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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.

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## Figures

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. 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. 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. 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. 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. 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. 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')〉.

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. 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).

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