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Sheared stably stratified turbulence and large-scale waves in a lid driven cavity

N. Cohen, A. Eidelman, T. Elperin, N. Kleeorin, and I. Rogachevskii

The Pearlstone Center for Aeronautical Engineering Studies, Department of Mechanical Engineering,

Ben-Gurion University of the Negev, P.O.Box 653, Beer-Sheva 84105, Israel

(Dated: May 26, 2014)

We investigated experimentally stably stratified turbulent flows in a lid driven cavity with a non-zero vertical mean temperature gradient in order to identify the parameters governing the meanand turbulent flows and to understand their effects on the momentum and heat transfer. We foundthat the mean velocity patterns (e.g., the form and the sizes of the large-scale circulations) dependstrongly on the degree of the temperature stratification. In the case of strong stable stratification,the strong turbulence region is located in the vicinity of the main large-scale circulation. Wedetected the large-scale nonlinear oscillations in the case of strong stable stratification which can beinterpreted as nonlinear internal gravity waves. The ratio of the main energy-containing frequenciesof these waves in velocity and temperature fields in the nonlinear stage is about 2. The amplitude ofthe waves increases in the region of weak turbulence (near the bottom wall of the cavity), wherebythe vertical mean temperature gradient increases.

PACS numbers: 47.27.te, 47.27.-i

I. INTRODUCTION

A number of studies of turbulent transport in lid-driven cavity flow have been conducted in the past, be-cause the lid-driven cavity is encountered in many prac-tical engineering and industrial applications, and servesas a benchmark problem for numerical simulations. De-tailed discussions of the state of the art of the differ-ent studies of isothermal and temperature stratified lid-driven turbulent cavity flows have been published in sev-eral reviews (see, e.g. Refs. 1 and 2).Fluid flow and heat transfer in rectangular cavities

driven by buoyancy and shear have been studied nu-merically and experimentally in a number of publications(see, e.g., Refs. 324). In particular, three-dimensionallaminar lid-driven cavity flow have been studied experi-mentally and numerically (see Refs. 9 and 13), wherebythe Taylor-Gortler like (TGL) longitudinal vortices, aswell as other general flow structures, have been found.In the isothermal flow, both the number of vortex pairsand their average size increases as the Reynolds num-ber increases in spite of the lateral confinement of theflow. Different effects in a mixed convection in a liddriven cavity have been investigated in the past (see,Refs. 12, 16, 19, 20, 22, and 23). Excitation of an insta-bility in a lid-driven flow was studied in Ref. 11 and 24numerically and experimentally. In addition, lid-drivenflow in a cube filled with a tap water have been exper-imentally investigated in Ref. 24 to validate the numer-ical prediction of steady-oscillatory transition at lowerthan ever observed Reynolds number. The authors of

conim@post.bgu.ac.il eidel@bgu.ac.il elperin@bgu.ac.il; http://www.bgu.ac.il/me/staff/tov nat@bgu.ac.il gary@bgu.ac.il; http://www.bgu.ac.il/gary

Ref. 24 reported that their results agree with the numer-ical simulation demonstrating large amplitude oscillatorymotion overlaying the base quasi-two-dimensional flow inthe mid-plane.

There are several studies on lid driven cavity flow withstable stratification. Three-dimensional numerical sim-ulation in a shallow driven cavity have been conductedfor a stably stratified fluid heated from the top movingwall and cooled from below for a wide range of Rayleighnumbers and Richardson numbers (see Ref. 17). It wasfound that an increase of the buoyancy force preventsthe return flow from penetrating to the bottom of thecavity. The fluid is recirculated at the upper portionof the cavity, and the upper recirculation induces shearon the lower fluid layer and forms another weak recir-culated flow region. Multicellular flow becomes evidentwhen the Richardson number is larger than 1 and mayproduce waves that propagate along the transverse direc-tion. Strong secondary circulation, and separated floware evident for the Richardson number is about 0.1. Therate of the heat transfer increases as the Richardson num-ber decreases.

