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Numerical Heat Transfer, Part A:Applications: An International Journal ofComputation and MethodologyPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/unht20

Numerical Simulation of MixedConvection in a Two-DimensionalLaminar Plane Wall Jet FlowK. Kumar Raja a , P. Rajesh Kanna a & Manab Kumar Das aa Department of Mechanical Engineering , Indian Institute ofTechnology Guwahati, North Guwahati , Guwahati, Assam, IndiaPublished online: 14 Sep 2007.

To cite this article: K. Kumar Raja , P. Rajesh Kanna & Manab Kumar Das (2007) Numerical Simulationof Mixed Convection in a Two-Dimensional Laminar Plane Wall Jet Flow, Numerical Heat Transfer,Part A: Applications: An International Journal of Computation and Methodology, 52:7, 621-642, DOI:10.1080/10407780701339835

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NUMERICAL SIMULATION OF MIXEDCONVECTION IN A TWO-DIMENSIONAL LAMINARPLANE WALL JET FLOW

K. Kumar Raja, P. Rajesh Kanna, and Manab Kumar DasDepartment of Mechanical Engineering, Indian Institute of TechnologyGuwahati, North Guwahati, Guwahati, Assam, India

A two-dimensional, laminar, incompressible mixed convection with plane wall jet is simu-

lated numerically using the stream function–vorticity method. The buoyancy is assisting

the main flow. The flow and heat transfer study is carried out for Re ¼ 300–600,

Gr ¼ 103–107, and Pr ¼ 0.01–15. The streamlines, isotherm contours, similarity profiles,

vorticity at the walls, and the local and average Nu values are presented and analyzed.

In some cases, similarity behaviour is observed. The vorticity profile at the wall is similar

to boundary-layer-type flow. However, for high Gr, the wall vorticity increases in the down-

stream direction. The average Nusselt number increases when Re, Gr, and Pr are increased.

1. INTRODUCTION

Mixed-convection flow is encountered in many industrial situations. Flow andheat transfer characteristics studies are carried out in detail for buoyancy-assisting,two-dimensional, laminar, incompressible plane wall jet flow. This situation is com-mon in electronics cooling, defroster systems, paper industry fluid injection systems,heat exchangers, cooling of combustion chamber walls in gas turbines, automobiledemisters, and other applications.

Glauert [1] defined a plane wall jet as a stream of fluid blown tangentially alonga plane wall. A similarity solution for a plane wall jet as well as a radial wall jet forboth laminar and turbulent cases were presented with the introduction of Glauertconstant ‘F ’. Schwarz and Caswell [2] investigated the heat transfer characteristicsof a two-dimensional, laminar, incompressible wall jet. They found exact solutionsfor both constant-wall-temperature and constant-heat-flux cases. In addition, theysolved for variable starting length of the heated section at constant wall temperature.The solution was derived with the plate and jet regimes as nonconjugated.

Received 20 June 2006; accepted 6 January 2007.

The present address of K. Kumar Raja is IBM EMPNO: 008793, EGL Building, Business Park,

Second Floor, Block A, Bangalore, Karnataka, India.

The present address of P. Rajesh Kanna is Department of Mechanical Engineering, National

Taiwan University of Science and Technology, Taipei, Taiwan.

Address correspondence to Manab Kumar Das, Department of Mechanical Engineering, Indian

Institute of Technology Guwahati, North Guwahati, Guwahati 781039, Assam, India. E-mail: manab@

