An experimental study of the unsteady thermal flow around a thin fin on a sidewall of a differentially heated cavity

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International Journal of Heat and Fluid Flow 29 (2008) 1139–1153 www.elsevier.com/locate/ijhff

An experimental study of the unsteady thermal flow around a thin fin on a sidewall of a differentially heated cavity Feng Xu *, John C. Patterson, Chengwang Lei School of Engineering, James Cook University, Townsville, QLD 4811, Australia Received 7 August 2007; received in revised form 17 December 2007; accepted 11 January 2008 Available online 4 March 2008

Abstract The unsteady thermal flow around a thin fin on a sidewall of a differentially heated cavity is visualized using a shadowgraph technique and measured using fast-response thermistors. Experiments show that the transition of the unsteady thermal flow around the fin from initiation by sudden heating to a quasi-steady state undergoes a number of stages including the formation of a horizontal gravity current under the fin and a starting plume bypassing the fin, entrainment into the downstream thermal boundary layer, and separation and oscillations of the thermal flow above the fin. We present a series of flow visualization images to describe the transition of the unsteady thermal flow, and obtain the unsteady velocity scales of the fronts of the lower intrusion and starting plume as functions of the time and Rayleigh number, which are verified by the results of the flow visualization experiments. In the transition to the quasi-steady state, separation and oscillations of the thermal flow above the fin are observed. It is demonstrated that these oscillations, which are sensitive to the geometry of the fin, trigger instability of the downstream thermal boundary layer flow and thus enhance convection. It is found that the frequency of the oscillations is a linear function of the Rayleigh number. Ó 2008 Elsevier Inc. All rights reserved. Keywords: Unsteady thermal flow; Fin; Oscillations

1. Introduction Natural convection in a differentially heated cavity is one of the classic heat and mass transfer problems and has significance in fundamental fluid mechanics and many industrial applications such as electronic cooling systems, solar collectors, nuclear reactors, and crystal growth processes. One of the earliest studies of natural convection in a cavity was reported by Bachelor (1954a) in the context of a double-pane glass window, and the steady-state flow has been the focus of many early studies (see, e.g. Eckert and Carlson, 1961; Elder, 1965; Gill, 1966). However, since most buoyancy-induced flows in nature and industrial applications are unsteady, increasing attention has been given to unsteady natural convection resulting from sudden heating. Patterson and Imberger (1980) first discussed the *

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0142-727X/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.ijheatfluidflow.2008.01.001

features of unsteady natural convection in a cavity following sudden heating and cooling in terms of a thermal boundary layer flow adjacent to the sidewall, a horizontal intrusion flow under the ceiling or above the bottom and the flow in the core. These stages of the flow development are relevant to the present paper. There have been extensive studies of the starting behaviour of the thermal boundary layer on a suddenly heated vertical semi-infinite plate, beginning with the analytical work of Siegel (1958). Briefly, the development of the boundary layer following sudden isothermal or isoflux heating is described as follows: at any fixed position downstream of the leading edge of the plate the flow is described initially by the one dimensional flow and temperature fields described by, for example, Schetz and Eichorn (1962) and Goldstein and Briggs (1964). At some later time the flow at that point becomes two-dimensional and steady, as described by Ostrach (1964). The transition occurs over a non-zero time and the point of transition

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Nomenclature A aspect ratio df penetration distance of the intrusion front (m) f frequency (Hz) g acceleration due to gravity (m/s2) H, L, W height, length and width of the cavity (m) h half-length of the sidewall (m) Pr Prandtl number Ra Rayleigh number in terms of H Rah Rayleigh number in terms of h Re Reynolds number T0 initial temperature of the fluid (K) Tc, Th temperatures of the cold and hot sidewalls (K)

travels downstream at a speed determined by the parameters of the problem. This transition, referred to as the ‘leading edge effect’ (LEE), is characterized by the presence of an oscillatory component, and as the passage of the LEE appears to trigger the conversion from an unsteady to a steady boundary layer, its speed has been of some interest. Various previous analytical, experimental and numerical studies (e.g. Goldstein and Briggs, 1964; Brown and Riley, 1973; Mahajan and Gebhart, 1978; Joshi and Gebhart, 1987; Schladow, 1990) investigated various aspects of this behaviour, but all found that simple models of the speed of the LEE based on the boundary layer velocity under predicted the speed significantly. Armfield and Patterson (1992) in an analytical and numerical study demonstrated that the speed of the LEE was better estimated by the maximum speed of travelling waves initiated by the perturbation at the leading edge on start up. This was supported by the experiments of Patterson et al. (2002). Much of this work was in the context of isothermal wall boundary conditions. Further investigations of the properties of the transient boundary layer have since focused on isoflux boundary conditions, variation in the Prandtl number, the longer timescale development, and the interaction with the interior flow (e.g. Lin and Armfield, 2005; Xu et al., 2005; Lin et al., 2007). The relevance of the behaviour of the boundary layer on a semi-infinite plate to that on the heated wall of a cavity was identified numerically by Patterson and Armfield (1990), Schladow (1990) and Armfield and Patterson (1992), and experimentally by Scho¨pf and Patterson (1995) and Patterson et al. (2002). At the downstream end of the vertical thermal boundary layer on a heated sidewall, the ceiling of the cavity forces a horizontal hot intrusion into the interior, which separates from and then reattaches to the ceiling, resulting in a series of trailing waves (Patterson and Imberger, 1980; Ivey, 1984; Paolucci and Chenoweth, 1989; Le Quere, 1990; Patterson and Armfield, 1990; Ravi et al., 1994; Scho¨pf and Patterson, 1995). These intrusions are clearly gravity currents similar to those described by the review of Simpson

temperature difference between one sidewall and initial ambient fluid t time (s) time when the front bypasses the fin tip (s) tf u velocity of the lower intrusion front (m/s) v velocity of the plume front (m/s) x, y horizontal and vertical coordinates (m) b coefficient of thermal expansion (1/K) dI, dP, dT thicknesses of the lower intrusion, plume and thermal boundary layer (m) j thermal diffusivity (m2/s) m kinematic viscosity (m2/s) DT

