Research Article Numerical Simulation and Stability Study ... · made by Haaland and Sparrow [ ]....

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013, Article ID 198695, 12 pageshttp://dx.doi.org/10.1155/2013/198695

Research ArticleNumerical Simulation and Stability Study of Natural Convectionin an Inclined Rectangular Cavity

Hua-Shu Dou,1 Gang Jiang,1 and Chengwang Lei2

1 Faculty of Mechanical Engineering and Automation, Zhejiang Sci-Tech University, Hangzhou 310018, China2 School of Civil Engineering, The University of Sydney, NSW 2006, Australia

Correspondence should be addressed to Hua-Shu Dou; [email protected]

Received 18 December 2012; Accepted 22 January 2013

Academic Editor: Zhijun Zhang

Copyright © 2013 Hua-Shu Dou et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper examines the process of instability of natural convection in an inclined cavity based onnumerical simulations.The energygradient method is employed to analyze the physics of the flow instability in natural convection. It is found that the maximumvalue of the energy gradient function in the flow field correlates well with the location where flow instability occurs. Meanwhile,the effects of the flow time, the plate length, and the inclination angle on the instability have also been discussed. It is observedthat the locations of instabilities migrate right as the flow time increased. With the increase of plate length, the onset time of theinstability on the top wall of the cavity decreases gradually and the locations of instabilities move to the right side. Furthermore, thelocations of instability move left with the increase of the inclination angle in a certain range. However, these positions move rightas the accumulation of the heat flux is restrained in the lower left corner of the cavity once the inclination angle exceeds a certainrange.

1. Introduction

Transient natural convection flows in a cavity are commonin industrial applications such as in heat exchangers, solarcollectors, and nuclear reactors and in our daily life suchas in light emitting diode (LED) street lights, computers,and mobile phones. Actually, some engineering problemsare related to the cases with an inclined cavity. Yet, naturalconvection adjacent to an inclined plate has received lessattention than the classic cases of vertical and horizontalplates. Natural convection heat transfer is regarded as one ofthe three basic forms of heat transfer, and there exists somedifference of heat transfer efficiency between different man-ners of natural convection. The achieved results suggestedthat turbulent flowhas amore powerful ability to transfer heatin natural convection compared to laminar flow.

Patterson and Imberger [1] carried out extensive investi-gations on the transient behavior of natural convection of atwo-dimensional rectangular cavity in which the two oppos-ing vertical sidewalls are simultaneously heated and cooled byan equal amount. The authors present several flow regimes

of the flow development of the boundary layer which werenamed as “conduction regime,” “stable convection regime,”and “unstable convection regime,” respectively. These studiesare based on the relative values of the Rayleigh number Ra,the Prandtl number Pr, and the aspect ratio of the cavity. Itwas found that the flow of natural convection in an inclinedcavity loses its stability by forming longitudinal vortices.Sparrow and Husar [2] made experiments on inclined platesto reveal the presence of cellular secondary flows superposedupon the natural convection main flow and believed thatthese longitudinal vortices were the first stage of the laminar-turbulent transition process. Haaland and Sparrow [3] triedto use linear stability theory to predict a critical angle (orrange of angles) at which the disturbances change theircharacter from travelling waves to longitudinal vortices.Theyfinally found that it is difficult to predict the critical angletheoretically by treating each type of disturbance separately.

Lloyd and Sparrow [4]made some investigations with theaim of establishing the relationship between the inclinationangle and the nature of the instability.They tried to determineaccurate quantitative information on the angular dependence

2 Mathematical Problems in Engineering

of the Rayleigh number for instability.They found that waveswere the mode of instability for inclination angles of less than14 deg (relative to the vertical). Further, when the inclinationangle is beyond 17 deg, the instability was characterized bylongitudinal vortices. The range between 14 deg and 17 degwas a zone of continuous transition, with the two modes ofinstability coexisting.

Ganesan and Palani [5] proposed to study the naturalconvection effects on impulsively started inclined plate withheat and mass transfer by an implicit finite difference schemeof the Crank-Nicolson type. In order to access the accuracyof the numerical results, they compared their study withavailable exact solution of Moutsoglou and Chen [6] andachieved a good agreement. In addition, they observed thatlocal wall shear stress decreases as the angle of inclinationdecreases.

Xu et al. [7] researched transient natural convection flowsaround a thin fin on the sidewall of differentially heatedcavity, which illustrated that the fluid boundary layer adjacentto a vertical thermal wall included three sublayers, and thesesublayers were determined by different dynamic and energybalances. When the initial time 𝑡

1is sufficiently small, the

balance was determined by viscous term and buoyancy termwhich yields the inner viscous layer. Subsequently, as theflow time increases to 𝑡

2, the balance is still determined by

viscous term and buoyancy term, which yields a viscous layerwithin the thermal boundary layer but outside the innerviscous layer. At the same time, with the formation of innerviscous layer and viscous layer, there is a balance betweenthe conduction term and the unsteady term which yieldsthe thermal boundary layer. Saha et al. [8] studied naturalconvection of an inclined flat plate under a sudden coolingcondition, and they found that the cold boundary layeradjacent to the plate is potentially unstable, if the Rayleighnumber Ra exceeded a critical value. In other words, theboundary layer would be always stable if Ra was below athreshold.

Said et al. [9] made some numerical investigations onturbulent natural convection in a parallel-walled channelwhich is inclined with respect to gravity. They found thatthe channel overall average Nusselt number was reduced asthe inclination angle was increased. The rate of reductionin the overall Nusselt number decreases as the Rayleighnumber increases. Additionally, they observed that the localNusselt number was much higher along the lower wall of thehorizontal channel where cold air entered in comparison tothe upperwall where hot air existed at both channel openings.

