Vortex shedding in the near wake of a finite cylinder...

14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008

- 1 -

Vortex shedding in the near wake of a finite cylinder

Rosaria Giordano, Tommaso Astarita, Giovanni Maria Carlomagno

Department of Aerospace Engineering (DIAS), University of Naples “Federico II”, Italy, [email protected]

Abstract Vortex shedding of both finite and infinite cylinders have been investigated by means of the Particle Image Velocimetry (PIV) technique and compared. To obtain reliable turbulence statistics 5˙000 images have been taken for each test. Instantaneous and mean velocity fields are measured in many planes along the cylinder length, in order to study the flow evolution between the basis and the end of the cylinder. The Reynolds number investigated (Re = 16˙000) is within the Shear-Layer Transition Regime, indicated by Williamson (1996) for the infinite cylinder. This regime is characterized by the formation of Karman vortices. As regard the finite one, a counter-rotating pair of tip vortices forms at the free end, extends into the wake and interacts in a complex manner with Karman vortex shedding. For the aspect ratio used in this work (AR=8), Kawamura et al. (1984) showed that the Karman vortices are present only in a small region near the cylinder base. In order to detect the Karman vortex shedding, a phase averaged method has been chosen. This has been computed by means of the Proper Orthogonal Decomposition (POD) technique and it has been evidenced that this approach works properly for the infinite cylinder, as well as for some sections near the base of the finite one. As conclusion of this consideration and accordingly with Kawamura et al. (1984), it has been identified the region which shows the Karman vortex shedding for a finite cylinder with AR=8 . 1. Introduction In the literature it’s possible to find many works relative to the analysis of an infinite cylinder (2D) wake. Both the mean flow field and the phase averaged one have been widely investigated with different techniques, e.g. Hot Wire and Particle Image Velocimetry (PIV). Williamson (1996) showed 8 vortex shedding regimes, depending of the Reynolds number (Re). That of interest for the present investigation is the Shear-Layer Transition Regime, valid for Re varying from 1˙000 to 200˙000. In this regime, the separation of the shear layers from the sides of the body causes, by increasing the Reynolds number, the reduction of the Strouhal number (St) and the formation length of the mean recirculation region. Furthermore, for an increase of Re, the turbulent transition point moves upstream in the separating shear layers and the 2D Reynolds stresses magnitude increases. The latter effect is caused by the Kelvin-Helmholtz instability. Studies present in the literature found that the instability vortices, appearing in the shear layer, have a frequency higher than the von Karman ones. Lin et al. (1995) showed that the shear layer vortices amalgamate into von Karman one, for Re = 10˙000, while it seems that it’s impossible to find this kind of phenomenology below Re = 1˙000. Although strictly speaking any type of cylinder wake has 3D flow characteristics, really for a 2D cylinder the flow is practically bi-dimensional. Three dimensional features of the flow arise when the cylinder aspect ratio is small as well as the cylinder is finite, i.e. with a free end: the latter causes changes in the vortex formation length as well as the vortex shedding pattern. Budair et al. (1991) investigated the wake of a finite cylinder with Re=15˙000 and found the vortex shedding disappearance for an aspect ratio lower than 7. According to Kawamura et al. (1984), for high aspect ratios, the vortex shedding frequency may vary in a cellular manner along the cylinder height and each cell has a different frequency (or Strouhal number), whereas the vortex shedding is suppressed near the free end and the cylinder base. Park and Lee (2000) measured, by means of the hot wire technique, the vortex shedding frequency and the vortex formation length for aspect ratios


