Mixed Convection Flow And Heat Transfer In A Lid Driven Cavity Using SIMPLE Algorithm by IRJET Journal

Mixed Convection Flow And Heat Transfer In A Lid Driven Cavity Using SIMPLE Algorithm

Arti Kaushik

Department of Mathematics, Maharaja Agrasen Institute of Technology, Delhi. India

Abstract - In the present study mixed convection flow and heat transfer in steady 2-D incompressible flow through a lid driven cavity is investigated. The upper wall of cavity is moving with uniform velocity and is at higher temperature. The stationary lower wall is kept at lower temperature. The governing equations of the model are solved numerically using SIMPLE algorithm. A staggered grid system is employed for numerical computations for velocity, pressure and temperature. Under relaxation factors for velocity, pressure and temperature are used for the stability of the numerical solutions Bernoulli equation has been taken up to check the accuracy of the computed solutions. The significant findings from this study have been given under conclusion.

Keywords - incompressible flow, lid driven cavity, SIMPLE algorithm, staggered grid system, mixed convectionflow

1. INTRODUCTION

The phenomena of combined forced convection and natural convection in a lid driven cavity have many practical applications ([1],[2],[3]) The importance of this problemhasledtovariousresearches([4]-[9])

A numerical study of mixed convection heat transfer in a two-dimensional rectangular cavity with partially heated bottom wall and vertically moving sidewalls was done by Guo and Sharif [10]. Later, mixed convective heat transfer was studied in a lid driven cavity with aspect ratio 10, when the top of the lid is moving and is at higher temperature than the bottom wall [11]. Numerical simulationsweredonefor2-Dlid-drivensquareenclosure partitioned by a solid divider with a finite thickness and finite conductivity to study the mixed convection heat transfer taking two different orientations of left vertical wall of enclosure [12]. Saha et al numerically studied two dimensional mixed convection in a square enclosure with moving top lid and keeping both top and bottom wall at constant temperature[13]. They studied flow and heat transfer characteristics, streamlines, isotherms and average wall Nusselt number for different Richardson number.

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Using the finite volume method of the ANSYS FLUENT commercial CFD code laminar mixed convection characteristics were studied in a square cavity with a variable sized isothermally heated square blockage inside the cavity[14]. By keeping the blockage at a higher temperatureandfour surfacesof the cavity(including the lid) at a colder temperature, it was found that the blockage placed around the top left and the bottom right corners of the cavity results in the most preferred heat transfer. A study of steady laminarmixed convectioninside a lid-drivensquare cavityfilled with water, when both top and bottom walls are moving and are kept at cold andhot temperature respectively was done by Ismael et al [15]. They applied USRfinite differencemethodandshowedthatconvectionisdeclined for certain critical values for the partial slip parameter. Using ANSYS FLUENT commercial code based on a finite volume method numerical study was done for mixed convection laminar flow in a lid-driven square cavity in which top lid of the cavity is moving rightwards [16]. The cavity had two square isothermally heated internal blockages which were kept at hot temperature and the walls of the cavity are kept at a cold temperature. They observed that the location of the blockage as well as the separation distance between the two blockages significantly changes the average Nusselt number. The effect of Richardson number on the heat transfer in a differentially heated lid−driven square cavity when top and bottom moving walls are maintained at different constant temperatures was studied[17]. Finite element approach using characteristic based split (CBS) algorithm was applied for this study. A numerically study of twodimensional laminar mixed convection in a lid-driven square cavity filled with a nanofluid was done by Zeghbid and Bessaih[18].They studied the effect of the Rayleigh number, the Reynolds number and the volume fraction of the nonofluid on the average Nusselt number using finite volume method The movable top and bottom walls were kept at a local cold temperature and the nanofluid is constantly heated by two heat sources placed on the two verticalwalls.Undertheseconditionsitwasfoundthatthe increase in Rayleigh number and solid volume fraction of nanofluids results in increase in average Nusselt number

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dimensional conjugate heat transfer in a stepped liddrivencavityunderforcedandmixedconvectionwasdone by Janjanam et al[19]. They studied three different nanoparticle volume concentrations of pure water and Aluminium oxide /water nanofluid and observed that mixed convection is 24% higher than that of forced convection for lower values of Reynolds number and higher values of Grashof number. The mixed convection flow in a tall lid driven cavity for non –Newtonian power law fluid was studied by Kumar et al[20] They observed the effect of triangular surface corrugations on the flow and presented numerical results for different values of aspect ratio of cavity, Richardson numbers, Prandtl numbers (Pr), power-law indexes at a constant Grashof number. A laminar mixed convection in a square cavity with moving vertical wall and having a hot obstacle was studiedbyOuahouahetal[21].Theyinvestigatedtheeffect of the Richardson number and Reynolds number on both hydrodynamic and thermal characteristics and observed that high values of Richardson and Reynolds numbers resultsintheenhancedheattransfer.

