Exp Fluids (2010) 48:1059–1079 DOI 10.1007/s00348-009-0791-6
RESEARCH ARTICLE
An experimental investigation into the effect of vortex generators on the near-wake flow of a circular cylinder ¨ nal • Mehmet Atlar Ug˘ur Oral U
Received: 23 January 2009 / Revised: 19 November 2009 / Accepted: 20 November 2009 / Published online: 10 December 2009 Ó Springer-Verlag 2009
Abstract The effect of the streamwise vortex generators on the near-wake flow structure of a circular cylinder was experimentally investigated. Digital particle image velocimetry (DPIV) measurements were performed in a large circulating water tunnel facility at a Reynolds number of 41,300 where the flow around a bare cylinder was expected to be at the sub-critical flow state. In order to capture various flow properties and to provide a detailed wake flow topology, the DPIV images were analysed with three different but complementary flow field decomposition techniques which are Reynolds averaging, phase averaging and proper orthogonal decomposition (POD). The effect of the vortex generators was clearly demonstrated both in qualitative and in quantitative manner. Various topological features such as vorticity and stress distribution of the flow fields as well as many other key flow characteristics including the length scales and the Strouhal number were discussed in the study. To the best of the authors’ knowledge, the study presents the first DPIV visualization of the near-wake flow of a circular cylinder fitted with the vortex generators in the open literature.
¨ nal (&) U. O. U Faculty of Naval Architecture and Ocean Engineering, Istanbul Technical University, Istanbul, Turkey e-mail:
[email protected] URL: www.gidb.itu.edu.tr M. Atlar School of Marine Science and Technology, Newcastle University, Newcastle Upon Tyne, UK e-mail:
[email protected] URL: www.ncl.ac.uk/marine/
1 Introduction Because of its geometrical simplicity, the circular cylinder is one of the most commonly used structures in many engineering applications which are subjected to flow motion (e.g. chimneys, towers, cables, bridge supports, offshore structures). In spite of the simple geometry, flow around a circular cylinder is an extremely complex phenomenon that is highly dependent on the flow regime. Therefore, the investigation of this flow field has been one of the most interesting research topics occupying many aero- and hydrodynamic researchers during the last century. Amongst those, Coutanceau and Bouard (1977), Williamson (1996), Basu (1985) and Norberg (2003) presented major review studies on the properties of this most interesting flow. Zdravkovich (1997) also presented an excellent review of the topic covering whole range of the flow regimes while Zdravkovich (1990) and (1997) provided comprehensive reviews for the detailed classification of various flow regimes. It is a well known fact that boundary layer separation behind bluff bodies, including circular cylinders, results in excess energy losses as well as other flow induced problems. In these situations, it is possible to control the flow separation using various active or passive techniques (Gad-el-Hak and Bushnell 1991). One of the most effective techniques for preventing or retarding the flow separation is the use of passive vortex generators which can stimulate macro-vortical motions confined in the boundary layer and its close surroundings, hence, providing momentum enhancement in the vicinity of a wall. Simple geometrical properties of the passive vortex generators can provide relatively practical and cost effective solutions to complex flow separation phenomena. Hence, vortex generators are commonly used as flow control devices, especially in
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aerodynamic applications (Lin 2002). In addition, the effect of the different shapes of vortex generators on the separation delay has been widely studied for various geometries such as aerofoil, backward-facing step and flat plate in the open literature. Lin et al. (1989, 1990), Wendt et al. (1993) and Yao et al. (2002) in particular give representative studies of this continuing research with an emphasis on effective energy savings whilst Lin (1999) and (2002) give comprehensive studies focusing on the subboundary layer vortex generators. Whilst there are a large number of studies exploring the subject of flow around circular cylinders and that of the vortex generators separately, there are only a limited number of studies investigating these two subjects in combination. Amongst these studies, Igarashi (1985) examined the effect of ‘saw-blade’-type roughness devices on the flow around a circular cylinder over a Reynolds number (Re) range of 8.7 9 103 \ Re \ 6.37 9 104. In his study, Igarashi called these devices ‘vortex generators’. He positioned them at various locations around a cylinder and made a classification of the resulting flow based on a ‘‘roughness Reynolds number’’ which was dependent upon the height of the devices, their locations around the cylinder and the flow velocity at the outer boundary layer. In a similar study, Igarashi and Iida (1988) investigated the effect of differing vortex generator height on the flow and heat transfer around a circular cylinder in a Reynolds number range of 1.3 9 103 \ Re \ 5.2 9 104. Although the name ‘vortex generator’ was used by the previous authors, their proposed devices do not produce streamwise (well-defined) vortices, as typical conventional vortex generators do, but rather create waviness in the boundary layer which causes a transition to turbulence. Joubert and Hoffman (1962) conducted an experimental study to investigate the effect of the vortex generators on the drag of a circular cylinder. The vortex generators were placed on the cylinder at various angular positions relative to the first stagnation line. Significant drag reduction at Re = 1.3 9 105 was reported for all the angular positions varying from 10° to 70°, with an optimum position at 50°. A maximum drag reduction of 71% was achieved with the vortex generators at Re = 1.7 9 105 which was the beginning of the drag crisis in this study. The vortex generators lost their effectiveness after Re = 3 9 105 which presumably corresponds to ‘‘single bubble’’ regime of the ‘‘transition-in-boundary-layers’’ state (Zdravkovich 1997). This extremely sensitive flow regime displays an asymmetric separation bubble on one side of the cylinder, which causes a discontinuous fall of the drag. Joubert and Hoffman (1962) presented limited information on the measurements, and their study did not include the geometrical properties of the vortex generators used or the pictorial details of the wake flow. The only information
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given about the properties of the wake pattern was that it became narrower at the lower drag values. In a subsequent study, Johnson and Joubert (1969) investigated the effect of the vortex generators on the heat transfer properties of a circular cylinder, as well as the drag, in the range 5 9 104 \ Re \ 4 9 105. Similar to the previous study, a single vortex generator configuration was used in the investigation while its angular position relative to the first stagnation line was varied. Increasing the angular positions up to 60° resulted in an increasing effect of the vortex generators on the drag up to Re = 105 where the drag crisis was starting. Significant drag reduction was achieved for all cases up to Re = 3 9 105 including the ones with vortex generators fitted at 70° and 80°. As a quantified example, the vortex generators at an angular position of 70° provided a *22% drag reduction compared to the bare cylinder case for Re = 45,000, which is similar to the Reynolds number used in the present study. In their experiments, Johnson and Joubert visualized the flow pattern on the cylinder surface using an oil-film technique and noted that the vortex generators caused a wavy separation line. The study, however, did not include any information on the topological feature of the near-wake flow. The above review highlights the lack of experimental investigations into the effect of vortex generators applied to circular cylinders. The studies cited concentrated on the drag and heat transfer performance, without any detailed information on how the wake flow structure was affected by these devices using the state-of-the-art flow measurement techniques. Such information is not only essential for understanding the physics of the flow phenomena involving vortex generators but also important for the provision of support for accurate numerical modelling of these flow control devices. In order to address at this shortcoming, a ¨ nal recent postgraduate study by the Principal Author (U 2007) investigated the effect of the vortex generators on the flow around a circular cylinder using experimental and computational methods. In this study, conventional flat plate-type vortex generators of rectangular cross section were fitted longitudinally on a circular cylinder, which was vertically oriented across the flow. A set of physical model tests were conducted in a large circulating water tunnel to capture the detailed near-wake flow structures using the digital particle image velocimetry (DPIV) technique. This information supported the unsteady RANS-based numerical modelling which offered further analysis of the complex flow field under the effect of the vortex generators. This paper is based on the experimental part of the ¨ nal 2007). The Reynolds number conabove research (U sidered throughout these experiments was Re = 41,300. According to Zdravkovich (1997)’s classification, this flow regime corresponds to the upper part of the ‘‘transition-inshear-layers’’ state. The unsteadiness and vortex shedding
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Top view of the cylinder with VG Side view of VG
Y
l= 3.2 α α
h= 1.6
Flow direction
R
5 =3
X (0;0)
Front view of VG 6.4
1.6
s= 6.4 t= 0.2
which occur in the wake field of the circular cylinder were investigated using a two-dimensional DPIV which is a state-of-the-art flow measurement system to determine the general flow topology including the vorticity structure and stress fields. The experiments were conducted at the Emerson Cavitation Tunnel at the University of Newcastle, UK. Taking advantage of the application of different powerful flow decomposition techniques, many aspects of the near-wake flow of the cylinder have been clearly demonstrated and discussed. To the authors’ knowledge, this study presents the first DPIV visualization of the nearwake flow of a circular cylinder in the presence of vortex generators in the open literature.
