Table Of ContentTransition phenomena in unstably stratified turbulent flows
M. Bukai,∗ A. Eidelman,† T. Elperin,‡ N. Kleeorin,§ I. Rogachevskii,¶ and I. Sapir-Katiraie∗∗
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: January 25, 2011)
We study experimentally and theoretically transition phenomena caused by the external forcing
from Rayleigh-B´enard convection with the large-scale circulation (LSC) to the limiting regime of
unstablystratifiedturbulentflowwithoutLSCwherebythetemperaturefieldbehaveslikeapassive
scalar. In the experiments we use the Rayleigh-B´enard apparatus with an additional source of
1 turbulence produced by two oscillating grids located nearby the side walls of the chamber. When
1 thefrequencyofthegridoscillationsislargerthan2Hz,thelarge-scalecirculation(LSC)inturbulent
0 convection is destroyed, and the destruction of the LSC is accompanied by a strong change of the
2 mean temperature distribution. However, in all regimes of the unstably stratified turbulent flow
n the ratio (cid:2)(ℓx∇xT)2+(ℓy∇yT)2+(ℓz∇zT)2(cid:3)/hθ2i varies slightly (even in the range of parameters
wherebythebehaviourofthetemperaturefieldisdifferentfrom thatofthepassivescalar). Hereℓ
a i
J are the integral scales of turbulence along x,y,z directions, T and θ are the mean and fluctuating
parts of the fluid temperature. At all frequencies of the grid oscillations we have detected the
2
long-term nonlinearoscillations ofthemean temperature. Thetheoretical predictionsbased on the
2
budget equations for turbulent kinetic energy, turbulent temperature fluctuations and turbulent
heat flux,are in agreement with theexperimental results.
]
n
y PACSnumbers: 47.27.te, 47.27.-i
d
-
u
I. INTRODUCTION Numerous experiments demonstrate that the mean
l
f temperature distribution in turbulent convection is
.
s Variousaspectsofformationoflarge-scalecirculations stronglyinhomogeneousandanisotropicduetothemean
c (LSC) and other types of coherent structures in a tur- wind (see, e.g., [7, 15, 18, 21, 40]), and the mean tem-
i
s bulent convection at large Rayleigh numbers have been perature gradient in the horizontal direction inside the
y
discussedinanumberofstudiesfortheatmosphericflows LSC can be significantly larger than in the vertical di-
h
p [1–4],laboratoryexperimentsintheRayleigh-B´enardap- rection, i.e., the vertical turbulent heat flux inside the
[ paratus [5–30] and direct numerical simulations [31–34]. LSC is very small. The dependencies of the dissipation
Amean-fieldtheory[4,35,36]hasbeenproposedforthe rate(normalizedbythemoleculartemperaturediffusion)
3 explanationofformationofcoherentstructuresinturbu- of the temperature fluctuations hθ2i as a function of the
v
lentconvection. Itisknownfromlaboratoryexperiments Rayleigh number and different statistical characteristics
8
8 and numerical simulations [7, 14, 15, 21] that the coher- of this normalizeddissipation rate have been studied ex-
5 ent structures in the Rayleigh-B´enard turbulent convec- perimentallyforturbulentRayleigh-B´enardconvectionin
2 tion are not driven by the turbulent Reynolds stresses, [29].
2. associated with the tilting plumes at the upper and the On the other hand, in a forced turbulent convection
1 lower horizontal walls. If the mean flow is established, without a mean wind (i.e., in unstably stratified turbu-
9 the temperature of the fluid is larger at one side of the lent flows) the temperature field behaves like a passive
0 LSC and smaller at the other side of the LSC, and the scalar (see, e.g., [37, 41–43]). It is not clear why in tur-
v: mean flow is driven by the mean buoyancy force at the bulent convection with LSC the vertical gradient of the
i side walls of the chamber. meantemperaturefieldissmallinsideLSC,andtheverti-
X
Theeffectofcoherentstructures(meanwind)onglobal cal temperature gradient can even change the direction,
r properties of turbulent convectionis a subject of numer- i.e., it can be directed upwards. A detailed study of a
a
ous discussions (see, e.g., reviews [37–39]). It has been transition from one regime with the mean wind to an-
recently found in laboratory experiments in [40], that other regime without the mean wind due to additional
in turbulent convection with the mean wind the bulk forcinginunstablystratifiedturbulentflow,canelucidate
production of turbulence is due to the large-scale shear the effects of the mean wind on turbulent convection.
caused by the mean wind rather than by buoyancy. Different effects in turbulent convectionwith a forcing
have been studied previously (see, e.g., [41, 44, 46, 47]).
In particular, an experimental study of a turbulent con-
vection with forcing in the form of periodical pulses has
∗ [email protected]
been performed in [44]. In these experiments, in spite
† [email protected]
of the forcing, a large-scale circulatory flow was not de-
‡ [email protected];http://www.bgu.ac.il/me/staff/tov
§ [email protected] stroyed, and the heat transfer in this system was en-
¶ [email protected];http://www.bgu.ac.il/˜gary hanced in comparison with constant and sinusoidally
∗∗ [email protected] modulated energy inputs. It was found in [44] that the
2
enhancement of the heat transfer depends on the syn-
chronization of the kicking period of energy input with
the intrinsic time scale of the turbulent flow. In other
type of experiments [46, 47] a relation between convec-
tive heat flux and temperature gradient was determined
inaverticalchannelfilledwithwater,theaveragevertical
mass flux being zero. It was shown in these experiments
that when the LSC is suppressed by physically blocking
itspath,thereexistsastrongverticaltemperaturegradi-
entinthebulkoftheRayleigh-B´enardapparatusandthe
temperature remains active, rather than passive field.