(1982), for which a wide literature exists. The interaction of the intrusion with the far end sidewall and the interior flow may result in a cavity-scale oscillation (Patterson and Imberger, 1980). Detailed analyses and experimental investigations of gravity currents with variable inflow are described by Huppert (1982) and Maxworthy (1983). As the flow approaches a steady or quasi-steady state, the interior fluid becomes stratified (Eckert and Carlson, 1961), and a double-layer structure of the vertical thermal boundary layer is formed (Xu et al., 2005). This double-layer structure was also predicted for the isolated boundary layer by Gill (1966) and others. At this stage, travelling waves in the thermal boundary layer may be observed if the Rayleigh number of the flow is sufficiently high. These evidently are a manifestation of a convective instability in the boundary layer flow (Scho¨pf and Patterson, 1996; Xu et al., 2005). Apart from the above brief description of the development of an understanding of the basic flows in a differentially heated cavity, methods of controlling the heat transfer through the cavity have attracted considerable attention due to their relevance to industrial applications. One simple passive way to control heat transfer is to place a fin on the sidewall. Studies of natural convection flows induced by a fin on a heated or cooled sidewall of a cavity have been extensively reported in the literature (e.g. Shi and Khodadadi, 2003; Tasnim and Collins, 2004; Bilgen, 2005), most of which were aimed at the effects of the geometry, material properties, and position of the fin. Nag et al. (1994) demonstrated that, as the thickness of a fin with a finite conductivity located at the mid height of the hot sidewall is reduced, heat transfer through the cavity initially decreases until a critical fin thickness is reached. A further reduction in fin thickness results in a subsequent increase of heat transfer. The fin length is also an important factor affecting heat transfer. If the length of a poorly conducting fin is sufficiently large, secondary circulations arise at the upper and lower base corners of the fin (Frederick, 1989; Nag et al., 1993). It was reported that heat transfer through the finned sidewall is reduced as the fin length increases

F. Xu et al. / Int. J. Heat and Fluid Flow 29 (2008) 1139–1153

are characterized in Section 4. Section 5 summarizes the major findings from the study. The role of the fin in controlling heat transfer however is not determined in this paper. That will be the subject of a further paper in which computational methods, validated by the present experimental results, will be utilized to explore the dependencies of heat transfer on the properties of the fin. 2. Experimental apparatus and approaches The present flow visualizations and temperature measurements were performed in the model cavity sketched in Fig. 1, which is similar to that described in Xu et al. (2006) and an earlier paper by the same authors (e.g. Xu et al., 2005). Briefly, the cavity contains initially motionless and isothermal water as the working fluid and is 1-m long (L), 0.24-m high (H) and 0.5-m wide (W) with a fin of cross-section 40-mm  2-mm attached at the mid height of the hot sidewall (on the right). All the cavity walls and the fin are constructed from Perspex sheets except the cooled and heated sidewalls, which are made of 1.13-mm thick gold-coated copper plates. Since the thermal conductivity of the Perspex walls is about 0.2 W m1 K1, much smaller than that of the cooper sidewalls and also smaller than that of water (0.6 W m1 K1), the Perspex walls and the fin may be approximately regarded as insulated. Under the typical experimental conditions relevant to this investigation, the heat loss through the Perspex walls is estimated to be less than 1% of the heat transfer through the cooper sidewalls, and thus has a negligible effect on the experiments. The sidewall with the fin is heated by the hot water in a water bath, and the opposing sidewall cooled by the cold water in another water bath. Each water bath, connected to a refrigerated/heating circulator, is initially separated from the sidewall by a pneumatically operated gate with an air gap between the gate and the sidewall. At the start of the experiments, the gates are rapidly lifted, allowing the hot and cold water respectively in the two water baths to flush against the copper sidewalls. This process is completed within a fraction of a second, and thus a close to instantaneous isothermal boundary condition is established at each end of the cavity (also see Patterson and Armfield, 1990). During the experiments, the temperatures of the

Tc

y

x

Cavity

T5 T4 T3

T2 Fin T1

Th h = H/2

H = 0.24 m

since natural convection flows adjacent to the finned sidewall are depressed (Nag et al., 1993). The vertical location of the fin may also change the flow in the cavity and thus affect heat transfer. Nag et al. (1993) showed that heat transfer is dramatically reduced as the fin is moved from a lower to an upper position. A fin on the sidewall depresses convective flows adjacent to the finned sidewall at low Rayleigh numbers, and thus reduces the convective heat transfer through the finned sidewall (Bejan, 1983). However, if the fin is perfectly conducting, it apparently increases the heating or cooling surface area and therefore increases conduction. As a consequence, the total heat transfer through the finned sidewall is determined by the combined effects of conduction and convection (Bilgen, 2005). Steady natural convection flows in a cavity with a fin on the heated wall at relatively low Rayleigh numbers (e.g. Ra < 107) have been the focus of previous studies (see e.g. Bilgen, 2005). However, since unsteady natural convection in the cavity also has extensive applications in industry, recently, an experimental investigation of the transient thermal flow in a cavity with a small square fin on the heated wall at high Rayleigh numbers (Ra > 109) was performed by the present authors (Xu et al., 2006). It was found that the small square fin blocks the natural convection boundary layer flow and forces an intrusion to form beneath the fin. The intrusion bypasses the short fin, and subsequently reattaches to the sidewall. Consequently, the presence of fin significantly changes the early transient thermal boundary layer flow following sudden heating. In the quasi-steady stage, a double-layer structure adjacent to the finned sidewall is formed and no clear separation of the flow around the fin is observed, which is consistent with the observations in Shakerin et al. (1988). Although Bilgen (2005) has demonstrated that the fin depresses natural convection flows adjacent to the finned sidewall at low Rayleigh numbers, the applicability of their result to higher Rayleigh numbers (e.g. Ra > 107) needs to be examined since the Rayleigh number plays an important role in the natural convection flows induced by a fin (Shi and Khodadadi, 2003). Accordingly, in the present paper an experimental investigation at high Rayleigh numbers is described in an attempt to find a way to trigger instabilities of the thermal flow using a fin and in turn generate unstable flows adjacent to the finned sidewall. In contrast to the experiments reported in Xu et al. (2006), a larger thin fin with a cross-section of 0.04 m  0.002 m is attached at the mid height of the heated sidewall with the expectation that an instability in the boundary layer downstream of the fin will be generated as the flow bypasses the fin and reattaches to the boundary layer. In the remainder of this paper, the experimental set-up and procedures are described in Section 2. The unsteady thermal flow around the fin is visualized and the dynamic mechanisms discussed in Section 3. The temperature measurements of the unsteady flow around the fin are presented and the oscillatory features of the temperature time series

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L=1m Fig. 1. Schematic of the experimental cavity with a thin fin: Points T1–T5 denote Thermistor locations, which are corresponding to Thermistor 1 (0.498 m, 0.06 m), Thermistor 2 (0.458 m, 0 m), Thermistor 3 (0.495 m, 0.004 m), Thermistor 4 (0.498 m, 0.06 m), and Thermistor 5 (0.498 m, 0.09 m), respectively.