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Fig. 1 Two-dimensional schematical view of the vortex generators (dimensions are in mm)
2 Experimental set-up 2.1 The circulating water tunnel facility The Emerson Cavitation Tunnel is a closed circuit depressurized tunnel, which has a measuring section of 3.10 m 9 1.22 m 9 0.81 m and a contraction ratio of 4.271:1. The tunnel contains 60 tonnes of water which is circulated using a 300 kW DC motor driving a 1.4-mdiameter-four-bladed impeller. The maximum attainable water speed in the measuring section is 8 m/s. The longitudinal and transverse turbulence intensity of the tunnel between 1 and 7 m/s flow speeds are about 1.75–2.05% and 2.00–2.75%, respectively (Korkut 1999; Korkut and Atlar 2002). The large observation windows on the walls and floor of this facility give good laser and camera access. More detailed information about the facility can be found in Atlar (2000) or Korkut and Atlar (2002). 2.2 The test cylinder and vortex generators Test cylinders used in the experiments were made from PVC tubes with a circular cross section. The tubes had a constant outer diameter (D) of 70 mm, a height (H) of 806 mm and a wall thickness of 5 mm. Conventional, rectangular-type vortex generators (VG) manufactured from stainless steel flat plates with a 0.2 mm thickness (t) were used in the experiments. In order to determine the boundary layer thickness of the cylinder and hence the height of the VG, three-dimensional unsteady RANS-based computational simulations by using different two-equation turbulence models were performed prior to the experiments. According to the predicted boundary layer of the circular cylinder at a Reynolds number of Re = 41,300, the heights of the VG were set at 1.6 mm. The dimensions of the VG were determined according to the conventional rules given in Johnson and Joubert (1969) and Gad-el-Hak and Bushnell (1991). In this way, the height/length ratio
Fig. 2 Three-dimensional representation of the cylinder with vortex generators
(h/l) and spacing/height (s/h) ratio of the VG were 0.5 and 4.0, respectively. The general arrangement details and dimensions of the VG can be seen in Fig. 1. The VG were positioned with an angle of ±10° to the central symmetry axis, which was parallel to the flow direction, as seen in Figs. 1 and 2 in two and three-dimensional view, respectively. The VG were symmetrically fitted on two rows on either side of the cylinder in vertical orientation, and the angular position (a) of the rows from the first stagnation line was systematically varied (see Fig. 1). Taking into account the distance between the VG and the separation line, the a angles were determined at 50°, 60°, 65° and 70° with the corresponding test cases denoted as VG50, VG60, VG65 and VG70, respectively. The bare cylinder case in the absence of the VG was denoted by CC throughout this study. 2.3 DPIV specifications The two-dimensional DPIV system used in the experiments consisted of a pulsed laser, digital camera and synchronizer sub-systems. A personal computer controlled the system
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and to execute the required analysis to determine the velocity field. A New Wave Gemini double-pulsed Nd-YAG laser system was used for the illumination of the measurement area. The system produced a 532 nm wave length laser beam provided by energy of a 120 mJ in 10 ns period which corresponded to a power of 12 MW. The two independent pulsating laser generators had a maximum frequency of 15 Hz. A high sensitivity digital camera, ‘‘DANTEC 80C60 HiSense’’, captured pairs of images during the experiment. The camera could resolve 4,096 steps of light intensity at 12 bit data width. The camera had a maximum resolution of 1,280 9 1,024 pixels and a maximum operation frequency of 4.5 Hz while operating in double-frame mode. Accordingly, both the camera and the laser system were operated at a frequency of 4.5 Hz during the experiments. A Nikon f:2.8 Macro objective lens was used on the camera. Silver-coated hollow glass spheres manufactured by Potter Industries were used as the seeding material in the experiment. The representative size of the particles was a nominal 14 lm in diameter. 2.4 Experiments As stated earlier, all of the tests were performed at a Reynolds number of 41,300 defined based on the cylinder diameter (D) and inflow velocity (U?). The test cylinders were vertically fixed in the midway of the measuring section beam of the tunnel, spanning between the top and bottom walls. The centre of the cylinders was located at 1.12 m downstream of the entrance to the measuring section. The measurements were taken at the spanwise midsection of the cylinders, which corresponded to a 403 mm distance from the bottom window of the test section. A schematic view of the set-up and the general coordinate system used in the experiments are shown in Fig. 3.
Fig. 3 Schematic view of the PIV set-up and the general coordinate system
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According to this configuration, the measurement area was under the effect of counter-rotating VG which can be seen readily in three-dimensional view of the VG arrangement shown in Fig. 2. Based on this arrangement, the tunnel blockage ratio and the cylinder aspect ratio were approximately 0.057 and 11.51, respectively. As far as the blockage and aspect ratios of the experimental study are concerned, the former may be ignored— as it is lower than 0.1—according to Zdravkovich (2003) for the ‘‘transition-in-shear-layers’’ and upper flow regimes. On the other hand, Norberg (1994) reports that an aspect ratio H/D C 25 is required for independent conditions at the spanwise midsection for 104 B Re B 4 9 104. However, in the same study, for Re [ 2 9 104, very slight variations were observed from the quasi-infinite cylinder results in terms of the Strouhal number and base pressure coefficient for H/D = 10. Szepessy and Bearman (1992) measured a converging value of fluctuating lift for H/D = 9 and 11 in lieu of the highly depended results obtained for smaller aspect ratios at Re = 43,000. Therefore, the aspect ratio of the present study can be considered to be ineffective on the measurements taken at the spanwise midsection of the cylinder. On the other hand, according to the previous flow survey of the Emerson Cavitation Tunnel, the tunnel boundary layer thickness (dw) at the location of the cylinder centroid is about 20–30 mm, thus, the effect of H/dw can be ignored (Zdravkovich 2003). However, in order to prevent the three-dimensional wall effects from being amplified, a practical clearance of 50 mm was provided at each end of the cylinder between the VG and sides. The size of the measurement area was set at 160 mm 9 128 mm. This size corresponded to an area of 2.29D 9 1.83D, where D is the cylinder diameter. The ratio of the measurement area to CCD image area was 18.6. The camera lens was set to the largest aperture (f:2.8) for the maximum light permeability. In order to enhance the sensitivity of the velocity measurement, the camera was slightly defocused after the initial set-up. To eliminate (or reduce) the influence of the instantaneous flow deviations, five sets of measurement were performed for each test case during the experiments. At each measurement set, 382 pairs of images (which is the upper limit of the DPIV system buffer memory) were recorded. This corresponded to 1,910 pairs of images for each test run. As mentioned earlier, both the camera and the laser system were operated at a frequency of 4.5 Hz. Thus, the duration of the measurement for each run was nearly 85 s. This is approximately equivalent to 130 vortex shedding period for the bare cylinder test case (CC). The velocity fields were obtained by a spatial crosscorrelation of the instantaneous PIV images captured with a very small time delay (Keane and Adrian 1992). The
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large number of arithmetic operations which is required for the procedure was facilitated using a Fast Fourier Transform (FFT; Smith 2003; Willert and Gharib 1991). In this process, the digital image data, which had a resolution of 1,280 9 1,024 pixels, were divided to 32 9 32 pixels interrogation areas with a 50% overlap. This yielded 79 (longitudinal) 9 63 (transverse) velocity vectors (4,977 in total) for each measured flow field with a spatial resolution of approximately 2 mm. The result of the spurious vector determination procedure (Nogueira et al. 1997) was never presented more than 50–100 bad vectors which were about a 1–2% of the total vector quantity. In order to check the basic flow properties of the tunnel, two sets of preliminary measurements were taken in the absence of the test cylinder by capturing 382 pairs of images in each measurement set for the tunnel flow field. The measured free stream velocity (U?) of 0.588 m/s, which corresponds to Re = 41,300, showed slight variations (*1%) around the global mean and the tunnel had a negligible cross-flow, around 0.5%. The measured free stream turbulence intensity was found to be approximately 1.73% which was in a good agreement with Korkut and Atlar (2002). The integral length scale of the free stream, which is a highly influential parameter for the flow around a bluff body, was calculated as 0.016 m. This length corresponded to 0.22D in respect of the cylinder diameter. It should be noted here that, as described earlier in details, the spanwise midsection of the cylinder, where the measurements were performed, was under the effect of the counter-rotating pairs of the VG. Since a symmetrical VG configuration with respect to the spanwise midsection was adopted in this study, which can be seen in Figs. 1 and 2, the wake structures would show variations at other sections than the measurement plane. The uncertainty analyses of the tests were performed according to the procedures and formulae given in Bene¨ nal dict and Gould (1996) as described in full details by U (2007). The uncertainties in time-averaged streamwise and transverse velocities were estimated to be ±0.18 and ±2.2% for CC and ±0.11 and ±1.78% for VG70, respectively.
