Effects of leading edge geometry on the vortex shedding frequency of an elongated bluff body at high Reynolds numbers

J. Wind Eng. Ind. Aerodyn. 128 (2014) 66–75

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Effects of leading edge geometry on the vortex shedding frequency of an elongated bluff body at high Reynolds numbers Zachary J. Taylor a, Roi Gurka b, Gregory A. Kopp a,n a b

Boundary Layer Wind Tunnel Laboratory, Faculty of Engineering, University of Western Ontario, London, Canada School of Coastal and Marine Systems Science, Coastal Carolina University, Conway, SC, USA

art ic l e i nf o

a b s t r a c t

Article history: Received 6 October 2013 Received in revised form 10 March 2014 Accepted 13 March 2014 Available online 3 April 2014

Measurements have been performed in a large scale wind tunnel on an elongated bluff body with a chord-to-thickness ratio of 7 over the Reynolds number range Re¼4.0–7.5  104. Six different leading edge separation angles were created by altering the leading edge geometry. Time-resolved, synchronized, surface pressure and Particle Image Velocimetry data allow for detailed characterization of the flow around the body and in the recirculation region. The results show a linear decrease in the shedding frequency of nearly 40% as the leading edge separation angle is increased from 01–901. The PIV data are phase averaged in the recirculation region and the convection speed of the vortices is characterized. From the phase averaged data, the velocity outside of the recirculation region is observed to decrease markedly as the leading edge separation angle is increased, which is suggested to be responsible for the observed changes in the shedding frequency. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Bridge aerodynamics Elongated bluff bodies Particle image velocimetry Vortex shedding Separation-reattachment

1. Introduction Long-span bridges have been susceptible to vortex-induced instabilities, with examples including Rio-Niteroi Bridge (Battista and Pfeil, 2000), Storebaelt Bridge (Larsen et al., 2000), and Tokyo Bay Bridge (Fujino and Yoshida, 2002). Understanding the factors that controlled vortex shedding for these particular bridges was of critical importance to the development of mitigation strategies. However, a complete understanding of vortex shedding from such elongated bluff bodies remains incomplete. For an elongated bluff body, the flow separates at the leading edge and reattaches along the chord of the body before separating once again at the trailing edge. Thus, the decks of nearly all longspan suspension and cable-stayed bridges can be classified as elongated bluff bodies. The leading edge separating-reattaching flow adds complexity to the case of classical von Kármán vortex shedding in the wake of shorter bluff bodies (i.e., those with no reattachment along the body). With the susceptibility of longspan suspension and cable-stayed bridges to these aerodynamic instabilities, it is important to understand the characteristics that distinguish elongated bluff bodies from the well-known case of shorter bluff bodies. For a review of the mechanisms associated with shorter bluff bodies the reader is referred to Zdravkovich (1997).

n

Corresponding author. E-mail address: [email protected] (G.A. Kopp).

http://dx.doi.org/10.1016/j.jweia.2014.03.007 0167-6105/& 2014 Elsevier Ltd. All rights reserved.

In his study on circular cylinder wakes, Roshko (1954) anticipated the shedding frequency based on certain parameters including vortex convection speed and base pressure; however, we have previously shown that the role of pressure in the recirculation region is reduced for elongated bluff bodies (Taylor et al., 2011). It has also been shown that, for elongated bluff bodies, the mechanism governing shedding frequency is dependent on the Reynolds number, which is typically defined using the thickness, or the cross-stream dimension, of the body, t. At lower Reynolds numbers (Reo2000), Nakamura and Nakashima (1986) discovered that the vortex shedding frequency of some elongated bluff bodies is controlled by the separated shear layer impinging on the trailing edge corner. Naudascher and Wang (1993) extended the definition of this instability and renamed it the Impinging Leading Edge Vortex (ILEV) instability since they found that the shedding frequency of the leading edge vortices can be controlled by a feedback loop of leading edge vortices impinging on the trailing edge corner – not necessarily the impingement of the leading edge separated shear layer. As a vortex passes the sharp trailing edge corner, a pressure pulse is experienced by the leading edge separation-reattachment and another leading edge vortex is shed from this location. Hourigan et al. (2001) suggested that the shedding frequency of the leading edge vortices in the ILEV instability was strongly influenced by the ‘preferred’ trailing edge shedding frequency (i.e., that determined using a leading edge of elliptical cross-section). For elongated bluff bodies at Re 42000 there are comparably fewer studies and the factors governing vortex shedding frequency are unclear.

