On mixed convection in a cavity with sinusoidally heated moving lid and uniformly heated and cooled side walls
 Deborah S Ducasse^{1} and
 Precious Sibanda^{1}Email author
DOI: 10.1186/16872770201383
© Ducasse and Sibanda; licensee Springer. 2013
Received: 18 September 2012
Accepted: 18 March 2013
Published: 10 April 2013
Abstract
In this study a steady state twodimensional mixed convection problem in an air filled cavity is investigated. The effects of linearly heated and uniformly cooled walls on flow and heat transfer characteristics within the cavity are determined. The strength of the fluid circulation within the cavity is found for both heated and cooled walls for different Reynolds and Grashof numbers. The nonlinear coupled equations are solved numerically using the penaltyGalerkin finite element method. Stream function and isotherm results are obtained for different Reynolds and Grashof numbers. The results for the heat transfer rate are presented in terms of both the local and the average Nusselt number. In general, the strength of the circulation is stronger for the case of cooled walls, and the anticlockwise circulation is significantly stronger for cooled walls, while the clockwise circulation is only slightly stronger for the cooled walls and the difference in strength decreases both with increasing Reynolds and Grashof numbers. Both the local and average Nusselt numbers are generally higher for the case of cooled side walls than that for heated side walls.
MSC:34B15, 65N30, 76M20.
1 Introduction
Over the last decade mixed convection in a square cavity has become an increasingly attractive field of study. The popularity of such studies is due in part to the many applications of such flows in industrial and natural settings. These applications include, for example, the thermalhydraulics of nuclear reactors, drying technologies, the dynamics of lakes, and food processing. Mixed convection in a liddriven cavity has been extensively studied. Torrance et al. [1] investigated the fluid motion in a liddriven cavity where the upper wall was maintained at a different temperature to that of the other walls in the cavity. They observed that an increase in the aspect ratio led to an increase in secondary circulations in the lower part of the cavity.
Kawaguti [2] was one of the first to investigate the effect of the Reynolds number on forced convection in a square cavity. Results were found using finite difference methods for values of the Reynolds number Re between 0 and 64 for different ratios of lengths of the cavity. An attempt was made to find results for $Re=128$, but a convergent solution was not found. By modifying Kawaguti’s method, Burggraf [3] was able to attain stable solutions for forced convection for a Reynolds number up to 1,000. Heinrich and Marshall [4] used the penalty finite element method to investigate liddriven cavity flow up to $Re=400$. For a low Reynolds number, their results were in excellent agreement with published results. As the Reynolds number increased, however, their results were less accurate in comparison to other results. Early studies of natural convection in a cavity were of an experimental nature. An example of an early experimental study is that of Eckert and Carlson [5]. They made an experimental investigation into the flow and heat transfer in an enclosed cavity. The top and bottom plates were heated to different temperatures. Their results showed that, contrary to previous beliefs, it was very unlikely that a core of uniform temperature could exist at large Rayleigh numbers unless the height to width aspect ratio was very small. Ostrach [6] gives an excellent review of early experimental studies of convection in a cavity. Recently, Prasad and Koseff [7] investigated mixed convection in a liddriven cavity through experimental means where the lower surface was heated and the upper surface cooled. The lid speed and values of the Reynolds and Grashof numbers were varied. Their results indicated that the overall heat transfer rate was a very weak function of the Grashof number for the range of Reynolds numbers examined.
Moallemi and Jang [8] investigated flow in a liddriven cavity with the bottom wall heated. They studied the effects of small Prandtl numbers on the flow and heat transfer for various values of the Richardson number. Sivakumar et al. [9] investigated mixed convection in a liddriven cavity with a cooled right wall and a heat source in the left wall. Three different lengths of the heat source were examined. The finite volume method was used to solve the resulting equations. It was found that for low values of the Richardson number, reducing the heating portion length had no effect on the fluid flow. However, on increasing the Richardson number, it was seen that the flow depended heavily on the length of the heat source. It was also found that reducing the length of the heated portion of the wall resulted in a better heat transfer rate. Nithyadevi et al. [10] studied the case where the side walls were partially heated or partially cooled while the rest of the cavity was well insulated. They used the finite volume method to solve the equations. They studied nine different cases where the position of the heating and cooling sources was moved between the top, bottom and middle of the wall of the cavity. They found that the heat transfer rate was enhanced when there was a cooling source near the top of the cavity. For a combination of a bottom top heating source, the heat transfer rate was found to be high, while for the combination of a top bottom heating source, the heat transfer rate was low. Related studies have been done by, among others, Paraconi and Corvaro [11] and Pesso and Piva [12], Sathiyamoorthy et al. [13].
