Hajime NAKAMURA and Tamotsu IGARASHI

Transcription

1 622 Heat Transfer in Separated Flow behind a Circular Cylinder for Reynolds Numbers from 120 to (2nd Report, Unsteady and Three-Dimensional Characteristics) Hajime NAKAMURA and Tamotsu IGARASHI This paper describes the unsteady and three-dimensional characteristics of heat transfer from a circular cylinder to the cross-flow of air for Reynolds numbers from 120 to An infrared camera was used to measure the time-spatial characteristics of heat transfer on the cylinder surface, heated under a condition of constant heat flux. Fluctuating heat transfer was measured using a heat flux sensor. The heat transfer in the separated flow region had a spanwise nonuniformity, the wavelength of which agreed well with that of the streamwise vortices formed in the near-wake. In particular, the streamwise vortices formed at approximately Re = 200 due to mode-a instability effectively enhanced the heat transfer at the rear of the cylinder. For Reynolds numbers greater than 6 000, the heat transfer at the rear face was markedly increased by the alternating reattaching flow, caused by the rolling-up of the separated shear layers. Key Words: Forced Convection, Heat Transfer, Wake, Unsteady Flow, Three-Dimensional Flow, Circular Cylinder, Reynolds Number 1. Introduction As investigated in Part I of this study (1), the dependency of Nusselt number on Reynolds number varies greatly in the separated flow region of a circular cylinder, corresponding to the change in the length of the vortex formation region behind the cylinder. The timeaveraged value of Nusselt number at the rear stagnation point, Nu r /Re 0.5, has a maximum at Re 200, corresponding to the shortening of the vortex formation region, and has a minimum at approximately Re = , corresponding to the lengthening of the vortex formation region. The vortex formation region shortens again in the range of < Re < with increasing Reynolds number, leading to a sharp increase in Nusselt number at the rear of the cylinder, and for Re > up to the end of the subcritical regime, the vortex formation length is nearly constant regardless of the Reynolds number, resulting in an increase in Nusselt number in the separated flow region in proportion to the Reynolds number raised to the Received 28th January, 2004 (No ). Japanese Original: Trans. Jpn. Soc. Mech. Eng., Vol.69, No.681, B (2003), pp (Received 14th August, 2002) Department of Mechanical Engineering, National Defense Academy, Hashirimizu, Yokosuka, Kanagawa , Japan. nhajime@nda.ac.jp power of 2/3. The above facts indicate that the characteristics of the heat transfer in the separated flow region change greatly with Reynolds number. Many studies have been carried out to investigate the heat transfer from a circular cylinder in terms of crossflow. However, only few attempts have so far been made to investigate the unsteady and three-dimensional characteristics in the heat transfer. Boulos-Pei (2) investigated the fluctuation in heat transfer around a circular cylinder using a heat flux sensor in the range of Re = The fluctuating pattern in the separated flow indicated the existence of two distinct regions, the main and the secondary vortex regions. Although Boulos-Pei attempted to clarify the mechanisms of the heat transfer in these regions, their understanding was not complete because of the onepoint measurement and the inadequate response time of the heat flux sensor. Kumada et al. (3) arranged a number of heat flux sensors in the circumferential and spanwise directions of a circular cylinder in order to investigate the mechanism of the heat transfer in the separated flow at Re = They confirmed that the fluctuation of the heat transfer around the rear stagnation point was interlocked by the vortex shedding and found the existence of the spanwise nonuniformity of the heat transfer. However, the understanding by Kumada et al. of the mechanism was incomplete because of the inadequate area of