Lin [10] presented a numerical experiment for the onsetand its linear development of longitudinal vortices in naturalconvection over inclined plates. It was observed that thecritical Grashof number increases with the increase of theinclination angle while the effect of inclination angle on theNusselt number is less pronounced when the value of theinclined angle increases.

Iyer and Kelly [11] did not agree with the conclusionsmade by Haaland and Sparrow [3]. They found that theseexperiments in [3] were not sensitive enough to detectthe first instabilities predicted by theoretical analysis. Iyerand Kelly used a spatial linear stability analysis with the

parallel flow assumption to examine the formation andgrowth of both wave instabilities and longitudinal vorticesand attempted to find a correlation between experimental andtheoretical results by finding the total amplification betweenthe earliest disturbances and the observed disturbances.

Thus, there are many factors to affect the natural con-vection heat transfer efficiency, in particular, the differenceof laminar flow and turbulence. In order to achieve a flowfield where the flow is in turbulent state, it is helpful toascertain the locations where instabilities could occur andwhere instability would occur firstly.

After almost 20 years of work, Dou and co-authors [12–18] suggested a new approach to analyze flow instability andturbulence transition based on “energy gradient method.”This approach is different from the linear stability theory,the weak nonlinear stability theory, the secondary instabilitytheory, and the energy method. This approach explains themechanism of flow instability from physics and derives thecriteria of turbulence transition.The theoretical results are inagreement with the experimental data of the pipe Poiseuilleflow, plane Poiseuille flow, planeCouette flow, Taylor-Couetteflow, boundary layer flow, and so on.

In this study, numerical simulation is used to obtainthe flow field at various geometrical and flow parameters.Then, the energy gradient method is used to investigate thephysical mechanism of flow instability in natural convection.The paper is divided into two sections. In the first section,the same computational geometries and numerical schemeof Saha et al. [8] are used to validate the numerical methodused in this study. In the second section, energy gradientmethod is briefly introduced to calculate the value of 𝐾 inthe whole flow field and make some further investigations onthe effects of the flow time, plate length, and the inclinationangle on flow instability of natural convection.This is the firsttime for the energy gradient method to be used in naturalconvection.

2. Computational Geometry andNumerical Procedures

2.1. Computational Geometry. The computational geometryis shown in Figure 1. We consider the top wall as a cooledinclined flat plate where the temperature is fixed at 𝑇

𝑐.

Initially, the fluid temperature in the domain is 𝑇0which is

higher than𝑇𝑐.The four sidewalls of the domain are rigid and

nonslip. Except for the top plate, all the three otherwalls of therectangle cavity are assumed to be insulated. The length andthe width of the cavity are defined as L and W, respectively.The inclination angle of the cavity is expressed with 𝜃.

2.2. Numerical Procedures

2.2.1. Governing Equations. The development of naturalconvection adjacent to an inclined cavity is governed bythe following two-dimensional Navier-Stokes and energy

Mathematical Problems in Engineering 3

𝑦𝐿

𝑇0

𝑊

𝑥

𝜃

𝑜

𝑇𝑐

Figure 1: Schematic of the inclined cavity.

equations, and these equations are based on the Boussinesqapproximation:

𝜕𝑢

𝜕𝑥+𝜕V

𝜕𝑦= 0,

𝜕𝑢

𝜕𝑡+ 𝑢𝜕𝑢

𝜕𝑥+ V𝜕𝑢

𝜕𝑦

= −1

𝜌

𝜕𝑝

𝜕𝑥+ 𝜈(

𝜕2𝑢

𝜕𝑥2+𝜕2𝑢

𝜕𝑦2) + 𝑔𝛽 sin 𝜃 (𝑇 − 𝑇

0) ,

𝜕V

𝜕𝑡+ 𝑢𝜕V

𝜕𝑥+ V𝜕V

𝜕𝑦

= −1

𝜌

𝜕𝑝

𝜕𝑦+ 𝜈(

𝜕2V

𝜕𝑥2+𝜕2V

𝜕𝑦2) + 𝑔𝛽 cos 𝜃 (𝑇 − 𝑇

0) ,

𝜕𝑇

𝜕𝑡+ 𝑢𝜕𝑇

𝜕𝑥+ V𝜕𝑇

𝜕𝑦= 𝑘(

𝜕2𝑇

𝜕𝑥2+𝜕2𝑇

𝜕𝑦2) ,

(1)

where𝑥 and𝑦 are the horizontal and vertical coordinateswithorigin at the lower left corner of the cavity, 𝑡 is the time, 𝑇 isthe temperature, 𝑝 is the pressure, 𝑢 and V are the velocitycomponents in the 𝑥 and 𝑦 directions, 𝑔 is the accelerationdue to gravity, 𝛽 is the coefficient of thermal expansion, 𝜌is the fluid density, 𝑘 is the thermal diffusivity, and 𝜐 is thekinematic viscosity.

2.2.2. Numerical Scheme. The governing equations (1) areimplicitly solved using a finite-volume SIMPLE scheme, withthe QUICK scheme approximating the advection term. Thediffusion terms are discretized using central differencingwithsecond-order accurate. A second-order implicit time-march-ing schemewill be used for the unsteady term.Thediscretizedequations are iterated with specified underrelaxation factors.The boundary condition of the left wall and the right wallis 𝜕𝑇/𝜕𝑥 = 0, 𝑢 = V = 0; the boundary condition of thebottom is: 𝜕𝑇/𝜕𝑦 = 0, 𝑢 = V = 0. In addition, we should notethat the flow is unsteady, and all the plots depend on the time.

