at early times to a non-

linear dominant and approximately incompressible one at later times. The transition occurs

around the highest peak of the nonlinear growth rate. This validates the physical picture on

which our theory was based.

In Fig. 6, we compare the predictions of the matched asymptotics given by (56) which is

derived from the matching technique proposed by Prandtl, the matched nonlinear theory given by

(57), the full numerical simulation as well as the compressible linear theory and the impulsive

model (1) for an air-SF6 unstable interface. Figure 6(a) is for the overall growth rate and Fig. 6(b)

is for the amplitude. Since the incident shock is weak, the prediction of the impulsive model is

close to the asymptotic velocity of the linear theory. Figures 6(a) and 6(b) shows that the results


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- 17 -

of (56) and (57) indeed give same order of accuracy. The reader may compare Fig. 6(a) with Fig.

5(b) to distinguish the prediction of (56) from that of (57). Since the expression of the matched

nonlinear theory given by (56) is simpler than the one given by (57), we use (56)-(58) for the

predictions of the growth rates.

We emphasize that our theory provides the quantitative growth rates from linear to non-

linear stages for all Atwood numbers. Although we have demonstrated that the range of the vali-

dity of the Pade approximant is significantly larger than that of primitive perturbation expansion,

the range of the validity of the Pade approximant is still not infinity. Therefore, our theory may

not applicable at asymptotic large times. In fact, at the time when v bb = 0, our theory is no longer

valid. In reality, the unstable system becomes turbulent at very late times. The physics of fluid

turbulence involves much more than just the nonlinearity [26].

5. Quantitative Predictions

In this section we present the quantitative predictions of our nonlinear theory for the overall

growth rate and the growth rates of the bubble and spike.

Figure 6(b) shows that our analytical prediction is in remarkable agreement with the result

of the full non-linear numerical simulation from linear to nonlinear regimes. In experiments, it

was difficult to measure the growth rate directly. Instead, one measured the amplitude of the dis-

turbed interfaces, i.e. the half of the longitudinal distance between the spike and bubble tips. One

assumed that the amplitude was a linear function of time and applied a linear regression analysis

to determine the overall growth rate of the unstable interfaces. The overall growth rate deter-

mined from the experimental data was 9.2 m/s over the time period 310-750 s, (see Fig. 6(b)).

When we applied the linear regression to the amplitude predicted by our theory and to the ampli-

tude determined from numerical solution of full Euler equations, we found the identical growth

rate 9.3 m/s for over that time period. Therefore, the prediction of our theory is in excellent

agreement with the experimental result, as well as with the full nonlinear numerical simulation.

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.

A comparison for the growth rate of Kr-Xe unstable interface is shown in Fig. 7. The inter-

face is accelerated by a strong shock of Mach number 3.5 moving from Kr to Xe. The reflected

wave is a shock. The initial amplitude of the perturbation is a 0(0) = 5 mm, the wave length is

36 mm, and the pressure ahead of the shock is 0.5 bar. These physical parameters are taken from


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- 18 -

Zaytsevs recent experiments. The post-shock Atwood number is A = 0.184. The dimensionless

perturbation amplitude a 0(0)k is 0.87. This initial perturbation amplitude is about two times

larger than the amplitude a 0(0)k = 0.40 given in Figs. 5 and 6 for air-SF6. Figure 7 shows that

the predictions of (57) is still in quite good agreement with the results of the full non-linearnumerical simulation. The experimental results are not yet published.

The physical systems shown in Figs. 5, 6 and 7 belong to the parameter regime

a02k2 > A 2

2

1 . Our theoretical predictions were based on P0

2 Pade approximation in that

parameter regime. A comparison between the prediction of our nonlinear theory based on P10

construction (for the case of a02k2 < A 2

2

1 ), and the result from full numerical simulation can

be found in Fig. 3 of reference [18]. As one expected, the theoretical prediction based on P10 con-

struction of Pade approximant is less accurate. However, our theoretical prediction is still quite

good and is significantly better than the predictions of linear theory and impulsive model [18].

