Heat Transfer Research, 2011, Vol. 42, No. 3 Numerical Study of Free Convection Heat Transfer in a Square Cavity with a Fin Attached to Its Cold Wall SAEID JANI, 1* MEYSAM AMINI, 2 and MOSTAFA MAHMOODI 2 1 Faculty of Engineering, Golpayegan College of Engineering, Golpayegan, Iran 2 Department of Mechanical Engineering, University of Kashan, Kashan, Iran Fluid flow and natural convection heat transfer in a differentially heated square cavity with a fin attached to its cold wall is investigated numerically. The top and the bottom horizontal walls of the cavity are insulated while the left and right vertical walls of the cavity are maintained at a constant temperature T h and T c, with T h > T c, respectively. The governing equations written in terms of the primitive variables are solved numerically using the finite volume method and the SIMPLER algorithm. Using the developed code, a parametric study is performed, and the effects of the Rayleigh number, length of the fin and its position on the flow pattern and heat transfer inside the enclosure are investigated. The results show that for high Rayleigh numbers, a longer fin placing at the middle of the right wall has a more remarkable effect on the flow field and heat transfer inside the cavity. * * * Keywords: natural convection, numerical method, square cavity, thin fin 1. INTRODUCTION Free convection fluid flow and heat transfer occur in many industrial and engineering systems such as heat exchangers, home ventilation systems, refrigeration units, fire prevention, etc. (Ostrach, 1988). Buoyancy-driven flow in partitioned enclosures has received considerable attention in recent years because of its modern applications * Address all correspondence to SAEID JANI E-mail: jani@gut.ac.ir 251 ISSN1064-2285 2011 Begell House, Inc.
to cooling of electronic equipment, solar energy collectors, building construction, nuclear reactors, etc. For example, in the study of solar collectors the determination of the lost of overall heat transfer rates towards the ambient is very important. The rate of heat transfer through a fluid layer in an enclosure may be controlled by attaching fins or partitions to one of the active walls of the enclosure. Properly placing fins on the active wall(s) of an enclosure, the heat transfer rate may be reduced or enhanced. Studies of various aspects of this problem have been carried out by many researchers. Frederick (1989) studied numerically laminar free convection in an air-filled differentially heated inclined enclosure with a thin partition placed at the middle of its cold wall. Decreasing heat transfer of up to 47 percent in comparison to the cavity with no partition was observed in this study. Frederick and Valencia (1989) studied free convection heat transfer in a square cavity with a conducting partition located at the middle of its hot wall using numerical simulation. They observed that for a low value of the partition-to-fluid thermal conductivity ratio and for Rayleigh numbers from 10 4 to 10 5 a reduction in heat transfer relative to the case of cavity with no partition occurs. Scozia and Frederick (1991) studied free convection in tall cavities with multiple conducting fins on the cold wall. They numerically investigated the effects of the Rayleigh number, angle of inclination of the cavity, cavity aspect ratio, number of fins, and fin lengths on the heat transfer rate. They observed that the average Nusselt number increases with increasing number of fins and reaches a maximum value, then it decreases with further increase in the number of fins at all Rayleigh numbers greater than 10 3. Nag et al. (1993) investigated the effect of a horizontal thin partition positioned on the hot wall of a horizontal square cavity. They observed that for a partition of infinite thermal conductivity, the Nusselt number on the cold wall is greater than in the case with no fin. Lakhal et al. (1997) studied numerically free convection in rectangular enclosures with perfectly conducting fins attached to the hot wall. Yucel and Turkoglu (1998) numerically analyzed the natural convection heat transfer in vertical slender cavities with conducting fins attached to the cold wall, using the control volume approach. They investigated the effects of the fin length, number of fins, and the Rayleigh number on the heat transfer rate. However, the results were presented at different Rayleigh numbers in the range of 10 3 and 10 4 ; it was observed that the heat transfer rate through an enclosure can be controlled by attaching fins to its active wall(s). Bilgen (2002) investigated laminar and turbulent free convection in cavities with partitions positioned on the insulated horizontal walls. His study covered various geometrical parameters such as: aspect ratio of the cavity, number of partitions, position of partitions, length of partitions, and the Rayleigh number. He found that the heat transfer was reduced when two partitions were used instead of one, the aspect ratio was made smaller, and the position of partitions was farther away from the hot wall. Shi and Khodadadi (2003) reported results of a numerical study of laminar free convection in a differentially heated square cavity due to a perfectly conducting thin fin on its hot wall. They found that heat transfer on the cold wall without fins can be promoted for high Rayleigh numbers and with fins 252
