study of natural convection flows in a …2 natural convection in a tilted trapezoidal enclosure for...

1 Graduate Student, Department of Mechanical Engineering, University of Illinois at Urbana-Champaign,1206, W Green St. Urbana-61801, IL, USA. E-mail: [email protected]

2 Department of Mechanical Engineering, Bangladesh University of Engineering & Technology (BUET),Dhaka-1000, Bangladesh. E-mail: [email protected]

* Corresponding author

STUDY OF NATURAL CONVECTION FLOWS IN A TIL TEDTRAPEZOIDAL ENCLOSURE WITH ISOFLUX HEA TINGFROM BELOW

Hasib Uddin1* and Sumon Saha2

Received: Apr 8, 2008; Revised: Sept 17, 2008; Accepted: Sept 23, 2008

Abstract

Laminar steady state natural convection in a two-dimensional symmetrical trapezoidal enclosure hasbeen studied using a finite element method. In this investigation, the top wall is considered adiabatic,both inclined sidewalls are maintained at a constant cold temperature and an isoflux heat source isprovided at the bottom surface. The pressure-velocity form of the Navier-Stokes equations and energyequation are used to represent the mass, momentum and energy conservations of the fluid medium inthe enclosure. Galerkin weighted residual method of finite element formulation with triangularmesh elements is employed. The fluid investigated here is air of Prandtl number fixed at 0.7. TheRayleigh number is varied from 103 to 106 while the sidewall inclination angle is varied from -15° to45°. The results are presented in terms of streamline and isotherm plots as well as the variation ofaverage Nusselt number with Rayleigh number for different base wall tilt angles of 0°, 15°, and 30°.The results show that the average Nusselt number increases with the increase of Rayleigh numberand the effect of the sidewall inclination angle on heat transfer is significantly reduced at higherRayleigh number. Effects of sidewall inclination angle on convection heat transfer characteristicsdecrease with the increase of base wall tilt angle at higher Rayleigh number and Rayleigh numberequal to 105 can be considered as a critical limit for the present.

Keywords: Natural convection, trapezoidal enclosure, finite element, isoflux

Suranaree J. Sci. Technol. 15(3):...-...

Introduction

Natural convection flows are particularlycomplex to control because they depend onseveral parameters among which the geometryconcerned and the thermo-physical characteris-tics of the fluid are the most important. Naturalconvection in enclosures of different geometry

has drawn greater attention for intensive researchefforts in recent years because of its extensiveapplication in wide range of engineering areassuch as thermal insulation, geothermal reservoirs,nuclear waste management, solar collector,crystal growth and design of cooling systems

2 Natural Convection in a Tilted Trapezoidal Enclosure

for various electronic equipments. The enclosuresphenomena can loosely be organized into twolarge classes (Bejan, 1984); heated from side andheated from below. The study of both flowclasses is also relevant to our understanding ofnatural circulation in the atmosphere, the hydro-sphere, and the molten core of the earth. Thepresent investigation concentrates attention onisoflux heating from below, which mainly focusesthe application of many practical transportdevices such as commercially refrigeratingcavities, furnaces, electronics cooling, processingequipments and others.

Actual enclosures occurring in practiceoften have the shapes differing from rectangularcover. Thus, various channels of constructions,panels of electronic equipment and solar energycollectors are of nonrectangular form. Moreover,a smaller number of previous studies haveconsidered the trapezoidal geometry, which isencountered in several practical applications,such as attic spaces in buildings, greenhousesor sun drying of crops. Solution of this type ofproblem is not simple to obtain because of thesloping walls. In general, the mesh nodes do notlie along the sloping walls and consequently froma program-using and computational point of view,the effort required for determining the flowcharacteristics increases significantly.

