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On the identification of flow instabilities in a differentially-heated closed cavity: Non-intrusive measurements
International Journal of Heat and Mass Transfer xxx (xxxx) xxx
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On the identification of flow instabilities in a differentially-heated closed cavity: Non-intrusive measurements Vimal Kishor a, Suneet Singh a, Atul Srivastava b,⇑ a b
Department of Energy Science and Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India Department of Mechanical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
a r t i c l e
i n f o
Article history: Received 11 September 2019 Received in revised form 19 October 2019 Accepted 20 October 2019 Available online xxxx Keywords: Natural convection Differentially-heated fluid layer Flow instabilities Non-intrusive measurements
a b s t r a c t Transition from laminar to turbulent flow in differentially heated closed cavities has been studied quite extensively using numerical simulations, however, experimental studies are rather limited. In one of the numerical works Henkes and Hoogendoorn (1990), it has been indicated that two different mechanisms simultaneously lead to instabilities in the flow, which, in turn, are identified in the form of two frequencies in flow oscillations. The aim of the present work is to identify and investigate these two mechanisms using non-intrusive experiments. As a definite advancement over the conventional ways of predicting flow instabilities using physical probes, which often lead to undesirable flow perturbation(s), the present measurements have been performed in a complete non-intrusive manner. To achieve this, the experimental test section has been mapped using a laser-based Mach Zehnder interferometer. Interferometry-based results are further corroborated using smoke visualization technique. The experiments are conducted for three Rayleigh numbers in the transition regime. The dynamics of the flow is analyzed near one of the corners of the cavity where instability is likely to appear first. Using spectral analysis of the real-time temperature field, frequencies of oscillations have been identified. It is noted that as one moves inside the boundary layer, the amplitude corresponding to one of the two frequencies becomes quite dominant, which indicates towards the presence of Tollmien-Schlichting (TS) instability. Outside the boundary layer, another frequency, along with the above-mentioned frequency, is observed. This additional frequency is due to the instability caused by the hydraulic jump. With further increment in Rayleigh number, more frequencies are found in the flow, which lead to sudden enhancement in heat transfer from the thermally active walls of the cavity. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction Natural convection inside the differentially heated cavities has been studied numerically and experimentally quite extensively. Several two-dimensional numerical studies have been reported in the literature in which natural convection in air has been studied in the laminar and turbulent region for square cavities [1–5]. The Boussinesq approximation was used to solve such types of numerical problems [1,3,6–10]. These studies have been further extended to a different aspect ratio of the cavity, different inclinations of the cavity and for different fluids. In some of the numerical studies in differentially heated cavity, the boundary conditions were maintained in such a way that two vertical walls were kept isothermal and other walls were kept adiabatic and vice versa. Paolucci and Chenoweth [7] studied the problem in a closed rectangular ⇑ Corresponding author. E-mail address: [email protected] (A. Srivastava).
differentially heated cavity to identify the first transition to oscillatory flow from laminar. The authors reported the route to chaos by discussing the instabilities which appear in the flow whenever there is a transition of flow from laminar to periodic for air (Pr = 0.71). Further, more numerical studies were performed with air in two dimensional differentially heated cavities in the transition region [8,11,12]. In these studies, it was reported that when the flow is periodic, the interaction between the vertical boundary layer and horizontal flow causes the internal wave instability in the flow. These studies were performed for different Prandtl numbers (Pr 1) and aspect ratios 1. Later, work was done to identify the presence of first oscillation in the flow whenever there is a transition of flow from laminar to periodic [13,14]. The study to understand the transition in the flow was extended to threedimensional numerical work where the instabilities in the flow were observed [15,16]. In some numerical studies, the route to turbulence is discussed after the transition from steady state to periodic [4,17–19].
https://doi.org/10.1016/j.ijheatmasstransfer.2019.118933 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: V. Kishor, S. Singh and A. Srivastava, On the identification of flow instabilities in a differentially-heated closed cavity: Non-intrusive measurements, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118933
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Nomenclature h k Nu Pr q‘‘ Ra
heat transfer coefficient, W/m2/K thermal conductivity of fluid, W/mK Nusselt number, hL/k Prandtl number, m/a heat flux, hDT, W/m2 Rayleigh number, gbDTL3/ ma
Experimental investigations of natural convection in a differentially heated cavity also have been reported in the literature. Seki et al. [20] performed flow visualization experiments in a differentially heated cavity by varying the Prandtl number of the working fluid and the aspect ratio of the cavity (by changing the width of the cavity). Authors discussed the effect of Prandtl number and aspect ratio on the flow behavior qualitatively. Srivastava et al. [21] also presented an experimental study of Rayleigh-Bernard natural convection (top wall of the cavity is maintained at a relatively lower temperature than the temperature of the bottom wall and the other two walls were kept insulated) in a rectangular cavity. Experiments were carried out for water and air as a working fluid and results from three different optical techniques (interferometry, shadowgraph, and schlieren) were compared. For low Rayleigh number, results from three techniques were comparable, but at higher Rayleigh number, due to the presence of closely-spaced fringes near the thermally active wall, it becomes difficult to resolve such fringes resulting in possible loss of fringes and, therefore, the accuracy of interferometry results reduces. Experiments in vertical differentially heated cavity of high aspect ratio were also performed for high Rayleigh number (order of 1011) [22]. Authors reported the effect of temperature difference between the vertical walls and the wall emissivity on the vertical temperature gradients. Salat et al. [23] performed an experimental and numerical study in a differentially heated square cavity for higher Rayleigh number (1.5 109) with air as the working fluid. They measured the temperature at the mid-depth of the cavity using micro-sized thermocouples. The work concluded that there were some physical ingredients lacking in the study which were important for a better understanding of the discrepancy on the thermal stratification. Tian and Karayiannis [24] conducted experiments in differentially heated cavity to study natural convection in low turbulence region for air. Intrusive probe technique was employed to measure the temperature. The study claimed that experiments were carried out with high accuracy and these experimental results can be used as a benchmark for further Computational Fluid Dynamics (CFD) code validation. Arnold et al. [25] carried out an experimental study in differentially heated cavities for different aspect ratio. They reported the results as a variation of heat transfer rates with inclination angles of the cavity. Ivey [26] performed experiments to visualize the flow inside a differentially heated square cavity for high Rayleigh number for which the flow was found to be periodic. Experiments were carried out with a mixture of water and glycol to vary the Prandtl number and Rayleigh number. The author observed the hydraulic jump near the corner and concluded to have high heat transfer in this region. Wright et al. [27] performed experiments in a vertical differentially heated cavity of high aspect ratio. Smoke pattern visualization and interferometry were employed to conduct these experiments. It was concluded that beyond a Rayleigh number (104), the flow becomes periodic and three-dimensional motion of air was observed in the cavity. Kishor et al. [28] conducted natural convection experiments in a differentially heated cavity of aspect ratio three using a non-intrusive optical technique (MachZehnder Interferometry). Authors reported the comparison of
Greek symbols a thermal diffusivity, m2/s b volumetric coefficient of thermal expansion, 1/K m kinematic viscosity, m2/s q fluid density, kg/m3
experimental data obtained in the form of whole field temperature distribution and Nusselt number on the thermally active walls with numerically simulated results. Experimentally obtained results showed a good agreement with numerical results. Saury et al. [29] reported an experimental work in which they studied the problem in the transition region (laminar to turbulent transition). Authors performed the experiments using an intrusive probe technique in a differentially heated cavity of aspect ratio four for different inclination angle to identify the transition in the flow. They proposed a unique modified critical Rayleigh number (beyond which flow becomes oscillatory) for different inclination using modified characteristic length. There have been limited studies on the transition to turbulence for buoyancy driven flows in cavities, especially of high aspect ratios. Moreover, most of these studies are based on numerical simulations. The flow characteristics in the transition region are susceptible to type and size of perturbations, and, hence there is a significant discrepancy in the numerical and experimental results. However, the experimental studies on the transition to turbulence in these type of cavities are almost non-existent. One such recent study was carried out using intrusive probes [29]. The use of probes may lead to enhancement in perturbations, thereby affecting the results significantly. Moreover, the comparison between the flow transients at different points of the cavity is not carried out in these reported work. The experimental work reported in the present manuscript is an effort to bridge some of the above-mentioned gaps in the literature and to identify the flow instabilities in differentially-heated cavities of high aspect ratios in a purely non-intrusive manner. The buoyancy-induced convective field in the cavity has been mapped using one of the variants of classical interferometry technique, namely Mach Zehnder interferometer (MZI). The MZI yields the temperature information in a completely non-intrusive and real time manner, which can then be used to understand the flow phenomena. In order to further confirm the flow patterns, smokebased flow visualization experiments are carried out. The temperature difference across the two vertical walls of the cavity has been applied and the rest of the two walls (horizontal) have been kept insulated. The top corner of the cavity near the cold wall has been identified as the most critical region where the flow was found to be periodic. The analysis of the experimental results is carried out inside and outside of the boundary layer region to understand the different mechanisms of flow instabilities. Effect of periodicity on heat transfer rates has been presented using the local Nusselt number profiles on the top of the cold wall.
