NUMERICAL STUDIES OF TRANSITION FROM STEADY TO UNSTEADY COUPLED THERMAL BOUNDARY LAYERS

International Journal of Computational Methods Vol. 11, Suppl. 1 (214) 13442 (15 pages) c World Scientific Publishing Company DOI: 1.1142/S2198762134427 NUMERICAL STUDIES OF TRANSITION FROM STEADY TO UNSTEADY COUPLED THERMAL BOUNDARY LAYERS F. XU,,Q.YANG and P. YU School of Civil Engineering Beijing Jiaotong University Beijing 144, China Data Storage Institute, Agency for Science Technology and Research (A*STAR) DSI Building, 5, Engineering Drive 1, 11768 Singapore fxu@bjtu.edu.cn Received 15 May 213 Revised 26 August 213 Accepted 19 October 213 Published 21 April 214 The transition from the steady to the unsteady coupled thermal boundary layers around a partition in a differentially heated partitioned cavity is investigated numerically. The partitioned cavity is filled with water, which is imposed a temperature difference between the two sidewalls of the cavity. The numerical results are consistent with the corresponding experiments. The development of the unsteady coupled thermal boundary layers following a sudden temperature difference between the fluids at the two sides of the partition at a high Rayleigh number of Ra = 1 11 is described. The transition from the steady to the unsteady coupled thermal boundary layers over a wide range of Rayleigh numbers from Ra = 1 9 to 1 11 is observed. The dependence of traveling waves in the unsteady coupled thermal boundary layers on the Rayleigh number and heat transfer through the partitioned cavity are characterized and quantified. Keywords: Unsteady coupled thermal boundary layers; natural convection; differentially heated partitioned cavity. 1. Introduction Fluid motions may appear on both sides of a vertical wall if there is a temperature difference between the fluids at the two sides of the wall due to heat transfer through the wall. This phenomenon is relevant to many situations in nature and engineering. In particular, such coupled thermal boundary layers are extensively present on both sides of the vertical wall of heat exchangers in which e.g., gas at one side of the vertical wall is liquefied by liquid nitrogen of a low temperature at the other side. Corresponding author. 13442-1

F. Xu, Q. Yang & P. Yu Accordingly, a differentially heated partitioned cavity has been paid considerable attention. Natural convection in a differentially heated non-partitioned cavity was earlier presented by Batchelor [1954]. Later on, transient natural convection in the cavity was investigated by Patterson and Imberger [198] in which natural convection in the cavity is quantified through a scaling analysis. Studies [e.g., Xu et al. (28)] further showed that transient natural convection following sudden heating and cooling may be classified into three stages, including an initial stage, a transitional stage and a steady or quasi-steady stage during which the leading edge effect (LEE) may appear in the initial stage and the interior fluid becomes stratified in the fully developed stage. For the Rayleigh number of 1 3, natural convection in the nonpartitioned cavity is weak and heat transfer is dominated by conduction [Patterson and Imberger (198)]. As the Rayleigh number increases, natural convection in the cavity becomes stronger and heat transfer is dominated by convection [Gill (1966)]. As the Rayleigh number increases further, fully developed flows in the cavity could be unstable [Paolucci and Chenoweth (1989); Janssen and Armfield (1996); Armfield and Janssen (1996)]. The study by Le Quere and Behnia (1998) has demonstrated that if the Rayleigh number is larger than the critical value of 1.84 1 8 for the Prandtl number of.71, fully developed flows in a square cavity are unsteady. If the Rayleigh number is sufficiently large, fully developed flows in the cavity could be even turbulent [Paolucci (199)]. Recently, increasing studies have also focused on natural convection in a differentially heated partitioned cavity due to the extensive application in industry. The study of the transient natural convection in the partitioned cavity was performed by Xu et al. [29]. The development of the coupled thermal boundary layers around the partition following a sudden temperature difference between the fluids at the two sides of the partition at a low Rayleigh number of 1.84 1 9 was observed. Following the sudden temperature difference, intrusion flows discharged from the downstream end of the coupled thermal boundary layers appear and oscillate between the partition and sidewalls, which finally results in fully developed flows and the stratified environment in the partitioned cavity. The study by Xu et al. [29] further showed that the temperature condition on the partition changes from initially isothermal to approximately isoflux at the fully developed state. Previous studies showed that the partition, even though perfectly conducting, depresses natural convection in the partitioned cavity, and thus heat transfer through the cavity is reduced [e.g., Nishimura et al. (1988); Cuckovic-Dzodzo et al. (1999)]. The relationship between heat transfer and the Rayleigh number may be quantified by the scale of Nu Ra 1/4 [Anderson and Bejan (1981)]. The above-mentioned studies [e.g., Xu et al. (29)] showed that fully developed flows in the square partitioned cavity are steady for the Rayleigh number of 1.84 1 9 and the Prandtl number of 6.63. However, as the Rayleigh number increases, e.g., larger than 1.4 1 1 for the Prandtl number of 7.5, fully developed flows in the 13442-2

