A k model for turbulent mixing in shocktube flows induced byRayleigh–Taylor instabilitySerge Gauthier and Michel Bonnet Citation: Phys. Fluids A 2, 1685 (1990); doi: 10.1063/1.857576 View online: http://dx.doi.org/10.1063/1.857576 View Table of Contents: http://pof.aip.org/resource/1/PFADEB/v2/i9 Published by the American Institute of Physics. Additional information on Phys. Fluids AJournal Homepage: http://pof.aip.org/ Journal Information: http://pof.aip.org/about/about_the_journal Top downloads: http://pof.aip.org/features/most_downloaded Information for Authors: http://pof.aip.org/authors

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A k-e model for turbulent mixing in shock-tube flows induced by Rayleigh­Taylor instability

Serge Gauthier and Michel Bonneta)

CEA, Centre d'Etudes de Limeil- Va/enton, 94195 Villeneuve-St. -Georges, Cedex, France

(Received 17 August 1989; accepted 14 May 1990)

A k-€ model for turbulent mixing induced by Rayleigh-Taylor instability is described. The classical linear closure relations are supplemented with algebraic relations in order to be valid under strong gradients. Calibrations were made against two shock-tube experiments {Andronov et al. [Sov. Phys. JETP 44,424 (1976); Sov. Phys. Dokl. 27, 393 (1982)] and Houas et al. [Proceedings o/the 15th International Symposium on Shock Waves and Shock Tubes (Stanford U.P., Stanford, CA, 1986)]) using the same set of constants. The new interpretation of the experimental data of Brouillette and Sturtevant [Physica D 37, 248 (1989)], where the mixing length is discriminated from the wall jet, requires a different numerical value for the Rayleigh-Taylor source term coefficient. A detailed physical study is given in both cases. It turns out that the spectrum is narrower in the Brouillette and Sturtevant case than in the Andronov et al. case but the small length scales are of the same magnitude.

I. INTRODUCTION Rayleigh-Taylor and Richtmyer-Meshkov instabilities

occurring within the context of inertial confinement fusion (lCF) have been the object of strong interest in recent years. While the linear growth of small perturbations on the inter­face between layers of different densities is relatively well understood, the nonlinear stage and the subsequent turbu­lent phase are. poorly understood. Some understanding of the nonlinear growth can be gained by using numerical simu­lations as opposed to the fully developed turbulent phase where models are needed. Because of the space-time scale tinniness involved in an ICF experiment, the basic mecha­nisms of the instability are usually studied on Rayleigh-Tay­lor-induced mixing in a shock tube although the effects of sphericity, high temperatures, and small Prandtl numbers are disregarded. Such experiments were first carried out in the Soviet Union in the early 1970s by Andronov et al. 1,2 and Zaitsev et al.,3 and more recently by Houas et al. in Mar­seilles,4 by Benjamin5 in Los Alamos, and at Caltech by Brouillette and Sturtevant.6 In this latter work the authors give two interpretations of their experimental data. The first one agrees with the pioneering results of Andronov et al., 1.2

but they claimed it takes into account the wall effects. Ignor­ing boundary layer effects, a second interpretation leads to a mixing-length thickness one order of magnitude smaller than the one of the previous interpretation.

The two-equation version of the Andronov et al. model 1

has also been used in an ICF experiment in order to study the effects of turbulent mixing on the neutron yield. 7 These au­thors later published a more sophisticated modef with evo­lution equations for every second-order quantity. At the same time, turbulent mixing models became fashionable in the West· and simple models8

•9 and later more elaborate

models were built. 8-

12 Nevertheless, it is not yet clear what information is brought by n-equation models. In that re­spect, it should be mentioned that in the experiments we are interested in we only have access to the mixing-zone thick­a) Deceased, 16 July 1990.

ness and some mean flow quantities such as concentration or temperature. The behavior of these quantities are usually well reproduced, and sometimes better described, by simple k-L or k-€ models. Before setting up sophisticated models, more information should be obtained from the experiments in order to motivate the development of complex models. For that purpose, much attention should be paid to multi­scale models because they account for the spectral character of the turbulence and it turned out that they predict accu­rately fast evolving and nonequilibrium turbulent flows more accurately.

In this context, the challenge was to build a one-dimen­sional (I-D) mixing model imbedded in a I-D production compressible hydrocode. As in the work of Andronov et al., we used closure models which are probably the only way out, despite the weakness of such an approach. As a matter of fact, when using closure models in unsteady situations, we have to face some major difficulties. The first one is the need of giving an initial turbulent state. This necessity comes from the basic assumption where variables are split into a mean and a fluctuating part. In other words, statistical models are unable to describe transition mechanisms. Related to this inability, the resulting averaged equations are known not to produce chaotic states. As a result, the final state is slightly sensitive to the initialization procedure. A second difficulty arises when simulating the interaction of turbulence with a shock wave. The numerical treatment of strong discontinui­ties requires a numerical dissipation in order to smear out shock waves and smooth gradients, which makes them artifi­cially less efficient in producing turbulent kinetic energy. It follows a dependence on the spatial resolution. A third diffi­culty is encountered when making the closure assumptions. They often assume a weak dependence of the nth correlation on the (n - 1)th one. Under shock wave influence, such a hypothesis is no longer valid and closure assumptions have to be supplemented with some relationships to produce meaningful results. Last but not least, closure assumptions cannot account for inhomogeneity.

