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Temperature oscillations in a differentially heated cavity with and without a fin on the sidewall
International Communications in Heat and Mass Transfer 37 (2010) 350–359
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International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / i c h m t
Temperature oscillations in a differentially heated cavity with and without a fin on the sidewall☆ Feng Xu ⁎, John C. Patterson, Chengwang Lei School of Civil Engineering, The University of Sydney, Sydney, NSW 2006, Australia
a r t i c l e
i n f o
Available online 2 February 2010 Keywords: Fin Oscillation Thermal boundary layer
a b s t r a c t Temperature measurements of the thermal flows in a differentially heated cavity with and without a fin on the sidewall were performed. The oscillatory behaviours in the transition from an initially isothermal state to an eventually stratified interior fluid ambient are characterized. It is revealed that, for the case without a fin, the amplitude of the oscillations in the thermal boundary layer increases as the stratification of the interior fluid increases. For the case with a fin, the temperature measurements show that the oscillations are induced by an unstable fluid layer above the fin. The spectral analysis reveals that the frequency properties of the oscillations are different between the cases with and without a fin. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Natural convection flows are present in many industrial systems such as solar collectors and nuclear reactors. In particular, transient flows in a differentially heated cavity are frequently encountered in industrial applications, and thus have been investigated extensively in the literature [1–3]. Since heat is transferred into and out of the differentially heated cavity through the sidewalls, the thermal boundary layer flows adjacent to the sidewalls are naturally the focus of investigations. Previous studies (e.g. [1]) showed that following sudden heating, a growing thermal boundary layer forms and a one-dimensional vertical flow is driven by buoyancy forces. This one-dimensional flow persists for a short period of time, and then perturbations, which are referred to as the leading edge effect (LEE), arise in the vicinity of the leading edge of the vertical wall and travel downstream [4–6]. After the LEE travels away, the local boundary layer flow becomes steady and two-dimensional. It is worth noting that the interior fluid in the cavity following start-up is usually isothermal. The top and bottom walls of the cavity force horizontal intrusions into the interior, which separate from and then reattach to the top and bottom walls respectively [7,8]. The interior fluid in the cavity eventually becomes stratified. Understanding of the instability of the boundary layer adjacent to the sidewall of a differentially heated cavity is also important since the flow regime (which could be either stable or unstable) determines heat transfer through the sidewall. Indeed, the instability of the thermal boundary layer has been widely investigated for over five ☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (F. Xu). 0735-1933/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2010.01.004
decades [9,10]. It was demonstrated that only a narrow band of the frequencies of the oscillations in the thermal boundary layer is significantly amplified downstream. Most of the studies of the thermal boundary layer instability considered the case with a thermal vertical wall placed in an isothermal fluid [11,12]. However, natural convection often occurs in a thermally stratified fluid ambient and thus the effect of the stratified ambient on the instability of the thermal boundary layer was also investigated [13,14]. Improving or depressing natural convection flows in the cavity and in turn controlling heat transfer through the cavity are of considerable practical importance in industrial applications. A simple way to manipulate the flow regimes adjacent to the sidewall and in turn to enhance or depress heat transfer is to place a fin on the sidewall of the cavity. Investigations of the thermal flow induced by a fin on the sidewall have been extensively reported in the literature (e.g. [15,16]). Previous studies have considered the effects of the material properties and geometry of the fin. Clearly, the fin may be either conductive or adiabatic. It is expected that a conductive fin attached to the sidewall increases the surface area for heat transfer, and in turn increases the overall heat transfer through the cavity. Accordingly, the effect of the thermal conductivity of different fin materials on heat transfer through the finned sidewall was investigated [17–19]. An adiabatic or poorly conducting fin may also have a significant impact on natural convection flows in a differentially heated cavity and the heat transfer through the cavity, and in particular the geometry of the fin may play an important role. It was demonstrated that, as the thickness of a poorly conductive fin located at the mid height of the hot sidewall is reduced, the heat transfer through the cavity initially decreases until a critical thickness is reached. If the thickness is reduced further below the critical thickness, the heat transfer through the cavity increases again [15]. The fin length is another factor affecting natural convection flows in the cavity. If the length of an
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Nomenclature A f g H, L, W ΔH N Pr Ra t S T, Tfit T0 TUS TDS ΔT x, y β α σ ν
Aspect ratio Frequency (Hz) Acceleration due to gravity (m/s2) Height, length and width of the cavity (m) Height difference between T2 and T1 or Tf2 and Tf1 locations (m) Number of data samples used to calculate the moving root mean square temperature Prandtl number Rayleigh number Time (s) Stratification coefficient Measured and fitted temperatures (K) Initial temperature of the fluid (K) Fitted temperature from T1 or Tf1 (K) Fitted temperature from T2 or Tf2 (K) Temperature difference between sidewalls (K) Horizontal and vertical coordinates (m) Coefficient of thermal expansion (1/K) Thermal diffusivity (m2/s) Moving root mean square temperature (K) Kinematic viscosity (m2/s)
adiabatic fin is sufficiently large, secondary circulations arise at the upper and lower base corners of the fin [20]. It is shown in [20] that heat transfer through the finned sidewall is reduced as the fin length increases. Steady natural convection flows induced by an adiabatic fin have been the focus of previous studies (e.g. [20]). However, since unsteady natural convection in a cavity has many applications in industry, investigations of the transient thermal flows around an adiabatic fin have been performed by the present authors [21,22]. It was demonstrated that for different Rayleigh numbers the fin may play different roles in the thermal flow adjacent to the finned wall [20,23]. For low Rayleigh numbers, the thermal flow around the fin is laminar
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Table 1 Experimental parameters for temperature measurements.
No fin
Fin
Run
Initial temperature of water in cavity (T0, K)
Temperature difference between sidewalls (ΔT, K)
Ra (× 109)
Pr
1 2 3 4 5 6 7 8
294.45 294.45 295.55 294.15 296.65 296.55 295.55 295.55
4 8 16 32 4 8 16 32
0.88 1.75 3.77 6.87 1.01 2.00 3.77 7.54
6.83 6.83 6.63 6.81 6.45 6.47 6.63 6.63
and heat transfer through the finned wall is depressed [20]. On the other hand, for high Rayleigh numbers, the fin may trigger instabilities of the thermal flow around the fin, and thus enhance heat transfer through the finned wall, in comparison with that through the wall without a fin [23]. However, understanding of the instabilities induced by the fin is still incomplete. Furthermore, for the case without a fin, little attention has been paid to the instability properties of the thermal boundary layer in the transition from an initially isothermal state to an eventually stratified interior. Accordingly, temperature measurements using fast-response thermistors, aimed at studying the oscillations in the transition from an initially isothermal state to a stratified interior fluid in the cases with and without a fin on the sidewall, were performed in the present study. 2. Experimental setup In this study, the temperature measurements were carried out in the model cavity shown in Fig. 1. The rectangular cavity, containing water as the working fluid, is 1 m long, 0.24 m high, and 0.5 m wide. For the experiments with a fin, the fin of a cross-section of
Fig. 1. Schematic of the experimental model with a fin on the sidewall.