Numerical Studies of Transition from Steady to Unsteady partitioned cavity are unsteady and the critical Rayleigh number of the transition from steady to unsteady flows in the partitioned cavity depends on the aspect ratio [Williamson et al. (212)]. Williamson et al. [212] also indicated that unsteady flows at a fully developed state are absolutely unstable for which perturbations in the coupled thermal boundary layers may feed each other through the partition. Indeed, as the Rayleigh number increases further for small Prandtl numbers (e.g., Pr =.71), fully developed flows in the partitioned cavity are even turbulent [refer to Hanjalic et al. (1996)]. Although the transition to the unsteady flows in the square partitioned cavity was observed by Williamson et al. [212] based on few cases around the critical values (e.g., Ra = 1.4 1 1 and 1.6 1 1 ), the features of the transition from the steady to the unsteady flows in the partitioned cavity over a wide range of Rayleigh numbers are still unclear. Accordingly, in this study, the transition from the steady to the unsteady coupled thermal boundary layers in the partitioned cavity is numerically investigated over a wide of Rayleigh numbers from Ra = 1 9 to 1 11. The flow structures and traveling waves in the unsteady coupled thermal boundary layers and heat transfer are observed and discussed. The numerical results are also validated in comparison with the recent experiments in Xu [212]. 2. Numerical Procedures In this paper, a differentially heated cavity with a partition vertically placed at the middle is considered, refer to Xu et al. [29]. Figure 1 shows a schematic of the partitioned cavity with the isothermal sidewalls, the horizontal insulated walls and the conducting partition. The development of natural convection flows in the partitioned cavity is governed by the following 2D dimensionless Navier Stokes and (-.5,.5) T / y = (.5,.5) P3 Thermal boundary layers T c P2 T h Partition P1 P4 (-.5, -.5) T / y = (.5, -.5) Fig. 1. Schematic of computational domain and boundary conditions. P1 (.4,.4), P2 (.4, ), P3 (.4,.4) and P4 (.4,.4) are recording points used in the sequent figures. 13442-3

F. Xu, Q. Yang & P. Yu energy equations with the Boussinesq approximation [Xu et al. (29)]: u x v =, (1) y u t u u x v u y = p x Pr ( 2 ) u Ra 1/2 x 2 2 u y 2, (2) v t u v x v v y = p y Pr ( 2 ) v x 2 2 v y 2 PrT, (3) T t u T x v T y = 1 Ra 1/2 Ra 1/2 ( 2 T x 2 2 T y 2 ), (4) where x and y are the horizontal and vertical coordinates with origin at the center of the cavity, t is the time, T is the temperature, T h and T c are the temperatures of the hot and cold sidewall (see Fig. 1), p is the pressure, u and v are the velocity components in the x and y directions. The quantities in Eqs. (1) (4) are nondimensional. The Rayleigh number (Ra), the Prandtl number (Pr) and the aspect ratio (A) are defined as: Ra = gβ TH 3, (5) νκ Pr = ν κ, (6) A = H L, (7) where, g is the acceleration due to gravity, T is the temperature difference between the two sidewalls, H is the height of the cavity, κ is the thermal diffusivity, ν is the kinematic viscosity and β is the coefficient of thermal expansion. The quantities in the equations are normalized by the following scales: the length scale H, thetime scale H 2 /κ/ra 1/2, the temperature scale T,thevelocityscaleκRa 1/2 /H [also see Xu et al. (29)]. Note that the dimensionless is adopted for all quantities in this paper. As seen in Fig. 1, the top and bottom walls of the cavity are adiabatic; the two sidewalls are isothermal; the partition of a zero thickness is diathermal for which only horizontal heat transfer is considered; and all interior walls and the partition are rigid and no-slip. The working fluid is initially quiescent. At t =, the temperature of the fluid on the left side of the partition is.5, and that on the right side of the partition is.5. The SIMPLE algorithm was used for the pressure velocity coupling. All second derivatives and linear first derivatives in Eqs. (2) (4) were approximated by a second-order center-differenced scheme. The advection terms were discretized by using the QUICK scheme. The time integration is by a second-order backward difference scheme. The discretized equations are iterated with under-relaxation factors [also see Xu et al. (29); Saha et al. (212a, 212b); Saha and Gu (212)]. 13442-4