- 2 -

(L/D) of 6, 10 and 13 and Re=20˙000. Apart from the von Karman vortices frequency, they measured a 24Hz frequency component different from the periodic vortex shedding frequency that is not affected by the aspect ratio of the finite cylinder. Afgan et al. (2007) validated these results by means of a Large Eddy Simulation (LES): they computed both the length of the vortex formation region and the frequency of the vortex shedding. In the present work, since the frequency used for the image acquisition is 7Hz and the von Karman vortices have a frequency almost 10 times bigger, a phase averaging method has to be used to reconstruct the vortices evolution. In the literature various approaches are present for the phase averaging. Lin et al. (1995) used the hot wire signal to trigger the sampling of the vortex shedding phases, whereas Braza et al. (2006) used a pressure signal. Ben Chiekh et al. (2004) proposed to use the first two principal modes obtained with the POD technique, in order to reconstruct the vortices evolution, whereas Perrin et al. (2007) compared the phase averaged flow fields obtained with a trigger originated from pressure signal with the POD-based approach. Perrin et al. (2007) found that an enhancement in the averaged velocity fields is obtained with the POD approach, since the phase angles is determined directly from the velocity fields to be averaged. Konstantinidis et al. (2005) proposed an other approach to compute a posteriori phase averaged. It consists in to cross-correlate all the flow fields and to check the ones that correlate well with each other. By this way, various flow field ensembles can be computed, all of them representative of a particular phase of the vortex shedding. The present authors after having investigated both the approaches based on POD and cross-correlation have chosen to use the former. The reason of this choice is in the less computational effort and better phase location of the POD approach. 2. Experimental apparatus and method The used experimental set-up is composed of an aspirated subsonic open circuit wind tunnel which has a low turbulence intensity level (0.1%) and a rectangular test section of 300×400mm2 (Fig. 1). Two different cylinders have been used for measurements: a finite cylinder and a infinite one.

Fig. 1. Sketch of the experimental apparatus.


- 3 -

The former has a diameter of 26mm and a length of 208mm giving an aspect ratio AR=L/D=8 and a blockage coefficient D/H=0.045; the latter has a diameter of 20mm and spans the width of the test section, giving an aspect ratio AR=20 and a blockage coefficient D/H=0.067. The free-stream velocity is measured by using a differential pressure transducer and the Reynolds number, based on the cylinder diameter D and the free stream velocity U∞, is equal to 16˙000. The flow fields are measured with the Particle Image Velocimetry technique. Seeding, injected into the flow, is used in the form of oil droplets, about 1µm in diameter. The light sheet, which is generated by a double cavity Nd-YAG laser, has a thickness of about 1mm, a pulse duration of 6ns, a wavelength of 532nm and a maximum energy per pulse of about 200mJ. To display, acquire and record digital images, the following items are used: a video camera Kodak Megaplus model ES 1.0 with a CCD sensor (1008×1018 pixels, 256 grey levels) and a PC with a Matrox Genesis frame grabber. The PC is also equipped with a counter board, which permits to synchronize the system with the pulsed laser. The acquired images are interrogated with a high accuracy PIV algorithm. This algorithm first calculates the cross-correlation of two homologous windows with the classical FFT method. Once the cross-correlation map is obtained, a peak detection operation is performed over the map to precisely locate the peak through a sub-pixel interpolation. In this way, it is possible to compute a preliminary displacement field of all the windows that is used to perform translation, rotation and deformation of each interrogation area. In turn, this displacement is corrected by means of an iterative procedure, in which the velocity is calculated with an TH interrogation window with linear dimension equal to 32 pixels. Finally, at each step of the iteration process, the quality of the measured vectors is controlled with a data validation criterion. In order to obtain reliable turbulence statistics 5˙000 images have been taken for each test. Once obtained the flow fields by means of the PIV technique, since the acquisition frequency is lower than the vortex shedding one, a phase averaged method has to be used. As mentioned before, the method used here is the one proposed by Ben Chiekh et al. (2004). For this purpose, for each investigation plane the snapshoot approach (see Sirovich 1987) of the POD technique has been applied to 2˙000 flow fields. By following this approach, it’s possible to find the temporal POD eigenfunctions a(n)(tk) and then the POD coefficients Φ(n)(xi), so that each instantaneous flow field can be expressed as:

( ) ( ) ( )( )( )i

nM

1nk

nki xtat,xu ∑

== Φ (1)

By changing the point of view, Φ(n)(xi) are the normalised modes of the flow field and a(n)(tk) are the coefficients. If the first two modes contain a large part of the flow field energy, each snapshot flow field is well approximated by a combination of them and, then, the first two coefficient a(1)(tk) and a(2)(tk) are enough for the description of it. Therefore the vortex shedding phase can be computed as:

( )( ) ( )( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛=

k1

k2

2

1k ta

taλλ

arctantα (2)

where λ1 and λ2 are the two first eigenvalues obtained by the POD relative to the first two temporal POD eigenfunctions a(1)(tk) and a(2)(tk).


