the effects on heat transfer of unsteady flow around two square bars mounted staggered in a plane...

ht. Comm. Heat Mass Tnzmfh Vol. 29. No. 2. pp. 2X%242. 2002

pw4-Qn dopyright Q 2002 J&vier S&ace Ltd Printed in the USA. All ri@a resend

0735-1933~ front matter

PII: So73s.1933@2)tM314-7

THE EFFECTS ON HEAT TRANSFER OF UNSTEADY FLOW ABOUND TWO SQUARE BARS MOUNTED STAGGERED IN A PLANE CHANNEL

Alvaro Valencia

Department of Mechanical Engineering Universidad de Chile

Casilla 2777, Santiago, Chile e-mail: [email protected]

(Communicated by J.P. Hartnett and W.J. Minkowycz)

ABSTRACT The unsteady larninar flow and heat transfer characteristics in a plane channel with two square bars mounted staggered to the approaching flow are numerically analyzed. A finite volume method is applied with a fine grid and time resolution. The longitudinal and transverse separation distances between the bars are varied, whereas tbe bar height to channel height d/H, and the Reynolds number based on channel height Re, are d/H=118 and 800 respectively, Complex flow patterns develop in the channel due the interaction between the two staggered square bars. For the cases with IJdl 2 shedding of vortices with only one frequency am present in the flow, and biased flow with frequency modulation of the vortices shedding for the cases with LJd<2 are found. o 2992 Ekvier Sdence Ltd

The technical background for this investigation is the cooling of electronic components. An analysis

shows that the Reynolds numbers by air cooled components are small. Often channel Reynolds numbers

between 100 and 1000 are of interest, which implies a nominal laminar flow. The two dimensional flow

in a plane channel with two square bars arranged staggered to the approaching flow is considered in the

present investigation. The bars induce unsteady transverse vortices to augment fluid mixing. Transverse

vortices have their axes transverse to the flow and are consistent with two dimensional flow.

Breuer et al. [ 11 investigated in detail the confined flow around a square bar mounted inside a plane

channel with a blockage ratio of 118 by two entirely different numerical techniques, namely a lattice-

Boltzmann automata and a finite volume method. Accurate computations were canied out on grids with

different resolutions. Stroubal numbers S were determined for the entire Reynolds number range. Both

methods provide a local maximum of S at R-150. Compared with the scattered data in the literatme, the

233

234 A. Valencia Vol. 29, No. 2

deviations between both methods are almost negligibIe. The results of [l] on drag coefficient, variation of lift

coefficient and Strouhal number will be used as benchmark in the present work. The unsteady flow around a

square bar can still be considered as two dimensional only for bar Reynolds numbers Re&OO, therefore, two

dimensional numerical simulations should be carried out considering this limit. Valencia [2] presented

numerical studies of the flow and heat transfer in a channel with a built-in tandem of rectangular bars. Data

are presented for channel Reynolds numbers ranging from 100 to 400, and longitudinal bar separation

distances Ud ranging from 3 to 9. The key conclusion is that for longitudinal bar separation distances IJ& 5

the mean heat transfer enhancement is constant. A numerical investigation conducted Rosales et al. [3] to

analyze the unsteady flow field and heat transfer characteristics for a tandem pair of square bars in a laminar

channel flow. They studied the drag, lift and heat transfer coefficients from the downstream heated bar due to

inline and offset eddy-promoting bars. The results show that the drag coefficient and bar Nusselt number

decrease as the heated bar approaches the wall.

Williamson [4] studied the flow behind a pair of bluff bodies placed side by side in a stream using a

variety of flow visualization methods. Above a critical separation distance between the bodies, vortex

shedding synchronization occurs (2cT/d<6), either in phase or in anti-phase. Below a critical separation

distance between the bluff bodies the flow becomes asymmetric, the bulk flow between the two cylinders

deflects, the deflection to one side or the other can equally take place. Williamson observes in this regime

certain harmonic modes of vortex shedding whereby the shedding frequency on one side of the wake is a

multiple of that on the other. The flow pattern show a bi-stable nature. D. Sumner et al. [5] identified nine

flow patterns of the flow around two circular cylinder of equal diameter, arranged in a staggered

configuration. They observed shear layer reattachment, induced separation, vortex pairing and

synchronization, and vortex impingement for the different arrangements. They revealed that vortex

shedding frequencies are associated with individual shear layers; more specifically, the two shear layers

from the downstream cylinder often shed vortices at different frequencies.

