Numerical study of natural convection in a cavity of

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng (in press) Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2139

Numerical study of natural convection in a cavity of high aspect ratio by using the lattice Boltzmann method Mohammed Jami1 , Samir Amraqui1 , Ahmed Mezrhab1, ∗, † and Cherifa Abid2 2 Ecole

1 Facult´ e des Sciences, Département de Physique, Université Mohamed I, Oujda, Morocco polytechnique Universitaire de Marseille, IUSTI U.M.R. N 6595, Technopole Chteau Gombert, 5 Rue Enrico Fermi, 12453 Marseille cedex 13, France

SUMMARY A comprehensive numerical study has been conducted to investigate two-dimensional, steady heat transfer of natural convection in a divided enclosure of high aspect ratio. The vertical walls of the enclosure are maintained at different temperatures, while the horizontal walls are adiabatics. A numerical hybrid scheme with lattice Boltzmann for fluid velocity and finite difference for the temperature is adopted. Parametric studies of the effects of aspect ratio, number and length of partitions attached to the cold wall of the enclosure on heat transfer and fluid flow have been performed. Copyright q 2007 John Wiley & Sons, Ltd. Received 8 November 2006; Revised 31 May 2007; Accepted 1 June 2007 KEY WORDS:

high aspect ratio; lattice Boltzmann method; natural convection; partition

1. INTRODUCTION Enhancement of heat transfer by natural convection inside an enclosure is a major aim because of its practical importance in a large number of engineering applications including solar collector designs, electronic equipment cooling, building construction, nuclear engineering, etc. Many numerical and experimental works have provided insights into the flow phenomena in a simple enclosure. Partition may be used as an insert to reduce the heat exchange inside the cavity. Bajorek and Lloyd [1] carried out experimental studies by using a Mach–Zehnder interferometer for a square enclosure with two partial dividers, one attached to the top wall and the other to the bottom. The investigations were conducted for air- and carbon dioxide-filled enclosures both with and without partitions for Grashof numbers ranging between 1.7 × 105 and 3 × 106 . They found that the partitions reduce the heat transfer rate appreciably. Mach–Zehnder interferograms showed that the partitions have a noticeable impact on the isotherm pattern. Ampofo [2] experimentally studied turbulent natural ∗ Correspondence †

to: Ahmed Mezrhab, Faculté des Sciences, Département de Physique, Université Mohamed I, Oujda, Morocco. E-mail: [email protected], [email protected]

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2007 John Wiley & Sons, Ltd.