The goal of this paper is to study experimentally and
theoretically transition phenomena due to the external
forcing from Rayleigh-B´enard convection with LSC to
the limiting regime of unstably stratified turbulent flow
withoutLSC,wherebythetemperaturefieldbehaveslike FIG.1. Experimentalset-up: (1)-laserlightsheet;(2)-heat
a passive scalar. In the experimental study we use the exchangers; (3) - grid driver; (4) - digital CCD camera.
Rayleigh-B´enardapparatus with an additional source of
turbulence produced by the two oscillating grids located
nearby the side walls of the chamber. Additional forc- II. EXPERIMENTAL STUDY
ing for turbulence allows us to observe evolution of the
mean temperature and velocity fields during the transi-
Let us start with description of the experimental set-
tion from turbulent convection with the LSC (for very
up and the results of laboratory study of the transition
small frequencies of the grid oscillations) to the limiting
phenomena in unstably stratified turbulent flow.
regime of unstably stratified flow without LSC (for very
high frequencies of the grid oscillations). In the exper-
iments we use Particle Image Velocimetry to determine
A. Experimental set-up and instrumentation
theturbulentandmeanvelocityfields,andaspeciallyde-
signedtemperatureprobewithsensitivethermocouplesis
The experimentsontransitionphenomenainunstably
employed to measure the temperature field.
stratified turbulence have been conducted in rectangu-
lar chamber with dimensions 26 × 58 × 26 cm3 in air
Note that this experimental set-up with the turbulent flow with the Prandtl number Pr =0.71. The side walls
convection and the external forcing has been previously of the chambers are made of transparent Perspex with
used in studies of turbulent transport of particles (see, the thickness of 10 mm. In the experiments we use the
e.g., [45, 48–50]). Comprehensive investigationof turbu- Rayleigh-B´enardapparatus with an additional source of
lencestructures,meanvelocityandtemperaturedistribu- turbulence produced by two oscillating grids. Pairs of
tions,velocityandtemperaturefluctuationscanelucidate vertically oriented grids with bars arranged in a square
a complicated physics related to particle clustering and array (with a mesh size 5 cm) are attached to the right
formation of large-scale inhomogeneities in particle spa- and left horizontal rods (see Fig. 1). The grids are posi-
tial distributions in unstably stratified turbulent flows. tionedatadistanceoftwogridmeshesfromthechamber
Notably, turbulent convection with the external forcing walls and are parallel to the side walls. Both grids are
is used in chemical industrial devices to improve mixing operated at the same amplitude of 61 mm, at a random
(see, e.g., [51–53]). Another example of turbulent con- phase and at the same frequency which is varied in the
vection with forcing is atmospheric wind superimposed range from 0.5 Hz to 16.5 Hz. Here we use the following
on turbulent convective boundary layer. systemofcoordinates: z is the verticalaxis,the y-axisis
perpendiculartothe gridsandthe xz-planeisparallelto
the grids.
The paper is organizedas follows. SectionII describes Averticalmeantemperaturegradientinthe turbulent
the experimentalset-up,instrumentationandthe results airflowwasformedbyattachingtwoaluminiumheatex-
of laboratory study of the transition phenomena in un- changers to the bottom and top walls of the test section
stably stratified turbulent flow. The theoretical analysis (a heatedbottom anda cooledtopwallofthe chamber).
and comparison with the experimental results are per- In order to improve heat transfer in the boundary lay-
formed in Section III. Finally, conclusions are drawn in ersatthe bottom andtopwallsweusedheatexchangers
Section IV. In Appendix we present the detailed deriva- withrectangularfins3×3×15mm3 (seeFig.1)whichal-
tions of the theoretical model for a passive scalar that lowedusto formameantemperaturegradientupto 115
used for the explanation of the observed effects. K/m at a mean temperature of about 308 K when the
3
frequency of the grid oscillations f > 10 Hz. The thick- era with a single-axis Scheimpflug adapter. The angle
nessofthe aluminiumplateswiththe finsis2.5cm. The between the optical axis of the camera and the front
topplateisabottomwallofthetankwithcoolingwater. view plane as well as the angle between the optical axis
Temperature of water circulating through the tank and and the probed cross-section was approximately 45 de-
the chiller is kept constant within 0.1 K. Cold water is grees. The perspective distortion was compensated us-
pumped into the cooling system through two inlets and ing Stereoscopic PIV system calibrationkit whereby the
flows out through two outlets located at the side wall correction was calculated for a recorded image of a cal-
of the cooling system. The bottom plate is attached to ibration plate. The corrections were introduced in the
the electrical heater that provides constant and uniform recorded in the probed cross-section images before their
heating. The voltageofa stable power supply applied to processing using a cross-correlation technique for deter-
the heater varies up to 200 V. The power of the heater mining velocity fields.
varies up to 300 W. An incense smoke with sub-micron particles (ρ /ρ ∼
p
The temperature field was measured with a tem- 103), as a tracer was used for the PIV measurements.
perature probe equipped with twelve E-thermocouples Smoke was produced by high temperature sublimation
(with the diameter of 0.13 mm and the sensitivity of of solid incense grains. Analysis of smoke particles us-
≈65µV/K)attachedto a verticalrodwitha diameter 4 ing a microscope (Nikon, Epiphot with an amplification
mm. Thespacingbetweenthermalcouplesalongtherod of560)andaPM-300portablelaserparticulateanalyzer
was22mm. Eachthermocouplewasinsertedintoa1mm showedthattheseparticleshaveanapproximatelyspher-
diameter and 45 mm long case. A tip of a thermocouple ical shape and that their mean diameter is of the order
protruded at the length of 15 mm out of the case. The of 0.7µm. The probability density function of the par-
temperature was measured for 12 rod positions with 23 ticle size measured with the PM300 particulate analyzer
mm intervals in the horizontal direction, i.e., at 144 lo- was independent of the location in the flow for incense
cations in a flow. The exact position of eachthermocou- particle size of0.5−1µm. The maximum tracerparticle
plewasmeasuredusingimagescapturedwiththeoptical displacement in the experiment was of the order of 1/4
system employed in PIV measurements. A sequence of of the interrogation window. The average displacement
600 temperature readings for every thermocouple at ev- of tracer particles was of the order of 2.5 pixels. The av-
ery rod position was recorded and processed using the erage accuracy of the velocity measurements was of the
developed software based on LabView 7.0. Temperature order of 4% for the accuracy of the correlation peak de-
maps were plotted using Matlab 7. tection in the interrogation window of the order of 0.1
In order to decrease mean flow generated by oscil- pixel (see, e.g., [54–56]).
