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Numerical analysis of laminar pulsating flow at a backward facing step with an upper wall mounted adiabatic thin fin
Computers & Fluids 88 (2013) 93–107
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Computers & Fluids j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p fl u i d
Numerical analysis of laminar pulsating flow at a backward facing step with an upper wall mounted adiabatic thin fin Fatih Selimefendigil a,⇑, Hakan F. Öztop b a b
Department of Mechanical Engineering, Celal Bayar University, Manisa, Turkey Department of Mechanical Engineering, Technology Faculty, Fırat University, Elazıg˘, Turkey
a r t i c l e
i n f o
Article history: Received 4 October 2012 Received in revised form 15 July 2013 Accepted 29 August 2013 Available online 10 September 2013 Keywords: Pulsating flow Adiabatic fin Backward facing step
a b s t r a c t The effect of an upper wall mounted adiabatic thin fin on laminar pulsating flow in a backward facing step has been investigated numerically. Study is performed for different Reynolds numbers (based on the step height) in the range of 10 and 200 and for the expansion ratio of 2. The working fluid is air with the Prandtl number of 0.71. The governing equations are solved with a general purpose finite volume based solver, FLUENT. The effects of various pertinent parameters, Reynolds number, fin length and pulsating frequency on the fluid flow and heat transfer characteristics are numerically studied. It is observed that fin alters the flow field and thermal characteristics. In the steady flow case, heat transfer enhancement is obtained with the installation of the fin on the upper wall and increases with increasing fin length and increasing Reynolds number. Heat transfer enhancement of 188% is obtained for fin length of Lf = 1.5H at Reynolds number of 200. In the pulsating flow case, time-spatial averaged Nusselt number along the bottom wall downstream of the step normalized with spatial averaged Nusselt number in the steady flow case versus excitation Strouhal number shows a resonant type behavior; first an increase in the value is seen up to St = 0.05, then a decrease is seen with the increasing values of the frequency of the pulsation for the case without fin. Adding a fin shifts the maximum value of the normalized Nusselt number from St = 0.05 to St = 0.1. Compared to steady flow with no-fin case, adding a fin is not advantageous for heat transfer enhancement in pulsating flow. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Flow separation and its subsequent reattachment is important in many engineering applications such as flow around airfoils, buildings, combustors and collectors of power systems and therefore a vast amount of literature is dedicated to that subject. 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 Abu-Mulaweh [1] for laminar mixed convection over vertical, horizontal and inclined backwardand forward-facing steps studied in the open literature. In this review, the effects of pertinent parameters such as Reynolds number, Prandtl number and expansion ratio on the fluid flow and thermal characteristics is also presented. Barkley et al. [2] have numerically studied the 3D linear stability analysis of flow over a backward-facing step for Reynolds numbers between 450 and 1050. Erturk [3] has numerically studied the 2D flow over a backward-facing step for Reynolds number between 100 and 3000 based on the channel height. He reported that the outflow ⇑ Corresponding author. E-mail addresses: [email protected] (F. Selimefendigil), hfoztop1@ gmail.com (H.F. Öztop). 0045-7930/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compfluid.2013.08.013
boundary condition and the location of the exit have effect on the accuracy of the numerical solution. Nie and Armaly [4] have reported the wall temperature distributions, Nusselt number and the friction coefficient for the laminar 3D flow adjacent to backward-facing step in a rectangular duct. Iwai et al. [5] have numerically investigated the 3D flow over a backward facing step at low Reynolds number for various duct aspect ratios. They reported that the maximum Nusselt number did not appear on the centerline, but near the two side walls in every case. Barbosa-Saldana and Anand [6] have numerically studied the 3D laminar flow over a horizontal forward-facing step. The expansion ratio and aspect ratio of two and four is considered. Effects of the Reynolds number on the locations of the reattachment line, velocity distribution and averaged Nusselt number are discussed. Experimental studies have also been conducted for the flow over a backward facing or forward facing step. Armaly et al. [7] have reported the velocity distribution and reattachment length using Laser-Doppler measurement for flow downstream of a backward facing step in a two-dimensional channel. Their results showed separation length varies with Reynolds number and various flow regimes are characterized by variations of the separation length. Abu-Mulaweh [8] has reported the measurements of heat transfer and flow over an vertical forward-facing step for turbulent
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Nomenclature H h k L n Nu p Pr Re T u, v x, y
step size (m) local heat transfer coefficient (W/m2 K) thermal conductivity (W/m K) fin length (m) unit normal vector local Nusselt number, hH/k pressure (Pa) Prandtl number, am Reynolds number, u0mH temperature (K) x–y velocity components (m/s) Cartesian coordinates (m)
mixed convection using two-component LDV and a cold wire anemometer. He reported that when the step height increases the turbulence intensity of the streamwise and transverse velocity fluctuations and the intensity of temperature fluctuations downstream of the forward-facing step increase. Terhaar et al. [9] have experimentally studied the unsteady heat transfer downstream over a backward-facing step in the laminar flow regime for pulsating flow conditions at Reynolds number of 300 (based on the hydraulic diameter of the channel). They showed that the Nusselt number increases up to a certain excitation Strouhal number and then degrades as the pulsation frequency increases. Sherry et al. [10] have made an experimental investigation for the recirculation zone formed downstream of a forward facing step immersed in a turbulent boundary layer. In their study, the mechanisms effecting the reattachment distance is discussed. Stuer et al. [11] have experimentally studied the separation ahead of a forward facing step under laminar flow conditions using the hydrogen bubble technique. 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 [12]. But, in the literature there exist also cases where pulsating flow does not effect [13] or even deteriorate heat transfer enhancement [14]. That means 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 has received less attention in the literature. Yilmaz and Oztop [15] have numerically investigated the turbulent forced convection over double forward facing step by using two adiabatic steps with different lengths and heights. They studied the effects of step heights, step lengths and Reynolds numbers on heat transfer and fluid flow. They reported that the second step can be used to control the fluid flow and heat transfer characteristics. Oztop et al. [16] have numerically performed turbulent forced convection over double forward facing step with obstacles of rectangular cross-sectional area. They studied the effect of step height, Reynolds number and obstacle aspect ratio on fluid low and heat transfer. They reported that heat transfer rate is enhanced with increasing aspect ratio of the obstacle. Kumar and Dhiman [17] have numerically studied the heat transfer enhancement in laminar forced convection flow over a backward facing step with the insertion of an adiabatic circular cylinder. They considered different cross-stream positions of the circular cylinder for the Reynolds number between 1 and 200. They obtained heat transfer enhancement up to 155 percent compared to no-cylinder case. Khanafer et al. [18] have studied the mixed convection in laminar pulsating flow over a backward-facing step. They considered the Reynolds
Greek characters a thermal diffusivity (m2/s) h non-dimensional temperature, m kinematic viscosity (m2/s) q density of the fluid (kg/m3)
TT c T h T c
Subscripts c cold f fin h hot wall
number between 100 and 1000, non-dimensional pulsating frequency between 0.1 and 5 and Richardson number between 0.000178 and 10. Their simulation results showed that average Nusselt number increases with an increase in both Reynolds number and Grashof number and decrease with an increase in the pulsating frequency. Recently, Selimefendigil and Oztop [19] have numerically studied the forced convection in pulsating flow at a backward facing step with a stationary circular cylinder subjected to nanofluid. They studied the effects of pulsating frequency, nanoparticle volume fraction and Reynolds number on the heat transfer characteristic of the backward facing step flow. Based on the literature survey and to the best of authors’ knowledge a study of laminar forced convection over a backward facing step in a channel with the upper wall mounted adiabatic fin in pulsating flow has never been studied in the literature. The main objective of this study is to numerically study the pulsating flow over a backward facing step in the presence of an upper wall mounted adiabatic thin fin. The effects of various pertinent parameters such as fin length, Reynolds number, oscillating frequency on the fluid flow and heat transfer characteristics over a backwardfacing step will be investigated in detail and optimal combinations of parameters for the maximum heat transfer augmentation will be determined using numerical simulations. 2. Numerical simulation 2.1. Problem description A schematic description of the physical problem considered in this study is shown in Fig. 1. A channel with a backward facing step is considered. The step size of backward facing step is H and channel height is 2H. At the inlet of the channel, a uniform velocity with a sinusoidal time dependent part (U = 1 + A sin(2pft)) and a uniform temperature (h = 0) is imposed. Generally parabolic
Fig. 1. Geometry with boundary conditions.
