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Effects of adiabatic extensions on heat transfer through a differentially heated square cavity
International Communications in Heat and Mass Transfer 37 (2010) 1221–1225
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i c h m t
Effects of adiabatic extensions on heat transfer through a differentially heated square cavity☆ Chengwang Lei a,⁎, Andrew O'Neill b a b
School of Civil Engineering, The University of Sydney, Sydney, NSW 2006, Australia Reliability Engineering, Rio Tinto Aluminium — Yarwun, Gladstone DC, QLD 4680, Australia
a r t i c l e
i n f o
Available online 4 August 2010 Keywords: Natural convection Differentially heated cavity Adiabatic extension
a b s t r a c t This study is concerned with transient and steady state heat transfer by natural convection in a differentially heated cavity. The purpose is to evaluate a passive approach for enhancing heat transfer through the cavity. In this study, the effects of three different corner geometries (including sharp, round and straight corners) and adiabatic extensions of various dimensions on natural convection heat transfer are investigated numerically. The numerical results show marginal variations of the heat transfer rates among the three different corner shapes and a strong dependence of heat transfer enhancement on the Rayleigh number. For a given Rayleigh number, the enhancement of heat transfer by adiabatic extensions is limited. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction An effective means of dissipating heat is required for a wide variety of situations in both industrial and real life applications. For instance, semi-conductor devices produce a large amount of heat relative to their small sizes. It is a common practice to use fans in electronic devices to dissipate the heat generated by semi-conductors. Despite that fans provide an effective way for dissipating the heat, the adoption of fans has two immediate drawbacks in addition to consuming extra power. Firstly, the operation of fans produces undesirable noise; and secondly, the reliability of the fans and in particular of their bearing systems needs to be addressed. As the semi-conductors become more and more powerful, fans have to operate at ever increasing speeds in order to cope with the increasing heat generated by electronic devices. Running fans at very high speeds escalates the noise and reliability problems associated with fans. Ideally, heat should be dissipated naturally without the need for fans, for example, by a natural convection process. However, heat dissipation by natural convection in general cannot achieve a rate of cooling that is comparable with that by forced convection. This has motivated numerous studies aiming at enhancing heat transfer by natural convection (see for example [1–4]). A particular problem that has received considerable attention is a differentially heated cavity, which is an appropriate model for many industrial and domestic heat exchangers and heat dissipation systems in confined spaces. Numerous investigations of the differentially heated cavity [5–8] have been carried out since the pioneer work of Bachelor
☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (C. Lei). 0735-1933/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2010.06.031
[9], and a number of techniques have been developed for enhancing heat transfer through the differentially heated cavity. In general, the applied techniques for manipulating heat transfer through a differentially heated cavity can be classified into two categories, that is, active and passive approaches. The active approaches utilise external energy to change the state of the natural convection flow and in turn enhance heat transfer. Examples of active approaches include enhancement using sound waves [10], oscillating wall temperatures [2,11,12], ultrasonic vibration [3], etc. In contrast to the active approaches, the passive approaches do not require external energy supply. Instead, they rely on internally induced changes of the state of the natural convection flow to enhance heat transfer. Examples of passive enhancement include the use of heat sinks [13–15] and horizontal fins [16–21]. Another passive approach for manipulating heat transfer through the differentially heated cavity is to place inserts at the corners of the cavity. Costa et al. [22] demonstrated that the overall heat transfer through the cavity can be either enhanced or depressed, depending on the thermo-physical property of the inserts. It is found that by placing triangular inserts which are 1000 times more conductive than the fluid in each corner of the cavity, the overall heat transfer can be enhanced by as much as 36.9% for a Rayleigh number of 106. Placing two inserts in diagonally opposing corners of the cavity also results in enhancement of heat transfer, but to a lesser extent. Moreover placing the inserts in the leading-edge corners of the cavity gives better enhancement than placing inserts in the trailing-edge corners. Costa et al. [22] also found that heat transfer through the differentially heated cavity is depressed by placing inserts that are of the same thermal conductivity as the working fluid. Accordingly, they concluded that the enhancement in heat transfer with conductive corner inserts was primarily due to the increase of the surface area for heat transfer due to the presence of the highly conductive inserts.
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C. Lei, A. O'Neill / International Communications in Heat and Mass Transfer 37 (2010) 1221–1225
Nomenclature A Ae g H k L Le Nu P Pr Ra T T0 TH TC t u v x y
Aspect ratio Adiabatic extension ratio Gravitational acceleration (m/s2) Cavity height (m) Thermal conductivity (W/mK) Cavity length (m) Adiabatic extension length (m) Nusselt number Pressure (N/m2) Prandtl number Rayleigh number Temperature (K) Initial water temperature (K) Temperature of the hot wall (K) Temperature of the cold wall (K) Time (s) Velocity in the x-direction (m/s) Velocity in the y-direction (m/s) Horizontal coordinate (m) Vertical coordinate (m)
Greek symbols β Thermal expansion coefficient (1/K) ΔT Temperature difference (K) κ Thermal diffusivity (m2/s) ν Kinematic viscosity (m2/s) ρ Fluid density (kg/m3)
In this study, we demonstrate another passive way for manipulating heat transfer through the differentially heated cavity, that is, by modifying the geometry of the cavity. The base model is a differentially heated square cavity. With the heating and cooling lengths of the sidewalls fixed, the cavity is expanded by adding adiabatic corners of different shapes. The effect of the corner shape on the overall heat transfer is examined through numerical simulations. Furthermore, the differentially heated cavity with adiabatic extensions of various lengths is also investigated numerically to determine the effect of the adiabatic extension on the overall heat transfer. 2. Problem formulation and numerical method The base model considered in this study is a two-dimensional (2D) square cavity of a height H and a width L = H, giving an aspect ratio of A = H/L = 1. The cavity contains water as the working fluid which is initially isothermal (with an initial temperature of T0) and stationary. At start-up, the two opposing sidewalls of the cavity are heated to TH = T0 + ΔT and cooled to TC = T0 – ΔT respectively, where ΔT is the absolute temperature difference between the sidewalls and the mean interior temperature. The top and bottom surfaces are adiabatic, and all the internal surfaces are rigid and non-slip. Two sets of extension to the base square cavity model are considered in this study. Firstly, the square cavity is extended in all directions by adding adiabatic corners of different shapes; and secondly, the square cavity is extended in the vertical directions by adding adiabatic extensions to the heating and cooling sidewalls. Details of the extended models will be presented in the following section. The development of the temperature and flow structures within the cavity, which is determined by a natural convection process, can
