Laminar unsteady flow and heat transfer in confined channel flow past square bars arranged side by...
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Laminar unsteady flow and heat transfer in confined channel flow
past square bars arranged side by side
University of Notre DameTuesday, December 4, 2001
Professor Alvaro Valencia
Universidad de ChileDepartment of Mechanical Engineering
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Motivation Laminar flow in a channellow heat transfer Heat transfer Enhancement in channels:
Q=AhT h with fluid mixing transverse vortex generators
Streaklines around a square bar for Re=250, and Re=1000Davis, (1984)
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Turbulent flow near a wall, Re=22000, experimental results, Bosch( 1995)
Numerical results, k- turbulence model
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Anti-phase and in-phase vortex shedding around cylinders
Re=200G/d=2.4
Williamson, (1985)
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Wake interference of a row of normal flat plates arranged side by side in a uniform flow, Hayashi, (1986)
a) G/Hc=0,5 Rec=59b) G/Hc=1,0 Rec=100c) G/Hc=1,5 Rec=100d) G/Hc=2,0 Rec=100
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Numerical simulation of laminar flow around two square bars arranged side by side with free flow condition. Bosch (1995)
Rec=100G/Hc=0,21 bar behavior
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Rec=100G/Hc=0,75Bistable vortex shedding
For G/d >1.5 synchronization of the vortex shedding in anti-phase or in-phase
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Geometry of the computational domain
ReH=800 (Rec=100) Pr=0,71 (Air) Transverse bar separation distance, G/H or G/Hc
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Mathematical formulation
0
yv
xu
uxP
yuv
xuu
tu 2
vyP
yvv
xvu
tv 2
2
2
2
2
yT
xTk
yTv
xTu
tTCp
Continuity
Navier Stokes equations (momentum)
Thermal energy
The variables were non-dimensionalized with Uo, H, and To.
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Boundary Conditions Inlet:
Fully developed parabolic velocity profile Constant temperature To
Walls:
Constant wall temperature Tw=2To
Thermal entrance region
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Boundary conditions
Outlet: wake equation to produce little reflection of the unsteady vortices at the exit plane
00
xU
t
,,VU
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Numerical solution technique
Differential equations were solved with an iterative finite-volume method described in Patankar( 1980).
The convection terms were approximated using a power-law sheme
The method uses a staggered grid and handles the pressure-velocity coupling with the SIMPLEC algorithm, van Doormal (1984).
A first-order accurate fully implicit method was used for time discretization in connection with a very small time step. 1.5Uot/x=0.1
A tipical run of 70.000 time steps with the 192x960 grid points takes about 4 days in a personal computer Pentium III.
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Grid selection The confined flow around a square bar mounted
inside a plane channel was chosen for evaluate the numerical method and grid size.
A lot of data was found in the literature for the confined laminar flow past a square bar, it was found also a great dispersion of the results.
M. Breuer et al presented accurate computations of the laminar flow past a square cylinder based on two different methods, (2000).