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velocity profiles at the inlet are used in numerical simulation studies of backward facing step flow. But with the given inflow length of 10 channel heights and the low Reynolds number of the current study, the simulation results are similar for uniform and parabolic velocity profiles for the same flow rate which have been confirmed from the numerical simulation results at the highest Reynolds number of 200. 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 h = 1, while the other walls of the channel are assumed to be adiabatic. An adiabatic thin fin with length Lf is mounted on the top wall of the channel. The streamwise downstream location of the fin where it is mounted on the top wall is H starting from the edge of the step. In the present study, it is expected that different fin lengths change the heat transfer and fluid flow characteristic of the backward facing step. Working fluid is air with a Prandtl number of Pr = 0.71. It is assumed that thermophysical properties of the fluid is temperature independent. The flow is assumed to be two dimensional, Newtonian, incompressible and in the laminar flow regime. The three dimensional effects of the flow over a backward facing step have been studied by [20,21]. Williams and Baker [20] have noted the side walls effect on the development of the three dimensional flow for Reynolds number grater than 400. They have numerically investigated the complex interaction of a wall jet with the main stream for the mechanism of the three dimensionality. Chiang and Sheu [21] have also numerically showed the wall induced three dimensional vortical flow with increasing Reynolds number for a laminar backward facing step flow. The three dimensional effects and the possible effect of the presence of side walls on flow field and heat transfer are out of the scope of the present paper. 2.2. Governing equations and solution method By using the dimensionless parameters,
ðU; V Þ ¼
ðu; v Þ ; u0
ðX; Y Þ ¼
ðx; yÞ p T Tc tH ; P ¼ 2; h¼ ; k¼ ; H u0 Th Tc qu0 ð1Þ
for a two dimensional, incompressible, laminar and steady case, the continuity, momentum and energy equations can be expressed in the non-dimensional form as in the following:
@U @V þ ¼ 0; @X @Y
ð2Þ
! @U @U @U @P 1 @2U @2U þU þV ¼ þ þ ; @k @X @Y @X Re @X 2 @Y 2
ð3Þ
! @V @V @V @P 1 @2V @2V ; þU þV ¼ þ þ @k @X @Y @Y Re @X 2 @Y 2
ð4Þ
! @h @h @h 1 @2h @2h ; þU þV ¼ þ @k @X @Y PrRe @X 2 @Y 2
ð5Þ
Reynolds number (Re) based on the step height and non-dimensional frequency, Strouhal number (St) can be written as
Re ¼
u0 H
m
;
St ¼
fH : u0
ð6Þ
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 + A sin(2pStk), V = 0, h = 0. At the bottom wall, downstream of the step, temperature is constant h = 1. At the channel exit, gradients of all variables in the x-direc@T tion are set to zero, @U ¼ 0; @V ¼ 0; @X ¼ 0. @X @X On the channel walls (except the downstream of the step) and on the adiabatic thin fin, adiabatic wall with no-slip @h boundary conditions are assumed, U ¼ 0; V ¼ 0; @n ¼ 0, where n denotes the surface normal direction. Local Nusselt number is defined as
Nux;t ¼
hx;t L @h ¼ ; k @n S
ð7Þ
where hx,t represent the local heat transfer coefficient and k denote the thermal conductivity of air. Spatial averaged Nusselt number is obtained after integrating the local Nusselt number along the bottom wall downstream of the step as
Nut ¼ 1L
RL 0
ð8Þ
Nux;t dx:
Time and spatial averaged Nusselt number is obtained after integrating spatial averaged Nusselt number along the bottom wall downstream of the step for one period of the oscillation s as
Num ¼ s1
Rs 0
Nut dt:
ð9Þ
Eqs. (2)–(5) along with the boundary and initial conditions are solved with Fluent (a general purpose finite volume solver [22]). The convective terms in the momentum and energy equations are solved using QUICK scheme and SIMPLE algorithm is used for velocity– pressure coupling. The system of algebraic equations are solved with Gauss-Siedel point by point iterative method and algebraic multigrid method. The convergence criteria for continuity, momentum and energy equations are set to 103, 105 and 106, respectively. The steady solutions are used as the initial conditions for the unsteady computations. The time step size is chosen as 1/50th of the period of the pulsating flow. The unstructured body-adapted mesh of appropriate size consists of only triangular elements. The computational domain is divided into 15872 triangular elements for no fin case at Reynolds number of 100. The mesh is finer near the walls 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 also carried out to obtain an optimal grid distribution with accurate results and minimal computational time at Re = 100 for no-fin case. Four different grid densities are tested and the convergence in the length-averaged Nusselt number (along the bottom wall downstream of the step) is checked. The results at Reynolds number of 100 for the case without fin is tabulated in Table 1. From this table, grid size of 15872 is decided to be fine enough to resolve the flow and thermal field for the case without fin at Re = 100. Fig. 2 shows the local Nusselt number distribution along the bottom wall for
Table 1 Grid sensitivity study for the case without fin at Reynold number of 100. Nusselt numbers are the length averaged values along the bottom wall downstream of the step. Case
Number of cells
Nusselt number
G1 G2 G3 G4
3968 8928 15,872 24,800
1.0369 1.0353 1.0347 1.0345
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Table 2 Reported values for the reattachment lengths XR at Reynolds number 100 (Expansion ratio of 2).
G1 G2 G3 G4
Nu
10
5
0
0
5
10
15
20
25
30
35
x/H Fig. 2. Local Nusselt number distribution along the bottom wall downstream of the step with fin of length Lf = 1.0H and Re = 200 for different grid densities.
different grid sizes (G1 = 6428, G2 = 10694, G3 = 32130, G4 = 49766) at the highest Reynolds number, Re = 200 in the presence of the fin Lf = 1.0H. From this plot, the result of grid size G3 is close to the result of finest grid. Grid independence studies are also conducted for the subsequent computations for different Reynolds numbers and fin lengths parameter combinations. Fig. 3 depicts the time step size independence study result for the case with fin length Lf = 1.0H and Re = 200 at the excitation Strouhal number of 0.01. Four different time step sizes (dt = s/25, dt = s/50, dt = s/100, dt = s/150) are considered for the period of the oscillation s.Considering both accuracy and computational time of the simulation a time step size of dt = s/50 is chosen in this study. The numerical code is first checked against the benchmarked results of backward facing step reported in the literature [23–27]. Table 2. shows the results of the reattachment length divided by step height at Reynolds number of 100 for expansion ratio of 2. Minimum deviation for the percentage in the error is obtained for the results of Acharya et al. [23] which is 2.93 percent. The agreement between the other sources is less than 5 percent, only 6.83 percent error is obtained for the results of El-Refaee et al. [26]. 3. Results and discussion As stated earlier, the overall purpose of this study is to investigate the effects of an upper wall mounted inclined fin on the heat transfer and fluid flow characteristics for different fin lengths and pulsating frequencies in the laminar range of Reynolds number between 10 and 200. The main parameters that effect the fluid flow and thermal characteristics are Reynolds number, step height,
dt= τ/25 dt= τ/50 dt= τ/100 dt= τ/150
3.2 3 2.8 2.6
Nu
2.4 2.2 2 1.8 1.6 1.4 1.2 900
1000
1100
1200
1300
1400
1500
time Fig. 3. Time histories of space-averaged Nusselt number along the bottom wall downstream of the step with fin of length Lf = 1.0H and Re = 200 for different time step sizes at the excitation Strouhal number of 0.01.
XR/S
Error (%)
Acharya et al. [15] Lin et al. [16] Dyne et al. [17] El-Refeee et al. [18] Cochran et al. [19]
4.97 4.91 4.89 4.77 5.32
2.93 4.1 4.49 6.83 3.9
Present
5.12
0
distance between the step edge to the channel exit downstream of the step, distance between the inlet to the step edge, fin height, fin inclination angle, streamwise distance of the fin to the step edge, expansion ratio, oscillating frequency, amplitude of the forcing at the inlet and Prandtl number. In the present study, expansion ratio is 2, streamwise distance of the fin to the step edge is H (step height) and fin inclination angle is 90 degrees. The distance between the inlet to the step edge is 10H (for Re P 100, the value can be set to any value with parabolic velocity imposed at the inlet [28]) and the distance between the step edge to the channel exit downstream of the step is 35H. The effect of varying forcing amplitude is investigated for the case corresponding to excitation Strouhal number of 0.1. The simulations are performed for Reynolds number between 10 and 200, fin height between 0 and 1.5H and excitation Strouhal number between 0.01 and 2. 3.1. Steady inflow 3.1.1. Streamlines and isotherms Streamline plots showing the flow patterns at Reynolds number of 10, 100 and 200 for no-fin case and for various fin lengths Lf = 0.5H, Lf = H and Lf = 1.5H are shown in Fig. 4. For the case without fin as the Reynolds number is increased, the flow at the edge of the step separates and a recirculation zone is observed behind the step. The size of this zone increases with an increase in the Reynolds number. When a fin is attached at the upper wall of the channel at Reynolds number of 10, the flow field in the vicinity of the fin disturbed and at fin length of Lf = 1.5H, a vortex is formed behind the fin. At Reynolds number of 100, this vortex behind the cell is already formed at fin length of Lf = 0.5H and gets larger in size and strength with increasing fin lengths. At this Reynolds number, for fin length of Lf = 1.5H, a small cell behind the step and a larger cell whose core is located downstream of the core of the cell formed behind the fin appear. At Reynolds number of 200, for the same fin lengths, the core of the cells formed behind the step moves downstream and the cells gets larger in strength and more distorted. From the streamline plots, it is seen that fin redirects the flow movement and the flow behavior behind the step (length and intensity of the recirculation zone behind the step) is considerably effected with the installation of an adiabatic thin fin on the upper wall of the channel. Isotherm plots for different fin lengths at Reynolds number of 10, 100 and 200 are shown in Fig. 5. The thermal boundary layer along the bottom wall downstream of the step has a steep temperature gradient around the flow reattachment point. Adding an adiabatic fin alters the temperature contours as can be seen from the plots. Comparing the isotherms for the case with the fin with those for the case without fin, it is seen that there is significant change on the clustering of the isotherm patterns and the location where this steep temperature gradient occurs especially at Reynolds number of 100 and 200. At Reynolds number of 100, for fin lengths of Lf = H and Lf = 1.5H, temperature contours from the tip of the fin extending downstream of the step are formed. The size and the strength of these contours increase when the Reynolds number
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97
Fig. 4. Streamlines for no-fin case and for various fin lengths at Reynolds number of 10 (first box), 100 (second box) and 200 (third box).
Fig. 5. Isotherms for no-fin case and for various fin lengths at Reynolds number of 10 (first box), 100 (second box) and 200 (third box).
increases. Variation of the normalized reattachment lengths (normalized with reattachment length for no-fin case) versus Reynolds number are depicted in Fig. 6. It is seen that increasing the fin length decreases the reattachment length compared to no-fin case and this effect is more pronounced at the higher Reynolds numbers.
3.1.2. Local and average Nusselt numbers Local Nusselt number distribution along the bottom wall downstream of the step for various fin heights at Reynolds number of 10, 100 and 200 are shown in Fig. 7. The horizontal axis denotes the distance from the edge of the step to the exit normalized by the step height. In these plots, a peak in the Nusselt number is seen
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3.2. Pulsating inflow
0.8 Lf=0.5H
0.7
L =1.0H f
The aim of the present section is to numerically investigate the effects of various pertinent parameters (pulsating amplitude, pulsating frequency, Reynolds number, fin length) on the transient evolution of the fluid flow and heat transfer characteristics. First, the effects of amplitude are investigated for amplitudes between 0.25 and 1 while keeping the excitation Strouhal number at fixed value of 0.1. Then, effect of frequency is investigated for excitation Strouhal numbers between 0.01 and 2 while keeping the amplitude of the forcing at A = 1. In these numerical studies, Reynolds number and fin length vary between 10–100 and 0.0–1.5H, respectively.
Lf=1.5H
/ Xw, R
0.4
X
0.5
f, R
0.6
0.3 0.2 0.1
0
20
40
60
80
100
120
140
160
180
200
Re Fig. 6. Normalized reattachment length (with value corresponding to no fin) versus Reynolds number for different fin lengths.
which corresponds to a location very close to the reattachment point where steep temperature gradient occurs. Adding a fin at Reynolds number of 10, does not change the location of the maximum Nusselt number, but peak value of Nusselt number increases with increasing fin length. At Reynolds number of 100 and 200, the location of the maximum Nusselt number moves upstream close to the step with the installation of the fin compared to the case without fin. At Reynolds number of 200, for fin lengths of Lf = H and Lf = 1.5H, secondary peaks are seen in the local Nusselt number distribution. The secondary peak corresponds to the separation area of the secondary vortex present for fin at lengths of Lf = H and Lf = 1.5H. Length averaged Nusselt number along the bottom wall downstream of the step and normalized averaged Nusselt number (normalized by the averaged Nusselt number for the no-fin case) are shown in Fig. 8 on top and bottom, respectively. From these plots, it seen that Nusselt number increases with increasing Reynolds number and with increasing fin length. The plot on the bottom also shows the heat transfer enhancement (HTE) performance of the system with the installation of the fin at various fin lengths. The HTE values are 1.51 and 1.88 at Reynolds number of 200 for the fin lengths of Lf = H and Lf = 1.5H, respectively. With the installation of an adiabatic thin fin on the upper wall at length Lf = 1.5H at Reynolds number of 200, heat transfer enhancement is 188 percent for the bottom hot wall as compared to the nofin case.