placed closer to the insulated walls. Free convection heat transfer in differentially heated square cavities with a horizontal thin fin has been numerically studied by Bilgen (2005). The cavity was formed by vertical isothermal walls and adiabatic horizontal walls while a thin fin was attached to its hot wall. It is found that the Nusselt number is an increasing function of the Rayleigh number, and a decreasing function of a fin length and relative conductivity ratio. Recently, Ben-Nakhi and Chamkha (2006) investigated the effects of the length and inclination of a thin fin placed in the middle of a hot wall on free convection in a square cavity using numerical simulation. They found that the Rayleigh number, thin fin inclination, and the fin length have significant effects on the average Nusselt number of the heated wall including the fin of the enclosure. Mezrhab et al. (2006) carried out a numerical study to investigate the effect of a single and multiple partitions on heat transfer phenomena in an inclined square cavity, differentially heated, for a range of Rayleigh numbers from 10 3 to 10 6. The partitions were attached to the cold wall of the cavity; in the case of a single partition, it was located in the middle of the cold wall. It was found that the mean Nusselt number decreases when the number of partitions increases. Oztop and Bilgen (2006) studied free convection in a differentially heated square cavity with hot vertical walls, insulated horizontal walls, and with a cold partition on the bottom wall. They found that in the presence of a cold partition, the heat transfer reduces and the heat reduction gradually increases with increase in the partition height and thickness. Frederick (2007) studied numerically free convection of air in a cubical enclosure with a thick partition fitted vertically on the hot wall. Sheikhzadeh et al. (2007) investigated numerically free convection heat transfer inside a differentially heated square cavity with two perfectly conductive thin fins attached to the isothermal walls. They found that when the dimensionless length of fins is 0.5, and the Rayleigh number ranges from 10 4 to 10 7, the Nusselt number remains constant. More recently, Kasayapanand (2009) investigated numerically free convection in a finned cavity under electric field. He found that the flow and heat transfer enhancements are the decreasing function of the Rayleigh number. The problem of transition to a periodic flow in a differentially heated cavity with a thin fin on its side wall was investigated numerically by Xu et al. (2009). They found that the unstable temperature configuration above the fin results in intermittent plumes at the leeward side of the fin. A detailed review of the existing literature reveals that the effects of the fin length and the location of the fin, or different combinations of these parameters on the fluid flow and heat transfer characteristics within a differentially heated square cavity with a fin attached to its cold wall have not yet been fully investigated. In this study, fluid flow and natural convection heat transfer in a differentially heated square cavity which is formed by horizontal adiabatic walls and vertical isothermal walls, with a fin attached to its cold wall is simulated numerically using the finite volume method. The governing equations are solved using the SIMPLER algo- 253
