Natural convection for large temperature gradients around 3 a square solid body within a rectangular...

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International Journal of Heat and Mass Transfer xxx (2006) xxx–xxx

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Natural convection for large temperature gradients arounda square solid body within a rectangular cavity

M. Bouafia, O. Daube *

LMEE – Universite d’Evry, CE 1455 Courcouronnes, 91020 Evry cedex, France

Received 6 October 2005; received in revised form 2 May 2006
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A numerical study of natural convection in cavity filled with air has been carried out under large temperature gradient. The flowsunder study are generated by a heated solid body located close to the bottom wall in a rectangular cavity with cold vertical wallsand insulated horizontal walls. They have been investigated by direct simulations using a two-dimensional finite volume numerical codesolving the time-dependent Navier–Stokes equations under the low Mach number approximation. This model permits to take intoaccount large temperature variations unlike the classical Boussinesq model which is valid only for small temperature differences. We wereparticularly interested in the first transitions which occur when the Rayleigh number is increased for flows in cavities of aspect ratioA = 1,2,4. Starting from a steady state, the results obtained for A = 1 and A = 4 show that the first transition occurs through a super-critical Hopf bifurcation. The induced disturbances determined for weakly supercritical regimes indicate the existence of two instabilitytypes driven by different physical mechanisms: shear and buoyancy-driven instabilities, according to whether the flow develops in asquare or in a tall cavity. For A = 2, the flow undergoes a pitchfork bifurcation leading to an asymmetric steady state which in turnbecomes periodic via a supercritical Hopf bifurcation point. In both the cases, the flow is found to be strongly deflected towards onevertical wall and instabilities are found to be of shear layers type.� 2006 Published by Elsevier Ltd.
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1. Introduction

The study of buoyant plumes induced by temperaturedifferences between a fluid and its surroundings is of impor-tance not only to scientific understanding but also to engi-neering practice. Mechanisms of buoyancy-inducedconvection become more complex when there are interac-tions of the flow with solid boundaries. This paper dealswith the analysis of a transitional plane plume originatingfrom a square obstacle immersed inside a rectangular cav-ity. Three aspect ratios have been considered in this study.It should be noticed that all the configurations present areflection symmetry and that the bifurcations which occurwhen the Rayleigh number increases will be analyzedthrough this symmetry breaking.

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0017-9310/$ - see front matter � 2006 Published by Elsevier Ltd.

doi:10.1016/j.ijheatmasstransfer.2006.05.013

* Corresponding author.E-mail address: [email protected] (O. Daube).

Laminar and turbulent plane plumes arising from hori-zontal line sources in infinite or semi-infinite medium havedrawn the attention of many investigations in the past:from classical self-similar solutions [1–3] to linear stabilityanalysis [4–7] for earlier theoretical studies, added to sev-eral experimental investigations carried out during thesame period and relating to plume oscillations [4,6,8–11].

Many observations of laminar plume flows have indi-cated a regular swaying motion in a plane perpendicularto the axis of the source [10,11]. Following these observa-tions, the stability problem of a freely rising plume was firststudied by Pera and Gebhart [4] using the linear stabilitytheory based on quasi-parallel flows. This theory predictsthe amplification characteristics of small periodic distur-bances as a function of frequency b and Grashof numberGrx. The authors obtained a neutral stability curve forPr = 0.7 but failed to find a critical Grashof number. Theyalso made experimental observations using a Mach–Zender

mailto:[email protected]

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Nomenclature

Greek symbols

� = DT/2T0 normalized temperature differencej thermal conductivityl dynamic viscosityp reduced pressureq densitycp, cv specific heat capacitiesa ¼ j

qcpthermal diffusivity

Subscripts and superscripts0 values at temperature difference

* Adams–Bashforth extrapolationh hotc coldA = H/L aspect ratioH height

L widthL0 reference lengthP mean thermodynamic pressureP 0 ¼ q0V 2

0 reference pressure

Pr ¼ l0

q0a0Prandtl number

Ra ¼ 2q0egL30=l0a0 Rayleigh number

t0 = L0/V0 reference timeT non-dimensional temperatureT0 = (Th + Tc)/2 reference temperature

V 0 ¼ l0

q0L0

ffiffiffiffiffiffiRap

reference velocity

u horizontal component of the non-dimensionalvelocity

v vertical component of the non-dimensionalvelocity

2 M. Bouafia, O. Daube / International Journal of Heat and Mass Transfer xxx (2006) xxx–xxx



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interferometer to observe the disturbances as they wereconvected downstream. The experimental results show thatsufficiently high frequency disturbances are stable as theyare convected downstream. Later, calculations of the sameproblem have been carried out by Haaland and Sparrow [5]taking into account non-parallel and higher-order effects ofthe base flow in the linear stability analysis. The authorsobtained a lower branch of neutral curve and then a criticalGrashof number. Their results show that the unstableregion is smaller than that obtained from the quasi-paralleltheory . The stability problem of non-parallel flows werealso investigated by Wakitani [7] using the method of mul-tiple scales. This method considers that the various distur-bance quantities have different amplification rates. Theirresults relating to the amplification rate of disturbanceswithin unstable regions show a substantial deviation fromthat predicted by the quasi-parallel theory.

The determination of the critical Grashof number corre-sponding to the transition from laminar to turbulent in afreely rising plume has been investigated experimentallyby Bill and co-workers [8,10] among others. The authorsdefined a local Grashof number based on the vertical dis-tance along the plume and the heat rate input. The onsetof transition considered as the initial appearance of the tur-bulent bursts, occurred at a Grashof number ofGrx = 5 · 108 according to [10] and at Grx = 11.2 · 108

for [8]. The later attributed the discrepancy to the weakprecision of the interferometer for small local disturbances.Moreover, several studies have examined the interactionbetween buoyant plumes and neighbouring vertical sur-faces and have observed a deflection of plume towardsthe surface [12,13]. This Coanda effect is seen to be a con-sequence of constrained entrainment of the fluid feedingthe plume due to the presence of the vertical surface.