2. Apparatus and instrumentation 2.1. Test cell In this section, the description of the experimental cavity is given. Experiments are performed in a differentially heated closed cavity of aspect ratio three, which is shown in Fig. 1(a). The photographic view of the cavity is also shown in Fig. 1(b). The main focus
Please cite this article as: V. Kishor, S. Singh and A. Srivastava, On the identification of flow instabilities in a differentially-heated closed cavity: Non-intrusive measurements, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118933
V. Kishor et al. / International Journal of Heat and Mass Transfer xxx (xxxx) xxx
3
4 100m
2 3
5
300m
1
600m Laser Beam Fig. 1. (a) Schematic of the differentially heated cavity [28]; (1) Aluminium sheets, (2) Bakelite sheets, (3) Electric heaters (Strips heater), (4) Thermocouples, (5) Cooling bath chamber; (b) Photographic view of the test cavity.
of this study is to understand the flow in the transition region when the flow becomes turbulent from laminar. Therefore, dimensions of the cavity are chosen on the basis of numerical simulations which were performed for different aspect ratios so that the transition region can be achieved in the cavity. In the literature, it is also reported that with the increment in the aspect ratio of the cavity, critical Rayleigh number (beyond which the flow becomes oscillatory) decreases. On the basis of simulations, the aspect ratio of the cavity was fixed as three. The dimensions of the cavity are 300 mm (height) 100 mm (width) 600 mm (depth). Two vertical walls of the cavity which are maintained at two different temperatures are made of aluminium (because of its high thermal conductivity). The temperature on the hot wall is maintained using six electric strip heaters of size 600 mm 48 mm 4 mm. These heaters are further connected to six different temperature controllers, which take feedback from the six thermocouples. These thermocouples are inserted in the aluminium plate at different heights (covering the full face of the vertical plate) to measure the temperature at different heights of the wall. The temperature on the hot wall is measured with an accuracy of ’±0.2 °C. On the back side of heaters, Bakelite is used to minimize the loss of heat to the surroundings. Another vertical wall is maintained at a relatively lower temperature than the hot wall. A cooling chamber is attached on the back side of the wall which has enough number of baffles so that it can provide a zig-zag path for the circulating water through it. It also ensures maximum contact of the thermostated water with the wall to maintain a uniform temperature. The other two horizontal walls of the cavity are kept insulated and the insulation is provided using Bakelite (due to low thermal conductivity) sheet of thickness 15 mm. The other two faces are open to ambient through which the laser beam is passed to record the depth-averaged interferograms inside the cavity. The experiments are performed with air (Pr = 0.71) as the working fluid in the cavity. The temperature differences between the vertical walls are kept at 10 °C, 14 °C, and 18 °C. To achieve the temperature difference, the cold wall of the cavity is maintained at 20 °C and the hot wall is maintained at three different temperatures of 30 °C, 34 °C, and 38 °C.
2.2. Mach Zehnder interferometer Experimental measurements reported in this paper are performed using a non-invasive technique. To measure the convective flow which sets up in the cavity due to natural convection after applying the temperature potential between the two vertical walls, the laser-based non-intrusive technique (Mach Zehnder interferometer) has been employed. The isometric view of Mach Zehnder interferometer (MZI) with the differentially heated closed cavity is shown in Fig. 2. In the setup of MZI, two mirrors and two beam splitters have been used, which are aligned at an angle of 45° to the propagation of the laser beam, as shown in the figure. All the optical components used in the MZI are achromatic in nature. The optical components (mirrors, beam splitter, etc.) used in setup are of diameter 100 mm which give a collimated beam of size ’ 60 mm. It is ensured that the light beam always falls on the center of the optical components to avoid any type of aberrations in the light beam. As shown in Fig. 2, a He-Ne laser of wavelength 632.8 nm is used as the coherent light source. The coherent light coming from the laser is passed through a collimating lens to have a collimating beam of light. This collimated beam is allowed to fall on a beam splitter (BS1) where the beam divides into two parts (two low-intensity beams). One of these beams reflects from the mirror (M1) and reaches to beam splitter (BS2) as shown in the figure. This section of the interferometer (from BS1 to BS2 via M1) is called the reference arm where the compensation chamber is kept. In this study, experiments are performed for air. Therefore, there is no need for the compensation chamber. Another section is known as the test arm where the test cavity is placed (from BS1 to BS2 via M2). The reference beam and the test beam superimpose on the beam splitter (BS2) and create constructive and destructive interference patterns. These interference patterns are recorded using a Charged Coupled device (CCD) camera, which is connected to a personal computer. The MZI works in two different modes, i.e. infinite and wedge fringe setting mode. In the infinite mode of MZI, the fringes represent the temperature contours in the convective field. At reference temperature condi-
Please cite this article as: V. Kishor, S. Singh and A. Srivastava, On the identification of flow instabilities in a differentially-heated closed cavity: Non-intrusive measurements, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118933
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Fig. 2. Isometric view of Mach-Zehnder interferometer with differentially heated cavity.
tion, the constructive maxima (full bright fringe) confirm that the interferometer is in infinite setting mode. The infinite setting mode is achieved by aligning the two arms of the interferometer in such a way that path lengths of the two arms are almost equal under reference condition of temperature. While a slight misalignment in the optical component is made to get horizontal or vertical straight equidistant fringes to obtain the wedge setting under reference conditions. After applying the temperature potential across the vertical wall of the cavity, the convective field is recorded in the personal computer. The whole Mach Zehnder interferometer setup is placed on the pressurized pneumatic vibration isolation optical table to minimize external disturbances such as people movement, sudden floor vibrations, etc. The whole field temperature data has been extracted from the interferometric images recorded by MZI under wedge fringe setting. The details of the data reduction technique to extract data from these images can be found in a recent publication [28]. The accuracy and feasibility of interferometry-based measurement technique in the context of temperature and heat transfer measurements in differentially-heated fluid layer has also been discussed in [28]. To obtain accurate quantitative data, a large number of data reduction techniques are reported in the literature [30–32]. The validation of these techniques for different applications also has been carried out in some of our earlier works [33–35]. After getting the whole field temperature distribution at four corners of the cavity, the local Nusselt number on the wall can be calculated. Using the principle of energy balance, energy balance in mathematical terms near the thermally active walls can be written as:
@T k ¼ hðT c T h Þ @x x¼0orx¼L
ð1Þ
Here, k is thermal conductivity of the fluid (air for present experiments) and @T=@x represents the near wall temperature gradients. Finally, the local Nusselt number (non-dimensionalized heat transfer rate) has been calculated using the following formula:
Nu ¼
hL k
ð2Þ
Here, L is the distance between the hot and cold wall of the differentially heated cavity.
2.3. Smoke-based flow visualization arrangement Fig. 3(a) shows the schematic of the setup for the smoke-sbased flow visualization experiments. A photographic view of the smoke generator (Fog Generator SAFEX 195 SG D) has been shown in Fig. 3(b). For smoke-based visualization, necessary modifications have been made in the test cavity. The top Bakelite sheet, that was used in interferometry-based experiments, has been replaced with an acrylic sheet, as shown in the figure. This acrylic sheet allows the diffuse light to pass through it and also has low thermal conductivity to act as an insulator wall. The front face of the cavity has been covered with an acrylic sheet because of its transparency and the back face of the cavity is kept dark so that a good contrast can be produced to capture good quality smoke patterns. Once the flow is setup inside the cavity after applying the temperature potential across the two vertical walls of the cavity, an optimum quantity of smoke produced by the fog generator (shown in Fig. 3(b) is put inside the cavity through its front face and thereafter, the front face is covered with acrylic sheet. The diffuse light source is kept outside the top wall and it illuminates the smoke inside the cavity, as schematically depicted in the figure. Once the flow is established inside the cavity, smoke patterns are recorded using a DLSR camera (Canon EOS 200D 24.2 MP) operating at a frame rate of 50 frames per second.
3. Results and discussion In this section, transient analysis of time-dependent temperature in the flow field is presented for different Rayleigh numbers. The temperature difference across the vertical thermally active walls has been varied from 10 °C to 18 °C in the interval of 4 °C and the corresponding Rayleigh numbers are 9.7 105, 1.3 106 and 1.7 106.1 Experiments are performed for air as the working fluid (Pr = 0.71). The top left corner of the cavity (top region near the cold wall of the cavity) is selected for observation where the instability is expected to appear first. This transient study has been performed on the whole field temperature data retrieved from the experiments performed using Mach-Zehnder Interferometer (MZI). 1 The choice of cavity temperature differences (Rayleigh numbers) employed in the reported experiments has been made so as to make sure that the flow field is in the transition regime (from laminar to turbulent). The working flow regime was ascertained through detailed numerical simulations and also based on the previously reported works available in the literature.
Please cite this article as: V. Kishor, S. Singh and A. Srivastava, On the identification of flow instabilities in a differentially-heated closed cavity: Non-intrusive measurements, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118933
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Dark background
Diffuse light
Test cell
Acrylic sheet DSLR camera (Canon EOS 200D 24.2 MP) (a)
Smoke generator (Fog Generator SAFEX 195 SG D) (b)
Fig. 3. (a) Schematic showing the relative positions of diffuse light, image acquisition system and experimental test section for smoke-based flow visualization inside the cavity; (b) Smoke generator.
