International Journal of Heat and Fluid Flow 24 (2003) 685–697

www.elsevier.com/locate/ijhff

Turbulent flow and heat transfer from a slot jet impinging ona moving plate

Himadri Chattopadhyay a, Sujoy K. Saha b,*

a Heat Power Engineering Group, Central Mechanical Engineering Research Institute, Durgapur 713 209, Indiab Department of Mechanical Engineering, Bengal Engineering College, Howrah, West Bengal 711 103, India

Received 27 November 2001; accepted 21 February 2003

Abstract

The flow field due to an impinging jet over a moving surface at a moderately high Reynolds number, emanating from a rect-

angular slot nozzle has been computed using the large eddy simulation technique. A dynamic subgrid-scale stress model has been

used for the small scales of turbulence. The velocity of the impinging surface perpendicular to the jet velocity has been varied up to

two times the jet velocity at the nozzle exit. Turbulence quantities such as kinetic energy, production rate of turbulent kinetic energy

and the Reynolds stresses are calculated for different surface velocities. It has been observed that, while the turbulent kinetic energy

increases with increasing velocity of the impinging surface, production rate of turbulence initially increases with increasing surface

velocity and then comes down. By analyzing the components of turbulent production it was found that P33 is the dominant term up

to the surface velocity of one unit and when the surface velocity is two times the jet velocity at the nozzle exit, the major contribution

to turbulence production comes from P13 and partly from P11. Heat transfer from the plate initially increases with non-dimensional

surface velocity up to 1.2 and then comes down.

� 2003 Elsevier Inc. All rights reserved.

Keywords: Impinging jet; Heat transfer; Moving surface; Turbulence; Large eddy simulation

1. Introduction

Flow and heat transfer due to impinging jets is a

subject of considerable interest for its theoretical as wellas practical values. Impinging jets are widely used for

heating, cooling and drying in several industrial appli-

cations. The jet impingement is also a common concern

for the researchers dealing with a VTOL aircraft.

Though in many situations the impinging jet is directed

to a moving surface, relatively few studies have reported

the effect of surface motion on the flow field due to an

impinging jet.Earlier studies (Chen et al., 1994; Zumbrunnen, 1991)

have shown that the moving impingement surface

strongly influences the flow field and thus heat transfer.

While in industrial practice the plate velocity can be as

high as ten times the jet velocity (Zumbrunnen et al.,

*Corresponding author. Tel.: +91-33-6684561/62/63; fax: +91-33-

6682916/6684564.

E-mail addresses: chimadri@hotmail.com (H. Chattopadhyay),

suoyks@mech.becs.ac.in (S.K. Saha).

0142-727X/$ - see front matter � 2003 Elsevier Inc. All rights reserved.

doi:10.1016/S0142-727X(03)00062-6

1992), the lack of literature forces the industry to rely

solely on operating experiences and rules of thumb that

may not be desirable in processes where quality is de-

termined by the amount and uniformity of heat transfer,e.g., metals manufacturing (Kohring, 1985).

Subba Raju and Schlunder (1977) have studied ex-

perimentally heat transfer from a single jet on a moving

belt. They have reported that with increasing plate ve-

locity, heat transfer initially increases and then reduces.

Huang et al. (1984) performed numerical modeling of a

turbulent, planar air jet on a rectangular duct with

surface motion effects. They have found that at higherplate speeds, Nusselt number was smaller at the loca-

tions where the surface motion opposed the dividing jet

flow, and Nusselt number was higher where the surface

motion and dividing jet flow are in same direction.

The work of Zumbrunnen et al. (1992) on an ana-

lytical model of a single planar jet impinging on a con-

stant heat flux and constant temperature moving surface

in the laminar regime showed that boundary layer de-velopment away from the jet is slowed by surface

motion which makes convective heat transfer more

Nomenclature

B slot nozzle width

C model parameter of eddy viscosity model

Cs Smagorinsky�s coefficient

H height of the computational domain

h nozzle-to-plate spacing

h0 heat transfer coefficient

k turbulent kinetic energy

L nozzle lengthNu Nusselt number

p pressure

P production rate of turbulence

Pr Prandtl number

q Heat flux between impinging plate and fluid

Re Reynolds number

T temperature

t timeu velocity component in x-directionv velocity component in y-directionw velocity component in z-direction

win nozzle exit velocity

W width of the computational domain

x, y, z spatial coordinates

Greeks

m kinematic viscosity

s turbulent shear stress

Subscripts

1 ambience

av average

in inlet (at nozzle exit)

w walls impinging surface

Superscripts

n current time step

� intermediate level0 small scale component00

fluctuating component

686 H. Chattopadhyay, S.K. Saha / Int. J. Heat and Fluid Flow 24 (2003) 685–697

effective. Chen et al. (1994) developed a numerical

model, taking fully into account the effect of neighbor-

ing jets, to determine convective heat transfer distribu-

tions in the laminar regime for an array of submerged

planar jets impinging on a uniform heat flux or constant

temperature moving surface where the surface motion is

directed perpendicular to jet planes. Unlike the prior

related work, they could simulate the moving surfacewithout imposing cross flow. With increasing plate

speed, flow separation near the flow-merging plane did

not occur and heat transfer distributions became more

uniform but the total heat transfer was reduced. Thus,

the results suggest that neglecting the surface motion

effect may lead to overestimates of heat transfer.

