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

International Journal of Heat and Mass Transfer 132 (2019) 394–406

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Mixed convection heat transfer correlations in shallow rectangular cavities with single and double-lid driven boundaries A. Louaraychi a, M. Lamsaadi a,⇑, M. Naïmi a, H. El Harfi a, M. Kaddiri a, A. Raji a, M. Hasnaoui b a b

Sultan Moulay Slimane University, Faculty of Sciences and Technologies, Laboratory of Flows and Transfers Modelling (LAMET), B.P. 523, Beni-Mellal, Morocco Cadi Ayyad University, Faculty of Sciences Semlalia, Laboratory of Fluid Mechanics and Energetics (LMFE), B.P. 2390, Marrakech, Morocco

a r t i c l e

i n f o

Article history: Received 27 March 2018 Received in revised form 18 November 2018 Accepted 30 November 2018

Keywords: Mixed convection Lid-driven cavities Parallel flow Finite volume method Heat transfer

a b s t r a c t 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. Ó 2018 Elsevier Ltd. All rights reserved.

1. 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 ? 1 correspond to strongly dominating forced and natural convection flows, respectively. Note that the Prandtl number, Pr, which characterizes the

⇑ Corresponding author. E-mail address: [email protected] (M. Lamsaadi). https://doi.org/10.1016/j.ijheatmasstransfer.2018.11.164 0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.

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 [6–11] or vertical [12] wall moves uniformly [6–10,12] or non-uniformly [11,13] in its plane, while the horizontal or vertical walls are differentially heated by a constant [8–12] or nonconstant temperature [6,7]. In the second type, both the horizontal [14–16] 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 [21–23] 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

395

A. Louaraychi et al. / International Journal of Heat and Mass Transfer 132 (2019) 394–406

Nomenclature A C g Gr H0 L0 Nu Nu Pe Pr p q0 Ra Re t T T00

aspect ratio of the rectangular enclosure, Eq. (16) dimensionless temperature gradient in the horizontal direction (x-direction) acceleration due to gravity (m/s2) Grashof number, (17) height of the enclosure (m) length of the rectangular enclosure (m) local Nusselt number, Eqs. (17) and (18) mean Nusselt number, Eq. (19) Peclet number, Eq. (16) Prandtl number, Eq. (16) dimensionless pressure [= p0 /qu02 0] constant heat flux density (W/m2) Rayleigh number, Eq. (16) Reynolds number, Eq. (17) dimensionless time [= t0 u00 =H0 ] dimensionless temperature [=(T0 T00 ) k/q0 H0 ] reference temperature (K)

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 jGrj 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

(u, v) u00 (x,y)

dimensionless axial [=(u0 , v0 )/u00 ] lid-velocity (m/s) dimensionless axial [= (x0 ,y0 )/H0 ]

and

and

transverse

transverse

velocities

coordinates

Greek symbols a thermal diffusivity of fluid at reference temperature (m2/s) b thermal expansion coefficient of fluid at reference temperature (1/K) k thermal conductivity of fluid at reference temperature (W/m C) q fluid density at reference temperature (kg/m3) w dimensionless stream function, (=w0 /a) Superscript 0 dimensional variables

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, 1 6 Ra 6 107 ; 0:1 6 Pe 6 500 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. 2. 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 H0 and length L0 , filled with a Newtonian fluid and submitted to a uniform density of heat flux, q0 , 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 direction of the imposed heat flux) with constant velocity u00 ; while the other walls are motionless. In the second case, both the horizontal walls move in opposite directions with the same uniform velocity u00 ; whereas the vertical ones are motionless. These cases correspond to single and doublelid driven cavities, respectively. The main assumptions made in this study are those commonly used, i.e.,  The fluid velocities are small enough to consider the flow as laminar. In fact, in most buoyancy driven motions, the fluid circulation is slow due to moderate temperature gradients [32].  The fluid is incompressible. For pressures close to atmospheric, liquids can be considered as incompressible with a good approximation.

396

A. Louaraychi et al. / International Journal of Heat and Mass Transfer 132 (2019) 394–406

y ,v

u '0 H

q

q L

x ,u

au '0 Fig. 1. Schematic view of the geometry and coordinates system. (Scale is not respected).

 The contribution of the viscous dissipation is negligible compared with that of the imposed heat flux.  The physical properties are considered independent of temperature except for the density in the buoyancy term, which obeys the Boussinesq approximation, whose systematic derivation from the equations of compressible fluid dynamics is quite tedious and requires careful ordering of several limiting processes. More details are given in the paper of Gray and Giorgini [33].  The third dimension of the cavity is large enough so that the problem can be considered as two dimensional. This assumption is generally relatively well satisfied and provides insight into the more complicated three dimensional flows [32]. Then, taking into account the above-mentioned assumptions, the equations describing the conservation of mass (1), momentum (2), (3), and energy (4), written in terms of velocity components (u0 , v0 ), pressure (p0 ) and temperature (T0 ), are:

@u0 @v0 þ ¼0 @x0 @y0

ð1Þ

" # 0 0 @u0 1 @p0 @ 2 u0 @ 2 u0 0 @u 0 @u þu þv ¼ þm þ @t0 @x0 @y0 q @x0 @x02 @y02

ð2Þ

" # 0 0   @v0 1 @p0 @ 2 v0 @ 2 v0 0 @v 0 @v þu þv ¼ þm þ 02 þ gb T0  T00 2 0 @t0 @x0 @y0 q @y0 @x @y

ð3Þ

" # @T @ 2 T0 @ 2 T0 0 @T 0 @T þ u þ v ¼ a þ @t0 @x0 @y0 @x02 @y02 0

0

0

u0 ¼ v0 ¼ 0 and

u0 þ au00 ¼ v0 ¼ 0 and u0  u00 ¼ v0 ¼ 0 and

ð8Þ

" # @u @u @u @p Pr @ 2 u @ 2 u þ þu þv ¼ þ @t @x @y @x Pe @x2 @y2 " # @v @v @v @p Pr @ 2 v @ 2 v RaPr þ þu þv ¼ þ þ T @t @x @y @y Pe @x2 @y2 Pe2 " # @T @T @T 1 @2T @2T þu þv ¼ þ @t @x @y Pe @x2 @y2 u ¼ v ¼ 0 and

