Heat and Mass Transfer 39 (2003) 721–728 DOI 10.1007/s00231-002-0356-1

Laminar flow and heat transfer in confined channel flow past square bars arranged side by side Alvaro Valencia, Ronald Paredes

721 Abstract A numerical investigation was conducted to analyze the unsteady laminar flow field and heat transfer characteristics in a plane channel with two square bars mounted side by side to the approaching flow. A finite volume technique is applied with a fine grid and time resolution. The transverse separation distance between the bars (G/d) is varied from 0 to 5, whereas the bar height to channel height is d/H=1/8, and the channel length is L=5H. Different flow regimes develop in the channel due the interaction between the two mounted square bars, steady flow, flow with vortex shedding synchronization either in phase or in anti-phase, or biased flow with low frequency modulation of vortex shedding are found. Results show that the pressure drop increase and heat transfer enhancement are strongly dependent of the transverse separation distance of the bars and the channel Reynolds number.

Nu(x/H) Nu p P Per Pr Re St T To Tw Tb u

Keywords Vortex shedding, Square bars, Laminar flow, Heat transfer

U0 U

List of symbols Cf skin friction coefficient, sw/(1/2qU02) drag coefficient, D/(1/2qU02 d) CD d bar height, m D drag, N/m f eddy shedding frequency, Hz apparent friction factor = (H/2L) Dp/(1/2qU02) fapp G transverse bar separation distance, m h(x) local heat transfer coefficient, W/m2 K H channel height, m k thermal conductivity, W/mK L channel length, m

Received: 26 October 2001 Published online: 22 August 2002
Springer-Verlag 2002 A. Valencia (&), R. Paredes Department of Mechanical Engineering, Universidad de Chile, Casilla 2777, Santiago, Chile E-mail: [email protected] Tel.: 056-2-6784386 Fax: 056-2-6988453 The financial support received of FONDECYT Chile under grant No 1010400 is gratefully acknowledged.

v

V X Y

local Nusselt number average Nusselt number pressure, Pa non-dimensional pressure, p/qU02 period, d/(StH) Prandtl number, m/a channel Reynolds number, U0H/m Strouhal number, fd/U0 temperature, K inlet fluid temperature, K channel wall temperature, K bulk temperature, K Cartesian velocity component in the x direction, m/s Cartesian velocity component in the y direction, m/s channel-averaged velocity at the inlet, m/s non-dimensional Cartesian velocity component, u/U0 non-dimensional Cartesian velocity component, v/U0 non-dimensional Cartesian coordinate, x/H non-dimensional Cartesian coordinate, y/H

Greek symbols a thermal diffusivity, m2/s D difference h non-dimensional temperature, T/To m kinematic viscosity, m2/s s non-dimensional time, tU0/H wall shear stress, N/m2 sw

1 Introduction There have been numerous numerical and experimental investigations of unsteady flow past bluff bodies in different configurations. These investigations are relevant to many practical engineering applications such as electronics cooling and compact heat exchangers. Compact heat exchangers currently operate in the laminar regime. This is because the flow velocities and passage sizes are small and consequently the Reynolds numbers are low. Furthermore, the passage lengths are large, so the flow is fully developed on most of the channel. In order to

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enhance the heat transfer rates over the laminar fully developed values, devices that mix the main flow are introduced. A variety of such interruption devices have been proposed in the past. These have included placement of staggered and in-line baffles; louvers; and transverse vortex generators in the form of cylinders, grooved channels or square bars. Suzuki et al. [1] computed the flow around a square bar in a channel for bar Reynolds numbers ranging from 37.5 to 150 and blockage ratios ranging from 0.05 to 0.5. The computation reveals that the vortex street shows a different pattern of motion from its counterpart formed behind a bar placed in a uniform flow. The blockage ratio is indicated to be a major factor governing the conditions for the appearance of crisscross motion of the vortex. Breuer et al. [2] investigated in detail the confined flow around a square bar mounted inside a plane channel with a blockage ratio of 1/8 by two entirely different numerical techniques, namely a lattice-Boltzmann automata and a finite volume method. Accurate computations were carried out on grids with different resolutions. Strouhal numbers were determined for the entire Reynolds number range. Both methods provide a local maximum of St at Red
150. Compared with the scattered data in the literature, the deviations between both methods are almost negligible. The results of Breuer et al. on drag coefficient, variation of lift coefficient and Strouhal number will be used as benchmark in the present work. The unsteady flow around the square bar can still be considered as two dimensional only for bar Reynolds numbers Red £ 300, therefore, two dimensional numerical simulations should be carried out considering this limit. Tatsutani, et al. [3] studied tandem of square bars in a channel for bar Reynolds numbers between 200 and 1600 based on the downstream bar height. They observed distinct flow patterns that are dependent on a critical inter-bar spacing, kc, given by kc=168 Red–2/3. Below the critical spacing, two counter-rotating eddies formed in the gap between the square bars and vortex shedding was only observed for the downstream bar. At the critical spacing, eddy shedding was initiated for the upstream bar. A numerical investigation conducted Rosales et al. [4] 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. Valencia [5] computed 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 bar separation distances ranging from 1 to 4 channel height. The key conclusion is that for longitudinal bar separation distances L/H‡2 the mean heat transfer enhancement is constant. Williamson [6] 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 gap size

