3.1. Drag coefficient and Strouhal number

Figure 3 shows the dependence of the computed drag coefficient $\overline {C_d}$, the blockage-corrected drag coefficient $\overline {C_{d_c}}$, and $St_d$ on $\beta$ (also reported in table 4). Here, the correction for the drag coefficient due to domain blockage was done using the method of Maskell (Reference Maskell1965) (see Appendix C for details). It is readily observed that there is excellent qualitative agreement with the experiments of Castro (Reference Castro1971) despite the large difference in $Re$ between the studies. Figure 3(a) shows the monotonic decrease in $\overline {C_d}$ with increasing $\beta$. However, its sharp fall is observed to be occurring much earlier than the $\beta \approx 20\,\%$ reported by Castro (Reference Castro1971). In the current DNS, an apparent reduction in drag is observed to start at $\beta \approx 4\,\%$. The exact location of the sudden drop cannot be determined due to a limited number of data points. Such a phenomenon arises because the vortex street formation region, where the separated shear layers begin to interact, is pushed further downstream as a result of the bleed flow through the plate (explained further in § 3.2). On the other hand, figure 3(b) shows that the increase in $St_d$ with respect to $\beta$ followed by a sudden drop at $\beta \approx 20\,\%$ is well predicted by the current DNS. Here, the blockage effect could have an influence on $St_d$ values reported in both studies. Note that the blockage ratio is almost the same in both studies. Additionally, the present DNS shows a higher $St_d$ as opposed to the experiments of Castro (Reference Castro1971). Such an increase in $St_d$ is attributed to a reduction in $Re_d$. Nevertheless, contrary to the observations of Castro (Reference Castro1971), it is quite interesting to observe that the sudden drop in $\overline {C_d}$ occurs at different $\beta$ in the present DNS, which could be an effect of the large difference in $Re$. Therefore, figure 3 demands further investigation to determine the mechanism for such a behaviour. Since the wake dynamics is believed to be following the general trend at intermediate porosities, and also for brevity, only $\beta =0\,\%$, 9 %, 16 % and $25\,\%$ have been discussed in detail in the following sections.

Figure 3.

Comparison of the mean quantities obtained in the current DNS with the experiments of Castro (Reference Castro1971). (a) Drag coefficient $\overline {C_d}$: open triangle denotes current DNS at $Re_d = 250$ ($\overline {C_d}=2.223$ at $\beta =0\,\%$); grey triangle denotes blockage corrected DNS at $Re_d = 250$ ($\overline {C_{d_c}}=1.961$ at $\beta =0\,\%$); grey square denotes wake traverse method at $Re_d = 9 \times 10^4$ ($\overline {C_d}=1.85$ at $\beta =0\,\%$); open square denotes drag balance method at $Re_d = 9 \times 10^4$ ($\overline {C_d}=1.89$ at $\beta =0\,\%$) (Castro Reference Castro1971). (b) Strouhal number $St_d$: open triangle denotes current DNS at $Re_d = 250$ at $15d$ downstream of the plate ($St_d=0.164$ at $\beta =0\,\%$); open diamond denotes $Re_d = 2.5 \times 10^4$; open square denotes $Re_d = 9 \times 10^4$ ($St_d=0.14$ at $\beta =0\,\%$) (Castro Reference Castro1971). Note that the lines are used to represent the trend.

Table 4.

Mean quantities. The peak spectral energy of the $w'$ velocity ($E_{peak}$), $v_{rms}$ and $w_{rms}$ are all calculated in the ‘far’ wake at $15d$ downstream along the top edge of the plate ($y/d=0$, $z/d=8.5$). All spectral data are taken for 60 shedding cycles. Since the wake characteristics are believed to be following the general trend at intermediate porosities, only $\beta =0\,\%$, 9 %, 16 % and $25\,\%$ have been analysed in detail.