Fig. 1. A schematic view of a square cavity with a thin fin and the boundary conditions. rithm. A parametric study is performed, and the effects of the Rayleigh number, the fin length and fin position on the right wall of the cavity on the fluid flow and heat transfer inside the enclosure are investigated. 2. PROBLEM DEFINITION A schematic view of the cavity considered in the present study and the coordinates are shown in Fig. 1. The cavity is square of side H. The left vertical wall of the cavity is kept at a constant temperature T h, while its right vertical wall is maintained at a temperature T c with T h > T c. The top and the bottom horizontal walls are insulated. A highly conductive thin fin is attached to the cold wall. The dimensionless length and the location of the thin fin on the right wall are shown by L and S, respectively. It is assumed that the fin is highly conductive and is maintained at the same temperature of the wall to which it is attached. The cavity is filled with water with the Prandtl number of 6.8. The fluid is considered to be incompressible, and the two-dimensional flow is assumed to be steady and laminar. The properties of the fluid are assumed to be constant with the exception of the density which varies according to the Boussinesq approximation. 3. MATHEMATICAL FORMULATION In order to express the governing equations in a dimensionless form, the following dimensionless variables are defined: X = x H, Y = y H, U = uh α, (1) V = vh α, P = ph2 ρα 2, 254 θ = T T c T h T c,
where u(u) and v(v) are the velocity components in the x(x)- and y(y)-directions, T(θ) is the temperature, and p(p) is the pressure. Also, α, v, and ρ are the thermal diffusivity, the kinematic viscosity, and the density, respectively. Using the above dimensionless variables, the steady-state continuity, momentum, and energy equations for the laminar natural convection fluid flow and heat transfer inside the cavity in a dimensionless form are given by Shi and Khodadadi (2003): U X V + = 0, Y (2) 2 2 U U P U U U + V = + Pr +, 2 2 X Y X X Y 2 2 V V P V V U + V = + Pr + Ra Pr, 2 2 + θ X Y Y X Y 2 2 θ θ θ θ U + V = + X Y X Y 2 2, (3) (4) (5) where the Rayleigh number Ra and the Prandtl number Pr are defined as Ra = gβ(t h T c )H 3, Pr = ν αν α, (6) where β is the thermal expansion coefficient. The dimensionless boundary conditions are On the right wall (X = 1) U = V = 0, θ = 0, On the left wall (X = 1) U = V = 0, θ = 1, On the top wall (Y = 1) U = V = 0, θ Y = 0, (7) On the bottom wall (Y = 1) U = V = 0, θ Y = 0, On the fin U = V = 0, θ = 0. The relation for the local Nusselt number can be written as Nu = θ x X=0. (8) The wall-averaged Nusselt number can then be obtained from Nu 1 = NudY. 0 255
4. NUMERICAL APPROACH The governing equations are discretized using the finite volume method and the SIMPLER algorithm. The diffusion terms in the equations are discretized by a second-order central difference scheme, while a hybrid scheme (a combination of the central difference scheme and the upwind scheme) is employed to approximate the convection terms. The set of discretized equations is solved by the TDMA line-byline method (Patankar, 1980). The computations are carried out for the Rayleigh number between 10 3 and 10 6. To validate the numerical procedure, the results obtained by the present code for a differentially heated square cavity filled with air are compared with the results of Davis (1983). Table 1 shows a comparison for the average Nusselt number and the maximum values of velocity components, namely, U max and V max, obtained by the present work in the case of no-fin cavity, filled with air, with the corresponding results of Davis (1983) for different Rayleigh numbers. As can be observed from the table, very good agreement exists between the two results. Table 1. Comparison of the present results in the case of no-fin cavity filled with air (Pr = 0.7) with the corresponding results of Davis (1983) Ra = 10 3 Ra = 10 4 Ra = 10 5 Ra = 10 6 Present study Davis (1983) Nu 1.113 1.118 U max 0.142 0.136 V max 0.139 0.138 Nu 2.244 2.243 U max 0.189 0.192 V max 0.241 0.243 Nu 4.507 4.519 U max 0.149 0.153 V max 0.259 0.261 Nu 8.791 8.799 U max 0.080 0.079 V max 0.259 0.262 Table 2. Average Nusselt number of a hot vertical wall of cavity filled with water (Pr = 6.8) for different grid sizes; Ra = 10 6, L f = 0.5, S f = 0.5 Grid Size 21 21 41 41 61 61 81 81 101 101 Nu 0.561 0.763 0.993 1.182 1.184 256
In order to conduct a grid independence study, the natural convection in a square cavity with the respective fin length and location equal to 0.5 and 0.5, filled with water at Ra = 10 6 was considered. Five different uniform grids, namely, 21 21, 41 41, 61 61, 81 81, and 101 101 are employed for the numerical calculations. The average Nusselt number of the hot vertical wall of the cavity for each of the five grids is obtained and it was found that a uniform 81 81 grid is sufficiently fine for numerical calculations (Table 2). All the results to be presented in what follows have been obtained by using this grid. 5. RESULTS AND DISCUSSION Having validated the numerical procedure via solving the test case, the code is employed to investigate the natural convection fluid flow and heat transfer inside the cavity with a fin attached to its cold wall. The results are presented for a range of Ra from 10 3 to 10 6 with a fin at different lengths, L f = 0.25, 0.5, and 0.75, and