Modelling of the homogeneous turbulence dynamics of stably stratified media

ht. 1. Hear Mars Tmm/eer. Printed in Great Britain

Vol. 36. No. 7. pp. 1953-1968.

1993

0017-9310/93f6.00+0.00 Pergamon Press Ltd

Modelling of the homogeneous turbulence dynamics of stably stratified media B. A. KOLOVANDIN and V. U. BONDARCHUK A. V. Luikov Heat and Mass Transfer Institute, 220072,Minsk, Belarus and C. MEOLA and G. DE FELICE, Facolta di Engegneriadell Universita di Napoli, Napoli, Italy (Received

30 December

1991)

Abstract-Numerical simulation of the dynamics of homogeneousturbulence of a stably stratified fluid in the presence of a vertical constant-density gradient was carried out. The second-order model which is universal with respect to turbulent Reynolds and Pecletnumbers and molecular Prandtl number is applied to the numerical simulation of the dynamics of the velocity and density field parameters up to the final stage of decay. At small evolution times NT, the results of similation are compared with the familiar experimentaldata. The structure of a ‘relict’ turbulencewas investigatedfor the molecularPrandtl numbers corresponding to air and sea water.

1. INTRODUCTION

A KNOWLEDGE of the laws governing the evolution of the homogeneous turbulence in a stably stratified liquid is of great theoretical and practical interest. Theoretically, this interest is motivated by the possibility of a more detailed (than in the case of the general form of shear turbulence) study of the role of density gradient as a turbulence generator in an actively stratified medium. From the practical point of view, the homogeneous turbulence of a stably stratified fluid is quite a realistic model of the upper atmosphere or of the thermohalocline in the ocean. Over the past years considerable experimental efforts have been devoted to the elucidation of the fundamental aspects of the evolution of stably stratified homogeneous turbulence for both liquid (salt solution) and air. In the paper by Dickey and Mellor [l] which seems to be the first statistical-parametric experimental investigation of turbulence in a stably stratified fluid, a wavy change in the dispersion of vertical velocity fluctuations has been detected on the attainment of a certain dimensionless time Nt (here N = ((g/p) dp/dz) ‘/’ is the Brunt-VIisH11 frequency) which presumably is indicative of the transition of turbulent fluctuations into internal waves. Moreover, at this very value of Nt, a jumpwise decrease in the rate of kinetic energy dissipation of disturbances E, was detected confirming the presence of transition to a weakly dissipative wave field (in subsequent experiments such a behaviour of E, was not detected). The wave behaviour of the energy of lateral velocity fluctuations has been confirmed for the first time by Riley et al. [2] who numerically investigated the evolution of homogeneous turbulence in a stably stratified

medium. However, in this case no jumpwise change in the parameter E, has been detected which is indicative of the absence of the dominating contribution of internal waves into the perturbed velocity field, at least, for the range of NT values studied. In more detailed (than in ref. [l]) experimental works of Stillinger ef al. [3] and Itsweire et al. [4], a great number of ‘energy’ properties of velocity and density fields, different characteristic length scales, relating to velocity and density fields and also the transverse turbulent density flux were studied. The latter parameter has turned out to be extremely important for interpreting the specific features associated with the role of gravitation in the evolution of homogeneous turbulence. It has been shown that the considered parameter falls down to zero (the so-called collapse of turbulence), then, as NT increases, it acquires negative values, i.e. the counter-gradient density transfer is observed. When in the earlier studies it was assumed that the condition uZp = 0 signifies a complete suppression of vertical velocity fluctuation and transition of disturbed field to the so-called twodimensional ‘fossil’ turbulence, it has been demonstrated in refs. [3, 41 that gravitation slows down (as compared with the case of passive stratification) the decay of vertical fluctuations, i.e. downstream at the ‘collapse’ point they remain even more substantial than in the absence of the effect of gravitation. It was shown in those works that after the point where u,p = 0, a wavy change in all the measured parameters was observed ; moreover a ‘discontinuity’ in the law of decay of the energy of vertical fluctuations was noted : the values of & averaged over the amplitude of ‘waves’, degenerate self-similarly after the collapse point, just as before it, but with a smaller

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NOMENCLATURE 0”

;:: d: cij

D/j 4, Fr

F, Ri K

Lh L M

N n,.r P P pij

F-1 R

R* sij

T BV t

vi

absoluteconstant,equation (2.1) Greek symbols variable coefficient, equation (2.I) variable coefficient,.equation (2.3) variable coefficient,equation (2.1) deviator of Pij tensor 2 variable coefficient,equation (2.1) variable coefficient,equation (2.1) YU coefficientin parameterd,, deviator of D,, tensor 6” A Laplaceoperator second-ranktensor,equation (2. I) parameterof turbulent Reynoldsnumber E,, rate of kinetic energy turbulence dissipation Froude number, NM/U rate of temperaturefluctuation ‘interaction function’, equation (2.4) 6 gravity accelerationvector degradation moleculardiffusion kinetic energy of vertical K Taylor microscaleof scalefield fluctuations, (1/2)p,ut 4 Taylor microscaleof velocity field buoyancy scale,G/N 2 4, ‘overturning’ scale,p”‘/(d&Jdx,) kinematicviscosity vector of distancebetweentwo points sizeof grid cell ; .. . . densityfluctuations Brunt-Vaisala frequency, P meandensity P (WPO) WW “2 molecularPrandtl number,V/K variable coefficient,equation (2.3) 0 potential energyof turbulence, T time (1/2)~o(glpo)~/(dp/d.~2) TU time scaleof velocity field, g/e,, pressurefluctuations Tslr time scaleof density field, 7/.sI,. second-ranktensorof Reynoldsstresses Subscripts production, equation (2.1) , a asymptotic value turbulencePecletnumber,~“-A& time scaleratio parameter,r,/rL,,, s condition of strong turbulence turbulenceReynoldsnumber,q2 -Au/v T relatesparameterto turbulized medium t relatesfunction to temperaturefield second-ranktensorof meanshear u relatesfunction to velocity field Brunt-VaisllH period, 2n/N temperaturefluctuations W condition of weak turbulence meanvelocity vector. 0 absenceof meanshear.