iitg.ernet.in

621

Numerical Heat Transfer, Part A, 52: 621–642, 2007

Copyright # Taylor & Francis Group, LLC

ISSN: 1040-7782 print=1521-0634 online

DOI: 10.1080/10407780701339835

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Schafer et al. [3] solved the conjugate convection=conduction equations todetermine the flow and heat transfer phenomena associated with using a singlelaminar slot jet to cool a linear array of discrete heat sources. Flow was restrictedto a channel formed by an impingement surface, within which the heat sources wereflush-mounted, and a parallel confinement surface, through which the jet was dis-charged. The effects of the jet velocity profile and pertinent dimensionless para-meters on flow and heat transfer conditions were calculated. Hong et al. [4]reported the effect of inclination angle and Pr of a laminar mixed-convection flowin a duct with a backward-facing step. Their study covered both buoyancy-assistingand-opposing flow conditions. They found that the reattachment length is increasedwhen inclination angle is increased from 0� to 180�, but this decreases the wall fric-tion coefficient and Nusselt number. Fluid having a low Prandtl number approachesa fully developed condition at a shorter distance and vice versa. The velocity profilesat various downstream location were given for the comparison of forced and mixedconvection to understand the effect of buoyancy. Correlations were given for primaryand secondary vortex reattachment length, location of the peak Nusselt number, andmaximum Nusselt number. Potthast et al. [5] studied the mixed-convection flowfields and heat transfer of partially enclosed axial and radial laminar jets impingingon a heated flat plate. They observed that free convection may increase the heattransfer by more than 200%.

Buoyancy-opposing, laminar, mixed-convection plane wall jet flow wasreported by Higuera [6]. The flow detachment due to an adverse pressure gradientin the downstream direction was studied analytically and numerically by consideringtwo different cases, ambient fluid passing over a cold plate and cold fluid passing

NOMENCLATURE

g gravitational acceleration, m=s2

Gr Grashoff numberð¼ h3gbDT=n2Þh inlet slot height, m

i x-direction grid point

j y-direction grid point

L length of the plate, m

n normal direction

Nu local Nusselt number ð¼ �qh=qyÞNu average Nusselt number

Pr Prandtl number ð¼ n=aÞRa Rayleigh number ð¼ Gr� PrÞRe Reynolds number ð¼ �uuh=nÞRi Richardson number ð¼ Gr=Re2Þt nondimensional time

tt dimensional time

Tw constant bottom wall temperature, �CT1 constant ambient temperature, �Cu; v dimensionless velocity components

along ðx; yÞ axes

uu; vv dimensional velocity components along

ðx; yÞ axes, m=s

�uu inlet mean velocity, m=s

Umax maximum velocity in normal directions,

m=s

x; y dimensionless Cartesian coordinates,

ðxx; yyÞ=h

xx; yy dimensional Cartesian coordinates along

and normal to the plate, m

a thermal diffusivity, m2=s

b coefficient of expansion

d height where u ¼ Umax=2

DT temperature difference ð¼ Tw � T1Þ; �Ce convergence criterion

g similarity variable ð¼ y=dÞh nondimensional temperature

n kinematic viscosity, m2=s

q density, kg=m3

w dimensionless stream function

x dimensionless vorticity

Subscripts

w wall

1 ambient condition

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over an adiabatic plate. Angirasa [7] studied a laminar buoyant wall jet and reportedthe effect of velocity and the width of the jet during convective heat transfer from thevertical surface. Rao et al. [8] presented results for conjugate, laminar mixed con-vection with surface radiation from a vertical plate having a heat source. Aissiaet al. [9] presented numerical solutions of the Navier-Stokes equations governingthe buoyant round laminar jet. The influence of the exit conditions at the nozzleexit on the dynamic and thermal parameters of the vertical jet flow were reported.A numerical study was conducted by Singh and Sharif [10] to investigate mixed-convective cooling of a two-dimensional rectangular cavity with differentiallyheated side walls. The horizontal walls were assumed to be adiabatic. Cold fluidwas blown into the cavity from an inlet in the side wall of the cavity and exitedthrough an outlet in the opposite side wall. Various placement configurations ofthe inlet and outlet were examined for a range of Reynolds number and Richardsonnumber. For a given Reynolds number, the Richardson number was varied from 0,representing pure forced convection, to 10, which implied a dominant buoyancyeffect. Injection of air at the top and bottom of hot and cold walls was comparedand the results presented in the form of isotherms, streamlines, cooling efficiency,average temperature, and local and average Nusselt numbers at the hot wall. Itwas observed that maximum cooling effectiveness was achieved if the inlet waskept near the bottom of the cold wall while the outlet was placed near the top ofthe hot wall.