between the bodies, vortex-shedding synchronization occurs (1
2 Governing equations The flow and temperature fields are governed by the continuity, non-steady two dimensional Navier-Stokes and energy equations. The fluid properties are assumed to be constant and the viscous dissipation in the energy equation is neglected. The non-dimensional governing equations for the flow and energy transport can be written as: @U @V þ ¼0 @X @Y

ð1Þ


 @U @U 2 @UV @P 1 @2U @2U þ ¼ þ þ þ @s @Y @X Re @X 2 @Y 2 @X

ð2Þ

723 Fig. 1. Computational domain


 @V @UV @V 2 @P 1 @2V @2V þ þ þ ¼ þ @s @X @Y Re @X 2 @Y 2 @Y
2  @h @Uh @Vh 1 @ h @2h þ þ ¼ þ @s @X @Y Re Pr @X 2 @Y 2

ð3Þ

compatible that setting the first derivatives in the axial direction equal to zero with the physics at the exit plane, i.e.

ð4Þ

@/ @/ þ U0 ¼0 @t @x

The velocity components have been normalized with the average velocity U0 and the lengths with the channel height H. Re is the channel Reynolds number of the flow and Pr is the Prandtl number of the fluid. Also the time and the temperature were normalized with H/U0 and with the inlet fluid temperature To respectively. The geometry studied is shown schematically in Fig. 1. The streamwise length of the channel was set equal to L=5H, and two square bars of height d=0.125H were placed side by side to the approaching flow. The computations were made for twelve non-dimensional transverse bar separation distances ranging G/d from 0 to 5. The air flow, Pr=0.71, is hydrodynamically fully developed at the inlet, and the channel Reynolds number Re was first set to 800. After that we have varied the channel Reynolds number from 200 to 1000 for the case with G/d=0.75. The two dimensional laminar flow and heat transfer around a square bar mounted centered inside the plane channel of length L=5H shown in Fig. 1 was also investigated. The bar height was also d=0.125H, the channel Reynolds number was Re=800, and the inflow length was 1.5H. This geometry was the same used by Breuer et al. [2]. We use this case as reference, and we compare our Strouhal numbers, drag coefficients, and variation of lift coefficients with the results of [2] to validate our numerical method. The temperatures at the top and bottom channel walls are constant and set to TW=2To, and the inlet temperature To is uniform. The bars do not have imposed temperature and their thermal conductivity are the same as of the fluid. The redevelopment of the thermal boundary layer on the bar surfaces has a significant impact on heat transfer enhancement, if the heating is applied to the vortex generators. 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. We found that the wave equation was more

ð5Þ

where the variable / is the independent variable u, v, or T. Eq. (5) is enforced at the exit plane for the momentum and energy equations, Eqs. (2), (3), and (4). Local heat transfer on the channel walls were evaluated with the local Nusselt numbers calculated with the following equation: NuðxÞ ¼

hðxÞH ð@T=@yÞwall H ¼ k ðTb ðxÞ  Tw Þ

ð6Þ

The bulk temperature was calculated using the velocity and the temperature distribution with the equation: RH uT dy Tb ðxÞ ¼ R0 H ð7Þ 0 u dy The flow losses are evaluated with the apparent friction factor defined as: 1 d d fapp ¼ ðCf 1 þ Cf 2 Þ þ CD1 þ CD2 ð8Þ 2 2L 2L where Cf1 and Cf2 are the skin friction coefficients on the channel walls, CD1 and CD2 are the drag coefficients of the Table 1. Averaged values for different grid sizes in a channel with one mounted square bar. *: Values computed with the maximum velocity Umax=1.5U0