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water in the water baths are kept constant by the circulators with a temperature stability of ±0.01 K. Natural convection flows in the cavity are visualized using the shadowgraph technique with a zigzagged optical set-up (refer to Scho¨pf and Patterson, 1995; Xu et al., 2005), which is briefly described below. A point light source is placed at the focal point of a spherical mirror with a diameter of 0.3 m and a focal length of 2.4 m, which generates a parallel light beam. When the parallel light beam is passed through the cavity filled with water with non-uniform temperatures, it is deflected by the water due to variations of the refractive index of the water. The refractive index of water depends on its density, and therefore in turn on its temperature. As a consequence, a non-uniform pattern with varying light intensity, which is referred to as shadowgraph, results when the light beam exits the cavity. The exiting light beam is then projected onto another identical spherical mirror and refocused to a black-and-white CCD camera located at the focal point of the mirror. The shadowgraph images are sampled at an adjustable sampling rate of up to 5 frames/s and digitized into an array of 576  768 pixels with an 8-bit frame grabber. Since the diameter of the mirror is only 0.3 m, the parallel light beam generated by the mirror cannot fully cover the entire cavity of 1-m length. Considering that the emphasis here is on the unsteady thermal flow around the fin, we focus the flow visualization on the region surrounding the fin. It is worth noting that, as described in Merzkirch (1974), the shadowgraph light intensity is sensitive to changes of the second spatial derivative of the refractive index (and thus the temperature). Accordingly, bright strips in the shadowgraph image correspond approximately to the maxima or minima of the second derivative of the temperature in the direction normal to the strips. In these experiments, fast-response thermistors (Thermometrics FP07 with a time constant of 0.007 s in water) are used to measure the temperature of the water in the cavity. The body of the thermistor has a maximum diameter of 2.2 mm and a length of 12.6 mm, and is mounted in an insulated tube in order to minimize thermal disturbances. The temperature measurements in this study were performed separately from the flow visualization experiments so that high-quality undisturbed shadowgraph images were obtained for qualitative descriptions of the flow. The total runtime of each experiment is dependent on the development of natural convection flows in the cavity, which is determined by the three dimensionless parameters which govern the flow: the Rayleigh number (Ra), the Pra-

ndtl number (Pr) and the aspect ratio (A). These are defined as follows: gbðT h  T c ÞH 3 ; mj m Pr ¼ ; j H A¼ : L

Ra ¼

Rah ¼

gbDTh3 ; mj

ð1Þ ð2Þ ð3Þ

Note that Rah (=Ra/16) is defined in terms of a halflength of the sidewall and is adopted in the scaling analysis in this paper. Furthermore, for the description of physical locations, the origin of the coordinates is at the center of the cavity as shown in Fig. 1. 3. Visualizations of the unsteady thermal flow around a thin fin A total of four experiments with temperature differences between the sidewalls varying from 4 K to 32 K were performed in order to visualize the unsteady thermal flow around the thin fin at different Rayleigh numbers. The corresponding ranges of Ra and Pr are 9.5  108–7.7  109 and 6.6–6.8, respectively, as listed in Table 1. The flow visualization results presented below are for the case with Ra = 3.8  109 and Pr = 6.6, unless specified otherwise. In the following subsections, the transient flow structures during the transition of the thermal flow around the fin from sudden heating to a quasi-steady state are described. The transition undergoes a number of stages, including a horizontal gravity current, a starting plume, entrainment of the plume into the thermal boundary layer downstream of the fin, and separation and oscillations of the thermal flow above the fin. The flow development may be broadly classified into an early stage, a transitional stage, and a quasi-steady stage as described below. 3.1. Lower intrusion, starting plume and turbulent mixing in the early stage Following sudden heating, if the fin length is larger than the thickness of the thermal boundary layer (which scales with j1/2t1/2, where t is the time from initiation; refer to Patterson and Imberger, 1980), the fin blocks the upstream thermal boundary layer flow and causes the heated fluid to accumulate underneath the fin. As a result, a lower intrusion front and a head are formed, as seen in Fig. 2a. As time increases, the lower intrusion front moves horizontally

Table 1 Experimental parameters for flow visualizations Number

Lab temperature (K)

Initial temperature of water in cavity (K)

Temperature difference between sidewalls (K)

Ra (108)

Pr

1 2 3 4

294.3 294.5 294.2 294.2

295.8 294.9 295.55 295.7

4 8 16 32

9.5 18 38 77

6.6 6.8 6.6 6.6

F. Xu et al. / Int. J. Heat and Fluid Flow 29 (2008) 1139–1153

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show that with this variable flux the initial balance in the intrusion is between the buoyancy-induced pressure gradient and viscous forces, but that the balance quickly becomes one between the buoyancy-induced pressure gradient (gbDTdI/ut, also see Patterson and Imberger, 1980) and inertia (u/t). Accordingly, considering the variable flux of the intrusion (udI  j5/2Raht3/2/h3) and the balance between the pressure gradient and inertia (gbDTdI/ ut  u/t), we are able to obtain a velocity scale: 2=3

u

j3=2 Pr1=3 Rah 1=2 t : h2

ð4Þ

Further, the penetration distance (df) of the intrusion front referenced to the origin of the intrusion may be calculated by the following scale: 2=3

d f  ut 

0.4

0.3

0.2

+

Rah = 5.9 107 = 1.1 108 = 2.4 108 +

+

0.1

= 4.8 108 +

toward the tip of the fin due to the continuous accumulation of the heated fluid (Fig. 2b–d). For the present Rayleigh number (Ra = 3.8  109), the lower intrusion front arrives at the tip of the fin after approximately 20 s, as indicated in Fig. 2d. At this time, the thermistor measurements in the thermal boundary layer reported later in the paper (Section 4) indicate that the thermal boundary layer is still unsteady. Since the fin length is sufficiently large in the present case, the horizontal motion of the lower intrusion front under the fin is clearly visualized. As expected, the horizontal motion of the lower intrusion front is synchronized with that under the ceiling, and the flow structures of the two fronts and heads are almost identical before the lower intrusion front bypasses the fin. Indeed, both intrusion flows are typical horizontal gravity currents (see e.g. Simpson, 1982). However, the intrusion heads observed here exhibit features which are characteristic of low Reynolds number gravity currents (see e.g. Britter and Simpson, 1978; Simpson, 1982) in that they remain stable and laminar with no clear mixing behind the heads. The Reynolds number achieved in the present case is O(60), where the Reynolds number is defined by Re = udI/m with dI  15 mm and u  4 mm/s (estimated based on shadowgraph images). The gravity current is the result of the flux of heated fluid from the still unsteady thermal boundary layer. Simple scaling arguments for the unsteady thermal boundary layer (Patterson and Imberger, 1980) indicate that the (two-dimensional) flux of heated fluid into the intrusion is O(j5/2Raht3/2/h3). Scaling arguments (Huppert, 1982)

ð5Þ

Based on the shadowgraph images (see e.g. Fig. 2), the penetration distance of the lower intrusion front at different times is measured. This process is also repeated for different Rayleigh numbers. The measurement results for all experimental cases are shown in Fig. 3, which plots the normalized penetration distance df/h of the lower intrusion front against the time scale Pr1/3Rah2/3(tj/h2)3/2. The clear linear correlation between the normalized penetration distance and the time scale confirms the scaling relations (4) and (5). For the present case, the maximum velocity magnitude measured using the visualization images (see e.g. Fig. 2) is approximately 4 mm/s near the tip of the fin, which is consistent with the speed of the intrusion under the ceiling without a fin at a Rayleigh number of 7.2  108, reported by Scho¨pf and Patterson (1995). After the lower intrusion front arrives at the tip of the fin, buoyancy causes it to rise vertically, passing the end of the fin. Fig. 4 shows the subsequent development of the thermal flow around the fin. Fig. 4a exhibits the curling up of the lower intrusion front driven by the buoyant force.

df /h

Fig. 2. Formation of the lower intrusion front and head: (a) t = 14 s, (b) t = 16 s, (c) t = 18 s, and (d) t = 20 s.

j3=2 Pr1=3 Rah 3=2 t : h2

0

0

0.5

1

1.5

Pr1/3Rah2/3(tκ/h2)3/2 Fig. 3. Dependence of the normalized penetration distance of the lower intrusion front on the Rayleigh number and time.