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At Re42000, the ILEV instability is known to be suppressed for rectangular cylinders (Mills et al., 2003). Parker and Welsh (1983) performed the benchmark study for rectangular cylinders at Reynolds numbers of order O(104) over a wide range of elongation ratios up to c/t ¼52. They found that, with the ILEV instability suppressed, there was a wide range of elongation ratios (7.6oc/ to 25) for which they could not detect periodic vortex shedding in the wake. However, several different studies have shown that with acoustic resonance or external forcing, the leading edge separated shear layer can be forced to once again shed vortices periodically at high Reynolds numbers (e.g., Parker and Welsh, 1983; Welsh et al., 1984; Stokes and Welsh, 1986; Mills et al., 2002). While many of these studies were focused on the susceptibility of elongated bluff bodies to acoustic forcing, the results obtained in unforced flow suggest that the preferred vortex shedding mode is that created by the ILEV instability and that the feedback between the leading edge and trailing edge shedding is disrupted at higher Reynolds numbers. Therefore, it is unclear how the vortex shedding frequency is controlled at these Reynolds numbers. Another critical aspect of elongated bluff bodies to consider, which is particularly relevant to bridge aerodynamics, is how changes to geometry affect the vortex shedding frequency. For example, it remains unclear whether or not the ILEV instability is suppressed for geometries other than the rectangular cylinder at the higher Reynolds numbers that are the focus of the current study. In general, previous studies on elongated bluff bodies have concluded that changes in the leading edge geometry affect the shedding frequency more significantly than changes to the trailing edge geometry (Welsh et al., 1984; Stokes and Welsh, 1986; Nguyen and Naudascher, 1991). Likewise, changes in the leading edge separation-reattachment have been shown to create markedly different levels of turbulent kinetic energy and near wake structure (Taylor et al., 2013). A force balance analysis in the near wake (Taylor et al., 2011) showed that larger leading edge separation-reattachment increases the role of the turbulent stresses in the recirculation region. The current study focuses on how the size of the leading edge separation-reattachment affects the shedding frequency variation of elongated bluff bodies through changes in the leading edge geometry. Furthermore, departures from the parametric trends established by Roshko (1954) for shorter bluff bodies are discussed.

2. Details of the experiments 2.1. Model details The model has been designed to accommodate different forebodies of constant cross-section that can be fit to the leading edge.

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Six different forebodies were used including the rectangular forebody, which is part of the base model, as shown in Fig. 1. The base model is of rectangular cross-section with a thickness, t, of 76.2 mm and a chord-to-thickness ratio of c/t ¼7. The chordto-thickness (elongation) ratio is kept constant across all tests; however, the length of the forebody is not considered in the measurement of the chord. The forebodies used in the current study include one of elliptical cross-section (3:1 axis ratio) and four of triangular cross-section with half interior angles, θ, ranging from 301 to 901 at 151 increments defined in Fig. 1. The streamlined forebody of semi-elliptical cross-section provides baseline behavior for when the wake is not influenced by the leading edge separation-reattachment. The streamwise location x ¼0 corresponds with the fixed separation points for all of the geometric configurations except for the one of elliptical cross-section (which has no leading edge separation). Roshko (1993) emphasized two extrinsic characteristics that alter the three-dimensionality of a given experiment: aspect ratio and end plates. In the present experiments, rectangular end plates are used which extend 0.3 m from the surface of the model, normal to the top and bottom surfaces, 0.41 m upstream from the leading edge and 0.57 m downstream into the wake from the trailing edge (Fig. 2). These dimensions ensure that they protrude into the wake for at least one complete vortex shedding wavelength. Cherry et al. (1984) review the aspect ratios (span-tothickness) used in various studies involving leading edge separation showing that the range in previous experiments is 2–18. Thus, the aspect ratio in the current study is relatively high with a spanto-thickness ratio of 24 (span-to-chord ratio is 3.4). In the present study, the model is instrumented to assess the three-dimensional nature of the flow, and the surface pressures show that the average pressure distribution is constant across the span for each case. 2.2. Wind tunnel tests The tests were performed in a large-scale wind tunnel in the Boundary Layer Wind Tunnel Laboratory at the University of Western Ontario. The tunnel is of closed-circuit design and includes a test section measuring 3.35 m wide by 1.83 m high giving a blockage ratio of 4.1%. The length of the test section is 39 m, and the testing was all performed approximately 2 m from the inlet. At this location, hot-wire anemometry data show the longitudinal turbulence intensity to be less than 1% with no dominant peaks in the spectrum to avoid forcing of the convectively unstable leading edge separated shear layers (e.g., Chaurasia and Thompson, 2011; Thompson, 2012). The vertical and horizontal velocity profiles are uniform to within 1%, away from the walls. The susceptibility to acoustic resonance of these phenomena has