be described by the following set of unsteady governing equations, for which the Boussinesq assumption has been made: ∂u ∂v + =0 ∂x ∂y
ð1Þ 2
2
∂u ∂u ∂u 1 ∂p ∂ u ∂ u + +u +v =− +ν ρ ∂x ∂t ∂x ∂y ∂ x2 ∂ y2 2
2
∂v ∂v ∂v 1 ∂p ∂ v ∂ v + +u +v =− +ν ρ ∂y ∂t ∂x ∂y ∂ x2 ∂ y2
! ð2Þ
! + gβ ðT−T0 Þ
! ∂T ∂T ∂T ∂2 T ∂2 T + +u +v =κ : ∂t ∂x ∂y ∂ x2 ∂ y2
ð3Þ
ð4Þ
In the above equations, all the fluid properties are evaluated at the reference water temperature T0. For a cavity model of a fixed geometry (the aspect ratio is fixed), the properties of the flow and heat transfer in the cavity are characterized by two non-dimensional parameters, the Prandtl number (Pr) and the Rayleigh number (Ra), which are defined as: Pr =
ν κ
Ra =
gβ ΔT H : νκ
ð5Þ 3
ð6Þ
The above governing equations along with the specified boundary and initial conditions are solved implicitly using a Finite Volume Method. The SIMPLE scheme [23] is adopted for pressure–velocity coupling. The spatial discretization of all diffusion terms is by a secondorder central-differencing scheme, and that of all advection terms is by a second-order upwind scheme. For transient flow calculations, time marching is by a second-order backward differencing scheme. An iterative procedure with under-relaxation is applied to determine the flow solution at each time step. Non-uniform meshes with grid nodes concentrated toward all interior surfaces are constructed for all models in this study. A mesh dependence test has been conducted using the base model (i.e. a square cavity with heated and cooled sidewalls). The test is carried out for two Rayleigh numbers of Ra= 2.22× 108 and 1.1 × 109 respectively with a fixed Prandtl number Pr= 6.62. Two meshes with a total number of approximately 16,000 and 50,200 cells respectively are constructed and tested. The numerical results are compared in Table 1, which shows the calculated steady-state Nusselt numbers on the heated sidewall using the two different meshes and for the two different Rayleigh numbers. Here the Nusselt number is defined as Nu =
1 H ∂T H ∫0 k ∂x dy
kΔT = H
=
∫H0 ∂T dy ∂x ΔT
:
ð7Þ
It is seen in Table 1 that refining the mesh from 16,000 cells to 50,200 cells has a negligible effect on the calculated Nusselt number for both Rayleigh numbers. Therefore, the coarse mesh is adopted for all the
Table 1 Calculated steady-state Nusselt numbers with different meshes (Pr = 6.62). Mesh
Ra = 2.22 × 108
Ra = 1.11 × 109
Mesh 1 (16,000 cells) Mesh 2 (50,200 cells) Variation
93.72 93.74 0.02%
142.19 142.23 0.03%
C. Lei, A. O'Neill / International Communications in Heat and Mass Transfer 37 (2010) 1221–1225
calculations with the base model in this study. For the extended models (refer to the next section), the mesh is constructed in accordance with the mesh required for the base model. Since the adiabatic extensions do not provide additional driving force to the natural convection flow, no separate mesh dependence tests are conducted for the extended models. For unsteady calculations, the time step is usually determined based on a time-step dependence test. A meaningful time-step dependence test for the present study would involve calculations of the flow from the initial start-up through to a final steady state with different time steps. However, these calculations are extremely expensive in terms of the required computing resources. In this study, the time step is selected with reference to the time steps adopted in [24,25], which determined the time step based on tests of the start-up flow only. Since the Rayleigh numbers considered in this study are lower than those considered in [24,25], it is expected that an equivalent or smaller time step compared with those adopted in the above literature would provide sufficient accuracy for the unsteady calculations. 3. Numerical results Two sets of numerical simulations have been carried out in this study. First, steady-state calculations are conducted with different corner shapes including sharp, round and straight corners, all of which are assumed adiabatic. The numerical results are compared with the base model. In the second set of numerical simulations, both steady and transient calculations are conducted with adiabatic extensions of various dimensions. Details of the numerical models and the numerical results are presented below.
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Here we introduce a new dimensionless parameter, the adiabatic extension ratio Ae, to quantify the geometric change. The adiabatic extension ratio is defined as: Ae =
Le : H
ð8Þ
For the comparison of the effect of different corner shapes, the adiabatic extension ratio is fixed at Ae = 0.125. Steady-state calculations are conducted for three moderate Rayleigh numbers of Ra = 2.27 × 107, 1.13× 108 and 5.67× 108 respectively with a fixed Prandtl number of Pr= 6.62. The numerical results are presented in Fig. 2. Fig. 2(a) shows the calculated steady-state Nusselt numbers on the heated sidewall with different corner geometries. It is clear that, with all the geometric configurations including the base model, the calculated Nusselt number increases with the Rayleigh number. It is also noticeable that there is a slight increase in the steady-state Nusselt number with the modified corner geometries (i.e. with sharp, round or straight adiabatic corners) compared with the base model, suggesting that the steady-sate heat transfer through the hot sidewall is enhanced by adding the adiabatic corners. The extent of heat transfer enhancement due to the modifications to the corner geometry is shown in Fig. 2(b) for different Rayleigh numbers. Clearly, for all the three shapes of the corner geometry, the enhancement of heat transfer increases with decreasing Rayleigh number, and a better enhancement is achieved with either sharp or
3.1. Effect of corner shapes In order to examine the effect of corner geometry on heat transfer through the differentially heated cavity, the base model of the square cavity is extended equally in all directions (left, right, top and bottom) by a length of Le, and corners of different shapes including sharp (1), round (2) and straight (3) corners respectively are added (see Fig. 1). The shaded region enclosed by dotted lines in Fig. 1 represents the original base model. All the added corners are assumed adiabatic, and thus, the lengths over which heat is transferred to and from the cavity remain unchanged, resulting in the same Rayleigh number for different corner shapes.
