شبیه سازی عددی همرفت مختلط آرام دو–پخشی diffusive laminar در حفره کم عمق شیبدار

ABSTRACT Numerical simulation of double diffusive laminar mixed convection in shallow inclined cavities with moving lid

In the past three decades a lot of interest has been given to the heat transfer in fluid filled cavities. Numerous investigations have been conducted in the past on lid-driven cavity flow and heat transfer considering various combinations of the imposed temperature gradients and cavity configurations. This is because the driven cavity configuration is encountered in many practical engineering and industrial applications. Convection plays a dominant role in crystal growth (e.g., [1], [2], [3], [4]) in which it affects the fluid-phase composition and temperature at the phase interface that results in a single crystal since poor crystal quality is due to turbulence. It is the foundation in modern electronics industry to produce pure and perfect crystals to make transistors, lasers rods, microwave devices, infrared detectors, memory devices, and integrated circuits. The analysis of lid-driven cavity flow and heat transfer considering various combinations of the imposed temperature gradients and cavity configurations is important because the resulting flow however, is rather complex even when the flow is purely shear driven for the isothermal case without any temperature gradient. When a temperature gradient is imposed such that the shear driven and buoyancy effects are of comparable magnitude then the resulting flow falls under the mixed convection regime and the interaction and coupling of these effects makes the analysis more complex. The combination of temperature and concentration gradients in the fluid will lead to double-diffusive flows. This has an important influence on the solidification process in a binary system [5] and oceanography [6]. Hyun and Lee [7] have reported numerical solutions for double-diffusive convection in a rectangular enclosure with aiding and opposing temperature and concentration gradients. Their solutions were compared favorably with reported experimental results. Most of the studies performed on natural convection heat transfer inside enclosed spaces are related to unidirectional heat flows across rectangular cavities, wherein the buoyancy is induced by imposing a heat flux or a temperature difference either horizontally or vertically from below, in examining conventional convection or thermal instabilities, respectively. Detailed reviews on this topic are easily available in the literature, such as those by Ganzarolli and Milanez [8] and Basak et al. [9]. The natural convection in a two dimensional cavity with vertical cavity walls are maintained at constant and uniform different levels of temperature and two adiabatic walls which analyzed by Loa et al. [10] and Costa [11]. Aydin et al. [12] investigated the effect of Rayleigh number and aspect ratio in two-dimensional enclosure isothermally heated from the left and cooled from the ceiling while the other side wall and the floor are perfectly insulated. Iwatsu and Hyun [13] also carried three-dimensional numerical simulations of fluid flow and heat transfer in a lid-driven cavity and analyzed the results for different values of Ri and Pr = 0.71. They observed that when Ri ⩾ 0, the stable temperature distribution tends to suppress the vertical fluid motion and as a result much of the fluid motion takes place in the vicinity of the top sliding lid and the bulk of the cavity region is nearly stagnant. Pesso and Piva [14] investigated the influence of Prandtl number and Rayleigh number on average Nusselt number. The study shows that the average Nusselt number increases with the Prandtl number and, in particular, its effect on the Nusselt number is more evident at high Rayleigh numbers. Mohamad and Viskanta [15] they reported on the onset of instability in a shallow lid-driven cavity heated from below. They carried out a linear stability analysis and found that Pr influences the conditions for the initiation of the mixed convection regime. Amiri et al. [16] have analyzed the effects of mixed convection heat transfer in lid-driven cavity with sinusoidal wavy bottom surface. Moallemi and Jang [17] numerically studied mixed convective flow in a bottom heated square driven cavity and investigated the effect of Prandtl number on the flow and heat transfer process. They found that the effects of buoyancy are more pronounced for higher values of Prandtl number. Mansour and Viskanta [18] studied mixed convective flow in a tall vertical cavity where one of the vertical sidewalls, maintained at a colder temperature than the other, was moving up or downward thus assisting or opposing the buoyancy. They observed that when shear assisted the buoyancy a shear cell developed adjacent to the moving wall while the buoyancy cell filled the rest of the cavity. Oztop and Dagtekin [19] performed numerical analysis of mixed convection in a square cavity with moving and differentially heated sidewalls. Prasad and Koseff [20] performed a series of experiments in a cavity filled with water and measured heat flux at various locations over the hot cavity floor for a range of Re and Gr. However, in the aforementioned works attention has been addressed to the un-tilted geometry, although the inclination effects may be of interest in many science and engineering applications. In fact, as the buoyancy force components change with orientation, transitions between different flow patterns may occur, and the heat transfer rates may change drastically as, e.g., found by Sharif [21] who carried out the analysis of mixed convective flow in a lid-driven inclined shallow rectangular cavity with a top hot lid and cooled from bottom with inclination angle γ in the range between 0° and 30° of aspect ratio 10 with Rayleigh numbers ranging from 105 to 107 keeping the Reynolds number fixed at 408.21. He found that the local Nusselt number at the heated moving lid starts with a high value and decreases rapidly and monotonically to a small value towards the right side. Chamkha and Al-Naser [22] investigated the double-diffusive convective flow of a binary fluid mixture in inclined square and rectangular cavities. They expanded the range of inclination angle to be from 0° to 90°. Their study was focused on the effect of the buoyancy ratio and the rotation on the evolution of the flow field, temperature and solute field in the cavity. Cianfrini and Corcione [23] expanded the study range of the tilting angle from 0 ⩽ γ ⩽ 90° to 0° ⩽ γ ⩽ 360°. It is found for any tilting angle γ < 45°, the heat transfer rate is the same as that for angle (90°–γ). For any tilting angle γ > 225°, the heat transfer rate is the same as that for angle (450°−γ). Benhadji and Vasseur [24] study the Darcy model with the Boussinesq approximations in double diffusive convection in a shallow porous cavity saturated with a non-Newtonian fluid, a power-law model is used to characterize the non-Newtonian fluid behavior. Recently, Teamah et al. [25], [26], [27], [28], [29] studied numerically the mixed convection in a rectangular lid-driven cavity under the combined buoyancy effects of thermal and mass diffusion with moving upper surface, both upper and lower surfaces are being insulated and impermeable, constant different temperatures and concentration are imposed along the vertical walls of the enclosure, steady state laminar regime is considered. The transport equations for continuity, momentum, energy and spices transfer are solved. The numerical results are reported for the effect of Richardson number, Lewis number, and buoyancy ratio on the iso-contours of stream line, temperature, and concentration. In addition, the predicted results for both local and average Nusselt and Sherwood numbers are presented and discussed for various parametric conditions. This study was done for 0.1 ⩽ Le ⩽ 50 and Prandtl number Pr = 0.7. Throughout the study the Grashof number and aspect ratio are kept constant at 104 and 2 respectively and −10 ⩽ N ⩽ 10, while Richardson number has been varied from 0.01 to 10 to simulate forced convection dominated flow, mixed convection and natural convection dominated flow.

This study is a parametric study and extension of Sharif [21] study. A wide range of buoyancy ratio is studied from −10 ⩽ N ⩽ 10. In addition, a wide range of Lewis number is studied from 0.1 ⩽ Le ⩽ 10. Moreover, the top lid moves in the upward and downward directions thus the lid-driven shear may be assisting or opposing the buoyancy and the resulting flow. Thermal and mass fields as well as the heat and mass transfer processes will be very different from those of the horizontal configuration case.