The two-dimensional and three-dimensional numericalsimulations in a driven cavity have been conducted fora stably stratified fluid heated from the top moving wallover broad ranges of the parameters (see Refs. 14 and18). It was found that when the Richardson number isvery small, the gross flow characteristics are akin to theconventional driven-cavity flows, as addressed by earlierstudies. In this case the isotherm surfaces maintain afair degree of two-dimensionality. When the Richardsonnumber increases, the primary and meridional flows areconfined to the upper region of the cavity. In the middleand lower parts of the cavity, fluid tends to be stagnant,and heat transfer is mostly conductive.

There are only a few experimental studies on lid driventurbulent cavity flow with stable stratification. In par-ticular, stably-stratified flows in a three-dimensional lid-

2driven cavity have been experimentally studied in orderto examine the behavior of longitudinal Taylor-Gortler-like vortices (see Refs. 2, 58). It was found that theTaylor-Gortler vortices appear in all situations and theyare generated in the region of concave curvature of theflow above a surface of separation in the shear flow. Inthe stably stratified flow, the vortices appear to enhancethe mixing as they convolute the interface. In the unsta-bly stratified flow, where the forced and free convectioneffects are approximately in balance, the Taylor-Gortlervortices are still formed. However, there are a few Taylor-Gortler vortex pairs and their size is small. The Taylor-Gortler vortices arise in the region of the downstreamsecondary eddy and corner vortices along the end-walls.At higher Reynolds numbers ( 104) the flow is unsteadyin the region of the downstream secondary eddy and ex-hibits some turbulent properties.Stably stratified sheared turbulent flows are of a great

importance in atmospheric physics. Since Richardson(1920), it was generally believed that in stationary homo-geneous atmospheric flows the velocity shear becomes in-capable of maintaining turbulence when the Richardsonnumber exceeds some critical value (see, e.g., Refs. 2527). The latter assertion, however, contradicts to at-mospheric measurements, experimental evidence and nu-merical simulations (see, e.g., Refs. 2833). Recentlyan insight into this long-standing problem has beengained through more rigorous analysis of the turbu-lent energetics involving additional budget equations forthe turbulent potential energy and turbulent heat flux,and accounting for the energy exchange between turbu-lent kinetic energy and turbulent potential energy (see,Refs. 3437). This analysis opens new prospects to-ward developing consistent and practically useful turbu-lent closures for stably stratified sheared turbulent flows.The main goal of this study is to investigate experi-

mentally stably stratified turbulent flows in a lid drivencavity in order to identify the parameters governing themean and turbulent flows and to understand their effectson the momentum and heat transfer. This paper is or-ganized as follows. Section II describes the experimentalset-up and instrumentation. The results of laboratorystudy of the stably stratified sheared turbulent flow andcomparison with the theoretical predictions are describedin Section III. Finally, conclusions are drawn in SectionIV.

II. EXPERIMENTAL SET-UP

The experiments have been carried out in a lid-driventurbulent cavity flow generated by a moving wall in rect-angular cavity filled with air (see Fig. 1). An internalpartition which is parallel to the XZ-plane is inserted inthe cavity in order to vary the aspect ratio of the cavity.Here we introduce the following system of coordinates:Z is the vertical axis, the Y -axis is a direction of a longwall and the XZ-plane is parallel to a square side wall

FIG. 1. Scheme of the experimental set-up with shearedtemperature stratified turbulence: 1-Rectangular cavity; 2-Heated top wall; 3-Cooled bottom wall; 4-Internal partition;5-Plate heating elements; 6-Gear wheels; 7-Sliding rings; 8-Tank with cold water; 9-Chiller; 10-Electric motor; 11-Gearbox; 12-Rigid steel frame; 13-Laser light sheet optics; 14-CCDcamera; 15-Generator of incense smoke; 16-Pump.

of the cavity. A top wall of the cavity moves in a Y -axisdirection and generates a shear flow in the cavity. Theexperiments have been conducted in the cavity with thedimensions 242924 cm3. Heated top wall and cooledbottom wall of the cavity impose a temperature gradientin the flow which causes temperature stratification of theair inside the cavity.