Fig. 1. Extended square cavity with different corner shapes.
Fig. 2. Steady-state results for different corner geometries and different Rayleigh numbers. (a) Calculated Nusselt numbers with different corner shapes for different Rayleigh numbers. (b) Enhancement of heat transfer for different Rayleigh numbers obtained with different corner shapes.
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Fig. 3. Differentially heated cavity with adiabatic extensions.
round corners. For the lowest Rayleigh number considered here (i.e. Ra = 2.27 × 107), the maximum enhancement is 3.5% with the sharp and round corners, and the enhancement reduces to 2.8% with the straight corners. Both sharp and round adiabatic corners have similar effect on the calculated Nusselt number on the hot wall, and they result in a better enhancement of heat transfer compared with the straight corners. Hence, in the following section, the effect of the adiabatic extension ratio on heat transfer is examined based on the sharp corners only due to their relative simplicity for mesh generation.
Nusselt number. The maximum enhancement of the steady-state heat transfer is about 1.9% for the current Rayleigh number. Next, the effect of the adiabatic extension on transient heat transfer is investigated for a fixed adiabatic extension ratio of Ae = 0.1 and for the same Rayleigh and Prandtl numbers as the steady-state calculations. The parameters are selected based on a physical model with a height of H = 0.24 m. The reference temperature is set at T0 = 295 K, and the temperature difference is fixed at ΔT = 1 K, resulting in a Rayleigh number of Ra = 2.22 × 108. A dimensional time step of 0.1 s is adopted for the unsteady calculations. The model is calculated with and without the adiabatic extensions for comparison purposes. The numerical results are shown in Fig. 5. Plotted in Fig. 5(a) are the time histories of the calculated heat transfer rate through the hot sidewall in a linear time scale. It is seen in Fig. 5(a) that, for both cases with and without the adiabatic extensions, the calculated heat transfer rate undergoes a transition and gradually approaches a constant at the steady state. In order to clearly demonstrate the heat transfer features at the early start-up and transitional stages, the calculated heat transfer rate through the hot sidewall is replotted in a logarithmic time scale in Fig. 5(b).
3.2. Effect of the adiabatic extension ratio In this section, the effect of the adiabatic extensions of various dimensions on heat transfer through the sidewalls is investigated. For this purpose, the sidewalls are extended vertically in two opposing directions while the width of the differentially heated cavity remains unchanged (see Fig. 3). Similarly, the extended sections of the sidewalls are assumed adiabatic, and thus the Rayleigh number remains the same for this investigation. Both steady and transient features of heat transfer through the sidewalls are considered. First, a series of steady-state calculations are carried out for adiabatic extension ratios ranging from Ae = 0.01 to Ae = 0.3. The Rayleigh number is fixed at Ra = 2.22 × 108 and the Prandtl number is fixed at Pr= 6.62. The calculated Nusselt numbers on the hot wall for different adiabatic extension ratios are presented in Fig. 4. It is clear in Fig. 4 that, as the adiabatic extension ratio increases from zero, the calculated Nusselt number initially increases, indicating that heat transfer through the sidewall is enhanced with the addition of the adiabatic extension. However, as the adiabatic extension ratio is increased beyond 0.1, there is no further increase of the calculated
Fig. 4. Calculated steady-state Nusselt number on the hot wall for various adiabatic extension ratios.
Fig. 5. Effect of the adiabatic extension on transient heat transfer through the hot sidewall. (a) and (b) Time history of the heat rate on the hot sidewall plotted in (a) a linear time scale; (b) a logarithmic time scale. (c) Time history of the enhancement of heat transfer.
C. Lei, A. O'Neill / International Communications in Heat and Mass Transfer 37 (2010) 1221–1225
Clearly, the major features of the flow in the early and transitional stages are the same with and without the adiabatic extensions. In the early stage, the calculated heat transfer rate with the addition of the adiabatic extensions surpasses that obtained with the base model. The comparison of the heat transfer rate is reversed after about 200 s, which approximately corresponds to the time when the intrusion flows arrive at the opposing sidewalls. This remains to be the case until around 1900 s when the heat transfer rate calculated with the adiabatic extensions again surpasses that without the extensions. It is noticed that the second switch of the comparison between these two models occurs after the perturbations induced by the arrival of the intrusion flow from the cold wall die out. Fig. 5(c) shows quantitative information of the enhancement of transient heat transfer due to the addition of the adiabatic extensions. It confirms that heat transfer through the hot wall is enhanced from the start-up until the arrival of the intrusion from the cold wall (around 200 s). A maximum heat transfer enhancement of 4.75% is observed at 137 s after the start-up. Associated with the flow oscillations induced by the arriving intrusion (during the period from approximately 200 to 1900 s), heat transfer through the hot wall is depressed due to the presence of the adiabatic extensions. After the flow oscillations die out, heat transfer is again enhanced with the adiabatic extensions. This remains to be the case for the rest of the flow transition until the final steady state, at which heat transfer is enhanced by about 2.1%. Note that the enhancement of the steady-state heat transfer calculated using the unsteady model (2.1%) is slightly higher than that predicted by the steady model (1.9%). This may be attributed to two factors. Firstly, the final state of the unsteady model (i.e. at 8000 s) is not at a truly steady state since the transition to the steady state is an extremely slow process. To cover the complete transition to the final steady state would require significant computing resources, and is not feasible. Therefore, the present calculation is terminated at 8000 s. Secondly, for unsteady calculations, an additional numerical error associated with the finite time step is inevitable. It can be estimated from Fig. 5(c) that the overall effect of adding the adiabatic extensions is a net 1% enhancement of heat transfer through the hot wall for the period covered by the calculation. It is also expected that the net effect of the adiabatic extensions on transient heat transfer depends on both the Rayleigh number and the adiabatic extension ratio. 