Although there are many studies in infinite or semi-infi-nite medium, few information is available about thermal

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Pplumes in confined geometry where influence of boundariesand initial conditions must be taken into account. Thebehaviour of plumes in confined geometry can be simulatednumerically using direct numerical simulations (DNS)which allows for parametric studies on such flows for engi-neering purposes. The few numerical studies found in theliterature consider flows under the Boussinesq approxima-tion. Among them, one can notice works carried out byDesrayaud and Lauriat [14] where considerable researchefforts have been devoted to unsteady thermal plumes inmany geometric configurations. The authors consideredtwo-dimensional flows generated by a horizontal linesource inside a rectangular cavity with adiabatic sidewallsand cold top and bottom walls. By varying the depth ofimmersion, the aspect ratio and the Rayleigh number, theauthors have shown the existence of three different physicalmechanisms leading to chaotic flows by a succession ofbifurcation points. All the configurations considered showa base flow which consists of two symmetric counter-rotat-ing rolls around an ascending thermal plume arising abovethe line source.

A first scenario is found for aspect ratio equal to 1 anddepths of immersion smaller than the width. The underly-ing mechanism of the onset of instabilities is the destabili-zation of a significant mass of fluid below the line source.The periodic flow appears through a supercritical Hopfbifurcation characterized by a frequency proportional tothe height of the fluid under the source. A second scenariois found only if the aspect ratio A 6 1 and if the layer offluid below the source is small enough and the plumereaches the top wall. In this case, the plume is boundedbelow by a stable conducting layer of fluid. The flow under-goes a supercritical Hopf bifurcation leading to a periodicmotion with a high fundamental frequency. This motion isfollowed by a quasi-periodic regime before a weakly turbu-lent regime arises via an intermittent route to chaos. A

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third scenario is found in cavities of aspect ratio A = 2 andfor depths of immersion of the source greater than thewidth. The symmetric flow undergoes a pitchfork bifurca-tion leading to a deflection of the plume towards one ver-tical wall. The symmetry breaking is driven by thedestabilization of an upper unstable layer of motionlessfluid above the plume. An increase of the Rayleigh numberleads to a periodic motion with a very low frequency via asubcritical Hopf bifurcation.

Two-dimensional convection flows have also been inves-tigated numerically by Zia et al. [15] in a differentiallyheated cavity containing a heat source above the bottomsurface of the cavity, and by Horvat et al. [16] for a low-vis-cosity flow with zero thermal diffusivity. In this paper, ourapproach is close to the one used by Desrayaud and Lau-riat [14]. The difference lies in taking into account largetemperature differences between the source and the verticalwalls of the cavity. Because of the large density variationinvolved, the usual Boussinesq approximations are nolonger valid and the compressible Naviers–Stokes equa-tions have to be considered. However, since these equationscontain acoustic waves, the numerical stability criterion istoo much restrictive because of the wide disparity betweenthe time scales associated with convection and the propaga-tion of acoustic waves. A solution to this problem consistsof the filtering of these waves using the low Mach approx-imation of the Navier–Stokes equations [18]. This modeldecouples pressure fluctuations from density fluctuationsmaking the resulting equations similar to the incompress-ible case. In particular, acoustic waves are excludedwhereas gravity waves may still occur.

The Low-Mach Number equations (LMN) have beenused in many studies to investigate the effects of large tem-perature variations in natural convection flows [19–21] andalso for near-critical fluids [22,23]. In the present study, theLMN equations supplemented by the state equation forideal gases are used to investigate the flows in 2D-cavitiesof aspect ratio A = 1,2,4. The plumes arise from a hotsquare solid immersed in the vertical centerline of the cav-ity which have isothermal vertical walls and insulated hor-izontal walls. This paper is organized as follows. The nextsection recalls the governing equations and the numericalmethod. Section 3 is devoted to results obtained in steadyand oscillatory regimes for each configuration. The paperends with a summary of major conclusions.

2. Problem formulation

2.1. Governing equations

Consider a two-dimensional cavity of width L andheight H filled with a viscous fluid. The gravity is directeddownwards the y-axis. A solid body of square section isimmersed on the vertical centerline of the cavity close tothe bottom wall. Its dimensions in horizontal and verticaldirections are lx = ly = L/8. The body is considered as aheating source with a uniform temperature Th. The vertical

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walls of the cavity are maintained at the cold temperatureTc (Tc < Th), while the top and the bottom walls are ther-mally insulated. The working fluid is air which is initiallyat a uniform temperature T0 = (Th + Tc)/2 and pressureP0. It is assumed to be an ideal gas with constant specificheats cp and cv of ratio c = 1.4. Its dynamic viscosity land thermal conductivity j are allowed to depend ontemperature.

Since we are primarily interested in flows induced bylarge temperature differences, we have adopted the lowMach number model described by Chenoweth and Pao-lucci [19] and used later by Le Quere [20]. In this model,the complete Naviers–Stokes equations are expanded inpowers of the small parameter M2 (M is the Mach num-ber). Paolucci [18] showed that the pressure may be dividedinto two parts: a mean thermodynamic pressure whichdepends only on time and a dynamic pressure not coupledwith the density fluctuations preventing the propagation ofthe acoustic waves. The equations are made dimensionlessby scaling length, time, temperature, pressure and velocityby reference quantities: L0 = L, t0 = L0/V0, T0 = (Th +Tc)/2, P 0 ¼ q0V 2

0, V 0 ¼ l0

q0L0


. The thermophysical

properties (density, dynamic viscosity, thermal conductiv-ity, thermal diffusivity) are scaled by q0, l0, j0, a0 wherethe subscript 0 denotes values at the reference temperatureT0. Finally, according to Le Quere et al. [20], the governingequations in the dimensionless form are

oqotþr � ðqVÞ ¼ 0 ð1Þ

qoV

otþ ðV � rÞV

� �¼ �rPþ ðRaÞ�1=2r � ��s� Pr�1 q� 1

2ey

ð2Þ

qoTotþ ðV � rÞT

� �¼ 1

Pr Ra1=2r � ðjrT Þ þ c� 1

cdPdt

ð3Þ

P ¼ qT ð4Þ

Note that we have five equations for six unknowns q, u, v,T, P and P . The necessary additional equation is given bycalculating the term dP=dt . Mass conservation equation(1), combined with energy equation (3) and state equation(4) permit to write the divergence of velocity in the form gi-ven by Eq. (5). When integrated over the fluid domain, thisequation leads to Eq. (6) for dP=dt

r � V ¼ r � ðjrT ÞPr Ra1=2P

� 1

cP

dPdt

ð5Þ

dPdt¼ c

A Pr Ra1=2

ZC

joTon

dS ð6Þ

V is the velocity vector of components (u,v), ��s ¼lðrV þ ðrVÞt � 2

3ðrVÞI the viscous stress tensor, P ¼

ðP � P þ q0gyÞ=q0V 20 the reduced pressure, P ðtÞ the mean

thermodynamic pressure, e = DT/2T0 the normalized tem-perature difference where DT is the temperature difference(Th � Tc) between the obstacle and the sidewalls andA = H/L0 the aspect ratio. The independent dimensionless

