[q.zhang s.sohn]-an analytical nonlinear theory of richtmyermeshkov instability(1996)

7/27/2019 [Q.zhang S.sohn]-An Analytical Nonlinear Theory of Richtmyermeshkov Instability(1996)

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AN ANALYTICAL NONLINEAR THEORY OF RICHTMYER-

MESHKOV INSTABILITY

Qiang Zhang and Sung-Ik SohnDepartment of Applied Mathematics and Statistics

SUNY at Stony Brook

Stony Brook, NY 11794-3600

ABSTRACT

Richtmyer-Meshkov instability is a fingering instability which occurs at a

material interface accelerated by a shock wave. We present an analytic, explicit

prediction for the growth rate of the unstable interface. The theoretical predic-

tion agrees, for the first time, with the experimental data on air-SF 6, and is in

remarkable agreement with the results of recent full non-linear numerical simula-

tions from early to late times. Previous theoretical predictions of the growth rate

for air-SF6 unstable interfaces were about two times larger than the experimental

data.

PACS numbers: 47.20.Ma, 47.20.Ky

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. It plays an impor-

tant role in studies of supernova and inertial confinement fusion (ICF). The occurrence of this ins-

tability was first predicted theoretically by Richtmyer [1] in 1960, and ten years later, confirmed

experimentally by Meshkov [2]. Since then several experiments [3-4] and numerical simulations

[5-11] on the growth of the RM unstable interfaces have been performed. Several theories have

been developed by different approaches [1,12-15]. Most of previous theoretical work focused on

the asymptotic growth rate of the linearized Euler equations. However, the growth of the RM

unstable interface is nonlinear and so far theories failed to give a quantitatively correct prediction

for the growth rate of RM unstable interface in the nonlinear regime. Previous theoretical predic-

tions were about twice as large as the experimental data on air-Sf6 .

In this paper, we present a non-linear theory for Richtmyer-Meshkov instability for

compressible fluids in the case when the reflected wave is a shock. Our result is analytic and


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expressed explicitly in terms of the solution of the linear theory. Our theory provides a theoreti-

cal prediction which, for the first time, agrees with the experimental data. The theory shows that

the decay of the growth rate of the instability which has been observed in recent full non-linear

numerical simulations for compressible fluids [8] is due to the effect of non-linearity, rather than

the effect of compressibility. The effect of compressibility dominates at the early stage of the

development of the instability. The effect of non-linearity dominates at intermediate and late

times. Once the shock moved away from the material interface, the fluids can be treated as

approximately incompressible.

Richtmyers impulsive model is a widely used theoretical model for the growth rate of RM

unstable interfaces [1]. It predicts

v imp = u2 + 1

2 1 ka(0+).

Herev imp is the growth rate.uis the difference between the shocked and unshocked mean inter-

face velocities. i is the postshocked density in material i. Richtmyer showed three examples in

which the predictions of the impulsive model agree quite well with the asymptotic solutions of

the linear theory [1]. A more extensive comparison over a large parameter space showed thedomains of agreement and disagreement [14]. 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 impor-

tant and the linear theory is no longer valid [8].

Recently, Hecht et al. have considered a potential flow model for Richtmyer-Meshkov ins-

tability [15]. They approximate the shape of the interface near the tip of each bubble as a parabola

and derived a set of equations for the positions and curvatures of the bubbles. Their model

predicts that for Atwood number A = 1, the asymptotic bubble velocity is 2/3kt.

In this paper we construct a nonlinear theory for the growth rate of the unstable RM inter-

face. The following three steps are key to our analysis: A perturbation theory (Taylors series in

the amplitude of the initial perturbation) truncated at finite order (n = 3 here); the incompressible

approximation to simplify the functional form of the Taylors series, to make its term-by-term

evaluation feasible (the physical reason for this incompressible approximation will be discussed

shortly); and of Pade approximation of the finite Taylors series to extend its time of validity

beyond that of Taylors series itself. The result of three steps is an approximate expression for

the nonlinear growth rate of the form

v=F(vlin). (1)


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Here vlin is the solution of the linear theory for a compressible fluid. The functional form of F

needs to be determined. The mathematical theory we use to construct F is the theory of Pade

approximants [16].

We adopt the following physical picture for the dynamics of the RM unstable interfaces.

