Characterization of the velocity fluctuations in the wake of a triangular prism of moderate aspect-ratio

Characterization of the velocity fluctuations in the wake of a triangular prism of moderate aspect-ratio Guido Buresti, Giacomo Valerio Iungo Department of Aerospace Engineering, University of Pisa, Italy E.mail: g.buresti@ing.unipi.it, giacomo.iungo@ing.unipi.it Keywords: Bluff body aerodynamics, wakes, flow fluctuations, wavelet-hilbert procedures. SUMMARY: The velocity fluctuations experimentally detected in the wake of a prism having triangular cross-section and an aspect ratio h/w = 3. are characterized. It is shown that, besides the fluctuations induced by an alternate vortex shedding with Strouhal number St.15, further components at St.5 and St.9 are present, with different relative intensities in different wake regions. The single components are extracted from the velocity signals through procedures based on the wavelet and Hilbert transforms, and their time-varying amplitudes are statistically analysed. All the extracted components are found to be highly modulated, but with no clear evidence of a positive or negative correlation between the fluctuations of their amplitudes. This indicates that the various components may be connected with the dynamics of different flow structures, and suggestions in this regard are given, using also the information obtained from previously available numerical visualizations of the vorticity field. 1. INTRODUCTION The prediction of the unsteady aerodynamic loads on three-dimensional bluff bodies is of interest for many engineering applications, as, for instance, the design of buildings under the action of the wind. These unsteady loads are produced by velocity fluctuations connected with the dynamics of the vorticity structures present in the wake. Therefore, in order to achieve a satisfactory predictive capability of the aerodynamic loads on these types of bodies, the present understanding of wake flows must be increased through an accurate analysis of the velocity fluctuations and of their connection with the dynamic behaviour of the vorticity structures. In particular, considering finite cylinders and prisms in cross-wind, the wakes are dominated by the velocity fluctuations induced by the alternate vortex shedding from the body sides, but further fluctuations at different frequencies may appear in the upper region of the wake, which are connected with the dynamics of the vorticity structures originated by the flow passing over the body free-end; these structures are considerably influenced by the shape of the body cross-section and by the wind orientation. In previous investigations (Buresti et al. 1998, Buresti & Lombardi 2, 22, 23) the wake flow fluctuations due to vortex shedding were characterized experimentally for the case of triangular prisms of moderate aspect-ratio having two different isosceles cross-sections and placed vertically on a plane with the wind normal to the base of the cross-section, or orientated against its apex edge. Four values of the aspect-ratio (i.e. the ratio between the model height, h, and the width of its cross-section base, w) were analysed, viz. h/w = 1, 1.5, 2, 3. Alternate vortex shedding from the lateral sides of the models was found to exist for all aspect ratios, even if a decrease in its regularity with decreasing aspect-ratio was apparent, particularly when the incoming flow was directed against the apex edge of the prisms. For this wind orientation, the presence of a couple of counter-rotating stream-wise vortices over the freeends of the bodies was ascertained, and further fluctuations were also found at lower frequencies than those of the lateral vortex shedding. In particular, for an equilateral triangular prism a fluctuation at approximately 1/3 of the vortex shedding frequency was detected above and aside