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For that purpose, a one-equation turbulent model has been developed at Limeil and successfully used to numeri­cally simulate Rayleigh-Taylor-induced turbulent mixing at an interface. 8 In this model the characteristic length was assumed to be the mixing length, which is a global quantity. In order to remove these unsatisfactory global quantities we added an equation for the dissipation rate E of the turbulent kinetic energy.

In Sec. II we present the one-equation model while the k-E model is detailed in Sec. III. A physical study on the experimental data of Andronov et al. and Brouillette and Sturtevant is carried out in Sec. IV. The initial condition, the spatial resolution, and the source term coefficient sensitivity are reported in Sec. V.

II. THE ONE-EQUATION MIXING MODEL

As a first step we develop a one-equation model similar to the Andronov et al. model. The two basic thermodynami­cal assumptions are ( 1) the miscibility of the two fluids and (2) the existence of an equilibrium temperature everywhere. The mixing model is then simply defined by

P =PI +P2' (1)

P(PI ,P2,n = PI (ppn + P2 (p2,n, (2)

e(PI,P2,n =cel (PI,n + (1-c)e2(P2,n, (3)

where P, P, and e are, respectively, the total density, pres­sure, and internal energy. The indices refer to the partial corresponding quantities of fluids 1 and 2. The concentra­tion of one species is defined by c = PI Ip. We then assume that the turbulence is statistically isotropic and homoge­neous in the plane normal to the shock propagation direc­tion. It follows that the mixing flow can be described only by the mean density P, the mean temperature T, the mean pres­sure P, the mean velocity u, and the mean concentration c. As a consequence, physical variables depend only on the time t and the space coordinate r. Note, however, that we do take into account the fluctuating parts of the three velocity components. By means of the classical Favre average for all variables but the pressure and density, it is straightforward to derive evolution equations for the mean quantities P, c, u, e, and k. We will hereafter omit tildes and bars over mean quantities except for the correlations of fluctuating quanti­ties. The splitting procedure introduces correlations between fluctuating parts from which evolution equations can be derived. They can also be closed algebraically by the first gradient law (Fick's law). In this model we retain only an evolution equation for the turbulent kinetic energy de­fined by

pk = ~ pU;2 + pu'i, (4)

where u; and u;. are, respectively, the radial and the trans­verse fluctuating parts of the velocity. The evolution equa­tion takes the form

P Dk = _ pU;2 aaurr _ u; ap + ~ (akL(t)pk 112 ak)

Dt ar ar ar

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k 312

- ap L(t)'

Phys. Fluids A. Vol. 2. No.9. September 1990

(5)

where the bars denote the ensemble average and where the turbulent kinetic energy flux has been closed by the first gradient law; L(t) is the mlxmg length and D IDt = a lat + Ur alar; and ak and a are two dimension­less constants. The first term on the right-hand side in Eq. (5) is the source term of the mean flow velocity gradient. The Reynolds stress is modeled by the classic linear relation

--2 4 112 aUr 2 pu; = -3avL(t)k Pa;+3Pk. (6)

However, the first-gradient law holds for small gradients. Under a shock wave, within the Richtmyer-Von Neumann scheme, the velocity gradient becomes meaningless and so does the linear relation (6). This closure relation has to be replaced by a nonlinear equation or supplemented with some relationships. We choose to simply bound the radial turbu­lent kinetic energy by the following double inequality:

(7)

where Am = 0.1 and AM = 1.25. This relation simply means that the radial turbulent kinetic energy is always strictly positive and never greater than the total turbulent kinetic energy. Notice that the relation (7) is probably the simplest

way of making the closure of pU;2 nonlinear. The last term in Eq. (5) is a simple modeling of the

turbulence dissipation. More interesting is the second term. This term is actually a source term for k and provided the first gradient law is used to calculate the mass flux, it has the same form as the Rayleigh-Taylor instability criterion for an incompressible fluid. We naturally interpret this term as the main source term in a Rayleigh-Taylor-induced turbulent flow. Following this interpretation we retain only positive contributions satisfying the instability criterion,

-, ap 0 'f J'P 0 . h Du Ur a;= I a;r< Wit r=Iit' (8)

This assumption implies that a given level of turbulence may only decrease by diffusion or dissipation but not by a source term. Of course, this statement must be weakened because a slight decrease in turbulence by a source term may arise, for example, if the source term is proportional to the mean velocity gradient. However, in our case, numerical ex­periments show a strong decrease or even a disappearance of the turbulent kinetic energy, in an expansion, for example, and lead us to this empirical procedure. Moreover, because products of strong gradients are oflittle significance we may linearize the Rayleigh-Taylor source term by using the mean density gradient over the mixing zone:

-,ap_ L(t)kl12PA-PBaP4(1 ) u --a -- -- c -c r ar r P L(t) ar '

if ap r>O with r Du ar = Dt '

17 ap = 0, otherwise, r ar

(9)

(10)

where (PA - PB )IL(t) gives the order of magnitude of the density gradient while 4c( 1 - c) mimics its shape.