4. Summary It is not uncommon to speculate that heat transfer through a differentially heated cavity is controlled by the extent of the sidewalls over which heat is transferred to and from the cavity. The heating and cooling lengths of the sidewalls also determine the Rayleigh number of this problem, which, along with the Prandtl number and aspect ratio, characterizes natural convection in the cavity. However, the present study has demonstrated that heat transfer through the cavity can be enhanced by modifying the geometry while keeping the heating and cooling lengths of the sidewalls unchanged. It is revealed in this study that adding adiabatic extensions to the differentially heated cavity enhances steady-state heat transfer to a certain extent (refer to Fig. 4). Further extending the domain provides no additional benefit in terms of heat transfer. The enhancement of heat transfer due to the adiabatic extensions depends strongly on the Rayleigh number (refer to Fig. 2b). The lower the Rayleigh number, the better enhancement is achievable. It is also found in this study that the geometry of the corners on an extended domain has a minor effect on heat transfer (Fig. 2). However, a slightly reduced enhancement of heat transfer is obtained with the straight corners compared to the sharp and round corners. The numerical results based on an unsteady model further demonstrate that better enhancement of heat transfer can be achieved
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at the early stage of the flow development following the start-up (refer to Fig. 5c). However, heat transfer is depressed during the early transitional stage following the arrival of the intrusion from the opposing sidewall. During the late transition toward the final steady state, the addition of the adiabatic extensions again enhances heat transfer. It is worth investigating the enhancement of transient heat transfer with different adiabatic extension ratios and for different Rayleigh numbers in future. Acknowledgement The first author is grateful for the financial support of the Australian Research Council. References [1] T. Fujii, M. Fujii, M. Takeuchi, Influence of various surface roughness on the natural convection, Int. J. Heat Mass Transfer 16 (1973) 629–640. [2] H.S. Kwak, K. Kuwahara, J.M. Hyun, Resonant enhancement of natural convection heat transfer in a square enclosure, Int. J. Heat Mass Transfer 41 (1998) 2837–2846. [3] H.Y. Kim, Y.G. Kim, B.H. Kang, Enhancement of natural convection and pool boiling heat transfer via ultrasonic vibration, Int. J. Heat Mass Transfer 47 (2004) 2831–2840. [4] E. Bilgen, Natural convection in cavities with a thin fin on the hot wall, Int. J. Heat Mass Transfer 48 (2005) 3493–3505. [5] A.E. Gill, The boundary-layer regime for convection in a rectangular cavity, J. Fluid Mech. 26 (1966) 515–536. [6] J.C. Patterson, J. Imberger, Unsteady natural convection in a rectangular cavity, J. Fluid Mech. 100 (1980) 65–86. [7] B.V.K.S. Sai, K.N. Seetharamu, P.A.A. Narayana, Solution of transient laminar natural convection in a square cavity by an explicit finite element scheme, Numer. Heat Transfer, Part A Appl. 25 (1994) 593–609. [8] E.V. Kalabin, M.V. Kanashina, P.T. Zubkov, Heat transfer from the cold wall of a square cavity to the hot one by oscillatory natural convection, Numer. Heat Transfer Part A Appl. 47 (2005) 609–619. [9] G.K. Bachelor, Heat transfer by free convection across a closed cavity between vertical boundaries at different temperatures, Q. Appl. Math. 12 (1954) 209–233. [10] P. Vainshtein, M. Fichman, C. Gutfinger, Acoustic enhancement of heat-transfer between 2 parallel plates, Int. J. Heat Mass Transfer 38 (1995) 1893–1899. [11] E.V. Kalabin, M.V. Kanashina, P.T. Zubkov, Natural-convective heat transfer in a square cavity with time-varying side-wall temperature, Numer. Heat Transfer, Part A Appl. 47 (2005) 621–631. [12] R. El Ayachi, A. Raji, M. Hasnaoui, A. Bahlaoui, Combined effect of radiation and natural convection in a square cavity differentially heated with a periodic temperature, Numer. Heat Transfer, Part A Appl. 53 (2008) 1339–1356. [13] J.R. Welling, C.N. Wooldridge, Free convection heat transfer coefficients from vertical fins, J. Heat Transfer 87 (1965) 439–444. [14] C.W. Leung, S.D. Probert, M.J. Shilston, Heat exchanger: optimal separation for vertical rectangular fins protruding from a vertical rectangular base, Appl. Energy 19 (1985) 77–85. [15] B. Yazicioglu, H. Yuncu, Optimum fin spacing of rectangular fins on a vertical base in free convection heat transfer, Heat Mass Transf. 44 (2007) 11–21. [16] A. Bejan, Natural-convection heat-transfer in a porous layer with internal flow obstructions, Int. J. Heat Mass Transfer 26 (1983) 815–822. [17] A. Nag, A. Sarkar, V.M.K. Sastri, Natural-convection in a differentially heated square cavity with a horizontal partition plate on the hot-wall, Compu. Meth. Appl. Mech. Eng. 110 (1993) 143–156. [18] X. Shi, J.M. Khodadadi, Laminar natural convection heat transfer in a differentially heated square cavity due to a thin fin on the hot wall, J. Heat Transfer 125 (2003) 624–634. [19] F. Xu, J.C. Patterson, C. Lei, Experimental observations of the thermal flow around a square obstruction on a vertical wall in a side-heated cavity, Exp. Fluids 40 (2006) 364–371. [20] F. Xu, J.C. Patterson, C. Lei, An experimental study of the unsteady thermal flow around a thin fin on a sidewall of a differentially heated cavity, Int. J. Heat Fluid Flow 29 (2008) 1139–1153. [21] F. Xu, J.C. Patterson, C. Lei, Transition to a periodic flow induced by a thin fin on the sidewall of a differentially heated cavity, Int. J. Heat Mass Transfer 52 (2009) 620–628. [22] V.A.F. Costa, M.S.A. Oliveira, A.C.M. Sousa, Control of laminar natural convection in differentially heated square enclosures using solid inserts at the corners, Int. J. Heat Mass Transfer 46 (2003) 3529–3537. [23] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, New York, 1980. [24] J.C. Patterson, S.W. Armfield, Transient features of natural convection in a cavity, J. Fluid Mech. 219 (1990) 469–497. [25] F. Xu, J.C. Patterson, C. Lei, Oscillations of the horizontal intrusion in a side-heated cavity, in: M. Behnia, W. Lin, G.D. McBain (Eds.), Proceedings of the Fifteenth Australasian Fluid Mechanics Conference (CD-ROM), The University of Sydney, 2004, Paper AFMC00243.
