Proceedings of the International Heat Transfer Conference IHTC14 August 8-13, 2010, Washington, DC, USA
IHTC14-22269
ON THE CORRELATION BETWEEN VORTICITY STRENGTH AND CONVECTIVE HEAT TRANSFER
Charbel Habchi1,2, Thierry Lemenand1, Dominique Della Valle1,3 and Hassan Peerhossaini1,*
1LTN, CNRS UMR 6607, École Polytechnique – Université de Nantes, rue Christian Pauc, 44306 Nantes, France
2ADEME, 20 avenue du Grésillé, 49004 Angers, France
3ONIRIS, rue de la Géraudière, 44322 Nantes, France
* Address all correspondence to this author. Email: hassan.peerhossaini@univ- nantes.fr
ABSTRACT
Vorticity is an inherent feature of fluid flow and has an essential role in the convective heat and mass transfer. The present study aims at determining quantitatively the relationship between the streamwise vorticity flux Ω, and the convective heat transfer, characterized by Nusselt number Nu . Physical vortices are created by using a vorticity generator inserted on the wall of a heated straight channel. It is shown that the streamwise variations of Ω and Nu are related through a power law of the type Nu≈α
(
Ω−C1)
β +C2.NOMENCLATURE
D h Channel hydraulic diameter, m h Vorticity generator (tab) height, m J Streamwise vorticity flux, 1/s k Turbulent kinetic energy (TKE), m2/s2
Re Reynolds number based on channel hydraulic diameter, Re=WDh/υ
Re h Reynolds number based on the tab height, υ
/ h W Reh =
Nu Nusselt number
Nu 0 Nusselt number for straight channel S Cross section, area m2
T w Wall temperature, K T b Flow bulk temperature, K
U Convective heat transfer coefficient, W/m2.K
W Mean flow velocity, m/s
z , y ,
x Cartesian coordinates, m y , + y ∗ Dimensionless wall distance Greek symbols
ε TKE dissipation rate, m2/s3 ϕw Wall heat flux density, W/m2 λ Fluid thermal conductivity, W/m.K υ Kinematic viscosity, m2/s
ωz Streamwise vorticity, 1/s
Ω Dimensionless streamwise vorticity flux INTRODUCTION
In many situations of practical interest vorticity is artificially generated to enhance heat and mass transfer (1, 2), but also vorticity exists naturally in many types of fluid flow such as in the near-wall region of turbulent boundary layer (3) or surfaces with curvature and/or rotation (4). Physical understanding of the mechanism of heat transfer by vorticity is therefore crucial for active and passive control in numerous technological applications.
In the presence of vorticity generators, Chang et al. (5) suggest the use of the cross-averaged absolute streamwise vorticity flux to characterize the intensity of the secondary flow produced by vorticity generators. It was shown qualitatively (5) that a strong relation is observed between the longitudinal variation of the streamwise vorticity flux and the Nusselt number downstream from the vortex generators. However, this relationship is qualitative; no quantitative expression is defined that relates the vorticity strength to the Nusselt number, which is
the main focus of the present study. Moreover, it was shown ( 1, 6, 7) that most of the heat transfer enhancement is caused essentially by the streamwise vortices, while the transverse stationary vortices, e.g. recirculation flow behind the obstacles, enhance slightly the heat transfer in the regions near the vortex generator. Therefore, neglecting the effects of these transverse vortices on the heat transfer, the cross-section averaged absolute streamwise vorticity flux, can be used to describe the intensity of the secondary flow.
In the present work, pressure driven longitudinal vorticity is generated in a turbulent duct flow by using a vorticity generator. Numerical simulations are performed to establish an expression relating the convective heat transfer to the vorticity strength. This flow represents the main features of many heat transfer devices such as multifunctional heat exchangers- reactors (2).
NUMERICAL PROCEDURE Physical domain
Two flow configurations are studied here: the first is an empty square duct flow of 7.62 cm each side, i.e. hydraulic diameterDh=7.62cm, and 33.15 cm long, and in the second geometry a vorticity generator of trapezoidal shape is inserted on the duct wall with an inclination angle of 24.5° relative to the wall plane. The dimensions of the vortex generator and the duct taken here are adapted from Yang et al. (8) and Dong and Meng ( 9) for the sake of comparison and validation of the present numerical results with these previous studies. The dimensions of the physical domain and of the vorticity generator are schematically shown on Figure 1 in the Cartesian frame of reference. In the following sections all spatial scales are scaled with the tab height h .
