1. Introduction
Turbulent wakes are of fundamental importance for a wide variety of engineering and environmental applications, including wind turbine and automobile aerodynamics, flows over buildings and hills and many others. The physics of turbulent wakes are particularly complex as they involve a combination of statistical non-stationarity (i.e. they evolve in a Lagrangian sense), inhomogeneity (shear) and self-organisation in the form of coherent structures, such as the well-known Kármán vortex street (Von Kármán Reference Von Kármán1911). An assumption that greatly facilitates the modelling of turbulent wakes (and turbulent shear flows in general) is that of self-similarity in the flow statistics (Tennekes & Lumley Reference Tennekes and Lumley1972; Townsend Reference Townsend1976; Pope Reference Pope2000). Under an appropriate transformation of the flow coordinates and variables, the resulting average field becomes invariant with respect to the streamwise distance. At this point, the flow can be thought to have reached a ‘moving equilibrium’ (as stated by Townsend Reference Townsend1976), which lays the basis for the theoretical treatment of turbulent wakes (and other turbulent shear flows) in an asymptotic sense (Townsend Reference Townsend1976; George Reference George1989). A question that naturally arises is, under which conditions does self-similarity appear in turbulent flows? In a series of monographs, Townsend (Reference Townsend1956, Reference Townsend1970, Reference Townsend1976) postulated that self-similarity is a result of the flow ‘forgetting’ the initial conditions that gave rise to turbulence, presumably occurring sufficiently far downstream of the bluff body. In turbulent wakes, however, this far-wake region is characterised by the presence of large-scale coherent structures. This observation might pose a certain contradiction, given that coherent structures retain their coherence for a large spatial extent (Taneda Reference Taneda1959; Cimbala, Nagib & Roshko Reference Cimbala, Nagib and Roshko1988; Huang & Keffer Reference Huang and Keffer1996): if these structures are formed in the near vicinity of the body, they will then carry within them the memory of the initial conditions into the self-similar region of the flow, i.e. in a region where such memory is assumed to be forgotten.
This contradiction can be resolved by an additional hypothesis proposed by Townsend (Reference Townsend1976) (see also Bevilaqua & Lykoudis Reference Bevilaqua and Lykoudis1978) where coherent structures that appear in the self-similar region of turbulent wakes are assumed to be formed locally by the mean shear, and independently of the initial conditions of the flow. Townsend postulated a particular shape for these theoretical structures, i.e. the Townsend–Grant pair eddies (Grant Reference Grant1958; Townsend Reference Townsend1976), but their physics or even existence remains an open topic. Nevertheless, the subsequent experimental results of Cimbala et al. (Reference Cimbala, Nagib and Roshko1988) indicated that structures which very closely resemble the Townsend–Grant eddies indeed emerge far downstream of bluff bodies, in a region of the flow governed by the so-called secondary vortex street (SVS) instability. As noted by Taneda (Reference Taneda1959) and Cimbala et al. (Reference Cimbala, Nagib and Roshko1988), this is a flow phenomenon that occurs in the far wake of bluff bodies after the primary Kármán vortex street has decayed, and the wake is reorganised into larger-than-Kármán vortical structures which grow in size and persist over long distances, and with Strouhal numbers which are lower than the conventional primary street (Cimbala et al. Reference Cimbala, Nagib and Roshko1988).
The above brief discussion underlines the importance of identifying the flow mechanisms that generate the SVS, which have been studied extensively at relatively low Reynolds numbers (Durgin & Karlsson Reference Durgin and Karlsson1971; Karasudani & Funakoshi Reference Karasudani and Funakoshi1994; Vorobieff, Georgiev & Ingber Reference Vorobieff, Georgiev and Ingber2002; Saha Reference Saha2007; Kumar & Mittal Reference Kumar and Mittal2012; Thompson et al. Reference Thompson, Radi, Rao, Sheridan and Hourigan2014; Dynnikova, Dynnikov & Guvernyuk Reference Dynnikova, Dynnikov and Guvernyuk2016; Jiang & Cheng Reference Jiang and Cheng2019; Jiang Reference Jiang2021; Ju & Jiang Reference Ju and Jiang2022; Singh & Narasimhamurthy Reference Singh and Narasimhamurthy2022) and more limitedly at high Reynolds numbers (Taneda Reference Taneda1959; Valensi Reference Valensi1974; Louchez Reference Louchez1986; Cimbala et al. Reference Cimbala, Nagib and Roshko1988; Huang & Keffer Reference Huang and Keffer1996; Kopp & Keffer Reference Kopp and Keffer1996; He et al. Reference He, Draper, Ghisalberti, An, Branson, Cheng and Ren2022; He Reference He2023). Assuming that the Townsend–Grant eddies coincide with the secondary street, the question of the validity of Townsend’s hypothesis can be partially reduced to whether the SVS is inherently connected to the initial flow conditions. One could consider that, since the SVS follows the primary (Kármán) street, the latter is the direct cause of the former. This postulation has found support in the literature (Papailiou & Lykoudis Reference Papailiou and Lykoudis1974; Williamson & Prasad Reference Williamson and Prasad1993), with some studies suggesting that the SVS forms through the merging of the Kármán street vortices (Matsui & Okude Reference Matsui and Okude1983a , Reference Matsui and Okudeb ; Meiburg Reference Meiburg1987; Ju & Jiang Reference Ju and Jiang2022). However, as shown by Cimbala et al. (Reference Cimbala, Nagib and Roshko1988), the Kármán street is not a necessary prerequisite for the existence of the SVS, as the latter can be observed even in wakes where the Kármán street is completely suppressed, as in the case of highly porous plates (see, for instance, figure 1). In that case, there are two conflicting views in the literature regarding the origins of the SVS.
On one hand and in agreement with Townsend’s postulation, there is the view that the SVS is indeed a local phenomenon (i.e. not explicitly dependent on the initial conditions of the flow) generated instead by the mean strain of the flow at a far downstream position. Cimbala et al. (Reference Cimbala, Nagib and Roshko1988) arrived at that conclusion by performing a linear stability analysis of the mean streamwise velocity of the far wake of bluff bodies, which yielded frequencies that matched, approximately, the ones that were determined experimentally for the SVS. On the other hand, contrary to Townsend’s postulation, there is the view that the SVS is in fact intimately connected to the initial conditions of the flow, in the form of very near-wake structures which merge and eventually become the SVS. Huang & Keffer (Reference Huang and Keffer1996) postulated the above after performing hot-wire measurements in the wake of a porous mesh, and observing that the corresponding power spectral density plots yielded, initially, a small spectral peak in the near wake, which grew as the distance from the mesh increased, eventually becoming the well-defined broad peak which is characteristic of the SVS.
Figure 1. Schematic of the wake behind a porous plate with high porosity, based on the simulation conducted in this study. Here,
$\overline {u_b}$
denotes the mean bleeding velocity through the porous plate.
