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Effect of Reynolds number on flow and heat transfer in incompressible forced convection over a 3D backward-facing step
Accepted Manuscript Title: Effect of reynolds number on flow and heat transfer in incompressible forced convection over a 3d backward-facing step Author: J.H. Xu, S. Zou, K. Inaoka, G.N. Xi PII: DOI: Reference:
S0140-7007(17)30148-2 http://dx.doi.org/doi: 10.1016/j.ijrefrig.2017.04.012 JIJR 3613
To appear in:
International Journal of Refrigeration
Received date: Revised date: Accepted date:
15-12-2016 11-4-2017 11-4-2017
Please cite this article as: J.H. Xu, S. Zou, K. Inaoka, G.N. Xi, Effect of reynolds number on flow and heat transfer in incompressible forced convection over a 3d backward-facing step, International Journal of Refrigeration (2017), http://dx.doi.org/doi: 10.1016/j.ijrefrig.2017.04.012. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Effect of Reynolds Number on Flow and Heat Transfer in Incompressible Forced Convection over a 3D Backward-Facing Step
J.H. Xua, S. Zoua, K. Inaokab, G.N. Xia,* a
b
School of Mechanical Engineering, Nantong University, Nantong 226019, China
School of Mechanical Engineering, Doshisha University, Kyotanabe 610-0321, Japan * Corresponding author. Nantong University, 9 Se Yuan Rd., Nantong 226019, China. Tel:+8618762850738. Email address: [email protected] (G.N. Xi).
Highlights
Periodic instability on heat transfer are clarified.
The heat transfer is greatly enhanced with the increase of Re.
Taking away the hot fluid and bringing cold fluid are identified.
Abstract A three-dimensional incompressible numerical model for the case of the 3D backward-facing step flow is established to investigate the characteristics of fluid flow and heat transfer in the low and middle Reynolds number ranges (200≤Re≤1400). The governing equations, including continuous, unsteady Navier-Stokes and energy equations, are solved by the finite volume method in FLUENT. The simulation results show that the time averaged reattachment length reaches the peak value at Re = 1000, and subsequently decreases as the increase of Re. The formation of secondary peak Nu influenced by flow instability has a better contribution to the heat transfer at the center area. Taking away the hot fluid and carrying the cold fluid into the floor wall caused by periodic instability has positive effects on heat transfer enhancement.
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Keywords: 3D backward-facing step; forced convection; heat transfer; periodic instability
Nomenclature A
surface area of the floor wall, [m2]
AR
aspect ratio, AR W / S
c
Courant number
C
f
skin friction coefficient, C f
2 w
u0
2
1
Cf
time averaged friction coefficient, C f
C
time-spatial averaged friction coefficient, C fs
fs
Cp
specific heat capacity, [J kg-1 K-1]
ER
expansion ratio, ER H / h
f
frequency, [Hz]
Gr
Grashof number
h
inlet channel height, [m]
H
exit channel height, [m]
j
Colburn’s j-factor, j
Nu
local Nusselt number, Nu
Nu
time averaged Nusselt number, Nu
0
C f dt
1 A
A
0
C f d
Nu s Re Pr
1/ 3
qwS
T w T 0 1
Nu dt 0
Nu s
time-spatial averaged Nusselt number, Nu s
Pr
Prandtl number
1 A
A
Nu d
0
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P
pressure, [Pa]
qw
wall heat flux, [W·m-2]
Re
Reynolds number, R e R e D
Re S
Reynolds number, R e S
Reg
grid Reynolds number
S
step height, [m]
t
time, [s]
T
temperature, [K]
T
time-averaged temperature, [K]
u0
averaged velocity component in the x direction, [m s-1]
u, v, w
velocity components in x, y and z directions, [m s-1]
u
streamwise fluctuating velocity, [m s-1]
v
transverse fluctuating velocity, [m s-1]
- u v
shear stress, [m2 s-2]
v
turbulent heat flux, [m s-1 K]
W
channel width, [m]
XR
time averaged reattachment length, [m]
x
streamwise coordinate, [m]
y
transverse coordinate, [m]
z
spanwise coordinate, [m]
2 u0h
u0S
Greek letters
unit area, [m2]
fluctuating temperature, [K]
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thermal conductivity, [W m-1 K-1]
dynamic viscosity, [kg m-1 s-1]
kinematic viscosity, [m2 s-1]
density, [kg m-3]
sampling time, [s]
w
wall shear stress, w u y w y 2
2
Subscripts max
maximum
0
inlet value
w
wall value
x, y, z
values along x, y, and z coordinates
1. Introduction Flow separation and reattachment of backward-facing step caused by abrupt expansion are well applied into many heat exchanging devices, such as energy system equipment, turbine blades, dump combustors, and many other heat-exchanging devices. For the sake of improving the performance of these applications, investigating the mechanism of flow and heat transfer enhancement is very crucial. The benchmark problem of laminar and turbulent flow has been extensively studied by experimental (Armaly et al., 1983; Lee et al., 1998; Terhaar et al., 2010; Kapiris et al., 2014) and numerical (Vogel and Eaton, 1985; Moser et al., 1999; Kozel et al., 2005; Barrios et al., 2012; Zhong et al., 2014) methods. The experimental and numerical research carried out by Armaly et al. (1983) on the three-dimensional backward-facing step flow has arisen much more attention. The laminar (ReD＜1200),the transitional (1200＜ReD＜6600 ), and the turbulent (ReD＞6600) regimes of the flow are
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identified by the primary reattachment length based on the experimental research with LaserDoppler measurements. Since then, a lot of three-dimensional laminar flow investigations (Armaly et al., 2002; Nie and Armaly, 2003; Nie and Armaly, 2004; Chen et al., 2006; Nie et al., 2009) have been published. The effects of side walls caused much attention in fluid dynamics community for many numerical and laboratory experiments. Williams and baker (1997) did the research on the three-dimensionality with the effects of side walls and found that the complex jet-like flow was located at the step wall near the sidewalls. The complex flow structure near the side walls were also reported by Chiang and Sheu (1999). In their work, the limiting streamlines near roof, floor, and end walls at Re = 800 were investigated to illustrate the spanwise width (2h≤ B ≤100h) effects on the flow field. Tylli et al. (2002) investigated the effects on the primary and secondary recirculation zones, and explained the differences between experimental and 2D simulations in the transitional and turbulent regimes. Biswas et al. (2004) reported the spatial evolution and three-dimensionality of jetlike flows with different expansion ratios (ER=1.9423, 2.5, 3.0). The studies on heat transfer characteristics of backward-facing step flow were focus on laminar and turbulent flow. Iwai et al. (2000) investigated the effects of the duct aspect ratio with a fixed step height in laminar regime (125≤ ReS ≤375). The results revealed the 2D region can be obtained with an aspect ratio of at least 16. The maximum Nusselt number appeared near the side walls increased with the increasing of aspect ratio. The effects of step height on distribution of Nu, Cf and flow structures at ReD = 343 were studied by Nie and Armaly (2002). The results showed that the maximum Nusselt number increased as the step height increased. He also found that the three-dimensional behavior and sidewall effects increased with increasing step height. Kitoh et al. (2007) studied the time mean reattachment length and the distribution of Nu and Cf on the floor wall at unsteady flow state with the effects of expansion ratios by the methodology of DNS. In his work, the flow structure in the