The top moving wall of the chamber consists of identi-cal rectangular plate heating elements with a width of 16cm which are connected by hinges to the two adjustmentheating elements. Twenty connected heating elementsform a closed conveyer belt which is driven by two ro-tating gear-wheels with hinges. At each moment 6 mov-ing heating elements are located in the plane and formthe moving top wall of the cavity. Each heating elementcomprises the aluminum plate with the attached electri-cal heater and a temperature probe and is insulated witha textolite cap. The heaters and the temperature probesare electrically connected to a power supply unit and tothe measuring device through a set of sliding rings. Abottom stationary cold wall of the cavity is manufac-tured from aluminum and serves as a top wall of a tankfilled with water which circulates through a chiller witha controlled temperature.

All moving parts of the experimental set-up includingan electrical motor and a gear-box are attached to a rigidsteel frame in order to minimize vibrations of the cavitythat is connected to the moving wall with a soft flexi-ble sealing. Perspex walls are attached to the frame andenclose the moving conveyer belt consisting of heating el-ements in order to reduce heat transfer from the heating

3elements. The experiments have been conducted at dif-ferent velocities of the top wall and aspect ratios of thecavity, and at different temperature differences betweenthe top and the bottom walls of the cavity. This experi-mental set-up allows us to produce sheared temperaturestratified turbulence.

The turbulent velocity field have been measured us-ing a digital Particle Image Velocimetry (PIV) system(see, e.g., Refs. 3840) with LaVision Flow Master III. Adouble-pulsed Nd-YAG laser (Continuum Surelite 2170mJ) is used for light sheet formation. Light sheet op-tics comprise spherical and cylindrical Galilei telescopeswith tuneable divergence and adjustable focus length.We employ a progressive-scan 12 Bit digital CCD cam-era (pixels with a size 6.7m 6.7m each) with dualframe technique for cross-correlation processing of cap-tured images. The tracer used for PIV measurements isincense smoke with sub-micron particles (with the mate-rial density tr 1 g/cm

3), which is produced by hightemperature sublimation of solid incense particles. Ve-locity measurements were conducted in two perpendicu-lar cross-sections in the cavity, Y Z cross-section (frontalview) and XZ cross-section (side view).

We have determined the mean and the r.m.s. veloci-ties, two-point correlation functions and an integral scaleof turbulence from the measured velocity fields. Series of520 pairs of images, acquired with a frequency of 1 Hz,have been stored for calculating velocity maps and forensemble and spatial averaging of turbulence character-istics. The center of the measurement region coincideswith the center of the chamber. We have measured ve-locity in the probed cross-section 240 290 mm2 with aspatial resolution of 20482048 pixels. This correspondsto a spatial resolution 142 m / pixel. This probed regionhas been analyzed with interrogation windows of 32 32or 16 16 pixels, respectively.

In every interrogation window a velocity vector havebeen determined from which velocity maps comprising3232 or 6464 vectors are constructed. The mean andr.m.s. velocities for every point of a velocity map (1024 or4096 points) have been calculated by averaging over 520independent maps, and then they are averaged over thespace. The two-point correlation functions of the velocityfield have been calculated for every point of the velocitymap inside the main vortex (with 16 16 vectors) byaveraging over 520 independent velocity maps, and thenthey will be averaged over 256 points. An integral scale0 of turbulence has been determined from the two-pointcorrelation functions of the velocity field.

The temperature field has been measured with a tem-perature probe equipped with twelve E-thermocouples(with the diameter of 0.13 mm and the sensitivity of65V/K) attached to a vertical rod with a diameter 4mm. The spacing between thermocouples along the rodis 22 mm. Each thermocouple is inserted into a 1 mmdiameter and 45 mm long case. A tip of a thermocou-ple protrudes at the length of 15 mm out of the case.The mean temperature is measured for 10 rod positions

with 25 mm intervals in the horizontal direction, i.e., at120 locations in a flow. The exact position of each ther-mocouple is measured using images captured with theoptical system employed in PIV measurements. A se-quence of temperature readings (each reading is averagedover 50 instantaneous measurements which are obtainedin 20 ms) for every thermocouple at every rod positionis recorded and processed using the developed softwarebased on LabVIEW 7.0. The measurements from 12 ther-mocouples are obtained every 0.8 s. Similar experimen-tal technique and data processing procedure were usedpreviously in the experimental study of different aspectsof turbulent convection, stably stratified turbulent flows(see Refs. 4143) and in Refs. 4448 for investigating aphenomenon of turbulent thermal diffusion (see Refs. 49and 50).