The above views of Cimbala et al. (Reference Cimbala, Nagib and Roshko1988) and Huang & Keffer (Reference Huang and Keffer1996) are not conclusive given that they are not based on direct visualisations of the SVS dynamics which may unambiguously reveal its formation mechanism, as this was not possible neither with the smoke visualisation measurements of Cimbala et al. (Reference Cimbala, Nagib and Roshko1988), nor with the point measurements of Huang & Keffer (Reference Huang and Keffer1996). The goal of this study is to resolve this debate by determining the physical picture behind the formation mechanism of the turbulent SVS when the primary Kármán vortex street is absent. This is achieved by combining observations from time-resolved particle image velocimetry (PIV) measurements and large eddy simulations (LES). We also test the dependence of the SVS instability when varying the initial conditions and free-stream turbulence of the flow. The structure of the paper is as follows: § 2 details the experimental and numerical methodologies employed in the study. Section 3 presents the investigation results on the identification and formation of turbulent SVS under varying initial and free-stream turbulence intensity conditions. Finally, § 4 draws the conclusions of this work.
2. Methodology
2.1. Experimental details
Experiments were conducted in a free-surface water channel, as depicted in figure 2, whose test section has a cross-section of
$0.6\times0.6\;\textrm{m}^2$
and that extends 9 m in length. The free-stream velocity was measured as
$U_\infty = 0.19\, \mathrm{m\,s}^{-1}$
(or
$0.21\, \mathrm{m\,s}^{-1}$
for one case, as discussed later) in the absence of the mesh/plate, and was characterised by a turbulence intensity (
$TI = \sqrt{\overline{u'^2} + \overline{v'^2}}/U_{\infty}$
) of 1.1 % at that flow speed, where
$ u'$
and
$ v'$
represent the velocity fluctuations in the
$x$
and
$y$
-directions, respectively.
Figure 2. Mesh and plate configurations (left) and experimental set-up schematic in the water channel (right). Flow direction is from left to right in the channel.
The wakes of a coarse stainless steel mesh (with a hole diameter of 1.67 mm and a wire thickness of 0.45 mm) and a porous plate with round holes (1.35 cm in diameter) were tested (see figure 2). Both the plate and mesh extended across the entire height of the channel and had a width of
$D = 0.03$
m, resulting in a blockage ratio (plate width over channel width) of 5 %. The geometric porosities of the mesh and plate, denoted as
$\beta$
(defined as the open area of the mesh or plate over their gross area), were slightly different and were chosen to achieve an almost identical effective porosity (defined as the average bleed velocity over the free-stream velocity,
$u^* = \overline {u_b}/U_\infty$
), thus creating two bluff bodies that had the same wake bleeding, but different homogeneity (i.e. porosity distribution over their surface). The reasoning behind that choice was that, if the initial conditions indeed have an effect on the SVS formation, important differences would be observed in the far wake when changing the distribution of fluid bleeding from the body, even if the total bleeding remained the same.
A trial-and-error procedure indicated that
$\beta$
values of 54 % for the mesh and 50 % for the plate led to
$u^* = 0.63$
for both (see table 1). We note that, at that amount of fluid bleeding, the Kármán vortex street is completely suppressed in the turbulent wake (Steiros & Hultmark Reference Steiros and Hultmark2018; Steiros, Bempedelis & Ding Reference Steiros, Bempedelis and Ding2021; Cicolin et al. Reference Cicolin, Chellini, Usherwood, Ganapathisubramani and Castro2024). The numerical experiment that is described in § 2.2 was also designed to have the same effective porosity. The details of the calculation of
$u^*$
are provided in Appendix A. In addition to the mesh and plate cases, a third experimental case (mesh-g case) was designed to test the effect of a different condition – namely, increased free-stream turbulence – on the SVS formation. A turbulence grid was placed 19 cm upstream of the mesh to raise the turbulence intensity (
$\textit{TI}$
) to 12 % at the mesh location (the channel velocity was measured 0.21 m s
$^{-1}$
in that case), while maintaining all other experimental settings unchanged. Detailed characteristics of the grid and its flow properties are provided in Appendix B.
Table 1. Details of the cases including geometric porosity (
$\beta$
), effective porosity (
$u^*$
), drag coefficient (
$C_D$
) and free-stream turbulence intensity (
$\textit{TI}$
%).
Time-resolved planar PIV measurements were conducted using a Litron laser as the light source, combined with mirrors and optical lenses. The schematic in figure 2 shows a streamwise horizontal laser sheet positioned 35 cm above the channel floor. Measurements were taken at various locations relative to the mesh/plate using a Cartesian coordinate system with
$x$
in the streamwise direction,
$y$
along the width and
$z$
in the vertical (spanwise) direction of the mesh/plate (see figure 2). The corresponding velocity components are
$u$
,
$v$
and
$w$
, respectively. The origin was set at the centre of the mesh/plate. The Reynolds number was 5700 for the mesh and plate, and 6300 for the mesh with the grid (mesh-g), based on free-stream velocities 0.19 and 0.21 m s
$^{-1}$
, respectively, and the width of the mesh/plate. The field of view (FOV) was
$24 \times 35\,\textrm{cm}^2$
, covering approximately 8
$D$
in the
$x$
-direction and 12
$D$
in the
$y$
-direction, with an acquisition frequency of 50 Hz. The time intervals between the images (
$\Delta t$
) ranged from 2000 to 6000 microseconds, adapting to the flow dynamics in the near and far wake. The PIV snapshots were taken using a 4-megapixel Phantom VEO 640 camera, synchronised with a DG645 Digital Delay Generator, over 60 s, covering approximately 76 SVS cycles (based on the SVS Strouhal number,
$St_{\textit{SVS}} \approx 0.2$
, and
$ St = \textit{fD}/U_{\infty }$
, where
$ f$
is the shedding frequency). A total of 3000 vector fields were processed using PIVlab, a free MATLAB toolbox (Thielicke & Sonntag Reference Thielicke and Sonntag2021). For preprocessing, the background image was removed by subtracting the average intensity from each image before analysis. Velocity fields were obtained using a standard multi-pass cross-correlation method (four passes), with final interrogation windows of
$16 \times 16$
pixels (px) and 50 % overlap for the mesh and plate cases, and
$24 \times 24$
px with 50 % overlap for the mesh-g case. For postprocessing, a
$5 \times 5$
px local median filter was applied to detect and correct spurious velocity vectors, following the method of Westerweel & Scarano (Reference Westerweel and Scarano2005). The percentage of corrected spurious vectors in the processed vector fields was found to be less than 1 % in all cases.