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This one-equation model has been used to successfully simulate the shock-tube experiment of Andronov et al. where an air-helium contact surface is accelerated and de­celerated by a shock wave of initial Mach number 1.3 in air.I.2 The results of the mixing-zone thickness have already been reported9 and showed good agreement between experi­mental data and numerical simulation. By using the same set of model constants, a c = a k = a e = 0.1, au = 0.05, a = 4, and a r = 1.75, where a c and a e are the coefficients of diffu­sion for the concentration and the internal energy, respec­tively, we have also simulated the cylindrical implosion of a cylinder of helium surrounded by air. Details of the simula­tion which correctly reproduced the mixing-length behavior are also given in Ref. 9.

However, this one-equation model suffers from some drawbacks. It contains unsatisfactory global quantities such as the mixing length and the mean density gradient over the mixing zone. Moreover, one-equation models are not widely used and, consequently, associated coefficients have not been the object of careful studies as opposed to more popular models such as the k-E models.

III. THE TWO-EQUATION MIXING MODEL: k-e MODEL

A. Model equations

We next developed a k-E model by adding an equation for the dissipation rate E of the turbulent kinetic energy. The basic hydrodynamical and thermodynamical assumptions detailed in the previous section are still used. As in the one­equation model, we take advantage of the simplicity of Favre's averaging by casting the constant density closures into density weighted ones. 13 The additional equation for E is usually built by comparison with the k equation. Following Launder and Spalding,14 we assume that source terms are equal to the k-equation source terms up to a dimensional factor of the form CS'Elk. Adding a diffusion term and a production term resulting from the compressibility effect, we finally obtain an evolution equation for E. The complete set of equations are the continuity equation,

Dp aU r -= -p-; Dt ar

(11)

the concentration equation,

Dc a (CD k2

ac) p Dt = ar -;;:P~ ar ;

(12)

the velocity equation,

DU r ap a pU;2 p--= ------;

Dt ar ar (13)

the energy equation,

De _pau r + -, ap a [CD k 2 p-= U r -+- -p-

Dt ar ar ar U e E

X:r(e+ ;)] +pE; (14)

the turbulent kinetic energy equation,

1687 Phys. Fluids A. Vol. 2. No.9. September 1990

-,-2 aU r pUr a;- -ap

U'­r ar

+i.(CD p~ ak) -pE; ar Uk E ar

and the dissipation rate equation

DE E --2 aU r E - ap p Dt = - CEI k pu; a; - C.o k u; a;

(15)

a (CD k2

aE) C aU r +- -p-- -pCE2 -- CdPE-. ar U E E ar k ar

(16)

As assumed in the standard "k-E," turbulent fluxes are computed from first-gradient law where diffusion coeffi­cients are built from the two basic local quantities k and E. As already stated, on the shock wave some fluxes have to be limited by algebraic relations in order to satisfy realizability conditions such as the positivity of the radial turbulent kinet­ic energy. This procedure has also been used in plasma phys­ics for the thermal flux. For example, the radial turbulent kinetic energy has to satisfy the obvious inequalities

0< pU;2 < 2pk.

Therefore, on strong gradients, we use the closure

4 aU r 2 pU'2 = -3'/1, a; + 3'pk,

with Ampk < pU;2 <AMpk, (17a)

where Am and AM' respectively, have been chosen equal to 0.10 and 1.25 and

(17b)

In the same way, approximating the turbulent mass flux

p'u' by the density gradient may lead to very high unphysi­cal values under a shock wave. Instead of linearizing the Rayleigh-Taylor source term, we supplement the closure relation with an algebraic relation. To this end, we note that the Schwarz inequality yields an upper bound of the mass flux,

(18)

The momentum mass flux is bounded by the total turbu­lent kinetic energy following relation (7) and the variance of density fluctuations is assumed to be bounded by the mean density itself,15 we finally write the turbulent mass flux as

p'u' = - Dp gradp,

with the realizability condition

I p'u' I <apA ~2pk 112,

where ap is chosen to be equal to 0.67 and Dp is defined by Dp = (CDlup)/(k 2IE). The constant up has to be adjusted in order to fit the experimental data. Finally, the main pro­duction term in Eqs. (15) and (16) reads

-ap PRT = - u;­

ar

and is computed as

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1 CD k 2 ap ap ap ------, if Ip'u'I<lp'u'llim andif r-ar>O,

p CTp € ar ar

PRT = sign(- ap)a (pU;2)112 ap, if I p'u'I>1 P'U'llim andif r ap >0 ar p p ar ar

0, if r ap <0, ar

The closure relations for t/J = e,€,k and the internal energy are now written as

CD k 2

pt/J'U' = -pDq, gradt/J, with Dq, =--. (19) CTq, €

The set of equations (11 )-( 16) is closed by giving the equation of state of each species,

R* 1 R* PI =-PIT, el =---T, (20) /11 rl - 1 /11

R* 1 R* P2=-P2T, e2=---T, (21) /12 r2 - 1 /12

and the mixing model which gives the total density, pressure, and internal energy as functions of the partial quantities,

P=PI +P2' (22)

P(PI ,P2,T) = PI (PI ,T) + P2 (p2,T), (23)

e(PI ,P2,T) = eel (PI ,T) + (1 - e)e2 (P2,T)· (24)

The boundary conditions are of a Dirichlet type for the velocity, internal energy, turbulent kinetic energy, and its dissipation rate. The initial condition for both the turbulent kinetic energy and its dissipation rate is a triangular profile of height kinit (€init) and width L init detailed below.

