3.1. Preliminary results and identifying vortex shedding

Mean drag force data are presented in figure 1. The open symbols are the present results obtained at $Re=65\,000$ (in the wind tunnel) and $Re=25\,000$ (in the flume). Given differences in the various parameters mentioned above, agreement with data from the literature is acceptable. As the porosity $\beta$ increases, the mean drag falls and in the region where no vortex shedding is expected the theoretical model of Steiros & Hultmark (Reference Steiros and Hultmark2018) provides a reasonable fit to the data, given the lack of any attempt to correct for blockage. However, it is not clear from these data when vortex shedding might begin to disappear or over what range of $\beta$ this transition between shedding and no shedding occurs.

In order to explore the transition region, it is necessary to identify the occurrence of shedding in an accurate way. For a solid bluff body, vortex shedding is the dominant flow structure and the spectral energy associated with it is usually at least one order of magnitude larger than the energy in other frequency components in the spectrum. Identifying vortex shedding in such a case is thus somewhat trivial and possible to do in a number of ways. Perhaps the most common approach is to use hot-wire anemometry, positioning a probe in or near one of the separated shear layers once they have begun to interact (or even outside the wake further downstream as Castro Reference Castro1971, did). Figure 5(a) shows spectra of the axial velocity obtained in this way, for three typical $\beta$ values (0.12, 0.27 and 0.36). Clear shedding peaks (with frequency $f_s$) are apparent for $\beta =0.12$ and even 0.27, from which it is straightforward to deduce the Strouhal number, defined in the usual way as $St=f_sD/U_\infty$. The results are shown in figure 5(b) for four Reynolds numbers. There is an obvious fall in $St$ with increasing $Re$, as one might expect, because transition in the separated shear layers presumably moves closer to the separation points as $Re$ increases, leading to somewhat thicker shear layers in the vortex formation region (and, thus, lower maximum vorticity there). The fall in $St$ with $Re$ (which was also noted by Castro Reference Castro1971) is relatively small however, and does not hide the clear trend with $\beta$. This shows an initial rise in $St$ up to $\beta =0.12$, although details below this $\beta$ are not known because no plates of intermediate porosity were available. Beyond $\beta =0.12$, there is a steady fall in $St$ until it became more difficult to identify any obvious main peak in the spectrum. For $\beta =0.27$, the presence of a peak is unequivocal (see figure 5a) but it is clear that the peak is both much weaker and also broader. Beyond about $\beta =0.28$ clear peaks were much more uncertain, so $St$ values shown in that range are subject to significant uncertainty. Again, the results are broadly in line with those of Castro (Reference Castro1971).

Figure 5.

(a) Power spectral density (PSD) of the axial velocity within one of the shear layers, for three different values of $\beta$. (b) Strouhal number for all plates and for four different Reynolds numbers. The data are from the wind tunnel and are scattered in the grey region, because of the absence of a clear spectral peak.

An alternative means of deducing shedding frequency is to examine the time series of the drag force. For $\beta =0$, the drag signal spectrum shows a clear oscillating pattern associated with vortex shedding, reflected in a well-defined peak in the drag spectrum, as shown in figure 6. The peak frequency is twice the Strouhal frequency, because two vortices are shed per cycle, one from each side of the plate. The lift force often also provides a direct measure of the vortex shedding frequency for bluff bodies but in this case the plates are very thin, so they produce negligible lift forces when compared with drag. It is interesting that when $\beta \neq 0$, the drag spectra show little evidence of a shedding peak even in the range of $\beta$ where Kármán shedding definitely occurs (as evidenced not least by the velocity spectra discussed above). Recall that a Kármán vortex street can only form once the two separated shear layers can interact. For a solid plate, this happens around the end of the mean flow recirculating ‘bubble’ behind the plate. Incidentally, it is recognised that no drag peak would be evident if the shedding were oblique. However, such shedding usually occurs at very low Reynolds numbers (in laminar conditions) and for bodies where the separation points are not fixed by the geometry – e.g. in the classic case of a circular cylinder. Such oblique shedding has never (to our knowledge) been observed from sharp-edged bodies at high Reynolds numbers so, although we did not obtain spanwise correlation data, we are confident that oblique shedding does not occur in the present cases.

