Page 1

Vol. 26, NO. 6, pp. 869-878, 1999 Copyright Š 1999Elsevier ScienceLtd Printed in the USA. All rights reserved 0735-1933/99/S-see front matter

Int. Comm. Heat Mass Transfer,

Pergamon

P H S0735-1933(99)00075-5

S I M U L A T I O N OF T U R B U L E N T F L O W AND H E A T T R A N S F E R AROUND R E C T A N G U L A R BARS

Alvaro Valencia and Carolina Orellana Department of Mechanical Engineering Universidad de Chile Casilla 2777, Santiago CHILE

( C o m m u n i c a t e d by J.P. Hartnett and W.J. M i n k o w y c z ) ABSTRACT Numerical investigations on the turbulent fluid flow and the local heat transfer from rectangular bars for three different aspect ratios were carried out with Re=22000 and Pr=-0.71. The standard k-e turbulence model and a modified version were used in conjunction with the Reynolds-averaged momentum and energy equations for the simulations. The predictions of drag, lift, local and global heat transfer coefficients on the front, side and on the rear face of the rectangular bars are compared with available experimental results. Š 1999 Elsevier Science Ltd

Introduction

The flow past slender, bluff structures is frequently associated with periodic vortex shedding causing dynamic loading on the structures. For the design of such structures, the unsteady loading forces must be known and hence methods for predicting the flow and the forces are of great practical importance. Rectangular bar is also one of the most interesting bluff body in connection with the question of heat transfer mechanism in the separated, reattached flow region. For situations with high Reynolds numbers, which usually occur in practice, stochastic three-dimensional turbulent fluctuations are superimposed on the periodic vortex-shedding motion. A resolution of these motions in a direct simulation is not feasible at present. Hence, there is still a need for economic calculation methods based on the use of a turbulence model for simulating the influence of the stochastic fluctuations on the periodic vortex-shedding motion.

Measurements of the velocity characteristics for the turbulent flow around a square cross-section bar mounted in a water channel for Re=14000 show that in the zones of highest velocity oscillations the energy associated with the turbulent fluctuations is about 40% of the total energy, [1]. Franke and Rodi 869


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Vol. 26, No. 6

[2] calculated with the standard k-e turbulence model the flow past a square bar at Re=22000 studied experimentally by Lyn et al. [3] with a two-component laser-Doppler, they obtained a steady solution and no vortex shedding due to the relatively coarse grid used. Bosch and Rodi [4] reported calculations with the standard k-e model and a modification attributable to Launder and Kato [5] for the flow past a square bar at Re=22000 placed at various distances from an adjacent wall. They obtained the unsteady shedding motion around the bar and the modified version yields reasonable predictions of the experimentally observed flow motion, [6]. However the comparison of time-mean total fluctuating energy shows that the k-e turbulence models underpredict severely the fluctuation level behind the bar.

Rodi [7] reviews calculations of vortex-shedding flows performed with various turbulence models, the k-~ model yielded low shedding frequency and drag coefficient, while the Reynolds-stress model predicted better these parameters. However the prediction of the pressure distribution along the bar surface with the k-e model was better than with the second-moment closure models. The predictions of unsteady separated flows were improved in [8] with a hybrid model that blends elements from both the large-eddy methodology and the standard eddy-viscosity approaches.

Igarashi [9,10] carded out experimental studies on fluid flow and heat transfer around rectangular bars with different ratios of the width to height, c/d=0.33-1.5, and angles of attack. The heat transfer coefficients changed drastically as the width to height ratio was varied. Mass transfer coefficients obtained for rectangular bars with naphthalene sublimation technique [11,12] show good agrement in average transfer rates and the trends of the data is similar in local transfer rates. The Nusselt number on the front face increases from the stagnation point to the edges. On the side faces, the Nusselt number first deceases in the leading edge separation bubble and then increases thereafter up to the rearward separation comer. On the rear face the heat transfer coefficient is almost uniform and higher than that on the front face. The purpose of the present study is to compare simulated with k-e turbulence models, local and global heat transfer rates on rectangular bars in the two-dimensional flow region with available experimental results.