Number of CV

St*

Cd *

1000·DCd* DCl*

fapp

Nu

160·32 240·48 320·64 400·80 480·96 560·112 640·128 720·144 800·160 880·176 960·192 1040·208 Breuer [2]

0,000 0,118 0,124 0,128 0,131 0,133 0,135 0,137 0,138 0,139 0,139 0,140 0,145

3,060 1,462 1,495 1,479 1,465 1,451 1,438 1,427 1,417 1,407 1,400 1,393 1,36

0,00 0,19 5,82 8,93 11,96 14,58 16,76 18,64 20,17 21,52 22,54 23,39 25

0,0479 0,0489 0,0507 0,0508 0,0511 0,0513 0,0517 0,0520 0,0524 0,0527 0,0531 0,0536 –

8,257 8,399 8,428 8,450 8,473 8,485 8,496 8,499 8,506 8,508 8,518 8,524 –

0,000 0,128 0,285 0,363 0,427 0,475 0,510 0,539 0,562 0,581 0,595 0,606 0.6

3 Numerical method The differential equations introduced above were solved numerically with an iterative finite volume method, details of which can be found in Patankar, [9]. The convection terms in the equations were approximated using a powerlaw scheme. The method uses staggered grids and Cartesian velocity components, handles the pressure-velocity coupling with the SIMPLEC algorithm in the form given by Van Doormaal and Raithby, [10], and solves the resulting system of equations iteratively with a tridiagonal matrix

algorithm. A first-order accurate fully implicit method was used for time discretization in connection with a very small time step Ds=DtU0/H=0.0004 to capture the complex unsteady flow with a grid size of 960·192 control volumes. The time step satisfied the Courant condition, C=UmaxDs/DX=0.1, for this condition we have considered Umax=1.5U0. The calculation with different grids sizes were performed with different time steps, in such a way that the Courant number of the flow was constant C=0.1. In some calculated cases approximately 3600 time step were needed for one time period. A run of 2.5·105 time steps with 960·192 grid points takes about 160 hours on a personal computer with a Pentium IV processor. To determine means values the program should be run until a unsteady but periodic state is reached, and then the values of all fields in each 1/16 of one period are saved.

Fig. 2. Instantaneous maps of velocity vectors for four cases: (a) G/d=4.5, (b) G/d=4, (c) G/d=2, (d) G/d=1.5

Fig. 3. Time series of velocity vectors for G/d=0.75 in one period Per: (a) s=0, (b) s=0.25Per, (c) s=0.5Per, (d) s=0.75Per, Per=d/(StH)=1.464

two bars mounted side by side to the approaching flow in the channel. In addition, since the flows of concern are time dependent, an initial condition is required, the initial condition in the present calculations is a parabolic velocity field.

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4 Results and discussion To check grid independence in this work the case with one mounted square bar in the channel axis was simulated with Re=800, d/H=0.125, L/H=5. This case corresponds to the work of Breuer et al. [2]. We use this case as benchmark, and we compare our results of Strouhal number, drag coefficient, variation of drag and lift coefficient with the results of [2] to validate our numerical method. Values of integral parameters as the Strouhal number of the flow, mean drag coefficient, fluctuation of drag and lift coefficients, apparent friction factor, and mean Nusselt number on the heated channel walls are compared for twelve different grid sizes in Table 1. One can observe that with the grid of 960·192 control volumes the differences in all the integral parameters compared with the grid of 1040·208 control volumes are smaller than 3.6%. The differences among our numeric results with the grid of 960·192 control volumes and those reported in Breuer et al. [2] are also small. Therefore the grid of 960·192 control volumes with a fine time step of 0.0004 will be used for the simulation of the unsteady laminar flow around the square bar arrangements in the channel. 5 Variation of bar separation distance G/d for constant Re=800 The structure of the flow in the computational domain will be discussed for different transverse bar separation