lating grids and to increase the range of homogeneous Wedeterminedthemeanandther.m.s.velocities,two-
and isotropic turbulence we use partitions located in xz point correlation functions and an integral scale of tur-
plane between grids and side walls of the chamber (of bulence from the measured velocity fields. Series of 520
the width 85 mm). The aspect ratio of the chamber pairs of images acquired with a frequency of 1 Hz, were
A=Hy/Hz =1.52, where Hy is the size of the chamber storedforcalculatingvelocitymapsandforensembleand
along y-axis between partitions and Hz is the hight of spatial averaging of turbulence characteristics. The cen-
the chamber z-axis,respectively. terofthe measurementregioninyz-planecoincideswith
The velocityfields weremeasuredusingaStereoscopic the center of the chamber. We measured velocity in a
Particle Image Velocimetry (PIV), see [54–56]. In the flow domain 256×256 mm2 with a spatial resolution of
experiments we used LaVision Flow Master III system. 1024×1024 pixels. This corresponds to a spatial reso-
A double-pulsed light sheet was provided by a Nd-YAG lution 250 µm / pixel. The velocity field in the probed
laser (Continuum Surelite 2×170 mJ). The light sheet regionwasanalyzedwithinterrogationwindowsof32×32
opticsincludessphericalandcylindricalGalileitelescopes or 16×16 pixels. In everyinterrogationwindow a veloc-
with tuneable divergence and adjustable focus length. ityvectorwasdeterminedfromwhichvelocitymapscom-
We used two progressive-scan 12 bit digital CCD cam- prising 32×32 or 64×64 vectors were constructed. The
eras (with pixel size 6.7µm ×6.7µm and 1280×1024 mean and r.m.s. velocities for every point of a velocity
pixels) with a dual-frame-technique for cross-correlation map were calculated by averaging over 520 independent
processing of captured images. A programmableTiming maps, and then they were averagedover the central flow
Unit (PC interface card) generated sequences of pulses region.
to control the laser, camera and data acquisition rate. Thetwo-pointcorrelationfunctionsofthevelocityfield
The software package LaVision DaVis 7 was applied to weredeterminedforeverypointofthecentralpartofthe
controlallhardwarecomponentsandfor32bitimageac- velocity map (with 16× 16 vectors) by averaging over
quisition and visualization. This software package com- 520 independent velocity maps, which yields 16 correla-
prises PIV software for calculating the flow fields using tion functions in y and z directions. Then the two-point
cross-correlationanalysis. correlation function was obtained by averaging over the
In order to obtain velocity maps in the central region ensembleofthesecorrelationfunctions. Anintegralscale
of the flow in the cross-section parallel to the grids and ofturbulence, ℓ, wasdetermined fromthe two-pointcor-
perpendicular to a front view plane, we used one cam- relationfunctionsofthevelocityfield. Intheexperiments
4
weevaluatedthevariabilitybetweenthefirstandthelast
20 velocity maps of the series of the measured velocity
field. Since very small variability was found, these tests
showed that 520 velocity maps contain enough data to
obtain reliable statistical estimates. We found also that
the measured mean velocity field is stationary.
The characteristic turbulence time in the experiments
τ =0.28−0.62seconds,whilethecharacteristictimepe-
z
riodfor the large-scalecirculatoryflow (∼10s) is by 1.5
ordersofmagnitudelargerthanτ . Thesetwocharacter-
z
istic times are much smaller than the time during which
the velocity fields are measured (∼ 600 s). The size of
theprobedregiondidnotaffectourresults. Thetemper-
aturedifferencebetweenthetopandbottomplates,∆T,
in all experiments was 50 K (i.e., the global Rayleigh
number based on molecular transport coefficients, was
Ra =0.73×108). Similar experimental set-up and data
processing procedure were used previouslyin the experi-
mentalstudy of differentaspects of turbulent convection
[22,40]andin[45,48–50]forinvestigatingaphenomenon
of turbulent thermal diffusion [57, 58].
B. Experimental results
Let us start the analysis of the obtained experimental
results from the dynamics of the mean velocity field. In
Fig. 2 we show the mean flow patterns obtained in the
experimentswithunforcedturbulentconvection(Fig.2a)
andwithforcedturbulentconvectionatdifferentfrequen-
cies of the grid oscillations (Figs. 2b and 2c). The solid
lines in Fig. 2 are stream lines in the cental yz-plane for
x = 13 cm. The center of Figs. 2a, 2b, 2c, which has
coordinates y = 10 cm and z = 10 cm, coincides with
the center of the chamber. When the frequency of the
grid oscillations is larger than 2 Hz, the large-scale cir-
culation(LSC)inturbulentconvectionintheyz-planeis
notobserved(seeFigs.2band2c). Ontheotherhand,a
complicatedmeanflowofthe spiralforminthe xz-plane
with a non-zero mean vorticity in the y direction exists
when the frequencies of the grid oscillations 2 < f < 7
Hz.