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order to obtain a better understanding of the unsteady flow phenomena and to present a detailed topology, the phaseaveraging technique, which allows for decomposition of the coherent and random motions, is also required. On the other hand, better extraction of a finer-scale random contribution can be provided effectively by the time averaging of the different components calculated by using POD technique which is an energy-based flow decomposition method. 3.1 Reynolds-averaged flow fields 3.1.1 Method description From the point of view of the Reynolds averaging, the flow can be considered as a combination of a global mean (time~ , and a fluctuating averaged) velocity component, U ~ velocity component, U (Tennekes and Lumley 1972). By ~ , can be represented definition, the total velocity variable, U as: ~ ðx ~ ðx ~ ðx ~; tÞ ¼ U ~Þ þ U ~; tÞ U
ð1Þ
where ~ x and t denote the spatial coordinate vector and the time, respectively. It should be noted that if the flow under consideration is unsteady and periodical, as in the present ~ ) includes the contribution of study, the fluctuating part (U both the periodic (organized, coherent) motion and the random motion of the flow. In the Reynolds averaging analysis, global mean values were calculated by the inclusion of 1910 pairs of PIV images. In Fig. 4, the mean streamlines are shown for the bare cylinder (CC) case. As expected, the Reynolds averaging procedure sweeps out the alternating vortex passage, and hence the result was composed of two vortices symmetrically taken place on either side of the wake centreline. 0.8 0.6 0.4
3 Analyses and discussion
Y/D
0.2 0 -0.2
In this section, the experimental measurements are analysed and discussed based on three complementary flow decomposition techniques: Reynolds averaging, phase averaging and proper orthogonal decomposition (POD). This procedure was adopted because whilst the classical Reynolds averaging technique based on global time averaging of all flow fields provides valuable data, it does not help to capture the effect of the unsteadiness of the flow. In
-0.4 -0.6 -0.8 0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
X/D
Fig. 4 Mean streamlines for the bare cylinder case (CC)
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The short and thick wake structure was an expected geometrical feature for this flow regime as highlighted in Zdravkovich (1997).
0.8 0.6 0.4
3.1.2 Critical values along the wake centreline The evaluation of the non-dimensional recirculation bubble length, lc/D, which is one of the most critical parameters of the wake flow along centreline, gave a value of 1.39 for the CC case. This is within an approximate ±1% accuracy which is valid for all length scale measurements. The compilation of Noca et al. (1998) shows that the recirculation bubble length steadily decreases from Re = 1,500 up to approximately 15,000 and presents slight variation around 1.4 thereafter. According to Zdravkovich (1997), from about Re = 105 recirculation bubble length increases with increasing Reynolds number. Djeridi et al. (2004) reported a value of 1.51 at Re = 1.4 9 105 measured using DPIV with a similar free stream turbulence intensity of 1.6% to that of the present study (1.73%). Accordingly, the results from the current study may be considered as in good agreement with the literature. Another critical length scale, non-dimensional vortex formation length, lf/D, is perhaps the most cited flow parameter in both computational and experimental studies in the literature since it is directly related to the forces on the cylinder and the Strouhal number. The distance between the cylinder centroid and the location of the maximum root-mean-square (RMS) streamwise velocity fluctuation along the wake centreline was used as the measure of vortex formation length in the present study, by following e.g. Griffin and Votaw (1972) while many other definitions are also mentioned in the literature, e.g. Bloor and Gerrard (1966), Anagnostopoulos (1997). The measured non-dimensional value of this length scale for CC, lf/D = 1.40, corresponds to a point slightly downstream of the mean wake closure point. According to the literature (e.g. Norberg 1998), vortex formation length is a very close value to the recirculation bubble length while the latter is also occasionally used as a measure of the former. Peltzer and Rooney (1985) reported slightly oscillating lf/D values around 1.32 in the range of 104 B Re B 1.6 9 105 which is in good agreement with the present study. Another characteristic length scale, which can be obtained from the global mean streamwise velocities along the wake centreline, is lu/D which is the non-dimensional distance from the cylinder centroid to the location of the minimum streamwise velocities along the wake centreline. The measured value of lu/D was 0.87 in the present study for CC. Norberg (1998) reports decreasing values of lu/D from 1.75 to 1.04 with increasing Reynolds number in the range of 1,500 B Re B 10,000, and this trend also appears to be conserved in the present study.
