European Journal of Mechanics B/Fluids 41 (2013) 29–45
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European Journal of Mechanics B/Fluids
journal homepage: www.elsevier.com/locate/ejmflu
Heat flow visualization analysis on natural convection in rhombicenclosures with isothermal hot side or bottom wallR. Anandalakshmi, Tanmay Basak ∗
Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai - 600036, India
h i g h l i g h t s
• Rhombic enclosures with horizontal or vertical thermal gradients are considered.• Heat flow and thermal mixing inside the rhombic enclosures of various angles (ϕs’) have been analyzed.• Heatlines are demonstrated to explain thermal gradients and thermal mixing.• Conduction as well as convection dominant solutions exists for ϕ = 90° in Rayleigh–Benard convection.• Suitable heating strategies have been proposed for enhanced convective heat transfer with all Pr .
a r t i c l e i n f o
Article history:Received 20 February 2012Received in revised form25 February 2013Accepted 21 March 2013Available online 12 April 2013
The heat flow visualizationmethod based on heatfunction has been adopted to analyze natural convectionvia differential heating (case 1) and Rayleigh–Benard convection (case 2) in fluid filled (Prandtl number,Pr = 0.015–1000) rhombic enclosures with various inclination angles, ϕ, using the Galerkin finiteelement method for the range of Rayleigh number, Ra = 103–105. An accurate prediction of the flowstructure and heat distribution in such configurations is of great importance due to various engineeringapplications such as solar energy systems, cooling of electronic devices, combustion, etc. The strength offluid and heat flow increases with ϕ and Ra due to an enhanced convection effect and the maximummagnitude of streamfunction (ψmax) and heatfunction (Πmax) is observed at ϕ = 90° in both cases.Conduction based static fluid solutions are also found at ϕ = 90° associated with convection baseddynamic fluid solutions in case 2 for all Pr . Heating strategies are compared via heatlines and also basedon local (Nu) and average Nusselt numbers (Nu). It is also shown that ϕ = 30° shows higher Nu in case 2compared to case 1, whereas ϕ = 90° shows higherNu in case 1 compared to case 2 for all Pr . There existsa critical rhombic angle (ϕ = 50°) such that differential heating may be the profound heating situation atϕ ≥ 50° for materials involving Pr = 1000. Also, the rhombic cavity may be an alternative geometricaldesign in convective thermal processing of fluids with vertical thermal gradient (case 2) as it establishesonly convection based dynamic solutions for all possible angles (ϕ < 90°).
Internal heat transfer and energy conservation are two impor-tant aspects of several engineering and environmental problems.A few significant applications of such an analysis are of great im-portance in combustion, bio-thermal convection, solar energy sys-tems, electronic equipment cooling, geothermal reservoirs, etc.[1–5]. These applications mainly depend on the heat transfer pro-cesses which are operative in and on the enclosure space. In turn,heat transfer rates across those enclosure spaces are determined by
themode of heat transfer on the enclosure spaces, geometrical con-figurations and thermal boundary conditions. Natural convection isan attractive system in thermal control as the transport process influid is controlled by the difference in density with a gravitationalfield. Natural convection in enclosures has been the subject of re-cent research due to its low cost, reliability and simplicity in use.Even though investigators have concentrated more on numericalinvestigations on natural convection in complex enclosures in re-cent past [6–9], a large number of studies are still carried out tolearn even more about rectangular and square geometrical config-urations [10–20].
Natural convection in differentially heated square enclosureshas been studied extensively by Davis [21]. Davis and Jones [22]obtained a benchmark numerical solution of buoyancy driven flowin a square cavity with vertical walls at different temperaturesand adiabatic horizontal walls at the range of Rayleigh numbers
30 R. Anandalakshmi, T. Basak / European Journal of Mechanics B/Fluids 41 (2013) 29–45
Nomenclature
g Acceleration due to gravity, m s−2
L Side of the rhombic cavity, mN Total number of nodesn Normal vector to the planeNu Local Nusselt numberNu Average Nusselt numberp Pressure, PaP Dimensionless pressurePr Prandtl numberR Residual of weak formRa Rayleigh numberT Temperature of the fluid, KTh Temperature of hot wall, KTc Temperature of cold wall, Ku x component of velocity, m s−1
U x component of dimensionless velocityv y component of velocity, m s−1
V y component of dimensionless velocityX Dimensionless distance along the x coordinatex Distance along the x coordinate, mY Dimensionless distance along the y coordinatey Distance along the y coordinate, m
ϕ Inclination angle with the positive direction of the Xaxis
Φ Basis functionsψ Dimensionless streamfunctionΠ Dimensionless heatfunctionϵ Error in heat balance within the cavity
Subscripts
avg Spatial averageb Bottom walli Global node numberk Local node numberl Left wallr Right walls Side wallt Top wall
between 103 and 106. Aydin and Yang [23] numerically investi-gated natural convection of air in a rectangular enclosure locallyheated from below and symmetrically cooled from the two ver-tical sides. Ntibarufata et al. [24] analyzed the natural convectionin partitioned enclosures with localized heating from below. Con-vective motion in a bottom heated square enclosure was studiedby Oosthuizen [25] and the results were used to determine theconditions under which convective motion develops. Numericalsimulations of laminar, steady, two-dimensional natural convec-tion flows in a square enclosure with discrete heat sources on theleft and bottom walls were presented by Chen and Chen [26]using a finite-volume method. Recently, stochastic bifurcationanalysis within two-dimensional square enclosures have been in-vestigated by Venturi et al. [27] for Rayleigh–Benard convection
using stochastic modeling approaches. Kahveci and Oeztuna [11]presented a linear stability analysis for the onset of natural con-vection in a horizontal nanofluid layer. They concluded that theoscillatory instability is possible when nanoparticles concentratenear the bottom of the layer and the density gradient caused bysuch a bottom-heavy nanoparticle distribution competes with thedensity variation caused by heating from the bottom. They furtherestablished that the instability is almost purely a phenomenon dueto buoyancy coupled with the conservation of nanoparticles.
Even though several complicated enclosure models have beendesigned for investigating and describing natural convection pro-cesses in energy related applications [28–31], considerable effortsare still needed to obtain a unique enclosure design for efficientheat transfer processes. For example, one of the prime objectivesin electronic industries exists in developing the most appropriatecooling technology to neutralize heat dissipation in devices. Therole of constant-flux heat sources flush-mounted on the bottomwall of the horizontal and planar square cavity was quantitativelydepicted by Banerjee et al. [4]. Quantitative predictions on heaterlength and heater strength ratios help in determining the range inwhich devices operate within the specified thermal limit [4]. Theeffects of aspect ratio, inclination angles, and heat source lengthon the convection and heat transfer process in rectangular cavi-ties with a constant-flux heat source symmetrically embedded atthe bottom wall were analyzed by Sharif and Mohammad [32] forcooling of electronic components.
In view of the various applications of energy efficient processes,a comprehensive understanding of heat transfer and flow circula-tions within non-rectangular cavities is very much essential forindustrial development. Rhombic enclosures have attracted ourattention as various shapes including squares can be formed basedon inclination angles. Also, rhombic cavities are simple geometricalshapes which naturally evolve in various applications such asbuilding structures, solar collectors, electronics thermal controlgeothermal applications, etc. Although a few number of numericalinvestigations have been carried out in rhombic enclosures [33,34], the detailed analysis of heat flow was poorly understood dueto an improper visualization tool. Moukalled and Acharya [33]studied the effect of vertical eccentricity on natural convection ina rhombic annulus using the relationship among the controllingparameters such as enclosure gap values, eccentricity and Rayleighnumber on flow strength and heat transfer. Moukalled et al. [34]also analyzed heat transfer and flow patterns in an enclosurebetween two isothermal concentric cylinders of rhombic crosssections with various enclosure gaps and three different rhombicangles.
The present article attempts to analyze natural convectionwithin rhombic cavities with various inclination angles of sidewalls based on the heatline approach. The heat flow visualizationin the case of convection heat transfer through a two-dimensionaldomain is non-trivial as heatflux lines are non-orthogonal to theisotherms to analyze the direction and intensity of the convectivetransport processes. In order to analyze the convection heat trans-fer, there is a need for a proper tool by which one can visualize theheat flow distributions, and heatlines have been used to illustratethe flow of heat within cavities.
The ‘heatline’ concept was first developed by Kimura and Be-jan [35] to visualize convective heat transfer. Bejan [36] alsoanalyzed the heatline approach for various physical situations.Heatfunctions are energy analog of streamfunctions in such a waythat the former intrinsically satisfies the thermal energy equationfor energy flows, while the latter plays the same role in the masscontinuity equation to explain fluid dynamics. Heatlines are math-ematically represented by heatfunctions which are in turn relatedto the Nusselt number based on a proper dimensionless form. Afew number of articles were presented using this heatline conceptfor various physical situations [37–44].