III. EXPERIMENTAL RESULTS AND

COMPARISON WITH THE THEORETICAL

PREDICTIONS

We start the analysis of the experimental results withthe mean flow patterns obtained in the experiments con-ducted at different values of the temperature differenceT between the top and bottom walls. A set of meanvelocity fields obtained in the central Y Z plane is shownin Figs. 2 and 3. These experiments demonstrate strongmodification of the mean flow patterns with an increas-ing temperature difference T between the hot top walland the cold bottom wall of the cavity whereby the topwall moves in the left direction.Figures 2 and 3 demonstrate the major qualitative

changes in the mean flow patterns as the bulk Richardsonnumber, Rib = gTHz/U

20 , encompasses a wide range.

Here is the thermal expansion coefficient, U0 = 118cm/s is the lid velocity, Hz = 24 cm is the vertical heightof the cavity and g is the gravitational acceleration. Theprimary mean circulation (the main vortex), shown inthe upper panel of Fig. 2, occupies the entire cavity.Two weak secondary mean vortexes are observed at thelower corners of the cavity. The qualitative character ofthe mean flow for small T (or Rib 1) is similar tothe conventional lid driven cavity flow of a non-stratifiedfluid.When T (or Rib) is gradually increased, a change of

the mean flow is observed already at a relatively low tem-perature difference [compare the middle panel of Fig. 2that corresponds to T = 11 K (or Rib = 0.06), with thebottom panel of Fig. 2 that is for T = 21 K, Rib = 0.12].The position of the main vortex is shifted to the left,while the position of the right weak secondary vortex isshifted upwards and its size increases. At a further in-crease of the temperature difference T between the topand bottom walls, 33 T 54 K (0.19 Rib 0.29),the main vortex is pushed upwards, its size decreases, andthe weak secondary mean flow is also strongly changed(see Fig. 3).

4FIG. 2. Mean flow patterns obtained in the experiments inY Z cross-section at the different temperature differences be-tween the top and bottom walls: T = 0 K (upper panel);T = 11 K (middle panel); T = 21 K (lower panel). Coor-dinates y and z are measured in mm.

FIG. 3. Mean flow patterns obtained in the experiments inY Z cross-section at the different temperature differences be-tween the top and bottom walls: T = 33 K (upper panel);T = 44 K (middle panel); T = 54 K (lower panel). Coor-dinates y and z are measured in mm.

5Strong effect of stratification on the flow pattern inthe cavity can be also seen by inspecting mean veloc-ity maps in XZ cross-section obtained for different tem-perature difference between the top and bottom walls.The mean velocity fields shown in Fig. 4 are measuredin the cross-section close to the center of the main vor-tex. The frontal and side velocity maps demonstrate acomplex three-dimensional flow in the lid-driven cavitywhich comprises the main vortex seen in frontal view,and several secondary vortices (seen in frontal and sideviews).

It is possible to distinguish between three regions inthe mean flow for T 33 K or Rib 0.19 (see theupper panel of Fig. 3): a relatively strong mean flowin the upper part of the cavity including a main vortexin its left side, a mean sheared flow with a lesser meanvelocity in its right side, and a very weak mean flow inthe rest of the cavity. Therefore, when the strength ofstable stratification increases and the bulk Richardsonnumber, Rib, increases up to 0.3, the main vortex tendsto be confined to a small zone close to the sliding top lidand to the left wall of the cavity.

The stable stratification suppresses the vertical meanmotions, and, therefore, the impact of the sliding top wallpenetrates to the smaller depth into the fluid. As seen inFig. 3, when Rib is not small, the mean flow in the middleand lower parts of the cavity interior is weak, and muchof the fluid remains almost stagnant. The lower panel ofFig. 3 demonstrates this trend. The mean flow is almoststagnant in the bulk of the cavity interior excluding theregion close to the sliding top wall.