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i c h m t
Effects of adiabatic extensions on heat transfer through a differentially heated square cavity☆ Chengwang Lei a,⁎, Andrew O'Neill b a b
School of Civil Engineering, The University of Sydney, Sydney, NSW 2006, Australia Reliability Engineering, Rio Tinto Aluminium — Yarwun, Gladstone DC, QLD 4680, Australia
a r t i c l e
i n f o
Available online 4 August 2010 Keywords: Natural convection Differentially heated cavity Adiabatic extension
a b s t r a c t This study is concerned with transient and steady state heat transfer by natural convection in a differentially heated cavity. The purpose is to evaluate a passive approach for enhancing heat transfer through the cavity. In this study, the effects of three different corner geometries (including sharp, round and straight corners) and adiabatic extensions of various dimensions on natural convection heat transfer are investigated numerically. The numerical results show marginal variations of the heat transfer rates among the three different corner shapes and a strong dependence of heat transfer enhancement on the Rayleigh number. For a given Rayleigh number, the enhancement of heat transfer by adiabatic extensions is limited. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction An effective means of dissipating heat is required for a wide variety of situations in both industrial and real life applications. For instance, semi-conductor devices produce a large amount of heat relative to their small sizes. It is a common practice to use fans in electronic devices to dissipate the heat generated by semi-conductors. Despite that fans provide an effective way for dissipating the heat, the adoption of fans has two immediate drawbacks in addition to consuming extra power. Firstly, the operation of fans produces undesirable noise; and secondly, the reliability of the fans and in particular of their bearing systems needs to be addressed. As the semi-conductors become more and more powerful, fans have to operate at ever increasing speeds in order to cope with the increasing heat generated by electronic devices. Running fans at very high speeds escalates the noise and reliability problems associated with fans. Ideally, heat should be dissipated naturally without the need for fans, for example, by a natural convection process. However, heat dissipation by natural convection in general cannot achieve a rate of cooling that is comparable with that by forced convection. This has motivated numerous studies aiming at enhancing heat transfer by natural convection (see for example [1–4]). A particular problem that has received considerable attention is a differentially heated cavity, which is an appropriate model for many industrial and domestic heat exchangers and heat dissipation systems in confined spaces. Numerous investigations of the differentially heated cavity [5–8] have been carried out since the pioneer work of Bachelor
☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (C. Lei). 0735-1933/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2010.06.031
[9], and a number of techniques have been developed for enhancing heat transfer through the differentially heated cavity. In general, the applied techniques for manipulating heat transfer through a differentially heated cavity can be classified into two categories, that is, active and passive approaches. The active approaches utilise external energy to change the state of the natural convection flow and in turn enhance heat transfer. Examples of active approaches include enhancement using sound waves [10], oscillating wall temperatures [2,11,12], ultrasonic vibration [3], etc. In contrast to the active approaches, the passive approaches do not require external energy supply. Instead, they rely on internally induced changes of the state of the natural convection flow to enhance heat transfer. Examples of passive enhancement include the use of heat sinks [13–15] and horizontal fins [16–21]. Another passive approach for manipulating heat transfer through the differentially heated cavity is to place inserts at the corners of the cavity. Costa et al. [22] demonstrated that the overall heat transfer through the cavity can be either enhanced or depressed, depending on the thermo-physical property of the inserts. It is found that by placing triangular inserts which are 1000 times more conductive than the fluid in each corner of the cavity, the overall heat transfer can be enhanced by as much as 36.9% for a Rayleigh number of 106. Placing two inserts in diagonally opposing corners of the cavity also results in enhancement of heat transfer, but to a lesser extent. Moreover placing the inserts in the leading-edge corners of the cavity gives better enhancement than placing inserts in the trailing-edge corners. Costa et al. [22] also found that heat transfer through the differentially heated cavity is depressed by placing inserts that are of the same thermal conductivity as the working fluid. Accordingly, they concluded that the enhancement in heat transfer with conductive corner inserts was primarily due to the increase of the surface area for heat transfer due to the presence of the highly conductive inserts.
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C. Lei, A. O'Neill / International Communications in Heat and Mass Transfer 37 (2010) 1221–1225
Nomenclature A Ae g H k L Le Nu P Pr Ra T T0 TH TC t u v x y
Aspect ratio Adiabatic extension ratio Gravitational acceleration (m/s2) Cavity height (m) Thermal conductivity (W/mK) Cavity length (m) Adiabatic extension length (m) Nusselt number Pressure (N/m2) Prandtl number Rayleigh number Temperature (K) Initial water temperature (K) Temperature of the hot wall (K) Temperature of the cold wall (K) Time (s) Velocity in the x-direction (m/s) Velocity in the y-direction (m/s) Horizontal coordinate (m) Vertical coordinate (m)
Greek symbols β Thermal expansion coefficient (1/K) ΔT Temperature difference (K) κ Thermal diffusivity (m2/s) ν Kinematic viscosity (m2/s) ρ Fluid density (kg/m3)
In this study, we demonstrate another passive way for manipulating heat transfer through the differentially heated cavity, that is, by modifying the geometry of the cavity. The base model is a differentially heated square cavity. With the heating and cooling lengths of the sidewalls fixed, the cavity is expanded by adding adiabatic corners of different shapes. The effect of the corner shape on the overall heat transfer is examined through numerical simulations. Furthermore, the differentially heated cavity with adiabatic extensions of various lengths is also investigated numerically to determine the effect of the adiabatic extension on the overall heat transfer. 2. Problem formulation and numerical method The base model considered in this study is a two-dimensional (2D) square cavity of a height H and a width L = H, giving an aspect ratio of A = H/L = 1. The cavity contains water as the working fluid which is initially isothermal (with an initial temperature of T0) and stationary. At start-up, the two opposing sidewalls of the cavity are heated to TH = T0 + ΔT and cooled to TC = T0 – ΔT respectively, where ΔT is the absolute temperature difference between the sidewalls and the mean interior temperature. The top and bottom surfaces are adiabatic, and all the internal surfaces are rigid and non-slip. Two sets of extension to the base square cavity model are considered in this study. Firstly, the square cavity is extended in all directions by adding adiabatic corners of different shapes; and secondly, the square cavity is extended in the vertical directions by adding adiabatic extensions to the heating and cooling sidewalls. Details of the extended models will be presented in the following section. The development of the temperature and flow structures within the cavity, which is determined by a natural convection process, can
