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AIAA 91-1311 A Compatibility-Satisfying Method for Laminar Channel Flows with a Heated Fence S. M. Liang National Cheng Kung University Tainan, Taiwan, R. 0. C. W. B. Tsai Far East Engineering College Tainan, Taiwan, R. 0. C.

AIAA 26th Thermophysics Conference June 24-26, 1991 / Honolulu, Hawaii

For permission to copy or republish, contact the American Inst ie of Aeronaufics and AstrOnOutiCS 370 L'Enfant Romenade, S.W., Washington, D.C. 20024

AIM-91-1311-CP A COMPATIBILITY-SATISFYING METHOD FOR LAMINAR CHANNEL

FLOWS WITH A HEATED FENCE

W

Shen-Min Liang*

National Cheng Kung IJniversity, Tainan, Taiwan, R. 0. C

and

Wen-Bin ’hi**

Far East Engineering Collegc, Tninnn, Taiwan, R.O.C.

A b s t r a c t

The laminar forccd convection associated with its hcat

transfer of a two-dimensional channel with a heated fence

is Ynvestigated numerically. The Reynolds number, IZen,

based on thc channel height, I € , ranges from 100 to 800

and thc blockage ratio, h j H , from 0.25 to 0.4. TIic re-

sults were obtained with air as the working fluid. The

incompressible Navicr-Stokes equations governing the flow

are solvcd on non-staggered grids by using the primitive-

variable approach. The Neumann boundary condithn de-

rived from the normal momentum equation i s prescribed ror

solving thc pressure Poisson equation in which the source

term is corrccted at each iteration The multiple soltions of

pressure resulted from the Neumann condition i s uniquely

determined by fixing the pressure to be unity at the first

point on the upper wall. In order to verify the accuracy

of the prcscnt method, thc computed results arc compared

with the cxperiniental data of Tropea and Gackstatter for

those Ikyno lds numbers at which the flow maintains its

two-dimensionality in experiment. The comparison is sat-

isfactory. The heat transfer ratc of the channel flow is es-

tablished by correlating the overall average Nusselt number,

Nut, thc Reynolds number, Ren, and the blockagr ratio,

h j N , of the form: K t 7 0 . 6 7 4 ( R e ~ ) 0 ~ 3 ’ G ( ( A / H ) ~ ~ ~ ~ ~ ~ z .

‘I Associate professor, Instit,utc of Aeronautics and

‘U

-

.. ~~~ ~ _ ~ _ _

Astronautics. Mcrnbcr A I A A . * * Lecturer.

Copyright 01991 by authors. Published by the American

Institute of Aeronautics and Astronautics, Inc. with per-

mission. -i

Nommclat~llrQ

=::fence height and channel height, rrspcctively

= channel length

L reattachment length

7 local and average Nusselt numbers, respec-

tively

:: pressure of fluid

Prandtl number

= Reynolds number based on channel height

and fence height, respectively

= time and artificial time, respectively

= dimensional temperature

= dimensional temperature on heated elerncnt

:= dimensional temperature of inflow

= maximum velocity at upstreani

= dimensionless velocity components in Cnrte-

sian coordinates

:= Cartesian coordinates

: fence width

=: thermal diffusivity of fluid

= viscosity of fluid

: density of fluid

-~ dimensionless temperature, (T T m ) / ( T w

To3 1

In t roduc t ion

Separated flow near obstacles or steps has brrn a re-

search subject for several decadcs. This problem has rele-

vancc in a multitude of engineering applications, arid simple

geometry also provides a good meams for testing thoorct-

ical or numerical methods [l-91. In recent ycars, the dc-

sign of high-performance heat exchangers and the cooling

of electronic eqnipments require accurate quantitative in-

formation on hcat transfer rates in separated flow regions.

Numerical simulation can economically meet the need it correct mathematical model could be established.