3. Criteria of Instability Based onthe Rayleigh Number

The natural convection boundary layer adjacent to aninclined cavity is subjected to sudden cooling boundarycondition which yields a thermal boundary layer along thewall. At the same time with the formation of the thermalboundary layer, the viscous boundary layer is formed whichis determined by the balance between viscous term andinertial term. Meanwhile, the velocity inside the boundarylayer develops, governed by the balance of viscous and inertialterms with the buoyancy term. It is known from [7, 19] thatthe thickness of thermal boundary layer, the velocity insidethe thermal boundary layer, and the thickness of the viscousboundary layer are related to the balance between conductionand advection terms.

We can find in [8] that, when the balance betweenconduction and advection terms plays a leading role inexchange of momentum and energy, the steady-state scalesof the boundary layer (𝑡

𝑠) can be achieved:

𝑡𝑠∼

(1 + Pr)1/2(1 + tan2𝜃)1/2

tan 𝜃 ⋅ Ra1/2Pr1/2(𝐿2sin2𝜃𝑘) , (2)

where Ra is the Rayleigh number, Ra = 𝑔𝛽Δ𝑇(𝐿 sin 𝜃)3/𝜐𝑘and Pr is the Prandtl number, Pr = 𝜐/𝑘. In the present case,the thermal boundary layer is bounded by a rigid surfaceof the plate and a cold air layer, which is equivalent to thefree-rigid boundary configuration [20–24], in which there isa critical Rayleigh number Ra

𝑐= 1106.5. Also, there exists a

critical time scale 𝑡𝐵for the onset of thermal layer instability

at a given Ra. If 𝑡 > 𝑡𝐵, the instability will set in before

the growth of the thermal boundary layer completes. On theother hand, if 𝑡 < 𝑡

𝐵, the instability will never occur no

matter how much time it iterates. The critical time scale 𝑡𝐵

is described as follows [8]:

𝑡𝐵= (

Ra𝑐

Ra)

2/3𝐿2sin2𝜃𝑘. (3)

The ratio between the growth time of thermal boundary layer𝑡𝑠and the critical time scale 𝑡

𝐵described as follows [8],

𝑡𝑠

𝑡𝐵

= [(1 + Pr)3(1 + tan2𝜃)3Ra

Pr3tan6𝜃 ∙ Ra4𝑐

]

1/6

. (4)

Absolutely, if Ra > Pr3tan6𝜃 ∙ Ra4𝑐/(1 + Pr)3(1 + tan2𝜃)3, that

is, 𝑡𝑠> 𝑡𝐵, the instability will set in before the growth of the

thermal boundary layer completes, or else, if Ra < Pr3tan6𝜃 ∙Ra4𝑐/(1 + Pr)3(1 + tan2𝜃)3, that is, 𝑡

𝑠< 𝑡𝐵, the instability will

never occur.

4. Validation of Numerical Methods

4.1. Numerical Scheme Test. In order to verify the accuracy ofthe numerical scheme used in this study, the computationalgeometry of Saha et al. [8] is considered. In the followingnumerical simulations, the condition Pr < 1 should be


Table 1: Values of configuration dimensions, Ra, 𝑡𝑠, and 𝑡

𝐵shown in

Figure 6.

Runs Configuration dimension Ra 𝑡𝑠

𝑡𝐵

(a1) 𝑊(0.5m) × 𝐿 (5m) 5.33 × 104 815.6 s 915.5 s(b1) 𝑊(0.5m) × 𝐿 (5m) 5.33 × 105 257.9 s 197.2 s(c1) 𝑊(0.5m) × 𝐿 (2m) 6.83 × 103 364.7 s 576.8 s(d1) 𝑊(0.5m) × 𝐿 (15m) 2.88 × 106 998.9 s 576.8 s

satisfied. If Pr > 1, Pr3tan6𝜃∙Ra4𝑐/(1+Pr)3(1+ tan2𝜃)3 is very

large. Thus, it is difficult to observe instability if the value ofRa is low.

Figure 2 shows the simulated temperature contours withthree different cavity dimensions at 1000 s, which is takenfrom Saha et al. [8].The lengths are 5.4m, 10.8m, and 16.2m,respectively, and the width is 0.6m.

Figure 3 presents the current results of numerical simu-lations at 1000 s for the cavity sizes in Figure 2. ComparingFigure 3 with Figure 2, it can be found that the presentsimulations are in agreementwith those in [8], which demon-strates that the numerical scheme is reliable and accurate.

4.2. Grid Independence Test. Grid independence will beexamined with three different mesh sizes for the samedomain. The three mesh sizes are as follows (Figure 4): (a)100× 300with 30000 cells, 60400 faces, and 30401 nodes; (b)150×450with 67500 cells, 135600 faces, and 60801 nodes; (c)200 × 600 with 120000 cells, 240800 faces, and 120801 nodes.

The simulation results of temperature contours at 300 swith these three meshes are shown in Figure 4. The temper-ature versus time with three different mesh sizes is shown inFigure 5, which is recorded at the samemonitor point (2, 1). Itcan be seen from Figures 4 and 5 that mesh convergence hasbeen achieved with these meshes.

4.3. Test of Simulation Results. Four different configurationsare listed in Table 1, and substituting these correspondingdata into (2) and (3), we will get 𝑡

𝑠and 𝑡𝐵, respectively. All the

numericalmodels in Figure 5 satisfy the following conditions:Pr = 0.72, 𝜃 = 5.71∘, and 𝑘 = 2.04×10−5. Figures 6(a2), 6(b2),6(c2), and 6(d2) are partial enlarged drawings of the middleportion of the plate in Figures 6(a1), 6(b1), 6(c1), and 6(d1),respectively.

By comparing these four groups of pictures in Figure 6,some conclusions may be written as follows.

(1) It is observed that there is an “end effect” at thetop right corner of the enclosure in each case, andthis phenomenon is affected by the configuration andinclination angle of the cavity.