Marzouk et al. [11] studied numerically a heated pulsed asymmetric jet in thelaminar region. They solved the governing equations by the finite-difference method.They noticed that the pulsation accelerates the initial development of the jet andimproves diffusion of the entrainment flow of the ambient fluid near the nozzle.Jeddi et al. [12] studied the mixed-convection flow from a hot vertical impingingjet on a colder horizontal disk. The geometry was analogous to a conventionalburning gas cooktop. A numerical simulation of the system was carried out to studythe dependence of fluid flow and heat transfer on the geometric, thermal, and fluidflow parameters. Results showed that heat transfer efficiency versus several para-meters such as inlet velocity magnitude and flue gas temperature has an optimumvalue, in which heat transfer efficiency is maximum. Sahoo and Sharif [13] investi-gated the flow and heat transfer characteristics in the cooling of a heated surfaceby impinging slot jets. The principal objective of this study was to investigate theassociated heat transfer process in the mixed-convective region. Kanna and Das[14] studied the conjugate laminar plane wall jet and reported closed-form solutionsfor interface temperature, local Nusselt number, and average Nusselt number. Theconjugate heat transfer of a laminar offset jet was reported by Kanna and Das[15]. The governing equation was solved by the alternating direction implicit(ADI) method, and clustered Cartesian grids were used for the computations. Kannaand Das found that the local Nusselt number increases to a peak value andthen decreases in the downstream direction. In another study, the same authors[16] carried out a steady-state heat transfer study for a two-dimensional, laminar,incompressible, plane wall jet over a backward-facing step. The heat transfercharacteristics of the jet as functions of Reynolds number, Prandtl number, and stepgeometry (step length and step height) were reported in detail. Results werepresented in the form of isotherm, Nusselt number, and average Nusselt number.

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In some cases, the computed results were compared with the results when the steplength was zero.

Though a large number of studies have been carried out on laminar plane walljet flow, the buoyancy-assisted laminar plane wall jet has not been reported in detail.The present study is concerned with the cooling of heated slabs by a laminar planewall jet flow. This type of flow situation occurs in electronics cooling, refrigerated aircurtains, the paper industry, electrical motor cooling, etc.

2. PROBLEM DEFINITION

A two-dimensional, incompressible, laminar mixed convection in a laminarplane wall jet flow is considered for the study. The governing equations for a con-stant-property fluid with the Boussinesq approximation are available in Bejan [17].The schematic diagram of the physical problem and the boundary conditions ofthe problem under consideration are shown in Figures 1a and 1b, respectively. A sur-face of length 30 times the height of the slot in the streamwise direction is maintainedat a temperature Tw, which is higher than the ambient temperature T1. The gravityforce is acting opposite to the streamwise direction, x. A parabolic velocity profile ofthe jet is located at the leading edge of the surface. The temperature at the inlet is thesame as the ambient temperature. The width of the jet is h, which is equal to the slotheight. An adiabatic wall is assumed to be present in the y direction, as shown inFigure 1b.

The unsteady stream function–vorticity equations governing the incompress-ible mixed–convection laminar flow in nondimensional form are as follows.

Vorcity transport equation:

qxqtþ qðuxÞ

qxþ qðvxÞ

qy¼ 1

Reðr2xÞ � Gr

Re2

qhqy

ð1Þ

where Reynolds number Re = �uuh=n and Gr= h3gbDT=n2.The vorticity x is given by

x ¼ qv

qx� qu

qyð2Þ

Figure 1. Buoyancy-assisting mixed convection in laminar plane wall jet flow.

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The relations between velocities and stream function are

u ¼ qwqy

v ¼ � qwqx

ð3Þ

The Poisson equation for the stream function is

q2wqx2þ q2w

qy2¼ �x ð4Þ

The energy equation is

qhqtþ qðuhÞ

qxþ qðvhÞ

qy¼ 1

Re Prr2h ð5Þ

The nondimensional variables used for Eqs. (1)–(5) are

x ¼ xx

hy ¼ yy

hu ¼ uu

�uuv ¼ vv

�uuw ¼ ww

h�uux ¼ hxx

�uuh ¼ T � T1

Tw � T1

t ¼ tt

ðh=�uuÞ

with the hat indicating a dimensional variable and u; h denoting the average jet velo-city at the jet exit and the jet width, respectively.

The boundary conditions needed for the numerical simulation are prescribed asshown in Figure 1b, The inlet slot height is assumed to be 0.05.

At the jet inlet, along AE (Figure 1b),

uðyÞ ¼ 120y� 2;400y2 xðyÞ ¼ 4;800y� 120 wðyÞ ¼ 60y2 � 800y3 ð6aÞ

Along ED and AB, due to the no-slip condition,

u ¼ v ¼ 0 ð6bÞ

Along CD,

qu

qy¼ 0 and

qv

qy¼ 0 ð6cÞ

At the downstream boundary, the condition of zero first derivative has beenapplied for velocity components. This condition implies that the flow has reacheda developed condition. Thus, at BC,

qu

qx¼ qv

qx¼ 0 ð6dÞ

Similar boundary conditions have been used by Gu [18] and Al-Sanea [19, 20].

