Research Journal of Recent Sciences ________________________________________________ ISSN 2277 - 2502
Vol. 1(ISC-2011), 304-309 (2012)
Res.J.Recent.Sci.

Thermal and Hydrodynamic Analysis of the Impingement
Cooling inside a Backward Facing Step Flow
Khudheyer S. Mushatet
College of Engineering, Thiqar University, Nassiriya, IRAQ

Available online at: www.isca.in
(Received 16th September 2011, revised 11th January 2012, accepted 25th January 2012)

Abstract
The heat transfer and fluid flow of multiple confined impinging jets impinge normally to the backward facing step cross flow has
been numerically investigated. Different sizes of impinging jets were tested while the channel contraction ratio(SR) was ranged
from 0.25 to 0.5. The continuity, Navier-Stockes and energy equations were solved numerically. The discretized form of these
equations was obtained by using finite volume method with staggered grid technique. A Fortran built home computer code
depending on SIMPLE algorithm was developed to obtain the numerical results. The standard k-ε model is used to treat the
effect of turbulence while the wall functions laws were used to treat the regions near the solid walls. The aim of this study is to
show how multiple confined impinging jets can be a controlling factor to enhance the rate of heat transfer form the hot surface
of the channel backward facing step flow. The conducted results show that the heat transfer is enhanced significantly when
using multiple impinging jets. The highest heat transfer was found closer to the region of the facing step. Also the results show
that the rate of heat transfer is increased as the jets sizes increase.
Keywords : Impingement cooling, duct flow, backward facing step.

Introduction
Flow separation and reattachment phenomena are widely
encountered in multiple engineering applications such as
cooling of electronic devices, combustion chambers and
cooling of turbine blades. Although the geometry of the
channel backward facing step is simple, but the resulting
structure of the flow and heat transfer over this step include a
high degree of complexity. In the region just after the step,
the heat losses occur and this phenomena is extended to
reattachment point. So, the present study aims to enhance the
heat transfer in this type of the flow geometry even after the
reattachment point by using multiple impinging slot jets
distributed on the upper channel wall. These jets can remove
a large amount of heat transfer from the hot stepped wall. To
the knowledge of the author, there is no study documented
on this particular flow geometry up to date, Consequently
this study will assist to promote the research area and giving
new aspects to enhance the rate of heat transfer. There are
many studies about the turbulent flow over a backward
facing step. Kasagi and Matsunaga1 investigated the
turbulent flow in a channel with a backward facing step. 3-D
particle tracking velocimeter was used as a measurement
technique. They found that the Reynolds normal and shear
stresses had the maximum values upstream of the reattachment. Their study was compared with numerical
simulation. Lio an Hwang2 performed a numerical study on
turbulent flow in a duct with a backward-facing step. The
turbulent flow and heat transfer in a channel with rib
turbulators was investigated by Lio and chen3, Rau et al.4 and
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Hane and Park5.The main objective of these studies was to
obtain the heat transfer characteristics and friction factor.
ABE and Kondah6 presented a new turbulent model for
predicting fluid flow and heat transfer in separating and
reattaching flows. The presented model was modified from
low-Reynolds number k-ε model. They demonstrated that the
used model was efficient in separating and reattaching flows
downstream of backward facing step. Jovice and Driver 7
presented an experimental study on the turbulent flow over a
backward facing step at low Reynolds number. The aim was
to validate the numerical simulation which was performed by
Stanford/NASA center for turbulence research. Ichimiya and
Hosaka8 performed an experimental study to study the
characteristics of impingement heat transfer caused by three
impinging jets. They conducted that there was two peaks of
the local Nusselt number behind the second nozzle. Zhang et
al.9 developed a stochastic separation flow model to simulate
the sudden expansion of particle-laden flows. Wang et al.10
used large eddy method and Lagrangian techniques to
simulate the turbulent flow over a backward-facing step. The
study verified that the particles follow a path when the
vorticity of the gas phase is small. Thangam and Knight 11
and Nie and Armaly12 investigated the effect of step height
on the separation flow for convective flow adjacent to a
backward-facing step. Rhee and Sung13 adopted a diffusive
tensor heat transfer model instead of the familiar constant
Prandtl number model.