In addition to the above, the drag coefficient of the plate was measured using two force transducers (model ATI Mini 40) mounted on the top and bottom of the plate, using a similar procedure to Steiros, Bempedelis & Cicolin (Reference Steiros, Bempedelis and Cicolin2022). More details on the drag measurement technique can be found in Steiros et al. (Reference Steiros, Bempedelis and Cicolin2022). A reliable drag coefficient for the mesh is not feasible to measure directly due to its flexible structure. However, an estimated value based on a control volume approach yields comparable results (0.84
$\pm$
0.12 for the plate and 0.86
$\pm$
0.13 for the mesh with the standard deviation representing variation along the streamwise exit boundary in the far wake), suggesting an almost identical drag of the mesh and the plate. We note that the directly measured drag of the plate does not exactly match the inferred one from the far-wake measurements due to blockage effects far downstream.
2.2. Numerical details
A LES was carried out using the open-source flow solver Xcompact3d (Bartholomew et al. Reference Bartholomew, Deskos, Frantz, Schuch, Lamballais and Laizet2020). The simulation used sixth-order compact schemes for spatial discretisation (Laizet & Lamballais Reference Laizet and Lamballais2009) and a third-order Adams–Bashforth method for time integration. The Reynolds number based on the plate width and free-stream velocity was
${Re} = 6000$
, and the dynamic Smagorinsky model (Germano et al. Reference Germano, Piomelli, Moin and Cabot1991) was used for turbulence modelling. The computational domain spanned 120 × 20 × 10 plate widths and was discretised with a uniformly spaced Cartesian grid consisting of 1921
$\times$
321
$\times$
160 points. Free-slip conditions were applied at the normal (
$y\hbox{-}$
direction) boundaries, and periodic conditions were enforced in the spanwise boundaries (
$z\hbox{-}$
direction). A schematic of the computational domain is shown in figure 3.
Figure 3. Schematic of the computational domain: x–y plane (left) and y–z plane (right).
The intention was for the LES to represent an ‘idealised’ experiment, where the plate porosity is perfectly homogeneous and the free-stream turbulence intensity is effectively zero, enabling the study of SVS formation under conditions that cannot be achieved in a laboratory setting. To this end, the porous plate was parametrised with an actuator-type method, which represents it as a uniformly distributed momentum sink. The implementation follows Calaf, Meneveau & Meyers (Reference Calaf, Meneveau and Meyers2010), and the reader is referred to Bempedelis, Laizet & Deskos (Reference Bempedelis, Laizet and Deskos2023) and Jané-Ippel et al. (Reference Jané-Ippel, Bempedelis, Palacios and Laizet2023, Reference Jané-Ippel, Bempedelis, Palacios and Laizet2024) for details on implementation in Xcompact3d and validation against experimental data and other solvers. A mesh sensitivity study and a validation study specific to this work are provided in Appendix C.
In the actuator disk/plate method, the total force acting on the flow is
\begin{equation} F_T=-\frac {1}{2}\rho C_T^\prime \widetilde {u}^2 A, \end{equation}
where
$\rho$
is the density,
$C_T^\prime$
is the ‘modified’ or ‘local’ thrust coefficient (defined following classical actuator theory, see Calaf et al. (Reference Calaf, Meneveau and Meyers2010)),
$\widetilde {u}$
is the average velocity through the disk/plate and
$A$
is the area of the body. The total force
$F_T$
is then distributed across the area covered by the body. In the present work, the ‘modified’ thrust coefficient was set to
$C_T^\prime =2.56$
, leading to an average effective porosity
$u^*$
of 64 % and a drag coefficient
$C_D=1.03$
(in agreement with extensions of actuator theory accounting for confinement effects; Garrett & Cummins Reference Garrett and Cummins2007; Steiros et al. Reference Steiros, Bempedelis and Cicolin2022), as shown in table 1. This is higher than the plate drag measured in the experiments, likely due to a combination of the idealised conditions of the numerical simulation, the involvement of actuator theory and end effects in the experiments. However, both values are within the range of values for the drag coefficient of porous plates at that porosity reported in the literature (Graham Reference Graham1976; Steiros & Hultmark Reference Steiros and Hultmark2018).
The simulation was conducted using 4096 cores on ARCHER2, the UK’s national supercomputing facility. Following an initialisation period spanning approximately 60 SVS cycles (corresponding to 2.5 flow-through cycles), 10 000 snapshots of the flow at the mid-span plane were extracted over a period of approximately 84 SVS cycles.
3. Results and discussion
3.1. Instantaneous vorticity fields
To illustrate the flow topology from the near to the far wake and the evolution of the SVS for each tested case, figure 4 presents the non-dimensional instantaneous
$z$
-vorticity fields. The plate/mesh is located at
$x/D = 0$
, extending from
$y/D = -0.5$
to
$0.5$
. Videos of the non-dimensional instantaneous
$z$
-vorticity and streamwise
$u$
-velocity fields are provided in the supplementary movies 1–4.
Figure 4. Instantaneous non-dimensional
$z$
-vorticity fields: (a) LES, (b) mesh, (c) plate and (d) mesh-g. Each window in the experimental cases shows a snapshot taken at different times.
Qualitatively similar wake evolution is observed across the LES and mesh cases, as shown in figures 4(a) and 4(b). Immediately downstream of the body (
$x/D = 0$
), strong velocity gradients cause the formation of shear layers and the initiation of Kelvin–Helmholtz instabilities (KHIs). A low-velocity ‘buffer’ region initially separates the shear layers, until the onset of their interaction, which occurs at
$x/D \approx 11$
in the LES case and
$x/D \approx 7$
in the mesh case. Following this interaction, the shear layers exhibit oscillations. In the LES case, these oscillations are initially weaker in the range
$x/D \approx 11{-}25$
and become more pronounced in the range
$x/D \approx 25{-}40$
. Similarly, in the mesh case, weaker oscillations are observed in the range
$x/D \approx 7{-}22$
, strengthening in the range
$x/D \approx 22{-}40$
. These strong oscillations correspond to the SVS (see also the smoke visualisations of Taneda Reference Taneda1959 and Cimbala et al. Reference Cimbala, Nagib and Roshko1988). Beyond
$x/D \gt 40$
, the regularity of these oscillations gradually diminishes for both cases (see also supplementary movies 1–2).
Figure 5. Instantaneous non-dimensional
$z$
-vorticity fields showing the SVS in the far wake of the (a) mesh, (b) plate and (c) mesh-g cases.
Figure 6. Evolution of the non-dimensional mean streamwise velocity (
$U_m/U_{\infty }$
) (left axis) and turbulence intensity (
$TI(\%)= \sqrt{\overline{u'^2} + \overline{v'^2}}/U_{\infty}$
) (right axis) along the wake centreline (
$y/D = 0$
) for all cases.