be described by the following set of unsteady governing equations, for which the Boussinesq assumption has been made: ∂u ∂v + =0 ∂x ∂y
ð1Þ 2
2
∂u ∂u ∂u 1 ∂p ∂ u ∂ u + +u +v =− +ν ρ ∂x ∂t ∂x ∂y ∂ x2 ∂ y2 2
2
∂v ∂v ∂v 1 ∂p ∂ v ∂ v + +u +v =− +ν ρ ∂y ∂t ∂x ∂y ∂ x2 ∂ y2
! ð2Þ
! + gβ ðT−T0 Þ
! ∂T ∂T ∂T ∂2 T ∂2 T + +u +v =κ : ∂t ∂x ∂y ∂ x2 ∂ y2
ð3Þ
ð4Þ
In the above equations, all the fluid properties are evaluated at the reference water temperature T0. For a cavity model of a fixed geometry (the aspect ratio is fixed), the properties of the flow and heat transfer in the cavity are characterized by two non-dimensional parameters, the Prandtl number (Pr) and the Rayleigh number (Ra), which are defined as: Pr =
ν κ
Ra =
gβ ΔT H : νκ
ð5Þ 3
ð6Þ
The above governing equations along with the specified boundary and initial conditions are solved implicitly using a Finite Volume Method. The SIMPLE scheme [23] is adopted for pressure–velocity coupling. The spatial discretization of all diffusion terms is by a secondorder central-differencing scheme, and that of all advection terms is by a second-order upwind scheme. For transient flow calculations, time marching is by a second-order backward differencing scheme. An iterative procedure with under-relaxation is applied to determine the flow solution at each time step. Non-uniform meshes with grid nodes concentrated toward all interior surfaces are constructed for all models in this study. A mesh dependence test has been conducted using the base model (i.e. a square cavity with heated and cooled sidewalls). The test is carried out for two Rayleigh numbers of Ra= 2.22× 108 and 1.1 × 109 respectively with a fixed Prandtl number Pr= 6.62. Two meshes with a total number of approximately 16,000 and 50,200 cells respectively are constructed and tested. The numerical results are compared in Table 1, which shows the calculated steady-state Nusselt numbers on the heated sidewall using the two different meshes and for the two different Rayleigh numbers. Here the Nusselt number is defined as Nu =
1 H ∂T H ∫0 k ∂x dy
kΔT = H
=
∫H0 ∂T dy ∂x ΔT
:
ð7Þ
It is seen in Table 1 that refining the mesh from 16,000 cells to 50,200 cells has a negligible effect on the calculated Nusselt number for both Rayleigh numbers. Therefore, the coarse mesh is adopted for all the
Table 1 Calculated steady-state Nusselt numbers with different meshes (Pr = 6.62). Mesh
Ra = 2.22 × 108
Ra = 1.11 × 109
Mesh 1 (16,000 cells) Mesh 2 (50,200 cells) Variation
93.72 93.74 0.02%
142.19 142.23 0.03%
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calculations with the base model in this study. For the extended models (refer to the next section), the mesh is constructed in accordance with the mesh required for the base model. Since the adiabatic extensions do not provide additional driving force to the natural convection flow, no separate mesh dependence tests are conducted for the extended models. For unsteady calculations, the time step is usually determined based on a time-step dependence test. A meaningful time-step dependence test for the present study would involve calculations of the flow from the initial start-up through to a final steady state with different time steps. However, these calculations are extremely expensive in terms of the required computing resources. In this study, the time step is selected with reference to the time steps adopted in [24,25], which determined the time step based on tests of the start-up flow only. Since the Rayleigh numbers considered in this study are lower than those considered in [24,25], it is expected that an equivalent or smaller time step compared with those adopted in the above literature would provide sufficient accuracy for the unsteady calculations. 3. Numerical results Two sets of numerical simulations have been carried out in this study. First, steady-state calculations are conducted with different corner shapes including sharp, round and straight corners, all of which are assumed adiabatic. The numerical results are compared with the base model. In the second set of numerical simulations, both steady and transient calculations are conducted with adiabatic extensions of various dimensions. Details of the numerical models and the numerical results are presented below.
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Here we introduce a new dimensionless parameter, the adiabatic extension ratio Ae, to quantify the geometric change. The adiabatic extension ratio is defined as: Ae =
Le : H
ð8Þ
For the comparison of the effect of different corner shapes, the adiabatic extension ratio is fixed at Ae = 0.125. Steady-state calculations are conducted for three moderate Rayleigh numbers of Ra = 2.27 × 107, 1.13× 108 and 5.67× 108 respectively with a fixed Prandtl number of Pr= 6.62. The numerical results are presented in Fig. 2. Fig. 2(a) shows the calculated steady-state Nusselt numbers on the heated sidewall with different corner geometries. It is clear that, with all the geometric configurations including the base model, the calculated Nusselt number increases with the Rayleigh number. It is also noticeable that there is a slight increase in the steady-state Nusselt number with the modified corner geometries (i.e. with sharp, round or straight adiabatic corners) compared with the base model, suggesting that the steady-sate heat transfer through the hot sidewall is enhanced by adding the adiabatic corners. The extent of heat transfer enhancement due to the modifications to the corner geometry is shown in Fig. 2(b) for different Rayleigh numbers. Clearly, for all the three shapes of the corner geometry, the enhancement of heat transfer increases with decreasing Rayleigh number, and a better enhancement is achieved with either sharp or
3.1. Effect of corner shapes In order to examine the effect of corner geometry on heat transfer through the differentially heated cavity, the base model of the square cavity is extended equally in all directions (left, right, top and bottom) by a length of Le, and corners of different shapes including sharp (1), round (2) and straight (3) corners respectively are added (see Fig. 1). The shaded region enclosed by dotted lines in Fig. 1 represents the original base model. All the added corners are assumed adiabatic, and thus, the lengths over which heat is transferred to and from the cavity remain unchanged, resulting in the same Rayleigh number for different corner shapes.