Fig. 1. Schematic of the model showing the pressure tap layout ( þ symbols) and the different leading edge attachments used in the experiments.

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been emphasized by Parker and Welsh (1983) and the current setup was found to be free of any significant sources of acoustic resonance. The model was mounted in the horizontal and vertical center of the tunnel cross-section. The free stream speed was adjusted to yield 8 different Reynolds numbers, based on thickness (Re¼tU1/υ), ranging between 4.0  104 and 7.5  104 in increments of 0.5  104. For each of the geometric configurations (6 in total), and each Reynolds number, the pressure data were sampled for 120 s. After analysis of these data in terms of fluctuating lift, drag, and shedding frequency (Fig. 3), the results were observed to be independent of this Reynolds number range, and a Reynolds number of 5.0  104 (U1 ¼ 9.9 m/s) was selected for the synchronized pressure and Particle Image Velocimetry (PIV) experiments. The PIV and pressure data acquisition systems were synchronized at 500 Hz and the data were sampled for 64 s. 2.3. Details of the pressure measurements The model was fitted with 512 pressure taps (Fig. 1). Each spanwise row of taps consisted of 67 taps spaced 25.4 mm apart towards the edge of the model and spaced at 15.9 mm closer to the center. To obtain sectional pressure data, three streamwise loops made up of the taps on the leading and trailing edge surfaces, as well as 27 taps on the top and bottom surfaces, were used. These taps were spaced at 15.9 mm within approximately 3.5t of the leading edge and 25.4 mm from this location until the trailing edge. The pressure taps were connected to multiplexing pressure scanners with 12 taps connected to each scanner. The tubing system used to connect each tap to the scanners comprises plastic tubing with an inner diameter of 1.6 mm and a length of approximately 60 cm. A brass restrictor with a small bore was used to divide the length of the tubing and add damping to the resonant system, which has been tested to have a frequency response that is flat to approximately 200 Hz. The vortex shedding frequency of each body varied between 22 and 34 Hz and the data have been low-pass filtered at 180 Hz to remove any high frequency tube resonance. The pressure data are interpolated in time at each time step, which is necessary due to the multiplexed nature of the measurement system. Phase lag tests have been performed on this system to ensure that effects from this procedure are negligible. The reader is referred to Ho et al. (1999) for more details on the pressure scanning system.

al., 2010). The cameras are of CMOS type with a resolution of 1024  1024 pixels2 at 10-bit depth. The illumination comes from an 80 W dual head Nd:YLF laser emitting green light with a wavelength of 527 nm and an average pulse energy of 22 mJ in each half of an image pair. The wind tunnel was seeded with atomized olive oil, which yields mean particle diameters of 1 μm (Echols and Young, 1963). With such a large-scale facility, seeding proved to be the most significant experimental challenge; however, the data are observed to have satisfactory signal-to-noise ratios in the region of interest. To maximize the quality of the data as close to the body as possible, the trailing edge corner (x/t, y/t)¼(7,  0.5) was near the center of camera's field-of-view. This technique limits the amount of reflection captured by the camera, which ordinarily limits imaging close to the body surface. The field of view captured by the PIV images was 2.5t  2.5t with nearly one quarter of the image sacrificed to capture close to the body. However, due to the combination of available pulse energy and seeding, only 1.6t  1.2t of this area was used, which was sufficient to contain the entire recirculation region. The delay between images in an image pair was set to 100 μm which yielded particle window displacements of approximately 6 pixels at the edge of the recirculation region. The data were cross-correlated by OpenPIV (Taylor et al., 2010) using a Fast Fourier Transform (FFT) based cross-correlation scheme with 32  32 pixels2 interrogation windows and 50% overlap. Erroneous vectors were identified (typicallyo5%) and rejected using global and local filters, the missing data were replaced by local interpolation and a Gaussian filtering kernel was used to smooth the data.