R. Anandalakshmi, T. Basak / European Journal of Mechanics B/Fluids 41 (2013) 29–45 31
a b
Fig. 1. Schematic diagram of a rhombic cavity for various inclination angles ϕ in (a) case 1: differential heating and (b) case 2: Rayleigh–Benard convection.
The objective of this article is to analyze the thermal energy andfluid flow through the rhombic enclosures of insulated horizontalwalls with isothermally heated left wall and isothermally cooledright wall (case 1: differential heating) and insulated vertical wallswith isothermally heated bottomwall and isothermally cooled topwall (case 2: Rayleigh–Benard convection) for various engineer-ing applications. In addition, the heatline concept is used for thecurrent investigation to visualize the path and magnitude of heatflow, which can provide a better understanding of the heat energydistribution within the rhombic enclosures. This study also docu-ments in great detail on how the heat transfer for a specific caseis affected by the competition among various controlling parame-ters such as rhombic angle, heating strategy, Rayleigh number, andPrandtl number via heatlines, local and average Nusselt numbers.
2. Mathematical modeling and simulation
2.1. Governing equations, boundary conditions and solution proce-dure
The physical domain of a rhombus shaped cavity with side Land the left wall inclined at an angle ϕ with the X axis is shownin Fig. 1(a)–(b) for various boundary conditions (case 1 and case 2).The fluid is considered as Newtonian and the fluid properties areassumed to be constant except the density for the body force termwhere the density varies linearly with temperature according tothe Boussinesq approximation. Incompressible and laminar fluidflowwith no slip is assumed along the solid boundaries of rhombiccavities.
Under these assumptions, the governing equations for steadytwo-dimensional natural convection flow in a rhombic cavity usingconservation of mass, momentum and energy can be written interms of the following dimensionless variables or numbers:
X =xL, Y =
yL, U =
uLα, V =
vLα, θ =
T − TcTh − Tc
P =pL2
ρα2, Pr =
ν
α, Ra =
gβ(Th − Tc)L3Prν2
(1)
as∂U∂X
+∂V∂Y
= 0, (2)
U∂U∂X
+ V∂U∂Y
= −∂P∂X
+ Pr∂2U∂X2
+∂2U∂Y 2
, (3)
U∂V∂X
+ V∂V∂Y
= −∂P∂Y
+ Pr∂2V∂X2
+∂2V∂Y 2
+ Ra Pr θ, (4)
U∂θ
∂X+ V
∂θ
∂Y=∂2θ
∂X2+∂2θ
∂Y 2. (5)
The boundary conditions for case 1 and case 2 are written asfollows.Case 1: Differential heating
U = 0, V = 0,∂θ
∂Y= 0,
for Y = 0, 0 ≤ X ≤ 1 on AB,U = 0, V = 0, θ = 1,
for X sin(ϕ)− Y cos(ϕ) = 0, 0 ≤ Y ≤ sin(ϕ) on AD ,U = 0, V = 0, θ = 0,
for X sin(ϕ)− Y cos(ϕ) = sin(ϕ), 0 ≤ Y ≤ sin(ϕ) on BC,
U = 0, V = 0,∂θ
∂Y= 0, for Y = sin(ϕ),
cos(ϕ) ≤ X ≤ 1 + cos(ϕ) on DC.(6)
Case 2: Rayleigh–Benard convectionU = 0, V = 0, θ = 1,
for Y = 0, 0 ≤ X ≤ 1 on AB,
U = 0, V = 0,∂θ
∂n= 0,
for X sin(ϕ)− Y cos(ϕ) = 0, 0 ≤ Y ≤ sin(ϕ) on AD,
U = 0, V = 0,∂θ
∂n= 0,
for X sin(ϕ)− Y cos(ϕ) = sin(ϕ), 0 ≤ Y ≤ sin(ϕ) on BC,
U = 0, V = 0, θ = 0, for Y = sin(ϕ),cos(ϕ) ≤ X ≤ 1 + cos(ϕ) on DC.
(7)
Note that in Eqs. (1)–(7), X and Y are the dimensionlesscoordinates varying along the horizontal and vertical directions,respectively; U and V are the dimensionless velocity componentsin the X and Y directions, respectively; θ is the dimensionlesstemperature; P is the dimensionless pressure; Ra and Pr are theRayleigh and Prandtl numbers, respectively; Th and Tc are thetemperatures at hot and cold walls, respectively; L is the each sideof the rhombic cavity; ϕ is the inclination angle with the positivedirection of the X axis.
Themomentum and energy balance equations (Eqs. (3)–(5)) aresolved using the Galerkin finite element method. The continuityequation (Eq. (2)) has been used as a constraint due to mass con-servation and this constraint may be used to obtain the pressuredistribution. In order to solve Eqs. (3) and (4), we use the penaltyfinite element method where the pressure (P) is eliminated by apenalty parameter (γ ) and the incompressibility criteria given byEq. (2) which results in
P = −γ
∂U∂X
+∂V∂Y
. (8)
32 R. Anandalakshmi, T. Basak / European Journal of Mechanics B/Fluids 41 (2013) 29–45
The continuity equation (Eq. (2)) is automatically satisfied forlarge values of γ . Typical values of γ that yield consistent solutionsare 107. Using Eq. (8), the momentum balance equations (Eqs. (3)and (4)) reduce to
U∂U∂X
+ V∂U∂Y
= γ∂
∂X
∂U∂X
+∂V∂Y
+ Pr
∂2U∂X2
+∂2U∂Y 2
, (9)
and
U∂V∂X
+ V∂V∂Y
= γ∂
∂Y
∂U∂X
+∂V∂Y
+ Pr
∂2V∂X2
+∂2V∂Y 2
+ Ra Pr θ. (10)
The system of equations (Eqs. (5), (9) and (10)) with boundaryconditions are solved by the Galerkin finite element method [45].Expanding the velocity components (U, V ) and temperature (θ)using the basis set Φk
Nk=1 as
U ≈
Nk=1
Uk Φk(X, Y ), V ≈
Nk=1
Vk Φk(X, Y ), and
θ ≈
Nk=1
θk Φk(X, Y ).
(11)
The Galerkin finite element method yields the nonlinear resid-ual equations for Eqs. (5), (9) and (10) at the nodes of the inter-nal domain Ω . The detailed solution procedure is given in earlierworks [19,46].
2.2. Streamfunction, Nusselt number and heatfunction
2.2.1. StreamfunctionThe fluid motion is displayed using the streamfunction (ψ)
obtained from velocity components (U and V ). The relation-ships between streamfunction and velocity components for two-dimensional flows are [47,48]
U =∂ψ
∂Yand V = −
∂ψ
∂X, (12)
which yield a single equation
∂2ψ
∂X2+∂2ψ
∂Y 2=∂U∂Y
−∂V∂X. (13)
The no slip condition is valid at all boundaries as there is nocross flow. Hence, ψ = 0 is used in the residual equation at theboundary nodes. Using the above definition of the streamfunction,the positive sign of ψ denotes anti-clockwise circulation and theclockwise circulation is represented by the negative sign of ψ . Ex-panding the streamfunction (ψ) using the basis set Φk
Nk=1 asψ =N
k=1 ψkΦk(X, Y ) and the relationships for U, V from Eq. (11), theGalerkin finite elementmethod yields the linear residual equationsfor Eq. (13) and thedetailed solutionprocedure to obtainψs at eachnode point is given in earlier works [19,46].
2.2.2. Nusselt numberThe heat transfer coefficient in terms of the local Nusselt num-
ber (Nu) is defined by
Nu = −∂θ
∂n, (14)
where n denotes the normal direction on a plane. The normalderivative is evaluated by the biquadratic basis set in the ξ − η
domain. The local Nusselt numbers at the bottom wall (Nub), topwall (Nut), left wall (Nul) and right wall (Nur) are defined as
Nub =
9i=1
θi∂Φi
∂Y(15)
Nut = −
9i=1
θi∂Φi
∂Y(16)
Nul =
9i=1
θi
sinϕ
∂Φi
∂X− cosϕ
∂Φi
∂Y
(17)
and
Nur = −
9i=1
θi
sinϕ
∂Φi
∂X− cosϕ
∂Φi
∂Y
. (18)
The average Nusselt number at the bottom, top and inclined sidewalls are given by
Nub =
10 Nub dX
X |10
=
1
0Nub dX,
Nut =
1+cosϕ
cosϕNut dX,
Nul =
1
0Nul dS1,
and
Nur =
1
0Nur dS2. (19)
Here dS1 and dS2 are the small elemental lengths along the left andrightwalls, respectively. The averageNusselt numbers are also use-ful to benchmark overall heat transfer rates within the cavity. Notethat Nul = Nur for case 1 and Nub = Nut for case 2.