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Research Journal of Recent Sciences ____________________________________________________________ ISSN 2277 - 2502
Vol. 1(ISC-2011), 85-92 (2012)
Res.J.Recent.Sci
In this paper, an attempt is made to incorporate the effect of
impinging slot jets flow into the problem of a channel
backward-facing step and the turbulent flow and heat transfer
of the combined new configuration (figure 1) is numerically
simulated. Different parameters, such as contraction ratio
(SR), size and number of impinging jets, and jet and channel
Reynolds numbers are tested.

Methodology
Mathematical Formulation: The working fluid (air) is
assumed to be Newtonian, Incompressible and the thermo
physical properties are constants. The continuity, NavierStockes and energy equations were used to model the
considered problem. These equations be described as
follows.

u v

0
x y

(1)

v v p   v    v    u 
u  v     eff   2  eff    eff  (3)
x y y x  x  y  y  x  y 
T
T

T   
T 

 v
  eff
   eff
x
y x 
x  y 
y 
eff    t
 t
eff ,T 

Pr Pr
where eff is the combined laminar and turbulent

k in  0.05U in , Tin = Tc = 25  C
2

 in  kin1.5
Rein 

h

,

  0.005

U in h

(12)



kin ,

U in , Tin

are the turbulent kinetic energy,

velocity and temperature at a channel inlet respectively.
At the walls, no slip conditions are imposed; u =v = 0., k = 0.,


 0., Tw  Th = 50  C
y

To treat the large steep gradient near the walls of the channel
and step, wall function laws used by Versteege15 is adopted.
The local Nusselt number along the bottom wall is expressed

Nu 


, at y=0. Zero gradients are imposed on the
Y

channel exit for considered variables.

Turbulence Model: The standard turbulence k-є model
proposed by Launder and Spalding14 is used here to handle
the effect of turbulence in the flow. This model includes two
transport equations, one for turbulent kinetic energy and the
other for the rate of dissipation of turbulent kinetic energy.

  u 2  v 2  u v 2 
Where G  t 2   2   
  
  x 
 y   y x  

The model coefficients are ( σk ;σЄ ; C1є ; C2є ; Cµ) = (1.0,
1.3 , 1.44 , 1.92 , 0.09 ) respectively. The flow parameters at
inlet are described as follows:

(5)

viscosity

(7)

(8)

(9)

(10)

(11)



as

(6)

k
k  
k   
k 
 v   eff ,k    eff ,   G  
x
y x 
x  y 
y 

        

2
u  v   eff ,    eff ,   C1 G  C2
x
y x  x  y  y 
k
k

t  C

k2

(4)

and eff is an effective exchange coefficient.

u


t
, eff ,   

t

the eddy viscosity is obtained by the following formula:

where

u
u p   u    u    v 
u  v    2  eff    eff    eff  (2)
x
y x x  x  y  y  y  x 

u

eff , k   

Numerical solution: In this paper, the numerical simulations
were performed on non-uniform staggered grid system by
using finite volume method (FVM). This gives a system of
discretization equations which means that the system of
governing elliptic partial differential equations is transformed
in to a system of algebraic equations. Then, the solution of
these transformed equations is performed by an implicit line
by line Gaussian elimination scheme. An elliptic finite
volume computer code was developed to attain the results of
the numerical procedure by using pressure-velocity coupling
(SIMPLE algorithm) according to Versteege (1995). This
code is based on a hybrid scheme. Because of the strong
coupling and non-linearity that are inherent in these
equations, relaxation factors are needed to ensure
convergence. To ensure that the turbulent fluid flow solutions
are not significantly affected by the mesh, the numerical
simulations are examined under different grid sizes that range
from 62×28 to 82×52. Adding grid points beyond 62×28 did
not significantly affect the results.

Results and Discussion
The computed results are presented for two-dimensional
turbulent flow through a channel backward-facing step along