various positions of the fin, S f, namely, 0.25, 0.5, and 0.75. In order to observe the effect of a fin on the flow and temperature fields inside the cavity, the streamlines and isotherms for the case of a no-fin differentially heated square cavity, filled with water, are presented in Figs. 2a and 2b, respectively. Figures 3a and 3b show the streamlines and the isotherms for a square cavity, filled with water, with a fin at its lower position (S f = 0.25), and for various Ra and different lengths of the fin, L f = 0.25, 0.5, and 0.75, respectively. The plots are arranged going from left to right with increase of the Rayleigh number. For all values of Ra, a large clockwise (CW) rotating cell is observed for all fin lengths (Fig. 3a). The fluid that is heated next to the hot wall (left wall) rises and replaces the cold fluid next to the Fig. 2. Streamlines and isotherms inside a no-fin square cavity, filled with water (Pr = 6.8), for various Ra: (a) streamlines; (b) isotherms. 257
Fig. 3. Streamlines and isotherms inside the square cavity, filled with water (Pr = 6.8), for various Ra and different fin lengths for S f = 0.25: (a) streamlines; (b) isotherms. 258
Fig. 4. Streamlines and isotherms inside the square cavity, filled with water (Pr = 6.8), for various Ra and different fin lengths for S f = 0.5: (a) streamlines; (b) isotherms. 259
cold wall (right wall) that is falling, thus giving rise to a CW rotating vortex (also called the primary vortex). As the Rayleigh number increases, the flow pattern and the temperature distribution change from conduction to the convection-dominated regime for all L f values. It seems that various lengths of the fin not only change the flow fields near the fin, but also the strength of the CW vortex. This is because the fin blocks the movement of the fluid and weakens the CW vortex. The streamlines and isotherms of the flow and temperature fields in the cavity filled with water are presented in Figs. 4a and 4b for the middle position of a fin (S f = 0.5), and Figs. 5a and 5b for a fin at its upper position (S f = 0.75), respectively. Note that the plots are arranged going downward with the ascending L f value. With increasing the Rayleigh number, a similar trend corresponding to the flow (Figs. 4a and 5a) and temperature (Figs. 4b and 5b) distribution can be seen in these figures and Figs. 3a and 3b, respectively. By comparing all the cases in Figs. 3a, 4a, and 5a, it is possible to see that a fin attached to the middle of the wall has the most remarkable effect on the fluid flow in the cavity. A fin redirects the movement of the fluid and weakens the intensity of the fluid motion within the area under the fin for S f < 0.5, whereas this phenomenon occurs within the area above the fin when S f 0.5. In general, the presence and the character of the primary CW rotating vortex is unaltered, with a longer fin bringing about more changes to the flow compared to a shorter fin. For Ra = 10 4, the streamlines and the isotherms near the fin exhibit similar trends as those for Ra = 10 3, and in some instances, the flow field exhibits two local minima that may be reminded for the common cases of a cavity with no fin. At moderate Rayleigh numbers (Ra = 10 5 ), the boundary layer regime is developed towards the cavity walls and for higher Ra, the convective mode of heat transfer is dominated throughout the cavity. Moreover, the more packed stream function contours indicate that the fluid moves faster as the Rayleigh number increases. For the case of Ra = 10 4 (convection not being strong compared to conduction), the presence of the fin brings about resistance to the motion of the CW vortex and it has the most remarkable effects on the flow field when it is placed at the middle of the left wall. Also placing a fin near the right upper corner can enhance the strength of the primary vortex somewhat. This is because the fluid moves slowly in this area and a cold fin placed at S f = 0.75 cools the hot fluid coming toward it from the left wall. For such a situation, a longer fin results in a stronger primary vortex. It is observed that for the case of Ra = 10 5 (convection dominating conduction), a fin can block the flow, thus weakening the primary vortex, but at the same time a long enough cold fin can cool the fluid and makes it lighter resulting in enhancement of the CW vortex. These two mechanisms are found to certainly counter balance each other for nearly the fin middle position, regardless of the fin length. For Ra = 10 6, given the strong effect of natural convection implies that placing a fin of any length can always enhance the strength of the primary vortex regardless of its position. It should be noted that the cold fin effect of blocking the fluid motion is more dominant than the effect of cooling the fluid to enhance the strength of the primary vortex in this case. 260
