Turbulent Jet Flow

ORIGINAL

Conjugate heat transfer study of incompressible turbulentoffset jet flows

E. Vishnuvardhanarao Æ Manab Kumar Das

Received: 27 June 2008 / Accepted: 26 March 2009 / Published online: 28 April 2009

� Springer-Verlag 2009

Abstract In the present case, the conjugate heat transfer

involving a turbulent plane offset jet is considered. The

bottom wall of the solid block is maintained at an iso-

thermal temperature higher than the jet inlet temperature.

The parameters considered are the offset ratio (OR), the

conductivity ratio (K), the solid slab thickness (S) and the

Prandtl number (Pr). The Reynolds number considered is

15,000 because the flow becomes fully turbulent and then it

becomes independent of the Reynolds number. The ranges

of parameters considered are: OR = 3, 7 and 11, K =

1–1,000, S = 1–10 and Pr = 0.01–100. High Reynolds

number two-equation model (k–e) has been used for

turbulence modeling. Results for the solid–fluid interface

temperature, local Nusselt number, local heat flux, average

Nusselt number and average heat transfer have been

presented and discussed.

List of symbols

Cp Specific heat at constant pressure

Ce1, Ce2

and Cl

Turbulence model constants

h Width of the jet

H Offset height

k Turbulent kinetic energy

K Ratio of thermal conductivity of solid

and fluid

OR Offset ratio, H/h

p Static pressure

p0 Ambient pressure

P Non-dimensional static pressure

Pr Prandtl number

Re Reynolds number, U0 h/mT Dimensional temperature

Tin Inlet temperature

T? Ambient temperature

U0 Average inlet jet velocity

u; v Dimensional mean velocities in x,

y-directions, respectively

U;V Non-dimensional velocities in X,

Y-directions, respectively

x, y Dimensional co-ordinates

X, Y Non-dimensional co-ordinates

Greek symbols

a, at Laminar and turbulent thermal diffusivities,

respectively

e Dissipation

h Non-dimensionalized temperature.

m, mt Laminar and turbulent kinematic viscosity

q Density of fluid

rk, re Turbulence model constants

DT Reference temperature difference

E. Vishnuvardhanarao � M. K. Das (&)

Department of Mechanical Engineering,

Indian Institute of Technology Guwahati,

Guwahati, Assam 781039, India

e-mail: [email protected]

E. Vishnuvardhanarao

e-mail: [email protected]

Present Address:E. Vishnuvardhanarao

Fluidyn Software and Consultancy Pvt. Ltd,

Bangalore 560 102, India

Present Address:M. K. Das

Department of Mechanical Engineering,

Indian Institute of Technology,

Kharagpur, West Bengal 721302, India

123

Heat Mass Transfer (2009) 45:1141–1152

DOI 10.1007/s00231-009-0486-9

1 Introduction

Wall jet and offset jet are commonly used for cooling

applications. Examples where these kinds of jet cooling are

utilized represent the flows like heat exchangers, fluid

injectors, environmental dischargers, combustion cham-

bers, cooling systems and others. A schematic diagram of

an offset jet used for cooling a heated solid block is shown

in Fig. 1. When the height t = 0, the offset jet becomes a

wall jet. Asymmetric entrainment on both sides of the jet

causes the jet to deflect towards the plate and finally atta-

ches to it. This is called the Coanda effect [1]. The jet is

mainly divided into three regions, viz. the recirculation

region, the impingement region and the wall jet region (as

shown in Fig. 1).

The details of the turbulent offset jet flow were studied

by several authors. Hoch and Jiji [2] have considered the

case of an offset jet in the presence of a free-stream motion.

