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Control of vortex shedding by thermal effect at low Reynolds numbers
Experimental Thermal and Fluid Science 21 (2000) 227±237
www.elsevier.nl/locate/etfs
Control of vortex shedding by thermal eect at low Reynolds numbers J.-C. Lecordier a, L.W.B. Browne b, S. Le Masson
a,1
, F. Dumouchel
a,2
, P. Paranthoen
a,*
a b
UMR 6614 C.N.R.S., Universit e de Rouen, 76821 Mont Saint Aignan Cedex, France Department of Mechanical Engineering, University of Newcastle, NSW 2308, Australia Received 26 May 1999; accepted 7 March 2000
Abstract An experimental study has been made of the control of vortex shedding in the wake of two two-dimensional blu bodies, a circular cylinder and a ¯at ribbon. The study has shown that this control, easily realized by heating the blu body, depends on the nature of the ¯uid. In the absence of buoyancy eects, related to the temperature dependence of the dynamic viscosity, the heating is found to stabilize the wake in air while the opposite result is obtained in water. Detailed measurements of the velocity ®elds in air, in isothermal and in heated body, show that this control is linked to slight modi®cations of the ¯ow in the near wake and can be taken into account by the eective Reynolds number approach. The measurements also show that the degree of instability can be related to the level of interaction between the two initial shear layers at the end of the recirculation zone. Ó 2000 Elsevier Science Inc. All rights reserved. Keywords: Wake; Shear-layer instability; Shedding control; Laser-doppler measurements
1. Introduction Extensive experiments carried out at Rouen and Nizhny Novgorod have shown that a very simple means of controlling vortex shedding from a cylinder at low Reynolds numbers, 45 < Re U1 d=mg < 90, can be obtained by heating or cooling the cylinder, [1±3]. Here U1 , d and mg are the upstream velocity, the cylinder diameter and the kinematic viscosity of the ¯uid at the temperature of the upstream ¯ow, respectively. In air, by applying heat input to the cylinder, the characteristics of vortex shedding can be signi®cantly altered and total suppression can be achieved by increasing the heat input suciently. This experimental result was earlier found by Uberoi and brie¯y reported
* Corresponding author. Tel.: +33-02-35-14-65-80; fax: +33-02-3570-83-84. E-mail address: [email protected] (P. ParanthoeÈn). 1 Present address: Technopole Anticipa, Lab QFE ESP, 2 Avenue P. Marzin, 22307 Lannion Cedex, France. 2 Present address: Laboratoire de Mecanique, Universite du Havre, 25 rue Ph. Lebon, BP 540 76058 Le Havre Cedex, France.
in 1965 at a Symposium on the concentrated vortex motions in ¯uids, see [4]. The control of the transition from a periodic to a laminar wake by applying a heating current within a line source was also mentioned by Crum and Hanratty [5]. Berger and Schumm [6], Schumm et al. [7] reported similar results in an experiment mainly devoted to the proof that the Landau equation exactly governs the bifurcation process. As proposed at ®rst by Uberoi, this eect has been related to the increase of the kinematic viscosity of the gas and to the corresponding decrease of the eective Reynolds number. However the precise value of the effective temperature to be used was not clear, see for example Sreenivasan et al. [8]. Consideration of this eective temperature by Lecordier et al. [1], Ezerski [2], Paranthoen et al. [9], Dumouchel et al. [10] indicated that, in air ¯ow, it was never the usual ®lm temperature de®ned as the arithmetic mean between the cylinder temperature Tw and the air ¯ow temperature Te . By considering the temperature dependence of the kinematic viscosity mg , this eective temperature Teff for a heated cylinder in air was found by LeMasson [11] to be: Teff Te 0:24 0:02DTw ; where DTw Tw ÿ Te ; 1 a value close to the hot recirculation zone temperature.
0894-1777/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved. PII: S 0 8 9 4 - 1 7 7 7 ( 0 0 ) 0 0 0 0 7 - 8
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Nomenclature U longitudinal velocity, m/s V transverse velocity, m/s d diameter of the cylinder, m L length of the cylinder, m g gravitational acceleration, m/s2 T temperature, °C DT temperature excess, °C x coordinate for horizontal axis, m y coordinate for vertical axis, m s length of the wake bubble P electric power, W I interaction term, I D1 dU D=2 oU oy max Nu Nusselt number Re Reynolds number Gr Grashof number Greek symbols a non-dimensional parameter, a Gr=Re2
Conversely Yu and Monkewitz [12] argued that the main eect of heating in air is a reduction of the density in the near wake which leads to a reduction of absolute instability. In their stability analysis, using similar pro®les for velocity and temperature, calculations for constant or temperature-dependent dynamic viscosity did not lead to signi®cant dierences. These authors concluded that ``...the change in the stability characteristics brought about by heating the cylinder was the result of a subtly modi®ed interaction between the two mixing layers via the inertial terms and not a viscous eect''. When the level of heating is increased and is high enough to induce buoyancy eects, mixed-convection or free-convection regimes have to be considered. Under these circumstances the buoyancy eects, which appear in the near wake become predominant in comparison with variable ¯uid properties eects. A large number of experimental or numerical studies have reported the in¯uence of such buoyancy eects on the development of the vortex shedding phenomenon in the vertical con®guration, see [13±19]. In this con®guration when the level of heating increases, the vortex shedding is found at ®rst to increase owing to the positive buoyancy force and then to break down. In the horizontal con®guration, except for the numerical studies by Socolescu et al. [20] and Nakabe et al. [18], most of the previous investigations were limited to the study of the change of the eective Nusselt number with mixed convection, see [21,22]. The limit between the forced-convection regime and the mixed-convection regime is usually characterized by a critical value of the parameter a Gr=Re2 depending on the geometry of the ¯ow. Here Gr is the Grashof number de®ned as Gr gbDTw d 3 =m2g , where g is the acceleration due to the gravity, b the temperature coecient for volume expansion, DTw is the temperature dierence between the cylinder and the upstream ¯ow. Based on the change of the eective Nusselt number
b d D mg r
temperature coecient for volume expansion, °Cÿ1 shear layer thickness, m shear layer spacing, m kinematic viscosity, m2 /s RMS value of ¯uctuations
Subscripts w wall c critical e refers to conditions at eective temperature ®lm refers to conditions at arithmetic mean ®lm temperature, Tfilm Tw Te =2 max maximum value e free stream Superscript * normalized value
with mixed convection to the Nusselt number with forced convection alone, the limiting value of a is about 0.05 in the cross-¯ow situation as shown by Fand and Keswani [21] and Morgan [22]. In our experimental con®guration where the horizontal cylinder is located transversely in a cross-¯ow of uniform velocity, the range of heating for the forcedconvection regime, deduced from the critical value of a mentioned above, and for the laminar vortex street range studied here (Re 50), needed to be such that the temperature dierence between the cylinder and the ¯ow was Tw ÿ Te < 12:5
m2g : bd 3
2
The motivation of our experimental investigation was to provide some new information on the in¯uence of changes in the density and viscosity for both air and water on the ¯ow regime downstream of a heated blu body in the absence of buoyancy eects. For this purpose, near-wake heating experiments were carried out in ¯uid, air and water, the physical properties of which are not temperature dependent in the same way. While the in¯uence of heating on density is identical in liquids and gases, the in¯uence of heating on the dynamic viscosity is opposite. The dynamic viscosity of gases increases and that of liquids decreases. The eects related to the in¯uence of viscosity and density may then be separated. Particular attention was made to avoid a level of heating high enough to generate buoyancy eects in order that the heated blu body always operated in a forced convection regime. Furthermore, this experiment was limited to a study of the near wake over a range of values of the Reynolds numbers just above the critical Reynolds number where the oblique modes of vortex shedding are not present, see [23±27].
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After a recall of the basic phenomenon in air and water ¯ows, the characteristics of the velocity pro®les in the near wake of two two-dimensional blu bodies heated or not are presented by using the eective Reynolds number concept as described by Lecordier et al [1] and Dumouchel et al. [10]. These measurements are used to determine the level of interaction between the two initial shear layers at the end of the recirculation zone and consequently in the instability conditions, previously studied in isothermal wakes by Paranthoen et al. [28].