(2) The iterative time reaches 𝑡 = 𝑡𝑠both in Figures 6(a1)

and 6(c1), and it is very clear that the flow is still stablewhich can be seen from the partial enlarger of themiddle portion of the plate.

(3) The oscillation of the curves from the isothermsin Figures 6(b1) and 6(d1) indicates the onset ofinstability. It is clearly seen that the thermal boundary

layer travels in waveform rather than in a smoothmanner.

(4) It can be found that 𝑡𝑠< 𝑡𝐵is satisfied for the

simulated results shown in Figures 6(a1) and 6(c1),and it is observed that the thermal boundary layer isstable at these cases.While, in Figures 6(b1) and 6(d1),𝑡𝑠> 𝑡𝐵, the stability criterion is violated. Thus, the

flow instability occurs at a proper time. These resultsare in good agreement with the predictions of criteriaof instability based on Ra.

Nevertheless, we observe some unique phenomena fromthese previous instability cases. Firstly, instabilities do notoccur in the whole flow field. Secondly, there exists a distinc-tive time difference of instabilities at different locations. Inother words, it is expected to employ a theory to predict thelocations where instabilities could occur and positions whereinstability would take place firstly. In the following study, wewill briefly introduce the energy gradient method to analyzethe above phenomena observed.

5. Application of Energy Gradient Method

5.1. Energy Gradient Method. From the classical theory ofthe Brownian motion, the fluid particles exchange energyand momentum all the time via collisions. The fluid particlewill collide with other particles in transverse directions asit flows along its streamline, and this particle would obtainenergy expressed as Δ𝐸 after many cycles; at the same time,the particle would drop energy due to viscosity along thestreamline; with the same periods, the energy loss expressedas Δ𝐻 would be considerable. Consequently, there exists acritical value of the ratio of Δ𝐸 and Δ𝐻, above which theparticle would leave its equilibrium by moving to a newstreamline with higher energy or lower energy and belowwhich the particle would not leave its streamline for itsoscillation would be balanced by the viscosity along thestreamline. Making reference to [13–18], we can express thecriteria of instability as follows:

𝐹 =Δ𝐸

Δ𝐻=

((𝜕𝐸/𝜕𝑛) (2𝐴/𝜋))

((𝜕𝐻/𝜕𝑠) (𝜋/𝜔𝑑) 𝑢)

=2

𝜋2𝐾𝐴𝜔𝑑

𝑢=2

𝜋2𝐾V

𝑚

𝑢< Const,

(5)

where

𝐾 =𝜕𝐸/𝜕𝑛

𝜕𝐻/𝜕𝑠. (6)

Here, 𝐹 is a function of coordinates which expresses theratio of the energy gained in a half period by the particleand the energy loss due to viscosity in the half period. 𝐾 isa dimensionless field variable (function) and expresses theratio of transversal energy gradient and the rate of the energyloss along the streamline.𝐸 = 𝑝+1/2𝜌𝑉2 is the kinetic energyper unit volumetric fluid, 𝑠 is along the streamwise direction,and 𝑛 is along the transverse direction. 𝐻 is the loss of the


(a) Length = 5.4m (b) Length = 10.8m

(c) Length = 16.2m

Figure 2: Temperature contours of three different geometries in [8].

(a) L = 5.4m (b) L = 10.8m

(c) L = 16.2m

Figure 3: Temperature contours of three different geometries (present calculation).

total mechanical energy per unit volumetric fluid along thestreamline for finite length, which can be calculated from theNavier-Stokes equations. Further, 𝜌 is the fluid density, 𝑢 isthe streamwise velocity ofmain flow,𝐴 is the amplitude of thedisturbance distance, 𝜔

𝑑is the frequency of the disturbance,

and V𝑚= 𝐴𝜔𝑑is the amplitude of the disturbance of velocity.

Equation (5) represents the criteria of instability; that is,if 𝐹 exceeds its threshold, the fluid particle would lose itsstability or the instability would never set in. Equation (6)represents characteristic of instability; that is, it shows themost dangerous positions in a flow field, and it indicatesthat instability would occur firstly at the position with themaximum value of𝐾 once instability sets in.

5.2. Criterion of Instability Based on Energy Gradient inNatural Convection. Dou and Phan-Thien [25] proposed anenergy gradient theory which describes the rules of fluidmaterial stability from the viewpoint of energy field and canbe considered as a supplement to the Newtonian mechanics.They claimed that the instability of natural convection couldnot be resolved by Newton’s three laws, for the reason thata material system moving in some cases is not simply dueto the role of forces. This method does not attribute theRayleigh-Benard problem to forces, but to energy gradient. Itpostulates that when the fluid is placed on a horizontal plateand it is heated from below, the fluid density in the bottombecomes low which leads to energy gradient 𝜕𝐸/𝜕𝑦 > 0along 𝑦-coordinate. Only when 𝜕𝐸/𝜕𝑦 is larger than a criticalvalue, will the flow become unstable, and then fluid cells ofvorticities will be formed. This conclusion is in accordancewith the former criteria of instability.

When the fluid is placed on an inclined plate or a box, thecriterion of natural convection can be written as (Figure 7)

𝐾 = √(𝜕𝐸

𝜕𝑥)

2

+ (𝜕𝐸

𝜕𝑦)

2

. (7)

In present study, neglecting the influence of the gravity, weget 𝐸 ∼ 𝑝

0.

Besides the ability to predict whether instability couldoccur in natural convection [25], energy gradient methodhas another two functions in natural convection wheninstabilities had occurred. It can predict the locations whereinstabilities could occur and the position where instabilitywould take place firstly via the value of 𝐾. In the proposedmethod, the flow is expected to be more unstable in the areawith high value of 𝐾 than that in the area with low value of𝐾. This is the distinctive difference between energy gradienttheory and the criteria of instability based on the Rayleighnumber and is also the focus of investigation in this paper.