The present numerical results were compared with their results
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Grid size
CV on bar
St* Cd* 1000xCd*
Cl* Nu 1000x f
32x160 4 0.000 3.06 0.00 0.00 8.26 47.948x240 6 0.118 1.46 0.19 0.13 8.40 48.964x320 8 0.124 1.50 5.82 0.29 8.43 50.780x400 10 0.128 1.48 8.93 0.36 8.45 50.896x480 12 0.131 1.47 11.96 0.43 8.47 51.1112x560
14 0.133 1.45 14.58 0.48 8.49 51.3
128x640
16 0.135 1.44 16.76 0.51 8.50 51.7
144x720
18 0.137 1.43 18.64 0.54 8.50 52.0
160x800
20 0.138 1.42 20.17 0.56 8.51 52.4
176x880
22 0.139 1.41 21.52 0.58 8.51 52.7
192x960
24 0.139 1.40 22.54 0.60 8.52 53.1
208x1040
26 0.140 1.39 23.39 0.61 8.52 53.6
*: Strouhal numbers St, Drag coefficient and Lift coefficient are based here on the maximum flow veliocity
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Grid size
Strouhal number
0.100
0.105
0.110
0.115
0.120
0.1250.130
0.135
0.140
0.145
0.150
6 8 10 12 14 16 18 20 22 24 26
CV on bar
St*
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Grid size
Drag Coefficient
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4 6 8 10 12 14 16 18 20 22 24 26CV on bar
Cd*
Variation of Drag coefficient
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
6 8 10 12 14 16 18 20 22 24 26CV on barCd*
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Grid size
Variation of Lift coefficient
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
6 8 10 12 14 16 18 20 22 24 26
Cv on bar
Cl*
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Conclusion on grid selection
St
* Cd* Cl
*
Present studys 0,1394 1,400 0,595
Breuer et al. (2000) 0,1450 1,364 0,628
Error 3,9% 2,6% 5,2%
The grid with 192x960 control volumes CV was chosen because delivery good results with a reasonable calculation time
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Cases studied
The computations were made for 11 transverse bar separation distances
Re=800 Pr=0.71 air flow Hc/H=1/8 bar height L/H=5 channel length
Case G
1 0,5Hc 0,0625H 2 0,75Hc 0,09375H 3 1,0 Hc 0,1250H 4 1,5 Hc 0,1875H 5 2,0 Hc 0,2500H 6 2,5 Hc 0,3125H 7 3,0 Hc 0,3750H 8 3,5 Hc 0,4375H 9 4,0 Hc 0,5000H 10 4,5 Hc 0,5625H 11 5,0 Hc 0,6250H
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Flow pattern (11 – 4)
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Flow pattern (3)
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Flow pattern (2)
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Flow pattern (1)
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Instantaneous temperature field Case 1
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Instantaneous local skin friction coefficient on the channel walls. Case 1 Cf= / (1/2Uo**2) : wall shear stress
Inferior wall Superior wall
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Local skin friction coefficient on theinferior channel wall. Cases 11 to 6
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Local skin friction coefficient on the channel walls.Cases 5 to 1
Superior wall Inferior wall
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Local Nusselt numbers:Cases 11 to 6
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Local Nusselt numbers:Cases 5 to 1Inferior wall Superior wall
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Frequency: Case (2)Velocity U, Position: 2Hc behind the bar Inferior bar
Superior bar
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Frequency: Case (2)Velocity V, Position: 2Hc behind the bar Inferior bar
Superior bar
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Frequency: Case (2)Drag coefficients Inferior bar
Superior bar
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Frequency: Case (2)
Lift Coefficients Inferior bar
Superior bar
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Strouhal numbers and Frequencies
Case G Frequency F lower bar
Stc lower bar
Frequency F superior bar
Stc superior
bar Dominant frequency
1 0,0625H 1,488 0,186 1,302 0,163 0,419 2 0,09375H 1,395 0,174 2,047 0,256 0,698 3 0,1250H 1,674 0,209 1,674 0,209 1,674 4 0,1875H 1,795 0,224 1,795 0,224 1,795 5 0,2500H 2,000 0,250 2,000 0,250 2,0 6 0,3125H 1,895 0,237 1,895 0,237 1,895 7 0,3750H 1,840 0,230 1,840 0,230 1,840 8 0,4375H 1,774 0,222 1,776 0,222 1,774 9 0,5000H 1,687 0,211 1,688 0,211 1,688
10 0,5625H 0 0 0 0 0 11 0,6250H 0 0 0 0 0
St=fd/Uo Struhal number F=fH/Uo non dimesional frequency
F: frequency of Velocity V St=F/8
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Dominant frequency of the flowlow frequency modulation in cases: G=0.0625, 0.09375, and 0.125H
f G/H=0 = 1.14
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Skin friction coefficient on channel wall Cf= / (1/2Uo**2) : wall shear stress
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Drag coefficients for the lower and superior bar Cd=D/(1/2Uo**2)d
Cd G/H=0 =5
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Lift coefficients: lower bar, superior bar Cl=L/(1/2Uo**2)d
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Mean Nusselt number : inferior wall and superior wall Nu=hH/k q=hT wall heat flux
nu G/H=0 =11
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Apparent friction factor f=PH/(Uo**2)L
f G/H=0 = 0.164
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Mean Heat Transfer enhancement and Pressure drop increaseNuo and fo for a plane channel without built-in square bars
0
2
00
2
0 PP
ff
Q
QNuNup
Nu0= 7,68 and f0= 0,01496
Nu with 1 square bar=8.52 f with 1 square bar =0.053
G/H
0.8
0.9
1.0
1.1
1.2
1.3
1.4
Nu/Nu0
0,0625
0,09375
0,1250
0,1875
0,2500
0,3125
0,3750
0,4375
0,5000
0,5625
0,62505
6
7
8
9
10
fapp/fapp0
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Conclusions The effect of two square bars placed side by
side in a laminar flow in a plane channel on pressure drop and heat transfer was numerically investigated.