-10 0 15.5
0 6
y/h
24.5°
z/h
h= 1.3 cm Flow
x/h
-3 3
0 6
y/h 1.04
7.6°
(a) (b)
Figure 1. Vorticity generator and duct dimensions represented on (a) the symmetry plan of the tab and (b) on
a cross sectional plan.
Solution method and turbulence model
The CFD code Fluent® 6.3 is used in the present study to perform the numerical simulations. The computational mesh is a cell-centered finite volume discretization. The conservation equations for mass, momentum, and energy are solved sequentially with double precision, segregated and second-order accuracy. Pressure-velocity coupling is performed by finite volumes with the SIMPLE algorithm. The RSM model (10)
associated with a two-layer model for the wall region computation is used in the present study. The flow in the near- wall region is computed by using a two-layer model. Following this model, in the viscous sub-layer, the one-equation model of Wolfstein (11) is used, where only the turbulent kinetic energy transport equation is solved and the turbulent viscosity and energy dissipation rate are computed from empirical correlations based on length scales.
Boundary conditions
The heat conduction on the vorticity generator surfaces and thickness is taken into account by using the coupled option of two-sided walls model. The thermal conductivity in the tab is taken as constant and equal to 16.27 W/m.K corresponding to the conductivity of the steel.
The physical properties of the working fluid, which is water, especially the viscosity and the thermal conductivity, change significantly with the temperature and therefore, they are assumed piecewise linear functions of temperature, as proposed by Rahmani et al. (12). The specific heat and density are assumed to be constant for the temperature range used here and their values are respectively set at 4182 J/kg.Kand 998 kg.m3 (12).
No-slip boundary conditions are applied to the solid surface of the tab and of the duct wall. At the computational domain inlet, a fully developed turbulent flow velocity profile is used; the TKE and the turbulence energy dissipation rate at the inlet are fixed by the turbulence intensity I derived from the equilibrium turbulent flow (13). The fluid temperature at the inlet is set constant at 298 K.
Flow and heat-transfer simulations are carried out in a steady turbulent flow for Reynolds numbers based on the tab height h of Reh=2080 and for a constant temperature of
K
=370
Tw prescribed on all channel walls in both flow configurations: straight channel with and without vortex generator.
Meshing and numerical solution accuracy
A non-uniform unstructured three-dimensional mesh with hexahedral volumes is built and refined at all solid boundaries (using the software Gambit®). Mesh size is controlled by adjusting the number of the nodes in the (x, y) directions, on the duct periphery, and on the axial coordinate z.
The mesh density is increased until no effect is observed on the quality of the results. The mesh with the lowest node density yielding high-quality results is used to simulate the entire geometry. The final mesh size after refinement is 1,800,566.
The maximum value of the wall dimensionless distance y in + the first grid point after refinement does not exceed 4.14.
Hence, sincey+<4, it is guaranteed that the viscous sub-layer is properly modeled. The present geometry, the maximum value of y is found to be about 4.70 which is ∗ <30 guarantying accurate simulations for the heat transfer near the wall as mentioned by Defraeye et al. ( 14).
Series of flow simulations are carried out for testing several stop-criteria values ranging from 10-3 to 10-9. It is found that beyond the value 10-6, no significant changes are observed in the temperature field and the turbulence kinetic energy, thus the value 10-6 is retained as the convergence criterion for the simulations.
RESULTS AND DISCUSSION Experimental validation
The present numerical simulations are performed in the same hydrodynamic conditions as previous PIV (8) and DNS ( 9) studies. Therefore, the results of these studies are used here to validate the present numerical results as shown in Figure 2.
Figure 2 (a) represents the y component, i.e. normal to the wall, of the mean velocity at y/h= 0.5 and z/h= 4. It is shown that the present numerical results reproduce fairly the experiments and DNS data. It is observed that the maximum normal velocity occurs at the tab symmetry plane due to the presence of the common up-flow induced by two neighbor counter-rotating vortices as it is detailed in the next section.