The dominant effects of the compressibility occur near the shock. This effect influences the

material interface when the shock hits the interface. At that time the amplitude of the disturbance

is small and the linear theory is valid. As time evolves, the transmitted shock and the reflected

wave move away from the material interface, and the effects of compressibility are reduced. As

the amplitude of the disturbance increases with time, the nonlinearity starts to play a dominant

role in the interfacial dynamics. >From this physical picture, we can consider incompressible

fluids to construct an approximate functional form F. Furthermore, We show that the results

from this approach are remarkably successful.

The governing equations for inviscid, irrotational, incompressible fluids are

2(x,z,t)=0, 2(x,z,t)=0, (2)

t

x

x

+

z

=0 atz = , (3)

t

x

x

+

z

=0 atz = , (4)

()g t

+

t

+

2

1 [(

x

)2 +(

z

)2]

2

1 [(

x

)2 +(

z

)2]=0 atz = .(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 interface at time t. and

are densities. The impulsive force, g = u(t), approximates the effect of the incident shock.

t= 0 is the time at which the shock hits the material interface. and are the velocity poten-tials. The velocity fields are given by v= and v = . The initial material interface is

given byz = (x,0)=a (0)cos(kx). Herea (0) is the amplitude of the initial (unshocked) distur-

bance. Now we derive the generating series for Pade approximants. We expand all quantities in

terms of powers ofa 0 , i. e. f= fn . Here fn= n ,n andn , are proportional to (a 0)n . The ini-

tial conditions are given by n(t= 0)= .

n(t= 0)=0 for n 1, and 1(t= 0)= a 0cos(kx).

.

1(t=0) is determined from the strength of the impulsive force. a 0 = a (0+) is the post-shocked

perturbation amplitude.


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We have determined the first-, second-, third-order quantities analytically. For , we have

1 = (1+ t)a 0cos(kx), (6)

2 =2

1 Aka0

2 2t2cos(2kx), (7)

3 = 24

1 k2a0

3 2[(4A 2 +1)t3 +3t2]cos(kx)+8

1 k2a0

3 2[(4A 2 1)t3 3t2]cos(3kx). (8)

HereA =( )/( + ) is the postshocked Atwood number, and = ukA. We note that the

first order solution given by (6) is identical to Richtmyers impulsive model.

The velocities at the tip of a spike and at the tip of a bubble are given by vs =t

| kx = 0

and vb =t | kx = respectively. The growth rate for the overall unstable interface is defined as

v=2

1 (vs vb) and can be determined analytically from (6) - (8),

v = a 0 2a0

3k2t+ (A 2 2

1 )3a0

3k2t2 +O (a05 ). (9)

The even order terms do not contribute to the growth rate of the overall unstable interface given

by (9). 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. We divide the parameter space in to two subspaces:

a02k2 A 2 +

2

1 and a0

2k2 < A 2 +2

1 . We consider the parameter regime a0

2k2 A 2 +2

1 first.

From the generating series given by (9), one can construct three possible Pade approximants:

P02 (t), P 1

1 (t) and P 20 (t). However onlyP 2

0 (t) has the property, found in the recent full nonlinear

numerical simulation, that the growth rate decays at late times. Therefore P 20 (t) is the only candi-

date for our physical system and the result is

v= P20 (t)=

1+ a02k2t+ 2a0

2k2(a02k2 A 2 +

2

1 )t2

a 0 (10)

fora 02k2 A 2 +

2

1 . Equation (10) is not applicable in the parameter regimea0

2k2 A 2 +2

1 . In

parameter regime a02k2 A 2 +

2

1 , we construct a lower order Pade approximant. In this case,

we can construct either P 01 (t) orP 1

0 (t). Only P 10 (t) has the physical property of decaying and the

result is

v = P 10 (t)=

1+ a02k2t

a 0 (11)

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for a 02

k

2

A2

+ 2

1

. Since P 10

(t) approximation is based on a partial contribution of third order

term rather than the full contribution, the theoretical prediction given by (11) will be less accu-

rate.

Equations (10) and (11) can be combined together to form a single expression

v= P20 (t)=

1+ a02k2t+ max{0, a0

2k2 A 2 +2

1 }2a0

2k2t2

a 0 =F(a 0). (12)

Equation (12) determines the functional relationship between the the nonlinear growth rate v and

linear growth ratea 0 for the impulsive model.

Once we have determined the functional form ofFfrom (10), equation (1) becomes

v=F(vlin)=

1+vlina 0k2t+max{0, a 0

2k2 A 2 +2

1 }vlin

2 k2t2

vlin . (13)

Therefore we have completed our goal of constructing an approximate theory for the nonlinear

growth rate of overall RM unstable interface for compressible fluid for the case of reflected

shock. In the small amplitude or short time limits, (13) recovers the result of linear theory.