3.2. Behaviour of the vortex street

Figures 4(a,c,e,g) and 4(b,d, f,h) show instantaneous spanwise vorticity contours in side and bottom view, respectively. It appears that perforation induces the shear layers emanating from either side of the plate to interact farther downstream, thereby creating a larger formation region when compared to non-perforated plates. This demarcates the ‘near’ wake region from the ‘far’ wake region, where the ‘near’ wake is characterized by the ‘bleeding’ jet flow through the holes, and the ‘far’ wake is characterized by the unsteady vortical structures. It can be inferred that the delayed interaction of the shear layers with increasing $\beta$ causes the monotonic decrease in $\overline {C_d}$, as shown in figure 3(a). Further, a strong ‘bleed’ flow at $\beta =16\,\%$ suppresses the vortex shedding at the plate, rendering the vortex-dominated flow quasi-periodic with reduced flow three-dimensionality. The vortex topology is demonstrated further by the iso-contours of the instantaneous $\lambda _2$ criterion in figure 5. Here, $\lambda _2$ is defined as the second-largest eigenvalue of the tensor $S_{ij}S_{ij}+\varOmega _{ij}\varOmega _{ij}$, where $S_{ij}$ and $\varOmega _{ij}$ are the symmetric and antisymmetric parts of the velocity gradient tensor, respectively (Jeong & Hussain Reference Jeong and Hussain1995). At higher $\beta =25\,\%$, it is observed that the spanwise vortex tubes undergo flow instabilities in the form of helical twisting and stretching (not evident at this time instant, but discussed in subsequent sections). Figure 5 is further striking and is indicating clearly that an already ‘turbulent’ state behind a non-perforated plate is tending towards relaminarization and is pushed further into a transitional state by the presence of perforation at the chosen $Re$. The instability modes will be discussed in further detail in § 3.6.

Figure 4.

Instantaneous spanwise vorticity $\omega _y=\pm (0.004\unicode{x2013}0.005)$: (a,c,e,g) side view depicting wake width; (b,d, f,h) bottom views, at (a,b) $\beta = 0\,\%$, (c,d) $\beta = 9\,\%$, (e, f) $\beta = 16\,\%$, and (g,h) $\beta = 25\,\%$.

Figure 5.

Instantaneous $\lambda _2$ from $-3\times 10^{-6}$ to $-4\times 10^{-6}$: (a) $\beta = 0\,\%$, (b) $\beta = 9\,\%$, (c) $\beta = 16\,\%$, (d) $\beta = 25\,\%$.

Next, the vortex characteristics of the first pair of counter-rotating vortex rollers shown in figures 4(a,c,e,g) are studied. At a given spanwise location, the circulation $\varGamma _y$ and the vortex centre $X^c$ of a vortex with vorticity $\omega _y$ are calculated by

(3.1)$$\begin{gather} \varGamma _y= \int\int_D \omega _y \,{{\rm d}\kern0.06em x} \,{\rm d}z, \end{gather}$$

(3.2)$$\begin{gather}X^c= \frac{1}{\varGamma _y}\int\int_D X \omega _y \,{{\rm d}\kern0.06em x} \,{\rm d}z. \end{gather}$$

Here, the domain of integration $D$ signifies the region enclosing the vortex, and $X$ is the coordinate of the surface element ${{\rm d}\kern0.06em x}\, {\rm d}z$ having vorticity $\omega _y$. The vortex separation distance $b$ between counter-rotating vortices is obtained from

(3.3)\begin{equation} b = |X_+^c -X_-^c|. \end{equation}

The subscripts ‘$+$’ and ‘$-$’ denote the signs of the two counter-rotating vortices. The vortex separation is further decomposed into streamwise component $b_x$, and cross-stream component $b_z$.

These vortical quantities are averaged along the span and reported in table 4, where $E_{peak}$ denotes the peak spectral energy associated with the $\hat {w}$ velocity signal, and $v_{rms}$ is used as a sufficient measure of three-dimensionality in the flow (Karniadakis & Triantafyllou Reference Karniadakis and Triantafyllou1992). It is observed that the variation of $E_{peak}$ with respect to $\beta$ follows a trend similar to $St_d$, where it increases up to $\beta =16\,\%$ followed by a sharp drop at $\beta =25\,\%$. In contrast, $b_z$ and $v_{rms}$ follow an inversely related trend. It can be deduced that as the three-dimensionality in the flow reduces, the counter-rotating vortices are brought closer to each other in the cross-stream direction $b_z$, causing increased spectral energy $E_{peak}$. This leads to a larger streamwise separation $b_x$, with a higher shedding frequency, or $St_d$.

