0
Research Papers: Natural and Mixed Convection

# Effect of Temperature Dependent Fluid Properties on Heat Transfer in Turbulent Mixed Convection

[+] Author and Article Information
Francesco Zonta

e-mail: francesco.zonta@uniud.it

Alfredo Soldati

e-mail: soldati@uniud.it

Center for Fluid Mechanics and Hydraulics,
DiEGM, University of Udine,
Udine 33100, Italy

1Also at Department of Fluid Mechanics, CISM, Udine 33100, Italy.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received January 22, 2013; final manuscript received July 10, 2013; published online November 7, 2013. Assoc. Editor: James A. Liburdy.

J. Heat Transfer 136(2), 022501 (Nov 07, 2013) (12 pages) Paper No: HT-13-1035; doi: 10.1115/1.4025135 History: Received January 22, 2013; Revised July 10, 2013

## Abstract

The effect of the uniform fluid properties approximation (Oberbeck-Boussinesq (OB)) in turbulent mixed convection is investigated via direct numerical simulation (DNS) of water flows with viscosity (μ) and thermal expansion coefficient (β) both independently and simultaneously varying with temperature (non-Oberbeck-Boussinesq conditions (NOB)). Mixed convection is analyzed for the prototypical case of Poiseuille-Rayleigh-Bénard (PRB) turbulent channel flow. In PRB flows, the combination of buoyancy driven (Rayleigh-Bénard) with pressure driven (Poiseuille) effects produce a complex flow structure, which depends on the relative intensity of the flow parameters (i.e., the Grashof number, Gr, and the shear Reynolds number, Reτ). In liquids, however, temperature variations induce local changes of fluid properties which influence the macroscopic flow field. We present results for different absolute values of the shear Richardson numbers ($Riτ=|Gr/Reτ2|$) under constant temperature boundary conditions. As Riτ is increased buoyant thermal plumes are generated, which induce large scale thermal convection that increases momentum and heat transport efficiency. Analysis of friction factor (Cf) and Nusselt number (Nu) for NOB conditions shows that the effect of viscosity is negligible, whereas the effect of thermal expansion coefficient is significant. Statistics of mixing show that (i) mixing increases for increasing Riτ (and decreases for increasing Reτ) and (ii) the effect of thermal expansion coefficient on mixing increases for increasing Riτ (and decreases for increasing Reτ). A simplified phenomenological model to predict heat transfer rates in PRB flows has also been developed.