@T ¼ 1 for x ¼ 0 and x ¼ A; @x

ð9Þ

ð10Þ

ð11Þ

ð12Þ

u þ a ¼ v ¼ 0 and

@T ¼ 0 for y ¼ 0; @y

ð13Þ

u  1 ¼ v ¼ 0 and

@T ¼ 0 for y ¼ 1 @y

ð14Þ

In addition, to analysis the flow structure, the stream function,

w, related to the velocity components via:

ð4Þ

u¼

ð5Þ

@T0 ¼ 0 for y0 ¼ 0; @y0

ð6Þ

@T0 ¼ 0 for y0 ¼ H0 ; @y0

ð7Þ

where k stands for the thermal conductivity and ‘‘a” a parameter, which sets equal to 0 and 1 for single and double-lid driven cavities, respectively. 0 0 On the other hand, using the characteristic scales H0 ; qu02 0 ; H =u0 , 0 0 0 u0 and q H =k, corresponding to length, pressure, time, velocity, and temperature, respectively, the dimensionless governing equations and the corresponding boundary conditions are:

@w @y

and v ¼ 

@w ðw ¼ 0 on all boundariesÞ @x

ð15Þ

is used [13–32]. From the above equations the following parameters:

A¼

where q, m, g, b and a are the fluid density, the kinematic viscosity, the gravitational acceleration, the thermal expansion coefficient and the thermal diffusivity, respectively. The thermal and kinematic boundary conditions associated to the governing equations are:

@T0 q0 þ ¼ 0 for x0 ¼ 0 and x0 ¼ L0 ; @x0 k

@u @v þ ¼0 @x @y

L0 ; H0

Pr ¼

m u0 H0 gbq0 H04 ; Pe ¼ 0 and Ra ¼ a a mka

ð16Þ

emerge. They correspond to the aspect ratio of the enclosure, the Prandtl, Pr, Peclet, Pe, and Rayleigh, Ra, numbers, respectively: Note that

Pe ¼ RePr and Ra ¼ GrPr

ð17Þ

where Re and Gr are the Reynolds and Grashof numbers, respectively. On the other hand, to characterize the local heat transfer, through the filled-fluid cavity, it is useful to introduce the local Nusselt number defined by:

NuðyÞ ¼

q0 L 0 1 ¼ kDT0 ðDT=AÞ

ð18Þ

where DT ¼ TðA; yÞ  Tð0; yÞ is the side to side dimensionless local temperature difference. This definition is, however, notoriously inaccurate owing to the uncertainty of the temperature values evaluated at the two vertical walls (edge effects). Instead, Nu is calculated on the basis of a temperature difference between two vertical sections, far from the end sides, as suggested by Lamsaadi et al. [27] for natural convection.

397

A. Louaraychi et al. / International Journal of Heat and Mass Transfer 132 (2019) 394–406

Thus, by analogy with Eq. (18), and considering two infinitesimally close sections, Nu can be expressed by:

NuðyÞ ¼ lim dx=dT ¼ lim

1

dX!0 dT=dx

dX!0

¼

1 ð@T=@xÞx¼A=2

ð19Þ

where dx is an infinitesimal distance between two symmetrical sections with respect to the central one. The average Nusselt number, representing the overall heat transfer, can thus be deduced just like:

Z Nu ¼

1

NuðyÞdy

ð20Þ

0

105 and 103, depending on the values of the governing parameters and on the type driven cavity, while starting the numerical code with (u = v = T = 0) as initial conditions. Typical numerical results, in terms of streamlines and isotherms, are presented in Figs. 2 and 3, obtained, for A ¼ 24, Ra = 104 and various values of Pe with single and double-lid driven cavities. As can be seen, the flow is parallel to the horizontal boundaries of the shallow enclosure and the temperature is linearly stratified in the x-direction independently of the values of Ra and Pe and the kinematic boundary conditions. As it will be discussed in the next section, this is related to the nature of the thermal boundary conditions, which are of Neumann type. The approximate analytical solution, developed in the next section, relies on such observations.

3. Numerical approach 4. Parallel flow approach The system of Eqs. (8)–(11) associated with the boundary conditions (12)–(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)–(11). A line-by-line tridiagonal matrix algorithm with relaxation is used in conjunction with iterations to solve the nonlinear discretized equations. The convergence is reached when  P  kþ1 P  kþ1  k  k  f i;j  < 105 i;j f i;j ; where f i;j stands for the value of u, i;j f i;j v, p or T at the kth iteration level and the grid location (i, j) in the plane (x, y). The mesh size is chosen so that a best compromise between running time and accuracy of the results may be found. The procedure is based on grid refinement until the numerical results agree, within a reasonable accuracy, with the analytical parallel flow ones, developed in the next section. Hence, in the limit of the values selected for A, Pe, Ra and a, as shown in Tables 1 and 2, a uniform grid of 381  121 is considered appropriate to model accurately the flow and temperature fields within a cavity of A = 24 (found as the smallest value of A beyond which mixed convection heat transfer does not change). The values of timestep sizes considered for the simulations are selected between

While referring to Figs. 2 and 3, the following simplifications, in the central part of the cavity, can be made:

uðx; yÞ ¼ uðyÞ; vðx; yÞ ¼ 0; wðx; yÞ ¼ wðyÞ and Tðx; yÞ ¼ Cðx  A=2Þ þ hðyÞ

ð21Þ

where C is an unknown constant temperature gradient in xdirection (see for instance [27,35]). Accordingly, the system (8)– (11), associated with the boundary conditions (12)–(14), becomes: 3

d uðyÞ Ra C ¼ dy3 Pe

ð22Þ

2

d hðyÞ ¼ CPe uðyÞ dy2 uþa¼

ð23Þ

dhðyÞ dhðyÞ ¼ 0 for y ¼ 0 and u  1 ¼ ¼ 0 for y ¼ 1 dy dy

ð24Þ

with

Table 1 Convergence tests of Nu for A = 24, Ra = 104 and various values of Pe for a single-lid driven cavity (a = 0). Pe

0.1 0.5 5 25 50 100 150 200

Numerical solution

Analytical solution

Grids(341  121)