The fluid flow solution has been provided by them. Uti-

lizing this fluid flow solution, Hoch and Jiji [3] later on

have provided the analytical solution for the temperature

distribution for the same geometry. In a study, Pelfrey and

Liburdy [4] have provided the details of the mean flow and

turbulent characteristics and showed how entrainment,

local pressure and turbulent energy components are affec-

ted by the jet curvature in the reattachment region. In this

study, the plate parallel to the jet was considered as adia-

batic. For large Reynolds numbers generally larger than

104, the impingement distance becomes a function of the

offset ratio only. Holland and Liburdy [5] presented the

thermal characteristics of offset jets in a condition similar

to that of Pelfrey and Liburdy [4] for different offset ratios.

In the study, they examined the surface temperature dis-

tribution, the maximum temperature decay and the tem-

perature variations in three regions. Kim et al. [6] have

conducted an experimental study for Reynolds number

6,500 to 39,000. They have reported that the time-averaged

reattachment points are found to coincide with the maxi-

mum Nusselt number locations.

Yoon et al. [7] and Koo and Park [8] have performed the

numerical simulation and Nasr and Lai [9] have done an

experimental study of the offset jet. It was found that

although three turbulence models predict quantitatively,

the standard k–e turbulence model predicts better the

reattachment length with the experimental value. Shuja

et al. [10] studied the jet impingement on a surface where

air is considered as the impinging gas. Four turbulence

models, including the standard k–e, low Reynolds number

k–e, and two Reynolds stress models, are introduced to

account for the turbulence.

Kanna and Das [11, 12] have carried out a steady-

state heat transfer study for a two-dimensional, laminar,

incompressible, plane wall jet flow for different geome-

tries. Results are presented in the form of isotherm,

Nusselt number, and average Nusselt number for a range

of Reynolds and Prandtl number. In another paper,

Vishnuvardhanarao and Das [13] have presented the

numerical solution of fluid flow and heat transfer for

turbulent offset jet. The heat transfer study under adia-

batic wall boundary condition has also been compared

with the experimental results. In order to understand the

mechanism and its behavior, the detailed velocity and

temperature distribution in the domain are discussed. The

conjugate heat transfer studies of a turbulent wall jet

Fig. 1 Schematic diagram of

the offset jet and a solid bock

1142 Heat Mass Transfer (2009) 45:1141–1152

123

under constant wall temperature and constant heat flux

conditions have been presented by the same authors in

references [14] and [15], respectively. Vishnuvardhanarao

and Das [16] have also reported the conjugate heat

transfer study of offset jet under constant heat flux

boundary condition.

In the present case, a conjugate heat transfer from a

solid block heated with a constant wall temperature is

considered. It is being cooled by a turbulent plane offset

jet. In the laminar flow regime, many publications are

devoted to conjugate heat transfer on flat plate details of

which may be found in Kanna and Das [17]. However,

the conjugate heat transfer study involving a turbulent

flow has received little attention. Some of the conjugate

heat transfer works published in literature (involving

turbulent flow) are by Iaccarino et al. [18], Yilbas [19],

Kassab et al. [20] and Hsieh and Lien [21].

In the conjugate heat transfer approach, the conduc-

tion in the solid and the convection equations in the fluid

regions are solved simultaneously. The near-wall treat-

ments of turbulence models are the key factors to yield

an accurate wall heat transfer predictions. In the standard

high Reynolds number k–e models, wall functions are

commonly employed to bridge the turbulent and near-

wall viscous regions. In the present case, the conjugate

heat transfer involving a turbulent plane wall jet is

considered. The bottom of the solid slab is heated by a

constant temperature. The parameters considered are the

offset ratio, the conductivity ratio (solid/fluid), the solid

slab thickness and the Prandtl number. The Reynolds

number considered is 15,000 because the flow becomes

fully turbulent and then it becomes independent of the

Reynolds number [5]. Power–law scheme is used for

discretizing the convective terms. In order to have the

same numerical accuracy for the fluid flow solution as

reported in Koo and Park [8], higher grid size is used

and non-uniform grids are used along the walls and at

the entrance of jet. The geometry is similar to that of

Pelfrey and Liburdy [4].