Governing Eauations

At the high Reynolds number considered here, stochastic three-dimensional turbulent fluctuations are superimposed on the mainly two-dimensional periodic vortex-shedding motion around the bar. An instantaneous quantity can be separated into a mean value, the periodic fluctuation and the stochastic turbulent fluctuation. Replacing in the momentum equations is obtained averaged equations that they contain products of turbulent velocity fluctuations. These Reynolds stresses appearing in the momentum


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FLOW AND HEAT TRANFSER AROUND BARS

871

equations are simulated by the statistical k-• turbulence model. The continuity, the averaged momentum equations and the energy equation are used then to describe the incompressible unsteady separated flow and heat transfer around the bar in the computational domain. Continuity:

OUl=0

(I)

Momentum: p

DU, 01, a . . . . or, or,. 2 .. = -----" + - - t t t t +t t p t - - +--.-~)-- p~oq ] z~ a,, a,I a,j a,t 3

(2)

Energy:

Dr_ O [(r+r )0T] P'b-F-N

(3)

where the turbulent viscosity p, and the turbulent dynamic thermal diffusivity r, are given by: k2

_ IAt

~,,:p c~ --~

r, - g .

(4)

The turbulent kinetic energy k and its dissipation rate e are computed from the standard k-e model of Launder and Spalding [13]: Dk

p~

:

,.~

¼, ,..2lr

~t

Ot O[gl

[tit+

)

÷ o

-

] +

(s>

c2p../_

(6)

The standard version of the k-E model calculates the production term G of k from:

To reduce the excessive production of k in stagnation regions (e.g., in front of the bar) due to an unrealistic simulation of the normal turbulent stresses in eddy-viscosity models, Launder and Kato [5] proposed to replace the production expression in Equation 5 by:

c

I1.ou,

0e .,


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A. Valencia and C. Orellana

Vol. 26, No. 6

Launder and Kato [5] obtained significantly improved flow predictions with their modification for the square bar, in this work we studied principally the capacity of the models for the prediction of heat transfer around one heated rectangular bar. The standard constants are employed: C,=0.09, C1=1.44, C2=1.92, (rk=l.0, ~=1.3, Prt=0.9.

The computational domain and the boundary conditions are sketched in Figure 1. At the inlet, the flow enters with only a streamwise component U 0 equivalent to a Reynolds number of Re=22000, and a turbulence level of Tu= uVU0=2% is prescribed as in the experimental condition, [3]. The experimental blockage is 0.07, however the computational domain imposes a blockage of d/H=0.15, due to the use of uniform meshes. A ratio of lat/la=10 is assumed to compute the inflow value of the dissipation rate e. At the outlet the streamwise gradients of all variables are set to zero. At the symmetry planes the normal velocity component and the normal derivatives of all other variables are set to zero.

Wall functions given by Launder and Spalding [13] are employed to prescribe the boundary conditions along the bar faces. The wall functions are applied in terms of diffusive wall fluxes. For the wall-tangential moment these are the wall shear stresses and the non-dimensional wall distance y+ defined as:

pu.c:l.:..

(9)

ln(Ey ") The subscript p refers to the grid point adjacent to one wall. The production rate of k and the averaged dissipation rates over the near-wall cell for the k-equation as well as the value of e at the point p are computed respectively from the following equation:

(lo)

For the temperature boundary condition, the heat flux to one wall is derived from the thermal wall function:

qw=

(r.~t0n(~y')/~+/')

(11)

where the empirical function P is specified as: _ ~/4 e-~(7)

A ~ Pr Prt x/4 (~t-1)(~)

and the wall temperature T w is a constant in the present w o r k , Tw= 2T..

(12)


Vol. 26, No. 6

FLOW AND HEAT TRANFSER AROUND BARS

873

Detoil A

-o ?'m

I{ o Eo LI. T,. lp

5,33 0

/7 ii :I

~/~D~ oil A at)

L

:

]B.G3 d

FIG. 1 Computational domain

Numerical Solution Technioue

The differential equations introduced above were solved numerically with an iterative finite-volume method, details of which can be conveniently found in Patankar, [14]. The convection terms in the momentum equations were approximated using a Power-Law scheme. The method uses a nonstaggered grid and Cartesian velocity components, handles the pressure-velocity coupling with the SIMPLEC algorithm in the form given by Van Doormaal and Raithby, [15], and solves the resulting system of difference equations iteratively with a tridiagonal-matrix algorithm. A first-order accurate fully implicit method was used for time discretization in connection with a relatively small time step Ax=AtU0/d<0.005. A typical run of 25000 time steps with 449x161 grid points takes more than

10 4 CPU

minutes on a IBM

RISC 6000 397 workstation (peak performance 0.5 GFLOPS).

Results and Discussion

Values of integral parameters (mean and amplitude of the oscillation in drag, amplitude of the oscillation in lift, and Strouhal number) predicted with the standard k-e model and with the modified version according to Launder and Kato (LK) for different grids are compared in Table 1 with values from the literature. The simulations for different grids with the k-e model and with the LK version yield too low mean drag coefficient and a small oscillation in the drag coefficient. The Strouhal numbers are higher than the experimental due the blockage ratio used in the simulations, [8]. The results agree well with previous numerical results with the k-e model. With the LK version the simulated amplitude of the


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A. Valencia and C. Orellana

Vol. 26, No. 6

oscillation in lift is almost twice as with the standard k-e model. The reason for this difference arises from the excessive k production in the stagnation region created by the standard k-e model.