Figure 6.

(a) Power spectral density of the drag force measured on porous plates in the flume for different values of $\beta$. The only visible peak occurs for the solid plate ($\beta =0$). (b) Mean streamlines for the plates $\beta =0$ and 0.16. $Re=25\,000$. Red lines emphasise the streamlines surrounding the recirculation bubbles.

Figure 6(b) shows the mean streamlines behind the plate, deduced from the PIV data, for $\beta =0$ and 0.16. In the latter case, the mean flow bubble has moved downstream, with its closure occurring nearly twice as far from the plate as it does for $\beta =0$. It seems that once there is a significant mass flux through the plate, the region where the Kármán vortices are formed is sufficiently farther downstream such that the fluctuating pressure signal associated with their formation is relatively weak at the plate location. Perhaps this weakening is also partly a result of the flow fields generated by the numerous individual, small-scale jets of fluid resulting from the porosity. These may reduce the possibility that the pressure fluctuations generated by the vortex formation process can be transmitted to the back of the plate in the way that they are when $\beta =0$, not least perhaps because of a feedback interference between the instantaneous pressure at the back of the plate and the flow through the holes. (Note again that no tests were done for $0<\beta <0.12$; it seems very likely that shedding peaks would occur in the fluctuating drag signal over some lower part of that range.) It should also be mentioned that, as noted above, vortex shedding becomes increasingly weaker as $\beta$ increases, because the shear layers are significantly thicker by the time they can interact, so that the maximum mean velocity gradient and, thus, vorticity within them is smaller.

Mean streamlines are shown in figure 7 for $Re=25\,000$ and six values of $\beta$. It is clear that the recirculation bubble moves further downstream as the porosity increases, reducing its size monotonically for $\beta \geq 0.16$ until it fades away somewhere between $\beta =0.28$ and 0.33. The bubble height ($h$) and length for the solid plate agree with the values observed in the literature (e.g. Roshko Reference Roshko1954; Castro Reference Castro1971). For $\beta \neq 0$, the figures also confirm the trends for the vertical distance between the shear layers ($H$) found by Steiros et al. (Reference Steiros, Bempedelis and Ding2021). As the porosity increases, both $h$ and $H$ decrease up to the point where the bubble disappears beyond $\beta \approx 0.28$.

Figure 7.

Mean streamlines for different values of $\beta$. Background colour plots indicate the streamwise component $u$ (parallel to $U_\infty$) with blue and red positive and negative values, respectively; white arrows indicate the streamlines and yellow lines emphasise those bounding the recirculation bubbles. $Re=25\,000$. Results are shown for (a) $\beta =0$, (b) $\beta =0.12$, (c) $\beta =0.16$, (d) $\beta =0.21$, (e) $\beta =0.28$, ( f) $\beta =0.33$.

It is classically argued (e.g. Bearman Reference Bearman1967; Leal & Acrivos Reference Leal and Acrivos1969) that a recirculating bubble is formed when the entrainment needs of the two separated shear layers can only be met by some extra fluid that has to be provided from further downstream – hence, the necessary ‘closure’ of the two separating mean streamlines, with some of the fluid between the two shear layers thus having negative mean axial velocity, having been injected from around the rear stagnation point. When there is sufficient bleed fluid through the porous plate to provide the shear layers’ entrainment needs, there is no need for further fluid to be added to the near wake (from further downstream) so that a recirculating region does not form. It seems likely that at that stage the shear layers are sufficiently diffuse to prevent the usual interaction between them leading to Kármán shedding. But it is not really clear if such shedding can still occur without the bubble or, indeed, whether shedding always occurs if there is a bubble. More recent analyses, like that of Steiros et al. (Reference Steiros, Bempedelis and Ding2021), generally assume the flow is steady, so make no direct link between the existence of a bubble and the Kármán shedding process. In fact, there must be a specific porosity marking the boundary between bubble and no-bubble appearance, whereas such a firm boundary between shedding and no-shedding seems less likely. It is therefore important to study the major unsteady features of the wake in more detail and this is addressed in § 3.2.