The equations of the model are solved by means of a finite-difference method within a Lagrangian description. The mesh is moved with the velocity u r so that the cell mass is constant. The diffusion terms of the k and € equations are handled in a coupled implicit manner while Rayleigh-Tay­lor source terms are discretized in an explicit way. The Richtmyer-Von Neumann artificial viscosity is used for the shock waves. 16

B. Model constants

The k-€ model constants are quite well known from the relevant literature. All of them, however, are determined either by numerical optimization or by comparison with some key experiments carried out with incompressible fluids. In Ref. 12 we have made an attempt to connect C.o and CTp ' which are the coefficients of the Rayleigh-Taylor source term in the k and € equations. By considering turbu­lence decaying in a stratified stationary incompressible fluid the following relation has been found:

C.o = I + 0.2CTp • (25)

1688 Phys. Fluids A, Vol. 2, No.9, September 1990

This relation shows that C.o is always greater than one as opposed to the classical value used in Refs. 7 and 10, C.o = 0.85. Both values will be used in the simulations re­ported below and we will find two sets of constan ts (C.o ,CT p )

that calibrate the experimental data. Conventional choices are made for the other constants.

We shall use CEI = 1.47, CE2 = 1.90, CTk = 0.87, CTE = 1.30, CTe = 0.90, CD = 0.09 and AM = 1.25, Am = 0.1, and a = 0.67 for all runs reported in this paper; C.o is fixed either by Eq. (25) or at the value 0.85. Moreover, we used Cd = O. This choice does not markedly influence the results since the term div u is almost always and almost everywhere equal to zero for I-D slab geometry applications.

Finally, the model constants and the initial conditions used for numerical applications are given in Table I.

c. Diffusion term self-similar behavior

The contributions of the right-hand side ofEq. (15) fall into three classes: production, diffusion, and dissipation terms. As we shall see below, in a shock-tube experiment the Rayleigh-Taylor source term acts only on the shock waves, while the characteristic time scale of dissipation is relatively long compared with the duration of the experiment. There­fore the main terms responsible for the spreading of the mix­ing zone are the diffusion terms. They are nonlinear and consequently the solution has a compact support increasing with time. To see this, consider the diffusion part of the k-€ model:

ak a (CD k2

ak) at = ar -;;: -;- a; , (26)

a€ a (CD k2

a€) at = ar -;;: -;- ar '

(27)

of which the self-similar solution, in terms of the reduced variable S = xt - 113, is

K(S) = t -1I3(S~ - S2)a,/(2ak -a<>, (28)

(29)

TABLE I. Coefficient values and initial conditions for the simulation of An and BS experiments. The BS result has been reproduced by two sets of con­stants (C""up )'

Experiment C", up

An BS BS

1.06 0.85 1.18

0.3 2.0 0.9

klllli L inil

0.OO28U' O.lOcm 0.OO28U' 0.03 cm 0.OO28U' 0.03cm

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Defining the supports of K and E by 5 ~ = 52, the self­similar front evolution is then given by

(30)

and the local behavior of the diffusion coefficients around the front is ofthe form

k2/€~5~ -52, or for 5=50 +85, k2/€~85· (31)

It follows that diffusion coefficients are regular when both k and € vanish at the boundaries of the mixing zone. Equations (28) and (29) show that the maximum of the turbulent kinetic energy and its dissipation rate are located at 5 = 0, and they decrease as time proceeds according to the power law

k(O,t) 'Zt - 1/3, €(O,t) 'Zt - 1/3. (32)

D. Model initialization

Statistical models for fully developed turbulence are mostly used in steady situations where the initial conditions are subject only to numerical constraints. Indeed, initial con­ditions are chosen close enough to the expected solutions in order to achieve convergence and to minimize computing cost. The situation is quite different in unsteady problems. As closure models do not calculate the initial phases of an instability, a turbulent state has to be initialized with phe­nomenological rules and experimental data if available. This fact comes from the splitting of physical variables into a mean and a fluctuating part. In other words, we have as­sumed a fully turbulent state for closing the turbulent fluxes by a first-gradient law of the corresponding mean quantities.

The mixing thickness at the very beginning of the inter­action process may be appreciated from the 2-D numerical simulations carried out by Youngs 17 and is modelized by the following relation: 18

Lo(t) =hl +h2 = (1 +{3)h l , (33)

where hI (respectively, h2 ) is the penetration of fluid 2 (re­spectively, 1) into fluid 1 (respectively, 2), hI = 2aA t U(t - to) and f3 = (P2IPI ) 1/3. The empirical constant a is evaluated by Read's experiments, let a = 0.070 ± 0.007; 19 At is the Atwood number, U is the mean velocity, and to is the time of shock wave-interface interaction.