On the basis of real-time transient temperature data obtained from the experiments, spectral analysis is carried out. A total of 500 frames, recorded at 30 frames per second, are analyzed. In the selected region (top left corner) of the cavity, the time variation of temperature variation at different points is recorded and Fast Fourier Transform (FFT) has been performed to identify the frequency(ies) of oscillations present in the flow. Later, transient analysis has also been performed using smoke-based flow visualization. Using FFT of intensity variation with time, which is obtained through smoke visualization, instabilities in the flow are studied. 3.1. Flow instability Experiments are performed for varying Rayleigh numbers in the cavity. For Ra = 9.7 105, four points are chosen near the cold wall in the top left region of the cavity to identify the dominant frequency in the boundary layer region, i.e. in the vicinity of the thermally active walls. The non-dimensionalized locations (with respect to the width of the cavity) of these points are given in Table 1. The spectral analysis of time-varying fluid temperature, as retrieved through interferometry technique, is performed on these points. The whole field temperature contours obtained through quantitative analysis of the interferometric images corresponding to the four corners of the cavity have been presented in the form of Fig. 8(a–d) in one of the later sections of this manuscript. In order to provide a direct quantitative evidence of the
Table 1 Coordinates of the points (non-dimensionalized with respect to the width of the cavity) where FFT of time variation of temperature has been performed. The top left corner is taken as the reference point. Figure
4
6
7
(a) (b) (c) (d)
(0.05,0.1) (0.13,0.1) (0.21,0.1) (0.35,0.1)
(0.35,0.2) (0.4,0.4) – –
(0.05,0.1) (0.35,0.25) – –
oscillations in the flow field, one of the representative time histories of temperature at a select point lying inside the boundary layer region in the top left corner of the cavity has also been shown in Fig. 8(e). The FFT plots of temperature variations are shown in Fig. 4 for a cavity temperature difference of 10 °C at four different locations (spatial coordinates of these points have been provided in Table 1). One can see that two peaks of different amplitudes are present in Fig. 4(a), corresponding to two different frequencies. However, as one goes towards the wall (from Fig. 4(a)–(d)), one of these two peaks starts dominating. The points closer to the wall are inside the boundary layer and at these points, the oscillations corresponding to one of the frequencies almost get vanished (Fig. 4 (d)). These selected points are indeed inside the boundary layer, which can be seen in Figs. 8 and 9 later in this section. Further details of the discussion on the identification of the boundary layer region using interferometry technique can be found in earlier work [28]. The frequency which is seen at all the locations is approximately 0.42 Hz while the other frequency is 0.72 Hz. For higher Rayleigh number also, the dominance of a single frequency has been observed in the boundary layer region. Further, flow visualization for the cavity has also been performed using the smoke-based technique to observe the flow patterns inside the cavity and the associated instabilities in the flow. Representative smoke patterns recorded in the top portion of the cavity for a temperature difference of 10 °C (Ra = 9.7 105) are shown in Fig. 10. The transient analysis of the intensity values captured through smoke-based flow visualization has been performed. The FFT analysis has been carried out on the intensity variation with time in the cavity at different points. Fig. 5 shows the results of the spectral analysis for two points (one near the boundary of the cavity and another in the core of the cavity). As observed earlier in the spectral analysis of the temperature data obtained from the MZI near the boundary, intensity variation spectrum plot (Fig. 5(a)) also shows the dominance of single frequency near the boundary. While in the core of the cavity, the presence of multiple frequencies is to be observed. Near the
Please cite this article as: V. Kishor, S. Singh and A. Srivastava, On the identification of flow instabilities in a differentially-heated closed cavity: Non-intrusive measurements, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118933
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Fig. 4. Amplitude spectrum of temperature variation with time near the top portion of the left (cold wall) boundary for temperature difference of DT = 10 °C.
Fig. 5. Amplitude spectrum of intensity variation with time in smoke-based flow visualization for temperature difference of DT = 10 °C (a) near the left boundary, (b) away from the left boundary.
Please cite this article as: V. Kishor, S. Singh and A. Srivastava, On the identification of flow instabilities in a differentially-heated closed cavity: Non-intrusive measurements, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118933
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Fig. 6. Amplitude spectrum of temperature variation with time at two different points away from the left boundary (cold wall) for temperature difference of DT = 10 °C.
Fig. 7. Amplitude spectrum of temperature variation with time (a) near the left boundary, (b) away from the left boundary for temperature difference of DT = 10 °C.
boundary, the frequency which is dominating is 0.3 Hz and away from the boundary, multiple frequencies of 0.36, 0.66, and 0.9 Hz have been identified. Thus, in addition to interferometry-based approach, the presence of instabilities in the flow has also been confirmed using an independent experimental approach based on the concept of smoke visualization. The presence of an additional frequency may be attributed to the fact that the front and back surfaces of the cavity were open for the experiments performed using the interferometry technique, while they were closed by the acrylic sheet for smoke visualization. Moreover, it is also seen that the amplitude corresponding to this frequency is relatively quite weak. With the smoke-based visualization experiments also confirming the presence of flow instabilities in the cavity, further studies have been carried out using the whole field temperature data recorded by employing Mach-Zehnder interferometry. Following this technique, the interferometric images have been quantitatively analysed to retrieve the whole field temperature distribution in the fluid layer. Details of the data reduction methodology
followed to retrieve the temperature field distribution from the recorded interferograms may be seen in some of the earlier works of the authors [28,30–35]. In order to verify that the experimentally observed frequencies near the corner are present throughout the cavity, the FFT analysis of temperature data is performed at two different spatial locations in the core of the cavity (the coordinate of these locations are given in Table 1). The spectrum plots at these locations are shown in Fig. 6. The constant presence of two dominant frequencies is seen throughout the cavity, which is the same as those observed near the wall (shown earlier in Fig. 5(a)). As the temperature difference between the thermally active walls is increased to 14 °C (Ra = 1.3 106), though the number of frequencies present in the flow remain the same, the amplitude of oscillations increases due to an increase in the strength of driving potential (buoyancy force). When the temperature difference is further increased to 18 °C (Ra = 1.7 106), two more frequencies appear in the flow, indicating the increased level of
Please cite this article as: V. Kishor, S. Singh and A. Srivastava, On the identification of flow instabilities in a differentially-heated closed cavity: Non-intrusive measurements, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118933
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Fig. 8. (a, b, c and d) Experimentally obtained whole field temperature contours at the four corners of the differentially heated cavity for DT = 10 °C (Ra = 9.7 105); (e) Transient history of temperature at one of the points located inside the boundary layer region in the top left corner of the cavity. Full field temperature contours for this region have been shown in sub-figure (a).
flow instabilities in the fluid medium. The amplitude spectrum plots corresponding to two different spatial locations in the flow domain for 18 °C temperature difference have been shown in Fig. 7 (coordinates of the spatial locations are given in Table 1). As can be seen from the Fig. 7(b), the spectral analysis of the transient temperature field of the spatial locations away from the boundary shows the presence of a total of four different frequencies of the flow oscillations (0.36 Hz, 0.78 Hz, 1.2 Hz, and 1.56 Hz). On the other hand, three frequencies are to be seen for the point which is nearer to the wall or in the boundary layer region, as presented in Fig. 7(a).
3.2. Discussion on flow instabilities As discussed in the previous section, periodicity in the flow is distinctly observed and the presence of dominant frequencies has been found using spectral analysis of the temperature variation with time. The two frequencies observed at Ra = 9.7 105 (DT = 10 °C) confirm the presence of two types of instabilities in the flow. It is observed that in the boundary layer region, the flow is unstable due to the presence of a single dominant frequency. This instability in the flow arises due to the disturbance in the boundary layer region near the thermally active walls of the cavity.
Please cite this article as: V. Kishor, S. Singh and A. Srivastava, On the identification of flow instabilities in a differentially-heated closed cavity: Non-intrusive measurements, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118933
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Cold wall
Top left corner
(b)
(c)
Right lower corner (d)
Hot wall
(a)
Fig. 9. Interferograms recorded at 10C difference under infinite fringe setting (a) top corner of cold wall, (b) top corner of hot wall, (c) bottom corner of cold wall and (d) bottom corner of hot wall.