The flow field of an impinging jet is complex and may

contain laminar, transitional and turbulent zones. Adirect numerical simulation (DNS) is undoubtedly the

best approach to reveal the details of such complex

flows. However, in DNS the number of grid points

needed is of the order of Re9=4 so that all scales of mo-

tion are resolved. Large eddy simulation (LES) is a

technique intermediate between the direct simulation of

turbulent flows and the solution of the Reynolds-aver-

age equations through closure approximations. In LES,the contribution of the large scale structures to the

momentum and energy transfer is computed exactly and

the effect of the smallest scales of turbulence is modeled.

Since the small scales are more homogeneous and uni-

versal and less affected by the boundary conditions than

the large eddies, the modeling effort is less. However, it

still requires reasonably fine meshes. At the same time, it

can be used at much higher Reynolds number than

DNS. Several prior studies have demonstrated the in-

adequacy of standard turbulence models in handling the

complex flow field due to jet impingement (see Craft

et al., 1993; Hosseinalipour and Mujumdar, 1995). In a

recent study Shuja et al. (2001) reported that the stan-

dard k–e model predicts excessive kinetic energy gener-

ation in the vicinity of the stagnation region, while the

low Re k–e model performs somewhat better. Anotherstudy on slot jet impingement by Tzeng et al. (1999) has

demonstrated that prediction by the turbulence model

depends strongly on choice of numerical scheme used in

spatial discretization and grid distribution. A survey of

recent literature suggested that a large number of pub-

lications are reported on impinging jets where LES is

being used. Some notable examples include the works of

Rizk and Menon (1988), Jones and Wille (1996), Czieslaet al. (1997), Voke and Gao (1998) and Yuan et al.

(1999).

Very recently Cziesla et al. (2001) have reported the

flow structure of an impinging slot jet using a dynamic

subgrid model of LES. Turbulent quantities such as

kinetic energy, production rate and its components were

calculated. The stress budget captures the locally nega-

tive turbulence production rate successfully. Their re-sults compared favorably with available experimental

data, particularly in the stagnation zone. They have

demonstrated that in the stagnation region the produc-

tion rate is dominated by P11 and away from the stag-

nation region P13 becomes dominant.

In the present work, LES of the flow field of an axial

jet emanating from a rectangular slot and impinging on

a moving surface has been performed. The movement of

H. Chattopadhyay, S.K. Saha / Int. J. Heat and Fluid Flow 24 (2003) 685–697 687

the plate is perpendicular to the jet motion at the exit of

the nozzle. While the Reynolds number for the present

study was 5800 as in the study of Cziesla et al. (2001),

the surface velocity of the impingement surface wasvaried up to two times the jet velocity at the nozzle exit.

It was expected that the results of the present investi-

gation would elucidate the flow structure and form a

database of turbulent quantities for such a configura-

tion. Generating such data from experimentation is

quite demanding in terms of planning, facilities and cost

of experimentation. A CFD tool can thus become quite

useful in obtaining a first estimate of the flow structure.

2. Mathematical formulation

2.1. Basic equations

Fig. 1 shows the geometry of interest. It consists of a

semi-enclosed rectangular slot jet of width B and lengthLy . The non-dimensional length of the impingement

plate, Lx, based upon the characteristic dimension B, is10 and the distance between the impingement plate and

the top wall is h ¼ Lz ¼ 8.

For the present work, we have assumed that the effect

of temperature variation on the properties of the fluid is

negligible. The flow medium is air, which is a Newtonian

fluid. We also assume negligible dissipation effect ontemperature and negligible volume expansion. All ve-

locity variables were non-dimensionalized by Win. If the

Navier–Stokes equations are approximated by a finite-

difference (/finite volume) scheme, then an approxima-

tion filter (top hat filter) is introduced which filters out

all subgrid-scales with scales smaller than �DD, where �DD is

the filter size. In finite-difference procedure

�DD ¼ ð�DD1�DD2

�DD3Þ1=3 ð1Þ

where �DDi is the grid size in x-, y- and z-direction.

Fig. 1. Computational domain for a single jet element.

The filtered Navier–Stokes and continuity equations

for an incompressible flow will assume the form of Eqs.

(2)–(4) which are the conservative, non-dimensional

and incompressible continuity, momentum and energyequations, written for the large scales which are denoted

by an overbar.

o�uuioxi

¼ 0 ð2Þ

o�uuiot

þ o�uui�uujoxj

¼ � o�ppoxi

þ 1

Reo2�uuiox2j

� osijoxj

ð3Þ

oTot

þ o�uujToxj

¼ 1

RePro2Tox2j

� oqjoxj

ð4Þ

with

sij ¼ uiuj � uiuj ð5Þand

qj ¼ �u0jT 0 ð6Þ

The term sij and qj in Eqs. (5) and (6) are the contri-butions of small scales to the large scale transport

equation, which have to be modeled. Following heat

and momentum transfer analogy we can write:

qj ¼2

PrtmT

oToxj

ð7Þ

While it is possible to calculate the value of subgrid-scale

Prandtl number using a dynamic model, in this study,

Prt was taken as unity. The earlier work of Cziesla et al.

(2001) on the stationary surface uses a similar assump-tion and thus the results for a moving surface should be

compared vis-�aa-vis the stationary case using the same

model.

