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

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)

Turbulent flow near a wall, Re=22000, experimental results, Bosch( 1995)

Numerical results, k- turbulence model

Anti-phase and in-phase vortex shedding around cylinders

Re=200G/d=2.4

Williamson, (1985)

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

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

Rec=100G/Hc=0,75Bistable vortex shedding

For G/d >1.5 synchronization of the vortex shedding in anti-phase or in-phase

Geometry of the computational domain

ReH=800 (Rec=100) Pr=0,71 (Air) Transverse bar separation distance, G/H or G/Hc

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.

Boundary Conditions Inlet:

Fully developed parabolic velocity profile Constant temperature To

Walls:

Constant wall temperature Tw=2To

Thermal entrance region

Boundary conditions

Outlet: wake equation to produce little reflection of the unsteady vortices at the exit plane

00

xU

t

,,VU

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.

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

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

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*

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*

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*

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

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

Flow pattern (11 – 4)

Flow pattern (3)

Flow pattern (2)

Flow pattern (1)

Instantaneous temperature field Case 1

Instantaneous local skin friction coefficient on the channel walls. Case 1 Cf= / (1/2Uo**2) : wall shear stress

Inferior wall Superior wall

Local skin friction coefficient on theinferior channel wall. Cases 11 to 6

Local skin friction coefficient on the channel walls.Cases 5 to 1

Superior wall Inferior wall

Local Nusselt numbers:Cases 11 to 6

Local Nusselt numbers:Cases 5 to 1Inferior wall Superior wall

Frequency: Case (2)Velocity U, Position: 2Hc behind the bar Inferior bar

Superior bar

Frequency: Case (2)Velocity V, Position: 2Hc behind the bar Inferior bar

Superior bar

Frequency: Case (2)Drag coefficients Inferior bar

Superior bar

Frequency: Case (2)

Lift Coefficients Inferior bar

Superior bar

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

Dominant frequency of the flowlow frequency modulation in cases: G=0.0625, 0.09375, and 0.125H

f G/H=0 = 1.14

Skin friction coefficient on channel wall Cf= / (1/2Uo**2) : wall shear stress

Drag coefficients for the lower and superior bar Cd=D/(1/2Uo**2)d

Cd G/H=0 =5

Lift coefficients: lower bar, superior bar Cl=L/(1/2Uo**2)d

Mean Nusselt number : inferior wall and superior wall Nu=hH/k q=hT wall heat flux

nu G/H=0 =11

Apparent friction factor f=PH/(Uo**2)L

f G/H=0 = 0.164

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

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.

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.

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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