the upper wake. This finding seems to be in agreement with what described by Kitagawa et al. (1997), who found velocity fluctuations with a dominating frequency at 1/3 of the usual vortex shedding frequency in the upper portions of the wake of a finite circular cylinder with h/d = 25; these fluctuations produced significant cross-flow oscillations when their frequency coincided with the natural one of the cylinder. A low-frequency peak in the velocity spectra was also detected by Park & Lee (2) aside the upper wake of finite circular cylinders; the frequency of this peak was found to be constant for models with different aspect-ratios (h/d = 6, 1 and 13), in spite of the fact that they were characterized by values of the vortex shedding frequency that increased with aspectratio. On both the latter articles it was suggested that this low frequency might be related to oscillations of the tip vortices. A numerical simulation using a LES code was then carried out by Camarri et al. (24) to study the velocity and vorticity field in the wake of an equilateral triangular prism with aspectratio h/w = 3, positioned with its apex edge against the incoming stream. The simulation provided values of the rms wake fluctuations that were in good agreement with those obtained from the hotwire measurements of Buresti & Lombardi (23). Furthermore, it allowed the shape of the upper boundary of the near wake to be characterized, giving useful indications on the dynamics of the vorticity structures originating from the free-end of the body and on their possible connection with the wake fluctuating flow field. In particular, it was ascertained that the vorticity sheets shed from all the edges of the prism interact in a non-trivial way to give rise to a complex topology, with striking differences in the positions of the middle and lateral portions of the upper wake boundary. Furthermore, the low-frequency fluctuations were shown to be associated with a vertical, in-phase, oscillation of the vorticity structures detaching from the free-end, and the possibly important role of the vorticity component in the lateral direction being shed from the rear edge of the free-end was pointed out. These numerical results are invaluable for the interpretation of the experimental data and may serve as a guide for new experimental campaigns. In the present paper the experimental wake velocity signals for the same configuration studied numerically by Camarri et al. (24) are analysed through time-frequency processing techniques based on the wavelet and Hilbert transforms, which allow the signal components associated with the most significant frequencies present in the fluctuations to be extracted and analysed. In particular, the regions where the various components are predominant will be described, and the time-variation of the frequency and amplitude of each extracted component will be characterized through the analysis of its associated complex analytic signal. 2 EXPERIMENTAL AND ANALYSIS PROCEDURES 2.1. Experimental set-up The tests were carried out in the subsonic wind tunnel of the Department of Aerospace Engineering of the University of Pisa, which has a free-stream turbulence level of.9%. The experimental set-up is sketched in Fig. 1, where the used reference frame is also shown. The tested model has an equilateral triangular cross-section, an aspect ratio h/w = 3 (with a base width w = 9 mm), and is positioned vertically on a plane, where the boundary layer thickness is below 1 mm, and may thus confidently be assumed not to affect the flow in the upper wake region. The tests were carried out at a Reynolds number Re = Uw/ = 1.5 15. Time signals of the velocity fluctuations were obtained through hot-wire anemometry, using single-wire probes placed in different parts of the flow, inside and outside the wake, for stream-wise positions x/w 4.. The hot-wire signals consisted of 32768 samples acquired at 2 Hz; 5 measurements were carried out in each position. All the velocity signals analysed in the following were acquired in positions where no hot-wire rectification could occur, as was ascertained through the comparison between the experimental signals and the numerical ones described in detail in Camarri et al. (24).

Fig. 1 - Experimental lay-out 2.2. Signal analysis The considered wake flow is characterized by the presence of several dominating components in the fluctuations, as may be derived from the analysis of the various frequency peaks present in the spectra of the velocity signals in different regions. However, the complexity of the flow is such that these components are characterized by different levels of regularity and by significant modulations, both in amplitude and frequency; this is due to the fact that the observed fluctuations may be associated with the passage or the oscillation of vorticity structures, whose size, shape and velocity, even when they have clearly dominating features, are never exactly constant. It is then quite understandable that in these conditions the analysis of the wake velocity signals using only conventional Fourier spectral methods becomes largely insufficient, and time-frequency processing techniques should be used. In the present work the experimental velocity signals are analysed by means of a wavelet- Hilbert procedure, introduced and described in detail in Buresti et al. (24), and whose main aspects will be now summarized. As basic time-frequency analysis technique, allowing a rapid assessment of the time variation of the energy associated with the different frequencies present in the signals, the continuous wavelet transform is used, with a complex Morlet wavelet (t) = e i o t e t2 /2 (1) If the central frequency parameter o 6., this wavelet is admissible and progressive (i.e. with ˆ ( ) = for <, where ˆ ( ) is the Fourier transform of (t)), and was chosen because of its versatility; in particular, its frequency resolution may easily be varied by changing o. In the following this wavelet transform is also advantageously used to produce frequency power spectra, which are analogous but smoother than Fourier spectra, and provide a first assessment of the main frequency components present in the signal. In order to extract the single significant frequency components, the wavelet maps, produced with a high value of the wavelet central frequency (e.g. o = 6 ) in order to increase the frequency resolution and thus to reduce the interference effects between adjacent components, are then filtered, by neglecting the wavelet coefficients outside a band around the dominating spectral peak frequency, and an inverse transform procedure is applied. The extracted signal may then be subtracted from the original one, and the procedure may be repeated until all the detectable components are obtained. The resulting extracted components are narrow-band signals whose