2 Mathematical formulations

The flow is assumed to be steady, two-dimensional, tur-

bulent and the fluid is incompressible. Body forces are

neglected and the properties are assumed to be constant.

Reynolds averaged Navier–Stokes (RANS) equations are

used for predicting the turbulent flow. Boussinesq

approximation is used to link the Reynolds stresses to the

velocity gradients. The standard k–e model is used for

calculating the turbulent viscosity (mt). The dimensionless

variables are defined as:

U ¼ u

U0

; V ¼ v

U0

; X ¼ x

h; Y ¼ y

h; h¼ T � T1

Th � T1

P ¼ p� p0

qU20

; kn ¼k

U20

; en ¼e

U30=h

; tt;n ¼ tt

t; at;n ¼

at

a

ð1Þ

The non-dimensionalized equations are:

Continuity equation:

oU

oXþ oV

oY¼ 0 ð2Þ

x-momentum equation:

oðUÞ2

oXþ oðUVÞ

oY¼� o

oXPþ 2

3kn

� �þ 1

Re

o

oX1þ tt;n

� �oU

oX

� �

þ 1

Re

o

oY1þ tt;n

� �oU

oY

� �ð3Þ

y-momentum equation:

oðUVÞoX

þ oðVÞ2

oY¼ � o

oYPþ 2

3kn

� �

þ 1

Re

o

oX1þ tt;n

� � oV

oX

� �

þ 1

Re

o

oY1þ tt;n

� � oV

oY

� �ð4Þ

h-equation:

oðUhÞoX

þ oðVhÞoY

¼ 1

Re � Pr

o

oX1þ at;n

� � ohoX

� �

þ 1

Re � Pr

o

oY1þ at;n

� � ohoY

� �ð5Þ

Turbulent kinetic energy (kn):

oðUknÞoX

þ oðVknÞoY

¼ 1

Re

o

oX1þ tt;n

rk

� �okn

oX

� �

þ 1

Re

o

oY1þ tt;n

rk

� �okn

oY

� �þ Gn � en

ð6Þ

Rate of dissipation (en):

oðUenÞoX

þ oðVenÞoY

¼ 1

Re

o

oX1þ tt;n

re

� �oen

oX

� �

þ 1

Re

o

oY1þ tt;n

re

� �oen

oY

� �

þ C1een

knGn � C2e

e2n

knð7Þ

Production (Gn):

Gn ¼tt;n

Re2

oU

oX

� �2

þ2oV

oX

� �2

þ oU

oXþ oV

oX

� �2" #

ð8Þ

Heat Mass Transfer (2009) 45:1141–1152 1143

123

Eddy viscosity (tt,n):

tt;n ¼ Cl � Re � k2n

enð9Þ

Eddy diffusivity (at,n):

at;n ¼tt;n

Prð10Þ

3 Numerical scheme and method of solution

In the present work, the dimensionless governing equations

are discretized using the control volume method [22].

Power–law scheme is used to discretize the convective

terms and central difference is used for diffusive terms due

to the stability of the solution. To avoid the fine mesh

required to resolve the viscous sub-layer near the boundary,

wall function method [23] has been used which is appro-

priate for high Reynolds number flows. SIMPLE [22]

algorithm is followed to solve the finite difference equa-

tions. Pseudo-transient approach [24] is used to under-relax

the momentum and the turbulent equations. An under-

relaxation of 0.2 is used for pressure.