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parameters are the Prandtl number and the Rayleigh num-ber defined at the reference temperature T0

Pr ¼ l0

q0a0

; Ra ¼ 2q0egL30=l0a0 ð7Þ

where a0 = j0/q0cp is the thermal diffusivity. The system ofequations is closed by Sutherland’s law for the dynamic vis-cosity l

lðT Þ ¼ T 3=2 1þ Sl

T þ Slð8Þ

and since the Prandtl number is assumed constant in thisstudy, the thermal conductivity j is given by

jðT Þ ¼ lðT Þ ð9Þ

where Sl = 0.368 for e < 0.6 and T0 2 [210 K, 673 K]. Thecomputational domain is defined for a rectangular cavityX = [0, 1] · [0, A]. At the boundaries, the no-slip conditionsfor velocities are coupled to temperature conditions

T ¼ 1� e at x ¼ 0

T ¼ 1� e at x ¼ 1

T ¼ 1þ e on the obstacleoTon ¼ 0 at y ¼ 0 and y ¼ H=L

8>>><>>>:

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2.2. Numerical modelling

The governing equations are solved numerically by afinite-volume scheme using the usual staggered arrange-ment on a non-uniform grid. The resulting mesh leads toa grid layout strongly refined close to the source and thewalls of the cavity. The time derivative is approximatedusing a second order backward Euler scheme in whichthe diffusive and viscous linear terms are implicitly treatedwhile the convective nonlinear terms are explicitly treatedusing an Adams–Bashforth extrapolation. For instance,the fully time discretized formulations for the energy equa-tion read

qn 3T nþ1 � 4T n þ T n�1

2Dtþ ðV :rT Þ�

� �

¼ Ra�1=2rðj�rT nþ1Þ þ c� 1

c

� �dP �

dt

where * refers to Adams–Bashforth extrapolation (f* =2fn � fn�1).

A similar approach is then applied to the momentumequations. This process results in Helmholtz-type equa-tions for Tn+1, un+1 and vn+1

ðrT I�r � j�rÞT nþ1 ¼ Sn;n�1T ð11Þ

ðrV I�r � lnþ1rÞunþ1 ¼ �ffiffiffiffiffiffiRap

rPnþ1 þ Sn;n�1u ð12Þ

ðrV I�r:lnþ1rÞvnþ1 ¼ �ffiffiffiffiffiffiRap

rPnþ1 þ Sn;n�1v ð13Þ

r � V ¼ Sn;n�1D ð14Þ

PR

OO

F

where I is the identity tensor, rT ¼ 3qnPrffiffiffiffiffiffiRap

=2Dt andrV ¼ 3qnþ1


=2Dt . Sn;n�1T , Sn;n�1

u , Sn;n�1v and Sn;n�1

D aresource terms depending on time steps n and n � 1.

2.3. Time marching procedure

Assuming the flow is known at time levels (0, . . . ,n), thedifferent stages to compute the flow at (n + 1) arerespectively:

• Determination of the temperature field from (3) usingextrapolated values of j* and ðdP=dtÞ�:

• Calculation of the new physical properties jn+1 andln+1.

• Calculation of P nþ1 using mass conservation (see [20])and of ðdP=dtÞnþ1 using Eq. (6). The divergence of veloc-ity Snþ1

D in Eq. (13) can then be computed at time level(n + 1).

• Calculation of qn+1 by using Eq. (4).• Resolution of the momentum equations by using a ‘‘pro-

jection’’ method derived from the classical projectionmethods for incompressible flows [24]:

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• Prediction. Computation of a provisional velocity fieldsatisfying the following equation over the computationaldomain X of boundary C:

ðrV I�r � lnþ1rÞV� ¼ �Ra1=2rPn þ Sn;n�1 ð15Þ• Projection. Computation of an auxiliary function /

satisfying

qnþ1ðVnþ1 � V�Þ ¼ rU in X

r � Vnþ1 ¼ Dn;n�1 in X

Vnþ1 � n ¼ 0 on C

8><>: ð16Þ

This leads to the resolution of the diffusive problem

r � ð 1qnþ1rUÞ ¼ Sn;n�1

D �r � V� in XoUon ¼ 0 on C

(ð17Þ

The resolution of the Poisson equation (16) is donethrough a multigrid acceleration procedure. The veloc-ity V and the pressure P are then calculated from theexpressions

Vnþ1 ¼ Vn þ 1

qnþ1rU ð18Þ

Pnþ1 ¼ Pn � 3U2DT

ð19Þ

2.4. Validation of the code

The code has been tested in some relevant benchmarkcases corresponding to the non-Boussinesq convection ina differentially heated square cavity [29]. Tests have beencarried out for a normalized temperature differencee = 0.6 and for Ra = 106 and 107. For these values of theparameters, the flow tends to a steady state. Two cases

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have been studied, the first for constant fluid properties, thesecond using the Sutherland’s law for viscosity and a con-stant Prandtl number. Results obtained by our code reveala very good agreement with the reference solutions result-ing from this benchmark.

The present study is devoted to investigate unsteadyflows generated by a square source located in the mid-planeof cavities with aspect ratios A = 1,2,4. The influence ofthe grid size has been analyzed by testing several non-uni-form meshes. For all geometries, the grid layout wasrefined close to the walls and around the heat source byusing Chebychev collocation points in both horizontaland vertical directions. After a number of trial numericalexperiments, a (296 · 296) grid for the square cavity wasconsidered as a good compromise between stability oraccuracy requirements on one hand and computationalcosts on the other hand.

The influence of the time step has also been considered.Different values were tested by comparing the solutions interms of amplitude and frequency in periodic regimes.When considering both these aspects as well as the relativelylong computer simulations times involved, it was finallydecided to use a time step Dt = 0.004 for values of Ra closeto the first bifurcation point (Ra < 7 · 106 in square cavity).We have verified that decreasing the time step to Dt = 0.002gave a decrease of amplitude of 4% and an increase of fre-quency of 1.5% for Ra = 6.8 · 106. When decreasing morethe time step from Dt = 0.002–5 · 10�4, the results in termsof frequency and amplitude remain constant.

