vortex dynamics in the wake of wall-mounted cylinders...

15th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 05-08 July, 2010

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Vortex dynamics in the wake of wall-mounted cylinders:

experiment and simulation

Thomas Uffinger1, Stefan Becker2, Irfan Ali3

1: Institute of Process Machinery and Systems Engineering, University of Erlangen, Germany, [email protected] 2: Institute of Process Machinery and Systems Engineering, University of Erlangen, Germany, [email protected]

3: Institute of Fluid Mechanics, University of Erlangen, Germany, [email protected] Abstract In the present paper the three-dimensional flow fields around three different wall-mounted cylinder stump geometries are evaluated. A combined approach which uses experimental as well as numerical investigations is applied to provide an extensive data base. The ability to compare measured and computed data increases the reliability of the deduced conclusions. For the experimental investigations laser-Doppler anemometry is used to determine the average velocity vectors and the RMS-values of the velocities. Concerning computational fluid dynamics three different simulation techniques are applied. A shear stress transport (SST) turbulence model represents RANS approaches. A hybrid simulation technique combining Reynolds averaged Navier-Stokes (RANS) and large eddy simulation (LES) features is realized by the scale adaptive simulation (SAS) model. The third type of numerical simulation is a large eddy simulation (LES), that is able to capture the unsteady flow field in the entire computational domain. Experimental and numerical results are presented and compared with each other. 1. Introduction The investigation of the flow field around cylinders helps to understand the fundamental physical phenomena of bluff body aerodynamics. Furthermore cylinders or cylinder-like geometries can be found in many technical applications, such as for example antennas and attaching parts of vehicles, high-rise buildings or supports in internal and external flows. Due to the importance for fundamental research in aerodynamics as well as due to the broad spectrum of technical applications, the topic has received much attention in literature. The flow around circular cylinders is investigated by Naudascher and Rockwell [12], Zdravkovich [18] and Norberg [14]. Square cylinders are covered in publications of Bearman [1], Knisely [9], Norberg [13], Fujita [7] or Dutta [6]. Nearly all of the mentioned papers focus on two-dimensional flow fields. The three-dimensional flow field around wall-mounted cylindrical objects of finite length is investigated in a much lower number of works. Examples are the papers of Sakamoto [15], Becker et al. [2] and Wang [16]. Despite the availability of numerous publications, there are still open questions concerning the physical phenomena in flow fields around cylindrical structures, which require further studies. For this work, a basic cylinder stump geometry with a length L and a side length D as depicted in Figure 1 is modified by attaching additional geometries in front or behind the square cylinder. The aim is to study the influence of the geometry modifications on the flow features in the wake of the cylindrical structure. Of special interest are the size of the recirculation region, the shape of the roof vortex and its influence on the wake of the cylinder and the amount of turbulent kinetic energy in the flow field. Additionally the anisotropy of turbulence as well as the unsteady flow features are investigated. Therefore, the study not only helps to better understand the physical phenomena of aerodynamics in flows around cylinder stump geometries but also provides information of possible sources of flow-induced sound, which is important for aeroacoustic investigations that are for example topic of the work of Becker et al. [3]. Furthermore, experimental results are compared to numerical data.


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Fig. 1 Basic flow configuration: wall-mounted cylinder in a cross-flow 2. Basic features of the flow around cylinders in a cross-flow The flow field around wall-mounted cylinders strongly depends on the aspect ratio of the cylinder L/D and on the ratio between the cylinder length and the thickness of the boundary layer (see Sakamoto [15], Becker et al. [2] and Wang [16]). For aspect ratios larger than 5 (smooth geometries), periodic spanwise vortices, which form a Kármán vortex street behind the cylinder, appear along most of the cylinder length. Close to the wall and in the roof region this vortex street is disturbed. Due to the adverse pressure gradient in the boundary layer upstream of the cylinder, a flow separation is observed that stretches around the base of the cylinder and forms a horseshoe vortex system. In the roof region helix-like vortices are formed which interact with the Kármán vortex street. The frequency of the Kármán vortex shedding is influenced by the ratio between the cylinder length and the boundary layer thickness (see Sakamoto [15]). The larger the ratio, the larger the Strouhal number St and consequently the shedding frequency gets. Figure 2 shows an illustration of the flow field around a wall-mounted square cylinder stump with the three above mentioned flow features (Kármán vortex street, roof vortices and horseshoe vortex system).