3.2.1. Instantaneous fluctuation of kinetic energy

Figure 6 shows iso-contours of instantaneous fluctuation kinetic energy, defined as $k=\widehat {u_i}\widehat {u_i}/2$. Here, the occurrence of high-intensity spots in the ‘near’ wake at $\beta >0\,\%$ corresponds to the length of the ‘bleed’ jet core. The spanwise coherence of $k$ in the ‘far’ wake signifies the location of shear layer roll-up. The distribution of fluctuation kinetic energy in the ‘near’ and ‘far’ wakes shows clearly that the primary source of fluctuation energy in the ‘near’ wake is the jets emanating from perforation holes (see figures 6b,d, f,h). These jets are separated by wakes from the solid portion of the plate, resulting in an anisotropic and inhomogeneous flow. Such a jet–wake interaction produces a significant amount of shear. It appears that the length of the primary vortex formation around the ‘bleed’ jet varies with $\beta$, similar to the trend followed by $St_d$. Here, it is observed to increase from $\beta =9\,\%$ to $\beta =16\,\%$, followed by a reduction at $\beta =25\,\%$. At $\beta =16\,\%$, the length of the ‘bleed’ jet core nearly coincides with the location of shear layer roll-up, indicating the jet–wake coupling close to equilibrium. On the other hand, the coherent or unsteady motion is responsible for the high-fluctuation kinetic energy in the ‘far’ wake. Additionally, it could be seen here that the turbulence in the ‘near’ as well as ‘far’ wake (also see $v_{rms}$ in table 4) is suppressed gradually as $\beta$ increases towards $16\,\%$ with a corresponding rapid ascent of $St_d$ (see table 4). Because of the changing nature of the small-scale flows through the holes, the turbulence re-emerges at the higher $\beta$ of $25\,\%$. This is accompanied with a fall in $St_d$ from its elevated values at $\beta =9\,\%$ and $16\,\%$.

Figure 6.

Instantaneous fluctuation kinetic energy at mid $z/d$ plane: (a,c,e,g) whole domain; (b,d, f,h) the ‘bleed’ jet region (‘near’ wake). Note that $x/d=5$ indicates the plate location. Observe that the colour bar ranges differ between panels.

Further, the local Reynolds numbers are shown in table 5. Here, $Re_m$ is defined based on the margin $m$ between two adjacent holes (see figure 1b), and $Re_h$ is defined based on the width of the hole, $h$. The local Reynolds number of the jet, $Re_{jet}$, is defined as $Re_{jet}=\bar {u}_{jet,max} h/\nu$, where $\bar {u}_{jet}$ indicates the jet centreline velocity, and subscript $max$ signifies maximum value along the centreline. At lower porosities, flow retardation is expected to be high, with vortex street dominating over ‘bleed’ flow, leading to $Re_{jet}< Re_m$. At higher porosities, ‘bleed’ flow controls the vortex street due to increased momentum with $Re_{jet}>Re_m$. At $\beta =16\,\%$, the ‘bleed’ flow is just enough to prevent the shear layers from interacting in the ‘near’ wake with $Re_{jet}\approx Re_m$. Thus it could be argued that the peculiar behaviour of quasi-laminar flow with weak three-dimensionality at $\beta =16\,\%$ is related to the fact that $Re_{jet}\approx Re_m$. Here, the ‘bleed’ jet is marginally longer with least three-dimensionality, as seen by the relatively lower $k$ in the ‘near’ wake (see figures 6b,d, f,h). At slightly higher $\beta =25\,\%$, $k$ revives to a larger value, indicating transition of the laminar ‘bleed’ jet to an irregular state. This could be due to proximity interference effects of adjacent jets when $h\gtrsim m$.

Table 5.

Various Reynolds numbers shown here are defined as $Re_h=U_o h/\nu$, $Re_{m}=U_o m/\nu$ and $Re_{jet}=\bar {u}_{jet,max} h/\nu$. Here, $m$ is also equal to $2s$ (cf. figure 1b).

Although the variation of $St_d$ with $\beta$ in the current DNS follows a trend very similar to high $Re$ cases of Castro (Reference Castro1971), it is quite possible that the wake and ‘bleed’ flow vortex dynamics might be different at much higher $Re$. Nevertheless, it was presumed by Castro (Reference Castro1971) that the flow through the holes themselves affects $\overline {C_d}$ and $St_d$. The observations in this section indicate clearly that the local Reynolds number in the ‘near’ wake plays a primary role in determining the overall three-dimensionality of the flow, and the ‘near’ wake affects the behaviour of large vortical structures in the ‘far’ wake.