For the rectangular cavity of aspect ratio A = (H/L) = 4, a similar study leads us to take a (296 · 440) non-uniform grid. Using this mesh, the flow was found to beperiodic at Ra between 1.2 · 104 and 1.3 · 104. In view ofthe moderate values of the Rayleigh number at which insta-bilities occurs, time steps used are Dt = 0.01 forRa < 1.4 · 104. For the cavity with aspect ratio A = 2, thecomputations have been performed with a (296 · 296)non-uniform grid and time steps varying from 0.007 to0.004 for values of Ra within the range [4 · 104, 1.5 · 106].

2.5. Methodology

The numerical experiments were carried out as follows.For each considered value of the aspect ratio, the cavity isfilled with a fluid at uniform reference temperature T0 =(Th + Tc)/2. At instant t = 0, the body and the lateral wallsare set to their final temperatures Th and Tc and a firstcomputation is performed for a sufficiently low value ofthe Rayleigh number in order to reach a steady flow. Thissteady state is in turn used as an initial condition for a sub-sequent computation at a larger value of Ra and the proce-dure is resumed for increasing values of the Rayleighnumber. When the flow becomes unsteady, its temporalcharacteristics are analyzed from time series of temperatureand velocity values at selected sampling points.

It must be noticed that all the studied configurationspresent a reflection symmetry with respect to the vertical

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centerline of the cavity. The bifurcations which occur whenthe Rayleigh number increases will thus be analyzedthrough this symmetry breaking. When it was found neces-sary, a symmetric base flow (possibly unstable) has beencomputed by considering the half of the domain andimposing symmetric boundary conditions on the centralvertical axis.

3. Results and discussion

The parameters governing the plume formation are theRayleigh number, the aspect ratio A, the location of theheating source, the normalized temperature difference eand the initial conditions. A quite extensive analysis wouldbe needed to cover all the possible effects of each parame-ter. In this study, the following values were maintainedconstant for all the configurations. The source center islocated at position Ms(0.5,0.25) and the normalized tem-perature difference is e = 0.2. This value corresponds toan actual temperature difference DT = 120 K between thesource and the vertical walls and to a reference temperatureT0 = (Th + Tc)/2 = 300 K. This temperature difference ismuch larger than the commonly admitted limits of validityfor the Boussinesq approximation which is about 30 K forair in the standard atmospheric conditions. To validate thispoint, Boussinesq computations reported in Section 3.2.2show that the nature of the first bifurcation is stronglyaffected by non-Boussinesq effects. However, we did notmake an extensive study of larger values of � though thebenchmark for the differentially heated cavity was com-puted for � = 0.6.

In all the considered configurations, the expected baseflow consists first of a hot fluid rising above the heatingsource in the form of an ascending thermal plume. In orderto satisfy the zero mass flux condition, the flow thenreturns to the source forming two counter-rotating recircu-lations movements. In addition, secondary flows may occurdepending on the aspect ratio as will be shown later.

3.1. Cavity A = 1

3.1.1. Symmetric steady flow

Flow structure: For sufficiently low values of Ra, theflow reaches an asymptotic steady state exhibiting a sym-metric motion about the vertical centerline of the cavity.Typical velocity fields and isotherms are displayed inFig. 1 for Ra = 5 · 106. The flow pattern is characterizedby a primary flow which consists of two counter-rotatingrecirculations flows delimited by a vertical thermal plume.In this primary flow (zone A in Fig. 1(a) and (b)), the hotfluid rises above the source until it reaches the top wall,then moves outwards along the horizontal wall beforemoving downwards along the sidewalls inducing boundarylayers in which it is cooled. When it arrives at the bottomwall, the fluid returns to the source, almost horizontallyto it. The core of the primary flow is stratified in its lowerpart (zone B in Fig. 1(a) and (b)) and contains two small

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A

A

B B

C C

D D

(a)

A

A

A

A

A

B B

C C

D

(b)

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-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Vel

ocity

v

Horizontal position

Ra=5.×106 ; A=1

y=0.66 y=0.48 y=0.41

(c)

0.8

0.85

0.9

0.95

1

1.05

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Tem

pera

ture

Horizontal position

Ra=5.×106 ; A=1

y=0.66 y=0.48 y=0.41

(d)

Fig. 1. (a, b) Velocity field and isotherms at steady state in the square cavity for Ra = 5.0 · 106, (c, d) the corresponding horizontal profiles of verticalvelocity and temperature.




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(zone C in Fig. 1(a) and (b)). The primary flow delimits afluid layer below the source which is almost at rest and uni-form in temperature. Moreover, the primary flow is distin-guished by marked sheared zones located in the plume andthe boundary layers. These shear layers characterized bylarge velocity gradients are clearly visible in the horizontalprofiles of temperature and vertical velocity plotted for var-ious heights above the source (see Fig.1(c) and (d)).

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-0.1

-0.05

0

0.05

0.1

0.15

2500 2520 2540 2560 2580 2600

u

Time

Ra=6.8×106 ; A=1

y=0.66 y=0.83

(a)

Fig. 2. (a) Time evolutions of the horizontal velocity at points M1(0.5,0.6M2(0.5,0.83).

3.1.2. Unsteady periodic flow

The flow becomes periodic beyond a critical value of theRayleigh number located in the range [5.5 · 106, 6 · 106].For instance, Fig. 2(a) shows the time evolutions of thehorizontal component u of the velocity for Ra = 6.8 · 106

at two points on the vertical centerline. These signals areclearly periodic and a Fourier analysis yields a fundamen-tal frequency f1 = 0.197 with its two first odd harmonics

1e-06

1e-05

0.0001

0.001

0.01

0.1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

u

Frequency

Ra=6.8×106 ; A=1

f1

3f1

5f1

(b)

6) and M2(0.5,0.83) and (b) power spectrum of the u-velocity at point

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(Fig. 2(b)). This behaviour is related to the breaking of thereflection symmetry associated with the bifurcation.

Moreover, computations not reported here have shownthat the closer the Rayleigh number is to its critical value,the longer is the time for attaining the asymptotic flow.This suggests that the onset of unsteady solutions is dueto a supercritical Hopf bifurcation. This is confirmed bythe fact that no hysteresis effect could be found when theRayleigh number was decreased from Ra = 6 · 106 toRa = 5 · 106.