Fig. 5. Streamlines and isotherms inside the square cavity, filled with water (Pr = 6.8), for various Ra and different fin lengths for S f = 0.75: (a) streamlines; (b) isotherms. 261
Table 3. Average Nusselt number of a hot wall for the cavity with no fin filled with water with Pr = 6.8 Nu (no fin) Ra = 10 3 1.144 Ra = 10 4 2.314 Ra = 10 5 4.797 Ra = 10 6 9.317 In order to compare the temperature fields for all cases, Figs. 3b, 4b, and 5b are considered again. The value of θ on the right wall and the fin is 0, whereas the value of θ on the left wall is 1. Comparing these figures for Ra = 10 6, a fin with L f = 0.25 at most positions only changes the temperature distribution locally and the rest of the cavity remains unaffected. This is because the CW vortex has not altered too much upon introduction of a 0.2H long fin and the fin only changes the velocity distribution locally. As mentioned before, a fin at S f < 0.5 can weaken the fluid motion in the area below the fin and thus a decreased heat transfer capability is expected. Moreover, at the higher Rayleigh numbers, the temperature contours above the fin are more packed than those under the fin. This implies better heat transfer on the top surface of the fin than on the bottom surface for nearly all mentioned values of S f. For longer fins, the temperature contours to the left of the fin are also affected by the introduction of the fin. Higher temperature gradients next to the left wall are observed while placing the fin close to two ends of the right wall compared to placing a fin at the middle of the right wall. In order to study the effect of the fin on the average heat transfer rate in the cavity, a variable called the Nusselt Number Ratio (NNR) is introduced. This variable is defined as: Nu with a fin NNR = (10) Nu without a fin Thus, NNR can be obtained according to (10). A value of NNR greater than 1 indicates that the heat transfer rate is enhanced in the cavity, whereas reduction of heat transfer is indicated when NNR is less than 1. Thus, the average Nusselt number can be obtained from the product of NNR and average Nusselt number results for a nofin cavity in Table 3. Figures 6a, 6b, and 6c show the variations of NNR with respect to the fin position for three different lengths of the fin, L f = 0.25, 0.5, and 0.75 for Ra = 10 4 to 10 6, respectively. Based on these figures, it is observed that placing a fin on the right wall always increases the heat transfer in the cavity, except that for Ra = 10 4, when L f = 0.5, and 0.25 < S f 0.5. This is because at low Rayleigh numbers when convection is not dominant, the isothermal fin is the main source of energy that is brought into 262
Fig. 6. Variation of the NNR along the heated wall of the cavity, filled with water (Pr = 6.8), with respect to position of the fin: (a) Ra = 10 4, (b) Ra = 10 5, and (c) Ra = 10 6. the cavity. But, on the other hand, the fin blocks the flow motion within the cavity. It seems that the latter effect becomes more prominent than the former one, when the fin is placed in the middle of the cold wall. For Ra = 10 4, the effect of the fin length on the average heat transfer is more remarkable for longer fin lengths, regardless of the fin position S f. Upon comparing the diagrams of Fig. 6, it is possible to see that the effect of the fin length on NNR becomes less remarkable with the rise of the Rayleigh number due to the fact that these three curves come closer. This is because the cooling effect of the cold fin on the flow and enhancing the primary vortex can compensate the effect of the fin blocking the flow. Moreover, it can clearly be seen 263
from Fig. 6c, that for Ra = 10 6, placing a fin at the middle of the right wall has a more remarkable effect on the mean Nusselt number compared to placing it on the two ends of the left wall. 6. CONCLUSIONS Using the finite volume method, fluid flow and natural convection heat transfer inside a square cavity with a fin attached to its cold wall was studied numerically. The numerical procedure was validated by comparing the average Nusselt number for a differentially heated square cavity filled with air obtained by the code with the existing results in the literature. Very good agreement was observed between them. Subsequently, a parametric study was performed and the effects of the Rayleigh number, length of the fin and its position on the flow pattern and heat transfer were investigated. It was observed that, placing an isothermal horizontal fin on the right cold wall of a differentially heated cavity generally modifies the clockwise rotating