ThedestructionoftheLSCisaccompaniedbyastrong
changeofthemeantemperaturedistribution. Thedistri-
bution of the measured mean temperature field depends
strongly on the frequency f of the grid oscillations (see
Figs. 3-5). In particular, for the very low frequency f
the thermal structure inside the LSC in turbulent con-
vectionis inhomogeneousandanisotropic. The hotther-
mal plumes concentrate at one side of LSC, and cold
plumes accumulate at the opposite side of LSC [18, 40].
In the centralpart of the flow the verticalmean temper- FIG.2. Meanflowpatternsobtainedintheexperimentswith
ature gradient ∇zT changes its sign depending on the unforced turbulentconvection (upperpanel) and with forced
frequency of the grid oscillations (see Fig. 3). turbulent convection at the frequencies f = 2.2 Hz (middle
panel) and f =16.5 Hz (lower panel) of the grid oscillations
At all frequencies of the grid oscillations we have ob-
for the unstably stratified turbulent flow at the temperature
served the long-term nonlinear oscillations of the mean
difference between the top and bottom walls ∆T = 50 K.
temperature with the period that is of the order of 10
Coordinates y and z are measured in mm.
s. Note that the temperature was measured at 144 lo-
5
∇ T ∇ T
z x
0.5 1
0.8
0
0.6
−0.5 0.4
0.2
−1
f
0 5 10
0 5 10 f
FIG. 5. Horizontal gradient of the mean temperature ∇ T
x
(squares)andtheamplitudeofthenonlinearlong-term oscil-
FIG. 3. Vertical gradient of the mean temperature ∇ T
z lations of this gradient (triangles) versus the frequency f of
(squares) and the amplitude of the nonlinear long-term os- thegridoscillationsfortheunstablystratifiedturbulentflow.
cillationsofthisgradient(triangles)versusthefrequencyf of The mean temperature gradient is measured in K cm−1 and
thegridoscillationsfortheunstablystratifiedturbulentflow.
thefrequency f is measured in Hz.
The mean temperature gradient is measured in K cm−1 and
thefrequency f is measured in Hz.
term variations of the vertical mean temperature gradi-
entδ(∇ T)=∇ T−∇ T duetothenonlinearoscillations
∇ T i i i
y ofthemeantemperature,wherei=x,y,z. Thebarhere
denotes additional time averaging. Note that the sep-
aration distance of 22 mm between thermal couples is
0.4 sufficient to measure the gradients of the mean temper-
ature. Indeed, the integralscale of turbulence is about 2
cm and the characteristic length scale of the mean tem-
0.2 perature field is much larger than the integral scale of
turbulence. Note also that for large frequencies of the
grid oscillations the role of the thermal boundary layer
0 in turbulent convection diminishes. It must be empha-
sized that in all our experiments rectangular fins were
attached at the bottom and top walls of the chamber in
−0.2 order to improve heat transfer in the boundary layers.
In Fig. 6 we show time dependencies of the instan-
0 5 10 f taneous temperature Ttot = T +θ, the long-term vari-
ations of mean temperature δT = T −T and the long-
termvariationsoftheverticalmeantemperaturegradient
FIG. 4. Horizontal gradient of the mean temperature ∇yT δ(∇zT) = ∇zT −∇zT due to the nonlinear oscillations
(squares)andtheamplitudeofthenonlinearlong-termoscil- of the mean temperature. In Figs. 3-5 we also show by
lations of this gradient (triangles) versus the frequency f of trianglesthe amplitude, {[δ(∇ T)]2}1/2, of the nonlinear
i
thegridoscillationsfortheunstablystratifiedturbulentflow. long-term oscillations of the gradients of the mean tem-
The mean temperature gradient is measured in K cm−1 and perature, where i = x,y,z. The dependencies shown in
thefrequency f is measured in Hz. Figs. 3-6 are obtained at the central region of the cham-
ber with the size of 10×10×10 cm3.
InFig.7weshowtheresultsofaFourieranalysisofthe
cations in a flow (the spacing between thermal couples signalδT =T −T. Inspectionof Fig.7 shows thatthere
was 22 mm). We perform the sliding averaging of the are two main maxima in the spectrum with the periods
instantaneous temperature field over time of 3 s (the 12.5 s and 20 s. Other smaller maxima in the spectrum
characteristic turbulence time in the experiments was are at the frequencies which are multiples of these main
0.3 - 0.6 s) in order to determine the mean temperature frequencies ortheir sums anddifferences. These are typ-
field T at 144 locations. Then we determine the long- ical features of nonlinear oscillations. The theory that
6
rms
T(tot) u
y,z
315
10
310
8
δT 2
0
−2
6
δ(∇ T)
z
1
0 4
−1
t
0 100 200 300 400 500
0 5 10 f
FIG.6. Timedependenciesoftheinstantaneoustemperature
Ttot=T+θ,thevariationsofmeantemperatureδT =T−T FIG. 8. Components uryms = phu2yi (triangles) and urzms =
and thevariations of thevertical mean temperaturegradient phu2i (squares) of the r.m.s. velocity versus the frequency
z
δ(∇ T) = ∇ T −∇ T due to the long-term nonlinear oscil- f of thegrid oscillations for theunstably stratified turbulent
z z z
lations of the mean temperature (with the period ∼ 10 s). flow. The turbulent velocity is measured in cm s−1 and the
The bar denotes additional time averaging. These time de- frequency f is measured in Hz.
pendencies are measured in thecenter of the chamber at the
frequency f =10 Hz of the grid oscillations for the unstably
stratified turbulent flow. These temperature characteristics
weshowthefrequencydependenciesofthefollowingmea-
are measured in K.