2 623 the spanwise region to be investigated and the inadequate response time of the heat flux sensors. Scholten-Murray (4) investigated the simultaneous measurement of the fluctuating heat transfer and velocity using a hot-film sensor and a Doppler laser anemometer, respectively, in the range Re = They confirmed that the intensity of the fluctuation of the heat transfer in the separated flow region was very high. However, the mechanism of the heat transfer in the separated flow region was again unclear as a result of the one-point measurement. In the present study, an infrared camera was used to investigate the time-spatial characteristics of heat transfer in the separated flow region of the cylinder. The measurement using an infrared camera was convenient to obtain the instantaneous distribution of the heat transfer and its fluctuating pattern on the cylinder surface. Moreover, the infrared camera had the advantage of being able to carry out measurements at low Reynolds numbers (Re < 10 3 ). Measurements using a heat flux sensor have never been performed at such low Reynolds numbers due to its inadequate sensitivity and inadequate spatial resolution. The present measurements clarified the unsteady and threedimensional characteristics of the heat transfer at the rear of the cylinder, which changes with Reynolds number according to the flow regimes. Nomenclature d : diameter of circular cylinder [m] or [mm] f : frequency [Hz] f s : vortex shedding frequency [Hz] Gr d : Grashof number based on d = d3 gβ(t w T 0 ) ν 2 g : acceleration of gravity [m 2 /s] h : instantaneous heat transfer coefficient [W/(m 2 K)] h : time-averaged value of heat transfer coefficient [W/(m 2 K)] L : length of circular cylinder [m] or [mm] L f : length of vortex formation region from the center of the cylinder [m] or [mm] Nu : instantaneous Nusselt number = hd/λ Nu : time-averaged value of Nusselt number = hd/λ q c : heat conduction from heated surface to the inside of the cylinder [W/m 2 ] q in : input heat flux to stainless-steel sheet [W/m 2 ] q mes : measured heat flux [W/m 2 ] q rad : radiative heat flux [W/m 2 ] q sus : heat flux due to heat conduction through stainlesssteel sheet [W/m 2 ] q : convective heat flux [W/m 2 ] Re : Reynolds number = u 0 d/ν r : radius of circular cylinder [m] or [mm] St : Strouhal number T 0, T w : freestream temperature, wall temperature [K] t : time [s] u, u 0 : velocity, freestream velocity [m/s] β : thermal expansion coefficient [1/K] φ : angle from the forward stagnation point of the cylinder [ ] λ : thermal conductivity of fluid [W/(mK)] ν : kinematic viscosity of fluid [m 2 /s] Subscripts f, r : forward stagnation point, rear stagnation point x, y, z : streamwise, vertical, and spanwise coordinate: x = y = 0, center of the cylinder; z = 0, midspan of the cylinder 2. Experimental Apparatus and Procedure Table 1 lists the circular cylinders used for the measurement of heat transfer by an infrared camera. Each heated cylinder was placed horizontally, as shown in Fig. 1 (a) for cylinders 1 4 (from Table 1), and in Fig. 1 (b) for cylinder 5 (from Table 1). The wind tunnel shown in Fig. 1 (a) has a working section of 400 mm in height and 300 mm in width. The freestream velocity u 0 ranged from 0.25 to 16 m/sec, and the turbulent intensity of the freestream in this range was approximately 1%. The infrared camera was positioned either below or downstream from the circular cylinder. Figure 1 (b) shows the wind tunnel used for cylinder 5 with the largest aspect ratio of L/d = The wind tunnel has a working section of 150 mm in height and 400 mm in width. The freestream velocity u 0 ranged from 0.17 to 16 m/sec, and the turbulent intensity of the freestream in this range was approximately 0.5%. Cylinder 5 was placed 30 mm downstream from the exit of the wind tunnel. Square end plates, 150 mm in length, were attached to both ends of the cylinder in order to suppress the effect of the boundary layers formed Table 1 Circular cylinders for heat transfer measurement using the infrared camera (a) Cylinders 1 4 (b) Cylinder 5 Fig. 1 Locations of circular cylinder and infrared camera