The present work concerns with a fully-developed, lam-

inar airflow past through a heated fence in a two-dimensional

channel, as shown in Fig. 1. The heat source is maintaincd

a t a constant temperature which is higher than thr up-

stream flow temperature. The fence is considered to be

thin, hence its width is fixed at one-tenth ol the channel

height. The geometry and flow parameters arc the fence

blockage ratio, h / H , and the Reynolds number, Re", based

on the the channel height. The objective of this work is to

develop an accurate numerical method for establishing a

heat transfer correlation of the flow. r . I h c flow over a rectangular obstaclc can, in gcneral,

induce two or three separated flow regions. Uergeles and

Athanmsiadis 11) have observed that a recirculation region

developed ahead of and behind the fence and a third rc-

circulation region with consatnt length is on Lhr top sur-

face of the obstacle, if the obstacle width-to-height ratio,

w l h , is greater than 1. They also showed that the length

of the downstream recirculation region behind the obsta-

cle increases almost linearly with the obstaclc width for

w / h < 4. Later, Tropea and Gackstatter [ZI further in-

vestigated this flow in a fully developed chanrlel Row alld

measured the locations and sizes of recirculation regions for

Reynolds number ranging from 150 to 4500 (bascd on the

channel half-width). They found that an abrupt decreac

in the length of the downstream recirculation region in the

case of high blockago ratio (2 0.5). The abrupt dccreasp

was attributed to an appearance of a recirculation region

on the upper wall. Their results for those Reynolds num-

bers at which the flow maintains its two-dimcnsionality in

experiment are nscd to verify the present numerical results.

Thc hcat transfcr rate in a laminar flow past a surface-

mounted rib was studied numerically by Hsieh and Ilnang

using thc wcll-known SIMPI,E/IID codc /IO]. I n tlrcir stzldy

the temperature on the lower wall and the rib mountcd on

it was niaitained at a constant temperature. A hcat thilris-

fer correlation was established as the form: Nui O . I I ~ : ~ I - W

for Reh 5 2 5 4 and w l h e: I ,

relates t,he overall average Nusselt number, tlici Itrynolds

number and the obstacle aspect ratio. However, thcir corn-

puted data are largely deviated from the corrclution f<jr

large obstacle aspect ratio, ( w l h 2 2 ) . Thus, an a c r ~ d r

method for calculating the incompressible now assoc i ;~ i , ,~ l

with its hcat transfer is desirable.

Recently, Gresho and Sani [ l l ] have analyacd thr wcII-

posed problem of the Navier-Stokes cquations associat,od

with appropriate initial and boundary conditions. mrcy

showed that the Dirichlct and Neumann boundary condi-

tions for the pressure Poisson equation derivcd rrorn t izF

momentum equations give thc same solution. Abclallah

arid Dreyer 112) confirmed the r e d l of Gresho aud Smi

on the inviscid stagnation point flow problem. J l o w e ~ ~ ,

the prescription of a Dirichlet boundary condition hr t , h

pressure Poisson equation seems to be difficult for a ~ l ~ ~ r ~ , ~ , ~ l

Row with ai obstacle. On the other hand, the prrscripi,ion

of the Neumann boundary condition can praducc r n u l t i p l v

solutions for pressure.

W

In this study, the Neumann boundary condition dc-

rived from the normal momentum equation is prescribed.

Th? multiple solutions of pressure resultcd from thc Ntrtr-

mann condition are uniquely determined by assuming thc

upstream pressure to be unity at the first point on the up

per wall. As Abdallah I131 pointed ou t that t h r compati-

bility condition which relates the source term i n tlic prcs-

sure Poisson equation and the Neumann condition rimy not

he automatically satisfied on a non-staggered grid without

using consistent schemes. In order lo satisfy the compati-

bility condition, a uniform correction for tlic source term,

introduced by Briley [ I I ] , is implemented during iter;rt.iou

process.

LIy solving the energy equation, tbc authors have fuoltrlil

that from past numerical experiments, upwind cliffcrcn<-

ing for convective terms was more stable, bnt prrrd~lrnrl

much numerical damping for the thermal plume b c h i u r l .J

the heated obstacle, compared to central differencing. On

the other hand, ccntral differencing could produce a large

dispersive effect in truncation errors, and hcncc gcncrates

oscillating solutions, leading to negative values of tempera-

ture. Since truncation errors are proportional to mesh sizes,

a uniformly distributed grid with small mesh sizes was uscd

to reduce to truncation errors and hcncc the associated dis-

persive effect.

'W'

Based on the above arguments, central differencing is

used for approximating all the spatial derivatiws, convcc-

tivc terms included, in thc momentum, energy and pressure

Poisson equations. The pressure Poisson equation, modi-

fied by addition of a false-transient term, is integrated in an

artificial time. These discretized equations were solved hy

the two-step Alternating Direction lniplicit (ADI) mathud

1151 as done by Ghia et al. 1161. Steady-state solutions are

obtained through an itcrativc procedure.