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3. NUMERICAL PROCEDURE

The unsteady vorticity transport equation (1) in time is solved by the alternatedirection implicit (ADI) scheme. The central differencing scheme is followed forthe convective as well as the diffusive and the buoyancy terms. It is formally-first-order-accurate in time and second-order-accurate in space OðDt;Dx2;Dy2Þ, and isunconditionally stable. Initially, boundary conditions are imposed, and then, using anx-direction sweep for the energy equation and solving with the tridiagonal matrixalgorithm (TDMA), temperature is obtained at the nþ (1=2) level. Temperaturevalues thus obtained at the nþ (1=2) level are plugged in the x-direction sweep forthe vorticity equation, and x at nþ (1=2) level is obtained at the interior of the domainby using the TDMA. The same procedure is repeated for the y-direction sweep, andvorticity values at the nþ 1 level are obtained. Using the newly obtained vorticityvalues at the nþ 1 level, stream function values are obtained by solving Poisson’sequation (4). The Poisson equation (4) is solved explicitly by the five-point Gauss-Seidel method. The velocities are obtained from Eq. (3). Again, the vorticity values atthe boundary are evaluated using updated values of the stream function, and the pro-cedure is repeated until convergence is achieved. Constant time step 0.001 is used forthe entire calculation. It has been observed that for coarse grids a larger time stepcan be used, whereas for fine grids the solution diverges with large time step.

The solution approaches steady state asymptotically while the time reachesinfinity. The computational domain considered here is clustered Cartesian grids.For unit length, the grid space at the ith node is [21]

xi ¼i

imax� jW

siniW

imax

� �� �ð7Þ

where W is the angle and j is the clustering parameter. W ¼ 2p stretches both endsof the domain, whereas W ¼ p clusters more grid points near one end of the domain.j varies between 0 and 1. When it approaches 1, more points fall near the end. Thedetails of constructing the grids can be found in Kanna and Das [14].

While selecting j, the time step also needs to be considered. The maximum vor-ticity error behavior is complicated as explained by Roache [22]. While marching intime for the solution, it has been observed that the maximum vorticity error gradu-ally decreases. It then increases drastically and finally decreases asymptotically,leading to a steady-state solution. The convergence criterion is to be set in such away that it should not terminate at a false stage. At steady state, the error reachesthe asymptotic behavior. Here it is set as the sum of vorticity error reduced to theconvergence criterion e.

Ximax; jmax

i; j¼1

jðxtþ4ti; j � xt

i; jÞj < e ð8Þ

The convergence criterion for temperature is given by

Ximax; jmax

i; j¼1

ðhtþ4ti; j � ht

i; jÞ < e ð9Þ

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The vorticity boundary condition of Thom [23] in used to obtain the vorticity at thewall and is given below.

xw ¼ �2ðwwþ1 � wwÞ

Dn2ð10Þ

where Dn is the grid space normal to the wall.At the bottom wall and the left side wall, constant stream lines are assumed

based on inlet flow. At the outlet in the downstream direction, streamwise gradientsare assumed to be zero. At the entrainment boundary, normal velocity gradient iszero [24]. The detailed boundary conditions are given below (Figure 1b).

Along ED; u ¼ v ¼ 0 w ¼ 0:050 x ¼ � 2ww � 2wwþ1

Dx2

qhqx¼ 0 ð11aÞ

Along AB; u ¼ v ¼ 0 w ¼ 0 x ¼ � 2wwþ1

Dy2h ¼ 1:0 ð11bÞ

Along BC ðexit boundaryÞ; qu

qx¼ 0

qv

qx¼ 0

qhqx¼ 0 ð11cÞ

q2wqx2¼ 0

qxqx¼ 0 ð11dÞ

Along CD (entrainment boundary);qv

qy¼ 0 h ¼ 0 ð11eÞ

A detailed discussion of the boundary conditions related to the entrainmentand exit boundary conditions may also be found in Kanna and Das [25]. Theentrainment boundary is the surface through which fluid enters into the domainby the action of free shear flow. The exit boundary is the surface through whichthe fluid is goes out of the domain.