3 624 (a) Cylinders 1 and 5 Fig. 3 Schematic diagram of circular cylinder for measurement of fluctuating heat transfer Fig. 2 (b) Cylinders 2 4 Schematic diagrams of circular cylinders for measurement of time and space-resolved heat transfer on both sidewalls of the wind tunnel. The infrared camera was located above or downstream of the circular cylinder. Figure 2 (a) and (b) show the schematic views of the circular cylinders for the measurement of the time-spatial characteristics of heat transfer. Test cylinders 1 and 5 were fabricated from a balsa wood rod, and test cylinders 2 4 were fabricated from an acrylic resin pipe. Each cylinder has a semicircular section, 200 mm in length for cylinders 1 4 and 270 mm in length for cylinder 5, removed from the center of the cylinder. A stainless-steel sheet of 0.01 mm in thickness covered the cylinder surface including the removed section. The surface of the cylinder was heated by applying a direct current to the stainlesssteel sheet under a condition of constant heat flux. The temperature differencebetweenthe heatedsurfaceandthe freestream was C. The temperature fluctuation caused by the unsteady heat transfer was observable because the heat capacity of the heated surface is very low in the removed section. Both electrodes on both ends of the cylinder were heated by heaters in order to suppress the axial heat conduction loss. For cylinder 4, having the largest diameter of 40 mm, a partition was attached to the inner wall of the acrylic pipe, 5 mm inside from the stainless-steel sheet, in order to suppress the natural convection inside of the cylinder. The surface of the cylinder was coated using black paint in order to enhance the emissivity of the infrared radiation. The infrared camera used in the present study (TVS-8502, Avio) can capture images of instantaneous temperature distribution at 120 frames per second, and a total of frames with a full resolution of pixels. The measurement of fluctuating heat transfer was performed using a heat flux sensor in order to accurately investigate the intensity of the fluctuation. Figure 3 shows an aluminum cylinder having a diameter of d = 50 mm and a length of L = 150 mm, which was set horizontally in a wind tunnel with a working section 400 mm high and 150 mm wide. The freestream velocity u 0 ranged from 1 to 9m/sec, and the turbulent intensity of the freestream was approximately 0.5%. The cylinder was heated from the inside by a nichrome heater to achieve a constant-surfacetemperature condition. The temperature difference between the heated surface and the freestream was approximately 25 C. A heat flux sensor (HFM-7E/L, Vatell) was attached at the center of the cylinder axis. The size of the end face of the sensor was 6.32 mm in diameter, the thermopile region diameter of which was approximately 3.5 mm. The time response of the heat flux sensor was evaluated using a flashlight. The result showed that no attenuation occurred up to 500 Hz, which was much higher than the vortex shedding frequency behind the circular cylinder (< 38 Hz). 3. Evaluation of Heat Transfer Coefficient The heat transfer coefficient h measured using an infrared camera was evaluated as q h = = q in q rad q c q sus, (1) T w T 0 T w T 0 where q is the convective heat flux from the heated surface to air, q in is the input heat flux to the stainless-steel sheet and q rad is the radiative heat flux calculated using the Stefan-Boltzmann equation. The time-averaged value of the heat conduction from the heated surface to the inside of the cylinder, q c, was estimated by the heat conduction analysis based on the temperature distribution around the cylinder. The circumferential heat conduction through the stainless-steel sheet, q sus, can be estimated using q sus = λ sus 1 r 2 d 2 T w dφ 2 t sus, (2) where λ sus and t sus are the thermal conductivity and thickness of the stainless-steel sheet, respectively. The thermophysical properties of Reynolds number and Nusselt number were calculated based on film temperature, that is, the mean temperature between the freestream and the heated surface. The experimental uncertainty of the Nusselt number for the measurement using the infrared camera was caused primarily by the uncertainty of the measured temperature, including error due to the inclination angle of the heated surface to the infrared camera and that due to the nonuniformity of the emissivity on the heated surface. The uncertainty in the estimation of the radiation loss, q rad, was also one of the main reasons