2.2. Subgrid closure model

The most commonly used subgrid-scale model is

based on the gradient transport hypothesis which cor-relates sij to the large scale strain-rate tensor

sij ¼ �2mT Sij þdij

3skk ð8Þ

where dij is Kronecker delta, skk ¼ u0ku0k and Sij is given

by

Sij ¼1

2

o�uuioxj

þ o�uuj

oxi

!ð9Þ

Lilly (1992) proposed an eddy viscosity proportional to

local large scale deformation

mT ¼ ðCSDÞ2jSj ð10Þ

Here CS is a constant, D is the grid filter scale andjSj ¼ ð2SijSijÞ1=2. Invoking Eq. (10) in (8) yields

688 H. Chattopadhyay, S.K. Saha / Int. J. Heat and Fluid Flow 24 (2003) 685–697

sij �dij

3skk ¼ �2CD2jSjSij ð11Þ

The quantity C is the square of the Smargorinsky co-

efficient, which depends on the type of flow under con-

sideration. Lilly (1992) and Germano et al. (1991)

suggested a method to calculate C for each time step and

grid point dynamically from the flow field data. In ad-

dition to the grid filter that signifies the resolved and

subgrid scales, a test filter is introduced for computationof C. The width of the test filter DD is larger than the grid

filter width. In the present computation, the ratio of the

test filter and the grid filter size is two.

The test filter defines a new set of stresses Tij given by

Tij �dij

3Tkk ¼ �2CDD2jbSS jbSS ij ¼ �2Caij ð12Þ

Germano et al. (1991) suggested that consistency be-tween Eqs. (11) and (12) depends on a proper choice of

C. This is achieved by subtraction of the test-scale av-

erage of ssij from Tij, to obtain

Lij ¼ ‘ij �dij

3‘kk ¼ Tij � ssij ¼ �2Caij þ 2

dCbz}|{Cbz}|{

ij ð13Þ

with

bij ¼ �DD2jSjSij ð14Þ

There are various formulations to obtain C from the

Eqs. (11)–(14). Following Piomelli (1993) we have used

C ¼ � 1

2

ðLij � 2dC � bzffl}|ffl{C � bzffl}|ffl{

ijÞaij

amnamnð15Þ

Piomelli and Liu (1995) indicated that there is no sig-

nificant difference between zeroth- and first-order

approximation for estimating C� (see Eq. (16)). The

present computation uses zeroth-order approximation

through the value at the previous time step

C� ¼ Cn�1 ð16Þ

To avoid ill-conditioning of C, we have used the local

averaging procedure of Zang et al. (1993). After aver-

aging, the following additional constraint was imposed

on the averaged C:

C P 0 ð17Þ

i i

2.3. Boundary conditions

At the exit of the jet, constant velocity (top hat)

profile and at the upper wall no-slip conditions are as-

sumed. At the lower (impingement) plate, the velocity usis prescribed. In the present investigation four sets of

values of us, i.e., 0.1, 0.5, 1.0 and 2.0 were used (Di-

richlet type boundary condition). At the upper wall thelogarithmic velocity profile for calculation of the time

averaged (denoted by h i) wall shear stress distribution

has been used. The approach described in Schumann

(1975) implies to set

swUp

¼ hswihUpi

ð18Þ

for the instantaneous wall shear stresses. Up is the re-sultant velocity at the near wall cell. Periodic boundary

conditions are used at y ¼ 0 and 2. At the exit planes

(x ¼ �5 and 5) a convective boundary condition (Or-

lanski, 1976) has been employed. The non-reflective

convective boundary condition is given by

ouiot

þ ue

ouiox

¼ 0 ð19Þ

where ue is the average velocity at the exit plane.

The impinging surface has been assumed to be iso-

thermal at non-dimensional temperature of unity.

3. Method of solution

The fractional step finite-difference method of Kimand Moin (1985) has been used to solve the set of Eqs.

(2) and (5). All variables were arranged on a staggered

grid arrangement (Harlow and Welch, 1965). The dis-

cretization scheme uses central difference formulation

and is second order in space. The two most important

factors in the choice of the spatial differencing strategy

are the formal order of accuracy and the global con-

servation properties of the numerical scheme. While theconservation property improves the stability of the

scheme and the physical realism of the predicted flow

field, the order of accuracy relates the accuracy of the

solution. The temporal and spatial accuracy are ex-

tremely important for LES. The numerical scheme must

be of fourth order on a collocated grid and second order

on a staggered grid. A recent study of Mittal and Moin

(1997) has demonstrated that the second-order centraldifferencing with a staggered grid arrangement provides

the energy spectra which matches excellently with its

experimental counterpart.

In the present work, for the convective terms the

Adams–Bashforth method is used in order to ensure

second-order accuracy in time. The viscous terms are

discretized by the Crank–Nicholson scheme, in semi-

implicit manner.The time discretization of the Navier–Stokes equa-

tions can be written as

unþ1i � uni

Dtþ Dpnþ1 ¼ � 3

2convn þ 1

2convn�1

þ 1

2Reðdiffn þ diffn�1Þ ð20Þ

Replacement of the velocity of the following time step

unþ1i by an intermediate velocity u�i

u� ¼ unþ1 þ DtDpnþ1 ð21Þ

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4y

11

21

31

41

Nu

exp. dataLES

Fig. 2. Comparison of numerical results with the experiment of

Schl€uunder et al. (1970).

Z

<wm>

0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

us=0.1us=0.5us=1.0us=2.0

Fig. 3. Decay of time-mean velocity in the center line.