When the temperature potential is increased across the two walls of the cavity, the velocity of the fluid in the boundary layer region increases and causes instability in the flow.2 The thus induced instability in the boundary layer region corresponds to the Tollmien-Schlichting instability.3 This type of instability is usually seen in the boundary layers for various flow situations and its importance has been reported in the available literature quite extensively [1,36]. In the present work, the frequency corresponding to Tollmien-Schlichting instability has been identified as 0.42 Hz on the basis of purely non-intrusive measurement technique and has been observed in the boundary layer region of the cavity. The
2 The fluid medium employed in the present work is air with Prandtl number almost of the order of 1 (Pr = 0.71). In view of this, the scales of thermal and velocity boundary layers (momentum and thermal diffusivities) are expected to be almost same. Thus, the instabilities induced in the flow field due to the increased temperature potential across the two vertical walls of the differentially heated cavity would also be reflected in the temperature field. This justifies the present approach of identifying the flow oscillation frequencies in the boundary layer region using the time-dependent thermal field instead of directly using the velocity field. Frequency corresponding to Tollmien-Schlichting instability in the boundary layer region has been shown in Fig. 4(d). 3 It is widely believed that Tollmien-Schlichting instability is primarily associated with Blasius boundary layer. However, one of the numerical works reported by Henkes and Hoogendoorn [1] has reported the existence of this type of instability in the boundary layer region in the context of natural convection in differentially-heated cavities. The identification of this instability under similar configuration, as reported in the present work based on non-intrusive measurements, provides a direct experimental support to the observations made by Henkes and Hoogendoorn [1].
boundary layer can be identified in the temperature contours and interferograms in the four corners of the cavity, as shown in Figs. 8 and 9, respectively. It is important to discuss the physics of the flow inside the cavity for understanding other instabilities present in the flow. The interferograms recorded under wedge fringe setting mode of Mach-Zehnder interferometer have been used to obtain the whole field temperature data (in the field of view), as shown in Fig. 8. While the wedge fringe setting of the interferometer is useful for quantitative analysis, the thermally induced flow field can be directly interpreted on the basis of the interferograms recorded under infinite fringe setting mode of the interferometer. Under this setting, the fringe patterns are the representation of the lines of constant temperature (isotherms) and hence the region of strong temperature gradients can be identified from the recorded images. Thus, with the experimental data available from two different approaches, the physics of the flow inside the cavity has been interpreted by combining the quantitative data on whole field temperature distributions (temperature contours obtained from the wedge interferograms, Fig. 8) and the real-time interferograms recorded under infinite fringe setting mode of the Mach-Zehnder interferometer (Fig. 9). From Fig. 9, the boundary layer region in the close vicinity of the cavity walls can be distinctly identified. With reference to the image shown in Fig. 9(d), the thermal gradients are expected to be maximum in the lower right corner of the cavity as the fluid comes in contact with the hot wall in this region. Corresponding
Please cite this article as: V. Kishor, S. Singh and A. Srivastava, On the identification of flow instabilities in a differentially-heated closed cavity: Non-intrusive measurements, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118933
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temperature contours (shown in Fig. 8(d)) also support this observation in the quantitative terms. Closely-spaced temperature contours (Fig. 8(d)) and fringe patterns (Fig. 9(d)) reflect towards the presence of strong temperature gradients in this region and hence one expects high heat transfer rates in the bottom right corner of the fluid domain. After taking heat from the hot wall, the fluid density decreases and it rises along the hot wall and strikes the horizontal wall of the cavity on the top. While moving up, the hot fluid transfers some heat to the core fluid as well. The boundary layer formation on the hot wall can be seen in the figures (Figs. 8 and 9). This relatively hot fluid changes its direction and now comes in contact with the cold side wall and transfers maximum thermal energy to the cold wall in the top region of the cavity (Figs. 8(a) and 9(a)). The closely-spaced temperature contours plot (Fig. 8(a)) in this region confirm the phenomenon of maximum heat transfer. After transferring the thermal energy to the cold wall, the fluid density increases again and it starts to move down along the cold wall. The fluid strikes the horizontal wall on the bottom and further changes its direction (Figs. 8(c) & 9(c)) and this cycle continues in the cavity and heat transfer from the hot to the cold wall takes place. The fluid movement inside the cavity, as interpreted based on the interferometry experiments, has also been corroborated using smoke-based flow visualization. One of the representative images of the fluid flow in the top region of the cavity has been shown in Fig. 10. Based on the above presented discussion on the fluid movement along the two vertical walls of the cavity (fluid rising along the right (hot) wall and descending along the left (colder) wall), it can be inferred that the fluid moving in the vertical direction
Hot wall
Cold wall
Top right corner
Fig. 10. Representative image of fluid movement inside the cavity recorded using smoke-based flow visualization technique for cavity temperature difference of 10 °C. For guide to eye, possible fluid movement in the cavity has also been schematically indicated using white line.
Table 2 Comparison of frequencies of oscillations identified through the present work with those reported in the literature in the transition region. Frequency (Hz)
f1 f2 f3 f4
Present work (Non-intrusive measurements)
DT = 10 °C
DT = 14 °C
DT = 18 °C
0.36 0.69 – –
0.36 0.69 – –
0.36 0.69 1.2 1.56
Saury et al. [29]
0.35 0.65 0.92 –
strikes the horizontal surfaces (walls) of the cavity. This phenomenon of the fluid striking the walls of the cavity is commonly known as hydraulic jump. When the velocity of the fluid striking the wall is relatively high, it may lead to disturbance in the flow. The instability that arises due to the hydraulic jump can be seen in the corners where vertically moving fluid interacts with the horizontally moving fluid. The energy which is generated due to the hydraulic jump gets released to the fluid in the form of periodicity leading to the instability in the flow. The plausible evidence of the hydraulic jump-induced instability has been discussed in the previous section, wherein, for Ra = 9.7 105, two distinct frequencies were identified in the flow fields which, in turn, indicate towards the presence of two types of flow instabilities. Out of these two frequencies, the oscillation in the flow corresponding to frequency 0.69 Hz is due to the instability caused by the hydraulic jump. The observations made in the form of the presence of two distinct frequencies find support in the work reported by Saury et al. [29] wherein the authors identified multiple frequencies (three) in the flow for a cavity of aspect ratio four. Table 2 summarizes the various frequencies as a function of increasing temperature potential (DT) applied across the two thermally active walls of the cavity, as found in the present work. For a direct comparison, frequencies reported by Saury et al. [29] have also been tabulated in the last column. As can be seen, the experimentally determined frequencies in the present work are reasonably close to the values reported in the literature. It is worth mentioning here that while Saury et al. [29] determined these frequencies using an intrusive way of measurements, the experimental analysis performed in the present work has been carried out in a purely non-intrusive manner. This, in turn, highlights the importance of the present work as the usage of probe of finite physical dimensions leads to undesirable perturbation(s) in the fluid medium. This limitation gets completely overcome through non-intrusive measurement techniques, as employed in the present experimental study. 3.3. Signatures of the instabilities in the heat transfer profiles The experimental findings discussed in the previous section established the presence of instabilities (multiple flow oscillation frequencies) in the flow based on the smoke visualization experiments as well as on the basis of temperature field obtained through interferometry. The onset of flow instabilities directly influences the heat transfer phenomenon as the heat transfer rates are expected to significantly increase as the instability(ies) appear in the flow field. Following this, heat transfer rates in terms of Nusselt number have been calculated in the top left corner of the cavity (near the cold wall) using the whole field temperature data retrieved through interferometric analysis. The variation of the local Nusselt number along the length of the cold wall that corresponds to 1/6th of the total height of the cavity has been shown in Fig. 11 for three different temperature potentials (DT = 10 °C, 14 °C & 18 °C) applied across the two thermally active walls of the cavity. Nusselt number variations over 1/6th of the height of the cavity have been shown as the interest is in the corner flows where the instabilities in the flow field were detected. It is worth noting here
Please cite this article as: V. Kishor, S. Singh and A. Srivastava, On the identification of flow instabilities in a differentially-heated closed cavity: Non-intrusive measurements, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118933
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oscillates predominantly with single frequency while amplitude corresponding to the other frequency is quite weak. The dominance of this instability in the boundary layer indicates towards the existence of the well-known Tollmien-Schlichting instability. It is important to note that this type of instability is commonly observed in boundary layers for various flow configurations. On the other hand, second instability is present away from the boundary layer and corresponds to the hydraulic jump, which is caused by the flow striking the horizontal wall of the cavity. The fluid which moves vertically along the thermally active walls strikes the horizontal wall and causes this instability. Moreover, it is concluded that when there is inclusion of more frequencies in the flow due to increment in the Rayleigh number, the instabilities cause a sudden enhancement of heat transfer on the thermally active walls.