modulations in amplitude and frequency may then be characterized through a demodulation technique based on the Hilbert transform, which is now briefly recalled. To any time-varying real signal x(t) it may be associated a complex analytic signal (i.e. with zero Fourier components for negative frequencies) in the following form: Z x (t) = A x (t) e i x (t) = x(t)+ i x H (t) (2) where x H (t) is the Hilbert transform of x(t), so that the real and imaginary parts are orthogonal. The instantaneous frequency of x(t) may now be defined as x (t) = 1 2 d x (t) (3) dt and this definition (always permissible from a mathematical point of view) acquires a sensible physical meaning if the analysed signal is an asymptotic one, i.e. a sinusoidal signal that is slowlymodulated in amplitude and frequency, x(t) = A(t) cos[ (t) t + o ]. In this case, provided the modulation frequencies are sufficiently lower than the fundamental frequency, the time variations of the signal amplitude and frequency may directly be recovered from the time variations of the modulus and of the phase derivative of the associated analytic signal. This technique is applied to all the extracted fluctuating velocity components, in order to obtain their mean frequency values and to statistically characterize their time-varying amplitude through the analysis of the modulus of their associated analytical signals. As pointed out in Buresti et al. (24), to obtain a reliable statistical analysis of the instantaneous frequency a thresholding procedure must be used (e.g. by considering only the time intervals in which the modulus of the associated analytic signal is above 75% of its average value), in order to avoid spurious large fluctuations of the instantaneous frequency. This does not apply to the extraction and analysis of the signals representing the time-variation of the modulus; however, in all cases it is advisable to discard the tails of the signals to avoid any possible end-effect inaccuracy introduced by the Hilbert transform numerical procedure (in the present case, 1 s long portions were discarded at the beginning and at the end of the signals). 3. RESULTS AND DISCUSSION The frequencies appearing in the velocity fluctuations will be described in terms of their associated Strouhal number St = fw/u, where f is the considered frequency, and U is the free-stream velocity (in the present case U = 25 ). The velocity signals acquired aside the wake are characterized by the presence of a spectral peak around St.15, which was shown in Buresti & Lombardi (23), through the analysis of the correlation between signals simultaneously acquired on the two sides of the wake, to correspond to an alternate vortex shedding from the lateral vertical edges of the prism. The regions where this frequency (which will be referred to as HF in the following) is particularly evident are those just outside the lateral wake boundary; the latter may adequately be detected by moving the hot-wire probe from high to low absolute values of y/w, and finding the positions corresponding to a local maximum in the kurtosis 1 of the velocity signals. The HF peak decreases in intensity moving up in the vertical direction and increases moving downstream until x/w 2.5, which may then be assumed to roughly correspond to the formation length of the vortices. An example of wavelet spectrum obtained in a point in which this peak dominates is shown in Fig. 2 a). 1 In the present paper the kurtosis Ku is defined as the fourth-order central moment non-dimensionalized through the standard deviation, which implies that Ku = 3 for a Gaussian process.