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separated and reattachment regions was clearly shown at 300≤ ReS ≤1000. Khanafer et al. (2008) studied numerically the laminar mixed convection heat transfer with the effect of pulsating flow over a backward-facing step. The results illustrated that the flow structure, heat transfer and the average wall friction coefficient were both affected by Reynolds number, Richardson number and the dimensionless oscillation frequency. Lan et al. (2009) reported the simulations of three-dimensional turbulent flow over a backward-facing step. The - - - f turbulent model was used to improve effectively the heat transfer predictions in the high Reynolds number range of 20000-50000. The results showed that the increasing of Reynolds number or aspect ratio leads to the higher bulk Nusselt number. Xie and Xi (2017) numerically investigated the heat transfer of separation and reattachment flow over a 2D backward-facing step by DNS. The results illustrated that the isotherms were more unsteady as the increase of Reynolds number and the size of primary recirculation zone increased with the decrease of expansion ratio. The combined effects of flow and heat transfer instability resulted in the dissimilarity between Nu and Cf at ReS = 1000. In the transitional flow regime, Chiang et al. (1999) numerically investigated the vortical evolution in a 2D backward-facing step. He clarified how the recirculating bubble containing flow reversals is torn into smaller eddies and discussed the eddy distortion and the merging of eddies in his study. Flow structures in a 3D backward-facing step channel was careful investigated by Rani et al. (2007) to indicate that the flow becomes unsteady along with effects of the Kelvin-Helmholtz instability and Taylor-Görtler-Like vortices at ReD=1000 and 2000. The backward-facing step flow in the late transitional regime was investigated based on DNS by Schäfer et al. (2009). The calculated results indicated that the oscillations of the primary reattachment and the upper wall separation zones are dominated by bubble structures in the late transitional regime (ReD = 6600). Tihon et al. (2012) experimentally and numerically studied the transitional flow (water) with different expansion ratios and inlet flow
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conditions at moderate Reynolds numbers. The results suggested that the inlet pulsatile flow plays significant roles on flow structures, reattachment length and upper wall recirculation zones. Xie and Xi (2017) mainly examined heat transfer enhancement with the effects of the vortical structure in the transitional flow over a 2D backward-facing step. Periodic instability of vortical structure destroyed the thermal boundary layer and the heat transfer was greatly enhanced at the bottom wall, particularly in the range 6≤ x/H ≤14. It is worth noting that there are a lot of laminar and turbulent investigations on the flow structures and reattachment in 2D and 3D backward-facing step flow but little attention on heat transfer characteristics has been done in transitional backward-facing step flow. Further more, the mechanisms leading to heat transfer enhancement in transitional regime were rarely investigated in a 3D backward-facing step channel. The aim of this paper is to study the heat transfer enhancement of 3D backward-facing step in transitional regime. In particular, the heat transfer enhancement influenced by periodic instability are mainly discussed. 2. Numerical methodology 2.1 Governing equations In this paper, the value of Gr/Re2 is lower than 0.1 at Re = 1000, so that the buoyancy effect can be ignored. The governing equations including the continuity, Navier-Stokes and energy equations for three dimensional, transitional, incompressible and unsteady case are written as follows: ui xi
u i t
T t
0
u iu j x j
u jT x j
(1)
1 P
xi
2T 2 C p x j
u i u j x j x j xi
(2)
(3)
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Where the Cartesian velocity components are denoted ui, (i, j = 1, 2, 3 for x, y, z, respectively), xi is coordinates components, is density, is kinematic viscosity, P is pressure, Cp is specific heat, λ is thermal conductivity, T is temperature, t is time. In this study, the physical properties of fluid (air) are assumed as constants, that is, ρ = 1.247 kg·m-3,
= 1.76×10-5 m2·s-1, Cp = 1005 J·kg-1·K-1, λ = 0.0251 W·m-1·K-1 and Pr = 0.705. 2.2 Computational domain and boundary conditions The computational domain is presented in Fig. 1. The step height S is 0.015m, channel height H is 0.030m and channel’s width W is = 0.24m in the geometry. The duct expansion ratio, ER = H/h, is set as 2. The step aspect ratio is defined by AR = W/S = 16. The channel length downstream of the step is chosen as Ld = 50S so that the flow can be treated as fully developed at exit plane. The boundary conditions in this study can be expressed as follows:
The inlet flow is assumed as laminar, fully developed and isothermal. The temperature of incoming flow is maintained at uniform temperature, Tin = 283K. The transverse velocity (v) and spanwise velocity (w) are both set as 0. The distribution of the streamwise velocity component (u) is considered to be the approximation by Shah and London (1978). Therefore, the distribution of u is summarized: n m y S H S 2 z m 1 n 1 1 1 u 0 m n H S 2 W 2
u
H S W *
m 1 . 7 0 . 5 ( ) *
(4)
(5) 1 .4
(6)
1 2 3 n (7) 1 1 1 2 0 . 3 3 3 ２
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The outflow boundary condition, at the channel exit, assumes fully developed and the streamwise gradient of flow variables is 0, that is,
u
v w T 0， 0， 0， 0 . x x x x
All wall surfaces meet the no-slip condition. The downstream floor wall of the backwardfacing step is heated at uniform temperature, Tw = 313K. Meanwhile, the rest of the walls are treated to be adiabatic.
2.3 Numerical solution procedure All these simulations are calculated by FLUENT. The finite volume method is used for the spatial discretization of Eqs. (1)–(3). The convection terms in the unsteady Navier-Stokes equations are evaluated using QUICK scheme. Energy equation is solved with second-order upwind difference scheme SIMPLE algorithm, proposed by Patankar and Spalding (1972), is utilized for the coupling of pressure and velocity. The initial conditions of velocity components and temperature are set as u = u0, v = 0,w = 0, and Tin = 283K. Max iterations per time step is set to be 5. The CPU (eight-core processor) is Intel Xeon E5-2687W 0 @ 3.10GHz and the RAM is 32.0 GB. Also, the simulations are carried out in a serial process. All calculations are performed on the dell station and the CPU time for Re = 1000 is around 20 days. The mesh is generated by using preprocessor ICEM. Fig. 2(a) exhibits the nonuniform Cartesian grid structure of the whole computational domain. As shown in the Fig. 2(b), the magnified view to the z-axis, the minimum grid spacing near the all walls is set to ensure the accuracy of the calculation results. 2.4 Grid independence The density of grid plays an important role on the calculation results. Also, the grid refinement is made in the regions around walls, where the velocity gradient is high. Numerical calculation accuracy is reduced with sparse grid, resulting in the obvious deviation from the true solution. On the contrary, the superfine grid consumes much more memory and
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CPU time. Therefore, the design of grid plays an important role on the calculated accuracy and consuming time. Grid Reynolds number (Reg), defining the minimum grid spacing ( x min ) as the reference length, reflects the grid density. The expression is used: Re g
u 0 x min
(8)
Grid independence study is carried out with three different cases. Fig. 3 shows the time averaged Nusselt number ( Nu x ) and time averaged streamwise friction coefficient ( C fx ) along the center line for different Reg, respectively. The CPU time of Reg = 8, Reg = 10, and Reg = 12 is around 25 days, 14 days, and 6 days, respectively. It can be clearly seen that the values of Nu x and C fx , in the range 0≤ x/S ≤12, are in good agreement. On the other hand, in the range 12≤ x/S ≤50, the small deviation exists. The deviation is calculated based on the data of Reg = 12, the minimum deviation of these values is about 0.2% while the maximum deviation is about 1%, meeting the requirement of the grid independence. Therefore, the value of Reg in this simulations is set equal to 10. In Eqs. (9), the time step ( t ) between two successive iterations is set to get a Courant number (c) less than 1, evaluated with the smallest grid spacing ( x min ), and the maximum velocity (umax= 1.5u0). c