TABLE 1 Comparison of present results for a square bar with literature Ref.

CD

ACI~

AC L

St

101x281 k-e

1.706

0.0056

0.210

0.171

126x351 k-e

1.683

0.0033

0.161

0.176

161x449 k-e

1.707

0.0060

0.123

0.172

201x561 k-e

1.747

0.0010

0.126

0.165

161x449 k-e LK

1.766

0.0077

0.253

0.177

[3] experiment

2.14

0.09

0.135

[8] d/H--0.19

2.37

0.20

0.178

[5] k-e k-E LK

1.66

2.05

0.0 0.03

0.10 1.16

0.127 0.145

Figure 2 shows velocity vectors and contours of turbulent kinetic energy k around a square bar calculated with the standard k-e turbulence model for the grid 161x449 and Re=22000. The velocity field shows the unsteady vortex shedding behind the bar and the separation zones on the side faces of the bar. The contours lines of k show that the kinetic energy follows the shedding of the unsteady vortices. The k contours on the front and on the side faces indicate high heat transfer rates in these regions. The model does not show high k on the rear face of the bar, and therefore underpredict the high experimentally observed periodic fluctuations in this zone, [3]. The flow in the rear region is highly anisotropic and cannot be correctly described by a eddy-viscosity model.

FIG. 2 Velocity vectors and k contours at one phase around a square bar ( it is shown only a part of the computational domain).


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FLOW AND HEAT TRANFSER AROUND BARS

875

Distributions of the local Nusselt number on a square bar at four phases are shown in Fig. 3. The Nusselt number increases from the stagnation point to the edges. This trend is in contrast with that of a circular cylinder in which the local transfer rates decrease from stagnation to the separation point. On the side faces the Nusselt number is almost uniform, in contrast with experimental results in which the local' transfer rates increase up to the rearward separation comer, [11]. The Nusselt number is uniform on the rear face and lower than that on the front face, experimental results show that in this region the Nusselt number is higher than that on the front face due to the active turbulent eddy motion in the wake region.

l째k

On the side and rear faces ( zones of turbulent separated flow ) are observed the greatest deficiencies in the numeric prediction of local heat transfer around the bar with the k-e turbulence model.

130

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

120 t10

P h m 1/4 T Ptlase 1/2 T Phlme 3/4 T P h l l e 1/1 T

100

9O 9O 70 8O 5O

40

front

side

rear

side

J 0,0

0.1

0.2

03

0.4

05

0.6

x/d

FIG. 3 Local Nusselt number distribution around a square bar, Grid 161x449 k-e

Table 2 shows the average Nusselt number on every face around the square bar calculated with the four grids and with the modified version of the k-e turbulence model. The amplitude of the oscillation of average Nusselt number on the side faces is about 1%. With the finest grid the average values increase about 6% compared with the 161x449 grid. The modified version (LK) reduces the level of turbulent kinetic energy around the bar and therefore reduces the prediction of Nusselt numbers, Eq. 11. The flow characteristics and the heat transfer around the bar are strongly dependent of the bar aspect ratio (c/d), Table 3. The frequency of vortex shedding increases with c/d, then for c/d=0.65 the interaction between the shedding vortex in the superior and inferior part of the bar is lower as in the case with higher c/d. The mean value and the amplitude of the oscillation in drag are reduced considerably with an increase in c/d.


876

A. Valencia and C. OreUana

Vol. 26, No. 6

TABLE 2 Average Nusselt Number for every face for a square bar Grid

front face

side face

rear face

side face

Mean

Mean Nu Ref.

101x281 k-e

63.6

55.4 +_0.7

46.8

55.4 _+0.7

55.3

73.3 [16]

126x351 k-E

68.2

61.1 +-0.5

46.4

61.1 ± 0.5

59.2

74. [12]

161x449 k-E

72.8

67.8 +_ 0.6

47.5

67.8 ± 0.6

63.9

103 [11]

161x449 k-e LK

55.5

60.6 +_ 1.1

43.9

60.6 +_ 1.1

55.2

110. [9]

201x561 k-e

76.8

72.1 _+0.9

51.9

72.1 ± 0.9

68.3

The Nusselt number on the bar faces decreases with c/d, then with higher c/d is the region with boundary layer type flow on the side faces longer. For the rectangular bar with c/d=0.65 is the amplitude of the oscillation of Nusselt number on the side faces almost 8%, for the rectangular bar with c/d=1.53 is the Nusselt number practically constant on the side faces. The mean heat transfer around the rectangular bar varies drastically with c/d.