- 4 -

3. Results and discussion In this section the results obtained with both finite and infinite cylinders will be shown and compared. First the mean flow field will be discussed; then the results obtained with the POD approach to compute the phase averaged flow fields will be shown. The measurement zone, extended behind the cylinder, is 1.6·D high and 3·D long, for both cylinders. Fig. 2 shows the reference frame used.

Fig. 2. Sketch of the reference frame.

With regard to the infinite cylinder, the measurement plane is located in the middle of the cylinder length, i.e. z = 20mm, whereas 8 measurement planes have been investigated for the finite cylinder, as explained in Tab. 1.

No. plane z/L z/D

1 0.14 1.15 2 0.29 2.31 3 0.43 3.46 4 0.58 4.62 5 0.67 5.38 6 0.77 6.15 7 0.87 6.92 8 0.96 7.69

Tab. 1. Measurement planes investigated for the finite cylinder.

3.1. Mean flow fields The mean flow fields have been obtained by averaging 5˙000 PIV images. Fig. 3 shows mean vortical fields for the infinite cylinder (2D) and the finite one in z/D =0.43. The vorticity (Ω) has been computed on a central cross stencil with 4 points and multiplied by (D/U∞), so that it is dimensionless. The anti-symmetry of all mean vortical fields demonstrates the goodness of the averaging process. Maps relative to both finite and infinite cylinders show the same vorticity contour legend. This means that the order of magnitude of the vorticity relative to the shear layer is the same for both cylinders. A substantial difference can be seen in the maps of Fig. 3: for the finite cylinder the length of the mean vortices is higher than the one of the infinite cylinder. Furthermore, it has been measured that for the finite cylinder this length changes a lot by varying the measurement planes along the cylinder length.


- 5 -

2D z/L=0.43 Fig. 3. Mean vortical field.

Several researches (e.g. Bloor 1964, Park and Lee 2000 and Saad et al. 2007) have measured the vortex formation length as the point that has the maximum value of the turbulence intensity measured along the symmetry section. According to these authors, also here the turbulence intensity has been computed on the symmetry section as a function of x/D (see Fig. 4). This has been calculated as:

( )2

22

U'v'uIT

∞

+= (3)

where u′ 2 and v′ 2 are the two available mean normal Reynolds stresses.

Fig. 4. Turbulence intensity distribution on y/D = 0 section.

The infinite cylinder turbulence intensity is higher than the one relative to the finite one, whereas the vortex formation length is smaller. The latter decreases by approaching the free end of the finite cylinder and assumes about the same value of the 2D cylinder for z/D = 0.87. Probably this is due to the downwash flow along the free end of the finite cylinder. Fig. 4 is qualitatively similar to Fig. 8 in Park and Lee (2000); nevertheless a quantitative comparison can’t be done since in work of Park and Lee (2000) the Reynolds number is 20˙000 and L/D is 10.


- 6 -

z/L = 0.43 2D Fig. 5. Mean Reynolds stresses (u′u′, v′v′ and u′v′), mean turbulent kinetic energy (k) and turbulent production (p).

Fig. 5 shows the maps of the Reynolds stress tensor components relative to the 2D cylinder as well as the z/L = 0.43 section of the finite cylinder. All these components of the Reynolds stress tensor have been divided by U∞

2, so that they are dimensionless. To facilitate the comparison between


- 7 -

finite and infinite cylinders, for each component the same contour legend has been chosen. For all components, the values of finite cylinder are smaller than the infinite one, as expected by Fig. 4. The u′u′ maps have a two-lobe structure with a maximum value located at abut x/D ≈ 2.15 and y/D ≈ ±0.50 for the finite cylinder and in x/D ≈ 0.35 and y/D ≈ ±0.25 for the 2D cylinder. The v′v′ maps have a one-lobe structure with the maximum value located on the symmetry section and in x/D ≈ 2.9 and x/D ≈ 0.63 for finite and infinite cylinders, respectively. Finally, the u′v′ maps have a two-lobe structure, but the maximum values are shifted downstream: x/D ≈ 2.60 and y/D ≈ ±0.50 for the finite cylinder and x/D ≈ 0.58 and y/D ≈ ±0.25 for 2D one. The last two rows of Fig. 5 show the mean kinetic turbulent energy k and the turbulent production p. They are computed respectively as: vvuuk ′′+′′= (4)

yvvv

yuvu

xvvu

xuuup

∂∂′′+

∂∂′′+

∂∂′′+

∂∂′′= (5)