First, however, given that it is well known that the energy in the fluctuating motions in the wake (particularly the cross-stream component) is largest in the region of the vortex formation – around the mean streamline closure marking the end of the bubble – we show in figure 8 the flow fluctuations on the wake centreline ($y=0$). These data are derived from the PIV measurements. Figures 8(a) to 8(e) show the spectra of the fluctuating velocity ($v'$) along the $x$ direction for different values of $\beta$, whilst figure 8( f) shows the fluctuating energy level, measured as $(\overline {u'^2} + \overline {v'^2})/U_\infty ^2$, in the region up to about $x/D=20$. A dominant frequency is clear in most cases, with $fD/U_\infty \approx 0.15$, although close inspection shows that this Strouhal number is not fixed but decreases a little with increasing $\beta$, consistent with figure 5(b). Note that the first appearance of a clear peak in the spectrum occurs at increasing $x/D$ as $\beta$ increases; even at $\beta =0.12$ a peak does not appear until around $x/D=1.0$. This seems consistent with the fact that peaks in drag spectra do not appear once $\beta \ge 0.12$, for the possible reasons discussed above – i.e. the shedding is not initiated until further downstream once $\beta >0$ and the flows through the holes in the plate might be expected to influence the fluctuating pressure field.

Figure 8.

(a–e) Spectra of the $v'$ component along the centreline ($y/D=0$). The colour bar for the spectral plots indicates PSD levels with an arbitrary scale, running from the highest (red) to the lowest (yellow/white) level, for which the frequency resolution is $\Delta f D/U_\infty \approx 0.007$. ( f) Fluctuation energy along the centreline; solid circles indicate the $x/D$ location of the downstream end of the bubble, if it exists – see figure 7 – and, likewise, these locations are shown by the vertical dashed lines in (a–e). All data are from the PIV measurements. Results are shown for (a) $\beta =0.12$, (b) $\beta =0.16$, (c) $\beta =0.21$, (d) $\beta =0.28$, (e) $\beta =0.33$. ( f) Fluctuating energy on $y=0$.

Roshko (Reference Roshko1954) proposed a universal Strouhal number based on the maximum distance between the separated shear layers (i.e. the wake width, $H$) and the velocity just outside the shear layers at the separation points. The latter is related to the difference between the free-stream velocity and the centreline values near the plate and clearly decreases with increasing $\beta$. The wake width is related to the thickness of the shear layers (the ‘diffusion’ length, as argued by Gerrard Reference Gerrard1966). Figure 7 shows that this wake width $H$ also decreases with $\beta$, so for a constant value for the universal Strouhal number (about 0.16 according to Roshko Reference Roshko1954), only small changes in shedding frequency $f_s$ might be expected – again consistent with the present data. The frequency signature for $\beta =0.33$ suggests a much broader spectral peak, making it difficult to identify any particular peak, as discussed earlier in the context of the hot-wire data.

The centreline energy levels shown in figure 8( f) have, at least for the lower values of $\beta$, a clear peak around the location of the end of the bubble in each case (cf. figure 7), consistent with that being the region of vortex formation, as observed for solid bluff bodies (Szepessy & Bearman Reference Szepessy and Bearman1992). There is a relatively rapid fall in peak energy, particularly beyond $\beta \approx 0.16$, with more than an order of magnitude difference in the peak values at $\beta =0$ and 0.33. Based on these data one might speculate that Kármán shedding ceases beyond about $\beta =0.2$ but, again, details of any transition region are unclear so, in the next section, we explore this further by using coherence and phase measurement within the wake.