Several geometrical configurations, moreor less complex, have been examined under theo-retical, numerical or experimental approaches.Nithyadevi et al. (2007) had investigated naturalconvection in rectangular cavity with partiallyactive side walls. Heat transfer experiments intriangular enclosures were first reported by Flacket al. (1980). Effect of non-uniform heat flux onconvection heat transfer in trapezoidal channelwas experimentally studied by Remley et al.(2001). Kumar (2004) experimentally investigatednatural convective heat transfer in trapezoidalenclosure. He focused on the performance of abox type solar cooker and evaluated the naturalconvective heat transfer coefficient. Ganzarolliand Milanez (1995) performed numerical studyof steady natural convection in rectangularenclosures heated from below and symmetricallycooled from the sides. The size of the cavity wasvaried from square to shallow. Natural convec-

tion in tilted parallelepipedic cavities for largeRayleigh numbers was studied numerically andexperimentally by Baïri et al. (2007). The majorityof researches involving convection study isrestricted to the cases of simple geometry likerectangular, square, cylindrical and sphericalcavities. But the configurations of actualcontainers occurring in practice are often far frombeing simple. Research works on naturalconvection dealing with trapezoidal porousenclosures were reported by Baytas and Pop(2001) while Kumar and Kumar (2004) carriedout parallel computation for various values offlow and geometric parameters both underDarcian and non-Darcian assumptions on theporous model. The heat and mass transfer byfree convection in a trapezoidal enclosure heatedby its lower base and cooled by its inclinedupper side was studied by Boussaid et al. (1999).They solved the momentum, energy and massequations by finite volume method and observedthe influences of the geometrical parameters too.Kuyper and Hoogendoorn (1995) investigatedlaminar natural convection flow in trapezoidalenclosures to study the influence of the inclina-tion angle on the flow and also the dependenceof the average Nusselt number on the Rayleighnumber. Iyican and Bayazitoglu (1980); McQuainet al. (1994); Van der Eyden et al. (1998); Reynoldset al. (2004); Papanicolaou and Belessiotis (2005)and several other researchers had been madeseveral attempts to understand the basic heattransfer and fluid flow characteristics inside atrapezoidal cavity. The work similar to thecurrent one was done by Natarajan et al. (2008).They considered a regular trapezoidal enclosureand maintained 30° angle of inclination of thesidewalls and investigated in detail naturalconvection flows with uniform and non-uniformheating of the bottom wall. They used penaltyfinite element method to obtain smooth solutionsin terms of stream functions and isothermcontours. For convection dominant heat transfermode, in the case of uniform heating at Rayleighnumber, Ra = 105, the heat transfer rate graduallydecreased from the left of the bottom wall andattained minimum at the center of the bottom walland increased at the right. They found thatthe non-uniform heating exhibited greater heat

3Suranaree J. Sci. Technol. Vol. 15 No. 3; July - September 2008

transfer at the center of the bottom wall thanwith uniform heating case for all Rayleighnumber regimes. They also showed that forPr = 0.7 and Ra = 105, the average Nusseltnumber during uniform heating from belowwas greater than that for non-uniform heating.As expected, the trend was increasing for bothcases with the increase of Rayleigh number.

The current investigation demonstratesthe effects of natural convection in a regulartrapezoidal enclosure having isoflux heat source,q at the bottom wall under different base walltilt angles. The inclined sidewalls are cooledby means of constant temperature (Tc) bath andthe top wall is well insulated. No previous workhas been found considering isoflux heating frombelow in the present physical model. TheGalerkin finite element method (Reddy, 1993) hasbeen used to obtain the velocity componentsand temperature of the working fluid. Parametricstudies are carried out for Ra = 103, 104, 105, 106

and Pr = 0.7. Afterwards, the angle of inclinationof the vertical sidewalls is varied from 45° to -15°while the base wall is considered to be tilted at0°, 15°, and 30° to observe the heat transferperformance in terms of average Nusselt numberwith the variation of Rayleigh number.

Mathematical Formulation

A trapezoidal cavity of length L and height H isconsidered in Figure 1 having the sidewallsinclined at an angle Ø with the y-axis. The basewall is considered to be tilted at ψ = 0°, 15°, and

30° respectively with the horizontal x-axis. Thetrapezoid is symmetrical about the verticalcentral axis and has equal length and height(L = H). The fluid is assumed to be incompres-sible, Newtonian and laminar. The Boussinesqapproximation is adopted to account for thevariations of temperature as a function ofdensity and to couple in this way the tempera-ture field to the flow field. The dissipation effectdue to the viscous term is neglected and no heatgeneration is considered. Then the governingequations for steady natural convection usingconservation of mass, momentum and energycan be written in the dimensionless form asfollows:

U V0

X Y

∂ ∂+ =

∂ ∂ (1)

(2)

(3)

2 2

2 2U V

X Y X Y

∂θ ∂θ ∂ θ ∂ θ+ = +

∂ ∂ ∂ ∂ (4)

where U and V are the velocity components inthe X and Y directions respectively, θ is thetemperature, P is the pressure, Ra and Pr arethe Rayleigh number and Prandtl numberrespectively.