exponent in the power law (of course,all the above relates to relatively large turbulent Reynolds and Peclet numbersthat correspondto the experiments mentioned). In a more comprehensiveexperimental work of Lienhard and Van Atta [S], carried out for a temperature-stratified air, the concept of homogeneous turbulence of a stably stratified medium has been developedasan essentiallytwo-scaleprocessin which large-scaleperturbations are controlled by buoyancy forcesand the fine-scalestructure by viscosity. In the light of this concept it hasbeenshownthat the condition u,P = 0 doesnot mean the disappearanceof the active turbulent mixing but isthe manifestationof an important trend in the stratified flow turbulence which wasdiscoveredin the work, i.e. that at a certain value of the ‘phase’N7 the vertical massfluxes,associated with large and smallvortices, are equal in magnitude and oppositein sign.The changein the signof the massflux (counter-gradienttransfer) is associated exclusively with large-scaleturbulent motions and resultsfrom restratification, i.e. the consequenceof the motion of large vortices which were ‘thrown’ by

turbulence from the region of high to the region of small density, under the action of buoyancy to the equilibrium and then to the region of higher density. As to the massflux associatedwith fine vortices, it is down-gradientat any time instant. Thus, the scalesat which turbulent mixing takes placeand the scaleson which the generationof internal waves occurs and which manifestsitself in the alternation of the massflux sign,aregreatly spacedin the spectrumof the scales.As to the mechanismof energy transfer from relatively fine-scaleturbulence to large-scale wave motion, then it is not asyet entirely clear. In order to elucidatethis extremely important problem, a number of attempts at direct numerical simulation of the evolution of homogeneousturbulenceof a stratifiedfluid have beenundertaken.The first effort has been made by Riley et al. [2]. The authors of that work have shown that even though stratification increasesthe decay rate of the dissipation of the kinetic energy disturbances,this parameterremainsrather high for the turbulenceenergy to be regardedas fully converted into the energy of internal waves.Thus, it follows from that work that

1955

Homogeneous turbulence dynamics

at leastin the studiedrangeof the dimensionless time N7, the internal gravitation waves coexist with the

turbulenceproper aswith a purely stochasticfield. A detailednumericalinvestigationof the stochastic and wave modes in the turbulence of a stratified mediumwas carried out in a recent work of Metais and Herring [6]. The results obtained in the above-mentioned numerical studies show that the development of a homogeneous turbulencein a stably stratified medium have been studied at least for moderate turbulent Reynoldsand Pecletnumbers.In practical situations existing in the atmosphereand ocean, the conditions of smallvalues of R1 and PA numbers(atmosphere) and small R1 and moderate or even large PL (the thermohalocline in the ocean) are being realized. Therefore, when the above-mentioned numerical works have somerelevanceto the atmosphere,they practically do not have direct bearing on the ocean, sincethe casePA >>R1 is as yet inaccessible for direct numerical simulation. Moreover, great evolution timesat which RL --) 0 are alsoinaccessible asyet for direct numericalsimulation. In the presentwork an attempt hasbeenmadeto apply the second-ordermodel, developedby one of the authors [7], to the study of the dynamicsof the stably stratified homogeneousturbulencein a Boussinesqfluid. In so doing the following problemswere posed: l

l

l

duction at the expenseof the meanvelocity shear; &ii s v(-A$$),=,

= f(dUa,+6ij)e,

is the model relation for the rate of dissipationof Reynolds stresses ; E,= sii is the dissipationof tur-1 bulencekinetic energy; aij = 3uiuj/q -6, is the deviator of the fi;tii tensor;

= - 41 -d”)aij&u-b,[a,q’Sij+

(Bubij +y,cij)Pk,]

-~Bgk[2~6ij-3(uit6jk+uitsn)]

isthe modelrelation for the second-ranktensorof the interaction of pressurefluctuationswith the gradients of velocity fluctuations the three constitutive parts of which model, respectively, the ‘slow’ return to the isotropy, ‘rapid’ deformation of turbulenceby mean shear and contribution of gravitation; &= u,? is the doubled kinetic energy of turbulence; S, = : (dU,/ax,+dU,/ax,) is the second-ranktensor of meanshear; b, = 3Pij/Pkk-6, is the deviator of the Pij tensor; cij = 3D,lP,,-6, is the deviator of the D, tensor; D,= -G dUkldxj+a dU,Jdx,); /I = -(l/fi,)(dj/dT), is the thermal expansion coefficientof the medium; a,, b,, d,, a,, flu, and yy are the coefficientsof the modelwhich generallyare some functions of governing parameters.For the homothe study of the dynamicsof relatively strong tur- geneousturbulenceconsideredtheseparametersare: the bulenceat different values of the generalFroude the turbulent Reynolds number Rl = 2”21,/v; ratio of the turbulent kinetic energy production rate number,Fr = NM/U; the study of the effect of molecularPrandtl number to the rate of its dissipationp” = Pkk/2&,; the Taylor on the evolution (decay) of moderately strong microscaleof the length I, which in the considered homogeneous anisotropicturbulenceisdeterminedby stratified turbulence; the relation the study of the transition of moderately strong stratified turbulenceto a weakrelict turbulenceand 1,’ = %2/E, = 5V7”, the elucidationof the relative role of internal waves and of the turbulenceproper at a very largetime of where 7” = ?/.suis the time scalerelating to the turevolution. bulent velocity field. 2. GOVERNING

EQUATIONS, PARAMETERS

SCALES

AND

2.2. An exact equation for the mean square of scalar fluctuations

The second-order model of homogeneoustur3+2(1-13)E, = 0, bulenceof a stably stratified Boussinesq fluid consists of the following differential equations (the details is the rate of relating to the allowancefor gravitation in different where E,= rc(dtlaX,)2 E K(-A~N’)+~ ‘smearing’ of scalar fluctuations ; p, = P,,/2-e, is the modelequationsare omitted). ratio of the rate of scalarfluctuation production due to the vector of the gradient of its mean value 2.1. Model equation for the tensor of Reynolds stresses P,, = -2ti;;s aT/ax, to the rate of destructionE,. This is given by 2.3. The model equation for the scalar quantity flux vector

where

This is given by Pij = - (uiuk dU,/dx, +uju, dU,/dx,)

is the second-rank tensor of Reynolds stresspro-

+t-Pi,+E;,-Q)i,

Y.

= -/!lgiTz,

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The right-hand side of the equation considered simulatesthe effect of gravitation on the vorticity of Pi, = - (uiuk d T/dx, + u,t dCJi/dxk) velocity fluctuation field. The coefficientF;f*, associatedwith the total effect isthe vector of turbulent flux production by the gradients of the mean values of the velocity and scalar of vorticity destruction and stretching of isotropic quantity, Ed,= d,l/z,,,&t is the model relationshipfor velocity field. is modelledby the turbulent Reynolds the rate of flux z ‘smearing’ numberfunction where

C*(&) = - a,,,( 1- d,) ;

3 + : (4dUJdx, - dU,/dxi)ui t

“I.,

+ :Bg&Lai~ +&IF is the model relation for the vector of pressurewith the gradientsof a scalarinteraction (the relationsfor (DiF)and a$” are exact), av,ru$ = [4 - F, + n,; ’ (Z/F)

R-

P,]r;

’

is the modelrelationshipfor the mixed time scale7,,, for R1 >> 1, P1 >> 1, P, = const., ls, = const.; 7,,,!

=

[2(a,,.,

+ 3/5)R/Rm.o

- pu]7;

= F:,*(l -dJ+C3,,

whereF$* = 1l/3 and F,*,,?= 14/Sare the asymptotic valuesof Fz* for Ri >>1 and R, c 1 which are found analytically by the known procedure. The parameter d,, is the empiric function of the turbulent Reynoldsnumber.In the presentstudy it is specifiedin the form d,, = 1-2(l+J(l+&/Rf))-‘.