2.4. Details of the PIV measurements The time-resolved PIV system used in this study is capable of long recording times at a sampling frequency of 500 Hz (Taylor et

Fig. 3. Variation of the Strouhal number, St ¼ fst/U1, with leading edge separation angle θ, and Reynolds number.

Fig. 2. Rendering of the model showing the elliptical forebody and the extent of the end plates.

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Fig. 4. Variation of (a) the sectional drag coefficient Cd ¼Fd/0.5ρU21tL and (b) the base pressure Cb with leading edge separation angle θ, and Reynolds number.

Fig. 5. Examples of the time series' used to phase average the data for leading edge separation angles of (a) θ¼ 01 and (b) θ¼ 901, both for Re ¼5  104. The dark lines are the Fourier series fit to the data from which the phase average is constructed.

2.5. Phase averaging The time signal used for phase averaging is obtained from one of the surface pressure taps on the base of the model. There are noticeable changes in the amplitude, frequency content and turbulence-induced randomness between different geometries (Fig. 5). In addition, there are slight changes of these three features in the time record of a given geometry. Thus, phase binning of the data required a windowed approach. These windows consisted of 1024 points or, equivalently, 2.048 s. A FFT is performed on each window and the coefficients of the dominant frequencies (fpeak 7 4.8 Hz) are used to reconstruct the time trace (Fig. 5). The phase of each data point is then identifiable from the reconstructed signal, and each time step is classified into one of 16 phase bins, each one having a length of 1/16 of the shedding period. The number of bins was chosen to balance (i) the number of realizations in each bin for convergence of the phase averaged statistics and (ii) the resolution in time over the shedding cycle. The number of bins selected in the current study is consistent with previously published studies of similar type (e.g., Cantwell and Coles, 1983), and each bin contained approximately 2000 samples.

3. Results 3.1. Vortex shedding frequency The emphasis of this paper is on the variation of the vortex shedding frequency in the wake as the influence of the leading edge separation-reattachment is altered through changes to the leading edge geometry. We express the change in the leading edge geometry using the leading edge separation angle, θ, which is assumed to be tangent to the leading edge forebody at the fixed

separation point (i.e., 301o θr901). The elliptical forebody does not cause the flow to separate at this location, and it represents a bounding case with θ¼ 01. To obtain the time series of the sectional aerodynamic forces, the pressure data were integrated around the streamwise loops of pressure taps (Fig. 1) at each time step. The computed time series of the sectional lift coefficient for each model was used to obtain the vortex shedding frequency, which is normalized using the thickness-based Strouhal number, St¼ft/U1, and plotted in Fig. 3. The Strouhal number is plotted for all eight Reynolds numbers for which data were taken and observed to decrease monotonically with increased separation angle. For shorter bluff bodies, the drag and base pressure coefficients reveal significant changes in the vortex shedding behavior. A thorough explanation of the vortex shedding regime changes for a circular cylinder based on the base pressure coefficient can be found in Roshko (1993). The drag and base pressure coefficients from all the six forebodies and eight Reynolds numbers are shown in Fig. 4. As with the shedding frequency, the drag and base pressure do not vary significantly with Reynolds number. The drag coefficient is observed to increase monotonically as the forebody becomes increasingly bluff. Correspondingly, the base suction is observed to be greater for forebodies with higher shedding frequency. However, the minimal variation with Reynolds number (over the range tested), and the trend with separation angle, suggest that any transition between aerodynamic instabilities is gradual rather than abrupt. 3.2. Flow along the body 3.2.1. Leading edge reattachment length The determination of flow reattachment has inherent difficulties because of the need to measure on, or close, to the surface of

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reattachment point, the pressure data are observed to have significant variation since larger reattachment lengths are relatively closer to the trailing edge, altering the thickness and development length of the boundary layer, as well as turbulence levels in the region downstream of reattachment. Since the pressure data have been shown to match the mean pressure characteristics of separation-reattachment from other studies, in the remainder of the paper we focus on the development of the flow along the surface of the body (after separation-reattachment) and in the recirculation region.

Fig. 6. Variation of the leading edge reattachment length, xr, with leading edge separation angle, θ for Re¼ 5  104.