M. JAMI ET AL.

convection of air in a non-partitioned or partitioned cavity with differentially heated vertical and conducting horizontal walls. To measure the local velocity and temperature at different locations in the cavity, he used a laser Doppler anemometer and a microdiameter thermocouple. He showed that the partitions tend to reduce the heat transfer rates along the hot wall compared with similar cavities without partitions. Also, due to the high heat transfer input from the hot partitioned wall, all the flow and thermal field quantities including the turbulence quantities obtained in the cold wall boundary layer of the partitioned cavity have higher peaks and thicker boundary layer structure than the corresponding values in the non-partitioned cavity. Arquis and Rady [3] numerically studied natural convection heat transfer and fluid flow characteristics from a horizontal fluid layer with finned bottom surface. They showed that the heat transfer rates for low values of fin height may be decreased by the insertion of fins. For high values of fin height, the finned surface effectiveness is greater than one for a wide fin space. For low values of Rayleigh number and high values of fin height, the finned surface effectiveness increases linearly with the decrease in fin spacing. Probert and Ward [4] experimentally studied the heat transfer behavior in a divided cavity with aspect ratios of 18.2 and 26.4. They found that the insertion of comparatively short and mounted baffles in the top and bottom of the cavity reduced the overall, steady-state heat transfer across the cavity. They also studied the effect of the restriction of height of a vertical cavity by attaching horizontal partitions to its cold wall. Their aim was to inhibit the transition from laminar to a turbulent boundary layer and hence achieve a lower heat transfer rate across the cavity. For this study, they chose a cavity of height and width equal to 91 and 2.5 cm, respectively. The partitions did not completely bridge the 2.5-cm-thick cavity space, 1.3 mm gaps remaining between the hot and the end faces. They found that increasing the number of partitions from 2 to 10 produced an increase in the heat transferred between the isothermal walls. Furthermore, this increase is particularly noticeable at low temperatures. Chang et al. [5] numerically investigated laminar natural convection in a two-dimensional partitioned enclosure. An upwind finite-difference scheme was used to solve this problem. A study was conducted for Grashof numbers of 103 –108 . It was shown that the increasing partition size, caused by enlarging its height or width, decreased the average heat transfer across the enclosure. When the partitions were set close to the cold wall, the total heat transfer was reduced. Maximum heat transfer occurred when the partitions were set slightly off the center of the enclosure towards the hot wall. Mezrhab and Bchir [6] investigated numerically the effect of adding a thick partition located vertically close to the hot wall of a differentially heated cavity, forming a narrow vertical channel in which the flow is controlled by vents at the bottom and top of the partition. They showed that the radiation has a significant influence on the flow and heat transfer in the channel. The purpose of this study is to investigate numerically the natural convection in a differentially heated enclosure of high aspect ratio and with several partitions attached to its cold wall. We propose to use a numerical scheme with coupling between the lattice Boltzmann method (LBE) [7–10] for the fluid flow and finite difference for the temperature. A complete parametric study is made for different Rayleigh numbers, partition lengths, partition numbers and aspect ratios.

2. MATHEMATICAL MODEL Details of the geometry are shown in Figure 1. The flow is assumed to be incompressible, laminar and two-dimensional in an enclosure of square cross-section; the horizontal walls are perfectly insulated, while the vertical walls are maintained at two different temperatures, Th and Tc . Copyright q


Int. J. Numer. Meth. Engng (in press) DOI: 10.1002/nme

NUMERICAL STUDY OF NATURAL CONVECTION

Figure 1. The geometry of the divided enclosure.

The working fluid is air and its physical properties are assumed to be constant at the average temperature T0 , except for the density whose variation with the temperature is allowed for in the buoyancy term. 2.1. Velocity field The components of the velocity are determined by the LBE that was developed by Frisch et al. [11]. This method is a relatively new numerical scheme used to simulate various hydrodynamic systems. It has found several interesting applications in complicated situations such as two-phase flows and flows through porous media [12]. The basic principle is to solve the evolution equation of the distribution functions of fictitious fluid particles colliding and moving on a uniform square grid [13]. The particles are placed on the nodes of the grid. They move from node x, where they are located, towards the neighboring node x + ci between instants t and t + 1, according to their velocity ci . The modules of discrete velocities are ⎧ 0 if i = 0 ⎪ ⎪ ⎨ if i = 1, 2, 3, 4 |ci | = 1 ⎪ √ ⎪ ⎩ 2 if i = 5, 6, 7, 8 Copyright q



M. JAMI ET AL.

In this study, we give a brief overview of the LBE and consider a two-dimensional case with nine discrete velocities ci (i = 0, . . . , 8), on a uniform, square grid (Figure 2). This model is characterized by two basic steps: • the propagation of particles from nodes to their neighbors; • the collisions or redistributions between the various velocities at each lattice node. Time, space and momentum are discretized and coherently coupled together. The dynamics of the system follows the Lattice Boltzmann equation: f i (x + ci , t + 1) = f i (x, t) + (F)i

(1)

where f is the particle distribution function and is the collision term. We consider the LBE-MRT (multi-relaxation time) approach [14]; the state of the system at each lattice node is represented by the vector F = { f 0 , f 1 , . . . , f 8 }T and a set of moments that is derived from F by a linear transformation M = {m 0 , m 1 , . . . , m 8 }T = AF

(2)