um a t x xm
1
(9)
i n
2.5 Validation of the model In order to validate the reliability of numerical results, the time averaged reattachment length, compared with other credible results, is shown in Fig. 4. The present results of 2D simulations show excellent agreement with numerical results published by Iwai et al. (2000)
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and Sugawara et al. (2005). In 3D simulations, for a Reynolds number range 100≤ Re ≤ 800, the results of reattachment length obtained from this study are in excellent agreement with the results published by Iwai et al. (2000) and Armaly et al. (1983). However, in a Reynolds number range 800＜ Re ≤1400, the time averaged reattachment length is greatly influenced by aspect ratios. Fig. 5 can show that local Nusselt number Nu and streamwise friction coefficient Cfx along the center line at Re = 500. These results of the 3D simulation are in perfect agreement with that of Iwai et al. (2000). As a result, the accuracy and reliability of present simulations are confirmed. 3. Results and discussion 3.1 Flow properties The entire limiting streamlines plotted on the floor, roof, and side walls of the channel can be clearly observed from Fig.6 to analyze the flow structure in the separated and reattachment regions. The reattachment lines of the primary recirculation zone along the floor wall are obvious shown at Re = 600, 800, 1000, and 1200. It is found from this picture that the reattachment lines are the approximate parabolic. The reattachment flow region increases and moves further downstream up to Re = 1000, but decreases and moves upstream at Reynolds number range 1000≤ Re ≤1200. On the roof wall, two symmetrical eddies formed from the side walls extend all the way to channel center plane as the increase of the Re. Due to the effects of flow instability, two symmetrical eddies along the roof wall are destroyed at Re = 1200. The flow is steady up to Re = 800, but becomes gradually unsteady along with the apparent three-dimensionality with the increase of Re. 3.2 Time-spatial averaged heat transfer In Fig. 7, the averaged heat transfer coefficient is plotted in the form of j-factor against Reynolds number. Overall, the values of j-factor decreases with the increase of Re. In the
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Reynolds number range, 200≤ Re ≤1000, the slope of the plotted data is close to that of the Re-0.5. This manifests that the heat transfer is dominated by the steady, flat-plate boundary layer type flow. However, in the Reynolds number range 1000≤ Re ≤1400, the Reynolds number dependency of the j-factor begins to show smaller slope compared with that of Re-0.5. This suggests that the heat transfer is dominated by the unsteady flow. Therefore, the heat transfer coefficient becomes higher than the values in the laminar range. It is thus reasonable to conclude that the unsteady flow at higher Reynolds number range is likely to be the cause of the obvious heat transfer enhancement on the floor wall of the backward-facing step. As is shown in Fig. 8, time-spatial averaged Nusselt number ( Nu s ) and time-spatial averaged friction coefficient ( C fs ) are plotted against the Re from 200 to 1400. With the increase of the Reynolds number, the value of time-spatial averaged Nusselt number increases with the linear rule from 0.847 to 2.179, but the value of time-spatial averaged friction coefficient decreases rapidly in laminar flow, and then tends to be steady in transitional flow. According to the tendency of the two evaluation parameters, the heat transfer performance on the floor wall is enhanced with the increase of Re. 3.3 Time averaged heat transfer The contours of time averaged (calculated with 20000 time steps) Nusselt number ( Nu x , z ) along with the xu-line on the floor wall are illustrated in Fig. 9. The dashdotted line (xu-line) on which represents the shear stress on the floor wall ( u / y
y0
) is 0. The black
dots in each of Fig. 9 represent the maximum of time averaged Nusselt number ( Nu max ). According to the distributions of xu-line, it is also found that the reattachment flow region increases and moves further downstream up to Re = 1000, but decreases and moves upstream at Re = 1200. It is common that the location of Nu max appear symmetrically near the two
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side walls. The locations of Nu max can be seen in Table 1. As the Re increases, streamwise location (xmax /S) of Nu max moves downstream from the step, but the spanwise location (zmax/S) of Nu max has no obvious change. In order to intuitively compare with the variation of Nu x , z at different Re, three-dimensional contours of Nu x , z (calculated with 20000 time steps) are shown in Fig. 10 with bird’s eye sight. At Re =1000, 1200, Nu at the center area is obvious enhanced. Therefore, the fluid flow and heat transfer at the center area are studied in the following section. 3.4 Periodic instability The flow becomes gradually unsteady in a Reynolds number range, 1000≤ Re ≤1400. Two monitor points P1 (x/S = 10, y/S = 1, z/S = 0) and P2 (x/S = 40, y/S = 1, z/S = 0) are set to monitor the instantaneous transverse velocity (v) in the flow field. Power spectral density of v can be obtained based on the v data of 20000 time steps by the method of fast Fourier transform (FFT). Fig. 11 shows the Instantaneous transverse velocity and power spectral density of v at Re =1000, 1200, 1400. It is very clear that the variation of v changes with obvious rule and the periodic characteristic of v at Re = 1000 is stable (f1 = f2 = 4.58) at points both P1 and P2. Sheu and Rani (2006) numerically found that the fundamental frequency of the variation is f =4.77, which is close to the value f1 = 4.58. As the Re is increased, in point P1, the fundamental frequency is also increased. On the other hand, with the increase of Re, the periodic regime lost in point P2, located far away from the inlet. In order to discuss the effects of periodic instability on heat transfer enhancement near the center plane, we mainly analyze the flow field and heat transfer on the floor wall at Re = 1000. The instantaneous contours of Nu distribution along with the xu-line ( u / y
y0
) on the
floor wall at Re = 1000 can be clearly seen from Fig. 12. In a cycle, it is clear that the position
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and formation of the xu-line are changing periodically at the center area. In contrast, the position of xu-line without any change can be seen near the side walls. In Fig. 13, it can be concluded that intensity of instantaneous streamwise velocity variation at center plane (z/S=0) is higher than the other spanwise planes. Therefore, flow instability at center plane is discussed in the following parts. Fig. 14 shows the streamline and vector plots of the center plane at Re = 1000 in a cycle. It can also be observed that the Kelvin-Helmholtz instability occurs in the shear layer of the primary recirculation zone. Importantly, the vortexes shed constantly and periodically has positive effects on the flow instability in the region downstream from the primary recirculation zone. As a result, in that region, the thermal boundary layer are destructed with the effect of flow instability. Fig. 15 shows that Nu distribution at the center line on the floor wall at Re = 1000. The first peak of Nu is caused by the primary recirculation zone. The occurrence of secondary peak of Nu influenced by flow instability has a great influence on the heat transfer enhancement. The mechanisms resulting in the secondary peak of Nu are studied in detail. The instantaneous mapping of fluctuating velocity and temperature at center plane is plotted in Fig. 16, two fluctuating velocity components of u and v and fluctuating temperature is defined by: u u u
(10)
v v v
(11) T T
(12)
where u and v are the time averaged streamwise velocity and transverse velocity, T is time averaged temperature.