The remaining coefficientsof the model have the form (for detailsseeref. [7]) : b,=2{l+[(cr,JF~*-2)(1-d,) + (cr,./F:,t-

’

is the modelrelationshipfor the mixed time scalefor Ri CC1, PL c 1, pU= const., is, = const.;

3’. = 2(fi 4 -+A

B. = 4 4,

2)d,lp”}

Y. = -&

- ’,

4,>

a, N 3, a,, N 2800, a,TN 1, a,,.N 0.5. 0

7ko= (3/10)&q[*-(2J’] 2.5. A model equation for the rate of ‘smearing’

of

isthe asymptotic(for a largeevolution time) turbulent scalar quantityfluctuations Prandtl numberfor R1 CC1, P,K 1, ls$= const., pU--* 0 (Deissler’srelation [S]); PI = ?“-A,/K is the turD 1 1 +2dP, bulent Peclet number, A,?= ~K?/E,= 6~7, is the EE, = 2dP,,s,L 1+cs,= 7” squaredTaylor microscaleof length relating to a sca- (F:, + F:2 R)E, ; lar field ; 7, = F/s, is the time scalerelating to a scalar field ; wherethe first two termson the right-hand sideof the equation model the effect of E,production by mean velocity shearand by the gradient of meanvalue of the scalarquantity (in strong turbulencethis effect is absent); the secondtwo termssimulatethejoint effect of the production of E,by deformation of vortex filais the parameterof the time scales7J7, ratio of homo- mentsof velocity field and associatedscalarfield ‘forgeneous turbulence for R, c 1, P1 << 1. p,, = 0, mations’ and also the destruction of parameterE,by pi = const. (exact solution of Dunn and Reid A). moleculardiffusion. The coefficientsF:, and FF2 are modelledby the 2.4. A model equation for the rate of turbulent kinetic following functions of turbulent Reynoldsand Peclet energy dissipation numbersand alsoof the parametersP, and P, : This is given by ;~,+(F$*-3P.)e:,~

= -2d,Jg,-

l-

1 + 0 7ttm.

u,t,

where (F:* - 3~J&,f/;;i

= GEN+ DIS+

IN

is the model relation for the sum of three effects: production of s, by meanshear(GEN = 0 for R1 >> 1), destruction of the parameter E, (or of vorticity z = (l/v)&,) and changein 2 by meansof stretching (compression)of vortex filaments(seefor detailsref. [71).

Homogeneous

turbulence

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1957

The coefficients F:,* and FF2* model the aboveThis estimateshowsthat at smallevolution times indicated effectsin the caseof the velocity and scalar Fr, <<1, i.e. gravitation doesnot influenceturbulence. field isotropy in the form of the following functions For the time 7 N N- ’ the parameterFr, attains the of the turbulent Reynoldsnumber valueof the order of unity, i.e. the effectsof turbulence and buoyancy becomecommensurable. At thesetimes Ff,*= F**-2-jd 7 N of evolution, wavy-like changesof the statisticalparFf2*= 2+jd. ametersof turbulenceshouldbegin. Experimentaldata obtainedin recentyearsfor the turbulence of stably stratified fluid [3-S], which 3. DYNAMICS OF TURBULENCE IN A STABLY STRATIFIED FLUID evolves in the wake behind a grid. show that wavy downstreamvariation of time-averageparametersis Basedon the consideredsecond-ordermodel, the one of the specificfeaturesof the evolution of stably present section investigatesthe dynamics of homostratified homogeneous turbulence.As a result of the geneousturbulence of a stably stratified fluid in the indicated experimentsa numberof conclusionshave absenceof meanvelocity shear.It is assumedthat in beendrawn about both the phenomenonof transition an infinite fluid volumewith a constantvertical mean from three-dimensionalturbulence to a presumably density gradientdp/dx2 = const. (the coordinatex2 is quasi-two-dimensional turbulenceplusinternal gravidirected vertically, opposite to the gravity acceltational wavesat relatively small distancesfrom the eration) the evolution of turbulenceis going on from grid and interaction betweenturbulenceand internal the initial state correspondingto a three-dimensional waves after the collapse.However, the question of homogeneousisotropic turbulence. whether or not gravitation causesthe transition of As is known, in a stably stratified fluid disturbed three-dimensionalturbulence to quasi-two-dimenfrom equilibriumfluctuationsoriginate internal gravisional or to a gravitational wave or to a supertational waveswith the Brunt-ViisIlii frequency positional field of turbulence and internal waves remainsopenup to now, sincethis can be elucidated N_ g dP “* only at high evolution times which as yet are inacPOdx2> ’ -( cessibleeither for direct modellingof physicalexperIn the caseof a turbulized fluid, the elementsof the iment. In the present work an attempt has been made fluid with a wide spectrumof turbulent fluctuations to apply an alternative approach to direct numerical becomeinvolved into disturbancesof gravitational nature. The turbulent modes which have the fre- simulation-to apply a second-ordermodel which quenciescommensurablewith the Brunt-V&ala fre- describesthe dynamics of the coupled velocity and quency turn out to beaffectedby the buoyancy forces. density fields for studying the transition of a threeIn the vertical direction the displacementof the tur- dimensionalturbulencein a stably stratified medium, bulized elementis estimatedby the so-calledover- to the final stageof evolution with a view to elucidate the structure of the field at this stage: whether it is a turning scale two-dimensionalturbulence (as is predicted in some L, = ~‘“/(d~,/dx,), theoreticalworks) or this is the field of internal waves where 2 are the mean-squareddensity fluctuations. or the superpositionof turbulent and wave modes. In this situation, quite a reasonablequestionmay This length scalecharacterisesthe size of large turarise: what is the basis for the belief about the bulent modesin the density stratified fluid. The ‘adjustment’of the turbulence, developingin adequate prediction on the basisof the proposed time, to the internal gravitational wavescan be easily modelof the behaviour of stably stratified turbulence illustrated on the example of grid-generatedturbu- at high evolution times?It can be answeredwith cerlence.At smallevolution timeswhenturbulenceis still tainty that, first, becauseof the satisfactoryagreement virtually isotropic, the well-known relations for the betweenthe predictedandexperimentalresultsfor the turbulenceenergyand Taylor lengthscalearefulfilled caseof strongturbulencein a non-stratifiedfluid (see ref. [IO]). Second,becausethe model matchesexact asymptotics[9] with weak non-stratified turbulence, z - 7- ‘, 1” - T”2. i.e. at very large evolution timesand, finally, on the It can beeasilyseenthat the Froude number(herethe trivial account for the buoyancy effect in the differquantity is usedwhich is reciprocal of that generally ential equationsof the model. usedbecauseof the fact that in the absenceof graviIn what follows we shall try to interprete the tation the value of the parameterwhich determines numerical resultsobtained, to correlate them, when the effect of gravitation is equalto zero). possible,with the experimentaldata of other authors; to investigatethe degenerationof the stably stratified Fr,. = N11,j~“2 turbulenceafter the collapsehaving explainedthe role of the molecularPrandtl number; to investigatethe will evolve following the ‘law’ character of the dynamics of stably stratified turbulenceat smallturbulenceReynoldsandPecletnumFrT - NT.