2. Experimental setup 2.1. Experiments in air¯ow These experiments were carried out in air, in the potential core of a laminar plane jet. The plane jet facility consists of a variable speed blower supplying air to a two-dimensional rectangular 10:1 contraction. The jet exits normally to an end plate (17 35 cm2 ) from a slit of width b 15 mm and span 15 cm centrally located in this plate. To minimize air turbulence, large chambers with baes and sound absorbing material were used between the fan and the contraction. On the centerline at the nozzle exit the turbulence intensity ru is approximately 0.4%. The vortex shedding blu body was successively, a smooth stainless steel 1 mm diameter tube and a 1 mm 55 lm ribbon mounted horizontally in the middle of the jet close to the exit plane. Their total length was 15 cm (L=d 150). These bodies could be heated by Joule eect by means of direct current. The heat input per unit length was calculated from the supplied voltage and current. The temperature of the 1 mm diameter circular cylinder was measured by means of a thermocouple located within the tube. In order to avoid vibrations, the circular cylinder and the ribbon were damped with pieces of foam located at their ends and suciently tensioned. Furthermore the use of four parallel 100 lm diameter wires separated by about 1 mm and located on each edge of the jet exit plane perpendicular to the cylinder, was found to prevent oblique shedding. With the selected diameter, d 1 mm, Reynolds numbers Re from 34 up to 75 could be obtained by varying the upstream velocity U1 between 50 and 115 cm/s. The 1 mm cylinder temperature excess could reach a maximum of 250°C leading to a maximum value of a of 0.02 corresponding to the forced-convection regime.
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2.2. Experiments in water ¯ow Similar experiments were carried out in water. A stainless steel tube of 3 mm diameter or a nichrome ribbon (3:2 mm 85 lm) were either located in a 50 mm diameter water tunnel at 100 mm from an ASME nozzle or in the middle of a water channel with a free surface. The working depth of the water channel was approximately 13 cm. In order to prevent oblique shedding we used, as did Hammache and Gharib [29], two upstream circular cylinders positioned normal to the obstacle. The tube was heated by an isolated constantan wire twisted on a ceramic cylinder placed inside the tube and driven by a direct current. The ribbon was directly heated by a direct current. In most of the experiments the temperature excess was limited to 4°C in order that a be less than 0.05 and so avoid buoyancy eects. With the 3 mm circular cylinder the eddies were seen by injecting ink through a 0.4 mm hole in the tube, located on the centerline of the tunnel and facing the ¯ow. The vortex frequency was then obtained by measuring the time needed for approximately 50 eddies to be produced. During these experiments the upstream velocity and temperature were carefully measured with a BROOKS electro-magnetic ¯owmeter and a thermometer in order to control the Reynolds number. As shown in Fig. 1 some visualizations of the ¯ow were obtained in the water channel by using very small particles generated by electrolytic etching of silver wires located in a test section upstream of the cylinder. 2.3. Experimental techniques Due to limitations of the hot-wire for velocity measurements in the air near wake, see [28], velocity measurements were made by means of an LDA system. We used an LDA TSI system incorporating a 1.5 W spectra physics laser, an integrated optical transmission unit and a light collection using the forward-scattering mode. The optical measuring volume was 0:08 0:08 1:4 mm3 with the major axis parallel to the cylinder. Directional ambiguity was avoided by the use of the frequency shifting technique with Bragg cells. Velocity measurements were found to be not very sensitive to the sampling rate and the seeding conditions. The location of the center of the measuring volume was accurately obtained by studying the light scattered by a 20 lm wire on which the measuring volume was adjusted. The location of the wire was known by displacing it from the surface of the cylinder after electric contact.
Fig. 1. Visualization of the ¯ow in water at Re > Rec .
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Fig. 2. Experimental setup.
For the heating conditions in air, some problems exist with the LDA system due to the absence of particles in the center of the near wake. Even at a moderate level of temperature (Tw ÿ Te 50°C), it appears that particles are missing in a symmetrical manner in the center of the wake bubble. This eect could be due to the thermophoresis eect already mentioned by Talbot et al. [30] in LDA measurements made in a strongly heated boundary layer ( 700°C). In our situation even with moderate temperature dierences the small dimension of the 1 mm diameter cylinder induces close to the cylinder surface large temperature gradients ( 20 000°C/m). This eect prevents particles from seeding the recirculation zone. Due to the small time constant of the particles this eect only occurs in the wake bubble and accurate velocity measurements are possible at all other positions, see [11]. The same two components LDA system was used for velocity measurements in the water ¯ow. Uncertainties in measured mean and ¯uctuating velocities were 2% and 5% in air and water, respectively. Measurements of the temperature ®eld were made in water using thermocouples built with 70 lm diameter Chromel and Alumel wires. The frequency response of the thermocouple was about 50 Hz and sucient to measure the temperature ¯uctuations lower than 1 Hz generated in the heated wake in water as shown by Dumouchel [31]. As shown in Fig. 2, the origin of the coordinate system was taken at the center of the cylinder or the ribbon. The x-axis was measured in the direction of the ¯ow, the y-axis was perpendicular to the ¯ow and the zaxis coincided with the cylinder axis. U and V are the mean values of longitudinal (along x) and transverse (along y) velocities respectively. ru and rv are the RMS values of longitudinal and transverse velocity ¯uctuations, respectively. In our study, the asterisk always denotes normalization. The lengths are non-dimensionalized by the diameter of the cylinder or the width of ribbon, d, and the velocities are normalized by the upstream velocity U1 . 3. Description of the basic phenomenon The ®rst results concern the possibility of the control of vortex shedding downstream from a horizontal heated cylinder located in air¯ow or water¯ow by means of a heating current applied to the cylinder. Examples of the suppression of vortex shedding are shown in Fig. 3.
Fig. 3. Example of suppression of transverse velocity ¯uctuations linked to a stabilization of the wake in air with heating; (x 10, y 0; Re ÿ Rec 2; 4; 6; 8; 10).
Measurements of the intensity of the transverse velocity rv were obtained on the centerline at 10 diameters. As shown in this ®gure, the intensity of the velocity ¯uctuations is practically zero when the heat input per unit length reaches a critical value (P/L)crit . In this experiment the Reynolds numbers Re, calculated with the dynamic viscosity of the non-heated upstream ¯ow, were always larger than the critical Reynolds number Rec . This critical Reynolds number Rec , corresponding to the transition from a two-dimensional steady to a two-dimensional periodic wake. For this reason there are velocity ¯uctuations in the heated wake, due to the vortex shedding, for the lowest heating conditions. When the power per unit length increases, the intensity of velocity ¯uctuations decreases and in due course reaches close to zero. The wake is then similar to a steady laminar wake. In parallel, the frequency of the vortex shedding, not shown here, is found to be dependent on the heat input in the cylinder, see [1]. This frequency decreases monotonically with the increase of the heat input before the sudden vanishing of vortices. The same phenomenon is observed for the case of the heated ribbon. The level of velocity ¯uctuations and the frequency are also found to decrease with increasing heat input. This indicates that this eect is not simply linked to a modi®cation of the location of the separation point due to temperature dependence of the dynamic viscosity l close to the cylinder surface as suggested by Lecordier et al. [1]. When the same experiment was carried out in the water channel with a heated horizontal blu body, the opposite trend was found. Measurements of temperature ¯uctuations were obtained with the thermocouple located 8 diameters downstream of the 3 mm diameter heated ribbon and slightly o the centerline. As shown in Fig. 4, the level of temperature ¯uctuations increases with increasing heat input. Just below the critical Reynolds number, heating the ¯at ribbon initiates the
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Fig. 4. Example of enhancement of temperature ¯uctuations linked to a destabilization of the wake in water with heating; (x 8, y 0:5, Re ÿ Rec ÿ1:7; ÿ1; ÿ0:3).
instability. In parallel, the frequency of vortices, not presented here, is found to increase monotonically with the increase of the heat input, see [11]. However, in these experiments made in water¯ows with a 3 mm diameter heated body the range of heating where the eects of free convection may be ignored is narrower and restricts the observation of this phenomenon to a very limited range of Reynolds numbers. This opposite behavior in liquids and gases also found numerically by Socolescu et al. [20], suggests that this thermal control, in the absence of free convection eects, is due to both changes of dynamic viscosity and density with temperature and is not simply linked to density changes as claimed by Yu and Monkewitz [12]. The important role of viscosity was mentioned by Acrivos et al. [32]. Yang and Zebib [33] observed that the viscous term in the Orr±Sommerfeld equation has a profound eect in the near wake and considered the dual eect of viscosity and density on stability. The viscous in¯uence at low Reynolds numbers has been shown to change mode selection and associated growth rate in the in¯ectional instability of shear layers, see [34]. 4. In¯uence of heating on the velocity characteristics in the near wake of a blu body In this section, some experimental results are presented of the velocity ®eld downstream of a heated cylinder or a ¯at ribbon in air and in water in order to investigate the modi®cation of the velocity ®eld due to the heating. 4.1. Velocity results in the near wake of a heated cylinder in air The velocity distributions were measured in the wake of the cylinder (1 < x < 15) for 9 values of the
231
Fig. 5. Mean longitudinal velocities measured in the wake of the cylinder in air in isothermal and non-isothermal conditions; (Re ÿ Rec 10; P =L 0 and 20 W/m).