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From the qualitative analysis, due to this periodic vortexes motion, as shown in Fig. 16, the vortexes carry the cold ( ＜ 0 ) fluid into the floor wall, which has positive effects on the heat transfer. On the other hand, the vortexes carry the hot ( ＞ 0 ) fluid away from the floor wall, which also has positive effects on the heat transfer. The red arrows in this picture represent the position of Numax, where is located at the cold fluid. Overall, cold fluid moves into the floor wall and hot fluid moves away from the floor wall has positive effects on heat transfer. Discussion of the heat transfer mechanism is based on this picture. From the quantitative analysis, one monitor point P3 (x/S = 32, y/S = 0.1, z/S = 0) is set to get the time variation of fluctuating velocity ( u , v ) and temperature ( ). Fig. 17 illustrates time variation pattern of the fluctuating velocity field in five cycles. It can be seen from the picture that the number density of the plotted points in u - v plane is mainly in the first and third quadrants compared with those in other two quadrants. Table 2 shows the number of the data sets assigned into each quadrant and the frictional contributions to - u v and v
from each quadrant of u - v plane. - u v represents the fluctuating velocity
cross-corelation and v is the cross-corelation of fluctuating velocity and temperature. It is found that the fractional contributions to - u v and v in the first and third quadrants are much larger than that of the other quadrants. So, there are two main effects on heat transfer. Firstly, hot ( ＞ 0 ) fluid in the first quadrant and the cold ( ＜ 0 ) fluid in the third quadrant contribute positively to v . Secondly, cold ( ＜ 0 ) fluid in the first quadrant and hot ( ＞ 0 ) fluid in the third quadrant contribute negatively to v . In general, the first effect occupies the dominant role by the comparison. Through the analysis of statistics, it is concluded that taking away the hot fluid and carrying the cold fluid into the floor wall lead to the formation of the secondary peak of Nu. Thus, the heat transfer on the floor wall is greatly enhanced. 4. Conclusions
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A three-dimensional incompressible numerical study is conducted to clarify the heat transfer mechanism. The governing equations with boundary conditions are solved in FLUENT. The main results calculated to study the effects of various Reynolds numbers are as follows:
With the increase of the Re, the value of time-spatial averaged Nusselt number increases accompanying with the decrease of the time-spatial averaged friction coefficient. It is very clear that the heat transfer is constantly enhanced with the increase of the Re by the j-factor.
The reattachment flow region increases and moves further downstream up to Re = 1000, but decreases and moves upstream with increase of Re.
The first peak of Nu is caused by the primary recirculation zone. The formation of secondary peak of Nu influenced by flow instability has a great influence on the heat transfer enhancement.
Taking away the hot fluid and carrying the cold fluid into the floor wall caused by periodic instability has positive effects on heat transfer enhancement.
Acknowledgements This Project is supported by the National Natural Science Foundation of China, No.51476080.
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step flow. Int. J. Heat Mass Transf. 47, 4713–4720. Chen, Y.T., Nie, J.H., Hsieh, H.T., Sun, L.J., 2006. Three-dimensional convection flow adjacent to inclined backward-facing step. Int. J. Heat Mass Transf. 49, 4795–4803. Nie, J.H., Chen, Y.T., Hsieh, H.T., 2009. Effects of a baffle on separated convection flow adjacent to backward-facing step. Int. J. Therm. Sci. 48, 618–625. Williams, P.T., Baker, A. J., 1997. Numerical simulations of laminar flow over a 3D backward-facing step. Int. J. Numer. Methods Fluids 24, 1159–1183. Chiang, T.P., Sheu, T.W.H., Fang, C.C., 1999. Numerical investigation of vortical evolution in a backward- facing step expansion. Appl. Math. Modeling. 23, 915–932. Tylli, N., Kaiktsis, L., Ineichen, B., 2002. Sidewall effects in flow over a backward-facing step: Experiments and numerical simulations. Phys. Fluids. 14, 3835–3845. Biswas, G. Breuer, M., Durst, F., 2004. Backward-Facing Step Flows for Various Expansion Ratios at Low and Moderate Reynolds Numbers. J. Fluids Eng. 126, 362–374. Iwai, H., Nakabe, K., Suzuki, K., 2000. Flow and heat transfer characteristics of backwardfacing step laminar flow in a rectangular duct. Int. J. Heat Mass Transf. 43, 457–471. Nie, J.H., Armaly, B.F., 2002. Three-dimensional convective flow adjacent to backwardfacing step - Effects of step height. Int. J. Heat Mass Transf. 45, 2431–2438. Kitoh, A., Sugawara, K., Yoshikawa, H., Ota, T., 2007. Expansion Ratio Effects on ThreeDimensional Separated Flow and Heat Transfer Around Backward-Facing Steps. J. Heat Transf. 129, 1141–1155. Khanafer, K., Al-azmi, B., Al-shammari, A., Pop, I., 2008. International Journal of Heat and Mass Transfer Mixed convection analysis of laminar pulsating flow and heat transfer over a backward-facing step. Int. J. Heat Mass Transf. 51, 5785–5793.
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Lan, H., Armaly, B.F., Drallmeier, J.A., 2009. International Journal of Heat and Mass Transfer Three-dimensional simulation of turbulent forced convection in a duct with backward-facing step. Int. J. Heat Mass Transf. 52, 1690–1700. Xie, W.A., Xi, G.N., 2017. Fluid flow and heat transfer characteristics of separation and reattachment flow over a backward-facing step. Int. J. Refrigeration. 74, 177–189. Chiang, T., Sheu, T., 1999. A numerical revisit of backward-facing step flow problem. Phys. fluids. 11, 862–874. Rani, H.P., Sheu, T.W.H., Tsai, E.S.F., 2007. Eddy structures in a transitional backwardfacing step flow. J. Fluid Mech. 588, 43–58. Schäfer, F., Breuer, M., Durst, F., 2009. The dynamics of the transitional flow over a backward-facing step. J. Fluid Mech. 623, 85–119. Tihon, J., Pěnkavová, V., Havlica, J., Šimčík, M., 2012. The transitional backward-facing step flow in a water channel with variable expansion geometry. Exp. Therm. Fluid Sci. 40, 112–125. Xie, W.A., Xi, G.N., Zhong, M.B., 2017. Effect of the vortical structure on heat transfer in the transitional flow over a backward-facing step. Int. J. Refrigeration. 74, 465–474. Shah, R.K., London, A.L., 1978. Laminar Flow Forced Convection in Ducts: A Source Book for Compact Heat Exchanger Analytical Data. Adv. Heat Transf. Patankar, S. V., Spalding, D.B., 1972. A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Int. J. Heat Mass Transf. 15, 1787–1806. Sugawara, K., Kaihara, E., Yoshikawa, H., Ota, T., 2005. DNS of Three-Dimensional Unsteady Separated Flow and Heat Transfer Around a Downward Step. ASME Summer Heat Transfer Conference. 3, 17–22. Sheu, T.W.H., Rani, H.P., 2006. Exploration of vortex dynamics for transitional flows in a three-dimensional backward-facing step channel. J. Fluid Mech. 550, 61–83.