The percentage of error (ϵ) in heat balances within the cavitycan be calculated using the average Nusselt numbers at the left,right, bottom and top walls as follows.Case 1
ϵ =|Nul − Nur |
min[Nul, Nur ]× 100. (20)
Case 2
ϵ =|Nub − Nut |
min[Nub, Nut ]× 100. (21)
2.2.3. HeatfunctionThe heat flow within the enclosure is displayed using the heat-
function (Π ) obtained from conductive heatfluxes (− ∂θ∂X ,−
∂θ∂Y ) as
well as convective heatfluxes (Uθ, Vθ ). The heatfunction satisfiesthe steady energy balance equation (Eq. (5)) [35] such that
∂Π
∂Y= Uθ −
∂θ
∂X,
−∂Π
∂X= Vθ −
∂θ
∂Y(22)
which yield a single equation
∂2Π
∂X2+∂2Π
∂Y 2=
∂
∂Y(Uθ)−
∂
∂X(Vθ) . (23)
Using the above definition of the heatfunction, the positive signofΠ denotes anti-clockwise heat flow and the clockwise heat flow
R. Anandalakshmi, T. Basak / European Journal of Mechanics B/Fluids 41 (2013) 29–45 33
Table 1Comparison of the average Nusselt number [Nul (case 1) or Nub (case 2)] for variousgrid systems for case 1 and case 2 at Ra = 105 and Pr = 0.7with various inclinationangles (ϕ).
is represented by the negative sign ofΠ . Expanding the heatfunc-tion (Π ) using the basis set Φk
Nk=1 asΠ =
Nk=1ΠkΦk(X, Y ) and
the relationship for U, V and θ from Eq. (11), the Galerkin finite el-ement method yields the linear residual equations for Eq. (23) andthe detailed solution procedure to obtainΠs at each node point isgiven in earlier works [19,46].
The residual equation (Eq. (23)) is further supplemented withvarious Dirichlet and Neumann boundary conditions in order toobtain a unique solution of Eq. (23). The Neumann boundarycondition ofΠ obtained from Eq. (22) is specified as
n · ∇Π = 0 (for isothermal hot/cold wall in both cases). (24)
The top and bottom insulated walls may be represented by theDirichlet boundary condition for case 1 as obtained from Eq. (22)which is simplified into ∂Π
∂X =∂θ∂Y = 0 for an adiabatic wall and a
reference value ofΠ = 0may be assumed at Y = 0which leads toΠ = 0 ∀X at Y = 0 andΠ = Nu ∀X at Y = sinϕ for case 1. Sim-ilarly, the left and right insulated walls may be represented by theDirichlet boundary condition for case 2 as obtained from Eq. (22)which is simplified into ∂Π
∂t =∂θ∂n = 0 for an adiabatic wall and a
reference value ofΠ = 0may be assumed atX sin(ϕ)−Y cos(ϕ) =
sinϕ which leads toΠ = 0 ∀Y at X sin(ϕ)− Y cos(ϕ) = sinϕ andΠ = Nu ∀Y at X sin(ϕ) − Y cos(ϕ) = 0 for case 2. It may also benoted that most of the earlier works [35,37,39,44] are also limitedwithin two adiabatic walls where the Dirichlet boundary conditionis either 0 or Nu at the adiabatic walls for fluids confined withinsquare cavities.
3. Results and discussion
3.1. Numerical procedure and validation
The computational grid within the rhombus is generated viamapping the rhombus into a square domain in the ξ−η coordinatesystem as mentioned in our earlier works [49,50]. The current so-lution scheme produces grid invariant results as shown in Table 1for cases 1 and 2. The computational grid in ξ −η coordinates con-sists of 28×28 biquadratic elements which correspond to 57×57grid points. The biquadratic elements with fewer number of nodessmoothly capture the nonlinear variations of the field variableswhich are in contrast with finite difference/finite volume solu-tions [51–53]. In order to assess the accuracy of our numerical pro-cedure, we have tested our algorithm based on the grid size (57 ×
57) for a square enclosure filled with air (Pr = 0.71) subject to hotleft wall and cold right wall in the presence of insulated horizontalwalls at Ra = 103, 104 and 105 as carried out in an earlierwork [44]and the results are in good agreement as shown in Fig. 2. Further,Table 2 shows that |ψ |max and Nu agree with the results of ear-lier researchers [21,44] for a range of Rayleigh numbers (Ra =
103–105).Detailed computations were carried out for various values of
Pr (Pr = 0.015, 0.7, 7.2 and 1000), inclination angles (ϕ = 30°,45°, 75° and 90°) and Ra (Ra = 103–105). The Galerkin finiteelement approach as used in the current work also offers specialadvantage on evaluation of the local Nusselt number at the side
Table 2Comparisons of the present results with the benchmark resolutions of G.D.Davis [21] and Deng et al. [44] for natural convection in an air (Pr = 0.71) filledsquare cavity with 28 × 28 biquadratic elements.
Ra Present G.D. Davis [21] Deng et al. [44]|ψ |max Nu |ψ |max Nu |ψ |max Nu
walls and horizontal walls as element basis functions have beenused to obtain heatflux, whereas finite difference/finite volumebasedmethods involve interpolation functions to calculate Nusseltnumbers at the surface [51–53].
In order to validate heatfunction contours or heatlines, we havecarried out simulations for a range of Rayleigh numbers (Ra =
103–105) and Prandtl numbers (Pr = 0.015–1000). At low Ra, thefluid is almost stagnant and heat transfer is conduction dominant.Under these conditions, heatlines essentially represent ‘heatflux’lines, which are commonly used for conductive heat transport. Inaddition, heatflux lines, by definition, are perpendicular to isother-mal surfaces and parallel to adiabatic surfaces. It is observed thatthe heatlines emanate from a hot surface and end on a cold surfacewhich are perpendicular to isothermal surfaces, similar to heat-flux lines, during a conduction dominant regime. As they approachthe adiabatic wall, they slowly bend and become parallel to thatsurface. Also, the heatlines and isotherms are found to be smoothcurves, without any distortion in the presence of dominant con-ductive heat transport. These features are discussed in Sections 3.2and 3.3.
The sign of heatfunction needs special mention. The sign ofheatfunction is governed by the sign of the ‘non-homogeneous’Dirichlet condition. In the current situation, the negative sign ofheatlines represents the clockwise flow of heat, while the pos-itive sign refers to the anti-clockwise flow. This assumption isin accordance with the sign convention for streamfunction. Thestreamfunction and heatfunction have identical signs for convec-tive transport. The detailed discussion on heat transport based onheatlines for various cases is presented in the following sections.
3.2. Case 1: differential heating
Figs. 3–5 show the effects of Ra = 103–105 and Pr = 0.015–1000 on fluid flow and heat transfer in rhombic cavities for variousϕs (30°, 45°, 75° and 90°) induced by the hot isothermal left wall.Fig. 3(a)–(d) illustrate the distribution of the streamfunction (ψ),heatfunction (Π ) and temperature (θ ) contours for Ra = 103 andPr = 0.015. The fluid flow for various ϕs in rhombic cavities maybe explained with streamlines (ψ) for Pr = 0.015 and Ra = 103
(Fig. 3(a)–(d)). Due to buoyancy and imposed temperature gradientbetween hot isothermal left wall and cold isothermal right wall,hot fluid rises from the left wall and cold fluid flows down alongthe right wall with a clockwise unicellular flow pattern withinthe cavity. Note that the fluid circulation cell is not circular andelongated diagonally due to geometrical asymmetry with respectto the central vertical line for ϕ = 30° and 45° (Fig. 3(a)–(b)).The non-circular flow is further induced due to higher inclinationsof the side walls with the bottom wall. As ϕ increases, the fluidcirculation shape gradually approaches to be circular and it isalmost circular for 90° (Fig. 3(c)–(d)). The strength of the fluidcirculation cell tends to become stronger with ϕ. Overall, theflow strength within the cavity is very weak due to conductiondominant heat transfer at Ra = 103. Note that |ψ |max is 0.16,0.43, 1.04 and 1.13 for ϕ = 30°, 45°, 75° and 90°, respectively (seeFig. 3(a)–(d)).
34 R. Anandalakshmi, T. Basak / European Journal of Mechanics B/Fluids 41 (2013) 29–45
a
b
c
Fig. 2. Streamfunction (ψ), heatfunction (Π ) and temperature contours (θ ) for the hot left wall, cold right wall with adiabatic horizontal walls at Pr = 0.71, ϕ = 90°and (a) Ra = 103; (b) Ra = 104; (c) Ra = 105 [44]. Clockwise and anti-clockwise flows are shown via the negative and positive signs of streamfunction and heatfunction,respectively.
