control of laminar pulsating flow and heat transfer in backward-facing step by using a square...

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Accepted Manuscript Not Copyedited Control of laminar pulsating flow and heat transfer in backward facing step by using a square obstacle Fatih Selimefendigil * Assistant Professor Department of Mechanical Engineering Celal Bayar University Manisa, 45140, Turkey Email: [email protected] Hakan F. Oztop Professor Department of Mechanical Engineering Technology Faculty, Fırat University Elazı ˘ g, 23119, Turkey Email: [email protected] ABSTRACT In the present study, laminar pulsating flow over a backward facing step in the presence of a square obstacle placed behind the step is numerically studied to control the heat transfer and fluid flow. The working fluid is air with a Prandtl number of 0.71 and the Reynolds number is varied from 10 and 200. The study is performed for three different vertical positions of the square obstacle and different forcing frequencies at the inlet position. Navier- Stokes and energy equation for a 2D laminar flow are solved using a finite-volume based commercial code. It is observed that properly locating the square obstacle the length and intensity of the recirculation zone behind the step are considerably affected and hence it can be used as a passive control element for heat transfer augmentation. Enhancements in the maximum values of the Nusselt number of 228% and 197% are obtained for two different vertical locations of the obstacle. On the other hand, in the pulsating flow case at Reynolds number of 200, two locations of the square obstacle are effective for heat transfer enhancement with pulsation compared to the case without obstacle. Keywords: pulsating flow, square obstacle, backward facing step Nomenclature a horizontal location of the obstacle b vertical location of the obstacle c length of the square obstacle f frequency of the oscillation H step size h local heat transfer coefficient k thermal conductivity h local heat transfer coefficient L length of the bottom wall * Corresponding author Journal of Heat Transfer. Received October 08, 2012; Accepted manuscript posted April 04, 2014. doi:10.1115/1.4027344 Copyright (c) 2014 by ASME Downloaded From: http://heattransfer.asmedigitalcollection.asme.org/ on 04/13/2014 Terms of Use: http://asme.org/terms

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Control of laminar pulsating flow and heattransfer in backward facing step by using a

square obstacle

Fatih Selimefendigil ∗

Assistant ProfessorDepartment of Mechanical Engineering

Celal Bayar UniversityManisa, 45140, Turkey

Email: [email protected]

Hakan F. Oztop

ProfessorDepartment of Mechanical Engineering

Technology Faculty, Fırat UniversityElazıg, 23119, Turkey

Email: [email protected]

ABSTRACTIn the present study, laminar pulsating flow over a backward facing step in the presence of a square obstacle

placed behind the step is numerically studied to control the heat transfer and fluid flow. The working fluid is air witha Prandtl number of 0.71 and the Reynolds number is varied from 10 and 200. The study is performed for threedifferent vertical positions of the square obstacle and different forcing frequencies at the inlet position. Navier-Stokes and energy equation for a 2D laminar flow are solved using a finite-volume based commercial code. It isobserved that properly locating the square obstacle the length and intensity of the recirculation zone behind thestep are considerably affected and hence it can be used as a passive control element for heat transfer augmentation.Enhancements in the maximum values of the Nusselt number of 228% and 197% are obtained for two differentvertical locations of the obstacle. On the other hand, in the pulsating flow case at Reynolds number of 200, twolocations of the square obstacle are effective for heat transfer enhancement with pulsation compared to the casewithout obstacle.

Keywords: pulsating flow, square obstacle, backward facing step

Nomenclaturea horizontal location of the obstacleb vertical location of the obstaclec length of the square obstaclef frequency of the oscillationH step sizeh local heat transfer coefficientk thermal conductivityh local heat transfer coefficientL length of the bottom wall

∗Corresponding author

Journal of Heat Transfer. Received October 08, 2012;Accepted manuscript posted April 04, 2014. doi:10.1115/1.4027344Copyright (c) 2014 by ASME

Downloaded From: http://heattransfer.asmedigitalcollection.asme.org/ on 04/13/2014 Terms of Use: http://asme.org/terms

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n unit normal vectorNu local Nusselt numberp pressureP non-dimensional pressurePr Prandtl numberRe Reynolds numberSt Strouhal numbert timeT temperatureu,v x-y velocity componentsU, V non-dimensional x-y velocityx, y Cartesian coordinatesX, Y non-dimensional coordinatesXr reattachment lengthα thermal diffusivityβ thermal expansion coefficientθ non-dimensional temperatureν kinematic viscosityρ density of the fluidτ nondimensional time

1 Introduction

Separation and its subsequent reattachment for fluid flow is important in many engineering applications such as flow

around buildings, airfoils, combustors and collectors of power systems and hence a large amount of literature is dedicated to

the studies related to this topic [1–5]. The flow over a backward facing or forward facing step is a benchmark problem where

flow separation and reattachment occur. A comprehensive review is presented in ref. [6] for laminar mixed convection over

vertical, horizontal and inclined backward- and forward-facing steps. In this review, the effects of pertinent parameters such

as Reynolds number, Prandtl number and expansion ratio on the fluid flow and thermal characteristics are also presented.