Hayashi and Sakurai [6] performed experimental investigations on the wake interference of a row of

normal flat plates, consisting of two, three or four plates arranged side by side in a uniform flow with

Reynolds numbers of about 10’. When the transverse separation distance between the two flat plates is

less than three, the flows through the slits are biased either upward or downward in a stable way (except

for a two-plate row with a transverse separation distance of 2.75 which shows an unstable biased flow),

leading to multiple flow patterns for a single separation distance. The flow pattern for equal sized square

bars in side by side arrangement ate categorized into three regimes: single vortex street, bi-stable flow

and two vortex streets, Bosch [7]. For transverse separation distances between the two bars, T/d< 1.4, a

single vortex street is formed as in the case of a single bluff body. At the critical spacing of 1.44TIdC2.4,

bi-stable flow is found. The flow is biased to one side and intermittently flips to the other side. The flow

Vol. 29, No. 2 FLOW AROUND TWO SQUARE BARS 235

changes from the biased pattern to two symmetric vortex streets at T/d 2 2.4 and either in-phase or anti-

phase vortex streets are found.

In this study, attention is concentrated on the flow and heat transfer in a plane channel with two square

bars mounted staggered to the approaching flow. The longitudinal and transverse separation distances

between the bars L/d and T/d are varied, whereas the bar height to channel height d/H and the channel

Reynolds number Re are constant. The different flow regimes are characterized an their influence on

pressure drop and heat transfer on the channel walls are quantified.

Mathematkal Model and Gapmetry

The flow is assumed to be unsteady, two-dimensional, and laminar. The relevant conservation equations

describing the flow and temperature fields am the continuity, the timedependent NavierStokes equations,

and the energy equation. The fluid is assumed to be Newtonian with constant properties and the dissipation

terms in the energy equation are neglected. The velocities were made dimensionless with the averaged

velocity II0 at the inlet, all lengths were made dimensionless with the channel height H, the pressure with

~U,J*, and the temperature with the inlet tempemture T, respectively. Also the time was made dimensionless

with I-I/U& The Reynolds number Re is defined as #J&I/p,.

The geometry studied is shown schematically in Fig. 1. The streamwise length of the channel was set

equal to 5H, where H is the channel height, and two square bars of height d=O.l25H were placed staggwed

to the approaching flow. The channel Reynolds number Re was set to 800, and the air flow (FzO.71) is

hydrodynamically fully developed at the inlet. The tempemtums at the top and bottom channel walls are

constant and are equal to T*2T, and the inlet temperature T, is uniform. The bars do not have imposed

temperature and their thermal conductivity are the same as of the fluid. The bars here only generate

transverse vortices and the heat transfer surface is the same as in the plane channel without mounted bars.

The exit boundary conditions are chosen to minimize the distortion of the unsteady vortices shed from the

bars and to reduce perturbations that reflect back into the domain. It was found that the wave equation was

more compatible that setting the filst derivatives in the axial direction equal to zero with the physics at the

exit plane.

The computations wem made for six dimensionless longitudinal bar separation distances IJd ranging from

0 to 4 for a constant transverse bar separation distance T/d=l.75, for the 1ongitudinaI bar sepamtion distance

Ud=O the cases with T/d=0 and T/d=1 were also simulated as limit cases of one square and mctanguku bar

mounted on the channel axis respectively. The case with Ud=O and T/d=0 was the sang caIculated by Bmuer


et al. [l], this case was used as reference. The Strouhal number, drag coefficient, and variation of lift

coefficient were compared with the results of [l] to validate the grid size used in the present work.

FIG. 1 Computational domain

The flow losses are evaluated with the apparent friction factor defined asf~=1/2(C~+C/1+C~,f~~~~~+

&(&OH), where cfi and Co are the skin friction coefficients on the channel walls, Co, and Cm are the drag

coefficients on the two bars mounted staggered to the approaching flow in the channel.

Numerical Method

‘Ibe differential equations introduced above were solved numerically with an iterative finite-volume

method, details of which can be found in Valencia [2]. The convection terms in the equations were

approximated using a power-law scheme. The method uses staggered grids and Cartesian velocity

components, handles the pressure-velocity coupling with the SIMPLEC algorithm. A first order accurate

implicit Euler method was used for time discretization in connection with a very small time step

Ar=AtU&I=O.ooo4 to capture the complex unsteady flow with a grid size of 960x192 control volumes. The

time step satisfied the Courant condition, C=U,Ar/AX=O.I. The calculation with the different grids sixes

were performed with different time steps, in such a way that the Courant number of the flow was constant

C=O.l.

In the cases with low frequency modulation of the flow approximately 5000 time step were needed for

one period of vortex shedding. In these cases also the computations are made for mom than twenty periods to

obtain representative mean values. To determine these means values the program run until a unsteady but

periodic state is reached, and then the values of all fields in each 1116 of one period are saved. A typical run

of 2x10’ time steps with 960x192 grid points takes about 100 hours on a personal computer with a Pentium

lV processor.