Numerical Studies of Transition from Steady to Unsteady T.5 198 23 398 43 798 83 Initial LEE Tranisional Fully developed Power spectrum 8 4-4 198 23 398 43 798 83 peak frequecy by the coasest mesh -.5 1-1 1 1 1 1 2 t -8 5 1 15 f (b) Fig. 2. Temperature time series and spectra at P3 (sketched in Fig. 1) in the coupled thermal boundary layers. Temperatures calculated using different grid systems. (b) Power spectra of the temperatures in the fully developed stage. A grid system was constructed with coarse grids in the core and fine grids concentrated in the proximity of all wall boundaries with the grid expanding at a constant rate from the wall to the interior edge. A grid dependence test was conducted on the three grid systems (23 198, 43 398 and 83 798). In order to evaluate the effect of the three grid systems on the numerical solution, Fig. 2 plots temperature time series and spectra at P3 (see Fig. 1) in the coupled thermal boundary layers calculated using the three grid systems for Ra = 1 11 (the highest Rayleigh number considered in this paper). Note that a logarithmic time scale is adopted here in order to show the early behaviors following a sudden temperature difference between the fluids at the two side of the partition. It may be seen from this figure that the temperatures calculated using the three grid systems are all able to characterize the main flow features in the three stages (an initial stage, a transitional stage and a fully developed stage). Clearly, the LEE is the remarkable phenomenon of the transient flows in the initial stage; subsequently, the amplitude and frequency of traveling waves in the coupled thermal boundary layers change with time in the transitional stage; and finally, the amplitude and frequency of traveling waves approach constant values in the fully developed stage. To observe further the effect of the grid on the coupled thermal boundary layers, Fig. 2(b) presents the spectra of the temperatures calculated using the three grid system. It is seen from this figure that the spectra of the temperatures calculated using the two finer grid systems are consistent well but different from that calculated using the coarsest grid system. Particularly, the peak frequency of traveling waves calculated using the coarsest grid system (f = 1.53) is considerably different from those calculated using the two finer grid systems (around f =1.1). This means that the numerical solution is not sensitive to the two finer grid systems but to the coarsest grid system, suggesting that either of the two grid systems may be used to characterize the features of the flow development and traveling waves. 13442-5

F. Xu, Q. Yang & P. Yu In consideration of the computing time, the grid system of 43(H) 398(L) was adopted in this study. Additionally, a time-step-dependence test was also conducted with twodimensionless time-steps of.2 and.1, and it is demonstrated that the calculated results are not sensitive to the two time-steps. Therefore, the time-step of.2 is considered to be sufficiently small to capture the major flow features and was adopted in this study. 3. Results The experiments described in Xu [212], whose experimental model is sketched in Fig. 1, are considered. The experimental cavity of.5 m wide (L) and.5 m high (H) is filled with water. Accordingly, the above-mentioned numerical procedures were performed at the specified aspect ratio (A = 1) and different Prandtl numbers and different Rayleigh numbers for comparisons with the corresponding experiments in Xu [212]. The parameters of numerical simulations are listed in Table 1. Note that the computational domain and boundary conditions are shown in Fig. 1. 3.1. Natural convection flows in a partitioned cavity An important feature of natural convection flows in the partitioned cavity is traveling waves around the partition at high Rayleigh numbers. Figure 3 presents a set of numerical results and shadowgraph images about the unsteady coupled thermal boundary layers in the fully developed stage at Ra = 2.6 1 1. Note that the shadowgraph images of the coupled thermal boundary layers here were cut from the original shadowgraph images in Xu [212] and the two neighboring shadowgraph images have the same time interval of.18 as that of numerical results. It is seen from Fig. 3 that there are clear traveling waves in the downstream of the thermal boundary layer at the left of the partition. Corresponding to the numerical solution in Fig. 3, the shadowgraph image at the same Rayleigh number is shown in Fig. 3(b). Clearly, the bright edge of the thermal boundary layer at the left of the partition is wavy, and these traveling waves in the experiment are corresponding to Table 1. Parameters for numerical simulations. Number Ra Pr Frequency (f) at P2 1 1 9 7.8 2 2.6 1 9 7.8 3 7.9 1 9 7.8 4 1 1 7.8 5 1.3 1 1 7.8 6 1.8 1 1 7.8 1.53 7 2.6 1 1 7.8 1.82 8 4.1 1 1 7.4 1.8 9 1 11 6. 1.82 13442-6