The choice of the (non-singular) matrix A is based on that in kinetic theory of gases and takes advantage of the symmetries of the problem. It is convenient if the density (sum of the f i ) and the flux of momentum are included in M. In the LBE-MRT approach, the advection substep of Equation (1) is performed in F space, whereas the collision substep is performed in M space. For simplicity, the collision step is defined as a linear relaxation of moments towards their equilibrium value, following the set of equations that applies for all moments that are modified in collisions (that is, other than density and cartesian components of the momentum for Navier–Stokes or other than the density, if one wishes to simulate the diffusion equation), eq

bc bc m ac j = m j + s j (m j − m j )

(3) eq

bc where m ac j is the moment after collision, m j is the moment before collision, m j is the equilibrium value of the non-conserved moment which can be chosen at will, provided the symmetry of the problem is respected [10, 15], and s j is the relaxation rate. For stability reasons, s j , must be chosen between 0 and 2. It can be shown that the kinematic viscosity of the system is given by

= 13 (1/s X Y − 12 )

(4)

where s X Y is the relaxation rate of the off-diagonal stress tensor. It is beyond the scope of this paper to present an analysis of the LBE technique and of the accuracy of its predictions for simple situations using techniques used, for instance, in [10]. When a body force is added [16, 17], the conservation of the linear momentum is modified in the collision step, and we obtain: u ac = u bc + f x v ac = v bc + f y

(5)

where f x and f y are the components of the body force that must not vary too quickly to justify such simple expressions, u ac and v ac are the components of the velocity after collision and u bc and v bc are the components of the velocity before collision. Copyright q




Figure 2. Computational node.

Figure 3. Bounce-back boundary condition.

In our case, the temperature T provides a buoyancy force in the y direction. Therefore, as indicated in Equation (5), we increment the x and y components of the linear momentum, respectively, by f x = 0 and f y = −g(T (x, t) − T0 ), where x is the position of the air node that has temperature T (x, t) at time t. The bounce-back condition is used at the solid boundaries. This type of condition supposes that the postcollision distribution function at a grid site xs with a velocity −ci directed towards xf is set equal to the postcollision distribution function at the grid site xf with a velocity ci directed towards xs [18], as shown in Figure 3. The wall is placed at mid-distance between xs and xf . If this wall is stationary, the equation of evolution in xs becomes f j (xs , t + 1) = f i (xs + c j , t + 1)

(6)

where j is such that c j = −ci . 2.2. Temperature field The energy equation is discretized by the finite-difference method (with a fourth-order stencil for space derivatives). The dimensionless temperature is considered and the energy equation for the present system can be expressed by the following two-dimensional equation: 2 2 * * * * u (7) + 2 + v = *x *y *x 2 *y If N x × N y is the number of fluid points in the numerical grid, according to the bounce-back boundary condition and LBE techniques [19], the effective boundaries will be at 12 , N x + 12 and N y + 12 . We may use for the finite-difference scheme the same grid points as for the LBE scheme. Copyright q



M. JAMI ET AL.

In that case, we must use the following as the boundary conditions for the temperature: 1 1 1 N x + , y = h = 0.5 for yN y + 2 2 2 1 1 1 , y = c = −0.5 for yN y + 2 2 2 1 1 1 * =0 for xN x + x, 2 2 2 *y * 1 1 1 =0 for xN x + x, N y + 2 2 2 *y At the partition, the conditions of continuity of temperature and temperature gradient are imposed. These conditions are written as p+ = p− ,

* * = *z p− *z p+

(8)