4 625 for the lower flow velocity of u 0 < 1m/sec. The uncertainty in the estimation of the heat conduction losses q c and q sus was not negligible for the smaller cylinders and for lower flow velocities. The uncertainty due to natural convection around the cylinder was negligible because the value of the Grashof number based on d, Gr d /Re 2 was 0.09 at maximum under the present experimental conditions. Based on the above considerations, the experimental uncertainty of the time-averaged local Nusselt number Nu for the largest cylinder of d = 40 mm was within 3%. For the smallest cylinder of d = 6.4 mm, the uncertainty of Nu was relatively high, ranging from 3% to 10%, corresponding to the freestream velocities of u 0 = 16 m/sec to 0.17 m/sec. The fluctuation of the heat transfer measured using the infrared camera was considerably attenuated, mainly due to the heat capacity of the stainlesssteel sheet. In particular, at the rear stagnation point of the cylinder, where the fluctuation and the axial nonuniformity are highest, the r.m.s. value of the heat transfer as measured by infrared camera was approximately 10% that measured by the heat flux sensor. Therefore, the fluctuating heat transfer measured by the infrared camera was used to discuss the qualitative characteristics of the unsteady heat transfer. For the measurement using the heat flux sensor, at a constant wall temperature, the heat transfer coefficient was evaluated as q h = = q mes q rad, (3) T w T 0 T w T 0 where q mes is the heat flux measured by the sensor. The experimental uncertainty of the local and instantaneous Nusselt number, Nu, caused by the uncertainties of the measured heat flux was within 2% under the present experimental conditions. 4. Experimental Results and Discussions 4. 1 Wake transition regime (120 < Re < 400) Previous studies have shown that the wake behind a circular cylinder undergoes a transition to threedimensional flow with the formation of streamwise vortices at Re = , as reported by Williamson (5) and Brede et al. (6), among others. Figure 4 shows flow visualizations behind the cylinder obtained by Williamson (7).A circular cylinder is positioned at the bottom of each photograph, and the flow direction is from bottom to top. In the range of < Re < , streamwise vortices appear behind the cylinder as shown in Fig. 4 (a). The spanwise wavelength of the streamwise vortices is 3 5 diameters. This is caused by mode-a instability, which is characterized by the appearance of streamwise vortices forming vortex patches, for which the sign of rotation alternates with the alternating vortex shedding. For Re > , the streamwise vortices change their structure as shown in Fig. 4 (b). This is caused by mode- (a) Mode-A (Re = 200) (b) Mode-B (Re = 270) Fig. 4 Flow visualization from Williamson, Ref. (7), Fig. 8, Cambridge University Press B instability, which is characterized by the finer structure of streamwise vortices, whose spanwise wavelength is approximately one diameter, forming continuous vortex tubes, of which the sign of rotation does not alternate. The existence of these structures was also confirmed through numerical simulations by Zhang et al. (8) and Persillon-Braza (9). In the following, we investigate the effect of these structures on the heat transfer in the separated flow behind a circular cylinder. Figure 5 (a) (c) show instantaneous temperature distributions at the rear of the cylinder as measured by the infrared camera (top) and the time histories of spanwise temperature distribution at the rear stagnation point of φ = 180 (bottom), for Reynolds numbers of Re = 117 and 213 and Re = 347, respectively. Figure 6 (a) (c) show time-averaged and fluctuating Nusselt numbers along the span at the rear stagnation point corresponding to the above three Reynolds numbers. At Re = 117, in the laminar shedding regime, the instantaneous temperature exhibits almost uniform distribution along the span, as reflected by the two-dimensional flow. The fluctuation level is very low, as shown in Fig. 6 (a). The good symmetry of the instantaneous temperature along the vertical direction, shown in Fig. 5 (a), indicates the negligible effect of natural convection. At Re = 213, a strong spanwise nonuniformity appears, as reflected by the three-dimensional flow behind the cylinder. The instantaneous distribution shows a periodical spanwise nonuniformity, the wavelength of which is approximately 3 4 diameters, corresponding to the spanwise wavelength of the streamwise vortices observed in mode-a. According to Williamson (7),aspanwise vortex tube deforms into a wavy structure, and is then pulled out in part which then extends into a reverse flow region behind the cylinder, forming a pair of streamwise vortices. This process is likely to induce spanwise nonuniformity in the heat transfer. Moreover, mode-a instability causes vortex dislocation (10) at certain spanwise locations, as investigated by Williamson (7). The vortex dis-