To characterize the change in the mean flow patternwith the increase of the stratification, we show in Fig. 5the maximum vertical size Lz of the main (energy con-taining) large-scale circulation versus the temperaturedifference T between the bottom and the top walls ofthe chamber obtained in the experiments. Inspection ofFig. 5 shows that the maximum vertical size Lz of large-scale circulation is nearly constant when T < 25 K. Onthe other hand, when T > 25 K, the maximum verticalsize Lz decreases with T as Lz 1/T .

This scaling can be understood on the base of thebudget equations for the mean velocity and temperaturefields. Indeed, using the budget equation for the meankinetic energy EU = U

2/2, we obtain that the change ofthe mean kinetic energy EU is of the order of the work ofthe buoyancy force, U2/2 (T )Lz, where = g/T0is the buoyancy parameter, T is the mean temperaturewith the reference value T0, and T is the change of themean temperature over the size of the large-scale circula-tion Lz. On the other hand, the budget equation for thesquared mean temperature, T 2, shows that the change ofthe mean temperature T over the size of the large-scalecirculation Lz is of the order of the temperature differ-ence T between the bottom and the top walls of thechamber. This yields the following scaling:

Lz U2/T. (1)

FIG. 4. Mean flow patterns obtained in the experiments inXZ cross-section at the different temperature differences be-tween the top and bottom walls: T = 0 K (upper panel);T = 21 K (middle panel); T = 40 K (lower panel). Coor-dinates x and z are measured in mm.

60 10 20 30 40 500

5

10

15

20

25

T

Lz

FIG. 5. Maximum vertical size Lz of large-scale circulationversus the temperature difference T between the bottomand the top walls of the chamber obtained in the experiments.Dashed lines correspond to fitting of the experimental points,while solid line corresponds to the theoretical estimates. Thetemperature difference T is measured in K, while the sizeLz is measured in cm.

0 10 20 30 40 50

1

2

3

4

T

Uy, u

y

FIG. 6. Characteristic horizontal mean Uy (squares) and tur-bulent uy (snowflakes) velocities versus the temperature dif-ference T between the bottom and the top walls of the cham-ber obtained in the experiments. The temperature differenceT is measured in K, while the velocities are measured incm/s.

For the largest stratification (T = 54 K) obtainedin our experiments, the cavity can be separated into tworegions: the region with strong turbulence (the left up-per region) and weak turbulence region. In particular,the turbulent kinetic energy in the first region is by twoorders of magnitude larger than that in the other area.To characterize the velocity fields obtained in the exper-iments, in Figs. 6 and 7 we show the characteristic hori-zontal and vertical mean, Uy,z, and turbulent, uy,z, r.m.s.velocities (measured in the region with a strong turbu-lence) versus the temperature difference T between thebottom and the top walls of the chamber. Inspection ofFig. 6 shows when T < 25 K, the horizontal mean ve-locity, Uy, is larger than the turbulent velocity uy, whilefor T > 25 K, Uy < uy. This tendency is related to thefact that for T > 25 K, the size of the main (energy

0 10 20 30 40 50

1

2

3

4

T

Uz, u

z

FIG. 7. Characteristic vertical mean Uz (squares) and turbu-lent uz (snowflakes) velocities versus the temperature differ-ence T between the bottom and the top walls of the chamberobtained in the experiments. The temperature difference Tis measured in K, while the velocities are measured in cm/s.

containing) large-scale circulation decreases with T .This implies that the size of the shear-produced tur-

bulence region due to the main vortex decreases withincrease of the stratification. In this region, the meanvelocity Uy is nearly constant. Substituting the verticalsize of the turbulence region, Lz, determined by Eq. (1)into the mean shear S = dUy/dz Uy/Lz, we obtain

dUy/dz UyT/U2 T/U, (2)

where we have taken into account that Uy U . There-fore, the value of shear increases with T , and, conse-quently, the shear production rate =

