Mixed convection heat transfer correlations in shallow rectangular cavities with single and double-lid driven boundaries

https://doi.org/10.1016/j.ijheatmasstransfer.2018.11.164Get rights and content

Highlights

  • The analytical and numerical results (finite volume method) agree well.

  • Ra/Pe3 is the parameter determining the three observed convective regimes.

  • For single-lid driven cavity, mixed convection exists for 0.01612 < Ra/Pe3 < 1156.2824.

  • For double-lid driven cavity, mixed convection exists for 0.0988 < Ra/Pe3 < 9150.0833.

  • The kinematic boundary conditions affect notably mixed convection regime limits.

  • Pr effect does not appear since this parameter is incorporated in Pe and Ra.

Abstract

Mixed convection in single and double-lid driven horizontal rectangular cavities filled with a Newtonian fluid and subjected to uniform heat flux along their vertical short sides is studied numerically and analytically. The finite volume method with the SIMPLER algorithm is used to solve the full governing equations for which the Boussinesq approximation is adopted while the analytical approach lies on the parallel flow assumption, valid in the case of shallow enclosures. A good agreement between the two approaches is observed within the explored ranges of Peclet and Rayleigh numbers. The effects of such parameters on the flow and heat transfer characteristics are analyzed for both kinds of driven cavities. The zones characterizing the dominance of natural and forced convections as well as when the two phenomena compete (mixed convection) are delineated. It is found that the transition from one dominated regime to another depends on the ratio Ra/Pe3.

Introduction

In recent years, mixed convection in lid-driven cavities has received considerable attention from researchers. This phenomenon is commonly encountered in many engineering applications including cooling of electronic devices, food processing, float glass production [1], thermal hydraulics of nuclear reactors [2], dynamics of lakes [3], crystal growth, flow and heat transfer in solar ponds [4], and lubrication technologies [5].

In closed cavities, mixed convection flow is induced by both the shear force, caused by the movement of the wall, and the buoyancy one, produced by the thermal gradient due to the temperature difference between non-isothermally heated boundaries, which are of comparable magnitudes. In general, such a phenomenon is governed by some dimensionless parameters such as the Reynolds number, Re, and Grashof one, Gr, expressing shear and buoyancy effects, respectively. The competition between the two phenomena can be incorporated in the modified Richardson number (also called the mixed convection parameter) as follows: Ri = Gr/Ren, where n depends on the geometrical characteristics and physical conditions. The limiting cases Ri → 0 and Ri → ∞ correspond to strongly dominating forced and natural convection flows, respectively. Note that the Prandtl number, Pr, which characterizes the physical proprieties of the fluid, is of importance and must be taken into account.

Remember that most of the investigations, in such a situation, are related to square cavities. These can be classified in two types, according to the number of moving walls. In the first type, one horizontal [6], [7], [8], [9], [10], [11] or vertical [12] wall moves uniformly [6], [7], [8], [9], [10], [12] or non-uniformly [11], [13] in its plane, while the horizontal or vertical walls are differentially heated by a constant [8], [9], [10], [11], [12] or non-constant temperature [6], [7]. In the second type, both the horizontal [14], [15], [16] or vertical [17], [18] walls are moving with a uniform velocity in their planes and have different constant temperatures, while the other ones are adiabatic. In the works cited before, mixed convection has been analyzed numerically for various values of Pr, Re and Ri or Gr. These studies have been extended to cubic cavities [19], [20] and inclined single [21], [22], [23] and double [24], [25] lid-driven square ones.

On the other hand, only a little interest has been accorded to the rectangular enclosures, as in the present case, which may reveal something different as reported in the study conducted in the past by Cormack et al. [26] and the work of Lamsaadi et al. [27], where all walls are motionless. In fact, these authors have observed a flow parallelism and a thermal stratification, from a threshold value of the aspect ratio. However, the effect of the Prandtl number has been found to be negligible because of the strong domination of the momentum diffusion on the thermal one. In contrast, in the study of Karimipour et al. [28], the parameter Pr seems to affect notably mixed convection heat transfer in a rectangular inclined lid-driven cavity with short walls thermally insulated, while the long ones are considered isothermal. In addition to the analytical and numerical works, several experimental investigations have been conducted. In this regard, it is interesting to report to the study by Rossby [29], where it has been found that a linear temperature distribution will generate a single convicting cell of marked asymmetric structure.