(421  121)

(381  121)

(381  161)

(381  81)

6.8773 6.9422 7.7514 13.7977 30.1091 99.5491 218.0696 384.9385

6.8773 6.9422 7.7514 13.7977 30.0829 99.4279 217.8088 384.5569

6.8773 6.9422 7.7514 13.7977 30.0936 99.4741 217.9231 384.7260

6.8761 6.9410 7.7514 13.7962 30.0926 99.4747 217.9258 384.7312

6.8808 6.9457 7.7551 13.8018 30.0966 99.4724 217.9152 384.7107

6.8745 6.9394 7.7485 13.7944 30.0642 99.2643 217.3456 383.5063

Table 2 Convergence tests of Nu for A = 24, Ra = 104 and various values of Pe for a double-lid driven cavity (a = 1). Pe

0.1 0.5 5 25 50 100 150 200

Numerical solution

Analytical solution

Grids(341  121)

(421  121)

(381  121)

(381  161)

(381  81)

6.8934 7.0244 8.7912 27.5924 87.9333 337.0214 754.4207 1348.0362

6.8934 7.0244 8.7912 27.5924 87.8298 336.7170 753.9114 1347.1129

6.8934 7.0244 8.7912 27.5924 87.8727 336.8514 754.1381 1347.5314

6.8922 7.0232 8.7898 27.5910 87.8717 336.8508 754.1368 1347.5147

6.8969 7.0279 8.7951 27.5966 87.8754 336.8529 754.1417 1347.579

6.8906 7.0216 8.7881 27.5892 87.7604 336.1068 752.1875 1335.225

398

A. Louaraychi et al. / International Journal of Heat and Mass Transfer 132 (2019) 394–406

Fig. 2. Streamlines (left) and isotherms (right) for A = 24, Ra = 104 and various values of Pe: (a) Pe = 0.1, (b) Pe = 0.5, (c) Pe = 5, (d) Pe = 25, (e) Pe = 50, (f) Pe = 100 and (g) Pe = 150 for a single-lid driven cavity (a = 0). (Scale not respected).

Z

1

uðyÞdy ¼ 0

ð25Þ

0

can be imposed in that direction [36]. In particular, in the parallel flow region and with the application of conditions (12), Eq. (30) becomes:

Z

and

Z

C þ PrRe

1

hðyÞdy ¼ 0

as return flow and mean temperature conditions, respectively. Using such an approach, the solutions of (22) and (23), satisfying (24)–(26), for single (a = 0) and double (a = 1) lid-driven cavities, are generalized in the following form:

     Ra C  3 2y  3y2 þ y þ 3y2  2y  a 3y2  4y þ 1 uðyÞ ¼ 12 Pe hðyÞ ¼

ð27Þ

 Ra C2  12y5  30y4 þ 20y3  1 1440    Pe C  15y4  20y3 þ 2  a 15y4  40y3 þ 30y2  3 þ 60 ð28Þ

The expression of the stream function, w(y), can be deduced by integrating Eq. (15), taking into account of the corresponding boundary conditions and Eq. (27), which gives:

     Ra C  4 y  2y3 þ y2 þ y3  y2  a y3  2y2 þ y wðyÞ ¼ 24 Pe

ð29Þ

uðyÞhðyÞdy ¼ 1

ð31Þ

0

ð26Þ

0

1

which, when substituted to Eqs. (27) and (28), gives the following transcendental equation:

" #  Ra2 Ra Pe Pe2  3 2 2 ða þ 1ÞC þ C  2 þ 3a þ 2a þ 1 C þ 1 ¼ 0 3360 362; 880 210 ð32Þ whose solution, via Newton-Raphson method, for given a, Pe and Ra, leads to C. Finally, taking into account of Eqs. (19)–(21), the mean Nusselt number can be expressed as

Nu ¼ 

1 C

ð33Þ

Note that, in pure natural convection (u (0) = u(1) = 0), the governing equations are defined by (8)–(11) while replacing Pe by 1. In such a situation, the parallel flow approach leads the following transcendent equation:

Ra2 C3 þCþ1¼0 362; 880

ð34Þ

whose approximate solution is: On the other hand, the parallel flow approximation is only valid in the core of the layers. Flow in the end regions is much more complicated and cannot be approximated in such a simple manner. For this reason, the thermal boundary condition in the x direction Eq. (12) cannot be reproduced exactly with this approximation. Nevertheless, an equivalent energy flux condition:

Z

1

 0

@T dy þ Pe @x

Z

Z

1

uTdy ¼ 0

0

1

  @T  @x x¼0

dy or A

ð30Þ



13 2 1 1 þ 362;880 Ra3 C¼

13 2 

13 2  1 1 1 þ 362;880 Ra3 1 þ 362;880 Ra3

ð35Þ

which agrees well with that obtained via Newton-Raphson method (see Fig. 4). While referring to Eq. (33), the mean Nusselt number, in pure natural convection, can be expressed as:

399

A. Louaraychi et al. / International Journal of Heat and Mass Transfer 132 (2019) 394–406

Fig. 3. Streamlines (left) and isotherms (right) for A = 24, Ra = 104 and various values of Pe: (a) Pe = 0.1, (b) Pe = 0.5, (c) Pe = 5, (d) Pe = 25, (e) Pe = 50, (f) Pe = 100 and (g) Pe = 150 for a double-lid driven cavity (a = 1). (Scale not respected).

1

C

10

-1

Solution of Eq. (34) via Newton-Raphson metoth Approximate solution of Eq. (34)

10

-2

10

-3

1

10

10

2

10

3

10

4

10

5

Ra

10

6

10

7

Fig. 4. Comparison between the solutions of Eq. (34).