Vol. 29, No. 2 FLOW AROUNLI TWO SQUARE BARS 237

Results and Discussion

To check grid independence, numerical simulations of the case with one mounted square bar on the

channel axis, L/d=0 and T/d=O, were performed with ten different grids sixes ranging from 320x64 until

1040x208 control volumes, and the results were compared with the presented in [ 11. The mean and variation

of drag coefficient, the variation of lift coefficient, and Strouhal number or eddy-shedding frequency of

the flow were compared. The differences on these parameters calculated with the 960x192 grid size with

the results of [l] were inferior to 5%. Therefore the grid size with 960x192 control volumes with a fine

dimensionless time step of 0.0004 will be used for the correct simulation of the unsteady laminar flow

with two bars mounted staggered in the channel.

The structure of the flow in the computational domain will be discussed. It will be illustrated through the

use of instantaneous velocity vectors. Figure 2 shows insta&neou s maps of fluctuating velocity vectors for

three cases Ud=O, 0.5 and 2 with T/d=l.75. The structure of the flow in the channel changes dramatically

with the longitudinal bar separation distance Ud. For the case with L/d=2 unsteady flow with shedding of

vortices with only one frequency or Strouhal number of S=O.182 was found, also the top bar in this

longitudinal position stabilizes the flow behind the bottom bar and inhibits the shedding of unsteady

vortices from the bottom bar, Fig. 2 (c).

For the cases with Ud=O, 0.5 and 1 complex vortices structures were found, with a low frequency

modulation ‘of the flow. The flow has a bi-stable behavior, leading to multiple flow patterns for a single

longitudinal bar separation distance. Figures 2 a) and b) show velocity vectors for the cases Ud=O and 0.5

characterized with non-dimensional low frequencies of ZM.085 and S=O.O52 respectively. The velocity

vectors of Fig. 2 (a) show that the vortices shedding from the bottom bar are bigger than the shedding

from the top bar, the opposite is observed in the case with Ud=O.5, Fig. 2 (b). This behavior can change

with the low frequency that modulate this flow, the flow biased either upward or downward in a periodic

form. Since the origin of the biasing is not clear, it is interesting to see that also occurs in laminar, low

Reynolds number flows. A similar behavior was found for the case with Ud=l with a low frequency

modulation of this flow of S=O.O7.4.

Fig. 3 shows the power density spectrums of drag and lift coefftcients for the bottom and top bars in

one case with frequency modulation of the flow, Ud=O.5 and T/d=1.75. The frequencies in the Figure 3

have been normalized with the channel height H and the mean velocity U,, and therefore they do not

correspond to the Strouhal number S. The signals were processed by means of the Fast Fourier

Transform, (FFl’).


FIG. 2 Velocity vectors for the cases Ud=O, Ud=OS, and LJd=2 with T/d=1.75 in the region 1.2%x/H<3.0

The characteristic dimensionless frequency fHAJ, of vortex shedding around a square bar in a channel

is around 1.5, however this frequency dominates only in the lift coeffkient of the top bar, in the lift

coefficient of the bottom bar appears a harmonic around the value 2.2, Fig. 3. The low frequency

modulation of the vortex shedding around the bars is more clear to see in the spectrum of drag

Vol. 29, No. 2 FLOW AROUND TWO SQUARE BARS 239

coefftcients around the bars. In these spectrums appear the frequency of 0.4 and the frequency of 1.5 is

completely suppressed in these spectrums. The power spectrums indicate that the flow has a transitional

character with this longitudinal separation distance L/d. A similar behavior were found for the cases with

L/d=0 and 1.

(a)

09

PIG. 3 Power density spectrum of drag and lift coefficient: (a) bottom bar,

(b) top bar, case T/d=l.75, Ud=O.5

The effects of the longitudinal separation distance L/d on mean values of integral parameters of the

unsteady flow around the two square bars mounted staggered in the channel are shown in Table 1. The

Strouhal numbers were calculated with the dominant frequency present in the time signals of the drag and

lift coefficients of the two bars, see Fig. 3. The cases with one rectangular bar ( T/d=l, L./d=0 ) and with

one square bar ( T/d=O, Ud=O ) mounted on the channel axis were also computed, and the mean values

are shown in Table 1 for comparison. The bars have a repulsion force in the transverse direction of the

flow for L/d<3 represented with the mean values of the lift coefficients CL. The forces on the bars in the

longitudinal direction represented with the drag coeffkients Cn decrease with Ud. The drag coefficients

in the different arrangements with two bars can be not exactly compared with the case with only one

square bar mounted in the channel ( T/d=O, Ud=O ), because the velocity profile upstream the bars is


TABLE 1

Mean values of Strouhal number, drag and lift coefficients on the bottom and top bar, and skin friction coefficient on the channel walls

Figure 4 compares the time averaged Nusselt number distributions on the top channel wall for the

eight cases. The local Nusselt numbers take a maximum at the inserted position of the bars, and other

smaller local maximums. The first one results from flow acceleration due to the blockage effect of the

two mounted bars, while the other local maximums are caused by the shedding of vortices from the bars.