Reynolds number: Re ÿ Rec 2; 4; 6; 8; 10; 12:7; 15; 20; 30 and for several values of heating. Examples of the pro®les measured for Re ÿ Rec 10, P =L 0 and 20 W/m, are shown in Fig. 5. With the heating, the total shape of the longitudinal velocity pro®les is about identical with that observed in the isothermal case. The in¯uence of the heating is characterized by only slight changes of the velocity ®eld. The heated pro®les being slightly wider than the isothermal pro®les. This behavior is also characterized at each section by a decrease of the centerline mean longitudinal velocity in the heated case. The measured transverse velocities V , not plotted here, are always antisymmetric and present only slight differences with those obtained in the isothermal case. The distributions of the RMS values of the transverse velocities, with and without heating, measured for the same Reynolds number, Re ÿ Rec 10, are plotted in Fig. 6. In air the in¯uence of heating always results in a decrease of the level of the ¯uctuations i.e., heating the cylinder is equivalent reducing the Reynolds number. An increase of the power density per unit length results in an ampli®cation of the changes described above. This stabilization of the wake in air due to the heating can be taken into account by the decrease of an eective Reynolds number Reeff , as shown by Dumouchel et al. [10]. For all the experimental situations mentioned above, an eective Reynolds number Reeff was calculated using relation (1) and the cylinder temperature measured by a thermocouple located inside the heated tube. By this method, it appears that some characteristics of the heated or unheated wake are in close agreement when these quantities are related to the eective Reynolds number. For example, consider Fig. 7 where the length of the wake bubble s has been plotted versus (Reeff ÿ Rec ), heated and unheated results are quite similar. The eect of heating is also well taken into account by the use of Reeff when velocity ¯uctuations in the wake
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Fig. 6. RMS transverse velocities measured in the wake of the cylinder in air in isothermal and non-isothermal conditions; (Re ÿ Rec 10; P =L 0 and 20 W/m).
Fig. 8. Centerline evolution of RMS values of the transverse velocities in air in isothermal and non-isothermal conditions at the same eective Reynolds number: Re ÿ Reeff 6; cylinder case.
are considered. As presented in Fig. 8, the centerline evolution of rv for the same value of the eective Reynolds number, Reeff Rec 6, is similar in both isothermal and non-isothermal cases. rv increases in the near wake before reaching, at the same location, the same maximum value and then decreases further downstream. As shown in Fig. 9 in log±log plot rv max and xmax are independent of whether the body is heated or unheated. rv max is the maximum value of the RMS transverse velocities and xmax is the location of rv max . rv max increases when the eective Reynolds number increases while the opposite behavior is found for xmax . rv max is found to increase with the threshold dierence (Reeff ÿ Rec ) with 0:34 \an exponent of 0.34 i.e., rv max A Reeff ÿ Rec while
Fig. 9. Evolution of xmax and rv;max in air as a function of the eective Reynolds number; cylinder case.
xmax decreases with the threshold dierence (Reeff ÿ Rec ) with an exponent of ÿ0.4. When the non-normalized rv max is plotted as a function of (Reeff ÿ Rec ) using log±log coordinates, as in Fig. 10, the results obtained in heated and unheated conditions for the cylinder are not the same. However, in both cases the exponent relating rv max and (Reeff ÿ Rec ) is then close to +0.5, in agreement with the Landau theory, see [35,36]. 4.2. Velocity results in air and in water in the near wake of a heated ribbon Fig. 7. Mean length of the standing eddies in air in isothermal and non-isothermal conditions as a function of the eective Reynolds number; cylinder case.
Experimental results obtained for the heated or unheated ribbon in air were similar to the ones determined
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Fig. 10. Evolution of rv;max in air as a function of the eective Reynolds number; cylinder case.
for the cylinder and are not presented here. Comparisons of velocity results in air and in water are presented using the eective Reynolds number concept. In the case of air, the ribbon temperature was not known and the eective Reynolds number Reeff could not be deduced from relation (1). The eective Reynolds number was then deduced from the power density per unit length P/L. It was assumed, as checked for the cylinder, that the ratio between (Re ÿ Reeff ) and (Re ÿ Rec ) is equal to the ratio between P/L and (P/L)crit . For the case of water, the level of heating was very low in order to avoid buoyancy eects and the eective Reynolds numbers was approximately equal to the usual Reynolds number, i.e., calculated in isothermal condition. The small amount of heating did not allow comparisons to be made between the heated and unheated cases. In order to have signi®cant change of thermal properties of the ¯uid in the near wake, without buoyancy eects, higher cylinder temperature could be chosen by selecting smaller diameter cylinders according to relation (2). However this choice precludes the possibility of spatially resolved velocity measurements in the near wake. Comparisons of pro®les of the longitudinal mean velocity and the RMS values of the transverse velocities measured in air and in water for the same value of the eective Reynolds number, Reeff Rec 7:7, are shown in Figs. 11 and 12. Reeff Rec 7:7 was obtained under the following conditions Re Rec 7:7, P =l 0; Re Rec 12; 8, P =l 20 W/m in air and Re Rec 7:7, P =l 0 W/m; Re Rec 7:7, P =L 20 W/m in water. Results obtained in the same ¯uid at the same eective Reynolds number are very close. Furthermore, results for velocity measurements in air and in water at the same eective Reynolds number agree rather well. Nevertheless it appears that, near the ribbon, the maximum value of the longitudinal mean velocity is always higher in water than in air. This result
233
Fig. 11. Comparisons of pro®les of the longitudinal mean velocity measured in air and in water for the same value of the eective Reynolds number: Reeff Rec 7:7; ribbon case.
Fig. 12. Comparisons of pro®les of the RMS values of the transverse velocities measured in air and in water for the same value of the effective Reynolds number: Reeff Rec 7:7; ribbon case.
could be due to the dissimilar experimental setups used for these two blu bodies. The centerline evolution of rv is also found to be similar in air and in water, the intensity increasing in the near wake before reaching a maximum and decreasing further downstream. As shown in Fig. 13(a) and (b) in log±log plots rv max and xmax are again independent of the heated and unheated conditions. rv max is found to be increasing with the threshold dierence (Reeff ÿ Rec ) with the exponent 0.46 in air, 0.40 in water, while xmax decreasing with the exponent ÿ0.5 in both air and water. When the non-normalized rv max is plotted as a function of (Reeff ÿ Rec ), as in Fig. 14, the results obtained in heated and unheated conditions for the ribbon are again quite similar. There are large dierences however between the air and water results. Nevertheless, as already
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Fig. 13. (a) Evolution of xmax and rv;max in air as a function of the eective Reynolds number; ribbon case. (b) Evolution of xmax and rv;max in water as a function of the eective Reynolds number; ribbon case.
Fig. 15. De®nitions of D; d and oU . oy max Fig. 14. Evolution of rv;max in air and in water as a function of the eective Reynolds number; ribbon case.
found for the circular cylinder, the exponent for the relationship between rv max and (Reeff ÿ Rec ), for both air and water, is then +0.6 close to the value +0.5 predicted by the Landau theory, see [35,36].
5. Discussion These extensive measurements of velocity carried out downstream of a heated circular cylinder or ribbon show the in¯uence of heat input on the velocity ®eld in the near wake. They con®rm that over the studied Reynolds numbers range, control of vortex shedding, in air, can be realized using the heat input to the cylinder or ribbon.