 Nu ¼

1 362; 880

13

2

Ra3 þ 1þ



1

13

1 362;880

Ra

2 3

ð36Þ

It is advisable to recall that, the case Ra = 0 corresponds to pure forced convection for which the buoyancy forces are absent. In such a case, the transcendent equation is:



2 þ 3a þ 2a2 2 Pe þ 1 C þ 1 ¼ 0 210

ð37Þ

and C can be explicitly deduced as:

1 C ¼ 2þ3aþ2a2 2 Pe þ 1 210

ð38Þ

leading to the mean Nusselt number, in pure forced convection:

Nu ¼

2 þ 3a þ 2a2 2 Pe þ 1 210

ð39Þ

400

A. Louaraychi et al. / International Journal of Heat and Mass Transfer 132 (2019) 394–406

In the present study it is to notice that, in natural and forced regimes, the boundary layer regime can be reached for large Rayleigh and Peclet numbers, respectively, and it is readily to find from Eqs. (36) and (39) that:

 Nu ¼

1 362; 880

13

2

Ra3

ðNatural convectionÞ

ð40Þ

2 þ 3a þ 2a2 2 Pe 210

ðForced convectionÞ

ð41Þ

and

Nu ¼

In the following section, the order of magnitude obtained by (40) and (41) of the heat transfer rate, as function of Ra and Pe for such regimes, is confirmed by a scale analysis. 5. 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 Pr P 1: For pure natural convection, related details can be found in the article by Alloui et al. [37]. In the boundary-layer region of interest, where most fluid motion is restricted to a thin layer dx << A along each vertical wall, let du be the order of the velocity magnitude within the core region of the cavity and DT the vertical temperature difference changes. Eqs. (8)–(11) and (13) and (14) require the following balance:

du  Ra C and du C  DT ðNatural convectionÞ C

DT Pe

ðForced convectionÞ

ð42Þ ð43Þ

So, in our study, to examine the situation it is more appropriate to use the couple (Ra, Pe), in which Pr is incorporated, instead of (Gr, Re), which reduces the governing parameters number and makes it possible to explicitly ignore the role of Pr and not to need its value. In the following sections, the value of A, satisfying the large aspect ratio approximation and the flow and heat transfer characteristics related to natural, mixed and forced convective regimes are determined according to Ra and Pe values, whose effects are clearly examined and discussed for single and double-lid driven cavities. 6.1. Determination of the value of A satisfying the large aspect ratio approximation In this subsection, the goal is to determine the lower value of A, leading to convection heat transfer results in agreement with those obtained analytically. Thus, the effect of A, on Nu, evaluated numerically, is illustrated in Figs. 5 and 6, for Ra = 104 and various values of Pe. As can be seen, an increase of A leads to an asymptotic behavior of Nu, which seems less precocious for a double-lid driven cavity than for a single-lid driven one, due to the horizontal kinematic boundary conditions, and this more especially for forced convection (Pe P 100). Hence, for all the considered values of Pe and for both cases of driven cavity, the asymptotic analytical limits are largely achieved for A = 24, which reduces to 8 in pure natural convection as has been shown by Lamsaadi et al. [27], thus highlighting the delaying role of the driven walls vis-a-vis the asymptotic state. Moreover, according to Tables 1 and 2, the analytical and numerical results agree perfectly, which validates each analytical approach and calculation code and justifies the choice of the asymptotic value of A. In addition, a well conformation, of such a fact can be obtained from the comparison between the numerical and analytical curves depicted below, in the limit of the explored

On the other hand, from Eq. (12), and the equivalent energy flux condition, it is easy to find that:

values of (Ra 6 5  106 ; Pe 6 500).

duDT  1 ðNatural convectionÞ

ð44Þ

6.2. Determination of the parameters controlling mixed convection

ð45Þ

Several geometric configurations have been considered in the past and the corresponding works have been reported in the literature. It has been found that the parameter Gr/Ren (modified Richardson number or the mixed convection parameter), where the exponent n depends on the geometry, thermal boundary conditions and fluid, is systematically appropriate to delimit the convective regimes. For example, Eckert and Diaguila [39] have studied mixed convection in a vertical tube filled with air and have showed that the heat transfer differs with about 10% from that related to pure natural convection when Gr/Re2.5 > 0.007, while the same deviation from forced convection has been observed for Gr/Re2.5 < 0.0016. Siebers et al. [40] have studied mixed convection along a vertical plate heated by a uniform flux and have observed that the heat transfer is mainly ensured by forced convection/(natural convection) when Gr/Re2 < 0.7/(Gr/Re2 > 10). For the flows on insulated flat plates, Sparrow and Gregg [41] found that the heat is mainly transported by forced convection when Gr/Re2 < 0.15  f(pr), with 0.01 < Pr < 10. On the other hand, for confined media, Turki et al. [42] have found that, for flows of a Newtonian fluid filling a square cavity, the exponent ‘‘n” and the limits of mixed convection regime are sensitive to any change in the Pr value. Thus, to delimit such a regime, the criteria 0.5 < Gr/Re1.9 < 31 and 24.4 < Gr/ Re1.2 < 293.4 have been established for Pr = 1 and Pr = 6.97, respectively. In this work, the ratio Ra/Pe3 appears clearly as a parameter delimiting the three regimes whatever the value of Pr, by matching

DT 

1 Pe

ðForced convectionÞ

Finally, solving Eqs. (42)–(45) gives:

du  Ra1=3 ; Nu  Ra

DT 

2=3

1 ; Pe

DT  Ra1=3 ;

C  Ra2=3

and

ðNatural convectionÞ C

1 Pe2

and Nu  Pe2 ðForced convectionÞ

ð46Þ ð47Þ

6. 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 for all the explored values of Ra and Pe. On the other hand, the Prandtl number, Pr, seems to affect strongly mixed convection heat transfer as shown in Refs. [28,38]. Note that, in contrast, a variation of Pr does not produce any change in natural convection, as has been observed by Lamsaadi et al. [27] and Alloui et al. [35], provided that Pr P 1.

401

A. Louaraychi et al. / International Journal of Heat and Mass Transfer 132 (2019) 394–406

900

Analytical solution

Nu

Numerical solution Pe = 200

Pe = 150

Pe = 100

100

Pe = 50

Pe = 25

10

Pe=5

6 2

4

6

8

10

12

14

16

18

20

22

A

24

26

28

Fig. 5. Evolution of Nu with the aspect ratio in a single-lid driven cavity for Ra = 104 and various values of Pe.