Numerical Studies of Transition from Steady to Unsteady.5 -.2 -.1.4.3.2.1 (b) (c) (d) (e) (f) (g) (h) (i) (j) Fig. 3. Numerical isotherms with an interval of.25 and shadowgraph images in the upper section of the cavity in the fully developed stage at Ra = 2.6 1 1., (c), (e), (g) and (i) Isotherms with a time interval of.18. (b), (d), (f), (h) and (j) Shadowgraph images with a time interval of.18. those from numerical results in Fig. 3. Waves travel downstream with the passage of time, as seen in Figs. 3(d), 3(f), 3(h) and 3(j) in which one traveling wave is marked by a short black line and the corresponding numerical results are shown in Figs. 3(c), 3(e), 3(g) and 3(i). Traveling waves in the numerical simulation are consistent with those in the experiment. The further numerical results show that the flow structures of natural convection in the partitioned cavity are dependent on the Rayleigh number (in particular traveling waves in the unsteady coupled thermal boundary layers around the partition). Figure 4 presents a set of isotherms at different Rayleigh numbers in order to further observe the dependence of natural convection flows on the Rayleigh number. Figure 4 shows that the flow is steady for long time (that is, the fluid in the coupled thermal boundary layers around the partition smoothly flows from upstream to downstream) if the Rayleigh number is lower than a critical value, which is around Ra = 1.4 1 1 [Williamson et al. (212)]. However, with the increase of the Rayleigh number, the flow becomes unsteady and traveling waves occur in the unsteady coupled thermal boundary layers around the partition, as shown in Fig. 4(b), although the oscillations of the unsteady flow are weak at Ra = 1.8 1 1. As the Rayleigh number increases further, the unsteady flow around the partition becomes clearer and traveling waves become larger (Figs. 4(c) and 4(d)). Additionally, it may be seen from Fig. 4 that the feature of traveling waves such as the peak frequency changes with the Rayleigh number, which will be confirmed in the subsequent figures. 13442-7

F. Xu, Q. Yang & P. Yu Traveling waves Traveling waves -.1 -.5.5.1 -.1 -.5.5.1 Traveling waves Traveling waves -.1 -.5.5.1 (c) Traveling waves (b) Traveling waves -.1 -.5.5.1 Fig. 4. Isotherms with an interval of.25 in the fully developed stage (t = 75) at different Rayleigh numbers. Ra = 1 1.(b)Ra=1.8 1 1.(c)Ra=4.1 1 1.(d)Ra=1 11. (d) For the purpose of quantifying natural convection flows in the partitioned cavity, the volumetric flow rate of the horizontal convection through the cavity is defined below [see Lei and Patterson (22)], 1/2 Q = 1 u dy, (8) 2 1/2 where the volumetric flow rate is calculated across the vertical line at x =.25 at different Rayleigh numbers and plotted in Fig. 5. The oscillations of the volumetric flow rates in the early stage are the result of internal gravity waves (have a different peak frequency from traveling waves around the partition), as indicated in Xu et al. [29] and Williamson et al. [212]. As time increases, the oscillations 13442-8