3.1 Boundary conditions

In this case, the flow of jet is emanating into a quiescent

ambient fluid. The schematic diagram of the typical flow is

shown in Fig. 1. The cases computed here are for offset

ratio 3, 7 and 11. The non-dimensionalized boundary

conditions are provided as input to the solution. At the inlet

of the jet (EF) U ¼ 1:0;V ¼ 0:0 are the boundary condi-

tions for the velocities. For the turbulent kinetic energy

equation, the boundary condition at inlet is kn = 1.5I2

where I the turbulence intensity and is equal to 0.02. For

the dissipation equation, the boundary condition is en ¼

k3=2n C3=4

l

� �=l; where l = 0.07h is considered. For the solid

wall (i.e. AB, AE and FG), no slip boundary condition is

considered for velocity. Neumann boundary conditions are

provided for the top boundary (i.e. entrainment side, GH)

and at the exit boundary (i.e. BH), a developed condition of

q//qn = 0 is considered where / ¼ U;V; h; kn and en:

It has been ensured that the first grid point near the wall

falls in the logarithmic region, i.e. 30 \ Y? \ 100 where

Y? = yus/t, us being the friction velocity.

Once the solution is obtained for velocity, it is used for

solving the energy equation, since the flow is incom-

pressible. In the present case, conjugate heat transfer

consisting of the solid block attached to the wall is con-

sidered. The inlet of the jet and quiescent ambient fluid

temperatures are at h = 0. The left wall (i.e., AE and FG)

are considered adiabatic. On the entrainment side (GH)

X

θ

0 10 20 30 40 50 60 700.98

0.985

0.99

0.995

1

OR=3OR=7OR=11

X

θ

0 10 20 30 40 50 60 700.5

0.6

0.7

0.8

0.9

1

OR=3OR=7OR=11

X

θ

0 10 20 30 40 50 60 700.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

OR=3OR=7OR=11

(a)

(b)

(c)

Fig. 2 Interface temperature distribution at the solid–fluid interface at

different offset ratios. a Pr = 0.01, S = 10 and K = 1,000. b Pr = 1.0,

S = 10 and K = 1,000. c Pr = 100.0, S = 10 and K = 1,000


123

X

θ

0 10 20 30 40 50 60 700

0.2

0.4

0.6

0.8

1

Pr=0.01Pr=0.1Pr=1Pr=10Pr=100

X

θ

0 10 20 30 40 50 60 700.5

0.6

0.7

0.8

0.9

1

S=1S=2.5S=5S=7.5S=10

X

θ

0 10 20 30 40 50 60 70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

K=1K=10K=50K=100K=500K=1000

(a)

(b)

(c)

Fig. 3 Interface temperature distribution at the solid–fluid interface

at OR = 7. a S = 10 and K = 1,000. b Pr = 1 and S = 10. c Pr = 1

and K = 1,000

X

Nu

x

0 10 20 30 40 50 60 700.6

1

1.4

1.8

2.2

OR=3OR=7OR=11

X

Nu x

0 10 20 30 40 50 60 700

10

20

30

40

50

60

70

OR=3OR=7OR=11

X

Nu x

0 10 20 30 40 50 60 7060

140

220

300

OR=3OR=7OR=11

(a)

(b)

(c)

Fig. 4 Local Nusselt number distribution at different offset ratios.

a Pr = 0.01, S = 10 and K = 1,000. b Pr = 1, S = 10 and

K = 1,000. c Pr = 100, S = 10 and K = 1,000


123

X

Nu x

0 20 40 60

100

101

102

103

Pr=0.01Pr=0.1Pr=1Pr=10Pr=100

X

Nu x

0 10 20 30 40 50 60 700

10

20

30

40

50

K=1K=10K=50K=100K=500K=1000

X

Nu x

0 10 20 30 40 50 60 700

10

20

30

40

50

S=1S=2.5S=5S=7.5S=10

(a)

(b)

(c)

Fig. 5 Local Nusselt number distribution and at OR = 7.

a S = 10 and K = 1,000. b Pr = 1 and S = 10. c Pr = 1 and

K = 1,000

X

q x

0 10 20 30 40 50 60 700.005

0.0075

0.01

0.0125

0.015

OR=3OR=7OR=11

X

q x

0 10 20 30 40 50 60 700

0.0006

0.0012

0.0018

0.0024

0.003

OR=3OR=7OR=11

X

q x

0 10 20 30 40 50 60 701E-05

2E-05

3E-05

4E-05

5E-05

6E-05

OR=3OR=7OR=11

(b)