Wall temperature distributions, Nusselt number and the friction coefficient for the laminar 3D flow adjacent to backward-

facing step in a rectangular duct have been reported in [7]. A numerical study of 3D linear stability analysis of flow over

a backward-facing step for Reynolds numbers between 450 and 1050 was presented in [8]. Experimental studies have also

been conducted for the flow over a backward facing or forward facing step [9–13]. When pulsations are applied to the

flow system, heat transfer may be enhanced due to the change of the thickness of the boundary layer and thus the thermal

resistance [14–17]. But, in the literature there exist cases where pulsating flow does not affect [18] or even deteriorate heat

transfer enhancement [19]. The flow parameters and geometry of the problem may also have an effect on the heat transfer

enhancement along with the pulsation. Heat transfer and fluid flow characteristics over a backward or forward facing step

in a channel with the insertion of obstacles in pulsating flow have received less attention in the literature [20, 21]. Recently,

pulsating flow at a backward facing step in the presence of a stationary circular cylinder located downstream of the step with

nanofluids has been studied in [22]. The effects of different combinations of parameters on the heat transfer argumentation

were shown. A numerical study of heat transfer enhancement in laminar forced convection flow over a backward facing

step with the insertion of an adiabatic circular cylinder for the Reynolds numbers between 1 to 200 was presented in [23].

Heat transfer enhancement up to 155 % compared to no-cylinder case was attained. This study is however different from the

Journal of Heat Transfer. Received October 08, 2012;Accepted manuscript posted April 04, 2014. doi:10.1115/1.4027344Copyright (c) 2014 by ASME

Downloaded From: http://heattransfer.asmedigitalcollection.asme.org/ on 04/13/2014 Terms of Use: http://asme.org/terms

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studies in [22, 23] since a square obstacle is used at different locations in pulsating flow to control the fluid flow and heat

transfer. Based on the above literature survey and to the best of authors’ knowledge, a study of laminar forced convection

in pulsating flow over a backward facing step in the presence of a square obstacle has never been studied in the literature.

The main objective of the current study is to explore the effects of various parameters such as vertical location of the square

obstacle, Reynolds number and oscillating frequency on the fluid flow and heat transfer characteristics over a backward-

facing step.

2 Numerical Simulation

2.1 Problem description

A schematic description of the considered physical problem is depicted in Fig. 1. A channel with a backward facing step

is considered. The step size of the backward facing step is H and channel height is 2H. At the inlet, a uniform velocity with

a sinusoidal time dependent part (U = 1+ sin(2πStτ)) and a uniform temperature ( θ = 0) are imposed. The downstream

length starting from the edge of the step to the exit of the channel is 35H to ensure that the recirculation length downstream

of the step is independent of the computational domain. The downstream bottom surface of the backward facing step is

maintained at θ = 1, while the other walls of the channel are assumed to be adiabatic. A square obstacle is placed behind

the step for three different vertical locations. The length of the square is c = 0.5H. The center of the square object is located

at (a,b) where a = 0.75H and b = 0.5H,1.0H,1.5H. In the present study, it is expected that different obstacle locations

change the heat transfer and fluid flow characteristic of the backward facing step as it has been shown for a circular cylinder

in ref. [23]. Working fluid is air with a Prandtl number of Pr=0.71. It is assumed that thermo-physical properties of the fluid

are temperature independent. The flow is assumed to be two dimensional, Newtonian, incompressible and in the laminar

flow regime.

2.2 Governing equations and solution method

By using the dimensionless parameters,

(U,V ) =(u,v)

u0,(X ,Y ) =

(x,y)H

,τ =u0

Ht, P =

pρu02 ,θ =

T −Tc

Th −Tc, (1)

for a two dimensional, incompressible, laminar and unsteady case, the continuity, momentum and energy equations can be

expressed in the following format:

∂U∂X

+∂V∂Y

= 0, (2)

Journal of Heat Transfer. Received October 08, 2012;Accepted manuscript posted April 04, 2014. doi:10.1115/1.4027344Copyright (c) 2014 by ASME

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∂U∂τ

+U∂U∂X

+V∂U∂Y

=− ∂P∂X

+1

Re

(∂2U∂X2 +

∂2U∂Y 2

), (3)

∂V∂τ

+U∂V∂X

+V∂V∂Y

=−∂P∂Y

+1

Re

(∂2V∂X2 +

∂2V∂Y 2

), (4)

∂θ

∂τ+U

∂θ

∂X+V

∂θ

∂Y=

1PrRe

(∂2θ

∂X2 +∂2θ

∂Y 2

), (5)

The boundary conditions for the considered problem in dimensionless form can be expressed as:

At the channel inlet, velocity is unidirectional and sinusoidal, temperature and velocity are uniform, (U = 1+0.9sin(2πStτ), V =

0, θ = 0.) At the bottom wall, downstream of the step, temperature is constant (θ = 1.) At the channel exit, gradients of all

variables in the x-direction are set to zero, ( ∂U∂X = 0, ∂V

∂X = 0, ∂θ

∂X = 0.) On the channel walls (except the downstream of the

step) and on the square obstacle, adiabatic wall with no-slip boundary conditions are assumed, (U = 0, V = 0, ∂θ

∂n = 0),

where n denotes the surface normal direction.