In Fig. 8, we compare the effects of different initial amplitude. We consider Kr-Xe unstable

interface with the same physical parameters used in Fig. 7. We choose the two initial amplitudes,

a 0(0)k = 0.87 and 0.6. The solid curves are results from the linear theory and nonlinear theory,

and the dashed curves are ones from the numerical simulations. The index (i) corresponds to

a 0(0)k = 0.87 and the index (ii) a 0k(0) = 0.6. Figure 8 shows that the predictions of (57) are in

good agreement with the results of the numerical simulation for both cases and the nonlinear

effect is more pronounced in the larger initial amplitude case.

In Fig. 9, we compare our theoretical predictions for the the bubble and spike, i.e. (58) and

(59), with the results from full numerical simulations as well as the predictions of the linear

theory and Richtmyers impulsive model for air-SF6 interface. The physical parameters are same

as the ones used in Fig. 5. Figures 9(a) and 9(b) are for the growth rate and the amplitude of the

bubble, respectively. Figure 9(c) and 9(d) are for the growth rate and the amplitude of the spike,

respectively. The amplitudes of the bubble and spike are determined by integrating (58) and (59)

over time, respectively. In Fig. 10, we consider a Kr-Xe interface. The physical parameters are

same as the ones used in Fig. 7. Figures 10(a) and 10(b) are for the growth rate and the amplitude

of the bubble, respectively. Figures 10(c) and 10(d) are for the growth rate and the amplitude of

the spike, respectively.

Figures 9 and 10 show that our theoretical predictions are in good agreement with the

results from full numerical simulations, while the predictions of the linear theory for


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- 19 -

compressible fluids and the linear impulsive model are qualitatively incorrect at later times. We

comment that our theory is still valid at the early stage after the formations of the roll-ups,

mushroom-shaped vortex structures, at the spike. The reason is that our theory is for the tips of

the spike and bubble. The roll-ups are secondary structures which do not have significant effectson the growth rates at the early stages of roll-ups. At the later stage of roll-ups, the secondary

structures do affect the growth rate of the spike. This is why the prediction for the spike growth

rate becomes less accurate than that for the bubble at later times. For further improvement of the

theory for RM instability, it may be necessarily to include the rotational effects in the dynamics

of the spike.

6. Conclusion

In this paper, a nonlinear theory for Richtmyer-Meshkov instability in compressible fluids

has been presented. The theory is applicable from the linear (small amplitude) regime to the

moderately large amplitude nonlinear regime of the development of RM instability for systems

with no indirect phase inversion. Most of full numerical simulations and experiments were con-

ducted over this time range. Our theory may not be applicable at very late time. Our theory

shows that the dynamics of RM instability is compressible and approximately linear at early time,

and nonlinear and approximately incompressible at late times. This physical picture has been

validated by the results from full numerical simulations.

Our theory provides analytical predictions (expressed explicitly in terms of the solution of

the linear theory) for the overall growth rate, as well as the growth rates of the spike and bubble,

of RM unstable interface with all density ratios. Our theoretical predictions are in excellent agree-

ment with the results from full numerical simulations for compressible fluids, and the results from

experiments on air-SF6 unstable interface without any adjustable parameters.

In the process of deriving our nonlinear theory, we have developed the procedure to obtain

the nonlinear perturbation solutions for incompressible fluids with arbitrary density ratio and

arbitrary velocity distribution along the initial interface. We give an analytical expression for the

n th order solution. Our solution procedure is recursive, - the higher order solutions are

expressed in terms of the lower order quantities. The nonlinear perturbation solution for the

impulsive model are given explicitly through fourth order in this paper.

The nonlinear theory presented in this paper is for Richtmyer-Meshkov instability in two

dimensions for the case of a reflected shock wave. Our theoretical approach can also be extended


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- 20 -

to the case of reflected rarefaction wave and the three dimensions.

Appendix A: Derivation of the First and Second Order Quantities

In this appendix, we derive the first order quantities (1), (1) and (1), and second orderquantities (2), (2) and (2).