TABLE 3 Comparison between three rectangular bars: Grid 161x449 k-E c/d

CD

ACL

St

Nu front face

Nu side face

Nu rear face

Nu side face

Mean Nu

Mean Nu Ref.

0.65

2.237±0.067

1.277

0.156

79.3

80.5+_6

77.7

80.5±6

79.6

128 [10]

1.0

1.707+-0.006

0.123

0.172

72.8

67.8_+0.6

47.5

67.8±0.6

63.9

110[10]

1.53

1.419±0.002

0.04

0.207

66.8

58.3±0.1

37.1

58.3_+0.1 55.1

99 [10]

Conclusions

The flow and heat transfer around three different rectangular bars were numerically simulated with the standard k-e turbulence model and one modified version for different grid sizes. The modified version of the k-e model yields to lower local and global Nusselt number predictions than the standard version. The oscillation of the Nusselt number on the side faces decreases strongly with c/d. The prediction of local heat transfer rates is poor in separated regions compared with experimental results around the bar. In this work the differences on predicted mean Nusselt number with available experimental results were quantified.


Vol. 26, No. 6

FLOW AND HEAT TRANFSER AROUND BARS

877

Acknowledeements

The financial support received of CONICYT CHILE under grant N 째 1980695 is gratefully acknowledged.

Nomenclature

A

Van Driest's constant (=26)

c

bar length

C D drag coefficient C L lift coefficient Cp specific heat at constant pressure C~, C t, C~ k-e turbulence model constants d

bar diameter

E

c o n s t a n t , (=9).

f

frequency

G

generation of turbulent energy

H

computational domain height

h

local heat transfer coefficient

k

turbulent kinetic energy

L

computational domain length

Nu local Nusselt number, h(x) d / k p

pressure

Pr Prandtl number ~/et (=0.71) Pr t turbulent Prandtl number (=0.9)

Re Reynolds number, Uo d //.t St

Strouhal number, f d / U 0

T . inlet fluid temperature T w bar wall temperature U 0 average velocity at the inlet y* wall coordinate F

thermal diffusivity

F t turbulent thermal diffusivity Kronecker delta e

dissipation rate of turbulent kinetic energy von Karman constant, (=0.4)


878

la

A. Valencia and C. Orellana

Vol. 26, No. 6

molecular viscosity

lat turbulent viscosity p

density

a k, a~ x

k-e turbulence model constants

dimensionless time (=tUo/d)

References

1. D . F . G . Dur~o, M. V. Heitor and J. C. F. Pereira, Exp. Fluids, 6, 298 (1988). 2.

R. Franke and W. Rodi, Proc. 8th Symp. on Turbulent Shear Flows, pp. 20-1-1-20-1-6, Munich, Germany (1991).

3.

D . A . Lyn, S. Einav, W. Rodi and J.-H. Park, J. Fluid Mech. 304, 285 (1995).

4.

G. Bosch and W. Rodi, Int. J. Heat and Fluid Flow, 17, 267 (1996).

5.

B.E. Launder and M. Kato, Unsteady Flows, FED-Vol. 157, pp. 189, ASME (1993).

6.

G. Bosch, M. Kappler and W. Rodi, Exp. Thermal and Fluid Science, 13, 292 (1996).

7.

W. Rodi, J. Wind Eng. Ind. Aero. 46-47, 3 (1993).

8.

P. Koutmos and C. Mavridis, Int. J. Heat Fluid Flow, 18, 297 (1997).

9.

T. Igarashi, Int. J. Heat Mass Transfer, 29, 777 (1986).

10. T. Igarashi, Int. J. Heat Mass Transfer, 30, 893 (1987). 11. R. J. Goldstein, S. Y. Yoo and M. K. Chung, Int. J. Heat Mass Transfer, 33, 9 (1990). 12. C. H. Chung, M. K. Chung and S. Y. Yoo, Experimental Heat Transfer, Fluid Mechanics and Thermodynamics, Vol. 1, p. 539. Elsevier Science Publishers B V, 1993. 13. B. E. Launder and D. B. Spalding, Comp. Meth. App. Mech. Engng. 3, 269 (1974). 14. S. V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, New York, (1980). 15. J. P. Van Doormaal and G. D. Raithby, Numer. Heat Transfer, 7, 147 (1984). 16. J. P. Holman, Heat Transfer, 8th edn, p.307, McGraw-Hill, New York, (1997).

Received April 19, 1999

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