At the present time, it is not clear which part of the near wake is the more in¯uenced by changes in the physical properties of the ¯uid that in turn lead to ¯ow ®eld conditions responsible for the control. However, comparisons of experimental results show that in similar eective Reynolds numbers conditions, the velocity pro®les at the end of the recirculation zone are roughly the same. To show this, some characteristics of the heated or unheated wake, given in Fig. 15 namely the maximum velocity gradient oU =oy max , the shear layer thickness d and the shear layer spacing D determined at the end of the mean recirculation zone downstream of the cylinder in air, were studied as a function of (Reeff ÿ Rec ) and are shown in Fig. 16. When the effective Reynolds number is used, the results with and without heating are similar. oU =oy max increases with increasing (Reeff ÿ Rec ) while D and d decrease. Following the analysis of Abernathy and Kronauer [37] which assumes that the shedding vortex street
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Fig. 16. Maximum mean velocity gradient, shear layer thickness and shear layer spacing at the end of the recirculation zone in air in isothermal and non-isothermal conditions as a function of the eective Reynolds number; air, cylinder case.
phenomenon could result from the dynamic interaction between the initial vortex sheets, we have already characterized in the isothermal situation this interaction using an interaction term I, see [28]. This term, I, can be calculated for each experiment, at the end of the recirculation zone, as the ratio between the rate of circulation in the shear layer and the shear layer spacing D. dC oU oy D max dU 3 I dt 2 D D This simple analysis assumes that the interaction between the two initial shear layers is proportional to the strength of each layer and inversely proportional to the distance between these layers. In order to compare the results obtained with the cylinder and the ribbon in air and in water, the interaction term has been normalized by using the frequency of the vortex shedding and the velocity U at D=2 determined at the critical Reynolds number. This normalized interaction term Iadim is presented as a function of (Reeff ÿ Rec ) in Fig. 17. From this ®gure the results obtained from cylinder and ribbon experiments agree reasonably well and the value of this normalized interaction term is about 5.5 at the transition. Furthermore, suppression of the vortex shedding phenomenon by heating is obtained when the interaction term decreases to the value obtained at the transition without heating. Some scatter exists for some points corresponding to the larger values of heating in air. The same values of heating lead also to some scatter in Fig. 7 related to the length of the wake bubble s . This could be due to the existence of ¯exural modes as shown by Ezersky and Ermoshin [3] for a given range of heating conditions. The physical mechanism of the control of the instability that we have described is close to the one described
235
Fig. 17. Variation in air and in water in isothermal and non-isothermal conditions of the normalized interaction term with (Reeff ÿ Rec ); circular cylinder and ribbon cases.
by Gerrard [38] and Strykowski and Sreenivasan [39]. This analysis is not in contradiction with the basis of the analysis of Yu and Monkewitz [12] who assume that the control of vortex shedding is related to slight changes of the interaction between the two initial mixing layers. These authors relate this behavior to the eect of density, due to the fact that results for constant and temperature-dependent viscosity were not signi®cantly dierent. However in their approach it seems that the pro®le-shape parameter N and the velocity-ratio parameter K of the selected velocity pro®les are not dependent on heating conditions. As we have shown in this paper, the initial velocity pro®les in the wake are always coupled with the temperature ®eld. In these conditions the main eect of heating would be to change viscosity and density around the cylinder leading to slight changes in the shape of the velocity pro®les in the near wake and consequently in stability conditions. The increase of the damping eect of viscosity with heating in air would be only an additional secondary eect. 6. Practical signi®cance The ¯ow past a heated blu body has many applications in various ®elds of engineering including heat exchangers and combustion devices. As shown in this study, the ¯ow regime downstream of a heated blu body just above the critical Reynolds number is strongly dependent on the variations of the ¯uid properties with temperature. 7. Conclusion From these experiments carried out over the Reynolds numbers range corresponding to the two-dimensional periodic wake, it appears that the control of
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vortex shedding downstream of a horizontal cylinder or ¯at ribbon can be easily realized using heat input to the cylinder or ribbon. In air ¯ow, increase of the heat input to the cylinder or ribbon leads to a suppression of the vortex shedding while in a liquid ¯owing around a heated cylinder or ribbon the eect is the opposite. However in liquids the range of heating conditions where the buoyancy eects can be ignored is more limited. The opposite behavior found in liquids and gases suggests that this control is due to both changes of dynamic viscosity and density with temperature and is not only dependent on density changes as claimed by Yu and Monkewitz [12]. This opposite behavior found in air and in water suggests that the eect of the change of viscosity is predominant. The changes of the density and the viscosity due to the heating of the near wake lead to slight changes in velocity pro®les and consequently in stability conditions and amplitude of the shed vortices. Extensive velocity measurements in air of the near wake in isothermal or non-isothermal conditions show that the in¯uence of heat input on the velocity ®eld downstream of the heated circular cylinder or ribbon can be taken into account using the eective Reynolds number approach. Furthermore, these measurements show that the control of the instability in heated or unheated conditions can be linked to a critical value of an interaction term related to the strength of the shear layers and the shear layer spacing at the end of the recirculation zone.
References [1] J.-C. Lecordier, L. Hamma, P. Paranthoen, The control of vortex shedding behind heated circular cylinders at low Reynolds numbers, Exp. Fluids 10 (1991) 224±229. [2] A.B. Ezerski, Detached ¯ow around a heated cylinder at small Mach numbers, Prikladnaya Mekhanika i Tekhn. Fizika 5 (1990) 56±62. [3] A.B. Ezerski, D.A. Ermoshin, The instability of density strati®ed vortices, Eur. J. Mech. B 14 (1995) 617±628. [4] N.D. K uchemann, Symposium on concentrated vortex motion, J. Fluid Mech. 21 (1965) 1±20. [5] G.F. Crum, T.J. Hanratty, Dissipation of a sheet of heated air in a turbulent ¯ow, Appl. Sci. Res. 15 (1965) 177±195. [6] E. Berger, M. Schumm, Untersuchungen der instabilituetsmechanismen im nachlauf von zylindern. Rep. no. Be343/18-1 Tech. Univ. Berlin, Germany, 1988. [7] M. Schumm, E. Berger, P. Monkewitz, Self-excited oscillations in the wake of two-dimensional blu bodies and their control, J. Fluid Mech. 271 (1994) 17±53. [8] K.R. Sreenivasan, S. Tavoularis, R. Henry, S. Corrsin, Temperature ¯uctuations and scales in grid generated turbulence, J. Fluid Mech. 100 (1980) 597±621. [9] P. Paranthoen, L.W.B. Browne, S. Le Masson, J.-C. Lecordier, Control of vortex shedding by thermal eect at low Reynolds number, Rapport interne MT1, Universite de Rouen, Janvier, 1995. [10] F. Dumouchel, J.-C. Lecordier, P. Paranthoen, The eective Reynolds number of a heated cylinder, Int. J. Heat Mass Transfer 41 (1998) 1787±1794.