.12 a).4 b).9.3 Pu Pu.6.2.3.1.5.1 St.15.2.25.4 c).3 Pu.2.4.3 Pu.2.5.1 St.15.2.25 d).1.1.5.1 St.15.2.25.5.1 St.15.2.25 Fig. 2 - Examples of the wavelet spectra obtained in various regions around the wake ( o = 6 ). a) x/w = 2.5, y/w = 1., z/h =.5; b) x/w = 4., y/w =.5, z/h = 1.1; c) x/w = 2.5, y/w =, z/h =.5; d) x/w = 2.5, y/w = 2.5, z/h =.3 As already pointed out, a lower frequency at St.5 (LF in the following) was also detected in the upper part of the wake, and becomes the dominating one above z/h 1. and from x/w 1.5 to x/w = 4. (see Fig. 2 b)). Both experimental and numerical analyses (Camarri et al. 24) showed that this frequency is associated with a vertical, in-phase, oscillation of the vortical structures present in that region, i.e. both the two counter-rotating axial vortices detaching from the body free-end and the upper part of the vorticity layer that originates from the body sides and is dragged around the above-mentioned vortices. However, the examination of all the velocity signals showed the presence, in various regions inside and around the wake, of another spectral peak at an intermediate frequency around St.9 (IF in the following). This frequency becomes dominating in the symmetry plane y =, in positions corresponding to different stream-wise coordinates x/w depending on the vertical coordinates; as an example, Fig. 2c) shows the spectrum obtained at z/h =.5 for x/w = 2.5. In the following the origin of the fluctuation at this frequency will be analysed in more detail. The three above described frequencies may also be found together in the velocity spectra of signals acquired aside the wake, sufficiently far from its boundary, as can be seen in Fig. 2d). Indeed, in such positions the fluctuations due to the vortex shedding are reduced in such a way that

the peaks at the remaining frequencies become detectable. This suggests that, even if the fluctuations at the different frequencies are probably produced by different flow phenomena, and thus are most evidenced close to the regions where these phenomena take place, they all contribute to the global oscillation of the wake, which is indeed best felt further away from its boundaries. In order to better associate the lower and, specially, the intermediate frequency with the dynamics of the wake vorticity structures, it may be useful to refer to the numerical results of Camarri et al. (24), and in particular to the visualizations of the time-averaged vorticity field in the near upper wake, some of which are shown in Fig. 3. In particular, in fig 3 a) the vorticity x- component is shown, which allows the axial counter-rotating vortices detaching from the free-end to be visualized. However, particularly interesting is the visualization of the vorticity y-component being shed from the rear edge of the free-end, shown in Fig. 3 b), which forms a layer delimiting the central upper boundary of the wake, and whose intensity and shape is influenced by the velocity field induced by the axial vortices. A clearer idea of the evolution of the vorticity field may be gained through Fig. 3 c), which shows the modulus of the component lying on the y-z plane at x/w =.8. As can be seen, the upper parts of the sheets of z-vorticity shed from the sides of the prism are advected upwards around the axial vortices, and this causes them to bend, change direction and reconnect with the y-vorticity shed from the rear edge of the free-end. Fig. 3 - Numerical visualization of the upper-wake vorticity field. a) x-component; b) y-component; c) modulus of the component lying on the y-z plane at x/w =.8. (From Camarri et al. 24)

It should be pointed out that the numerical visualizations showed also that the sheet of transversal y-vorticity rapidly bends downwards inside the wake, increasing the width of its lower part with respect to the upper region, and in practice corresponds to the boundary of the recirculation region behind the body, as could be determined by the sign of the x-component of the velocity. Furthermore, from the time-history of the instantaneous images, it could be ascertained that this vorticity sheet tends to oscillate, with increasing amplitude with increasing distance from the body free-end, like a flag with a constraint at the rear edge of the body tip. If we now consider the regions where the IF fluctuations were prevalent in the experimental velocity signals, we may reasonably deduce that they are closely connected with the oscillations of this sheet of transversal vorticity. As for the axial counter-rotating vortices and the sheets of vorticity wrapped around them that can be seen in fig. 3 c), they also tend to bend downwards, but less than the above-mentioned y- vorticity sheet, and seem to completely dominate the upper part of the wake more downstream. Therefore, this suggests that their oscillation is the origin of the LF fluctuations. Having established with reasonable confidence the association between the fluctuations at the different frequencies and the dynamics of the vortical structures present in the wake, in the following the characteristics of the single fluctuations will be analysed in more detail. In Fig. 4 examples are given of 2 s long portions of the different components extracted in positions where they are dominant. In each case the modulus of the associated analytic signal is also shown, to demonstrate that it describes with satisfactory accuracy the time variation of the component s amplitude. As can be seen, in all cases the amplitude of the fluctuations varies significantly in time, in a more or less irregular way. In Table 1 the values of the average frequencies of the three components, obtained from the Hilbert analysis (with a threshold at 75% of the mean value of the modulus) of at least 4 measurements for each component, are given together with their standard deviations. HF IF LF St (Average).1516.95.498 St (Stand. Dev.).13.12.6 Table 1 - Average and standard deviation values for the three frequencies evaluated over multiple measurements. HF /m Sk Ku Average.482.486 3.244 Stand. Dev..47.324.876 IF /m Sk Ku Average.515.584 3.77 Stand. Dev..29.173.455 LF /m Sk Ku Average.527.662 3.37 Stand. Dev..34.254.856 Table 2 - Statistical central moments of the modulus of associated analytical signals for the three frequencies. The average and standard deviation values are evaluated over multiple measurements.