4500

Analytical solution

Nu

Numerical solution Pe = 200

1000

Pe = 150

Pe = 100

100

Pe = 50

Pe = 25

Pe=5

10 6 2

4

6

8

10

12

14

16

18

20

22

24

A

26

28

Fig. 6. Evolution of Nu with the aspect ratio in a double-lid driven cavity for Ra = 104 and various values of Pe.

the two asymptotic relations (40) and (41). This is illustrated in the figures below. Note that, it is practically interesting in heat transfer computations to distinguish the conditions under which a given convection may be regarded as pure (either natural or forced) from those under which it may be considered as mixed. The importance of this matter stems from the fact that often, only results for a pure convection are available. In this work, a convection will be considered to be effectively pure (either natural or forced) if the heat transfer deviates by no more than 5% from the value associated with the completely pure convection. The 5% criterion is used to determine

the parameters characterizing mixed convection in single and double-lid driven rectangular cavities filled with a Newtonian fluid. Thus, in order to differentiate the three convective regimes (forced, natural and mixed convection), the following relative differences are introduced:

ef ¼

  Nu  Nuf  Nuf

and

en ¼

  Nu  Nun  Nun

ð48Þ

where and Nun are the mean Nusselt numbers corresponding to pure forced and natural convections, respectively. It is assumed that

402

A. Louaraychi et al. / International Journal of Heat and Mass Transfer 132 (2019) 394–406

forced convection/(natural convection) is predominant when both ef < /5%/(en < 5%). In the opposite case, the convective regime is qualified as mixed. On the basis of Eq. (48), the calculations performed for Pe and Ra, varying in their explored ranges, allow to construct the diagram shown in Fig. 7. For both cases of driven cavity, the locations of the points (log (Ra), log(Pe)), obtained analytically (solid lines) and numerically (symbols) and corresponding approximately to ef = en = 5%, are organized in almost parallel straight lines. These results are correlated in the following mathematical forms:

¼ gn

and

Ra Pe3

¼ gf

ð49Þ

corresponding, respectively, to natural and forced convections, where gn and gf are coefficients depending on the nature of the kinematic boundary conditions applied to the horizontal surfaces of the fluid layer. Their values are reported in Table 3 and illustrated in Fig. 7 with dashed lines. As can be seen, the agreement between the parallel flow solution and the numerical one is excellent with a maximum deviation not exceeding 2%. In addition, the straight lines defined by Eq. (49) separate the (Ra, Pe) plane into three zones for each type of driven cavity. The first zone, located below the straight line (1), is characterized by the dominance of forced convection. The second zone, located above the straight line (2), corresponds to the natural convection dominating regime. Finally, the third zone, delimited by the two straight lines, is related to the situation where both the phenomena compete (mixed convection regime). This last zone is translated towards higher values of Pe, while passing from double to single-lid driven cavity. This clearly shows that the transition from one regime to another, for the same value of Pe, requires a high buoyancy force as the number of moving horizontal walls doubles. To summarize, the mixed convection regime in a shallow rectangular cavity, filled with a Newtonian fluid and submitted to a uniform density of heat flux from its short vertical sides, is defined by the criteria:

10

Convective regime Single-lid driven cavity Double-lid driven cavity

0:01612 <

Ra Pe3

Natural convection

Forced convection

gn

gf

1156.2824 9105.0833

0.01612 0.09880

< 1156:2824 and 0:0988 <

for single-lid respectively.

driven

cavity

and

double-lid

Analytical solution Numerical solution Single-lid driven cavity Double-lid driven cavity

5

10

4

dom ina nt n atu ral con vec tion

Correlating relation Eq. (49)

10

Mi xed con vec tion

(2)

(1)

10

3

0.1

1

cavity,

ancy effect, giving rise to dynamical and thermal patterns virtually identical to those obtained in pure natural convection for Ra = 104 (see Fig. 8), where the flow and the thermal fields are centro-

(1) Upper limit of dominant forced convection

6

driven

Typical streamlines (left) and isotherms (right) are displayed in Fig. 2 (single-lid driven cavity) and 3 (double-lid driven cavity) for Ra = 104 and various values of Pe. It is interesting to note that the flow is unicellular and clockwise, as a result of cooperating roles of buoyancy and shear effects, which act together from left to right. Also, except for the end sides where the flow undergoes a rotation of 180°, the dynamical and thermal fields exhibit parallel and linearly stratified aspects according to the horizontal direction inside the core region of the enclosure. On the other hand, for relatively high values of Pe, streamlines and isotherms seem to be more sensitive to the horizontal kinematic boundary conditions, since the centro-symmetry of the convective cell disappears and the isotherms inclination, with respect to the vertical direction, increases while passing from identical boundary conditions to mixed ones. However, for a low value of Pe, the effect of such conditions on the centro-symmetry of the isolines does not clearly appear. Ra Hence, a decrease of Pe (ie. when Pe 3 > gn ), makes stronger buoy-

(2) Lower limit of dominant natural convection

10

< 9150:0833

6.3. Dynamical and thermal structures

7

Ra

Ra Pe3

ð50Þ

dom ina nt f orc ed con vec tion

Ra Pe3

Table 3 Values of gn and gf.

10

1

10

2

Fig. 7. Diagram characterizing the different convective regimes.

Pe

10

3

403

A. Louaraychi et al. / International Journal of Heat and Mass Transfer 132 (2019) 394–406

Fig. 8. Streamlines (left) and isotherms (right) for pure natural convection, Ra = 104). (Scale not respected).