[11] S. Le Masson, Contr^ ole de l'instabilite de Benard-von Karman en aval d'un obstacle chaue a faible nombre de Reynolds, These de doctorat de l'Universite de Rouen, 1991. [12] M.H. Yu, P.A. Monkewitz, The eect of nonuniform density on the absolute instability of two dimensional inertial jets and wakes, Phys. Fluids A 2 (1990) 1175±1181. [13] Y. Mori, K. Nijikata, T. Nobuhara, A fundamental study of symmetrical vortex generation behind a cylinder by wake heating or by splitter plate or mesh, Int. J. Heat Mass Transfer 29 (1986) 1193±1201. [14] K. Noto, R. Matsumoto, Numerical simulation on development of the K arm an vortex street due to the natural convection, Flow vizualization III, vol. 5, Pineridge, 1987, pp. 796±809. [15] S.K. Chang, J.Y. Sa, The eect of buoyancy on vortex shedding in the wake of a circular cylinder, J. Fluid Mech. 220 (1990) 253± 266. [16] V. Vilimpoc, R. Cole, P.C. Sukanek, Heat transfer in Newtonian liquids around a circular cylinder, Int. J. Heat Mass Transfer 33 (1990) 447±456. [17] T. Tezduyan, M. Kawahara, T.J. Hughes, A numerical study of vortex shedding around a heated-cooled circular cylinder by the three-step Taylor±Galerkin method, Int. J. Numer. Meth. Fluids 21 (1995) 857±867. [18] K. Nakabe, H. Hasegawa, K. Matsubara, K. Suzuki, Heat Transfer from an heated cylinder in a ¯ow between parallel plates in a free-forced combined convection regime, ISTP-9, Singapore, June 1996, pp. 25±28. [19] N. Michaud-Leblond, M. Belorgey, Near wake behavior of an heated circular cylinder: viscosity±buoyancy duality, Exp. Therm. Fluid Sci. 15 (1997) 91±100. [20] L. Socolescu, I. Mutabazi, O. Daube, S. Huberson, Etude de l'instabilite du sillage bidimensionnel derriere un cylindre faiblement chaue, C.R. Acad. des Sciences Paris 322 (IIb) (1995) 203± 208. [21] R.M. Fand, K.K. Keswani, Combined natural and forced convection heat transfer from horizontal cylinders to water, Int. J. Heat Mass Transfer 16 (1973) 1175±1191. [22] V.T. Morgan, The overall convective heat transfer from smooth circular cylinders, Adv. Heat Transfer 11 (1975) 199±263. [23] D. Gerrich, H. Eckelmann, In¯uence of end plates and free ends on the shedding frequency of circular cylinders, J. Fluid Mech. 122 (1982) 109±121. [24] C.W. Van Atta, M. Gharib, Ordered and chaotic vortex streets behind circular cylinder at low Reynolds numbers, J. Fluid Mech. 174 (1987) 113±133. [25] C.H.K. Williamson, Oblique and parallel modes of vortex shedding in the wake of a circular cylinder at low Reynolds numbers, J. Fluid Mech. 206 (1989) 579±627. [26] C.H.K. Williamson, Three-dimensional wake transition, J. Fluid Mech. 328 (1996) 345±407. [27] M. K onig, H. Eisenlohr, H. Eckelmann, The ®ne structure in the Strouhal±Reynolds relationship of the laminar wake of a circular cylinder, Phys. Fluids A 2 (1990) 1607±1614. [28] P. Paranthoen, L.W.B. Browne, S. Le Masson, J.-C. Lecordier, Characteristics of the near wake of a cylinder at low Reynolds numbers, Eur. J. Mech. B 18 (1999) 659±674. [29] M. Hammache, M. Gharib, A novel method to promote parallel vortex shedding in the wake of circular cylinders, Phys. Fluids A 1 (10) (1989) 1611±1614. [30] L. Talbot, R.K. Cheng, R.W. Scheer, D.R. Willis, Thermophoresis of particles in a heated boundary later, J. Fluid Mech. 101 (4) (1980) 737±759. [31] F. Dumouchel, Etude experimentale des champs dynamiques et thermiques de l'ecoulement de Benard-von Karman en aval d'un obstacle chaue dans l'air et dans l'eau, These de doctorat de l'Universite de Rouen, 1997.
J.-C. Lecordier et al. / Experimental Thermal and Fluid Science 21 (2000) 227±237 [32] A. Acrivos, L.G. Leal, D.D. Snowden, F. Pan, Further experiments on steady separated ¯ows past blu objects, J. Fluid Mech. 34 (1968) 25±48. [33] X. Yang, A. Zebib, Absolute and convective instability of a cylinder wake, Phys. Fluids A 1 (1989) 689±696. [34] E. Villermaux, On the role of viscosity in shear instabilities, Phys. Fluids A 10 (2) (1998) 368±373. [35] M. Provansal, C. Mathis, L. Boyer, Benard-von K arm an instability: transient and forced regimes, J. Fluid Mech. 182 (1987) 1±22.
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[36] B.J.A. Zielinska, J.E. Wesfreid, On the spatial structure of global modes in wake ¯ow, Phys. Fluids A 7 (6) (1995) 1418± 1424. [37] F.H. Abernathy, R.E. Kronauer, The formation of vortex street, J. Fluid Mech. 13 (1962) 1±20. [38] J.H. Gerrard, The mechanics of the formation region of vortices behind blu bodies, J. Fluid Mech. 25 (1966) 401±413. [39] P.J. Strykowski, K.R. Sreenivasan, On the suppression of vortex shedding at low Reynolds numbers, J. Fluid Mech. 218 (1990) 71± 107.
www.elsevier.nl/locate/etfs
Control of vortex shedding by thermal eect at low Reynolds numbers J.-C. Lecordier a, L.W.B. Browne b, S. Le Masson
a,1
, F. Dumouchel
a,2
, P. Paranthoen
a,*
a b
UMR 6614 C.N.R.S., Universit e de Rouen, 76821 Mont Saint Aignan Cedex, France Department of Mechanical Engineering, University of Newcastle, NSW 2308, Australia Received 26 May 1999; accepted 7 March 2000
Abstract An experimental study has been made of the control of vortex shedding in the wake of two two-dimensional blu bodies, a circular cylinder and a ¯at ribbon. The study has shown that this control, easily realized by heating the blu body, depends on the nature of the ¯uid. In the absence of buoyancy eects, related to the temperature dependence of the dynamic viscosity, the heating is found to stabilize the wake in air while the opposite result is obtained in water. Detailed measurements of the velocity ®elds in air, in isothermal and in heated body, show that this control is linked to slight modi®cations of the ¯ow in the near wake and can be taken into account by the eective Reynolds number approach. The measurements also show that the degree of instability can be related to the level of interaction between the two initial shear layers at the end of the recirculation zone. Ó 2000 Elsevier Science Inc. All rights reserved. Keywords: Wake; Shear-layer instability; Shedding control; Laser-doppler measurements
1. Introduction Extensive experiments carried out at Rouen and Nizhny Novgorod have shown that a very simple means of controlling vortex shedding from a cylinder at low Reynolds numbers, 45 < Re U1 d=mg < 90, can be obtained by heating or cooling the cylinder, [1±3]. Here U1 , d and mg are the upstream velocity, the cylinder diameter and the kinematic viscosity of the ¯uid at the temperature of the upstream ¯ow, respectively. In air, by applying heat input to the cylinder, the characteristics of vortex shedding can be signi®cantly altered and total suppression can be achieved by increasing the heat input suciently. This experimental result was earlier found by Uberoi and brie¯y reported
* Corresponding author. Tel.: +33-02-35-14-65-80; fax: +33-02-3570-83-84. E-mail address: [email protected] (P. ParanthoeÈn). 1 Present address: Technopole Anticipa, Lab QFE ESP, 2 Avenue P. Marzin, 22307 Lannion Cedex, France. 2 Present address: Laboratoire de Mecanique, Universite du Havre, 25 rue Ph. Lebon, BP 540 76058 Le Havre Cedex, France.
in 1965 at a Symposium on the concentrated vortex motions in ¯uids, see [4]. The control of the transition from a periodic to a laminar wake by applying a heating current within a line source was also mentioned by Crum and Hanratty [5]. Berger and Schumm [6], Schumm et al. [7] reported similar results in an experiment mainly devoted to the proof that the Landau equation exactly governs the bifurcation process. As proposed at ®rst by Uberoi, this eect has been related to the increase of the kinematic viscosity of the gas and to the corresponding decrease of the eective Reynolds number. However the precise value of the effective temperature to be used was not clear, see for example Sreenivasan et al. [8]. Consideration of this eective temperature by Lecordier et al. [1], Ezerski [2], Paranthoen et al. [9], Dumouchel et al. [10] indicated that, in air ¯ow, it was never the usual ®lm temperature de®ned as the arithmetic mean between the cylinder temperature Tw and the air ¯ow temperature Te . By considering the temperature dependence of the kinematic viscosity mg , this eective temperature Teff for a heated cylinder in air was found by LeMasson [11] to be: Teff Te 0:24 0:02DTw ; where DTw Tw ÿ Te ; 1 a value close to the hot recirculation zone temperature.