Otherwise, the majority of investigations concerning rectangular driven cavities has been dealt with Dirichlet boundary conditions of temperature. It is appropriate here to mention a work by Prasad et al. [30], where mixed convection inside a rectangular cavity has been studied numerically using a finite-volume method for Re = 100, Gr = 0, ±104 and ±106, A = 0.5, 1 and 2 (A = height/width) and Pr = 1. The two vertical walls are maintained at cold temperature, T = 0. In one case, the top-moving wall is maintained at hot temperature, T = 1, and the bottom is cold, T = 0. In the other case, the top is cold, T = 0, and the bottom is hot, T = 1. These authors have observed that, a strong convection for Gr < 0 as ∣Gr∣ is increased for A = 0.5 and 1. However, for A = 2, the flow undergoes a Hopf bifurcation for Gr  −105 and the flow does not remain steady any longer and becomes transient. For this value of Gr, the authors have obtained a periodic oscillation of the total kinetic energy, which does not keep periodic when Gr tends to −106. For this kind of configuration and boundary conditions imposed by these authors, interesting behaviors of the flow and thermal fields with increasing inclination have been observed by Sharif [31], in the case of shallow inclined driven cavities with hot top moving lid and cooled bottom. Hence, for A = 10 (width/height) and Pr = 6, 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. However, at the cold wall, this parameter exhibits oscillatory behavior near the right side owing to the presence of separation bubble at the cold surface in that location. In this study, it has been concluded that, the average Nusselt number increases mildly with the cavity inclination for a prevailing forced convection (Ri = 0.1), while it increases much more rapidly with that inclination when domination returns to natural convection (Ri = 10).

As it is known, the problem of mixed convection heat transfer of Newtonian fluids in a lid driven enclosure subjected to thermal boundary conditions of Neumann type (i.e. imposed heat fluxes to the boundaries) is not yet examined. So, in order to know more about the effect of the boundary conditions kind on flow and heat transfer, the present paper deals with such a problem within a single and double-lid driven horizontal rectangular cavity filled with a Newtonian fluid. The enclosure is submitted to constant heat fluxes from its short vertical edges, while its long horizontal boundaries are insulated. In what follows, a numerical solution of the full governing equations is obtained, for a wide range of the governing parameters, whose influence on the mixed convection heat transfer is amply discussed, for the two considered configurations. Moreover, an analytical solution, valid for stratified flows in slender enclosures, is derived on the basis of the parallel flow concept. The computations are limited to values of governing parameters within the ranges, 1Ra107, 0.1Pe500 and A = 24. Useful correlating relations between Ra and Pe to realize the contribution of mixed convection to heat transfer are also proposed, in both cases of driven cavity under consideration.

Section snippets

Physical problem and governing equations

A schematic of the physical problem and the associated boundary conditions are shown in Fig. 1. It consists of a shallow horizontal rectangular cavity of height H′ and length L′, filled with a Newtonian fluid and submitted to a uniform density of heat flux, q′, from its short vertical sides, while the long horizontal ones are insulated. Two cases of kinematic boundary conditions are considered in this study. In the first case, the top wall is assumed to slide from left to right (i.e. in the

Numerical approach

The system of Eqs. (8), (9), (10), (11) associated with the boundary conditions (12), (13), (14), are numerically solved using a finite volume method and SIMPLER algorithm in a staggered uniform grid system [34]. A second order backward finite difference scheme is employed to discretize the temporal terms appearing in (9), (10), (11). A line-by-line tridiagonal matrix algorithm with relaxation is used in conjunction with iterations to solve the nonlinear discretized equations. The convergence

Parallel flow approach

While referring to Fig. 2, Fig. 3, the following simplifications, in the central part of the cavity, can be made:u(x,y)=u(y),v(x,y)=0,ψ(x,y)=ψ(y)andT(x,y)=C(x-A/2)+θ(y)where C is an unknown constant temperature gradient in x-direction (see for instance [27], [35]). Accordingly, the system (8), (9), (10), (11), associated with the boundary conditions (12), (13), (14), becomes:d3u(y)dy3=RaPeCd2θ(y)dy2=CPeu(y)u+a=dθ(y)dy=0fory=0andu-1=dθ(y)dy=0fory=1with01u(y)dy=0and01θ(y)dy=0as return flow and

Scaling analysis

In this section, a scaling analysis is performed to predict the flow behavior and heat transfer, depending on Ra (pure natural convection) and Pe (pure forced convection), while assuming the existence of a boundary-layer regime in the regions adjacent to the vertical walls, a thermal stratification in the core region for A >> 1 and negligible inertia terms when Pr1. For pure natural convection, related details can be found in the article by Alloui et al. [37].

In the boundary-layer region of

Results and discussion

The fact of imposing uniform heat flux, as thermal boundary conditions, leads to flow characteristics independent on the aspect ratio, A, when this one is large enough. The approximate solution, developed in the preceding section, on the basis of the parallel flow assumption, is thus valid asymptotically in the limit of A >> 1. Therefore, numerical tests are performed to determine the smallest value of A leading to results reasonably close to those of large aspect ratio approximation, commonly

Conclusion

The problem of mixed convection in a two-dimensional single and double-lid driven enclosures filled with a Newtonian fluid is investigated analytically and numerically in the case of imposed uniform heat fluxes on their short vertical walls. Numerical and analytical solutions are obtained for various combinations of the controlling parameters, which are the Peclet (0.1Pe500) and Rayleigh (1Ra107) numbers. The analytical solution is derived on the basis of the parallel flow assumption, valid

Conflict of interest

The authors declare that there is no conflict of interest related to such a study.

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