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bers,i.e. to elucidatewhether or not it really tendsto a two-dimensionalstate asit follows from a number of theoreticalinvestigations. 3.1. Dynamics of moderately stably stratifiedfluid

strong

turbulence

of

This sectiondealswith the analysisof the resultsof numericalsimulationfor the dynamicsof moderately strong turbulence in a stably stratified medium as implementedin the experimentsof Itsweire et al. [4] for water and Lienbard and Van Atta [5] for air. In particular, out of the multitude of experiments made for water, three have been selectedthat correspond to different values of the Froude number (Fr = NM/U = 15.2x 10e2; 6.3 x 10e2; 3.7 x lo-‘) with the grid cell measuringM = 3.81x 10v2 m. Of the experimentswith air [5], that one was selected which correspondedto the following parameters: Fr = 2.4 x lo-*, M = 5.08 x 10m2m. The numericalresultsgiven belowwereobtainedas the solution of the Cauchy problem for a systemof FIG. 2. Evolutionof thekineticenergyof transverse velocity ordinary differential equations that describe the fluctuations, rr;/U*. For designations seeFig. 1. ensembleaveragedturbulenceparameters: 3, 2, E,, 7. sp, Uzp; for air the latter three parametersare replacedby the quantities 7, E, and q The initial as conditions were borrowed from the corresponding Tz 3.5N-‘, experiments. The numericalresultsshowthat a turbulent flux of irrespectiveof the Froude number value. -. the scalar substanceu,p (Fig. 1) which determines Oscillationsof the function u,p drrectly causeoscilthe sourcegravitation terms in the equationsfor the lations of lateral velocity fluctuations, 3 (seeFig. 2) parametersz, p and sl performs oscillationsnear and, via this parameter,of the kinetic energy of turthe position where 6 = 0, with the first crossingof bulence(Fig. 3). Lessnoticeablearethe oscillationsof timeaxis at Nr z 2.5 ; the oscillationperiod isassessed the averagedvalue of horizontal velocity fluctuations, u: (Fig. 4) sincegravitation effect for them manifests itself only through pressurefluctuations. 0006..

FIG. 1. Evolution of a transversemass(or heat) flux.

2

q = m/Uh4(-dp/dxJ. *, Fr = NIU/U = 0.037; x, Fr = 0.063; +, Fr = 0.152;-, experimentfor water [4]; 0, Fr = 0.024 ; -, experiment for air [5j ; solidlines,numerical FIG.3. Evolutionof the full doubledkineticenergyof tur-

simulation.

bulence, a/U*. For designations seeFig. 1.

Homogeneous turbulence dynamics

1959

of non-gradienttype (which is not naturally present in this equation) but through the rate of density fluctuation production by the meandensity gradient Ppp= -2u,p dp/dx,.

IO’

I00

I02

NT

FIG. 4. Evolution of the kinetic energy of longitudinal velocity fluctuations, z/U2. For designations see Fig. 1.

When NT < 1, the function 2 evolvesin the fashion similar to the meansquareof the passivescalarfluctuations (see,for example,ref. [lo]). At NT N 2.5, the decayof p2 beginswhich qualitatively is similarto the decay of the isotropic passivescalar. A specificfeature different from the isotropic case is the wavy-like oscillationin densityfluctuationswith a period equal to that of the transversedensity flux oscillations. As is known, the wavy-like oscillationsof the turbulence parameters in a stably density-stratified medium indicatesthe presenceof internal waves in the field excited by gravitational forces.As is known, the indicator of the transition of turbulenceto internal gravitational waves is the value of the ratio of the kinetic energyof vertical velocity fluctuations ~=fp,~=$,~*(i$/N2) to the potential energy of the densityfield

Note that the absenceof vertical turbulent mass(or heat) transfer occurring at NT = 30 (seeFig. 5) does not at all mean the suppressionof vertical velocity fluctuations,which, asfollows from Fig. 2, decaywith an averaged (over the amplitude oscillations) rate which doesnot exceedthe rate of decay of the nonstratified fluid velocity fluctuations. The oscillations of the parameter 2 u p show up directly alsoon the behaviour of density fluctuations (Fig. 5) but not through the sourcegravitational term

.

or the ratio of characteristiclength scales K -= P

-L,,' 0L,'

where Lb = (z/N’) ‘I2 is the buoyancy scale, L, = --j 112 p /(dfi/dx) is the ‘overturning’ scale or the characteristic distancealong the vertical over which the elementof a turbulized fluid may displacefrom the equilibriumposition.The upperlimit of the parameter L, is the scaleL,, determinedby the turbulenceinertia. It isobvious that at smallevolution timesof strong turbulencethe valueof the ratio (Lb/L,) exceedsunity. In this casethe contribution of the internal wavesinto the turbulent field is insignificant.The condition K p=l

, lo-’

I@

IO' NT

FIG. 5. Evolution of mean squared density (or temperature) fluctuations, 0 = ~/Mz(d+x2)*. For designations see Fig. 1.

testifies to the ‘parity’ of turbulent velocity disturbancesin the vertical direction and internal waves; sometimes,this condition is consideredto be the condition for the transition of turbulence to internal waves,assumingthat finally (at very large evolution time) turbulencewill entirely passover into the internal waves; in what follows it will be shownthat such a transition is not alwayspossible. The curves presentedin Fig. 6 show that in the regionof a relatively strongturbulence(the turbulence Reynoldsand Pecletnumbersare presentedin Figs. 7 and 8, respectively) the ratio Lb/L, similarly to the transversemassflux is an oscillatingfunction attaining the asymptote. At small values of the dimensionlesstime, L,,/L, > 1 indicating the prevailingcon-

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10-I

IOU

IO’ NT

NT

FIG. 6. Evolution of the ratio of the kinetic energy of vertical velocity fluctuations to the potential energy, K/P = (?JN2)/@l/(dj3/dx2)2 = (L+./L,J2. For designations see Fig. 1.

tribution of turbulence into a perturbed field. The condition L,/L, = 1 correspondsto the coordinate NT N 2.

The minimum value of the function K/P correspondsto the coordinate Ns N 2.5, i.e. to the point where the transversemassflux vanishesfor the first time. The asymptotic value which is attained by the ratio K/P is equalapproximately to 0.7. Since, as was shown above, the contribution of

IO2

FIG.