The heat flow distribution for various ϕs in rhombic enclosuresis explained with heatlines. It may be noted that the heat flowoccurs from hot to cold wall, and the negative heatfunction cor-responds to clockwise circulatory heat flow. It is interesting toobserve that the heatlines are slightly distorted for ϕ = 30° and45° (Fig. 3(a)–(b)) and the wall-to-wall heatlines in the top portionof the cavity are compressed towards the top wall for ϕ = 75° and90° (Fig. 3(c)–(d)) due to the presence of closed loop heatlines. Theclosed loop heatlines denote convection heat circulation due to theonset of convection at higher ϕs (ϕ = 75° and 90°). It is interestingto note that the heatlines are distorted at the center due to convec-tion dominant heat transport based on |ψ |max values for all ϕs atPr = 0.015 and Ra = 103 (Fig. 3(a)–(d)). It may also be notedthat the characteristic features of heatlines in all configurationssignify convectionheat transfer despite its lowmagnitudes (|Π |maxis 1.06, 1.04, 1.10 and 1.11 for ϕ = 30°, 45°, 75° and 90°, respec-tively) as the heatlines are distorted enough even at Ra = 103 forPr = 0.015 (see Fig. 3(a)–(d)). It is also found that the heatlines aredistorted at the center due to high convective heat flow. Hence,isotherms are distorted much at the center of the cavity for all ϕs.The boundary layer thickness is highnear the lower left portion andthe top right portion of the cavity as indicated by disperse heatlinesat ϕ = 30° and 45°. It is also interesting to note that themajor heatflow occurs from a small portion of the upper part of the left hot
wall to the cold rightwall, whereas the lower half portion of the hotleft wall gives lesser heat to the lower portion of the cold right wallfor ϕ = 30° and 45° and therefore a large portion of the lower leftwall is stagnant with the hot fluid, whereas a large portion of thetop rightwall is stagnantwith the cold fluid (see Fig. 3(a)–(b)). Asϕincreases to 90° (see Fig. 3(d)), the boundary layer thickness nearthe lower left portion and the top right portion of the cavity de-creases due to significant heat flow near those regions and thatis indicated by high heatfunction gradient 0.1 ≤ |Π | ≤ 0.5 and0.5 ≤ |Π | ≤ 1 near the lower left portion and the upper rightportion of the cavity, respectively (see Fig. 3(d)).
Dominant convective transport is clearly illustrated by stream-lines and heatlines at Ra = 105 irrespective of ϕ. Fig. 4(a)–(d)shows the effect of enhanced convection at Ra = 105 due tostronger buoyancy forces over viscous forces. A number of mul-tiple circulation cells are observed near the top right corner andbottom left corner of the cavity. The intensity of fluid motion isgreatly increased compared to that of Ra = 103 as indicated by|ψ |max at Ra = 105. It may be noted that |ψ |max = 3.24, 4.38, 6.28and 7.48 forϕ = 30°, 45°, 75° and 90°, respectively, due to the pri-mary circulation cell at Ra = 105, whereas |ψ |max is 0.16, 0.43, 1.04and 1.13 for ϕ = 30°, 45°, 75° and 90°, respectively, within theprimary circulation cell at Ra = 103 (Figs. 4(a)–(d) and 3(a)–(d)).As ϕ increases, the central primary circulation cell grows bigger
R. Anandalakshmi, T. Basak / European Journal of Mechanics B/Fluids 41 (2013) 29–45 35
a
b
c
d
Fig. 3. Streamfunction (ψ), heatfunction (Π ) and temperature contours (θ ) for case 1 with Pr = 0.015 and Ra = 103 for (a) ϕ = 30°, (b) ϕ = 45°, (c) ϕ = 75° and(d) ϕ = 90°. Clockwise and anti-clockwise flows are shown via the negative and positive signs of streamfunction and heatfunction, respectively. White and black shades inθ represent the hot and cold fluid inside the cavity, respectively.
and fills the entire part of the cavity at Ra = 105 and Pr = 0.015(Fig. 4(b)–(d)). It is also interesting to note thatmultiple circulationcells are found to occur near all corners of the cavity at ϕ = 90°(Fig. 4(d)).
It is interesting to observe that the trend on heat flow patternsvia heatline cells and heat flow distribution within the cavity isgreatly enhanced from Ra = 103 to Ra = 105 corresponding toeach specific angle. It is also interesting to observe that heatlinecirculation cells which are indicative of convective transport aremuch stronger for Ra = 105 than those for Ra = 103. Also, asϕ increases, the convective heat transport in the central portionof the cavity gradually becomes stronger for Ra = 105. Note that|Π | = 0.5, 1, 1.6 and 2 occur for ϕ = 30°, 45°, 75° and 90°, re-spectively, near the core of the heat circulation cells at Ra = 105. Itis interesting to observe that convective heatline cells and stream-lines are almost identical in shape at the core of the cavity. As aresult, the isotherms are distorted largely at the core of the cavityespecially for ϕ = 75° and 90°. It is interesting to note that local-ized heat circulation occurs near the bottom left corner of the cav-ity due to the presence of tiny heat circulation cell in that regimefor ϕ = 30° and 45° (see Fig. 4(a)–(b)).
It is also found that secondary heat circulation cells correspond-ing to secondary fluid circulation cells are observed near the lowerleft portion of the cavity with |Π | = 0.1 and that is absent nearthe top right portion of the cavity despite the presence of secondaryfluid circulation cells atϕ = 30° and 45°. This is due to the fact thatthe strong primary heat circulation cells further obstruct the flowof heat from the lower left wall to the top right wall and thereforesecondary heat circulation cells are absent near the top right wallas convection is not strong enough in that region for ϕ = 30° and
45°. Further,wall-to-wall heatlines also show that the heat transferis due to the conduction dominant effect near the top right portionof the cavity for ϕ = 30° and 45° (see Fig. 4(a)–(b)). As a result ofhigh heat transport to the cold rightwall, the boundary layer thick-ness along the side walls is reduced for all ϕs (see Fig. 4(a)–(d)).
It is also interesting to note that closed loop heatlines convecta large amount of heat along the core. Hence, the amount of heattransfer from the hot left wall to the cold right wall is higher nearthe region where the closed loop heatline cells are present. This isillustrated by highly dense heatlines near those regions for all ϕs(see Fig. 4(a)–(d)). It may be noted that 0.1 ≤ |Π | ≤ 0.5 and 0.1 ≤
|Π | ≤ 1.5 occur for ϕ = 30° and 45°, respectively, near the lowerportion of the cold right wall (see Fig. 4(a)–(b)). This effect is morepronounced asϕ increases since the size of the closed loop heatlinecells increases with ϕ. Consequently, 0.5 ≤ |Π | ≤ 2.0 occur forϕ = 75° and 90° near themiddle portion of the cold right wall (seeFig. 4(c)–(d)). This is also illustrated by isotherms that are largelycompressed near those regions and the region of compressionincreases with ϕ. It may be noted that θ ≤ 0.3 are observed nearthe lower right portion of the cavity for ϕ = 30°, whereas θ ≤ 0.6are observed near the top right portion of the cavity forϕ = 90°. Asϕ increases, the region of heat flow moves from the lower portionto the upper portion along the cold right wall and therefore, theboundary layer thickness near the lower portion of the cold rightwall increases with ϕ (see Fig. 4(a)–(d)). An increase in the sizeof the closed loop heat circulation cells further compresses thewall-to-wall heatlines near the left hot wall of the cavity. As ϕincreases, the zone of highly dense heatlines moves towards thelower portion of the hot left wall and therefore the boundary layerthickness near the top portion of the hot left wall increases with
36 R. Anandalakshmi, T. Basak / European Journal of Mechanics B/Fluids 41 (2013) 29–45
a
b
c
d
Fig. 4. Streamfunction (ψ), heatfunction (Π ) and temperature contours (θ ) for case 1 with Pr = 0.015 and Ra = 105 for (a) ϕ = 30°, (b) ϕ = 45°, (c) ϕ = 75° and(d) ϕ = 90°. Clockwise and anti-clockwise flows are shown via the negative and positive signs of streamfunction and heatfunction, respectively. White and black shades inθ represent the hot and cold fluid inside the cavity, respectively.
ϕ (see Fig. 4(a)–(d)). It may be noted that 0.5 ≤ |Π | ≤ 1.5 occurnear the upper portion of the hot left wall for ϕ = 30°, whereas0.5 ≤ |Π | ≤ 2.0 occur near the lower portion of the hot left wallforϕ = 90° (see Fig. 4(a)–(d)). Due to strong convective closed loopheatline cells, isotherms are highly distorted in the middle portionof the cavity for all ϕs (see Fig. 4(a)–(d)).