For the first order equations, (16)-(18) and (31) give

0i1 Si(1)(t)cos(ikx) =

0i1 Si(1)(t)cos(ikx) =

0i1 Ti(1)(t)cos(ikx) =

0i1 R i(1)(t)cos(ikx) = 0.

It follows that S i(1)(t) = Si(1)(t) = Ti(1)(t) = Ri(1) = 0. Then from (25)-(27), we have the first order

solution (34) and

(1) = ka 0 e kzcos(kx), (1) = ka

0 e kzcos(kx). (A1)

For the second order equation, (16)-(18) and (31) give

i =02

Si(2)(t)cos(ikx) =

z 22(1)

(1) +x

(1)

x(1)

at z = 0, (A2)

i =02

Si(2)(t)cos(ikx) = z 2

2(1) (1) +

x(1)

x(1)

at z = 0, (A3)

i =02

Ti(2)

(t)cos(ikx) = ( tz2(1)

tz2(1)

)(1)

21

[( x

(1)

)2

+ ( z

(1)

)2

]

+2

1 [(

x(1)

)2 + (z

(1) )2] at z = 0, (A4)

i =02

Ri(2)cos(ikx) = z(1)

z

(1) at z = 0, t = 0. (A5)

The right hand sides of (A2)-(A5) can be evaluated analytically from the first order solution given

by (34) and (A1). Then we can determine the Fourier coefficients. The non-zero Fourier

coefficients are

S2(2) = a0

2k(t + 1), S2(2) = S2(2), T0(2) =2

1 ( )2a02 , R0(2) = R2(2) =

2

1 ( )a02 .

From (25)-(27) and (31), we obtain the second order solution (35) and

(2) =2

1 a0

2 [(1 + A)t + 1]e 2kzcos(2kx) 2

1 Aa0

2 (t + 1), (A6)

(2) =2

1 a0

2 [(1 A)t + 1]e 2kzcos(2kx) +2

1 Aa0

2 (t + 1). (A7)


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- 21 -

Appendix B: Derivation of the Third Order Quantities

In this appendix, we derive the third order quantities (3),(3) and (3). From (16)-(18), thesource terms for the third order equations are given by

i =13

Si(3)(t)cos(ikx) = (

z 22(1)

(2) +2

1

z 33(1)

(1)2 +z 2

2(2) (1))

+x

(1)

x(2)

+ (xz

2(1) (1) +

x(2)

)x

(1) at z = 0, (B1)

i =13

Si(3)(t)cos(ikx) = (z 2

2(1) (2) +

2

1

z 33(1)

(1)2 +z 2

2(2) (1)) (B2)

+x

(1)

x(2)

+ (xz

2(1) (1) +

x(2)

)x

(1) at z = 0,

i =13

Ti(3)(t)cos(ikx) = ( tz

2(2)

tz2(2)

)(1)

+2

1 (

tz 23(1)

tz 23(1)

)(1)2 + ( tz2(1)

tz

2(1) )(2)

( x(1)

xz2(1)

+z

(1)

z 22(1)

)(1) + (x

(1)

xz2(1)

+z

(1)

z 22(1)

)(1)

( x(1)

x(2)

+z

(1)

z(2)

) + (x

(1)

x(2)

+z

(1)

z(2)

) at z = 0.

(B3)

and initial condition (31) is

i =13

Ri(3)cos(ikx) = [ z

(2)

z(2)

]cos(kx) +2

1 [

z 22(1)

z 2

2(1) ][cos(kx)]2 . (B4)

From the first order solution given by (34) and (A1), and the second order solution given by (35)

and (A6)-(A7), we can evaluate the right hand sides of (B1)-(B4) explicitly. From these equa-

tions we can determine Si(3), S

i(3), Ti

(3), and R

i(3), The non-zero source terms are

S1(3) (t) =

8

1 k2a0

3 [(2A + 3)2t2 + (4A + 6)t + 3],

S3(3) (t) =

8

1 k2a0

3 [(18A + 9)2t2 + (12A + 18)t + 9],

Si(3)(t) = Si(3)(t, A A), i = 1, 3,

T1(3) (t) =

2

1 ka0

3 2[( )A(3t + 1) + ( + )(t + 1)],


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- 22 -

T3(3) (t) =

2

1 ka0

3 2[( )A(t + 1) + ( + )(t + 1)],

R1(3) =

8

1 ka0

3 ( + ),

R3(3) =

8

3 ka0

3 ( + ).

