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Characterization of fluid flow patterns and heat transfer in horizontal channel mixed convection

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Abstract

Two mechanisms of roll initiation are highlighted in a horizontal channel flow, uniformly heated from below, at constant heat flux (Γ = 10, Pr = 7, 50 ≤ Re ≤ 100, 0 ≤ Ra ≤ 106). The first mechanism is the classical one, it occurs for low Rayleigh numbers and is initiated by the lateral wall effect. The second occurs for higher Rayleigh numbers and combines the previous effect with a supercritical vertical temperature gradient in the lower boundary layer, which simultaneously triggers pairs of rolls in the whole zone in between the two lateral rolls. We have found that in the present configuration, the transition between the two roll initiation mechanisms occurs for Ra/Re 2 ≈ 18. Consequently, the heat transfer is significantly enhanced compared to the pure forced convection case owing to the flow pattern responsible of the continuous flooding the heated wall with cold fluid.

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Abbreviations

b :

channel width (m)

g :

gravitational acceleration constant (m s−2)

h :

channel height (m)

k :

thermal conductivity of water (W m−1 K−1)

L :

channel dimensionless length

l :

channel length (m)

l e :

development length (m)

Le :

dimensionless development length

n :

outward unit vector

Nu :

Nusselt number, q h /k(T sa−T av)

P :

dimensionless dynamic pressure

Pr :

Prandtl number, ν/α

q :

constant heat flux density at the bottom wall (W m−2)

Ra :

Rayleigh number, gβh 4 q/kvα

Re :

Reynolds number, u av h/ν

T :

temperature (K)

U, V, W :

dimensionless velocity components, in X, Y, Z directions, u/u av, v/u av , w/u av

u av :

mean longitudinal velocity (m/s)

X, Y, Z :

dimensionless coordinates, x/h, y/h, z/h

α :

thermal diffusivity (m2 s−1)

β :

volumetric thermal expansion coefficient (K−1)

Γ :

spanwise aspect ratio, b/h

ρ :

fluid density (kg m−3)

μ :

dynamic viscosity of water (kg m−1 s−1)

ν :

kinematic viscosity of water (m2 s−1)

θ :

dimensionless temperature, (T − T i)k/(q h)

av:

average value

c:

central

cr:

critical

i:

inlet quantities

ref:

reference

sa:

spanwise average

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Acknowledgments

The authors thank The CNRS for providing substantial computer resources on its IBM SP4 high performance computer at IDRIS, Orsay, France.

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Batna, Rue Boukhlouf Mohamed el Hadi, 05000, Batna, Algeria

    A. Benderradji

  2. IUSTI-CNRS UMR 6595, Polytech’Marseille, Technopôle de Château-Gombert, 5 rue Enrico Fermi, 13453, Marseille, France

    A. Haddad, R. Taher, M. Médale, C. Abid & F. Papini

Authors
  1. A. Benderradji
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  2. A. Haddad
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  3. R. Taher
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  4. M. Médale
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  5. C. Abid
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  6. F. Papini
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Corresponding author

Correspondence to C. Abid.

Appendix : Determination of appropriate mesh resolution

Appendix : Determination of appropriate mesh resolution

To save computational solution time the spatial convergence analysis has been conducted for a steady state configuration, which corresponds to the following set of control parameters: Pr = 6.7, Re = 50 and Ra = 1.28 × 104. Taking advantage that this configuration leads to a symmetric longitudinal roll pattern with respect to the longitudinal vertical median plane, one can therefore consider a computational domain restricted to only one half of the physical domain. We have considered for this spatial convergence analysis three meshes discretizing one half of the physical domain, while considering symmetry boundary conditions over the longitudinal vertical median plane. The three meshes are made up of tri-quadratic hexahedral finite elements, uniformly distributed along the X, Y and Z directions, respectively. Their individual description is given in Table 1.

Table 1 Considered meshes in the spatial convergence analysis
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Let us compare the results obtained for three meshes, qualitatively and quantitatively. First of all, one can observe the isotherms at the lower heated wall compare qualitatively well between the coarsest and finest meshes in Fig. 15 or between the intermediate and finest meshes in Fig. 16. Moreover, we have also compared quantitatively the results obtained from the three considered meshes. The spanwise temperature profile is also plotted at three longitudinal locations, cf. Fig. 17a, and finally the spanwise vertical velocity component is also plotted for the three meshes, at mid channel height at three longitudinal locations, cf. Fig. 17b. Finally, we have compared for higher Rayleigh number (Re = 50, Ra = 5.13 × 104), the spanwise average Nusselt number versus the longitudinal direction for two mesh resolutions (M1 and M3). One can observe that the coarser mesh slightly overestimates the Nusselt number (less 4%) and underestimates the roll development length (less than 6%). One can conclude from theses Figs. 15–18 that the three meshes are qualitatively able to represent the present mixed convection and moreover spatial convergence has been already reached for mesh M2, since no substantial difference can be observed compared to the finest mesh M3.

Fig. 15
figure 15

Computed thermal field over the heated wall. Comparison between the isotherms computed with the finest mesh M3 (upper part) and coarser mesh M1 (lower part)

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Fig. 16
figure 16

Thermal field over the heated wall. Comparison between the isotherms computed with the finest mesh M3 (upper part) and intermediate mesh M2 (lower part)

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Fig. 17
figure 17

a Computed spanwise plot of the non-dimensional temperature at heated lower wall, for three longitudinal abscissas and for the three meshes; b Computed spanwise plot of the non-dimensional vertical velocity component at mid channel height, for three longitudinal abscissas and for the three meshes

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Fig. 18
figure 18

Computed spanwise average Nusselt number at lower wall for Re = 50, Ra = 5.13 × 104 and two spatial resolution (M1 = 375 × 100 × 10 and M3 = 375 × 200 × 20)

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Benderradji, A., Haddad, A., Taher, R. et al. Characterization of fluid flow patterns and heat transfer in horizontal channel mixed convection. Heat Mass Transfer 44, 1465–1476 (2008). https://doi.org/10.1007/s00231-008-0379-3

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  • DOI: https://doi.org/10.1007/s00231-008-0379-3

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