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Fig.1 Schematic of the computational domain
(a)
(b)
Fig. 2 The whole grid and the magnified view of z-plane
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(a)
(b)
Fig. 3 Time averaged Nu and Cf along the center line at Re = 600
Fig. 4 Comparison of time averaged reattachment length
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(a)
(b)
Fig. 5 Validation of local Nusselt number Nu and streamwise friction coefficient Cfx along the center line at Re = 500
(a) Re=600
(b) Re=800
(c) Re=1000
(d) Re=1200
Fig. 6 Instantaneous limiting streamlines on the three walls
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Fig. 7 j-factor as a function of Re
Fig. 8 Time-spatial averaged Nusselt number and friction coefficient
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(a) Re=600
(b) Re=800
(c) Re=1000
(d) Re=1200
Fig. 9 Contours of time averaged Nu x , z distribution along the xu-line on the floor wall
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(a) Re=600
(c) Re=1000
(b) Re=800
(d) Re=1200
Fig. 10 3D contours of time averaged Nu x , z distribution on the floor wall
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Fig. 11 Instantaneous transverse velocity and power spectral density of v P1 (x/S = 10, y/S = 1, z/S = 0), P2 (x/S = 40, y/S = 1, z/S = 0)
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wt = 0
wt = 1/2π
wt = 1/8π
wt = 5/8π
wt = 1/4π
wt = 3/4π
wt = 3/8π
wt = 7/8π
Fig. 12 Instantaneous Nu distribution with xu-line on the floor wall at Re = 1000
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Fig. 13 Instantaneous streamwise velocity contours at different z-planes at Re = 1000
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Fig. 14 Streamlines and vector at the center plane at Re =1000
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First peak
Secondary peak
Fig. 15 Nu distribution at the center line on the floor wall at Re = 1000
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Fig. 16 Instantaneous mapping of fluctuating velocity( u and v ) and fluctuating temperature ( ) at Re =1000
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Fig. 17 Hysteresis of velocity fluctuation illustrated in u v plane at Re =1000
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Table 1 Locations and values of Nu max . Re
xmax /S
600 800 1000 1200 1400
8.6 10.7 12.2 13.5 14.3
zmax/S 6.96 7.09 7.11 7.15 7.16
Table 2 Frictional contributions to - u v and v from each quadrant of u v plane n
Quadrant
1 2 3 4
Signs of u and v u ＞ 0 v ＞ 0 u ＜ 0 v ＞ 0 u ＜ 0 v ＜ 0 u ＞ 0 v ＜ 0
Signs of ＞0 ＜0 ＞0 ＜0 ＞0 ＜0 ＞0 ＜0
Number of data (n) 657 468 0 144 465 665 165 0
u v i 1
n
i
v
i
i 1
N u v
N v
0.333 0.165 0 -0.002 0.182 0.326 -0.004 0
3.138 -2.249 0 -0.352 -2.527 3.468 -0.478 0
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S0140-7007(17)30148-2 http://dx.doi.org/doi: 10.1016/j.ijrefrig.2017.04.012 JIJR 3613
To appear in:
International Journal of Refrigeration
Received date: Revised date: Accepted date:
15-12-2016 11-4-2017 11-4-2017
Please cite this article as: J.H. Xu, S. Zou, K. Inaoka, G.N. Xi, Effect of reynolds number on flow and heat transfer in incompressible forced convection over a 3d backward-facing step, International Journal of Refrigeration (2017), http://dx.doi.org/doi: 10.1016/j.ijrefrig.2017.04.012. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Effect of Reynolds Number on Flow and Heat Transfer in Incompressible Forced Convection over a 3D Backward-Facing Step
J.H. Xua, S. Zoua, K. Inaokab, G.N. Xia,* a
b
School of Mechanical Engineering, Nantong University, Nantong 226019, China
School of Mechanical Engineering, Doshisha University, Kyotanabe 610-0321, Japan * Corresponding author. Nantong University, 9 Se Yuan Rd., Nantong 226019, China. Tel:+8618762850738. Email address: [email protected] (G.N. Xi).
Highlights
Periodic instability on heat transfer are clarified.
The heat transfer is greatly enhanced with the increase of Re.
Taking away the hot fluid and bringing cold fluid are identified.
Abstract A three-dimensional incompressible numerical model for the case of the 3D backward-facing step flow is established to investigate the characteristics of fluid flow and heat transfer in the low and middle Reynolds number ranges (200≤Re≤1400). The governing equations, including continuous, unsteady Navier-Stokes and energy equations, are solved by the finite volume method in FLUENT. The simulation results show that the time averaged reattachment length reaches the peak value at Re = 1000, and subsequently decreases as the increase of Re. The formation of secondary peak Nu influenced by flow instability has a better contribution to the heat transfer at the center area. Taking away the hot fluid and carrying the cold fluid into the floor wall caused by periodic instability has positive effects on heat transfer enhancement.
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Keywords: 3D backward-facing step; forced convection; heat transfer; periodic instability
Nomenclature A
surface area of the floor wall, [m2]
AR
aspect ratio, AR W / S
c
Courant number
C
f
skin friction coefficient, C f
2 w
u0
2
1
Cf
time averaged friction coefficient, C f
C
time-spatial averaged friction coefficient, C fs
fs
Cp
specific heat capacity, [J kg-1 K-1]
ER
expansion ratio, ER H / h
f
frequency, [Hz]
Gr
Grashof number
h
inlet channel height, [m]
H
exit channel height, [m]
j
Colburn’s j-factor, j
Nu
local Nusselt number, Nu
Nu
time averaged Nusselt number, Nu
0
C f dt
1 A
A
0
C f d
Nu s Re Pr
1/ 3
qwS
T w T 0 1
Nu dt 0
Nu s
time-spatial averaged Nusselt number, Nu s
Pr
Prandtl number
1 A
A
Nu d
0
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P
pressure, [Pa]
qw
wall heat flux, [W·m-2]
Re
Reynolds number, R e R e D
Re S
Reynolds number, R e S
Reg
grid Reynolds number
S
step height, [m]
t
time, [s]
T
temperature, [K]
T
time-averaged temperature, [K]
u0
averaged velocity component in the x direction, [m s-1]
u, v, w
velocity components in x, y and z directions, [m s-1]
u
streamwise fluctuating velocity, [m s-1]
v
transverse fluctuating velocity, [m s-1]
- u v
shear stress, [m2 s-2]
v
turbulent heat flux, [m s-1 K]
W
channel width, [m]
XR
time averaged reattachment length, [m]
x
streamwise coordinate, [m]
y
transverse coordinate, [m]
z
spanwise coordinate, [m]
2 u0h
u0S
Greek letters
unit area, [m2]
fluctuating temperature, [K]
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thermal conductivity, [W m-1 K-1]
dynamic viscosity, [kg m-1 s-1]
kinematic viscosity, [m2 s-1]
density, [kg m-3]
sampling time, [s]
w
wall shear stress, w u y w y 2
2
Subscripts max
maximum
0
inlet value
w
wall value
x, y, z
values along x, y, and z coordinates
1. Introduction Flow separation and reattachment of backward-facing step caused by abrupt expansion are well applied into many heat exchanging devices, such as energy system equipment, turbine blades, dump combustors, and many other heat-exchanging devices. For the sake of improving the performance of these applications, investigating the mechanism of flow and heat transfer enhancement is very crucial. The benchmark problem of laminar and turbulent flow has been extensively studied by experimental (Armaly et al., 1983; Lee et al., 1998; Terhaar et al., 2010; Kapiris et al., 2014) and numerical (Vogel and Eaton, 1985; Moser et al., 1999; Kozel et al., 2005; Barrios et al., 2012; Zhong et al., 2014) methods. The experimental and numerical research carried out by Armaly et al. (1983) on the three-dimensional backward-facing step flow has arisen much more attention. The laminar (ReD＜1200),the transitional (1200＜ReD＜6600 ), and the turbulent (ReD＞6600) regimes of the flow are