From our solution formulae (25) we obtain (36) for (3), and from (26) and (26), we have

(3) =8

1 ka0

3 [( 4A 2 + 2A + 2)2t2 + (4A + 4)t + 1]e kzcos(kx)

+24

1 ka0

3 [(12A 2 + 18A + 6)2t2 + (12A + 12)t + 9]e 3kzcos(3kx), (B5)

(3)

= 81

ka03

[(4A2

+ 2A 2)2

t

2

+ (4A 4)t 1]ekz

cos(kx)

+24

1 ka0

3 [( 12A 2 + 18A 6)2t2 + (12A 12)t 9]e 3kzcos(3kx). (B6)

Appendix C: Derivation of the Fourth Order Quantities

In this appendix, we derive the fourth order quantities (4),(4) and (4). From (16)-(18),

the source terms for the fourth order equations are given by

i =14 Si

(4)(t)cos(ikx) = (z 2

2(1) (3) +z 3

3(1) (1)(2) +61

z 4

4

(1)

(1)3 +z 2

2

(2)

(2)

+2

1

z 33(2)

(1)2 +z 2

2(3) (1)) +

x(1)

x(3)

+ (xz

2(1) (1) +

x(2)

)x

(2)

+ (xz

2(1) (2) +

2

1

xz 23(1)

(1)2 +xz

2(2) (1) +

x(3)

)x

(1) at z = 0,

i =14

Si(4)(t)cos(ikx) = (z 2

2(1) (3) +

z 33(1)

(1)(2) +6

1

z 44(1)

(1)3 +z 2

2(2) (2)

+2

1

z 33(2) (1)2 +

z 22(3) (1)) +

x(1)

x(3)

+ (xz

2(1) (1) +x

(2) )x

(2)

+ (xz

2(1) (2) +

2

1

xz 23(1)

(1)2 +xz

2(2) (1) +

x(3)

)x

(1) at z = 0,

i =14

Ti(4)(t)cos(ikx) = ( tz

2(2)

tz2(2)

)(2)

+2

1 (

tz 23(2)

tz 22(2)

)(1)2 + ( tz2(3)

tz

2(3) )(1)


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- 23 -

[2

1

xz2(1)

(1)

2

+2

1

x(2)

2

+x

(1)

xz2(1)

(2)

+

x

(1)

xz

2(2) (1) +

x

(1)

x(3)

+xz

2(1)

x(2)

(1) +2

1

x(1)

xz2

3(1) (1)2]

+ [2

1

xz2(1)

(1)

2

+2

1

x(2)

2

+x

(1)

xz2(1)

(2)

+x

(1)

xz2(2)

(1) +x

(1)

x(3)

+xz

2(1)

x(2)

(1) +2

1

x(1)

xz 23(1)

(1)2]

[2

1

z 22(1)

(1)

2

+2

1

z(2)

2

+z

(1)

z 22(1)

(2)

+z

(1) z 2

2(2) (1) + z(1) z

(3) +z 2

2(1) z(2) (1) +

21

z(1)

z 33(1) (1)2]

+ [2

1

z 22(1)

(1)

2

+2

1

z(2)

2

+z

(1)

z 22(1)

(2)

+z

(1)

z 22(2)

(1) +z

(1)

z(3)

+z 2

2(1)

z(2)

(1) +2

1

z(1)

z 33(1)

(1)2]

at z = 0. From (31) we have

i =04

R i(4)cos(ikx) = [ z

(3) z

(3) ]cos(kx) +2

1 [

z 22(2)

z 22(2) ][cos(kx)]2

+6

1 [

z 22(1)

z 2

2(1) ][cos(kx)]3 .