Recently, Basak et al. [14, 15] studied mixed convection in a liddriven cavity. The moving upper lid was insulated in both studies. In Basak et al. [14] they investigated cooled side walls with a heated (uniformly and nonuniformly) bottom wall. In Basak et al. [15] they looked at a uniformly heated bottom wall with linearly heated left side wall. The right side wall was heated either linearly or cooled uniformly. Both papers used the penalty finite element method to solve the governing equations of motion and heat transfer. Both studies found that the strength of convection increased with increasing Grashof numbers and that liddriven flow was dominant for $Gr={10}^{3}$. Their work showed that the heat transfer rate for a uniformly heated bottom wall was higher than that for the nonuniformly heated bottom wall and that the heat transfer rate was larger for the case of one cooled right wall than it was for the case when both walls were linearly heated.
Research has also been done into mixed convection with more than one moving wall in a cavity. Oztop and Dagtekin [16] used the finite volume method to investigate three cases of moving walls with adiabatic top and bottom wall and cooled left wall and heated right wall. They observed that the fluid flow and the heat transfer in the cavity depended on both the Richardson number Ri and the direction of motion of the walls.
Cheng and Liu [17] investigated four cases of mixed convection in a square cavity. In the first case, the side walls were well insulated while the top moving wall was heated and the bottom wall cooled. In the second case, the bottom wall was heated and the top wall cooled. In the third and fourth cases, the top and bottom walls were well insulated and one side wall was cooled while the other was heated. They found that for the first case, when the Richardson number was greater than 1, the heat transfer was mainly through conduction. When the Richardson number was less than 1, forced convection dominated the fluid flow as was found by Oztop and Dagtekin [16]. In contrast to the findings of Oztop and Dagtekin, in the second case, when the Richardson number was greater than 1, the lower half of the cavity showed natural convection while the upper part was dominated by forced convection. When the Richardson number was less than 1, the fluid flow was similar to that for case 1. For the third case when the Richardson number was greater than 1, heat transfer was shown to be by conduction near the side walls and by convection in the center. Again when the Richardson number was less than one, the fluid flow was similar to that for cases 1 and 2. For the fourth case, a large portion of the cavity was dominated by natural convection for all values of the Richardson number.
Corcione [18] investigated the effect of the Rayleigh number and the width to height aspect ratio of the cavity on steady laminar natural convection in an air filled cavity. The cavity was heated from below and cooled from above while six different cases of side wall heating and cooling were investigated. They showed that the heat transfer rate from a cooled or heated boundary increased as the Rayleigh number increased. In the case with insulated side walls, the heat transfer rate from the heated bottom wall or top cooled wall increased as each of the side walls was replaced by a cooled or heated side wall. Wong [19] investigated mixed convection in a square cavity. The left wall was maintained at a constant velocity. The other walls were kept stationary. The top and bottom walls were insulated and the left moving wall was heated while the right side wall was cooled. A numerical solution was found using the consistent splitting scheme and the finite element method. Wong fixed the Reynolds number at 100 and varied the Grashof number to measure the effects of the Richardson number. When the Richardson number was 0.001, the flow was found to be strongly influenced by forced convection. At $Ri=1$, mixed convection was found to occur. At $Ri=100$, the fluid flow was mainly dominated by natural convection. These findings are similar to those of Oztop and Dagtekin [16] even though only one side wall was moving in this case. Chinyoka [20] investigated nonisothermal twodimensional unsteady fluid flow problems in a rectangular channel.
In this study mixed convection in a twodimensional cavity is investigated using the penaltyGalerkin finite element method to solve the governing momentum and heat equations. Much work has been done on mixed convection with uniformly heated or cooled walls. Since little work has been done on sinusoidal and linearly heated walls and even less work on a combination of these, the aim of this investigation is to combine linear and sinusoidal heating of different walls of the cavity with a moving lid to study the resulting mixed convective flow.
2 Mathematical modeling
A twodimensional square cavity of length L is considered for the current investigation. The cavity lid is sinusoidally heated while moving at a constant speed from left to right, and the cavity is heated at a constant rate from the bottom. Two cases of side wall heating are investigated. In the first instance, the cavity side walls are linearly heated, while in case 2 the side walls are uniformly cooled.
where u and v are the velocity components in the x and y directions respectively, α is the thermal diffusivity, β is the coefficient of thermal expansion, ν is the kinematic viscosity, ρ is the mass density, g is the acceleration due to gravity, ${T}_{h}$ is the highest temperature attained along the bottom wall and the middle of the top wall and ${T}_{c}$ is the lowest temperature attained at the top corners in case 1 and along the side walls in case 2.