5 626 (a) Re = 117 (b) Re = 213 (mode-a) (c) Re = 347 (mode-b) Fig. 5 Instantaneous temperature distributions in the separated flow region as measured using the infrared camera (top) and the time histories of spanwise temperature distribution at the rear stagnation point of φ = 180 (bottom); d = 6.4mm (a) Re = 117 (b) Re = 213 (mode-a) (c) Re = 347 (mode-b) Fig. 6 Time-averaged and fluctuating Nusselt numbers along span at the rear stagnation point of φ = 180 ; d = 6.4mm location, accompanied with an energetic reverse flow to the rear of the cylinder, probably enhances the heat transfer at the rear of the cylinder, as seen in the time history for Re = 213 at t f s = and z/d = 4.5, 2.5. At Re = 347, the spanwise nonuniformity has a finer structure, the wavelength of which is roughly one diameter, corresponding to the streamwise vortices which appeared in mode-b. The spanwise structure caused by the streamwise vortices in both mode-a and mode-b is considerably stationary for several tens to several hundreds shedding cycles, as can be seen in Fig. 5 (b) and (c). The above feature occurs due to a feedback mechanism in the formation of the streamwise vortices, as investigated by Williamson (7). As indicated in Part I of this study (1),asmalldiscrepancy exists in the correlation between the Nusselt number at the rear stagnation point and vortex formation length, i.e., the Nusselt number at the rear stagnation point, Nu r /Re 0.5, has a maximum at Re 200, whereas the vortex formation length is nearly constant (L f /d = ) in the range of Re = This is considered to be caused by the difference in the characteristics of the streamwise vortices in mode-a and mode-b. According to Brede et al. (6), the value of the circulation of the streamwise vortices in mode-a is approximately twice that in mode-b. Moreover, the mode-a instability accompanies an energetic reverse flow due to the vortex dislocation, as described above. Therefore, it is reasonable to say that the mode-a instability that occurs at Re 200 effectively enhances the heat transfer at the rear of the cylinder, resulting in a higher value of Nu r /Re 0.5 than that in mode-b (Re > 260). Figure 7 shows the variation in Strouhal number with Reynolds number for 50 < Re < 400. The shedding frequency f s was measured using a hot-wire anemometer set at x = 3d, y = d and z = 0. For Reynolds numbers below 160, the Strouhal number measured in the present study was lower than that for the parallel shedding reported by Roshko (11) and Williamson (12) and was nearly equal to that for the oblique shedding reported by König (13). This indicates that the vortex shedding for the present experiment is oblique in this range. However, as shown in Figs. 5 (a) and 6 (a), the oblique shedding induces only a slight spanwise nonuniformity in the heat transfer. For Reynolds numbers above 170, the Strouhal number for the present experiment (L/d = 56.3) agrees well with that obtained by Norberg (14) at similar aspect ratios (L/d = 40 and 100). For the parallel

6 627 Fig. 7 Relationship between Reynolds number and Strouhal number for 50 < Re < 400; d = 6.4mm Fig. 9 (a) φ = 158 (b) φ = 180 Power spectrum of temperature fluctuation in the range of Re = ; d = 6.4mm (a) Re = 213 (b) Re = 347 Fig. 8 Power spectrum of temperature fluctuation at various angular positions; d = 6.4mm shedding condition (Williamson (12) and Norberg (14) ), the Strouhal number discontinuously decreases in the range of mode-a. However, this change was not observed in the present study, because the aspect ratio of the cylinder used was not sufficiently large (L/d = 56.3). Figure 8 (a) and (b) show the power spectrums of the temperature fluctuation