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F. Xu et al. / Int. J. Heat and Fluid Flow 29 (2008) 1139–1153

Fig. 4. Starting plume bypassing the thin fin: (a) t = 24 s, (b) t = 30 s, (c) t = 33 s, (d) t = 36 s, and (e) t = 41 s.

As a consequence, a new front and a head, similar to those of starting plumes (see e.g. Campbell et al., 1989), are formed on the downstream side of the fin, as seen in Fig. 4b. Fig. 4 also demonstrates that, as the starting plume moves upwards, the plume head becomes increasingly unstable, as indicated by the strip-like patterns near the front in Fig. 4c. The front shape here is slightly different from those observed at larger Prandtl numbers and lower Rayleigh numbers, which are more stable (see e.g. van Keken, 1997; Kaminski and Jaupart, 2003). As the plume head ascends further, the strip-like patterns become more complex (Fig. 4d). The chaotic strip patterns in Fig. 4e indicate that the plume head breaks up before it strikes the intrusion under the ceiling. These strip-like patterns may be characteristic of three dimensional flows. Since shadowgraph presents images which are essentially transverse integrals of the flow, it is not possible to confirm from these images that the flow is two-dimensional. However, what is important here is the indication of instability of the plume as it interacts with the upper intrusion. Further,

since the scaling analysis below shows consistent results with the experimental measurements, the presence of a small region of the three dimensional flow at the head and in the final stages of the plume ascent will have little effect on the overall flow development. The time for the plume front to travel from the fin to the ceiling is approximately 20 s for the present Rayleigh number. It is worth noting that the plume head initially moves directly upwards rather than reattaching to the downstream thermal boundary as observed in the case with a smaller square fin (Xu et al., 2006). This is because the plume head is further away from the downstream thermal boundary layer compared with that with a smaller square fin. It would not be possible to experimentally determine the maximum fin length for which immediate reattachment occurs. However a numerical investigation of this aspect is presently underway which will be reported separately. Starting plumes have been extensively studied (see e.g. Bachelor, 1954b; Turner, 1962; Shlien and Thompson, 1975; Moses et al., 1993; Kaminski and Jaupart, 2003).

F. Xu et al. / Int. J. Heat and Fluid Flow 29 (2008) 1139–1153

v

Rah j2 t: h3

ð6Þ

The corresponding vertical position relative to its origin therefore scales with y  v(t  tf) (Rahj2t(t  tf)/h3, where tf is the time when the lower intrusion arrives at the fin tip). The measured positions of the plume front, non-dimensionalised by h, for all the experimental cases

1

0.8

0.6

y/h 0.4 +

Rah = 5.9 107 +

= 1.1 108

0.2

8

+

= 2.4 10

8

= 4.8 10 +

Kaminski and Jaupart (2003) pointed out that the development of the starting plume is usually classified into four distinct stages: a conductive heating stage, an accelerating stage, a steady stage and a decelerating stage when the plume approaches an upper wall or a free surface. In the present case, the starting plume is generated by a time varying volume flux O(j5/2Raht3/2/h3), and almost all of the plume’s initial ascent is unsteady. Simple scaling arguments show that, for this initial part of the starting plume in the absence of entrainment, the ascent is governed by a balance between the viscous forces (mv/dP2) and the buoyancy forces (gbDT). Together with the imposed volume flux above (vdP  j5/2Raht3/2/h3), this balance yields a scale of the vertical velocity given by

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0

0

5

10

15

20

25

30

Rahκ 2t(t-t f)/h 4 Fig. 5. Dependence of the vertical position of the plume front on the Rayleigh number and time.

are plotted against Rahj2t(t  tf)/h4 in Fig. 5. The good linear fit to the plot again demonstrates the correctness of the scaling relation (6).

Fig. 6. Interaction between the plume flow and the thermal boundary layer flow: (a) t = 48 s, (b) t = 56 s, (c) t = 78 s, and (d) t = 88 s.

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After the plume front strikes the intrusion, the broken plume head ultimately merges into the intrusion and the downstream thermal boundary layer, as shown in Fig. 6. Due to the strong interaction of the plume head with the downstream thermal boundary layer flow and the horizontal intrusion under the ceiling, the broken plume head spreads over the corner region of the downstream thermal boundary layer, as seen in Fig. 6a. The chaotic structure in this region indicates that the mixing is thorough and displays features of turbulence. As a consequence, convection in the downstream thermal boundary layer is intensified, which may be expected to enhance the heat transfer through the hot sidewall. With the passage of time, entrainment by the downstream thermal boundary layer draws the plume flow closer to the hot sidewall, as seen in Fig. 6b. A clockwise circulation forms adjacent to the downstream sidewall between the plume flow and the fin, and the plume flow behind the head (the ‘stem’ of the starting plume, Moses et al., 1993) still retains strip-like patterns. As time increases further, the plume flow is drawn closer toward the fin by entrainment (Fig. 6c and d). It is also clear that the perturbations induced by the broken plume head are eventually convected away (Fig. 6d). In summary, the early-stage thermal flow around the fin undergoes distinct stages of development including a horizontal gravity current, a starting plume, turbulent mixing in the downstream corner region and reattachment to the downstream thermal boundary layer. These early flow phenomena are apparently different from those observed in the case with a smaller square fin (Xu et al., 2006). In particular, the apparently stronger mixing in the present case is expected to enhance the heat transfer through the section of the sidewall downstream of the fin. 3.2. Flows in the transitional stage After the perturbations from the plume head are convected away, the thermal flow around the fin enters a slow transitional process. Fig. 7 shows the subsequent development of the thermal boundary layer flow after the cold intrusion coming from the far cooled wall strikes the hot wall upstream of the fin. The location of the incoming intrusion in the upstream corner in Fig. 7a is indicated by a solid line overdrawn in the figure. For the purpose of observing clearly the development of the double-layer structure of the boundary layer (Xu et al., 2005), the images shown in this figure have been processed by subtracting a background image recorded immediately before the experiment was started. As a result of the image processing, the hot sidewall and the fin can be easily identified. Prior to the arrival of the intrusion, the outer edge of the thermal boundary layer upstream of the fin is clearly visible as a continuous bright strip. Downstream of the fin, the thermal boundary layer has been significantly affected by the entrainment of the former plume, and a highly disturbed flow is evident. Clearly, the former plume has been