Flow structure: This transition is characterized by abreaking of the reflection symmetry which induces a sway-ing motion of the thermal plume. To illustrate this behav-iour, results for Ra = 6.8 · 106 are presented. Typicalinstantaneous velocity fields are shown in Fig. 3(b)–(f)evenly distributed over one time period. From these plots,it can be seen that the upper part of the rising plume oscil-lates horizontally once to the right once to the left side ofthe cavity. During one period, the secondary upper rollsinside the primary flow (see Section 3.1.1), break into twosmaller counter-rotating rolls and afterwards merge again(see Fig. 3(b), (d) and (f)). The fluid layer below the sourceremains almost at rest although the convective motionextends slightly to the bottom of the cavity. Time averagedvelocity field showing a symmetric pattern about the verti-cal centerline, is plotted in Fig. 3(a) by integrating instan-taneous fields over a sufficiently long time (400 non-dimensional time units).

Base flow and flow fluctuations: In order to examine theinstability mechanisms, we have first calculated the sym-metric unstable base flow by imposing symmetry boundaryconditions on the vertical centerline and then computed thetemporal fluctuations with respect to this base flow by sub-

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(a) mean flow (b) t = t

(d) t = t0+T/2 (e) t = t0+

Fig. 3. (a) Mean velocity field, (b–f) instantaneous fields a

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tracting the symmetric base solutions from instantaneousfields. Before considering the fluctuations, it may benoticed that the symmetric unstable base flow is slightlydifferent from the symmetric time averaged flow.

The spatial distributions of the fluctuations are dis-played in Fig. 4 for Ra = 6.8 · 106 at eight instants overone oscillation period. The temperature field correspondingto the base solution is also presented in Fig. 4(e). Theseplots indicate that the maximum of disturbances are con-fined within the plume and to a lesser extent in the bound-ary layers and the core of the primary flow. Thedisturbances consist of structures of alternate sign distrib-uted simultaneously in the two halves of the cavity whereasthe layer of fluid below the source remains still stable.

The instabilities which occur in zones with large gradientvelocity with an inflection point may be identified withshear instabilities and move along the primary flow direc-tion. They first arise just above the source in the form ofdetached blobs, then are convected in the plume until theupper surface is reached. At the top wall, they spread outhorizontally to the left and to the right, then sink alongthe sidewalls before vanishing when approaching the bot-tom wall. The results for Ra = 6.8 · 106 indicate a rangeof temperature fluctuations between [�0.03, +0.03] withmaximum values located within the plume.

3.2. Cavity A = 2


As usual, a symmetric steady flow is found for low val-ues of the Rayleigh number Ra, in this case forRa < 4.5 · 104. The main characteristics of the flow forRa = 4.0 · 104 are deduced from Fig. 5(a)–(c) in which typ-

0 (c) t = t0+T/4

3T/4 (f) t = t0+T

t five instants over one time period for Ra = 6.8 · 106.

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506507508

509510511512513514515516517518519520521

522523

524525526527528529530531532533534535536537538539540

(a) t = t0 (b) t = t0+T/8 (c) t = t0+T/4

(d) t = t0+3T/8 (e) base flow (f) t = t0+5T/8

(g) t = t0+3T/4 (h) t = t0+7T/8 (i) t = t0+T

Fig. 4. (a–d) and (f–i) Contours of temperature fluctuations at eight instants over one time period, (e) base flow temperature; thin (thick) lines for positive(negative) values; variation range [�0.03,+0.03] for Ra = 6.8 · 106.




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played, and from Fig. 5(d) and (e) in which horizontal pro-files of temperature and vertical velocity Vy are plotted.

• The flow consists essentially of two symmetric recircula-tion zones delimited by an ascending thermal plumewhich extends up to the upper wall of the cavity(Fig. 5(b)). There exists a small layer of quiet fluid justbelow the heating source in which heat transfer is essen-tially conductive.

• Unlike the square cavity case, no stratification may beobserved in the flow (Fig. 5(c)).

• The temperature and v profiles are typical of shear layersdeveloping along the sidewalls, at least in the upper part(y > 1). These layers are rather thick because of the lowvalue of the considered Rayleigh number (see Fig. 5(d)and (e)).

3.2.2. Asymmetric steady flow

When the Rayleigh number is increased beyond a criti-cal value Rac, the steady symmetric flow looses its stability,yielding an asymmetric steady solution. This transitionappears to be a pitchfork bifurcation whose threshold liesat a critical value within the range [4.5 · 104, 5 · 104].

Flow structure: The structure of the bifurcated flow isillustrated in Fig. 6(a)–(c) by velocity, temperature fieldsand streamlines for Ra = 1.1 · 105. Two large recirculationzones may still be observed, but they are no longer symmet-ric with respect to the central axis. The main point is theonset of the well known Coanda effect which induces adeflection of the thermal plume towards one or the othersidewall, depending on the initial conditions. To check thispoint, the symmetric steady flow – within roundoff andconvergence errors – for Ra = 5 · 104 was flipped withrespect to the vertical centerline, yielding another initial

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(a) (b) (c)

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0.2

0.4

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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ocity

v

Horizontal position

Ra=4×104 ; A=2

y=1.15 y=0.72 y=0.56

(d)

0.8

0.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Tem

pera

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Horizontal position

Ra=4×10 4 ; A=2

y=1.15 y=0.72 y=0.56

(e)

Fig. 5. (a–c) Velocity field, streamlines and isotherms at steady state in cavity A = 2 for Ra = 4. · 104, (d) and (e) the corresponding temperature andvertical velocity profiles.

(a) (b) (c) (d)

Fig. 6. Asymmetric steady state for Ra = 1.1 · 105 in cavity A = 2. Velocity field (a), temperature contours (b), streamlines (c, d) for two different initialconditions.




Uconditions for computations at Ra = 1.1 · 105. The asym-metric steady state which was obtained for this value ofRa can be deduced from the previous one by symmetry(Fig. 6(d)).

From a more physical point of view, we would like toemphasize two points:

• After a slight deviation towards one of the sidewalls, theupper end of the thermal plume moves back up to thecenter of the upper horizontal wall. When the Rayleigh

number is increased, this behaviour evolves towards aconfiguration exhibiting a strongly deflection leadingto an instability process (see Section 3.2.3).