Declaration of Competing Interest The authors declared that there is no conflict of interest. Fig. 11. Variation of local Nusselt number on the cold wall in the top region (top left corner) of the cavity for different temperature differences.
that the flow is found to be in the transition regime for the three Rayleigh numbers employed in the experiments. It can be observed from Fig. 11 that the profile of the experimentally obtained local Nusselt number values for DT = 18 °C shows a steeper slope in comparison with that seen in the profile corresponding to DT = 14 °C. However, this sudden change in the slope of the Nusselt number profile is not to be observed when the temperature difference is increased from 10 °C to 14 °C. Using the spatial variations, average heat transfer rates in terms of Nusselt number for three temperature differences (DT = 10 °C, 14 °C & 18 °C) have been calculated. The average Nusselt numbers have been found to be 8.81 (DT = 10 °C), 9.63 (DT = 14 °C) and 12.02 (DT = 18 °C). Thus, while for DT = 10 °C and 14 °C, the average value of Nusselt numbers almost comparable with each other, one observes a significant increase in the average Nusselt number as the cavity temperature difference is increased from 14 °C to 18 °C (~25%). The sudden jump in the heat transfer rates when the temperature difference is increased from 14 °C to 18 °C can be attributed to the appearance of two additional oscillation frequencies in the flow field when the temperature potential across the fluid layer is 18 °C. On the other hand, when the temperature difference is changed from 10 °C to 14 °C, the number of frequencies do not change. Hence, the appearance of additional two frequencies in the flow field leads to a significant enhancement in the heat transfer rates, which, in turn, indicate towards the additional instabilities in the flow field. 4. Conclusions Laminar to turbulent transition has been studied experimentally for a differentially heated cavity of aspect ratio three. The study has been carried out using non-intrusive techniques, i.e. Mach Zehnder Interferometer and smoke-based visualization technique. Using real-time temperature measurements inside the cavity, the instabilities in the flow have been distinctly identified. Primary attention has been paid in understanding the convective field in the four corners of the cavity where the flow is expected to become periodic first. For the experiments performed for 10 °C temperature difference between the thermally active walls, the flow was in the transition region and the oscillations with two frequencies were observed. Near the thermally active walls, fluid
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On the identification of flow instabilities in a differentially-heated closed cavity: Non-intrusive measurements Vimal Kishor a, Suneet Singh a, Atul Srivastava b,⇑ a b
Department of Energy Science and Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India Department of Mechanical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
a r t i c l e
i n f o
Article history: Received 11 September 2019 Received in revised form 19 October 2019 Accepted 20 October 2019 Available online xxxx Keywords: Natural convection Differentially-heated fluid layer Flow instabilities Non-intrusive measurements
a b s t r a c t Transition from laminar to turbulent flow in differentially heated closed cavities has been studied quite extensively using numerical simulations, however, experimental studies are rather limited. In one of the numerical works Henkes and Hoogendoorn (1990), it has been indicated that two different mechanisms simultaneously lead to instabilities in the flow, which, in turn, are identified in the form of two frequencies in flow oscillations. The aim of the present work is to identify and investigate these two mechanisms using non-intrusive experiments. As a definite advancement over the conventional ways of predicting flow instabilities using physical probes, which often lead to undesirable flow perturbation(s), the present measurements have been performed in a complete non-intrusive manner. To achieve this, the experimental test section has been mapped using a laser-based Mach Zehnder interferometer. Interferometry-based results are further corroborated using smoke visualization technique. The experiments are conducted for three Rayleigh numbers in the transition regime. The dynamics of the flow is analyzed near one of the corners of the cavity where instability is likely to appear first. Using spectral analysis of the real-time temperature field, frequencies of oscillations have been identified. It is noted that as one moves inside the boundary layer, the amplitude corresponding to one of the two frequencies becomes quite dominant, which indicates towards the presence of Tollmien-Schlichting (TS) instability. Outside the boundary layer, another frequency, along with the above-mentioned frequency, is observed. This additional frequency is due to the instability caused by the hydraulic jump. With further increment in Rayleigh number, more frequencies are found in the flow, which lead to sudden enhancement in heat transfer from the thermally active walls of the cavity. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction Natural convection inside the differentially heated cavities has been studied numerically and experimentally quite extensively. Several two-dimensional numerical studies have been reported in the literature in which natural convection in air has been studied in the laminar and turbulent region for square cavities [1–5]. The Boussinesq approximation was used to solve such types of numerical problems [1,3,6–10]. These studies have been further extended to a different aspect ratio of the cavity, different inclinations of the cavity and for different fluids. In some of the numerical studies in differentially heated cavity, the boundary conditions were maintained in such a way that two vertical walls were kept isothermal and other walls were kept adiabatic and vice versa. Paolucci and Chenoweth [7] studied the problem in a closed rectangular ⇑ Corresponding author. E-mail address: [email protected] (A. Srivastava).
differentially heated cavity to identify the first transition to oscillatory flow from laminar. The authors reported the route to chaos by discussing the instabilities which appear in the flow whenever there is a transition of flow from laminar to periodic for air (Pr = 0.71). Further, more numerical studies were performed with air in two dimensional differentially heated cavities in the transition region [8,11,12]. In these studies, it was reported that when the flow is periodic, the interaction between the vertical boundary layer and horizontal flow causes the internal wave instability in the flow. These studies were performed for different Prandtl numbers (Pr 1) and aspect ratios 1. Later, work was done to identify the presence of first oscillation in the flow whenever there is a transition of flow from laminar to periodic [13,14]. The study to understand the transition in the flow was extended to threedimensional numerical work where the instabilities in the flow were observed [15,16]. In some numerical studies, the route to turbulence is discussed after the transition from steady state to periodic [4,17–19].
https://doi.org/10.1016/j.ijheatmasstransfer.2019.118933 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: V. Kishor, S. Singh and A. Srivastava, On the identification of flow instabilities in a differentially-heated closed cavity: Non-intrusive measurements, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118933
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Nomenclature h k Nu Pr q‘‘ Ra
heat transfer coefficient, W/m2/K thermal conductivity of fluid, W/mK Nusselt number, hL/k Prandtl number, m/a heat flux, hDT, W/m2 Rayleigh number, gbDTL3/ ma
Experimental investigations of natural convection in a differentially heated cavity also have been reported in the literature. Seki et al. [20] performed flow visualization experiments in a differentially heated cavity by varying the Prandtl number of the working fluid and the aspect ratio of the cavity (by changing the width of the cavity). Authors discussed the effect of Prandtl number and aspect ratio on the flow behavior qualitatively. Srivastava et al. [21] also presented an experimental study of Rayleigh-Bernard natural convection (top wall of the cavity is maintained at a relatively lower temperature than the temperature of the bottom wall and the other two walls were kept insulated) in a rectangular cavity. Experiments were carried out for water and air as a working fluid and results from three different optical techniques (interferometry, shadowgraph, and schlieren) were compared. For low Rayleigh number, results from three techniques were comparable, but at higher Rayleigh number, due to the presence of closely-spaced fringes near the thermally active wall, it becomes difficult to resolve such fringes resulting in possible loss of fringes and, therefore, the accuracy of interferometry results reduces. Experiments in vertical differentially heated cavity of high aspect ratio were also performed for high Rayleigh number (order of 1011) [22]. Authors reported the effect of temperature difference between the vertical walls and the wall emissivity on the vertical temperature gradients. Salat et al. [23] performed an experimental and numerical study in a differentially heated square cavity for higher Rayleigh number (1.5 109) with air as the working fluid. They measured the temperature at the mid-depth of the cavity using micro-sized thermocouples. The work concluded that there were some physical ingredients lacking in the study which were important for a better understanding of the discrepancy on the thermal stratification. Tian and Karayiannis [24] conducted experiments in differentially heated cavity to study natural convection in low turbulence region for air. Intrusive probe technique was employed to measure the temperature. The study claimed that experiments were carried out with high accuracy and these experimental results can be used as a benchmark for further Computational Fluid Dynamics (CFD) code validation. Arnold et al. [25] carried out an experimental study in differentially heated cavities for different aspect ratio. They reported the results as a variation of heat transfer rates with inclination angles of the cavity. Ivey [26] performed experiments to visualize the flow inside a differentially heated square cavity for high Rayleigh number for which the flow was found to be periodic. Experiments were carried out with a mixture of water and glycol to vary the Prandtl number and Rayleigh number. The author observed the hydraulic jump near the corner and concluded to have high heat transfer in this region. Wright et al. [27] performed experiments in a vertical differentially heated cavity of high aspect ratio. Smoke pattern visualization and interferometry were employed to conduct these experiments. It was concluded that beyond a Rayleigh number (104), the flow becomes periodic and three-dimensional motion of air was observed in the cavity. Kishor et al. [28] conducted natural convection experiments in a differentially heated cavity of aspect ratio three using a non-intrusive optical technique (MachZehnder Interferometry). Authors reported the comparison of
Greek symbols a thermal diffusivity, m2/s b volumetric coefficient of thermal expansion, 1/K m kinematic viscosity, m2/s q fluid density, kg/m3
experimental data obtained in the form of whole field temperature distribution and Nusselt number on the thermally active walls with numerically simulated results. Experimentally obtained results showed a good agreement with numerical results. Saury et al. [29] reported an experimental work in which they studied the problem in the transition region (laminar to turbulent transition). Authors performed the experiments using an intrusive probe technique in a differentially heated cavity of aspect ratio four for different inclination angle to identify the transition in the flow. They proposed a unique modified critical Rayleigh number (beyond which flow becomes oscillatory) for different inclination using modified characteristic length. There have been limited studies on the transition to turbulence for buoyancy driven flows in cavities, especially of high aspect ratios. Moreover, most of these studies are based on numerical simulations. The flow characteristics in the transition region are susceptible to type and size of perturbations, and, hence there is a significant discrepancy in the numerical and experimental results. However, the experimental studies on the transition to turbulence in these type of cavities are almost non-existent. One such recent study was carried out using intrusive probes [29]. The use of probes may lead to enhancement in perturbations, thereby affecting the results significantly. Moreover, the comparison between the flow transients at different points of the cavity is not carried out in these reported work. The experimental work reported in the present manuscript is an effort to bridge some of the above-mentioned gaps in the literature and to identify the flow instabilities in differentially-heated cavities of high aspect ratios in a purely non-intrusive manner. The buoyancy-induced convective field in the cavity has been mapped using one of the variants of classical interferometry technique, namely Mach Zehnder interferometer (MZI). The MZI yields the temperature information in a completely non-intrusive and real time manner, which can then be used to understand the flow phenomena. In order to further confirm the flow patterns, smokebased flow visualization experiments are carried out. The temperature difference across the two vertical walls of the cavity has been applied and the rest of the two walls (horizontal) have been kept insulated. The top corner of the cavity near the cold wall has been identified as the most critical region where the flow was found to be periodic. The analysis of the experimental results is carried out inside and outside of the boundary layer region to understand the different mechanisms of flow instabilities. Effect of periodicity on heat transfer rates has been presented using the local Nusselt number profiles on the top of the cold wall.