We should clarify that the value of𝐾 derived from energygradient method can only be applied in situations whereinstabilities could occur. Thus, all the subsequent numericalsimulations should satisfy the condition that Ra is larger thanPr3tan6𝜃∙Ra4

𝑐/(1+Pr)3(1+tan2𝜃)3, in order to study the cases

of instability occurrence.

6. Results and Discussions

6.1. Results with New Geometries. We will use the samenumerical scheme as that described previously to simulatethe natural convection in a cavity. The numerical resultsand analysis are discussed as follows. In order to avoid theinfluence of “end effect” on numerical results, we choose


(a) 100 ∗ 300 𝑡 = 300 s (b) 150 ∗ 450 𝑡 = 300 s

(c) 200 ∗ 600 𝑡 = 300 s

Figure 4: Comparison of the calculated results with three different mesh sizes: (a) Δ𝑥 = 0.054, (b) Δ𝑥 = 0.036, and (c) Δ𝑥 = 0.027.

50 100 150 200 250 300

299.4

299.5

299.6

299.7

299.8

299.9

300

Grid (a)Grid (b)Grid (c)

𝑇(K

)

𝑡 (s)

Figure 5: Temperature profile calculated with three different gridsizes.

another computational domain as shown in Figure 8. Theaspect ratio of the domain is as L (16.2m) × W (2m). Thecalculated results at Ra = 2.1 × 108 are shown in Figure 8.According to the criterion based on the Rayleigh number,the flow is unstable. Figure 8(a) shows the total pressure con-tours. It is found from this picture that there are two areas ofunstable region in the domain.This phenomenon is explainedas follows. As the boundary conditions are defined, thetemperature on the top wall is lower than the temperature ofthe fluid, which leads the fluid with higher temperature in thecavity to move upwards and gather together. Consequently,pressure difference develops gradually due to the movementof the fluid in the cavity, and the irregular phenomena of the

total pressure contours indicate that the flow of the fluid isunstable.

Figure 8(b) shows the velocity contours along 𝑥-coordinate, the straight line m-m splits the cavity into aleft region and a right region, and the minus symbol ofvelocity means particles move along negative direction of𝑥-coordinate. It is noticeable that the velocity is distinctivein the left area, while the velocity in the right area tends tobe 0. However, we attribute this phenomenon to inclinationof the cavity. For a horizontal plate, the velocity profile issymmetrical in the whole domain. Furthermore, we observethat the negative speed appears mainly close to the top wall.When the heated fluid expands and moves upwards to thetop wall, it can be seen from Figure 8(a) that the pressureclose to the top wall is much higher than any other area,and this higher pressure would drive the fluid near the topwall to travel to the left side; therefore, the negative velocityis formed. At the same time, we notice that there existsa continuous area above the top wall where the value of𝑥-velocity (𝑢) is large. In [26], we obtain that the thermalboundary layer, containing lots of heat flux, travels along thebottom wall in the process of natural convection; hence, theflow, we observed, is actually the heat flux; furthermore, themovement direction of the heat flux accords well with thevalue of 𝑥-velocity (𝑢).

Figure 8(c) shows the velocity contours along 𝑦-coordinate; the straight line m-m splits the cavity into aleft region and a right region. Similar to Figure 8(b), speedis fierce in the left district and is faint in the right district.Here, we get some new discoveries. Firstly, in comparison toFigure 8(b), we can find that the district with high value of𝑦-velocity (V) is in good accordance with the district withhigh value of 𝑥-velocity (𝑢); however, the areas with highvalue of V are obviously much larger than the areas with thehigh value of 𝑢; this phenomenon confirms that lots of fluidparticles move upwards in this cavity, while, at the same time


Partial enlargement

(a1)

(a2)

Partial enlargement

(a1)

(a2)

Partial enlargement

(b1)

(b2)

Partial enlargement

(b1)

(b2)

Partial enlargement

(c1)

(c2)

Partial enlargement

(d1)

(d2)

(a) (b)

(c) (d)

Figure 6: Four groups of geometries with different Ra: (a1) Ra = 5.335𝑒4, (b1) Ra = 5.335𝑒5, (c1) Ra = 6.83𝑒3, and (d1) Ra = 2.88𝑒6.

𝜕𝑝0/𝜕𝑥𝑥

𝑦

𝑜

𝐾𝜕𝑝0/𝜕𝑦

Figure 7: Calculation of the value of 𝐾.

with the fluid moving upwards, the fluid also travels alongthe horizontal direction. Secondly, we find that the minusvelocity along the negative vertical direction still exists inthis cavity. For the temperature on the top wall is lowerthan that in the cavity, which leads to a large density of fluidnear the top wall, then the fluid with large density movesdownwards due to the buoyancy. Meanwhile, it is knownfrom Figure 8(a) that the pressure near the top wall is muchlarger than that of the bottom, and the pressure differencewould drive the fluid to move downwards. Consequently,the negative speed is produced under the action both of thebuoyancy and the pressure difference.

Figure 8(d) shows the temperature contours in this cavity;the straight line m-m splits the cavity into a left region anda right region. Firstly, it is surveyed that the locations ofinstabilities concentrate mainly in the left region close tothe top wall, while instabilities seldom occur in the rightregion of this cavity, and this is due to the inclination ofthe cavity resulting in lots of heat flux concentrated on thetop wall in the left region. In addition, we find a uniquelocation of instability above the bottom. It is known from [26]that the movement of thermal boundary layer leads to theaccumulation of heat flux above the bottom; with referenceto Figure 8(a), we know that there exists an obvious pressuredifference at the location where heat flux is accumulatedwhich leads to the fact that the fluid loses its stability andmoves upwards. At last, we observe that the locations ofinstabilities accord well with the locations with high valueof velocity. This illustrates that the instabilities result in thedramatic change of velocity.