Equations (12) and (13) are continuous at the boundary of parameter space, a02k2 =A 2 +

2

1 . We

emphasize that the range of validity of Pade approximants ((12) and (13)) is not limited to that of

the generating series (9). This is well known from the theory of Pade approximants. Therefore

we expect (13) provides a quantitative prediction for the growth rate of the RM instability from

linear regime to nonlinear regime.

Equations (12) and (13) are only applicable when there is no indirect phase inversion. An

indirect phase inversion is defined for the situation a 0(o +)vlin(t)


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In Figure 1, we compare the results of the non-linear theory given by (13), the linear

theory, Richtmyers impulsive model, and a full non-linear numerical simulation [8] for an air-

SF6 unstable interface. The interface is accelerated by a weak shock of Mach number 1.2 mov-

ing from air to SF6 . The reflected wave is a shock. The initial amplitude of the perturbation is

a(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 1(a) is for the growth rate and Figure 1(b) is for the amplitude. Figure 1

shows that our analytical prediction is in remarkable agreement with the result of the full non-

linear numerical simulation. In experiments, it was difficult to measure the growth rate directly.

Instead, one measured the amplitude of the disturbed interfaces, i.e. the half of the longitudinal

distance between the spike and bubble tips. One assumed that the amplitude was a linear func-

tion of time and applied a linear regression analysis to determine the overall growth rate of the

unstable interfaces. The overall growth rate determined from the experimental data was 9.2 m/s

over the time period 310-750 s, (see Figure 1b). When we applied the linear regression to the

amplitude predicted by our theory and to the amplitude determined from numerical solution of

full Euler equations, we found the identical results 9.3 m/s for the growth rate over that time

period for both our theory and for the full numerical solutions. Therefore, the prediction of our

theory is in excellent agreement with the experimental result, as well as with the full nonlinear

numerical simulation. In our theory the amplitude is obtained by integrating (13) over time.

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 Figure 2. The

interface 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)=5mm, the wavelength is 36mm, and the pressure ahead of the shock is 0.5 bar. These physical parameters are

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

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

than the amplitude a 0(0)k= 0.40 given in Figure 1 for air-SF6. Figure 2 shows that the predic-

tions of (13) is still in quite good agreement with the results of the full non-linear numerical

simulation. The experimental results are not yet published.

The physical systems shown in Figures 1 and 2 belong to the parameter regime

a02k2 > A 2 +

2

1 . Our theoretical predictions were based on P0

2 Pade approximation in that


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parameter regime. In Figure 3, we compare the results of the linear, non-linear, and impulsive

theories, and of a full non-linear numerical simulation for a system in the parameter regime

a02k2 < A 2

2

1 . A shock of Mach number 1.5 incidents from light fluid to heavy fluid. The

dimensionless pre-shocked and post-shocked interface amplitudes are 0.40 and 0.32 respectively.

The post-shock Atwood number is A = 0.836. The polytropic exponents of the light and heavy

fluids are = 5/3 and 7/5 respectively. The reflected wave is a shock. Figure 3(a) is for the

dimensionless growth rate and and Figure 3(b) is for the dimensionless amplitude. The dimen-

sionless time, growth rate and amplitude are defined as kc 0M0t, a./c 0M0 and a 0krespectively.

Here c 0 is the speed of sound ahead of the incident shock, M0 is the incident shock Mach

number. For this set of parameters, the theoretical prediction given by (14) is based on P10 con-

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

quite good and is significantly better than the predictions of linear theory and impulsive model.

One may replace vlin by its asymptotic limit vlin =

tlimvlin(t) in (13) to obtain an approxi-

mate expression for the non-linear growth rate,

vappr=1+v lin

a 0k2t+max{0, a 0

2k2 A 2 +2

1 }vlin

2 k2t2

vlin

. (14)

Note that (14) can also be obtained from (12) by replacing a 0 withv lin . In other words, (14) is a

nonlinear, incompressible theory with a corrected impulse strength. The difference between the

linear theory and (13) is due to the non-linear effects; while the difference between (13) and (14)

is due to the compressibility effects. We definetp as the time at which the nonlinear growth rate

reaches the highest peak. tp is an order parameter. For t < tp , the compressibility effects dom-

inate, and fortp < t, the non-linear effects dominate. Attp the system changes from a compressi-

ble, approximately linear one to a non-linear, approximately incompressible one. This conclusion

is illustrated for the case of air-SF6 in Figure 4. In Figure 4, tp is about 150s. Figure 4 shows

that the prediction of (14) is quite accurate even at intermediate time scales. For Kr-Xe case, the

results are similar. We comment that the highest peak of the nonlinear growth rate may not coin-

cide with the highest peak of the linear growth rate.