Fig. 7. Variation of the normalized pressure coefficient in the streamwise direction within and beyond the leading edge separation-reattachment as a function of leading edge separation angle, θ, for Re¼ 5  104.

the body, and it has been performed in many different ways. Some of the notable methods are: reversed flow intermittency (e.g., Kiya and Sasaki, 1983; Djilali and Gartshore, 1991); flow visualization (Ram and Arakeri, 1990); mean velocity profiles (Taylor et al., 2011); and surface oil visualization (Cherry et al., 1984). It is known from previous studies (Cherry et al., 1984; Ram and Arakeri, 1990) that flow reattachment occurs in close proximity to, and downstream of, the location where the pressure fluctuations are maximum. Furthermore, it is well known that the time-averaged surface pressure within separation-reattachment is similar, regardless of geometry, when scaled by the reattachment length (Roshko and Lau, 1965; Djilali and Gartshore, 1991; Ram and Arakeri, 1990). Ram and Arakeri (1990) have shown that the location of the peak pressure fluctuations can also be scaled by the reattachment length. Following these observations, we define the reattachment point in relation to the location of maximum pressure fluctuation by xr ¼ xjmaxðC 0p Þ =0:95 (Ram and Arakeri, 1990). The reattachment lengths estimated in this manner are shown in Fig. 6, where error bars mark the uncertainty caused by the spatial resolution of the pressure tap layout. Since the data are independent of Reynolds number over the range tested, the data shown in Fig. 6 and hereafter are taken from experiments performed at Re¼5.0  104. In order to validate both the current pressure data and the estimation of the parameter xr we have collapsed the data similarly to Roshko and Lau (1965). They showed that the timeaveraged pressure coefficients collapse when scaled using the minimum value of the pressure coefficient and when the streamwise direction is scaled by the reattachment length. We use the same approach for each leading edge separation angle (except θ¼01) by scaling the pressure coefficient as C np ¼ ððC p  minðC p ÞÞÞ= ð1  minðC p ÞÞ and the streamwise dimension by xn ¼ x/xr. The present results are observed to scale reasonably well (Fig. 7) thus confirming the estimate of the reattachment length, xr. Past the

3.2.2. Frequency content along the surface The ILEV instability at Reo2000 reveals the importance of the frequency content of the leading edge flow, and vortices are shed from the leading edge separation-reattachment periodically at low Reynolds numbers (Hourigan et al., 2001). Fig. 8 shows estimates of the power spectral density of pressure data taken from taps at the leading edge, the trailing edge and in the recirculation region, for each body. The lack of periodicity in the spectra plotted for x¼ xr demonstrates that vortices are not shed from the leading edge periodically for Reynolds numbers of order O(104) and that the ILEV instability is suppressed for each case. As the separation angle increases, the fluctuating energy in the pressure signal increases at lower frequencies, which agrees with the lowfrequency flapping of separating-reattaching shear layers found by others (Cherry et al., 1984). Near the trailing edge, the energy of the fluctuating pressure begins to center about the wake shedding frequency; however, the strength of the vortex shedding signal is found to weaken at this location as the leading edge separation angle is increased. It is also noted that the frequency content at the base of the model is nearly identical to that near the trailing edge highlighting the relationship between the trailing edge flow and that in the recirculation region. 3.3. Recirculation region The recirculation region, also referred to as the formation region, is where the wake vortices are formed. The formation process in the recirculation region of circular cylinders has been described by Gerrard (1966). Although there exist many different criteria to define the size of this region based on the type of flow data available (Griffin, 1995), its boundaries are most suitably defined by the time-averaged separating streamlines from the trailing edge when full flow field data are available (Balachandar et al., 1997). These streamlines converge in the wake creating a saddle point and closing the recirculation region (Balachandar et al., 1997; Taylor et al., 2011). The length of the recirculation region, LR, (shown in Table 1) is determined as the distance from the trailing edge to the saddle point in the time-averaged velocity field, which is accurate to within one PIV grid point. 3.3.1. Vortex identification To provide an overview of the phase averaged results, two samples of the phase averaged vorticity field are shown in Fig. 9 at every second phase, for θ¼01 and 901. The strength of both the vorticity and the pressure are observed to be markedly different in these two cases, and the following section provides quantitative analysis of the vortex characteristics. In order to quantify features of the trailing edge vortices, it is first necessary to establish a robust vortex identification technique. Chong et al. (1990) suggest a criterion for identification based on the second eigenvalue of the three-dimensional velocity gradient tensor – a method which is Galilean invariant. There exist many other vortex identification schemes in three-dimensions (e.g., Jeong and Hussain, 1995; Chakraborty et al., 2005); however,

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Fig. 8. Estimates of the normalized power spectral density, Sðf Þ=s2 , using data from three different pressure taps located at: x/t¼ 1, x/t¼ 6.17, and at the base of the model (x,y,z) ¼(7t,0,0), respectively. From top left to bottom right: θ ¼01, 301, 451, 601, 751, 901 for Re¼ 5  104.