M a t h e m a t i c a l Formula t ion

A. Governing Equations

Assuming there are no viscous dissipation and no cx-

ternal forces present, the equations governing the two- di-

mensional, incompressible, laminar channel flow with a

heated fence are the full Navier-Stokes equations:

v

4

Continuit.y Equation D - : V . V = O (1)

Momcntum Equation Vt + V . V V = -Vp+V2? /Rei, (2)

Energy Equation Bt t -?. V? = V 2 6 / ( R e 1 , P 7 ) (3)

-t 4 -->

Thr ahovc equations have been properly nondirnensional-

ked. The spaces variables x , t ~ have been normalizcd by I I ,

the velocity components u , v by l J , the pressure p hy p l J 2 .

Thc energy equation (3) is decoupled from other equations

and can be solved once the velocity distribution is obtained.

Instcad of directly solving Eqs. (1) and (2), an auxil-

iary equation was generally derived to replace the cont,inu-

ity equation by taking the divergence of Eq. (2):

v 2 p = s (4)

with

a S =: - [ - ( u u z + vuy) -1 ( u , ~ ~ -i- u, , ) /Re1~1

+ - [ - (uvz vvy) + ( v m ~t rea] Ilt

az a v aY

The pnrpvsc of retaining the x , y derivatives of u , I, and the

local dilatation D in the source term S is to damp out, the

niimcrical insatbilities during iteration process 1171, becallsc

D may not he zero at the incipient stage. If D is zero, the

continuity equation is satisfied, and Eq. (4) reduces t,o

V2P = 2(u,v, - .,%)

Thus, Rqs.

appropriate boundary conditions.

B. Boundary Conditions

(2)-(4) are the equations to be solvcd with

Since the upper and lower wall surfaces of the chan-

nel are assumed to be adiabat.ic, afJ/aylwa~t = 0. The di-

mensionless temperature on the heated fence is mity. The

boundary condition on the body surface is the no-slip con-

dition, u = v = 0. At the upstream boundary, a fully devel-

oped velocity profile is imposed and the inflow temperature

is zero. At the downstream boundary, zero gradients of

both the velocity and the ternperaturc are prescribed. The

boundary condition used for the pressure Poisson equation

is the normal momentum equation, except the first point

of the upper wall at which the pressure is imposed to he

unity.

The boundary conditions prescribed are summarized

below: (refer to Fig. 1)

1) u = v = 0,

P,Ps and P,P5;

2) u = 4y( l - y),

3) u = v = 0,

8 , = 0, p , = u , , / R e H , on sides P2P3,

v = 0, 0 = I , p = 1, on side P IP2 ;

8 = 1, p , = t ~ , , / R e r ~ , on sidcs PSPB and

p7ps;

4) VI - 1) = 0, 0 = 1,

5) u , = vl: .:- 0 , = 0,

p , - ayy /Re l i , on side P,P,;

p , := ( - u u , + u,,,,)/RerI, on side

p3p.1.

Numer ica l Method

A. Finite-Difference Approximation

Central differences are used to approximate all the spa-

tial derivatives, convective terms included, i n thr momen-

tum, pressure Poisson and energy equations. Thc pressure

Poisson equation is modified by addition of a false-trasient

term

As suggested by Harlow and Welch, 1171 the local dilatation

D"+' in S is forced to be zero at each iteration. Hence,

the approximation of a D / & reduces to ~ - D " / A t . The D"

ran eliminate numerical instabilities during iteration pro-

cess. The two-step alternating direction implicit method

of Peaceman and Rachford 1151 is used to solve the dis-

cretizewd momentum, pressure Poisson and energy equa-

tions. One-sided difference formaulas of first-order accu-

racy are used for first derivatives. Fictitious points outside

the walls were introduced when central differences were

used to approximate second derivatives. The pressure at

the boundary except at one point is linearly extrapolated

from the known quantities at interior points through the

normal momentum equation

El. Compatibility Condition

Since the pressure Poisson equation is solved for the

pressure, the divergence theorem implies that the compat.

ihility condition:

( 5 )

which relates the source term S and thc Nrumann rondi-

tion. Abdalla has showed that the compatibility conditionn

( 5 ) may not be automatically satisfied on a non-staggered

grid without using a consistent scheme; consequently, a

non-convergence process can occur. Ilence, a correction,

A S , introduced by Briley, is computed at each itcration.