Figure 8(e) shows the contours of the value of 𝐾; thestraight line m-m splits the cavity into a left region and aright region. It is easy to get some similar and new discoveriesby comparing Figure 8(d) to Figure 8(e). First, the area withhigh value of 𝐾 is mainly concentrated in the left side of thiscavity, and this is similar to the previous discoveries. Second,the area with high value of 𝐾 is in excellent agreementwith the locations of instabilities. This result can verify theaccurate prediction of energy gradient method. Third, themanifestation of instability is the formation of vorticity, andpositions with the formation of vorticities accord well withthe locations with high value of 𝐾; however, there exists aregion with low value of 𝐾 inside the vorticity, as the bluearea shows in the contours of the value of 𝐾 in Figure 8(e).Fourth, there exists an area with high value of 𝐾 on the


The inclination

00.

005

0.01

0.01

50.

020.

025

0.03

0.03

50.

040.

045

0.05

0.05

50.

060.

065

0.07

0.07

50.

080.

085

0.09

angle of the cavity is 5.71deg, 𝐿 = 16.2m,𝑊 = 2m, Ra = 2.1𝑒8, Pr = 0.72, 𝑡 = 240 s

(a) Total pressure

LeftRight

m

m The inclination

−0.

08−

0.07−

0.06−

0.05−

0.04−

0.03−

0.02−

0.01 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09 0.

1


(b) 𝑥-velocity

−0.

065

−0.

06−

0.05

5−

0.05

−0.

045

−0.

04−

0.03

5−

0.03

−0.

025

−0.

02−

0.01

5−

0.01

−0.

005 0

0.00

50.

010.

015

0.02

0.02

50.

030.

035

0.04

0.04

50.

050.

055

Left Right

m

m The inclinationangle of the cavity is 5.71deg, 𝐿 = 16.2m,𝑊 = 2m, Ra = 2.1𝑒8, Pr = 0.72, 𝑡 = 240 s

(c) 𝑦-velocity

Left Right

m

m

Partial enlargement rotates 270 deg

290.

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129

1.5

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329

3.5

294

294.

529

529

5.5

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9.5

The inclinationangle of the cavity is 5.71deg, 𝐿 = 16.2m,𝑊 = 2m, Ra = 2.1𝑒8, Pr = 0.72, 𝑡 = 240 s

(d) Temperature

Left Right

m

m

0.00

50.

0074

0.00

980.

0122

0.01

460.

017

0.01

940.

0218

0.02

420.

0266

0.02

90.

0314

0.03

380.

0362

0.03

860.

041

0.04

340.

0458

0.04

820.

0506


(e) Value of𝐾

Figure 8: Calculated results of various parameters: (a) total pressure contours, (b) 𝑥-velocity contours, (c) 𝑦-velocity contours, (d)temperature contours, and (e)𝐾 contours.

left wall, yet no vorticity is formed. We can observe fromthe partial enlargement of isotherms in Figure 8(d) that thethermal boundary layer has lost its stability for it travels ina waveform. The reason why it does not form a vorticity isthat the thermal boundary travels along the wall, and theheat flux is accumulated above the bottom which leads tothe onset of instability. Except for the above discoveries, thereexit two problems to be solved: the first one is that we cannotdetermine the critical value of 𝐾 which can predict whetherinstability could occur in natural convection.The other is thatwe do not comprehend so far why the area with high value of𝐾 and the area with the low value of𝐾 coexist at the locationsof instabilities.

6.2. Effect of Flow Time and Discussions. In order to investi-gate the effect of the flow time, we choose the same geometry

and numerical scheme to achieve some numerical simula-tions. Figures 9 and 10 show the contours of temperatureand the value of 𝐾 at 270 s and 300 s, respectively. The basicparameters are showed in the corresponding figures. Com-paring Figures 9 and 10 with Figures 8(d) and 8(e), we canobserve the following characteristics. First, the regions withhigh value of 𝐾 coincide with the locations of instabilities.Second, with the flow time accumulated, the positions andareas of instabilities are increasing. Third, the area with highvalue of 𝐾 and the area with the low value of 𝐾 coexist atthe locations of instabilities, and the reason is still unknowntill now. Fourth, the locations of instabilities spread rightalong the top wall with the increase of the flow time. This isbecause the accumulation of heat flux above the bottom wallis restricted; in other words, the movement of the thermalboundary layer on the top wall is restrained. Thus, lots ofheat flux is accumulated on the top wall near the rightside which leads to the occurrence of instabilities. Fifth, the


The inclination

290.

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9.5


(a) Temperature

0.00

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048

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(b) Value of𝐾

Figure 9: Calculated results at 270 s: (a) temperature contours and (b) 𝐾 contours.

The inclination

290.

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(a) Temperature

0.00

50.

0071

0.00

920.

0113

0.01

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0.01

760.

0197

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0.03

020.

0323

0.03

440.

0365

0.03

860.

0407

0.04

280.

0449


(b) Value of𝐾

Figure 10: Calculated results at 300 s: (a) temperature contours and (b)𝐾 contours.

fluid in the cavity tends to flow in a turbulent manner. It issuggested in energy gradient method that the amplificationof local turbulence phenomenonwould lead to that the wholedomain loses its stability. In the current case, the naturalconvection cannot be restricted so that the fluid tends to beturbulence, and this phenomenon is in good agreement withenergy gradient method.