Eqns. (1) - (4) are normalized using thefollowing dimensionless scales:

2 4c

2T Tx y uL vL pL qL g qL

X ,Y ,U ,V ,P , , T ,RaL L T k

−= = = = = θ = ∆ = =

α α ∆ αυκρα

Figure 1. Schematic diagram of the physical system


2 4x y uL vL pL qL g qLX ,Y ,U ,V ,P , , T ,Ra

L L T k

β= = = = = θ = ∆ = =

α α ∆ αυκ and

Here ρ, β, υ, α and g are the fluid density,coefficient of volumetric expansion, kinematicviscosity, thermal diffusivity, and gravitationalacceleration, respectively. The correspondingboundary conditions for the above problem aregiven by:

All walls: U = V = 0, Top wall: , Inclined

sidewalls: θ = 0, Bottom wall: (5)

The average Nusselt number can be written as,

(6)

where θ S(X) is the local dimensionless tempera-ture of the heated surface.

Numerical Procedure

The governing equations are solved numericallyusing a Galerkin weighted residual method offinite element formulation. A mixed finite element(FE) model is implemented with two types oftriangular Lagrange elements: an element withlinear velocity and pressure interpolations forthe continuity and momentum equations andan element with a quadratic basis velocityand temperature interpolations for the energyequation. A stationary nonlinear solver is usedtogether with Direct (UMFPACK) linear systemsolver. The relative tolerance for the error criteriais considered to be 10-6. As the dependentvariables vary greatly in magnitude, manualscaling of the dependent variables is used toimprove numerical convergence. The manualscaling values are kept constant and selected insuch a way that the magnitudes of the scaleddegrees of freedom become one. The nonlinearequations are solved iteratively using Broyden’smethod with an LU-decomposition precon-ditioner, always starting from a solution for anearby Rayleigh number. The numerical simula-tions are performed by varying the number ofelements in order to increase the accuracy andefficiency for the solutions. Non-uniform gridsof triangular element are employed with denser

grids clustering in regions near the heat sourcesand the enclosed walls. It may be noted that asimilar finite element method has been used tosolve fluid flow and heat transfer problems inrecent investigations by Lo et al. (2005); Royand Basak (2005) and Asaithambi (2003).

Grid Sensitivity Check

Test for the accuracy of grid sensitivity is exam-ined for the arrangements of five different non-uniform grid systems with the following numberof elements within the trapezoidal enclosure:3,376; 4,930; 6,006; 9,006 and 11,232. The resultsare shown in Table 1. From these comparisons,it is suggested that 9006 non-uniform elementsare sufficient to produce accurate result.

Code Validation

Since the validation against experimental data isnot possible in the present case, the computa-tional code is validated with the results obtainedfor natural convection flows in a trapezoidalenclosure as mentioned by Natarajan et al.(2008). The comparison of the result is shownin Figure 2 and it has been observed that thepresent numerical solution is almost in completeagreement with the aforementioned one in termsof the isotherm plot. Therefore, it can be decidedthat the current code can be used to predictthe flow and thermal field for the presentproblem accurately. However, almost similarexperimental results for square straight enclo-sure with discrete bottom heating obtainedby Corvaro and Paroncini (2008) are comparedby the present code and reported in Table 2.The agreement is found to be excellent whichvalidates the present computations and lend usconfidence for the use of the present code.

Results and Discussion

In the present research work, the working fluidinside the trapezoidal cavity is chosen as air withPrandtl number, Pr = 0.7. The sidewall inclinationangles (Ø) are taken as 45°, 30°, 15°, 0°, and -15°.So the top adiabatic wall of the trapezoid getsshorter and for Ø = 0°, the trapezoid becomes


square in shape and finally it tends to be atriangle. The Rayleigh number (Ra) is variedfrom 103 to 106 and the tilt angle of the base wall(ψ) is varied from 0° to 30°. Flow and tempera-ture fields are presented in terms of streamlineand isotherm contours respectively. Effects ofthe Rayleigh number, the inclination angle of thesidewalls and the tilt angle of the base wallon the heat and fluid flow phenomenon areobserved. Later, heat transfer performance isexamined in terms of average Nusselt number(Nu) to predict the characteristics of naturalconvection under different geometric conditions.