sured turbulence parameters: the r.m.s. velocity fluctu-
ations hu2iand hu2i,andthecorrespondingintegral
En(f˜) 0.08 Hz scales oqf turybulencpe alozng horizontal y and vertical z di-
0.05 Hz
rections (ℓ and ℓ ), where u are the fluctuations of the
y z
10 .03 Hz .06 Hz fluid velocity. The mean, r.m.s. velocities and the two-
0.10 Hz pointcorrelationfunctionsofthevelocityfieldhavebeen
0.16 Hz
calculated by averaging over 520 independent velocity
5
0.13 Hz
maps,andthentheyhavebeenaveragedoverthecentral
0.18Hz
flow region. The integralscales of turbulence, ℓ and ℓ ,
0 y z
0.05 0.1 0.15 0.2 0.25 f˜ havebeendeterminedfromthenormalizedtwo-pointlon-
gitudinal correlation functions of the velocity field, e.g.,
FIG. 7. The normalized spectrum function E (f˜) = Fy(y˜)=huy(r0)uy(r0+y˜ey)i/hu2y(r0)i[andsimilarlyfor
|(δT)˜|2/R |(δT)˜|2df˜ of the signal δT = T(t)−Tn, where Fz(z˜)afterreplacementinthe aboveformulay byz],us-
f f L
in the Fourier space (δT)˜= R δT exp[−if˜t]dt and f˜is the ing the following expression: ℓy = h 0 Fy(y˜)dy˜iS [and
f similarly for ℓ after replacementin tRhe above formula y
frequencyof thenonlinearlong-term oscillations ofthemean z
temperature. by z], where L = 10 cm is the linear size of the probed
flow region, e is the unit vector in the y direction, and
y
h...i is the additional averagingover the yz plane.
S
explains the mechanism ofthese nonlinear oscillationsof Inspection of Fig. 9 shows that the turbulent length
the mean temperature field, has not been developed yet. scales are weakly dependent on the frequency f of the
One may hypothesize that there are two possible mech- grid oscillations in the unstably stratified turbulent flow
anisms for such oscillations. One mechanism could be for f > 3 Hz, while the turbulent velocities and turbu-
related to the large-scale Tollmien-Schlichting waves in lenttimescalesvarystronglywiththefrequencyf. Note
sheared turbulent flows (see [59]), which can cause the thatτ f andτ f tendtoaconstant(∼3)forhigherfre-
y z
nonlinearoscillationsofthemeantemperaturefield. An-
quencies(f >5Hz)ofthe forcing,whereτ =ℓ / hu2i
other mechanism of the nonlinear oscillations could be y y q y
related to the generation of small-scale kinetic helicity andτz =ℓz/ hu2ziarethecharacteristicturbulenttimes
due to large-scale shear flows in the system. The large- in horizontalpand vertical directions. Note also that the
scale shear generates large-scale helicity, and since the ratio of the characteristic values of the mean velocity to
total helicity is conserved, a non-zero small-scale helic- the turbulent velocity in the centralpart of the chamber
ityis produced. Ourpreliminaryanalysisshowsthatthe varied from 1 at f = 0 to 0.5 when the frequency f of
small-scalehelicitycancausethenonlinearoscillationsof the grid oscillations f > 2 Hz. In particular, the char-
the mean temperature field. acteristic values of the mean velocity at the frequencies
Inordertoavoidsideeffectsofthegridswepresentthe of the grid oscillations f =0;2.2;16 Hz are as following:
experimentalresultsrecordedinthecentralregionofthe 7.5 cm/s; 3 cm/s and 6 cm/s, respectively.
chamber with the size of 10×10×10 cm3. In Figs. 8-9 In Fig. 10 we show the frequency dependence of the
7
∗ rms
ℓ ,ℓ u /u
y z z z
40 1
0.8
30
0.6
20
0.4
10
0.2
0
0 5 10 f
0 5 10 f
FIG. 9. Horizontal and vertical integral scales of turbulence FIG. 10. Ratios u∗/urms (squares) and u∗/u˜ (circles) ver-
z z z z
ℓy (triangles) and ℓz (squares) versus the frequency f of the sus the frequency f of the grid oscillations. Here urzms is
gridoscillationsfortheunstablystratifiedturbulentflow. The the r.m.s. of the vertical component of velocity fluctuations
turbulentlengthscalesaremeasuredinmmandthefrequency in the unstably stratified turbulent flow, u∗ is the r.m.s. of
z
f is measured in Hz. theverticalcomponentofvelocityfluctuationsintheisother-
mal turbulence, u˜ is the vertical component of the effective
z
turbulent velocity, u˜z = [(u∗z)2+4ℓzβphθ2i]1/2, that takes
into account the production of the turbulence by buoyancy.
ratio u∗z/urzms, where urzms is the r.m.s. of the vertical The velocity is measured in cm s−1 and the frequency f is
componentof velocity fluctuations in the unstably strat- measured in Hz.
ified turbulent flow and u∗ is the r.m.s. of the vertical
z
componentofvelocityfluctuationsinthe isothermaltur-
θ
bulence. In Fig. 10 along with the ratios of the r.m.s. of rms
theverticalcomponentofvelocityfluctuations,u∗/urms,
z z
in isothermal and unstably stratified turbulent flows, we
1
also plot the ratio u∗/u˜ , where u˜ is the vertical com-
z z z
ponent of the effective turbulent velocity, 0.8
0.6
u˜ =[(u∗)2+4ℓ β hθ2i]1/2 , (1)
z z z
p 0.4
β = g/T∗ is the buoyancy parameter, T∗ is a reference 0.2
value of the mean absolute temperature and g is the ac-
celeration of gravity. This definition of the effective ve- f
0 5 10
locity is derived from the budget equation (8) for the
turbulentkinetic energyinSect.III. Thiseffectiveveloc-
FIG. 11. The r.m.s. of temperature fluctuations θ =
ity takes into account the production of the turbulence rms
phθ2i versus the frequency f of the grid oscillations for the
by buoyancy. Inspection of Fig. 10 shows that the val-
unstably stratified turbulentflow.