∂u/∂x

Z

-0.2 0 0.2 0.4 0.60

0.05

0.1

0.15

0.2

0.25

x=4.5x=5.5x=4.0x=6x=2.0x=8.0

(a) ∂u/∂x

Z

-0.2 0 0.2 0.4 0.60

0.05

0.1

0.15

0.2

0.25

x=4.5x=5.5x=4.0x=6x=2.0x=8.0

(b)

∂u/∂x

Z

-0.2 0 0.2 0.4 0.60

0.05

0.1

0.15

0.2

0.25

x=4.5x=5.5x=4.0x=6x=2.0x=8.0

(c) ∂u/∂x

Z

-0.2 0 0.2 0.4 0.6 0.80

0.05

0.1

0.15

0.2

0.25x=4.5x=5.5x=4.0x=6x=2.0x=8.0

(d)

Fig. 4. Distribution of mean horizontal velocity gradient in x-direction. (a) us ¼ 0:1, (b) 0.5, (c) 1.0 and (d) 2.0.

H. Chattopadhyay, S.K. Saha / Int. J. Heat and Fluid Flow 24 (2003) 685–697 689

∂u/∂z

Z

-40 -30 -20 -10 0 100

0.05

0.1

0.15

0.2

0.25

x= 4 .5x= 5 .5x= 4 .0x= 6x= 2 .0x= 8 .0

(a) ∂u/∂z

Z

-60 -50 -40 -30 -20 -10 0 100

0.05

0.1

0.15

0.2

0.25

x=4 .5x=5 .5x=4 .0x=6x=2 .0x=8 .0

(b)

∂u/∂z

Z

-60 -50 -40 -30 -20 -10 0 100

0.05

0.1

0.15

0.2

0.25

x= 4 .5x= 5 .5x= 4 .0x= 6x= 2 .0x= 8 .0

(d)∂u/∂z

Z

-60 -50 -40 -30 -20 -10 0 100

0.05

0.1

0.15

0.2

0.25

x= 4 .5x= 5 .5x= 4 .0x= 6x= 2 .0x= 8 .0

(c)

Fig. 5. Distribution of mean horizontal velocity gradient in z-direction. (a) us ¼ 0:1, (b) 0.5, (c) 1.0 and (d) 2.0.

690 H. Chattopadhyay, S.K. Saha / Int. J. Heat and Fluid Flow 24 (2003) 685–697

yields the following term:

u�i � uniDt

¼ � 3

2convn þ 1

2convn�1 þ 1

2Reðdiffn þ diffnþ1Þ

ð22Þ

Application of Eq. (2) to Eq. (21) gives finally the

Poisson equation for the pressure field

r2pnþ1 ¼ 1

DtDu�i ð23Þ

Once the pressure field is solved, Eq. (21) yields the finalvelocity field

unþ1i ¼ u�i � Dtrpnþ1 ð24Þ

4. Results and discussions

Computations have been performed with a grid of

152 22 79¼ 264,176 cells. The size of Dx is 0.05 be-

tween x ¼ �0:5 and +0.5. Outside this region, Dx has

been increased continuously by 1%. In the y-direction, auniform grid of y ¼ 0:1 has been used. In the z-direction,Dz ¼ 0:00469 has been used on the impingement plateand then Dz has been increased in the normal direction

continuously by a factor of 1.14. For all computations,

the distance between the centre of the near wall grid cells

and the impingement plate (zþp ¼ zpRes) was maintained

below unity, where Res is the Reynolds number based on

friction velocity vs ð¼ 2ss=qÞ. For time-averaging, 200

instantaneous fields over 40,000 time steps have been

used. To ensure stability, it was necessary to deploy finertime resolution (Dt � 0:005). Due to lack of data for jet

impingement flow on moving surface, the present code

was calibrated by comparing the axial velocity profile

and the heat transfer over the stationary impinging

surface. It can be observed from Fig. 2 that the distri-

bution of Nu at Re ¼ 6500 matches very satisfactorily

with the experimental results of Schl€uunder et al. (1970).

The code validation and the adequacy of sample size arediscussed comprehensively in Chattopadhyay and Saha

(2001). A grid sensitivity test was performed by com-

< u"2>

Z

0 0.02 0.04 0.06 0.080

0.5

1

1.5

2

2.5

3

X=4 .5X =5 .5X =4 .0X =6 .0X =2 .0X =8 .0

(a) < u"2>

Z

0 0.02 0.04 0.06 0.080

0.5

1

1.5

2

2.5

3

X = 4 .5X = 5 .5X = 4 .0X = 6 .0X = 2 .0X = 8 .0

(b)

< u"2>

Z

0 0.04 0.08 0.12 0.16 0.20

0.5

1

1.5

2

2.5

3

X =4 .5X =5 .5X =4 .0X =6 .0X =2 .0X =8 .0

(d)< u"2>

Z

0 0.02 0.04 0.06 0.080

0.5

1

1.5

2

2.5

3

X = 4 .5X = 5 .5X = 4 .0X = 6 .0X = 2 .0X = 8 .0

(c)

Fig. 6. Distribution of hu02i at different surface velocities. (a) us ¼ 0:1, (b) 0.5, (c) 1.0 and (d) 2.0.

H. Chattopadhyay, S.K. Saha / Int. J. Heat and Fluid Flow 24 (2003) 685–697 691

paring Nusselt number at two other meshes of 102 18 75 and 182 26 81. The average Nusselt numberdue to the present grid mesh differs from that of the

extrapolated grid-insensitive situation by less than 3%.