The ratio of the turbulent kinetic energy contained in the turbulent mixing zone (TMZ) with respect to the direct­ed energy has been estimated by Mikaelian 18 who extended the experimental results of Read to the impulsive accelera­tion case. He found

EJEc 'Z0.093A;,

where

E t = L. p(r)k(r)dr LUlil

and

1689 Phys. Fluids A, Vol. 2, No.9, September 1990

(34)

The shape of k(r,t = 0) is a triangle of height kini! given by Eq. (34) and width Lini! given by Eq. (33). The initial value of the dissipation rate is initially computed from 0.164k 3/21 L (to). The determination of the transition time between the early linear behavior and the late turbulent dif­fusion behavior is currently a delicate point. In situations where the initial width is too small with respect to the mesh we turn on the model at a later time. Since right after the shock wave went through the interface the behavior is essen­tially diffusive, the initial conditions are modified according to Eqs. (30) and (32).

IV. NUMERICAL APPLICATIONS

Both the experiments of Andronov et al. (An) and Brouillette and Sturtevant (BS) have been simulated with the k-€ model.

In the An experiment the organic film separating air from helium is located 16.9 cm from the right end wall of the tube. I The shock wave is produced by a constant pressure applied to the left boundary. In our simulation the interface is accelerated at 760 Ilsec and the computational grid is made of 360 zones. By using the guidelines given in Sec. III D the model is initialized at 800 Ilsec with a triangular k 1 u2 profile of height 0.0028 and width 0.10 cm. These values have been adjusted after some numerical optimization; by doing so, a better matching between simulations and experimental data is found especially during the early stages of the mixing. There is initially no species mixing but we let k produce it through the diffusion equation.

In the BS experiment the helium length is 10.5 cm. The computational grid is made of 366 zones, and the interface is accelerated at 1000 Ilsec. The model is initialized at 1040 Ilsec in the same way as in the previous case, the initial sup­port of k is 0.03 cm according to Eq. (33). Figure 1 shows the experimental data of the mixing length of both experi­ments in connection with our simulation. The An data are fairly well fitted for the following set of constants up = 0.3 and CoD = 1.06 while the BS experimental data are correctly reproduced either with up = 0.9 and CoD = 1.18 or up = 2.0 and CoD = 0.85. Figure 1 also shows the wall jet length given by BS. Their experimental data seem to give an almost-linear time evolution of the mixing length while the simulation is close to the t 1/3 self-similar diffusion behavior. We have dis­played in Figs. 2 and 3 the turbulent mixing-zone boundaries for the An and BS experimental data. It appears that the penetration of the turbulent mixing-zone boundary in air is slightly underestimated in the An experiments while the penetration in helium in the BS case is overestimated. The turbulent kinetic energy and its dissipation rate displayed in Figs. 4(a) and 4(b) reveal several strong enhancements of decreasing amplitude due to the successive interactions of the turbulent mixing zone with reshocks. For the An case the turbulent kinetic energy located at the old interface encom­passes two orders of magnitude when intercepted by the first reshock. In the BS case the first rise is only by a factor of 20. As shock waves become more and more attenuated after some reflections on the end wall, the enhancement of turbu­lent kinetic energy is smaller and smaller. By looking at any

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5.5

4.5

3.5

2.5

1.5

0.5

-o.S

L(em)

x Xx xxx XXx X X

x

I(S)

FIG. 1. Thickening of the air/helium TMZ: + : experimental data of An­dronov et al.; X: the experimental data of Brouillette and Sturtevant for the TMZ; 0: the experimental data of Brouillette and Sturtevant for the wall jet;-: results of the simulation for the Andronov et al. case with ai' = 0.3 and C", = 1.06;-: results of the simulation for the Brouillette and Sturte­vant case with ap = 0.9, C", = 1.18 and ap = 2.0, C", = 0.85.

typical quantity such as the turbulent kinetic energy, the dissipation rate or turbulent fluxes such as the flux of k in Fig. 5, we observe the same kind of behavior: the quantity is strongly enhanced by shock waves and decays over a large time scale.

17

" r(em)

15

14

13

12

11

10 <>00000000000000

+ +

+ +

+ + + + I(S)

1. 7 10

FIG. 2. Trajectory of the air/helium TMZ versus time for the Andronov et al. case.

1690 Phys. Fluids A, Vol. 2. No.9. September 1990

9.5 r(em)

•• 5

7.5

'.5

5.5

4.5

I(S)

FIG. 3. Trajectory of the air/helium TMZ versus time for the Brouillettt and Sturtevant case with ap = 0.9 and C", = l.I8.

o. 9

O ••

0.7

O ••

0.5

0.4

0.3

0.2

0

O. 9

O ••

0.7

O ••

0.5

0.4

0.3

0.2

1.1 10- 3

k(em2s·2) (a)

I(S)

1.310- 3 1. 5 10- 3

k(cm2s·2)

I(S)

FIG. 4. (a) Evolution of the turbulent kinetic energy (upper diagram) and its dissipation rate (lower diagram) at the old interface versus time for the Brouillette and Sturtevant case (ai' = 0.9 and C", = l.I8). For conven­ience k and E have been scaled, respectively. by its maximum (kn"" = 0.548x 107

) and twice its maximum (Ema, = 0.39\ X 10"). (b) Same curves in the lower rectangle for the Andronov et al. case. Here k and E have been scaled, respectively, by its maximum (kmax = 0.303 X lOx) and twice its maximum (Ema , = 0.660X 10"). Here k and Eexhibit peaks each time a reshock is going through the mixing zone. This is due to the Ray­leigh-Taylor production term which is the same in both k and E equations up to the factor C",t:!k.