Local Nusselt number, spatial and time-spatial averaged Nusselt Number along the bottom wall downstream of the step

can be defined as

Nux,t =hx,tL

k, Nut =

1L

∫ L

0Nux,tdx, Nu =

∫τ

0Nutdt. (6)

where hx,t , k and τ represent the local heat transfer coefficient, thermal conductivity of air and period of the oscillation,

respectively. Eqs. (2) (5) along with the boundary conditions are solved with Fluent (a general purpose finite volume

solver [24]). The convective terms in the momentum and energy equations are solved using QUICK scheme and SIMPLE

algorithm is used for velocity-pressure coupling.The pressure-based segregated algorithm was used in Fluent software. The

system of algebraic equations is solved with Gauss-Siedel point by point iterative method and algebraic multi-grid method.

The convergence criteria for continuity, momentum and energy equations are set to 10−4,10−5 and 10−6, respectively. The

computational domain is divided into 51030 triangular elements for the case with square obstacle at position b/H = 1.0

and at Reynolds number of 200. A detail of the mesh in the vicinity of the step and square obstacle for this case is shown

Journal of Heat Transfer. Received October 08, 2012;Accepted manuscript posted April 04, 2014. doi:10.1115/1.4027344Copyright (c) 2014 by ASME

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in Fig. 2. The mesh is finer near the walls of the channel and square obstacle to resolve the high gradients in the thermal

and hydrodynamic boundary layer and in the vicinity of the step for the recirculation region downstream of the step. Mesh

independence study is carried out for each case to obtain an optimal grid distribution with accurate results and minimal

computational time. Local Nusselt number distribution for four different grid densities (G1=17200, G2=36290, G3=51030,

G3=69982) are depicted in Fig. 3 for the case at b/H = 1.0 and Reynolds number of 200. Steady flow simulation results

are used as initial conditions for the unsteady computations. A time step size of dt = T/100 was used in the unsteady

computations where T denotes the period of the pulsating flow. Time step size independence study is conducted for the

case with square obstacle at position b/H = 1.0 for the case at Reynolds number of 200 and Strouhal number of 1. Fig. 4

demonstrates the space averaged Nusselt number along the bottom wall for different time step sizes. The result at time step

size dt = T/100 is close to the solution for dt = T/200.

The numerical code is first checked against the benchmarked results for the backward facing step reported in the literature

[25–29]. Table 1 shows the results of the reattachment length normalized by step height at Reynolds number of 100 for

expansion ratio of 2. Difference in the table indicates the percentage in the difference between the reattachment length

computed with the present code and computed in various studies. The agreement between the other sources is within 5%.

3 Results and Discussion

The main parameters that affect the fluid flow and thermal characteristics are Reynolds number, step height, distance

between the step edge to the channel exit downstream of the step, distance between the inlet to the step edge, obstacle length,

horizontal and vertical locations of the obstacle, expansion ratio, oscillating frequency, amplitude of the forcing at the inlet

and Prandtl number. In the present study, expansion ratio is 2, horizontal distance of the square obstacle to the step edge is

0.5H. The distance between the inlet to the step edge is 10H (for Re ≥ 100, the value can be set to any value with parabolic

velocity imposed at the inlet [30]) and the distance between the step edge to the channel exit downstream of the step is 35H.

Amplitude of the forcing at the inlet is set to 0.9. The simulations are performed for Reynolds number between 10 to 200,

vertical locations of the obstacle from the bottom wall of the channel 0.5H to 1.5H and Strouhal number between 0.1 to 2.

3.1 Steady case

Fig. 5 and Fig. 6 demonstrate the effects of square obstacle on the streamlines and isotherms for various Reynolds

numbers. In the absence of the obstacle as the Reynolds number is increased, the flow at the edge of the step separates and a

recirculation zone is observed behind the step (Fig. 5 (a,e,i)). The size of this zone increases with an increase in the Reynolds

number. When the obstacle is mounted behind the step for the location at b = 0.5H, the effect on the flow patterns on the

upstream side is negligible for all Reynolds number considered compared to the case without obstacle (Fig. 5 (b,f,j)) since

the flow coming from the upstream of the step is not considerably affected by the square obstacle. A similar behavior was

observed for the cylinder obstacle in ref. ( [23]). For the obstacle location at b = 1.0H, more flow is directed towards the

step and bottom wall downstream of the step, the reattachment occurs at the location upstream of the reattachment locations

compared to the case without obstacle as shown in Fig. 5 (g, k). At Reynolds number of 200, some portion of the fluid