Fig. 2. Schematic of the thermistor locations. (a) In the case without a fin, the thermistor locations are numbered as T1 at (0.4978 m, − 0.048 m), T2 at (0.4974 m, 0.0624 m), T3 at (0.4970 m, 0.0899 m) and T4 at (0.4869 m, 0.0868 m). (b) In the case with a fin, the thermistor locations are numbered as Tf1 at (0.4983 m, − 0.0624 m), Tf2 at (0.4978 m, 0.0628 m), Tf3 at (0.4974 m, 0.0866 m), Tf4 at (0.4956 m, 0.007 m), and Tf5 at (0.4572 m, − 0.0013 m) respectively.
Fig. 3. Temperatures in the upstream thermal boundary layer for different Rayleigh numbers in the cases with and without a fin. (a) Temperatures from T1 at (0.4978 m, − 0.048 m) in the case without a fin. (b) Temperatures from Tf1 at (0.4983 m, − 0.0624 m) in the case with a fin.
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0.04 m × 0.002 m is attached at the mid height of the hot sidewall. The cavity walls and the fin are all made from Perspex sheets except the cooled and heated sidewalls, which are made of 1.13 mm thick goldcoated copper plates. In the experiments, one sidewall (or the finned sidewall) is heated by the hot water in a water bath, and the opposing sidewall is cooled by the cold water in another water bath. Initially, each water bath is separated from the sidewall by a pneumatically operated gate with an air gap between the gate and the sidewall (see Fig. 1). At the start of each experiment, the gates are rapidly lifted, and the hot and cold water in the two water baths flushes against the copper sidewalls.
Since this process is completed within a fraction of a second, an approximately instantaneous isothermal boundary condition is established at each end of the cavity. During the experiments, the temperatures of the water in the water baths, connected to two independent refrigerated/heating circulators, are kept constant with a temperature stability of ± 0.01 K. This or similar experimental apparatus and the start-up procedures have previously been adopted in a series of differentially heated cavity experiments [3,7]. The temperature of the natural convection flows adjacent to the sidewall was measured using Thermometrics FP07 thermistors with an accuracy of ±0.01 K. All thermistors were placed in the vicinity of
Fig. 4. Temperatures in the downstream boundary layer for different Rayleigh numbers in the cases with a fin at (0.4978 m, 0.0628 m) and without a fin at (0.4974 m, 0.0624 m).
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the hot sidewall in both cases with and without a fin. The thermistor locations are approximately shown in Fig. 2. Note that for the description of the physical locations, the origin of the coordinates is at the center of the cavity, as shown in Fig. 1. The development of natural convection flows in the cavity is determined by three dimensionless parameters governing the flow: the Rayleigh number (Ra), the Prandtl number (Pr) and the aspect ratio (A). They are defined as follows,
Ra =
gβΔTH3 ; να
ð1Þ
Pr =
ν ; α
ð2Þ
A=
H : L
ð3Þ
In the case with a fin, the fin length is also an important parameter governing the natural convection flows in the cavity, although it is fixed in this study. For the purpose of examining the oscillations, two sets of experiments with and without a fin on the sidewall are presented in this paper. The temperature differences between the two sidewalls range from 4 to 32 K, and the corresponding ranges of Ra and Pr are 0.88–7.54 × 109 and 6.45–6.83 respectively. All experimental conditions are listed in Table 1. Note that part of the temperature data obtained in the case with a fin has been reported in [8] to support the transient flow visualizations reported there.
Fig. 5. Stratification coefficient (S) vs time for different Rayleigh numbers in the cases (a) without and (b) with a fin.
3. Results and discussion 3.1. Impact of the fin on the measured temperatures in the thermal boundary layer Fig. 3 shows the time series of the temperatures in the upstream boundary layer in the cases with and without a fin. As previously described [6,12], an important behaviour in the early development of the thermal boundary layer adjacent to a suddenly heated vertical wall is the leading edge effect, characterized by an overshoot and subsequent travelling waves. In order to show these early features, the horizontal axes in Fig. 3 are drawn on a logarithmic scale. It is seen in this figure that the temperature in the upstream boundary layer first increases slightly until an overshoot occurs, then remains approximately constant for a certain period of time (dependent on the Rayleigh number), and eventually decreases and approaches quasi-steady states for different Rayleigh numbers. Indeed, all the temperature time series are oscillatory in the transition from start-up to quasi-steady states although the amplitude of the oscillations is small. The comparisons of the temperatures presented in Fig. 3a and b show that the development of the temperatures in the cases with and without a fin at equivalent Rayleigh numbers is similar with small variations possibly resulting from the slightly different measurement locations. This implies that the fin has a negligible impact on the temperature structures upstream of the fin, which was also confirmed by corresponding numerical simulations [23]. For the purpose of observing the effect of the fin on the thermal boundary layer, Fig. 4 shows the measured temperatures at a point downstream of the fin and at a corresponding location in the case without the fin. The impact of the presence of the fin on the temperatures downstream of the fin can be seen by comparing the cases (Fig. 4e–h) with those without the fin (Fig. 4a–d) at similar Rayleigh numbers. It is clear in Fig. 4e–h that the impact of the fin in the early transitional stage depends strongly on the Rayleigh number. For the highest Rayleigh number of 7.54 × 109, strong oscillations (at a
relatively low frequency) occur before 1600 s, after which time the amplitude of the oscillations significantly reduces with time (Fig. 4h). However, this early transient feature is very weak for the lowest Rayleigh number of 1.01 × 109 (Fig. 4e). The shadowgraph visualizations in [8] show that these oscillations at relatively low frequencies are produced when a starting plume bypassing the fin reattaches to the downstream vertical boundary layer. The interaction between the plume and the downstream vertical boundary layer during the reattachment process is very complex and depends strongly on the Rayleigh number. That is, for higher Rayleigh numbers, the interaction is stronger and triggers strong low frequency oscillations, as seen in Fig. 4g and h. However, for the lowest Rayleigh number case, the interaction between the plume and the downstream boundary layer is
Fig. 6. Moving root mean square temperature (σ) from T2 for Ra = 1.75 × 109 in the case without a fin. (a) σ vs t. (b) σ vs S.