0894-1777/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved. PII: S 0 8 9 4 - 1 7 7 7 ( 0 0 ) 0 0 0 0 7 - 8
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Nomenclature U longitudinal velocity, m/s V transverse velocity, m/s d diameter of the cylinder, m L length of the cylinder, m g gravitational acceleration, m/s2 T temperature, °C DT temperature excess, °C x coordinate for horizontal axis, m y coordinate for vertical axis, m s length of the wake bubble P electric power, W I interaction term, I D1 dU D=2 oU oy max Nu Nusselt number Re Reynolds number Gr Grashof number Greek symbols a non-dimensional parameter, a Gr=Re2
Conversely Yu and Monkewitz [12] argued that the main eect of heating in air is a reduction of the density in the near wake which leads to a reduction of absolute instability. In their stability analysis, using similar pro®les for velocity and temperature, calculations for constant or temperature-dependent dynamic viscosity did not lead to signi®cant dierences. These authors concluded that ``...the change in the stability characteristics brought about by heating the cylinder was the result of a subtly modi®ed interaction between the two mixing layers via the inertial terms and not a viscous eect''. When the level of heating is increased and is high enough to induce buoyancy eects, mixed-convection or free-convection regimes have to be considered. Under these circumstances the buoyancy eects, which appear in the near wake become predominant in comparison with variable ¯uid properties eects. A large number of experimental or numerical studies have reported the in¯uence of such buoyancy eects on the development of the vortex shedding phenomenon in the vertical con®guration, see [13±19]. In this con®guration when the level of heating increases, the vortex shedding is found at ®rst to increase owing to the positive buoyancy force and then to break down. In the horizontal con®guration, except for the numerical studies by Socolescu et al. [20] and Nakabe et al. [18], most of the previous investigations were limited to the study of the change of the eective Nusselt number with mixed convection, see [21,22]. The limit between the forced-convection regime and the mixed-convection regime is usually characterized by a critical value of the parameter a Gr=Re2 depending on the geometry of the ¯ow. Here Gr is the Grashof number de®ned as Gr gbDTw d 3 =m2g , where g is the acceleration due to the gravity, b the temperature coecient for volume expansion, DTw is the temperature dierence between the cylinder and the upstream ¯ow. Based on the change of the eective Nusselt number
b d D mg r
temperature coecient for volume expansion, °Cÿ1 shear layer thickness, m shear layer spacing, m kinematic viscosity, m2 /s RMS value of ¯uctuations
Subscripts w wall c critical e refers to conditions at eective temperature ®lm refers to conditions at arithmetic mean ®lm temperature, Tfilm Tw Te =2 max maximum value e free stream Superscript * normalized value
with mixed convection to the Nusselt number with forced convection alone, the limiting value of a is about 0.05 in the cross-¯ow situation as shown by Fand and Keswani [21] and Morgan [22]. In our experimental con®guration where the horizontal cylinder is located transversely in a cross-¯ow of uniform velocity, the range of heating for the forcedconvection regime, deduced from the critical value of a mentioned above, and for the laminar vortex street range studied here (Re 50), needed to be such that the temperature dierence between the cylinder and the ¯ow was Tw ÿ Te < 12:5
m2g : bd 3
2
The motivation of our experimental investigation was to provide some new information on the in¯uence of changes in the density and viscosity for both air and water on the ¯ow regime downstream of a heated blu body in the absence of buoyancy eects. For this purpose, near-wake heating experiments were carried out in ¯uid, air and water, the physical properties of which are not temperature dependent in the same way. While the in¯uence of heating on density is identical in liquids and gases, the in¯uence of heating on the dynamic viscosity is opposite. The dynamic viscosity of gases increases and that of liquids decreases. The eects related to the in¯uence of viscosity and density may then be separated. Particular attention was made to avoid a level of heating high enough to generate buoyancy eects in order that the heated blu body always operated in a forced convection regime. Furthermore, this experiment was limited to a study of the near wake over a range of values of the Reynolds numbers just above the critical Reynolds number where the oblique modes of vortex shedding are not present, see [23±27].
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After a recall of the basic phenomenon in air and water ¯ows, the characteristics of the velocity pro®les in the near wake of two two-dimensional blu bodies heated or not are presented by using the eective Reynolds number concept as described by Lecordier et al [1] and Dumouchel et al. [10]. These measurements are used to determine the level of interaction between the two initial shear layers at the end of the recirculation zone and consequently in the instability conditions, previously studied in isothermal wakes by Paranthoen et al. [28].
2. Experimental setup 2.1. Experiments in air¯ow These experiments were carried out in air, in the potential core of a laminar plane jet. The plane jet facility consists of a variable speed blower supplying air to a two-dimensional rectangular 10:1 contraction. The jet exits normally to an end plate (17 35 cm2 ) from a slit of width b 15 mm and span 15 cm centrally located in this plate. To minimize air turbulence, large chambers with baes and sound absorbing material were used between the fan and the contraction. On the centerline at the nozzle exit the turbulence intensity ru is approximately 0.4%. The vortex shedding blu body was successively, a smooth stainless steel 1 mm diameter tube and a 1 mm 55 lm ribbon mounted horizontally in the middle of the jet close to the exit plane. Their total length was 15 cm (L=d 150). These bodies could be heated by Joule eect by means of direct current. The heat input per unit length was calculated from the supplied voltage and current. The temperature of the 1 mm diameter circular cylinder was measured by means of a thermocouple located within the tube. In order to avoid vibrations, the circular cylinder and the ribbon were damped with pieces of foam located at their ends and suciently tensioned. Furthermore the use of four parallel 100 lm diameter wires separated by about 1 mm and located on each edge of the jet exit plane perpendicular to the cylinder, was found to prevent oblique shedding. With the selected diameter, d 1 mm, Reynolds numbers Re from 34 up to 75 could be obtained by varying the upstream velocity U1 between 50 and 115 cm/s. The 1 mm cylinder temperature excess could reach a maximum of 250°C leading to a maximum value of a of 0.02 corresponding to the forced-convection regime.
229
2.2. Experiments in water ¯ow Similar experiments were carried out in water. A stainless steel tube of 3 mm diameter or a nichrome ribbon (3:2 mm 85 lm) were either located in a 50 mm diameter water tunnel at 100 mm from an ASME nozzle or in the middle of a water channel with a free surface. The working depth of the water channel was approximately 13 cm. In order to prevent oblique shedding we used, as did Hammache and Gharib [29], two upstream circular cylinders positioned normal to the obstacle. The tube was heated by an isolated constantan wire twisted on a ceramic cylinder placed inside the tube and driven by a direct current. The ribbon was directly heated by a direct current. In most of the experiments the temperature excess was limited to 4°C in order that a be less than 0.05 and so avoid buoyancy eects. With the 3 mm circular cylinder the eddies were seen by injecting ink through a 0.4 mm hole in the tube, located on the centerline of the tunnel and facing the ¯ow. The vortex frequency was then obtained by measuring the time needed for approximately 50 eddies to be produced. During these experiments the upstream velocity and temperature were carefully measured with a BROOKS electro-magnetic ¯owmeter and a thermometer in order to control the Reynolds number. As shown in Fig. 1 some visualizations of the ¯ow were obtained in the water channel by using very small particles generated by electrolytic etching of silver wires located in a test section upstream of the cylinder. 2.3. Experimental techniques Due to limitations of the hot-wire for velocity measurements in the air near wake, see [28], velocity measurements were made by means of an LDA system. We used an LDA TSI system incorporating a 1.5 W spectra physics laser, an integrated optical transmission unit and a light collection using the forward-scattering mode. The optical measuring volume was 0:08 0:08 1:4 mm3 with the major axis parallel to the cylinder. Directional ambiguity was avoided by the use of the frequency shifting technique with Bragg cells. Velocity measurements were found to be not very sensitive to the sampling rate and the seeding conditions. The location of the center of the measuring volume was accurately obtained by studying the light scattered by a 20 lm wire on which the measuring volume was adjusted. The location of the wire was known by displacing it from the surface of the cylinder after electric contact.
Fig. 1. Visualization of the ¯ow in water at Re > Rec .
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Fig. 2. Experimental setup.
For the heating conditions in air, some problems exist with the LDA system due to the absence of particles in the center of the near wake. Even at a moderate level of temperature (Tw ÿ Te 50°C), it appears that particles are missing in a symmetrical manner in the center of the wake bubble. This eect could be due to the thermophoresis eect already mentioned by Talbot et al. [30] in LDA measurements made in a strongly heated boundary layer ( 700°C). In our situation even with moderate temperature dierences the small dimension of the 1 mm diameter cylinder induces close to the cylinder surface large temperature gradients ( 20 000°C/m). This eect prevents particles from seeding the recirculation zone. Due to the small time constant of the particles this eect only occurs in the wake bubble and accurate velocity measurements are possible at all other positions, see [11]. The same two components LDA system was used for velocity measurements in the water ¯ow. Uncertainties in measured mean and ¯uctuating velocities were 2% and 5% in air and water, respectively. Measurements of the temperature ®eld were made in water using thermocouples built with 70 lm diameter Chromel and Alumel wires. The frequency response of the thermocouple was about 50 Hz and sucient to measure the temperature ¯uctuations lower than 1 Hz generated in the heated wake in water as shown by Dumouchel [31]. As shown in Fig. 2, the origin of the coordinate system was taken at the center of the cylinder or the ribbon. The x-axis was measured in the direction of the ¯ow, the y-axis was perpendicular to the ¯ow and the zaxis coincided with the cylinder axis. U and V are the mean values of longitudinal (along x) and transverse (along y) velocities respectively. ru and rv are the RMS values of longitudinal and transverse velocity ¯uctuations, respectively. In our study, the asterisk always denotes normalization. The lengths are non-dimensionalized by the diameter of the cylinder or the width of ribbon, d, and the velocities are normalized by the upstream velocity U1 . 3. Description of the basic phenomenon The ®rst results concern the possibility of the control of vortex shedding downstream from a horizontal heated cylinder located in air¯ow or water¯ow by means of a heating current applied to the cylinder. Examples of the suppression of vortex shedding are shown in Fig. 3.