Effects of Rayleigh Number on Heatand Fluid Flow Field

The evolution of flow and thermal fields withinthe trapezoidal enclosure for Ra = 103, 105, and106 and Ø = 45°, 30°, 15°, 0°, and -15° ispresented in Figures 3 and 4 when the base wallis considered horizontal (ψ = 0°). Due to thesymmetrical boundary conditions of the inclinedsidewalls, the flow and temperature fields aresymmetrical about the vertical central axis of theenclosure. As expected, hot fluid rises up fromthe central region as a result of buoyancy forces,

Figure 2. Comparison of the isotherm plots with Natarajan et al. (2008) at Ra = 105, Pr = 0.7,Ø = 30°°°°° and ψψψψψ = 0°°°°°

Table 2. Comparison between the experimental and numerical average Nusselt number fordiscrete isothermal heat source size = 0.2, Ø = 0°°°°° and ψψψψψ = 0°°°°°

Ra Experimental Data Numerical Data Error (%)Corvaro and Paroncini (2008) Present code

7.56 × 104 4.80 5.31 -10.631.38 × 105 5.86 6.07 -3.601.71 × 105 6.30 6.37 -1.111.98 × 105 6.45 6.58 -2.012.32 × 105 6.65 6.82 -2.562.50 × 105 6.81 6.94 -1.91

Table 1. Comparison of the results for various grid dimensions at Ra = 106, Ø = 45°°°°° and ψψψψψ = 0°°°°°

Elements 3,376 4,930 6,006 9,006 11,232

Nu 5.900 5.884 5.876 5.866 5.865


after that owing to the cold inclined walls, flowsdown along the walls forming two symmetric rolls(Figure 3) with clockwise and anticlockwiserotations inside the cavity. At Ra = 103, viscousforces are more dominant than the buoyancyforces and hence, heat transfer is essentiallydiffusion dominated and the shape of the stream-line tends to follow the geometry of the enclo-sure. For Ra = 105 and 106, it is interesting toobserve that the stream function contours nearthe walls tend to have neck formation due tostronger circulation at higher Rayleigh numberwhich contrasts the flow pattern for Ra = 103.The core of the circulating rolls moves upwardwith the increase of Ra indicating significantincrease of the intensity of convection.

For Ra = 103 as can be expected, heattransfer characteristics are essentially diffusiondominant as further indicated by the isothermpatterns shown in Figures 3, 5, and 7. As a repre-sentative case, for Ø = 45° and ψ = 0° (Figure 3),during diffusion dominant heat transfer, thetemperature contours with θ < 0.14 occursymmetrically near the side walls of theenclosure. The other isotherm lines with θ > 0.14are smooth curves symmetric with respect tothe vertical symmetric line. The distortion ofthe isotherm field increases with enhancedbuoyancy as Ra increases, where the heattransfer becomes increasingly advectiondominant. The circulations are greater nearthe center and least at the wall due to no-slipboundary condition. Due to the initiation ofadvection, the isotherms are significantlydistorted and pushed near the sidewalls forhigher Ra. Moreover, as a representative case,Figure 3 shows for Ø = 45° isotherm with θ = 0.06which breaks into two symmetric contour linesat Ra = 106. Consequently at Ra = 106, thetemperature gradients near the bottom and thesidewalls tend to be significant to develop thethermal boundary layer. Due to greater circula-tion at the top half of the enclosure, there aresmall gradients in temperature at the centralregime whereas a large stratification zone oftemperature is found at the vertical symmetryline due to the stagnation of flow. Formation ofthermal boundary layer is observed partially forRa = 103 and almost throughout the enclosure

for Ra = 105 and 106. As the nonlinearity of theisotherms increases with the increase of Ra(105 and 106), a mushroom profile is observedin the Figures 3, 5, and 7.