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Y/D
0.2 0 -0.2 -0.4 -0.6 -0.8 0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
X/D Fig. 5 Mean streamlines for the VG70 test case Table 1 Characteristic values along the wake centreline URMS VRMS lu/D lc/D lf/D U1
max
U1
CC
0.89
1.39
1.40
0.35
0.69
VG50
1.09
1.58
1.66
0.28
0.52
VG60
1.18
1.65
1.69
0.28
0.50
VG65
1.15
1.69
1.72
0.27
0.46
VG70
1.29
1.85
2.03
0.24
0.40
max
The mean streamlines for the test case VG70 are shown in Fig. 5. The difference in the streamlines compared to the bare cylinder case (see Fig. 4) was a clear demonstration of the qualitative effect of the VG in the wake region (vortex formation zone) of the cylinder. The reverse flow region lengthened and became more slender with a decreasing wake width. The comparative results from the measured data along the wake centreline are summarized in Table 1. From the table, it is clear that there are significant increases in the recirculation bubble length, lc/D, with the increase in the VG position angles. Similarly, although the disparity of the shortening at VG65 was unclear, the point of the minimum streamwise velocity along the wake axis was moving constantly in the direction of the X axis with increasing position angles. It should be noted also that the effect was less significant for the cases VG50, VG60 and VG65. The increase in lc/D between the test case CC and VG60 was about 21% while at VG70, lc/D and lu/D showed an increase of around 33 and 45%, respectively. In addition, the lf/D values increased in accordance with the recirculation bubble length as the locations of the VG approached the separation line. The vortex formation length was 44% shifted downstream at VG70 with respect to the bare cylinder case. In all cases, the highest value of the streamwise
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0.8
0.3
0.6
VRMS / U∞
URMS / U∞
Exp Fluids (2010) 48:1059–1079
0.2
CC VG50 VG60 VG65 VG70
0.1
Y/D=0 0
0
0.5
1
1.5
2
2.5
X/D
0.4
CC VG50 VG60 VG65 VG70
0.2
Y/D=0 3
0 0
0.5
1
1.5
2
2.5
3
X/D
Fig. 6 Non-dimensional RMS streamwise (left) and transverse velocity (right) profiles along the wake centreline
RMS velocity always occurred slightly downstream of the wake closure point. Moreover, the difference between two length scales increases with increasing values of the VG position angles. This suggested that the form of the growing vortex structure was more elongated and aligned with the wake centreline so that a specific streamwise distance occurred between the core of the vortex and the point that it crosses on the wake centreline. The RMS velocity fluctuations in X and Y directions along the wake centreline for all test cases are given in Fig. 6. In both groups of curves, the locations of the peaks are not well defined as in Szepessy and Bearman (1992). Both show a significant decrease as a result of the effect of the VG. This also indicates that the VG significantly suppresses the fluctuating motion of the flow. The maximum values obtained from the curves are also displayed in Table 1. The most pronounced jump occurred at VG70 where the VG were nearest to the separation line and the maximum values for both components decreased by 10 and 12% in respect to VG65; a similar situation also occurred for the length scales. This suggested that even if the VG were far away from the separation line they can still influence the boundary layer flow and as a result of this, slight differences may be identified between cases VG50, VG60 and VG65. However, at VG70, the separation line was located on the coverage area of the VG so that the influence on the wake zone became considerable. On the other hand, it should be noted that the VG had a stronger effect on the transverse fluctuations and the maximum streamwise and transverse RMS velocities decreased by 30 and 41%, respectively (VG70 case). Basu (1985) reported that the transition to turbulence in the boundary layer can reach to the separation point (or line) as a result of the increase in the Reynolds number. Further increase in the Reynolds number starts the transition to turbulence in the boundary layer. The specific flow regime that indicated the transition to turbulence in the boundary layer was distinguished with a slow movement of
the separation point (or line) in the streamwise direction. In the same regime, it was possible to observe an increase in the vortex formation length as well. The results obtained from the Table 1; Figs. 3 and 4 exhibit a similarity with this event. The increase in the vortex formation length, thus, designates a delay of the separation. 3.1.3 Vorticity and circulation The spatial derivation of the mean velocity components allows the calculation of the mean vorticity field as follows: oV oU ð2Þ fz ¼ ox oy The vorticity contours for both cases are obtained from Eq. 2, made non-dimensional using the inflow velocity and cylinder diameter, and are shown in Fig. 7. The symmetrical and opposite valued structure of the vorticity field and the spread of the shear layer along the X direction can be observed in the same figure. As the layer becomes thicker, the intensity of the vorticity decreases considerably. At VG70, the non-dimensional vorticity values, which are [4 or 5 (fz D=U1 C 4 or 5), spread to a narrower but much more extended region. With increasing values of the position angles, the vorticity was maintained for a longer time and began to diffuse further away from the cylinder. Based on Fig. 7, it may be possible to associate the elongation of the vortex formation length with the quantity of the circulation in the shear layers. The circulation starts to be generated from the first stagnation line and spreads to the near-wake from the separation line. Table 2 displays the global mean values of the circulation which were calculated from the integration of the nondimensional vorticity values throughout the flow field using the Stokes Theorem. Each column of this table presents the results based on different criteria for the value of the nondimensional vorticity. It can be seen in Table 2 that the
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0.5 8.0 5.3 2.7 0.0 -2.7 -5.3 -8.0
0
-0.5
Y/D
Y/D
0.5
0
-0.5
0.5
1
1.5
2
2.5
0.5
1
X/D
1.5
2
2.5
X/D
Fig. 7 CC (left) and VG70 (right) test cases, contours of time-averaged non-dimensional vorticity fz D=U1 (the plots are sharing the contour legend) Table 2 Global mean circulation (Y/D [ 0) C
Test case
fz D U1 [ 0
C
fz D U1 [ 1
C
CC
2.27
1.71
1.17
VG50
2.47
2.06
1.64
VG60
2.53
2.13
1.76
VG65
2.56
2.19
1.81
VG70
2.71
2.41
1.98
Table 3 Thickness of the shear layers Test case
X/D = 0.6
X/D = 0.8
fz D U1 [ 4
fz D U1 [ 2
X/D = 1.0
X/D = 1.5
CC
0.17
0.17
–
–
VG50
0.16
0.17
0.19
–
VG60
0.16
0.17
0.19
–
VG65
0.16
0.18
0.21
–
VG70
0.16
0.16
0.18
0.11
circulation in the measurement area increased by a maximum of 20% due to the effect of the VG, for fz D=U1 [ 0. By taking into account the nondimensional vorticity value [2, this increase reached 70%. In order to appreciate the spreading of the high shear stress region, one can observe the variation of the thickness (non-dimensionalized with D) of the shear layers with a non-dimensional vorticity value greater than that in e.g. 4, presented in Table 3. In the presence of the VG cases, the vortices form further away from the cylinder and as a result of this, the thickness of the shear layers was diminished at the cross section X/D = 0.6. However, at the cross section X/D = 1.0, whilst the bare cylinder case (CC) did not present any vorticity level higher than 4, the shear layer thickness at the other cases was even higher than that of CC at X/D = 0.6. Furthermore, at X/D = 1.5, the shear layers were apparent only for VG70 case.
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The above discussion suggested that the shear layers were more concentrated and resistant for the cases with VG. Since the vortex formation occurs at the region where the shear layer begins to diffuse with the effect of the opposite shear layer, a more elongated form of the vortex formation region is expectable. 3.1.4 Drag estimation Based on the time-averaged streamwise velocity profiles, it was possible to make an approximate estimation of the mean drag coefficient in accordance with the conservation of the momentum principle. However, a direct calculation from the momentum thickness values can lead to under predictions of the drag coefficients since this procedure does not take into account the existent pressure deficit in the near wake, which can be neglected in the far wake. According to Jones (from Schlichting 1960), the mean drag coefficient can be evaluated from the following integral expression at a cross section located downstream of the obstacle: 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 Z 2 U @ DP U A y cD ¼ 2 d 1 þ ð3Þ 2 2 U1 1=2 q U1 D U1 In this formula, DP represents the difference between the time-averaged pressure values measured along a cross section and the pressure at infinity. Since the pressure was not measured in the experiments, this information was obtained from the results of three-dimensional computer ¨ nal (2007). Table 4 presents the calculated simulations of U mean drag coefficients for all the test cases at X/D = 2.6 which was the downstream boundary of the measurement area. The value estimated for CC case was in close agreement with the literature for the current flow regime (e.g. Basu 1985). As seen from the table, the trend was clear and indicated a reduction in drag with increased position angles of the VG. For example, in the case of VG70, a decrease by about 28% in the drag value was
Exp Fluids (2010) 48:1059–1079 Table 4 Estimated mean drag coefficients
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Test case
cD
CC
1.14
VG50 VG60
Table 5 Variation of the Strouhal number Test case
Strouhal number
0.89
CC
0.186
–
–
0.85
VG50
0.262
*41
*41
VG65
0.84
VG60
0.250
*34
*-5
VG70
0.82
estimated. One should note here that the drag estimations were performed assuming a homogenous wake structure along the spanwise direction, which actually differs due to the wavy separation lines along the span. It is known that the drag coefficient is most affected by the pressure distribution following the separation (Zdravkovich 1997). This was particularly relevant for the present flow regime. The area between the first stagnation line and the separation line has a small effect on the drag. As the elongation of the formation length causes the vortices to form further away from the cylinder, the low pressure zone moves away from the cylinder as well. Accordingly, a severe decrease in the drag coefficient may be experienced. Similar characteristics are always detected on the transition between the flow regimes, and if the formation length increases, then the drag value decreases, or vice versa. Thus, it is possible to conclude that the drag of the cylinder decreased due to the effect of the VG and the effect became more pronounced as the position angles were increased. 3.1.5 Strouhal number It is expected that the VG have an effect on the nominal frequency of the wake zone and hence on the Strouhal number. The frequency of the wake zone depends upon the formation and the shedding periods of the vortices. The mechanism that determines the frequency is complex. Some of the factors that affect the nominal frequency of the wake zone are the entrainment in the shear layer and hence the intensity of the turbulence in it, the location of the formation of the shear layer, the thickness of the shear layer and the circulation spreading from the separation line. Occasionally, as a result of the mutual cancellation of the opposite effects, the value of the frequency may minimally change or may not change at all. Gerrard (1966) explained this mechanism in detail for the ‘‘transition-in-shear-layers’’ flow state where the formation zone shows a tendency to shorten rapidly but the associated Strouhal number remains approximately constant. This mechanism was validated by Zdravkovich (1997) with the exception of some minor differences.