ues ofthese ratiosareveryclose. The latter implies that
themeasuredturbulentvelocityintheunstablystratified
turbulent flow, urzms, is of the order of u˜z. III. THEORETICAL ANALYSIS AND
COMPARISON WITH EXPERIMENTAL
The dependence of the r.m.s. of the temperature fluc-
RESULTS
tuations hθ2iversusthefrequencyf ofthegridoscilla-
tions is sphown in Fig. 11. This dependence is caused by
the nontrivialbehaviour of the mean temperature gradi- Let us first analyze the frequency dependence of the
ents along x,y,z axes (see Figs. 3-5), where θ are fluctu- temperature fluctuations. To this end we consider the
ations of fluid temperature. We have also analyzed the budget equation for hθ2i:
experimentalresultsobtainedatthesmallertemperature
Dhθ2i
difference,∆T =25K,betweenthetopandbottomwalls +divΦ =−2(F·∇)T −2ε , (2)
θ θ
Dt
(i.e., smaller Ra), and found only minor changes in the
finalresults. Inthe nextsectionweanalyzethe obtained (see, e.g., [60–63]), where D/Dt = ∂/∂t+U·∇, u are
experimental results. the fluctuations of the fluid velocity, U is the mean ve-
8
locitythatdescribesthecoherentstructures,Φ =huθ2i thespectralτ approximationthatisvalidforlargePeclet
θ
is the third-order moment that determines the flux of numbers.
hθ2i, F = hu θi is the turbulent heat flux, θ are the Now let us discuss the generalization of Eq. (4). To
i i
temperature fluctuations, T is the mean temperature, this end we use the budget equation for the turbulent
ε =Dh(∇θ)2iisthe dissipationrateofhθ2i, whereD is heat flux F =hu θi:
θ i i
the coefficient of molecular temperature diffusion.
DF 1
LetususetheKolmogorov-Obukhovhypothesiswhich i +∇ Φ(F) =β hθ2i+ hθ∇ pi−hu u i∇ T
Dt j ij i ρ i i j j
allows to estimate the dissipation rate ε of hθ2i, as
θ
ε ≈ hθ2i/τ (see, e.g., [64, 65]), where τ = ℓ/u is −(F·∇)U −ε(F) , (5)
θ 0 0 0 i i
the characteristic turbulent time and u is the charac-
0 where β = βe , p are the pressure fluctuations, ρ is
teristic turbulent velocity at the scale ℓ. Indeed, ε ≡ i z
Dh(∇θ)2i = Dhθ2i kbk2E˜ (k)dk ≈ hθ2i/τ , whθere the fluid density, U is the mean velocity, ε(iF) is the
E˜ (k) = (q −1)k−1(Rkk/0k )−qθis the spectrum0function dissipation rate of the turbulent heat flux and Φ(ijF) =
θ 0 0 hu u θi + δ ρ−1hθpi/2 is the term that includes the
of the temperature fluctuations, q is the exponent of the i j ij
third-order moments. According to the estimate made
spectrum of the temperature fluctuations (e.g., q = 5/3
forthe Corrsin-Obukhovspectrum), k =ℓ−1, k =ℓ−1, in[62], βihθ2i+ρ−1hθ∇ipi≈2βihθ2i. Inasteady-state
0 b b case Eqs. (2) and (5) yield:
ℓ = ℓ/Pe1/(3−q) and Pe= u ℓ/D is the Peclet number.
b 0
The latter estimate implies that the main contribution ℓ ∇ T 1
∗ ∗
= =const, (6)
to the dissipation rate εθ arises from very small molec- hθ2i 2CθCF
ular temperature diffusion scales ℓb. The Kolmogorov- p
Obukhovhypothesishasbeenwidelyusedinatmospheric where
turbulenceinanumberofstudiesincludinginvestigations [ℓ ∇ T]2 = (ℓ ∇ T)2+(ℓ ∇ T)2+(ℓ ∇ T)2
∗ ∗ x x y y z z
ofatmosphericstablystratifiedandconvectiveboundary
layers. In a steady-state Eq. (2) yields: ×(cid:2) 1+4CθCFβτ02(∇zT) −1 , (cid:3) (7)
(cid:2) (cid:3)
hθ2i≈−2τ (F·∇)T and CF is an empirical constant. Here we take into
0
account that the dissipation rate of the turbulent heat
≈2 (ℓx∇xT)2+(ℓy∇yT)2+(ℓz∇zT)2 . (3) flux is ε(F) = F /C τ and τ ≈ τ ≈ τ = τ .
(cid:2) (cid:3) i i F 0 x y z 0
In Eqs. (6) and (7) we also have neglected the terms
In deriving Eq. (3) we take into account that the com-
∼O[ℓ3/L3;ℓ3/(L2L )], where L and L are the char-
ponentsoftheturbulentheatfluxandturbulenttemper- T T U T U
acteristic spatialscales of the mean temperature and ve-
ature diffusion coefficients in an anisotropic turbulence
locity field variations. Equation (6) is valid in a general
are given by F = −DT ∇ T and DT = C ℓ hu2i,
x x x x x x x case including the passive scalar regime. In the case of
and similarly for other components F ,F of thepturbu-
y z
the passivescalarregimewherebythereis onlythe verti-
lent heat flux and the turbulent temperature diffusion
coefficients DT,DT with the replacement in the above cal gradient of the mean temperature, Eq. (6) coincides
y z with Eq. (4).