The three cases with a strong low frequency modulation of the flow, L/d=O, 0.5 and 1 with T/d=1.75,

show higher local heat transfer coefficients due the transitional character of these flows.

To evaluate the heat transfer enhancement and pressure drop increase in the channel with staggered

square bars the mean Nusselt number and the apparent friction factor should be compared with the values

for a channel without vortex generators. The Nusselt number and apparent friction factor in a channel are

Nu,,=7.68 and f,=O.OlS respectively for a channel Reynolds number of Re=800. Fig. 5 shows the mean

Nusselt number and apparent friction factor for the eight studied cases. The cases with T/d=1.75 and

Ud=O, 0.5 and 1 have higher heat transfer enhancement and pressure drop increase compared with the

cases with T/d=1.75 and UdZ2. In those cases with more frequencies present ( T/d=1.75, L/d=O, 0.5 and

1) the flow has a transitional character and hence transfer more heat on the channel walls and generates

higher pressure drop. The decision about the optimal arrangement for heat transfer enhancement depend

of the constraints under which the comparison is made, in this work the effects of Ud on heat transfer and

pressure drop are quantified and shown in Fig. 5 for constant airflow rate.

Vol. 29, No. 2 F’LOW AROUND TWO SQUARE BARS 241

------- T-1.75 L-2 --- T-1.75 L-5 ._..._. . . T-,.75 W

FIG. 4 Time averaged local Nusselt number on the top channel wall

1.5

1.4

1.3

1.2

1.1

A

A

OO ..O

a 0

0 0

l 0

11

10

D

B

7

6

5

4

3

Mean heat transfer enhancement and pressure drop increase, 01 Nu/Nuo 0: f/fo with T/d=1.75.

A:Nu/Nuo A: f/fo for the cases T/d=1 and T/d=O.


com!lllalonq

Numerical simulations were used to explore the unsteady laminar fluid flow and heat transfer in a

plane channel with two square bars mounted staggered to the approaching flow. The effects of vortex

shedding have been captured by solving the continuity, the Navier-Stokes and the energy equations.

Computations were made for six longitudinal bar separation distances IJd for a constant transverse

separation distance T/d. Two cases were also simulated as limit cases of one square and rectangular bar

mounted on the channel axis. The numerical results reveal the complex structure of the flow as function of

LJd for a constant T/d=1.75. With 2.01uda.O unsteady flow with only one present frequency in the flow

was found. For the cases with oIud<l complex vortices structures with more frequencies present and

transitional character of these flows were found.

Acknowledmnents

The financial support received of CONICYT CHILE under grant number 1010408 is gratefully

acknowledged.

cf CD

d

D

f

fppP

h(x)

H

L

skin friction coefficient, rw /( 1/2pUoZ)

drag coefficient, D/( 1/2pU~d)

bar height, m

drag, N/m

eddy shedding frequency, Hz

apparent friction factor =QIQL)Apl( 1/2pIJ0~)

local heat transfer coefficient, W/m2K

channel height, m

longitudinal bar separation distance, m

Fr

Re

S

T

T0

T,

Tb

uo

T

Nu=h(x)H/k Nusselt number rw

Pmndtl number, u/a

channel Reynolds number, l&H/v

Strouhal number, fdiUc

transverse bar separation distance, m

inlet fluid temperature, K

channel wag temperature, K

bulk temperature, K

channel-averaged velocity at the inlet, mls

nondimensional time, tU&l

wall shear stress, N/m2

1.

2.

3.

4.

5.

6.

7.

M. Breuer, J. Bemsdorf, T. Zeiser, F. Durst, Inr. J. of Heat and Fluid Flow, 21, 186-l% (2000).

A. Valencia, Heat and Mass Traqfer, 33,465470 (1998).

J. L Resales, A. Ortega, J.A.C. Humphrey, Int. J. of Heat and Mass Tran.$er, 44.587-603 (2001).

C.H.K. Williamson, J. of Fluid Mechanics, 159, l-18 (1985).

D. Sumner, S.J. Price, M.P. Pdidoussis, J. of Fluid Mechanics, 411.263-303 (2000).

M. Hayashi, A. Sakurai, J. of Fluid Mechanics, 164, l-25 (1986).

G. Bosch, Experimentelle and theoretische Untersuchung akr instatiotiren Stimung um &ndrische strukturen, Ph.D. Dissertation, Universitat Fridericiana ru Karlsruhe, Germany, (1995).

Received January 14, .kO2

the effects on heat transfer of unsteady flow around two square bars mounted staggered in a plane...

Documents