In summary, the flow time affects numerical results to aconsiderable degree. Two features are distinctive particularly:the locations of instabilities migrate right on the top wall andthe fluid in the whole domain tends to flow in a turbulentmanner as the flow time increases.

6.3. Effect of Plate Length and Discussions. Figure 11 showsthe contours of temperature and the value of 𝐾, respectively,with a plate length of 24.3m.Thebasic parameters are showedin the corresponding figures. Comparing Figures 8(d) and8(e) with Figures 11(a) and 11(b), we can also make someconclusions. First, the regions with high value of 𝐾 coincidewith the locations of instabilities. Second, the area with highvalue of 𝐾 and the area with the low value of 𝐾 coexist atthe locations of instabilities. The above two viewpoints aresimilar to the former conclusions. Third, the onset time ofinstability is inversely proportional to the length of the plate;that is, the onset time of instability decreaseswith the increaseof 𝐿. Corcione [27] stated that the heat transfer rate from anyheated or cooled boundary surface of the enclosure increasesas the Rayleigh number increases. This phenomenon is ingood accordance with the results in [27]. In all the related

cases, Ra is actually proportional to the length of the cavity𝐿. Consequently, when we substitute the larger parameter of𝐿 into (2), it is easy to get a shorter time of 𝑡

𝑠which means

that the flow is easier to lose its stability. Fourth, the locationsof instabilities migrate right as the plate length increases.Because themigration length of the thermal boundary on thetop wall is prolonged, the heat flux transfer would be limitedas long as the instabilities set in, which in turn leads to theaccumulation of heat flux on the top wall near the right side,and the heat flux loses its stability.

In summary, the length of the cavity has an obviouseffect on numerical simulation. Two features are distinctiveparticularly: the onset time of instabilities will get shorter andthe locations of instabilities on the top wall will migrate rightwith the increase of the plate length.

6.4. Effect of InclinationAngle andDiscussions. Figures 12 and13 show the contours of temperature and the value of 𝐾 withinclination angle 𝜃 = 10∘ and 𝜃 = 15∘, respectively. The basicparameters are showed in the corresponding figures. Kurianet al. [28] studied laminar natural convection inside inclinedcylinders of unity aspect ratio and demonstrated that thereexits a threshold of inclination angle. As the inclination angleincreased from 0 deg to its threshold, convection increasedto a maximum. However, when the inclination angle wasgreater than its critical value, the convection effects and thedimensionless axial temperature gradient decreased relativelysmall with the increase of inclination angle. Comparing thesefigures with Figures 8(d) and 8(e), we can obtain some


The inclination angle of the cavity is

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9.5

5.71deg, Ra = 7.08𝑒8, Pr = 0.72, 𝑡 = 260 s,𝑊 = 2m, 𝐿 = 24.3m

(a) Temperature

The inclination angle of the cavity is

0.00

50.

0073

70.

0097

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10.

0144

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766

5.71deg, Ra = 7.08𝑒8, Pr = 0.72, 𝑡 = 260 s,𝑊 = 2m, 𝐿 = 24.3m

(b) Value of𝐾

Figure 11: Calculated results with length of 24.3m: (a) temperature contours and (b) 𝐾 contours.

The inclinationangle of the cavity is 10 deg,

290.

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𝐿 = 16.2m,𝑊 = 2m, 𝑡 = 560 s, Ra = 1.11𝑒8, Pr = 0.72

(a) Temperature


0.00

050.

0008

40.

0011

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0015

20.

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60.

0022

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288

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322

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662

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696

𝐿 = 16.2m,𝑊 = 2m, 𝑡 = 560 s, Ra = 1.11𝑒8, Pr = 0.72

(b) Value of𝐾

Figure 12: Calculated results with 10 degrees of angle: (a) temperature contours and (b)𝐾 contours.

similar and different observations. First, the regionswith highvalue of 𝐾 still coincide with the locations of instabilities.Second, the area with high value of 𝐾 and the area with thelow value of 𝐾 coexist at the locations of instabilities. Theabove two observations are similar to the former conclusions,while the following three conclusions are different. Third,the intensity of flow instability in the left region is muchstronger than that in the right region as the inclination angleincreases. Upton and Watt [29] made experimental study inan inclined rectangular enclosure, and the results showedthat the angle of inclination has a significant effect on theflow and heat transfer in natural convection in an enclosure.Buoyancy in the intrusion layer was found to be the mainfactor determining the character of these flows. When theangle of the inclined plate increases, more components ofbuoyancy will be imposed on the top wall which leads tothat more heat flux will be accumulated in the left region.Hence, it is easy to observe that the intensity of flow instabilityis much stronger in the left region. This result is in goodaccordance with the conclusion stated by Upton and Watt[29]. Fourth, the locations of instabilities would spread leftalong the top wall with the increase of inclination angle ina certain range. Due to the same reason as stated previously,more heat flux will be accumulated in the left region; thus,it is easy to observe that more instabilities would occur inthe left region as the inclination angle increases in a certainrange which in turn hinders the movement of the heat flux.Fifth, the locations of instabilities would migrate right oncethe inclination angle exceeds a certain range. Although theincreased component of buoyancy along the top wall canaccelerate themovement of heat flux to the left side, the widthof the cavity and the higher pressure difference will restrict

the accumulation of heat transfer severely in the lower leftcorner. As a result, more heat flux will be accumulated onthe top wall near the right side, and it is easy to observelots of instabilities on the right side of the top wall. At thesame time, we are surprised to find that the fourth and fifthviewpoints accord well with the results of Kurian et al. [28] tosome extent.