Forthestudyofflowinliddrivencavityawell-known technique is SIMPLE algorithm proposed by Patankar and Spalding which has been used extensively in literature [22]. Sivakumar et al studied mixed convection heat transfer and fluidflowinlid-drivencavities with different lengths and locations of the heating portion using SIMPLE algorithm[23] Using SIMPLE algorithm, an adaptive mesh refinementmethodwaspresentedbyLiandWoodfor2-D steady incompressible lid-driven cavity flows[24].Their method is applicable to mathematical model containingcontinuity equationsfor incompressible fluid, steady state fluid flows or mass and heat transfer A laminar two-dimensional lid driven cavity flow was investigated by Mohapatra with inclined side wall for different inclination angle using SIMPLE algorithm on staggered grid [25]. His results were in good agreement withthe benchmark solutions. ComparisonofSIMPLEand SIMPLER algorithm to study flow in a square cavity to analyze velocity and pressure distribution was done by Earn et al[26]. They showed that convergence rate of a numerical scheme is significantly affected by underrelaxation factors for velocity components and pressure. They also deduced that reduction of the heating portion resultsinincreasedheattransferrate. Alietalstudiedthe heattransferrateinadoublelid-drivenrectangularcavity keepingbottomwall iskeptatahightemperatureandthe top wall is kept at a low temperature using SIMPLE algorithm[27] They employed hybrid nanofluid and showed that the presence of hybrid nanofluid in the rectangularcavityincreasestheheattransfersignificantly.

In the view of above mentioned literature, it is clear thatSIMPLEalgorithmhasnotbeenappliedmuchtostudy mixed convective heat transfer in a lid driven cavity with movingupperwallwhichisathighertemperaturethanthe bottomwall.Ouraiminthispaperistoexaminethemixed convection and heat transfer in a lid-driven cavity under these conditions using SIMPLE algorithm. The results obtainedareshownthroughquiverplotandcontourplots forvariousconsidereddimensionlessparameters.

2. METHODOLOGY

We consider a two dimensional, incompressible, steady and laminar flow through a square cavity of height and length L. The walls of the cavity have no-slip condition excepttheupperwallwhichismovinginitsownplaneata constant speed U0. The cavity upper wall is kept at a high temperature Th whereas bottom wall is kept at a low temperatureTc,Theleftandrightwallsareassumedtobe adiabatic. All thermo-physical properties of the fluid are considered as constant except the density variation of the buoyancy term. The density is assumed to vary linearly withtemperatureas

,where

is thefluiddensity, gistheaccelerationduetogravityandβ is the coefficient of thermal expansion. The fluid is considered as Newtonian. We also assume that viscous dissipationisneglected.

Under these assumptions, the governing conservations equationsaregivenby:

where u and v are the fluid velocity components in the xand y directions, p the pressure, T the temperature, g is acceleration due to gravity ,β the volumetric coefficient of thermalexpansion, iskinematicviscosity,ρthedensity ofthefluidandαthethermaldiffusivity.

Introducingthenon-dimensionalquantities

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    cc gTT

0 uv xy    (1) 22 22 1 uupuu uv xyxxy      (2)   22 22 1 c vvpvv uvgTT xyyxy       (3) 22 22 TTTT uv xyxy     (4)

(1)-(4)innon-dimensionalformas

and μ. The SIMPLE algorithm has been implemented in MATLAB programming language. During the numerical process, different values of under relaxation factors for u velocity and v velocity and pressure were tested for all cases. The effect of thermal diffusivity α on v velocity is shown in Figure1a and 1b. Itcan be seen that increase in valueofαresultsinhigherv-velocity.Forlowervaluesof α, the v velocity has sudden increase near midpoints of eastboundary.

where Re, Pr and Gr are non- dimensionalised Reynolds number,PrandtlnumberandGrashofnumberrespectively andaredefinedas

The initial and boundary conditions in the non dimensionalisedform are

The governing equations along with the boundary conditions are solved using SIMPLE algorithm. The algorithm was originally put forward by Patankar and Spalding. It is a guess and correct procedure for the calculation of pressure and velocities. For the application of the technique the computational domain is discretised employing the staggered grid arrangement... Then using finite volume method the nonlinear governing partial differential equations are converted into a system of discretised equations. These discretised equations along withpressurecorrectionequationaresolvedtoobtainthe velocities, pressure and temperature at all node points of thegrid.