Journal of Heat Transfer. Received October 08, 2012;Accepted manuscript posted April 04, 2014. doi:10.1115/1.4027344Copyright (c) 2014 by ASME

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separated from the bottom wall again separates at the edge of the square obstacle as seen in Fig5 (k). When the square

obstacle is located at b = 1.5H, the length of the recirculation zone behind the step decreases compared to the no-obstacle

case. At Reynolds number of 100 and 200, the flow separated form the bottom wall is entrained into the wake of the square

obstacle and formation of the vortices is seen behind the square obstacle as shown in Fig. 5 (h, l). As it can be seen from the

streamline plots, the square obstacle redirects the flow movement and the flow behavior behind the step (length and intensity

of the recirculation zone behind the step) changes accordingly. The wake behind the square obstacle is also considerably

affected with the installation of the obstacle especially for the locations at b = 1.0H and b = 1.5H. In Fig. 5 (l), jet-like

flow is occurred between the edge of backward facing step and left bottom edge of square object. Thus, two circulating

cells are formed behind the square object. As it can be seen from Fig. 6, adding a square obstacle alters the temperature

contours. When the obstacle is located at position b = 0.5H, isotherm patterns are directed from the step towards the left

edge of the square obstacle for Reynolds number of 10 as shown in Fig.6(b). The heat transfer rate for the region below

the bottom edge of the square obstacle is low in that case. When the obstacle is located at b = 1.0H as shown in Fig. 6 (g,

k), the location where the steep temperature gradient occurs moves upstream when compared to no-obstacle case. For this

position more clustering of the isotherms is seen on the bottom wall downstream of the step. For the square obstacle position

at b = 1.5H, similar trends in temperature contours are seen as for the case at obstacle location b = 1.0H. Finally, the square

object can be used to control temperature distribution. The effect of varying horizontal position of the obstacle on the local

Nusselt number is demonstrated in Fig. 7 for various Reynolds numbers. In these plots, a peak in the Nusselt number is seen

which corresponds to a location very close to the reattachment point where steep temperature gradient occurs. There is large

temperature difference at that region. For the case (b = 0.5H for Re=10), a minimum in the Nusselt number is seen which

corresponds to the location under the square obstacle where isotherms are clustered from the step towards the left edge of

the obstacle (Fig. 6 (b)). For other locations at b = 1.0H and b = 1.5H, the peak value in the Nusselt number increases

and shifts towards the step indicating that the length of the reattachment deceases with the installation of the obstacle when

compared to the case without obstacle. Fig. 8 shows the effect of horizontal location of the square object on the maximum

value of Nusselt number (top figure) and location of the maximum value of the heat transfer (bottom) for various Reynolds

numbers. For all cases considered, peak value in the Nusselt number increases monotonically with increasing values of

Reynolds number. The peak values are greater for the obstacle position at b = 1.0H and b = 1.5H compared to no-obstacle

case and obstacle position at b = 0.5H. The location where this peak is seen increases with increasing Reynolds number

for the case without obstacle and for the case when the obstacle is located at position b = 0.5H. For other locations of the

obstacle at b = 1.0H and b = 1.5H, the location of the maximum heat transfer changes negligibly with increasing values of

Reynolds number. Similar observations have been reported for the cylinder obstacle in ref. [23]. Length averaged Nusselt

number along the bottom wall downstream of the step is shown in Fig. 9. Averaged heat transfer increases as the Reynolds

number increases. When the square obstacle is located at b = 1.0H, the average heat transfer does not change compared

to cases when the obstacle is located at b = 0.5H and without obstacle case. Averaged heat transfer enhancements of 13%

and 21% are obtained for the case with obstacle at b = 1.5H compared to no-obstacle case for Reynolds number of 100

and 200, respectively. Maximum heat transfer enhancements of 228% and 197% are attained at Reynolds number of 200 in

Journal of Heat Transfer. Received October 08, 2012;Accepted manuscript posted April 04, 2014. doi:10.1115/1.4027344Copyright (c) 2014 by ASME

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the presence of the square obstacle compared to no-obstacle case, when the obstacle is located at positions b = 1.5H and

b = 1.0H.

3.2 Transient case

The effect of the varying vertical location of the square object on length averaged Nusselt number along the bottom

wall downstream of the step divided by the steady state value for various Reynolds number at Strouhal number of 0.1 is

demonstrated in Fig. 10. There is negligible change for the time evolution of the Nusselt number for the case with obstacle

at position b = 0.5H when compared to the case without obstacle. As Reynolds number increases, a shift in the mean

Nusselt number for no-obstacle case and for obstacle case at positions b = 1.0H and b = 1.5H is observed. Distortion from

a pure sinusoid is seen with increasing Reynolds number. Fig. 11 shows the effect of varying vertical location of the square

obstacle on length averaged Nusselt number for Strouhal number of 2. As the frequency increases, the number of periods to

reach steady state generally increases for the same Reynolds number. Averaged Nusselt number decreases with increasing

frequency for the same Reynolds number.