A S = (// Sdxdy - / $ds)/ // dzdy

A C A

which is used to uniformly correct the source term S . When

the iteration process converges, the correction A S

approaches zero and the compatibility condilion is satis-

fied automatically.

C. Solution Procedure

The steps involved in the solution procedure are listed

below.

1) Make initial guesses for u, w , p .

2) Solve Eq. (2) for u, u at n + 1 time step.

3) Evaluate aplan on the boundary from the normal mo-

mentum equation and compute A S .

4) Correct S using A S to obtain new pressure from Eq.

(4).

5 ) Calculate the pressure at the boundary from known

quantities at interior points.

6) Check for convergence criterion

for all i , j , where r is a preassigned value. The proce-

dure is repeated from step 2 to step 6 until c o n d i t i o n

(6 ) is satisfied.

7) Solve Eq. (3) for the temperature from the convergcd

solution (u, u) using the same convergence criterion ( 6 ) .

Results and Discussion

Figure 2 shows the computed streamlines at / I F I l -

400 and 800 for h / H = 0.25 and at R ~ H = 200 and 400

for h / H = 0.4. It is seen that the sizes of rccirculation

regions ahead of and behind the fence increase with t h

Reynolds number. The recirculating intcrrsity of the recir-

culation flow behind the fence also grows with the Rcynolds

number, because more energy is entrained into the recircu-

lating flow from the main stream for highcr Rcynolds num-

ber. By comparing Figs. 2b and 2d, it is indicated t.liat

at the same Reynolds numbcr, the higher fence produces

larger recirculation regions because of stronger bluckagr ef-

fect. The length of the downstream recirculation rcgion,

called the reattachment length, L,, at, various Itcynolds

numbers are plotted in Fig. 3. Notc that thc reatt,acli-

ment Icngths have been normalized by the fcnce Iicight,

h. The comparison of the computed realtachincnt lengths

for h / H L 0.25 with the experimental data of Tropea and

Gackstatter is quite satisfactory. No comparison was rnadr

for the case of h / H = 0.4, since no experimcntal data were

avaiable. The almost linear variation of the reattachrncnt

length with Reynolds number for lower blockagc ratio is

well predicted.

Figurc 4 shows pressure distributions at dilli.rcnt

Reynolds numbers for h / H L 0.25. Note that thc value

of the pressure contour not indicated lies between bwo a&

jacent indicated-value contours. It is clearly seen that thew

are a high pressure region ahead of the fence and a low pres-

sure region developed behind the fencc, resultcd fro111 the

blockage effect. Since the flow is fastly accelcrated nrar

W

the upstream corner, the flow acceleration leads to a large

pressure gradient in the vicinity of the corner. In the case

of Re" = 200, constant pressures at, downstream stations

wcrc wcll predicted, as shown in Fig. 4a, which incans that

a constant pressurc gradient is formed and the flow is fully

redeveloped at downstream. It is expected that similar re-

sults could be obtained for other Reynolds numbers if thc

computational domain is taken large enough. Moreover, a

lower pressure region behind the fence is developed, and

extends to the upper wall with a positive pressure gradicnt

in the 3 o'clock direction.

W

Figurc 5 shows the surface pressure distribution along

the upper wall. It is seen that a locally low surface pressure

region occurs on the upper wall behind the fencc. Because

of the formation of the low pressure region, the pressure is

decreascd with increasing Rcynolds number and blockagc

ratio. The upper-wall boundary layer at the low pressure

region may results in a flow separation which has bcen ob-

served in experiments 121. The tendancy to flow separa-

tion is also indicatcd in the skin friction coefficient distri-

butions. Figure 6 shows thc skin friction coefficient distri-

bution along the upper wall. The skin friction coefficient

first obtains a maximum and sharply decreases to a mini-

mum which is closer t o zero for larger Reynolds numbers. If

the skin friction coefficient changes its sign from a positive

value to a negative value, a recirculation flow occurs. Fig-

ure 7 shows the lower-surface pressure distributions. The

lower-surface pressure distributions are qualitatively simi-

lar to those (Runs 15 and 17) obtained by Leone and Gresho

1181, using finite element methods.