The cases of the porous plate and mesh with grid (mesh-g), shown in figures 4(c) and 4(d), respectively, illustrate, qualitatively, how an increase in body inhomogeneity and free-stream turbulence may affect the wake evolution. While the mesh-g case alters the ambient flow by increasing its turbulence intensity, the plate case alters the inner near wake by introducing intense mixing immediately downstream of the plate due to its high inhomogeneity. Both cases result in thicker, more disorganised shear layers, leading to an earlier shear-layer interaction (
$\approx 6D$
for the plate and
$3D$
for the mesh-g) and a reduced buffer region. Both cases also exhibit reduced-amplitude SVS oscillations in the far wake (
$\gt 40D$
), compared with the strong oscillations observed in the LES and mesh cases (see figure 4; also figure 5 with a different scale and the supplementary movies 3–4). These preliminary observations will be investigated in more detail later on in the manuscript.
3.2. Wake centreline velocity and turbulence intensity
Figure 6 shows the evolution of the non-dimensional mean streamwise velocity (
$U_m/U_{\infty }$
) along the wake centreline for all cases. The LES and mesh cases exhibit a sharp drop in mean velocity immediately after the plate, attributed to the presence of the buffer region which separates their shear layers. The end of the buffer region approximately coincides with the location of the velocity minimum at the centreline (
$7D$
for the mesh and
$11D$
for the LES case), as beyond that point, mixing causes a gradual increase in wake velocity. Indeed, observations of instantaneous snapshots, together with inspection of time-averaged enstrophy plots (see Appendix D) indicate that the streamwise location where the shear layers start influencing the centreline (and thus where the buffer region ends) matches, approximately, the location of the mean streamwise velocity minima for these two cases.
Figure 6 shows that, in contrast to the LES and mesh cases, the plate and mesh-g cases lack a large initial region of velocity drop. Close inspection of the data reveals a velocity minimum very close to the bodies – at approximately
$x \approx 0.5D$
for the plate and
$x \approx 2D$
for mesh-g. Given the above discussion, this suggests that these two cases exhibit a very limited buffer region, presumably because the increased inhomogeneity and free-stream turbulence, respectively, accelerate mixing between the wake and the outer flow (see also figure 4
c–d). Downstream of
$20D$
, all experimental mean velocity profiles collapse, while the LES case exhibits slightly lower values. This small offset can be attributed to the LES case’s higher drag coefficient (
${C\!}_{D_{\textit{LES}}} = 1.03$
) compared with the plate’s (
$C_{D_{\textit{Plate}}} = 0.91$
), which generates slightly larger velocity deficits, and to its zero free-stream turbulence, which generates sharper velocity profiles.
The centreline turbulence intensity (
$\textit{TI}$
) distributions in figure 6 exhibit a roughly common pattern for the experimental cases: the
$\textit{TI}$
exhibits a sharp spike immediately downstream the mesh/plate, because the fluid that is passing through the holes of the body generates turbulence, much like the behaviour encountered in grid turbulence flows (Melina, Bruce & Vassilicos Reference Melina, Bruce and Vassilicos2016; Steiros Reference Steiros2022). The LES case does not exhibit this initial spike, because its perfect homogeneity prevents the formation of cross-wise velocity gradients in the immediate vicinity of the plate (excluding the shear layers), suppressing turbulence production (see also the mean enstrophy plot in Appendix D). Whether the initial spike exists or not, however, we note that for all cases the result is a region very close to the plate/mesh where
$\textit{TI}$
in the centreline assumes a local minimum, or very low values.
Further downstream (i.e. beyond
$5D$
),
$\textit{TI}$
gradually increases for all cases, reaching a well-defined peak for the LES case (at roughly
$18D$
), and a plateau for the mesh (between
$20D$
and
$30D$
) and the mesh-g cases (between
$7D$
and
$30D$
) (the plate case exhibits only a marginal peak at roughly
$10D$
). Inspection of the
$\textit{TI}$
contour plots shown in figure 7 shows that this increase coincides with the widening of the mean-flow shear layers which eventually become thick enough to affect the centreline. Beyond
$40D$
all cases in figure 6 exhibit a power-law-like decay of
$\textit{TI}$
.
Despite the many similarities among cases, important differences do exist: the increase of free-stream turbulence and plate inhomogeneity brings the
$\textit{TI}$
peak forward, as the shear layers grow thicker earlier. In the plate case, the large inhomogeneity in the porosity causes intense mixing in the immediate vicinity of the body (see figure 7), the shear layers are greatly diffused and the subsequent peak in
$\textit{TI}$
(see figure 6) is attenuated. In contrast, the ‘ideal’ LES case attenuates the initial mixing between the inner and outer wake due to its perfect homogeneity, and as a result produces the largest
$\textit{TI}$
peak when its shear layers eventually meet. The above significant differences in the average behaviour of the near and far wake hint that different SVS behaviours can be expected for the four tested cases.
Figure 7. Turbulence intensity fields: (a) LES, (b) mesh, (c) plate and (d) mesh-g.
3.3. Spectral analysis of the velocity time series
We analyse the pre-multiplied energy spectra of the
$v$
-velocity fluctuations of our PIV measurements by applying a Hann window with 50 % overlap across 9 segments to reduce spectral leakage. The analysis is conducted at the inflection points of the mean streamwise
$u$
-velocity profiles, across three wake regions:
$x/D = 1{-}15$
,
$x/D = 20{-}30$
and
$x/D = 40{-}65$
(figure 8). These regions broadly capture the key stages of wake development, i.e. the development and interaction of individual shear layers, the formation of the SVS vortices and the eventual decay of SVS vortices, although we note that the exact boundaries of these regions may vary among cases.
In the LES case (figure 8
a), the shear layers initially display distinct low-energy peaks at relatively high Strouhal numbers (see e.g. the
$x/D = 3$
curve, which gradually shifts to lower broadband frequencies downstream as the KHIs develop, reaching
$St \approx 0.2$
at
$x/D \approx 15$
). In figure 8(b) the dominant peak remains close to
$St \approx 0.2$
and becomes sharper, suggesting the formation of coherent SVS structures. Further downstream, the peak amplitude decreases, the frequency shifts to lower values and the peak broadens as the SVS starts to gradually decay (figure 8
c). These observations on the spectral properties of the SVS are in qualitative agreement with the previous hot-wire measurements of Cimbala et al. (Reference Cimbala, Nagib and Roshko1988) and Huang & Keffer (Reference Huang and Keffer1996). Although both the LES (figure 8
a–c) and mesh (figure 8
d, f) cases follow qualitatively similar trends, the LES case exhibits sharper peaks with higher amplitudes. The LES peaks persist over longer distances and show a smaller variability in their frequency compared with the mesh case, suggesting a more coherent SVS in ‘ideal’ conditions (i.e. when free-stream turbulence and body inhomogeneity are minimal). The effects of inhomogeneity and free-stream turbulence are particularly pronounced in the plate (figure 8
g–i) and mesh-g (figure 8
j–l) cases, where damped spectral peaks can be observed in the far wake (i.e. beyond
$40D$
) suggesting an attenuated SVS.