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Fig. 7. Temperatures and power spectra from T1, T2, T3 and T4 for Ra = 1.75 × 109 in the case without a fin. (a) Temperatures from T1, T2 and T3. (b) Temperature from T4. (c) Spectrum for T1. (d) Spectrum for T2. (e) Spectrum for T3. (f) Spectrum for T4.
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weak, and thus the low frequency oscillations are also weak (Fig. 4e). With the passage of time, this early transient flow feature disappears and the thermal boundary layer downstream of the fin approaches a quasi-steady state, in which high frequency oscillations originating from an unstable fluid layer above the fin dominate the wave features of the vertical boundary layer downstream of the fin. These features will be discussed below.
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these high frequency oscillations originate from an unstable fluid layer with an adverse temperature gradient above the fin, in which the temperature increases towards the fin surface, resulting in a Rayleigh–Benard-type instability. As a consequence, the oscillations originating from the intermittent plumes are recorded by the thermistors at the downstream side of the fin, as shown in Fig. 4e. 3.3. Temperature oscillations and spectra in the case without a fin
3.2. Stratification of the interior fluid and its impact on temperature oscillations It is clear from above that the temperatures decrease with time in the upstream thermal boundary layer (Fig. 3) and increase with time in the downstream thermal boundary layer (Fig. 4), indicating that the stratification of the interior fluid continuously increases with time [23]. Indeed, the transition from an initially isothermal to an eventually stratified fluid ambient is an important feature of the natural convection flows in the suddenly differentially heated cavity. In order to quantitatively describe this transition, a dimensionless stratification coefficient (S) is defined as follows, S=
TDS −TUS ΔH
= ΔTH :
ð4Þ
To eliminate the effect of the temperature oscillations on the calculated stratification coefficient, polynomial curve fitted values (TUS and TDS) of the temperature time series are adopted in Eq. (4). Here, TUS is the fitted temperature from T1 or Tf1; TDS is the fitted temperature from T2 or Tf2; and ΔH is the height difference between the T2 and T1 or Tf2 and Tf1 locations. Fig. 5a shows the stratification coefficients for different Rayleigh numbers in the case without a fin. Clearly, the stratification coefficient increases nonlinearly with time; that is, the stratification coefficient increases rapidly between 1000 s and 3000 s but relatively slowly for t N 4000 s for the present Rayleigh numbers. Furthermore, the stratification coefficient also increases with the Rayleigh number. Fig. 5b shows the stratification coefficients in the case with a fin. Similarly to those in the case without a fin (Fig. 5a), the stratification coefficient here also increases with time for different Rayleigh numbers. However, S is larger than that in the case without a fin in the early stage (e.g. t = 1000 s). This is because the early oscillations induced by the reattachment of the starting plume to the downstream boundary layer enhance the heat transfer through the downstream sidewall [23], and in turn increase the temperatures downstream of the fin (also see Fig. 4e–h). As a consequence, the temperature difference between the downstream and upstream boundary layers is larger in the case with a fin in the early stage. It is worth noting that the curves of the stratification coefficients for the three higher Rayleigh numbers almost overlap and are significantly different from that for the lowest Rayleigh number of 1.01 × 109. The similarity of the measured stratification coefficients for the three high Rayleigh numbers is likely the result of turbulence induced by the fin. The visualizations presented in [8] show that distinct turbulence structures appear downstream of the fin for the three high Rayleigh numbers, resulting in a similar dimensionless temperature difference (S) (between the downstream and upstream boundary layers) for these Rayleigh numbers. The flow at the lowest Rayleigh number of 1.01 × 109 is different in that no clear turbulence structure is observed. Note that a further discussion of the turbulence structure adjacent to the sidewall is out of the scope of the present investigation. A re-examination of the temperature oscillations in the downstream boundary layer presented in Fig. 4 reveals an interesting flow behaviour for the lowest Rayleigh number case with the fin, that is, strong oscillations occur at around 2400 s (Fig. 4e). These oscillations are not observed in the case without a fin for an equivalent Rayleigh number (Fig. 4a). Numerical simulation by Xu et al. [23] indicates that
Linear stability analysis [14] indicates that the stratification may change the stability properties of the thermal flow adjacent to a vertical wall. For example, as seen in Fig. 4a–d, the amplitude of the oscillations increases with the continuously increasing stratification of the interior fluid. In order to quantify the amplitude increase, a moving root mean square temperature (σ), which is a function of time, is calculated as follows vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u T−Tfit 2 u∑ t ΔT σ = : N
ð5Þ
Here, N = 1000 with 500 samples before and after a specified time respectively, and a time-dependent fitted value (Tfit, polynomial curve fitted value) of the temperature time series is used to calculate the moving root mean square temperature. The value of σ is therefore a measure of the amplitude of the temperature oscillations about Tfit. Fig. 6a shows the moving root mean square temperature obtained from T2 against time. Clearly, σ increases with time. Furthermore, σ against S is plotted in Fig. 6b. Similarly, σ increases with S; that is, the amplitude of the temperature oscillations increases with the stratification of the interior fluid, which is consistent with the above qualitative observations in Fig. 4b. The frequency of the oscillations is an important parameter describing the oscillatory behaviour in the transition from start-up to a quasi-steady state. A preliminary spectral analysis reveals that the
Fig. 8. Dominant frequencies of the oscillations at different heights for different Rayleigh numbers in the case without a fin. (a) f vs y. (b) f vs Ra.
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Fig. 9. Temperatures and power spectra from Tf1, Tf2, Tf3, Tf4 and Tf5 in the quasi-stage stage for Ra = 2 × 109 in the case with a fin. (a) and (b) Temperatures from Tf1, Tf2, Tf3, Tf4 and Tf5. (c), (d), (e), (f) and (g) Spectra for Tf1, Tf5, Tf4, Tf2 and Tf3 respectively.