4. GRID INDEPENDENCE STUDY

The computational domain used has a size of 30 times the slot height in thedownstream direction and 20 times the slot height in the normal direction. A seriesof grid independence studies has been done to find an acceptable number of gridpoints in both directions. The following grid systems were considered ¼ 61� 61,73� 73, 85� 85, 97� 97, 105� 105, 121� 121. It has been found that the Nu vari-ation for the 105� 105 grid is 0.06%, which is less than 1% as shown in Figure 2. Itis concluded that grid refinement level 5 (105� 105) will be sufficient for the entirecalculations. Figure 3 shows typical grids used for the computations.

5. VALIDATION OF THE CODE

The code developed by Kanna [26] based on the stream function–vorticityapproach following the ADI scheme, has been tested for various problems such as

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lid-driven cavity flow, backward-facing step flow, wall jet flow, offset jet flow, andwall jet flow over backward-facing step flow. This code has been modified to includenatural convection, leading to a mixed-convection flow problem.

Figure 2. Grid independence study: Re ¼ 500, Pr ¼ 0.71, Gr ¼ 104.

Figure 3. Typical clustered grids used for the computations.

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The present code is validated for natural-convection heat transfer by compar-ing the results of a buoyancy-driven laminar heat transfer in a square cavity with dif-ferentially heated side walls. The left wall is kept hot while the right wall is cooled.The top and bottom walls are insulated. In the present work, numerical predictionsusing the algorithm developed are obtained for Rayleigh numbers between 103 and106. The u velocity at the vertical mid-plane (x ¼ 0.5), v velocity at the horizontalmid-plane (y ¼ 0.5), and the temperature at the horizontal mid-plane (y ¼ 0.5) arecompared with the results of Wan et al. [27] and are shown in Figures 4a, 4b, and4c, respectively. The results are in very good agreement with the benchmark solutionfor the range of Rayleigh numbers considered.

Angirasa [7] validated buoyant wall jet computation by comparing to natural-convection results of Churchill and Chu [28]. In the present case, the buoyancy-assistingwall jet flow code is validated with Gr ¼ 0 case (nonbuoyant case). The flow becomesforced-convection heat transfer for Gr ¼ 0. The computed velocity and temperature

Figure 4. Comparison of results with similarity solution for Re ¼ 500, Gr ¼ 0.0, Pr ¼ 1.4. (a) Comparison

of horizontal velocity u at the mid-width x ¼ 0.5; (b) Comparison of vertical velocity v at the mid-height

y ¼ 0.5; (c) Comparison of temperature at the mid-height y ¼ 0.5.

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profiles are compared with the available experimental and similarity solutions. Theu-velocity similarity profile for different downstream locations is compared (Figure 5a)with the similarity solution of Glauert [1] and the experimental results of Quintana et al.[29]. The computed temperature similarity profile is compared (Figure 5b) with thesimilarity solution in a similar way as presented by Seidel [30]. It is observed that atdifferent downstream locations, good agreement among them is obtained.

6. RESULTS AND DISCUSSION

The two-dimensional, incompressible, buoyancy-aided, laminar plane wall jetflow problem is solved by stream function–vorticity formulation for three variables,Re,Pr, and Gr. The maximum Re considered for the study is 600. Transition isexpected [31] for Re > 700 or Gr ¼ 0. Computation is carried out for Reynolds num-bers ranging from 300 to 600. Prandtl numbers ranging from 0.01 to 15 are con-sidered, and Grashof number values used are 103–107. The flow characteristics arestudied with streamline contours, u-velocity contours, similarity profiles for horizon-tal velocity, and bottom wall vorticities. The heat transfer results are presented forisotherm contours, similarity profiles, and local Nusselt numbers. The mean Nusseltnumber is given for all the cases considered.

The effect of Re on the streamline contouns is shown in Figure 6 for Pr = 0.71and Gr ¼ 104. The main flow is discharged from a slot and spreads along the wall(Figure 6a). When Re is increased, the jet is deflected toward the bottom wall(Figures 6b–6d). The streamline contours are superimposed to visualize the effectof change in Re. The inertia force is less for flow with low Re. For this case, the flowcannot overcome the downstream friction [32]. As a result, the jet spreads more inthe normal direction. The effect of Grashof number on the streamline contours isshown in Figure 7 for Re ¼ 400 and Pr ¼ 0.71. When Gr is increased from 103 to107, it is noticed that the streamline contour is concentrated near the bottom wall.It is observed that at higher Gr, the fluid in contact with the hot wall carries more

Figure 5. Comparison of present results (lines) with those of Wan et al. [27] (with markers).

