Aspect ratio and boundary conditions effects on laminar natural convection of power-law fluids in a rectangular enclosure with differentially heated side walls
Introduction
Natural convection in rectangular enclosures with differentially heated vertical side walls have been analysed extensively for Newtonian fluids [1], [2], [3], [4], [5], [6] and an extensive review can be obtained in Ref. [5]. In comparison to the vast body of literature regarding the natural convection of Newtonian fluids, limited effort has been directed towards the fundamental understanding of natural convection of non-Newtonian power-law fluids in rectangular enclosures [7], [8], [9], [10], [11], [12], [13], [15], [16]. Kim et al. [11] analysed transient natural convection of power-law fluids with n ⩽ 1 in a square enclosure with differentially heated vertical side walls subjected to constant wall temperatures (CWT) and found that the mean Nusselt number increases with decreasing power-law index n for a given set of values of Rayleigh and Prandtl numbers. This result is consistent with the numerical findings of Lamsaadi et al. [8] and Ohta et al. [9] for natural convection of shear-thinning fluids in the Rayleigh-Bénard configuration. The strengthening of natural convection in rectangular enclosures for shear-thinning fluids was also confirmed by both experimental and numerical analyses by Inaba et al. [10] in the Rayleigh-Bénard configuration. Lamsaadi et al. [12], [13] addressed the effects of the power-law index on natural convection in the high Prandtl number limit for both tall [12] and shallow enclosures [13] where the sidewall boundary conditions are subjected to constant wall heat fluxes (CWHF). Lamsaadi et al. [12], [13] showed that the rate of convective heat transfer rate becomes significantly dependent on the nominal values of Rayleigh number Ra and the power-law index n for large values of aspect ratio and nominal Prandtl number Pr. The above findings were confirmed in previous analyses by the present authors for both CWT [15] and CWHF [16] boundary conditions for square enclosures with differentially heated vertical sidewalls. Turan et al. [15], [16] subsequently used the numerical simulation results to propose correlations for mean Nusselt number for square enclosures with differentially heated vertical sidewalls for 103 ⩽ Ra ⩽ 106; 0.6 ⩽ n ⩽ 1.8 and 1 ⩽ Pr ⩽ 104 for both CWT and CWHF boundary conditions.
The aspect ratio of the enclosure AR = H/L influences the thermal transport in rectangular enclosures with differentially heated side walls [4], [6] and interested readers are referred to Ref. [6] for an extensive review of such effects. Most previous studies on aspect ratio effects on natural convection in rectangular enclosures were carried out for CWT boundary conditions [6] but a recent analysis by Turan et al. [4] showed for Newtonian fluids that the effects of aspect ratio on the mean Nusselt number for the CWHF boundary condition are both qualitatively and quantitatively different from the CWT boundary condition. Recently, Khezzar et al. [17] analysed the effects of inclination angle on natural convection of power-law fluids with aspect ratios 1, 4 and 8 for different values of Rayleigh and Prandtl numbers.
It has been demonstrated by Turan et al. [15], [16] that Pr has a marginal influence on in the range 10 ⩽ Pr ⩽ 106 in the case of natural convection of power-law fluids in square enclosures with differentially heated vertical sidewalls. Practical power-law fluids such as aqueous solutions of commonly used polymers including carboxymethyl cellulose, polyethylene oxide, carbopol, polyacrylamide and xantham gum exhibit Prandtl numbers of the order of 103 or higher.
The present paper deals with numerical analysis of steady two-dimensional natural convection of non-Newtonian fluids obeying the power-law model in rectangular enclosures with differentially heated side walls. The differences between the effects of aspect ratio on natural convection in response to CWT and CWHF boundary conditions are specifically addressed and correlations for are proposed for both boundary conditions. A parametric study has been conducted with the power-law index nranging from 0.6 to 1.8 for a range of nominal Rayleigh numbers (104-106) and aspect ratios (i.e., AR = 1/8-8) and at a fixed value of Prandtl number Pr = 103 (definitions are provided in Section 2). The power-law exponents range 0.6–1.8 spans both shear-thinning, n < 1, (e.g., fluids such as non-drip paint, ketchup, lava, blood etc.) and shear-thickening, n > 1, (e.g., mixtures of corn starch and water, ethylene glycol, so-called “bullet-proof” custard etc.) fluids. A value of power-law exponent n smaller than 0.6 (i.e., n < 0.6) produces a fluid which is extremely shear thinning similar in some ways to some yield stress fluids. On the other hand convection becomes extremely weak and therefore cannot impart any influence on heat transfer within the enclosure for n ⩾ 1.8. In light of the above, the main objectives of the present paper are as follows:
- 1.
To indicate the differences between the effects of aspect ratio on natural convection of power-law fluids in response to CWT and CWHF boundary conditions.
- 2.
To propose correlations for which account for natural convection of power-law fluids in rectangular enclosures with various aspect ratios for both CWT and CWHF boundary conditions.
The rest of the paper will be organised as follows. The necessary mathematical background and numerical details will be presented in the next section, which will be followed by the scaling analysis. Following this analysis, the results will be presented and subsequently discussed. The main findings will be summarised and conclusions will be drawn in the final section of this paper.
Section snippets
Non-dimensional numbers
For the Ostwald-De Waele (i.e., power-law) model the viscous stress tensor τij is given by [8], [12], [13], [15], [16], [17]:where eij = 0.5(∂ui/∂xj + ∂uj/∂ xi) is the rate of strain tensor, K is the consistency and n is the power-law index. Following several previous analyses [8], [12], [13], [16] the norminal Rayleigh and Prandtl numbers are defined as:Using dimensional analysis it is possible
Scaling analysis
A scaling analysis is performed to elucidate the anticipated effects of aspect ratio, Rayleigh number, Prandtl number and power-law index on the Nusselt number for power-law fluids. The wall heat flux q can be scaled as: q ∼ kΔT/δth ∼ hΔT, which gives rise to the following relation:where the thermal boundary-layer thickness δth is related to the hydrodynamic boundary-layer thickness δ in the following manner: δ/δth ∼ f2(RaCWT,Pr,AR,n) and δ
Effects of nominal Rayleigh number RaCWT for the CWT configuration
The variations of non-dimensional temperature θCWT = (T − Tcen)/(TH − TC) and vertical velocity component V = u2L/α along the horizontal mid-plane (i.e., x2/H = 0.5) for RaCWT = 104, 105 and 106 at Pr = 103 are shown in Fig. 2, Fig. 3 respectively for both n = 0.6 and n = 1.8 fluids. Fig. 2a–c show that the variation of θCWT with x1/L remains linear for small values of aspect ratio (e.g., AR = 0.125) for RaCWT = 104, 105 and 106 for both n = 0.6 and 1.8 excluding the case given by AR = 0.125, RaCWT = 106 and n = 0.6. This
Conclusions
The influences of CWT and CWHF boundary conditions on the effects of aspect ratio (=H/L where H is the enclosure height and L is the enclosure length) on the heat transfer characteristics of steady laminar natural convection of power-law fluids in rectangular enclosures with differentially heated vertical side walls have been numerically analysed. It is found that for the CWT boundary condition exhibits a non-monotonic trend with aspect ratio AR for shear-thinning (n < 1), Newtonian (n = 1) and
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