F. Xu et al. / International Communications in Heat and Mass Transfer 37 (2010) 350–359
dominant frequency of the oscillations remains approximately constant in the transition from an initially isothermal state to an eventually stratified ambient. Accordingly, only the frequency properties of the oscillations in the quasi-steady stage (from 4000 s to 5000 s) are described here. Fig. 7a replots the temperature time series recorded by T1, T2 and T3 in the thermal boundary layer for a short time period from 4900 s to 5000 s, for Ra = 1.75 × 109. Clearly, the oscillations grow as they propagate downstream. Fig. 7b separately plots the temperature time series from T4 because of its very small amplitude compared with those at the other locations. As shown in Fig. 2, T4 is placed about 0.01 m further from the sidewall than the other thermistors, and thus the oscillations recorded by T4 are much weaker than others. The spectra of the temperatures from different thermistors are shown in Fig. 7c–f respectively. It is seen in Fig. 7 that the frequency properties of the temperatures are different at different locations, but the dominant frequencies at all locations are between 0.1 and 0.124 Hz. Fig. 7c–e show that as the oscillations propagate downstream, their dominant frequency bands are narrowed. This is consistent with theoretical analyses [14], which suggest that the boundary layer adjacent to a thermal vertical wall acts as a filter that may narrow a wide frequency band of the oscillations; that is, the oscillations with the frequency around the dominant frequency are amplified when they travel downstream. Fig. 7f also shows that except the dominant frequency around 0.11 Hz, there is a peak of the low frequency, which is a result of the oscillation of the intrusion flow discharged from the end of the thermal boundary layer. For the purpose of further examining the frequency properties, Fig. 8a shows the dominant frequency against height for different Rayleigh numbers. An interesting feature shown in this figure is that as the oscillations propagate downstream, their dominant frequencies slightly increase for low Rayleigh numbers (e.g. Ra ≤ 2 × 109) but decrease for high Rayleigh numbers. The mechanism responsible for this phenomenon is unable to be obtained, based on only the current temperature measurements. The dependence of the dominant frequency on the Rayleigh number is shown in Fig. 8b. Clearly, the dominant frequency increases with the Rayleigh number, which is consistent with the shadowgraph flow visualizations by Xu et al. [7].
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result of the occurrence of turbulence structures at the downstream side of the fin (see the visualizations in [23]); that is, the turbulence structures induced by the fin widen the frequency band of the temperature oscillations, as expected. Fig. 10a shows the dominant frequency against height for different Rayleigh numbers in the cases with a fin. It is seen in this figure that the dominant frequencies of the oscillations induced by the fin are rather different from and consistently lower than those in the case without a fin for the equivalent Rayleigh numbers (Fig. 8a). The dominant frequencies of the oscillations at the upstream side of the fin change less for different Rayleigh numbers and are considerably lower. This implies that the fin plays an important role in the oscillation propagation. The dependence of the dominant frequency on the Rayleigh number in the cases with a fin is shown in Fig. 10b. Evidently, the relationship between the frequency and Rayleigh number in the case with a fin is different from that in the case without a fin (Fig. 8b) although the dominant frequencies of the oscillations at the downstream side of the fin still increase with the Rayleigh number. However, at the upstream side of the fin, the dominant frequencies do not increase and change less as the Rayleigh number increases, which is consistent with the arguments in Fig. 8a. Fig. 11 presents the moving root mean square temperature (σ, calculated from 4000 s to 5000 s) at different heights for different Rayleigh numbers in the cases with and without a fin. The comparisons between the cases with and without a fin reveal that the oscillations induced by the fin are apparently stronger than those in the case without a fin except for the Rayleigh number of 7.54 × 109 and at the further downstream position for the Rayleigh number of 3.77 × 109. The exceptional examples (in Fig. 11c and d) are the result of the turbulent burst; that is, the distinct waves at the downstream side of the fin break down to turbulence and the amplitudes of these broken waves are smaller than those unbroken waves in the case without a fin for high Rayleigh numbers, which is also confirmed by the widening of the power spectra in the case with a fin.
3.4. Temperature oscillations and spectra in the case with a fin in a quasi-steady state In order to display further the oscillatory properties of the boundary layer in the case with a fin, Fig. 9a shows the temperatures from Tf2, Tf3 and Tf4 over a short period of 100 s in the quasi-steady stage. It is seen in this figure that the amplitude of the temperature oscillations at this stage is strongly dependent on the spatial location but less dependent on time. The temperature from Tf4 displays the oscillations in the leeward of the fin although these oscillations are very weak, and those from Tf2 and Tf3 display the oscillations in the downstream thermal boundary layer. The oscillations originating from the unstable fluid layer around the fin grow significantly as they propagate downstream. Furthermore, the quasi-steady state temperatures from Tf1 and Tf5 (refer to Fig. 2 for their locations) are replotted in Fig. 9b. These two temperature time series are also oscillatory although their amplitudes are small. The corresponding power spectra for Tf1 to Tf5 are shown in Fig. 9c–g respectively. It is clear in these spectra that the frequency distributions of the oscillations are considerably different at these locations. In particular, the temperature from Tf1 at the upstream side of the fin has a lower dominant frequency around 0.02 Hz, but the dominant frequencies of the temperatures near the fin and at the downstream side of the fin are around 0.07 Hz. It is worth noting that the spectra of the temperatures further downstream of the fin (e.g. those for Tf2 and Tf3) have a wider frequency band (different from those in the case without a fin in Fig. 7). This is a
Fig. 10. Dominant frequencies of the oscillations at different heights for different Rayleigh numbers in the case with a fin. (a) f vs y. (b) f vs Ra.
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Fig. 11. Root mean square (σ) of the temperatures at different heights for different Rayleigh numbers in the cases with a fin (solid line) and without a fin (dashed line).
4. Conclusions
Acknowledgment
In this study, the temperatures in the thermal flows adjacent to the sidewall of a differentially heated cavity with and without a fin were measured by a set of fast-response thermistors. The oscillatory behaviours were observed and characterized based on these temperature measurements. The temperature measurements show that, in the case without a fin, the stratification of the interior fluid following start-up increases nonlinearly with time. The temperature time series at different locations clearly characterize the development of the oscillations in the transition from an initially isothermal state to an eventually stratified fluid ambient. That is, as the stratification of the interior fluid increases, the amplitude of the oscillations increases. These oscillations are indeed convective instabilities and are consistent with the theoretical analysis [14]. The temperature measurements also show that, in the case with the presence of a fin on the sidewall, the unstable fluid layer above the fin triggers stronger oscillations in the downstream boundary layer in comparison with the case without a fin, which is consistent with the visualizations in Xu et al. (2008). These stronger oscillations, which are caused by a Rayleigh–Benard-type instability in the flow above the fin, are also confirmed by numerical results in [23]. Furthermore, the present spectral analysis reveals that the frequency properties of the oscillations induced by the fin with a lower dominant frequency are considerably different from those in the case without a fin. The oscillatory properties of the boundary layer are a key to developing processes by which the boundary layer may be triggered into transition to turbulence, and the consequent enhancement of the total heat transfer. This paper has outlined the wave properties during the development of the quasi-steady flow, and in particular has emphasised the differences brought about by the presence of a single horizontal non-conductive fin on the heated wall. This has clear implications for the manipulation of the boundary layer flows to enhance heat transfer.