(a)

(c)

Fig. 6 Interface heat transfer (qx) distribution for different offset

ratios. a Pr = 0.01, S = 10 and K = 1,000. b Pr = 1, S = 10 and

K = 1,000. c Pr = 100, S = 10 and K = 1,000


123

h = 0 is applied. To be more precise, the exit boundary

condition q2h/qX2 = 0 has been used. At the bottom of the

solid block (i.e. CD) h = 1.0 is applied. Finally, at the

interface two conditions are met. One is temperature at

the interface should be equal and the other the heat transfer

across the interface must be equal. The details are men-

tioned in Appendix A.

3.2 Code validation and grid independence study

To validate the code developed, results obtained from the

present computation are compared with the experimental

results given by Pelfrey and Liburdy [4] for OR = 7.

Comparisons have been made with the experimental

velocity profiles at different downstream locations of

X = 3 (recirculation region), 6 (recirculation region),

9 (impingement region), 12 (impingement region) and 15

(wall jet region). It is observed that very good agreement

with the experimental results have been obtained. The

details of the validation have been presented in Vishnu-

vardhanarao and Das [13].

After ensuring the code validation, a grid independence

study is carried out for all the cases and the effect of domain

size is also considered and tested. After doing considerable

numerical testing, it is found that a domain size of 75 9 35

and a grid size of 151 9 101 are satisfactory for all the

cases. Even though a grid size of 121 9 101 produces a

good solution, the grid size of 151 9 101 is considered for

all the cases. Similarly the grid independence test is also

done in the solid the block for one depth. For the other

depths, the grid size has been correspondingly increased,

which is reasonable in the solid block.

4 Results and discussion

In the present work, Re = 15,000 is chosen for all com-

putations. Three offset ratios 3, 7 and 11 are considered. At

each offset ratio, Prandtl number is varied from 0.01 to

100, solid thickness is varied from 1 to 10 and thermal

conductivity ratio is varied from 1 to 1,000. In order to

study the heat transfer characteristics, results are presented

in both graphical and tabulated forms.

4.1 Interface temperature

Figure 2a–c shows the effect of offset ratio on the interface

temperature (hi) for Prandtl numbers 0.01, 1 and 100,

respectively, keeping the solid thickness (S = 10) and the

thermal conductivity ratio (K = 1,000) constant. It is found

that the minimum temperature is found at the reattachment

point and it increases with the offset ratio. Interface

X

q x

0 10 20 30 40 50 60 70

10-4

10-3

10-2

Pr=0.01Pr=0.1Pr=1Pr=10Pr=100

X

q x

0 10 20 30 40 50 60 70

10-5

10-4

10-3

K=1K=10K=50K=100K=500K=1000

X

q x

0 10 20 30 40 50 60 700

0.001

0.002

0.003

S=1S=2.5S=5S=7.5S=10

(a)

(b)

(c)

Fig. 7 Interface heat transfer (qx) distribution and at OR = 7.

a S = 10 and K = 1,000. b Pr = 1 and S = 10. c Pr = 1 and

K = 1,000


123

temperature reduces rapidly up to the reattachment point

and increases in the same way in the development region. In

the similarity region, hi increases very slowly. It is observed

that the average temperature in the recirculation region

increases with the offset ratio for all Prandtl numbers. For

Pr = 0.01, in the developed and similarity regions for

OR = 11, average temperature falls below OR = 7. But,

for Pr = 1 and 100, average temperature in all the three

regions increases with the offset ratio. Figure 3a shows the

effect of Prandtl number on the interface temperature,

Table 1 Average Nusselt number Nu� �

at various Prandtl numbers for OR = 3

S (thickness of solid slab) Thermal conductivity

ratio K(ks/kf)

Nu (Pr = 0.01) Nu (Pr = 0.1) Nu (Pr = 1) Nu (Pr = 10) Nu (Pr = 100)