Fig. 3. Example of suppression of transverse velocity ¯uctuations linked to a stabilization of the wake in air with heating; (x 10, y 0; Re ÿ Rec 2; 4; 6; 8; 10).
Measurements of the intensity of the transverse velocity rv were obtained on the centerline at 10 diameters. As shown in this ®gure, the intensity of the velocity ¯uctuations is practically zero when the heat input per unit length reaches a critical value (P/L)crit . In this experiment the Reynolds numbers Re, calculated with the dynamic viscosity of the non-heated upstream ¯ow, were always larger than the critical Reynolds number Rec . This critical Reynolds number Rec , corresponding to the transition from a two-dimensional steady to a two-dimensional periodic wake. For this reason there are velocity ¯uctuations in the heated wake, due to the vortex shedding, for the lowest heating conditions. When the power per unit length increases, the intensity of velocity ¯uctuations decreases and in due course reaches close to zero. The wake is then similar to a steady laminar wake. In parallel, the frequency of the vortex shedding, not shown here, is found to be dependent on the heat input in the cylinder, see [1]. This frequency decreases monotonically with the increase of the heat input before the sudden vanishing of vortices. The same phenomenon is observed for the case of the heated ribbon. The level of velocity ¯uctuations and the frequency are also found to decrease with increasing heat input. This indicates that this eect is not simply linked to a modi®cation of the location of the separation point due to temperature dependence of the dynamic viscosity l close to the cylinder surface as suggested by Lecordier et al. [1]. When the same experiment was carried out in the water channel with a heated horizontal blu body, the opposite trend was found. Measurements of temperature ¯uctuations were obtained with the thermocouple located 8 diameters downstream of the 3 mm diameter heated ribbon and slightly o the centerline. As shown in Fig. 4, the level of temperature ¯uctuations increases with increasing heat input. Just below the critical Reynolds number, heating the ¯at ribbon initiates the
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Fig. 4. Example of enhancement of temperature ¯uctuations linked to a destabilization of the wake in water with heating; (x 8, y 0:5, Re ÿ Rec ÿ1:7; ÿ1; ÿ0:3).
instability. In parallel, the frequency of vortices, not presented here, is found to increase monotonically with the increase of the heat input, see [11]. However, in these experiments made in water¯ows with a 3 mm diameter heated body the range of heating where the eects of free convection may be ignored is narrower and restricts the observation of this phenomenon to a very limited range of Reynolds numbers. This opposite behavior in liquids and gases also found numerically by Socolescu et al. [20], suggests that this thermal control, in the absence of free convection eects, is due to both changes of dynamic viscosity and density with temperature and is not simply linked to density changes as claimed by Yu and Monkewitz [12]. The important role of viscosity was mentioned by Acrivos et al. [32]. Yang and Zebib [33] observed that the viscous term in the Orr±Sommerfeld equation has a profound eect in the near wake and considered the dual eect of viscosity and density on stability. The viscous in¯uence at low Reynolds numbers has been shown to change mode selection and associated growth rate in the in¯ectional instability of shear layers, see [34]. 4. In¯uence of heating on the velocity characteristics in the near wake of a blu body In this section, some experimental results are presented of the velocity ®eld downstream of a heated cylinder or a ¯at ribbon in air and in water in order to investigate the modi®cation of the velocity ®eld due to the heating. 4.1. Velocity results in the near wake of a heated cylinder in air The velocity distributions were measured in the wake of the cylinder (1 < x < 15) for 9 values of the
231
Fig. 5. Mean longitudinal velocities measured in the wake of the cylinder in air in isothermal and non-isothermal conditions; (Re ÿ Rec 10; P =L 0 and 20 W/m).
Reynolds number: Re ÿ Rec 2; 4; 6; 8; 10; 12:7; 15; 20; 30 and for several values of heating. Examples of the pro®les measured for Re ÿ Rec 10, P =L 0 and 20 W/m, are shown in Fig. 5. With the heating, the total shape of the longitudinal velocity pro®les is about identical with that observed in the isothermal case. The in¯uence of the heating is characterized by only slight changes of the velocity ®eld. The heated pro®les being slightly wider than the isothermal pro®les. This behavior is also characterized at each section by a decrease of the centerline mean longitudinal velocity in the heated case. The measured transverse velocities V , not plotted here, are always antisymmetric and present only slight differences with those obtained in the isothermal case. The distributions of the RMS values of the transverse velocities, with and without heating, measured for the same Reynolds number, Re ÿ Rec 10, are plotted in Fig. 6. In air the in¯uence of heating always results in a decrease of the level of the ¯uctuations i.e., heating the cylinder is equivalent reducing the Reynolds number. An increase of the power density per unit length results in an ampli®cation of the changes described above. This stabilization of the wake in air due to the heating can be taken into account by the decrease of an eective Reynolds number Reeff , as shown by Dumouchel et al. [10]. For all the experimental situations mentioned above, an eective Reynolds number Reeff was calculated using relation (1) and the cylinder temperature measured by a thermocouple located inside the heated tube. By this method, it appears that some characteristics of the heated or unheated wake are in close agreement when these quantities are related to the eective Reynolds number. For example, consider Fig. 7 where the length of the wake bubble s has been plotted versus (Reeff ÿ Rec ), heated and unheated results are quite similar. The eect of heating is also well taken into account by the use of Reeff when velocity ¯uctuations in the wake
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Fig. 6. RMS transverse velocities measured in the wake of the cylinder in air in isothermal and non-isothermal conditions; (Re ÿ Rec 10; P =L 0 and 20 W/m).
Fig. 8. Centerline evolution of RMS values of the transverse velocities in air in isothermal and non-isothermal conditions at the same eective Reynolds number: Re ÿ Reeff 6; cylinder case.
are considered. As presented in Fig. 8, the centerline evolution of rv for the same value of the eective Reynolds number, Reeff Rec 6, is similar in both isothermal and non-isothermal cases. rv increases in the near wake before reaching, at the same location, the same maximum value and then decreases further downstream. As shown in Fig. 9 in log±log plot rv max and xmax are independent of whether the body is heated or unheated. rv max is the maximum value of the RMS transverse velocities and xmax is the location of rv max . rv max increases when the eective Reynolds number increases while the opposite behavior is found for xmax . rv max is found to increase with the threshold dierence (Reeff ÿ Rec ) with 0:34 \an exponent of 0.34 i.e., rv max A Reeff ÿ Rec while
Fig. 9. Evolution of xmax and rv;max in air as a function of the eective Reynolds number; cylinder case.
xmax decreases with the threshold dierence (Reeff ÿ Rec ) with an exponent of ÿ0.4. When the non-normalized rv max is plotted as a function of (Reeff ÿ Rec ) using log±log coordinates, as in Fig. 10, the results obtained in heated and unheated conditions for the cylinder are not the same. However, in both cases the exponent relating rv max and (Reeff ÿ Rec ) is then close to +0.5, in agreement with the Landau theory, see [35,36]. 4.2. Velocity results in air and in water in the near wake of a heated ribbon Fig. 7. Mean length of the standing eddies in air in isothermal and non-isothermal conditions as a function of the eective Reynolds number; cylinder case.