Effects of Inclination Angle of the SideWalls on Heat and Fluid Flow Field

The development of the flow and thermal fieldsin the cavity with increasing inclination angleof the sidewalls are shown in Figures 3 - 8. Forhorizontal base wall (Figure 4), at Ra = 103, 105

and 106, as the inclination angle, Ø, decreasesthe streamline contour tends to take circularshape indicating weaker flow. The circulationstrength of the trapezoid with Ø = -15° isobserved to be the most and it exceeds theintensity of the square cavity. During diffusiondominant heat transfer, for Ra = 103 and Ø = 45°,(Figure 3) temperature contours with θ > 0.16 arenearly smooth curves which span from the middlebottom of the enclosure and they are generallysymmetric with respect to the vertical center line.With the decrease of Ø, the sidewalls continueto come closer to the heat flux boundary and asa result, for Ø = -15° (Figure 3), the similarpattern of isotherm is observed at θ > 0.02 andno isothermal line is found to be broken intosymmetry indicating better heat transfer perfor-mance. At Ra = 106, for advection dominant heattransfer regime, the effect of inclination angle isless significant on heat transfer compared to thediffusion dominant regime as the isothermpatterns remains almost identical.

Effects of Tilt Angle of the Base Wallon Heat and Fluid Flow Field

The base wall of the trapezoidal enclosure isconsidered to be tilted at an angle ψ = 0°, 15°,and 30° to investigate fluid flow and heat trans-fer characteristics. The flow and thermal fields inthe cavity with different tilt angles are shown inFigures 3 to 8. At Figure 4, it is observed thatfor the horizontal cavity (ψ = 0°), where thebuoyancy force is acting only in y direction, tworecirculation cells are formed and the streamlineplot is symmetric about the vertical midlineowing to the symmetry of the problem geometry


and the boundary conditions. For the inclinedenclosures (Figures 5 - 8) this symmetry is com-pletely destroyed due to the buoyancy forcecomponents acting in both X and Y directions.The effect of the base wall tilting is clearlyvisible on the flow patterns for all cases. Thoughsimilar effect is observed on isotherm contoursinside the cavity for higher Ra, the contoursremain unaltered for Ra = 103 at diffusiondominant regime. From Figure 8 for ψ = 30°, in

case of square cavity (Ø = 0°), it is observed thatthe left recirculation vortex becomes dominatingin the enclosure, the right vortex is squeezedthinner and ultimately forms minor vortices orlocal eddies at the corners. This circulationinside the cavity is greater near the centerand least at the wall due to no slip boundarycondition. Eddy formation is observed for thetrapezoid with Ø = -15° while ψ = 15° (Figure 6)and no eddy is created in the square cavity

Figure 3. Isotherm plots in the trapezoidal cavity for different Rayleigh numbers and sidewallinclination angles at base wall tilt angle, ψψψψψ = 0°

Ra = 103 Ra = 105 Ra = 106

Ø =

-1

5°

Ø =

0°

Ø =

15

Ø =

30°

Ø =

45°


(Ø = 0°) with this base wall tilt condition. Inter-estingly, for larger sidewall inclination angles(Ø = 15°, 30°, and 45°), no eddy formation isobserved inside the enclosure indicating lesseffect of base wall tilting. The isotherms arealso adjusted according to the change in theflow fields and pushed towards the lower partof the right inclined sidewall indicating thepresence of a large temperature gradient there.

Heat Transfer Characteristics

Finally, the present investigation concentrateson the influence of the sidewall inclination angle(Ø) and base wall tilt angle (ψ) for the variationsof average Nusselt number of the heated wallwith the Rayleigh number. In order to observethe results, the variations of average Nusseltnumber with Ra = 103 to 106 and Ø = -15°, 0°, 15°,

Figure 4. Streamline plots in the trapezoidal cavity for different Rayleigh numbers and sidewallinclination angles at base wall tilt angle, ψψψψψ = 0°°°°°

Ra = 103 Ra = 105 Ra = 106

Ø

= -

15°

Ø =

0°

Ø =

15°

Ø

= 3

0°

Ø =

45°


30°, and 45° are plotted for base wall tilt angleψ = 0°, 15° and 30° in Figures 9(a-c) respectively.Figure 9(a) indicates that the average Nusseltnumber remains invariant up to a certain value ofRa and then increases rapidly with increasingRa. At lower Ra, the curves maintain a flat trendindicating little change in Nu and the least slopeis found for the least side wall inclination anglewhich indicates low temperature gradients.

Diffusion dominant heat transfer is demonstratedat Ra = 103 and significant effect of advection isinitiated from Ra = 104. For higher value ofsidewall inclination angle, advection is found tobe initiated earlier.