3. RESULTS AND DISCUSSION

The governing equations are solved numerically to obtain unknownvariablesu,v,pandTforvariousvaluesofα,β,ρ

In Figure 2a and 2b, the variation in temperature with change in α can be seen. For lower values of α, the temperature start decreasing from north boundary and then remains same in the cavity. For higher values of α, this pattern remains same but then a sudden increase in temperature can be seen near south east corner of the cavity.

0 UV XY    (6) 22 22 1 Re UUPUU UV XYXXY    (7) 22 222 1 ReRe UVPVVGr UV XYYXY      (8) 22 22 1 RePr UV XYXY     (9)

canwriteEqn.

3 0 2 Re3,Pr, UL gTL Gr    

1,0,101,1 0,0,001,0 0,001,0&1 UVXY UVXY UVYXX X        

Fig.1a.Contourplotforvvelocityforα=10 Fig. 1b.Contourplotforvvelocityforα=50

Y-velocityContoursforlid-drivencavityflow 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Y-velocityContoursforlid-drivencavityflow 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

The effect of coefficient of thermal expansion β on the u velocity is shown in Figure 3a and 3b and the effect of coefficient of thermal expansion β on the v velocity is showninFigure4aand4b.Itcanbeseenthatdecreasein valueofβresultsinhigheru-velocityandv-velocity.The values of u velocity increase slowly from north boundary till midpoints of the cavity and then decrease rapidly towards the south boundary. This pattern remains same for all values of β. The v-velocities are highest near midpoints along east boundary and lowest near west boundary.

Fig 2a.Temperaturecontoursforα=10 Fig.2b.Temperaturecontoursforα=50 Fig.3aContourplotforuvelocityforβ=0.00001 Fig.3bContourplotforuvelocityforβ=0.0001

TemperatureContoursforlid-drivencavityflow 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 TemperatureContoursforlid-drivencavityflow 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X-velocityContoursforlid-drivencavityflow 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X-velocityContoursforlid-drivencavityflow 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Y-velocityContoursforlid-drivencavityflow 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig 4aContourplotforvvelocityforβ=0.00001

In Figure 5a and 5b, the pressure contours for different valuesofβcanbeseen.Thepressureishighforallpointin cavitybutdecreasesnearthesouthboundaryofthecavity. Notethatforlowervalueof βthepressureinthecavity is higher. For higher value of β the pressure decreases slightlyneareastandwestboundary.

Thechangeintemperatureinsidethecavitywithchangein β is minimal. This can be seen in Figure 6a and 6b. Figure 7a,7band7cshowstheeffectofdensityoffluidρontheuvelocity.Theeffectofdensityonv-velocitycanbeseenin Figure8a,8band8c

Fig 4bContourplotforvvelocityforβ=0.0001 Fig 5aPressureContourplotforβ=0.00001 Fig.5bPressureContourplotforβ=0.0001 Fig. 6aTemperaturecontoursforβ=0.0001 Fig 6b.Temperaturecontoursforβ=0.00001

Y-velocityContoursforlid-drivencavityflow 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 PressureContoursforlid-drivencavityflow 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 PressureContoursforlid-drivencavityflow 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 TemperatureContoursforlid-drivencavityflow 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 TemperatureContoursforlid-drivencavityflow 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X-velocityContoursforlid-drivencavityflow 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig.7aContourplotuvelocityforρ=1

Itcanbeseenthatincreaseinvalueofρresultsinhigheru -velocityandv-velocity.Forhighervaluesofρ,uvelocity increase slowly from north boundary till midpoints of the cavity and then decrease rapidly towards the south boundary. For lower values of ρ, u velocity increase from north boundary till midpoints of the cavity and then decrease towards the south boundary in a symmetric manner. The v-velocities are higher near midpoints along east and west boundaries and lowest near south east corner.

Fig 7bContourplotuvelocityforρ=10 Fig 7c Contourplotuvelocityforρ=50 Fig 8aContourplotvvelocityforρ=1 Fig.8bContourplotvvelocityforρ=10 Fig 8cContourplotvvelocityforρ=50 Fig9aPressureContourforρ=1

X-velocityContoursforlid-drivencavityflow 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X-velocityContoursforlid-drivencavityflow 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Y-velocityContoursforlid-drivencavityflow 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Y-velocityContoursforlid-drivencavityflow 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Y-velocityContoursforlid-drivencavityflow 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 PressureContoursforlid-drivencavityflow 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 PressureContoursforlid-drivencavityflow 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig9bPressureContourforρ=10

InFigure9a,9band9c,wecansee,asexpected,thatthe pressureinthecavitydecreaseswithdecreasingdensity. Fig10aand10bshowsthatthetemperatureatsoutheast cornerincreasesinmagnitudeasthedensityρincreases.