3.3. Spatio-temporal analysis

Figure 7 shows the spatio-temporal plot of cross-stream velocity $w/U_o$ for various $\beta$. Here, the horizontal axis corresponds to the normalized time $tU_o/d$, and the vertical axis is the spanwise extent of the plate, $y/d$. The data in figures 7(a,c,e,g) are recorded from the ‘near’ wake at $1d$ downstream of the plate, while the data in figures 7(b,d, f,h) stem from the ‘far’ wake at $15d$ downstream of the plate. The spanwise alternate bands in the ‘far’ wake signify a Kármán-like vortex street. It is evident that the coherence in the spanwise bands in the ‘far’ wake increases with $\beta$ up to 16 %. In contrast, the incoherence increases from $\beta =16\,\%$ to $25\,\%$. On the other hand, it is observed that the spanwise vortex structures in the ‘near’ wake transition to streamwise vortex structures with an increase in $\beta$. Here, the streamwise alternate bands signify shear layer undulations under the influence of ‘bleed’ flow. Note that the wavelength of these waves is approximately equal to the geometric pitch of the perforations. Such a flow state in the ‘near’ wake indicates vortex street suppression at higher $\beta$, causing delayed shear layer interaction leading to monotonic decrease in $\overline {C_d}$.

Figure 7.

The $w$ velocity time trace sampled in (a,c,e,g) the ‘near’ wake ($1d$ downstream along the top edge of the plate, $z/d=8.5$), and (b,d, f,h) the ‘far’ wake ($15d$ downstream along the top edge of the plate, $z/d=8.5$).

Figures 8 and 9 show the corresponding time trace of the spanwise and cross-stream velocity signals in the ‘near’ and ‘far’ wakes, respectively. Figure 8 further supports the vortex street suppression at higher $\beta$, demonstrated by reduced irregularities in the $v$ and $w$ velocity signatures. The signals at $\beta =16\,\%$ are nearly quiescent, indicating vortex street suppression leading to undulating shear layers in the ‘near’ wake. Here, the quiescent flow also suggests quasi-laminar characteristics. It has already been observed by fluctuation kinetic energy, and it will also be shown later by ‘bleed’ flow characteristics and secondary instabilities, that a weak three-dimensionality exists at $\beta =16\,\%$. The revival of three-dimensionality at $\beta =25\,\%$ could be due to local $Re$ effects where the neighbouring ‘bleed’ jets interact with each other when $h\gtrsim m$. Further, the ‘bleed’ jets undergo meandering instability due to proximity interference effects (discussed further in § 3.7). On the other hand, figure 9 demonstrates the vortex street behaviour in the ‘far’ wake. The $w$ velocity signal at $\beta =0\,\%$, coupled with higher fluctuations in the $v$ velocity, indicates the irregular behaviour of the vortex street. With increasing $\beta$, the irregularities reduce, leading to a quasi-periodic state with weak three-dimensionality at $\beta =16\,\%$. At $\beta =25\,\%$, the $v$ velocity fluctuations increase, indicating revival of three-dimensionality in the flow. Overall, it is clear that the low-$Re$ turbulent state at $\beta =0\,\%$ is pushed back to a transitional state by the perforations. A similar phenomenon is also reported in the wakes of circular cylinders, where addition of dilute concentrations of polymer additives causes turbulence suppression (Richter, Iaccarino & Shaqfeh Reference Richter, Iaccarino and Shaqfeh2010, Reference Richter, Iaccarino and Shaqfeh2012).

Figure 8.

Velocity time trace sampled in the ‘near’ wake ($1d$ downstream of the plate along the top edge of the plate, $y/d=0$, $z/d=8.5$): (a,c,e,g) $v/U_o$, (b,d, f,h) $w/U_o$.

Figure 9.

Velocity time trace sampled in the ‘far’ wake ($15d$ downstream of the plate along the top edge of the plate, $y/d=0$, $z/d=8.5$): (a,c,e,g) $v/U_o$, (b,d, f,h) $w/U_o$.