At the diffusion dominant regime, thedifference between the values of Nu for differentsidewall inclination angles is significant whereasat Ra = 106, for the advection dominant regime

Figure 5. Isotherm plots in the trapezoidal cavity for different Rayleigh numbers and sidewallinclination angles at base wall tilt angle, ψψψψψ = 15°

Ra = 103 Ra = 105 Ra = 106

Ø

= -

15°

Ø

= 0

°

Ø =

15°

Ø

= 3

0°

Ø =

45°


the difference of the values of Nu is found to bereduced (Figure 9(a)). The Nusselt number is thehighest for the lowest inclination angle. It can bedecided that as the sidewall inclination anglesdecrease the trapezoid tends to get rectangulargeometry and the constant temperature sidewallsacting as the heat sinks come closer to the heatsource resulting improved heat transfer insidethe cavity.

From Figures 9(a-c), it is evident that there

is no effect of base wall tilt angle on averageNusselt number at lower Ra. But at Ra > 104,when the heat transfer mechanism is essentiallydominated by advection, effect of ψ increaseson the heat transfer characteristics. However,trapezoidal enclosures with positive Ø are lessaffected with the change in ψ compared to thesquare cavity and the trapezoidal cavity withnegative Ø. Figure 9(b) indicates that at Ra = 105

to 106 for ψ = 30°, change in the sidewall inclina-


Ra = 103 Ra = 105 Ra = 106

Ø

= -

15°

Ø =

0°

Ø =

15°

Ø

= 3

0°

Ø =

45°


tion angle after Ø = 15° does not influence theheat transfer performance so much while thetrapezoid with Ø = -15° shows the highest valueof Nu and the square cavity shows the lowestvalue. Trapezoidal enclosures with positive valueof Ø remain in between without any significantchange in the heat transfer performance. There-fore, Ra = 105 can be considered as a critical limitfor the current problem. Finally, it is observedthat the overall heat transfer performance dete-

riorate in terms of Nu with the increase of thebase wall tilt angle.

Conclusions

Natural convection in a two-dimensionaltrapezoidal enclosure, where the top wall isconsidered adiabatic, two inclined sidewalls aremaintained at constant low temperature andan isoflux heat source is applied on the bottom

Figure 7. Isotherm plots in the trapezoidal cavity for different Rayleigh numbers and sidewallinclination angles at base wall tilt angle, ψψψψψ = 30°°°°°

Ra = 103 Ra = 105 Ra = 106

Ø

= -

15°

Ø =

0°

Ø =

15°

Ø

= 3

0°

Ø =

45°


wall has been analyzed numerically using finiteelement method under different base wall tiltconditions. The main parameters of interest arethe Rayleigh number (Ra), the sidewall inclina-tion angle (Ø), and the base wall tilt angle (ψ). Ithas been observed that the diffusion dominantheat transfer modes are found for Ra < 103 andthe circulation is so weak that the viscous forcesare dominant over the buoyancy forces. At theadvection dominant regime, for Ra > 105 the

compression of isotherms and the formation ofthermal boundary layer increase due to increasedcirculation intensity. The average Nusseltnumber is higher for lower sidewall inclinationangle indicating higher heat transfer. At advec-tion dominant regime for higher Rayleighnumber, the variation of average Nusseltnumber tends to be significantly reduced withthe change of the sidewall inclination angle.

Heat transfer performance in terms of


Ra = 103 Ra = 105 Ra = 106

Ø

= -

15°

Ø =

0°

Ø =

15°

Ø

= 3

0°

Ø =

45°


Figure 9. Variation of average Nusselt number with Rayleigh number for different inclinationangles at base wall tilt angle, (a) ψψψψψ = 0°°°°°, (b) ψψψψψ = 15°,°,°,°,°, and (c) ψψψψψ = 30°°°°°

(a)

(b)

(c)

average Nusselt number inside the cavityincreases as the cold sidewalls acting as the heatsinks come closer to the hot bottom heat source.Effects of sidewall inclination angle decrease withthe increase of base wall tilt angle at higherRayleigh number. At the diffusion dominantregime, for Ra = 103 no significant influence ofthe variation of the base wall tilt angle isobserved. With the increase of the tilt angleoverall heat transfer rate is found to be decreased

inside the trapezoidal enclosure at the advectiondominant regime.

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study of natural convection flows in a …2 natural convection in a tilted trapezoidal enclosure for...

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