Fig. 2 Flow structures around a wall-mounted cylinder stump following Wang [16] 3. Measurement setup and used facilities and equipment The measurements were carried out in the aerodynamic wind tunnel of the university, which is a closed return type of tunnel with an open test section. The dimensions of the nozzle exit are 1.87 x 1.40 m². The contraction ratio of the nozzle is 5:1. With an installed fan power of 400 kW speeds of up to 55 m/s are possible. The turbulence level of the tunnel does not exceed 0.12 %. To avoid thick boundary layers around the cylinder geometries, the stumps are not directly mounted onto the wind tunnel floor, but onto a plate that is installed at a distance of 500 mm from the tunnel floor (see Figure 3).


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Fig. 3 Experimental setup in the aerodynamic wind tunnel At the upstream face of the plate a NACA 0001 profile is attached. The distance between the exit of the tunnel nozzle and the leading edge of the plate is about 500 mm. The cross-sections of the three investigated cylinder geometries are shown in Figure 4.

Fig. 4 Cross sections of the three investigated cylinder geometries (dimensions in mm): a) square cylinder, b) cylinder with elliptical afterbody, c) cylinder with wedge in front

The base configuration is a square cylinder with an edge length D of the cross-section of 20 mm. For the second geometry, an elliptical afterbody with a radius of 30 mm is attached behind the square cylinder. The third configuration contains a wedge in front of the cylinder. The cylinder length L of all three cases is 120 mm, which results in an aspect ratio L/D of 6. As measurement technique laser-Doppler anemometry (LDA) is used. To provide the particles necessary for applying LDA, diethyl hexyl sebacate (DEHS) is atomized and added to the circulating air in the wind tunnel. As light source for the utilized two-component LDA system serves an Argon-ion laser. In the transmitter box the laser beam is spitted in two green (514,5 nm) and two blue (488 nm) beams. The frequency of one of the beams of each color is shifted 40 MHz. The four beams are coupled into monomode glass fibres and guided to the LDA probe which has an outer diameter of 60 mm. The diameter of the measuring volume is approximately 75 µm. The beam path is deflected 45° by a mirror to reduce the area of the probe, which is exposed to the flow. The working distance of the probe is about 400 mm. To automatically scan the measurement grid, a 3D-traverse system is available. All components of the LDA setup are represented in Figure 5. Since all three velocity components have to be evaluated, each measuring point has to be sampled twice. In the first run, the U and W velocity components are determined. Then the probe is rotated 90° and the remaining velocity component V is measured. In order to get a sufficient number of statistically independent samples, the measuring time per point and probe orientation was set to 120 s. For a 95 % confidence interval, the statistical uncertainty of the main velocity is less than 0.01 % in the undisturbed flow regions, following Bendat and Piersol [4]. In areas, where low velocities and high turbulence intensities are achieved, the computation of the uncertainty of measured data is more difficult. An estimation for the 95 % confidence interval gives 1 % for the mean values and 1.5 % for moments of first order. Additionally, the uncertainty of the measurement system must be considered. In the undisturbed flow regions, the LDA system detects turbulence intensities down to 0.8 %. Containing uncertainties due to data processing, wind tunnel velocity fluctuations or vibrations of the traverse system, this value can be regarded as a lower threshold for measured turbulence intensities.


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The free stream velocity U∞ is 10 m/s. With the cylinder edge length D of 20 mm and air as flow medium, a Reynolds number Re of around 12,500 is achieved. The cylinder height L is more than 20 time larger than the boundary layer thickness. Therefore, the influence of the boundary layer on the flow around the cylindrical structures can be neglected (see also Becker [2]). In a distance of 10 D upstream of the center line of the cylinder the boundary layer was investigated. The displacement thickness at this position is 0.788 and the momentum thickness is 0.357, which gives a form factor H12 of 2.207, representing a laminar boundary layer.