3.2. Using coherence data to diagnose vortex shedding

When regular vortex shedding occurs, a measurement of the coherence in the axial velocity between two locations on opposite sides of the wake will yield near-unity coherence with a $180^\circ$ phase difference at the shedding frequency (because vortices are shed alternately from each side). In the laminar flow regime (with a Reynolds number of, say, around 100) the vortex street continues regularly all the way downstream from its inception; there must then also be a perfect coherence at twice the shedding frequency, but with zero phase difference. Only the action of diffusion will eventually weaken the coherence at the higher harmonics. In the present case of high Reynolds number, for which the flows are always fully turbulent, these higher harmonic coherence peaks might be expected to weaken considerably. Even at the shedding frequency, there will not be perfect coherence because the background turbulence inevitably broadens the spectrum somewhat – there is ‘jitter’ in the signals.

Two hot-wire probes were used for the present measurements, each placed around the centre of the two shear layers at the same distance downstream, as shown in figure 9. The data presented below were obtained for an averaging period of 240 s (typically equivalent, at $Re=26\,000$, to some 3200 shedding cycles in cases where Kármán shedding occurred), with the probes located at $x/D=6.2$ and $y/D=\pm 0.8$. It was found that changes in these locations had little effect on the resulting data.

Figure 9.

Position of the probes for the wind tunnel experiments with respect to the recirculation bubbles for several values of $\beta$. Black dots at $x/D=6.2$ show the probe locations. The recirculation bubbles are indicated by the bounding mean streamlines, as identified from the PIV data shown in figure 7.

We denote the coherence by $C_{u_1u_2}$ and the corresponding phase by $\phi$, determined in the usual way from the cross-spectra between the two signals $u'_1$ and $u'_2$. The plots in figure 10 show the results for four plates in the range $0.1\le \beta \le 0.33$. Coherence values close to unity indicate a high level of synchronisation whereas lower values indicate weaker synchronisation. For $\beta =0.1$ and 0.19, there is strong coherence at $f_sD/U_\infty \approx 0.14$, with a phase shift of $\phi \approx 180^\circ$, indicating that the two signals are closely in anti-phase, consistent with classical vortex shedding. For $\beta =0.24$, a similar peak appears at about the same frequency, with similar values of $\phi$, also consistent with vortex shedding, but there is no sign of a second peak at the frequency of the first harmonic. A second peak, the first harmonic, does occur for $\beta =0.1$ and $\beta =0.19$ with a phase shift close to $0^\circ$, indicating the two signals are in-phase at this frequency, as expected. In both these cases there is also a hint of the second harmonic, at three times the shedding frequency. For all the other frequencies, the coherence value is near zero and the phase shift shows scattered values, indicating a lack of coherence, caused entirely by the turbulence fluctuations at each location. For $\beta =0.33$, figure 10(d), there is no sign of any peak.

Figure 10.

Coherence between two velocity signals at each shear layer ($C_{u_1u_2}$) and its phase shift ($\phi$) for four different porosity levels $\beta$. $Re=65\,000$. Results are shown for (a) $\beta =0.10$, (b) $\beta =0.19$, (c) $\beta =0.24$, (d) $\beta =0.33$.

To provide further insight, we have evaluated spectral proper orthogonal decompositions (SPODs) from the PIV fields acquired in the second field of view (FoV) (figure 4b), from $5.5< x/D<13$. Details about SPOD and the algorithm used can be found in Towne, Schmidt & Colonius (Reference Towne, Schmidt and Colonius2018). Each data set had 3000 fields, comprising a time scale equivalent to at least 200 periods of vortex shedding. Eight SPOD modes have been computed for each plate. Figures 11(a) and 11(c) show the SPOD energy levels of each mode versus frequency for two cases, $\beta =0.16$ and 0.21. Only the first four modes are shown because the energy contents of the other four are negligible. In both cases the first mode is more energetic and contains the only peaks observable. Interestingly, the energy level plots for the first mode have similarities to the coherence plots, suggesting that they are related to the same physical mechanism in the flow. The first peak is clearly associated with vortex shedding, as the mode shapes in figures 11(b) and 11(d) illustrate. The pair of lobes with alternate velocity fluctuation levels are consistent with the pair of vortices of a typical von Kármán street and are also consistent with the phase shift $\phi \approx 180^\circ$ found at the first peak in figure 10. As for the second peak, the shapes reveal that it constitutes a harmonic of the vortex shedding. The signal of the velocity fluctuations at $y/D=\pm 0.8$ are in-phase, explaining the phase shift of $\phi \approx 0^\circ$ observed for the second peak.