as measured using the infrared camera at various angular positions for Re = 213 and 347, respectively. Figure 9 (a) and (b) show the power spectrums of the temperature fluctuation for various Reynolds numbers at φ = 158 and 180, respectively. The f s was measured using a hot-wire anemometer. The power spectrum was averaged in the spanwise direction ( z /d < 8). A distinctive peak was found to exist in the region of φ = at f s, indicating the formation of surface flow interlocked by vortex shedding. This peak consistently exists in the range for 120 < Re < 500, as shown in Fig. 9 (a), regardless of the difference in the structure of the streamwise vortices. This peak becomes unclear in the ranges of Re < 200 and Re > 400, corresponding to the lengthening of the vortex formation region. In contrast, no distinct peak exists around the rear stagnation point (φ 180 ) for this Reynolds number range. This fact indicates that the fluctuation related to the vortex shedding is weak near the rear stagnation point. However, the timeaveraged Nusselt number shows a maximum at the rear stagnation point, as shown in Part I, corresponding to an intense fluctuation of Nusselt number at the rear stagnation point, shown in Fig. 6 (b) and (c). This contradiction can be explained by the existence of velocity fluctuation related to the streamwise vortices, which was not interlocked by the vortex shedding Near-wake lengthening regime (400 < Re < 1 500) As shown in Part I, the vortex formation length increases in the range of 400 < Re < and reaches a maximum at Re = , corresponding to the minimum value of Nu r /Re 0.5. The instantaneous temperature distribution and its fluctuating pattern at the rear of the cylinder at Re = is shown in Fig. 10. Figure 11 shows the time-averaged and fluctuating Nusselt numbers at the rear stagnation point. The finer spanwise structure is found in the instantaneous temperature distribution, as shown in Fig. 10 (top), with a spanwise wavelength that is roughly equal to the cylinder diameter. This is likely caused by the formation of the streamwise vortices equivalent to mode-b. A similar spanwise nonuniformity, whose wavelength of 0.7d, was previously reported by Nakamura-Igarashi (15) at Re = According to the flow visualization by Bays-Muchmore and Ahmed (16),the spanwise wavelength of the streamwise vortices behind a circular cylinder was approximately equal to the cylinder diameter for a wide range of Reynolds number (Re = ). This indicates that the spanwise nonuniformity of the heat transfer with a wavelength of roughly one diameter, which is caused by the streamwise vortices in mode-b, persists up to at least Re = A large spanwise structure having a wavelength of approximately 10d appeared in the range of 700 Re 3 000, as shown in Fig. 10 (bottom). This structure was particularly noticeable at Re = , at which the vortex formation length reaches a maximum. The form

7 628 Fig. 10 Fig. 11 Instantaneous temperature distribution in the separated flow region as measured using the infrared camera (top) and the time history of spanwise temperature distribution at the rear stagnation point of φ = 180 (bottom); Re = 1 520, d = 6.4mm Time-averaged and fluctuating Nusselt numbers along span at the rear stagnation point of φ = 180 ; Re = 1 520, d = 6.4mm of this large structure changes only slightly with time, resulting in the periodical spanwise nonuniformity of the time-averaged Nusselt number, as shown in Fig. 11. The origin of this structure may be due to the instability of the elongated shear layer, although its mechanism is unclear. There have been remarkably few studies on the unsteady and three-dimensional flow in this Reynolds number range, resulting in a poor understanding of the heat transfer mechanism Near-wake shortening regime (3 000 < Re < ) The vortex formation region shortens in the range of < Re < , leading to a sharp increase in the value of Nu r /Re 0.5. The mechanism of the heat transfer in this range is investigated in the following. At first, we investigate the mechanism of the heat transfer for Re > , at which the vortex formation region was fully shortened. Figure 12 shows the fluctuating Nusselt number and its power spectrum as measured by Fig. 12 Fluctuating Nusselt number