Streamlines and isotherms for the three time instances during the acceleration phase of a half period (corresponding

to minimum, middle and maximum values of the Nusselt number) when the system reaches the periodic steady state are

demonstrated in Fig. 12 (Reynolds number of 100, Strouhal number of 0.5, without obstacle (w), with obstacle at different

locations (P1: b = 0.5H, P2: b = 1.0H, P3: b = 1.5H)) and Fig. 13. When the Nusselt number is maximum, the fluid

accelerates towards the region adjacent to the step, and the core of the vortex formed behind the step moves upward for the

case without obstacle. The presence of the obstacle at locations b = 1.0H and b = 1.5H affects the main flow adjacent to step

and recirculation region formed behind the step. Two vortices are formed ( behind the step and adjacent to the right lower

corner of the square obstacle) at the time instance when the Nusselt number is minimum for the case when the obstacle is

at location b = 1.5H,. These cells disappear during the acceleration phase of the fluid. There is a significant change during

the time instance when the minimum and maximum values of the Nusselt number are reached between the streamlines of

no-obstacle case and streamlines of obstacle case at position b = 1.5H. These plots show that a considerable change occurs

in the flow field during the pulsation cycle when the obstacle installed at positions b = 1.0H and b = 1.5H. Fig. 13 shows the

isotherms for the three time instances during the acceleration phase of a half period at Reynolds number of 100 and Strouhal

number of 0.5. Temporal evolution of the local Nusselt number along the bottom wall for three time instances within half

a period is depicted in Fig. 14. There is a slightly change of the isotherms on top part of the step for the case without

obstacle. This behavior is also seen in the local Nusselt number distribution shown in Fig. 14. The trend is similar to that

of the case without obstacle for the case with obstacle located at b = 0.5H. The isotherms change sightly for other obstacle

locations. The peak value of the Nusselt number increases for the obstacle at location b = 1.0H and shifts downstream of

the step at location b = 1.5H for the time instance when the maximum value of the Nusselt number is reached as it can

be seen from Fig. 14. The effects of square object and its location on the time-spatial averaged Nusselt number along the

bottom wall downstream of step for vaiouts Strouhal and Reynolds number are demonstrated in Fig. 15. In these plots,

vertical axis is normalized by the spatial averaged Nusselt number in the steady state case. Heat transfer enhancement with

Journal of Heat Transfer. Received October 08, 2012;Accepted manuscript posted April 04, 2014. doi:10.1115/1.4027344Copyright (c) 2014 by ASME

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the pulsation is observed for the case with obstacle located at b = 1.5H at Strouhal numbers greater than 0.5 at Reynolds

number of 10. The normalized Nusselt number decreases with increasing frequency at Reynolds numbers of 100 and 200.

This is expected, since the response (maximum value of the Nusselt number) decreases with increasing frequency as stated

earlier. Heat transfer enhancement with pulsation is effective for the case when the obstacle located at b = 1.0H at Reynolds

number of 100. At Reynolds number of 200, two locations of the square obstacle b = 1.0H and b = 1.5H are effective for

heat transfer enhancement with pulsation compared to the case without obstacle.

A measure for the heat transfer enhancement of pulsating flow in comparison to the steady flow with obstacle located at

different positions and with no-obstacle case is HTEw,o =NumNus

/NuwmNuws

with subindex m,s,wm and ws denoting the spatial-time

averaged, steady state, spatial-time averaged without obstacle case and steady state without obstacle case, respectively. This

will indicate the combined effect of pulsating flow and installation of the obstacle on the average heat transfer enhancement

along the bottom wall downstream of the step. As stated earlier, in the steady case, adding a square obstacle at locations b =

1.0H and b = 1.5H will enhance the maximum value of the heat transfer with increasing Reynolds number, but enhancement

of the length averaged-Nusselt number is seen for the case with obstacle located at b = 1.5H. In the pulsating flow case,

HTEw,o value exceeds one for the for Strouhal number of 1 when the obstacle is located at position b = 1.0H and at Strouhal

number of 1 and 2 when the obstacle is located at position b = 1.5H for Reynolds number of 200 as can be seen in Fig. 16.

When the average heat transfer is of interest, installation of the square obstacle behind step is advantageous only for these

cases in the pulsating flow compared to no-obstacle case. The variation of the HTE with Strouhal number is closely related

to the changes in the flow topology during the pulsation cycle. At Re=200, for low Strouhal number, the time snapshots of

streamlines during the acceleration phase of pulsating cycle are depicted in Fig. 17 (without obstacle case and for the case

when square obstacle is located at b/H = 1.5). In this figure, it is seen that for both cases, at time instance t5 (it is the time

when the maximum value in the Nusselt number is achieved), a large recirculation region is seen in the upper wall. At this

time instance, deflection of the streamlines is larger when they impinge the bottom wall downstream of the step. In this case,

pulsation in the flow generates obstacle like structures that appear and disappear periodically which modify the streamlines

and heat transfer rate as it has also been shown in [15]. Adding a square obstacle can even complicate the process (formation

of the vortices behind the square obstacle and its interaction with the recirculation region in the upper wall, modification of

the streamline pattern behind the step) and could either contribute to (obstacle location at b/H = 1.0, Re=200, low Strouhal

number) or degrade (obstacle location at b/H = 1.0, Re=200, high Strouhal number) the heat transfer enhancement.