Figure 8. Pre-multiplied spectra of non-dimensional
$v$
-velocity fluctuations measured at the inflection points of mean streamwise velocity profiles at different downstream locations:
$1{-}15D$
(left),
$20{-}30D$
(centre) and
$40{-}65D$
(right). The spectra are shown for: (a)–(c) LES, (d)– (f) mesh, (g)–(i) plate and ( j)–(l) mesh-g. Grey dashed lines indicate the location of peak values of the curves.
Next, we examine the cross power spectral density (CPSD) of the non-dimensional
$v$
-fluctuations. The CPSD is computed using a Hamming window of 100 samples with 50 % overlap, and the Fast Fourier Transform length is set equal to the window length. The analysis is performed using the time series at the location of the two inflection points of the mean streamwise velocity profile for each streamwise location. The CPSD allows for the assessment of the correlation of the velocity fluctuations between the two points in the frequency domain (Huang & Keffer Reference Huang and Keffer1996). An example of the produced CPSD plots is shown in figure 9 for
$30D$
downstream, where it is shown that the characteristic peak around
$St \approx 0.2$
is much larger for the LES and mesh cases, compared with the mesh-g and plate cases, suggesting a much more coherent SVS for the former.
To gain an understanding of the downstream evolution of the frequencies where the inflection points of the velocity profile interact, we extracted the peak CPSD value and corresponding frequency at which this occurs for each tested case at each downstream position. The results are plotted in figure 10. Figure 10(b) shows that beyond
$15D$
the peak interaction frequency lies in the range
$0.1\lt St\lt 0.3$
for all cases. Given our previous discussion on the power spectral density (PSD) results, this Strouhal number suggests the presence of large-scale SVS vortices. Figure 10(a) shows that beyond
$15D$
the coherence of the SVS vortices is larger for the LES and mesh cases, in agreement with the PSD plots (figure 8) and instantaneous visualisation of the flow (figure 4). In the far wake the SVS coherence decays, but is retained to some degree up to approximately
$80D$
, which is the streamwise limit of the far wake in this study.
Figure 9. Cross-spectral power density of non-dimensional
$v$
-velocity fluctuations at the two inflection points of mean
$u$
-velocity profiles at
$30D$
.
Figure 10. (a) The peak value of CPSD of non-dimensional
$v$
-velocity fluctuations at the two inflection points of mean
$u$
-velocity profiles at different streamwise locations; (b) corresponding non-dimensional frequencies.
In the near wake (i.e. below
$10D$
) the SVS vortices are absent and the cross-spectrum of the two inflection points can be thought to be a measure of the interaction between the individual shear layers. Figure 10(b) shows that such interaction occurs at high Strouhal numbers, suggesting the correlation of small structures within opposite shear layers (see also discussion in Huang & Keffer Reference Huang and Keffer1996 whose hot-wire measurements point to the same conclusion). Figure 10(a) shows that the mesh-g and plate cases exhibit higher peak CPSD values immediately downstream of the plate/mesh, compared with the LES and mesh cases, presumably due to the high initial mixing that these cases induce, which facilitates interaction between the two shear layers.
3.4. Modal decomposition analysis
Our previous discussion suggests that in all tested cases the flow follows a roughly common pattern, where small near-wake structures give place to large SVS vortices further downstream. In turn, the SVS structures gradually diffuse and decay in the far wake. The above picture is clearer in the more ‘idealised’ LES and mesh cases, but retains its validity (with some differences) when free-stream turbulence and body inhomogeneity are increased (mesh-g and plate cases). In this section we utilise modal decomposition techniques to visualise the SVS structures, and elucidate their formation mechanism, placing an emphasis on the LES and mesh cases, where the phenomenon is clearer.
3.4.1. Spectral proper orthogonal decomposition
Spectral proper orthogonal decomposition (SPOD) is applied to decompose the flow into modes ranked by their energy content, representing dominant spatial flow structures at specific frequencies (Lumley Reference Lumley1967; Towne, Schmidt & Colonius Reference Towne, Schmidt and Colonius2018; Schmidt & Colonius Reference Schmidt and Colonius2020) (see Appendix E for further details on the SPOD method). Spectral proper orthogonal decomposition was performed separately on both the
$z$
-vorticity and
$v$
-velocity components. The first
$v$
-velocity modes were found to be more energetic than the first
$z$
-vorticity modes, as quantified by
$ \sum _f \lambda _1(f) / \sum _f \sum _n \lambda _n(f)$
, where
$ \lambda _n(f)$
denotes the SPOD eigenvalue of mode
$ n$
at frequency
$ f$
, although both share similar dominant frequencies and spatial modes in terms of structure size, location and strength. We thus chose to base our SPOD analysis on the
$v$
-velocity component, with the details of SPOD parameters provided in Appendix E.
First, SPOD is applied to the wake region (
$0\!-\!80D$
) of the LES case. Figure 11 displays the SPOD mode spectra, ranking the energy of each mode across different frequencies. A clear gap between the leading and subsequent modes highlights a dominant flow mechanism primarily captured by the leading modes. For instance, the first mode is energetically significant at
$St \approx 1$
, showing a distinct gap between it and the less energetic modes. Additionally, the first four modes maintain significant energy across a broadband frequency range of approximately
$0.1\!-\!0.3$
, with a peak near a central frequency of
$St \approx 0.2$
. Similar peaks can be found in the power spectra of the inflection points (see figure 8
a–c).
Figure 11. The SPOD mode spectra of the
$ v$
-component from the LES case for the region between
$ 0$
and
$ 80D$
. The coloured curves, from black to light, represent the energy levels (
$ \lambda$
) of each mode, ranging from the first (most energetic, darkest curve) to the last (least energetic, lightest curve) of the SPOD analysis of the LES. The blue line indicates the sum of the energy of all modes at each frequency. The red dashed lines show the different frequencies.
Figure 12. The SPOD first spatial modes of the
$v$
-component of the LES at the different non-dimensional frequencies:
$St$
$\approx$
(a) 1, (b) 0.3, (c) 0.2, (d) 0.1.
The first SPOD spatial modes at different frequencies, across the distinct gap frequency (
$St \approx 1$
) and the broadband range (
$St \approx 0.1\!-\!0.3$
), are illustrated in figure 12. These modes exhibit coherent wavepacket structures whose wavelength and spatial extent increase as the frequency decreases, corresponding to different stages in the evolution of the wake. As shown in figure 12(a), the spatial mode corresponding to the spectrum peak at
$St \approx 1$
shows wavepackets in the shear layers. These wavepackets are associated with KHIs, in agreement with previous experimental observations and theoretical predictions over the range
$0.3 \lesssim St \lesssim 1$
, based on SPOD and quasi-parallel stability theory in turbulent annular jet shear layers (Suzuki & Colonius Reference Suzuki and Colonius2006; Gudmundsson & Colonius Reference Gudmundsson and Colonius2011; Schmidt et al. Reference Schmidt, Towne, Rigas, Colonius and Bres2018; Towne et al. Reference Towne, Schmidt and Colonius2018). It is noteworthy to mention that the CPSD analysis shown in figure 10(b) shows peaks for the LES at
$St\approx 1$
in the very near wake, suggesting interaction of the shear-layer instabilities in the two shear layers. Further downstream, the wavepackets of the mode at
$St \approx 0.3$
(figure 12
b) indicate closely spaced top- and bottom-shear-layer structures. Subsequently, the mode at
$St \approx 0.2$
depicts clearly separated SVS structures which span the entire width of the wake (figure 12
c). By
$40D$
(figure 12
d), these structures have become large and relatively diffused, characterised by a smaller
$St \approx 0.1$
.