Figure 11.

Spectral POD at $Re=35\,000$ for (a,b) $\beta = 0.16$ and (c,d) $\beta = 0.21$. The mode shapes in (b,d) correspond to the frequencies identified and marked with symbols in (a,c).

The key question is to understand why the second peak appears only if $\beta$ is small enough. We noted above that this second peak can only appear if vortex shedding is sufficiently regular (by which we mean, continuous in time). If, for whatever reason, the vortex shedding process is intermittent and, thus, does not continue for long enough periods of time, then the first peak will be weaker and the second peak (the harmonic) is rather less likely to appear – it may simply become submerged in the background spectral energy. Figure 12 shows examples of typical instantaneous axial velocity and spanwise vorticity fields for three cases, taken from the PIV data. Figure 12(a,d) ($\beta =0.12$) shows a pattern indicative of periodic vortex shedding and typical of a process yielding a coherence plot with two peaks. In figure 12(b,e) ($\beta =0.26$) the patterns show evidence of some shear layer interactions but without evidence of the usual shedding structure at that particular time. The coherence plot for this case, however, shows one peak, suggesting that shedding does occur from time to time; figure 10(c) shows the coherence for a case with a very similar $\beta$. Figure 12(c, f) ($\beta =0.33$) shows a case completely devoid of shedding, so that the coherence plot has no peaks at all – recall figure 10(d) for a coherence plot of a closely similar case.

Figure 12.

Three instantaneous fields of the total mean velocity (a–c) and vorticity $\omega$ (d–f), characteristic of the three states observed for $0\leq \beta \leq 0.35$: (a,d) regular and periodic vortex shedding, (b,e) irregular vortex shedding and (c, f) no vortex shedding. Results are shown for (a,d) $\beta =0.12$, (b,e) $\beta =0.26$, (c, f) $\beta =0.33$.

3.4. The steady wake at large $\beta$

Given that once $\beta$ exceeds around 0.28 no Kármán vortex shedding appears, it is pertinent to consider to what extent the wake resembles classical self-similar wakes far downstream of two-dimensional bluff bodies. We therefore present some wake data for the $\beta =0.33$ case, deduced from the PIV data that covers the wake region up to $x/D\approx 20$. Figure 16 shows cross-stream axial velocity and turbulence energy profiles, normalised in the standard way using $u_o$, the difference between the free-stream velocity ($U_\infty$) and the velocity on the wake centreline, and $L$, the width of the wake measured as (one half of) the distance between the points where the velocity difference is one half of its maximum. Self-similarity for a two-dimensional wake requires that

(3.1a,b)\begin{equation} \frac{U_\infty^2}{u_o^2}=A\left(\frac{x-x_o}{2D}\right) \quad\textrm{and} \quad \frac{L^2}{D^2}=B\left(\frac{x-x_o}{2D}\right), \end{equation}