In summary, the inclination angle of the cavity has adistinctive affection on numerical simulations.Three featuresare distinctive particularly. First, the intensity of flow insta-bility in the left region is much stronger than that in the rightregion with the increase of inclination angle. In addition, thelocations of instabilities will migrate left with the increaseof inclination angle in a certain range. At last, the locationsof instabilities will migrate right once the inclination angleexceeds a certain range.

7. Conclusions

Numerical simulations on natural convection in an inclinedcavity have been carried out using the unsteadyNavier-Stokesequations for various parameters such as the domain lengthand the inclined angle. The main conclusions are as follows.

(1) The energy gradient method is successfully employedto study the instability of the thermal boundary layer.It is found that instability occurs firstly at the positionwhere the energy gradient function gets its maximum𝐾max. The regions with high value of 𝐾 coincidewith the locations of instabilities. These observationsaccord well with the energy gradient theory.


290.

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129

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298

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529

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9.5


𝐿 = 16.2m,𝑊 = 2m, 𝑡 = 600 s, Ra = 3.68𝑒8, Pr = 0.72

(a) Temperature


0.00

050.

001

0.00

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95

𝐿 = 16.2m,𝑊 = 2m, 𝑡 = 600 s, Ra = 3.68𝑒8, Pr = 0.72

(b) Value of𝐾

Figure 13: Calculated results with 15 degrees of angle: (a) temperature contours and (b)𝐾 contours.

(2) The fluid within the whole domain tends to flow in aturbulent state with the increase of flow time if Ra islarger than its critical value.

(3) With the increase of plate length, the onset time ofinstability decreases and the locations of instabilitiesmigrate right along the top wall.

(4) The intensity of instabilities in the left region of thestudied inclined cavity is much stronger than that inthe right region.When the inclined angle increases ina certain range, the locations of instabilities along thetop wall migrate left.

(5) Once the inclined angle exceeds a certain range, thepositions of instabilities would move right since theaccumulation of the heat flux in the lower left cornerof the cavity is restrained.

Nomenclature

Ra: Rayleigh numberPr: Prandtl number𝑡: Time𝑡𝑠: Steady-state time scale𝑡𝐵: Critical time scale𝑇: Temperature𝑇𝑐: Temperature of the top wall𝑇0: Temperature of the fluid𝐿: Length of the cavity𝑊: Width of the cavity𝑥, 𝑦: Coordinatescavity𝑝: Pressure𝑝0: Total pressure𝑢, V: Velocity components in 𝑥, 𝑦 directions,

respectively𝑔: Gravity acceleration𝑘: Thermal diffusivityRa𝑐: Critical Rayleigh number

𝑛: Transverse direction𝑠: Streamwise direction𝐾: Dimensionless function expresses the ratio

of transversal energy gradient andstreamwise energy gradient

𝐴: Amplitude of the disturbance distance𝜔𝑑: Frequency of the disturbance

V

𝑚: Amplitude of the disturbance of velocity.

Greek Symbols

𝜃: Inclination angle𝛽: Coefficient of thermal expansion𝜌: Fluid density𝜐: Kinematic viscosityΔ𝐸: Energy difference along transverse directionΔ𝐻: Energy difference along streamwise directionΔ𝑇: Temperature difference.

Acknowledgment

Thiswork is supported by the Science Foundation of ZhejiangSci-Tech University (ZSTU) under Grant no. 11130032241201.

References

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[2] E.M. Sparrow andR. B.Husar, “Longitudinal vortices in naturalconvection flow on inclined plates,” Journal of Fluid Mechanics,vol. 37, no. 2, pp. 251–255, 1969.

[3] S. E. Haaland and E. M. Sparrow, “Vortex instability of naturalconvection flow on inclined surfaces,” International Journal ofHeat and Mass Transfer, vol. 16, no. 12, pp. 2355–2367, 1973.

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[8] S. C. Saha, J. C. Patterson, and C. Lei, “Scaling of naturalconvection of an inclined flat plate: Sudden cooling condition,”


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[19] S. C. Saha, Natural convection adjacent to an inclined flat plateand in an attic space under various thermal forcing conditions[Ph.D. thesis], James Cook University, Queensland, Australia,2008.

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[21] Y. M. Chen and A. J. Pearlstein, “Stability of free-convectionflows of variable-viscosity fluids in vertical and inclined slots,”Journal of Fluid Mechanics, vol. 198, pp. 513–541, 1989.

[22] C. Lei and J. C. Patterson, “Unsteady natural convection ina triangular enclosure induced by absorption of radiation,”Journal of Fluid Mechanics, vol. 460, pp. 181–209, 2002.

[23] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability,Dover, New York, NY, USA, 1961.

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[25] H.-S. Dou and N. Phan-Thien, “Instability of fluid materialsystems,” in Proceedings of the 15th Australasian FluidMechanicsConference, pp. 13–17, The University of Sydney, Sydney, Aus-tralia, December 2004.

[26] F. Xu, J. C. Patterson, and C. Lei, “Transient natural convectionflows around a thin fin on the sidewall of a differentially heatedcavity,” Journal of Fluid Mechanics, vol. 639, pp. 261–290, 2009.

[27] M. Corcione, “Effects of the thermal boundary conditions atthe sidewalls upon natural convection in rectangular enclosuresheated from below and cooled from above,” InternationalJournal of Thermal Sciences, vol. 42, no. 2, pp. 199–208, 2003.

[28] V. Kurian, M. N. Varma, and A. Kannan, “Numerical studieson laminar natural convection inside inclined cylinders of unityaspect ratio,” International Journal of Heat and Mass Transfer,vol. 52, no. 3-4, pp. 822–838, 2009.

[29] T. D. Upton and D. W. Watt, “Experimental study of transientnatural convection in an inclined rectangular enclosure,” Inter-national Journal of Heat and Mass Transfer, vol. 40, no. 11, pp.2679–2690, 1997.

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