Figure 11a and 11b shows the effect of density of fluid on the direction field of the flow. As the density of the fluid increases,thedirectionfieldoftheflowchangesdrastically

Theeffectofviscosityofthefluidonuvelocitycanbeseen inFigure12a,12band12c. Itcanbeseenthatthepattern ofincreaseofuvelocityfromnorthboundaryanddecrease towards south boundary remains same, but varies as the viscosity changes. For higher viscosity the maximum uvelocityliesnearmidpointsofeastandwestboundaries.

Fig9cPressureContourforρ=50 Fig.10a.Temperatureatallnodepointsforρ=1 Fig10b.Temperatureatallnodepointsforρ=10 Fig11aDirectionfieldoftheflowforρ=10 Fig11bDirectionfieldoftheflowforρ=50

PressureContoursforlid-drivencavityflow 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 0 0.5 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 1 2 3 4 5 0 0.5 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 -1 0 1 2 3 4 5 6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -1 0 1 2 3 4 5 6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 X-velocityContoursforlid-drivencavityflow 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig12aContourplotsforuvelocityforμ=0.01

Theeffectofviscosityofthefluidonvvelocitycanbeseen inFigure13a,13band13c.Forvvelocity,wecanseethat it has minimum values near west boundaries. For higher viscosity,thevalueofvvelocityalsohaslowervaluesnear northboundaries.

The variation of pressure inside the cavity with varying viscositycanbeseeninFigure14a,14band14c. Itcanbe clearly seen that the pressure increases with increasing viscosity of the fluid. The change in temperature with changeinviscosityisminimal.

Fig12bContourplotsforuvelocityforμ=0.05 Fig12cContourplotsforuvelocityμ=0.1 Fig13aContourplotsforvvelocityforμ=0.01, Fig13b Contourplotsforvvelocityfor μ=0.05 Fig13c Contourplotsforvvelocityforμ=0.1 Fig14aPressureContourplotsforμ=0.01 Fig14bPressureContourplotsfor μ=0.05

Fig14cPressureContourplotsforμ=0.1

The effect of velocity of the top boundary on u- velocity can be seen in Figure 15a,15b and 15c As the velocity of top boundary increases, u- velocity decreases. For higher velocity of top boundary, the maximum values of uvelocityshiftstonorthboundaryofthecavityfrom centre of the cavity. On the other hand, the v velocity first decreasesandthenstartsincreasingagain,withincreasing velocityoftopboundary.ThisisshowninFigure 16a,16b and16c

InFigure17a,17band17c,wecanseethatthepressurein the cavity decreases with increase in the velocity of top boundary.

Fig15aContourforuvelocityforwallvelocity0.1 Fig15b Contourforuvelocityforwallvelocity 0.25 Fig15c Contourforuvelocityforwallvelocity0.5 Fig16aContourforvvelocityforwallvelocity0.1 Fig16bContourforvvelocityforwallvelocity0.25 Fig16c Contourforvvelocityforwall0.5

X-velocityContoursforlid-drivencavityflow 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X-velocityContoursforlid-drivencavityflow 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X-velocityContoursforlid-drivencavityflow 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Y-velocityContoursforlid-drivencavityflow 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Y-velocityContoursforlid-drivencavityflow 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Y-velocityContoursforlid-drivencavityflow 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 PressureContoursforlid-drivencavityflow 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig17aPressurecontourforwallvelocity0.1

Thispaperpresentsanumericalstudyofmixedconvection flow and heat transfer in steady 2-D incompressible flow through a lid driven cavity. Numerical solutions for the governingequationsareobtainedusingSIMPLEalgorithm. The numerical computations for u-velocity, v velocity and pressure were conducted using staggered grid system. It was seen that the solution converges for few values of underrelaxationfactorforpressure.Itwasseenthattheuvelocityandv-velocityincreasewithincreaseindensityof the fluid, but decreases with increase in the value of coefficient of thermal expansion. Also the location of maximum values of u –velocity and v velocity changes drastically when the viscosity of the fluid changes. The directionfieldsoftheflowwerealsodiscussedforvarious cases. It was found that the direction field of the flow changessignificantlywhenthedensityofthefluidchanges orthevelocityoftop wall increases. It wasobservedthat the pressure inside the cavity increases with increase in densityorviscosityofthe fluid,whereasitdecreases with increaseinthevalueofcoefficientofthermalexpansion.It wasalsoobservedthatthetemperaturevarieswithchange inthermalconductivityanddensityoffluid,butnochange can be seen with change in coefficient of thermal expansionandviscosityofthefluid.

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PressureContoursforlid-drivencavityflow 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 PressureContoursforlid-drivencavityflow 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -1 0 1 2 3 4 5 6 0 0.2 0.4 0.6 0.8 1 1.2 1.4

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