Numerical Studies of Transition from Steady to Unsteady.12 Q.8.4 Ra=1 9 Ra=1 1 Ra=1 11 2 x 1-3 7 75 1-1 1 1 1 1 2 t 1 11 QRa 5/4 1 1 1 9 1 8 1 8 1 9 1 1 1 11 1 12 Ra (b) Fig. 5. Dependence of the volumetric flow rate on time and Rayleigh number. Q versus t. (b) QRa 5/4 versus Ra in the fully developed stage (the volumetric flow rate in the fully developed stage is averaged on t = 6 75). of the volumetric flow rates become small and approach different values, depending on the Rayleigh number. Further, the flow rate may even approach an oscillatory value at a high Rayleigh number of e.g., Ra = 1 11. A further examination indicates that the frequency of the oscillations in the fully developed stage is similar to that in the earlier stage. This means that internal gravity waves are still present in the partitioned cavity in the fully developed stage if the Rayleigh number is sufficiently large. In order to observe the effect of the Rayleigh number, QRa 5/4 is plotted in Fig. 5(b). The study by Patterson and Imberger [198] showed that the flow rate in the cavity is proportional to κra 1/4 and thus the nondimensional flow rate Q Ra 1/4 here; that is, QRa 5/4 Ra. The good linear relation at low Rayleigh numbers (e.g., lower than 2 1 1 )demonstratesthatthenumerical results are consistent with the prediction by Patterson and Imberger [198]. However, the flow rate deviates from the prediction at high Rayleigh numbers. This means that the unsteady flow adjacent to the partition increases the horizontal flow rate across the cavity in comparison with the prediction by Patterson and Imberger [198] based on a steady state. 13442-9

F. Xu, Q. Yang & P. Yu 3.2. Temperatures in coupled thermal boundary layers For the purpose of illustrating the overall development of the temperatures in the coupled thermal boundary layers from an initial to a fully developed stage, Fig. 6 presents time series of the calculated temperatures at different points (see Fig. 1) for Ra = 1 11. The development still includes the three stages at different points.5 198 23 398 43 798 83 P4 T P3 T Power spectra -.5 1-1 1 1 1 1 2 t.2 198 23 P4 398 43 798 83 P3 -.5 73 735 74 745 75 t 4 P4 P31.1 1.82-4 P2 (b) P2 P1 P1 P2 P1-8 5 1 15 f (c) Fig. 6. Temperature time series and spectra at different points (positions of P1, P2, P3 and P4 shown in Fig. 1) for Ra = 1 11.T versus t. (b)t versus t from t = 73 75. (c) Spectra of the temperatures in the fully developed stage. 13442-1

Numerical Studies of Transition from Steady to Unsteady [also see Fig. 2 and Xu et al. (29)], and the LEE is the important phenomenon in the initial stage although it has the variance in quantity between different points. Additionally, due to the interior stratification in the cavity, the temperatures finally approach different values at different points with the varying amplitude of the oscillations. That is, traveling waves grow up when they travel downstream in the coupled thermal boundary layers, as seen in Fig. 4. In order to observe the feature of traveling waves, Fig. 6(b) plots the temperatures time series in the fully developed stage. Clearly, when waves travel downstream, their amplitudes increase. It is worth noting that the temperature time series at P3 and P4 show the symmetrical development because the two points are symmetrical about the cavity center (see Fig. 1). To obtain further insights into traveling waves, a spectral analysis was performed. Figure 6(c) shows the spectra of the temperatures at different points. Since the points P3 and P4 are symmetric about the cavity center, the spectra of the temperatures at the two points are similar (also see Fig. 6). However, the spectral distributions of the temperatures at P1, P2 and P3 vary. This also confirms that traveling waves change with a varying peak frequency when traveling downstream. The above-mentioned spectral analysis was repeated for the temperature time series at different points and different Rayleigh numbers, and the corresponding peak frequencies were obtained. Figure 7 shows the dependence of the peak frequency on the Rayleigh number. Note that the peak frequency in Fig. 7 is normalized by the frequency of convective instability of the thermal boundary layer in Gebhart and Mahajan [1975]. In this study, the normalized peak frequency is.155, which is also consistent with those in Williamson et al. [212]. This means that traveling waves around the partition are instability of the thermal boundary layer, different from the internal gravity waves with lower frequencies shown in Fig. 5. Additionally, it may be seen in this figure that the calculated frequencies are slightly larger than those from the experiments in Xu [212]. This is because the experimental Rayleigh numbers are usually smaller than those calculated based 1-1 f/ra 1/6 /Pr 1/3 1-2 1-3 P2 Experiments in Xu [212] Simulations in Williamson et al. [212] 1 1 1 11 Ra Fig. 7. Peak frequency versus Rayleigh number. 13442-11