4 Conclusions

In this study, laminar pulsating flow over a backward facing step with a square obstacle installed is numerically studied.

The effects of Reynolds number, obstacle location and pulsating frequency on the fluid flow and heat transfer characteristics

are numerically investigated. In the steady case, properly locating the square obstacle redirects the flow movement. Length

and intensity of the recirculation zone behind the step are considerably affected with the installation of the obstacle. Adding

the square obstacle alters the isotherm plots. It is seen that there is significant change on the clustering of the isotherm

patterns and the location where this steep temperature gradient occurs in the steady flow case with the installation of the

Journal of Heat Transfer. Received October 08, 2012;Accepted manuscript posted April 04, 2014. doi:10.1115/1.4027344Copyright (c) 2014 by ASME

Downloaded From: http://heattransfer.asmedigitalcollection.asme.org/ on 04/13/2014 Terms of Use: http://asme.org/terms

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square object. Pulsation in the flow generates obstacle like structures that appear and disappear periodically which modify

the streamlines and heat transfer rate. Adding a square obstacle can even complicate the process and could either contribute

to or degrade the heat transfer enhancement. In the steady flow case, averaged heat transfer enhancements of 13% and 21%

are obtained for the case with obstacle at b = 1.5H compared to no-obstacle case for Reynolds number of 100 and 200,

respectively. Enhancements of 228% and 197% are attained in the maximum heat transfer in the steady flow case when

the obstacle is located at locations b = 1.5H and b = 1.0H, respectively. In the pulsating flow, as the pulsating frequency

increases, the number of periods to reach steady state generally increases for the same Reynolds number and the response

(Nusselt number) decreases. At Reynolds number of 100, heat transfer enhancement with pulsation is effective for the

case when the obstacle located at b = 1.0H. At Reynolds number of 200, two locations of the square obstacle b = 1.0H and

b= 1.5H are effective for heat transfer enhancement with pulsation compared to the case without obstacle. When the average

heat transfer is of interest, installing a square obstacle is advantageous only at Strouhal number of 1 when the obstacle is

located at position b = 1.0H and at Strouhal numbers of 1 and 2 when the obstacle is located at position b = 1.5H in the

pulsating flow compared to no-obstacle case for Reynolds number of 200.

References

[1] Saldana, J. G. B., and Anand, N. K., 2008. “Flow over a three-dimensional horizontal forward-facing step”. Numerical

Heat Transfer, Part A, 53, pp. 1–17.

[2] Iwai, H., Nakabe, K., and Suzuki, K., 2000. “Flow and heat transfer characteristics of backward-facing step laminar

flow in a rectangular duct”. International Journal of Heat and Mass Transfer, 43, pp. 457–471.

[3] Saldana, J. G. B., Anand, N. K., and Sarin, V., 2005. “Numerical simulation of mixed convective flow over a three-

dimensional horizontal backward facing step”. Journal of Heat Transfer, 127, pp. 1027–1036.

[4] Selimefendigil, F., and Oztop, H. F., 2013. “Numerical analysis of laminar pulsating flow at a backward facing step

with an upper wall mounted adiabatic thin fin”. Computers and Fluids, 88, pp. 93–107.

[5] Selimefendigil, F., and Oztop, H. F., 2014. “Effect of a rotating cylinder in forced convection of ferrofluid over a

backward facing step”. International Journal of Heat and Mass Transfer, 71, pp. 142–148.

[6] Abu-Mulaweh, H., 2003. “A review of research on laminar mixed convection flow over backward- and forward facing

steps”. International Journal of Thermal Sciences, 42, pp. 897–909.

[7] Nie, J., and Armaly, B., 2004. “Convection in laminar three-dimensional separated flow”. International Journal of

Heat and Mass Transfer, 47, pp. 5407–5416.

[8] Barkley, D., Gomes, M. G. M., and Henderson, R. D., 2002. “Three-dimensional instability in flow over a backward-

facing step”. J. Fluid Mech., 473, pp. 167–190.

[9] Stuer, H., Gyr, A., and Kinzelbach, W., 1999. “Laminar separation on a forward facing step”. Eur. J. Mech. B/Fluids,

18, pp. 675–692.

[10] Abu-Mulaweh, H., 2005. “Turbulent mixed convection flow over a forward-facing step - the effect of step heights”.

International Journal of Thermal Sciences, 44, pp. 155–162.

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[11] Armaly, B. F., Durst, F., Pereier, J. C. F., and Schonung, B., 1983. “Experimental and theoretical investigation of

backward-facing step flow”. J . Fluid Mech., 127, pp. 473–496.