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I(S)

FIG. 5. The turbulent kinetic energy flux at the old interface versus time in the Andronov et al. case. It exhibits peaks at the same times as the the turbu­lent kinetic energy itself.

We have checked the decaying law (32) for the turbu­lent kinetic energy. By using the formula k(t) = C(t - to)a + ko, we have computed the exponent a after the first reshock passage. In the An case a is found to be in the range 0.36-0.37, while for the BS case we found a approximately equal to 0.40.

The turbulence time scale k lEis of the order of 500 J.lsec for both cases at the middle of the mixing layer. It increases slightly with time. This characteristic time is much larger (104 J.lsec) at the boundaries of the turbulent region where the level of turbulence is very low.

The concentration of air against the abscissa r is dis­played in Fig. 6 at different times. The gradient is weakened as time evolution proceeds and the mixing spreads.

In order to prove the model's ability to simulate differ­ent Atwood numbers and Mach numbers, we also carried out computations on the experimental data of Houas et a/. This experiment belongs to the Andronov et al. interpreta­tion and we are able to match the experimental data by slightly varying the constant up' The k-E model has also been used to simulate the experiment of Cavailler and Rodri­guez20 at the Centre d'Etudes de Vaujours (France). This experiment is quite similar to that of Brouillette and Sturte­vant and belongs to the second interpretation. By using the set of constants associated to that class, the simulations fit the experimental results well. These results are not repro­duced here but they can be found in Refs. 12 and 20.

We have already pointed out that turbulence, and conse­quently mixing, are produced by the pressure gradient as opposed to shear flows where turbulence is produced by the velocity gradients. It is then of interest to quantify the rela­tive contributions of the pressure effects with respect to the

1691 Phys. Fluids A, Vol. 2, No.9, September 1990

\

c

0.8

O ••

0.4

0.2

4.5

\ I.

I " \ \ \ ; \ ; \ \ \ ~ \ , \ ".

\ ; \ \

\

5.5 •• 5

FIG. 6. Concentration of air against r, at four different times (1200, 1500, 1800, and 2100 Ilsec) of the mixing process in the BS case.

shear effects. In a turbulent flow this quantity is measured by the flux Richardson number defined as

RJ

= (p:!"!!)(ap lar)

pU,2 (auJar) (35)

In our simulations, RJ reaches values of the order of 103_104 on the shocks and zero elsewhere. Of course these high values are overestimated by the I-D character of the model but they are several orders of magnitude greater than those obtained in shear layers, i.e., 10- 1_10- 2

•

Several Mach numbers may be defined in a compressible turbulent flow. The first one is the flow Mach number de­fined by

(36)

where uJ is the flow velocity and c, the sound velocity com­puted from

c; = (ap) = R. T(C ~ + (1 - c) ~) . (37) ap s J.lI J.l2

For the gases under study Yair = 1.40,J.lair = 29'YHe = 1.67, J.lHe = 4.

The shock Mach number is defined as

Ms (r,t) = u, (r,t)lc, (r,t), (38)

where Us is the shock wave velocity. We can also define a turbulent Mach number based on the velocity fluctuations,

M t (r,t) = k I12(r,t)/c, (r,t). (39)

The maximum values of these Mach numbers are summar­ized in Table II, which shows that the flow Mach number is always smaller than unity.

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TABLE II. Maxima of the flow Mach number, the shock Mach number, and the turbulent Mach number in the simulation of An and BS experiments at different times.

t(lIsec) Flow Mach number

An exp 1000 0.60 1200 0.20 1600 0.05

BSexp 1200 0.60 1600 0.20 2000 0.05

The shock wave Mach number is greater than one for the two first reshocks, afterwards it is of the order of unity. The Mach number based upon the fluctuating velocity is always very small. In other words, the turbulence is strongly subsonic. A posteriori, it gives a justification to the simple modeling used.

We have displayed in Fig. 7(a) the interface velocity gradient profile. It exhibits some peaks on the shock waves showing that the velocity gradient is applied for a short time-between 5 and 10 ,usec-compared with the diffusion time scale which is of the order of 700 ,usec and the turbu­lence time scale (500 ,usec). The tensor deviator of the Reyn­olds stress computed from

brr = pu;2/2pk-1

is displayed in Fig. 7 (b) for the BS configuration. In the An case the return time to isotropy is larger than 10 ,usec.

The return time to isotropy is roughly equal to the veloc­ity gradient time scale. This is obviously due to the algebraic closure which ignores memory effects. In that connection, it is noted that using an evolution equation for the radial part of the Reynolds stress tensor would lead to the following equation:

D pU;2 --au Plli-p-= _2pu;2 a; + ...