Fig. 7. Development of the thermal boundary layer after the cold intrusion strikes the hot sidewall: (a) t = 327 s, (b) t = 741 s, (c) t = 2139 s, and (d) t = 11,786 s.

drawn much closer to the fin compared with that at the earlier stage (see e.g. Fig. 6d). It is worth noting that, apparently due to the strong perturbations in the thermal boundary layer downstream of the fin, distinct separation and trailing waves on the horizontal intrusion under the ceiling, which were observed in the case without the fin (Xu et al., 2005), do not appear in the present case. After the arrival of the cold intrusion, a second bright strip outside the thermal boundary layer appears in the vicinity of the upstream corner, as seen in Fig. 7b. At this time, the flow in the downstream corner is still characterized by a completely destabilized flow structure. As time increases, the lower bright strip outside the thermal boundary layer upstream of the fin extends toward the fin (Fig. 7c), and the chaotic strip-like patterns in the downstream corner become weaker. In the meantime, a short discontinuous bright strip, which originates from the downstream corner, extends downwards, and an additional horizontal bright strip is formed outside the thermal flow above the fin. At a much later time (t = 11786 s), the two bright strips downstream of the fin become clearer, and the bright strip from the top corner continuously extends downwards, as seen in Fig. 7d. However, the upper bright strip from the top corner does not extend to the fin even after a long run of the experiment (more than three hours); that is, no continuous outer layer over the total length of the heated sidewall is formed. This is because the thermal flow around the fin breaks the stratification above the fin

F. Xu et al. / Int. J. Heat and Fluid Flow 29 (2008) 1139–1153

and thus leads to the broken outer bright strip shown in the shadowgraph images (refer to Xu et al., 2005 for details of the shadowgraph image). 3.3. Separation and oscillations in the quasi-steady stage As time increases further, the thermal boundary layer flow approaches a quasi-steady state (Fig. 7d). The experimental measurements show that the thicknesses of the inner and outer layers (the distances between the bright strips and the sidewall) on the upstream side of the fin are 2.5 mm and 6.8 mm respectively at y = 0.052 m, similar to those in the case without the fin (2.3 mm and 7.4 mm, see Xu et al., 2005). This confirms that the fin has little impact on the flow development of the thermal boundary layer on the upstream side of the fin. As noted above, the interaction of the thermal flow around the fin with the thermal boundary layer immediately downstream of the fin results in a disturbed structure in that region. Fig. 8 shows a sequence of shadowgraph images, beginning at t = 11,861 s and ending at t = 11,870 s, separated by 1 s. The disturbed structure in the thermal boundary layer has the appearance of a regular travelling wave structure, and the overdrawn arrows in Fig. 8 identify the position of the wave front with time. The waves appear to be the result of a disturbance originating in the thermal flow near the juncture of the fin and the wall (Fig. 8a), which evidently triggers an instability in the thermal boundary layer. It is clear in Fig. 8 that a potentially unstable temperature configuration is present in the region above the fin with the water temperature increasing toward the upper

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surface of the fin. The scale for the steady intrusion thickness below the fin is O(h/Pr1/3Rah1/6) (Patterson and Imberger, 1980). Assuming that the same thickness scale applies to the thermal flow above the fin, and that the temperature difference across the intrusion is O(DT), the local Rayleigh number based on the intrusion thickness is O(Rah1/2/Pr). For the case shown here, this is 2.3  103, above the critical condition for a Rayleigh–Bernard-type instability (1.1  103, see Drazin and Reid, 1981). Thus we speculate that the thermal flow above the fin are in fact intermittent rising plumes generated by Rayleigh-Bernard instability, and that its interaction with the thermal boundary layer triggers the travelling waves in the downstream thermal boundary layer. It is worth noting that separation and intermittent plumes (resultant oscillations) were not observed in the case with a smaller square fin (Shakerin et al., 1988; Xu et al., 2006). This implies that these phenomena are sensitive to the geometric parameters of the fin and there exists a critical length of the fin; that is, clear separation and intermittent plumes appear if the fin length is larger than the critical value. Furthermore, the temperature measurements in this region, discussed below in Section 4, indicate that the frequency of the disturbance is constant for a given Ra. Based on the positions of the wave fronts in the downstream thermal boundary layer, the propagation velocity of the travelling waves may be estimated. Fig. 9 shows the vertical positions (y) of five consecutive wave fronts with time in the quasi-steady stage (see Fig. 8). Clearly, the downstream waves all travel at an approximately constant velocity, measured to be 5.1 mm/s which is of the same order as the results reported in Scho¨pf and Patterson

Fig. 8. The position of a disturbance shown by the overdrawn arrow at (a) t = 11,861 s to (j) t = 11870 s, with an interval of 1 s between images.

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F. Xu et al. / Int. J. Heat and Fluid Flow 29 (2008) 1139–1153 0.1

y (m)

0.08

0.06

0.04

0.02 11698

11713

11728

11743

t (s) Fig. 9. Measured locations of five consecutive travelling wave fronts vs time.

(1996) and Xu et al. (2005) for the speed of the travelling waves in a thermal boundary layer with no fin, at comparable Rayleigh numbers. The quasi-steady state flow structures around the fin depend on the Rayleigh number, which is demonstrated in Fig. 10. It is seen in this figure that, as the Rayleigh number increases, the double-layer structure under the fin becomes more distinct and the unstable flow structure around the fin becomes much stronger. At a Rayleigh num-

ber of Ra = 9.54  108, the bright strip in the upper part of the cavity is barely identifiable (Fig. 10a), and separation of the thermal flow above the fin is less distinct. However, for the case of Ra = 1.8  109, the upper outer bright strip may be observed and the local flow separation in the leeward side of the fin is distinct with perturbations to the downstream thermal boundary layer, as seen Fig. 10b. When the Rayleigh number increases to 7.7  109, the perturbations in the downstream thermal boundary layer flow, triggered by the thermal flow around the fin, become more easily identifiable (see Fig. 10d). It is worth noting that in Fig. 10d an outer bright strip above the fin extends horizontally toward the core. This is associated with the vertical temperature profile above the fin. As noted above, the vertical distribution of the temperature in the thermal flow immediately above the fin is unstable. On the other hand, the interior has become stably stratified. That is, as the height from the upper surface of the fin increases, the temperature of the fluid first decreases and then increases. Such a temperature profile yields extrema of the second derivative of the temperature above the fin, which in turn leads to an outer bright strip extending horizontally toward the core. This effect becomes more apparent at high Rayleigh numbers, as seen in Fig. 10d. In summary, the present flow visualizations display distinctive features of the unsteady thermal flow around the

Fig. 10. Thermal boundary layer flows at different Rayleigh numbers at t = 7200 s: (a) Ra = 9.5  108, (b) Ra = 1.8  109, (c) Ra = 3.8  109, and (d) Ra = 7.7  109.