3 Numerical solution
Equations (8)(11) were solved using the penaltyGalerkin finite element method (PGFEM). This method has the advantage of eliminating pressure as a dependent variable while still satisfying the continuity equation.
where λ is the penalty parameter. Mass conservation is satisfied in the limit $\epsilon \to 0$ or $\lambda \to \mathrm{\infty}$. Here λ is chosen to be 10^{7} as this has been shown to give consistent results (Basak et al. [14]). If required, the pressure field can be calculated using equation (20). This generally requires a higher level of convergence than is necessary for the velocity field. In this investigation we have chosen to restrict our results to the stream function and temperature contours. As an alternative to the penalty function technique above, the semiimplicit method for pressure linked equations (SIMPLE) technique may be used; see, for instance, Chinyoka [20–22].
where ${U}_{j}$, ${V}_{j}$ and ${\theta}_{j}$ are the nodal values of U, V and θ at the j th node.
The set of nonlinear equations is solved using the Gaussian quadrature and reduced integration to prevent locking, that is, the deterioration in performance of the numerical scheme as $\lambda \to \mathrm{\infty}$ (see, for instance, Qi et al. [23] and the references therein).
3.1 The stream function and the Nusselt number
4 Results and discussion
Results have been found for $20\times 20$ quadratic elements with 361 inner nodes, $Pr=0.7$ (the value for air), ${10}^{3}\le Gr\le {10}^{5}$, and $Re=1,10,100\text{and}1,000$. Convergence was found to be generally poor for $Re>1,000$.
Figure 3 shows that for $Re=1$, $Gr={10}^{3}$, two counter rotating circulations are formed in the cavity. The circulation on the lefthand side of the cavity is counterclockwise, while the circulation on the righthand side of the cavity is clockwise. The clockwise flow is significantly stronger than the anticlockwise flow and thus it can be observed that the flow is dominated by forced convection due to the moving lid. As was expected, the isotherms for $Re=1$, $Gr={10}^{3}$ reach a maximum along the bottom wall and at the center of the top wall, while reaching a minimum at the top corners of the cavity. At the bottom of the cavity, isotherms span the entire width of the cavity. When $\theta \le 0.6$, the isotherms instead curve upwards towards the lid of the cavity. Hot isotherms at the top of the cavity are restricted to the center of the top wall.
As Gr increases to 10^{4}, the anticlockwise circulation increases in size and in strength so that two symmetric rotations are formed within the cavity. Natural convection is now as equally dominant as the forced convection regime. The isotherm pattern is similar to that of $Gr={10}^{3}$, although the heat from the bottom wall has risen slightly higher into the middle of the cavity.
At $Gr={10}^{5}$, the clockwise circulation in the cavity increases in size and dominates most of the cavity. The circulation is slanted towards the top right and bottom left corners. A small third anticlockwise circulation is formed on the bottom righthand side of the cavity. It can be seen that the isotherms are compressed in the top corners of the cavity and in the middle of the bottom wall. The isotherms are widely dispersed in the lefthand corner of the cavity. This is due to the shape and angle of the clockwise circulation. Cool fluid is brought down on the righthand side of the cavity, while hot fluid is transported upwards towards the middle of the cavity on the lefthand side of the bottom wall. The strength of the circulations increase with increasing Grashof numbers.
As Gr increases to 10^{4}, the clockwise circulation increases in size to form two counter rotating circulations of a similar size and the natural convection is almost equally dominant to that of forced convection. The centers of the circulations are now at the same height. The temperature contours become slightly more compressed towards the side walls as the Grashof number increases. The hotter contour lines from the top wall are seen to be wider than that of $Gr={10}^{3}$ and are found to be narrower in the center towards the bottom wall. As Gr increases to 10^{5}, the strength of the stream function increases and the size and position of the two circulations are identical. It can be seen that the hot contour lines have widened even further along the top wall causing the cooler contour lines to become compressed towards the top of the cavity. Towards the bottom of the cavity, the cool isotherms (those where $\theta \le 0.4$) are widely dispersed towards the center of the cavity while the warmer isotherms are compressed towards the bottom wall. This is particularly the case towards the bottom corners of the cavity.
As Gr increases to 10^{4}, the anticlockwise circulation grows in size although the clockwise rotating circulation is still larger in size and in strength indicating that the flow is still slightly dominated by forced convection. Again the contour lines are similar in shape to those for $Re=1$, and a slight compression towards the top righthand corner can be seen. As Gr increases further to 10^{5}, natural convection becomes equal to forced convection. There is no observable difference in the shape of the temperature contours from that of $Re=1$.
In general, the circulation is stronger for the case of cooled side walls than that for linearly heated walls. The anticlockwise circulation is significantly stronger for the cooled walls than for the linearly heated walls while the clockwise circulation is only slightly stronger for the cooled walls. For $Re=1\text{and}10$, the cooled walls produce two identical counter rotating circulations in contrast to the three tilted circulations resulting from the linearly heated walls. For $Gr={10}^{5}$, the stream function pattern for the cooled walls shows a slightly stronger clockwise circulation indicating a slight domination by forced convection, but again the flow pattern is different from that of the linearly heated side walls.