primary vortex that is established due to natural convection. The cooling effect of the cold fin on the flow and enhancing the primary vortex becomes more marked with the rise of the Rayleigh number. For Ra 10 5, placing a longer fin at the top corner of the cold wall has a more remarkable effect on the heat transfer inside the cavity compared to placing it on the other two locations of the right wall, while at Ra = 10 6, the best condition for heat transfer occurs when a medium length fin is placed at the top corner of the cold wall. NOMENCLATURE g gravitational acceleration, m/s 2 H square cavity side, m k fluid thermal conductivity, W/m K L length of the fin, m L f dimensionless length of the fin; L f = L/H NNR Nusselt number ratio Nu local Nusselt number p pressure, Pa P dimensionless pressure Pr Prandtl number Ra Rayleigh number S position of the fin, m S f dimensionless position of the fin; = S/H T temperature, K T c temperature of the right (cold) wall, K T h temperature of the left (hot) wall, K u, v velocity components along x- and y-axes, respectively, m/s 264
U, V dimensionless velocity components along x- and y-axes, respectively x, y Cartesian coordinates, m X, Y dimensionless Cartesian coordinates Greek symbols α thermal diffusivity, m 2 /s β thermal expansion coefficient, 1/K θ dimensionless temperature υ fluid kinematic viscosity, m 2 /s ρ density, kg/m 3 Subscripts f fin max maximum Superscript average. REFERENCES Ben-Nakhi, A. and Chamkha, A. J., Effect of length and inclination of a thin fin on natural convection in a square enclosure, Numer. Heat Transfer. A, vol. 50, pp. 381 399, 2006. Bilgen, E., Natural convection in enclosures with partial partitions, Renewable Energy, vol. 26, pp. 257 270, 2002. Bilgen, E., Natural convection in cavities with a thin fin on the hot wall, Int. J. Heat Mass Transfer, vol. 48, pp. 3493 3505, 2005. Davis, G. V., Natural convection of air in a square cavity, a benchmark numerical solution, Int. J. Numer. Meth. Fluids, vol. 3, pp. 249 264, 1983. Frederick, R. L. and Valencia, A., Heat transfer in a square cavity with a conducting partition on its hot wall, Int. Commun. Heat Mass Transfer, vol. 16, pp. 347 354, 1989. Frederick, R. L., Natural convection in an inclined square enclosure with a partition attached to its cold wall, Int. J. Heat Mass Transfer, vol. 32, pp. 87 94, 1989. Frederick, R. L., Heat transfer enhancement in cubical enclosures with vertical fins, Appl. Thermal Eng., vol. 27, pp. 1585 1592, 2007. Kasayapanand, N., A computational fluid dynamics modeling of natural convection in finned enclosure under electric field, Appl. Thermal Eng., vol. 29, pp. 131 141, 2009. 265
Lakhal, E. K., Hasnaousi, M., Bilgen, E., and Vasseur, P., Natural convection in inclined rectangular enclosures with perfectly conducting fins attached on the heated wall, Heat Mass Transfer, 32, pp. 365 373, 1997. Mezrhab, A., Jami, M., Abid, C., Bouzidi, M., and Lallemand, P., Lattice-Boltzmann modeling of natural convection in a inclined square enclosure with partitions attached to its cold wall, Int. J. Heat Fluid Flow, vol. 27, pp. 456 465, 2006. Nag, A., Sarkar, A., and Sastri, V. M. K., Natural convection in a differentially heated square cavity with a horizontal partition plate on the hot wall, Comput. Meth. Appl. Mech. Eng., 110, pp. 143 156, 1993. Ostrach, S., Natural convection in enclosures, ASME J. Heat Transfer, 110, pp. 1175 1190, 1988. Oztop, H. and Bilgen, E., Natural convection in differentially heated and partially divided square cavities with internal heat generation, Int. J. Heat Fluid Flow, vol. 27, pp. 466 475, 2006. Patankar, S. V., Numerical Heat Transfer and Fluid Flow, New York: Hemisphere Publishing Corporation, Taylor and Francis Group, 1980. Scozia, R. and Frederick, R. L., Natural convection in slender cavities with multiple fins attached on an active wall, Numer. Heat Transfer A, vol. 20, pp. 127 158, 1991. Sheikhzadeh, G. H., Pirmohammadi, M., and Ghassemi, M., A numerical study of natural convection in a cavity with two fins attached to its vertical walls, ASME Int. Mech. Eng. Cong. and Expos., Seattle, Washington, USA, 2007. Shi, X. and Khodadadi, J. M., Laminar natural convection heat transfer in a differentially heated square cavity due to a thin fin on the hot wall, ASME J. Heat Transfer, vol. 125, pp. 624 634, 2003. Xu, F., Patterson, J. C., and Lei, C., Transition to a periodic flow induced by a thin fin on the sidewall of a differentially heated cavity, Int. J. Heat Mass Transfer, vol. 52, pp. 620 628, 2009. Yucel, N. and Turkoglu, H., Numerical analysis of laminar natural convection in enclosures with fins attached to an active wall, Heat Mass Transfer, vol. 33, no. 4, pp. 307 314, 1998. 266