Fig. 9. Streamlines (left) and isotherms (right) for pure forced convection in a single-lid driven cavity: (a) Pe = 100 and (b) Pe = 150. (Scale not respected).

symmetric, with a parallelism and a stratification in the major part of the cavity. For both cases of driven cavity, when the shear effect is imporRa tant (ie. when Pe 3 < gf ), the increase of Pe leads to streamlines more crowded near the moving walls and to isotherms more tilted leading to the dynamical and thermal structures almost identical to those obtained in pure forced convection for Pe = 100 and 150 in single and double-lid driven cavities where the streamlines and the isotherms are asymmetric (Fig. 9) and centro-symmetric (Fig. 10). Furthermore, by progressively increasing Pe from its lower value, the isotherms tend to be more tightened at the level of the vertical walls. This expresses strong temperature gradients near these boundaries and supports the development of two boundary layers with decreasing thickness. The tightening of the isotherms is more visible particularly near the left wall in the case of single-lid driven cavity. With two moving horizontal walls, the temperature distribution becomes symmetrical with thermal boundary layers, thinner than for a single-lid driven cavity. At high value of Pe, and for both configurations, the isotherms are almost linear in the central zone of the cavity. 6.4. Horizontal velocity and temperature distributions along the vertical central section Variations of the horizontal velocity (left) and the temperature (right) along the y-axis at the mid-length of the cavity (x = A/2) are illustrated in Fig. 11 (single-lid cavity) and Fig. 12 (double-lid driven cavity), for Pe = 20 and various values of Ra. Here also the agreement between the numerical results (symbol) and the analytical ones (solid lines) appears to be quantitatively perfect. The presented profiles seem to be more sensitive to the horizontal kinematic boundary conditions, since they are asymmetric in single-lid driven cavity and centro-symmetric in double-lid driven cavity, when forced convection is predominant. Thus, the presence of two zones with positive (on the top) and negative (on the bottom) signs, depending on the horizontal kinematic boundary conditions, in the horizontal velocity profile, is compatible with the monocellular clockwise flow driven by both lid and buoyancy

cooperating effects. Moreover, the temperature profile presents, in general, two portions, with negative and positive signs, whose amplitude depends on Ra and on the type of driven cavity. In fact, the resulting clockwise flow makes warm the top, by transporting the heat from the left hot side, and brings the coldest currents towards the lower part of the cavity after discharging the heat across the right vertical wall. Besides, it is easy to note that the increase of the parameter Ra leads to an important amplification of the velocity extremum values for both lid-driven cavities. In the case of a single-lid driven cavity, the velocity profile presents a minimum which becomes more amplified by increasing Ra. For the higher considered value of Ra, the maximum value at a position appears to be shifted away from the moving upper wall. This peak, higher than that of the moving wall, indicates that the flow becomes stronger and faster in this region, which explains the dominant effect of the buoyancy force. For the double-lid driven cavity, the velocity profile is characterized by its linearity when for an important shear effect. For a dominant buoyancy force, the velocity profile exhibits simultaneously two extrema (maximum and minimum) near the moving boundaries. The onset of buoyancy-driven flow (Racr ), which results in such extrema, can be expressed as:

Racr ¼ 48Pe þ

152 3 Pe 105

and Racr ¼ 24Pe þ

124 3 Pe 105

ð51Þ

for single and double-lid driven cavities, respectively (see Fig. 12). 6.5. Heat transfer rate For a well detailed analysis, the effect of Pe and Ra on heat transfer rate, are investigated by varying Pe and fixing Ra and vice versa (Table 4 and Figs. 13 and 14). In Table 4, given below, are displayed the values of Nu, obtained numerically for Ra = 104 and various values of Pe for both considered configurations and different regimes. First of all, it seems clear that Nu increases with Pe as a beneficial consequence of shear effect o heat transfer. Also, it is interesting to note that for the same value of Pe, the heat transfer is more important in the case of double-driven enclosure. This is related to the cooperating effects

Fig. 10. Streamlines (left) and isotherms (right) for pure forced convection in a double-lid driven cavity: (a) Pe = 100 and (b) Pe = 150. (Scale not respected).

404

A. Louaraychi et al. / International Journal of Heat and Mass Transfer 132 (2019) 394–406

1.5

u

0.20

T

Analytical solution

Analytical solution

0.15

1.0

(a)

(b)

0.10

0.5

0.05 0.0

Numerical solution 3 Ra = 10 3 Ra = 5x10 Racr = 12540.95

-0.5 -1.0

Ra = 5x10 5 Ra = 10

-1.5 0.0

0.2

0.4

0.6

y

Numerical solution 3 Ra = 10 3 Ra = 5x10 Racr = 12540.95

0.00 -0.05

4

-0.10

0.8

1.0

Ra = 5x10 5 Ra = 10

-0.15 0.0

0.2

0.4

0.6

y

4

0.8

1.0

Fig. 11. Horizontal velocity (a) and temperature (b) profiles at mid-length of the cavity, along the vertical coordinate in single-lid driven cavity for Pe = 20 and various values of Ra.

1.5

u

0.12

Analytical solution

T

1.0

0.08

(a)

0.5

0.00

Numerical solution 3 Ra = 10 3 Ra = 5x10 Racr = 9927.62

-0.5 -1.0

Ra = 5x10 5 Ra = 10

0.2

(b)

0.04

0.0

-1.5 0.0

Analytical solution

0.4

0.6

y

0.8

Numerical solution 3 Ra = 10 3 Ra = 5x10 Racr = 9927.62

-0.04 -0.08

4

1.0

-0.12 0.0

Ra = 5x10 5 Ra = 10

0.2

0.4

0.6

y

0.8

4

1.0

Fig. 12. Horizontal velocity (a) and temperature (b) profiles at mid-length of the cavity, along the vertical coordinate in double-lid driven cavity for Pe = 20 and various values of Ra.

Table 4 Numerical values of Nu for A = 24, Ra = 104 and various values of Pe. Single-lid driven cavity Pe Mixed convection Forced convection

0.1 6.877 1.0001

1 7.025 1.001

10 8.846 1.953

50 30.094 24.837

100 99.474 96.446

150 217.923 215.861

Double-lid driven cavity Pe Mixed convection Forced convection

0.1 6.893 1.0003

1 7.194 1.033

10 11.555 4.333

50 87.873 84.440

100 336.851 335.076

150 754.138 752.902

of the moving walls, which act so that the fluid circulation enhances. Moreover, an increase in Pe causes Nu to be closer to its forced convection value when Ra/Pe3 < gf, while a decrease of Pe makes Nu very close to that obtained in natural convection when Ra/Pe3 > gn. A confirmation of such facts is given by Fig. 13, in which are plotted the variations of Nu with Pe for different Ra. Note that, for a range of small values of Pe and independently of the driven cavity type, the heat transfer seems to be ensured by natural convection for each Ra, since Nu does not change in this case whatever the configuration, after which the tendency is such that the increase of Nu is slow and tends to be fast with Pe until the transfer is practically governed by forced convection (linear evolution of Nu