3.2.1. Effects of pulsation amplitude Space-averaged Nusselt number along the bottom wall for different velocity amplitudes and various fin lengths at St = 0.1 are depicted in Fig. 9 for Re = 100 and in Fig. 10 for Re = 200. In all cases, the peak values of the Nusselt number increase with increasing values of velocity amplitudes. There is a shift in the mean with decreasing values of the fin lengths. The same trend is seen at Re = 200 and peak values of the Nusselt numbers increase compared to the case at Re = 100. At Re = 200, more distortions from a pure sinusoid is seen especially for the case at velocity amplitude of 1. Time-spatial averaged Nusselt number normalized with spatial averaged Nusselt number in the steady flow case along the bottom wall versus velocity amplitude plots are shown in Fig. 11 for various fin lengths at Reynolds number of 10, 100 and 200. Normalized averaged Nusselt number increases almost linearly with increasing values of pulsating velocity amplitudes at Re = 100 and Re = 200. Only at Re = 200, for fin length of Lf = 1.5, the tendency is different which may be due to the higher harmonic contributions from the flow reversal. At Re = 10, there is negligible change in HTE (heat transfer enhancement with respect to steady flow value) with the change in velocity amplitude. Dimensionless pressure drop coefficient is defined as the normalized difference between the averaged-pressure at the inlet and outlet sections [29],
dP ¼
pin pout : 0:5qu20
ð10Þ
Time evolution of the pressure drop coefficient for different velocity amplitudes and various fin lengths at St = 0.1 are depicted in Fig. 12 for Re = 200. The peak value in the pressure loss coefficient
(a) Re=10
(b) Re=100 15
2
Nu
Nu
3
5
1 0 0
10
5
10
x/H
0 0
15
5
x/H
10
15
(c) Re=200 25
Lf =0.0H
Nu
20
L =0.5H f
15
L =1.0H
10
L =1.5H
f f
5
0 0
5
x/H
10
15
Fig. 7. Effect of the fin length on local Nusselt number distribution for Reynolds number of 10, 100 and 200.
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L =0.0H f
L =1.0H
L =0.5H f
L =1.5H
f
f
3
Nus
2 1 0
0
20
40
60
80
100
120
140
160
180
200
120
140
160
180
200
Re
1.5
s
Nu / Nu
ws
2
1 0.5
0
20
40
60
80
100
Re Fig. 8. Top – Spatial-averaged Nusselt number versus Reynolds number and Bottom – Normalized spatial-averaged Nusselt number (with value corresponding to no fin) versus Reynolds number for various fin lengths.
(a) L f=0.0H
(b) L f=0.5H
1.5
1.2
Nu
Nu
1.4 1.1
1 70
1.3
80
90
100
110
120
70
80
Nondimensional time
90
100
110
120
Nondimensional time
(c) L f=1.0H
(d) L f=1.5H
A=0.25 A=0.5
2
1.7
A=1
Nu
Nu
1.6 1.5 1.4
1.5
1.3 70
80
90
100
110
120
1 70
80
Nondimensional time
90
100
110
120
Nondimensional time
Fig. 9. Effect of velocity amplitude on temporal variations of spatial-averaged Nusselt number when the steady periodic oscillations are reached at various fin lengths for Reynolds number of 100 and excitation Strouhal number of 0.1.
(a) L =0.0H
(b) L f=0.5H
f
2.2 2.1
1.6
Nu
Nu
1.7
1.5 1.4 70
2 1.9
80
90
100
110
120
1.8 70
(c) L =1.0H f
2.8
3.5
Nu
Nu
2.6 2.4
90
100
110
120
(d) L f=1.5H
A=0.25 A=0.5 A=1
3 2.5
2.2 2 70
80
Nondimensional time
Nondimensional time
80
90
100
110
Nondimensional time
120
2 70
80
90
100
Nondimensional time
Fig. 10. Effect of velocity amplitude on temporal variations of spatial-averaged Nusselt number when the steady periodic oscillations are reached at various fin lengths for Reynolds number of 200 and excitation Strouhal number of 0.1.
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(a) Re=10
(b) Re=100 1.15
Num / Nu
s
Num / Nus
1.05
1
0.95 0.2
0.4
0.6
0.8
1.1 1.05 1 0.2
1
0.4
0.6
A
0.8
1
A (c) Re=200
Nu / Nu m s
1.25
Lf=0.0H
1.2
Lf=0.5H L =1.0H
1.15
f
Lf=1.5H
1.1 1.05 0.2
0.4
0.6
0.8
1
A Fig. 11. Variation of normalized spatial-averaged Nusselt number versus amplitude of the forcing for various fin lengths at Reynolds number of 200 and excitation Strouhal number of 0.1.
increases with increasing values of velocity amplitudes. The averaged values of the pressure loss coefficient versus velocity amplitudes are shown in Fig. 13 for different fin lengths and Reynolds numbers. The averaged pressure drop coefficient is more sensitive with respect to a change in the velocity amplitude for the case at fin length Lf = 1.5H. The averaged pressure drop coefficient increases with the increasing values of fin length.
sinusoid) with increasing Reynolds number and increasing fin length. The difference between the peak value of the Nusselt numbers increases for various fin lengths with increasing frequency. The time when the maximum values are reached also change with increasing frequency indicating that there is a phase shift between the forcing velocity at the inlet and response (Nusselt number along the bottom wall downstream of the step). The phase shift for an excitation amplitude and Strouhal number can defined as
3.2.2. Effects of pulsation Strouhal number Length averaged Nusselt number along the bottom wall downstream of the step plots are shown in Fig. 14 for various fin lengths at Reynolds number of 10, 100 and 200 and at excitation Strouhal number of 0.025. The plots are shown after removing the initial transients. With increasing the Reynolds number, there is an increase in the mean Nusselt number with respect to the non-fin case, which is higher for high fin lengths. Figs. 15 and 16 show the Nusselt number plots at excitation Strouhal number of 0.1 and 0.5, respectively. With increasing the frequency, the number of periods to reach steady periodic flow conditions generally increases for the same Reynolds number. Another observation is the increasing level of nonlinearity (distortion from a pure
U ¼ arg
Nu : u ðA;StÞ
Table 3 shows the phase shift in degrees for various combinations of Reynolds number, excitation Strouhal number and fin lengths. At the lower pulsating frequencies, increasing values of the fin length decrease the phase shift and phase shift values increase with increasing values of Reynolds numbers. 3.2.3. Streamlines and isotherms Streamline plots for the three time instances within half a period when the system reaches the periodic steady flow according to
(b) L f=0.5H 40
20
20
dP
dP
(a) L f=0.0H 40
0
0
−20
−20
−40
−40 80
90
100
110
80
120
100
110
Nondimensional time
(c) L f=1.0H
(d) L f=1.5H
120
A=0.25 A=0.5
60
A=1
40
20
dP
dP
90
Nondimensional time
40
0
20 0
−20 −40 80
ð11Þ
−20 90
100
110
Nondimensional time
120
−40 80
90
100
110
120
Nondimensional time
Fig. 12. Effect of velocity amplitude on temporal variations of averaged-pressure drop when the steady periodic oscillations are reached at various fin lengths for Reynolds number of 200 and excitation Strouhal number of 0.1.
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(b) Re=100
(a) Re=10 30
60
20
dPm
dPm
70
50 40 0.2
10
0.4
0.6
0.8
0 0.2
1
0.4
0.6
0.8
1
A
A (c) Re=200 30
Lf=0.0H Lf=0.5H
dPm
20
L =1.0H f
10
Lf=1.5H 0 0.2
0.4
0.6
0.8
1
A Fig. 13. Variation of averaged-pressure drop versus amplitude of the forcing for various fin lengths at Reynolds number of 10, 100, 200 and excitation Strouhal number of 0.1.
2.5
0.3
2
Nu
Nu
(a) Re=10 0.4
0.2
1.5
0.1 0 100
(b) Re=100
1 150
200
0.5 100
250
150
Nondimensional time
200
250
Nondimensional time
(c) Re=200 4
Lf=0.0H
3
Nu
Lf=0.5H L =1.0H
2
f
L =1.5H f
1 100
150
200
250
Nondimensional time Fig. 14. Effect of fin length on temporal variations of spatial-averaged Nusselt number when the steady flow periodic oscillations are reached at Reynolds number of 10, 100 and 200 for excitation Strouhal number of 0.025 and excitation amplitude of A = 1.
(a) Re=10
(b) Re=100
0.3
2
Nu
Nu
0.25 0.2
1.5
0.15 0.1 150
160
170
1 0
180
20
Nondimensional time
40
60
80
100
Nondimensional time
(c) Re=200 L =0.0H f
Lf=0.5H
3
Nu
Lf=1.0H L =1.5H
2 1 0
f
20
40
60
80
100
Nondimensional time Fig. 15. Effect of fin length on temporal variations of spatial-averaged Nusselt number when the steady flow periodic oscillations are reached at Reynolds number of 10, 100 and 200 for excitation Strouhal number of 0.1 and excitation amplitude of A = 1.
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(b) Re=100
(a) Re=10 0.28
0.24
Nu
Nu
1.6
1.2
0.2 30
1.4
32
34
36
38
1 50
40
52
54
56
58
60
Nondimensional time
Nondimensional time
Nu
(c) Re=200 3
L =0.0H
2.5
Lf=0.5H
2
Lf=1.0H
f
L =1.5H f
1.5 1 50
52
54
56
58
60
Nondimensional time Fig. 16. Effect of fin length on temporal variations of spatial-averaged Nusselt number when the steady flow periodic oscillations are reached at Reynolds number of 10, 100 and 200 for excitation Strouhal number of 0.5 and excitation amplitude of A = 1.