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identified by the primary reattachment length based on the experimental research with LaserDoppler measurements. Since then, a lot of three-dimensional laminar flow investigations (Armaly et al., 2002; Nie and Armaly, 2003; Nie and Armaly, 2004; Chen et al., 2006; Nie et al., 2009) have been published. The effects of side walls caused much attention in fluid dynamics community for many numerical and laboratory experiments. Williams and baker (1997) did the research on the three-dimensionality with the effects of side walls and found that the complex jet-like flow was located at the step wall near the sidewalls. The complex flow structure near the side walls were also reported by Chiang and Sheu (1999). In their work, the limiting streamlines near roof, floor, and end walls at Re = 800 were investigated to illustrate the spanwise width (2h≤ B ≤100h) effects on the flow field. Tylli et al. (2002) investigated the effects on the primary and secondary recirculation zones, and explained the differences between experimental and 2D simulations in the transitional and turbulent regimes. Biswas et al. (2004) reported the spatial evolution and three-dimensionality of jetlike flows with different expansion ratios (ER=1.9423, 2.5, 3.0). The studies on heat transfer characteristics of backward-facing step flow were focus on laminar and turbulent flow. Iwai et al. (2000) investigated the effects of the duct aspect ratio with a fixed step height in laminar regime (125≤ ReS ≤375). The results revealed the 2D region can be obtained with an aspect ratio of at least 16. The maximum Nusselt number appeared near the side walls increased with the increasing of aspect ratio. The effects of step height on distribution of Nu, Cf and flow structures at ReD = 343 were studied by Nie and Armaly (2002). The results showed that the maximum Nusselt number increased as the step height increased. He also found that the three-dimensional behavior and sidewall effects increased with increasing step height. Kitoh et al. (2007) studied the time mean reattachment length and the distribution of Nu and Cf on the floor wall at unsteady flow state with the effects of expansion ratios by the methodology of DNS. In his work, the flow structure in the
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separated and reattachment regions was clearly shown at 300≤ ReS ≤1000. Khanafer et al. (2008) studied numerically the laminar mixed convection heat transfer with the effect of pulsating flow over a backward-facing step. The results illustrated that the flow structure, heat transfer and the average wall friction coefficient were both affected by Reynolds number, Richardson number and the dimensionless oscillation frequency. Lan et al. (2009) reported the simulations of three-dimensional turbulent flow over a backward-facing step. The - - - f turbulent model was used to improve effectively the heat transfer predictions in the high Reynolds number range of 20000-50000. The results showed that the increasing of Reynolds number or aspect ratio leads to the higher bulk Nusselt number. Xie and Xi (2017) numerically investigated the heat transfer of separation and reattachment flow over a 2D backward-facing step by DNS. The results illustrated that the isotherms were more unsteady as the increase of Reynolds number and the size of primary recirculation zone increased with the decrease of expansion ratio. The combined effects of flow and heat transfer instability resulted in the dissimilarity between Nu and Cf at ReS = 1000. In the transitional flow regime, Chiang et al. (1999) numerically investigated the vortical evolution in a 2D backward-facing step. He clarified how the recirculating bubble containing flow reversals is torn into smaller eddies and discussed the eddy distortion and the merging of eddies in his study. Flow structures in a 3D backward-facing step channel was careful investigated by Rani et al. (2007) to indicate that the flow becomes unsteady along with effects of the Kelvin-Helmholtz instability and Taylor-Görtler-Like vortices at ReD=1000 and 2000. The backward-facing step flow in the late transitional regime was investigated based on DNS by Schäfer et al. (2009). The calculated results indicated that the oscillations of the primary reattachment and the upper wall separation zones are dominated by bubble structures in the late transitional regime (ReD = 6600). Tihon et al. (2012) experimentally and numerically studied the transitional flow (water) with different expansion ratios and inlet flow
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conditions at moderate Reynolds numbers. The results suggested that the inlet pulsatile flow plays significant roles on flow structures, reattachment length and upper wall recirculation zones. Xie and Xi (2017) mainly examined heat transfer enhancement with the effects of the vortical structure in the transitional flow over a 2D backward-facing step. Periodic instability of vortical structure destroyed the thermal boundary layer and the heat transfer was greatly enhanced at the bottom wall, particularly in the range 6≤ x/H ≤14. It is worth noting that there are a lot of laminar and turbulent investigations on the flow structures and reattachment in 2D and 3D backward-facing step flow but little attention on heat transfer characteristics has been done in transitional backward-facing step flow. Further more, the mechanisms leading to heat transfer enhancement in transitional regime were rarely investigated in a 3D backward-facing step channel. The aim of this paper is to study the heat transfer enhancement of 3D backward-facing step in transitional regime. In particular, the heat transfer enhancement influenced by periodic instability are mainly discussed. 2. Numerical methodology 2.1 Governing equations In this paper, the value of Gr/Re2 is lower than 0.1 at Re = 1000, so that the buoyancy effect can be ignored. The governing equations including the continuity, Navier-Stokes and energy equations for three dimensional, transitional, incompressible and unsteady case are written as follows: ui xi
u i t
T t
0
u iu j x j
u jT x j
(1)
1 P
xi
2T 2 C p x j
u i u j x j x j xi
(2)
(3)
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Where the Cartesian velocity components are denoted ui, (i, j = 1, 2, 3 for x, y, z, respectively), xi is coordinates components, is density, is kinematic viscosity, P is pressure, Cp is specific heat, λ is thermal conductivity, T is temperature, t is time. In this study, the physical properties of fluid (air) are assumed as constants, that is, ρ = 1.247 kg·m-3,
= 1.76×10-5 m2·s-1, Cp = 1005 J·kg-1·K-1, λ = 0.0251 W·m-1·K-1 and Pr = 0.705. 2.2 Computational domain and boundary conditions The computational domain is presented in Fig. 1. The step height S is 0.015m, channel height H is 0.030m and channel’s width W is = 0.24m in the geometry. The duct expansion ratio, ER = H/h, is set as 2. The step aspect ratio is defined by AR = W/S = 16. The channel length downstream of the step is chosen as Ld = 50S so that the flow can be treated as fully developed at exit plane. The boundary conditions in this study can be expressed as follows:
The inlet flow is assumed as laminar, fully developed and isothermal. The temperature of incoming flow is maintained at uniform temperature, Tin = 283K. The transverse velocity (v) and spanwise velocity (w) are both set as 0. The distribution of the streamwise velocity component (u) is considered to be the approximation by Shah and London (1978). Therefore, the distribution of u is summarized: n m y S H S 2 z m 1 n 1 1 1 u 0 m n H S 2 W 2
u
H S W *
m 1 . 7 0 . 5 ( ) *
(4)
(5) 1 .4
(6)
1 2 3 n (7) 1 1 1 2 0 . 3 3 3 ２
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The outflow boundary condition, at the channel exit, assumes fully developed and the streamwise gradient of flow variables is 0, that is,
u
v w T 0， 0， 0， 0 . x x x x
All wall surfaces meet the no-slip condition. The downstream floor wall of the backwardfacing step is heated at uniform temperature, Tw = 313K. Meanwhile, the rest of the walls are treated to be adiabatic.