8. Evolution of the turbulence Peclet number Pi = A,~I”‘/h-. For designations see Fig. 1.

internal waves into the superposition field at N7 = 2 becomes equal to that of the turbulenceproper, then starting from this point one should await the specific

featuresof the dynamicsof the consideredparameters inherent in a stratified fluid. Indeed, from the data given in Fig. 7, it is seen that at N7 2: 2.5 correspondingto the collapseof the transversemassflux, a jumpwisedecreasein the rate of turbulenceinertia decay is observed.The reasonfor this seemsto be internal waveswith a large period (proportional to the Brunt-VBisIla period, T,, = 2x/N) ; this is indicated by a jumpwise change in the increasein the Taylor macroscale of velocity field (Fig. 9) at N7

N 2.5.

Sincethe internal random wavesare lessdissipative structures than turbulence, then, starting from the samecoordinate(N7 N 2.3, a slowerdecayof velocity fluctuations is observed(seeFigs. 24). 3.2. Eflect of molecular 4

IO'

I

100

IO’ NT

FIG. 7. Evolution of the turbulence Reynolds number, RA = &~“*/v. For designations see Fig. 1.

Prandtl number

Realizations of the experiment consideredin the present work and the results of simulation, correspondingto them, relate to a moderately strong turbulenceof yelocity field, asindicatedby the pattern of turbulenceReynoldsnumberevolution (Fig. 7). At the sametime the scalarfield turbulenceis moderately strong for air and quite strong for water (seeFig. 8). In view of this, it can be expectedthat even at almost the samevalues of the global Froude number, and different values of molecular Prandtl number the dynamics of the turbulence parametersshould be different. As follows from the above-mentionedfigures,in the near region (N7 c 1), wherethe effect of buoyancy forceson the velocity field is negligible,the

Homogeneous turbulence dynamics

.

I 10-I

100

IO'

4 I 02

NT

FIG.9. Evolutionof theTaylor macroscale of velocityfluctuations,EU= LJM. For designations seeFig. 1.

1961

which in experimentsfor air and water differed from each other. In other words, the difference of the exponentsin the law of the evolution of the scalar field turbulenceparametersin the regionof Nr < 1 is due not to the differencein the valuesof u, but rather to the initial value of the scaleratio parameterR”. As noted above, in the ‘collapse’region,a ‘trough’ in the lawsof the evolution of turbulenceparameters occurs which is conditioned by the appearanceof internal waves.Downstreamof this ‘trough’ the parametersof the field which is the superpositionof the turbulence itself and random internal waves, are evolving conditionally self-similarly,i.e. self-similarly for the parameterssmoothed with respect to the amplitude of oscillations.However, the rate of selfsimilar evolution of the correspondingparameters differs substantiallyfor air and water. Thus, from Fig. 3 it follows that the exponent of decay rate averaged over the amplitudeof turbulencekinetic energy oscillationsn in the power ‘law’ q* N (NT)-”

is approximately equal to 0.8 for water and to about 1.63for air. Such a substantialdifferencein the rate of kinetic energydecayafter the collapsecan be attriturbulenceparametersbehavesimilarlyto the caseof a buted, on the one hand, to a different contribution of passivescalar(see,for example,ref. [lo]) : the velocity random internal long wavesinto the superpositional field parametersevolve self-similarlyand, naturally, field: the small decay rate of the parameter4’5for independentlyof u ; the dynamicsof the scalarfield water indicates a great contribution into this parparameters,in particular, of the quantities ~9and L, ameterof the energyof randominternal gravitational (Fig. 10) dependson the initial values of the scale wavesfavouring the production of large-scaleweakly dissipatingstructures. ratio parameter On the other hand, a relatively large rate of turbulent kinetic energydecay in air, ascomparedwith water and also with the case of the non-stratified medium,cannot be indeed associatedwith the presenceof internal waves.It seemsto be attributed to the relatively low inertia of the scalarfield (the evolution r of the turbulence Pecletnumber is presentedin Fig. 1 8). As is known, to a weakerturbulenceinertia there correspondsa more rapid decay of it. Therefore, an earlier transition to the final stageis observedin air than in a fluid. Thus, the influenceof the molecularPrandtl number on the evolution of the velocity field parameters showsup through the inertia of the scalar field, i.e. turbulence Peclet number PA: the higher the value of this parameterthe greater the contribution of the internal wavesinto the perturbed velocity field. IO’ -. From Fig. 5 it follows that the molecularPrandtl mm number exerts an influencealso on the degeneration of the meansquareof scalarpulsations.Starting from the value of Nr correspondingto the ‘collapse’,the rates of decay of the amplitude-averagedvalues of us and p* (or 7 for air) turn out to be identical for a given value of cr, so that the ratio of the kinetic to : rl potential energy,beingan oscillatingfunction of time, 10-I I00 IO' 02 NT hasa constantmeanvalue which isindependenteither FIG.10.Evolutionof theTaylor macroscale of densityfluc- of the global Froude number or of the molecular tuations,&, = L,/M. For designations seeFig. 1. Prandtl number(seeFig. 6).

1

1962

B.

A.

KOLOVANDIN

et

al.

It seemsthat this fact (which was not covered in earlier relevant publicationsof other authors) has a greatimportancefor correlating thedynamicsof moderately strong homogeneousturbulence of stably stratified fluid. 3.3. Transition of the stably stratifiedfluid to the final evolution stage

turbulence

As is known from the dynamics of the homogeneousturbulence of a non-stratified fluid, and in compliancewith the resultsof the presentwork, the parametersRA and PA decay at a largeevolution time (seeFigs. 7 and 8), i.e. during its evolution the turbulenceof a stably stratified fluid losesits inertia and goes over to the final stage of degenerationor, in conformity with the oceanographicterminology, to the stateof ‘fossil’turbulence. Of principal importance for understanding the character of ‘fossil’ turbulence is the study of the evolution of turbulencein the transition region, i.e. at moderatevaluesof RL and PL parameters. To carry out a comprehensive study of the evolution of stably stratified turbulence within this transition rangeof the parametersRA and PA, the authors analysedthe resultsof numericalsimulationof two experimental realizations which differed by molecular Prandtl numbersand had almost the sameglobal Froude numbers: the experimentof Itsweireet al. for o N 900, Fr = 3.7 x 10e2and that of Lienhard and Van Atta for u = 0.73, Fr = 2.4 x lo-‘. It follows from the data given in Figs. 1I and 12,that the stratified turbulence approachesthe final stage at substantially different rates: for liquid the state of moderately strong turbulence persistsup to N7 N 10’ ‘,

23

FIG.

12. Evolution of turbulence Peclet number at the transition and final stages. For designations see Fig. 1.

while for air the final stageof decay setsin already at N7 ‘v 104. As the inertias of the velocity and scalar fields decreasedifferently in time, the turbulence energy (Fig. 13) also decays differently in the media considered.At the final stageof degenerationof q2 for air, setting in at N7 N 104,the degenerationrate exponent in the power law is equal to 5/2 just as for non-stratified medium. For liquid, the degeneration rate exponent of g at the final stage,which setsin at

10-IIO’ I@Id IO’ I09 lo” IO”10’5 IO lols102’ I 23 NT

FIG. 11. Evolution of the turbulence Reynolds number at the transition and final stages. Numerical simulation for water (1) and air (2). For designations see Fig. 1.