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International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / i c h m t
Temperature oscillations in a differentially heated cavity with and without a fin on the sidewall☆ Feng Xu ⁎, John C. Patterson, Chengwang Lei School of Civil Engineering, The University of Sydney, Sydney, NSW 2006, Australia
a r t i c l e
i n f o
Available online 2 February 2010 Keywords: Fin Oscillation Thermal boundary layer
a b s t r a c t Temperature measurements of the thermal flows in a differentially heated cavity with and without a fin on the sidewall were performed. The oscillatory behaviours in the transition from an initially isothermal state to an eventually stratified interior fluid ambient are characterized. It is revealed that, for the case without a fin, the amplitude of the oscillations in the thermal boundary layer increases as the stratification of the interior fluid increases. For the case with a fin, the temperature measurements show that the oscillations are induced by an unstable fluid layer above the fin. The spectral analysis reveals that the frequency properties of the oscillations are different between the cases with and without a fin. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Natural convection flows are present in many industrial systems such as solar collectors and nuclear reactors. In particular, transient flows in a differentially heated cavity are frequently encountered in industrial applications, and thus have been investigated extensively in the literature [1–3]. Since heat is transferred into and out of the differentially heated cavity through the sidewalls, the thermal boundary layer flows adjacent to the sidewalls are naturally the focus of investigations. Previous studies (e.g. [1]) showed that following sudden heating, a growing thermal boundary layer forms and a one-dimensional vertical flow is driven by buoyancy forces. This one-dimensional flow persists for a short period of time, and then perturbations, which are referred to as the leading edge effect (LEE), arise in the vicinity of the leading edge of the vertical wall and travel downstream [4–6]. After the LEE travels away, the local boundary layer flow becomes steady and two-dimensional. It is worth noting that the interior fluid in the cavity following start-up is usually isothermal. The top and bottom walls of the cavity force horizontal intrusions into the interior, which separate from and then reattach to the top and bottom walls respectively [7,8]. The interior fluid in the cavity eventually becomes stratified. Understanding of the instability of the boundary layer adjacent to the sidewall of a differentially heated cavity is also important since the flow regime (which could be either stable or unstable) determines heat transfer through the sidewall. Indeed, the instability of the thermal boundary layer has been widely investigated for over five ☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (F. Xu). 0735-1933/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2010.01.004
decades [9,10]. It was demonstrated that only a narrow band of the frequencies of the oscillations in the thermal boundary layer is significantly amplified downstream. Most of the studies of the thermal boundary layer instability considered the case with a thermal vertical wall placed in an isothermal fluid [11,12]. However, natural convection often occurs in a thermally stratified fluid ambient and thus the effect of the stratified ambient on the instability of the thermal boundary layer was also investigated [13,14]. Improving or depressing natural convection flows in the cavity and in turn controlling heat transfer through the cavity are of considerable practical importance in industrial applications. A simple way to manipulate the flow regimes adjacent to the sidewall and in turn to enhance or depress heat transfer is to place a fin on the sidewall of the cavity. Investigations of the thermal flow induced by a fin on the sidewall have been extensively reported in the literature (e.g. [15,16]). Previous studies have considered the effects of the material properties and geometry of the fin. Clearly, the fin may be either conductive or adiabatic. It is expected that a conductive fin attached to the sidewall increases the surface area for heat transfer, and in turn increases the overall heat transfer through the cavity. Accordingly, the effect of the thermal conductivity of different fin materials on heat transfer through the finned sidewall was investigated [17–19]. An adiabatic or poorly conducting fin may also have a significant impact on natural convection flows in a differentially heated cavity and the heat transfer through the cavity, and in particular the geometry of the fin may play an important role. It was demonstrated that, as the thickness of a poorly conductive fin located at the mid height of the hot sidewall is reduced, the heat transfer through the cavity initially decreases until a critical thickness is reached. If the thickness is reduced further below the critical thickness, the heat transfer through the cavity increases again [15]. The fin length is another factor affecting natural convection flows in the cavity. If the length of an
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Nomenclature A f g H, L, W ΔH N Pr Ra t S T, Tfit T0 TUS TDS ΔT x, y β α σ ν
Aspect ratio Frequency (Hz) Acceleration due to gravity (m/s2) Height, length and width of the cavity (m) Height difference between T2 and T1 or Tf2 and Tf1 locations (m) Number of data samples used to calculate the moving root mean square temperature Prandtl number Rayleigh number Time (s) Stratification coefficient Measured and fitted temperatures (K) Initial temperature of the fluid (K) Fitted temperature from T1 or Tf1 (K) Fitted temperature from T2 or Tf2 (K) Temperature difference between sidewalls (K) Horizontal and vertical coordinates (m) Coefficient of thermal expansion (1/K) Thermal diffusivity (m2/s) Moving root mean square temperature (K) Kinematic viscosity (m2/s)
adiabatic fin is sufficiently large, secondary circulations arise at the upper and lower base corners of the fin [20]. It is shown in [20] that heat transfer through the finned sidewall is reduced as the fin length increases. Steady natural convection flows induced by an adiabatic fin have been the focus of previous studies (e.g. [20]). However, since unsteady natural convection in a cavity has many applications in industry, investigations of the transient thermal flows around an adiabatic fin have been performed by the present authors [21,22]. It was demonstrated that for different Rayleigh numbers the fin may play different roles in the thermal flow adjacent to the finned wall [20,23]. For low Rayleigh numbers, the thermal flow around the fin is laminar
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Table 1 Experimental parameters for temperature measurements.
No fin
Fin
Run
Initial temperature of water in cavity (T0, K)
Temperature difference between sidewalls (ΔT, K)
Ra (× 109)
Pr
1 2 3 4 5 6 7 8
294.45 294.45 295.55 294.15 296.65 296.55 295.55 295.55
4 8 16 32 4 8 16 32
0.88 1.75 3.77 6.87 1.01 2.00 3.77 7.54
6.83 6.83 6.63 6.81 6.45 6.47 6.63 6.63
and heat transfer through the finned wall is depressed [20]. On the other hand, for high Rayleigh numbers, the fin may trigger instabilities of the thermal flow around the fin, and thus enhance heat transfer through the finned wall, in comparison with that through the wall without a fin [23]. However, understanding of the instabilities induced by the fin is still incomplete. Furthermore, for the case without a fin, little attention has been paid to the instability properties of the thermal boundary layer in the transition from an initially isothermal state to an eventually stratified interior. Accordingly, temperature measurements using fast-response thermistors, aimed at studying the oscillations in the transition from an initially isothermal state to a stratified interior fluid in the cases with and without a fin on the sidewall, were performed in the present study. 2. Experimental setup In this study, the temperature measurements were carried out in the model cavity shown in Fig. 1. The rectangular cavity, containing water as the working fluid, is 1 m long, 0.24 m high, and 0.5 m wide. For the experiments with a fin, the fin of a cross-section of
Fig. 1. Schematic of the experimental model with a fin on the sidewall.