Distributions of streamlines, heatlines and isotherms for Pr =
7.2 and Ra = 105 are displayed in Fig. 5(a)–(d). The intensity offluid circulation is found to be stronger for Pr = 7.2 comparedto Pr = 0.015 as seen from the maximum values of streamfunc-tions (|ψ |max) near the core of the cavity for all ϕs (Figs. 5(a)–(d)and 4(a)–(d)). Further, they expand and take the shape of the cavitynear the cavity walls due to intense convection unlike in the pre-vious case with Pr = 0.015. In contrast to the previous cases withPr = 0.015, streamline cells near the core of the cavity split intotwo circulation cells for higher ϕs (ϕ = 75°–90°) (Fig. 5(c)–(d)).The multiple fluid circulation cells near the corners of the cavity asobserved for Pr = 0.015 are absent for Pr = 7.2 (Fig. 5(a)–(d)).Closed loop heatlines follow the shape of the cavity for Pr = 7.2in contrast to the earlier case (Pr = 0.015) for all ϕs. Due to largeheatline convective cells, the trajectories of heatlines from the hotleft wall take long paths to reach the cold right wall for all ϕs (seeFig. 5(a)–(d)). Further, larger portions of the hot wall deliver heatto the cold wall based on dense heatlines at higher Pr (Pr = 7.2).Also, heat transfer to the top right portions of the cavity is en-hanced as seen froma large thermal gradient based onhighly denseheatlines. As ϕ increases, the zone of highly dense heatlines movesfrom the middle portion to the lower portion of the hot left wall,whereas it moves from the middle portion to the upper portion ofthe cold right wall. This is illustrated by highly dense heatlines as
0.1 ≤ |Π | ≤ 1.5 and 1 ≤ |Π | ≤ 1.5 occuring near the mid-dle portion of the hot left wall and cold right wall, respectively, forϕ = 30° whereas 0.1 ≤ |Π | ≤ 4 and 0.5 ≤ |Π | ≤ 4 occuringnear the lower portion of the hot left wall and the upper portionof the cold right wall, respectively, for ϕ = 90° (see Fig. 5(a)–(d)).Note that the amount of heat transfer is high near the regionwherethe highly dense heatlines are present as illustrated by largelycompressed isotherms near those regions. As a result of largelycompressed isotherms due to the larger portion of highly denseheatlines, the boundary layer thickness is small near the lower por-tions of the hot left wall and the upper portions of the cold rightwall compared to the previous casewith Pr = 0.015 (Fig. 4(a)–(d)).In contrast to Pr = 0.015, as ϕ increases, the heat circulation cellnear the core of the closed loop heatline cells moves towards thehot left wall illustrating that more convective heat transfer takesplace from the hot left wall to the cold right wall for Pr = 7.2and Ra = 105 due to convection induced by flow circulation asshown in Fig. 5(a)–(d). High heat flow is also represented by heat-lines of high magnitude of heatfunctions for Pr = 7.2 comparedto Pr = 0.015. Note that |Π |max is 2.45, 3.53, 4.60, and 4.77 forϕ = 30°, 45°, 75° and 90°, respectively, for Pr = 7.2, whereas|Π |max is 1.82, 2.41, 3.21, and 3.64 for ϕ = 30°, 45°, 75° and 90°,respectively, for Pr = 0.015 at Ra = 105 (see Figs. 4(a)–(d) and5(a)–(d)). As discussed earlier, closed loop heatlines convect a largeamount of heat from the hot wall to the cold wall and thereforethermal mixing is high near those regions. In contrast to the pre-vious case with Pr = 0.015, thermal mixing is improved for thepresent case with Pr = 7.2 as the closed loop heatlines occupyalmost 60%–75% of the cavity for all ϕs (Fig. 5(a)–(d)). The qualita-tive trends of the streamlines and heatlines at higher Pr (=1000)
R. Anandalakshmi, T. Basak / European Journal of Mechanics B/Fluids 41 (2013) 29–45 37
a
b
c
d
Fig. 5. Streamfunction (ψ), heatfunction (Π ) and temperature contours (θ ) for case 1 with Pr = 7.2 and Ra = 105 for (a) ϕ = 30°, (b) ϕ = 45°, (c) ϕ = 75° and (d) ϕ = 90°.Clockwise and anti-clockwise flows are shown via the negative and positive signs of streamfunction and heatfunction, respectively. White and black shades in θ representthe hot and cold fluid inside the cavity, respectively.
are similar to that of Pr = 7.2 for all ϕs and results for higher Prare not shown for brevity of the manuscript.
3.3. Case 2: Rayleigh–Benard convection
Figs. 6(a)–(d)–8(a)–(d) illustrate the streamlines, heatlines andisotherms for Rayleigh–Benard convection with Pr = 0.015 and1000. Although streamlines exhibit qualitatively similar trends ofthe differential heating case for Pr = 0.015 and Ra = 103 as seenin Figs. 3 and 6, both magnitudes of streamfunctions and intensityof circulations are found to be higher for the differential heatingcase (case 1) with all ϕs due to larger heating effects. Similar to thedifferential heating case (case 1), heatlines are slightly distorted inthe middle portion of the cavity for ϕ ≤ 75° (see Fig. 6(a)–(c)),but parallel to adiabatic walls at ϕ = 90° for Ra = 103 and Pr =
0.015 (see Fig. 6(d)). It is also found that disperse heatlines areobserved due to significant conduction dominant heat transfer forall ϕs as depicted in Fig. 6(a)–(d). It is interesting to note that largeamount of fluid especially near the top right portion of the cavity ismaintained cold for lower ϕ (ϕ = 30°) compared to higher ϕ (ϕ =
90°) (see Fig. 6(a)–(d)). This is mainly due to low heatfunctiongradient near the top right portion of the cavity (0 ≤ Π ≤ 0.2)compared to the bottom right portion of the cavity for ϕ = 30°.Similarly, disperse heatlines of low heatfunction gradient (0.8 ≤
Π ≤ 1.1) lead to stagnant hot fluid near the lower left corner ofthe cavity. Dense heatlines of high heatfunction gradient near thelower right and top left portion of the cavity illustrate the heat flowfrom the hot bottom wall to the cold top wall for ϕ = 30° (seeFig. 6(a)). As ϕ increases further (ϕ = 75°), heatlines are almostuniformly distributed with slight distortion in the center and that
results in uniform temperature gradients throughout the domainexcept near the top right and bottom left corners of the cavitywhere the poor heat distribution is observed (see Fig. 6(a)–(c)).It may also be noted that the isotherms are almost parallel toisothermal walls signifying conduction dominant heat transfer atRa = 103 for ϕ = 75° (see Fig. 6(c)). At ϕ = 90°, the heatlinesperfectly satisfy the conduction situation, which stated that duringa conduction dominant regime, heatlines represent heatflux lines.Based on the definition of heatflux lines, which are perpendicularto the isothermal surface and parallel to the adiabatic surfaceduring conduction dominant heat transfer, there is no fluidmotion(−→V ≈ 0) during Ra = 103 and ϕ = 90° (see Fig. 6(d)). As a result,
isotherms are perpendicular to the adiabatic surface and parallel tothe isothermal surface, which is essentially a conduction dominantregime and that is depicted in Fig. 6(d). Less distorted heatlines andisotherm contours illustrate comparatively less heat transfer ratesin case 2 compared to case 1 for all ϕs as displayed in heatline andisotherm contours (see Figs. 6(a)–(d) and 3(a)–(d)). The conductiondominant static fluid solution is the characteristic of the squaredomain for Rayleigh–Benard convection. Thus, the rhombic cavitymay be an alternative choice for efficient heat transfer even atRa = 103 in case 2.
It is interesting to note that the conduction based static fluidsolution is still observed in addition to the convection based dy-namic solution for ϕ = 90° irrespective of Pr at Ra ≥ 2 × 103
in case 2 (see Figs. 7(a)–(d) and 8(a)–(d)). The conduction basedstatic solution occurs only for ϕ = 90° and it is observed that theconduction based static solution disappears with a slight pertur-bation of ϕ at higher Ra (Ra ≥ 2 × 103) irrespective of Pr (see
38 R. Anandalakshmi, T. Basak / European Journal of Mechanics B/Fluids 41 (2013) 29–45
a
b
c
d
Fig. 6. Streamfunction (ψ), heatfunction (Π ) and temperature contours (θ ) for case 2 with Pr = 0.015 and Ra = 103 for (a) ϕ = 30°, (b) ϕ = 45°, (c) ϕ = 75° and(d) ϕ = 90°. Clockwise and anti-clockwise flows are shown via the negative and positive signs of streamfunction and heatfunction, respectively. White and black shades inθ represent the hot and cold fluid inside the cavity, respectively.