Then, the time decaying law would be

pU,2 [ (au ) - I ] -:-;::;;exp - 2 a; t,

where the time scale is still ( au Jar) - I.

We conclude that shock waves act as a rapid distortion that is essentially governed by linear effects21 and the domi­nant behavior is essentially a diffusion.

In order to compute the characteristic quantities such as the Taylor or the Kolmogorov scales, we have to determine the molecular viscosity coefficient of the mixing. We assume a temperature-dependent dynamic viscosity for each species given by Sutherland's law,

11 (T) = cst [ T 3/2/( T + 114)],

where Tis expressed in kelvins. The mixing viscosity coeffi­cient is then computed with an harmonic average formula22

lI;i~ = ailll- 1 + a~1I2-1, where a l •2 are the mole fraction of species 1 and 2. They are related to the concentration by

1692 Phys. Fluids A, Vol. 2, No.9, September 1990

Shock Maeh number Turbulent Mach number

1.30 0.060 1.16 0.070 1.00 0.050 1.30 0.028 1.10 0.012 1.00 0.008

~(S'I)

(a)

-3 10"

-1 10"

-s 10 '"

-6 10'"

-7 10'"

-81.0"

-9 10' I(S)

0.6

(b) 0.5

o ••

o ••

0.1

.~~~~~~----------~ -0.1

-0.2

t(s) -0.3

FIG. 7. (a) The velocity gradient profile at the old interface versus time in the BS case. It gives the time scale of the distortion-IO IIsee. (b) Evolution of the tensor deviator of the Reynolds stress versus time in the BS case. The return time to isotropy is just equal to the velocity gradient time scale, let 10 IIsee. The first five or six reshocks are perceptible in the profile.

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and

where J.lI.2 are the molecular weights of species 1 and 2. In our case 17air = 170.8X 10- 6 cgs and 17He = 186.0X 10- 6

cgs. The Kolmogorov and Taylor scales are, respectively,

defined by

LK = (~iJ£)1/4, AT = [vrnix(k/£)]1/2, (40)

where the kinematic viscosity coefficient V rnix = 17rnix/P. Let us recall that the Kolmogorov length gives the scales of the dissipative structures and the Taylor microscale is the spac­ing between the vortex filaments. Therefore they should sat­isfy

LK <AT <Le'

where Le = k 312/£. Of course, in this unsteady, nonisotro­pic, and inhomogeneous situation we do not expect the Kol­mogorov spectrum, even corrected by the compressibility effects, to be valid but we do expect L K I, L c- I, and A i I to give an order of magnitude of the spectrum extension. In Figs. 8 and 9 the time evolution of these three length scales have been plotted. Very quickly Le reaches an almost steady state, meaning that the size of largest eddies saturate as shock waves attenuate. Therefore the characteristic length Le and the mixing length are not proportional to each other

Lk- I• )..,-1, Le-1 (cm-I)

::- -::

I~ ~

-::

-::

:r -::

Ir \

l(s)

FIG. 8. The characteristic wave number L ,~ " and the Taylor and the Kol­mogorov wave numbers in the An case. They give the order of magnitude of the spectrum extension.

1693 Phys. Fluids A, Vol. 2, No.9, September 1990

,:-- .___.t-----______ _ ------.. _----

....... -~J ----- -----_ ------_.

t(s)

FIG. 9. The characteristic wave number L, ',and the Taylor and the Kol­mogorov wave numbers in the BS case. They give an order of magnitude of the spectrum extension which is smaller by one decade than in the previous case. Both sets of constants give roughly the same results.

as assumed in a one-equation model. The spectrum is much narrower in the BS case than the An interpretation. We con­clude that the turbulence is less developed in this case. This conclusion is enforced by computing the Reynolds number of the turbulence, based on the turbulent kinetic energy which gives the typical fluctuating velocity and the charac­teristic length Le. It reads

(41 )

This Reynolds number is related to the ratio of the larger scale Lc to the smallest one (the Kolmogorov scale) by

(42)

Applied to the cases of An and BS we, obtained, respectively,

ReAn =0.15X105, ReBs =0.4X103

•

Notice that with the above definition the Reynolds number is also, up to the factor CD' the ratio between the turbulent and molecular viscosity coefficients. It must be noted that even the turbulent kinetic energy and its dissipation rate are different, the small scales and the Taylor scales in the An­dronov et al. and Brouillette and Sturtevant interpretations are very close to each other.

The quantities L K I, L e- I, and A i I have also been plot­ted for up = 0.9 and C.o = 1.18. The results, not reproduced here, are very close to the up = 2 and C.o = 0.85 cases.

V. INITIAL CONDITION, SPATIAL RESOLUTION, AND SOURCE TERM COEFFICIENT SENSITIVITY

As we have already discussed above, the model results are inherently sensitive to the initialization procedure.

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Moreover, because we are interested in turbulence produc­tion by a shock wave, the final result is also slightly depen­dent on spatial resolution. We have investigated these depen­dences on the An case. By varying the amplitude of the initial turbulent kinetic energy and its support, we have obtained the results listed in Table III were Lstand stands for the mix­ing length at t = 1700 Ilsec obtained in Sec. IV.