Table 3 Phase shift between the velocity and length averaged Nusselt number along the bottom wall downstream of the step for various flow parameters. St
0.01 0.025 0.05 0.1 0.25 0.5 1 2
Re = 10
Re = 100
Re = 200
Lp = 0.0H
Lp = 0.5H
Lp = 1.0H
Lp = 1.5H
Lp = 0.0H
Lp = 0.5H
Lp = 1.0H
Lp = 1.5H
Lp = 0.0H
Lp = 0.5H
Lp = 1.0H
Lp = 1.5H
32.2 50.1 59.6 69.4 90.5 94.1 89.9 87.3
29.5 46.7 50.7 60.2 82.6 82.5 66.4 59.5
27.6 38.9 43.3 49.1 67.2 86.5 87.3 60.8
22.1 30.6 29.8 33.2 42.8 59.1 74.2 83.1
43.1 64.2 87.8 119.9 113.2 132.1 144.4 108.3
41.5 55.3 62.3 60.9 47.3 33.7 57.5 74.7
37.2 45.5 52.4 61.6 97.8 69.5 12.5 1.4
30.1 35.7 38.8 43.9 0 83.6 62.3 24.6
50.7 77.1 102.4 132.9 115.9 159.5 109.7 99.2
45.4 53.1 72.1 67.7 54.6 30.3 77.6 66.4
40.6 47.2 56.6 61.4 96.7 44.8 82.1 6.84
36.1 42.7 45.5 54.6 64.7 95.8 87.9 2.1
Fig. 17 are shown in Fig. 18 at Reynolds number of 10, 100 and 200 for the case without fin. The first and last value of Fig. 17 represent the maximum and minimum values of Nusselt numbers, respectively. When the Nusselt number becomes minimum, the cell which is formed at the beginning of the cycle behind the step disappears for all Reynolds numbers. Another observation is that at Reynolds number of 100 and 200, for the maximum Nusselt
Re=10
0.23
X: 1040 Y: 0.224
Nu
Re=100
1.095
X: 1025 Y: 0.2336
X: 1835 Y: 1.091
Nu
0.24
number according to Fig. 17 secondary recirculation zone is seen at the upper wall. As stated earlier, with the installation of the fin on the upper wall, the fin redirects the flow movement and changes the streamlines behind the step for the steady case. For the unsteady case, streamline plots for the three time instances within half a period when the system reaches the periodic steady flow condition
X: 1845 Y: 1.088
1.09
0.22 X: 1050 Y: 0.2194
0.21 950
1050
1000
Number of time steps 1.505
1.085 1760
X: 1855 Y: 1.087
1780
1800
1820
1840
1860
Number of time steps
Re=200
Nu
X: 5035 Y: 1.502 X: 5045 Y: 1.5
1.5 X: 5055 Y: 1.498
1.495 4950
5000
5050
Number of time steps Fig. 17. Time instance within half a period when the system reaches periodic steady flow conditions for no-fin case at Reynolds number of 10, 100 and 200 for excitation Strouhal number of 0.5.
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(a)
(b) (c)
(d)
(e) (f)
(g)
(h) (i)
Fig. 18. Streamlines for no-fin case at the time instances according to Fig. 16 at Reynolds number of 10 (first box), 100 (second box) and 200 (third box), St = 0.5.
according to Fig. 19 are shown in Fig. 20 at Reynolds number of 10, 100 and 200 for the case with fin length of Lf = H. The first and last value of Fig. 19 represent the minimum and maximum values of Nusselt numbers, respectively. At the onset of the cycle when the Nusselt number is minimum, a large recirculation zone is exhibited behind the fin and increases in size and strength with increasing Reynolds number. At Reynolds number of 200, during the acceleration phase, the recirculation zone appeared on the bottom wall downstream of the step also gets larger in size. These plots show that a considerable change occurs in the flow field during the pulsation cycle when the fin is installed on the top wall. Isotherm plots for the three time instances within half a period when the system reaches the periodic steady flow condition according to Figs. 17 and 19 are shown in Figs. 21 and 22. for St = 0.5 at Reynolds number of 10, 100 and 200 without fin case
Re=100 X: 1960 Y: 1.406
X: 3260 Y: 0.2313
0.24
1.4
0.23
Nu
Nu
X: 3270 Y: 0.2487
Re=10
0.25
and with fin length of Lf = H. From the plots, it is seen that increasing the Reynolds number decreases the thermal boundary layer along the bottom wall downstream of the step and hence increases the heat transfer rate. There is considerable change for shape of the isotherms in the vicinity of the fin especially at Reynolds number of 100 and 200. In this case, the isotherm plots at Reynolds number of 10 show that, only a small portion of the isotherms in the vicinity of the fin distorted. When the flow motion increases, more distortion of the isotherms from the fin tip at the bottom to the downstream of the step and also more distortion form the upper wall towards the fin is seen. Time-spatial averaged Nusselt number along the bottom wall downstream of the step versus excitation Strouhal number plots are shown in Fig. 23 for various fin lengths at Reynolds number of 10, 100 and 200. In these plots, the vertical axis is normalized
0.22
X: 3245 Y: 0.208
0.21 3180
X: 1945 Y: 1.327
1.35
X: 1950 Y: 1.367
1.3
3220
3280
1800
1850
1900
1950
Number of time step
Number of time step
Re=200
Nu
2.2
X: 8030 Y: 2.213
2.1
X: 8025 Y: 2.124 X: 8020 Y: 1.977
2 7950
8000
8050
Number of time step Fig. 19. Time instance within half a period when the system reaches periodic steady periodic flow condition for fin length of Lf = H at Reynolds number of 10, 100 and 200 for excitation Strouhal number of 0.5.
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(a)
(b) (c)
(d)
(e) (f)
(g)
(h) (i)
Fig. 20. Streamlines for fin length of Lf = H at the time instances according to Fig. 18 at Reynolds number of 10 (first box), 100 (second box) and 200 (third box), St = 0.5.
(a)
(b) (c)
(d)
(e) (f)
(g)
(h) (i)
Fig. 21. Streamlines for no-fin case at the time instances according to Fig. 16 at Reynolds number of 10 (first box), 100 (second box) and 200 (third box), St = 0.5.
with spatial averaged Nusselt number in the steady flow case. In these plots, the normalized Nusselt number decreases with increasing frequency after St = 0.1. At Reynolds number of 10, there is negligible change in the normalized Nusselt number with increasing the excitation Strouhal number. For fin lengths of Lf = 0.5H and Lf = H, normalized Nusselt number values at Reynolds number of 100 and 200 become closer with increasing frequencies. According to Fig. 8, the fin length on the length-averaged Nusselt values in the steady flow case is more effective with increasing Reynolds number. The normalized Nusselt number versus excitation Strouhal number plot as in Fig. 23 shows a resonant type
behavior as similar in [9], the value first increases with an increase in the excitation Strouhal number (up to St = 0.05), then a decrease is seen with the increasing values of the frequency of the pulsation for the case without fin. Adding a fin shifts the maximum value of the normalized Nusselt number from 0.05 to 0.1. This may be due to the influence of the fin on the flow movement and its effect on the adaption time of the system to new flow condition in the pulsating flow case. A measure for the heat transfer enhancement for pulsating flow in comparison to the steady flow with different fin lengths and with no-fin length case can be given as,
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(a)
(b) (c)
(d)
(e) (f)
(g)
(h) (i)
Fig. 22. Isotherms for fin length of Lf = H at the time instances according to Fig. 16 at Reynolds number of 10 (first box), 100 (second box) and 200 (third box), St = 0.5.
Re=10 Re=100 Re=200
1.2 1.1 1 0.9 10
−2
10
−1
10
Num / Nus
Num / Nus
(b) L f=0.5H
(a) L f=0.0H
1.3
Re=10 Re=100 Re=200
1.1 1 0.9
0
10
−2
(d) L f=1.5H
1
10
−1
10
Num / Nus
s
Num / Nu
1.1
−2
10
(c) L =1.0H Re=10 Re=100 Re=200
10
−1
St
f
0.9
10
St
Re=10 Re=100 Re=200
1.1
1
0.9
0
0
10
−2
St
10
−1
10
0
St
Fig. 23. Variation of normalized Nusselt number (time and spatial-average Nusselt number in pulsating flow divided by the average in steady flow) for no-fin case and different fin lengths with excitation Strouhal number at Reynolds number of 10, 100 and 200.
HTEw;f ¼
Num Nus
Nuwm ; Nuws
ð12Þ
where subindex m, s, wm and ws denote the spatial-time averaged, steady flow case, spatial-time averaged without fin case and steady flow case without fin case, respectively. This will indicate the combined effect of pulsating flow and installation of the fin on the heat transfer augmentation. As stated earlier, in the steady flow case, adding a fin will increase the HTE in the direction of increasing Reynolds number and increasing fin length. In the pulsating flow case, as seen in Fig. 24, HTEw,f value is less than one (close to one at Re = 10) indicating that installation of a thin fin on the top wall is not advantageous in the pulsating flow case. In some cases (mostly at Re = 200), the values are even below 1. The least HTEw,f values are
obtained for the case at the excitation Strouhal number of 0.25 and at Reynolds number of 200 all considered fin lengths. The effects of the distance of the fin from the sudden expansion may have influences on the flow field and heat transfer characteristics, but it is out of scope of this paper. 4. Conclusions In this study, laminar pulsating flow over a backward facing step with an upper wall mounted adiabatic thin fin is numerically studied. The effects of various pertinent parameters, Reynolds number, fin length and pulsating frequency on the fluid flow and heat transfer characteristics are numerically investigated. The
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(a) L =0.5H
(b) L f=1.0H 1.1
1.05
1.05
HTEw,f
HTEw,f
f
1.1
1 0.95
1 0.95
0.9 10
−2
10
−1
10
0.9
0
10
−2
10
−1
10
0
St
St
(c) L f=1.5H 1.1
HTEw,f
Re=10 Re=100
1
Re=200 0.9 0.8 10
−2
10
−1
10
0
St Fig. 24. Variation of heat transfer enhancement for pulsating flow in comparison to the steady flow with different fin lengths and with no-fin length case (HTEw,f), with excitation Strouhal number at Reynolds number of 10, 100 and 200.
combined effects of the pulsation and installation of the fin on the heat transfer are investigated. Following results are obtained: In the steady case, fin redirects the flow movement and streamline plots show that length and intensity of the recirculation zone behind the step is considerably effected with the installation of an adiabatic thin fin on the upper wall of the channel. Adding an adiabatic fin alters the isotherm plots. Comparing the isotherms for the case with the fin with those for the case without fin, 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. For the steady flow conditions, Heat Transfer Enhancement (HTE) is increasing with an increase in the Reynolds number and fin length. At Reynolds number of 200 and fin length of Lf = 1.5H, heat transfer enhancement is 188 percent for the bottom hot wall as compared to the no-fin case. For the pulsating flow case, with increasing values of Reynolds number and increasing value of fin length, level of nonlinearity (distortion from a pure sinusoid) in the temporal variation of the Nusselt number increases. For St = 0.1, Normalized averaged Nusselt number increases almost linearly with increasing values of pulsating velocity amplitudes at Re = 100 and Re = 200 and at Re = 10, there is negligible change in heat transfer enhancement with respect to steady flow value with the change in velocity amplitude. For St = 0.1, The averaged pressure drop increases with the increasing values of fin length and the averaged pressure drop value is more sensitive with respect to a change in the velocity amplitude for the case at fin length of Lf = 1.5H. The normalized Nusselt number (normalized with respect to steady flow spatial averaged-Nusselt value) versus excitation Strouhal number shows a resonant type behavior. The normalized Nusselt value first increases with an increase in the excitation Strouhal number (up to St = 0.05), then a decrease is seen with the increasing values of the frequency of the pulsation for the case without fin. Adding a fin shifts the maximum value of the normalized Nusselt number from St = 0.05 to St = 0.1. Adding a fin is not advantageous for heat transfer enhancement in pulsating flow compared to steady flow with no-fin case.
The least HTEw,f values are obtained for the case at the excitation Strouhal number of 0.25 and at Reynolds number of 200 for the fin lengths considered.
Acknowledgment The authors are very grateful to the reviewers due their appropriate and constructive suggestions as well as their proposed corrections, which helped us improve the quality of the paper.