-

Figure 8 shows the temperature contours at various

Rcynolds numbers. The thermal plume is developed bchind

the heated fence, which is clearly seen in Figs. 8b-8d, in

particular, for larger Reynolds numbers. The zero-isotherm

stretches farther downstream as the Reynolds number in-

creases. The isotherm patterns are very similar for larger

Reynolds numbers.

Define the local heat transfer rate by the Nusselt num-

ber: as an

Nu = --

where n denotes the direction normal to Ibe fcnce surface,

and its average by the average Nusselt number:

- 1 Nu = - l'? Nuds

S

where s denotes the length of the fence surface. Figure 9

shows the distributions of the local Nusselt number along

the front, top and rear faces of the fencc. On the front

face, the local Nusselt number is found to increase with the

Reynolds number, and obtains its maximum at the top cor-

ner, The formation of the maximum local Nusselt number

is resulted from the flow acceleration near the corner where

convection dominates the heat transfer. On thc top face,

the local Nussclt number decreases along the surface. On

the rear face, the local Nusselt number changes very little,

sincc the heat transfer is primarily contributed by ronduc-

tion, not by convection. A similar result of local Nusselt

number distributions was obtained for h / H = 0.4. How-

ever, for h / W = 0.4, lhe peak value of the local Nusselt

number is larger than that for h / H == 0.25 at the samc

Reynolds number. The reason is that the flow accelera-

tion near the corner for h/H I 0.4 is fastcr than that for

h / H = 0.25. The variations of the average Nusselt number

on the front, top, rear faces and the overall average Nus-

selt number with the Reynolds number for h / H = 0.25 are

shown in Fig. 10. In general, the average Nussclt number

on each face increases with thc Reynolds number, except for

the rear face case. In this case, the average Nusselt number

obtains its maximum at Re 600. A result similar 1.0 that

for h l H := 0.25 was obtained for h / N == 0.4. Figure 1 1

shows the variations of the overall average Nusselt number,

Nut, versus the Reynolds number for two different values

of h / H . It is seen that the overall average Nusselt num-

ber for h / H I 0.25 is higher than that for h / H .I.. 0.4 at

a fixed Reynolds number. Namely, the lower the blockage

ratio, the higher the overall average Nusselt numbcr. By

taking the fcnce blockage ratio into account, a correlation

is obtained:

-

%, :: c ( R e , ) a ( h / H ) b for w = O.1H

where a = 0.376,b = - 0 . 2 9 2 , ~ = 0.674. Note that these

parameters may he functions of the fence width. The cor-

relation indicates that the overall average Nusselt uumhcr

is proportional to the Reynolds number, but inversely to

the blockage ratio.

Conclusion

A compatibility-satisfying method of the primitive-vari

able approach has been developed for solving two- dimen-

siona.1 incompressible channel flows with an isolated, heated

obstacle. The pressure Poisson equation, used to replace

the continuity equation, is solved with the prescription of

the Neumann boundary condition which is derivcd from

the normal momentum equation. The multiple solutions of

pressure resulted from the Neumann condition is uniquely

determined by fixing the pressure to be unity at the first

point of the upper wall. To ensure the satisfication of the

compatibility condition which relates thc source term in

the pressue Poisson equation and the Ncumann condition,

a correction process of Briley is implemented to uniformly

correct the source term on a non-staggcred grid. The ac-

curacy of the present method is verified hy comparing the

computed reattachment lengths with thc cxpcriniental data

of Tropes and Gackstatter. The heat transfcr rate of the

channel flow for w ~ 0.lH is established by the correlation:

%, :~ 0.674(Re~)0~37G(h/H)~~0~292.

Acknowledgement

This work was supported by National Sciencc Coun-

cil of R.O.C. under the contract of NSC 79-4001-3-006-48.

The authors are grateful to Dr. C. Gau for his helpful com-

ments.

References

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Surface-Mounted Obstaclc,” Trans. of the ASME, J .

Fluids Eng., Vol. 105, pp. 461-463, 1983.

2. C. D. Tropca and R. Gackstattcr, “Thr Flow over

Two-Dimensional Surface-Mounted Obstacles at Low

Reynolds Nnmbcrs,” Trans. of the /ISM?:. J . Fluids

Eng., Vol. 107, pp. 489-494, 1985.

3. B. F. Armaly, F. Durst, J. C. F. Pereira arid B.

Schonung, “Experimental and Theoretical Invcstiga-

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Vol. 127, pp. 473-496, 1983.