distances and constant Reynolds number. It will be illustrated through the use of instantaneous velocity vectors. Figure 2 shows instantaneous maps of fluctuating velocity vectors for four cases G/d= 4.5, 4, 2, and 1.5. The structure of the flow in the channel changes dramatically with the transverse bar separation distance G/d. For the cases with bar separation distances of 5 and 4.5 the flow is steady, because the proximity of the channels walls inhibits the flow instability in those cases, the flow behind the bars has a recirculation zone of about 2d, Fig. 2(a). With 2.0 £ G/d £ 4.0 unsteady flow with antiphase shedding of vortices and with only one frequency was found, Figs. 2(b) and (c). The intensity of the shed vortices increases with the increase of the gap between the square bars and the channel walls. The non-dimensional eddy shedding frequency or Strouhal number of the flow increases in these cases from 0.21 until 0.24. With G/d=1.5 the binary vortex street can be characterized as in phase vortex shedding, with only one present frequency in the flow or Strouhal number of 0.24, Fig. 2(d). The transverse bar separation distance G/d=1.5 produces an important change in the flow pattern. For the cases with G/d=0.5, 0.75 and 1 complex vortices structures were found, with a low frequency modulation of the flow. The flow has a bistable behavior, leading to multiple flow patterns for a single transverse bar separation distance. Figure 3 shows four velocity vectors for the case G/d=0.75 in one period characterized with a non-dimensional low frequency modulation of St=0.0854.

Fig. 4. (a) Time variation of drag coefficients for the inferior and superior bars, (b) power density spectrum of drag coefficient for the inferior bar, (c) power density spectrum of drag coefficient for the superior bar. G/d=0.75

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The velocity vectors of Fig. 3(a) show that the vortices shedding from the lower bar are bigger than the shedding from the upper bar. This behavior can change with the low frequency that modulate this flow, the flow can bias 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 G/d=0.5 with a low frequency modulation of the flow of St=0.051. The flow structure in the case with G/d=1 can be characterized as in phase vortex shedding also with a dominant low frequency modulation of the flow, St=0.06. Fig. 4(a) shows the time variation of drag coefficients for the inferior and superior bars for 5 dominant periods in the case with G/d=0.75. Figures 4(b) and (c) show the corresponding power density spectrum of drag coefficients for the inferior and superior bars. The frequencies in the figures have been normalized with the channel height H and the mean velocity U0. The time dependence of the coefficients shows the low frequency modulation of this unsteady laminar flow around the square bars. The variation of the mean drag and lift coefficients with the transverse bar separation distance G/H show Figures 5(a) and (b). For unsteady flow with shedding of vortices in antiphase or in phase and with one frequency

of the flow, 0.1875 £ G/H £ 0.5, the mean drag coefficients of the two bars are equals and decrease lineally with G/H. With low frequency modulation of the flow, 0.0625 £ G/H £ 0.125, the mean drag coefficients of the bars are not equal, due the biased flow showed in Fig. 3. In the arrangements with G/H<0.5 the mean lift coefficients of the bars increase with a decreases of G/H, and the square bars mounted in the channel have a repulsion force, Fig. 5(b). The flow accelerates in the gap between the bars and therefore they have a repulsive force in the transverse direction. We have calculated the case with a mounted rectangular bar in the channel axis for comparison, d/H =1/4, G/H=0, Re=800. In this case, the mean drag and lift coefficients of the bar are CD=5 and CL=0 respectively, the Strouhal number of the flow was St=0.295. Figure 6 compares time averaged Nusselt number distributions for eleven cases with two square bars mounted side by side to the approaching flow. Figure 6(a) shows the cases with 0.3125 £ G/H £ 0.625, and Fig. 6(b) shows the cases with 0.0625 £ G/H £ 0.25. The local Nusselt numbers take a maximum at the inserted position of the bars, and other smaller local maximum. The first one results from flow acceleration due to the blockage effect of the two mounted bars, while the other local maximum is caused by

Fig. 5. (a) Mean drag coefficients, and (b) mean lift coefficients for the inferior and superior bars as a function of the transverse bar separation distance G/H, (G/d=8G/H)

Fig. 6. (a) Time averaged local Nusselt number on the inferior channel wall for 0.3125 £ G/H £ 0.625, (b) Time averaged local Nusselt number on the inferior channel wall for 0.0625 £ G/H £ 0.25, (G/d=8G/H), Re=800

the shedding of vortices from the bars. The two cases with steady flow do not have second maximum, G/H=0.625 and 0.5625. The cases with low frequency modulation of the flow, 0.0625 £ G/H £ 0.125, show two local maximums due the low frequency modulated shedding of vortices. To evaluate the heat transfer enhancement and pressure drop increase in the channel with mounted square bars for constant Reynolds number the mean Nusselt number and the apparent friction factor are compared with the values in a channel without vortex generators. The Nusselt number and apparent friction factor in a channel are Nu0=7.68 and f0=0.015 respectively for Re=800. Fig. 7 shows the mean Nusselt number and apparent friction factor for twelve studied cases. For the two cases with steady flow reduction in heat transfer were obtained with an increase in the flow losses. The cases with antiphase or in phase vortex shedding, 0.1875 £ G/H £ 0.5, have a important heat transfer enhancement with moderate increase in the apparent friction factor. For the cases with 0 £ G/H £ 0.125 the heat transfer continues increasing with a reduction of G/H but the flow losses increase in exponential form.