[12] Terhaar, S., Velazquez, A., Arias, J., and Sanchez-Sanz, M., 2010. “Experimental study on the unsteady laminar

heat transfer downstream of a backwards facing step”. International Communications in Heat and Mass Transfer, 37,

pp. 457–462.

[13] Sherry, M., D.LoJacono, and J.Sheridan, 2010. “An experimental investigation of the recirculation zone formed down-

stream of a forward facing step”. J. Wind Eng.Ind.Aerodyn., 98, pp. 888–894.

[14] Habib, M., Said, S., Al-Farayedhi, A., Al-Dini, S., Asghar, A., and Gbadebo, S., 1999. “Heat transfer characteristics of

pulsated turbulent pipe flow”. Heat Mass Transfer, 34, pp. 413–421.

[15] Velazquez, A., Arias, J., and Mendez, B., 2008. “Laminar heat transfer enhancement downstream of a backward facing

step by using a pulsating flow”. International Journal of Heat and Mass Transfer, 51, pp. 2075–2089.

[16] Selimefendigil, F., and Oztop, H. F., 2014. “Numerical study and identification of cooling of heated blocks in pulsating

channel flow with a rotating cylinder”. International Journal of Thermal Sciences, 79, pp. 132–145.

[17] Khanafer, K., Al-Azmi, B., Al-Shammari, A., and Pop, I., 2008. “Mixed convection analysis of laminar pulsating flow

and heat transfer over a backward-facing step”. International Journal of Heat and Mass Transfer, 51, pp. 5785–5793.

[18] Mackley, M., and Stonestreet, P., 1995. “Heat transfer and associated energy dissipation for oscillatory flow in baffled

tubes”. Chem. Sci. Eng., 50, pp. 2211–2224.

[19] Hemida, H., Sabry, M., Abdel-Rahim, A., and Mansour, H., 2002. “Theoretical analysis of heat transfer in laminar

pulsating flow”. Int. J. Heat Mass Transfer, 45, pp. 1767–1780.

[20] Oztop, H. F., Mushatet, K. S., and ilker Yilmaz, 2012. “Analysis of turbulent flow and heat transfer over a double

forward facing step with obstacles”. International Communications in Heat and Mass Transfer, In Press.

[21] Yilmaz, I., and Oztop, H. F., 2006. “Turbulence forced convection heat transfer over double forward facing step flow”.

International Communications in Heat and Mass Transfer, 33, pp. 508–517.

[22] Selimefendigil, F., and Oztop, H. F., 2013. “Identification of forced convection in pulsating flow at a backward facing

step with a stationary cylinder subjected to nanofluid”. International Communications in Heat and Mass Transfer,, in

press.

[23] Kumar, A., and Dhiman, A. K., 2012. “Effect of a circular cylinder on separated forced convection at a backward-facing

step”. International Journal of Thermal Sciences, 52, pp. 176–185.

[24] FLUENT INC., 2005. FLUENT User’s Guide. Lebanon, NH.

[25] Acharya, S., Dixit, G., and Hou, Q., 1993. “Laminar mixed convection in a vertical channel with a backstep: a

benchmark study”. ASME HTD, 258, pp. 11–20.

[26] Lin, J., Armaly, B., and Chen, T., 1990. “Mixed convection in buoyancy-assisted vertical backward-facing step flows”.

Int. J. Heat Mass Transfer, 33, pp. 2121–2132.

[27] Dyne, B., Pepper, D., and Brueckner, F., 1993. “Mixed convection in a vertical channel with a backward-facing step”.

ASME HTD, 258.

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[28] El-Refaee, M., El-Sayed, M., Al-Najem, N., and Megahid, I., 1996. “Steady-state solutions of buoyancy-assisted

internal flows using a fast false implicit transient scheme (fits)”. Int. J. Numer. Meth. Heat Fluid Flow, 6, pp. 3–23.

[29] Cochran, R., Horstman, R., Sun, Y., and Emery, A., 1993. “Benchmark solution for a vertical buoyancy-assisted

laminar backward-facing step flow using finite element, finite volume and finite difference methods”. ASME HTD, 258,

pp. 37–47.

[30] Kaiktsis, L., Karniadakis, G. E., and Orszag, S. A., 1991. “Onset of three-dimensionality, equilibria, and early transition

in flow over a backward-facing step”. Journal of Fluid Mechanics, 231, pp. 501–528.