0 (non-conjugate) – 1.20773 7.10002 27.4381 63.4983 133.003

1 1 1.26125 7.53327 28.1117 63.8422 133.152

1 100 1.20915 7.14658 27.6411 63.667 133.102

1 1,000 1.20787 7.10515 27.4661 63.529 133.028

5 1 1.28284 7.56459 28.1249 63.8451 133.153

5 100 1.21388 7.25205 27.8785 63.7765 133.137

5 1,000 1.20839 7.12142 27.5423 63.5978 133.071

10 1 1.28659 7.5685 28.1264 63.8455 133.153

10 100 1.21837 7.31407 27.9545 63.8006 133.143

10 1,000 1.20893 7.13671 27.6013 63.6398 133.091




ratio K(ks/kf)


0 (non-conjugate) – 1.03974 6.32644 24.5308 56.619 118.506

1 1 1.07596 6.65482 25.0298 56.869 118.614

1 100 1.04063 6.35873 24.6713 56.7367 118.577

1 1,000 1.03983 6.32996 24.5495 56.6392 118.523

5 1 1.09243 6.68167 25.0405 56.8713 118.615

5 100 1.04364 6.4383 24.8561 56.8232 118.604

5 1,000 1.04015 6.34185 24.6048 56.6895 118.555

10 1 1.09546 6.68516 25.0418 56.8716 118.615

10 100 1.04651 6.48665 24.9184 56.8432 118.609

10 1,000 1.04049 6.35284 24.6472 56.7205 118.57




ratio K(ks/kf)


0 (non-conjugate) – 1.11616 6.32302 23.8996 54.6758 114.202

1 1 1.15479 6.68075 24.4173 54.928 114.31

1 100 1.11719 6.35867 24.0395 54.7891 114.27

1 1,000 1.11627 6.32691 23.918 54.6947 114.217

5 1 1.17064 6.70955 24.429 54.9306 114.311

5 100 1.12077 6.44792 24.2327 54.8783 114.299

5 1,000 1.11666 6.34041 23.9746 54.7441 114.249

10 1 1.17341 6.71329 24.4305 54.9309 114.311

10 100 1.12405 6.5016 24.2992 54.8995 114.304

10 1,000 1.11705 6.35269 24.0184 54.7754 114.264


123

keeping the S = 10 and K = 1,000 at OR = 7. It is found

that the interface temperature reduces as the Prandtl number

increases. Figure 3b and c shows the effect of thermal

conductivity ratio (K) and slab thickness (S). As expected

interface temperature reduces with the decrease in thermal

conductivity ratio and increase in the slab thickness.

Table 4 Heat transfer across the interface (Qi) at various Prandtl numbers for OR = 3


ratio K (ks/kf)

Qi (Pr = 0.01) Qi (Pr = 0.1) Qi (Pr = 1) Qi (Pr = 10) Qi (Pr = 100)

0 (non-conjugate) – 0.603864 0.355001 0.13719 0.03174 0.00665

1 1 0.275973 0.043828 0.00481 0.00049 4.96E-05

1 100 0.596952 0.331407 0.10669 0.0191 0.002806

1 1,000 0.603166 0.352491 0.13332 0.0297 0.005820

5 1 0.086138 0.009726 0.00099 9.97E-05 9.98E-06

5 100 0.570637 0.261842 0.05723 0.00752 0.000863

5 1,000 0.600376 0.342777 0.12001 0.02381 0.003931

10 1 0.046282 0.004930 0.00049 4.99E-05 5.00E-06

10 100 0.540632 0.207507 0.03635 0.00429 0.000463

10 1,000 0.59691 0.331391 0.10687 0.01915 0.002812



ratio K (ks/kf)