Fig. 5 Illustration of the laser-Doppler system 4. Numerical methods For the numerical investigations of this work, three different simulation techniques have been applied. The first approach uses the shear stress transport model (SST), which is a Reynolds averaged Navier-Stokes simulation technique. The second approach is based on a large eddy simulation (LES) that resolves most of the turbulent length scales. Just the small scale structures of turbulence are modeled. Finally, a hybrid method, called scale adaptive simulation (SAS), is applied. The SST model combines a k-ε model following Jones and Launder [8] and a k-ω model (see Wilcox [17]). In the k-ε model a differential equation is solved for the turbulent kinetic energy k and the dissipation ε, in the k-ω model for k and the turbulent frequency ω. The equation for k of the k-ε model can be transformed to the one of the k-ω model by substituting ε by use of k and ω. Doing the same for the ε-equation of the k-ε model leads to an additional term compared to the ω-equation of the k-ω model. The SST-approach blends this additional term depending on the distance to the wall and the local values of k and ω. This leads to a blending between the k-ε model, that is well


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suited for areas with some distance to walls, and the k-ω model, which gives good results in regions close to walls. Further details of the SST model can be found in the work of Menter [10]. In contrast to SST computations, large eddy simulations are necessarily unsteady. The basic idea of LES is, to directly resolve the large scales of turbulent flow fields, which is equivalent to a DNS for these scales. But while DNS also resolves the small scales, they are modeled in LES. The approach filters the Navier-Stokes equations spatially. This leads to an additional term in the momentum equation. This term containing the small scale turbulence has to be modeled. As small scale turbulence is more isotropic and homogeneous than large scale turbulence, the modeling in LES is much easier compared to RANS approaches. Consequently simpler modeling methods can be used. Therefore, the algebraic Smagorinsky model is often used for LES. Further information on the Smagorinsky model and LES can be found in the work of Breuer [5]. The SAS approach is an expansion of the SST model with additional terms in the transport equation for ω. The terms are activated by the ratio of the turbulent length scale Lt and the Kármán length LνK. Due to the introduction of the Kármán length the SAS model is able to adjust the size of resolved turbulent structures. Therefore, SAS can deliver results comparable to those of LES in unsteady regions. Details of the SAS approach can be found in the work of Menter and Egorov [11]. All three simulation approaches have been used to calculate the flow fields around the cylinder stump geometries presented in Section 3. 5. Results of the experimental investigations An overview of the measured flow field around the square cylinder and a schematic illustration of the flow structures is given in Figure 6.

(a) Velocity in the symmetry plane (y=0) and turbulent kinetic energy k in cross planes

(b) Schematic illustration of the main flow features

Fig. 6 Flow field around the square cylinder In the center plane (y=0) of the measurement data, velocity vectors are shown, in the other planes the distribution of the turbulent kinetic energy k. The flow field is strongly symmetric with two vortex cores forming behind the side faces of the cylinder. The separation of the flow in the roof area of the geometry and the large recirculation region in the wake of the cylinder is obvious. The


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flow in the center plane has a strong downward component in the upper half of the wake, which points out an extensive interaction of the roof vortices and the side vortices of the Kármán vortex street. Due to these findings Figure 2 has to be revised. The influence of the roof vortices on the flow field behind the cylinder is stronger than suggested by Figure 2, the Kármán vortex street develops along a smaller part of the cylinder height. A revised illustration of the flow structures around a wall-mounted square cylinder of finite length is given on the right hand side of Figure 6. The distribution of the velocity vectors and the turbulent kinetic energy k as well as a schematic illustration for the configuration with an elliptical afterbody is displayed in Figure 7.


(b) Schematic illustration of the main flow features

Fig. 7 Flow field around the square cylinder with an elliptical afterbody The general distribution is similar to the one of the square cylinder (Figure 6) but the downward component of the velocity as well as the size of the recirculation region is smaller and the influence of the roof vortex on the wake is weaker as Figure 7 shows. The amount of turbulent kinetic energy in the wake is significantly smaller compared to the square cylinder. An illustration of the flow structures around the cylinder with the elliptical afterbody is given on the right hand side of Figure 7. The Kármán vortex street clearly dominates the wake. The reattachment of the roof vortex on the top-side of the elliptical afterbody and the weak influence on the vortex street is shown, too. Measurement results for the third investigated configuration (cylinder with wedge in front) are displayed in Figure 8. In this case symmetry is no longer maintained. The mean velocity shows a three-dimensional flow field, instead. The reason for this behavior is the creation of an instability line along the front edge of the wedge. In combination with experimental conditions, never satisfying ideal inflow conditions that are perfectly symmetric, this leads to an asymmetric, three-dimensional flow field. According to Figure 8, the size of the recirculation region is in between the case of the square cylinder and the cylinder with the elliptical afterbody. The influence of the roof vortex on the flow field in the wake is strong. No separation of the flow on top of the cylinder is achieved.