5 Heat transfer at the walls
For all values of the Grashof number, the local Nusselt number is equal to one at the edge of the bottom wall on both sides due to the linear heating of the side walls. For $Gr={10}^{3}$, the local Nusselt number falls below 1 and reaches a local minimum between $x=0.4$ and $x=0.5$. There is not much variation in the Nusselt number across the cavity due to the relatively flat isotherms at the bottom of the cavity. When $Gr={10}^{4}$, the Nusselt number is of a similar value to that for $Gr={10}^{3}$ between $x=0$ and $x=0.5$. Beyond $x=0.6$, the graph rises above that of $Gr={10}^{3}$ reaching a maximum at $x=0.8$.
When $Gr={10}^{5}$, the Nusselt number is greater than 1 for the majority of the cavity due to the compressed isotherms. The Nusselt number drops below 1 between $x=0$ and $x=0.2$ due to the widely dispersed isotherms on the lefthand side of the cavity. The Nusselt number is at a maximum at $x=0.7$. The local Nusselt number at the edges of the top wall is equal to −1 for all values of the Grashof number due to the linearly heated walls. For $Gr={10}^{3}$, the Nusselt number rises and reaches a local maximum at $x=0.4$ and then gradually decreases. For $Gr={10}^{4}$, the Nusselt number follows a similar pattern to that for $Gr={10}^{3}$, although is lower than the graph of $Gr={10}^{3}$. At $Gr={10}^{5}$, the graph of the Nusselt number is close to being symmetric in shape. The Nusselt number drops to a local minimum at $x=0.1$ and then rises to a local maximum at the center of the cavity. The Nusselt number then drops again to a local minimum at $x=0.9$. When $Gr={10}^{3}$, the local Nusselt number at the righthand wall gradually increases as y increases. For $Gr={10}^{4}$, a similar occurrence happens, although the Nusselt number is below that for $Gr={10}^{3}$ until $y=0.8$ where the Nusselt number becomes greater than that of $Gr={10}^{3}$. For $Gr={10}^{5}$, the Nusselt number decreases below 0 attaining a local minimum at $y=0.3$ due to the compressed isotherms at the bottom corner of the right wall. The Nusselt number then increases reaching a local maximum at $y=0.9$.
The local Nusselt number at the lefthand wall for $Gr={10}^{3}$ and $Gr={10}^{4}$ is very similar in shape to that for the righthand wall due to a near symmetrical isotherm pattern at the two walls. However at $Gr={10}^{5}$, the local Nusselt number at the lefthand wall is above 0 for the entire height of the cavity and has an oscillatory pattern. The Nusselt number initially rises attaining a local maximum at $y=0.3$ due to the dispersed isotherms along the bottom half of the lefthand wall. Towards the top of the cavity, the isotherms become more dispersed once again and the Nusselt number again rises to a local maximum at $y=0.9$.
The graph at the lefthand wall is similar to that of Basak et al. [15] which also exhibits a slightly oscillatory shape. However, the graph in the present study has a local maximum near the top corner and then decreases in contrast to the graph in Basak et al. [15] which increases monotonically to the wall. This is due to the difference between the adiabatic and the heated lid.
The local Nusselt number decreases to a local minimum at $y=0.6$ for $Gr={10}^{3}$ and then increases slightly towards the top of the cavity. For $Gr={10}^{4}$, the Nusselt number decreases to a local minimum near $y=0.3$ and then slowly increases towards the top of the cavity. When $Gr={10}^{5}$, the local Nusselt number decreases to a local minimum near $y=0.2$ due to the dispersion of isotherms near the bottom corner(s). The Nusselt number then increases above that of $Gr={10}^{4}$ and $Gr={10}^{3}$ at $y=0.25$ and $y=0.3$, respectively, due to the more compressed isotherms in the top corners. The Nusselt number then increases to a local maximum near $y=0.8$. There is no oscillatory pattern when $Gr={10}^{5}$ as was observed in the case of the heated side walls.
6 Conclusion

In general, the strength of circulation was stronger for the case of cooled walls, and that the anticlockwise circulation was significantly stronger for cooled walls, while the clockwise circulation was only slightly stronger for cooled walls. The difference in strength decreased with increasing both Reynolds and Grashof numbers.

Both the local and average Nusselt numbers were generally higher for the case of cooled side walls than those for heated side walls.

The local Nusselt numbers at the left and right walls were similar in the case of cooled side walls due to symmetric patterns in the temperature isotherms. However, particularly for $Gr={10}^{5}$, the heated side walls had different local Nusselt numbers at the two different walls.

The local Nusselt number at the top wall was similar in shape for the two cases although the linearly heated case was lower in value than for the cooled side walls.
Declarations
Acknowledgements
The authors are grateful to the University of KwaZuluNatal for financial support.
Authors’ Affiliations
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