TS2 increases

with the increasing of the stratification, where T zuz

is the turbulent viscosity and z is the integral scale ofturbulence in the vertical direction. Now let us estimatethe turbulent kinetic energy, u2/2, using the budgetequation for this quantity:

u2/2 z/uz 2zS

2 2z(T/U)2, (3)

where u is the turbulent r.m.s. velocity. Since the meanvelocity is nearly constant for T > 25 K, and u zT[see Eq. (3)], the turbulent velocity increases with theincrease of the stratification. Here we have taken intoaccount that the integral scale of turbulence in verticaldirection does not change strongly with the change of Twhen T > 25 K (see Fig. 8).The internal gravity waves with the frequency =

Nkh/k can be excited in stably stratified flows. Here k isthe wave number, kh is the horizontal wave number andN = (zT )

1/2 is the Brunt-Vaisala frequency (see, e.g.Refs. 26, 33, 36, 5153). In our experiments in the re-gion of the cavity with a weak turbulence we observedthe large-scale internal gravity waves with the period ofabout 22 seconds. In particular, in Fig. 9 we show thenormalized one-point non-instantaneous correlation func-tion R() = T (z, t)T (z, t+)/T 2(z, t) of the large-scale temperature field determined for different z versus

70 10 20 30 40 50

5101520253035

y, z

T

FIG. 8. Integral scales of turbulence in horizontal y(snowflakes) and vertical z (squares) directions versus thetemperature difference T between the bottom and the topwalls of the chamber obtained in the experiments. The tem-perature difference T is measured in K, while the lengthsy,z are measured in mm.

30 20 10 0 10 20

0

0.5

1

R()

30 20 10 0 10 20

0

0.5

1

R()

FIG. 9. Normalized one-point non-instantaneous correlationfunction R( ) = T (z, t)T (z, t+ )/T 2(z, t) of the large-scale temperature field determined for different z: 2.5 cm (di-amonds), 5.1 cm (six-pointed stars), 7.4 cm (crosses), 11.9cm (snowflakes), 15.9 cm (squares), 18.9 cm (circles) shownin upper and lower panels, versus the time at the temper-ature difference T = 54 K between the bottom and thetop walls of the chamber obtained in the experiments, whereT = T T 0. The dashed fitting line corresponds to theLorentz function with 0 = 13 s and 0 = 0.286 s

1. Thetemperature difference T is measured in K.

the time , where T = T T 0, and T is the sliding av-eraged temperature (with 10 seconds window average),T 0 = T

(sa) and ...(sa) is the 10 minutes average. In-spection of Fig. 9 shows that the function R() has a formof the Lorentz function, R() = exp(/0) cos(0)with 0 = 13 s and 0 = 0.286 s

1, which corresponds tothe period of the wave 2/0 = 22 seconds. Note thatthe Fourier transform, R(), of the Lorentz function hasthe following form:

R() =2

0

([( 0)

2 + 20 ]1 + [( + 0)

2 + 20 ]1).

(4)

Such form of the correlation function R() indicates thepresence of the large-scale waves with random phases.The memory or correlation time for these waves is about11 s. Therefore, in our analysis the temperature field isdecomposed in three different parts: small-scale temper-ature fluctuations, the mean temperature field and thelarge-scale temperature field corresponding to the large-scale internal gravity waves.We also performed similar analysis for the vertical

large-scale velocity field. In our analysis the velocityfield is decomposed in three different parts: small-scalevelocity fluctuations, the mean velocity field and thelarge-scale velocity field corresponding to the large-scaleinternal gravity waves. In Fig. 10 we show the nor-malized one-point non-instantaneous correlation functionRu() = Uz(z, t)Uz(z, t+ )/U

2z (z, t) of the verti-

cal large-scale velocity field determined for different z ver-sus the time . Comparison of the normalized one-pointnon-instantaneous correlation function, Ru(), of the ver-tical large-scale velocity field with that of the tempera-ture field, R() (see Fig. 11) shows that for short timescales ( < 10 s) these correlation functions are different.This implies that for these time-scales the wave spectraof the large-scale velocity and temperature fields are dif-ferent. In particular, the velocity oscillations spectrumincludes also contributions from the double frequency, ascan be also seen from Fig. 12, where we show the spectralfunctions, Ru() = (2)