Fig. 2. Schematic of the thermistor locations. (a) In the case without a fin, the thermistor locations are numbered as T1 at (0.4978 m, − 0.048 m), T2 at (0.4974 m, 0.0624 m), T3 at (0.4970 m, 0.0899 m) and T4 at (0.4869 m, 0.0868 m). (b) In the case with a fin, the thermistor locations are numbered as Tf1 at (0.4983 m, − 0.0624 m), Tf2 at (0.4978 m, 0.0628 m), Tf3 at (0.4974 m, 0.0866 m), Tf4 at (0.4956 m, 0.007 m), and Tf5 at (0.4572 m, − 0.0013 m) respectively.
Fig. 3. Temperatures in the upstream thermal boundary layer for different Rayleigh numbers in the cases with and without a fin. (a) Temperatures from T1 at (0.4978 m, − 0.048 m) in the case without a fin. (b) Temperatures from Tf1 at (0.4983 m, − 0.0624 m) in the case with a fin.
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0.04 m × 0.002 m is attached at the mid height of the hot sidewall. The cavity walls and the fin are all made from Perspex sheets except the cooled and heated sidewalls, which are made of 1.13 mm thick goldcoated copper plates. In the experiments, one sidewall (or the finned sidewall) is heated by the hot water in a water bath, and the opposing sidewall is cooled by the cold water in another water bath. Initially, each water bath is separated from the sidewall by a pneumatically operated gate with an air gap between the gate and the sidewall (see Fig. 1). At the start of each experiment, the gates are rapidly lifted, and the hot and cold water in the two water baths flushes against the copper sidewalls.
Since this process is completed within a fraction of a second, an approximately instantaneous isothermal boundary condition is established at each end of the cavity. During the experiments, the temperatures of the water in the water baths, connected to two independent refrigerated/heating circulators, are kept constant with a temperature stability of ± 0.01 K. This or similar experimental apparatus and the start-up procedures have previously been adopted in a series of differentially heated cavity experiments [3,7]. The temperature of the natural convection flows adjacent to the sidewall was measured using Thermometrics FP07 thermistors with an accuracy of ±0.01 K. All thermistors were placed in the vicinity of
Fig. 4. Temperatures in the downstream boundary layer for different Rayleigh numbers in the cases with a fin at (0.4978 m, 0.0628 m) and without a fin at (0.4974 m, 0.0624 m).
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the hot sidewall in both cases with and without a fin. The thermistor locations are approximately shown in Fig. 2. Note that for the description of the physical locations, the origin of the coordinates is at the center of the cavity, as shown in Fig. 1. The development of natural convection flows in the cavity is determined by three dimensionless parameters governing the flow: the Rayleigh number (Ra), the Prandtl number (Pr) and the aspect ratio (A). They are defined as follows,
Ra =
gβΔTH3 ; να
ð1Þ
Pr =
ν ; α
ð2Þ
A=
H : L
ð3Þ
In the case with a fin, the fin length is also an important parameter governing the natural convection flows in the cavity, although it is fixed in this study. For the purpose of examining the oscillations, two sets of experiments with and without a fin on the sidewall are presented in this paper. The temperature differences between the two sidewalls range from 4 to 32 K, and the corresponding ranges of Ra and Pr are 0.88–7.54 × 109 and 6.45–6.83 respectively. All experimental conditions are listed in Table 1. Note that part of the temperature data obtained in the case with a fin has been reported in [8] to support the transient flow visualizations reported there.
Fig. 5. Stratification coefficient (S) vs time for different Rayleigh numbers in the cases (a) without and (b) with a fin.
3. Results and discussion 3.1. Impact of the fin on the measured temperatures in the thermal boundary layer Fig. 3 shows the time series of the temperatures in the upstream boundary layer in the cases with and without a fin. As previously described [6,12], an important behaviour in the early development of the thermal boundary layer adjacent to a suddenly heated vertical wall is the leading edge effect, characterized by an overshoot and subsequent travelling waves. In order to show these early features, the horizontal axes in Fig. 3 are drawn on a logarithmic scale. It is seen in this figure that the temperature in the upstream boundary layer first increases slightly until an overshoot occurs, then remains approximately constant for a certain period of time (dependent on the Rayleigh number), and eventually decreases and approaches quasi-steady states for different Rayleigh numbers. Indeed, all the temperature time series are oscillatory in the transition from start-up to quasi-steady states although the amplitude of the oscillations is small. The comparisons of the temperatures presented in Fig. 3a and b show that the development of the temperatures in the cases with and without a fin at equivalent Rayleigh numbers is similar with small variations possibly resulting from the slightly different measurement locations. This implies that the fin has a negligible impact on the temperature structures upstream of the fin, which was also confirmed by corresponding numerical simulations [23]. For the purpose of observing the effect of the fin on the thermal boundary layer, Fig. 4 shows the measured temperatures at a point downstream of the fin and at a corresponding location in the case without the fin. The impact of the presence of the fin on the temperatures downstream of the fin can be seen by comparing the cases (Fig. 4e–h) with those without the fin (Fig. 4a–d) at similar Rayleigh numbers. It is clear in Fig. 4e–h that the impact of the fin in the early transitional stage depends strongly on the Rayleigh number. For the highest Rayleigh number of 7.54 × 109, strong oscillations (at a
relatively low frequency) occur before 1600 s, after which time the amplitude of the oscillations significantly reduces with time (Fig. 4h). However, this early transient feature is very weak for the lowest Rayleigh number of 1.01 × 109 (Fig. 4e). The shadowgraph visualizations in [8] show that these oscillations at relatively low frequencies are produced when a starting plume bypassing the fin reattaches to the downstream vertical boundary layer. The interaction between the plume and the downstream vertical boundary layer during the reattachment process is very complex and depends strongly on the Rayleigh number. That is, for higher Rayleigh numbers, the interaction is stronger and triggers strong low frequency oscillations, as seen in Fig. 4g and h. However, for the lowest Rayleigh number case, the interaction between the plume and the downstream boundary layer is
Fig. 6. Moving root mean square temperature (σ) from T2 for Ra = 1.75 × 109 in the case without a fin. (a) σ vs t. (b) σ vs S.