• The streamline pattern around the heating source showsthat the obstacle is fully included in the largest recircu-lation zone (on the left side of the figure).

Nature of the bifurcation: By gradually decreasing theRayleigh number from Ra = 1.1 · 105 to Ra = 4. · 105,no hysteresis effect could be evidenced which is a first indi-

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560561562563564565566567568569570571572573574575576577578579580581582583584585586587588589590591592593594595596597598

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606607608609610611612613

614615

0

0.01

0.02

0.03

0.04

0.05

4x104 5x104 6x104 7x104 8x104 9x104

a2

Rayleigh number

Fig. 8. Squared amplitude of the horizontal component u of velocityversus the Rayleigh number at point M3(0.5,0.56) in asymmetric steadyregime.




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cation that the bifurcation is probably supercritical. Toconfirm this point, the steady asymmetric flow was com-puted for a value Ra = 5.0 · 104 slightly above Rac andthe unstable symmetric steady base flow for this same valueof Ra was also computed. The comparison between the twoflows are displayed in Fig. 7, where contours of the temper-ature difference and plots of the velocity differences are dis-played. The resulting perturbation fields clearly exhibit asymmetry opposed to the symmetry of the unstable baseflow as it could be expected since the bifurcation breaksthe natural symmetry of the problem. However, when Ra

is slightly increased and kept within the steady range, thisopposite symmetry is quickly no more visible. We believethat this is due to the competition with another instabilitymode similar to the mode that governs the first transition inthe cavity with aspect ratio A = 4. However, more workwould be needed to clarify this issue.

Finally, the variation of the squared amplitude of thehorizontal velocity u with respect to the Rayleigh numberRa at point M3(0.5,0.56) is plotted in Fig. 8. The linear pat-tern of the plot confirm the supercritical nature of thepitchfork bifurcation and allows us to compute by linearextrapolation an estimation of the critical value Rac ofthe Rayleigh number as Rac = 4.8 · 104.

Comparison with the Boussinesq case: The goal of thissection is to evidence the necessity of considering non-Boussinesq effects. For this purpose, calculations underthe Boussinesq approximation have also been carried outin the A = 2 configuration. Starting from Ra = 4 · 104

and gradually increasing the Rayleigh number, the firstbifurcation point corresponding to the transition from asymmetric steady state to an asymmetric steady stateappears at a critical Rayleigh number in the range7.0 · 104 < Rac < 7.5 · 104, therefore yielding a criticalvalue larger than in the non-Boussinesq case. Probablymore important is the onset of hysteresis effects in the vicin-ity of this point, revealing the subcritical nature of the tran-sition unlike in the non-Boussinesq case. To illustrate thispoint, a symmetry indicator IT for the temperature (equal

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Fig. 7. (a) Temperature field at asymmetric steady state, (b) the symmetricdisturbances for Ra = 5.0 · 104 in cavity A = 2, thin (tick) lines for positive (n

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with respect to the vertical centerline) was defined as

IT ¼Z Z

XðT ðx; yÞ � T ð1� x; yÞÞ2 dxdy

The variations of this indicator for increasing Ra and fordecreasing Ra are displayed in Fig. 9. Two main featuresare observed:

• When Ra is increased, the symmetric steady solutionjumps on a branch with finite amplitude consisting inan asymmetric solution, confirming the subcritical nat-ure of the bifurcation.

• When Ra is decreased from a point on the finite ampli-tude branch, the asymmetric solution persists for valuesof Ra down to 6 · 104, yielding multiple solutions in afinite range of Ra.

These results clearly show that even for relatively low val-ues of �, taking into account non-Boussinesq effects may benecessary.

(d) (e)

base flow, (c) temperature disturbances, (d) u-disturbances and (e) v-egative) fluctuations.

T617618619620621622623624625626627628629630631632633634635636637

638639640641642643644645646647648649650651652653654655656657658659660661662663664665666667668669670671672673674675676677

1e-10

1e-09

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

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0.1

6.0x104 6.5x104 7.0x104 7.5x104 8.0x104 8.5x104

Sym

met

ry in

dica

tor

Rayleigh number

Fig. 9. Variation of the symmetry indicator IT of the temperature fieldversus the Rayleigh number in the Boussinesq case.




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3.2.3. Unsteady periodic flow

The asymmetric steady solutions persist for values of Ra

up to 1.1 · 106. Beyond this value, the flow becomes peri-odic. For instance, Fig. 10(a) shows time evolution of thehorizontal component u at the sampling pointM3(0.5,0.56) for Ra = 1.3 · 106. It turns out that the signalis monochromatic. This is confirmed by its power spectrumwhich exhibits a fundamental frequency f1 = 0.145 and itsfirst and second harmonics, as it could be expected dueto the asymmetric nature of the base flow (see Fig. 10(b)).

When decreasing the Rayleigh number from Ra = 1.3 ·106 to Ra = 1.1 · 106, the flow reverts to a steady andasymmetric state indicating a supercritical Hopfbifurcation.

Flow structure: For this same Rayleigh number Ra =1.3 · 106, typical instantaneous velocity, streamlines andtemperature fields are displayed in Fig. 11(a)–(c). It shouldbe pointed out that the thermal plume remains deflectedtowards one opposite vertical wall and its oscillatorymotion develops around a deflected mean position. Inaddition, it can be noticed that the thermal plume behaves

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0.01

0.02

0.03

0.04

0.05

0.06

350 400 450 500

u

Time

Ra=1.3×106 ; A=2

(a) (b

Fig. 10. (a) Time evolutions of the horizontal component u at point M3(0

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like an oblique jet impacting on a vertical wall. After theplume impacts on the right wall, the flow is distributedon each side of the impact. One part of the flow movestowards the upper wall inducing a secondary small eddyin the right top corner of the cavity while the other partmoves down along the vertical wall. These two flows returnthen to the source, forming two primary recirculationzones, one of them including the heating source.

Temperature and vertical velocity profiles at differentheights above the source are plotted in Fig. 11(d) and (e).These plots clearly exhibit the development of shear layersin the plume and near the top and vertical walls. This phe-nomenon is simply the continuation of the trend that wasobserved in the previous sections, the shear layers becom-ing thinner and thinner.