Figs. 7(a)–(d)–8(a)–(d)). The multiple solutions involving the con-duction based static fluid solutions were also reported by Venturiet al. [27]. The present analysis is restricted to only the conductionbased static solution and the convection based dynamic solution atϕ = 90°.
As Ra increases to Ra = 105, the onset of convection showsintensive multiple clockwise streamline circulation cells for Pr =
0.015 and Ra = 105 for all ϕs in case 2. The maximum magni-tudes of the streamfunctions are comparatively higher than case1 for all ϕs (see Fig. 7(a)–(d(ii))). As mentioned earlier, both con-duction dominant (ψ = 0) and convection dominant fluid circu-lations (ψmax = 24.44) are observed for ϕ = 90° at Pr = 0.015and Ra = 105 (see Fig. 7(d)). But the conduction dominant heattransfer does not occur for other rhombic angles (ϕ ≤ 90°) (seeFig. 7(a)–(c)). As a result of conduction dominant heat transfer,there is no intense fluid motion at ϕ = 90° for Pr = 0.015 andRa = 105 (see Fig. 7(d(i))). It is also observed that the size andmagnitude of the primary circulation cells increase with ϕ as seenfrom Fig. 7(a)–(d(ii)). It is also interesting to note that the size andstrength of the secondary and tertiary circulation cells also increaseas ϕ increases from ϕ = 30° to 45° (see Fig. 7(a)–(b)). Further in-crease in ϕ results in multiple circulation cells near the corners ofthe cavity with the high magnitude single primary circulation cellnear the core of the cavity for Pr = 0.015 and Ra = 105 in case 2(see Fig. 7(c)–(d(ii))).
Similar to case 1, the size and magnitude of the closed loopheatline cells increase with ϕ. But the maximum magnitudes ofheatfunction (Πmax) and the intensity of closed loop heat circu-lation cells are comparatively higher than case 1 for all ϕs (seeFigs. 4(a)–(d)–7(a)–(d(ii))). As a result, isotherms are highly dis-torted for all ϕs compared to case 1. Due to closed loop heatline
cells, wall-to-wall heatlines are compressed near the left top andright bottomportions of the cavity and that further results in lesserboundary layer thickness near those regions due to high heat flowfor all ϕs as illustrated in Fig. 7(a)–(d(ii)). It may be noted that0.01 ≤ |Π | ≤ 0.5 and 1 ≤ |Π | ≤ 3 occur near the right portionof the upper cold wall and the left portion of the lower hot wall,respectively, for ϕ = 30° (see Fig. 7(a)). Isotherms are largely dis-torted in the central portion of the cavity due to highly dense closedloop heatlines in the central portion of the cavity for ϕ = 30° andthat further illustrates the high heat flow and thermal mixing inthat region. Also, the top right and bottom left portions of the cav-ity are filled with stagnant cold and hot fluid, respectively, due todisperse heatlines in those regions for ϕ = 30° (see Fig. 7(a)).As ϕ increases, the size and magnitude of the closed loop heat-lines increase and that results in high thermalmixing at the centralportion of the cavity (see Fig. 7(b)–(d(ii))). It is also interesting toobserve that as ϕ increases, bigger closed loop heatlines furthercompress the wall-to-wall heatlines and therefore, heatlines arefound to be dense near the top left and bottom right portions ofthe cavity due to the high heatfunction gradient near those regions(Fig. 7(b)–(d(ii))). This is clearly illustrated by the compression ofisotherms or high thermal gradient (Fig. 7(b)–(d(ii))). A large zoneof central core of the cavity ismaintained at a uniform temperatureof θ = 0.4–0.6 for the present case (case 2) in contrast to case 1where the core is maintained at θ = 0.3–0.7 for ϕ = 75° and 90°(see Figs. 7(c)–(d(ii)) and 4(c)–(d)). Overall, largely distorted heat-lines and isotherms illustrate the high heat transfer for case 2 com-pared to case 1 for identical parameters (Pr, Ra and ϕ). As a resultof conduction dominant heat transfer, heatlines and isotherms areperpendicular and parallel to the isothermalwalls, respectively, forϕ = 90° for at Pr = 0.015 and Ra = 105 (see Fig. 7(d(i))).
R. Anandalakshmi, T. Basak / European Journal of Mechanics B/Fluids 41 (2013) 29–45 39
a
b
c
d
Fig. 7. Streamfunction (ψ), heatfunction (Π ) and temperature contours (θ ) for case 2 with Pr = 0.015 and Ra = 105 for (a) ϕ = 30°, (b) ϕ = 45°, (c) ϕ = 75°,(d) (i) conduction at ϕ = 90° and (ii) convection at ϕ = 90°. Clockwise and anti-clockwise flows are shown via the negative and positive signs of streamfunction andheatfunction, respectively. White and black shades in θ represent the hot and cold fluid inside the cavity, respectively.
Fig. 8(a)–(d) illustrate the isotherms, streamlines and heatlinesfor Pr = 1000 at Ra = 105 in case 2. The intensity of flow andheat circulations are found to be stronger for Pr = 1000 thanthose with Pr = 0.015 (see Figs. 7(a)–(d(ii)), and 8(a)–(d(ii))). Butthe intensity of fluid flow near the core is higher for the presentcase (see Fig. 8(a)–(d(ii))) compared to that of case 1 for all ϕs atPr = 1000 and Ra = 105 (figures are not shown). In contrastto the previous case with Pr = 0.015, there is no multiple fluidcirculation cells for ϕ ≤ 75° (see Figs. 8(a)–(c) and 7(a)–(c)) andthere are multiple longitudinal fluid circulation cells (ψmax = 0.04for Pr = 1000 at Ra = 105) in the conduction dominant heattransfer at ϕ = 90° in case 2 due to higher momentum diffusiv-ity at Pr = 1000 compared to Pr = 0.015 (see Figs. 8(d(i)) and7(d(i))). It may also be noted that there is little distortion in theheatlines (Πmax = 1.22) at Pr = 1000 and Ra = 105 in con-trast to Pr = 0.015 in the conduction dominant heat transfer forϕ = 90° (see Fig. 8(d(i))). It is found that heatlines are highly dis-torted and the closed loop heatlines almost fill the entire cavityat larger ϕs. As a result, a large portion of the central zone of thecavity is maintained at a uniform temperature of 0.4–0.6 similar toPr = 0.015 for the same ϕ and Ra (see Figs. 7(d(ii)) and 8(d(ii))). It
may also be noted that the presence of closed loopheatlines furthercompresses the wall-to-wall heatlines and hence, heat flow is en-hanced near the top and bottomportions of the cavity as illustratedby highly compressed isotherms near those regions for all ϕs com-pared to low Pr values (see Figs. 7(d(ii)) and 8(d(ii))). This furtherresults in comparatively less boundary layer thickness near theisothermal walls for the present case compared to the low Pr case(see Figs. 7(d(ii)) and 8(d(ii))). Overall, the heat distribution andthermal mixing in Rayleigh–Benard convection (case 2) are higherthan that of the differential heating case (case 1) at larger values ofRa and that is clearly indicated by largemagnitudes of streamfunc-tions, heatfunctions and shades of isotherm contours in the grayscale plot.
3.4. Heat transfer rates: local and average Nusselt numbers
The variations of local heat transfer rates for the left wall (Nul)vs. the distance in case 1 and the bottomwall (Nub) vs. the distancein case 2 are presented in the plots of Fig. 9(a)–(f) for Pr = 0.015,0.7 and 1000 at Ra = 105. The heat transfer rates are shown only
40 R. Anandalakshmi, T. Basak / European Journal of Mechanics B/Fluids 41 (2013) 29–45
a
b
c
d
Fig. 8. Streamfunction (ψ), heatfunction (Π ) and temperature contours (θ ) for case 2 with Pr = 1000 and Ra = 105 for (a) ϕ = 30°, (b) ϕ = 45°, (c) ϕ = 75°,(d) (i) conduction at ϕ = 90° and (ii) convection at ϕ = 90°. Clockwise and anti-clockwise flows are shown via the negative and positive signs of streamfunction andheatfunction, respectively. White and black shades in θ represent the hot and cold fluid inside the cavity, respectively.