and its power spectrum as measured using the heat flux sensor at various angular positions; Re = , d = 50 mm the heat flux sensor at Re = In the laminar flow region of φ = 70, the Nusselt number fluctuates periodically following the vortex shedding behind the cylinder, and a distinct peak exists in the power spectrum at f s. This feature became prominent with the shortening of the vortex formation region. In the separated flow region, the power spectrum has a peak at f s in the region of 140 <φ<170, which is consistent with that for lower Reynolds numbers, as shown in Fig. 8. However, a distinct peak appears at 2 f s around the rear stagnation point in this flow regime. The double frequency of the power spectrum near the rear stagnation point is due to the alternating reattaching flow caused by the alternating rolling-up of the shear layers that have separated from the cylinder. This feature was clearly visualized by Igarashi (17), as shown in Fig. 13 (a). The visualization of the surface flow, as shown in Fig. 13 (b), indicates the existence of the time-averaged separation lines of the reattaching flow at φ 150. The reverse flow is formed at the rear face from φ = 170 to 150,interlocked by the vortex shedding, as indicated by Nakamura- Igarashi (15). A more detailed description concerning the unsteady heat transfer at Re was presented in a previous report by Nakamura-Igarashi (15). The transition of the heat transfer in the range of < Re < , in the near-wake shortening regime, is clearly indicated in Figs. 14 and 15, which show the fluctuating velocity in the vicinity of the rear stagnation point (x = 0.6 d, y = 0) and fluctuating Nusselt number at the rear stagnation point, respectively, measured using the heat flux sensor. At Re 3 000, the turbulence level of the fluctuating velocity is low due to the lengthening of the vortex formation region. This leads to an inactive feature in the heat transfer at the rear stagnation point. For < Re < , occasional peaks appear

8 629 (a) Smoke visualization around a circular cylinder at Re = (b) Oil-film pattern on the rear of the circular cylinder at Re = Fig. 13 Flow visualization from Igarashi (17) (a) Fluctuating velocity u /u 0 (b) Power spectrum Fig. 14 Fluctuating velocity and its power spectrum at x = 0.6 d, y = 0; Re = , d = 12 mm in the fluctuating velocity, and the peaks occur more frequently as the Reynolds number increases. These peaks are caused by the formation of the intermittent reattaching flow at the rear of the cylinder, which originated from the shortening of the vortex formation region. The appearance of a distinct peak in the power spectrum at double the frequency of the vortex shedding, 2 f s, indicates the alternating feature of the reattaching flow interlocked by the vortex shedding. These features are in agreement with the fluctuating Nusselt number, as shown in Fig. 15. For Re > 6 000, intermittent peaks appear in the fluctuating Nusselt number corresponding to the intermittent reattaching flow. The flow renewal caused by the reattaching flow significantly enhances the heat transfer around the rear stagnation point. This leads to a sharp increase in the Nusselt number around the rear stagnation point in the range of < Re < For Re > , at (a) Fig. 15 Fluctuating Nusselt number (b) Power spectrum Nu r /Re 0.5 Fluctuating Nusselt number at the rear stagnation point as measured using the heat flux sensor, and its power spectrum; Re = , d = 50 mm which the vortex formation region is fully shortened, the feature of the fluctuating velocity, having two peaks for almost every shedding cycle, is consistent regardless of the Reynolds number. In this region, the Nusselt number at the rear stagnation point is proportional to the Reynolds number raised to the power of 2/3, corresponding to data obtained in previous research studies (18), (19). 5. Conclusion The present investigation provides new insight into the unsteady, three-dimensional feature of the heat transfer in the separated flow behind a circular cylinder from the measurements of the time-spatial characteristics of heat transfer over a wide range of Reynolds numbers, from 120 to The primary results are as follows 1. Spanwise nonuniformity was observed in the heat