where z is the normal direction to the partition, p+ is the superior surface of the partition and p− is the lower surface of the partition. To evaluate the heat transfer at an isothermal wall of the cavity, the local and average Nusselt numbers based on the width of the cavity are defined as follows: * N u(xw , y) = −H (9) *x xw ,y −1 L * dy (10) N uw(xw ) = A 0 *x xw ,y where xw is the position of the hot or cold wall along the x-axis. 3. SENSITIVITY TO THE GRID SIZE Firstly, we have varied the grid (N x × N y ) to determine the optimum uniform grid (i.e. the best compromise between accuracy and computational costs). Figures 4(a) and (b) depict the variation of N uw according to N x × N y and the relative error resulting from this variation. From these figures, it can be observed that the change in the number of grid points in the horizontal and vertical directions from 41 × 839 to 51 × 1039 affects the average Nusselt number by less than 0.7 and 0.4% for Np = 9 and 39, respectively, and leaves the fluid flow structure unchanged. Consequently, a 41 × 839 grid seems to be sufficiently smooth for this numerical computation. 4. RESULTS AND DISCUSSION The code validation was already carried out in [20]. The mathematical model developed in the last section was used to investigate the natural convection in the geometry shown in Figure 1. Each case required the specification of five dimensionless parameters (A, Pr, Ra, L p , Np ), among which the Copyright q




Figure 4. The influence of mesh size on the accuracy of the results for Ra = 104 , A = 20 and L p = 0.5.

Figure 5. Representation domain of isotherms and streamlines.

Prandtl number is held fixed at Pr = 0.71. The ranges covered by the remaining parameters were 5A30 , 102 Ra2 × 104 , 0L p 1 and Np = 0, 9 and 39. 4.1. Effect of partition length The computations were conducted for a cavity aspect ratio, A = 20, divided by multiple partitions attached to its cold wall. The effect of the partition length was studied for Ra = 104 , Np = 39, and the dimensionless plate length L p was varied from 0 to 1 by step of 0.25. In order to have a clear presentation of the isotherms and streamlines, the cavity is not presented completely, but the part containing the central partition is selected for this reason, as shown in Figure 5. 4.1.1. Isotherms and streamlines. Streamlines and isotherms corresponding to two-cell flow patterns are shown in Figure 6. Let us note a perfect symmetry of the isotherms and streamlines with respect to the horizontal passing by the partition. When L p varies from 0 to 1, the partition blocks Copyright q



M. JAMI ET AL.

Streamlines

Isotherms

(a)

(b)

(c)

(d)

(e)

Figure 6. Streamlines and isotherms, Ra = 104 , A = 20 and Np = 39: (a) L p = 0; (b) L p = 0.25; (c) L p = 0.5; (d) L p = 0.75; and (e) L p = 1.

the fluid and consequently the flow becomes bi-cellular and the dimensionless stream function decreases; therefore, the circulation is reduced around the partition. The isotherms are modified and take the shape of the wave, they present a stratification. As can be seen, the isotherms and streamlines are affected in the same manner around each partition in the whole cavity. 4.1.2. Average Nusselt number. Figure 7 shows the variation of the average Nusselt number (N uw) with L p for Ra = 104 , A = 20 and Np = 39. Let us notice that the maximum of Nuw is obtained for an empty cavity (L p = 0). The average Nusselt number decreases with increase in L p until L p = 0.5, which presents the minimum of heat exchange. Indeed, in the case of the empty cavity, the movements’ convectifs are significant, which produces an increase in N uw. On the other hand, when the cavity is partitionned, the reduction of the movements’ convectifs causes a reduction in the average Nusselt number. However, when L p ranges between 0.75 and 1, N uw increases slightly as L p increases. In fact, the maximal temperature difference (Th − Tc ) becomes weaker. This is due to the conduction in all partitions, and in each cell, there is more air that can reach the cold wall. 4.2. Influence of number of partitions We considered an enclosure that approaches the collector geometry with an aspect ratio equal to 20 and divided by 9 and 39 partitions, respectively. Firstly, we investigated the cavity to show the effect of the number of partitions on streamlines and isotherms for a fixed Rayleigh number Ra = 104 . Secondly, we studied the heat transfer according to the Rayleigh number. Copyright q




Figure 7. Variations in the average Nusselt number with partition length for Ra = 104 , A = 20 and Np = 39.

(a)

(b)

(c)

Figure 8. Streamlines and isotherms, Ra = 104 , A = 20 and L p = 0.5: (a) Np = 0; (b) Np = 9; and (c) Np = 39.