The non-zero Fourier coefficients are

S2(4) (t) =

12

1 a0

4k3[(4A 2 + 18A + 2)3t3 + (12A 2 + 30A + 6)2t2 + (12A + 3)t + 4],

S4(4)

(t) = 61

a04

k3

[(30A2

+ 24A + 2)3

t3

+ (18A2

+ 36A + 6)2

t2

+ (12A + 15)t + 8],

Si(4)(t) = Si(4)(t, A A), i = 2, 4,

T0(4) (t) =

8

1 k2a0

4 2( )[(2A 2 + 2)2t2 + (4A 2 + 4)t + 3],

T2(4) (t) =

8

1 k2a0

4 2[( )((20A 2 + 2)2t2 + (8A 2+4)t + 5) + ( + )A(182t2 + 20t + 4)],


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- 24 -

T4(4) (t) =

4

1 k2a0

4 2[( )((7A 2 + 2)2t2 + (6A 2 + 4)t + 2) + ( + )A(92t2 + 10t + 2)].

R0(4) =

8

1 k2a0

4 ( ),

R2(4) =

24

5 k2a0

4 ( ),

R4(4) =

3

1 k2a0

4 ( ).

From our solution formulae (25) we obtain (37) for (4), and from (26) and (26), we have

(4) =24

1 k2a0

4 [( 16A 3 + 4A 2 + 18A + 2)3t3

+ (12A2

+ 30A + 6)2

t2

+ 3t A + 4]e2kz

cos(2kx)

+24

1 k2a0

4 [(16A 3 + 30A 2 + 16A + 2)3t3

+ (18A 2 + 24A + 6)2t2 + (15A + 15)t + 8]e 4kz cos(4kx)

+24

1 k2a0

4 [(2A 3 + 2A)3t3 + (6A 3 + 6A)2t2 + 9At + 3A ], (C1)

(4) =24

1 k2a0

4 [(16A 3 + 4A 2 18A + 2)3t3

+ (12A 2 30A + 6)2t2 + 3t + A + 4]e 2kzcos(2kx)

+24

1 k2a0

4 [( 16A 3 + 30A 2 16A + 2)3t3

+ (18A 2 24A + 6)2t2 + ( 15A + 15)t + 8]e 4kzcos(4kx)

24

1 k2a0

4 [(2A 3 + 2A)3t3 + (6A 3 + 6A)2t2 + 9At + 3A ]. (C2)

Appendix D: Proofs for (40)-(44)

Here we provide proofs for (40)-(44). From (20) and (21), it is easy to see that and haveopposite sign for their z dependence. We prove (40)-(44) by induction. From (34) and (35) as

well as the fact that Si(1) = Si(1) = Ti(1) = Ri(1) = 0, we can see (40)-(44) hold for k = 1.


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- 25 -

As an induction hypothesis, we assume (40)-(44) hold for all k n 1. Let us prove the

case k = n, for n 2. We prove (40) first. We examine the first term on the right hand side of

(17),

sum1 p !

1

zp +1p +1(a)(A)

i =1p (ni)(A)

= sum1 p !

1 (1)

a + p + 1 +i =1p

(ni + 1)

zp +1p +1(a)(A)

i =1p

(ni)(A)

= (1)(n +1)[ sum1 p !

1

zp +1p +1(a)(A)

i =1p

(ni)(A)]. (D1)

The first equality in (D1) follows from the induction hypothesis

(a)(A) = (1)a(a)(A) and (ni)(A) = (1)(ni + 1)(ni)(A).

for all a,ni < n, and the fact that the z dependence of(a) and (a) has an opposite sign (see (18)

and (31)). The second equality in (D1) follows from the facti =1p

ni + a = n, (see the definition of

sum1). Comparing the final expression of (D1) with the first summation on the right hand side of

(16), we see that the first summations from the right hand sides of (16) and (17) indeed satisfy the

relation (40). The proof for the relation between the second terms on the right hand sides of (16)

and (17) is similar (keeping in mind thati =1p

ni + a + b = n; see the definition of sum2). Therefore

we have proven that (40) is true for k = n. Following the similar procedure, we have checked that

(41) and (42) are hold for k = n also. There we need the facti =1p

ni +j =1q

mj + a + b = n. (See the

definition of sum3.)