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Contents lists available at ScienceDirect
Computers & Fluids j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p fl u i d
Numerical analysis of laminar pulsating flow at a backward facing step with an upper wall mounted adiabatic thin fin Fatih Selimefendigil a,⇑, Hakan F. Öztop b a b
Department of Mechanical Engineering, Celal Bayar University, Manisa, Turkey Department of Mechanical Engineering, Technology Faculty, Fırat University, Elazıg˘, Turkey
a r t i c l e
i n f o
Article history: Received 4 October 2012 Received in revised form 15 July 2013 Accepted 29 August 2013 Available online 10 September 2013 Keywords: Pulsating flow Adiabatic fin Backward facing step
a b s t r a c t The effect of an upper wall mounted adiabatic thin fin on laminar pulsating flow in a backward facing step has been investigated numerically. Study is performed for different Reynolds numbers (based on the step height) in the range of 10 and 200 and for the expansion ratio of 2. The working fluid is air with the Prandtl number of 0.71. The governing equations are solved with a general purpose finite volume based solver, FLUENT. The effects of various pertinent parameters, Reynolds number, fin length and pulsating frequency on the fluid flow and heat transfer characteristics are numerically studied. It is observed that fin alters the flow field and thermal characteristics. In the steady flow case, heat transfer enhancement is obtained with the installation of the fin on the upper wall and increases with increasing fin length and increasing Reynolds number. Heat transfer enhancement of 188% is obtained for fin length of Lf = 1.5H at Reynolds number of 200. In the pulsating flow case, time-spatial averaged Nusselt number along the bottom wall downstream of the step normalized with spatial averaged Nusselt number in the steady flow case versus excitation Strouhal number shows a resonant type behavior; first an increase in the value is seen up to St = 0.05, then a decrease is seen with the increasing values of the frequency of the pulsation for the case without fin. Adding a fin shifts the maximum value of the normalized Nusselt number from St = 0.05 to St = 0.1. Compared to steady flow with no-fin case, adding a fin is not advantageous for heat transfer enhancement in pulsating flow. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Flow separation and its subsequent reattachment is important in many engineering applications such as flow around airfoils, buildings, combustors and collectors of power systems and therefore a vast amount of literature is dedicated to that subject. 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 Abu-Mulaweh [1] for laminar mixed convection over vertical, horizontal and inclined backwardand forward-facing steps studied in the open literature. In this review, the effects of pertinent parameters such as Reynolds number, Prandtl number and expansion ratio on the fluid flow and thermal characteristics is also presented. Barkley et al. [2] have numerically studied the 3D linear stability analysis of flow over a backward-facing step for Reynolds numbers between 450 and 1050. Erturk [3] has numerically studied the 2D flow over a backward-facing step for Reynolds number between 100 and 3000 based on the channel height. He reported that the outflow ⇑ Corresponding author. E-mail addresses: [email protected] (F. Selimefendigil), hfoztop1@ gmail.com (H.F. Öztop). 0045-7930/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compfluid.2013.08.013
boundary condition and the location of the exit have effect on the accuracy of the numerical solution. Nie and Armaly [4] have reported the wall temperature distributions, Nusselt number and the friction coefficient for the laminar 3D flow adjacent to backward-facing step in a rectangular duct. Iwai et al. [5] have numerically investigated the 3D flow over a backward facing step at low Reynolds number for various duct aspect ratios. They reported that the maximum Nusselt number did not appear on the centerline, but near the two side walls in every case. Barbosa-Saldana and Anand [6] have numerically studied the 3D laminar flow over a horizontal forward-facing step. The expansion ratio and aspect ratio of two and four is considered. Effects of the Reynolds number on the locations of the reattachment line, velocity distribution and averaged Nusselt number are discussed. Experimental studies have also been conducted for the flow over a backward facing or forward facing step. Armaly et al. [7] have reported the velocity distribution and reattachment length using Laser-Doppler measurement for flow downstream of a backward facing step in a two-dimensional channel. Their results showed separation length varies with Reynolds number and various flow regimes are characterized by variations of the separation length. Abu-Mulaweh [8] has reported the measurements of heat transfer and flow over an vertical forward-facing step for turbulent
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Nomenclature H h k L n Nu p Pr Re T u, v x, y
step size (m) local heat transfer coefficient (W/m2 K) thermal conductivity (W/m K) fin length (m) unit normal vector local Nusselt number, hH/k pressure (Pa) Prandtl number, am Reynolds number, u0mH temperature (K) x–y velocity components (m/s) Cartesian coordinates (m)
mixed convection using two-component LDV and a cold wire anemometer. He reported that when the step height increases the turbulence intensity of the streamwise and transverse velocity fluctuations and the intensity of temperature fluctuations downstream of the forward-facing step increase. Terhaar et al. [9] have experimentally studied the unsteady heat transfer downstream over a backward-facing step in the laminar flow regime for pulsating flow conditions at Reynolds number of 300 (based on the hydraulic diameter of the channel). They showed that the Nusselt number increases up to a certain excitation Strouhal number and then degrades as the pulsation frequency increases. Sherry et al. [10] have made an experimental investigation for the recirculation zone formed downstream of a forward facing step immersed in a turbulent boundary layer. In their study, the mechanisms effecting the reattachment distance is discussed. Stuer et al. [11] have experimentally studied the separation ahead of a forward facing step under laminar flow conditions using the hydrogen bubble technique. 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 [12]. But, in the literature there exist also cases where pulsating flow does not effect [13] or even deteriorate heat transfer enhancement [14]. That means 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 has received less attention in the literature. Yilmaz and Oztop [15] have numerically investigated the turbulent forced convection over double forward facing step by using two adiabatic steps with different lengths and heights. They studied the effects of step heights, step lengths and Reynolds numbers on heat transfer and fluid flow. They reported that the second step can be used to control the fluid flow and heat transfer characteristics. Oztop et al. [16] have numerically performed turbulent forced convection over double forward facing step with obstacles of rectangular cross-sectional area. They studied the effect of step height, Reynolds number and obstacle aspect ratio on fluid low and heat transfer. They reported that heat transfer rate is enhanced with increasing aspect ratio of the obstacle. Kumar and Dhiman [17] have numerically studied the heat transfer enhancement in laminar forced convection flow over a backward facing step with the insertion of an adiabatic circular cylinder. They considered different cross-stream positions of the circular cylinder for the Reynolds number between 1 and 200. They obtained heat transfer enhancement up to 155 percent compared to no-cylinder case. Khanafer et al. [18] have studied the mixed convection in laminar pulsating flow over a backward-facing step. They considered the Reynolds
Greek characters a thermal diffusivity (m2/s) h non-dimensional temperature, m kinematic viscosity (m2/s) q density of the fluid (kg/m3)
TT c T h T c
Subscripts c cold f fin h hot wall
number between 100 and 1000, non-dimensional pulsating frequency between 0.1 and 5 and Richardson number between 0.000178 and 10. Their simulation results showed that average Nusselt number increases with an increase in both Reynolds number and Grashof number and decrease with an increase in the pulsating frequency. Recently, Selimefendigil and Oztop [19] have numerically studied the forced convection in pulsating flow at a backward facing step with a stationary circular cylinder subjected to nanofluid. They studied the effects of pulsating frequency, nanoparticle volume fraction and Reynolds number on the heat transfer characteristic of the backward facing step flow. Based on the literature survey and to the best of authors’ knowledge a study of laminar forced convection over a backward facing step in a channel with the upper wall mounted adiabatic fin in pulsating flow has never been studied in the literature. The main objective of this study is to numerically study the pulsating flow over a backward facing step in the presence of an upper wall mounted adiabatic thin fin. The effects of various pertinent parameters such as fin length, Reynolds number, oscillating frequency on the fluid flow and heat transfer characteristics over a backwardfacing step will be investigated in detail and optimal combinations of parameters for the maximum heat transfer augmentation will be determined using numerical simulations. 2. Numerical simulation 2.1. Problem description A schematic description of the physical problem considered in this study is shown in Fig. 1. A channel with a backward facing step is considered. The step size of backward facing step is H and channel height is 2H. At the inlet of the channel, a uniform velocity with a sinusoidal time dependent part (U = 1 + A sin(2pft)) and a uniform temperature (h = 0) is imposed. Generally parabolic
Fig. 1. Geometry with boundary conditions.
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velocity profiles at the inlet are used in numerical simulation studies of backward facing step flow. But with the given inflow length of 10 channel heights and the low Reynolds number of the current study, the simulation results are similar for uniform and parabolic velocity profiles for the same flow rate which have been confirmed from the numerical simulation results at the highest Reynolds number of 200. 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 h = 1, while the other walls of the channel are assumed to be adiabatic. An adiabatic thin fin with length Lf is mounted on the top wall of the channel. The streamwise downstream location of the fin where it is mounted on the top wall is H starting from the edge of the step. In the present study, it is expected that different fin lengths change the heat transfer and fluid flow characteristic of the backward facing step. Working fluid is air with a Prandtl number of Pr = 0.71. It is assumed that thermophysical properties of the fluid is temperature independent. The flow is assumed to be two dimensional, Newtonian, incompressible and in the laminar flow regime. The three dimensional effects of the flow over a backward facing step have been studied by [20,21]. Williams and Baker [20] have noted the side walls effect on the development of the three dimensional flow for Reynolds number grater than 400. They have numerically investigated the complex interaction of a wall jet with the main stream for the mechanism of the three dimensionality. Chiang and Sheu [21] have also numerically showed the wall induced three dimensional vortical flow with increasing Reynolds number for a laminar backward facing step flow. The three dimensional effects and the possible effect of the presence of side walls on flow field and heat transfer are out of the scope of the present paper. 2.2. Governing equations and solution method By using the dimensionless parameters,
ðU; V Þ ¼
ðu; v Þ ; u0
ðX; Y Þ ¼
ðx; yÞ p T Tc tH ; P ¼ 2; h¼ ; k¼ ; H u0 Th Tc qu0 ð1Þ
for a two dimensional, incompressible, laminar and steady case, the continuity, momentum and energy equations can be expressed in the non-dimensional form as in the following:
@U @V þ ¼ 0; @X @Y
ð2Þ
! @U @U @U @P 1 @2U @2U þU þV ¼ þ þ ; @k @X @Y @X Re @X 2 @Y 2
ð3Þ
! @V @V @V @P 1 @2V @2V ; þU þV ¼ þ þ @k @X @Y @Y Re @X 2 @Y 2
ð4Þ
! @h @h @h 1 @2h @2h ; þU þV ¼ þ @k @X @Y PrRe @X 2 @Y 2
ð5Þ
Reynolds number (Re) based on the step height and non-dimensional frequency, Strouhal number (St) can be written as
Re ¼
u0 H
m
;
St ¼
fH : u0
ð6Þ
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 + A sin(2pStk), V = 0, h = 0. At the bottom wall, downstream of the step, temperature is constant h = 1. At the channel exit, gradients of all variables in the x-direc@T tion are set to zero, @U ¼ 0; @V ¼ 0; @X ¼ 0. @X @X On the channel walls (except the downstream of the step) and on the adiabatic thin fin, adiabatic wall with no-slip @h boundary conditions are assumed, U ¼ 0; V ¼ 0; @n ¼ 0, where n denotes the surface normal direction. Local Nusselt number is defined as
Nux;t ¼
hx;t L @h ¼ ; k @n S
ð7Þ
where hx,t represent the local heat transfer coefficient and k denote the thermal conductivity of air. Spatial averaged Nusselt number is obtained after integrating the local Nusselt number along the bottom wall downstream of the step as
Nut ¼ 1L
RL 0
ð8Þ
Nux;t dx:
Time and spatial averaged Nusselt number is obtained after integrating spatial averaged Nusselt number along the bottom wall downstream of the step for one period of the oscillation s as
Num ¼ s1
Rs 0
Nut dt:
ð9Þ
Eqs. (2)–(5) along with the boundary and initial conditions are solved with Fluent (a general purpose finite volume solver [22]). The convective terms in the momentum and energy equations are solved using QUICK scheme and SIMPLE algorithm is used for velocity– pressure coupling. The system of algebraic equations are solved with Gauss-Siedel point by point iterative method and algebraic multigrid method. The convergence criteria for continuity, momentum and energy equations are set to 103, 105 and 106, respectively. The steady solutions are used as the initial conditions for the unsteady computations. The time step size is chosen as 1/50th of the period of the pulsating flow. The unstructured body-adapted mesh of appropriate size consists of only triangular elements. The computational domain is divided into 15872 triangular elements for no fin case at Reynolds number of 100. The mesh is finer near the walls 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 also carried out to obtain an optimal grid distribution with accurate results and minimal computational time at Re = 100 for no-fin case. Four different grid densities are tested and the convergence in the length-averaged Nusselt number (along the bottom wall downstream of the step) is checked. The results at Reynolds number of 100 for the case without fin is tabulated in Table 1. From this table, grid size of 15872 is decided to be fine enough to resolve the flow and thermal field for the case without fin at Re = 100. Fig. 2 shows the local Nusselt number distribution along the bottom wall for
Table 1 Grid sensitivity study for the case without fin at Reynold number of 100. Nusselt numbers are the length averaged values along the bottom wall downstream of the step. Case
Number of cells
Nusselt number
G1 G2 G3 G4
3968 8928 15,872 24,800
1.0369 1.0353 1.0347 1.0345
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F. Selimefendigil, H.F. Öztop / Computers & Fluids 88 (2013) 93–107 15
Table 2 Reported values for the reattachment lengths XR at Reynolds number 100 (Expansion ratio of 2).