at larger Ra (Ra = 105) as a representative situation of convec-tion dominant heat transfer for the brevity of the manuscript. Itis also noted that the local heat transfer rates are shown for bothconduction and convection dominant heat transfer for ϕ = 90° atRa = 105 in case 2. It is found that Nul is zero near the bottom cor-ner of the left wall for ϕ = 30° and 45° at Pr = 0.015 in case 1 (seeFig. 9(a)) as there is no significant heat flow from that region. Thisis evident from the heatfunction contours of Fig. 4(a)–(b) as tinysecondary heat circulation cells with low magnitudes occur nearthe bottom corner of the left wall and that is further illustrated bythe isotherms with a zone of stagnant hot fluid near the bottomcorner of the left wall (see Fig. 4(a)–(b)). As ϕ increases, signifi-cant amount of heat transfer is found near the bottom corner ofthe left wall (see Fig. 9(a)) due to a considerable amount of heatflow near that zone as shown from the heatlines emanating fromthe left wall to the right wall near the bottom corner of the left wallfor ϕ = 75° and 90° at Pr = 0.015 in case 1 (see Fig. 4(c)–(d)). Itis observed that Nul is found to increase with the distance fromthe bottom corner of the left wall and that shows maxima nearS = 0.8, S = 0.6, S = 0.45 and 0.35 for ϕ = 30°, 45°, 75° and
90°, respectively, at Pr = 0.015 in case 1 (see Fig. 9(a)). This isdue to the presence of highly dense heatlines with high heatfunc-tion gradients near those regions and that is further illustrated bythe largely compressed isotherms near those regions for all ϕs atPr = 0.015 in case 1 (Fig. 4(a)–(d)). Further,Nul decreaseswith dis-tance from its first maxima towards the top corner of the left wallfor allϕs (see Fig. 9(a)) due to the decrease in heat flow towards thetop corner of the left wall as illustrated from the disperse heatlinesof low heatfunction gradients as seen in Fig. 4(a)–(d). It may also benoted that Nul near the top corner of the left wall further increasesdue to high heat flow in that region as illustrated by highly denseheatlines for ϕ ≤ 75° (see Figs. 9(a) and 4(a)–(c)). The variations oflocal heat transfer rates along the right wall (Nur ) are found to bemirror image variations of Nul with distance. Hence, the detaileddiscussions on variations of Nur along the right wall are omittedfor the brevity of the manuscript.
Similar to the Pr = 0.015 case, ϕ = 75° and 90° show largerheat transfer rates for Pr = 0.7 in case 1 (see Fig. 9(c)) nearthe bottom corner of the left wall due to high heat flow in thatregime compared to ϕ = 30° and 45° cavities as illustrated from
R. Anandalakshmi, T. Basak / European Journal of Mechanics B/Fluids 41 (2013) 29–45 41
a b
c d
e f
Fig. 9. Variations of the local Nusselt numberwith distance for case 1 (differential heating) at (a) Pr = 0.015, (c) Pr = 0.7 and (e) Pr = 1000 and for case 2 (Rayleigh–Benardconvection) at (b) Pr = 0.015, (d) Pr = 0.7 and (f) Pr = 1000 are shown for various inclination angles ϕ = 30° (· · · ), ϕ = 45° (- - -), ϕ = 75° (– –) and ϕ = 90° (—) (forcase 2: convection at ϕ = 90° (- –) and conduction at ϕ = 90° (- • –)) at Ra = 105 .
the heatfunction contours (figures are not shown). The trend ofNul shows maxima near the middle portion of the left wall forϕ = 30° and 45° due to highly dense heatlines in that regime(see Fig. 9(c)). Similarly, the trend of Nul shows maxima near thelower portion of the left wall for ϕ = 75° and 90° (see Fig. 9(c)) asillustrated by highly dense heatlines (figures are not shown). The
heat transfer rate is minimum near the top portion of the left wallfor the cavities with ϕ ≥ 45° due to less heat flow as illustrated bythe disperse heatlines in that regime compared to ϕ ≤ 45° cavitiesat Pr = 0.7 in case 1 (see Fig. 9(c)). The trend ofNul for Pr = 1000 isqualitatively similar as that of the Pr = 0.7 case (Fig. 9(e)). Hence,similar explanations follow for the Pr = 1000 case.
42 R. Anandalakshmi, T. Basak / European Journal of Mechanics B/Fluids 41 (2013) 29–45
It is interesting to observe that the variations of Nub with dis-tance are almost constant for ϕ = 30° and 45° in case 2 (seeFig. 9(b)) in the left portion of the bottom wall due to heatlines ofconstant heatfunction gradient (Fig. 7(a)–(b)). Further, Nub showsincreasing trend with distance for ϕ = 30° and 45° due to anincrease in heatfunction gradient towards the right corner of thebottomwall as illustrated byhighly denseheatlineswithhighheat-function gradient near that regime (Fig. 7(a)–(b)). In contrast toϕ = 30° and 45°, variations of Nub for ϕ = 75° and 90° ex-hibit one maxima near the right middle portion of the bottomwall(see Fig. 9(b)) as illustrated by the highly dense heatlines with highheatfunction gradient in that regime (Fig. 7(c)–(d(ii))). Thereafter,Nub decreases due to disperse heatlines in the right portion of thebottomwall forϕ = 75° and 90° (Fig. 7(c)–(d(ii))). Further increasein Nub near the right corner of the bottom wall for ϕ = 75° is dueto comparatively higher heatfunction gradient near that regime forϕ = 75° compared to ϕ = 90° (Fig. 7(c)–(d(ii))). Similar to Nurin case 1, the trend of Nut is found to be the mirror image varia-tions of Nub with distance (figures not shown). Local heat transferrates (Nub and Nut ) show local maxima and minima due to the in-crease and decrease in heatfunction gradients, respectively, in con-vection dominant heat transfer. In contrast, during the conductiondominant mode at ϕ = 90°,Nub and Nut are maintained con-stant throughout the length of the bottom and top wall due toconstant heatfunction gradient as the heatlines are perpendicularto the isothermal walls for ϕ = 90° at Pr = 0.015 and Ra = 105 incase 2 (see Figs. 9(b) and 7(a)–(d(i))).
The variations of Nub are qualitatively similar for the Pr = 0.7and Pr = 1000 cases as that of the Pr = 0.015 case (see Fig. 9(b),(d) and (f)) except that the gradual increase of Nub occurs withdistance for ϕ = 30° and 45° at X ≥ 0.2 due to the increasein heatfunction gradients along the bottom wall at X ≥ 0.2 andthe maxima of Nub occurs at the extreme right end for ϕ = 75°,whereas a decreasing trend of Nub occurs for 90° due to less denseheatlines near the right corner of the bottom wall as illustrated inFig. 8(a)–(d(ii)) for Pr = 1000. ConstantNub is also observed for theconductiondominantmode atϕ = 90° for Pr = 0.7 and Pr = 1000as seen in Fig. 9(d) and (f).
The overall effects on heat transfer rates via the average Nusseltnumber for the left wall (Nul: case 1) and the bottom wall (Nub:case 2) vs. the logarithmic Rayleigh number for all the cases aredisplayed in the panel plots of Fig. 10(a)–(b). Fig. 10(a)–(b) showthe variations for Pr = 0.015 and 1000. It may be noted thatthe average Nusselt number on the cold wall (Nur or Nut ) refersto the amount of heat received by those cold walls, while theaverage Nusselt number on the hot wall (Nul or Nub) refers to theamount of heat taken away from the hot wall. The panel plots ofFig. 10(a)–(b) illustrate that the average Nusselt number increaseswith the Rayleigh number irrespective of ϕ and Pr . The averageheat transfer rates (Nub) are shown for both conduction andconvection dominant heat transfer at ϕ = 90° for both Pr in case 2.It may be noted that the average heat transfer rate is high at Ra =
105 compared to Ra = 103 (see the panel plots of Fig. 10(a)–(b))due to a high heatfunction gradient at higher Ra as high convectivemotion further enhances the heat flow within the cavity.
The average heat transfer rates at the left wall (Nul) aremaintained constant over a range of Ra (103
≤ Ra ≤ 5 × 103)for ϕ = 30° and 45° in case 1 due to dominant conduction heattransfer in that region. However, Nul starts increasing from Ra ≥
103 for ϕ = 75° and 90° due to convection dominant heat transferat even low Ra in case 1 (see the upper panel of Fig. 10(a)). This isfurther illustrated by the highly dense and closed loop heatlines atRa = 103 and Pr = 0.015 in Fig. 3(c)–(d). It was also found thatthere is no significant heat flow from the lower portion of the hotleft wall for ϕ = 30° and 45° compared to ϕ = 75° and 90° asillustrated by the disperse heatlines of low heatfunction gradient
at ϕ = 30° and 45° in case 1 for Pr = 0.015 and Ra = 105 (seethe upper panel of Figs. 10(a) and 4(a)–(d)). Overall, the maximumheat flow occurs from the hot left wall to the cold right wall due todense heatlines of high heatfunction gradient for ϕ = 90° in case 1at Ra = 105 and Pr = 0.015 (see the upper panel of Fig. 10(a)). Thequalitative trends of Nu for case 2 (Nub) are similar to that of case1 (Nul) except constant Nu occurs over a certain range of Ra due toconduction dominant heat transfer for allϕs except the conductionbased static fluid solution at ϕ = 90°. It is interesting to note thatconstant Nub at Ra ≤ 2 × 103 in the convection based dynamicsolution suggests that a critical Ra is required to initiate convectionfor rhombic cavities with any ϕ in case 2 for all Pr (see the lowerpanel of Fig. 10(a)–(b)). On the other hand, convection initiates forϕ = 75° and 90° at very low Ra (2 × 103
≤ Ra ≤ 104) in theconvection based dynamic solution for both cases (case 1 and 2)irrespective of Pr (see Fig. 10(a)–(b)). It may be noted that Nub isconstant throughout the range of Ra (103
≤ Ra ≤ 105) for theconduction based static fluid solution at ϕ = 90°, whereas Nubincreases with Ra at ϕ = 90° similar to other ϕs in the convectionbased dynamic solution for Pr = 0.015 in case 2 (see the lowerpanel of Fig. 10(a)). Similar to case 1, the maximum heat flowoccurs from the hot bottom wall to the cold top wall for ϕ = 90°in case 2 at Ra = 105 and Pr = 0.015 (see the lower panel ofFig. 10(a)). Overall, the heat transfer rate (Nul or Nub) increaseswith ϕ in both cases at Pr = 0.015 (see Fig. 10(a)).