2. Apparatus and instrumentation 2.1. Test cell In this section, the description of the experimental cavity is given. Experiments are performed in a differentially heated closed cavity of aspect ratio three, which is shown in Fig. 1(a). The photographic view of the cavity is also shown in Fig. 1(b). The main focus
Please cite this article as: V. Kishor, S. Singh and A. Srivastava, On the identification of flow instabilities in a differentially-heated closed cavity: Non-intrusive measurements, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118933
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3
4 100m
2 3
5
300m
1
600m Laser Beam Fig. 1. (a) Schematic of the differentially heated cavity [28]; (1) Aluminium sheets, (2) Bakelite sheets, (3) Electric heaters (Strips heater), (4) Thermocouples, (5) Cooling bath chamber; (b) Photographic view of the test cavity.
of this study is to understand the flow in the transition region when the flow becomes turbulent from laminar. Therefore, dimensions of the cavity are chosen on the basis of numerical simulations which were performed for different aspect ratios so that the transition region can be achieved in the cavity. In the literature, it is also reported that with the increment in the aspect ratio of the cavity, critical Rayleigh number (beyond which the flow becomes oscillatory) decreases. On the basis of simulations, the aspect ratio of the cavity was fixed as three. The dimensions of the cavity are 300 mm (height) 100 mm (width) 600 mm (depth). Two vertical walls of the cavity which are maintained at two different temperatures are made of aluminium (because of its high thermal conductivity). The temperature on the hot wall is maintained using six electric strip heaters of size 600 mm 48 mm 4 mm. These heaters are further connected to six different temperature controllers, which take feedback from the six thermocouples. These thermocouples are inserted in the aluminium plate at different heights (covering the full face of the vertical plate) to measure the temperature at different heights of the wall. The temperature on the hot wall is measured with an accuracy of ’±0.2 °C. On the back side of heaters, Bakelite is used to minimize the loss of heat to the surroundings. Another vertical wall is maintained at a relatively lower temperature than the hot wall. A cooling chamber is attached on the back side of the wall which has enough number of baffles so that it can provide a zig-zag path for the circulating water through it. It also ensures maximum contact of the thermostated water with the wall to maintain a uniform temperature. The other two horizontal walls of the cavity are kept insulated and the insulation is provided using Bakelite (due to low thermal conductivity) sheet of thickness 15 mm. The other two faces are open to ambient through which the laser beam is passed to record the depth-averaged interferograms inside the cavity. The experiments are performed with air (Pr = 0.71) as the working fluid in the cavity. The temperature differences between the vertical walls are kept at 10 °C, 14 °C, and 18 °C. To achieve the temperature difference, the cold wall of the cavity is maintained at 20 °C and the hot wall is maintained at three different temperatures of 30 °C, 34 °C, and 38 °C.
2.2. Mach Zehnder interferometer Experimental measurements reported in this paper are performed using a non-invasive technique. To measure the convective flow which sets up in the cavity due to natural convection after applying the temperature potential between the two vertical walls, the laser-based non-intrusive technique (Mach Zehnder interferometer) has been employed. The isometric view of Mach Zehnder interferometer (MZI) with the differentially heated closed cavity is shown in Fig. 2. In the setup of MZI, two mirrors and two beam splitters have been used, which are aligned at an angle of 45° to the propagation of the laser beam, as shown in the figure. All the optical components used in the MZI are achromatic in nature. The optical components (mirrors, beam splitter, etc.) used in setup are of diameter 100 mm which give a collimated beam of size ’ 60 mm. It is ensured that the light beam always falls on the center of the optical components to avoid any type of aberrations in the light beam. As shown in Fig. 2, a He-Ne laser of wavelength 632.8 nm is used as the coherent light source. The coherent light coming from the laser is passed through a collimating lens to have a collimating beam of light. This collimated beam is allowed to fall on a beam splitter (BS1) where the beam divides into two parts (two low-intensity beams). One of these beams reflects from the mirror (M1) and reaches to beam splitter (BS2) as shown in the figure. This section of the interferometer (from BS1 to BS2 via M1) is called the reference arm where the compensation chamber is kept. In this study, experiments are performed for air. Therefore, there is no need for the compensation chamber. Another section is known as the test arm where the test cavity is placed (from BS1 to BS2 via M2). The reference beam and the test beam superimpose on the beam splitter (BS2) and create constructive and destructive interference patterns. These interference patterns are recorded using a Charged Coupled device (CCD) camera, which is connected to a personal computer. The MZI works in two different modes, i.e. infinite and wedge fringe setting mode. In the infinite mode of MZI, the fringes represent the temperature contours in the convective field. At reference temperature condi-
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Fig. 2. Isometric view of Mach-Zehnder interferometer with differentially heated cavity.
tion, the constructive maxima (full bright fringe) confirm that the interferometer is in infinite setting mode. The infinite setting mode is achieved by aligning the two arms of the interferometer in such a way that path lengths of the two arms are almost equal under reference condition of temperature. While a slight misalignment in the optical component is made to get horizontal or vertical straight equidistant fringes to obtain the wedge setting under reference conditions. After applying the temperature potential across the vertical wall of the cavity, the convective field is recorded in the personal computer. The whole Mach Zehnder interferometer setup is placed on the pressurized pneumatic vibration isolation optical table to minimize external disturbances such as people movement, sudden floor vibrations, etc. The whole field temperature data has been extracted from the interferometric images recorded by MZI under wedge fringe setting. The details of the data reduction technique to extract data from these images can be found in a recent publication [28]. The accuracy and feasibility of interferometry-based measurement technique in the context of temperature and heat transfer measurements in differentially-heated fluid layer has also been discussed in [28]. To obtain accurate quantitative data, a large number of data reduction techniques are reported in the literature [30–32]. The validation of these techniques for different applications also has been carried out in some of our earlier works [33–35]. After getting the whole field temperature distribution at four corners of the cavity, the local Nusselt number on the wall can be calculated. Using the principle of energy balance, energy balance in mathematical terms near the thermally active walls can be written as:
@T k ¼ hðT c T h Þ @x x¼0orx¼L
ð1Þ
Here, k is thermal conductivity of the fluid (air for present experiments) and @T=@x represents the near wall temperature gradients. Finally, the local Nusselt number (non-dimensionalized heat transfer rate) has been calculated using the following formula:
Nu ¼
hL k
ð2Þ
Here, L is the distance between the hot and cold wall of the differentially heated cavity.
2.3. Smoke-based flow visualization arrangement Fig. 3(a) shows the schematic of the setup for the smoke-sbased flow visualization experiments. A photographic view of the smoke generator (Fog Generator SAFEX 195 SG D) has been shown in Fig. 3(b). For smoke-based visualization, necessary modifications have been made in the test cavity. The top Bakelite sheet, that was used in interferometry-based experiments, has been replaced with an acrylic sheet, as shown in the figure. This acrylic sheet allows the diffuse light to pass through it and also has low thermal conductivity to act as an insulator wall. The front face of the cavity has been covered with an acrylic sheet because of its transparency and the back face of the cavity is kept dark so that a good contrast can be produced to capture good quality smoke patterns. Once the flow is setup inside the cavity after applying the temperature potential across the two vertical walls of the cavity, an optimum quantity of smoke produced by the fog generator (shown in Fig. 3(b) is put inside the cavity through its front face and thereafter, the front face is covered with acrylic sheet. The diffuse light source is kept outside the top wall and it illuminates the smoke inside the cavity, as schematically depicted in the figure. Once the flow is established inside the cavity, smoke patterns are recorded using a DLSR camera (Canon EOS 200D 24.2 MP) operating at a frame rate of 50 frames per second.
3. Results and discussion In this section, transient analysis of time-dependent temperature in the flow field is presented for different Rayleigh numbers. The temperature difference across the vertical thermally active walls has been varied from 10 °C to 18 °C in the interval of 4 °C and the corresponding Rayleigh numbers are 9.7 105, 1.3 106 and 1.7 106.1 Experiments are performed for air as the working fluid (Pr = 0.71). The top left corner of the cavity (top region near the cold wall of the cavity) is selected for observation where the instability is expected to appear first. This transient study has been performed on the whole field temperature data retrieved from the experiments performed using Mach-Zehnder Interferometer (MZI). 1 The choice of cavity temperature differences (Rayleigh numbers) employed in the reported experiments has been made so as to make sure that the flow field is in the transition regime (from laminar to turbulent). The working flow regime was ascertained through detailed numerical simulations and also based on the previously reported works available in the literature.