Mean flow and fluctuations: The time averaged flow forthis same Rayleigh number was computed by integratinginstantaneous solutions over a sufficiently long time. Thestructure of this mean flow is similar to that one of theinstantaneous flow. In particular, it exhibits a thermalplume impacting on one lateral wall (see Fig. 12(a)).

The fluctuating fields were computed by subtracting themean field from instantaneous fields. Spatial distributionsof these fluctuations are displayed in Fig. 12 at fourinstants evenly spaced over one time period. The tempera-ture fluctuations are essentially located within the plume,near the top wall and along the vertical right wall towardswhich the plume is deviated. The disturbances consist ofpositive and negative elongated structures distributed inthe shear layers. The maximum amplitude of these fluctua-tions varying in the range [�0.01,0.01] are located withinthe plume at positions close to the source. Moreover, onecan notice that the left recirculation zone of the primaryflow is more stable than the right one. The fluctuations fillthe full domain of the right recirculation roll, whereas theyare essentially distributed at the top wall and along theplume in the largest left one.

Instability mechanisms: Examination of the instanta-neous streamlines suggests that the instability arises fromthe impact of the thermal plume on the sidewall. Actually,

1e-06

1e-05

0.0001

0.001

0.01

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0 0.2 0.4 0.6 0.8 1 1.2

u

Frequency

Ra=1.3×106 ; A=2

f1

2f1

3f1

)

.5,0.56), (b) power spectrum of u for Ra = 1.3 · 106 in cavity A = 2.

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Vel

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v

Horizontal position

Ra=1.3×106 ; A=2

y=1.15 y=0.72 y=0.56

(d)

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0.85

0.9

0.95

1

1.05

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Tem

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Horizontal position

Ra=1.3×106 ; A=2

y=1.15 y=0.72 y=0.56

(e)

Fig. 11. (a–c) Velocity field, streamlines and isotherms at periodic state in cavity A = 2, (d, e) the corresponding temperature and vertical velocity profilesfor Ra = 1.3 · 106.

(a) mean temperature (b) T0 (c) T0+T/4 (d) T0+T/2 (e) T0+T/3

Fig. 12. (a) Mean temperature field, (b–e) temperature fluctuations for Ra = 1.3 · 106 at four equally spaced times; thin (tick) lines for positive (negative)fluctuations, variation range [�0.01,+0.01], cavity A = 2.




we have checked that, in steady asymmetric case, thedeflection of the plume grows when the Rayleigh numberincreases. This phenomenon persists until the plumereaches the wall. The plume then splits in two parts asdescribed in the previous section, leading to the instabilityprocess.

3.3. Cavity A = 4


An asymptotic symmetric steady flow is found for Ray-leigh numbers lower than Ra = 1.3 · 104. Typical velocityand temperature fields are displayed for Ra = 1.2 · 104 in

689690691692693694695696697698

699700701702703704705706707

708709710711712713714715716717718719720721722723724725726727




Fig. 13(a) and (b). As seen in the previous cases, two sym-metric recirculation flows delimited by an ascending ther-mal plume can be observed. However, the plume in thiscase, does not extend to the top wall and a separated shearlayers regime is no longer observed. Horizontal profiles oftemperature and vertical velocity at three vertical positionsabove the source for the same Rayleigh number are dis-played in Fig. 13(d) and (e). These profiles are similar tothose of Poiseuille-type flows except in the upper part ofthe cavity where the fluid is at rest.

3.3.2. Unsteady periodic motion

When increasing the Rayleigh number beyondRa = 1.2 · 104, the flow looses its stability. The transitionoccurs at a critical value Rac in the range [1.2 · 104,1.3 · 104]. As it could be expected because of the larger dis-tance between the horizontal walls, the flow becomesunstable for lower values of the Rayleigh number than inthe previous cases A = 1 and A = 2. Time evolutions ofthe horizontal component u indicate that the flow under-

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0.2

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Vel

ocity

v

Horizontal position

Ra=1.2×104 ; A=4

y=3.05 y=2.15 y=0.85

(c)

Fig. 13. (a, b) Velocity field and isotherms at steady state in cavity A = 4,Ra = 1.2 · 104.

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goes a transition to a periodic state. The power spectrumof the horizontal velocity u indicates a very low value(f1 = 0.05) of the frequency and the absence of any har-monics (see Fig. 14).

Flow structure: Typical instantaneous velocity and tem-perature fields are displayed for this same Rayleigh numberin Fig. 15(a) and (b) and Fig. 15(f) and (g) at two instantstaken over one time period. We can see that the swayingmotion of the plume is of sinusoidal type as in the caseof a freely rising plume. Because of the low Rayleigh num-ber, the amplitudes of oscillations are very weak.

Mean and base flow: The time averaged temperature andvelocity fields are displayed in Fig. 15(c) and (h). Notewor-thy is the fact that the mean velocity field exhibits a slightlyasymmetric behaviour in the middle part of the cavitywhereas a possible asymmetry of the temperature field isbarely visible.

To clarify this point, it was found necessary to computethe unstable symmetric steady base flow and to examine thedifferences between this base flow and the time averaged

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pera

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Horizontal position

Ra=1.2×104 ; A=4

y=3.05 y=2.15 y=0.85

(d)

(d, e) the corresponding temperature and vertical velocity profiles for

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0.08

1600 1650 1700 1750 1800 1850

u

Time

Ra=1.4×103 ; A=4y = 1.23y = 3.05

(a)

0.0001

0.001

0.01

0.1

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u

frequency

Ra=1.4×104 ; A=4f1

(b)

Fig. 14. (a) Time evolutions of the horizontal velocity at points M4(0.5,1.23) and M5(0.5,3.05), (b) power spectrum of the u-velocity at point M(0.5,2.15).

Fig. 15. (a, b) Instantaneous velocity fields at instants t0 and t0 + T/2, (c) mean velocity field, (d) difference between the mean and base flow, (e) base flowvelocity, (f, g) instantaneous temperature fields, (h) mean temperature, (i) difference between the mean and base flow in the range [�0.007,0.004], (j) baseflow temperature; Ra = 1.4 · 104 and cavity A = 4.