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heat and, due to the higher buoyancy, the flow rate is increased. This effect is morepronounced at far downstream from the jet inlet (Figure 7e).

The effect of Gr on the isotherm contour is shown Figure 8 for Re ¼ 400and Pr ¼ 0.71. At Gr ¼ 103, the effect of the wall jet is more pronounced. Theisotherms are concentrated near the inlet and are spread far away in the outerfield in the downstream region (Figure 8a). When Gr is increased gradually(Figures 8b–8e), the effect of buoyancy force is increased. The isotherms are closelyspaced near the wall, with formation of a gradually thin boundary layer. AtGr ¼ 107 (Ri ¼ 62.5), the isotherms are clustered near the bottom wall (Figure 8e),with a resulting increase in heat transfer. The effect of Pr on the isotherm contoursis shown in Figure 9 for Re ¼ 400 and Gr ¼ 104. At Pr ¼ 0.01, the thermal dif-fusion is high and a thick boundary layer is observed (Figure 9a). The temperaturegradient in the streamwise direction is reduced. When Pr is increased, the zone ofclustered isotherms increases near the jet inlet (Figures 9b and 9c). At Pr ¼ 10.0,

Figure 6. Effect of Re on streamline contours for Pr ¼ 0.71, Gr ¼ 104. (a) Re ¼ 300; (b) Re ¼ 300 (solid

lines), Re ¼ 400 (dashed lines); (c) Re ¼ 300 (Solid lines), Re ¼ 500 (dashed lines); (d) Re ¼ 300 (Solid

lines), Re ¼ 600 (dashed lines).

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the isotherms are clustered near the heated wall AB (Figure 9d). A thin boundarylayer is observed at Pr ¼ 10.0.

The effect of Grashof number on u-velocity contours is presented in Figure 10.It is noticed that when Gr is increased, the buoyancy effect is dominating and theflow is governed by natural convection. Near the corner location C, Figure 1b, anegative u-velocity region is created, which increases in size at higher Gr. ThePrandtl number also affects the entrainment flow. The effect of Prandtl numberon u-velocity contours is presented in Figure 11. At Pr ¼ 0.01, it is noticed that

Figure 7. Effect of Gr on streamline contours for Pr ¼ 0.71, Re ¼ 400.

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entrainment occurs near the jet inlet (Figure 11a). When Pr is increased, the entrain-ment domain is increased (Figures 11b–11d).

The distribution of the bottom wall vorticity is shown in Figure 12. From Eq. (2),the vorticity at the bottom wall is equal to �ðqu=qyÞjy ¼ 0 (where qv=qx ¼ 0), whichis analogous to the skin friction coefficient ½cf ¼ mðqu=qyÞjy¼0=ð1=2Þq�uu2�. Near theinlet, the vorticity has a large value, and farther in the downstream direction it isreduced (Figure 12a). It approaches an asymptotic value in the downstream direction.This is characteristic of boundary-layer flow. For a given Re, when Gr is increased,

Figure 8. Effect of Gr on isotherm contours for Pr ¼ 0.71, Re ¼ 400.

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the skin friction is increased due to the increased buoyancy (Figure 12b). As a result,the vorticity is increased. It can be observed that for Gr up to 105, the vorticity isdecreasing in the downstream direction, similar to a boundary-layer flow. However,when Gr is increaed to 106, it is observed that after an initial drop, the vorticity isincreasing in the downstream direction. This implies that natural convection hasstarted dominating over forced convection because of the wall jet. At Gr ¼ 107, itis observed that the vorticity is gradually increasing in the downstream direction,implying that natural convection is the dominant mode of heat transfer. AtGr ¼ 106, the flow pattern changes drastically. It can be observed from the bottomwall vorticity (Figure 12b). It is due to the induced buoyancy at high Gr numbers.