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Dark background
Diffuse light
Test cell
Acrylic sheet DSLR camera (Canon EOS 200D 24.2 MP) (a)
Smoke generator (Fog Generator SAFEX 195 SG D) (b)
Fig. 3. (a) Schematic showing the relative positions of diffuse light, image acquisition system and experimental test section for smoke-based flow visualization inside the cavity; (b) Smoke generator.
On the basis of real-time transient temperature data obtained from the experiments, spectral analysis is carried out. A total of 500 frames, recorded at 30 frames per second, are analyzed. In the selected region (top left corner) of the cavity, the time variation of temperature variation at different points is recorded and Fast Fourier Transform (FFT) has been performed to identify the frequency(ies) of oscillations present in the flow. Later, transient analysis has also been performed using smoke-based flow visualization. Using FFT of intensity variation with time, which is obtained through smoke visualization, instabilities in the flow are studied. 3.1. Flow instability Experiments are performed for varying Rayleigh numbers in the cavity. For Ra = 9.7 105, four points are chosen near the cold wall in the top left region of the cavity to identify the dominant frequency in the boundary layer region, i.e. in the vicinity of the thermally active walls. The non-dimensionalized locations (with respect to the width of the cavity) of these points are given in Table 1. The spectral analysis of time-varying fluid temperature, as retrieved through interferometry technique, is performed on these points. The whole field temperature contours obtained through quantitative analysis of the interferometric images corresponding to the four corners of the cavity have been presented in the form of Fig. 8(a–d) in one of the later sections of this manuscript. In order to provide a direct quantitative evidence of the
Table 1 Coordinates of the points (non-dimensionalized with respect to the width of the cavity) where FFT of time variation of temperature has been performed. The top left corner is taken as the reference point. Figure
4
6
7
(a) (b) (c) (d)
(0.05,0.1) (0.13,0.1) (0.21,0.1) (0.35,0.1)
(0.35,0.2) (0.4,0.4) – –
(0.05,0.1) (0.35,0.25) – –
oscillations in the flow field, one of the representative time histories of temperature at a select point lying inside the boundary layer region in the top left corner of the cavity has also been shown in Fig. 8(e). The FFT plots of temperature variations are shown in Fig. 4 for a cavity temperature difference of 10 °C at four different locations (spatial coordinates of these points have been provided in Table 1). One can see that two peaks of different amplitudes are present in Fig. 4(a), corresponding to two different frequencies. However, as one goes towards the wall (from Fig. 4(a)–(d)), one of these two peaks starts dominating. The points closer to the wall are inside the boundary layer and at these points, the oscillations corresponding to one of the frequencies almost get vanished (Fig. 4 (d)). These selected points are indeed inside the boundary layer, which can be seen in Figs. 8 and 9 later in this section. Further details of the discussion on the identification of the boundary layer region using interferometry technique can be found in earlier work [28]. The frequency which is seen at all the locations is approximately 0.42 Hz while the other frequency is 0.72 Hz. For higher Rayleigh number also, the dominance of a single frequency has been observed in the boundary layer region. Further, flow visualization for the cavity has also been performed using the smoke-based technique to observe the flow patterns inside the cavity and the associated instabilities in the flow. Representative smoke patterns recorded in the top portion of the cavity for a temperature difference of 10 °C (Ra = 9.7 105) are shown in Fig. 10. The transient analysis of the intensity values captured through smoke-based flow visualization has been performed. The FFT analysis has been carried out on the intensity variation with time in the cavity at different points. Fig. 5 shows the results of the spectral analysis for two points (one near the boundary of the cavity and another in the core of the cavity). As observed earlier in the spectral analysis of the temperature data obtained from the MZI near the boundary, intensity variation spectrum plot (Fig. 5(a)) also shows the dominance of single frequency near the boundary. While in the core of the cavity, the presence of multiple frequencies is to be observed. Near the
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Fig. 4. Amplitude spectrum of temperature variation with time near the top portion of the left (cold wall) boundary for temperature difference of DT = 10 °C.
Fig. 5. Amplitude spectrum of intensity variation with time in smoke-based flow visualization for temperature difference of DT = 10 °C (a) near the left boundary, (b) away from the left boundary.
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Fig. 6. Amplitude spectrum of temperature variation with time at two different points away from the left boundary (cold wall) for temperature difference of DT = 10 °C.
Fig. 7. Amplitude spectrum of temperature variation with time (a) near the left boundary, (b) away from the left boundary for temperature difference of DT = 10 °C.
boundary, the frequency which is dominating is 0.3 Hz and away from the boundary, multiple frequencies of 0.36, 0.66, and 0.9 Hz have been identified. Thus, in addition to interferometry-based approach, the presence of instabilities in the flow has also been confirmed using an independent experimental approach based on the concept of smoke visualization. The presence of an additional frequency may be attributed to the fact that the front and back surfaces of the cavity were open for the experiments performed using the interferometry technique, while they were closed by the acrylic sheet for smoke visualization. Moreover, it is also seen that the amplitude corresponding to this frequency is relatively quite weak. With the smoke-based visualization experiments also confirming the presence of flow instabilities in the cavity, further studies have been carried out using the whole field temperature data recorded by employing Mach-Zehnder interferometry. Following this technique, the interferometric images have been quantitatively analysed to retrieve the whole field temperature distribution in the fluid layer. Details of the data reduction methodology
followed to retrieve the temperature field distribution from the recorded interferograms may be seen in some of the earlier works of the authors [28,30–35]. In order to verify that the experimentally observed frequencies near the corner are present throughout the cavity, the FFT analysis of temperature data is performed at two different spatial locations in the core of the cavity (the coordinate of these locations are given in Table 1). The spectrum plots at these locations are shown in Fig. 6. The constant presence of two dominant frequencies is seen throughout the cavity, which is the same as those observed near the wall (shown earlier in Fig. 5(a)). As the temperature difference between the thermally active walls is increased to 14 °C (Ra = 1.3 106), though the number of frequencies present in the flow remain the same, the amplitude of oscillations increases due to an increase in the strength of driving potential (buoyancy force). When the temperature difference is further increased to 18 °C (Ra = 1.7 106), two more frequencies appear in the flow, indicating the increased level of
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Fig. 8. (a, b, c and d) Experimentally obtained whole field temperature contours at the four corners of the differentially heated cavity for DT = 10 °C (Ra = 9.7 105); (e) Transient history of temperature at one of the points located inside the boundary layer region in the top left corner of the cavity. Full field temperature contours for this region have been shown in sub-figure (a).
flow instabilities in the fluid medium. The amplitude spectrum plots corresponding to two different spatial locations in the flow domain for 18 °C temperature difference have been shown in Fig. 7 (coordinates of the spatial locations are given in Table 1). As can be seen from the Fig. 7(b), the spectral analysis of the transient temperature field of the spatial locations away from the boundary shows the presence of a total of four different frequencies of the flow oscillations (0.36 Hz, 0.78 Hz, 1.2 Hz, and 1.56 Hz). On the other hand, three frequencies are to be seen for the point which is nearer to the wall or in the boundary layer region, as presented in Fig. 7(a).
3.2. Discussion on flow instabilities As discussed in the previous section, periodicity in the flow is distinctly observed and the presence of dominant frequencies has been found using spectral analysis of the temperature variation with time. The two frequencies observed at Ra = 9.7 105 (DT = 10 °C) confirm the presence of two types of instabilities in the flow. It is observed that in the boundary layer region, the flow is unstable due to the presence of a single dominant frequency. This instability in the flow arises due to the disturbance in the boundary layer region near the thermally active walls of the cavity.
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Cold wall
Top left corner
(b)
(c)
Right lower corner (d)
Hot wall
(a)
Fig. 9. Interferograms recorded at 10C difference under infinite fringe setting (a) top corner of cold wall, (b) top corner of hot wall, (c) bottom corner of cold wall and (d) bottom corner of hot wall.
When the temperature potential is increased across the two walls of the cavity, the velocity of the fluid in the boundary layer region increases and causes instability in the flow.2 The thus induced instability in the boundary layer region corresponds to the Tollmien-Schlichting instability.3 This type of instability is usually seen in the boundary layers for various flow situations and its importance has been reported in the available literature quite extensively [1,36]. In the present work, the frequency corresponding to Tollmien-Schlichting instability has been identified as 0.42 Hz on the basis of purely non-intrusive measurement technique and has been observed in the boundary layer region of the cavity. The
2 The fluid medium employed in the present work is air with Prandtl number almost of the order of 1 (Pr = 0.71). In view of this, the scales of thermal and velocity boundary layers (momentum and thermal diffusivities) are expected to be almost same. Thus, the instabilities induced in the flow field due to the increased temperature potential across the two vertical walls of the differentially heated cavity would also be reflected in the temperature field. This justifies the present approach of identifying the flow oscillation frequencies in the boundary layer region using the time-dependent thermal field instead of directly using the velocity field. Frequency corresponding to Tollmien-Schlichting instability in the boundary layer region has been shown in Fig. 4(d). 3 It is widely believed that Tollmien-Schlichting instability is primarily associated with Blasius boundary layer. However, one of the numerical works reported by Henkes and Hoogendoorn [1] has reported the existence of this type of instability in the boundary layer region in the context of natural convection in differentially-heated cavities. The identification of this instability under similar configuration, as reported in the present work based on non-intrusive measurements, provides a direct experimental support to the observations made by Henkes and Hoogendoorn [1].