F. Xu, Q. Yang & P. Yu on the temperature difference between the hot and cold waters in the water bath [the discussion may refer to Xu et al. (28)]. 3.3. Heat transfer through a partitioned cavity For the purpose of understanding heat transfer through the partitioned cavity, Fig. 8 plots the profiles of the Nusselt numbers on the partition and sidewall in the fully developed stage at different Rayleigh numbers. It is seen from this figure that the Nusselt number on the partition at low Rayleigh numbers of e.g., Ra = 1 9 and 1 1 is uniform and approximately constant over the full height except for the two ends (note that the coupled thermal boundary layers are steady). That is, the heat transfer rate through the partition does not vary with the height except for in the proximity to the top and bottom walls at Ra = 1 9 and 1 1. This means that the partition is approximately isoflux in the fully developed stage, also refer to Xu et al. [29]. However, if the Rayleigh number is larger than the critical value, the coupled thermal boundary layers are unsteady and traveling waves appear. As a result, the profile of the Nusselt number on the partition and even on the downstream sidewall is oscillatory at Ra = 1 11, as seen in Fig. 8. In order to examine the dependence of heat transfer through the cavity on time, time series of the Nusselt numbers on the sidewall at different Rayleigh numbers are illustrated in Fig. 9. It is seen from this figure that the Nusselt number is initially zero since the temperature difference between the sidewall and fluid is zero. As time increases, the Nusselt number may approach different values in the fully developed stage, depending on the Rayleigh number. Figure 9(b) shows the effect of the Rayleigh number. The normalized Nusselt number of Nu/Ra 1/4 =.154 is slightly smaller than that (.155) in Williamson et al. [212] but larger than that.5 Partition y Ra = 1 11 Ra = 1 Ra = 1 9 1 Right sidewall Ra = 1 11 -.5 1 2 3 Fig. 8. Nusselt number on partition and sidewall at different Rayleigh numbers. Nu 13442-12

Numerical Studies of Transition from Steady to Unsteady 1 8 6 198 23 398 43 Ra=1 11 798 83 Ra=1 1 Nu 4 2 Ra=1 9 1 2 3 4 5 6 7 t Nu/Ra 1/4.16.154.15.14 1 8 1 9 1 1 1 11 1 12 Fig. 9. Dependence of Nusselt number on time and Rayleigh number. Nu versus t. (b)nu/ra 1/4 versus Ra in the fully developed stage. (.148) in Nishimura et al. [1988] due to the varying Prandtl number and aspect ratio between this study and others. 4. Conclusions In this paper, natural convection flows in a differentially heated partitioned cavity were numerically investigated in a wide range of Rayleigh numbers (from Ra = 1 9 to 1 11 ) and compared with the corresponding experiments in Xu [212]. The numerical results here are consistent with the flow visualizations from the experiments. Both numerical and experimental results have also demonstrate that natural convection flows in the partitioned cavity approach an unsteady state at high Rayleigh numbers. Numerical results further show that the development of the unsteady coupled thermal boundary layers from start-up to an unsteady state may be classified into the three stages: an initial stage, a transitional stage and a fully developed stage. The remarkable flow phenomena involve the leading edge effect in the initial stage, perturbations from the gravity waves between the partition and sidewall in the transitional stage, and traveling waves in the fully developed stage. Through a spectral Ra 13442-13