0 (non-conjugate) – 0.51987 0.316322 0.12265 0.02831 0.005925

1 1 0.256775 0.043144 0.00479 0.00049 4.96E-05

1 100 0.514697 0.297312 0.09762 0.0177 0.002664

1 1,000 0.519348 0.31431 0.11954 0.0266 0.005257

5 1 0.084177 0.009691 0.00099 9.96E-05 9.98E-06

5 100 0.494899 0.239924 0.05444 0.00730 0.000848

5 1,000 0.517264 0.306531 0.10870 0.02181 0.003663

10 1 0.045711 0.004922 0.00049 4.99E-05 5.00E-06

10 100 0.472091 0.19345 0.03520 0.00421 0.000459

10 1,000 0.514676 0.297375 0.09781 0.01784 0.002672



ratio K(ks/kf)


0 (non-conjugate) – 0.55808 0.316151 0.11949 0.02733 0.005710

1 1 0.266138 0.043158 0.00479 0.00049 4.95E-05

1 100 0.552174 0.297238 0.09561 0.01739 0.002619

1 1,000 0.557486 0.314151 0.11655 0.02582 0.005089

5 1 0.085170 0.009692 0.00099 9.96E-05 9.98E-06

5 100 0.529653 0.239985 0.05378 0.00723 0.000843

5 1,000 0.555109 0.306407 0.10621 0.02123 0.003578

10 1 0.046002 0.004922 0.00049 4.99E-05 5.00E-06

10 100 0.503774 0.193531 0.03491 0.00419 0.000457

10 1,000 0.552154 0.297281 0.09576 0.01744 0.002625


123

4.2 Local Nusselt number

Figure 4a–c shows the effect of the offset ratio on the

local Nusselt number (Nux) distribution for Prandtl

numbers 0.01, 1 and 100, respectively, keeping the solid

thickness (S = 10) and thermal conductivity ratio

(K = 1,000) constant. The local Nusselt number is

maximum at the reattachment point and reduces with the

offset ratio. Nux increases rapidly up to the reattachment

point and decreases in the same way after the impinge-

ment region. It then decreases slowly in the similarity

region. Figure 5a shows the effect Prandtl number on

Nux at S = 10 and K = 1,000 for offset ratio 7. It

clearly shows that the local Nusselt number decreases

with the decrease in Prandtl number. It is found that, at

the same offset ratio, the peak Nusselt number occurs

always at the reattachment point. Figure 5b and c elu-

cidates that there is no effect of thermal conductivity

ratio and slab thickness on the local Nusselt number for

a conjugate heat transfer situation. This is because the

Nusselt number is governed by the fluid flow and thus is

unaffected by the solid properties.

4.3 Local heat flux

Figure 6a–c shows the variation of local heat transfer (qx)

across the interface for Prandtl number (Pr) 0.01, 1 and

100, respectively at constant solid thickness (S = 10) and

thermal conductivity ratio (K = 1,000) for different offset

ratios (OR). It is observed that the local heat transfer is

maximum (qmax) at the reattachment point and reduces

with the offset ratio. Heat transfer rapidly increases up to

the reattachment point and then decreases in the same way

in the impingement region. It then decreases gradually in

the wall jet region. Figure 7a shows the variation of qx at

various Pr at constant S = 10, K = 1,000 and OR = 7. qx

increases with the decrease in the Prandtl number. Though

qx decreases with increase in the Pr, heat transfer is max-

imum at the reattachment point. Figure 7b shows the effect

of thermal conductivity ratio (K) on the heat transfer at

constant S = 10, Pr = 1 and OR = 7. As expected, the

heat transfer reduces drastically with the decrease in K. It is

found that at low K, heat transfer is constant and the effect

of recirculating region, impingement region are negligible.

Figure 7c shows the effect of solid thickness (S) at constant

Pr = 1, K = 1,000 and OR = 7. It is found that there is a

considerable effect of the slab thickness on the heat transfer

and the heat transfer reduces as S increases.