formulas x by y or by z. We also neglected a small
In Fig. 12 we plot the non-dimensional ratio
term divΦ in Eq. (2) for nearly homogeneous turbu-
θ ℓ ∇ T/ hθ2i versus the frequency f of the grid oscilla-
lent convection. When horizontal gradients of the mean ∗ ∗
tions forpthe unstably stratified turbulent flow obtained
temperature are much smaller than the vertical gradi-
in our experiments, where we assumed that ℓ =ℓ and
ents, |∇ T|≪|∇ T|, Eq. (3) yields the following non- x y
x,y z
C C = 1/12. Inspection of Fig. 12 shows that this
dimensional ratio θ F
non-dimensional ratio varies slightly even in the range
ℓ |∇ T| ofparameterswherebythe behaviourofthe temperature
z z
=const. (4)
field is different from that of the passive scalar. Small
hθ2i
deviations of the experimental results from the theoreti-
p
Kraichnan (1968) derived Eq. (4) for a passive scalar calpredictions[seeEq.(6)]maybe causedbyanon-zero
regime using the Lagrangian-history direct-interaction term,divΦθ anddeviationsfromthesteady-statedueto
approximation for small Peclet numbers [66]. Equa- the long-term oscillations of the mean temperature.
tion(4) is inagreementwith the correspondingequation Now let us derive Eq. (1) for the measured turbulent
derived by means of the path integral approach in [67] velocity in the unstably stratified turbulent flow. We
for the passive scalaradvectedby the delta-correlatedin use the budget equation for the turbulent kinetic energy
time Gaussian smooth velocity field. Equation (4) can Ek =hu2i/2:
be also derived using spectrum of temperature fluctua- DE
tions found by dimensionalarguments in [68] and by the k +divΦk =−huiuji∇jUi+βFz +hu·ffi−εk,
Dt
renormalization procedure in [69]. In Appendix we also
(8)
deriveEqs.(3)and(4) using otherprocedures,namely a
quasi-linear approach (or a second order correlation ap- where Φ = ρ−1hupi+(1/2)huu2i is the term that in-
k
proximation) that is valid for small Peclet numbers and cludes the third-order moments, hu·f i is the production
f
9
ℓ∗∇∗T/θrms |Ep|/Ek
3
2
1.5
2
1
1 0.5
0
0 5 10 f
0 5 10 f
FIG. 13. The non-dimensional ratio |E |/E of thepotential
p k
to kinetic energies versus the frequency f of the grid oscilla-
FIG. 12. The non-dimensional ratio ℓ∗∇∗T/θrms versus the tionsfortheunstablystratifiedturbulentflow. Thehorizontal
frequencyf ofthegrid oscillations fortheunstablystratified line corresponds to |Ep|/Ek =0.1.
turbulentflow, where θrms=phθ2i.
(e.g., of the atmosphere as a whole), whereas the tur-
ofturbulencecausedbythegridoscillationsandε isthe bulent potential energy is a local quantity, i.e., it is de-
k
dissipation rate of the turbulent kinetic energy. In the termined in each point of turbulent flow. The buoyancy
steady-state Eq. (8) yields hu2i = 2τ [ν S2 +hu·f i+ term, βF , appears in Eqs. (8) and (10) with opposite
0 T f z
βFz], where νT is the turbulent viscosity. Let us intro- signsanddescribestheenergyexchangebetweenthetur-
ducethecharacteristicvelocityforisothermalturbulence bulentkinetic energyandthe turbulentpotentialenergy.
(u∗)2 = 2τ0[νT S∗2 +hu·ffi], where S∗ is the large-scale These two terms cancel in the budget equation for the
shear that can appear in isothermal oscillating grid tur- total turbulent energy, E =E +E :
k p
bulence. Note that for large frequencies of the grid os-
cillationsthelarge-scalecirculationscausedbyturbulent DE
+divΦ=P −hu u i∇ U +hu·f i−ε, (11)
p i j j i f
convectiondisappearandS →S . Sincetheverticaltur- Dt
∗
bulentheatfluxF =−D ∇ T andforlargefrequencies
z T z whereΦ=Φ +Φ andε=ε +ε . The conceptofthe
of the grid oscillations |∇ T| ∼ hθ2i/ℓ [see Eq. (4)], k p k p
z z total turbulent energy is very useful in analysis of strat-
we arrive at the following equatiopn:
ified turbulent flows. In particular, it allows to explain
urms ∼[(u∗)2+4ℓ β hθ2i]1/2 , (9) the physical mechanism for existence of turbulence for
z z z
p arbitrary values of the Richardson number, and abolish
where we take into account that the r.m.s. of the hori- the paradigm of the critical Richardson number in the
zontal component of velocity fluctuations in the unsta- stably stratified atmospheric turbulence (see [62, 63]).
bly stratified turbulent flow hu2hi ≃ u∗h and u∗h = Forverysmallhorizontalmeantemperaturegradients,
[(u∗)2−(u∗)2]1/2. This is thepreason that the frequency the source term P is much smaller than other terms
z p
dependencies of the ratios u∗z/urzms and u∗z/u˜z obtained in the right hand side of Eq. (11). In Figs. 13 and 14
in our experiments (see Fig. 10), are close. we showthe non-dimensionalratios|E |/E and|P |/ε
p k p k
Using Eq. (2) we obtain the evolutionary equation for versus the frequency f of the grid oscillations for the
the turbulent potential energy Ep =(β2/2N2)hθ2i: unstably stratified turbulent flow. Inspection of Figs. 13
and14showsthatthetemperaturefluctuationsaffectthe
DE
p +divΦ =P −βF −ε , (10) turbulent kinetic energy only in the vicinities of f = 0
p p z p
Dt
and f =5.2 Hz. Note that in the vicinity of f =5.2 Hz,
(see, e.g., [62, 63]), where N2 = β∇ T, Φ = the sign of ∇ T changes from positive to negative (see
z p z
(β2/2N2)Φ , P = −(β2/N2)(F ·∇)T is the source Fig. 3).