Increase (%)
Relative increase (%)
VG65
0.243
*31
*-3
VG70
0.226
*22
*-7
The Strouhal number was determined according to the velocity fluctuations in the near-wake field. Spectral analysis, which was performed for several spatial locations 1.0 \ X/D \ 2.2, -0.5 \ Y/D \ 0.5, pinpointed a near constant value for each selected location. The value of 0.186 obtained for CC was in very good agreement with Norberg (2003) who reported an emprical function giving a value around 0.185 for the present Reynolds number. In Table 5, the variation of the Strouhal number and its relative differences between the test cases are given. The Strouhal number exhibited a sharp increase (*41%) after which it gradually declined. This non-monotonous behaviour did not support the constant lengthening feature of the wake zone. This suggested that most of the frequency determining mechanisms, as Gerrard (1966) explained, operate in common. At the VG50 case, the shear layer became more concentrated indicating that the diffusion had decreased. With the decrease in the diffusion, the shear layer, located on the opposite side of the growing vortex, was easily able to penetrate into the growing vortex (Gerrard 1966). In addition, the distance between two shear layers diminished. As a result of these two effects, the Strouhal number showed a sudden increase. For the other test cases, the diffusion continued to decrease and two shear layers moved closer. However, the dampening of the fluctuation, which was clearly identified from the RMS velocity data, may indicate the weakening of the forming vortices. As a result, the required period for a shedding process and for a vortex to be able to draw the negative vorticity from the opposite side is expected to be increased. The decrease in the Strouhal number following its sudden increase may be explained with the common operation of this opposite tendentious mechanism. 3.2 Phase-averaged flow fields 3.2.1 Method description The organized and random motion in the turbulent wake of a circular cylinder can be decomposed via conditional averaging techniques which were used in many valuable studies such as Cantwell and Coles (1983), Antonia et al. (1987), Bisset et al. (1990) and Browne et al. (1986).
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Providing a better understanding of the basic characteristics of the coherent motion, phase-averaging technique can be advantageous where the velocity field can be decomposed into three components as follows (Hussain and Reynolds 1970): 0 ~ ~ ðx ~ ðx ~ ðx ~; tÞ ¼ U ~Þ þ Ue ðx ~; tÞ þ U ~; tÞ U ð4Þ 0 ~ ~ , Ue and U ~ represent the mean velocity where U component, which depends on the spatial variables, the statistical contribution of the organized motion and the random turbulent motion, respectively. Phase-averaged velocity is then defined as:
N 1 1X ~ ðx ~ ðx ~; tÞ ¼ lim ~; t þ nsÞ U U N!1 N n¼0
ð5Þ
where, N is the sample size and s is the signal wave period. Both the global mean and the fluctuating periodical components contribute to the phase-averaged velocity which is denoted by hi in the former. Thus, one can write: 0 ~ ðx ~ ðx ~ ðx ~; tÞ ¼ U ~; tÞ þ U ~; tÞ U ð6Þ In the present study, the velocity signals were used as a phase indicator (Sung and Yoo 2001). The required resolution of the reference signal was ensured by the Whittaker (or Shannon) signal reconstruction technique which provides the exact reconstruction of the original signal for the frequencies which are equal or lower than the half of the sampling frequency (Lourenco et al. 1997). The location of the reference signal was selected with care. Spectral analysis via Fast Fourier Transform was performed for various positions in the flow field to be aware of turbulence and of noise associated with other instabilities. To minimize the effect of the phase jitter, zero reference phase was assigned to each maximum of the signal (Sung and Yoo 2001). Consequently, the time periods between two consecutive maximums were considered in lieu of a constant wave period in the
phase-averaging process. This implies that ns is replaced Pn by m¼0 sm , with s0 = 0, in Eq. 5. A single vortex shedding cycle was expressed by 16 phase angles. Since the flow fields at the desired phase angles are not directly available from the PIV data, the required flow fields have to be reconstructed. This reconstruction was also carried out by the Whittaker signal reconstruction technique in the same manner by Lourenco et al. (1997). For the reconstruction of the flow field, the reference signals were determined for each measurement set. Each velocity field was reconstructed at each constant phase by a FORTRAN code, and 1910 flow fields in total were included in the computations. Since the output of the phase-averaging computations composed of many components and phase quantities, only the results of CC and VG70 cases are discussed in the present study. 3.2.2 Vorticity and circulation In order to obtain the vorticity at constant phase, the equation below can be used: h fz i ¼
Y/D
0.5
0
-0.5
0
-0.5
0.5
1
1.5
2
2.5
0.5
X/D Fig. 8 Q contours and streamlines at constant phase for CC (left) and VG70 (right)
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ð7Þ
where hi represents the values at constant phase while U and V denote streamwise and transverse velocity components. It should be noted here that the term ‘‘vorticity’’ does not represent the forming vortices in the wake region. Q criterion of Jeong and Hussain (1995) was used to identify the general vortex structure. When a new saddle point was formed at the phase-averaged streamlines and a split of the Q contours occurred, this instant was accepted as the first phase indicating the beginning of a new vortex formation. This stage is shown in Fig. 8 where the form of the Q contours along with the phase-averaged streamlines is displayed for both cases (CC and VG70). The general form of the streamlines and the vorticity contours at constant phases during one half of the shedding
0.5
Y/D
ohV i ohU i ox oy
1
1.5
X/D
2
2.5
Exp Fluids (2010) 48:1059–1079
cycle for both test cases are shown in Figs. 9 and 10 with p/ 8 radians phase angle steps. At the second half of the shedding cycle, the motion repeats itself and exhibits a perfect symmetry with respect to the wake centreline for all cases. The evolution of the half shedding period for both cases can be examined in Fig. 11, where the vortex circulations, hCi/U?D, and vortex trajectories at each phases are shown. For this purpose, the boundaries of the vortices were identified according to Q criterion. The generation of vortex was triggered with the formation of a saddle point for both cases. In Fig. 9, the interaction of two shear layers can be observed for the bare cylinder case (CC). When the lower part vortex becomes strong enough, spreading on a wide area around X/D = 1, it draws the other shear layer across the wake. Up to the 6th phase, while the saddle point moves downstream, upper part vorticity spreads to a wider area. At the 6th phase, the saddle disappears and the lower part vortex, which is at its 14th phase, begins to decay as it seen in Fig. 11. The upper side vortex, which does not encounter any opposite-signed vorticity from the other shear layer, rapidly grows up to the 8th phase. From Fig. 11, it can be seen that the circulation level increases up to this phase. At the 13th phase, the longitudinal coordinate of the centre of the closed streamlines X/D = 1.3 (Y/D = 0) nearly coincides with the wake closure point. At this stage, one can assume that the vortex formation process was completed. Thereafter, the circulation began to decrease rapidly and almost linearly. A similar but more complex mechanism was observed for the VG70 test case shown in Fig. 10 where more elongated form of the closed streamlines is apparent. The vertical location of the shear layers and narrowing form of the wake region suggested a clear separation delay phenomena and caused the shear layers to turn towards the centreline. The centroids of the closed streamlines were translated quite far away from the cylinder in comparison with the CC while they are closer to the wake centreline. High vorticity levels are observed on a larger streamwise extent but their values are comparable. The vorticity area downstream of the saddle at the first phase was more distinctive. The streamline curvature following the saddle was more pronounced for the first few phases at VG70. Another difference between the CC and VG70 cases is the existence of a saddle at all phases of the formation process for the latter. According to this property, it is likely that the circulation curve never exhibits a rapid decrease. The growth of the vortex continued until the 15th phase which was then followed by a slight decrease in the circulation level and shedding of the vortex. At the 16th phase, the longitudinal coordinate of the centre of the closed streamlines X/ D = 1.85 coincided with the wake closure point. As the same phenomenon was observed for CC, this may again indicate the termination of the vortex formation.