G1 G2 G3 G4
Nu
10
5
0
0
5
10
15
20
25
30
35
x/H Fig. 2. Local Nusselt number distribution along the bottom wall downstream of the step with fin of length Lf = 1.0H and Re = 200 for different grid densities.
different grid sizes (G1 = 6428, G2 = 10694, G3 = 32130, G4 = 49766) at the highest Reynolds number, Re = 200 in the presence of the fin Lf = 1.0H. From this plot, the result of grid size G3 is close to the result of finest grid. Grid independence studies are also conducted for the subsequent computations for different Reynolds numbers and fin lengths parameter combinations. Fig. 3 depicts the time step size independence study result for the case with fin length Lf = 1.0H and Re = 200 at the excitation Strouhal number of 0.01. Four different time step sizes (dt = s/25, dt = s/50, dt = s/100, dt = s/150) are considered for the period of the oscillation s.Considering both accuracy and computational time of the simulation a time step size of dt = s/50 is chosen in this study. The numerical code is first checked against the benchmarked results of backward facing step reported in the literature [23–27]. Table 2. shows the results of the reattachment length divided by step height at Reynolds number of 100 for expansion ratio of 2. Minimum deviation for the percentage in the error is obtained for the results of Acharya et al. [23] which is 2.93 percent. The agreement between the other sources is less than 5 percent, only 6.83 percent error is obtained for the results of El-Refaee et al. [26]. 3. Results and discussion As stated earlier, the overall purpose of this study is to investigate the effects of an upper wall mounted inclined fin on the heat transfer and fluid flow characteristics for different fin lengths and pulsating frequencies in the laminar range of Reynolds number between 10 and 200. The main parameters that effect the fluid flow and thermal characteristics are Reynolds number, step height,
dt= τ/25 dt= τ/50 dt= τ/100 dt= τ/150
3.2 3 2.8 2.6
Nu
2.4 2.2 2 1.8 1.6 1.4 1.2 900
1000
1100
1200
1300
1400
1500
time Fig. 3. Time histories of space-averaged Nusselt number along the bottom wall downstream of the step with fin of length Lf = 1.0H and Re = 200 for different time step sizes at the excitation Strouhal number of 0.01.
XR/S
Error (%)
Acharya et al. [15] Lin et al. [16] Dyne et al. [17] El-Refeee et al. [18] Cochran et al. [19]
4.97 4.91 4.89 4.77 5.32
2.93 4.1 4.49 6.83 3.9
Present
5.12
0
distance between the step edge to the channel exit downstream of the step, distance between the inlet to the step edge, fin height, fin inclination angle, streamwise distance of the fin to the step edge, expansion ratio, oscillating frequency, amplitude of the forcing at the inlet and Prandtl number. In the present study, expansion ratio is 2, streamwise distance of the fin to the step edge is H (step height) and fin inclination angle is 90 degrees. The distance between the inlet to the step edge is 10H (for Re P 100, the value can be set to any value with parabolic velocity imposed at the inlet [28]) and the distance between the step edge to the channel exit downstream of the step is 35H. The effect of varying forcing amplitude is investigated for the case corresponding to excitation Strouhal number of 0.1. The simulations are performed for Reynolds number between 10 and 200, fin height between 0 and 1.5H and excitation Strouhal number between 0.01 and 2. 3.1. Steady inflow 3.1.1. Streamlines and isotherms Streamline plots showing the flow patterns at Reynolds number of 10, 100 and 200 for no-fin case and for various fin lengths Lf = 0.5H, Lf = H and Lf = 1.5H are shown in Fig. 4. For the case without fin as the Reynolds number is increased, the flow at the edge of the step separates and a recirculation zone is observed behind the step. The size of this zone increases with an increase in the Reynolds number. When a fin is attached at the upper wall of the channel at Reynolds number of 10, the flow field in the vicinity of the fin disturbed and at fin length of Lf = 1.5H, a vortex is formed behind the fin. At Reynolds number of 100, this vortex behind the cell is already formed at fin length of Lf = 0.5H and gets larger in size and strength with increasing fin lengths. At this Reynolds number, for fin length of Lf = 1.5H, a small cell behind the step and a larger cell whose core is located downstream of the core of the cell formed behind the fin appear. At Reynolds number of 200, for the same fin lengths, the core of the cells formed behind the step moves downstream and the cells gets larger in strength and more distorted. From the streamline plots, it is seen that fin redirects the flow movement and the flow behavior behind the step (length and intensity of the recirculation zone behind the step) is considerably effected with the installation of an adiabatic thin fin on the upper wall of the channel. Isotherm plots for different fin lengths at Reynolds number of 10, 100 and 200 are shown in Fig. 5. The thermal boundary layer along the bottom wall downstream of the step has a steep temperature gradient around the flow reattachment point. Adding an adiabatic fin alters the temperature contours as can be seen from the plots. Comparing the isotherms for the case with the fin with those for the case without fin, it is seen that there is significant change on the clustering of the isotherm patterns and the location where this steep temperature gradient occurs especially at Reynolds number of 100 and 200. At Reynolds number of 100, for fin lengths of Lf = H and Lf = 1.5H, temperature contours from the tip of the fin extending downstream of the step are formed. The size and the strength of these contours increase when the Reynolds number
F. Selimefendigil, H.F. Öztop / Computers & Fluids 88 (2013) 93–107
97
Fig. 4. Streamlines for no-fin case and for various fin lengths at Reynolds number of 10 (first box), 100 (second box) and 200 (third box).
Fig. 5. Isotherms for no-fin case and for various fin lengths at Reynolds number of 10 (first box), 100 (second box) and 200 (third box).
increases. Variation of the normalized reattachment lengths (normalized with reattachment length for no-fin case) versus Reynolds number are depicted in Fig. 6. It is seen that increasing the fin length decreases the reattachment length compared to no-fin case and this effect is more pronounced at the higher Reynolds numbers.
3.1.2. Local and average Nusselt numbers Local Nusselt number distribution along the bottom wall downstream of the step for various fin heights at Reynolds number of 10, 100 and 200 are shown in Fig. 7. The horizontal axis denotes the distance from the edge of the step to the exit normalized by the step height. In these plots, a peak in the Nusselt number is seen
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3.2. Pulsating inflow
0.8 Lf=0.5H
0.7
L =1.0H f
The aim of the present section is to numerically investigate the effects of various pertinent parameters (pulsating amplitude, pulsating frequency, Reynolds number, fin length) on the transient evolution of the fluid flow and heat transfer characteristics. First, the effects of amplitude are investigated for amplitudes between 0.25 and 1 while keeping the excitation Strouhal number at fixed value of 0.1. Then, effect of frequency is investigated for excitation Strouhal numbers between 0.01 and 2 while keeping the amplitude of the forcing at A = 1. In these numerical studies, Reynolds number and fin length vary between 10–100 and 0.0–1.5H, respectively.
Lf=1.5H
/ Xw, R
0.4
X
0.5
f, R
0.6
0.3 0.2 0.1
0
20
40
60
80
100
120
140
160
180
200
Re Fig. 6. Normalized reattachment length (with value corresponding to no fin) versus Reynolds number for different fin lengths.
which corresponds to a location very close to the reattachment point where steep temperature gradient occurs. Adding a fin at Reynolds number of 10, does not change the location of the maximum Nusselt number, but peak value of Nusselt number increases with increasing fin length. At Reynolds number of 100 and 200, the location of the maximum Nusselt number moves upstream close to the step with the installation of the fin compared to the case without fin. At Reynolds number of 200, for fin lengths of Lf = H and Lf = 1.5H, secondary peaks are seen in the local Nusselt number distribution. The secondary peak corresponds to the separation area of the secondary vortex present for fin at lengths of Lf = H and Lf = 1.5H. Length averaged Nusselt number along the bottom wall downstream of the step and normalized averaged Nusselt number (normalized by the averaged Nusselt number for the no-fin case) are shown in Fig. 8 on top and bottom, respectively. From these plots, it seen that Nusselt number increases with increasing Reynolds number and with increasing fin length. The plot on the bottom also shows the heat transfer enhancement (HTE) performance of the system with the installation of the fin at various fin lengths. The HTE values are 1.51 and 1.88 at Reynolds number of 200 for the fin lengths of Lf = H and Lf = 1.5H, respectively. With the installation of an adiabatic thin fin on the upper wall at length Lf = 1.5H at Reynolds number of 200, heat transfer enhancement is 188 percent for the bottom hot wall as compared to the nofin case.