Figure 13. The SPOD mode spectra of the
$ v$
-component from the mesh case for the region between
$ 0$
and
$ 7D$
. The coloured curves, from black to light, represent the energy levels (
$ \lambda$
) of each mode, ranging from the first (most energetic, darkest curve) to the last (least energetic, lightest curve). The blue line indicates the sum of the energy of all modes at each frequency. The red dashed lines show the different frequencies.
Figure 14. The SPOD first spatial modes of the
$v$
-component of the mesh at the different non-dimensional frequencies:
$St$
$\approx$
(a) 3.5, (b) 1.5, (c) 0.85, (d) 0.44.
Figure 15. (a) The SPOD mode spectra of the first and second modes of the
$v$
-component for PIV cases in the region of
$x \approx 50{-}57D$
and first spatial modes of (b) mesh, (c) plate and (d) mesh-g cases corresponding to the peak frequency of their spectra. Each plot is scaled by its maximum and minimum values.
The SPOD analysis of the PIV cases exhibits qualitatively similar results to the SPOD of the LES, i.e. a shift from smaller structures and high frequencies close to the plate to larger structures and lower frequencies in the far wake (not shown here for the whole wake). Figure 13 shows the SPOD mode spectra in the near-wake region (
$x/D \approx 0{-}7$
) of the mesh case. The first mode is energetically dominant at distinct frequencies, namely
$St \approx 3.5, 1.5, 0.85, 0.44$
, with a clear gap separating it from the less energetic higher-order modes. Additionally, the first two modes retain significant energy across a broadband frequency range of approximately
$St \approx 1{-}0.2$
. Similar spectral peaks are also evident in the PSD and CPSD computed at the inflection points of the mean velocity profiles (see figures 8
d and 10
b). The corresponding spatial structures (figure 14
a) indicate that the high-frequency peak at
$St \approx 3.5$
is associated with small-scale fluctuations concentrated in the immediate-wake region
$(x/D \lt 2)$
. As the frequency decreases
$(St \approx 1.5, 0.85, 0.44$
; see figures 14
b, 14
c and 14
d), these structures evolve into larger-scale motions extending further downstream along the shear layers, consistent with the development of Kelvin–Helmholtz-type instabilities. Comparable high-frequency shear-layer dynamics was previously observed for mesh wakes by Cimbala et al. (Reference Cimbala, Nagib and Roshko1988) (with
$St \approx 2.2$
at
$x/D = 1)$
and Huang & Keffer (Reference Huang and Keffer1996) (with
$St \approx 1.8$
at
$x/D = 1$
and
$St \approx 1.0$
at
$x/D = 2$
), where similar downstream frequency shifts were observed.
Figure 15 presents the SPOD results for the mesh, plate and mesh-g cases in the
$50{-}57D$
region, where a clear vortex street is observed (i.e. an alternating velocity pattern). In this region, the eigenvalue spectra (with only the first and second modes shown for clarity) reveal a noticeable gap, resembling a bump around
$St \approx 0.2$
, between the first and the less energetic modes. As shown in the eigenvalue plot in figure 15 the characteristic peaks of the plate and mesh-g cases are attenuated compared with those in the LES and mesh cases, suggesting weaker vortex streets, this being in agreement to the previous PSD plots (see figures 8
g–i and 8
j–l).
3.4.2. Proper orthogonal decomposition-filtered dynamic evolution
Our discussion revealed that the SVS is preceded by shear-layer structures (see § 3.4.1), which are interacting within their corresponding shear layers (see § 3.3). However, what is not clear is how the SVS is formed; in particular, whether it is formed locally, in the same manner as the conceptual Townsend–Grant eddies (Townsend Reference Townsend1956, Reference Townsend1970, Reference Townsend1976), and is thus independent of initial conditions, or whether the SVS is a direct continuation of near-wake structures (Huang & Keffer Reference Huang and Keffer1996), shaped by initial conditions and carrying this influence into the wake (see also discussions of Bevilaqua & Lykoudis Reference Bevilaqua and Lykoudis1978; Steiros et al. Reference Steiros, Obligado, Bragança, Cuvier and Vassilicos2025). It is noteworthy to mention that mean shear could play a role in both of the above mechanisms, either by directly causing the SVS in the far wake (first mechanism), or by being a factor in the formation of the near-wake structures (second mechanism).
To clarify the above, one would need to observe the real-time evolution of these structures, but the multiscale nature of the turbulent wake makes it difficult to identify such processes (see for instance figure 4 and supplementary movies 1–4). We therefore use spatial proper orthogonal decomposition (POD), which decomposes the flow field into modes sorted in descending order of turbulent kinetic energy contribution (Mendez et al. Reference Mendez, Hess, Watz and Buchlin2020). By reconstructing the flow field using a limited number of POD modes, we obtain a ‘filtered’ version of the flow that preferentially captures the large-scale dynamics (see Weiss Reference Weiss2019 for a description of the methodology). Proper orthogonal decomposition is applied to both the full LES domain and each PIV FOV, as PIV measurements were not synchronised. The filtered snapshots are recreated using modes that capture 50 % of the POD energy of the velocity fluctuations, with the mean velocity added subsequently. Different thresholds are tested, and the 50 % level best captures key wake features such as shear layers, their interaction and the SVS in terms of size, structure and vorticity strength. Higher modes are primarily associated with small-scale turbulence in the wake and free stream and are effectively filtered out. Table 2 shows the number of modes needed to capture 50 % of the energy for each case. A representation of the POD-filtered dynamics is available in supplementary movies 5–8. For the mesh-g and plate cases the POD analysis was also performed at lower energy thresholds (30 % for the plate and 20 % for the mesh-g case) to enhance the clarity of the formation process (not plotted here, see movies 9–10).
Table 2. The number of the leading POD modes required to capture 50 % of the energy level for the full LES domain and each region of PIV cases.
Figure 16. Time evolution of instantaneous POD-filtered vorticity fields with 50 % energy level (from filtered data). Panels (a) to (e) show the POD-filtered vorticity fields of LES. Panels ( f) to ( j) show the vorticity fields of the mesh case. The time interval
$\Delta t$
between snapshots is non-dimensionalised, being 0.61 for LES and 0.36 for the mesh case.