Investigation has been made to explore the existence of a similarity profile ata downstream location. The profile at x ¼ 1:0 for u velocity and temperature forthe variables studied are presented in Figure 13. The similarity parameter for velo-city is with respect to Umax, where Umax is the maximum velocity in the y directionat a particular x location. The similarity variable in the normal direction isg ¼ y=d, where d is the height at which Umax=2:0 occurs. The effect of Re on the

Figure 9. Effect of Pr on isotherm contours for Gr ¼ 104, Re ¼ 400.

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velocity similarity profile is shown in Figure 13a. It is noticed that in the range ofRe considered, the similarity velocity profiles overlap each other. Since Gr ¼ 104,forced convection is the dominant mode. It is observed that in this range of Re,similarity temperature profiles also overlap each other (Figure 13b). The effect ofGr on u-velocity similarity profiles is presented in Figure 13c. For Gr in the range105–107, the u-similarity profiles are superimposed in the inner region (i.e., fromthe solid wall to the inflection point). However, in the outer region (i.e., fromthe inflection point to the free shear layer) there is a variation. For Gr below

Figure 10. Effect of Gr on u contours for Pr ¼ 0.71, Re ¼ 400.

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Figure 11. Effect of Pr on u contours for Gr ¼ 104, Re ¼ 400.

Figure 12. Bottom wall vorticity contours, Pr ¼ 0.71.

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105 (i.e., 103 and 104), the profiles as observed do not remain similar. The tempera-ture profile (Figure 13d), however, shows a similarity profile for the entire rangeconsidered here. The effect of Pr on the velocity similarity profile is shown inFigure 13e. It is observed that the location at which u=Umax occurs is shifted in

Figure 13. Effect of Re, Pr, and Gr on u profiles and temperature profiles. (a) u velocity similarly profile

for different Re. Pr ¼ 0.71, Gr ¼ 104; (b) Temperature similarity profile for different Re. Pr ¼ 0.71,

Gr ¼ 104; (c) u velocity similarity profile for different Gr. Pr ¼ 0.71, Re ¼ 400; (d) Temperature similarity

profile for different Gr. Pr ¼ 0.71, Re ¼ 400; (e) u velocity similarity profile for different Pr. Gr ¼ 104,

Re ¼ 400; (f) Temperature similarity profile for different Pr. Gr ¼ 104, Re ¼ 400.

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the normal direction at higher Pr. Above Pr ¼ 0:1, the velocity similarity profilesoverlap each other. At low Pr, momentum diffusivity is weak, and it is increasedwhen Pr is increased. The effect of Pr on the temperature profile is shown in Figure13f. At low Pr, heat transfer dominates and thus the profile varies linearly. How-ever, as the Pr is increased, a thermal boundary layer develops. The variation inthe temperature is less sensitive at Pr > 6:0; i.e., a near-similarity nature is observedin the temperature distribution.

Local Nusselt number distribution is presented in Figure 14. The effects ofvarious parameters on Nusselt number distribution have been studied consideringRe ¼ 400;Gr ¼ 104, and Pr ¼ 0:71 and varying one parameter at a time. The vari-ation of Nu for the range of Re is shown in Figure 14a, where Gr and Pr are keptconstant at 104 and 0.71, respectively. The Nusselt number is large near the inletand decreases in the downstream direction, which is a characteristics of flow overa flat plate. When Re is increased, cooling is greater and thus Nu is increased.The effect of Gr on Nu is presented in Figure 14b for Re ¼ 400 and Pr ¼ 0:71.

Figure 14. Local Nusselt number (Nu) for different parameters.

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Nu is higher for higher Gr value. A marked increase in Nu is observed when Gr isincreased above 104. It is less variant from Gr ¼ 103 to Gr ¼ 104. It reaches anasymptotic value in the downstream direction. When Pr is increased (for Re ¼ 400and Gr ¼ 104), Nu is increased (Figure 14c). It has large value near the inlet andfarther on it is reduced and approaches an asymptotic value. The improvement inNu is greater for Pr ¼ 0.71–6.0. After that, however, not much improvement isobserved with increase of Pr.