4.4 Average Nusselt number

Extensive computations are done in their respective ran-

ges and results of the average Nusselt number Nu� �

are

presented in Tables 1, 2 and 3 for offset ratios 3, 7 and

11, respectively. It shows clearly that Nu is a function of

Prandtl number only. The effect of solid thickness (S) and

thermal conductivity ratio (K) are negligibly small. It is

observed that Nu increases with the increase of Pr. It is

found that, for high Prandtl number fluids, average

Nusselt number reduces with the increase in offset ratio.

At Pr = 0.01, Nu of OR = 11 is higher than that of

OR = 7.

4.5 Average heat transfer

The average heat transfer (Qi) integrated over the surface

for various S, K and Pr are shown in Tables 4, 5 and 6 for

offset ratio 3, 7 and 11, respectively. The heat transfer for

the conjugate case is compared with the non-conjugate case

(S = 0). It is observed that as the solid thickness increases,

heat transfer decreases. However, as K is increasing, Qi

increases. For K = 1,000, Qi approaches almost equal to

the non-conjugate case. It is observed that at high Prandtl

number, the non-dimensionalized average heat transfer

reduces with the offset ratio. At Pr = 0.01, Qi of OR = 11

is higher than the OR = 7.

5 Conclusions

In the present case, the conjugate heat transfer study

involving a turbulent plane offset jet is considered. The

bottom of the solid block is maintained at a constant

isothermal temperature. The parameters considered are

the offset ratio, the conductivity ratio, the solid slab

thickness and the Prandtl number. The Reynolds number

considered is 15,000 because the flow becomes fully

turbulent and independent of the Reynolds number. It is

observed that the minimum interface temperature is found

at the reattachment point. The interface temperature

reduces rapidly up to the reattachment point and there-

after it increases. With the increase in OR, the interface

temperature increases. The interface temperature decrea-

ses with increase in Pr. It increases with K and decreases

with S. The local Nux decreases with the increase in OR.

It increases with the increase in Pr. However, Nux

remains nearly unaffected by the variation of K and S.

The non-dimensional local heat flux (qx) decreases with

the increase in OR. It decreases with the increase in Pr. It

increases with the increase in K and decrease with the

increase in S. The average Nusselt number data as tabu-

lated shows clearly that Nu is a function of Prandtl

number and OR only. As K is increasing, the average heat

transfer increases. For K = 1,000, the average heat

transfer approaches almost equal to the non-conjugate

case.


123

Appendix

Deriving the expression for heat flux in the fluid side

At the interface between the solid and fluid, the following

conditions are applied.

• hs = hf at the interface.

• Heat transfer across the interface must be equal.

Wall heat flux in the fluid side is given by:

qf ¼hw � hp;f

� �Clk3=2

n

Prt1j logðEYþÞ þ Pf

� � ð11Þ

where Pf pee-function, which is given by:

Pf ¼ 9:24Pr

Prt

� �3=2

�1

" #� 1þ 0:28 exp �0:007

Pr

Prt

� ��

ð12Þ

Wall heat flux in the solid side is given by:

qs ¼ �1

Re � Pr

ks

kf

� �ohoY¼ � 1

Re � Pr

ks

kf

� �hw � hp;s

DYð13Þ

Interface temperature is calculated by equating Eqs. 11

and 13. Where hp,f, hp,s are neighbor temperatures in the

fluid and solid regions.

Deriving the expression for Nusselt number calculation

We can write the above equation as:

Nux ¼hcH

k

¼ hc Tw � T1� �

� ta� 1

qCp� 1

U0ðTw � T1Þ� U0H

t

ð14Þ

Nux ¼Qw � Re � Pr

qCp Tw � T1� � ð15Þ

Finally,

We can write the above equation as:

Nux ¼Qw � Re � Pr

qCp Tw � T1� � � Th � T1

� �Th � T1� � ð16Þ

Finally,

Nux ¼Qw;n � Re � Pr

hw

ð17Þ

Since h? = 0. Which is used for calculating the Local

Nusselt number distribution. The average Nusselt number

is calculated as:

Nu ¼ZL

0

Nuxdx ð18Þ

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