FIG.

13. Evolution of total kinetic energy of turbulence at

the transitionand final stages. Numerical simulation for water (1) and air (2). For designations see Fig. 1.

Homogeneous turbulence dynamics

1963

fo-zi !!!!!!::! ] !!!) > !!!!!!j 1-J-1 10’Id 10sIO’I09 IO”10’3 10’5 IO”Id9 I@Id’ NT

FIG. 14. Evolution of density fluctuations at the transition and final stages. Numericalsimulationfor water(I) andair (2). For designations seeFig. I.

NTII IO”, has the value twice as small as that for air. The fluctuations of density(or temperature),shown in Fig. 14, decay in the far region qualitatively in the samefashion as the kinetic energy of turbulence. In the region of small parametersRl and PAthe decay rate exponentsfor the parameterp’;ifor water and I? for air are nearly equal to the correspondingvalues of the decay rate index of the parameterz,

FIG. 16.Evolutionof theTaylor macroscale of densityfluctuationsat the transitionandfinal stages. Numerical simulationfor water(I) andair (2). For designations seeFig. I.

The behaviour of the length scalesof energy containing vortices, i.e. of the Taylor macroscales L,, and L,, is of fundamentalimportance for understanding the character of velocity and scalar fields in the far region.As followsfrom Figs. 15and 16,the behaviour of the macroscalesof length for air and water is different in principle : when for air the parametersL,, and L, evolve similarly to the isotropiccase(or in the caseof a homogeneouspassivescalar field for the parameterL,), then for water the indicatedparameters increase infinitely. This fact can be presumably explainedby a different contribution of internal waves into the far field for the two casesconsidered:when N7 >>I, the effect of the internal waveson the turbulenceof air is insignificant,while in liquid the internal waves becomedominant. Even though such an assumptionhas an euristiccharacter, it is confirmed by the analysisof the evolution of other turbulence parameters.Most impressiveis the evolution of the ratio of the kinetic energy of vertical fluctuations to the total kinetic energyof the velocity pulsations(Fig. 17). As follows from the numerical results when N7 B 1 the vertical velocity fluctuationsin air become suppressed earlier than is the casewith longitudinal and lateral fluctuations, i.e. the velocity field becomes nearly two-dimensional.For water, the pattern is opposite: at large values of NT the vertical velocity fluctuations decay more slowly than the other two components,so that the parameterz/7 approaches unity, i.e. in the caseconsideredthe velocity field asymptotically approaches the ‘quasi-one-dimensional’field. FIG. 15. Evolution of the Taylor macroscale of velocity The above-describedpresumedpattern of the ‘fosfluctuationsat the transitionand final stages.Numerical simulationfor water(1) and air (2). For designations see sil’ turbulence structure is confirmedby the analysis of the evolution of the ratio betweenthe kinetic energy Fig. I.

1964

B. A. KOLOVANUIN

el al.

velocity field for Nr >>1. In the caseof an air at NT >>1 the parameter K/P asymptotically goesover from the region K/P < 1, where the contribution of 0 ?-internal waves into the superpositionvelocity field somewhatexceedsthe contribution of the turbulence 0 6-proper, to the region with the value K/P = 1 indicating parity of the energiesof vertical turbulent fluc0 s-tuations and internal waves.However, asthe data of Fig. 17 indicate, the contribution of vertical perturbations (both of turbulence and internal waves) into the three-dimensionalvelocity field is insignificant (turbulence tends to a two-dimensionalor horizontal one),i.e. in Fig. 18the information-bearing curve (in the senseof the analysisof the contribution of internal waves into the’ relict turbulence) is only that one which correspondsto liquid: it points to the dominating role of internal waves in the fossil oo-turbulenceof a stably stratified viscousfluid. L!:!r:rr !!!:I!II!!!I:I!! Thus, the analysis of the results obtained from 10-I 1-J’ p Id 107 109 Id’ 10’3 10’5 IO” 10’9 102’ numericalmodellingof turbulenceevolution in stably NT stratified mediashowsthat, dependingon the molecFIG. 17. Evolution of the ratio of the kinetic energy of transverse velocity fluctuations to the total kinetic energy ular Prandtl number, the fossil turbulence, i.e. the of turbulence at the transition and final stages. Numerical turbulenceof stably stratified mediawith RI -X 1 and simulation for water (I) and air (2). For designations see PA <<1, may representeither a quasi-two-dimensional Fig. I. field with the dominating contribution of random two-dimensionaldisturbancesand with a slight‘impurity’ of internal waves(for gaseousmedia), or ‘quasione-dimensional’field with the dominating conof vertical velocity disturbances and the potential tribution of vertical internal wavesand insignificant energy (Fig. 18). From this figure it follows that in impurity of three-dimensionalturbulence (for liquid the case of turbulence formation in liquid the par- media). ameter K/P asymptotically (when N7 + a) tendsto zero (or to a certain smallvalue). This testifiesto the dominatingrole of internal wavesin the superposition 0 a--

l

4. STRUCTURE STABLY

OF THE TURBULENCE STRATIFIED MEDIA

OF

The above analysisallows one to assumethat the perturbed field in stably stratified media is a superposition of the turbulenceproper and of the internal . gravitational waves,with the contribution of the inter2nal wavesinto the superpositionalfield beingdifferent 4 depending on the molecular Prandtl number. It should be noted, however, that the aforegoing conclusionabout the contribution of internal wavesinto perturbed velocity field at NT >>1 has a heuristic character,sincethe above-consideredfunctions in the far region are representedby smoothcurves without any signsof wavy motion either in the liquid, or in the gas(seeFigs. 17-19). Moreover, the evolution of a transversemassflux, servingasan indicator of wavy motion of the medium,demonstrates(seeFig. 19) the absenceof any signsof scalarfield anisotropy at all in the far region. In view of this, therearisesthe necessityof carrying out a detailedanalysisof the character of changein the processof time evolution of the perturbed field parameterswithin short intervalsof the argumentNz. FIG. 18. Evolutionof the ratio 0; the kineticenergyof This might confirm the existenceof internal waves verticalvelocityfluctuationsandthepotentialenergyat the transitionandfinal stages. Numericalsimulationfor water and would allow one to elucidatetheir contribution (1) andair (2).For designations seeFig. 1. to the perturbed velocity and density fields. r

.

1965

Homogeneous turbulence dynamics 1

Due to the coupling of the velocity and density fieldsin a stratified medium,the presenceof internal wavesin the superpositionfield must showup in the evolution of all the turbulenceparametersrelating to both velocity and density fields. In fact, as follows from the curves presentedin Fig. 20, it is already at the early stageof the evolution of a stably stratified turbulence,including even the region before the collapse, that wavy-like oscillations in the turbulence parametersbecomeobservable,as well as the oscillations of the function q about the zero value. The period of such oscillationat the initial stageof evolution is independentof the molecularPrandtl number and is approximately equalto T 1: O.S6T,,.