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(b) Velocity in a cross plane 2 D behind the cylinder

Fig. 8 Measured flow field around the square cylinder with wedge in front Above, just the amount of turbulent kinetic energy was considered, but the distribution of the fluctuation components Urms, Vrms and Wrms is of interest, too. A spatial averaging of the RMS-values of the fluctuations and the turbulent kinetic energy in the yz plane 2 D behind the cylinder is performed for the three configurations. The results are collected in Table 1.

Configuration Urms Vrms Wrms k Square cylinder 1.91 2.0 1.54 7.54 Cylinder with elliptical afterbody 1.95 1.57 1.56 6.2 Cylinder with wedge in front 1.26 1.3 1.14 4.32

Tab. 1 Amount of k and its distribution among the three fluctuation components Urms, Vrms and Wrms for the three different geometries (averaged over the yz plane in the wake, 2 D behind the cylinder)

The table shows that in case of the square cylinder the fluctuations in the main flow as well as in the spanwise direction are of the same order of magnitude and larger than the third component. For the cylinder with the elliptical afterbody, the fluctuations in x-direction are dominant. For the configuration with the wedge in front of the cylinder all three components are of the same order of magnitude. These results are consistent with the observations made above. If the influence of the roof vortex on the flow behind the cylinder is strong, then, the flow structures in the wake of the stump become three-dimensional. Therefore, coherent two-dimensional structures, especially due to the Kármán vortex street, are destroyed and smaller structures are observed under these conditions, which finally leads to a more isotropic turbulence behind the cylinder. Beside the average flow field, the unsteady behavior of the flow is also of special interest. Figure 9(a) shows the temporal evolution of the unsteady velocity u at the position x=0.06 m, y=0 m, z=0.12 m over a period of 2 seconds. Two frequencies are clearly visible: a low frequency of about 3 Hz and a much higher frequency. The high frequency is caused by the vortex shedding of the Kármán vortex street. The low frequency can be explained as follows: The region behind the cylinder is filled with fluid with a large amount of momentum over time. After some time the momentum in the wake is large enough and the fluid is washed out of the wake, which in


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consequence produces higher values of the unsteady velocity. After a while the velocity level drops back to a lower level and the wake is filled with fluid of high momentum content again. In Figure 9(b) the frequency spectra of the velocity u at two positions are shown. At the position in the center plane just the low frequency is visible, as the vortices do not come close enough to the symmetry plane. At the other position the shedding frequency of about 48 Hz can additionally be seen and the superelevation of the lower frequency is weaker, as the point is already located in the outer part of the wake.

(a) Temporal evolution of the velocity u over a time span of 2 seconds

(b) Frequency spectra of the velocities at two different positions in the wake of the cylinder

Fig. 9 Unsteady results for the square cylinder (positions in m) 6. Results of the numerical investigations For the square cylinder SST, SAS and LES simulations were carried out, for the two remaining configurations just SST and SAS computations were performed. Figure 10 shows a snapshot of the unsteady flow field in the xy plane 3 D above the wall and the xz symmetry plane computed by LES.

(a) xy plane 3 D above the wall

(b) xz plane

Fig. 10 Snapshot of the unsteady flow field computed by LES


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The vortex shedding of the Kármán vortex street is clearly visible in the xy plane. As in the experiments, the roof vortex strongly influences the wake of the cylinder which can be seen from the xz plane. A comparison of the vortex structures behind the square cylinder computed by SAS and LES can be seen in Figure 11. The coherent structures are shown as iso-surfaces, for the values of λsym = -1000, where λsym is the eigenvalue of the symmetric tensor S2 + Ω2. The terms S and Ω denote the symmetric and antisymmetric parts of the velocity gradient. In the results from the SAS computation iso-surfaces of Ω2 - S2=100000 s-2 colored with the velocity are used for the visualization of the turbulent structures, with Ω representing the vorticity and S the strain rate.

(a) SAS (b) LES

Fig. 11 Vortrex structures behind the square cylinder Although both simulations should be able to resolve the unsteady flow structures properly, noticeable differences are evident. Major difference is seen in the length of the vortex structures downstream, which for LES is longer. This can be attributed to the finer grid downstream in LES resolving the turbulent structures better. For the SAS computations the forces on the square cylinder were calculated. The temporal evolution of the forces in x- and y-direction are plotted in Figure 12(a), the frequency spectra of the signals are shown in Figure 12(b).