3.4. Spectral analysis

Fast Fourier transforms of the instantaneous $\hat {w}$ velocity signal sampled over 60 vortex shedding cycles are shown in figure 10. Figures 10(a,c,e) show the Fourier spectrum of the signal in the ‘near’ wake, whereas figures 10(b,d, f) show that of the ‘far’ wake. Here, $f_s$ denotes the Strouhal number corresponding to the dominant frequency. Figures 10(a,b) show that the dominant $St_d$ is the same in both the ‘near’ and ‘far’ wakes for $\beta =9\,\%$, signifying that the effect of the ‘bleed’ jet is not strong enough to push the formation of a vortex street downstream by a significant margin. In the $\beta =16\,\%$ case, distinct dominant frequencies emerge. Figures 10(c,d) show that in the ‘near’ wake spectrum, the most dominant frequency corresponds to $St_d = O[10^{-3}]$, which is insignificant. The next distinct value, $St_d = 0.185$, interestingly becomes the most dominant one in the ‘far’ wake. Note that a single dominant frequency was reported in Castro (Reference Castro1971), where the origin of the signal recording position is not particularly clear. (Note that Castro (Reference Castro1971) used circular holes and the experiments were conducted at higher $Re$.) In addition, figures 10(c,d) also show the presence of distinct higher harmonics of the Strouhal frequency containing a specific periodicity, further supporting the quasi-laminar behaviour described in the previous subsections.

Figure 10.

Fast Fourier transforms of the $\hat {w}$ velocity sampled at (a,c,e) the ‘near’ wake ($1d$ downstream along the top edge of the plate, $y/d=0$, $z/d=8.5$), and (b,d, f) the ‘far’ wake ($15d$ downstream along the top edge of the plate, $y/d=0$, $z/d=8.5$), for (a,b) $\beta = 9\,\%$, (c,d) $\beta = 16\,\%$, (e, f) $\beta = 25\,\%$. Here, $f_s$ denotes the dominant $St_d$.

3.5. Two-point correlation

Figure 11 shows the spanwise two-point correlation of all the velocity components in the ‘near’ and ‘far’ wakes. Here, the correlation coefficient is defined as $\rho _{\phi \phi }= \overline {\hat {\phi } (x,y,z,t)\,\hat {\phi } (x,y+\delta y,z,t)}/\overline {\hat {\phi } (x,y,z,t)^2}$, where $\hat {\phi }\in (\hat {u},\hat {v},\hat {w})$. The overline signifies time averaging over 75 vortex-shedding cycles. In the ‘near’ wake, the streamwise and cross-stream velocity components remain highly correlated at lower $\beta$ (see $\rho _{uu}$ and $\rho _{ww}$, respectively, in figures 11a,e) due to the formation of a spanwise-coherent Kármán vortex street (see figure 4). The correlation is lower at higher $\beta$ as the dominant ‘bleed’ flow suppresses the vortex shedding. Here, the oscillatory nature of correlation for perforation cases, i.e. at $\beta >0\,\%$, is due to the effect of periodically spaced ‘bleed’ flow through holes. For similar reasons, the uncorrelated nature of the spanwise velocity component at $\beta =0\,\%$ deviates marginally with increasing $\beta$ (see $\rho _{vv}$ in figure 11c). The irregular behaviour of the correlation functions along the span could be due to the limited number of samples considered for averaging (Singh & Narasimhamurthy Reference Singh and Narasimhamurthy2021). In the ‘far’ wake, the streamwise and cross-stream correlations remain correlated in all cases (see figures 11b, f) due to the presence of spanwise vortex tubes (see figure 4). Interestingly, $\rho _{uu}$ and $\rho _{ww}$ follow a trend similar to that of $St_d$, where the correlation increases with $\beta$ at 0 % to 16 %, followed by a slight drop at 25 %. Also, $\rho _{vv}$ shows a similar trend with negative correlation (see figure 11d). Thus it can be noted that all the velocity components show high correlation at $\beta =16\,\%$ due to ‘quasi-periodic’ vortex tubes in the ‘far’ wake (see figure 4).

Figure 11.

Two-point correlation $\rho$ at (a,c,e) the ‘near’ wake ($1d$ downstream along the top edge of the plate, $z/d=8.5$), and (b,d, f) the ‘far’ wake ($15d$ downstream along the top edge of the plate, $z/d=8.5$), for (a,b) $\widehat {uu}$, (c,d) $\widehat {vv}$, (e, f) $\widehat {ww}$.