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Accordingly, it seems likely that the formation of a vortex spread over 16 phases for VG70. The time elapsed during the process was compensated by the sudden shedding of the vortex and the generation of a new saddle point. Some studies suggest that the vortices are stronger at the end of the vortex formation region such as Anagnostopoulos (1997). As the vortex formation length was determined previously as lf/D = 1.40 at the Reynolds averaging analysis, this finding was validated for the CC case in the present study (see Fig. 11). However, the circulation curve for VG70 indicated a stronger vortex at X/D = 1.7 which conflicted with the vortex formation length calculated at the Reynolds averaging section. A plausible reason for this anomaly may be related to the geometrical form of the vortex. The geometrical difference of the vortex between two cases affected the point along the wake centreline where the RMS velocity fluctuations were a maximum. Since these velocities for VG70 did not have a well-defined peak, as shown in Fig. 11, the discrepancy in the vortex formation length should be considered as normal. In the same figure, it is also observed that the sharp rise in the first phases of the circulation curve for CC no longer existed in the circulation curve for VG70. In Fig. 11, the location of the vortex centroids (Y/D vs. X/D) at each phase is also shown. The vortices initially moved towards the centreline and then moved essentially parallel to the wake centreline. When the vortex shedding was finished at X/D & 1.4, the vortices begin a parallel movement as the centroid of the closed streamline reached the wake centreline. The centroid of the closed streamline and the vortex was located approximately on the same cross section at this stage. As far as the effect of the VG was concerned, the organized trajectory structure, which was observed in the CC test case, deteriorated in the VG70 case. The general trajectory feature of the vortex was similar to the one at CC but the vortex never reached the distance lf/D = 2.03 which was obtained for VG70 in the Reynolds averaging analysis. 3.2.3 Stress fields at constant phase In Fig. 12, comparative contours of the periodical com2 2 ~U~ =U1 ponents of the Reynolds stresses, U , V~V~ =U1 , 2 ~ ~ U V =U1 (which are the streamwise normal, transverse normal and shear stresses, respectively), at constant phase are shown for both CC and VG70 test cases. The phases indicated the instants when the distance from the centre of the cylinder to the centroid of the closed streamlines corresponded to the recirculation bubble length. In Fig. 12a, d, the phase-averaged streamlines were superimposed on the stress contours. The streamlines of the periodical component of the velocity field at constant phase are shown at Fig. 12c, f including the superimposed shear stress
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1070
Exp Fluids (2010) 48:1059–1079 2
1 0.5
Y/D
Y/D
0.5
0
-0.5
0
-0.5
0.5
1
1.5
2
2.5
0.5
1
X/D
1.5
2
3
4 0.5
Y/D
Y/D
0.5
0
-0.5
0
-0.5
0.5
1
1.5
2
2.5
0.5
1
X/D
1.5
2
6
0.5
0.5
Y/D
Y/D
2.5
X/D 5
0
-0.5
0
-0.5
0.5
1
1.5
2
2.5
0.5
1
X/D
1.5
2
2.5
X/D 7
8
0.5
0.5
Y/D
Y/D
2.5
X/D
0
-0.5
0
-0.5
0.5
1
1.5
2
2.5
0.5
X/D
1
1.5
2
2.5
X/D -6 -4 -2 0 2 4 6
Fig. 9 Streamlines and colour-coded non-dimensional vorticity hfz i D=U1 at constant phases for CC (with a p/8 rad step, instant 1 and 8 corresponding to phase angle 0 rad and 7p/8 rad, respectively)
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1071 2
1 0.5
Y/D
Y/D
0.5
0
-0.5
0
-0.5
0.5
1
1.5
2
0.5
2.5
1
1.5
2
3
4 0.5
Y/D
Y/D
0.5
0
-0.5
0
-0.5
0.5
1
1.5
2
2.5
0.5
1
1.5
2
5
6
0.5
0.5
Y/D
Y/D
2.5
X/D
X/D
0
-0.5
0
-0.5
0.5
1
1.5
2
2.5
0.5
1
X/D
1.5
2
2.5
X/D 7
8 0.5
Y/D
0.5
Y/D
2.5
X/D
X/D
0
0
-0.5
-0.5
0.5
1
1.5
2
2.5
0.5
X/D
1
1.5
2
2.5
X/D -6 -4 -2 0 2 4 6
Fig. 10 Streamlines and colour-coded non-dimensional vorticity hfz i D=U1 at constant phases for VG70 (with a p/8 rad step, instant 1 and 8 corresponding to phase angle 0 rad and 7p/8 rad, respectively)
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Exp Fluids (2010) 48:1059–1079 0.8
1.8
CC VG70
1.6 0.6
1.2
Y/D
〈Γ〉 / U∞ D
1.4
1
0.4
0.8 CC VG70
0.6
0.2
0.4 0
5
10
15
0 0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
X/D
Phase Fig. 11 Circulation curves (left) and vortex trajectories (right) for two test cases
contours. The periodical motion was a local rotation with respect to the global mean. The streamwise normal stresses reached a maximum value at the saddle point of the phased-averaged streamlines and above it, at a symmetrical location with respect to the centreline where the streamlines were bent towards the wake centreline for both test cases (Fig. 12a, d). The distances of the locations of the points, where the maximum streamwise normal stresses occurred, were around the vortex formation lengths, as expected. Accordingly, the distance from the cylinder centroid to these points was increased by 33% at VG70. Maximum stress points approached to the wake centreline by 40%. The contours clearly demonstrate that the streamwise periodical component of the stress was significantly decreased due to the effect of the VG. The maximum streamwise periodical stress was decreased by 75% at this constant phase. A similar situation was2 valid for transverse periodical stress component V~V~ =U1 contours (Fig. 12b, e). In both cases, the flow structure consisted of two closed contour areas which designated the large flow curvature in the velocity field. Accordingly, the maximum transverse stress points were located on the wake centreline. The transverse stress zones were translated in the streamwise direction while the maximum value of the transverse stress was decreased by 70% at VG70. Shear stress fields consist of circular shaped contour forms which were concentrated around the wake closure point (Fig. 12c, f). The saddle point of the stress contours coincided with the centroid of the streamlines of the periodical motion and that of the streamlines at constant phase for CC while it was located slightly lower and right side for VG70. The drastic reduction in the shear stresses can be observed similar to the other stress components at VG70. While the maximum non-dimensional shear stress value was 0.14 for CC, it was reduced to 0.05 for VG70. Figure 13 shows the comparative contours of the random 2 components of the Reynolds stresses hU 0 U 0 i=U1 , 0 0 2 0 0 2 hV V i=U1 , hU V i=U1 at constant phase for both CC and
123
VG70 test cases. It is clear that the random stress fields exhibited different topologies in comparison with the periodical ones. Random streamwise normal fluctuations were concentrated in the zones where the vorticity was strong for both cases. Large values of the random normal stress were observed with the forming vortex in CC, while such large values were observed with the vortex which had completed its formation for VG70. Another interesting feature for both cases was that the random stress values were even higher than the periodical ones. Whilst the maximum value of the streamwise random normal stress was higher than that of the streamwise periodical normal stress at CC, it was approximately twice the maximum value of the latter for VG70. However, the contour values indicated that the random motion was damped due to the effect of the VG in the VG70 test case. The location of the maximum random transverse stresses was in the middle of the closed streamlines of the periodical motion for the CC test case, whilst this location was at far downstream for the VG70 test case. The maximum value of the random transverse stress was equal to the periodical transverse stress for the VG70; however, this was not the case for CC where the latter was greater than the former. As far as the random shear stress field was concerned, the minimum shear stress was experienced with the vortex that formed whilst the maximum random shear stress was experienced with the growing vortex. Since the vortex shedding period was shorter for VG70, the extreme values took place on a narrower area. Similar to the other stress components, the extreme values of the shear stress showed a decrease of up to 50% at VG70. 3.3 POD flow fields 3.3.1 Method description Proper orthogonal decomposition is a powerful technique that effectively decomposes the organized and random fluctuating components of a turbulent velocity field based
