Vol. 26, NO. 6, pp. 869878, 1999 Copyright ÂŠ 1999Elsevier ScienceLtd Printed in the USA. All rights reserved 07351933/99/Ssee front matter
Int. Comm. Heat Mass Transfer,
Pergamon
P H S07351933(99)000755
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 ke turbulence model and a modified version were used in conjunction with the Reynoldsaveraged 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. ÂŠ 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 threedimensional turbulent fluctuations are superimposed on the periodic vortexshedding 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 vortexshedding motion.
Measurements of the velocity characteristics for the turbulent flow around a square crosssection 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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[2] calculated with the standard ke turbulence model the flow past a square bar at Re=22000 studied experimentally by Lyn et al. [3] with a twocomponent laserDoppler, 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 ke 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 timemean total fluctuating energy shows that the ke turbulence models underpredict severely the fluctuation level behind the bar.
Rodi [7] reviews calculations of vortexshedding flows performed with various turbulence models, the k~ model yielded low shedding frequency and drag coefficient, while the Reynoldsstress model predicted better these parameters. However the prediction of the pressure distribution along the bar surface with the ke model was better than with the secondmoment closure models. The predictions of unsteady separated flows were improved in [8] with a hybrid model that blends elements from both the largeeddy methodology and the standard eddyviscosity 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=0.331.5, 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 ke turbulence models, local and global heat transfer rates on rectangular bars in the twodimensional flow region with available experimental results.
Governing Eauations
At the high Reynolds number considered here, stochastic threedimensional turbulent fluctuations are superimposed on the mainly twodimensional periodic vortexshedding 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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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'bFN
(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 ke model of Launder and Spalding [13]: Dk
p~
:
,.~
¼, ,..2lr
~t
Ot O[gl
[tit+
)
÷ o

] +
(s>
c2p../_
(6)
The standard version of the kE 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 eddyviscosity models, Launder and Kato [5] proposed to replace the production expression in Equation 5 by:
c
I1.ou,
0e .,
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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 walltangential moment these are the wall shear stresses and the nondimensional 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 nearwall cell for the kequation 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 (~t1)(~)
and the wall temperature T w is a constant in the present w o r k , Tw= 2T..
(12)
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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 finitevolume method, details of which can be conveniently found in Patankar, [14]. The convection terms in the momentum equations were approximated using a PowerLaw scheme. The method uses a nonstaggered grid and Cartesian velocity components, handles the pressurevelocity 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 tridiagonalmatrix algorithm. A firstorder 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 ke 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 ke 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 ke model. With the LK version the simulated amplitude of the
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oscillation in lift is almost twice as with the standard ke model. The reason for this difference arises from the excessive k production in the stagnation region created by the standard ke model.
TABLE 1 Comparison of present results for a square bar with literature Ref.
CD
ACI~
AC L
St
101x281 ke
1.706
0.0056
0.210
0.171
126x351 ke
1.683
0.0033
0.161
0.176
161x449 ke
1.707
0.0060
0.123
0.172
201x561 ke
1.747
0.0010
0.126
0.165
161x449 ke LK
1.766
0.0077
0.253
0.177
[3] experiment
2.14
0.09
0.135
[8] d/H0.19
2.37
0.20
0.178
[5] ke kE 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 ke 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 eddyviscosity 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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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 ke 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 ke
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 ke 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.
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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 ke
63.6
55.4 +_0.7
46.8
55.4 _+0.7
55.3
73.3 [16]
126x351 kE
68.2
61.1 +0.5
46.4
61.1 ± 0.5
59.2
74. [12]
161x449 kE
72.8
67.8 +_ 0.6
47.5
67.8 ± 0.6
63.9
103 [11]
161x449 ke LK
55.5
60.6 +_ 1.1
43.9
60.6 +_ 1.1
55.2
110. [9]
201x561 ke
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 kE 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 ke turbulence model and one modified version for different grid sizes. The modified version of the ke 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.
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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~ ke 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
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A. Valencia and C. Orellana
Vol. 26, No. 6
molecular viscosity
lat turbulent viscosity p
density
a k, a~ x
ke 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. 20112016, 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, FEDVol. 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. 4647, 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, McGrawHill, New York, (1997).
Received April 19, 1999