In Figure 2 (b) the (x, y) Reynolds stress component at z/h=4.5 is presented and compared with PIV and DNS data. It is observed that results obtained from the RSM model are closer to PIV results than to the DNS results. The main difference occurs in the shear region at 1.5<y/h<2 where the Reynolds shear stresses are maximum. However, in the wake region 0<y/h<1.5, the DNS results are far from PIV measurements and the present RSM simulations are closer to the experimental results.
Finally, it is observed that the present numerical simulations reproduce fairly well the flow dynamics in the studied geometry. However, the discrepancy observed between the three methods, PIV, DNS and RSM can be attributed to the lack of information in the numerical models and/or to experimental errors in the measurements.
In the next sections the analyses of the flow structure and heat transfer are presented in details.
-3 -2 -1 0 1 2 3
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
RSM DNS PIV
V / W0
x / h
(a)
0 1 2 3 4
-0.03 -0.02 -0.01 0.00 0.01 0.02 0.03
<v.w> / W 2 0
RSM DNS PIV
y / h
(b)
Figure 2. Validation of the numerical results:
(a) y component of the mean velocity at y/h= 0.5 and z/h= 4.
(b) the (x, y) Reynolds stress component at z/h= 4.5.
PIV results are adapted from Yang et al. (8) and DNS results from Dong and Meng (9).
Secondary flow and thermal distribution
Figure 3 represents the flow streamlines on different cross sections downstream from the tab. For z/h= 2 in Figure 3 (a), a complex flow is observed consisting in a primary counter- rotating vortex pair (PCVP) in the wake of the tab, two secondary counter-rotating vortex pairs (SCVP) in the upper corners of the channel, and two corner vortices (CV) at the bottom corners of the channel.
The PCVPs are caused by the pressure difference between the high momentum fluid in the flow core, and the low momentum fluid in the tab wake. The velocity gradient induced by the rotating movement of the PCVP and the stagnant fluid in the upper channel corners generates a streamwise vorticity ωz in this region giving raise to the SCVP. The two CVs at the bottom corners are also induced by a similar mechanism. The streamwise evolution of these three types of longitudinal vortices can be observed by following Figure 3 (a) to (b). For z/h=15 represented in Figure 3 (b), and relative to z/h=2 represented in Figure 3 (a), these vortices are getting larger when they are convected downstream from the tab as shown.
Another fact is that the centers of the PCVP migrate towards the flow core as we move away from the tab. This mechanism is caused by the low pressure in the flow core which sucks up the PCVP.
-3 3 0
6
-3 3
0 6
(a) (b)
Figure 3. Streamlines in the duct cross sections at (a) z/h=2, (b) z/h=15 downstream from the tab.
These vortices play the role of internal agitators and their effects on the local convective heat transfer are shown in Figure 4, which represents the heat flux density ϕw profiles on the four channel walls. A minimum of ϕw is observed in the common upflow region of the PCVP and of the SCVP, and a maximum is observed in the downflow region that is caused by thinning of the thermal boundary layer. This phenomenon is always observed in the presence of streamwise vorticity, as mentioned by Fiebig (1). However, it is noted that the heat transfer enhancement by the downwash flow induced by the PCVP is 11% higher than that of the SCVP and 21% relative to the CV.
Figure 4. Streamlines and profiles of ϕw in the cross section at z/h=2.
Vortex strength and heat transfer
To quantify the intensity of the secondary flow, we refer to the absolute cross-averaged streamwise vorticity flux J defined by Chang et al. (5) and given in the following expression:
S dy dx
J S
∫∫
z= ω
(1)
where ωz is the streamwise vorticity and S the surface of the duct cross section.
Here, the dimensionless value of the absolute cross- averaged streamwise vorticity flux is obtained from the following equation:
h W
S
= J
Ω (2)
where W is the mean flow velocity.
The cross averaged Nusselt number is calculated through the expression below:
(
ϕ)
λλh w w b h D T T UD
Nu= = − (3)
where U is the convective heat transfer coefficient, λ the thermal conductivity of the working fluid, D the hydraulic h diameter of the square duct, and T the flow bulk temperature b calculated using a UDF (user-defined function) in Fluent.