boundary layer can be identified in the temperature contours and interferograms in the four corners of the cavity, as shown in Figs. 8 and 9, respectively. It is important to discuss the physics of the flow inside the cavity for understanding other instabilities present in the flow. The interferograms recorded under wedge fringe setting mode of Mach-Zehnder interferometer have been used to obtain the whole field temperature data (in the field of view), as shown in Fig. 8. While the wedge fringe setting of the interferometer is useful for quantitative analysis, the thermally induced flow field can be directly interpreted on the basis of the interferograms recorded under infinite fringe setting mode of the interferometer. Under this setting, the fringe patterns are the representation of the lines of constant temperature (isotherms) and hence the region of strong temperature gradients can be identified from the recorded images. Thus, with the experimental data available from two different approaches, the physics of the flow inside the cavity has been interpreted by combining the quantitative data on whole field temperature distributions (temperature contours obtained from the wedge interferograms, Fig. 8) and the real-time interferograms recorded under infinite fringe setting mode of the Mach-Zehnder interferometer (Fig. 9). From Fig. 9, the boundary layer region in the close vicinity of the cavity walls can be distinctly identified. With reference to the image shown in Fig. 9(d), the thermal gradients are expected to be maximum in the lower right corner of the cavity as the fluid comes in contact with the hot wall in this region. Corresponding
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temperature contours (shown in Fig. 8(d)) also support this observation in the quantitative terms. Closely-spaced temperature contours (Fig. 8(d)) and fringe patterns (Fig. 9(d)) reflect towards the presence of strong temperature gradients in this region and hence one expects high heat transfer rates in the bottom right corner of the fluid domain. After taking heat from the hot wall, the fluid density decreases and it rises along the hot wall and strikes the horizontal wall of the cavity on the top. While moving up, the hot fluid transfers some heat to the core fluid as well. The boundary layer formation on the hot wall can be seen in the figures (Figs. 8 and 9). This relatively hot fluid changes its direction and now comes in contact with the cold side wall and transfers maximum thermal energy to the cold wall in the top region of the cavity (Figs. 8(a) and 9(a)). The closely-spaced temperature contours plot (Fig. 8(a)) in this region confirm the phenomenon of maximum heat transfer. After transferring the thermal energy to the cold wall, the fluid density increases again and it starts to move down along the cold wall. The fluid strikes the horizontal wall on the bottom and further changes its direction (Figs. 8(c) & 9(c)) and this cycle continues in the cavity and heat transfer from the hot to the cold wall takes place. The fluid movement inside the cavity, as interpreted based on the interferometry experiments, has also been corroborated using smoke-based flow visualization. One of the representative images of the fluid flow in the top region of the cavity has been shown in Fig. 10. Based on the above presented discussion on the fluid movement along the two vertical walls of the cavity (fluid rising along the right (hot) wall and descending along the left (colder) wall), it can be inferred that the fluid moving in the vertical direction
Hot wall
Cold wall
Top right corner
Fig. 10. Representative image of fluid movement inside the cavity recorded using smoke-based flow visualization technique for cavity temperature difference of 10 °C. For guide to eye, possible fluid movement in the cavity has also been schematically indicated using white line.
Table 2 Comparison of frequencies of oscillations identified through the present work with those reported in the literature in the transition region. Frequency (Hz)
f1 f2 f3 f4
Present work (Non-intrusive measurements)
DT = 10 °C
DT = 14 °C
DT = 18 °C
0.36 0.69 – –
0.36 0.69 – –
0.36 0.69 1.2 1.56
Saury et al. [29]
0.35 0.65 0.92 –
strikes the horizontal surfaces (walls) of the cavity. This phenomenon of the fluid striking the walls of the cavity is commonly known as hydraulic jump. When the velocity of the fluid striking the wall is relatively high, it may lead to disturbance in the flow. The instability that arises due to the hydraulic jump can be seen in the corners where vertically moving fluid interacts with the horizontally moving fluid. The energy which is generated due to the hydraulic jump gets released to the fluid in the form of periodicity leading to the instability in the flow. The plausible evidence of the hydraulic jump-induced instability has been discussed in the previous section, wherein, for Ra = 9.7 105, two distinct frequencies were identified in the flow fields which, in turn, indicate towards the presence of two types of flow instabilities. Out of these two frequencies, the oscillation in the flow corresponding to frequency 0.69 Hz is due to the instability caused by the hydraulic jump. The observations made in the form of the presence of two distinct frequencies find support in the work reported by Saury et al. [29] wherein the authors identified multiple frequencies (three) in the flow for a cavity of aspect ratio four. Table 2 summarizes the various frequencies as a function of increasing temperature potential (DT) applied across the two thermally active walls of the cavity, as found in the present work. For a direct comparison, frequencies reported by Saury et al. [29] have also been tabulated in the last column. As can be seen, the experimentally determined frequencies in the present work are reasonably close to the values reported in the literature. It is worth mentioning here that while Saury et al. [29] determined these frequencies using an intrusive way of measurements, the experimental analysis performed in the present work has been carried out in a purely non-intrusive manner. This, in turn, highlights the importance of the present work as the usage of probe of finite physical dimensions leads to undesirable perturbation(s) in the fluid medium. This limitation gets completely overcome through non-intrusive measurement techniques, as employed in the present experimental study. 3.3. Signatures of the instabilities in the heat transfer profiles The experimental findings discussed in the previous section established the presence of instabilities (multiple flow oscillation frequencies) in the flow based on the smoke visualization experiments as well as on the basis of temperature field obtained through interferometry. The onset of flow instabilities directly influences the heat transfer phenomenon as the heat transfer rates are expected to significantly increase as the instability(ies) appear in the flow field. Following this, heat transfer rates in terms of Nusselt number have been calculated in the top left corner of the cavity (near the cold wall) using the whole field temperature data retrieved through interferometric analysis. The variation of the local Nusselt number along the length of the cold wall that corresponds to 1/6th of the total height of the cavity has been shown in Fig. 11 for three different temperature potentials (DT = 10 °C, 14 °C & 18 °C) applied across the two thermally active walls of the cavity. Nusselt number variations over 1/6th of the height of the cavity have been shown as the interest is in the corner flows where the instabilities in the flow field were detected. It is worth noting here
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oscillates predominantly with single frequency while amplitude corresponding to the other frequency is quite weak. The dominance of this instability in the boundary layer indicates towards the existence of the well-known Tollmien-Schlichting instability. It is important to note that this type of instability is commonly observed in boundary layers for various flow configurations. On the other hand, second instability is present away from the boundary layer and corresponds to the hydraulic jump, which is caused by the flow striking the horizontal wall of the cavity. The fluid which moves vertically along the thermally active walls strikes the horizontal wall and causes this instability. Moreover, it is concluded that when there is inclusion of more frequencies in the flow due to increment in the Rayleigh number, the instabilities cause a sudden enhancement of heat transfer on the thermally active walls.
Declaration of Competing Interest The authors declared that there is no conflict of interest. Fig. 11. Variation of local Nusselt number on the cold wall in the top region (top left corner) of the cavity for different temperature differences.
that the flow is found to be in the transition regime for the three Rayleigh numbers employed in the experiments. It can be observed from Fig. 11 that the profile of the experimentally obtained local Nusselt number values for DT = 18 °C shows a steeper slope in comparison with that seen in the profile corresponding to DT = 14 °C. However, this sudden change in the slope of the Nusselt number profile is not to be observed when the temperature difference is increased from 10 °C to 14 °C. Using the spatial variations, average heat transfer rates in terms of Nusselt number for three temperature differences (DT = 10 °C, 14 °C & 18 °C) have been calculated. The average Nusselt numbers have been found to be 8.81 (DT = 10 °C), 9.63 (DT = 14 °C) and 12.02 (DT = 18 °C). Thus, while for DT = 10 °C and 14 °C, the average value of Nusselt numbers almost comparable with each other, one observes a significant increase in the average Nusselt number as the cavity temperature difference is increased from 14 °C to 18 °C (~25%). The sudden jump in the heat transfer rates when the temperature difference is increased from 14 °C to 18 °C can be attributed to the appearance of two additional oscillation frequencies in the flow field when the temperature potential across the fluid layer is 18 °C. On the other hand, when the temperature difference is changed from 10 °C to 14 °C, the number of frequencies do not change. Hence, the appearance of additional two frequencies in the flow field leads to a significant enhancement in the heat transfer rates, which, in turn, indicate towards the additional instabilities in the flow field. 4. Conclusions Laminar to turbulent transition has been studied experimentally for a differentially heated cavity of aspect ratio three. The study has been carried out using non-intrusive techniques, i.e. Mach Zehnder Interferometer and smoke-based visualization technique. Using real-time temperature measurements inside the cavity, the instabilities in the flow have been distinctly identified. Primary attention has been paid in understanding the convective field in the four corners of the cavity where the flow is expected to become periodic first. For the experiments performed for 10 °C temperature difference between the thermally active walls, the flow was in the transition region and the oscillations with two frequencies were observed. Near the thermally active walls, fluid
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