3.2.1. Effects of pulsation amplitude Space-averaged Nusselt number along the bottom wall for different velocity amplitudes and various fin lengths at St = 0.1 are depicted in Fig. 9 for Re = 100 and in Fig. 10 for Re = 200. In all cases, the peak values of the Nusselt number increase with increasing values of velocity amplitudes. There is a shift in the mean with decreasing values of the fin lengths. The same trend is seen at Re = 200 and peak values of the Nusselt numbers increase compared to the case at Re = 100. At Re = 200, more distortions from a pure sinusoid is seen especially for the case at velocity amplitude of 1. Time-spatial averaged Nusselt number normalized with spatial averaged Nusselt number in the steady flow case along the bottom wall versus velocity amplitude plots are shown in Fig. 11 for various fin lengths at Reynolds number of 10, 100 and 200. Normalized averaged Nusselt number increases almost linearly with increasing values of pulsating velocity amplitudes at Re = 100 and Re = 200. Only at Re = 200, for fin length of Lf = 1.5, the tendency is different which may be due to the higher harmonic contributions from the flow reversal. At Re = 10, there is negligible change in HTE (heat transfer enhancement with respect to steady flow value) with the change in velocity amplitude. Dimensionless pressure drop coefficient is defined as the normalized difference between the averaged-pressure at the inlet and outlet sections [29],
dP ¼
pin pout : 0:5qu20
ð10Þ
Time evolution of the pressure drop coefficient for different velocity amplitudes and various fin lengths at St = 0.1 are depicted in Fig. 12 for Re = 200. The peak value in the pressure loss coefficient
(a) Re=10
(b) Re=100 15
2
Nu
Nu
3
5
1 0 0
10
5
10
x/H
0 0
15
5
x/H
10
15
(c) Re=200 25
Lf =0.0H
Nu
20
L =0.5H f
15
L =1.0H
10
L =1.5H
f f
5
0 0
5
x/H
10
15
Fig. 7. Effect of the fin length on local Nusselt number distribution for Reynolds number of 10, 100 and 200.
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L =0.0H f
L =1.0H
L =0.5H f
L =1.5H
f
f
3
Nus
2 1 0
0
20
40
60
80
100
120
140
160
180
200
120
140
160
180
200
Re
1.5
s
Nu / Nu
ws
2
1 0.5
0
20
40
60
80
100
Re Fig. 8. Top – Spatial-averaged Nusselt number versus Reynolds number and Bottom – Normalized spatial-averaged Nusselt number (with value corresponding to no fin) versus Reynolds number for various fin lengths.
(a) L f=0.0H
(b) L f=0.5H
1.5
1.2
Nu
Nu
1.4 1.1
1 70
1.3
80
90
100
110
120
70
80
Nondimensional time
90
100
110
120
Nondimensional time
(c) L f=1.0H
(d) L f=1.5H
A=0.25 A=0.5
2
1.7
A=1
Nu
Nu
1.6 1.5 1.4
1.5
1.3 70
80
90
100
110
120
1 70
80
Nondimensional time
90
100
110
120
Nondimensional time
Fig. 9. Effect of velocity amplitude on temporal variations of spatial-averaged Nusselt number when the steady periodic oscillations are reached at various fin lengths for Reynolds number of 100 and excitation Strouhal number of 0.1.
(a) L =0.0H
(b) L f=0.5H
f
2.2 2.1
1.6
Nu
Nu
1.7
1.5 1.4 70
2 1.9
80
90
100
110
120
1.8 70
(c) L =1.0H f
2.8
3.5
Nu
Nu
2.6 2.4
90
100
110
120
(d) L f=1.5H
A=0.25 A=0.5 A=1
3 2.5
2.2 2 70
80
Nondimensional time
Nondimensional time
80
90
100
110
Nondimensional time
120
2 70
80
90
100
Nondimensional time
Fig. 10. Effect of velocity amplitude on temporal variations of spatial-averaged Nusselt number when the steady periodic oscillations are reached at various fin lengths for Reynolds number of 200 and excitation Strouhal number of 0.1.
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(a) Re=10
(b) Re=100 1.15
Num / Nu
s
Num / Nus
1.05
1
0.95 0.2
0.4
0.6
0.8
1.1 1.05 1 0.2
1
0.4
0.6
A
0.8
1
A (c) Re=200
Nu / Nu m s
1.25
Lf=0.0H
1.2
Lf=0.5H L =1.0H
1.15
f
Lf=1.5H
1.1 1.05 0.2
0.4
0.6
0.8
1
A Fig. 11. Variation of normalized spatial-averaged Nusselt number versus amplitude of the forcing for various fin lengths at Reynolds number of 200 and excitation Strouhal number of 0.1.
increases with increasing values of velocity amplitudes. The averaged values of the pressure loss coefficient versus velocity amplitudes are shown in Fig. 13 for different fin lengths and Reynolds numbers. The averaged pressure drop coefficient is more sensitive with respect to a change in the velocity amplitude for the case at fin length Lf = 1.5H. The averaged pressure drop coefficient increases with the increasing values of fin length.
sinusoid) with increasing Reynolds number and increasing fin length. The difference between the peak value of the Nusselt numbers increases for various fin lengths with increasing frequency. The time when the maximum values are reached also change with increasing frequency indicating that there is a phase shift between the forcing velocity at the inlet and response (Nusselt number along the bottom wall downstream of the step). The phase shift for an excitation amplitude and Strouhal number can defined as
3.2.2. Effects of pulsation Strouhal number Length averaged Nusselt number along the bottom wall downstream of the step plots are shown in Fig. 14 for various fin lengths at Reynolds number of 10, 100 and 200 and at excitation Strouhal number of 0.025. The plots are shown after removing the initial transients. With increasing the Reynolds number, there is an increase in the mean Nusselt number with respect to the non-fin case, which is higher for high fin lengths. Figs. 15 and 16 show the Nusselt number plots at excitation Strouhal number of 0.1 and 0.5, respectively. With increasing the frequency, the number of periods to reach steady periodic flow conditions generally increases for the same Reynolds number. Another observation is the increasing level of nonlinearity (distortion from a pure
U ¼ arg
Nu : u ðA;StÞ
Table 3 shows the phase shift in degrees for various combinations of Reynolds number, excitation Strouhal number and fin lengths. At the lower pulsating frequencies, increasing values of the fin length decrease the phase shift and phase shift values increase with increasing values of Reynolds numbers. 3.2.3. Streamlines and isotherms Streamline plots for the three time instances within half a period when the system reaches the periodic steady flow according to
(b) L f=0.5H 40
20
20
dP
dP
(a) L f=0.0H 40
0
0
−20
−20
−40
−40 80
90
100
110
80
120
100
110
Nondimensional time
(c) L f=1.0H
(d) L f=1.5H
120
A=0.25 A=0.5
60
A=1
40
20
dP
dP
90
Nondimensional time
40
0
20 0
−20 −40 80
ð11Þ
−20 90
100
110
Nondimensional time
120
−40 80
90
100
110
120
Nondimensional time
Fig. 12. Effect of velocity amplitude on temporal variations of averaged-pressure drop when the steady periodic oscillations are reached at various fin lengths for Reynolds number of 200 and excitation Strouhal number of 0.1.
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(b) Re=100
(a) Re=10 30
60
20
dPm
dPm
70
50 40 0.2
10
0.4
0.6
0.8
0 0.2
1
0.4
0.6
0.8
1
A
A (c) Re=200 30
Lf=0.0H Lf=0.5H
dPm
20
L =1.0H f
10
Lf=1.5H 0 0.2
0.4
0.6
0.8
1
A Fig. 13. Variation of averaged-pressure drop versus amplitude of the forcing for various fin lengths at Reynolds number of 10, 100, 200 and excitation Strouhal number of 0.1.
2.5
0.3
2
Nu
Nu
(a) Re=10 0.4
0.2
1.5
0.1 0 100
(b) Re=100
1 150
200
0.5 100
250
150
Nondimensional time
200
250
Nondimensional time
(c) Re=200 4
Lf=0.0H
3
Nu
Lf=0.5H L =1.0H
2
f
L =1.5H f
1 100
150
200
250
Nondimensional time Fig. 14. Effect of fin length on temporal variations of spatial-averaged Nusselt number when the steady flow periodic oscillations are reached at Reynolds number of 10, 100 and 200 for excitation Strouhal number of 0.025 and excitation amplitude of A = 1.
(a) Re=10
(b) Re=100
0.3
2
Nu
Nu
0.25 0.2
1.5
0.15 0.1 150
160
170
1 0
180
20
Nondimensional time
40
60
80
100
Nondimensional time
(c) Re=200 L =0.0H f
Lf=0.5H
3
Nu
Lf=1.0H L =1.5H
2 1 0
f
20
40
60
80
100
Nondimensional time Fig. 15. Effect of fin length on temporal variations of spatial-averaged Nusselt number when the steady flow periodic oscillations are reached at Reynolds number of 10, 100 and 200 for excitation Strouhal number of 0.1 and excitation amplitude of A = 1.
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(b) Re=100
(a) Re=10 0.28
0.24
Nu
Nu
1.6
1.2
0.2 30
1.4
32
34
36
38
1 50
40
52
54
56
58
60
Nondimensional time
Nondimensional time
Nu
(c) Re=200 3
L =0.0H
2.5
Lf=0.5H
2
Lf=1.0H
f
L =1.5H f
1.5 1 50
52
54
56
58
60
Nondimensional time Fig. 16. Effect of fin length on temporal variations of spatial-averaged Nusselt number when the steady flow periodic oscillations are reached at Reynolds number of 10, 100 and 200 for excitation Strouhal number of 0.5 and excitation amplitude of A = 1.