Fig. 1. Extended square cavity with different corner shapes.
Fig. 2. Steady-state results for different corner geometries and different Rayleigh numbers. (a) Calculated Nusselt numbers with different corner shapes for different Rayleigh numbers. (b) Enhancement of heat transfer for different Rayleigh numbers obtained with different corner shapes.
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Fig. 3. Differentially heated cavity with adiabatic extensions.
round corners. For the lowest Rayleigh number considered here (i.e. Ra = 2.27 × 107), the maximum enhancement is 3.5% with the sharp and round corners, and the enhancement reduces to 2.8% with the straight corners. Both sharp and round adiabatic corners have similar effect on the calculated Nusselt number on the hot wall, and they result in a better enhancement of heat transfer compared with the straight corners. Hence, in the following section, the effect of the adiabatic extension ratio on heat transfer is examined based on the sharp corners only due to their relative simplicity for mesh generation.
Nusselt number. The maximum enhancement of the steady-state heat transfer is about 1.9% for the current Rayleigh number. Next, the effect of the adiabatic extension on transient heat transfer is investigated for a fixed adiabatic extension ratio of Ae = 0.1 and for the same Rayleigh and Prandtl numbers as the steady-state calculations. The parameters are selected based on a physical model with a height of H = 0.24 m. The reference temperature is set at T0 = 295 K, and the temperature difference is fixed at ΔT = 1 K, resulting in a Rayleigh number of Ra = 2.22 × 108. A dimensional time step of 0.1 s is adopted for the unsteady calculations. The model is calculated with and without the adiabatic extensions for comparison purposes. The numerical results are shown in Fig. 5. Plotted in Fig. 5(a) are the time histories of the calculated heat transfer rate through the hot sidewall in a linear time scale. It is seen in Fig. 5(a) that, for both cases with and without the adiabatic extensions, the calculated heat transfer rate undergoes a transition and gradually approaches a constant at the steady state. In order to clearly demonstrate the heat transfer features at the early start-up and transitional stages, the calculated heat transfer rate through the hot sidewall is replotted in a logarithmic time scale in Fig. 5(b).
3.2. Effect of the adiabatic extension ratio In this section, the effect of the adiabatic extensions of various dimensions on heat transfer through the sidewalls is investigated. For this purpose, the sidewalls are extended vertically in two opposing directions while the width of the differentially heated cavity remains unchanged (see Fig. 3). Similarly, the extended sections of the sidewalls are assumed adiabatic, and thus the Rayleigh number remains the same for this investigation. Both steady and transient features of heat transfer through the sidewalls are considered. First, a series of steady-state calculations are carried out for adiabatic extension ratios ranging from Ae = 0.01 to Ae = 0.3. The Rayleigh number is fixed at Ra = 2.22 × 108 and the Prandtl number is fixed at Pr= 6.62. The calculated Nusselt numbers on the hot wall for different adiabatic extension ratios are presented in Fig. 4. It is clear in Fig. 4 that, as the adiabatic extension ratio increases from zero, the calculated Nusselt number initially increases, indicating that heat transfer through the sidewall is enhanced with the addition of the adiabatic extension. However, as the adiabatic extension ratio is increased beyond 0.1, there is no further increase of the calculated
Fig. 4. Calculated steady-state Nusselt number on the hot wall for various adiabatic extension ratios.
Fig. 5. Effect of the adiabatic extension on transient heat transfer through the hot sidewall. (a) and (b) Time history of the heat rate on the hot sidewall plotted in (a) a linear time scale; (b) a logarithmic time scale. (c) Time history of the enhancement of heat transfer.
C. Lei, A. O'Neill / International Communications in Heat and Mass Transfer 37 (2010) 1221–1225
Clearly, the major features of the flow in the early and transitional stages are the same with and without the adiabatic extensions. In the early stage, the calculated heat transfer rate with the addition of the adiabatic extensions surpasses that obtained with the base model. The comparison of the heat transfer rate is reversed after about 200 s, which approximately corresponds to the time when the intrusion flows arrive at the opposing sidewalls. This remains to be the case until around 1900 s when the heat transfer rate calculated with the adiabatic extensions again surpasses that without the extensions. It is noticed that the second switch of the comparison between these two models occurs after the perturbations induced by the arrival of the intrusion flow from the cold wall die out. Fig. 5(c) shows quantitative information of the enhancement of transient heat transfer due to the addition of the adiabatic extensions. It confirms that heat transfer through the hot wall is enhanced from the start-up until the arrival of the intrusion from the cold wall (around 200 s). A maximum heat transfer enhancement of 4.75% is observed at 137 s after the start-up. Associated with the flow oscillations induced by the arriving intrusion (during the period from approximately 200 to 1900 s), heat transfer through the hot wall is depressed due to the presence of the adiabatic extensions. After the flow oscillations die out, heat transfer is again enhanced with the adiabatic extensions. This remains to be the case for the rest of the flow transition until the final steady state, at which heat transfer is enhanced by about 2.1%. Note that the enhancement of the steady-state heat transfer calculated using the unsteady model (2.1%) is slightly higher than that predicted by the steady model (1.9%). This may be attributed to two factors. Firstly, the final state of the unsteady model (i.e. at 8000 s) is not at a truly steady state since the transition to the steady state is an extremely slow process. To cover the complete transition to the final steady state would require significant computing resources, and is not feasible. Therefore, the present calculation is terminated at 8000 s. Secondly, for unsteady calculations, an additional numerical error associated with the finite time step is inevitable. It can be estimated from Fig. 5(c) that the overall effect of adding the adiabatic extensions is a net 1% enhancement of heat transfer through the hot wall for the period covered by the calculation. It is also expected that the net effect of the adiabatic extensions on transient heat transfer depends on both the Rayleigh number and the adiabatic extension ratio. 