The average Nusselt number is presented for different cases in Tables 1–3. It isnoticed that Nu is increased when Re, Pr, or Gr is increased. The effect of Re on Nuis shown in Table 1 (for Gr ¼ 104 and Pr ¼ 0.71). A base case of Re ¼ 300 is con-sidered. It is observed that for every 100 increase of Re, Nu is increased by 18%,35.47%, and 52%. It is observed that uniform improvement is obtained. The effectof Gr on Nu is shown in Table 2 (for Re ¼ 400 and Pr ¼ 0.071). A base case ofGr ¼ 0 is considered, which is a forced-convection situation. It is observed that upto Gr ¼ 104, the percent increase is only 10.83%. However, the improvementchanges drastically as Gr is further increased, and finally, at Gr ¼ 107, the percentincrease obtained is 240. The effect of Pr on Nu is given in Table 3 (for Re ¼ 300and Gr ¼ 104. A base case of Pr ¼ 0.71 (air) is considered. It is observed that as

Table 1. Average Nusselt number, effect of Re (Gr ¼ 104, Pr ¼ 0.71)

Re Nu % increase

300 10.249 0.00

400 12.103 18.09

500 13.884 35.47

600 15.589 52.10

Table 2. Average Nusselt number, effect of Gr (Re ¼ 400, Pr ¼ 0.71)

Gr Nu % increase

0.0 10.92 0.00

103 11.47 5.08

104 12.10 10.83

105 14.91 36.56

106 22.21 103.41

107 37.14 240.14

Table 3. Average Nusselt number, effect of Pr (Re ¼ 400, Gr ¼ 104)

Pr Nu % change from air (Pr ¼ 0.71)

0.01 3.96 �67.3

0.1 5.52 �54.35

0.71 12.103 0.00

7.1 26.083 þ115.50

10.0 29.118 þ140.58

15.0 33.181 þ174.15

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Pr is decreased to 0.01, the reduction is 67.3% since the conduction effect is greatercompared to the convection effect for this case. As Pr is increased, a market increasein Nu is observed. For Pr ¼ 15, an increase of 174% is observed, which is attributedto the increased convection effect.

7. A HEAT TRANSFER CORRELATION

The numerically calculated heat transfer results were correlated for mixedconvection in a wall jet. About 100 numerical data were generated, encompassingthe range of parameters 0:00625 � Ri � 62:5; 300 � Re � 600, and 0:01 � Pr � 15.A correlation, having a correlation coefficient of 0.99 and an error band of �6%,was evolved for Nu. It can be shown that the average Nusselt number is correlatedas a function of Richardson number, Reynolds number, and Prandtl number(Ri;Re;Pr) according to the following equation:

Nu ¼ 0:287ð1þ 0:51 Ri0:39ÞRe0:62 Pr0:33 ð12Þ

Figure 15 shows a comparisons of the computed values (on the x axis) and the valuesobtained from the correlation (on the y axis).

8. CONCLUSIONS

The flow and heat transfer study of mixed-convection flow of a wall jet hasbeen carried out for the case in which buoyancy is assisting the forced convection.The following conclusions may be drawn.

Figure 15. Comparisons of computed and the correlated values. Markers show the exact value. Line repre-

sents the equality of these two.

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1. For a given Gr and Pr, the streamlines are concentrated near the wall. The effectis greater near the entry and is similar to boundary-layer flow. The similarity pro-files for the u velocity and the temperature are observed to hold good. The wallvorticity, which is analogous mathematically to skin friction coefficient, increasesnear the entry, and an increase in the local Nu is observed. The average Nusseltnumber increases by 52% for Re ¼ 600 from the base case of Re ¼ 300.

2. The buoyancy effects start dominating over the forced convection from Gr ¼ 105.The streamline and the isotherms are more clustered near the wall at downstreamlocations. Similarity for u velocity is observed in this range and is absent when theGr value is reduced. However, similarity for temperature is observed in the rangeof Gr considered. The wall vorticity distribution is similar to a boundary-layertype for Gr up to 104. The wall vorticity increases in the downstream directionfor Gr ¼ 107. The local Nu increases considerably for high-Gr cases. The averageNu increases by 240% for Gr ¼ 107 from the base case of Gr ¼ 0.

3. As Pr increases, the heat transfer is governed by convection rather than conduc-tion. For high Pr, a relatively small thermal boundary layer is observed. Simi-larity in the u-velocity profile is observed for Pr ¼ 0:71 and above. Similarityin the temperature profile is not observed. The local Nu increases as Pr isincreased. From the base case of Pr ¼ 0:71, the average Nu increases by 174%when Pr is increased to 15 and decreases by 67% when Pr is decreased to 0.01.

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