where $A$ and $B$ are constants. It is well known that even a long way downstream of the wake generating body, although similarity in this form usually exists, there is wide variation in what ideally would be universal parameters. For example, the wake spreading rate, $\alpha =({U_\infty }/{u_o})({{\rm d}L}/{{\rm d}\kern0.7pt x})$, can vary widely depending on the body shape – initial conditions are not ‘lost’, at least at the distances that can normally be covered in experiments or computations. In the present case one would not really expect good similarity in the relatively short and early part of the wake that our experiments covered. Nonetheless, estimating $L/D$ and $u_o/U_\infty$ from the raw profiles led to reasonable collapse of the normalised axial velocity profiles, as seen in figure 16(a). The spreading rate can be expressed as $\sqrt {AB}/4$ that, using data from the best fit line to $L^2/D^2$ vs $x/D$, figure 16(c), leads to $\alpha =0.173$. This is very different to that found by Wygnanski, Champagne & Marasli (Reference Wygnanski, Champagne and Marasli1986) in their experiments on the wake of a 70 % solidity plate (i.e. $\beta =0.3$), which had $\alpha =0.09$. This difference is not surprising because their measurements were obtained (roughly) over the range $400<(x-x_o)/D<1600$. The very much higher turbulence intensities in the near wake must inevitably lead to a more rapid growth of the wake initially. Whilst reasonable local similarity is evident in the velocity profiles in our near-wake region, the normalised axial turbulence energy profiles, figure 16(b), are far from collapse although there is a clear trend towards such collapse as $x/D$ increases. Indeed, the centreline value of $u'^2/u_o^2$ at $x/D=19.5$ is not very far below the value found by Wygnanski et al. (Reference Wygnanski, Champagne and Marasli1986). Incidentally, their $\alpha$ for a solid flat plate was 0.072 compared with 0.09 for the 70 % solidity plate, illustrating the lack of universal behaviour in these wakes even at such long distances downstream. It is interesting that they concluded that the wake behind the solid plate (which no doubt shed a Kármán vortex street) reached a self-preserving state more rapidly than the wake behind the porous plate.

Figure 16.

(a) Transverse profiles of the axial velocity for $\beta =0.33$. (b) Corresponding profiles of the axial turbulence energy. The legends show values of $x'=x/D$. (c) Growth of the wake. (d) Variation of the centreline axial energy. The solid red symbols in (c,d) are for $\beta =0.15$, Note that slightly different values of $x_o$ have been used in (c,d).

Note that figures 16(c) and 16(d) include data obtained for a much lower porosity case, $\beta =0.15$. This is well within the vortex shedding regime and although, curiously perhaps, the mean velocity profiles at the four $x'$ locations (not shown) do happen to collapse when normalised using the $L/D$ and $u_o/U_\infty$ values appropriate for each, these values show clearly that this close to the plate even the mean flow is very far from any hint of local similarity. Importantly for the present context, Wygnanski et al. (Reference Wygnanski, Champagne and Marasli1986) also identified large-scale structures in their wakes. In particular, in the wake of a $\beta =0.7$ plate, they found that the predominant frequency at the most upstream location – $(x-x_o)/D\approx 10$ – was around $fD/U_r=0.3$. A very similar result was obtained by Cimbala, Nagib & Roshko (Reference Cimbala, Nagib and Roshko1988) at $x/D=10$ in the wake of a $\beta =0.47$ flat plate. This is about double the vortex shedding frequency of a solid plate, which suggests that the natural instability in the wake profile at that point downstream is actually not far from twice the vortex shedding frequency, whether or not such shedding actually occurs – so it is not in fact a ‘second harmonic’ at all. This may help to explain why, for $\beta >0.3$, the coherence peak at what seems to be the second harmonic can still appear and, indeed, appear more frequently than the first peak, as noted earlier. Recall also that figure 12(c, f) (for $\beta =0.33$) indicates the presence of large-scale structures, although these have not yet developed into the form typically seen in far wakes by, for example, Wygnanski et al. (Reference Wygnanski, Champagne and Marasli1986).

A related explanation for the broad coherence peak (when $\beta >0.3$) at a frequency significantly larger than the low-$\beta$ vortex shedding frequency is provided by the work of Huang & Keffer (Reference Huang and Keffer1996). They explored the wake in the range $1< x/D<20$ behind a two-dimensional mesh with $\beta =0.4$. By studying spectra of the transverse velocity ($v'$) and coherence from autospectra of two such signals obtained on either side of the wake, they showed that the small-scale structures that develop in the two initially independent shear layers merge to form quasi-periodic larger structures further downstream, having a characteristic frequency equivalent to $St\approx 0.19$. This began to be evident as early as $x/D=5$. It seems likely that the large-scale structures Huang & Keffer (Reference Huang and Keffer1996) identified are precisely those found by Wygnanski et al. (Reference Wygnanski, Champagne and Marasli1986), as indeed the former authors recognised. In any case, they are very different from the Kármán-type vortex structures that appear at lower porosities.