Now we prove that (43) is true for k = n. From (38) and (39) and the identities

+

=

2

1 (1 + A) and

+

=

2

1 (1 A), (D2)

(25) can be written as

ai(n)(A) =

2

1

0t

{ik(t t)[(1 + A)Fi(n)(A,t) (1 A)Fi(n)(A,t)]

+ (1 + A)Si(n)(A,t) + (1 A)Si(n)(A,t) + ik[(1 + A)Gi(n )(A) (1 A)Gi(n)(A)]}dt

=2

1

0t

{ik(t t)[[(1)n Fi(n )(A,t) Fi(n)(A,t)] + A[(1)n Fi(n )(A,t) + Fi(n)(A,t)]]


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[(1)n Si(n)(A,t) Si(n)(A,t)] A[(1)n Si(n)(A,t) + Si(n)(A,t)]

+ ik[(1)n Gi(n)(A) Gi

(n)(A)] + ikA[(1)n Gi(n)(A) + Gi

(n)(A)]}dt. (D3)

The second equality in (D3) follows from (40)-(42). From (D3) it is easy to see that

ai(n)(A) = (1)n +1ai

n (A). (D4)

From (D4) and (19), we see that (43) holds for k = n, n 2.

Finally, let us prove (44). From (38)-(42) and (D2), (26) can be expressed as

bi(n)(A) =

2

1

0t

{[(1)n Fi(n)(A,t) Fi(n)(A,t)] + A[(1)n Fi(n)(A,t) + Fi(n)(A,t)]}dt

+2

1 {[(1)n Gi

(n)(A) Gi(n)(A)] + A[(1)n Gi

(n)(A) + Gi(n)(A)]}

2ik

1 (1 + A)[(1)n Si

(n)(A) + Si(n)(A)]. (D5)

Similarly, (36) can be expressed as

bi(n)(A,t) = 2

1

0

t

{[(1)n Fi(n)(A,t) Fi(n)(A,t)] + A[(1)n Fi(n)(A,t) + Fi(n)(A,t)]}dt

2

1 {[(1)n Gi

(n)(A) Gi(n)(A)] + A[(1)n Gi

(n)(A) + Gi(n)(A)]}

2ik

1 (1 A)[(1)n Si

(n)(A) + Si(n)(A)]. (D6)

From (D5) and (D6), we see that bi(n)(A) = (1)n bi(n)(A). Then from (20) and (21), we see that(44) holds for k = n, n 2.

This completes our proofs for (40)-(44).

References

[1] R. D. Richtmyer, "Taylor instability in shock acceleration of compressible fluids", Comm.

Pure Appl. Math. 13, 297, (1960).


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- 27 -

[2] E. E. Meshkov, "Interface of two gases accelerated by a shock wave", Fluid Dyn. 4, 101,

(1969).

[3] A. N. Aleshin, E. V. Lazareva, S. G. Zaytsev, V. B. Rozanov, E. G. Gamalii and I. G.

Lebo, "Linear, nonlinear, and transient stages in the development of the Richtmyer-

Meshkov Instability", Sov. Phys. Dokl. 35, 159, (1990).

[4] R. Benjamin, D. Besnard, and J. Haas, "Shock and reshock of an unstable interface", LANL

report, LA-UR 92-1185, (1993).

[5] K. A. Meyer and P. J. Blewett, "Numerical investigation of the stability of a shock-

accelerated interface between two fluids", Phys. Fluids 15, 753, (1972).

[6] L. D. Cloutman, and M. F. Wehner, "Numerical simulation of Richtmyer-Meshkov instabil-

ities", Phys. Fluids A 4, 1821, (1992).

[7] T. Pham and D. I. Meiron, "A numerical study of Richtmyer-Meshkov instability in con-

tinuously stratified fluids", Phys. Fluids A 5, 344, (1993).