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Fig. 7. Temperatures and power spectra from T1, T2, T3 and T4 for Ra = 1.75 × 109 in the case without a fin. (a) Temperatures from T1, T2 and T3. (b) Temperature from T4. (c) Spectrum for T1. (d) Spectrum for T2. (e) Spectrum for T3. (f) Spectrum for T4.
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weak, and thus the low frequency oscillations are also weak (Fig. 4e). With the passage of time, this early transient flow feature disappears and the thermal boundary layer downstream of the fin approaches a quasi-steady state, in which high frequency oscillations originating from an unstable fluid layer above the fin dominate the wave features of the vertical boundary layer downstream of the fin. These features will be discussed below.
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these high frequency oscillations originate from an unstable fluid layer with an adverse temperature gradient above the fin, in which the temperature increases towards the fin surface, resulting in a Rayleigh–Benard-type instability. As a consequence, the oscillations originating from the intermittent plumes are recorded by the thermistors at the downstream side of the fin, as shown in Fig. 4e. 3.3. Temperature oscillations and spectra in the case without a fin
3.2. Stratification of the interior fluid and its impact on temperature oscillations It is clear from above that the temperatures decrease with time in the upstream thermal boundary layer (Fig. 3) and increase with time in the downstream thermal boundary layer (Fig. 4), indicating that the stratification of the interior fluid continuously increases with time [23]. Indeed, the transition from an initially isothermal to an eventually stratified fluid ambient is an important feature of the natural convection flows in the suddenly differentially heated cavity. In order to quantitatively describe this transition, a dimensionless stratification coefficient (S) is defined as follows, S=
TDS −TUS ΔH
= ΔTH :
ð4Þ
To eliminate the effect of the temperature oscillations on the calculated stratification coefficient, polynomial curve fitted values (TUS and TDS) of the temperature time series are adopted in Eq. (4). Here, TUS is the fitted temperature from T1 or Tf1; TDS is the fitted temperature from T2 or Tf2; and ΔH is the height difference between the T2 and T1 or Tf2 and Tf1 locations. Fig. 5a shows the stratification coefficients for different Rayleigh numbers in the case without a fin. Clearly, the stratification coefficient increases nonlinearly with time; that is, the stratification coefficient increases rapidly between 1000 s and 3000 s but relatively slowly for t N 4000 s for the present Rayleigh numbers. Furthermore, the stratification coefficient also increases with the Rayleigh number. Fig. 5b shows the stratification coefficients in the case with a fin. Similarly to those in the case without a fin (Fig. 5a), the stratification coefficient here also increases with time for different Rayleigh numbers. However, S is larger than that in the case without a fin in the early stage (e.g. t = 1000 s). This is because the early oscillations induced by the reattachment of the starting plume to the downstream boundary layer enhance the heat transfer through the downstream sidewall [23], and in turn increase the temperatures downstream of the fin (also see Fig. 4e–h). As a consequence, the temperature difference between the downstream and upstream boundary layers is larger in the case with a fin in the early stage. It is worth noting that the curves of the stratification coefficients for the three higher Rayleigh numbers almost overlap and are significantly different from that for the lowest Rayleigh number of 1.01 × 109. The similarity of the measured stratification coefficients for the three high Rayleigh numbers is likely the result of turbulence induced by the fin. The visualizations presented in [8] show that distinct turbulence structures appear downstream of the fin for the three high Rayleigh numbers, resulting in a similar dimensionless temperature difference (S) (between the downstream and upstream boundary layers) for these Rayleigh numbers. The flow at the lowest Rayleigh number of 1.01 × 109 is different in that no clear turbulence structure is observed. Note that a further discussion of the turbulence structure adjacent to the sidewall is out of the scope of the present investigation. A re-examination of the temperature oscillations in the downstream boundary layer presented in Fig. 4 reveals an interesting flow behaviour for the lowest Rayleigh number case with the fin, that is, strong oscillations occur at around 2400 s (Fig. 4e). These oscillations are not observed in the case without a fin for an equivalent Rayleigh number (Fig. 4a). Numerical simulation by Xu et al. [23] indicates that
Linear stability analysis [14] indicates that the stratification may change the stability properties of the thermal flow adjacent to a vertical wall. For example, as seen in Fig. 4a–d, the amplitude of the oscillations increases with the continuously increasing stratification of the interior fluid. In order to quantify the amplitude increase, a moving root mean square temperature (σ), which is a function of time, is calculated as follows vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u T−Tfit 2 u∑ t ΔT σ = : N
ð5Þ
Here, N = 1000 with 500 samples before and after a specified time respectively, and a time-dependent fitted value (Tfit, polynomial curve fitted value) of the temperature time series is used to calculate the moving root mean square temperature. The value of σ is therefore a measure of the amplitude of the temperature oscillations about Tfit. Fig. 6a shows the moving root mean square temperature obtained from T2 against time. Clearly, σ increases with time. Furthermore, σ against S is plotted in Fig. 6b. Similarly, σ increases with S; that is, the amplitude of the temperature oscillations increases with the stratification of the interior fluid, which is consistent with the above qualitative observations in Fig. 4b. The frequency of the oscillations is an important parameter describing the oscillatory behaviour in the transition from start-up to a quasi-steady state. A preliminary spectral analysis reveals that the
Fig. 8. Dominant frequencies of the oscillations at different heights for different Rayleigh numbers in the case without a fin. (a) f vs y. (b) f vs Ra.
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Fig. 9. Temperatures and power spectra from Tf1, Tf2, Tf3, Tf4 and Tf5 in the quasi-stage stage for Ra = 2 × 109 in the case with a fin. (a) and (b) Temperatures from Tf1, Tf2, Tf3, Tf4 and Tf5. (c), (d), (e), (f) and (g) Spectra for Tf1, Tf5, Tf4, Tf2 and Tf3 respectively.