Natural convection Ra = 104 6.8612 Natural convection Ra = 104 6.8612

with Pe). Such an increase starts precociously with a small Ra and the corresponding value of Pe can be correlated by (Ra/gn)1/3. Asymptotic limits of the results related to pure natural and forced convection flows, which are respectively defined by Eqs. (36) and (41), are indicated by broken lines. To get a more precise idea of the effect of Ra on heat transfer, the evolution of Nu with this parameter in the range 1  Ra  107, covering small and large values, is depicted in. Fig. 14. Here also an analogous trend can be observed, since Nu keeps constant (dominating forced convection heat transfer) for a range of small values of Ra, whose extent enhances with Pe, and beyond which Nu first undergoes a slight increase (mixed convection heat transfer), then another fast and linear (dominating natural convection heat trans-

405

A. Louaraychi et al. / International Journal of Heat and Mass Transfer 132 (2019) 394–406

10

4

Analytical solution

Nu

Forced convection asymptote Natural convection asymptote

Numerical solution Single-lid driven cavity Double-lid driven cavity

10

3

10

2

Ra = 105

10

Ra = 104

Ra = 103

1 0.1

1

10

1

10

2

Pe

10

3

Fig. 13. Evolution of the mean Nusselt number with Pe for various values of Ra in the case of single-lid driven cavity (a = 0) and double-lid driven cavity (a = 1).

10

4

Analytical solution

Nu

Natural convection asymptote

Numerical solution Single-lid driven cavity Double-lid driven cavity

10

3

10

2

Forced convection asymptote

Pe = 50

Pe = 25

10

Pe = 5

1 1

10

10

2

10

3

10

4

10

5

10

6

Ra

10

7

Fig. 14. Evolution of the mean Nusselt number with Ra for various values of Pe in single-lid driven cavity (a = 0) and double-lid driven cavity (a = 1).

fer). Needless to remind that Nu increases with Ra can be expected, since the buoyancy effects always act in order to promote convection heat transfer. Note that the transition to natural convection regime is found to be controlled by the parameter (gfPe3) and confirms well the criteria obtained with Eq. (50) for the mixed convection regime. The limiting dashed lines corresponding to dominating roles of forced and natural convections are defined by Eqs. (39) and (40), respectively. They appear to corroborate well the results obtained with both the adopted approaches.

7. 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 (0:1 6 Pe 6 500) and Rayleigh (1 6 Ra 6 107 ) numbers. The analytical solution is

406

A. Louaraychi et al. / International Journal of Heat and Mass Transfer 132 (2019) 394–406

derived on the basis of the parallel flow assumption, valid in the core region of the shallow enclosures and the numerical simulation is performed using a finite volume method. In the limit of variations of the governing parameters in their considered ranges, the analytical and numerical results are seen to be in good agreement. On other hand, the mixed convection parameter Ra/Pe3 is found to be the key for delineating the three convective flow regimes, where the limits of such regimes are found to be strongly dependent on the horizontal kinematic boundary conditions. Finally, the following criteria:

0:01612 <

Ra Pe3

< 1156:2824 and 0:0988 <

Ra Pe3

< 9150:0833

are established to define the mixed convection regime for single double-lid driven cavity, respectively. Outside the above ranges, the effect of natural/(forced) convection is dominant for higher/ (lower) values of the parameter Ra/Pe3. In perspective, the problem of boundary layers developed on the thermally active vertical walls is interesting to examine closely and will be the subject of a detailed study in the near future. At the same time, an asymptotic analysis of the problem, using the inverse of the aspect ratio, A, as an infinitely small parameter of the problem, will be done in order to quantify the role playing by A. Conflict of interest The authors declare that there is no conflict of interest related to such a study. References [1] L.A.B. Pilkington, Review lecture. The float glass process, Pro. R. Soc. London, A Math. Phy. Sci. 314 (1516) (1969) 1–25. [2] F.J.K. Ideriah, Prediction of turbulent cavity flow driven by buoyancy and shear, J. Mech. Eng. Sci. 22 (1980) 287–295. [3] J. Imberger, P.F. Hamblin, Dynamics of lake reservoirs and cooling ponds, A. Rev. Fluid Mech. 14 (1982) 153–187. [4] C.K. Cha, Y. Jaluria, Recirculating mixed convection flow for energy extraction, Int. J. Heat Mass Transf. 27 (10) (1984) 1801–1810. [5] R.K. Tiwari, M.K. Das, Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids, Int. J. Heat Mass Transf. 50 (9–10) (2007) 2002–2018. [6] T. Basak, S. Roy, P.K. Sharma, I. Pop, Analysis of mixed convection flows within a square cavity with linearly heated side wall(s), Int. J. Heat Mass Transf. 52 (2009) 2224–2242. [7] S. Sivasankaran, V. Sivakumar, P. Prakash, Numerical study on mixed convection in a lid-driven cavity with non-uniform heating on both side walls, Int. J. Heat Mass Transf. 53 (2010) 4304–4315. [8] V. Sivakumar, S. Sivasankaran, P. Prakash, J. Lee, Effect of heating location and size on mixed convection in lid-driven cavities, Comp. Math. With Appl. 59 (2010) 3053–3065. [9] M.A. Taher, S.C. Saha, Y.W. Lee3, H.D. Kim, Numerical study of lid-driven square cavity with heat generation using LBM, Am. J. Fluid Dyn. 3 (2) (2013) 40–47. [10] K. Yapici, S. OBUT, Laminar mixed-convection heat transfer in a lid-driven cavity with modified heated wall, Heat Transf. Eng. 36 (3) (2015) 303–314. [11] K.M. Khanafer, A.M. Al-Amiri, I. Pop, Numerical simulation of unsteady mixed convection in a driven cavity using an externally excited sliding lid, Eur. J. Mech B/Fluids 26 (2007) 669–687. [12] A.K. Hussein, S.H. Hussain, Mixed convection through a lid-driven air-filled square cavity with a hot wavy wall, Int. J. Mech. Mater. Eng. (IJMME) 5 (2) (2010) 222–235. [13] H.T. Rossby, Numerical experiments with a fluid heated non-uniformly from below, Tellus 50 (A) (1998) 242–257. [14] A.J. Chamkha, E. Abu-Nada, Mixed convection flow in single- and double-lid driven square cavities filled with water–Al2O3 nanofluid: effect of viscosity models, Eur. J. Mech. B/Fluids 36 (2012) 82–96. [15] M.A. Ismael, I. Pop, A.J. Chamkha, Mixed convection in a lid-driven square cavity with partial slip, Int. J. Therm. Sci. 82 (2014) 47–61.