Gutmark and Wygnansky (1976) indicate that the

mean axial velocity on the jet-axis decays linearly with

axial distance. Fig. 3 shows the variation of time aver-

aged value of the normal velocity component, hwmialong z at x ¼ 5 i.e., at the centreline of the domain for

varying surface velocities. It is readily seen that hwmiremains constant up to z ¼ 3:5. The decay of hwmi is

pronounced between z ¼ 1 and 0. This is in confirmation

with the experimental investigation of Gardon and

Akfirat (1965) where they observed that the reduction in

impingement velocity started at the distance of a slot

width away from the plate. The decay is slightly accel-

erated in the case of us ¼ 2:0.The time-mean gradient of the u component of ve-

locity is shown in Fig. 4. The gradient hou=oxi is zero on

the impinging surface and quickly reaches an almost-

asymptotic value by about z ¼ 0:05 up to us ¼ 0:5 and

by z ¼ 0:1 and 0.15 for us ¼ 1:0 and 2.0 respectively.

Thus the velocity gradient is steeper for lower levels of

surface velocity. The value is positive near the centerline,decreases away from the line and can be negative at the

edges of the domain. The value vanishes in the right side

of the domain indicating the flow no longer accelerates

but has reached a steady mean velocity. Values at two

locations at the same distance away from the center

show the downstream gradient is, in general, higher in

magnitude.

The time-mean gradient of horizontal velocity in thevertical direction, i.e., hou=ozi is shown in Fig. 5. The

quantity is mostly negative and is very steep on the left

side, i.e., the downstream of the impinging region. For

us ¼ 0:1 and 0.5, while the maximum value is as high as

)50 at 0.5 downstream of the center, the corresponding

value is about )5 in the downstream direction. Gener-

ally, the gradient is positive downstream indicating that

the tangential velocity increases with increasing height.However, at still higher us ¼ 1:0 and 2.0, the gradient is

negative even downstream. The implication is that the

tangential velocity all over the impinging surface always

decreases in the vertical direction. The asymptotic value

<w"2 >

z

0 0.02 0.04 0.060

0.5

1

1.5

2

2.5

3

x=4.5x=5.5x=4.0x=6.0x=2.0x=8.0

(a) <w"2>

Z

0 0.02 0.04 0.060

0.5

1

1.5

2

2.5

3

X= 4.5X= 5.5X= 4.0X= 6.0X= 2.0X= 8.0

(b)

<w"2>

Z

0 0.02 0.04 0.06 0.080

0.5

1

1.5

2

2.5

3

X =4 .5X = 5 .5X = 4 .0X = 6 .0X = 2 .0X = 8 .0

(c) <w"2>

Z

0 0.005 0.01 0.0150

0.5

1

1.5

2

2.5

3

X = 4 .5X = 5 .5X = 4 .0X = 6 .0X = 2 .0X = 8 .0

(d)

Fig. 7. Distribution of hw002i at different surface velocities. (a) us ¼ 0:1, (b) 0.5, (c) 1.0 and (d) 2.0.

692 H. Chattopadhyay, S.K. Saha / Int. J. Heat and Fluid Flow 24 (2003) 685–697

is reached by about z ¼ 0:05, 0.01 and 0.12 for us ¼ 0:5,1.0 and 2.0, respectively.

The non-dimensional turbulent kinetic energy k isdefined as

k ¼ 1

2hu00i u00i i ð25Þ

The turbulent kinetic energy has been averaged over the

width i.e., in the y-direction. The maximum value of k is

found to be 0.046, 0.049, 0.050 and 0.106 for the cor-responding surface velocity of 0.1, 0.5, 1.0 and 2.0, re-

spectively. An analysis of field data for kinetic energy

shows that while at us ¼ 0:1 and 0.5, the location of

maximum k is at a distance of 0.078 from the impinge-

ment surface at x ¼ 1:3 and 1.46, respectively (i.e., at the

left side of the computational domain), the location of

peak kinetic energy shifts upward, i.e., at a distance of

0.558 from the impingement surface at x ¼ 4:7 (almostbelow the impingement point) for us ¼ 1:0. For us ¼ 2:0,the location of peak k is at a distance of 0.08 from the

impingement surface at x ¼ 3:4, i.e., at the left side of

the impingement point.

A measure of the level of fluctuation is taken to be the

root mean square of the fluctuating quantity. Thus

(hu002iÞ1=2 is called the intensity of turbulence in the x-direction. The square of such quantities, which is called

turbulent normal stresses in respective directions, has

been presented in Fig. 6. The figure shows turbulent

intensities (/normal stresses) as a function of non-

dimensional distance from the impingement plate for a

Reynolds number of 5800 at different values of us. For

values of us up to 1.0, values of hu002i do not differ much

near the center of the impingement plate (x ¼ 4:5 and5.5) from that at one unit upstream and downstream

(x ¼ 4:0 and 6.0). However the distributions near the

boundaries (at x ¼ 2:0 and 8.0) vary considerably. The

peak values are observed in this region, which is about

0.08. For us ¼ 2:0, the peak value is about 0.19 and is

observed at x ¼ 4:0, i.e., near the impingement point.

The distribution of hw002i shows a similar pattern for

different values of us, as can be observed from Fig. 7.The peak values are observed near the impingement

point. For us ¼ 0:1 and 0.5 the value is about 0.04 and

0.05 and the location is at the right side of the im-

ox

H. Chattopadhyay, S.K. Saha / Int. J. Heat and Fluid Flow 24 (2003) 685–697 693

pingement point at a distance of z ¼ 1 from the im-

pingement plate. For us ¼ 1:0, the peak value of 0.075

occurs at both x ¼ 4:5 and 5.5 but generally the normal

stress is higher at the right side of the impingementpoint. For us ¼ 2:0, the location of peak value shifts

further away from the impingement plate (z ¼ 2:25) andthe magnitude is about 0.015 at the right side of the

impingement point. It is observed that, for us ¼ 0:1 and

0.5, the maximum value of hw002i is about twice the value

of hu002i; their value is of comparable magnitude for

us ¼ 1:0 and hw002i becomes negligible compared to hu002iat us ¼ 2:0. It should be mentioned here that for a sta-tionary impingement, the values of hu002i and hw002i were

quite close (0.02 and 0.022, respectively) and occurred

near the impingement point at z ¼ 2:0 (Cziesla et al.,

2001). Fig. 8 shows the shear stress ð�hu00w00iÞ dis-

tribution in the region z < 0:25. The highest value

of the quantity is within 0.003 in the impingement re-

gion.