Table 2. Averaged values for different Reynolds number for the case with G/d=0.75

Re

St

CD1

CD2

CL1

CL2

Cf

Nu/Nuo fapp/fo

200 400 600 800 1000

0. 0.188 0.0358 0.0854 0.0919

5.698 4.677 4.381 4.021 4.237

5.698 4.653 4.405 4.645 4.368

–2.389 –1.668 –1.288 –0.828 –0.943

2.389 1.6364 1.310 1.424 1.066

0.109 0.0554 0.0390 0.0305 0.0254

1.199 1.266 1.315 1.347 1.389

3.966 5.455 7.125 8.887 10.73

rear side of the bars. For Reynolds number of 400 the flow is unsteady with a well-defined periodicity in the wake, however the dominant frequency in each bar is different, for this reason this flow can not be classified as flow with in phase vortex shedding. With Reynolds numbers of 600 the flow has a very low dominant frequency modulation, Table 2. Biased flow with low frequency modulation appears for Re=800, see Fig.3. If the Reynolds number is increased at 1000 more frequencies appear, and the unsteady flow has a transitional character. The mean drag and lift coefficients of the two bars are different for Re‡400, indicating that the mean flow around the bars is not the same. The mean heat transfer 6 enhancement increases with the Reynolds number, Variation of Re for constant G/d=0.75 however the mean pressure drop in the channel increases The effects of the variation of channel Reynolds number Re in exponential form. on mean values of integral parameters of the flow and heat transfer in a channel with two mounted square bars with a 7 transverse separation distance of G/d=0.75 (G/H=0.09375) Conclusions are shown in Table 2. The Strouhal numbers were calcu- In this paper, we have employed numerical simulations to lated with the dominant frequency present in the time explore the unsteady laminar fluid flow and heat transfer signals of the drag and lift coefficients of the two bars. in a plane channel with two square bars mounted side by Numerical results show that for Re=200, the flow is side to the approaching flow. The effects of vortex shedsteady laminar with a recirculation zone attached to the ding have been captured by solving the continuity and the Navier-Stokes equations in two dimensions. Computations were made first for twelve transverse bar separation distances G/d for a constant Reynolds number, after that we have varied the Reynolds number for one case. The numerical results reveal the complex structure of the flow. With G/d=4.5 and 5 steady flow was found. With 1.5 £ G/d £ 4.0 unsteady flow with only one present frequency in the flow was found, and with G/d=1.5 the binary vortex street can be characterized as in phase vortex shedding. For the cases with G/d=0.5, 0.75 and 1 complex vortices structures were found with a bistable nature and low frequency modulation of the flow at Re=800. The flow was steady at Re=200, it became unsteady and periodic as the Reynolds number was increased, it showed low frequency modulation for Re>400, and the flow had a transitional character for Re=1000.

References

Fig. 7. Mean heat transfer enhancement and pressure drop increase as a function of the transverse bar separation distance G/H, (G/d=8G/H), Re=800

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4. Rosales JL; Ortega A; Humphrey JAC (2001) A numerical simula8. Bosch G (1995) Experimentelle und theoretische Untersuchung tion of the convective heat transfer in confined channel flow past der instationa¨ren Stro¨mung um zylindrische Strukturen, Ph.D. square cylinders: comparison of inline and offset tandem pairs. Dissertation, Universita¨t Fridericiana zu Karlsruhe, Germany International Journal of Heat and Mass Transfer 44: 587–603 9. Patankar SV (1980) Numerical heat transfer and fluid flow, 5. Valencia A (1998) Numerical study of self-sustained oscillatory Hemisphere, McGraw-Hill, New York flows and heat transfer in channels with a tandem of transverse 10. van Doormaal JP; Raithby GD (1984) Enhancements of the vortex generators. Heat and Mass Transfer 33: 465–470 SIMPLE method for predicting incompressible fluid flows. 6. Williamson CHK (1985) Evolution of a single wake behind a pair Numerical Heat Transfer 7: 147–163 of bluff bodies. Journal of Fluid Mechanics 159: 1–18 7. Hayashi M; Sakurai A; Ohya Y (1986) Wake interference of a row of normal flat plates arranged side by side in a uniform flow. Journal of Fluid Mechanics 164: 1–25

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