(a) Temporal evolution of the forces (b) Frequency spectra

Fig. 12 Forces acting on the square cylinder


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The force in y-direction shows a peak around 50-60 Hz, which corresponds to the frequency of vortex shedding. The frequency is slightly higher than in the experiments. In x-direction the double frequency can be found. The doubling of frequency is an expected behaviour, as the vortices separating on both sides of the cylinder generate forces of identical sign on the geometry in x-direction, while in y-direction the forces on each side have different signs and therefore result in half of the frequency compared to the x-direction. Besides the frequency due to vortex shedding, a much lower frequency can be seen in both time signals, but especially in the signal of the force in x-direction. The spectrum of this signal also shows the frequency, which is located around 2-3 Hz. The reason for this frequency is the same as for the experiments. A periodic process of filling the region behind the cylinder with fluid of high momentum content and washing it out after some time takes place in the wake of the cylinder. The good compliance of the experimental and the numerical investigations concerning this periodic phenomenon minimizes misinterpretations or errors and inaccuracies either in the measurements or in the simulations. To investigate the unsteady behavior of the flow field computed by LES, the temporal evolution of the pressure signal at the position x=0.06 m, y=0 m, z=0.12 m over a period of 2 seconds is presented in Figure 13(a). As in the experiments (see Figure 9) and in the SAS computation the signal shows low-frequency oscillations around 2-3 Hz. The low frequency also appears in the frequency spectra of Figure 13(b). For the point in the symmetry plane (y=0), no shedding frequency is visible, while the spectrum of the other point shows the shedding frequency as a peak around 60 Hz, i.e., a slightly higher frequency as in the SAS simulation.

(a) Temporal evolution of the pressure over a time span of 2 seconds

(b) Frequency spectra

Fig. 13 Pressure in the flow field calculated by LES for the square cylinder (positions in mm) 7. Comparison of experimental and numerical results As seen above, experiments and numerical computations using SAS or LES match very well concerning the unsteady behaviour of the flow field behind the square cylinder. The agreement of average values of the flow field for experiments and SST, SAS and LES computations is presented in the following. Figure 14 shows two profiles of the mean velocity U in the xz symmetry plane at two different distances behind the square cylinder.


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(a) D behind the square cylinder (b) 5 D behind the square cylinder

Fig. 14 Comparisons of the mean velocity U in the symmetry plane x=0 The LES results match the experimental data best. The recirculation region computed by SST and SAS is longer than in the measurements or in the LES, which can be seen from the lower velocity values of Figure 14(b). Profile plots for the turbulent kinetic energy k at the same positions as for the velocity profiles are presented in Figure 15. Due to reflections of the laser light, it was not possible to measure the velocity component in y-direction in the near-wall region. For the points near the wall the turbulent kinetic energy is therefore just computed from the two remaining velocity components, which explains the jump in the measured values at z=0.01 m. As for the mean velocity the results provided by the LES match experimental data best. The values of SST and SAS computations are considerably smaller than measured values. Although the amount of turbulent kinetic energy is overpredicted by LES in Figure 15(a), the overall agreement with experiments is much better than for the other simulation types. In Figure 15(b) numerical results by LES and experimental data match very well, while SST and SAS simulations underpredict the amount of turbulent kinetic energy.