θ p h
(or sink) of the turbulent potential energycaused by the In view ofthe aboveanalysis,we consider the passive-
horizontal turbulent heat flux F = hu θi, u is the likescalarbehaviourofthetemperaturefieldinthekine-
h h h
horizontal component of the velocity fluctuations and matic sense. In particular, if the temperature fluctua-
ε = (β2/2N2)ε . The turbulent potential energy is tions hθ2i do not affect the turbulent kinetic energy, the
p θ
analogoustotheavailablepotentialenergyintroducedby temperature field can be considered as a passive scalar.
Lorenz [70]: the available potential energy and the tur- In this case the evolution of the temperature field in
bulent potential energy are proportional to the squared a given turbulent velocity field is a kinematic problem,
temperature fluctuations. The principal difference be- whereby there is no dynamic coupling between the tem-
tweenthese two quantities is that the availablepotential perature fluctuations, hθ2i, andthe turbulent kinetic en-
energy is an integral property of the entire flow-domain ergy in spite of a non-zero turbulent heat flux, i.e., non-
10
flow the ratio (ℓ ∇ T)2 +(ℓ ∇ T)2 +(ℓ ∇ T)2 /hθ2i
|P |/ε x x y y z z
p k
varies slightly (cid:2)even in the range of parameters wh(cid:3)ereby
the behaviour of the temperature field is different from
that of the passive scalar. The experimental results ob-
10
tained in this study are in an agreement with the the-
oretical predictions based on the budget equations for
5 turbulent kinetic energy, turbulent potential energy and
turbulent heat flux.
We havealsofoundthatin the unstably stratifiedtur-
0 bulent flow the vertical component of the turbulent ve-
0 5 10 f locityfieldis determinedbyEq.(9). Thelatter equation
is derivedfromthe budget equationfor the turbulent ki-
netic energy. This implies that the buoyancy heat flux
contributestotheproductionoftheturbulentkineticen-
FIG.14. Thenon-dimensionalratio |P |/ε ofthesource(or
p k
sink)oftheturbulentpotentialenergycausedbythehorizon- ergy in turbulent convection in the absence of the LSC.
talturbulentheatflux,tothedissipationrateoftheturbulent However, the turbulent kinetic energy in large Rayleigh
kinetic energy versus the frequency f of the grid oscillations number turbulent convection with coherent structures
fortheunstablystratifiedturbulentflow. Thehorizontalline (LSC) is produced by shear, rather than by buoyancy
corresponds to |Pp|/εk =1. (see [40]).
The long-term nonlinear oscillations of the mean tem-
peraturehavebeendetectedforallregimes,fromthetur-
zero correlations of the velocity and temperature fluctu- bulentconvectionwith LSC tothe passivescalarregime,
ations. When the effect of the temperature fluctuations at all frequencies of the grid oscillations.
on the turbulent kinetic energy cannot be neglected, the
temperature is considered as an active field. This un-
derstanding of the passive or active behavior of the tem- Appendix A: Temperature fluctuations
perature field is different from that based on the scal-
ing behaviourof the temperature structure function (see Evolution of the instantaneous temperature field
[43]). T (t,r) in a turbulent flow is determined by the fol-
tot
Note that the turbulent convection studied in our ex- lowing equation:
periments is non-Boussinesq, i.e., div u 6= 0. In the
∂T
theoretical analysis we use the anelastic approximation, tot +(vˆ·∇)T +(γ−1)(∇·vˆ)T =D∇2T ,
tot tot tot
∂t
div (ρu)=0. Itshouldbenotedthatthedeviationfrom
(A1)
the Boussinesq approximation in our experiments is less
than 10% (i.e., spatialvariationsofthe fluid density are whereT (t,r)=T+θ, D isthecoefficientofmolecular
tot
δρ/ρ<0.1). temperature diffusion, γ = c /c is the ratio of specific
p v
heats, vˆ is the velocity field that satisfies to continu-
ity equation in anelastic approximation for a low-Mach-
IV. CONCLUSIONS number flow:
∇·(ρvˆ)=0. (A2)
In the present study we have investigated transition
phenomenacausedbytheexternalforcingfromRayleigh- Combining Eq. (A1) and (A2) we obtain the following
B´enard convection with LSC to the limiting regime of equation:
unstably stratified turbulent flow without LSC, whereby ∂T
the temperature field behaves like a passive scalar. The tot +(v·∇)Ttot =D∇2Ttot ,
∂t
external forcing of turbulence is produced by two oscil-
(A3)
lating grids. When the frequency of the grid oscillations
is larger than 2 Hz, the large-scale circulation (LSC) in where v = γvˆ and we take into account that for a low-
theyz-planeinturbulentconvectionisdestroyed. Onthe Mach-number flow without imposed external pressure
otherhand,acomplicatedmeanflowofthespiralformin gradient ∇ρ/ρ≈−∇Ttot/Ttot.
the xz-plane with a non-zero mean vorticity in the y di- AveragingEq.(A3)overensembleofturbulentvelocity
rectionexistswhenthefrequenciesofthegridoscillations fieldweobtaintheequationfortheevolutionofthemean
2 < f < 7 Hz. The destruction of the LSC is accompa- temperature field T(t,r):
nied by a strong change of the mean temperature distri-
∂T
bution. In the central part of the flow the vertical mean +(U·∇)T +∇·F=D∇2T , (A4)
∂t
temperature gradientchangesits directiondepending on
the frequency of the grid oscillations. We have shown whereU=γUˆ, Uˆ isthemeanfluidvelocity,F=huθiis
that for all regimes of the unstably stratified turbulent theheatflux,u=γuˆ anduˆ arethevelocityfluctuations.