[8] J. Grove, R. Holmes, D. H. Sharp, Y. Yang and Q. Zhang, Quantitative theory of

Richtmyer-Meshkov instability, Phys. Rev. Lett. 71, 3473, (1993).

[9] K. O. Mikaelian, "Growth rate of the Richtmyer-Meshkov instability at shocked interfaces",

Phys. Rev. Lett. 71, 2903, (1993).

[10] R. L. Holmes, J. W. Grove and D. H. Sharp, "A numerical investigation of Richtmyer-

Meshkov instability using front tracking", J. Fluid Mech. 301, 51, (1995).

[11] U. Alon, J. Hecht, D. Ofer and D. Shvarts, "Power laws and similarity of Rayleigh-Taylor

and Richtmyer-Meshkov mixing fronts at all density ratios", Phys. Rev. Lett. 74, 534,

(1995).

[12] D. L. Youngs, "Numerical simulation of turbulent mixing by Rayleigh-Taylor instability",

Laser and Particle Beams 14, 725, (1994).

[13] G. Fraley, "Rayleigh-Taylor stability for a normal shock wave-density discontinuity interac-tion", Phys. Fluids 29, 376, (1986).

[14] S. W. Haan, "Weakly nonlinear hydrodynamic instabilities in inertial fusion", Phys. Fluids

B 3, 2349, (1991).

[15] R. Samtaney and N. J. Zabusky, "On shock polar analysis and analytical expressions for

vorticity deposition in shock-accelerated density-stratified interfaces", Phys. Fluids A 5,

1285, (1993).


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[16] Y. Yang, Q. Zhang and D. H. Sharp, "Small amplitude theory of Richtmyer-Meshkov insta-

bility", Phys. Fluids A 6, 1856, (1994).

[17] J. Hecht, U. Alon and D. Shvarts, "Simple potential flow models of Rayleigh-Taylor and

Richtmyer-Meshkov bubble fronts", Phys. Fluids A 6, 4019, (1994).

[18] Q. Zhang and S.-I. Sohn, "An analytical nonlinear theory of Richtmyer-Meshkov instabil-

ity", Phys. Lett. A 212, 149, (1996).

[19] V. Rupert, "Shock-interface interactions: current research on the Richtmyer-Meshkov prob-

lem", in Shock Waves, Proceedings of the 18th International Symposium on Shock Waves,

edited by K. Takayama, (Springer-Verlag, Berlin, 1992).

[20] PCTM Workshops, 1988-1995: Proc. First Intl Workshop on the Physics of Compressible

Turbulent Mixing, Princeton, 1988, edited by W. P. Dannevik, A.C. Buckingham and C. E.

Leith; Proc. Second Intl Workshop on the Physics of Compressible Turbulent Mixing,

Pleasanton CA, 1989, compiled by V. Rupert; Proc. Third Intl Workshop on the Physics of

Compressible Turbulent Mixing, Abbey of Royaumont, France, 1991; Proc. Fourth Intl

Workshop on the Physics of Compressible Turbulent Mixing, Cambridge, England, 1993;

Proc. Fifth Intl Workshop on the Physics of Compressible Turbulent Mixing, SUNY at

Stony Brook, NY, 1995;

[21] D. Layzer, "On the gravitational instability of two superposed fluids in a gravitational field",

Astrophys. J. 22, 1, (1955).

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

Publishing Co., 1994).

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

Engineers, (McGraw-Hill, 1978).

[24] L. Prandtl, "Fluessigkeiten bei sehr kleimer reibung, III", International Math. Kongress,

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

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

[26] W. D. McComb, The Physics of Fluid Turbulence, (Oxford Science Publications, 1990).


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Captions

Figure 1. The predictions of the perturbation solutions through fourth order at time

t = 0.3, 0.6, 0.9 and 1.2. Solutions for different values of the Atwood number are shown

here. Figure 1(a), 1(b), 1(c) and 1(d) are for A = 0.1, 0.4, 0.7 and 1.0, respectively.