Table 1 Estimate of wake recirculation length as the function of leading edge separation angle, θ. θ [1]

LR/t ( 7 0.04)

0 30 45 60 75 90

0.91 0.95 0.95 0.98 1.02 1.05

many of these methods are equivalent when operating on twodimensional data. Thus, the calculation of swirling strength is often applied as a means of identification of vortices from PIV data (e.g., Adrian et al., 2000; Taylor et al., 2011) and is also used here. The swirling strength is defined as the imaginary portion of a complex pair of eigenvalues of the two-dimensional velocity gradient tensor (if the eigenvalues are real there is no vortex). All vortex identification methods are susceptible to falsely identifying vortex boundaries in areas of strong shear (Vernet et al., 1999); however, the swirling strength is found to satisfactorily identify the large Kármán-like vortices in the current study.

3.3.2. Vortex convection speed The convection speed of the vortex has been computed from an area average of the streamwise speed, u, within the phaseaveraged vortex and shown in Fig. 10. The magnitudes of the convection speeds found here are in reasonable agreement with previous studies (e.g., Balachandar et al., 1997; Taylor et al., 2011). Even though the convection speed of a vortex street is a

classical parameter in drag models (Bearman, 1967), there remains relatively little understanding of how vortices attain convection speed as they accelerate during their genesis. Balachandar et al. (1997) used results of a numerical experiment on a circular cylinder to show that there are distinct phases of vortex motion through the recirculation region where the vortex speeds up, then slows down to a minimum, before accelerating out of the wake. Elongated bluff bodies at high Reynolds number have shown similar behavior (Taylor et al., 2011). Due to the periodic nature of vortex shedding, the vortex shedding frequency should depend on the distance traveled by a vortex and the speed at which it travels that distance. As described previously by Gerrard (1966), it takes half of a vortex shedding cycle for a vortex to travel the length of the recirculation (formation) region. Since there is less than 15% difference in the recirculation lengths for the range of bodies examined here (Table 1), it is reasonable to assume that the observed change in the shedding frequency of nearly 40% is directly related to the speed at which a vortex passes through this region. As discussed by Griffin (1995), the length of the recirculation region, LR, can be approximated using many different techniques that yield approximately the same result. It is also worth noting that LR is timeinvariant since it is a time-averaged property. Thus, if we define an average convection speed for this purpose as the average speed in each phase of vortex growth, the shedding frequency must be related to, U c p 2LR f ¼ m2LR f

ð1Þ

In Eq. (1), the overbar denotes an average over all phases of the vortex shedding process and the constant of proportionality, m, should be of order O(1) due to the required relationship between speed, time and distance that the vortices travel in this region, as discussed above. From Fig. 10 it is clear that dUc/dta0, which,

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Fig. 9. Vorticity fields through the shedding cycle ϕ A ½0; 2π at intervals of π/4 for: (a) θ¼ 01 and (b) θ ¼901 at Re¼ 5  104. Open markers (○) are suction pressure coefficients plotted normal to the surface at 3 times the geometric scale, and lines are for ease of visualization only.

Fig. 10. Phase averaged convection speed of a vortex (yo 0) through all phases ϕ A ½0; 2π as a function of leading edge separation angle, θ, for Re ¼5  104.

Fig. 11. Schematic demonstrating the induced (Ui), outer (Uo) and convection (Uc) velocities.

two-dimensional vortex can be approximated as from the dispersion relation Uc ¼ λf, implies that the wavelength is changing in this region before attaining a relatively constant value in the developed wake. Yet, the mechanisms driving these changes in the convection speed through the growth process remain unknown. The convection, or bulk, speed of a vortex is properly defined as the spatial average of the vortex. Thus, if one assumes that the streamwise velocity at the left and right of the vortex are negligible, the resultant streamwise convection speed of a