Exp Fluids (2010) 48:1059–1079
1073
a
d 0.5
0.15 0.13 0.10 0.07 0.05 0.03 0.00
0
Y/D
Y/D
0.5
-0.5
0
-0.5
0.5
1
1.5
2
2.5
0.5
1
X/D
1.5
2
e
b 0.5
Y/D
Y/D
0.5
0
-0.5
0.40 0.33 0.27 0.20 0.13 0.07 0.00
0
-0.5
0.5
1
1.5
2
2.5
0.5
1
X/D
1.5
2
2.5
X/D
c
f
0.5
0.5 0.13 0.09 0.04 0.00 -0.04 -0.09 -0.13
0
-0.5
Y/D
Y/D
2.5
X/D
0
-0.5
0.5
1
1.5
2
2.5
X/D
0.5
1
1.5
2
2.5
X/D
2 2 2 Fig. 12 Periodical stress contours at constant phase U~U~ =U1 , V~V~ =U1 , U~V~ =U1 , respectively from top to bottom (a, b, c are for bare cylinder and d, e, f are for VG70, respectively. Collaterally placed plots are sharing the related contour legend)
on their energy levels. As a result of the improvement in digital visualization techniques, it has become a popular technique and has been widely used for the analysis of turbulent flow, e.g. Patte-Rouland et al. (2001), Ben Chiekh et al. (2004), Oudheusden et al. (2005). For a detailed mathematical background, review and analysis of the technique, the reader should refer to Sirovich (1987) and Berkooz et al. (1993). It is possible to decompose an unsteady flow field to a mean and fluctuating component as ~ ðx ~; tÞ, of in Eq. 1. POD decomposes the fluctuating part, U the velocity field into further modes as much as the number
of the instantaneous velocity fields, as represented in the following expression: N X ~ ðx ~ ðx ~ ðx ~ ðx ~Þ þ U ~; tÞ ¼ U ~Þ þ ~Þ ð8Þ ~; tÞ ¼ U an ðtÞ ~ /n ðx U n¼1
where ~ /n , an and N represent POD mode, weight coefficient of the mode and number of the velocity fields, respectively, while subscript ‘‘n’’ denotes the mode of the respective flow field (Sirovich 1987). By considering the common representation in Eqs. 1, 4 and 8, the fluctuating ~ ðx ~; tÞ, can be expressed as below. component, U
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Exp Fluids (2010) 48:1059–1079
a
d 0.5
0.17 0.14 0.11 0.09 0.06 0.03 0.00
0
Y/D
Y/D
0.5
-0.5
0
-0.5
0.5
1
1.5
2
0.5
2.5
1
1.5
2
e
b 0.5
Y/D
Y/D
0.5
0
-0.5
0.25 0.21 0.17 0.13 0.08 0.04 0.00
0
-0.5
0.5
1
1.5
2
0.5
2.5
1
1.5
X/D
2
2.5
X/D
c
f
0.5
0.5
Y/D
Y/D
2.5
X/D
X/D
0
-0.5
0.08 0.05 0.03 0.00 -0.03 -0.05 -0.08
0
-0.5
0.5
1
1.5
2
2.5
0.5
1
1.5
2
2.5
X/D
X/D
2 2 2 Fig. 13 Random stress contours at constant phase hU 0 U 0 i=U1 , hV 0 V 0 i=U1 , hU 0 V 0 i=U1 , respectively from top to bottom (a, b, c are for bare cylinder and d, e, f are for VG70, respectively. Collaterally placed plots are sharing the related contour legend)
X 0 ~ ~ ðx ~ ðx ~; tÞ ¼ Ue ðx ~; tÞ þ U ~; tÞ ¼ ~Þ U an ðtÞ ~ /n ðx N
ð9Þ
n¼1
In Eq. 9, the POD modes with high energy levels ~ ~ ðx ~; tÞ, of the correspond to the periodical component, U velocity field whilst the rest denotes the random one, 0 ~ ðx ~; tÞ. ~ ~Þ are the eigenfunctions of the two-point U /n ðx correlation matrix of the flow field which can be defined as: ~ ðx ~ ðx ~i ; t Þ U ~j ; tÞ Cij ¼ U
ð10Þ
Coefficients an in Eq. 9 are obtained from projection of the POD modes onto instantaneous velocity fields. The eigenvalues of the correlation matrix represent the
123
contribution of each POD mode in the fluctuation energy. The percentage energy contribution of modes can be calculated from the following ratio. %Energy ¼ kn =
N X
kn 100
ð11Þ
n¼1
In the present study, the POD analysis was applied to five sets of measurements covering 1910 flow fields, for each test case, using MATLAB software. The graphics and numerical data were obtained using time averaging of the total flow fields. Figure 14 shows the energy levels corresponding to the POD modes for all measurement cases tested. In order to
Exp Fluids (2010) 48:1059–1079
1075
Percentage of energy (%)
35 CC VG50 VG60 VG65 VG70
30 25 20 15 10 5 0 0
2
4
6
8
10
12
14
16
18
20
Modes
Fig. 14 Energy percentages corresponding to POD modes
produce the curves, the eigenmodes of the correlation matrix were calculated. The vertical scale of Fig. 14 represents the percentage of the energy calculated according to Eq. 11. It was clear that the first 5–6 modes, and in particular the first 2, showed a significant reaction to the presence of the VG. It was observed that the ratios of the first few modes, which represent the large-scaled periodical motion, to the total fluctuation energy, decreased with the increasing values of the VG position angles. It can be further suggested that the VG increased the contribution of the turbulence to the whole fluctuation, changing the balance between the large-scaled motion and the random one. Decomposing the fluctuating motion into its components, the first eight modes were considered to represent the large~ ~ ðx ~; tÞ. This selection was made scaled periodical motion, U in such a way that beyond the 8th mode, the energy level dropped to approximately 1.2% for each case. Accordingly, the rest of the POD modes contributed to the decomposi0 ~ ðx ~; tÞ. tion as the random component, U 3.3.2 Time-averaged stress fields In Fig. 15, comparative contours of the time-averaged 2 ~ 1 , periodical components of the Reynolds stresses, U~U=U 2 2 ~ 1 , U~V=U ~ 1 , for the CC and VG70 cases are shown. V~V=U As distinctive from the results of the phase averaging, it can be observed that the streamwise periodical stress components were effective on the wake centreline and form a saddle point around the wake closure point. For the cases with VG (only VG70 is included in Fig. 15), the streamwise periodical stresses were not considerable around the shear layers (Fig. 15a, c). The high stress fields covered the zones where the vortices were being fed by the shear layers and reached their maximum strengths. The contours of the transverse periodical stress form a single lobe structure which was symmetrically located at the wake centreline (Fig. 15b, e). The stress field at the back side of the cylinder was absent for the VG70 case. The significant stress region was considerably shifted downstream at VG70. The centre
of the shear stress fields, which displayed a set of extreme values on either side of the wake centreline, was located around the wake closure point (Fig. 15c, f). As a common feature of the comparative contour plots, it was clear that the periodical motion was dampened significantly and the stress fields were translated further away from the cylinder due to the effect of the VG. This validates the results found in the earlier discussed phase-averaging process. The characteristic values extracted from the contours are collected in the Table 6. According to the table, the extreme values of the periodical stress components show a constant decrease with the increasing values of the VG position angles. In harmony with the increase of the vortex formation length, in all cases, it is possible to observe (except at VG65 case) a downstream movement of the points of local extreme values. The extreme values of the three components were decreased by 61, 68 and 50%, respectively at VG70 with respect to CC. Figure 16 shows the comparative contours of the timeaveraged random components of the Reynolds stresses, 2 2 2 U 0 U 0 =U1 , V 0 V 0 =U1 , U 0 V 0 =U1 , for the CC and VG70 test cases. Highly disordered nature of the random stress fields does not allow clear identification of the peaks. Nevertheless, high levels of random fluctuations can be clearly identified. Significant values of the streamwise normal random stresses and shear stresses were found in the shear layers whilst the transverse component was in the middle of the wake region. For the streamwise component, a relatively high stress zone was also located between two shear layers around X/D = 1 for VG50, VG60 and CC test cases. This zone gradually moved away from the cylinder at the other cases. Since the stress components due to the random motion did not have smooth contour lines and an organized structure, the maximum values and their locations could not be clearly identified. However, it may be clearly observed that, along with the periodical stress components, the random stress components show a significant decrease for the test cases with VG. It should be noted that since the measurements in the present study covered a spectrum up to 2.25 Hz, the dampening of the random motion does not necessarily mean that the turbulence was decreased. The decrease in the random stress components may be associated with the strong three-dimensionality caused by the VG. The vortex stretching mechanism is a well-known feature of the three-dimensional flow motions (e.g. Davidson 2004). As a result of the vortex stretching, one can expect a decrease in the turbulent length scales which in turn causes the turbulent energy to be decreased in association with the dissipation of the turbulence in the smallest scales and hence production of heat. Gerrard (1966) quoted that the shear layer, with increasing turbulence level, has a more diffusive structure. So indeed, with