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Research Journal of Recent Sciences ____________________________________________________________ ISSN 2277 - 2502
Vol. 1(ISC-2011), 85-92 (2012)
Res.J.Recent.Sci
with orthogonal impinging slot jets flow. Different
contraction ratios and sizes of impinging jets are investigated
for different Reynolds numbers. The effect of contraction
ratio on the distribution of streamlines for three impinging
slot jets and H/B=11 is demonstrated in figure 2. It is evident
that the incoming cross flow affects the behaviour of
trajectories of the multiple impinging slot jets because the
potential core of each jet is distorted. The trajectories of the
impinging slot jet forced the cross channel flow towards the
bottom wall of the channel and reduced the reattachment
length just after the edge of the step. The flow struck the
bottom wall in the region between the step and the first
impinging jet and enhanced heat transfer, as shown in figure
4. This phenomenon occurred for all the studied cases and is
enhanced as the contraction ratio increase (the size and reattachment lengths are increased).
Figure 3 illustrates the non-dimensional axial velocity at
different stream wise stations. It is evident that the
dimensionless axial velocity increases as Re increases
because of increasing the inertia forces but this increase is
affected by the regions of the presence of impinging slot jets
or the facing step.
The effect of increasing the size of the impinging jets on the
distribution of the local Nusselt number is found in figure 4.
It is clear that the local Nusselt number increases as the slot
jet width increases (the ratio H/B decreases) because the
strength and size of the recirculation regions are larger behind
each jet and the step. The larger recirculation regions increase
the impinging flow towards the hot wall besides the
turbulence effects, which increases the rate of heat transfer.
Figure 5 demonstrate the comparison of Nusselt number
variation between the case of the backward facing step flow
with and without impingement cooling in addition to test of
the effect of number of impinging slot jets on the mentioned
variation. As the figure shows, the heat transfer from the
bottom hot wall is significantly increased when incorporating
imping cooling to the backward facing step flow. It can be
seen that the number of the impinging slot jets equal to three
achieved the maximum rate of heat transfer. When the
number of impinging slot jets exceeds three, the rate of heat
transfer is significantly decreased beyond x=0.1.
To validate the present numerical code, a test on some of the
published studies is performed and a good agreement was
obtained. Figure 6 represents an example for this comparison

with conventional backward-facing step problem. The
impinging slot jets affected the size of the recirculation
regions and the re-attachment length behind the facing step.
Also it is found that the heat transfer rate increases as the jet
size increases. The strength of recirculation regions and the
rate of heat transfer after the step and between the impinging
slot jet is enhanced as Reynolds number increases.
Nomenclature
B = slot jet width, m, G = generation term, Kg/m.sec3, H =
height of the channel, m, k = turbulent kinetic energy, m2/s2,
L = length of the channel, m, Nu = local Nusselt number, p =
pitch, m, P = pressure, N/m2, Pr = Prandtl number, -, Re =
Reynolds number,-, s = step height, m, SR = contraction ratio
(s/H),-, Tc = cold wall temperature, Ċ, Th = hot wall
temperature, Ċ, Uin = velocity at a channel inlet, Uj =velocity
at a slot jet inlet
Greek symbols: є = turbulence dissipation rate, m2/s3, µ
dynamic viscosity, N.s/m2, µt = turbulent viscosity, N.s/m2, ρ
air density, Kg/m3, eff = effective exchange coefficient,
kg/m.s

σk; σЄ

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turbulent Schmidt numbers, -

References
1.

Nobuhide Kasagi Akio Matsunaga, Three- Dimensional
particle – Tracking – Velocimeter velocimetry
measurement of Turbulennce Statistics and Energy
Budget in a backward-Facing Step Flow, Int. J. Heat
and Fluid Flow, 16, 477-485 (1995)

2.

Lio T., Hwang J., Developing Heat Transfer and
Friction in a Ribbed Rectangular Duct with Flow
Separation at Inlet, ASME. J. Heat Transfer, 114, 546573 (1992)

3.

Lio T.M., Hwang G.G. and Chen S.H., Simulation and
Measurements of Enhanced Turbulent Heat Transfer in
Channels With Periodic Ribs on One Principal Wall,
International Journal of Heat Mass Transfer, 36, 507507 (1993)

4.

Rau G., Cakan M., Moeller D. and Arts T., The Effect
of Periodic Ribs on The Local Aerodynamics and Heat
Transfer Performance of A Straight Cooling Channel,
ASME Journal of Turbomachinery, 120, 368-375
(1988)

5.

Han J.C, Heat Transfer and Friction Characteristics in
Rectangular Channels With Rib Turbulators, ASME
Journal of Heat Transfer, 110, 91-98 (1988)

Conclusion
A computational study for thermal and hydrodynamic
analysis of the impingement cooling inside a channel
backward facing step flow has been numerically performed.
It is found that the rate of the heat transfer is significantly
enhanced due to the presence of impinging slot jets compared

 T  Tc 
 ,  Th  Tc 

dimensionless temperature 



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Research Journal of Recent Sciences ____________________________________________________________ ISSN 2277 - 2502
Vol. 1(ISC-2011), 85-92 (2012)
Res.J.Recent.Sci
6.