Furthermore, both k and p have been made dimensionless. According to Braza et al. (2006), all the components of the Reynolds stress tensor have their maximum value near the vortices formation region and this explains why the turbulent kinetic energy has a one-lobe structure with the maximum located on the rear axes at x/D=0.6 and x/D=2.6, for both infinite and finite cylinders respectively. The turbulent production shows value very similar to the mean kinetic turbulent energy, but in this case there are two maxima for each map, located nearer the shear layer.

3.2. Phase averaged flow fields As above discussed, the phase averaging has been computed by combining the first two POD modes. This procedure is correct only if these modes contain most flow field energy. Many researchers have applied this procedure to a infinite cylinder (e.g. Ben Chiekh et al. 2004 and Perrin et al. 2007) and really it works properly for 2D cylinder, since in this case it’s very easy to find two predominant modes by means of POD. On the contrary, this doesn’t happen for the finite cylinder.

Fig. 6. Energy percentage associated to first 10 POD eigenvalues.


- 8 -

Fig. 6 shows the percentage energy associated to the first 10 eigenvalues respect to the overall energy for both finite and infinite cylinders. The curve relative to the 2D cylinder shows the first 2 eigenvalues having more than 30% of the overall energy, whereas all the others have values lower than 3%. This is the ideal condition to apply the POD technique to compute phase averaging, since the first two eigenvalues contain almost 65% of the flow field energy. By approaching the free end of the finite cylinder, the situation gets worse. For example, for z/L = 0.14 the first two eigenvalues contain almost 65% of overall energy, for z/L = 0.29 almost 57% and for z/L=0.43 almost 48%. It’s interesting to note that, contrary to the 2D cylinder, for the three sections of the finite cylinder nearer to the wall, the third eigenvalue owns a relatively high value of percentage energy, whereas the residual ones contain significantly smaller values. Curves relative to other measurement sections suggest the inapplicability of the POD approach. As a conclusion of these observations and accordingly with Kawamura et al. (1984), it can be affirmed that the region which shows the Karman vortex shedding for a finite cylinder with AR=8 extends from z/L≈0.14 until z/L≈0.43. Unfortunately, it wasn’t impossible to make a measurement nearer the cylinder base, because of too much reflections of the laser on the plane mounting the cylinder. Fig. 7 shows the first two POD modes relative to the 2D cylinder and the finite one for z/L=0.43. The former are very similar to the ones present in the literature (e.g. Perrin et al. 2007), while the latter, even if different from the 2D modes, are very similar to the one relative to the measurement planes at z/L=0.14 and z/L=0.29 (not shown here). For this reason, only the evolution of the vortices behind the finite cylinder at z/L = 0.43 has been shown in Fig. 8 and compared to the one of the 2D cylinder (Fig. 9). For each section only 4 phases have been illustrated. Besides the already widely shown different vortex formation length, the two sections show a substantial difference: in the 2D vortices evolution, when a vortex forms the other is almost disappeared, whereas in z/L = 0.43 section of the finite cylinder there are always two contemporaneous vortices. The same phenomenon happens for z/L = 0.14 and 0.29 (not shown here).

2D - mode 1 2D - mode 2

z/L = 0.43 - mode 1 z/L = 0.43 - mode 2

Fig. 7. First two POD modes.


- 9 -

z/L = 0.43 - ϕ = 0 z/L = 0.43 - ϕ = TK/4

z/L = 0.43 - ϕ = TK/2 z/L = 0.43 - ϕ = 3TK/4

Fig. 8: Vortices evolution phases in the wake of finite cylinder for z/L = 0.43.