<>

## References

Wang, M., Fu, S., and Zhang, G., 2002, “Large-Scale Spiral Structures in Turbulent Thermal Convection Between Two Vertical Plates,” Phys. Rev. E, 66, p. 066306.
Hartmann, D. L., Moy, L. A., and Fu, Q., 2001, “Tropical Convection and the Energy Balance at the Top of the Atmosphere,” J. Climate, 14, pp. 4495–4511.
Lumley, J. L., Zeman, O., and Siess, J., 2009, “The Influence of Buoyancy on Turbulent Transport,” J. Fluid Mech., 84, pp. 581–597.
Incropera, F. P., and Dewitt, D. P., 1985, Fundamentals of Heat and Mass Transfer, John Wiley and Sons Inc., New York.
Sugiyama, K., Calzavarini, E., Grossmann, S., and Lohse, D., 2009, “Flow Organization in Two-Dimensional Non-Oberbeck-Boussinesq Rayleigh-Benard Convection in Water,” J. Fluid Mech., 637, pp. 105–135.
Timchenko, V., 2012, “Eddie Leonardi Memorial Lecture: Natural Convection From Earth to Space,” ASME J. Heat Transfer, 134, p. 031014.
Zonta, F., Marchioli, C., and Soldati, A., 2011, “Time Behavior of Heat Fluxes in Thermally Coupled Turbulent Dispersed Particle Flows,” Acta Mech., 218, pp. 367–373.
Lee, J., Gharagozloo, P. E., Kolade, B., Eaton, J. K., and Goodson, K. E., 2010, “Nanofluid Convection in Microtubes,” ASME J. Heat Transfer, 132, p. 092401.
Arcen, B., Taniere, A., and Khalij, M., 2012, “Heat Transfer in a Turbulent Particle-Laden Channel Flow,” Int. J. Heat Mass Transfer, 55, pp. 6519–6529.
Ahlers, G., Grossmann, S., and Lohse, D., 2009, “Heat Transfer and Large Scale Dynamics in Turbulent Rayleigh-Benard Convection,” Rev. Mod. Phys., 81, pp. 503–537.
Verzicco, R., and Camussi, R., 2003, “Numerical Experiments on Strongly Turbulent Thermal Convection in a Slender Cylindrical Cell,” J. Fluid Mech., 477, pp. 19–49.
Xia, C., and Murthy, J. Y., 2002, “Buoyancy Driven Flow Transitions in Deep Cavities Heated From Below,” ASME J. Heat Transfer, 124, pp. 650–659.
Komori, S., Ueda, H., Ogino, F., and Mizushina, T., 1982, “Turbulence Structures in Unstably-Stratified Open-Channel Flow,” Phys. Fluids, 25, pp. 1539–1546.
Fukui, K., and Nakajima, M., 1985, “Unstable Stratification Effects on Turbulent Shear Flow in the Wall Region,” Int. J. Heat Mass Transfer, 28, pp. 2343–2352.
Fukui, K., Nakajima, M., and Ueda, H., 1991, “Coherent Structure of Turbulent Longitudinal Vortices in Unstably-Stratified Turbulent Flow,” Int. J. Heat Mass Transfer, 34, pp. 2373–2385.
Domaradzki, J. A., and Metcalfe, P. W., 1988, “Direct Numerical Simulations of the Effects of Shear on Turbulent Rayleigh-Benard Convection,” J. Fluid Mech., 193, pp. 499–531.
Iida, O., and Kasagi, N., 1997, “Direct Numerical Simulation of Unstably Stratified Turbulent Channel Flow,” ASME J. Heat Transfer, 119, pp. 53–67.
Zainali, A., and Lessani, B., 2010, “Large-Eddy Simulation of Unstably Stratified Turbulent Channel Flow With High Temperature Differences,” Int. J. Heat Mass Transfer, 53, pp. 4865–4875.
Zonta, F., Marchioli, C., and Soldati, A., 2012, “Modulation of Turbulence in Forced Convection by Temperature-Dependent Viscosity,” J. Fluid Mech., 697, pp. 150–174.
Zonta, F., Onorato, M., and Soldati, A., 2012, “Turbulence and Internal Waves in Stably-Stratified Channel Flow With Temperature-Dependent Fluid Properties,” J. Fluid Mech., 697, pp. 175–203.
Kerr, R., and Herring, J. R., 2000, “Prandtl Number Dependence of Nusselt Number in Direct Numerical Simulations,” J. Fluid Mech., 419, pp. 325–344.
Parodi, A., von Hardenberg, J., Passoni, G., Provenzale, A., and Spiegel, E. A., 2004, “Clustering of Plumes in Turbulent Convection,” Phys. Rev. Lett., 92, p. 194503. [PubMed]
Dean, R. B., 1978, “Reynolds Number Dependence of Skin Friction and Other Bulk Flow Variables in Two-Dimensional Rectangular Duct Flow,” ASME J. Fluid Eng., 100, pp. 215–223.
Sieder, E. N., and Tate, G. E., 1936, “Heat Transfer and Pressure Drop of Liquids in Tubes,” Ind. Eng. Chem., 28, pp. 1429–1435.
Perry, A., and Chong, M. S., 1987, “A Description of Eddying Motions and Flow Patterns Using Critical Point Concepts,” Annu. Rev. Fluid Mech., 9, pp. 125–155.
Schoppa, W., and Hussain, F., 2002, “Coherent Structure Generation in Near-Wall Turbulence,” J. Fluid Mech., 453, pp. 57–108.
Xi, H., Lam, S., and Xia, K., 2004, “From Laminar Plumes to Organized Flows: The Onset of Large Scale Circulation in Turbulent Thermal Convection,” J. Fluid Mech., 503, pp. 47–56.
Hetsroni, G., Yarin, L. P., and Kaftori, D., 1996, “A Mechanistic Model for Heat Transfer From a Wall to a Fluid,” Int. J. Heat Mass Transfer, 39, pp. 1475–1478.
Schlichting, H., 1979, Boundary Layer Theory, McGraw-Hill, New York.
Peltier, W. R., and Caulfield, C. P., 2003, “Mixing Efficiency in Stratified Shear Flows,” Annu. Rev. Fluid Mech., 35, pp. 135–167.
Fernando, H. J. S., 1991, “Turbulent Mixing in Stratified Fluids,” Annu. Rev. Fluid Mech., 23, pp. 455–493.
Lawrie, A. G. W., and Dalziel, S. B., 2011, “Rayleigh-Taylor Mixing in an Otherwise Stable Stratification,” J. Fluid Mech., 688, pp. 507–527.
Sameen, A., Verzicco, R., and Sreenivasan, K. R., 2009, “Specific Role of Fluid Properties in Non-Boussinesq Thermal Convection at the Rayleigh Number of 2 × 108,” Europhys. Lett., 86, p. 14006.