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dominant frequency of the oscillations remains approximately constant in the transition from an initially isothermal state to an eventually stratified ambient. Accordingly, only the frequency properties of the oscillations in the quasi-steady stage (from 4000 s to 5000 s) are described here. Fig. 7a replots the temperature time series recorded by T1, T2 and T3 in the thermal boundary layer for a short time period from 4900 s to 5000 s, for Ra = 1.75 × 109. Clearly, the oscillations grow as they propagate downstream. Fig. 7b separately plots the temperature time series from T4 because of its very small amplitude compared with those at the other locations. As shown in Fig. 2, T4 is placed about 0.01 m further from the sidewall than the other thermistors, and thus the oscillations recorded by T4 are much weaker than others. The spectra of the temperatures from different thermistors are shown in Fig. 7c–f respectively. It is seen in Fig. 7 that the frequency properties of the temperatures are different at different locations, but the dominant frequencies at all locations are between 0.1 and 0.124 Hz. Fig. 7c–e show that as the oscillations propagate downstream, their dominant frequency bands are narrowed. This is consistent with theoretical analyses [14], which suggest that the boundary layer adjacent to a thermal vertical wall acts as a filter that may narrow a wide frequency band of the oscillations; that is, the oscillations with the frequency around the dominant frequency are amplified when they travel downstream. Fig. 7f also shows that except the dominant frequency around 0.11 Hz, there is a peak of the low frequency, which is a result of the oscillation of the intrusion flow discharged from the end of the thermal boundary layer. For the purpose of further examining the frequency properties, Fig. 8a shows the dominant frequency against height for different Rayleigh numbers. An interesting feature shown in this figure is that as the oscillations propagate downstream, their dominant frequencies slightly increase for low Rayleigh numbers (e.g. Ra ≤ 2 × 109) but decrease for high Rayleigh numbers. The mechanism responsible for this phenomenon is unable to be obtained, based on only the current temperature measurements. The dependence of the dominant frequency on the Rayleigh number is shown in Fig. 8b. Clearly, the dominant frequency increases with the Rayleigh number, which is consistent with the shadowgraph flow visualizations by Xu et al. [7].
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result of the occurrence of turbulence structures at the downstream side of the fin (see the visualizations in [23]); that is, the turbulence structures induced by the fin widen the frequency band of the temperature oscillations, as expected. Fig. 10a shows the dominant frequency against height for different Rayleigh numbers in the cases with a fin. It is seen in this figure that the dominant frequencies of the oscillations induced by the fin are rather different from and consistently lower than those in the case without a fin for the equivalent Rayleigh numbers (Fig. 8a). The dominant frequencies of the oscillations at the upstream side of the fin change less for different Rayleigh numbers and are considerably lower. This implies that the fin plays an important role in the oscillation propagation. The dependence of the dominant frequency on the Rayleigh number in the cases with a fin is shown in Fig. 10b. Evidently, the relationship between the frequency and Rayleigh number in the case with a fin is different from that in the case without a fin (Fig. 8b) although the dominant frequencies of the oscillations at the downstream side of the fin still increase with the Rayleigh number. However, at the upstream side of the fin, the dominant frequencies do not increase and change less as the Rayleigh number increases, which is consistent with the arguments in Fig. 8a. Fig. 11 presents the moving root mean square temperature (σ, calculated from 4000 s to 5000 s) at different heights for different Rayleigh numbers in the cases with and without a fin. The comparisons between the cases with and without a fin reveal that the oscillations induced by the fin are apparently stronger than those in the case without a fin except for the Rayleigh number of 7.54 × 109 and at the further downstream position for the Rayleigh number of 3.77 × 109. The exceptional examples (in Fig. 11c and d) are the result of the turbulent burst; that is, the distinct waves at the downstream side of the fin break down to turbulence and the amplitudes of these broken waves are smaller than those unbroken waves in the case without a fin for high Rayleigh numbers, which is also confirmed by the widening of the power spectra in the case with a fin.
3.4. Temperature oscillations and spectra in the case with a fin in a quasi-steady state In order to display further the oscillatory properties of the boundary layer in the case with a fin, Fig. 9a shows the temperatures from Tf2, Tf3 and Tf4 over a short period of 100 s in the quasi-steady stage. It is seen in this figure that the amplitude of the temperature oscillations at this stage is strongly dependent on the spatial location but less dependent on time. The temperature from Tf4 displays the oscillations in the leeward of the fin although these oscillations are very weak, and those from Tf2 and Tf3 display the oscillations in the downstream thermal boundary layer. The oscillations originating from the unstable fluid layer around the fin grow significantly as they propagate downstream. Furthermore, the quasi-steady state temperatures from Tf1 and Tf5 (refer to Fig. 2 for their locations) are replotted in Fig. 9b. These two temperature time series are also oscillatory although their amplitudes are small. The corresponding power spectra for Tf1 to Tf5 are shown in Fig. 9c–g respectively. It is clear in these spectra that the frequency distributions of the oscillations are considerably different at these locations. In particular, the temperature from Tf1 at the upstream side of the fin has a lower dominant frequency around 0.02 Hz, but the dominant frequencies of the temperatures near the fin and at the downstream side of the fin are around 0.07 Hz. It is worth noting that the spectra of the temperatures further downstream of the fin (e.g. those for Tf2 and Tf3) have a wider frequency band (different from those in the case without a fin in Fig. 7). This is a
Fig. 10. Dominant frequencies of the oscillations at different heights for different Rayleigh numbers in the case with a fin. (a) f vs y. (b) f vs Ra.
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Fig. 11. Root mean square (σ) of the temperatures at different heights for different Rayleigh numbers in the cases with a fin (solid line) and without a fin (dashed line).
4. Conclusions
Acknowledgment
In this study, the temperatures in the thermal flows adjacent to the sidewall of a differentially heated cavity with and without a fin were measured by a set of fast-response thermistors. The oscillatory behaviours were observed and characterized based on these temperature measurements. The temperature measurements show that, in the case without a fin, the stratification of the interior fluid following start-up increases nonlinearly with time. The temperature time series at different locations clearly characterize the development of the oscillations in the transition from an initially isothermal state to an eventually stratified fluid ambient. That is, as the stratification of the interior fluid increases, the amplitude of the oscillations increases. These oscillations are indeed convective instabilities and are consistent with the theoretical analysis [14]. The temperature measurements also show that, in the case with the presence of a fin on the sidewall, the unstable fluid layer above the fin triggers stronger oscillations in the downstream boundary layer in comparison with the case without a fin, which is consistent with the visualizations in Xu et al. (2008). These stronger oscillations, which are caused by a Rayleigh–Benard-type instability in the flow above the fin, are also confirmed by numerical results in [23]. Furthermore, the present spectral analysis reveals that the frequency properties of the oscillations induced by the fin with a lower dominant frequency are considerably different from those in the case without a fin. The oscillatory properties of the boundary layer are a key to developing processes by which the boundary layer may be triggered into transition to turbulence, and the consequent enhancement of the total heat transfer. This paper has outlined the wave properties during the development of the quasi-steady flow, and in particular has emphasised the differences brought about by the presence of a single horizontal non-conductive fin on the heated wall. This has clear implications for the manipulation of the boundary layer flows to enhance heat transfer.
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