(a) D behind the square cylinder (b) 5 D behind the square cylinder

Fig. 15 Comparisons of the turbulent kinetic energy k in the symmetry plane y=0


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The above comparisons of profiles of the mean velocity and the turbulent kinetic energy between experiments and numerical results by SST, SAS and LES show, that LES provides the best results in comparison to the measurements. Although the SAS simulations should be able to capture the unsteady and three-dimensional structures of the flow field, the agreement of the results with experimental data is noticeably worse than for LES. Comparable results are achieved for the two remaining cylinder geometries. 8. Conclusion In this paper, three different wall-mounted cylindrical geometries were investigated both, experimentally and numerically. The measurements show a strong influence of the roof vortex on the flow field behind the cylinders. The cylinder with the elliptical afterbody is the only configuration with a Kármán vortex street developing over most of the cylinder height. For the square cylinder the roof vortices interact with the vortices of the Kármán vortex street. In case of the cylinder with the wedge in front, a line of instability is created at the leading edge of the wedge. Therefore, two-dimensionality of the vortices of the Kármán vortex street is destroyed. This process is supported by the interaction of the roof vortices with the wake of the cylinder. In consequence, smaller and three-dimensional vortex structures are found behind the cylinder with the wedge in front, while the flow field of the configuration with the elliptical afterbody is dominated by the two-dimensional Kármán vortex street. In case of small, three-dimensional structures the turbulence behind the cylinder is much more isotropic, which is of relevance for the acoustic behaviour of the geometry (see Becker et al. [3]). Furthermore, unsteady effects of very low frequencies around 3 Hz were found in the flow field. The phenomenon can be explained by a filling of the wake of the cylinder with fluid carrying a large amount of momentum. If enough momentum is collected in the recirculation region, fluid is washed out resulting in higher values of the unsteady velocity. After a while the velocity level drops back to a lower level and the process restarts from its beginning. The experimental results were validated numerically. Three types of simulations were used to compute the flow fields around the three cylindrical geometries: SST, SAS and LES. It was shown that both, SAS and LES capture the unsteady behavior of the flow. Nevertheless, the agreement of profiles of the mean veloctity and the turbulent kinetic energy between LES-based and measured data is better than for the other simulation approaches. The results corroborate the superior qualtity of LES-based numerical simulation approaches, which suggests the usage of such simulation techniques, although the computational effort is higher than for e. g. SST or SAS computations. References 1. Bearman, P. W., Vortex shedding from oscillating bluff bodies, Annual Review of Fluid

Mechnics, Vol. 16, pp. 195-222, 1984. 2. Becker, S., Lienhart, H. and Durst, F., Flow around three dimensional obstacles in boundary

layers, Journal of Wind Engineering and Industrial Aerodynamics, Vol. 90, pp. 265-279, 2002. 3. Becker, S., Hahn, C., Kaltenbacher, M. and Lerch, R., Flow-induced sound of wall-mounted

cylinders with different geometries, AIAA Journal, Vol. 46, pp. 2265-2281, 2008. 4. Bendat, J. S. and Piersol, A. G., Random data analysis and measurement procedures, John

Wiley & Sons, New York, 1986. 5. Breuer, M., Direkte Numerische Simulation und Large-Eddy Simulation turbulenter

Strömungen auf Hochleistungsrechnern, Shaker Verlag, Aachen, 2002. 6. Dutta, S., Influence of the orientation of a square cylinder on the wake properties, Experiments

in Fluids, Vol. 34, pp. 16-23, 2003.


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7. Fujita, H., Experimental investigations and prediction of aerodynamic sound generated from square cylinders, 19th AIAA Aeroacoustics Conference, Toulouse, France, June 2-4, AIAA-1998-2369, 1998.

8. Jones, W. P. and Launder, B. E., The prediction of laminarization with a two-equation model of turbulence, International Journal of Heat and Mass Transfer, Vol. 15, pp. 301-314, 1972.

9. Knisely, C. W., Strouhal numbers of rectangular cylinders at incidence: a review and new data, Journal of Fluids and Structures, Vol. 4, pp. 371-393, 1990.

10. Menter, F. R., Two-equation eddy-viscosity turbulence models for engineering applications, AIAA Journal, Vol. 32, pp. 1598-1605, 1994.

11. Menter, F. R. and Egorov, Y., A scale-adaptive simulation model using two-equation models, 43th AIAA Aerospace Science Meeting and Exhibit, Reno, Nevada, 2005.

12. Naudascher, E. and Rockwell, D., Flow-induced vibrations: an engineering guide, A.A. Balkema, 1994.

13. Norberg, C., Flow around rectangular cylinders: pressure forces and wake frequencies, Journal of Wind Engineering and Industrial Aerodynamics, Vol. 49, pp. 187-196, 1993.

14. Norberg, C., Fluctuating lift on a circular cylinder: review and new measurements, Journal of Fluids and Structures, Vol. 17, pp. 57-96, 2003.

15. Sakamoto, H., Vortex shedding from a rectangular prism and a circular cylinder placed vertically in a turbulent boundary layer, Journal of Fluid Mechanics, Vol. 126, pp. 147-165, 1983.

16. Wang, H. F., Flow structure around a finite-length square prism, 15th Australasian Fluid Mechanics Conference, September 13-17, University of Sydney, Australia, 2004.

17. Wilcox, D. C., Turbulence modeling for CFD, DWC Industries, Inc., La Canada, California, USA, 1993.

18. Zdravkovich, M. M., Flow around circular cylinders, Oxford University Press, 1997.

vortex dynamics in the wake of wall-mounted cylinders...

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