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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.
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