The flow pattern for equal sized square bars in side-by-side arrangements were categorized into three regimes: steady flow, in-phase vortex shedding and bistable vortex shedding.
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In the cases with vortex-shedding synchronization the frequency of the unsteady flow are almost four times that in the cases without synchronization of the periodic unsteady flow.
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The results show that the local and global heat transfer on the channel walls are strongly increased by the unsteady vortex shedding induced by the bars.
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References[1] H. Suzuki, Y. Inoue, T. Nishimura, K. Fukutani, k. Suzuki, Unsteady flow in a channel obstructed by a square rod (crisscross motion of vortex). International Journal of Heat and Fluid Flow 14 (1993) 2-9.[2] A. K. Saha, K. Muralidhar, G. Biswas, Transition and chaos in two-dimensional flow past a square cylinder, Journal of Engineering Mechanics, 126, (2000), 523-532.[3] M. Breuer, J. Bernsdorf, T. Zeiser, F. Durst, Accurate computations of the laminar flow past a square cylinder based on two different methods: lattice-Boltzmann and finite-volume, International Journal of Heat and Fluid Flow, 21, (2000), 186-196.[4] J. L Rosales, A. Ortega, J.A.C. Humphrey, A numerical simulation of the convective heat transfer in confined channel flow past square cylinders: comparison of inline and offset tandem pairs, International Journal of Heat and Mass Transfer, 44, (2001), 587-603.[5] K. Tatsutani, R. Devarakonda, J.A.C. Humphrey, Unsteady flow and heat transfer for cylinder pairs in a channel, International Journal of Heat and Mass Transfer, 36, (1993), 3311-3328.[6] A. Valencia, Numerical study of self-sustained oscillatory flows and heat transfer in channels with a tandem of transverse vortex generators, Heat and Mass Transfer, 33, (1998), 465-470.[7] D. Sumner, S.J. Price, M.P. Païdoussis, Flow-pattern identification for two staggered circular cylinders in cross-flow, Journal of Fluid Mechanics, 411, (2000), 263-303.[8] C.H.K. Williamson, Evolution of a single wake behind a pair of bluff bodies, Journal of Fluid Mechanics, 159, (1985), 1-18.[9] J.J. Miau, H.B. Wang, J.H. Chou, Flopping phenomenon of flow behind two plates placed side-by-side normal to the flow direction, Fluid Dynamics Research, 17, (1996), 311-328.[10] M. Hayashi, A. Sakurai, Wake interference of a row of normal flat plates arranged side by side in a uniform flow, Journal of Fluid Mechanics, 164, (1986), 1-25.[11] S.C. Luo, L.L. Li, D.A. Shah, Aerodynamic stability of the downstream of two tandem square-section cylinders, Journal of Wind Engineering and Industrial Aerodynamics, 79, (1999), 79-103.[12] G. Bosch, Experimentelle und theoretische Untersuchung der instationären Strömung um zylindrische Strukturen, Ph.D. Dissertation, Universität Fridericiana zu Karlsruhe, Germany, (1995).[13] S. Patankar, Numerical heat transfer and fluid flow, Hemisphere Publishing Co., New York, (1980).[14] J.P. van Doormaal, G.D. Raithby, Enhancements of the SIMPLE method for predicting incompressible fluid flows. Numerical Heat Transfer, 7, (1984), 147-163.