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a
d
0.5
0.5 0.18 0.15 0.12 0.09 0.06 0.03 0.00
0
Y/D
Y/D
Fig. 15 Time-averaged periodical stress contours 2 2 2 ~ 1 ~ 1 ~ 1 U~U=U , V~V=U , U~V=U , respectively from top to bottom (a, b, c are for bare cylinder and d, e, f are for VG70, respectively. Collaterally placed plots are sharing the related contour legend)
-0.5
0
-0.5
0.5
1
1.5
2
2.5
0.5
1
X/D
1.5
2
e
b 0.5
Y/D
Y/D
0.5
0
0.40 0.33 0.27 0.20 0.13 0.07 0.00
0
-0.5
-0.5
0.5
1
1.5
2
2.5
0.5
1
X/D
1.5
2
f 0.5
Y/D
Y/D
0.5
0
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X/D Table 6 Critical values of the time-averaged periodical stress components calculated with POD 2 ~ 1 U~V=U
Max. (X/D; Y/D) Max. (X/D; Y/D) Min.
(X/D; Y/D)
CC
0.20
1.13; 0.41
0.42
1.52; 0
-0.13 1.33; 0.36
VG50
0.14
1.33; 0.30
0.23
1.63; 0
-0.09 1.57; 0.26
VG60 VG65
0.12 0.10
1.45; 0.30 1.42; 0.27
0.21 0.16
2.00; 0 1.87; 0
-0.08 1.73; 0.27 -0.07 1.67; 0.25
VG70
0.08
1.79; 0.24
0.13
2.13; 0
-0.06 1.92; 0.23
an increase in the turbulence and hence of the turbulence viscosity, the diffusion shows an increase as well. From the identification of the vortices via Q criterion, it was found that the vortex forming at the CC test case spread to a much
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0.12 0.08 0.04 0.00 -0.04 -0.08 -0.12
0
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0.5
2 ~ 1 V~V=U
2.5
X/D
c
2 ~ 1 Test case U~U=U
2.5
X/D
2
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1
1.5
2
2.5
X/D
wider area than the one observed in the VG70 test case. Thus, the fact that the diffusion was less for the cases with VG can be associated with the dampening of the random motion. Furthermore, for the cases with VG, it can be seen from the presented figures and the tables that the periodical stress components show a decrease and the high-stress fields spread on a tighter region compared to the respective data for the CC test case. Based on the phase-averaging analysis results, the circulation of the vortex structure at VG70 was decreased; accordingly, the strength of the vortices was decreased. As mentioned earlier, the shear layers were more concentrated in the cases with VG. In addition, the layers gradually come closer along with the delay of the separation. Consequently, the required vorticity or circulation, regarding the fact that the forming
Exp Fluids (2010) 48:1059–1079
a
d
0.5
0.5 0.07 0.06 0.05 0.04 0.02 0.01 0.00
0
Y/D
Y/D
Fig. 16 Time-averaged random 2 , stress contours U 0 U 0 =U1 2 2 V 0 V 0 =U1 , U 0 V 0 =U1 , respectively, from top to bottom (a, b, c are for bare cylinder and d, e, f are for VG70, respectively. Collaterally placed plots are sharing the related contour legend)
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0
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1
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vortex is entrained by the other shear layer, should be decreased. Since the entraining layer contains concentrated opposite sign of vorticity, this would reduce the strength of the forming vortex as this can be seen in Fig. 11. Thus, weaker vortices are formed and accordingly the stresses due to the organized periodical motion are damped.
4 Conclusions This study presented the details and analysed results of the recent experimental investigation into the effect of a set of vortex generators fitted to a circular cylinder with a specific emphasis on the details of the near-wake flow. The flow measurements were taken using a two-dimensional DPIV
2
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system in the near-wake region of the circular cylinder with and without the vortex generators at a Reynolds number of 41,300. The instantaneous flow fields gathered from the measurements were analysed using three complementary techniques which are Reynolds averaging, phase averaging and POD. Various important flow properties such as vorticity, stress distribution and wake oscillation frequencies were obtained from these analyses and discussed in details. Based on the investigation, the followings are found. The comparative analyses of the results with and without the VG confirm that the vortex generators have a significant effect on the near-wake flow and hence the boundary layer of the cylinder. The Reynolds-averaged flow fields showed that the VG enforced the shear layers to get closer to each other, bend towards the centreline and
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decrease the width of the near-wake. These characteristics clearly indicated that the flow separation was delayed on the measurement plane. The characteristic length scales of the wake were considerably increased with the downstream motion of the separation location. The amount of the global mean circulation in the flow field was also increased because of the VG, whilst the width of the shear layers was decreased. Concentrated pattern of the shear layers caused the vortex formation process to occur at longer distances from the cylinder. The relative measure of comparative mean drag coefficients revealed that a significant drag reduction was possible with the vortex generators. The Strouhal number increased approximately 41% (at VG50) and subsequently decreased gradually due to the possible reduction in the strength of the forming vortices. The comparative phase-averaged flow field characteristics, providing detailed topological information and offering effective decomposition of stress fields at constant phase, indicated a similar but different vortex shedding cycle, demonstrating the weakening of the vortices and dissimilar circulation character due to the effect of the VG. The comparative POD analyses of the flow fields showed a serious decrease in the stresses for both random and periodical motions in the near-wake while the high stress regions move far downstream due to the effect of the VG. It is also identified that the VG have a more dominant effect on the large-scaled periodical motion than the random one. It is believed that the present study offers essential information to understand the physics of the bluff body near-wake flow phenomenon involving vortex generators as well as providing experimental support for accurate numerical modelling of this type flow control device. Acknowledgments The experimental work presented in this paper was conducted during the Principal Author’s research study visit to Newcastle University, which was sponsored by the Tinc¸el Culture Foundation, Istanbul University. The authors gratefully acknowledge ¨ mer Go¨ren and Assoc. Prof. Oks¸ an Prof. A. Yu¨cel Odabas¸ ı, Prof. O C¸etiner Yıldırım from Istanbul Technical University for their invaluable contributions to the study and Bo Beltoft Watz from Dantec Dynamics A/S for her support during the experiments. Special thanks are conveyed to Roderick Sampson for his help during the preparation of this paper and Ian Patterson for his efforts during the tests.
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