F. Xu, Q. Yang & P. Yu analysis of temperature time series, the peak frequency of traveling waves in the unsteady coupled thermal boundary layers in the fully developed stage is obtained. The dependence of the peak frequency on the Rayleigh number is quantified. Additionally, the horizontal convective flow rate in the partitioned cavity is gained and the dependence on the Rayleigh number is quantified. This study also presents the dependence of heat transfer through the partitioned cavity on time and Rayleigh number. It is demonstrated again [also see Xu et al. (29)] that heat transfer on the partition is approximately isoflux at low Rayleigh numbers for which the coupled thermal boundary layer flow is steady. Further, since the transition from a steady to an unsteady flow around the partition has been observed through the experiments [Xu (212)] and numerical simulations [Williamson et al. (212)], a stability analysis of the flow around the partition should be expected for further insights into the instability of this flow. Acknowledgments This study was supported by National Natural Science Foundation of China (Grant No. 1114215 and 1127245), the 111 project (Grant No. B132) and Fundamental Research Funds for the Central Universities (Grant No. 211JBM36). References Anderson, R. and Bejan, A. [1981] Heat transfer through single and double vertical walls in natural convection: Theory and experiment, Int. J. Heat Mass Transfer 24, 1611 162. Armfield, S. W. and Janssen, R. J. A. [1996] A direct boundary-layer stability analysis of steady-state cavity convection flow, Int. J. Heat Fluid Flow 17, 539 546. Batchelor, G. K. [1954] Heat transfer by free convection across a closed cavity between vertical boundaries at different temperatures, Quart. Appl. Math. 12, 29 233. Cuckovic-Dzodzo, D. M., Dzodzo, M. B. and Pavlovic, M. D. [1999] Laminar natural convection in a fully partitioned enclosure containing fluid with nonlinear thermophysical properties, Intl J. Heat Fluid Flow 2, 614 623. Gebhart, B. and Mahajan, R. [1975] Characteristic disturbance frequency in vertical natural convection flow, Int. J. Heat Mass Transfer 18, 1143 1148. Gill, A. E. [1966] The boundary-layer regime for convection in a rectangular cavity, J. Fluid Mech. 26, 515 536. Hanjalic, K., Kenjeres, S. and Durst, F. [1996] Natural convection in partitioned twodimensional enclosures at higher Rayleigh numbers, Int. J. Heat Mass Transfer 39, 147 1427. Janssen, R. J. A. and Armfield, S. [1996] Stability properties of the vertical boundary layers in differentially heated cavities, Int. J. Heat Fluid Flow 17, 547 556. Le Quere, P. and Behnia, M. [1998] From onset of unsteadiness to chaos in a differentially heated square cavity, J. Fluid Mech. 359, 81 17. Lei, C. and Patterson, J. C. [22] Unsteady natural convection in a triangular enclosure induced by absorption of radiation, J. Fluid Mech. 46, 181 29. Nishimura, T., Shiraishi, M., Nagasawa, F. and Kawamura, Y. [1988] Natural convection heat transfer in enclosures with multiple vertical partitions, Int. J. Heat Mass Transfer 31, 1679 1686. 13442-14

Numerical Studies of Transition from Steady to Unsteady Patterson, J. C. and Imberger, J. [198] Unsteady natural convection in a rectangular cavity, J. Fluid Mech. 1, 65 86. Paolucci, S. [199] Direct numerical simulation of two-dimensional turbulent natural convectioninanenclosedcavity, J. Fluid Mech. 215, 229 262. Paolucci, S. and Chenoweth, D. R. [1989] Transition to chaos in a differentially heated vertical cavity, J. Fluid Mech. 21, 379 41. Saha, S. and Gu, Y. T. [212] Free convection in a triangular enclosure with fluidsaturated porous medium and internal heat generation, The ANZIAM J. 53, C127 C141. Saha, S., Brown, R. and Gu, Y. T. [212a] Scaling for the Prandtl number of the natural convection boundary layer of an inclined flat plate under uniform surface heat flux, Int. J. Heat Mass Transfer 55(9 1), 2394 241. Saha, S., Gu, Y. T., Molla, M. M., Siddiqa, S. and Hossain, M. A. [212b] Natural convection from a vertical plate embedded in a stratified medium with uniform heat source, Desalin. Water Treat. 44(1 3), 7 14. Williamson, N., Armfield, S. W. and Kirkpatrick, M. P. [212] Transition to oscillatory flow in a differentially heated cavity with a conducting partition, J. Fluid Mech. 693, 93 114. Xu, F. [212] Travelling waves in the coupled thermal boundary layers around the partition in a differentially heated cavity, 7th Chinese Conference of Fluid Mechanics, Guilin. Xu, F., Patterson, J. C. and Lei, C. [28] On the double-layer structure of the boundary layer adjacent to a sidewall of a differentially-heated cavity, Int. J. Heat Mass Transfer 51, 383 3815. Xu, F., Patterson, J. C. and Lei, C. [29] Heat transfer through coupled thermal boundary layers induced by a suddenly generated temperature difference, Int. J. Heat Mass Transfer 52, 4966 4975. 13442-15