The production of turbulent kinetic energy sustainsthe turbulence. This term is given by

-<u"w">

Z

-0.005 -0.0025 0 0.00250

0.05

0.1

0.15

0.2

0.25

X = 4 .5X = 5 .5X = 4 .0X = 6 .0X = 2 .0X = 8 .0

(a) (

-<u"w">

Z

-0.005 -0.0025 0 0.00250

0.05

0.1

0.15

0.2

0.25

X =4 .5X =5 .5X =4 .0X =6 .0X =2 .0X =8 .0

(c) (d

Fig. 8. Distribution of hu00w00i at different surface vel

P ¼ � ðu00i u00j Þohuiioxj

�ð26Þ

An analysis of field data for P revealed that values of

maximum P increases and the location of maximum Pmoves upward up to us ¼ 1:0, and then the value of Pdrops considerably while the location shifts close to the

impingement surface at z ¼ 0:078. The maximum values

of P are 0.0043, 0.0071, 0.0109 and 0.0034 for us ¼ 0:1,0.5, 1.0 and 2.0, respectively. The distributions of tur-

bulence production are presented through Fig. 9(a)–(d)

for different us.

Four components of P can be expressed as

P11 ¼ �hu00u00i ohuiox

ð27aÞ

P13 ¼ �hu00w00i ohuioz

ð27bÞ

P31 ¼ �hu00w00i ohwi ð27cÞ

-<u"w">

Z

-0.005 -0.0025 0 0.00250

0.05

0.1

0.15

0.2

0.25

X = 4 .5X = 5 .5X = 4 .0X = 6 .0X = 2 .0X = 8 .0

b)

-<u"w">

Z

-0.004 -0.002 0 0.002 0.0040

0.05

0.1

0.15

0.2

0.25

X =4 .5X =5 .5X =4 .0X =6 .0X =2 .0X =8 .0

)

ocities. (a) us ¼ 0:1, (b) 0.5, (c) 1.0 and (d) 2.0.

P

Z

-0.001 0 0.001 0.002 0.003 0.0040

0.5

1

1.5

2

2.5

3

x=4.5x=5.5x=4.0x=6x=2.0x=8.0

(a) P

Z

-0.001 0 0.001 0.002 0.003 0.004 0.005 0.006 0.0070

0.5

1

1.5

2

2.5

3

x=4.5x=5.5x=4.0x=6x=2.0x=8.0

(b)

P

Z

0 0.002 0.004 0.006 0.008 0.010

0.5

1

1.5

2

2.5

3

x=4.5x=5.5x=4.0x=6x=2.0x=8.0

(c) P

Z

-0.001 0 0.001 0.002 0.0030

0.5

1

1.5

2

2.5

3

x=4.5x=5.5x=4.0x=6x=2.0x=8.0

(d)

Fig. 9. Distribution of turbulence production rate at different locations. (a) us ¼ 0:1, (b) 0.5, (c) 1.0 and (d) 2.0.

694 H. Chattopadhyay, S.K. Saha / Int. J. Heat and Fluid Flow 24 (2003) 685–697

P33 ¼ �hw00w00i ohwioz

ð27dÞ

For a stationary impingement surface it was reported

that in the stagnation region (4 < x < 6) the production

rate is dominated by P11 and away from the stagnationregion P13 becomes dominant (Cziesla et al., 2001). In

the present case, such generalization could not be made.

A detailed analysis of all the components of turbulent

production revealed that P33 is the dominant term for

us ¼ 0:1, 0.5 and 1.0. At us ¼ 2:0, major contribution to

turbulent production comes from P13 and also from P11.

Distributions of P33 for us ¼ 0:1, 0.5 and 1.0 are pre-

sented in Fig. 10; distributions of P13 and P11 for us ¼ 2:0are presented in Figs. 11 and 12, respectively.

At us ¼ 0:1, we observe from Figs. 6 and 7 that at the

location of maximum P , magnitude of hw002i is almost

twice the value of hu002i, while the value of hu0w0i is only

about )0.001 (Fig. 8). At us ¼ 0:5, at the location of

maximum production, value of hw002i is 0.05 while that

of hu002i is 0.015. The value of u0w0 is also too small

(about 0.001) to influence P13 or P31. For us ¼ 1:0, atx ¼ 4:5 and 5.5 the value of hw002i is almost seven times

that of hu002i. Thus for us ¼ 0:1, 0.5 and 1.0 major con-

tributions come from P33. However at us ¼ 1:0, the

contribution from P13 becomes significant at x ¼ 4:5(i.e., at the left of the impingement point) as the

occurrence of largest value of hu00w00i (about 0.0.003)

coincides with that of high ohui=oz.However, at us ¼ 2:0, the maximum value of hu002i is

much greater than that of hw002i. From Fig. 8 we also

observe that the value of hu00w00i is as large as )0.003near the impingement surface and higher values are

discerned only at the left side of the impingement point.