Table 3 Phase shift between the velocity and length averaged Nusselt number along the bottom wall downstream of the step for various flow parameters. St
0.01 0.025 0.05 0.1 0.25 0.5 1 2
Re = 10
Re = 100
Re = 200
Lp = 0.0H
Lp = 0.5H
Lp = 1.0H
Lp = 1.5H
Lp = 0.0H
Lp = 0.5H
Lp = 1.0H
Lp = 1.5H
Lp = 0.0H
Lp = 0.5H
Lp = 1.0H
Lp = 1.5H
32.2 50.1 59.6 69.4 90.5 94.1 89.9 87.3
29.5 46.7 50.7 60.2 82.6 82.5 66.4 59.5
27.6 38.9 43.3 49.1 67.2 86.5 87.3 60.8
22.1 30.6 29.8 33.2 42.8 59.1 74.2 83.1
43.1 64.2 87.8 119.9 113.2 132.1 144.4 108.3
41.5 55.3 62.3 60.9 47.3 33.7 57.5 74.7
37.2 45.5 52.4 61.6 97.8 69.5 12.5 1.4
30.1 35.7 38.8 43.9 0 83.6 62.3 24.6
50.7 77.1 102.4 132.9 115.9 159.5 109.7 99.2
45.4 53.1 72.1 67.7 54.6 30.3 77.6 66.4
40.6 47.2 56.6 61.4 96.7 44.8 82.1 6.84
36.1 42.7 45.5 54.6 64.7 95.8 87.9 2.1
Fig. 17 are shown in Fig. 18 at Reynolds number of 10, 100 and 200 for the case without fin. The first and last value of Fig. 17 represent the maximum and minimum values of Nusselt numbers, respectively. When the Nusselt number becomes minimum, the cell which is formed at the beginning of the cycle behind the step disappears for all Reynolds numbers. Another observation is that at Reynolds number of 100 and 200, for the maximum Nusselt
Re=10
0.23
X: 1040 Y: 0.224
Nu
Re=100
1.095
X: 1025 Y: 0.2336
X: 1835 Y: 1.091
Nu
0.24
number according to Fig. 17 secondary recirculation zone is seen at the upper wall. As stated earlier, with the installation of the fin on the upper wall, the fin redirects the flow movement and changes the streamlines behind the step for the steady case. For the unsteady case, streamline plots for the three time instances within half a period when the system reaches the periodic steady flow condition
X: 1845 Y: 1.088
1.09
0.22 X: 1050 Y: 0.2194
0.21 950
1050
1000
Number of time steps 1.505
1.085 1760
X: 1855 Y: 1.087
1780
1800
1820
1840
1860
Number of time steps
Re=200
Nu
X: 5035 Y: 1.502 X: 5045 Y: 1.5
1.5 X: 5055 Y: 1.498
1.495 4950
5000
5050
Number of time steps Fig. 17. Time instance within half a period when the system reaches periodic steady flow conditions for no-fin case at Reynolds number of 10, 100 and 200 for excitation Strouhal number of 0.5.
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(a)
(b) (c)
(d)
(e) (f)
(g)
(h) (i)
Fig. 18. Streamlines for no-fin case at the time instances according to Fig. 16 at Reynolds number of 10 (first box), 100 (second box) and 200 (third box), St = 0.5.
according to Fig. 19 are shown in Fig. 20 at Reynolds number of 10, 100 and 200 for the case with fin length of Lf = H. The first and last value of Fig. 19 represent the minimum and maximum values of Nusselt numbers, respectively. At the onset of the cycle when the Nusselt number is minimum, a large recirculation zone is exhibited behind the fin and increases in size and strength with increasing Reynolds number. At Reynolds number of 200, during the acceleration phase, the recirculation zone appeared on the bottom wall downstream of the step also gets larger in size. These plots show that a considerable change occurs in the flow field during the pulsation cycle when the fin is installed on the top wall. Isotherm plots for the three time instances within half a period when the system reaches the periodic steady flow condition according to Figs. 17 and 19 are shown in Figs. 21 and 22. for St = 0.5 at Reynolds number of 10, 100 and 200 without fin case
Re=100 X: 1960 Y: 1.406
X: 3260 Y: 0.2313
0.24
1.4
0.23
Nu
Nu
X: 3270 Y: 0.2487
Re=10
0.25
and with fin length of Lf = H. From the plots, it is seen that increasing the Reynolds number decreases the thermal boundary layer along the bottom wall downstream of the step and hence increases the heat transfer rate. There is considerable change for shape of the isotherms in the vicinity of the fin especially at Reynolds number of 100 and 200. In this case, the isotherm plots at Reynolds number of 10 show that, only a small portion of the isotherms in the vicinity of the fin distorted. When the flow motion increases, more distortion of the isotherms from the fin tip at the bottom to the downstream of the step and also more distortion form the upper wall towards the fin is seen. Time-spatial averaged Nusselt number along the bottom wall downstream of the step versus excitation Strouhal number plots are shown in Fig. 23 for various fin lengths at Reynolds number of 10, 100 and 200. In these plots, the vertical axis is normalized
0.22
X: 3245 Y: 0.208
0.21 3180
X: 1945 Y: 1.327
1.35
X: 1950 Y: 1.367
1.3
3220
3280
1800
1850
1900
1950
Number of time step
Number of time step
Re=200
Nu
2.2
X: 8030 Y: 2.213
2.1
X: 8025 Y: 2.124 X: 8020 Y: 1.977
2 7950
8000
8050
Number of time step Fig. 19. Time instance within half a period when the system reaches periodic steady periodic flow condition for fin length of Lf = H at Reynolds number of 10, 100 and 200 for excitation Strouhal number of 0.5.
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(a)
(b) (c)
(d)
(e) (f)
(g)
(h) (i)
Fig. 20. Streamlines for fin length of Lf = H at the time instances according to Fig. 18 at Reynolds number of 10 (first box), 100 (second box) and 200 (third box), St = 0.5.
(a)
(b) (c)
(d)
(e) (f)
(g)
(h) (i)
Fig. 21. Streamlines for no-fin case at the time instances according to Fig. 16 at Reynolds number of 10 (first box), 100 (second box) and 200 (third box), St = 0.5.
with spatial averaged Nusselt number in the steady flow case. In these plots, the normalized Nusselt number decreases with increasing frequency after St = 0.1. At Reynolds number of 10, there is negligible change in the normalized Nusselt number with increasing the excitation Strouhal number. For fin lengths of Lf = 0.5H and Lf = H, normalized Nusselt number values at Reynolds number of 100 and 200 become closer with increasing frequencies. According to Fig. 8, the fin length on the length-averaged Nusselt values in the steady flow case is more effective with increasing Reynolds number. The normalized Nusselt number versus excitation Strouhal number plot as in Fig. 23 shows a resonant type
behavior as similar in [9], the value first increases with an increase in the excitation Strouhal number (up to St = 0.05), then a decrease is seen with the increasing values of the frequency of the pulsation for the case without fin. Adding a fin shifts the maximum value of the normalized Nusselt number from 0.05 to 0.1. This may be due to the influence of the fin on the flow movement and its effect on the adaption time of the system to new flow condition in the pulsating flow case. A measure for the heat transfer enhancement for pulsating flow in comparison to the steady flow with different fin lengths and with no-fin length case can be given as,
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(a)
(b) (c)
(d)
(e) (f)
(g)
(h) (i)
Fig. 22. Isotherms for fin length of Lf = H at the time instances according to Fig. 16 at Reynolds number of 10 (first box), 100 (second box) and 200 (third box), St = 0.5.
Re=10 Re=100 Re=200
1.2 1.1 1 0.9 10
−2
10
−1
10
Num / Nus
Num / Nus
(b) L f=0.5H
(a) L f=0.0H
1.3
Re=10 Re=100 Re=200
1.1 1 0.9
0
10
−2
(d) L f=1.5H
1
10
−1
10
Num / Nus
s
Num / Nu
1.1
−2
10
(c) L =1.0H Re=10 Re=100 Re=200
10
−1
St
f
0.9
10
St
Re=10 Re=100 Re=200
1.1
1
0.9
0
0
10
−2
St
10
−1
10
0
St
Fig. 23. Variation of normalized Nusselt number (time and spatial-average Nusselt number in pulsating flow divided by the average in steady flow) for no-fin case and different fin lengths with excitation Strouhal number at Reynolds number of 10, 100 and 200.
HTEw;f ¼
Num Nus
Nuwm ; Nuws
ð12Þ
where subindex m, s, wm and ws denote the spatial-time averaged, steady flow case, spatial-time averaged without fin case and steady flow case without fin case, respectively. This will indicate the combined effect of pulsating flow and installation of the fin on the heat transfer augmentation. As stated earlier, in the steady flow case, adding a fin will increase the HTE in the direction of increasing Reynolds number and increasing fin length. In the pulsating flow case, as seen in Fig. 24, HTEw,f value is less than one (close to one at Re = 10) indicating that installation of a thin fin on the top wall is not advantageous in the pulsating flow case. In some cases (mostly at Re = 200), the values are even below 1. The least HTEw,f values are
obtained for the case at the excitation Strouhal number of 0.25 and at Reynolds number of 200 all considered fin lengths. The effects of the distance of the fin from the sudden expansion may have influences on the flow field and heat transfer characteristics, but it is out of scope of this paper. 4. Conclusions In this study, laminar pulsating flow over a backward facing step with an upper wall mounted adiabatic thin fin is numerically studied. The effects of various pertinent parameters, Reynolds number, fin length and pulsating frequency on the fluid flow and heat transfer characteristics are numerically investigated. The
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(a) L =0.5H
(b) L f=1.0H 1.1
1.05
1.05
HTEw,f
HTEw,f
f
1.1
1 0.95
1 0.95
0.9 10
−2
10
−1
10
0.9
0
10
−2
10
−1
10
0
St
St
(c) L f=1.5H 1.1
HTEw,f
Re=10 Re=100
1
Re=200 0.9 0.8 10
−2
10
−1
10
0
St Fig. 24. Variation of heat transfer enhancement for pulsating flow in comparison to the steady flow with different fin lengths and with no-fin length case (HTEw,f), with excitation Strouhal number at Reynolds number of 10, 100 and 200.
combined effects of the pulsation and installation of the fin on the heat transfer are investigated. Following results are obtained: In the steady case, fin redirects the flow movement and streamline plots show that length and intensity of the recirculation zone behind the step is considerably effected with the installation of an adiabatic thin fin on the upper wall of the channel. Adding an adiabatic fin alters the isotherm plots. Comparing the isotherms for the case with the fin with those for the case without fin, 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. For the steady flow conditions, Heat Transfer Enhancement (HTE) is increasing with an increase in the Reynolds number and fin length. At Reynolds number of 200 and fin length of Lf = 1.5H, heat transfer enhancement is 188 percent for the bottom hot wall as compared to the no-fin case. For the pulsating flow case, with increasing values of Reynolds number and increasing value of fin length, level of nonlinearity (distortion from a pure sinusoid) in the temporal variation of the Nusselt number increases. For St = 0.1, Normalized averaged Nusselt number increases almost linearly with increasing values of pulsating velocity amplitudes at Re = 100 and Re = 200 and at Re = 10, there is negligible change in heat transfer enhancement with respect to steady flow value with the change in velocity amplitude. For St = 0.1, The averaged pressure drop increases with the increasing values of fin length and the averaged pressure drop value is more sensitive with respect to a change in the velocity amplitude for the case at fin length of Lf = 1.5H. The normalized Nusselt number (normalized with respect to steady flow spatial averaged-Nusselt value) versus excitation Strouhal number shows a resonant type behavior. The normalized Nusselt value first increases with an increase in the excitation Strouhal number (up to St = 0.05), then a decrease is seen with the increasing values of the frequency of the pulsation for the case without fin. Adding a fin shifts the maximum value of the normalized Nusselt number from St = 0.05 to St = 0.1. Adding a fin is not advantageous for heat transfer enhancement in pulsating flow compared to steady flow with no-fin case.
The least HTEw,f values are obtained for the case at the excitation Strouhal number of 0.25 and at Reynolds number of 200 for the fin lengths considered.
Acknowledgment The authors are very grateful to the reviewers due their appropriate and constructive suggestions as well as their proposed corrections, which helped us improve the quality of the paper.
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