F. Xu et al. / Int. J. Heat and Fluid Flow 29 (2008) 1139–1153

thin fin in different stages. As can be seen in Fig. 7, the transition of the thermal boundary layer to the quasisteady state is a complex process. As the thermal flow around the fin approaches the quasi-steady state, separation and resultant oscillations of the thermal flow above the fin are observed. The oscillations of the thermal flow around the fin (see Figs. 8–10) improve the convection flow in the downstream side of the fin. This result is apparently different from those results reported for lower Rayleigh numbers (see e.g. Bilgen, 2005), in which the fin is found to depress the convection flow in the vicinity of the fin.

4. Temperature measurements of the unsteady thermal flow around a thin fin The temperature measurements of the unsteady thermal flow around the fin were performed at conditions

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(see Table 2) similar to those in the flow visualization experiments for the purpose of comparisons between the flow visualization and measurements. As can be seen in Table 2, four experiments with temperature differences between the two sidewalls ranging from 4 K to 32 K have been conducted. The corresponding ranges of Ra and Pr are 1.0–7.5  109 and 6.5–6.6, respectively. Similar to the flow visualizations described in Section 3, the results of the temperature measurements are described based on Ra = 3.8  109 (Experiment 7) in this section unless specified otherwise (Note that Experiment 7 has the same experimental conditions as Experiment 3 reported above). The thermistor locations for the temperature measurements are shown in Fig. 1. Since the thermal boundary layer flow is the focus of this paper, all five thermistors were placed in the vicinity of the fin and the hot sidewall. Three of the thermistors were placed at a distance of

Table 2 Experimental parameters for temperature measurements Number

Lab temperature (K)

Initial temperature of water in cavity (K)

Temperature difference between sidewalls (K)

Ra (109)

Pr

5 6 7 8

297.2 297.1 296.1 296.1

296.7 296.6 295.55 295.6

4 8 16 32

1.0 2.0 3.8 7.5

6.5 6.5 6.6 6.6

T (K)

298 296

5

294

T (K)

298

4

296 294

T (K)

298 296

3 294

T (K)

298 296

2 294

T (K)

298

1

296 294

0

5000

10000

15000

t (s)

Fig. 11. Time series of the temperatures at different locations shown in Fig. 1.

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F. Xu et al. / Int. J. Heat and Fluid Flow 29 (2008) 1139–1153

approximately 2 mm from the sidewall surface at different heights, with the other two located around the fin. Fig. 11 shows the time series of the temperatures measured at the five locations. The temperature fluctuations in the thermal boundary layer on the downstream side of the fin (Thermistors 3–5) are clear, and the time series of the temperatures at all the five locations eventually approach a quasi-steady state, which is consistent with the flow visualizations described previously. It is seen in this figure that the temperatures measured by Thermistors 3–5 in the upper part of the cavity increase with time, the temperature measured by Thermistor 2 at the mid height of the cavity weakly increases with time, and the temperature measured by Thermistor 1 in the lower part of the cavity decreases with time. As a consequence, the difference between the two time series of the temperatures by Thermistors 1 and 5 increases with time until the quasi-steady state is reached. This implies that the stable stratification of the fluid in the cavity is continuously increased (Eckert and Carlson, 1961). The transient flow features are demonstrated with a closer examination of these time series, as described in the following sections. 4.1. The leading edge effect (LEE) and perturbations from the starting plume in the early stage Fig. 12a shows the temperature time series measured by Thermistors 1 and 4 in the boundary layer at early times (up to 60 s). The features of the LEE are clear, which are characterized by an overshoot followed by a group of travelling waves (Patterson et al., 2002). Subsequently, the temperature measured by Thermistor 1 approaches an approximate constant, whereas the temperature measured by Thermistor 4 remains approximately constant for a short while until it is perturbed by the impact of the plume front from the thermal flow around the fin. It is worth noting that there is a distinct difference in the initial temperature growth recorded by Thermistors 1 and 4. It is expected

that the temperature signals taken at the same distance from the heated sidewall would be identical prior to the arrival of the LEE at the respective locations. The variation of the initial temperature growth at the two different locations may be attributed to a small variation of the actual distance from the wall, which could lead to a large variation of the measured temperature due to the existence of a very high lateral temperature gradient in the thermal boundary layer (Patterson et al., 2002). As seen in Fig. 2, after approximately 20 s, the lower intrusion front bypasses the fin, which is confirmed by the temperature signal recorded by Thermistor 2 near the tip of the fin (Fig. 12b). Subsequently, the perturbations from the starting plume are felt by Thermistor 4 (at approximately 36 s, Fig. 12a), although the shadowgraph image in Fig. 4(d) shows that the plume front does not directly impact on the thermal boundary layer. Clearly, since Thermistor 2 is on the direct path of the lower intrusion front and the resultant plume, the perturbations from the starting plume recorded by Thermistor 2 are much stronger than that recorded by Thermistor 4. It is clear that these perturbations from the starting plume have virtually no impact on the temperature measured by Thermistor 1, as it is upstream of the fin and there is no direct influence of the plume at that location. There is also no measurable influence on Thermistor 3 (at the leeward corner of the fin) as it is bypassed by the plume. Furthermore, since Thermistor 3 is located sufficiently close to the leading edge at the junction of the wall and the fin and is further away from the wall (outside the thermal boundary layer dT  y/Rah1/4, see Patterson and Imberger, 1980) compared to Thermistors 1 and 4, it does not detect the LEE. The early transient features of the temperature time series are dependent on the Rayleigh number. Fig. 13 shows the temperature time series measured by Thermistor 4 at two different Rayleigh numbers, relative to the initial temperature (T0) in each case. Clearly, as the Rayleigh number increases, the temperature in the thermal boundary layer

297.5

297.5

a

b 2

297

T(K)

T(K)

297

296.5 4 296 Overshoot Travelling wave

296.5

296 3

1 295.5

0

20

40

t (s)

60

295.5

0

20

40

60

t (s)

Fig. 12. Time series of the temperatures at different locations in the early stage: (a) temperatures measured by Thermistors 1and 4, and (b) temperatures measured by Thermistors 2 and 3.

F. Xu et al. / Int. J. Heat and Fluid Flow 29 (2008) 1139–1153

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perturbations to the downstream thermal boundary layer. This implies that the starting plume will play an important role in enhancing heat transfer through the downstream section of the sidewall in the early stage.