In conclusion, we have derived an analytical theory for the overall growth rate of

Richtmyer-Meshkov instability for compressible fluids. The theory is applicable when there is no

indirect phase inversion. Our derivation is based on the mathematical theory of Pade approxima-

tion. Our theoretical predictions are in excellent agreement with the results of full nonlinear


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numerical simulations and experimental data from linear to nonlinear regimes. Although the

range of the validity of the Pade approximant is significantly larger than that of primitive pertur-

bation expansion, it 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.

We would like to thank Dr. R. Holmes for helpful discussions and for providing the data

from his numerical simulations. We would like also to thank Drs. J. Glimm and D. H. Sharp for

helpful comments. This work was supported in part by the U. S. Department of Energy, contract

DE-FG02-90ER25084 and National Science Foundation, contract NSF-DMS-9301200.

References

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

[2] E. E. Meshkov, NASA Tech. Trans. F-13, 074, (1970).

[3] A. N. Aleshin, E. G. Gamalli, 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. Zaitsev,

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

[4] R. Benjamin, D. Besnard, and J. Haas, LANL report LA-UR 92-1185, (1993).

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

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

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

[8] J. Grove, R. Holmes, D. H. Sharp, Y. Yang and Q. Zhang, Phys. Rev. Lett. 71, 3473,

(1993).

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

[10] R. L. Holmes, J. W. Grove and D. H. Sharp, Preprint, SUNYSB-AMS-94-14, LA-UR-94-

2024, 1994.

[11] U. Alon, J. Hecht, D. Ofer and D. Sharts, Phys. Rev. Lett. 74, 534, (1995).

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

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

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


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

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

Publishing Co., (1994).


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Captions

Figure 1.Comparison of the results of the linear, non-linear, and impulsive theories, and of

a full non-linear numerical simulation for air-SF6 unstable interface. A shock of Mach

number 1.2 incidents from air to SF6 . The dimensionless preshocked interface amplitude is

0.40. (a) is for the growth rate and (b) is for the amplitude. For this set of parameters, the

theoretical prediction given by (14) is based on P 20 construction of Pade approximant.

Figure 2.Comparison of the results of the linear, non-linear, and impulsive theories, and of

a full non-linear 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. For this set of parameters, the

theoretical prediction given by (14) is based on P 20 construction of Pade approximant.

Figure 3. Comparison of the results of the linear, non-linear, and impulsive theories, and of

a full non-linear numerical simulation for a system in the parameter regime a02k2


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AirSF6

(a)

Growth Rate (m/s)

Time (s)0 200 400 600 800 1000

0

5

10

15

20

Linear Theory

Impulsive Model

Non-linear Theory

Numerical Simulation

ExperimentalTime Range

(b)

AirSF6

Amplitude (mm)

Time (s)0 200 400 600 800 1000

0

5

10

15

20

Linear Theory

Impulsive Model

Non-linearTheory

NumericalSimulation

Experimental Dataand Fitting Result

Figure 1


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KrXe

(a)

Growth Rate (m/s)

Time (s)0 100 200 300 400 500

0

10

20

30

40

50

Linear Theory

Impulsive Model

Non-linear Theory


(b)

KrXe

Amplitude (mm)

Time (s)0 100 200 300 400 500

0

5

10

15

20

25

Linear Theory

TheoryNon-linear

Impulsive Model

SimulationNumerical

Figure 2


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(a)

0 20 40 60 80 1000

0.02

0.04

0.06

0.08

Linear Theory

Impulsive Model

Nonlinear Theory


Growth RateDimensionless

DimensionlessTime

(b)

0 20 40 60 80 1000

2

4

6

8

Linear Theory

Impulsive Model

TheoryNonlinear

NumericalSimulation

AmplitudeDimensionless

DimensionlessTime

Figure 3


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Growth Rate (m/ s)

Time (s)0 200 400 600 800 1000

0

5

10

15

20

Linear Theory

Nonlinear Theory

Asymptotic Velocity

of Linear Theory

NonlinearIncompr. Approx.

Compressibility Effects Dominate

Time of the Highest Peak

Nonlinear Effects Dominate

Figure 4

[q.zhang s.sohn]-an analytical nonlinear theory of richtmyermeshkov instability(1996)

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