## Figures

Fig. 1

Sketch of the computational domain and flow conditions. The flow is driven along the horizontal direction (x) by an imposed pressure gradient. The bottom wall is kept at uniform higher temperature (TH) whereas the top wall is kept at uniform lower temperature (TC).

Fig. 2

(a) Normalized friction factor (Cf/Cf,0) and Nusselt number (Nub/Nub,0) for simulations of PRB turbulent channel flow: –▼– represents results obtained with OB assumptions; –•– represents results obtained assuming μ(T); –▪– represents results obtained assuming β(T). Results of Nub/Nub,0 obtained from our simplified model of heat transfer in PRB flows (open symbols) are also shown: -∇- for OB assumptions, –□– for β(T).

Fig. 3

Contour maps of the temperature field (panels a and c) and of the streamline rotation vector (panels b and d) on a xy plane parallel to the walls (and passing through the centerline of the channel) for simulations of channel turbulence at Reτ = 150. (a) PRB turbulent channel flow (Riτ = 498). (b) Forced convection turbulence.

Fig. 4

(a) Flow structures for PRB turbulent channel flow: structures are plotted on a specific region of the fluid domain using a dimensionless temperature isosurface (θ = 0.16). (b) The trace of temperature isosurface on a cross section (temperature contour) is shown to clarify the mechanisms of formation and merging of buoyant plumes. LSC: large scale circulation.

Fig. 5

Contour maps of the temperature field on a xy plane parallel to the walls (and passing through the centerline of the channel) for simulations of channel turbulence at Reτ = 150. (a) PRB turbulent channel flow (Riτ = 498). (b) Forced convection turbulence.

Fig. 6

Spanwise frequency spectra of fluid velocity fluctuations (Riτ = 498) for PRB turbulent channel flow: comparison between simulations with uniform thermophysical properties (filled circles) and simulations with temperature-dependent thermal expansion coefficient (open circles). Results from the simulation of forced convection turbulence (solid line) are also included. (a) Streamwise velocity fluctuations; (b) spanwise velocity fluctuations; (c) wall-normal velocity fluctuations.

Fig. 7

Statistics of mean fluid velocity and temperature for PRB turbulent channel flow at Riτ = 498 (Reτ = 150): comparison between simulations with uniform thermophysical properties (OB, solid line) and simulations with temperature-dependent viscosity (NOB, dashed line) and temperature-dependent thermal expansion coefficient (NOB, dash-dotted line). Results from the simulation of forced convection turbulence (symbols) are also included. (a) Mean fluid streamwise velocity 〈ux〉; (b) mean fluid temperature 〈θ〉.

Fig. 8

Wall-normal transport of momentum and heat for PRB turbulent channel flow at Riτ = 498 (Reτ = 150): comparison between simulations with uniform thermophysical properties (OB, solid line) and simulation with temperature-dependent thermal expansion coefficient (NOB, dash-dotted line). Results from the simulation of forced convection turbulence (symbols) are also included. (a) Wall-normal behavior of the total (τxztot), turbulent (τxzt), and viscous (τxzv) shear stresses. (b) Wall-normal behavior of the total (qadtot), turbulent (qadt), and conductive (qadv) heat fluxes.

Fig. 9

(a) Mixing efficiency (Rif) for PRB turbulent channel flow (assuming OB conditions) at Richardson number Riτ = 346 (Reτ = 180, solid line), Riτ = 498 (Reτ = 150, dotted line), and Riτ = 926 (Reτ = 110, dash-dotted line). (b) Production of TKE by mean shear, Pk. (c) Production of TKE by buoyancy, Bk. (d) Wall-normal behavior of the ratio between mixing efficiency for PRB turbulent channel flow assuming temperature-dependent thermal expansion coefficient (Rif,β) and the mixing efficiency for PRB turbulent channel flow assuming uniform fluid properties (Rif).

Fig. 10

Fluid velocity and temperature statistics for PRB turbulent channel flow at Riτ = 498 (Reτ = 150): comparison between simulations with uniform thermophysical properties (OB, solid line) and simulations with temperature-dependent viscosity (NOB, dashed line) and temperature-dependent thermal expansion coefficient (NOB, dash-dotted line). Results from the simulation of forced convection turbulence (symbols) are also included. (a) RMS of streamwise velocity fluctuations, 〈RMS(u' x)〉; (b) RMS of spanwise velocity fluctuations, 〈RMS(u' y)〉; (c) RMS of wall-normal velocity fluctuations, 〈RMS(u' z)〉; (d) RMS of temperature fluctuations, 〈RMS(θ')〉.

## Discussions

Some tools below are only available to our subscribers or users with an online account.

### Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related Proceedings Articles
Related eBook Content
Topic Collections