2.3 Numerical solution procedure All these simulations are calculated by FLUENT. The finite volume method is used for the spatial discretization of Eqs. (1)–(3). The convection terms in the unsteady Navier-Stokes equations are evaluated using QUICK scheme. Energy equation is solved with second-order upwind difference scheme SIMPLE algorithm, proposed by Patankar and Spalding (1972), is utilized for the coupling of pressure and velocity. The initial conditions of velocity components and temperature are set as u = u0, v = 0,w = 0, and Tin = 283K. Max iterations per time step is set to be 5. The CPU (eight-core processor) is Intel Xeon E5-2687W 0 @ 3.10GHz and the RAM is 32.0 GB. Also, the simulations are carried out in a serial process. All calculations are performed on the dell station and the CPU time for Re = 1000 is around 20 days. The mesh is generated by using preprocessor ICEM. Fig. 2(a) exhibits the nonuniform Cartesian grid structure of the whole computational domain. As shown in the Fig. 2(b), the magnified view to the z-axis, the minimum grid spacing near the all walls is set to ensure the accuracy of the calculation results. 2.4 Grid independence The density of grid plays an important role on the calculation results. Also, the grid refinement is made in the regions around walls, where the velocity gradient is high. Numerical calculation accuracy is reduced with sparse grid, resulting in the obvious deviation from the true solution. On the contrary, the superfine grid consumes much more memory and
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CPU time. Therefore, the design of grid plays an important role on the calculated accuracy and consuming time. Grid Reynolds number (Reg), defining the minimum grid spacing ( x min ) as the reference length, reflects the grid density. The expression is used: Re g
u 0 x min
(8)
Grid independence study is carried out with three different cases. Fig. 3 shows the time averaged Nusselt number ( Nu x ) and time averaged streamwise friction coefficient ( C fx ) along the center line for different Reg, respectively. The CPU time of Reg = 8, Reg = 10, and Reg = 12 is around 25 days, 14 days, and 6 days, respectively. It can be clearly seen that the values of Nu x and C fx , in the range 0≤ x/S ≤12, are in good agreement. On the other hand, in the range 12≤ x/S ≤50, the small deviation exists. The deviation is calculated based on the data of Reg = 12, the minimum deviation of these values is about 0.2% while the maximum deviation is about 1%, meeting the requirement of the grid independence. Therefore, the value of Reg in this simulations is set equal to 10. In Eqs. (9), the time step ( t ) between two successive iterations is set to get a Courant number (c) less than 1, evaluated with the smallest grid spacing ( x min ), and the maximum velocity (umax= 1.5u0). c
um a t x xm
1
(9)
i n
2.5 Validation of the model In order to validate the reliability of numerical results, the time averaged reattachment length, compared with other credible results, is shown in Fig. 4. The present results of 2D simulations show excellent agreement with numerical results published by Iwai et al. (2000)
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and Sugawara et al. (2005). In 3D simulations, for a Reynolds number range 100≤ Re ≤ 800, the results of reattachment length obtained from this study are in excellent agreement with the results published by Iwai et al. (2000) and Armaly et al. (1983). However, in a Reynolds number range 800＜ Re ≤1400, the time averaged reattachment length is greatly influenced by aspect ratios. Fig. 5 can show that local Nusselt number Nu and streamwise friction coefficient Cfx along the center line at Re = 500. These results of the 3D simulation are in perfect agreement with that of Iwai et al. (2000). As a result, the accuracy and reliability of present simulations are confirmed. 3. Results and discussion 3.1 Flow properties The entire limiting streamlines plotted on the floor, roof, and side walls of the channel can be clearly observed from Fig.6 to analyze the flow structure in the separated and reattachment regions. The reattachment lines of the primary recirculation zone along the floor wall are obvious shown at Re = 600, 800, 1000, and 1200. It is found from this picture that the reattachment lines are the approximate parabolic. The reattachment flow region increases and moves further downstream up to Re = 1000, but decreases and moves upstream at Reynolds number range 1000≤ Re ≤1200. On the roof wall, two symmetrical eddies formed from the side walls extend all the way to channel center plane as the increase of the Re. Due to the effects of flow instability, two symmetrical eddies along the roof wall are destroyed at Re = 1200. The flow is steady up to Re = 800, but becomes gradually unsteady along with the apparent three-dimensionality with the increase of Re. 3.2 Time-spatial averaged heat transfer In Fig. 7, the averaged heat transfer coefficient is plotted in the form of j-factor against Reynolds number. Overall, the values of j-factor decreases with the increase of Re. In the
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Reynolds number range, 200≤ Re ≤1000, the slope of the plotted data is close to that of the Re-0.5. This manifests that the heat transfer is dominated by the steady, flat-plate boundary layer type flow. However, in the Reynolds number range 1000≤ Re ≤1400, the Reynolds number dependency of the j-factor begins to show smaller slope compared with that of Re-0.5. This suggests that the heat transfer is dominated by the unsteady flow. Therefore, the heat transfer coefficient becomes higher than the values in the laminar range. It is thus reasonable to conclude that the unsteady flow at higher Reynolds number range is likely to be the cause of the obvious heat transfer enhancement on the floor wall of the backward-facing step. As is shown in Fig. 8, time-spatial averaged Nusselt number ( Nu s ) and time-spatial averaged friction coefficient ( C fs ) are plotted against the Re from 200 to 1400. With the increase of the Reynolds number, the value of time-spatial averaged Nusselt number increases with the linear rule from 0.847 to 2.179, but the value of time-spatial averaged friction coefficient decreases rapidly in laminar flow, and then tends to be steady in transitional flow. According to the tendency of the two evaluation parameters, the heat transfer performance on the floor wall is enhanced with the increase of Re. 3.3 Time averaged heat transfer The contours of time averaged (calculated with 20000 time steps) Nusselt number ( Nu x , z ) along with the xu-line on the floor wall are illustrated in Fig. 9. The dashdotted line (xu-line) on which represents the shear stress on the floor wall ( u / y
y0
) is 0. The black
dots in each of Fig. 9 represent the maximum of time averaged Nusselt number ( Nu max ). According to the distributions of xu-line, it is also found that the reattachment flow region increases and moves further downstream up to Re = 1000, but decreases and moves upstream at Re = 1200. It is common that the location of Nu max appear symmetrically near the two
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side walls. The locations of Nu max can be seen in Table 1. As the Re increases, streamwise location (xmax /S) of Nu max moves downstream from the step, but the spanwise location (zmax/S) of Nu max has no obvious change. In order to intuitively compare with the variation of Nu x , z at different Re, three-dimensional contours of Nu x , z (calculated with 20000 time steps) are shown in Fig. 10 with bird’s eye sight. At Re =1000, 1200, Nu at the center area is obvious enhanced. Therefore, the fluid flow and heat transfer at the center area are studied in the following section. 3.4 Periodic instability The flow becomes gradually unsteady in a Reynolds number range, 1000≤ Re ≤1400. Two monitor points P1 (x/S = 10, y/S = 1, z/S = 0) and P2 (x/S = 40, y/S = 1, z/S = 0) are set to monitor the instantaneous transverse velocity (v) in the flow field. Power spectral density of v can be obtained based on the v data of 20000 time steps by the method of fast Fourier transform (FFT). Fig. 11 shows the Instantaneous transverse velocity and power spectral density of v at Re =1000, 1200, 1400. It is very clear that the variation of v changes with obvious rule and the periodic characteristic of v at Re = 1000 is stable (f1 = f2 = 4.58) at points both P1 and P2. Sheu and Rani (2006) numerically found that the fundamental frequency of the variation is f =4.77, which is close to the value f1 = 4.58. As the Re is increased, in point P1, the fundamental frequency is also increased. On the other hand, with the increase of Re, the periodic regime lost in point P2, located far away from the inlet. In order to discuss the effects of periodic instability on heat transfer enhancement near the center plane, we mainly analyze the flow field and heat transfer on the floor wall at Re = 1000. The instantaneous contours of Nu distribution along with the xu-line ( u / y
y0
) on the
floor wall at Re = 1000 can be clearly seen from Fig. 12. In a cycle, it is clear that the position
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and formation of the xu-line are changing periodically at the center area. In contrast, the position of xu-line without any change can be seen near the side walls. In Fig. 13, it can be concluded that intensity of instantaneous streamwise velocity variation at center plane (z/S=0) is higher than the other spanwise planes. Therefore, flow instability at center plane is discussed in the following parts. Fig. 14 shows the streamline and vector plots of the center plane at Re = 1000 in a cycle. It can also be observed that the Kelvin-Helmholtz instability occurs in the shear layer of the primary recirculation zone. Importantly, the vortexes shed constantly and periodically has positive effects on the flow instability in the region downstream from the primary recirculation zone. As a result, in that region, the thermal boundary layer are destructed with the effect of flow instability. Fig. 15 shows that Nu distribution at the center line on the floor wall at Re = 1000. The first peak of Nu is caused by the primary recirculation zone. The occurrence of secondary peak of Nu influenced by flow instability has a great influence on the heat transfer enhancement. The mechanisms resulting in the secondary peak of Nu are studied in detail. The instantaneous mapping of fluctuating velocity and temperature at center plane is plotted in Fig. 16, two fluctuating velocity components of u and v and fluctuating temperature is defined by: u u u
(10)
v v v
(11) T T
(12)
where u and v are the time averaged streamwise velocity and transverse velocity, T is time averaged temperature.