1 0.8

Ra =3.8 x109

0

T-T (K)

0.6 0.4

4.2. Waves in the quasi-steady stage

LEE

0.2 0

Ra =1x 109

-0.2 0

30

60

90

t (s)

Fig. 13. Temperature time series at the point (0.498 m, 0.06 m, Thermistor 4) at different Rayleigh numbers.

grows more rapidly during the early stage, and the perturbations from the starting plume arrive earlier and are much stronger. In summary, as seen in Figs. 12 and 13, the starting plume originated from the lower intrusion induces strong

Following a slow transitional stage, the amplitudes of the temperature fluctuations recorded by all thermistors approach constant values (refer to Fig. 11). In order to display further the wave features of the thermal boundary layer flow downstream of the fin, Fig. 14a shows the temperature time series from Thermistors 3–5 for a short time period in the quasi-steady stage. It is seen in this figure that the amplitude of the temperature waves at this time is strongly dependent on the spatial location but less dependent on time. The temperature time series from Thermistor 3 (at the leeward side of the fin) displays oscillations which characterize the thermal flow around the fin. The time series from Thermistor 4 displays the travelling waves in the downstream thermal boundary layer induced by these oscillations and that from Thermistor 5 displays the travel-

298.5

a

b

0.095 Hz 10-2

T (K)

Two peaks

Power spectrum

298

5

297.5

4

297

10-4

10-6

3 10-8

296.5 12900

12950

0

13000

0.1

0.2

0.4

0.5

f (Hz)

t (s) 0.25

0.15

c

0.2 Hz

d

0.1 Hz

0.1 Hz Power spectrum

0.2

Power spectrum

0.3

0.15

0.1

0.1

0.2 Hz 0.05

0.05

0

0

0.1

0.2

0.3

f (Hz)

0.4

0.5

0

0

0.1

0.2

0.3

0.4

0.5

f (Hz)

Fig. 14. Temperatures and power spectra from Thermistors 3–5 in the quasi-stage stage: (a) temperature time series from Thermistors 3–5, (b) spectrum for Thermistor 3, (c) spectrum for Thermistor 4, and (d) spectrum for Thermistor 5.

F. Xu et al. / Int. J. Heat and Fluid Flow 29 (2008) 1139–1153 0.2

Temperature measurments

0.15 0.1 0.05 0

0

2

4

6

8

Ra ( 109) +

ling waves in the further downstream thermal boundary layer and the corner. The corresponding power spectra for Thermistors 3–5 are shown in Fig. 14b–d, respectively. The characteristics of the temperature waves are evidently different at these locations. Fig. 14b shows that, at Thermistor 3 (just downstream of the fin), the peak of the spectrum is at a very low frequency, with a noticeable hump around O(0.1 Hz) describing the oscillations of the thermal flow around the fin. For the purpose of illustrating clearly the peaks around the dominant frequencies, the power spectra for Thermistors 4 and 5 are plotted with a linear scale in Fig. 14c and d. It is seen in Fig. 14c that, at the Thermistor 4 location, there are two dominant frequencies, one around 0.1 Hz, and the other approximately double the former (i.e. around 0.2 Hz). The lower dominant frequency of 0.1 Hz is consistent with the frequency around the hump in Fig. 14b, which is amplified as it travels downstream, suggesting that this is the characteristic frequency of the travelling waves in the thermal boundary layer. The higher frequency wave may have been triggered by the impact of the separated thermal flow above the fin, which breaks up the travelling waves in the thermal boundary layer (two peaks in the temperature signal, see Fig. 14a) when it is entrained into the thermal boundary layer. Fig. 14d shows that the dominant frequency of the travelling waves is still significant at the further downstream location (close to the corner). However, the higher frequency signal has decayed by Thermistor 5. The measured travelling wave frequency in the downstream thermal boundary layer is consistent with the results of the numerical simulation and stability analysis at a comparable Rayleigh number reported in Armfield and Patterson (1992), which predicted a dominant frequency of 0.083 Hz for a Rayleigh number of 6  108. The Rayleigh number in Armfield and Patterson (1992) is calculated based on the full length of the heated sidewall, which has no fin on it. In the present case, the large thin fin has separated the thermal boundary layer into two distinct sections; an upper section and a lower section (refer to Fig. 2). For a meaningful comparison with the data in Armfield and Patterson (1992), the present Rayleigh number should be recalculated in terms of the half-length of the sidewall and the temperature difference between the hot and cold sidewalls, which gives a value of 4.8  108. This Rayleigh number is comparable with that in Armfield and Patterson (1992), and the measured travelling wave frequency of around 0.1 Hz is comparable with the previously reported 0.083 Hz. It has been demonstrated in Fig. 10 that separation and oscillations of the thermal flow around the fin depend on the Rayleigh number. Based on the spectral analysis of the temperature at Thermistor 4 for different Rayleigh numbers, the dominant frequency of the thermal flow around the fin may be obtained. Fig. 15 shows the dependence of the oscillatory frequency of the thermal flow around the fin on Ra (based on the full length of the side-

f (Hz)

1152

Fig. 15. Dependence of the oscillatory frequency of the thermal flow around the fin on the Rayleigh numbers.

wall). It is clear that the correlation between the oscillatory frequency and Ra is approximately linear in the range of the Rayleigh numbers considered here.

5. Conclusions In this paper, the unsteady thermal flow around a large thin fin placed at the mid height of the hot wall of a differentially heated cavity is visualized using the shadowgraph technique and measured by fast-response thermistors. The transient flow structures are observed and the flow features are characterized. The present flow visualizations reveal that, following sudden heating, a lower intrusion front (a horizontal gravity current) is formed underneath the fin and travels at the velocity scale of uh/j  PrRah2/3(tj/h2)1/2 in the early stage. After the lower intrusion bypasses the fin, the buoyancy effect results in a starting plume which ascends at a velocity scale of vh/j  Rahtj/h2 and eventually strikes the intrusion under the ceiling. The interaction among the starting plume, the intrusion flow under the ceiling and the downstream thermal boundary layer flow causes turbulent mixing in the vicinity of the downstream corner. After the perturbations induced by the plume head are convected away, entrainment by the thermal boundary layer draws the plume toward the heated sidewall. In the transition to a quasi-steady state, separation and resultant oscillations of the thermal flow around the fin take place, and the process is rather sensitive to the geometric parameters of the fin. The oscillations bring strong perturbations into the downstream thermal boundary layer, whereas a stable double-layer structure may be formed on the upstream side of the fin. The present temperature measurements show consistent flow features with the flow visualizations. The temperature time series from thermistors at different locations clearly characterize the LEE and other transient features of the thermal boundary layer flow in the transition from sudden heating to a quasi-steady state. Furthermore, the spectral analysis of the temperature time series has revealed the dominant frequency of the travelling waves in the thermal boundary layer. It is found that the dominant frequency is