1 Uo þ Ui U c ¼ ∬ udA  : A 2

ð2Þ

Uo is the speed of the flow outside of the recirculation region (which is distinct from the free-stream flow speed, U1) and Ui the speed of the induced flow between the two vortices within the recirculation region, shown schematically in Fig. 11. This expression is analogous to that of Bernal and Roshko (1986) who showed that the convection speed of vortices formed in a mixing layer is

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Fig. 12. The ‘outer’ velocity measured just outside of the recirculation region (y/t ¼  0.57), Uo, is plotted through all phases ϕ A ½0; 2πÞ as a function of leading edge separation angle, θ, (a) as measured for Re¼ 5  104, and (b) that predicted by Eq. (2) using the measured induced and convection velocities.

the average of the streamwise velocity of the two streams that form the mixing layer. The interface between the recirculation region and the ‘outer’ flow (i.e., the flow outside of the recirculation region) was conceptualized as a vortex sheet by Roshko (1954) and this formed the basis for the parameterization of vortex shedding from shorter bluff bodies. We have previously shown that for elongated bluff bodies the interface is not thin and should not be conceptualized as a thin shear layer due to the significant amount of turbulence generated from leading edge separation-reattachment which is present at the trailing edge separation (Taylor et al., 2011). Nevertheless, a significant portion of the convection speed of a vortex must be determined by the state of the outer velocity, which is shown in Fig. 12. The outer velocity is determined by taking a constant vertical position (y/t ¼  0.57) and by following the horizontal position of the vortex center. The observed speed-up (Uo/U1 41) for bodies with θ ¼01 and 301 near the end of the shedding cycle shows similarities to shorter bluff bodies where Roshko (1954) predicts speed-up along the separated shear layers due to strong base suction in the recirculation region. Note that the speed-up in the shear layer was initially observed in the vortex sheet measurements of Fage and Johansen (1928) who found that the speed-up decreased with increasing streamwise body length. In the current study, the outer velocity is observed to decrease as the leading edge separation angle is increased for a fixed body length. Since the ILEV instability is suppressed at these Reynolds numbers (Fig. 8) and vortices are not shed periodically from the leading edge, it is plausible to assume that the turbulence created by the leading edge separation-reattachment (quantified in Taylor et al., 2011) acts to disturb the vortex formation in the near wake. To measure the ‘induced’ velocity between the pair of vortices in the recirculation region, the data are sampled just outside of the phase-averaged vortex radius and plotted in Fig. 13. To confirm Eq. (2), the ‘outer’ velocity is predicted using the convection speed from Fig. 10 and the ‘induced’ velocity of Fig. 13, and the result is plotted in Fig. 12(b). The good agreement implies that, to the firstorder approximation, the convection speed through all of the phases of vortex growth in the recirculation region can be determined by the average, in each phase, of the ‘induced’ and ‘outer’ velocities as described by Eq. (2) and shown in Fig. 11.

4. Discussion Using the convection speed (Fig. 10) and the length of the recirculation region (Table 1), a constant of proportionality of m ¼0.9 is determined for Eq. (1). The value of the ‘induced’

Fig. 13. Speed induced by the opposing vortex, Ui, is plotted through all phases ϕ A ½0; 2πÞ as measured for Re¼5.0  104.

Fig. 14. Strouhal number variation with leading edge separation angle, θ.

velocity when averaged across all phases is U i ¼  0:1U 1 , which is common across all of the different leading edge geometries in the current study (Fig. 13). To determine the value of the outer velocity, U o , averaged over all phases, Fig. 12 can be used. These two velocities are averaged to estimate the shedding frequency via Eq. (1), as plotted in Fig. 14. The agreement between the approximation of Eq. (1) and the measured Strouhal numbers is generally good except in the case of θ ¼301. In the work of Roshko (1954) on a universal Strouhal number (StRn) he suggested that the diameter, or cross-stream dimension, of the body is not the correct length scale for shorter bluff bodies; rather, he agreed with the findings of Fage and Johansen (1928) that the distance between the shear layers, d0 , is a more appropriate length scale. Roshko's (1954) contribution was to incorporate the velocity on the free streamline in the wake, Us, such that StRn ¼St(U1/Us)(d0 /t). Roshko (1954) related the free streamline