Figure 5 represents the longitudinal evolution of the dimensionless vorticity flux and scaled Nusselt number. As is shown, the maximum of both Ω and Nu occur at aboutz/h≈−1. To avoid the effects of the inlet and outlet zones, only the fully developed flow region has been retained here for further analysis, i.e. −6<z/h<12.5.
-10 -5 0 5 10 15
0 1 2 3 4 5
Zone 1 Zone 2
z / h ΩΩΩΩ, Nu/Nu0
Vorticity flux,ΩΩΩΩ
Scaled Nusselt number, Nu/Nu 0 Exponential fitting curves
Figure 5. Longitudinal evolution of the cross section- averaged vorticity flux and scaled Nusselt number.
As is observed from Figure 5, this region can be divided into two sub-regions: the first zone upstream of the tab
(
−6<z/h<−1)
where both Ω and Nu are increasing functions of z/h, and the second zone downstream of the tab(
−1<z/h<12.5)
where both Ω and Nu decrease with z/h. The variation of Ω is typically exponential (15) and can be readily described by the following expression:1 1
1 C
h B exp z
A +
=
Ω (4)
However, to the best of our knowledge, no quantitative expression is given for the evolution of the Nusselt number.
Therefore, and regarding its exponential shape, it is assumed that the variation of the Nusselt number also follows the same expression as that of Ω, and hence:
2 2 2 0
h C B exp z Nu A
Nu +
= (5)
After fitting the computed results presented in Figure 5 with equations (6) and (7), we obtain the constants values presented in Table 1.
Since the variations of Nu and Ω follow the similar trends in zones 1 and 2, the relation between Nu and Ω can be established by using the generalized forms of equations (4) and (5). Hence, by rearranging equation (4), one can write:
−
=
1 1
1 A
ln C h B
z Ω (6)
and by injecting equation (6) in (5), we obtain:
(
1)
20
C Nu C
Nu =α Ω− β + (7)
with β=B1/B2 and α =A2/A1β.
After replacing in equation (7) the numerical values of the above coefficients, the relation between Ω and Nu can be readily obtained.
A1= 8.50 B1= 1.20 C1= 0.20 zone 1
A2= 20.00 B2= 0.55 C2= 1.10 A1= 2.20 B1= -6.00 C1= 1.00 zone 2
A2= 0.20 B2= -0.80 C2= 1.18 Table 1: Constant values of equations (4)-(7) The correlations of the scaled Nusselt number are plotted versus vorticity flux in Figure 6 and compared to the computed results in both zones 1 and 2.
0 1 2 3 4
0 1 2 3
P2
ΩΩ ΩΩ Nu/Nu0
Computed: Estimated:
Zone 1 Zone 1
Zone 2 Zone 2
P1
Figure 6. Relation between the cross section- averaged scaled Nusselt number and vorticity flux.
It can be observed that the estimated and computed results are in fair agreement in both zones. However, from Figure 6, it can be noticed that, for a given Ω we do not obtain the same value for the Nusselt number in both zones. This can be seen by comparing for example point P to point1 P on Figure 6. In fact, 2 since these two points correspond to two different axial positions
[
P1(
z/h=−1.9) (
,P2 z/h=6.9) ]
, i.e. two different cross sections, and as the vortex size and its relative position to the wall are not the same in these positions, the Nusselt number is thus not the same either.To stress this point, we have plotted on Figure 7 the streamlines for the points P and 1 P . We recall that these points 2 have the same vorticity flux. However, on the point P (Figure 2 7-b) the vorticity center is farther from the channel wall than on the point P (Figure 7-a) and therefore, the Nusselt number is 1 smaller. The correlation developed here does not take into account these topological characteristics of the secondary flow which have an essential importance on the convective heat transfer, thus the discrepancy in the values of the Nusselt number for a given vorticity flux on Figure 6 .
(a) (b)
Figure 7. Streamlines in the duct cross sections at (a) z/h= -1.92 and (b) z/h= 6.92 for Ω=1.7.
CONCLUDING REMARKS
A numerical study is performed here to investigate the relation between the vorticity strength Ω and the Nusselt number Nu . Streamwise vorticity is generated in a turbulent straight channel flow by using a trapezoidal vorticity generator.