Figure 16 shows the time evolution of a few instantaneous vorticity fields filtered by POD for both the LES and mesh cases. In the LES case (figure 16
a), distinct shear layers are visible from 0 to approximately
$11D$
. These shear layers begin to roll up into distinct vorticity ‘blobs’ (for example, see those denoted by A and B in figure 16
a), which are at the initial stage of shear-layer interaction. In figure 16(b–c), A and B start to ‘invade’ the opposite shear layer, while in subsequent time steps (figure 16
d–e) the blobs arrange themselves into a well-defined vortex street (i.e. the SVS). At the location of SVS formation (approximately
$15D$
) there is intense mixing of the opposing shear layers and opposite-signed vorticity is cancelled at a very high rate, as shown by the mean enstrophy in figure 17(a). A similar pattern can be observed in the mesh case (see filtered time evolution in figure 16
f–j and mean enstrophy in figure 17
b). The above flow picture indicates that the SVS is a continuation of small shear-layer vortices, i.e. not exclusively a ‘local’ phenomenon generated by mean shear, independent of the initial conditions of the wake.
Figure 16, and in particular figure 18, where instantaneous POD-filtered snapshots of all tested cases are compared, allow an assessment of the effects that increased free-stream turbulence and body inhomogeneity have on the shear-layer evolution. It can be seen that the formation of distinct vorticity blobs in the shear layers is brought upstream in all experimental cases, compared with the ‘ideal’ LES flow. The above is in agreement with our PSD analysis which showed that the PSD peaks within the shear layers in the very near wake (i.e. below
$7D$
) are much smaller for the LES compared with the other cases (see figure 8). Once formed, however, the vorticity blobs in the LES case are more coherent and generate a stronger SVS compared with the experimental cases, as shown in the instantaneous vorticity fields of figure 4 and the spectral analysis of § 3.3. We note that the far wake of the experimental cases in figure 18 shows somewhat more coherent vortices, compared with the LES case. However, this increase in coherence is artificial, caused by the POD procedure: POD was applied to the whole LES field, and therefore the low-energy structures of the far wake seem weak compared with the energetic ones of the near wake. On the other hand, POD was applied to each individual PIV window separately, thus showing the local coherent structures in a clearer manner. Still, we opted for that presentation of the LES POD case, as it depicts the evolution of the flow field without discontinuities.
Figure 17. Mean enstrophy fields (non POD-filtered): (a) LES, (b) mesh.
Figure 18. Instantaneous POD-filtered vorticity fields with 50 % energy level: (a) LES, (b) mesh, (c) plate and (d) mesh-g.
4. Summary and conclusions
This work presented a joint experimental and numerical characterisation of the SVS instability in the far wake of a bluff body, in fully turbulent conditions. In particular, the wakes of four highly porous plates/meshes were investigated, carefully designed to have a nearly identical effective porosity, which was sufficiently high to completely suppress the (primary) Kármán vortex street from the wake. The numerical (LES) case considered the idealised scenario of zero free-stream turbulence and perfect homogeneity in the distribution of the plate porosity, which cannot be realised experimentally. The three experimental (PIV) cases considered a (relatively) low free-stream turbulence –low porosity inhomogeneity case (mesh case), a low free-stream turbulence –high porosity inhomogeneity case (plate case) and a high free-stream turbulence –low porosity inhomogeneity case (mesh-g case). The study presented instantaneous and statistical quantities, spectral analysis and a modal decomposition of the flow.
The results revealed that the SVS formation was in all cases intimately connected to very near-wake coherent structures. In particular, the SVS was found to be a stage in the evolution of the near-plate shear-layer vortices which grew at a critical size, ‘invaded’ the opposite shear layer and were then arranged in a well-defined vortex street (i.e. the SVS). The SVS vortices subsequently grew and gradually diffused in the very far wake. Throughout this sequential evolution, the dominant frequencies of the velocity PSD plots shifted to lower values, while the size of the associated vortical structures increased. Based on the above flow picture our results are therefore in support of the postulation of Huang & Keffer (Reference Huang and Keffer1996) that the SVS is linked to the initial conditions of the flow, carrying their memory inside the self-similar region of the wake, and is not, exclusively, a local flow phenomenon, as postulated by Townsend (Reference Townsend1976) and Cimbala et al. (Reference Cimbala, Nagib and Roshko1988). Naturally, local flow conditions (e.g. ambient turbulence) can influence the evolution and characteristic properties of the vortex street as it is advected downstream, in accordance with previous observations (Cimbala et al. Reference Cimbala, Nagib and Roshko1988; Williamson & Prasad Reference Williamson and Prasad1993; Gupta & Wan Reference Gupta and Wan2019).
The sensitivity of the SVS to the initial conditions was also attested to by the large effect that a variation in the inhomogeneity of the body porosity had on the wake. In fact, the effects of increasing inhomogeneity (plate case) and free-stream turbulence (mesh-g case) were similar. Both variations resulted in thicker shear layers, with distinct vorticity ‘blobs’ forming earlier, compared with the ‘ideal’ LES and mesh cases. A cross-power spectrum analysis indicated that the interaction between the two shear layers was also brought forward in the plate and mesh-g cases. Moreover, the increased inner–outer wake mixing caused by these two variations led to the weakening of the vorticity blobs in the shear layers, which in turn resulted in weaker SVS vortices that decayed more rapidly far downstream, compared with the baseline LES and mesh cases. Such observations are particularly relevant to the wake modelling of wind turbines and porous disks, in which the well-known phenomenon of wake meandering shares many similarities with the SVS (Gupta & Wan Reference Gupta and Wan2019).
Supplementary movies
Supplementary movies are available at https://doi.org/10.1017/jfm.2025.10858.
Acknowledgements
E.B. sincerely appreciates Dr G. Rigas and Professor Y. Hwang for their valuable discussions.
Funding
K.S. acknowledges support from the ERC Starting Grant ONSET No. 101163321. This work used the ARCHER2 UK National Supercomputing Service (https://www.archer2.ac.uk), with access provided by the UK Turbulence Consortium (EP/R029326/1 and EP/X035484/1).
Declaration of interests
The authors report no conflict of interest.
Data availability statement
The data that support the findings of this study are available upon request.
Appendix A. Calculation of effective porosity
The effective porosity,
$ u^* = {\overline {u_b}}/{U_{\infty }}$
, represents the ratio of the mean streamwise velocity through the mesh/plate/actuator plate over the free-stream velocity. It is computed using a mass conservation budget for a two-dimensional incompressible flow.