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Table 1 : Reported values for the reattachment lengths XR at Reynolds number 100 (Expansion ratio of 2)

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Fig. 1 : Geometry (not drawn to scale) with boundary conditions

Fig. 2 : Distribution of the mesh in the vicinity of the step and square obstacle

Fig. 3 : Variation of the local Nusselt number distribution along the bottom wall for different grid sizes

Fig. 4 : Space averaged Nusselt number along the bottom wall at Re=200, St=1, b/H = 1.0 for different time step sizes

Fig. 5 : Streamlines for no-obstacle case and for square obstacle at various locations for Reynolds number of 10 (first

box), 100 (second box) and 200 (third box)

Fig. 6 : Isotherms for no-obstacle case and for square obstacle at various locations for Reynolds number of 10 (first

box), 100 (second box) and 200 (third box)

Fig. 7 : Effect of the obstacle and its position on the distribution of the local Nusselt number for Reynolds number of

10, 100 and 200

Fig. 8 : Variation of the maximum location of the Nusselt number (bottom) and maximum value of the Nusselt number

(top) versus Reynolds number for different obstacle positions and for the case without obstacle

Fig. 9 : Variation of the length-averaged Nusselt number along the bottom wall downstream of the step versus Reynolds

number for different obstacle positions and for the case without obstacle

Fig. 10 : Effect of the position of the obstacle on temporal variations of spatial-averaged Nusselt number when the

steady state periodic oscillations are reached at Reynolds number of 10, 100 and 200 for Strouhal number of 0.1.

Fig. 11 : Effect of the position of the obstacle on temporal variations of spatial-averaged Nusselt number when the

steady state periodic oscillations are reached at Reynolds number of 10, 100 and 200 for Strouhal number of 2.

Fig. 12 : Time evolution of the streamlines for no-obstacle case (first box) and for square obstacle at various locations

denoted by P1, P2 and P3 at Reynolds number of 100 for three different time instances corresponding to minimum (-a),

middle (-b-) and maximum(-c-) part of the cycle during an acceleration phase of a half period.

Fig. 13 : Time evolution of the isotherms for no-obstacle case (first box) and for square obstacle at various locations

denoted by P1, P2 and P3 at Reynolds number of 100 for three different time instances corresponding to minimum (-a),

middle (-b-) and maximum(-c-) part of the cycle during an acceleration phase of a half period .

Fig. 14 : Temporal evolution of the local Nusselt number for no-obstacle case and for square obstacle at various locations

- position 1 (b/H = 0.5), position 2 (b/H = 1.0) and position 3 (b/H = 1.5) at Reynolds number of 100, Strouhal number

of 0.5 for three different time instances corresponding to minimum (-a), middle (-b-) and maximum(-c-) part of the cycle

during an acceleration phase of a half period.

Fig. 15 : Variation of normalized Nusselt number (time and spatial-average Nusselt number in pulsating flow divided

by the average in steady flow) for no-obstacle case and for obstacle at different locations with Strouhal number at Reynolds

number of 10, 100 and 200.

Fig. 16 : Variation of heat transfer enhancement for pulsating flow in comparison to the steady state flow with obstacle

at different locations and with no-obstacle case (HTEw,o), with Strouhal number at Reynolds number of 10, 100 and 200

Fig. 17 : Snapshots of the streamlines for the case without obstacle and for the case at obstacle position P3: b/H = 1.5

during the acceleration phase of a half of the pulsating cycle. Time reads from t1 (min) to t5 (max).

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Table 1

XR/H Difference (%)

Acharya et al. [25] 4.97 -2.93

Lin et al. [26] 4.91 -4.1

Dyne et al. [27] 4.89 -4.49

El-Refaee et al. [28] 4.77 -6.83

Cochran et al. [29] 5.32 3.9

Present 5.12 0

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Fig. 1

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Fig. 2

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Fig. 3

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Fig. 4

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(a) no-obstacle (b) b/ H = 0.5

(c) b/ H = 1.0 (d) b/ H = 1.5

Re=10

(e) no-obstacle (f) b/ H = 0.5

(g) b/ H = 1.0 (h) b/ H = 1.5

Re=100

(i) no-obstacle (j) b/ H = 0.5

(k) b/ H = 1.0 (l) b/ H = 1.5

Re=200

Fig. 5

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(a) no-obstacle (b) b/ H = 0.5

(c) b/ H = 1.0 (d) b/ H = 1.5

Re=10

(e) no-obstacle (f) b/ H = 0.5

(g) b/ H = 1.0 (h) b/ H = 1.5

Re=100

(i) no-obstacle (j) b/ H = 0.5

(k) b/ H = 1.0 (l) b/ H = 1.5

Re=200

Fig. 6

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Fig. 7

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Fig. 8

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Fig. 9

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Fig. 10

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Fig. 11

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(a) -a- (b) -b-

(c) -c-

w

(d) -a- (e) -b-

(f) -c-

P1

(g) -a- (h) -b-

(i) -c-

P2

(j) -a- (k) -b-

(l) -c-

P3

Fig. 12

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(a) -a- (b) -b-

(c) -c-

w

(d) -a- (e) -b-

(f) -c-

P1

(g) -a- (h) -b-

(i) -c-

(j) -a- (k) -b-

(l) -c-

P3

Fig. 13

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Fig. 14

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Fig. 15

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Fig. 16

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(a) t1 (min) (b) t2

(c) t3 (d) t4

(e) t5 (max)

w

(f) t1 (min) (g) t2

(h) t3 (i) t4

(j) t5 (max)

P3

Fig. 17

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