Figure 2. Comparison of the shapes of a spike and bubble for A = 1. The solid curve is a

spike and the dashed curve is a bubble. Figure 2 is obtained from Fig. 1(d) by flipping the

bubble portion of the interface over the x-axis and translating it along the x-axis, so that the

symmetry axis of the spike and bubble coincide. Figures 2(a), 2(b), 2(c) and 2(d) are for the

time = 0.3, 0.6, 0.9 and 1.2, respectively.

Figure 3. Comparison of the shapes of a spike and bubble for different values of Atwood

number at time = 1.2. The solid curve is a spike and the dashed curve is a bubble. Figures

3(a), 3(b), 3(c) and 3(d) are for the A = 0.1, 0.4, 0.7 and 1.0, respectively.

Figure 4. Comparison of the perturbation solutions given by (47)-(50), the Pade Approxi-

mation given by (55), and the Layzer-type potential flow model [17] for A = 1. The asymp-

totic bubble growth rate of the Layzer-type model is based on the analytic expressions

2/3kt, while the finite time solution of the Layzer-type model is determined numerically.

The parameters here are a 0k = vlin = 1/2. The curves labeled a, b, c and d correspond to

.

1, .

1 .

2, .

1 .

2 + .

3, and .

1 .

2 + .

3 .

4 , respectively.

Figure 5. 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 . The dimensionless preshocked interface amplitude is 0.40. Figure 5(a) is the com-

parison of predictions of the perturbation solutions, .

1 and .

1 + .

3 , the prediction from the

Pade approximation given by (55), and the result from the full nonlinear numerical simula-

tion. Figure 5(b) is the comparison of the results from the compressible linear theory, the

incompressible nonliner theory given by (55), the matched nonlinear theory for compressi-

ble fluids given by (57), and a full nonlinear numerical simulation. Figure 5(b) shows thatthe RM unstable system indeed changes from a compressible and approximately linear sys-

tem to an a nonlinear and approximately compressible system at late times.

Figure 6. Comparison of the results of the matched asymptotics (56), the matched non-

linear theory given by (57), a full nonlinear numerical simulation, the linear theory, and

Richtmyers impulsive model for air-SF6 unstable interface. (a) is for the growth rate and

(b) is for the amplitude. The physical parameters here are identical to the ones in Fig. 5.


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The comparison shows that (56) and (57) have same accuracy.

Figure 7. Comparison of the results of the linear, nonlinear, and Richtmyers impulsive

theories, and of a full nonlinear numerical simulation for Kr-Xe unstable interface. A shock

of Mach number 3.5 incidents from Kr to Xe. The dimensionless preshocked interface

amplitude is 0.87. (a) is for the growth rate and (b) is for the amplitude.

Figure 8. Comparison of the effects of different initial amplitude for Kr-Xe unstable inter-

face. The physical parameters are same as the ones in Fig. 7 except initial amplitudes. The

solid curves are results from the linear theory and nonlinear theory, and the dashed curves

are ones from the numerical simulations. The curves labeled (i) correspond to a 0k = 0.87

and the curves labeled (ii) correspond to a 0k = 0.6.

Figure 9. Comparison of the predictions of the nonlinear theory, i.e. (58) and (59), and full

numerical simulations, for the growth rates and amplitudes of bubble and spike for air-SF6

interface, as well as the predictions of the linear theory and Richtmyers impulsive model.

The physical parameters are same as the ones used in Fig. 5. (a) and (b) are for the bubble.

(c) and (d) are for the spike.

Figure 10. Comparison of the predictions of the nonlinear theory, i.e. (58) and (59), and

full numerical simulations, for the growth rates of bubble and spike at Kr-Xe interface, as

well as the predictions of the linear theory and Richtmyers impulsive model. The physical

parameters are same as the ones used in Fig. 7. (a) and (b) are for the bubble. (c) and (d) are

for the spike.

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Qiang Zhang and Sung-Ik Sohn- Nonlinear Theory of Unstable Fluid Mixing Driven by Shock Wave

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