I!!!!!!!!!:!!!!:;!::!!!!

-I

IT IO’ IO’ 10~ IO’ IO= Ed’ rd3 10’~Id’ 10~IC? 102: NT

19.Evolutionof thetransverse mass(or heat)flux CJ at the transitionandfinalstages. For designations seeFig. 1.

FIG.

In accordancewith the data consideredin the previous sections, when NZ < 1, the stably stratified mediumturbulenceparametersevolve qualitatively in the samemannerasin the caseof independentvelocity and scalarfields. When Nr N 2.5-3.0, there occursa conventional collapseof turbulence which is manifestedby a vanishingtransversemassflux and which points to the formation of internal gravitational waves.

A distinctive feature of the developmentof turbulencein a stably stratifiedliquid in the initial period is the countergradientvertical massflux. In fact, as follows from Fig. 21(a), the averaged(over a number of ‘periods’) value of the parameterq is negative.In this very region of the dimensionlesstime Nr the consideredparameterfor the gasis on averageequal to zero. In the transition regiondeterminedby the moderate Reynoldsand Pecletnumbersand extendingover the intervals 10’“-1020for liquid and lo’--10” for air (see Fig. 18), the tendencyin the developmentof the parameter q (seeFig. 21) is preserved,i.e. the countergradienttransversemassflux is observedfor liquid and, on the average,zero heat flux for air. At the final stageof evolution determinedby small valuesof the turbulent Reynoldsand Pecletnumbers

turbulentmassflux in FIG. 20. Evolution of the transverse mass(or heat) flux FIG. 21. Evolutionof a transverse 4 = ii,p/r/M(-dfi/dx,) at the beginningof the transition the transition stage: (a) water, (NT), = 1014;(b) air, (NT), = 105. stage:(a) water,(NT), = 10.0;(b) air, (NT), = 10.0

1966

B. A. KOLOVANDIN et al.

do not play the dominating role in the superposition velocity field; the buoyancy effect is reduced to the suppressionof vertical velocity perturbations (both of internal wavesand proper turbulent fluctuations) so that with N7 + co the velocity field representsa quasi-two-dimensional (horizontal turbulence) with an insignificant ‘impurity’ of internal waves. I.!.:.:.!.!.!.!.!I 0 I2 (4

N=9

3

4 NT

5

6

7

8

5. CONCLUSION

In the presentpaper the results are presentedof modellingthe evolution of homogeneousturbulence in stably stratified media at any distancesfrom an arbitrary origin, for examplea grid beinga generator of the turbulent isotropic velocity field. The sourceof the scalar field turbulence is a constant transverse mean density (or temperature) gradient which generatesscalar quantity perturbations. The joint correlation of the transverse(in the direction of the -0166’ a C.!.!.!.! !I 0 I2 3 4 5 6 7 8 averaged density gradient) velocity and density NT fluctuationsor a turbulent transversemass(heat)flux, (b) N=Zl which is a sourceterm in the equationsfor turbulent essentiallydeterminesthe specificfeaturesof FIG. 22. Evolution of the transverse density (or heat) flux at stresses the final stage: (a) water, (NT)~ = 10”; (b) air, (NT), = 10i4. the evolution of turbulenceof a stably stratified fluid. I

I

5.1. Specificfeatures of the evolution of stably stratified turbulence in the initial period

1. The fundamentalaspectof the evolution of the consideredhomogeneousturbulence is a wavy-like -. alternating variation of the joint correlation u,p, I.e. the alteration in time of the ‘down-gradient’ and ‘countergradient’mass(or heat) fluxeswith the period TN 3.5N-’ N OST,,independentoftheFroudenumber. The resultsof the simulation carried out in the presentwork showthat this feature is typical of flows with any, including small,Brunt-VBisHlHfrequencies. However, the amplitudeof fluctuations of the abovementionedfunction decreases with the Froude numT ‘y O.lT,, ber Fr = NM/U. Thus, when consideringthe experimental confirmation of this feature, one shouldbear whereasfor air it virtually did not change. in mind that at smallFroude numbersthe amplitude Thus, both in the transition region and in the final of the oscillationsof the parameteru2pcan be rather stageof the evolution of density-stratifiedturbulence, small and located within the accuracy range of gravitation plays an important role in the formation measurementwhich makesit impossibleto reveal the of a perturbed velocity field : presenceof fluctuationsdirectly from the experiment. l at largemolecularPrandtl numberscharacteristic This situationseems to betypical for somerealizations for liquid mediathe internal wavesplay the domi- correspondingto smallFroude numbers,of the expernating role in vertical perturbations of the vel- imentsof Lienhardand Van Atta [5] and alsoof Yoon ocity field ; decayingmore slowly than turbulent and Warhaft. [ 1l] in which the changingof the signof velocity fluctuations (due to the smaller dis- the parameteru2pis not evident. sipative nature of large scalestructuresof wave 2. The sign of the dimensionless transversemass character), the internal waves lead to the for- flux averaged over a large interval of N7 depends mation of a ‘quasi-one-dimensional’ field with on the contribution of the internal waves into the NT--f co in which the intensity of vertical per- perturbed field: with the internal wavesplaying an turbations due to internal waves exceedsthe insignificantrole a down-gradientturbulent transport intensity of longitudinal fluctuationsby about an prevails; in the caseof a substantialcontribution of order of magnitude; the internal wavesthe countergradienttransport prel at molecular Prandtl numbers of the order of vails (the sign of the turbulent massflux is identical unity characteristicfor gases,the internal waves with that of the meandensity gradient). and extending over the region N7 > 10” liquid and N7 > IO’* for air, the wavy characterof the evolution of the transverseturbulent massor heat flux is preserved (see Fig. 22). Nevertheless,the qualitative differenceof the evolution of this function in the stage considered,in contrast to the previousstage,consists in the absence,on average, of turbulent massflux for liquid and in the recovery of the downgradient heat flux for air. The period of fluctuations of the function q(N7) for water somewhatincreased

Homogeneousturbulence dynamics 3. In the region considered, due to the wave-like oscillations of the transverse mass flux, all the ensemble averaged parameters of turbulence of both the velocity and scalar field evolve in a wavy manner with the same period as that of the function u,p. In this case the value uZp = 0, repeating with the above-mentioned period, does not at all mean the suppression of the vertical velocity perturbations. Moreover, a substantial contribution of internal waves, as evidenced by wavy oscillations in turbulence parameters, leads to a slower decay of the kinetic energy of velocity fluctuations than in the case of passive scalar. This is quite natural, since the superposition field with internal waves is less dissipative (of larger scale) than a purely turbulent velocity field. 4. The confirmation of the substantial contribution of internal waves into the superpositional field is a jumpwise change in the increasing Taylor macroscales of velocity and scalar fields starting from Nz N 2.5, i.e. from the instant when the transverse mass flux starts to perform alternating oscillations. 5. The differences in the molecular Prandtl numbers typical of the experiments in different media (air and liquid) lead to substantially differing turbulence Peclet numbers. This explains the difference in the rate of evolution of turbulence parameters in various experiments conducted with different media. 6. The amplitude-averaged value of the ratio between the kinetic energy of transverse velocity fluctuations and the potential energy of the density field in a strong turbulence is a universal constant independent either of the Froude number, or of the molecular Prandtl number and is approximately equal to