2D - ϕ = 0 2D - ϕ = TK/4

2D - ϕ = TK/2 2D - ϕ = 3TK/4

Fig. 9: Vortices evolution phases in the wake of infinite cylinder.


- 10 -

4. Conclusions PIV measurements have been done for both finite and infinite cylinders and compared. For the 2D cylinder the measurement section is in the middle of the cylinder length whereas, in order to study the flow evolution between the basis and the end of the finite cylinder, various sections along the cylinder length have been investigated. The Reynolds number investigated (Re = 16˙000) is within the Shear-Layer Transition Regime suggested by Williamson (1996). The mean flow fields and Reynolds stress component maps have been obtained by averaging 5˙000 images and showed the similarity between all the analysed sections. The substantial difference has been found in the vortex formation length. For this reason, the turbulence intensity has been computed in y/D = 0 section as a function of x/D and by varying the measurement section. The analysis showed that the vortex formation length for the 2D cylinder is smaller than the ones of all investigated sections of the finite cylinder. The sampling frequency smaller than the von Karman one made an a posteriori phase averaging method needed. For this reason, the Proper Orthogonal Decomposition technique has been used to find the principal modes of the flow field and, by combining them to reconstruct the vortices evolution. With this method, the Karman vortex shedding has been identified for the infinite cylinder as well as for the finite one, only for a small region near its base, accordingly with Kawamura et al. (1984). This region extends from z/L≈0.14 until z/L≈0.43, relatively to a finite cylinder with AR=8. References Afgan I, Moulinec C, Prosser R and Laurence D (2007). Large eddy simulation of turbulent flow for wall

mounted cantilever cylinders of aspect ratio 6 and 10. Int. J. Heat and Fluid Flow 28: 561–574. Ben Chiekh M, Michard M, Grosjean N and Bèra J (2004). Reconstruction temporelle d’un champ

aérodynamique instationnaire à partir de mesures PIV non résolues dans le temps. Proc 9e Congrès Francophone de Vélocimétrie Laser. Bruxelles, pp D.8.1-8.

Bloor MS (1964). The transition to turbulence in the wake of a circular cylinder. J. Fluid. Mech. 19: 290. Braza M, Perrin R and Hoarau Y (2006). Turbulence properties in the cylinder wake at high Reynolds

numbers. J. Fluids Struct. 22: 757-771. Budair M, Ayoub A and Karamcheti K (1991). Frequency Measurements in a Finite Cylinder Wake at a

Subcritical Reynolds Number. AIAA J., Vol. 29, No. 12, pp 2163-2168. Kawamura T, Hiwada M, Hibino T, Mabuchi I and Kumada M (1984). Flow around a finite circular cylinder

on a flat plate. Bull JSME 27: 2142–2151. Konstantinidis E, Balabani S and Yianneskis M (2005). Conditional averaging of PIV plane wake data using

a cross-correlation approach. Exp Fluids. Vol. 39, No. 1, pp 38-47. Lin JC, Towfighi J and Rockwell D (1995). Instantaneous structure of near-wake of a cylinder: on the effect

of Reynolds number. J. Fluids Struct. 9: 409-418. Park CW and Lee SJ (2000). Free end effects on the near wake flow structure behind a finite circular

cylinder. J. Wind Eng. Ind. Aer. 88: 231-246. Perrin R, Braza M, Cid E, Cazin S, Barthet A, Sevrain A, Mockett A and Thiele F (2007). Phase-averaged

measurements of the turbulence properties in the near wake of a circular cylinder at high Reynolds number by 2C-PIV and 3C-PIV. Exp Fluids. Vol. 42, No. 1, pp 93-109.

Saad M, Lee L and Lee T (2007). Shear layers of a circular cylinder with rotary oscillation. Exp Fluids 43: 569-578.

Sirovich L (1987). Turbulent and dynamics of coherent structures. Part I: coherent structures. Quart. Appl Math. Vol. 45, No 3, pp 561-571.

Williamson C H K (1996). Vortex dynamics in the cylinder wake. Annu. Rev. Fluid. Mech,, Vol. 28, pp 477-539.

Vortex shedding in the near wake of a finite cylinder...

Documents

Transcript of Vortex shedding in the near wake of a finite cylinder...