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
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
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
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.
The equations that describe the fluid flow and the heat transfer characteristics within the cavity, subject to the Boussinesq approximation are as follows:
subject to the boundary conditions
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,
We nondimensionalize the equations by using the following change of variables:
where
where the important parameters are the Prandtl number Pr, the Reynolds number Re, and the Grashof number Gr given, respectively, by
The corresponding boundary conditions are:
The geometry of the problem together with the associated boundary conditions is shown in Figure 1.
Figure 1. Schematic sketch of the cavity problem with, Case 1, linearly heated side walls and, Case 2, uniformly cooled walls.
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.
We introduce test functions
The continuity equation in its weighted form becomes
The PGFEM supposes that we may replace equation (18) with
where ϵ is an arbitrarily small parameter. Rearranging these equations, we find
where λ is the penalty parameter. Mass conservation is satisfied in the limit
Substituting equation (20) into momentum equations (15) and (16) leads to the penalized momentum equations:
To determine the finite element approximate solution of equations (17), (21) and (22),
we discretise the domain into 400 rectangular elements with a total of 441 nodes.
In principle, more elements may be used to ensure even greater accuracy of results,
but this greatly increases the computation time. Initially, one hundred elements were
used, but although the computation of such a formulation was efficient, the accuracy
of the solutions was impaired. Four hundred elements were found to give sufficient
accuracy without impairing the efficiency of the solution method. There are 361 nodes
which are not on a Dirichlet boundary. Associated with each node in the domain is
a basis function
where
By substituting these expressions for U, V and θ into equations (17), (21) and (22), we obtain (after manipulation and simplification)
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
3.1 The stream function and the Nusselt number
Results of the fluid motion are usually interpreted in terms of the stream function ψ, where
Expanding the stream function in the same way as we did for U, V and θ, we obtain
The heat transfer coefficient in terms of the Nusselt number is defined as
where n denotes the normal to the plane. The average Nusselt number is defined as
4 Results and discussion
Results have been found for
Benchmark results were found for uniformly heated bottom and cooled side walls with an adiabatic stationary lid. Figure 2 shows the stream function and temperature contours for the benchmark results. The results are in good agreement with those in the literature, particularly Basak et al.[24]. We note that in the figures below, a negative stream function value denotes clockwise flow while a positive value denotes anticlockwise flow.
Figures 35 show the results for the stream function and temperature contours for a moving sinusoidally heated lid with linearly heated side walls and a uniformly heated bottom wall for Grashof numbers between 10^{3} and 10^{5}.
Figure 3. Stream function and temperature contours for linearly heated side walls when
Figure 3 shows that for
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
At
For
Figure 4. Stream function and temperature contours for linearly heated side walls when
In Figure 5 at
Figure 5. Stream function and temperature contours for linearly heated side walls when
For
Figure 6. Stream function and temperature contours for linearly heated side walls when
Figures 79 show the results for the stream function and temperature contours for a moving sinusoidally
heated lid with cooled side walls and uniformly heated bottom wall for
Figure 7. Stream function and temperature contours for cooled side walls when
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
In Figure 8 it is clear that for
Figure 8. Stream function and temperature contours for cooled side walls when
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
Figure 9 shows that for
Figure 9. Stream function and temperature contours for cooled side walls when
Figure 10 shows that for
Figure 10. Stream function and temperature contours for cooled side walls when
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
5 Heat transfer at the walls
Figure 11 shows the effect of the Grashof number on the heat transfer at each of the walls in the cavity for the case of linearly heated side walls.
Figure 11. Local Nusselt number for linearly heated side walls with
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
When
The local Nusselt number at the lefthand wall for
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.
Figure 12 shows the average Nusselt number at the four walls of the cavity. At the bottom wall,
the Nusselt number is seen to increase as the Grashof number increases. The graph
of the average Nusselt number at the bottom wall is almost identical to the results
of Basak et al.[15]. At the top wall, the opposite is seen to happen. This is due to the fact that as
the Grashof number increases, the isotherms become more compressed in the top corners.
At the lefthand side wall, the average Nusselt number is lower for
Figure 12. The average Nusselt number for linearly heated side walls with
Figure 13. The local Nusselt number for cooled side walls with
The local Nusselt number decreases to a local minimum at
6 Conclusion
A numerical study has been performed using the penaltyGalerkin finite element method to analyze mixed convective heat transfer and fluid flow in an air filled square cavity. The effects of linearly heated and uniformly cooled walls on flow and heat transfer characteristics within the cavity have been studied. It was observed that
– 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
– 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.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
DSD carried out the numerical computations. PS participated in the design of the study and helped to draft the manuscript.
Acknowledgements
The authors are grateful to the University of KwaZuluNatal for financial support.
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