1961

time, long internal wavesin the perturbedfar field of a stably stratified liquid are dominating. 4. The previous statementis supportedby a number of factors which, in particular, includethe rate of turbulent energydecayin a gas(Fig. 13)the exponent of which in the power law is equal to (-5/2). For liquid it is twice as small thus indicating the dominating role of larger, i.e. lessdissipative,structuresin a perturbed field formed in a stably stratified liquid. 5. A fundamentally important feature of the evolution of stably stratified turbulencein various media is manifestedin the ratio of the kinetic energy of transversefluctuations to the total kinetic energy of velocity perturbations (seeFig. 17): for liquid this parametertendsto a value closeto unity; for air, to zero. This meansthat the ‘fossil’ turbulencein liquid is formed in the main by internal waves,while in air by a two-dimensional(horizontal) turbulence. 6. The evidenceof the dominating contribution of internal wavesinto the superpositionvelocity field in liquid at a high evolution time is the fact that the ratio of the kinetic energyof transversevelocity fluctuations to the potential energytendsto a certain valuesmaller than unity. 7. The qualitative differencebetweenthe perturbed velocity fieldsin liquid and gasat largeevolution times consists in the fact that the ‘fossil’ (or ‘relict’) turbulence(i.e. the turbulenceof stably stratifiedmedia at R1 K 1 and P, <<1) representseither quasi-twodimensionalrandom velocity perturbations (for gaseousmedia),or a quasi-one-dimensional velocity field causedby the dominating contribution of vertical internal waves.

0.65. 5.2. Specific features of density-stratified transition to thefinal stage

turbulence in

1. The transition zone is characterized by a substantial non-self-similarity of the evolution of turbulenceparameters. 2. The extensionof the transition region depends greatly on the molecularPrandtl number: the larger it is, the higher the value of Q.One fails to relate this fact only to a relatively greaterinertia of turbulencein liquid ; herethe dominating role is played by internal waves, the contribution of which into the superpositionalperturbedfield in liquidsismoresubstantial than in gases. 3. In the region consideredthe macroscales of the turbulence of velocity and density fields evolve differently for the two media considered:for air these are the functions with the maximumin the region of moderatevaluesof the parametersRA and P,, just as in the caseof a passivescalar; for a liquid theseare infinitely increasingfunctionswith respectto Nr. This fact is an indirect confirmation of the different contribution of internal wavesinto the superpositionperturbed field in the far regionfor liquidsand gases: in a stably stratifiedgasthe contribution of internal waves into the perturbed field is insignificant; at the same

5.3. The structure media

of turbulence

of stably

stratified

1. The oscillationchangein time of the parameters of turbulenceis the specificfeature of its evolution in a stably stratified mediumeven at an early stageof its developmentup to the time when the parameteru2 p converts into zero for the first time. 2. At the initial stageof turbulence development the sign of the transversemassflux, averagedover somenumber of the fluctuation periods,dependson the molecularPrandtl number: countergradientmass transfer prevails in liquid, whereasin gas there is virtually no transversemassflux. 3. The period of the oscillationsof functions in the initial stageof evolution is independentof the molecularPrandtl numberand isapproximatelyequal to TN 0.56T,,.

4. In the transition region the characterof change in time of turbulenceparametersis qualitatively similar to the evolution of turbulencein the initial period. 5. At the final stageof turbulence evolution, the wavy character of the turbulence parametersof the velocity and scalarfields is preserved.However, the

1968

B. A. KOL.OVANDIN

amplitude of the oscillations of these parameters for liquid is much in excess of that for air. 6. For liquid media the period of the perturbations of turbulence parameters at the final stage somewhat increases as compared with the previous, transition, zone. 7. Turbulent mass flux, averaged over a number of periods of oscillations is virtually absent for liquid media, whereas for gas it adopts a conventional sign inherent in down-gradient transfer. 5.4. Concluding remark on the role of stratljication in rhe formation of a perturbed field at high turbulence evolution times

The role of gravitation in the homogeneous turbulence of a stably stratified media manifests itself via the ‘participation’ of internal waves in the superposition perturbed velocity field which is reduced to the formation, at large evolution times, of a ‘quasione-dimensional’ perturbed field in a liquid with predominance of wave-type perturbations and of a quasitwo-dimensional (horizontal) field in a gas with predominance of random turbulence perturbations.

REFERENCES

1. T. D. Dickey and G. L. Mellor, Decaying turbulence in neutral and stratified fluids, J. Fluid Mech. 89, 13-31 (1980).

er al.

2. J. J. Riley, R. W. Metcalfe and M. A. Weissman, Direct numerical simulation of homogeneous turbulence in density stratified fluids. In Non-linear Properries of Inl. Waves, La Jolla Institute, AIP Conf. Proc. 76, pp. 79112 (1981). 3. D. S. Stillinger, K. N. Helland and C. W. Van Atta, Experiments on the transition of homogeneous turbulence to internal waves in a stratified fluid, J. Fluid Mech.

131,91-122

(1983).

4. E. C. Itsweire. K. N. Helland and C. W. Van Atta. The evolution of grid-generated turbulence in a stably stratified fluid, J. Fluid Mech. 162,299-338 (1986). 5. J. H. Lienhard and C. W. Van Atta, The decay of turbulence in thermally stratified flow, J. Fluid Mech. 210, 57-112 (1990). 6. 0. Metals and J. R. Herring, Numerical simulations of freely evoluting turbulence in stably stratified fluids, J. Fluid Mech.

202, 117-148

(1989).

7. B. A. Kolovandin, Modelling the dynamics of turbulent transport processes, In Advances in Heat Transfer 21, pp. 185-234 (1991). 8. R. Deissler, Turbulent heat transfer and temperature fluctuations in a field with uniform velocity and temperature gradients, Int. J. Hear Mass Transfer 6, 217270 (1963). 9. D. W. Dunn and W. H. Reid, Heat transfer in isotropic turbulence during the final period of decay, Nat1 Advis. Comm. Aeronaut. TN-4186 (1958). IO. V. U. Bondarchuk, B. A. Kolovandin and 0. G. Martynenko, Modelling of developing nearly homogeneous turbulence of velocitv and scalar fields. Inf. J. Hear Mass Transfer 34, 1 l-34 (1991). Il. K. Yoon and Z. Warhaft, The evolution of grid-generated turbulence under conditions of stable thermal stratification, J. Nuid Mech. 215,601-638 (1990).