These variations show that the larger the initial turbu­lent kinetic energy, the smaller the thickness at the end of the process. This result can be simply understood by noting that under the scaling k -+ ak, the quantity € scales as a 3

/2 due to the relation €(r,O) = 0.164 k(r,0) 3/2IL(t). It follows that the diffusion coefficients and damping are scaled as a l/2

while the Rayleigh-Taylor source term scales as a - 1/2. This explains the results reported on the first two lines of Table III.

It is also of interest to study the spatial resolution sensi­tivity. By halving and multiplying the number of mesh zones by 1.7 we obtained the results listed in Table III.

As expected, the more mesh zones we used, the larger the mixing zone. This dependence comes from the Richt­myer-Von Neumann numerical scheme where shock waves are stiffened and are therefore more efficient in producing turbulent kinetic energy when the space resolution is in­creased. We have also varied the source term coefficient. The coefficient AM has been varied from 1.25 to 1.60. The result means that an increasing value for the maximum of the ani­sotropy of the turbulence leads to a smaller value of the mix­ing length at the end of the process. In other words, the contribution of the mean velocity gradient term is globally negative.

VI. CONCLUSION

We have developed a k-€ model for handling situations characterized by high values of the turbulent flux Richard­son number. Because such cases involve a strong gradient, classic closure relations have been supplemented with alge­braic relations on strong gradients in order to satisfy realiza­bility conditions. Calibrations have been done by using ei­ther the interpretation of Andronov et al. or Brouillette and Sturtevant. The dominant behavior of such models is essen­tially of a diffusion type and we have shown that shock waves act as a rapid distortion that is governed by linear effects on the mean flow quantities. Physical studies of both interpreta­tions have been done by giving the behavior of some typical

TABLE III. Sensitivity of the mixing length at the end of the process with respect to initial conditions, number of mesh zones, and coefficients up and

AM'

kllH! = Q.5killil. .. tand

killil = 1.5killi t. ... 'and

Lil1ll = O.5L1I111 ... tund

Lilli. = l.5Lil11t.~lalid coarse mesh (halved) fine mesh (multiplied by 1.7) 1.38ap .. li,lld

0.620"p :-.tam\

AM = 1.60

L( 1700 #sec) - L" .... d

8.6% -4.3% -28%

29% -3.8%

7.6% -20%

20% -15%

L .. tand

1694 Phys. Fluids A, Vol. 2, No.9, September 1990

turbulent quantities such as the spectrum extension. In the Brouillette and Sturtevant case the turbulence is much less developed and non turbulent but nonlinear effects are of ma­jor importance especially during the early stages of the mix­ing.

Further progress in the understanding of Rayleigh­Taylor-induced turbulent mixing will probably be coming from careful experiments and nonlinear studies. In that re­spect, mean temperature and concentration profiles are wel­come in order to test the models more severely. Regarding theoretical studies, accurate viscous mixing of numerical codes both in two and three dimensions using instantaneous equations should be used instead of improving statistical models that do not treat the transition regime. In this way much may be learned of the early nonlinear stages of the mixing process and of the return to isotropy after a shock­interface interaction.

ACKNOWLEDGMENT

We would like to thank A. Grimaldi for her help during this work.

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3 S. G. Zaitsev, E. V. Lazareva, V. V. Chernukha, and V. M. Belyaev, SOY. Phys. Ook!. 30, 579 (1985).

4 L. Houas et al., in Proceedings 0/ the 15th International Symposium on Shock Waves and Shock Tubes, Berkeley, California, 1985, edited by O. Bershader and R. Hanson (Stanford V.P., Stanford, CA, 1986).

5 R. F. Benjamin, in Proceedings o/the International Workshop on the Phys­ics 0/ Compressible Mixing, Princeton, NJ, 1988, Lecture Notes in Engi­neering (Springer, Berlin, 1990).

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8 S. Gauthier, CEA LimeiJ-Valenton Internal Report 1982 (unpublished); S. Gauthier and A. Froger III, CEA Limeil-Valenton Internal Report, 1985 (unpublished); M. Bonnet, S. Gauthier, J. F. Haas, and P. Spitz IV, CEA Limeil-Valenton Internal Report, 1987 (unpublished).

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d'ete d' Analyse Numerique-Modelisation Numerique de la Turbulence, Clamart, Breau-Sans-Nappe, 1982 (unpublished).

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"w. Kollmann and O. Vandromme, Int. J. Heat Mass Transfer 22,1557 (1979).

I6R. O. Richtmyer and K. W. Morton, Difference Methods/or Initial Value Problems (Wiley-Interscience, New York, 1967).

17 0. L. Youngs, Physica 0 12, 32 (1984). 18 K. O. Mikaelian, Lawrence Livermore National Laboratory Report No.

VCRL-93499, 1985); Physica 0 36, 343 (1989). 19 K. I. Read, Physica 0 12, 45 (1984). 20G. Rodriguez, CEA Internal Report, 1989. 21 J. Mathieu, Turbulence et Approche Spectrale, Ecole d'ete d' Analyse Nu­

merique-Modelisation Numerique de la Turbulence, Clamart, Breau­Sans-Nappe, 1982.

22 J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory 0/ Gases and Liquids (Wiley, New York, 1965).

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