It is observed that the evolution of both Ω and Nu in the axial direction follow similar trends and can be estimated by exponential expressions. Two zones are identified: the first one, upstream of the vorticity generator, where both Ω and Nu increase and the second one, downstream from the vorticity generator trailing edge, where Ω and Nu decrease in the axial direction. Hence, from the exponential expressions describing the variations of Ω and Nu as functions of z , quantitative relations of the type Nu≈α
(
Ω−C1)
β +C2 have been established between Nu and Ω in both zones.However, it is found that for a given value ofΩ, we do not obtain the same value for Nu in both zones. This discrepancy is explained based on the argument that the size and position of the vorticity zone have also essential roles on the convective heat transfer. Since the relations found here do not take into account the vortex topology and its position, therefore the Nusselt number is not a single valued function ofΩ. Thus to obtain general correlations, it is important to introduce a vortex factor into the present analysis, which takes into account the size of the vorticity zone and the position of its epicenter.
ACKNOWLEDGMENTS
This work was partially supported by ADEME. Authors would like to acknowledge the continuous support of Dr. G.
Guyonvarch, the Ile de France Regional Director of ADEME and Dr. C. Garnier for monitoring this grant. C. Habchi gratefully acknowledges Dr. A. Ould El Moctar and Dr. M.
Khaled for fruitful and enlightening discussions.
REFERENCES
1. M. Fiebig, Vortices, generators and heat transfer, Chem.
Eng. Res. Des. 76 (1998), 108-123.
2. C. Habchi, T. Lemenand, D. Della Valle, H. Peerhossaini, Alternating mixing tabs in multifunctional heat exchanger- reactor, Chem. Eng. Process. (2009) in press, doi:10.1016/j.cep.2009.07.003
3. P.M. Le, D.V. Papavassiliou, A physical picture of the mechanism of turbulent heat transfer from the wall, Int. J. Heat Mass Transfer 52 (2009) 4873-4882.
4. R. Toé, A. Ajakh, H. Peerhossaini, Heat transfer by Görtler instability, Int. J. Haet Fluid Flow 23 (2002) 194-204.
5. L.M. Chang, L.B. Wang, K.W. Song, D.L. Sun, J.F. Fan, Numerical study of the relationship between heat transfer enhancement and absolute vorticity flux along main flow direction in a channel formed by a flat tube bank fin with vortex generators, Int. J. Heat Mass Transfer 52 (2009) 1794-1801.
6. J.M. Wu, W.Q. Tao, Numerical study on laminar convection heat transfer in a rectangular channel with longitudinal vortex generator, Part A: Verification of field
synergy principle, Int. J. Heat Mass Transfer 51 (2008) 1179- 1191.
7. T. Lemenand, C. Durandal, D. Della Valle, H. Peerhossaini, Turbulent direct-contact heat transfer between two immiscible fluids, Int. J. Therm. Sci., In press (2010).
8. W. Yang, H. Meng, J. Sheng, Dynamics of hairpin vortices generated by a mixing tab in a channel flow, Exp. Fluids 30 (2001) 705-722.
9. D. Dong, H. Meng, Flow past a trapezoidal tab, J. Fluid Mech. 510 (2004) 219-242.
10. B.E. Launder, G.J. Reece, W. Rodi, Progress in the development of a Reynolds stress turbulence closure, J. Fluid Mech. 68 (1975) 537-566.
11. M. Wolfstein, The velocity and temperature distribution of one-dimensional flow with turbulence augmentation and pressure gradient, Int. J. Heat Mass Transfer 12 (1969) 301- 318.
12. R.K. Rahmani, T.G. Keith, A. Ayasoufi, Numerical study of the heat transfer rate in a helical static mixer, J. Heat Transfer 128 (2006) 769-783.
13. J.O. Hinze, Turbulence, second ed., McGraw-Hill, New York, 1975.
14. T. Defraeye, B. Blocken, J. Carmeliet, CFD analysis of convective heat transfer at the surfaces of a cube immersed in a turbulent boundary layer, Int. J. Heat Mass Transfer 53 (2010) 297-308.
15. O. Lögdberg, J.H.M. Fransson and P.H. Alfredsson, Streamwise evolution of longitudinal vortices in a turbulent boundary layer, J. Fluid Mech. 623 (2009) 27-58.