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pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi velocity, Us, with the base pressure through U s ¼ U 1 1  C pb where C pb is the base pressure coefficient. Even though this length scale proved better at collapsing the data, a universal Strouhal number remains elusive. The current study on elongated bluff bodies highlights one of the major obstacles in the pursuit of a universal Strouhal number – the state of the separated shear layers is important. Roshko (1954) anticipated this possibility with his insistence on the importance of boundary layer history and the potential pitfalls of his universal Strouhal number when applied to bodies where the separating shear layers are not thin. Another notable attempt at a universal Strouhal number was made by Gerrard (1966) who not only used the size of the recirculation region but also an additional length scale, which he referred to as the ‘diffusion length’. The notion that he conveys with his ‘diffusion length’ is the importance of the state of the trailing edge boundary layer. In the search for a universal Strouhal number, the present study suggests to balance between the approaches of Roshko (1954) and Gerrard (1966). It is also important to distinguish between two similar assumptions made by Fage and Johansen (1928) and Roshko (1954). For shorter bluff bodies, the difference between the distance d0 of Roshko (1954) and the height between the edges of the shear layers (Fage and Johansen, 1928) is negligible. However, the distance d0 specified by Roshko (1954) referred to the distance between the free streamlines (i.e., those separating from the body at the trailing edge). For elongated bluff bodies, this distance is d0 /t  1 since the flow separates at each of the trailing edge corners (Fig. 9); yet, the thickness of the separated shear layers can be substantially greater than d0 /t  1 due to the leading edge separation-reattachment (Taylor et al., 2011; Taylor et al., 2013). Furthermore, the levels and scales of turbulence generated by the leading edge separation-reattachment change the nature of the separated shear layers considerably from the case of shorter of bluff bodies. The use of Roshko's definition (i.e., d0 /t 1) implies that the average value of the outer velocity, U o , in the present study is analogous to the speed along the free streamline, Us. Averaging across the six different bodies tested in the present study, the ‘universal’ Strouhal number found by setting d0 /t 1 and replacing Us with U o is Stn ¼0.25. This value is significantly higher than the range of StRn  0.15–0.16 found by Roshko (1954). Using the average value of Stn ¼0.25 it is possible to anticipate the shedding frequency of each body using St ¼ St n U o =U 1 , and the results are added to Fig. 14. The agreement with the measured values is observed to be reasonable for many of the cases. Furthermore, the current pressure data were used to verify the possibility of computing Us through the base pressure coefficient; however, this relationship predicted speeds along the separating streamline that were greater than the free stream speed for cases where this was not observed. Therefore, the problem remains of determining U o . The determination of the outer velocity is typically more prohibitive and more experimentally challenging than simply measuring the shedding frequency directly. Therefore, it is suggested that future studies focus on a ‘diffusion length’ approach (Gerrard, 1966) to anticipate U o through characterization of the state of the separated shear layers at the trailing edge as a function of the leading edge separation-reattachment.

5. Conclusions Measurements have been performed on six different leading edge geometries for the elongation ratio c/t ¼7. These measurements include surface pressure data synchronized at 500 Hz with PIV measurements of the recirculation region of each body. The synchronized pressure and the PIV data were phase averaged so

that the evolution of the vortex shedding process in the wake could be assessed. The frequency of the vortex shedding was observed to decrease linearly, and the reattachment length to increase monotonically, with increasing leading edge separation angle. The pressure data along the surface clearly demonstrate that the ILEV instability is suppressed for each of the bodies in the current study. Without the forcing of the ILEV instability, we have demonstrated that the periodic vortex shedding is the result of trailing edge vortex shedding. However, the trailing edge vortex shedding is strongly influenced by the leading edge separation-reattachment as demonstrated by the significant changes in the shedding frequency. The variation in the shedding frequency was shown to be due to the convection speed of vortices in the recirculation region since the length of the recirculation region is similar in each case. The convection speed variation is suggested to be due to two main features: (i) the velocity induced between the vortices in the recirculation region, and (ii) the outer velocity (i.e., that measured just outside of the recirculation region). Since the induced velocity was not observed to vary significantly, it was shown that the shedding frequency was most affected by changes in the outer velocity. The dependence on the outer velocity highlights the importance of the leading edge separation-reattachment. The magnitude of the outer velocity was observed to decrease markedly as the size of the leading edge separation-reattachment increased due to changes in the leading edge geometry of each forebody. Thus, it is suggested that future studies examine the specific factors leading to decreases in the outer velocity. These factors are expected to include increased turbulent kinetic energy and three-dimensionality of the trailing edge flow as the leading edge separation angle is increased.

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