Thus P11 and P13 are the major contributors to turbu-

lence production term. All the distributions for

P33

Z

-0.001 0 0.001 0.002 0.003 0.0040

0.5

1

1.5

2

2.5

3

x=4.5x=5.5x=4.0x=6x=2.0x=8.0

P33

Z

-0.001 0 0.001 0.002 0.003 0.004 0.005 0.006 0.0070

0.5

1

1.5

2

2.5

3

x=4.5x=5.5x=4.0x=6x=2.0x=8.0

P33

Z

-0.002 0 0.002 0.004 0.006 0.008 0.010

0.5

1

1.5

2

2.5

3

x=4.5x=5.5x=4.0x=6x=2.0x=8.0

(a) (b)

(c)

Fig. 10. Distribution of P33 component of turbulence production rate at different locations. (a) us ¼ 0:1, (b) 0.5 and (c) 1.0.

P13

Z

-0.0005 0 0.0005 0.001 0.00150

0.5

1

1.5

2

2.5

3

x=4.5x=5.5x=4.0x=6x=2.0x=8.0

Fig. 11. Distribution of P13 component turbulence production rate at

different locations. us ¼ 2:0.

P11

Z

-0.001 -0.0005 0 0.0005 0.001 0.0015 0.0020

0.5

1

1.5

2

2.5

3

x=4.5x=5.5x=4.0x=6x=2.0x=8.0

Fig. 12. Distribution of P11 component of turbulence production rate

at different locations. us ¼ 2:0.

H. Chattopadhyay, S.K. Saha / Int. J. Heat and Fluid Flow 24 (2003) 685–697 695

us

Nuav

0 0.5 1 1.5 230

35

40

45

Computed ValueBest Fit

Fig. 14. Variation of global Nusselt number with surface velocity.

696 H. Chattopadhyay, S.K. Saha / Int. J. Heat and Fluid Flow 24 (2003) 685–697

production terms show that the turbulence production is

usually confined within a very thin zone near the im-

pingement plate.

The heat transfer distribution over the impingementplate can be characterized by the Nusselt number. The

Nusselt number is defined as

Nu ¼ h0Bk0

ð28Þ

with the convective heat transfer coefficient h0 and the

thermal conductivity k0 of the fluid. Under the present

non-dimensional scheme, Nu can be evaluated straight

from the temperature gradient over the surface.

The distribution of the time-averaged local Nusselt

number is presented in Fig. 13. It can be seen from thisfigure that Nu at the upstream corner increases with

increasing surface velocity. A similar increase of Nu for a

moving plate was reported by Payvar and Majumdar

(1994). However, Nu at the impingement point decreases

with increasing surface velocity. While the value of Nuon a stationary impinging surface is 33.41, the values of

Nu are 34.09, 38.58, 41.25 and 39.22 for us ¼ 0:1, 0.5, 1.0and 2.0, respectively.

Fig. 14 presents the variation in the global Nu with

increasing surface velocity. The value increases up to the

surface velocity about 1.2 times the jet velocity at nozzle

exit and then reduces. The dependence of heat transfer

on the surface velocity is consistent with the level of

turbulent kinetic energy and turbulence production as

noted in the earlier sections. As these quantities increase

up to us of the order of 1.0 and then reduce, associatedheat transfer is also affected in a similar manner. The

result is in qualitative agreement with that of Subba

Raju and Schlunder (1977), who have also reported that

heat transfer from a moving surface initially increases

with plate movement and then reduces. Here, it is

noteworthy that while heat transfer from an array of jets

reduces due to the motion of the impinging surface

x

Nu

2 4 6 8 1010

20

30

40

50

60

70

us=0 .1us=0 .5us=1 .0us=2 .0

Fig. 13. Distribution of Nusselt number over the impingement surface.

(e.g., Chattopadhyay et al., 1999), in the case of a single

jet there is substantial enhancement in heat transfer upto a certain velocity of the surface.

5. Concluding remarks

Large eddy simulation has been used to simulate the

flow field of a rectangular impinging jet over a moving

surface. A localization procedure due to Piomelli andLiu (1995) has been used to implement the dynamic

eddy viscosity model. It has been demonstrated that

LES can provide an insight into the turbulent flow field

created by a jet impinging over a moving surface. The

surface velocity has a profound effect on the production

rate of turbulence and heat transfer which initially in-

creases up to us ¼ 1:2 and then decreases with increasing

value of surface velocity. The future studies should in-volve calculation at much higher Reynolds number and

the effect of dynamical calculation of Prs on prediction

of heat transfer.

Acknowledgements

The work was initiated at the Institut f€uur Thermo-dynamik, Ruhr Universit€aat, Bochum. Guidance and

advice from Prof. N.K. Mitra, who has sadly expired on

14th October 1999, are gratefully acknowledged.

References

Chattopadhyay, H., Biswas, G., Mitra, N.K., 1999. Heat transfer from

a moving surface due to impinging jets. In: Proc. ASME HTD 364-

1, pp. 261–270.

H. Chattopadhyay, S.K. Saha / Int. J. Heat and Fluid Flow 24 (2003) 685–697 697

Chattopadhyay, H., Saha, S.K., 2001. Numerical investigations of heat

transfer over a moving surface due to impinging knife-jets. Numer.

Heat Transfer, Part A 39, 531–549.

Chen, J., Wang, T., Zumbrunnen, D.A., 1994. Numerical analysis of

convective heat transfer from a moving plate cooled by an array of

submerged planar jets. Numer. Heat Transfer, Part A 26, 141–160.