The predicted flow patterns and temperature distributions in the partitioned enclosure are shown in Figure 8. These figures show that the variation of the number of partitions has a significant influence on the flow and the temperature fields. The increase of Np pushes the fluid to circulate in the lower part of the cavity. In fact, the cells become narrow and, consequently, the major part of the fluid circulates in the lower half of the cavity. It can be seen, for Np = 9, that the isotherms present a horizontal temperature gradient that becomes significant around the partition. This temperature gradient disappears in the case of Np = 39. Figure 9 shows the variation of the average Nusselt number as a function of Ra for L p = 0.5, A = 20 and for different partition numbers. As can be seen, the considerable increase in the number of partitions (Np = 39), causes a decrease in the heat transfer more efficiently. Nevertheless, for low partitions number (Np = 9), the opposite occurs. This phenomenon is more important for Ra>2 × 102 . It is also noted that for all values of Np , the increase in Ra provokes an increase in Nuw. 4.3. Aspect ratio The variations in the average Nusselt number with various aspect ratios (A) and Ra are plotted in Figure 10. In this case, we have considered nine partitions of L p = 0.5 inside the enclosure. Let Copyright q



M. JAMI ET AL.

Figure 9. Variations in the average Nusselt number with Ra for A = 20, L p = 0.5 and Np = 0, 9, 39.

Figure 10. Variations in the average Nusselt number with A for Ra = 102 –104 , L p = 0.5 and Np = 9.

us note that, for each value of A, Nuw increases with increasing Ra. For Ra = 103 and 104 , Nuw displays a maximum at A = 20; then the average Nusselt number decreases slightly as the aspect ratio of the enclosure increases. Nuw is more important for Ra = 104 , than for the other two Ra’s, showing the very important role of the aspect ratio when Ra becomes large. It is also noted that, for Ra = 102 , the effect of A on the average Nusselt number is negligible.

5. CONCLUSION The present numerical study treats the natural convection phenomena in a two-dimensional enclosure with partitions attached to its cold wall by means of numerical coupling between the lattice Boltzmann and the finite-difference method. Copyright q




This numerical study confirmed that the effects of the partitions in reducing heat transfer depend on L p , Np , Ra and A. It led to the following conclusions: 1. Independently of A, L p or Np , the increase in Ra causes an increase in Nuw. 2. For Ra = 104 , a maximum reduction in the heat transfer is observed at L p = 0.5. Beyond this last value, the variation in L p causes a little increase in Nuw. 3. The influence of A on heat transfer is more significant for A15 with large values of Ra. Besides, A = 20 gives the maximum value of Nuw for each value of Ra. 4. The flow becomes multicellular when L p increases, and consequently, Np + 1 cell flow is obtained at L p = 1. 5. For Ra5 × 102 , the influence of Np on heat transfer is negligible. However, for Ra>5 × 102 , and in comparison to the non-partitioned enclosure, Np = 39 significantly causes a decrease of the heat transfer in the cavity. However, for Np = 9, the trend is opposite.

NOMENCLATURE A g H k L lp Lp Nuw Pr Ra T T0 u, v x, y

aspect ratio (A = L/H ) acceleration of gravity (m s−2 ) enclosure width (m) thermal conductivity (W m−1 K−1 ) enclosure length (m) partition length (m) dimensionless partition length (lp/L) average Nusselt number Prandtl number (Pr = /) Rayleigh number (g(Th − Tc )H 3 /) temperature (K) mean temperature ((Th + Tc )/2) K dimensional velocity components (m s−1 ) Cartesian coordinates (m)

Greek symbols T

thermal diffusivity of the fluid (m2 s−1 ) volumetric expansion coefficient (K−1 ) maximal temperature difference (Th − Tc ) (K) kinematic viscosity of the fluid (m2 s−1 ) dimensionless temperature ((T − T0 )/T )

Subscripts p h c Copyright q

partition hot cold 2007 John Wiley & Sons, Ltd.


M. JAMI ET AL.

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