4. F. Durst and C. Tropea, “Flows over Two-Dirncll.iionaI

Backward-Facing Steps,” Structure of Comp1e:c Tirbu-

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mas and L. Fulachier (Eds). Berlin, Springer V<!rlag,

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i /

5 . W. Aung, “An Experimental Study of Laminar Ileal

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dency and Effect of Initial Shear-Layer Thickness i n

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7. C. Berner, F. Durst, and D. M. McEligat, ”Flow Around

Barnes, J . Heat Transfer,” Vol. 106, pp. 743-7119, 1984.

8. I,. P. Hackman, G. D. Raithby, and A. 1%. St,rong,

“Numerical Prediction of Flows Over Backward-Facing

Steps,” Int. J . Numer. Method i n Fluids, 1’01. 4, tip.

711-724, 1984.

9. S. S. Hsieh and D. Y. Hnang, ”Flow Charactcristirs of

Laminar Separation on Surface-Mounted Ribs,“ A / A A

J. , Vol. 2 5 , No. 6, pp. 819-823, 1987.

10. S. S. FIsieh and D. Y . Huang, “Numerical Computnlion

of Laminar Separated Forced Convection on Surfim-

Mounted Ribs,” Nurner. Heat T~ansjer-, Vol. 12, p p

335-348, 1987.

11. P. M. Gresho and R. I,. Sani, “On Pressure I iuunt lary

Conditions for the Incompressible Navier-Stokos ISqua-

t,ions,” Int. . I . Numer. Methods in Fluids, Vol. 7. p p

1111-1145, 1987.

12. S. Abdallah and .J . Dreyer, “Dirichlet and Ncumann

Boundary Conditions for the Pressure Poisson 1Sqiia-

tion of Incompressible Flow,” Int. J . Numer. Methods

in Fluids, Vol. 8 , pp. 1029-1036, 1988.

v

13. S. Abdallah, “Numerical Solutions for the Inrompross-

ible Navier-Stokes Equations in Primitivc Variables Us-

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70, pp. 193-202, 1987.

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Dimensional Steady Viscous Flow in Ducts,” .I. Corn-

put. Phys., Vol. 14 , pp. 8-28, 1974. i/

15. D. W. Peaceman and 11. FI. Rachford, “Thc N w n c r i c a l

A

I

I I 1 '7 % 77-,-r, , , , , , , , , , , , , , , ,- i-T-A ?z?

20.0

15.0

10.0

5.0

0.0

_t_ h/H = 0.40 . h,,, = 025 } Present results ___ h/H = 025. Experimental data [2]

--?--.--, 7 F--' 200 400 600 800

Reynolds Number, ReH I

Fig. 3 Comparison of computed reattachment lengths with experimental data.

Fig. 4 Pressure contours for diKerent Reynolds numbers, h / H = 0.25: (a) Re, = 200,

(b) Re, = 400, (c) Rei, = 600, (d) ReH = 800.

'd

w

1.2 -

~ Rei, -200 .- ......... Rei, = 400

Reir = 600 ...... Rei, = 800

, , , , , , 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 0.0 9.0 10.0 11.0 12.0

S, Distance along upper surface

Fig. 5 Pressure distributions along upper surface, h / H = 0.25

0

_ _ ~ _ . Reit = Z O O .... Reit = 400

Rei< = 600 ...... Reti = 000

...............

0.2 , I l I I / I

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 Distance along lower surface

Fig. 7 Pressure distributions along the bottom wall, h l H = 0.25.

1.0

/ , , I > ,

7 0.1 0.2 0 3 0.4 0 5 0.6

S, distance along fence surface

(b) R e , = 400, ( c ) Re , = 600, (d) Re,, = 800. r i ~ 9 Variation of local Nusselt number along

fence surface, h j H = 0.25

- --- Overall N u -- Front. Surface t_ Upper Surface A Rear Surface

P

/'.' ___I/

0 no0 500 700 c Reynolds Nirmher. Rei1

20.0

17.5

~ 15.0 a,

5 12.5

a

z a, 10.0

z 7.5

Y

m u,

- 5

a, w g 5.0

5 2.5

0.0

Fig. 10 Variations of average Nusselt numbers on front, top and rear faces and ovcrall average Nusselt number with

Reynolds number, h l H = 0.25.

Fig. 11 A heat transfer correlation for w / l I : 0 .1