5 a) HF Signal HF Modulus -5 7 7.5 8 t (s) 8.5 9 8 b) IF Signal IF Modulus -8 3 3.5 4 t (s) 4.5 5.4 c) LF Signal LF Modulus -.4 11 11.5 12 t (s) 12.5 13 Fig. 4 - Examples of the extracted components and of the moduli of the corresponding analytic signals. a) x/w = 2.5, y/w = 1., z/h =.5; b) x/w = 2.5, y/w =, z/h =.5; c) x/w = 4., y/w =.5, z/h = 1.1 In Table 2 the statistical analysis of the fluctuations of the moduli of the analytic signals associated with the various frequencies is reported in terms of standard deviation, (nondimensionalized with the mean value of the modulus), skewness, Sk, and kurtosis, Ku. As can be seen, the relative amplitudes of the modulus fluctuations are similar for all the components, and also the values of the skewness and kurtosis are comparable; however, rather high values of the standard deviations of the latter parameters are found, which may be due to the fact that the fluctuations of the moduli occur at too low frequencies to allow a complete stationarity of the higher moments to be achieved. In any case, no clear differences seem to emerge from the statistical analysis of the temporal behaviour of the amplitudes of the three components. Indeed, in all cases the fluctuations of the modulus are significant, and the skewness is moderately positive, indicating a tendency to intermittent bursts of high-amplitude oscillations of all the components.

.8 a) HF Signal IF Modulus -.8 3 3.2 3.4 t (s) 3.6 3.8 4.8 b) IF Signal IF Modulus -.8 3 3.2 3.4 t (s) 3.6 3.8 4.8 c) LF Signal LF Modulus -.8 3 3.2 3.4 t (s) 3.6 3.8 4 1.2 d) Original Sum -1.2 3 3.2 3.4 t (s) 3.6 3.8 4 Fig. 5- Example of extracted components at point x/w = 2.5, y/w = 2.5, z/h =.3. a) HF component; b) IF component; c) LF component; d) comparison between the sum of the three components and the original signal

The signals for which all the three components of the velocity fluctuations were detectable (which, as already pointed out, could be obtained aside the wake, sufficiently far from its boundary) were then used to study the possible correlations between the variations of the amplitude of the single components. In other words, the aim was to ascertain if any positive or negative correlation between the increase and decrease of the amplitude of the three components could be ascertained. In Fig. 5 an example is given of the time variation of the extracted components and of their moduli for a short time interval of one of such signals, together with the comparison between the sum of the three components and the original velocity signal. As can probably be perceived from Fig. 5 d), the residual signal is composed of high frequency and verylow frequency components, which may probably be also representative, at least in part, of the basic turbulence structure of the wind-tunnel free-stream. The particular point considered in Fig. 5 corresponds to a rather low position (z/h =.3), which explains why the LF fluctuation has lower mean value of the amplitude. However, the relative fluctuations of the modulus are again comparable for the three components. The signals representing the fluctuating part of the modulus of the three extracted analytic signals were then used to evaluate their correlation coefficients,. The latter were evaluated from the available multiple signals (approximately 7) for which all the three components could be extracted; the average and the standard deviation values of the correlation coefficients thus obtained are reported in Table 3. LF - HF LF - IF IF - HF (Average).16.24 -.54 (Stand. Dev.).72.82.66 Table 3 - Correlation coefficients between fluctuating parts of the moduli of the three components extracted from the same velocity signals. As is apparent from the values of Table 3, no significant correlation may clearly be observed between the fluctuations of the moduli of the three frequency components. This result agrees with what could be inferred from the visual analysis of the time variation of the moduli, shown e.g. in Fig. 6 for the same case reported in Fig. 5, but for a slightly longer time interval. It is also clear that for most of the time the three components are simultaneously present in the velocity signals. All the above findings substantiate the deduction that the three frequencies appearing in the wake velocity fluctuations originate from the dynamics of different vorticity structures..6.4 HF Modulus IF Modulus LF Modulus.2 2.5 3 3.5 t (s) 4 4.5 Fig. 6- Time variation of the moduli of the analytic signals associated with the three fluctuating components at point x/w = 2.5, y/w = 2.5, z/h =.3