Abe K. and Kondoh T., A new Turbulent Model for
Predicting Fluid Flow and Heat transfer in Separating
and Reattacing flows, 37, 139-151 (1994)

11. Thangam S. and Knight D., Effect of Step Height on
The Separated Flow Past a Backward Facing Step,
Phys. Fluids, 3, 604-606 (1989)

7.

Srba Jovice and David M. Driver, Backward Facing
Step Measurements at Low Reynolds Number,
Reh=500, NASA, California 94035-1000 (1994).

12. Nie J.H. and Armaly B.F., Three Dimensional
Convective Flow Adjacent to a Backward Facing Stepeffects of Step Height, Int. J. Heat Mass Transfer, 45,
2431-2438 (2002)

8.

Ichimiya K. and Hosaka N., Experimental Study of
Heat Transfer Characteristics Due To Confined
Impinging Two Dimensional Jet Exp. Thermal and
Fluid Science, 5, 803-807 (1992)

9.

13. Rhee G.H. and Sung H.G., Enhancement of Heat
Transfer in Turbulent Separated and Re-Attachment
Flow by Local Forcing, Numerical Heat Transfer, Part
A, l37, 733-735 (2000)

Zhang H.Q., Chan C.K. and Lau K.S, Numerical
Solution of Sudden Expansion Particle-Laden Flows
Using an Improved Stochastic Flow Model, Numerical
Heat Transfer, Part A, 4089-102 ( 2001)

14. Jones W.P. and Lunder B.E., The Prediction of
Laminarization with a Two Equation Model of
Turbulence, J. Heat Mass transfer (1972)

10. Wang B., Zhang H.Q. and Wang X.L., Large Eddy
Simulation of Particle Response to Turbulence along its
Trajectory in a Back Word-Facing Step Turbulent
Flow, Int. J. Heat Mass Transfer, 49, 415-420 (2006)

15. Versteege H.K. and Malalasekera W., An Introduction
of Computational Fluid Dynamics, Hemisophere
Publishing Corporation, United State of America (1995)

Tj = Tc

x1

B

P

h

H

s

L
Tw = Th
Figure-1
Schematic diagram of the considered problem, H=0.05m, L=0.4m, x1=0.0492m, H/B=11 and P/B=4

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Vol. 1(ISC-2011), 85-92 (2012)
Res.J.Recent.Sci

a. SR =0.5

b. SR =0.35

c. SR =0.25
Figure-2
Distribution of Stream lines for 3 jets and different values of contraction ratio;
Rej=13517, Rein=16896, H/B=11 and P/B=4

1
0.9

1

0.8

0.9

0.7

0.8
0.7

0.6

0.6

u-velosity

u-velosity

0.5
0.4
0.3
0.2

Re= 6210
Re= 13517
Re= 28127

0.1
0
-0.1

0.5
0.4
0.3
0.2
0
-0.1

-0.2

-0.2

-0.3

-0.3

-0.4

-0.4

-0.5

Re =6210
Re =13517
Re =28127

0.1

-0.5

-0.6

-0.6
0

0.25

0.5

0.75

1

0

0.25

0.5

y/H

y/H

a. x/L=0.0315

b. x/L=0.2

0.75

1

Figure-3
Variation of the dimensionless axial velocity (u/Uin) at different jet Reynolds numbers: H/B=11, P/B=4,
Rein=16896, and SR=0.5

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Vol. 1(ISC-2011), 85-92 (2012)
Res.J.Recent.Sci

100

100

90

90

H/B=11
H/B=2.5

H/B=11
H/B=2.5

80

70

70

60

60

Nu

Nu

80

50

50

40

40

30

30

20

20

10

10
0

0
0

0.1

0.2

0.3

0

0.4

0.1

0.2

0.3

0.4

x

x

Figure-4
Variation of local Nusselt number(on the hot wall) for SR=0.5, Re j=13517 and
Rein=16896
100
2 jets
3 jets
4 jets
without jets

90
80
70

Nu

60
50
40
30
20
10
0
0

0.1

0.2

0.3

0.4

x
Figure - 5
Comparison of variation of local Nusselt number (on the hot wall) with and without
impingement for SR=0.5, Rej=13517, Rein=16896, H/B=11 and P/B=4

100
present simulation
experiment

Nu

80
60
40
20
0

5

10

15

x

Figure-6
Comparison of the present results with published results of [8] for H/B=1 and Rej=6000

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