Craft, T.J., Graham, L.J.W., Launder, B.E., 1993. Impinging jet

studies for turbulence model assessment of the performance of four

turbulence models. Int. J. Heat Mass Transfer 36, 2685–2697.

Cziesla, T., Tandogan, E., Mitra, N.K., 1997. Large-eddy simulation

of heat transfer from impinging slot jets. Numer. Heat Transfer,

Part A 32, 1–17.

Cziesla, T., Biswas, G., Chattopadhyay, H., Mitra, N.K., 2001. Large-

eddy simulation of flow and heat transfer of an impinging slot jet.

Int. J. Heat Fluid Flow 22, 500–508.

Germano, M., Piomelli, U., Moin, P., Cabot, W.H., 1991. A dynamic

subgrid-scale eddy viscosity model. Phys. Fluids A 3, 1760–1765.

Gardon, R., Akfirat, J.C., 1965. The role of turbulence in determining

the heat transfer characteristics of impinging jets. Int. J. Heat Mass

Transfer 8, 1261–1272.

Gutmark, E.F., Wygnansky, I., 1976. The planar turbulent jet. J. Fluid

Mech. 73, 465–495.

Harlow, F.H., Welch, J.E., 1965. Numerical calculation of time-

dependent viscous incompressible flow of fluid with free surface.

Phys. Fluids 8 (12), 2182–2189.

Hosseinalipour, S.M., Mujumdar, A.S., 1995. Comparative evaluation

of different turbulent models for confined impinging and opposing

jet flows. Numer. Heat Transfer, Part A 28, 647–666.

Huang, P.G., Mujumdar, A.S., Douglas, W.J.M., 1984. Numerical

prediction of fluid flow and heat transfer under a turbulent

impinging slot jet with surface motion and crossflow, ASME paper

84-WA/HT-33.

Jones, W.P., Wille, M., 1996. Large-eddy simulation of a plane jet in a

cross flow. Int. J. Heat Fluid Flow 17, 296–306.

Kim, J., Moin, P., 1985. Application of a fractional-step method to

incompressible Navier Stokes equations. J. Comput. Phys. 59, 308–

323.

Kohring, F.C., 1985. Waterwall water-cooling system. Iron Steel Eng.

(June), 30–36.

Lilly, D.K., 1992. A proposed modification of the Germano subgrid-

scale closure method. Phys. Fluids A 4, 633–635.

Mittal, R., Moin, P., 1997. Suitability of upwind-biased finite

difference schemes for large-eddy simulation of turbulent flows.

AIAA J. 35, 1415–1417.

Orlanski, I., 1976. A simple boundary condition for unbounded flows.

J. Comput. Phys. 21, 251–269.

Payvar, P., Majumdar, P., 1994. Developing flow and heat transfer in a

rectangular duct with a moving wall. Numer. Heat Transfer, Part A

26, 17–30.

Piomelli, U., 1993. High Reynolds number calculations using

the dynamic subgrid-scale stress model. Phys. Fluids A 5, 1484–

1490.

Piomelli, U., Liu, J., 1995. Large-eddy simulation of rotating chan-

nel flows using a localized dynamic model. Phys. Fluids 7, 839–

848.

Rizk, M.H., Menon, S., 1988. Large-eddy simulations of axisymmetric

excitation effects on a row of impinging jets. Phys. Fluids 31, 1892–

1903.

Schumann, U., 1975. Subgrid scale model for finite difference

simulations of turbulent flows in plane channels and annuli.

J. Comput. Phys. 18, 376–404.

Shuja, S.Z., Yilbas, B.S., Budair, M.O., 2001. Local entropy gener-

ation in an impinging jet: minimum entropy concept evaluating

various turbulence models. Numer. Heat Transfer, Part A 190,

3623–3644.

Subba Raju, K., Schlunder, E.U., 1977. Heat transfer between an

impinging jet and a continuously moving surface. W€aarme-

Stoff€uubertr. 10, 131–136.

Schl€uunder, E.U., Kr€ootzsch, P., Hennecke, F.W., 1970. Gesetzm€aaßig-keiten der W€aarme- und Stoff€uubertragung bei der Prallstr€oomung aus

Rund- und Schlitzd€uusen. Chemie-Ing.-Techn. 42, 333–338.

Tzeng, P.Y., Soong, C.Y., Hsieh, C.D., 1999. Numerical investigation

of heat transfer under confined impinging turbulent slot jets.

Numer. Heat Transfer, Part A 35, 903–924.

Voke, P.R., Gao, S., 1998. Numerical study of heat transfer from an

impinging jet. Int. J. Heat Mass Transfer 41, 671–680.

Yuan, L.L., Street, R., Ferziger, J.H., 1999. Large-eddy simulation of

a round jet in crossflow. J. Fluid Mech. 379, 71–104.

Zang, Y., Street, R.L., Koseff, J.R., 1993. A dynamic subgrid-scale

model and its application to turbulent recirculating flows. Phys.

Fluids A 5 (12), 3186–3196.

Zumbrunnen, D.A., 1991. Convective heat and mass transfer in the

stagnation region of a laminar planar jet impinging on a moving

surface. ASME J. Heat Transfer 113, 563–570.

Zumbrunnen, D.A., Incropera, F.P., Viskanta, R., 1992. A laminar

boundary layer model of heat transfer due to a nonuniform

planar jet impinging on a moving plate. W€aarme-Stoff€uubertr. 27,

311–319.