Experimental results obtained for the heated or unheated ribbon in air were similar to the ones determined
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Fig. 10. Evolution of rv;max in air as a function of the eective Reynolds number; cylinder case.
for the cylinder and are not presented here. Comparisons of velocity results in air and in water are presented using the eective Reynolds number concept. In the case of air, the ribbon temperature was not known and the eective Reynolds number Reeff could not be deduced from relation (1). The eective Reynolds number was then deduced from the power density per unit length P/L. It was assumed, as checked for the cylinder, that the ratio between (Re ÿ Reeff ) and (Re ÿ Rec ) is equal to the ratio between P/L and (P/L)crit . For the case of water, the level of heating was very low in order to avoid buoyancy eects and the eective Reynolds numbers was approximately equal to the usual Reynolds number, i.e., calculated in isothermal condition. The small amount of heating did not allow comparisons to be made between the heated and unheated cases. In order to have signi®cant change of thermal properties of the ¯uid in the near wake, without buoyancy eects, higher cylinder temperature could be chosen by selecting smaller diameter cylinders according to relation (2). However this choice precludes the possibility of spatially resolved velocity measurements in the near wake. Comparisons of pro®les of the longitudinal mean velocity and the RMS values of the transverse velocities measured in air and in water for the same value of the eective Reynolds number, Reeff Rec 7:7, are shown in Figs. 11 and 12. Reeff Rec 7:7 was obtained under the following conditions Re Rec 7:7, P =l 0; Re Rec 12; 8, P =l 20 W/m in air and Re Rec 7:7, P =l 0 W/m; Re Rec 7:7, P =L 20 W/m in water. Results obtained in the same ¯uid at the same eective Reynolds number are very close. Furthermore, results for velocity measurements in air and in water at the same eective Reynolds number agree rather well. Nevertheless it appears that, near the ribbon, the maximum value of the longitudinal mean velocity is always higher in water than in air. This result
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Fig. 11. Comparisons of pro®les of the longitudinal mean velocity measured in air and in water for the same value of the eective Reynolds number: Reeff Rec 7:7; ribbon case.
Fig. 12. Comparisons of pro®les of the RMS values of the transverse velocities measured in air and in water for the same value of the effective Reynolds number: Reeff Rec 7:7; ribbon case.
could be due to the dissimilar experimental setups used for these two blu bodies. The centerline evolution of rv is also found to be similar in air and in water, the intensity increasing in the near wake before reaching a maximum and decreasing further downstream. As shown in Fig. 13(a) and (b) in log±log plots rv max and xmax are again independent of the heated and unheated conditions. rv max is found to be increasing with the threshold dierence (Reeff ÿ Rec ) with the exponent 0.46 in air, 0.40 in water, while xmax decreasing with the exponent ÿ0.5 in both air and water. When the non-normalized rv max is plotted as a function of (Reeff ÿ Rec ), as in Fig. 14, the results obtained in heated and unheated conditions for the ribbon are again quite similar. There are large dierences however between the air and water results. Nevertheless, as already
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Fig. 13. (a) Evolution of xmax and rv;max in air as a function of the eective Reynolds number; ribbon case. (b) Evolution of xmax and rv;max in water as a function of the eective Reynolds number; ribbon case.
Fig. 15. De®nitions of D; d and oU . oy max Fig. 14. Evolution of rv;max in air and in water as a function of the eective Reynolds number; ribbon case.
found for the circular cylinder, the exponent for the relationship between rv max and (Reeff ÿ Rec ), for both air and water, is then +0.6 close to the value +0.5 predicted by the Landau theory, see [35,36].
5. Discussion These extensive measurements of velocity carried out downstream of a heated circular cylinder or ribbon show the in¯uence of heat input on the velocity ®eld in the near wake. They con®rm that over the studied Reynolds numbers range, control of vortex shedding, in air, can be realized using the heat input to the cylinder or ribbon.
At the present time, it is not clear which part of the near wake is the more in¯uenced by changes in the physical properties of the ¯uid that in turn lead to ¯ow ®eld conditions responsible for the control. However, comparisons of experimental results show that in similar eective Reynolds numbers conditions, the velocity pro®les at the end of the recirculation zone are roughly the same. To show this, some characteristics of the heated or unheated wake, given in Fig. 15 namely the maximum velocity gradient oU =oy max , the shear layer thickness d and the shear layer spacing D determined at the end of the mean recirculation zone downstream of the cylinder in air, were studied as a function of (Reeff ÿ Rec ) and are shown in Fig. 16. When the effective Reynolds number is used, the results with and without heating are similar. oU =oy max increases with increasing (Reeff ÿ Rec ) while D and d decrease. Following the analysis of Abernathy and Kronauer [37] which assumes that the shedding vortex street
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Fig. 16. Maximum mean velocity gradient, shear layer thickness and shear layer spacing at the end of the recirculation zone in air in isothermal and non-isothermal conditions as a function of the eective Reynolds number; air, cylinder case.
phenomenon could result from the dynamic interaction between the initial vortex sheets, we have already characterized in the isothermal situation this interaction using an interaction term I, see [28]. This term, I, can be calculated for each experiment, at the end of the recirculation zone, as the ratio between the rate of circulation in the shear layer and the shear layer spacing D. dC oU oy D max dU 3 I dt 2 D D This simple analysis assumes that the interaction between the two initial shear layers is proportional to the strength of each layer and inversely proportional to the distance between these layers. In order to compare the results obtained with the cylinder and the ribbon in air and in water, the interaction term has been normalized by using the frequency of the vortex shedding and the velocity U at D=2 determined at the critical Reynolds number. This normalized interaction term Iadim is presented as a function of (Reeff ÿ Rec ) in Fig. 17. From this ®gure the results obtained from cylinder and ribbon experiments agree reasonably well and the value of this normalized interaction term is about 5.5 at the transition. Furthermore, suppression of the vortex shedding phenomenon by heating is obtained when the interaction term decreases to the value obtained at the transition without heating. Some scatter exists for some points corresponding to the larger values of heating in air. The same values of heating lead also to some scatter in Fig. 7 related to the length of the wake bubble s . This could be due to the existence of ¯exural modes as shown by Ezersky and Ermoshin [3] for a given range of heating conditions. The physical mechanism of the control of the instability that we have described is close to the one described
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Fig. 17. Variation in air and in water in isothermal and non-isothermal conditions of the normalized interaction term with (Reeff ÿ Rec ); circular cylinder and ribbon cases.
by Gerrard [38] and Strykowski and Sreenivasan [39]. This analysis is not in contradiction with the basis of the analysis of Yu and Monkewitz [12] who assume that the control of vortex shedding is related to slight changes of the interaction between the two initial mixing layers. These authors relate this behavior to the eect of density, due to the fact that results for constant and temperature-dependent viscosity were not signi®cantly dierent. However in their approach it seems that the pro®le-shape parameter N and the velocity-ratio parameter K of the selected velocity pro®les are not dependent on heating conditions. As we have shown in this paper, the initial velocity pro®les in the wake are always coupled with the temperature ®eld. In these conditions the main eect of heating would be to change viscosity and density around the cylinder leading to slight changes in the shape of the velocity pro®les in the near wake and consequently in stability conditions. The increase of the damping eect of viscosity with heating in air would be only an additional secondary eect. 6. Practical signi®cance The ¯ow past a heated blu body has many applications in various ®elds of engineering including heat exchangers and combustion devices. As shown in this study, the ¯ow regime downstream of a heated blu body just above the critical Reynolds number is strongly dependent on the variations of the ¯uid properties with temperature. 7. Conclusion From these experiments carried out over the Reynolds numbers range corresponding to the two-dimensional periodic wake, it appears that the control of
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vortex shedding downstream of a horizontal cylinder or ¯at ribbon can be easily realized using heat input to the cylinder or ribbon. In air ¯ow, increase of the heat input to the cylinder or ribbon leads to a suppression of the vortex shedding while in a liquid ¯owing around a heated cylinder or ribbon the eect is the opposite. However in liquids the range of heating conditions where the buoyancy eects can be ignored is more limited. The opposite behavior found in liquids and gases suggests that this control is due to both changes of dynamic viscosity and density with temperature and is not only dependent on density changes as claimed by Yu and Monkewitz [12]. This opposite behavior found in air and in water suggests that the eect of the change of viscosity is predominant. The changes of the density and the viscosity due to the heating of the near wake lead to slight changes in velocity pro®les and consequently in stability conditions and amplitude of the shed vortices. Extensive velocity measurements in air of the near wake in isothermal or non-isothermal conditions show that the in¯uence of heat input on the velocity ®eld downstream of the heated circular cylinder or ribbon can be taken into account using the eective Reynolds number approach. Furthermore, these measurements show that the control of the instability in heated or unheated conditions can be linked to a critical value of an interaction term related to the strength of the shear layers and the shear layer spacing at the end of the recirculation zone.
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