4. Summary It is not uncommon to speculate that heat transfer through a differentially heated cavity is controlled by the extent of the sidewalls over which heat is transferred to and from the cavity. The heating and cooling lengths of the sidewalls also determine the Rayleigh number of this problem, which, along with the Prandtl number and aspect ratio, characterizes natural convection in the cavity. However, the present study has demonstrated that heat transfer through the cavity can be enhanced by modifying the geometry while keeping the heating and cooling lengths of the sidewalls unchanged. It is revealed in this study that adding adiabatic extensions to the differentially heated cavity enhances steady-state heat transfer to a certain extent (refer to Fig. 4). Further extending the domain provides no additional benefit in terms of heat transfer. The enhancement of heat transfer due to the adiabatic extensions depends strongly on the Rayleigh number (refer to Fig. 2b). The lower the Rayleigh number, the better enhancement is achievable. It is also found in this study that the geometry of the corners on an extended domain has a minor effect on heat transfer (Fig. 2). However, a slightly reduced enhancement of heat transfer is obtained with the straight corners compared to the sharp and round corners. The numerical results based on an unsteady model further demonstrate that better enhancement of heat transfer can be achieved
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at the early stage of the flow development following the start-up (refer to Fig. 5c). However, heat transfer is depressed during the early transitional stage following the arrival of the intrusion from the opposing sidewall. During the late transition toward the final steady state, the addition of the adiabatic extensions again enhances heat transfer. It is worth investigating the enhancement of transient heat transfer with different adiabatic extension ratios and for different Rayleigh numbers in future. Acknowledgement The first author is grateful for the financial support of the Australian Research Council. References [1] T. Fujii, M. Fujii, M. Takeuchi, Influence of various surface roughness on the natural convection, Int. J. Heat Mass Transfer 16 (1973) 629–640. [2] H.S. Kwak, K. Kuwahara, J.M. Hyun, Resonant enhancement of natural convection heat transfer in a square enclosure, Int. J. Heat Mass Transfer 41 (1998) 2837–2846. [3] H.Y. Kim, Y.G. Kim, B.H. Kang, Enhancement of natural convection and pool boiling heat transfer via ultrasonic vibration, Int. J. Heat Mass Transfer 47 (2004) 2831–2840. [4] E. Bilgen, Natural convection in cavities with a thin fin on the hot wall, Int. J. Heat Mass Transfer 48 (2005) 3493–3505. [5] A.E. Gill, The boundary-layer regime for convection in a rectangular cavity, J. Fluid Mech. 26 (1966) 515–536. [6] J.C. Patterson, J. Imberger, Unsteady natural convection in a rectangular cavity, J. Fluid Mech. 100 (1980) 65–86. [7] B.V.K.S. Sai, K.N. Seetharamu, P.A.A. Narayana, Solution of transient laminar natural convection in a square cavity by an explicit finite element scheme, Numer. Heat Transfer, Part A Appl. 25 (1994) 593–609. [8] E.V. Kalabin, M.V. Kanashina, P.T. Zubkov, Heat transfer from the cold wall of a square cavity to the hot one by oscillatory natural convection, Numer. Heat Transfer Part A Appl. 47 (2005) 609–619. [9] G.K. Bachelor, Heat transfer by free convection across a closed cavity between vertical boundaries at different temperatures, Q. Appl. Math. 12 (1954) 209–233. [10] P. Vainshtein, M. Fichman, C. Gutfinger, Acoustic enhancement of heat-transfer between 2 parallel plates, Int. J. Heat Mass Transfer 38 (1995) 1893–1899. [11] E.V. Kalabin, M.V. Kanashina, P.T. Zubkov, Natural-convective heat transfer in a square cavity with time-varying side-wall temperature, Numer. Heat Transfer, Part A Appl. 47 (2005) 621–631. [12] R. El Ayachi, A. Raji, M. Hasnaoui, A. Bahlaoui, Combined effect of radiation and natural convection in a square cavity differentially heated with a periodic temperature, Numer. Heat Transfer, Part A Appl. 53 (2008) 1339–1356. [13] J.R. Welling, C.N. Wooldridge, Free convection heat transfer coefficients from vertical fins, J. Heat Transfer 87 (1965) 439–444. [14] C.W. Leung, S.D. Probert, M.J. Shilston, Heat exchanger: optimal separation for vertical rectangular fins protruding from a vertical rectangular base, Appl. Energy 19 (1985) 77–85. [15] B. Yazicioglu, H. Yuncu, Optimum fin spacing of rectangular fins on a vertical base in free convection heat transfer, Heat Mass Transf. 44 (2007) 11–21. [16] A. Bejan, Natural-convection heat-transfer in a porous layer with internal flow obstructions, Int. J. Heat Mass Transfer 26 (1983) 815–822. [17] A. Nag, A. Sarkar, V.M.K. Sastri, Natural-convection in a differentially heated square cavity with a horizontal partition plate on the hot-wall, Compu. Meth. Appl. Mech. Eng. 110 (1993) 143–156. [18] X. Shi, J.M. Khodadadi, Laminar natural convection heat transfer in a differentially heated square cavity due to a thin fin on the hot wall, J. Heat Transfer 125 (2003) 624–634. [19] F. Xu, J.C. Patterson, C. Lei, Experimental observations of the thermal flow around a square obstruction on a vertical wall in a side-heated cavity, Exp. Fluids 40 (2006) 364–371. [20] F. Xu, J.C. Patterson, C. Lei, An experimental study of the unsteady thermal flow around a thin fin on a sidewall of a differentially heated cavity, Int. J. Heat Fluid Flow 29 (2008) 1139–1153. [21] F. Xu, J.C. Patterson, C. Lei, Transition to a periodic flow induced by a thin fin on the sidewall of a differentially heated cavity, Int. J. Heat Mass Transfer 52 (2009) 620–628. [22] V.A.F. Costa, M.S.A. Oliveira, A.C.M. Sousa, Control of laminar natural convection in differentially heated square enclosures using solid inserts at the corners, Int. J. Heat Mass Transfer 46 (2003) 3529–3537. [23] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, New York, 1980. [24] J.C. Patterson, S.W. Armfield, Transient features of natural convection in a cavity, J. Fluid Mech. 219 (1990) 469–497. [25] F. Xu, J.C. Patterson, C. Lei, Oscillations of the horizontal intrusion in a side-heated cavity, in: M. Behnia, W. Lin, G.D. McBain (Eds.), Proceedings of the Fifteenth Australasian Fluid Mechanics Conference (CD-ROM), The University of Sydney, 2004, Paper AFMC00243.