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From the qualitative analysis, due to this periodic vortexes motion, as shown in Fig. 16, the vortexes carry the cold ( ＜ 0 ) fluid into the floor wall, which has positive effects on the heat transfer. On the other hand, the vortexes carry the hot ( ＞ 0 ) fluid away from the floor wall, which also has positive effects on the heat transfer. The red arrows in this picture represent the position of Numax, where is located at the cold fluid. Overall, cold fluid moves into the floor wall and hot fluid moves away from the floor wall has positive effects on heat transfer. Discussion of the heat transfer mechanism is based on this picture. From the quantitative analysis, one monitor point P3 (x/S = 32, y/S = 0.1, z/S = 0) is set to get the time variation of fluctuating velocity ( u , v ) and temperature ( ). Fig. 17 illustrates time variation pattern of the fluctuating velocity field in five cycles. It can be seen from the picture that the number density of the plotted points in u - v plane is mainly in the first and third quadrants compared with those in other two quadrants. Table 2 shows the number of the data sets assigned into each quadrant and the frictional contributions to - u v and v
from each quadrant of u - v plane. - u v represents the fluctuating velocity
cross-corelation and v is the cross-corelation of fluctuating velocity and temperature. It is found that the fractional contributions to - u v and v in the first and third quadrants are much larger than that of the other quadrants. So, there are two main effects on heat transfer. Firstly, hot ( ＞ 0 ) fluid in the first quadrant and the cold ( ＜ 0 ) fluid in the third quadrant contribute positively to v . Secondly, cold ( ＜ 0 ) fluid in the first quadrant and hot ( ＞ 0 ) fluid in the third quadrant contribute negatively to v . In general, the first effect occupies the dominant role by the comparison. Through the analysis of statistics, it is concluded that taking away the hot fluid and carrying the cold fluid into the floor wall lead to the formation of the secondary peak of Nu. Thus, the heat transfer on the floor wall is greatly enhanced. 4. Conclusions
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A three-dimensional incompressible numerical study is conducted to clarify the heat transfer mechanism. The governing equations with boundary conditions are solved in FLUENT. The main results calculated to study the effects of various Reynolds numbers are as follows:
With the increase of the Re, the value of time-spatial averaged Nusselt number increases accompanying with the decrease of the time-spatial averaged friction coefficient. It is very clear that the heat transfer is constantly enhanced with the increase of the Re by the j-factor.
The reattachment flow region increases and moves further downstream up to Re = 1000, but decreases and moves upstream with increase of Re.
The first peak of Nu is caused by the primary recirculation zone. The formation of secondary peak of Nu influenced by flow instability has a great influence on the heat transfer enhancement.
Taking away the hot fluid and carrying the cold fluid into the floor wall caused by periodic instability has positive effects on heat transfer enhancement.
Acknowledgements This Project is supported by the National Natural Science Foundation of China, No.51476080.
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Fig.1 Schematic of the computational domain
(a)
(b)
Fig. 2 The whole grid and the magnified view of z-plane
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(a)
(b)
Fig. 3 Time averaged Nu and Cf along the center line at Re = 600
Fig. 4 Comparison of time averaged reattachment length
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(a)
(b)
Fig. 5 Validation of local Nusselt number Nu and streamwise friction coefficient Cfx along the center line at Re = 500
(a) Re=600
(b) Re=800
(c) Re=1000
(d) Re=1200
Fig. 6 Instantaneous limiting streamlines on the three walls
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Fig. 7 j-factor as a function of Re
Fig. 8 Time-spatial averaged Nusselt number and friction coefficient
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(a) Re=600
(b) Re=800
(c) Re=1000
(d) Re=1200
Fig. 9 Contours of time averaged Nu x , z distribution along the xu-line on the floor wall
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(a) Re=600
(c) Re=1000
(b) Re=800
(d) Re=1200
Fig. 10 3D contours of time averaged Nu x , z distribution on the floor wall
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Fig. 11 Instantaneous transverse velocity and power spectral density of v P1 (x/S = 10, y/S = 1, z/S = 0), P2 (x/S = 40, y/S = 1, z/S = 0)
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wt = 0
wt = 1/2π
wt = 1/8π
wt = 5/8π
wt = 1/4π
wt = 3/4π
wt = 3/8π
wt = 7/8π
Fig. 12 Instantaneous Nu distribution with xu-line on the floor wall at Re = 1000
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Fig. 13 Instantaneous streamwise velocity contours at different z-planes at Re = 1000
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Fig. 14 Streamlines and vector at the center plane at Re =1000
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First peak
Secondary peak
Fig. 15 Nu distribution at the center line on the floor wall at Re = 1000
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Fig. 16 Instantaneous mapping of fluctuating velocity( u and v ) and fluctuating temperature ( ) at Re =1000
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Fig. 17 Hysteresis of velocity fluctuation illustrated in u v plane at Re =1000
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Table 1 Locations and values of Nu max . Re
xmax /S
600 800 1000 1200 1400
8.6 10.7 12.2 13.5 14.3
zmax/S 6.96 7.09 7.11 7.15 7.16
Table 2 Frictional contributions to - u v and v from each quadrant of u v plane n
Quadrant
1 2 3 4
Signs of u and v u ＞ 0 v ＞ 0 u ＜ 0 v ＞ 0 u ＜ 0 v ＜ 0 u ＞ 0 v ＜ 0
Signs of ＞0 ＜0 ＞0 ＜0 ＞0 ＜0 ＞0 ＜0
Number of data (n) 657 468 0 144 465 665 165 0
u v i 1
n
i
v
i
i 1
N u v
N v
0.333 0.165 0 -0.002 0.182 0.326 -0.004 0
3.138 -2.249 0 -0.352 -2.527 3.468 -0.478 0
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