F. Xu et al. / Int. J. Heat and Fluid Flow 29 (2008) 1139–1153

approximately linearly dependent on the Rayleigh number in the present parameter range. The present flow visualization and temperature measurements also indicate that the oscillations of the thermal flow around the fin trigger the instabilities of the downstream thermal boundary layer and reinforce the convection flow at the downstream side of the fin. As a consequence, it is expected that heat transfer through the finned sidewall is enhanced. A further numerical investigation to explore the effect of the fin on heat transfer is under way. Acknowledgements We would like to thank Mr Stuart Petersen for his assistance in the experiments, and the Australian Research Council for its financial support. References Armfield, S.W., Patterson, J.C., 1992. Wave properties of natural convection boundary layers. J. Fluid Mech. 239, 195–212. Bachelor, G.K., 1954a. Heat transfer by free convection across a closed cavity between vertical boundaries at different temperatures. Quart. Appl. Math. 12, 209–233. Bachelor, G.K., 1954b. Heat convection and buoyancy effects in fluids. Quart. J. Roy. Met. Soc. 80, 339–358. Bejan, A., 1983. Natural convection heat transfer in a porous layer with internal flow obstructions. Int. J. Heat Mass Transfer 26, 815–822. Bilgen, E., 2005. Natural convection in cavities with a thin fin on the hot wall. Int. J. Heat Mass Transfer 48, 3493–3505. Britter, R.E., Simpson, J.E., 1978. Experiments on the dynamics of a gravity current head. J. Fluid Mech. 88, 223–240. Brown, S.N., Riley, N., 1973. Flow past a suddenly heated vertical plate. J. Fluid Mech. 59, 225–237. Campbell, I.H., Griffiths, R.W., Hill, R.I., 1989. Melting in an archaean mantle plume: heads it’s basalts, tails it’s komatiites. Nature 339, 697–699. Drazin, P.G., Reid, W.H., 1981. Hydrodynamic Stability. Cambridge University Press. Eckert, E.R.G., Carlson, W.O., 1961. Natural convection in an air layer enclosed between two vertical plates at different temperatures. Int. J. Heat Mass Transfer 2, 106–120. Elder, J.W., 1965. Laminar free convection in a vertical slot. J. Fluid Mech. 23, 77–98. Frederick, R.L., 1989. Natural convection in an inclined square enclosure with a partition attached to its cold wall. Int. J. Heat Mass Transfer 32, 87–94. Gill, A.E., 1966. The boundary-layer regime for convection in a rectangular cavity. J. Fluid Mech. 26, 515–536. Goldstein, R.J., Briggs, D.G., 1964. Transient free convection about vertical plates and cylinders. J. Heat Transfer 86, 490–500. Huppert, H.E., 1982. The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface. J. Fluid Mech. 121, 43–58. Ivey, G.N., 1984. Experiment on transient natural convection in a cavity. J. Fluid Mech. 144, 389–401. Joshi, Y., Gebhart, B., 1987. Transition of transient vertical naturalconvection flows in water. J. Fluid Mech. 179, 407–438. Kaminski, E., Jaupart, C., 2003. Laminar starting plumes in high-Prandtlnumber fluids. J. Fluid Mech. 478, 287–298. Le Quere, P., 1990. Transition to unsteady natural convection in a tall water-filled cavity. Phys. Fluids 2, 503–515. Lin, W., Armfield, S.W., 2005. Unsteady natural convection on an evenly heated vertical plate for Prandtl number Pr < 1. Phys. Rev. E 72, 066309.

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Lin, W., Armfield, S.W., Patterson, J.C., 2007. Cooling of a Pr < 1 fluid in a rectangular container. J. Fluid Mech. 574, 85–108. Mahajan, R.L., Gebhart, B., 1978. Leading edge effects in transient natural convection flow adjacent to a vertical surface. J. Heat Transfer 100, 731–733. Maxworthy, T., 1983. Gravity currents with variable inflow. J. Fluid Mech. 128, 247–257. Merzkirch, W., 1974. Flow Visualization. Academic Press. Moses, E., Zocchi, G., Libchaber, A., 1993. An experimental study of laminar plumes. J. Fluid Mech. 251, 581–601. Nag, A., Sarkar, A., Sastri, V.M.K., 1993. Natural convection in a differentially heated square cavity with a horizontal partition plate on the hot wall. Comput. Method Appl. Mech. 110, 143–156. Nag, A., Sarkar, A., Sastri, V.M.K., 1994. Effect of thick horizontal partial partition attached to one of the active walls of a differentially heated square cavity. Numer. Heat Transfer Part A 25, 611–625. Ostrach, S., 1964. Laminar flows with body forces. In: Moore, F.K. (Ed.), Therory of Laminar Flows. Princeton University Press. Paolucci, S., Chenoweth, D.R., 1989. Transition to chaos in a differentially heated vertical cavity. J. Fluid Mech. 201, 379–410. Patterson, J.C., Armfield, S.W., 1990. Transient features of natural convection in a cavity. J. Fluid Mech. 219, 469–497. Patterson, J.C., Graham, T., Scho¨pf, W., Armfield, S.W., 2002. Boundary layer development on a semi-infinite suddenly heated vertical plate. J. Fluid Mech. 219, 467–497. Patterson, J.C., Imberger, J., 1980. Unsteady natural convection in a rectangular cavity. J. Fluid Mech. 100, 65–86. Ravi, M.R.R., Henkes, R.A.W.M., Hoogendoorn, C.J., 1994. On the high-Rayleigh-number structure of steady laminar natural-convection flow in a square enclosure. J. Fluid Mech. 262, 325–351. Schetz, J.A., Eichorn, R., 1962. Unsteady natural convection in the vicinity of a doubly infinite vertical plate. J. Heat Transfer 84, 334–338. Schladow, S.G., 1990. Oscillatory motion in a side-heated cavity. J. Fluid Mech. 213, 589–610. Scho¨pf, W., Patterson, J.C., 1995. Natural convection in a side-heated cavity: visualization of the initial flow features. J. Fluid Mech. 295, 279–357. Scho¨pf, W., Patterson, J.C., 1996. Visualization of natural convection in a side-heated cavity: transition to the final steady state. Int. J. Heat Mass Transfer 39, 3497–3509. Shakerin, S., Bohn, M., Loehrke, R.I., 1988. Natural convection in an enclosure with discrete roughness elements on a vertical heated wall. Int. J. Heat Mass Transfer 31, 1423–1430. Siegel, R., 1958. Transient free convection from a vertical flat plate. J. Heat Transfer 80, 347–360. Shi, X., Khodadadi, J.M., 2003. Laminar natural convection heat transfer in a differentially heated square cavity due to a thin fin on the hot wall. J. Heat Transfer 125, 624–634. Shlien, D.J., Thompson, D.W., 1975. Some experiments on the motion of an isolated laminar thermal. J. Fluid Mech. 72, 35–47. Simpson, J.E., 1982. Gravity currents in the laboratory, atmosphere, and ocean. Ann. Re. Fluid Mech. 14, 213–234. Tasnim, S.H., Collins, M.R., 2004. Numerical analysis of heat transfer in a square cavity with a baffle on the hot wall. Int. Commun. Heat Mass Transfer 31, 639–650. Turner, J.S., 1962. The ‘starting plume’ in neutral surroundings. J. Fluid Mech. 13, 356–368. Van Keken, P., 1997. Evolution of starting mantle plumes: a comparison between numerical and laboratory models. Earth Planet Sci. Lett. 148, 1–11. Xu, F., Patterson, J.C., Lei, C., 2005. Shadowgraph observations of the transition of the thermal boundary layer in a side-heated cavity. Exp. Fluids 38, 770–779. Xu, F., Patterson, J.C., Lei, C., 2006. Experimental observations of the thermal flow around a square obstruction on a vertical wall in a differentially heated cavity. Exp. Fluids 40, 363–371.