3.6. Modes of instability

Figure 12 shows the iso-contours of streamwise vorticity at all $\beta$ considered in the present study. The instability mode for the non-perforated plate is reported to have mode B vortex structure by Najjar & Balachandar (Reference Najjar and Balachandar1998), where reasonably well organized streamwise vortices with spanwise wavelength ${\approx }1.2$ are observed to extend in the braid region connecting the Kármán vortices. The strain field induced by the streamwise vorticity significantly distorts the spanwise Kármán vortices. Nevertheless, the spanwise and streamwise vortices are observed to be distinct (see figure 5). Although the vortex structure is observed to be slightly chaotic at lower porosities in the present study, the symmetry pattern at the higher porosity of $\beta =16\,\%$ appears similar to a mode B structure observed by Williamson (Reference Williamson1996b) for circular cylinder wakes, where mode B is defined as a secondary vortex structure with symmetric pattern of streamwise vortices from one braid region to the next. Figure 12 further indicates the effect of perforation on the secondary instabilities. Here, the vortical structures with spanwise wavelength ${\approx }1d$ in the ‘near’ wake of perforation cases correspond to the vorticity arising due to the effect of ‘bleed’ flow. In the ‘far’ wake, it appears that the prevailing secondary instability tends towards coherence and spatial compactness as $\beta$ is increased up to 16 %. The spanwise wavelength of this transition mode in the present perforated case is ${\approx }1d$ (i.e. six pairs along the span), in contrast to ${\approx }2d$ of the short-wavelength mode reported for non-perforated plates (Julien et al. Reference Julien, Lasheras and Chomaz2003, Reference Julien, Ortiz and Chomaz2004; Thompson et al. Reference Thompson, Hourigan, Ryan and Sheard2006) and ${\approx }1d$ of mode B reported for circular cylinders (Williamson Reference Williamson1996a). The spanwise coherence is also seen to increase for Kármán vortex structures. At $\beta = 16\,\%$, the wake instability (cf. figures 5c and 12c) appears to behave as quasi-laminar since the wake experiences periodic oscillations (cf. figure 9(b,d, f,h)). However, there exists the presence of fine-scale streamwise vortex structures. This case resembles closely the resonance state of the wake behind circular cylinders (Williamson Reference Williamson1996b). Further, figure 12(d) indicates that the prevailing secondary instability in the ‘far’ wake at $\beta = 25\,\%$ is signified by an antisymmetric pattern of streamwise vortices having spanwise wavelength ${\approx }2d$. This flow state has some resemblance to the mode A instability in the wake of circular cylinders.

Figure 12.

Three-dimensional iso-contours of streamwise vorticity $\omega _x =\pm (0.002\unicode{x2013}0.003)$, for (a) $\beta = 0\,\%$, (b) $\beta = 9\,\%$, (c) $\beta = 16\,\%$, (d) $\beta = 25\,\%$. Yellow and black colours indicate positive and negative values of $\omega _x$, respectively.

3.7. Bleed flow (jet) oscillation

In this study, we also observed another interesting phenomenon, where the jet or ‘bleed’ flow is experiencing an oscillatory motion along the spanwise direction (i.e. in the $x\unicode{x2013}y$ plane). This meandering jet instability is captured in the instantaneous rotation rate field, defined by $\varOmega _z=\partial u/\partial y-\partial v/\partial x$ (cf. figure 13). It appears that the meandering instability is profound as $\beta$ is increased to 25 %. It has been reported in the literature that if the pitch $c/h$ or the gap is sufficiently small, then the adjacent shear layers in a wake can experience proximity interference effects leading to engulfment or meandering oscillations and instabilities (Dadmarzi et al. Reference Dadmarzi, Narasimhamurthy, Andersson and Pettersen2018). The time trace of spanwise and cross-stream velocities sampled in the ‘near’ wake at the jet-central location (cf. figure 14) further substantiates this instability. Here, it is observed that the ‘bleed’ flow at $\beta = 9\,\%$ experiences low-frequency oscillations in the spanwise direction. In addition, it undergoes high-frequency oscillations in the cross-stream direction, indicating a strong influence of the vortex street. At $\beta =16\,\%$, the ‘bleed’ flow undergoes high-frequency oscillations in both the spanwise and cross-stream directions. The low amplitude of $v$ and $w$ velocity signals are in line with reduced three-dimensionality and vortex shedding suppression, respectively. At $\beta = 25\,\%$, high-frequency high-amplitude jet meandering is evident from the oscillatory high-frequency $v$ velocity signal. At the same time, a low-amplitude $w$ velocity signal signifies the absence of Kármán vortex shedding. The frequency of meandering is further quantified by the fast Fourier transform of the $\hat {v}$ velocity signal (cf. figure 15). The corresponding $St_d$ at $\beta =9\,\%$ is found to be negligibly small. On the other hand, the dominant meandering frequency is significantly higher at $\beta \geqslant 16\,\%$ with $St_d=0.187$.