4. CONCLUSIONS In the present paper the velocity fluctuations experimentally detected in the wake of a triangular prism of aspect ratio 3. placed vertically on a plane have been characterized using time-frequency signal processing procedures based on the wavelet and Hilbert transforms. The body shape and the direction of the flow were such that two strong counter-rotating vortices detached from the freeend, and significantly influenced all the upper-wake flow field. Fluctuations at three prevailing frequencies could be singled out, with different relative intensities depending on the wake regions. In particular, the frequency connected to the alternate vortex shedding originating from the lateral vertical edges of the prism, which corresponds to a Strouhal number St.15, dominates for vertical positions below z/h =.9, and is particularly strong in the regions just outside the lateral boundary of the wake. As already noticed in previous works, the amplitude of the vortex shedding fluctuations is strongly modulated, which is not surprising due to the lower regularity of this phenomenon with respect to the two-dimensional case. Indeed, the shed vortices bend in the downstream direction in the upper part of the wake (a phenomenon already documented, e.g., in Buresti et al., 1998), and it is reasonable to infer that they are also characterized by a wavy core. So modulations both in frequency and, specially, amplitude should be expected in the velocity signals derived from fixed-position hot-wire measurements. A lower-frequency, at St.5, was found to prevail in the velocity fluctuations on the whole upper part of the wake, in particular above z/h = 1. and for x/w 1.5, and may confidently be associated, in agreement with previous suggestions, with the oscillation of the counter-rotating axial vortices detaching from the body tip, and of the sheets of vorticity wrapped around them. Even if the cause of this oscillation is probably an instability of the couple of axial vortices, the fact that the frequency of the fluctuations is only slightly less than 1/3 of the vortex shedding frequency might be not casual, as it is plausible that the latter phenomenon may cause this particular frequency to be selected among the unstable ones for the counter-rotating vortices configuration. The thorough analysis of the wake velocity signals carried out in the present work allowed also the presence of fluctuations at an intermediate frequency, around St.9, to be detected, and the region where they prevail to be described. In particular, by using as a reference the numerical results of Camarri et a. (24), it was suggested that they may be caused by a flag-like oscillation of the sheet of transversal y-vorticity originating from the rear edge of the body free-end, and approximately lying along the downstream boundary of the recirculation region in the central part of the near wake. No immediate link is apparent between the frequency of these fluctuations and those of the previously described phenomena. The amplitudes of the different components have been shown to be characterized by time variations that are of comparable relative value. However, the wavelet-hilbert analysis applied to velocity signals acquired aside the wake, in positions where all the three frequencies could be detected, did not provide any clear indication of positive or negative correlation between the fluctuations of the amplitudes of the extracted components. This substantiates the present suggestion that they are connected with different physical mechanisms, and in particular with the dynamics of different vortical structures. Acknowledgements The present work was financially supported by the Italian National Research Council, C.N.R. and by the Italian Ministry of Education, University and Research, M.I.U.R.. Thanks are also due to Valentina Peselli for her help in the analysis of the velocity signals.

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