9 630 transfer at the rear of the cylinder for Reynolds numbers greater than 180, corresponding to the formation of streamwise vortices in the near-wake flow region. The fluctuation in these spanwise structures was independent of the vortex shedding behind the cylinder but was considerably stationary for several tens to several hundreds shedding cycles. 2. For 180 < Re < 260, the spanwise nonuniformity of the heat transfer had a wavelength of 3 4 diameters, corresponding to that of the streamwise vortices in the range of mode-a. The streamwise vortices in this range effectively enhanced the heat transfer at the rear of the cylinder, resulting in the formation of a lobe at Re 200 in the trend of Nusselt number with Reynolds number at the rear stagnation point. 3. For Re > 260, the spanwise wavelength of the heat transfer was roughly one diameter, corresponding to that of the streamwise vortices in the range of mode-b. This spanwise structure in the heat transfer persisted up to Re = In the range of < Re < , the heat transfer around the rear stagnation point increased markedly with Reynolds number. This was caused by the initiation of the alternating reattaching flow at the rear of the cylinder, corresponding to the shortening of the vortex formation region. References ( 1 ) Nakamura, H. and Igarashi, T., Heat Transfer in Separated Flow behind a Circular Cylinder for Reynolds Numbers from 120 to (1st Report, Time- Averaged Characteristics), Trans. Jpn. Soc. Mech. Eng., (in Japanese), Vol.68, No.675 (2002), pp ( 2 ) Boulos, M.I. and Pei, D.C.T., Dynamics of Heat Transfer from Cylinders in a Turbulent Air Stream, Int. J. Heat Mass Transfer, Vol.17 (1974), pp ( 3 ) Kumada, M., Ishihara, K. and Kato, M., Unsteady Characteristics of Heat Transfer on a Separated Region of a Circular Cylinder, Proc. 2nd JSME-KSME Thermal Eng. Conf., Kitakyushu, Japan, Vol.2, (1992), pp ( 4 ) Scholten, J.W. and Murray, D.B., Unsteady Heat Transfer and Velocity of a Cylinder in Cross Flow I. Low Freestream Turbulence, Int. J. Heat Mass Transfer, Vol.41 (1998), pp ( 5 ) Williamson, C.H.K., The Existence of Two Stages in the Transition to Three-Dimensionality of a Cylinder Wake, Phys. Fluids, Vol.31 (1988), pp ( 6 ) Brede, M., Eckelmann, H. and Rockwell, D., On Secondary Vortices in the Cylinder Wake, Phys. Fluids, Vol.8, No.8 (1996), pp ( 7 ) Williamson, C.H.K., Three-Dimensional Wake Transition, J. Fluid Mech., Vol.328 (1996), pp ( 8 ) Zhang, H.-Q., Fey, U., Noack, B.R., König, M. and Eckelmann, H., On the Transition of the Cylinder Wake, Phys. Fluids, Vol.7, No.4 (1995), pp ( 9 ) Persillon, H. and Braza, M., Physical Analysis of the Transition to Turbulence in the Wake of a Circular Cylinder by Three-Dimensional Navier-Stokes Simulation, J. Fluid Mech., Vol.365 (1998), pp (10) Williamson, C.H.K., The Natural and Forced Formation of Spot-Like Vortex Dislocations in the Transition of a Wake, J. Fluid Mech., Vol.243 (1992), pp (11) Roshko, A., On the Development of Turbulent Wakes from Vortex Streets, NACA Rep. 1191, (1954). (12) Williamson, C.H.K., Defining a Universal and Continuous Strouhal-Reynolds Number Relationship for the Laminar Vortex Shedding of a Circular Cylinder, Phys. Fluids, Vol.31 (1988), pp (13) König, M., Noack, B.R. and Eckelmann, H., Discrete Shedding Modes in the von Kármán Vortex Street, Phys. Fluids, A, Vol.5, No.7 (1993), pp (14) Norberg, C., An Experimental Investigation of the Flow around a Circular Cylinder: Influence of Aspect Ratio, J. Fluid Mech., Vol.258 (1994), pp (15) Nakamura, H. and Igarashi, T., Unsteady Heat Transfer in Separated Flow behind a Circular Cylinder, Proc. 12th Int. Heat Transfer Conf., Grenoble, France, Vol.2, (2002), pp (16) Bays-Muchmore, B. and Ahmed, A., On Streamwise Vortices in Turbulent Wakes of Cylinders, Phys. Fluids, A, Vol.5, No.2 (1993), pp (17) Igarashi, T., Fluid Flow and Heat Transfer in Separated Region of a Circular Cylinder, Proc. 4th ASME-JSME Thermal Eng. Joint Conf., Honolulu, Hawaii, Vol.3, (1983), pp (18) Richardson, P.D., Heat and Mass Transfer in Turbulent Separated Flows, Chemical Engineering Science, Vol.18 (1963), pp (19) Igarashi, T. and Hirata, M., Heat Transfer in Separated Flows Part 2: Theoretical Analysis, Heat Transfer Japanese Research, Vol.6, No.3 (1977), pp