Figure 13.

Instantaneous rotation rate at the wake centreline ($z/d=8$) showing the flapping mechanism of the ‘bleed’ flow through the holes (jet).

Figure 14.

Velocity time trace sampled at $y/d=2.5$, $z/d=8$ (i.e. along the centreline of the third hole), for (a,c,e) $v/U_o$, (b,d, f) $w/U_o$. Sampling across the streamwise direction is done at $1d$ downstream of the plate for $\beta = 9\,\%$ or $2d$ downstream of the plate for $\beta \geqslant 16\,\%$ due to differential jet length.

Figure 15.

(a,c,e) Fast Fourier transforms of the $\hat {v}$ velocity sampled at $y/d=2.5$, $z/d=8$ (i.e. along the centreline of the third hole), and (b,d, f) spectra along the span, at (a,b) $\beta =9\,\%$, (c,d) $\beta =16\,\%$, (e, f) $\beta =25\,\%$. Sampling along the streamwise direction is done at $1d$ downstream of the plate for $\beta = 9\,\%$ or $2d$ downstream of the plate for $\beta \geqslant 16\,\%$ due to differential jet length.

3.8. Mean primary and secondary flow

Figure 16 shows the mean streamwise velocity along the plate centreline, i.e. through the solid section of the plate, as well as along the hole centreline. The region along the $x$ axis with negative velocity signifies the presence of a recirculation zone. From figure 16(a), it is observed that the primary recirculation zone, i.e. the recirculation along the plate centreline, is present predominantly at all $\beta$. However, the secondary recirculation, i.e. recirculation along the hole centreline shown in figure 16(b), diminishes in size and eventually disappears at $\beta =25\,\%$. This is due to the fact that the ‘bleed’ flow pushes the vortex downstream with increasing $\beta$. Furthermore, the peak streamwise velocity with $\bar {u}/U_o>1$ shows the jet-like effect of the ‘bleed’ flow. It appears that the peak jet velocity reduces while its location is shifted downstream with increase in $\beta$ from 9 % to 25 % (also quantified earlier as $\bar {u}/{U_o}_{jet,max}$ in table 4).

Figure 16.

Mean primary flow along (a) the plate centreline, and (b) the hole centreline (averaged in time and along the pitch).

Additionally, a strong mean secondary flow (i.e. the spanwise velocity $\bar {v}/U_o$) is observed both upstream and downstream of the perforated plates. Figures 17(a,c) show contours of mean secondary flow upstream and downstream of the plate, respectively, at $\beta =9\,\%$. The corresponding mean pressure contours are shown in figures 17(b,d). Note that the data are averaged in time and along the pitch. It appears that the maximum secondary flow upstream of the plate occurs at $y/d\approx 0.35$ and $0.65$. This location corresponds to the edge of the square hole. The corresponding pressure contours show a high-pressure region away from the hole, indicating that the flow is driven by differential pressure towards the hole along the path of least resistance. In comparison, the maximum secondary flow downstream of the perforated plates is observed to be ${\approx }15\,\%$ of the inflow velocity. Similar observations can be made at higher $\beta$ (not shown here). Further, figure 18 shows the variation of mean secondary flow along the streamwise direction at various spanwise locations. It is clear from figure 18(b) that the mean secondary flow downstream of the perforated plates is significantly higher in the formation region. In contrast, the mean secondary flow is negligibly small for the non-perforated plate. In the ‘far’ wake, it reduces to a negligible value for both the perforated and non-perforated cases.

Figure 17.

Secondary flow (averaged in time and along the pitch) at $z/d=8$ (i.e. mid $z$ plane) for $\beta =9\,\%$: (a) $\bar {v}/U_o$ upstream of the plate; (b) $\bar {p}/(\rho U_o^2)$ upstream of the plate; (c) $\bar {v}/U_o$ downstream of the plate; (d) $\bar {p}/(\rho U_o^2)$ downstream of the plate.

Figure 18.

Secondary flow averaged in time and along the pitch, $\bar {v}/U_o$, at $z/d=8$ (i.e. mid $z$ plane): (a) upstream of the plate; and (b) downstream of the plate.