The chosen control volume, depicted in figure 19, extends from
$ y/D =-0.5$
to
$ y/D = 0.5$
along the body (
$d$
–
$a$
side). The exit boundary (
$b$
–
$c$
) is positioned
$2D$
downstream from the body. The boundaries
$a$
–
$b$
and
$c$
–
$d$
correspond to the top and bottom boundaries, respectively. The equation to calculate
$ \overline {u_b}$
is as follows:
\begin{equation} \overline {u_b}D = \int _{a}^{b} V_{\textit{top}} \,{\rm d}x + \int _{b}^{c} U_{\textit{exit}} \,{\rm d}y + \int _{c}^{d} V_{\textit{bottom}} \,{\rm d}x, \end{equation}
where
$D$
is the width of the body,
$U_{\textit{exit}}$
is the time- and space-averaged streamwise velocity through the exit boundary (
$b$
–
$c$
), while
$V_{\textit{top}}$
and
$V_{\textit{bottom}}$
are the time- and space-averaged
$v$
-velocities leaving the control volume through the top (
$a$
–
$b$
) and bottom boundaries (
$c$
–
$d$
), respectively.
Figure 19. Control volume and parameters for effective porosity calculation.
Appendix B. Turbulence grid characterisation used in the mesh-g case
The mesh-g case employs a regular grid placed 19 cm upstream of the mesh, aiming in an increase of the free-stream turbulence of the flow. The grid spans the flume width and has a cell size of 37.6 mm, bar thickness of 5.2 mm, while its blockage, including the encasing frame, is 33.94 %. The grid used is identical to the SRG 38 grid utilised in Kankanwadi & Buxton (Reference Kankanwadi and Buxton2020).
Figure 20 shows contours of the mean streamwise velocity (
$U_m/U_\infty$
) and turbulence intensity (
$\textit{TI}$
) downstream of the grid, while figure 21 plots the above quantities in the grid centreline (in both cases without the mesh). Both figures indicate the position of the mesh in the mesh-g case with dashed lines. The grid (without a downstream mesh) generates a
$\textit{TI}$
of 12 % at
$0D$
(mesh location), 5 % at
$15D$
and 2 % at
$70D$
downstream.
Figure 20. Grid characterisation: (a) non-dimensional mean streamwise velocity and (b) turbulence intensity fields.
Figure 21. Evolution of the turbulence intensity (
$\textit{TI}$
(%)) (left axis) and non-dimensional mean streamwise velocity (
$U_m/U_{\infty }$
) (right axis) along the wake centreline (
$y/D = 0$
) of the grid.
Appendix C. Large eddy simulation verification and validation
Results from a mesh sensitivity study are presented in figure 22. The computational set-up is the one described in § 2.2, with the only difference being in the discretisation, where three different levels are used. The ‘coarse’, ‘base’ and ‘fine’ grids discretise the domain with
$1153\times 193 \times 96$
,
$1537 \times 257 \times 128$
and
$1921 \times 321 \times 160$
points, respectively. Figure 22 shows very good agreement between the base and fine grids, with the latter being the one used for extracting the time-resolved data presented in the main text.
Figure 22. Mesh sensitivity study for the LES/actuator plate set-up. (a) Non-dimensional mean streamwise velocity along the wake centreline. Non-dimensional (b)
$\overline {u^\prime u^\prime }$
and (c)
$\overline {u^\prime v^\prime }$
Reynolds stresses at
$x/D=20$
and
$x/D=30$
.
A separate study (not shown here), using the coarse grid and a domain with double the spanwise extent (20 plate widths), was also conducted to confirm that the chosen spanwise domain size (10 plate widths) is sufficient.
To validate our LES, we conducted both experiments and simulations for a solid plate (zero porosity). Unlike the wake of porous plates, which as shown in our study, is highly sensitive to initial and ambient conditions, the solid plate offers a more controlled scenario: the porosity and its distribution can be perfectly matched, and the wake is dominated by primary Kármán vortex shedding, which is relatively insensitive to ambient turbulence (Kankanwadi & Buxton Reference Kankanwadi and Buxton2023). This controlled configuration enables the assessment of the LES’s ability to reproduce key flow features. The employed resolution corresponds to that used in the main manuscript (‘fine’ grid), but the domain size was reduced to
$60 \times 20 \times 5$
plate widths to ease computational resource requirements. Results are presented in figure 23. The LES is compared with both our PIV measurements and a direct numerical simulation (DNS) study conducted at a lower, yet still turbulent, Reynolds number (Narasimhamurthy & Andersson Reference Narasimhamurthy and Andersson2009). Good agreement is observed with both the experimental and numerical reference data, especially given the differences in ambient turbulence conditions and plate thickness.
Figure 23. The wake of a solid plate normal to the flow. (a) Non-dimensional mean streamwise velocity along the wake centreline. Non-dimensional (b)
$\overline {u^\prime u^\prime }$
and (c)
$\overline {u^\prime v^\prime }$
Reynolds stresses at
$x/D=5$
. The DNS data (at
${Re} = 750$
) are extracted from Narasimhamurthy & Andersson (Reference Narasimhamurthy and Andersson2009).
Appendix D. Time-averaged enstrophy
Figure 24 shows the evolution of the time-averaged square of non-dimensional
$z$
-vorticity (time-averaged enstrophy) along the wake centreline (
$y/D = 0$
). In the PIV cases, enstrophy peaks just downstream of the mesh/plate due to flow passage but decreases to a minimum shortly downstream while the LES case lacks this initial peak due to the perfect homogeneity of the actuator plate. In all cases, enstrophy begins to rise from this minimum as the edges of the growing shear layers reach the wake centreline. This minimum indicates the location just before the shear layers begin to influence the centreline and can be interpreted as the onset of shear-layer interaction.
Figure 24. Evolution of the time-averaged square of non-dimensional
$z$
-vorticity (time-averaged enstrophy) (from raw data) along the wake centreline (
$y/D = 0$
) and locations of minimum enstrophy for all cases.
Appendix E. The SPOD parameters
Spectral proper orthogonal decomposition decomposes stochastic flow data into spatial structures that are coherent in both space and frequency. The SPOD employs a second-order space–time formulation under the assumption of statistical stationarity, where the mean and variance remain constant (Lumley Reference Lumley1967, Reference Lumley1970). The SPOD process involves estimating the cross-spectral density tensor using Welch’s periodogram method, followed by computing its eigenvalues and eigenvectors, which represent energy ranking and spatial structures at specific frequencies. For further details, see Towne et al. (Reference Towne, Schmidt and Colonius2018) and Schmidt & Colonius (Reference Schmidt and Colonius2020).
To improve SPOD results, sampling parameters from Welch’s method – such as the number of blocks (
$N_{blk}$
), snapshots per block (
$N_{\textit{FFT}}$
), overlaps between blocks (
$N_{ovlp}$
) and the window function – are optimised based on recommendations by Welch (Reference Welch1967) and Schmidt & Colonius (Reference Schmidt and Colonius2020). These adjustments ensure sufficient frequency resolution and statistical convergence, reducing mode energy contamination. The optimised SPOD parameters for this study are summarised in table 3.
Table 3. Chosen values of SPOD parameters.
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