728729730731732733734735736737

738739740741742743744745746747




flow. These differences respectively are displayed inFig. 15(d) for the velocity field and in Fig. 15(i) for the tem-perature field. The velocity field exhibits a slight asymmet-ric behaviour whereas no asymmetry is visible for thetemperature. In Fig. 16(a) and (b) horizontal profiles ofthe u component are plotted at two vertical positions abovethe heating source. For locations sufficiently close to thesource, u-profile of the mean flow shows an asymmetricbehaviour with respect to the vertical centerline by compar-ison with the symmetric base solution. As one moves away

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0.004

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0.008

0.01

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Vel

ocity

u a

t alti

tude

y=

0.85

Horizontal position

Ra=1.4×104 ; A=4

mean field symmetric field

Vx =0

(a) (

Fig. 16. Horizontal profiles of u-velocity for Ra = 1.4 · 104: (a) at y = 0

Fig. 17. Temperature fluctuations (at the top row) and u fluctuations (below) ffor positive (negative) fluctuations. Variation range for temperature fluctuationcavity A = 4.

from the source, the difference between the base solutionand the average field vanishes (Fig. 16(b)). Concerningthe vertical component v of the mean flow, the horizontalprofiles (not shown here) remain symmetric and very closeto those of the base flow whatever the height.

Flow fluctuations: For the purpose of comparison withthe previous cases A = 1 and A = 2, spatial distributionsof temperature and velocity fluctuations with respect tothe symmetric base solution are displayed in Fig. 17 forRa = 1.4 · 104 at eight instants over one time period. The

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0

0.005

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0.02

0.025

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Vel

ocity

u a

t alti

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y=

1.23

Horizontal position

Ra=1.4×104 ; A=4

mean field symmetric field

Vx =0

b)

.85, (b) at y = 1.23, comparison between the mean and base flows.

or Ra = 1.4 · 104 at eight instants with a time step = T/8; thin (thick) liness [�0.03,+0.025], variation range for u velocity fluctuations [�0.1,+0.11],

T

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788

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803804805806

807808809810811812813814815816817818819820821822823824825826827828829830831832833834835836837838839840841842843844845846847

848

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851852853




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figures show that the instabilities patterns are substantiallydifferent from those observed in the previous cases. Actu-ally, the disturbances grow and spread out over the fullflow domain. In addition, they have a preferred downwardpropagation direction in spite of the flow direction in themiddle part of the cavity. Temperature and vertical velocityfluctuations (not shown here) are organized as large cellsconstituted of pairs of superimposed positive and negativeperturbations occupying the whole width of the cavity.Those of the horizontal velocity are structured as single‘‘circular’’ cells of positive and negative values alternatelysuperimposed in the vertical direction. Supported bydynamical sequences, it was found that the disturbancesoriginate from the unstable layer of fluid at the top ofthe cavity, then move down in the vertical direction untilthey reach the bottom wall. Their periodic motion is com-parable with a wave whose propagation speed is uniformand directed downwards.

The physical mechanism leading to this type of instabil-ity is different from that one found in the square cavityimplying shear layers with large velocity gradient. Here,the absence of sheared zones suggests that only the buoy-ancy forces are responsible for the disturbance develop-ment. This type of instability has been demonstrated tobe due to nonlinear density variation with temperature[25], this is why large disturbances are located in thehigh-density region. The buoyancy forces which act down-ward near the cold vertical walls where the fluid is densercause strong downward disturbances flow that explains apreferred direction of disturbances propagation: from topto bottom.

This second type of instability behaves like the so-called‘buoyancy-driven instabilities’ which were already reportedin previous works [26–28] for non-Boussinesq mixed con-vection in a tall vertical channel. The recent investigationscarried out by Suslov [28] show a stability diagram for awide range of Grashof and Reynolds numbers indicatingthe influence of the governing parameters (Gr, Re) on theappearance of the two distinct modes of instabilities: shearand buoyancy-driven instabilities.

4. Conclusions

In this paper, natural convection flows in a rectangulartwo-dimensional cavity have been numerically studied.The flows under consideration are generated by large tem-perature differences between a square solid obstacle kept ata constant hot temperature and walls kept at a constantcold temperature. In order to avoid too stringent stabilitycriterion in the numerics, the acoustic waves have been fil-tered out by considering the compressible Navier–Stokesequations under the low Mach number approximation.

Only moderate values of the Rayleigh number have beenconsidered. More specifically, we were interested in the firstbifurcation undergone by the flow, and possibly the sec-ond, when the Rayleigh number is increased. An importantpoint which was considered within this context, is that

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there exists a natural reflection symmetry with respect tothe mid vertical axis for this problem, whatever the valueof the aspect ratio A.

The main results may be summarized as follows:

• For sufficiently low values of the Rayleigh number, asteady symmetric flow is obtained whatever the valueof the aspect ratio A. This flow consists of two largerecirculation cells, one in each half of the cavity, whichisolate a quiet fluid layer, uniform in temperature, belowthe heating source.

• For sufficiently large values of the Rayleigh number, theflow becomes unsteady in a periodic manner. However,if the transition to unsteadiness occurs directly from theprevious symmetric steady flow through a supercriticalHopf bifurcation for A = 1 and A = 4, it is not the casefor A = 2 where the flow undergoes firstly a supercriticalpitchfork bifurcation leading to an asymmetric steadystate deflected towards one of the sidewalls. This asym-metric state then undergoes a supercritical Hopf bifurca-tion leading to an oscillatory motion around a deflectedmean flow.

• The instabilities which occur through the different Hopfbifurcations are not all of the same type, depending onthe aspect ratio A. For A = 1 and A = 2, they are shearlayer instabilities, whereas for A = 4 they are bulk insta-bilities which were referred by Suslov as buoyancy insta-bilities. In our case, they take the form of travellingwaves which fill the whole cavity and propagate down-wards with a well defined phase velocity.

• all the bifurcations that were encountered broke the nat-ural symmetry of the problem.

In these computations, the only parameter to be varied,apart from the Rayleigh number, was the aspect ratio A.Along these lines, computations will be carried out infuture considering larger values of the aspect ratio forwhich a wider range of instabilities may be expected. Inparticular, the results that were obtained by Weismanet al. [30] suggest that subcritical bifurcation could befound.

However, it is clear that the influence of other parame-ters have to be considered in future works. In particular,increasing the temperature difference, i.e. the parameter �,would likely lead to different flow structures andinstabilities.

5. Uncited reference

[17].

Acknowledgement

The computational time on a NEC-SX5 computer wasprovided by IDRIS, the computation center of the FrenchCNRS, under project #1102.

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