It is also interesting to note that ϕ = 30° shows a lower heattransfer rate in case 1 compared to case 2 at Pr = 0.015 andRa = 105 due to high intense convective heat transfer from the hotwall to the cold wall in case 2 compared to case 1 for ϕ = 30° asillustrated by highly dense heatlines of large heatfunction gradientat Pr = 0.015 and Ra = 105 (see Figs. 10(a), 4(a) and 7(a)). Onthe other hand, ϕ = 90° shows a higher heat transfer rate in case1 compared to case 2 at Pr = 0.015 and Ra = 105. This is dueto the presence of large high intense convective heat circulationcells for ϕ ≥ 75° in case 2 at Pr = 0.015 and Ra = 105 whichfurther obstruct the heat flow from the hot wall to the cold wall(see Figs. 10(a), 4(c)–(d) and 7(c)–(d)(ii)). However, the averageheat transfer rate is almost similar for ϕ = 45° in both cases(see Fig. 10(a)). Fig. 10(c) shows the variations of the average heattransfer rate with rhombic angles (ϕ) for case 1 and case 2 atRa = 105 for Pr = 0.015. The conduction based static fluid solutionin addition to the convection based dynamic fluid solution existsfor ϕ = 90° in case 2. It is interesting to note that Nu increaseswith ϕ in both cases in the convection based dynamic solution andcase 2 shows a higher heat transfer rate for all ϕs at Pr = 0.015(see Fig. 10(c)).
The variations of Nul and Nub for Pr = 1000 are qualitativelysimilar to those of Pr = 0.015 except the higher average heattransfer rates for Pr = 1000 at Ra = 105 due to higher momentumdiffusivity at Pr = 1000 (see Fig. 10(a)–(b)). The average heattransfer rate is higher for case 2 at ϕ ≤ 50°, whereas that ishigher for case 1 at ϕ ≥ 50° for Pr = 1000 and Ra = 105 (seeFig. 10(a)–(b)). Fig. 10(d) shows the variations of the average heattransfer rate with rhombic angles (ϕ) for both cases at Ra = 105
and Pr = 1000. Similar to Pr = 0.015, the conduction based staticfluid solution in addition to the convection based dynamic fluidsolution exists for ϕ = 90° in case 2. It is interesting to note thatNu increases with ϕ in the convection based dynamic solution forcase 1 in contrast to case 2,where that increaseswithϕ tillϕ ≤ 65°and afterwards that decreases at 65° ≤ ϕ ≤ 90°. But in contrast toPr = 0.015, case 1 shows a higher heat transfer rate for ϕ ≥ 50°at Pr = 1000 (see Fig. 10(c)–(d)). It is interesting to note that theconduction based static fluid solution is the unique characteristicof the square domain (ϕ = 90°). But, initiation of convection in thesquare domain (ϕ = 90°) leads to the convection based dynamicsolution and that shows a higher heat transfer rate compared to allother rhombic angles (ϕ ≤ 90°) for lower Pr (Pr = 0.015) in case2 (see Fig. 10(c)–(d)).
R. Anandalakshmi, T. Basak / European Journal of Mechanics B/Fluids 41 (2013) 29–45 43
a b
c d
Fig. 10. Variations of the average Nusselt number with the Rayleigh number for case 1 (differential heating) [Nul] and case 2 (Rayleigh–Benard convection) [Nub] at(a) Pr = 0.015 and (b) Pr = 1000 for various inclination angles ϕ = 30° (· · · ), ϕ = 45° (- - -), ϕ = 75° (– –) and ϕ = 90° (—) (for case 2: convection at ϕ = 90°(- –) and conduction at ϕ = 90° (- • –)). Plots for cases 1 and 2 are shown in the upper and lower panels, respectively. (c) Variations of the average Nusselt numbers withrhombic angles in the conduction and convection regime for cases 1 and 2 at Ra = 105 .
3.5. Conclusion
The heatlines and streamlines are used to analyze the natu-ral convection heat flow inside rhombic cavities to find the roleof inclination angle (ϕ) on the heat distribution and thermalmixing. The heatline concept has been implemented here for thevisualization of the heat flow inside the cavity for differential heat-ing (case 1) and Rayleigh–Benard convection (case 2) situations.Rayleigh–Benard convection at ϕ = 90° shows the conductionbased static fluid solution in addition to the convection based dy-namic solution for all Pr at Ra ≥ 2 × 103.
• At Ra = 103, heat transfer is conduction dominant for lower ϕs(ϕ = 30° and 45°) irrespective of Pr (Pr = 0.015–1000) in bothcases. Closed loop heatlines with slightly distorted isothermsare found at the core for ϕ = 75° and 90° for all Pr even atRa = 103 in case 1.
• At Ra = 105, multiple flow circulations are observed at Pr =
0.015 for all ϕs. Multiple circulation cells are suppressed andsingle flow circulation cells are found to occupy the entire cavityat higher Pr values (0.7, 7.2 and 1000) for all ϕs.
• Heatlines demonstrate that multiple circulation convectivecells transport heat locally in various zones at Pr = 0.015 andRa = 105. Interesting features on heatlines further illustratethat direct heat transport to the cold wall is rare, but the coldwall receives heat via convective heat circulation cells for allϕs especially at ϕ ≥ 45° in both cases irrespective of Pr atRa = 105.
• Conduction based static fluid solution as well as convectionbased dynamic solutions are found for ϕ = 90° in case 2 forall Pr . The conduction based static solution occurs only for ϕ =
90°, and it is observed that the conduction based static solutiondisappears with a slight perturbation of ϕ at higher Ra (Ra ≥
2 × 103) irrespective of Pr .• The heat distribution and thermal mixing are more effective in
Rayleigh–Benard convection (case 2) compared to differentialheating (case 1) and that is clearly indicated by large magni-tudes of streamfunctions, heatfunctions and largely distortedisotherms in case 2 at Ra = 105 irrespective of Pr .
• Themaxima in local Nusselt numbers are found to be correlatedwith large heatfunction gradients. The maximum average heattransfer rate from the hot wall to the cold wall (maximum Nul
44 R. Anandalakshmi, T. Basak / European Journal of Mechanics B/Fluids 41 (2013) 29–45
or Nub) occurs for higher ϕ in both cases irrespective of Pr atRa = 105.
• Average heat transfer rate is higher for case 2 at ϕ ≤ 50°,whereas that is higher for case 1 at ϕ ≥ 50° for higher Pr (Pr =
1000) at Ra = 105. Hence ϕ = 50° is the critical rhombic angleand the heating patterns (case 1 and 2) may be changed withϕ to achieve a higher heat transfer rate in various applicationsinvolving higher Pr (Pr = 1000) fluids at Ra = 105.
• The conduction dominant static fluid solution is the specialcharacteristic of the square cavity for Rayleigh–Benard convec-tion (case 2). Thus to avoid the conduction based static solution,the rhombic cavity (ϕ ≤ 90°)may be an alternative geometricaldesign for convective thermal processing of fluids with a verti-cal thermal gradient.Recently, Venturi et al. [27] reported the onset of convectiveinstability and multiple stable steady states arising within spe-cific range of Rayleigh numbers in Rayleigh–Benard convec-tion within two-dimensional square enclosures at Pr = 0.7.Systematic analysis of the multiplicity of steady states forRayleigh–Benard convection in rhombic cavities is yet to appearin the literature. Our future work will include the results for allpossible steady states at various Ra involving multiple circula-tions.
Acknowledgment
The authors would like to thank the anonymous reviewer forcritical comments which improved the quality of the manuscript.
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