1Ru() exp(i) d and

R() = (2)1

R() exp(i) d , of the normalized

one-point non-instantaneous correlation functions, Ru()and R(), of the vertical large-scale velocity and temper-ature fields. On the other hand, for larger time scales( 15 s) these correlation functions are nearly similar,except for their phases are shifted by 180 degrees.In Fig. 13 we show spatial vertical profiles of turbu-

lent temperature fluctuations rms and of the functionTrms T

2(z)1/2 of the large-scale temperature fieldat the temperature difference T = 33 K between thebottom and the top walls of the chamber obtained in theexperiments, where T = T T 0. Inspection of Fig. 13shows that the level of the intensities of turbulent tem-perature fluctuations are of the same order as the energyof the large-scale internal gravity waves. These turbulentfluctuations, rms , are larger in the lower part of the cav-ity where the mean temperature gradient is maximum.

830 20 10 0 10 20

0

0.5

1

Ru()

FIG. 10. Normalized one-point non-instantaneous correlationfunction Ru( ) = Uz(z, t)Uz(z, t + )/U

2

z (z, t) of thevertical large-scale velocity field determined for different zversus the time determined for different z versus the time at the temperature difference T = 54 K between the bottomand the top walls of the chamber obtained in the experiments,where Uz = Uz Uz0. The temperature difference T ismeasured in K.

30 20 10 0 10 20

0

0.5

1

R()

FIG. 11. Comparison of the normalized one-point non-instantaneous correlation function, Ru( ) (crosses), of thevertical large-scale velocity field with that of the large-scaletemperature field R( ) (squares) determined for different zversus the time at the temperature difference T = 54 Kbetween the bottom and the top walls of the chamber ob-tained in the experiments. The temperature difference T ismeasured in K.

0 0.2 0.4 0.6 0.8

0.01

0.02

0.03

0.04

Ru(), R

()

FIG. 12. The spectral functions, Ru() =(2)1

Ru( ) exp(i ) d (solid) and R() =

(2)1R( ) exp(i ) d (dashed-dotted), of the nor-

malized one-point non-instantaneous correlation functions,Ru( ) and R( ), of the vertical large-scale velocity andtemperature fields, where the function Ru( ) is shown inFigs. 10 and 11, while the function R( ) is shown in Fig. 9.

5 10 15 2000.20.40.60.81.01.2

z

rms

, Trms

FIG. 13. Spatial vertical profiles of turbulent tempera-ture fluctuations rms (dashed) and of the function Trms

T 2(z)1/2 of the large-scale temperature field (solid) at thetemperature difference T = 33 K between the bottom andthe top walls of the chamber obtained in the experiments,where T = T T 0. The temperature difference T is mea-sured in K, while the lengths z is measured in cm.

In the upper part of the cavity where the shear causedby the large-scale circulation is maximum, and the meantemperature gradient is decreased.

IV. CONCLUSIONS

We study experimentally stably stratified turbulenceand large-scale flows and waves in a lid driven cavitywith a non-zero vertical mean temperature gradient. Ge-ometrical properties of the large-scale vortex (e.g., itssize and form) and the level of small-scale turbulence in-side the vortex are controlled by the buoyancy (i.e., bythe temperature stratification). The observed velocityfluctuations are produced by the shear of the large-scalevortex. At larger stratification obtained in our exper-iments, the strong turbulence region is located at theupper left part of the cavity where the large-scale vor-tex exists. In this region the Brunt-Vaisala frequencyis small and increases in the direction outside the large-scale vortex. This is the reason of that the large-scaleinternal gravity waves are observed in the regions out-side the large-scale vortex. We found these waves byanalyzing the non-instantaneous correlation functions ofthe temperature and velocity fields. The observed large-scale waves are nonlinear because the frequency of thewaves determined from the temperature field measure-ments is two times smaller than that obtained from thevelocity field measurements. The measured intensity ofthe waves is of the order of the level of the temperatureturbulent fluctuations.

ACKNOWLEDGMENTS

We thank A. Krein for his assistance in construc-tion of the experimental set-up. This research was sup-ported in part by the Israel Science Foundation governed

9by the Israeli Academy of Sciences (Grant 1037/11), and by the Russian Government Mega Grant (Grant11.G34.31.0048).

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