[16] M.A. Sheremet, I. Pop, Mixed convection in a lid driven square cavity filled by a nanofluid: Buongiorno’s mathematical model, Appl. Math. Comput. 266 (2015) 792–808. [17] H.F. Oztop, I. Dagtekin, Mixed convection in two-sided lid-driven differentially heated square cavity, Int. J. Heat Mass Transf. 47 (2004) 1761–1769. [18] R.K. Tiwari, M.K. Das, Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids, Int. J. Heat Mass Transf. 50 (2007) 2002–2018. [19] N. Ouertatani, N. Ben Cheikh, B. Ben Beyaa, T. Lilia, A. Campo, Mixed convection in a double lid-driven cubic cavity, Int. J. Therm. Sci. 48 (2009) 1265–1272. [20] Nasreddine Benkacem, Nader Ben Cheikh, Brahim Ben Beya, Threedimensional analysis of mixed convection in a differentially heated liddriven cubic enclosure, J. Appl. Mech. Eng. 4 (2015) 159. [21] E. Abu-Nada, A.J. Chamkhac, Mixed convection flow in a lid-driven inclined square enclosure filled with a nanofluid, Eur. J. Mech. B/Fluids 29 (2010) 472– 482. [22] V. Sivakumar, S. Sivasankaran, Mixed convection in an inclined lid-driven cavity with non-uniform heating on both sidewalls, J. Appl. Mech. Tech. Phys. 55 (4) (2014) 634–649. [23] N.A. Bakar, R. Roslan, M. Ali, A. Karimipour, Mixed convection in an inclined lid-driven square cavity with sinusoidal heating on top lid, ARPN J. Eng. Appl. Sci. 12 (2017) 2539–2544. [24] M. Alinia, D.D. Ganji, M.G. Bandpy, Numerical study of mixed convection in an inclined two sided lid driven cavity filled with nanofluid using two-phase mixture model, Int. Comm. Heat Mass Transf. 38 (2011) 1428–1435. [25] M.R. Heydari, M.H. Esfe, M.H. Hajmohammad, M. Akbari, S.S.M. Esforjani, Mixed convection heat transfer in a double lid-driven inclined square enclosure subjected to Cu–water nanofluid with particle diameter of 90 Nm, Heat Transf. Res. 45 (1) (2014) 75–95. [26] D.E. Cormack, L.G. Leal, J. Imberger, Natural convection in a shallow cavity differentially heated end walls. Part 1. Asymptotic theory, J. Fluid Mech. 65 (1974) 209–229. [27] M. Lamsaadi, M. Naïmi, M. Hasnaoui, Natural convection heat transfer in shallow horizontal rectangular enclosures uniformly heated from the side and filled with non-Newtonian power law fluids, Energy Conv. Manage. 47 (15–16) (2006) 2535–2551. [28] A. Karimipour, A.H. Nezhad, A. D’Orazio, E. Shirani, The effects of inclination angle and prandtl number on the mixed convection in the inclined lid driven cavity using Lattice Boltzmann method, J. Theo. Appl. Mech. 51 (2) (2013) 447– 462. [29] H.T. Rossby, On thermal convection driven by non-uniform heating from below: an experimental study, Deep-Sea Res. 12 (1965) 9–16. [30] Y.S. Prasad, M.K. Das, Hopf bifurcation in mixed convection flow inside a rectangular cavity, Int. J. Heat Mass Transf. 50 (2007) 3583–3598. [31] M.A.R. Sharif, Laminar mixed convection in shallow inclined driven cavities with hot moving lid on top and cooled from bottom, Appl. Therm. Eng. 27 (2007) 1036–1042. [32] D.A. Siginer, A. Valenzuela-Rendon, On the laminar free convection and stability of grade fluids in enclosures, Int. J. Heat Mass Transf. 43 (2000) 3391– 3405. [33] D.D. Gray, A. Giorgini, The validity of the Boussinesq approximation for liquids and gases, Int. J. Heat Mass Transf. 19 (1976) 545–551. [34] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington, DC, USA, 1980. [35] I. Alloui, H. Benmoussa, P. Vasseur, Soret and thermosolutal effects on natural convection in a shallow cavity filled with a binary mixture, Int. J. Heat Fluid Flow 31 (2010) 191–200. [36] A. Bejan, The boundary layer regime in a porous layer with uniform heat flux from the side, Int. J. Heat Mass Transf. 26 (1983) 1339–1346. [37] Z. Alloui, J. Guiet, P. Vasseur, M. Reggio, Natural convection of nanofluids in a shallow rectangular enclosure heated from the side, Can. J. Chem. Eng. 90 (1) (2012) 69–78. [38] M.K. Moallemi, K.S. Jang, Prandtl number effects on laminar mixed convection heat transfer in a lid-driven cavity, Int. J. Heat Mass Transf. 35 (8) (1992) 1881–1892. [39] E.R.G. Eckert, A.J. Diaguila, Convective heat transfer for mixed free and forced flow through tubes, Trans. ASME 76 (1954) 497–504. [40] D.L. Siebers, R.G. Schwind, R.J. Moffat, Experimental mixed convection heat transfer from a large vertical surface in horizontal flow, Sandia Rept. Sand, Sandia Nat. Lab. Albuquerque, N., M., 1983, pp. 83–8225. [41] E.M. Sparrow, J.L. Gregg, Buoyancy effects in forced convective flow and heat transfer, J. Appl. Mech., Trans. ASME 26 (1959) 81–133. [42] S. Turki, Contribution to Numerical Study of Natural and Mixed Convection Heat Transfers in Confined Non-Newtonian Fluids, CNAM, Paris, France, 1990 (Ph.D. Thesis).