1. Introduction
Geometry-induced flow separation and reattachment occur in diverse engineering and environmental applications, and their inherent spatio-temporal dynamics can profoundly influence the structural integrity and performance of fluid engineering systems. For instance, periodically shed vortices from high-rise buildings, wind turbines and long-span bridges can exert unsteady aerodynamic loads on these structures, which can lead to vortex-induced vibrations and fatigue failure (Kawai Reference Kawai1992; Wu & Kareem Reference Wu and Kareem2012; Grinderslev et al. Reference Grinderslev, Houtin-Mongrolle, Sørensen, Pirrung, Jacobs, Ahmed and Duboc2023). Similarly, flow separation beneath an ice cover on cold-region rivers during winter can modify river-ice processes and local hydraulic conditions, which often result in a reduction in flow conveyance and the operational efficiency of hydroelectric power systems as well as extreme events, including flooding and property damage (Beltaos Reference Beltaos1983; Fang et al. Reference Fang, Dow, Tachie, Jarrod and Wang2023). Motivated by their widespread applications and fundamental importance in turbulent shear flows, significant research attention has been devoted to understanding the dynamics of flow separation and reattachment over various bluff bodies. These include flows over wall-mounted bluff bodies (Piirto et al. Reference Piirto, Saarenrinne, Eloranta and Karvinen2003; Fang & Tachie Reference Fang and Tachie2019; Li & Wang Reference Li and Wang2024) and bluff bodies that are partially submerged beneath a free surface (Fang et al. Reference Fang, Tachie and Dow2022; Agyei-Agyemang et al. Reference Agyei-Agyemang, Essel and Tachie2024; Israel et al. Reference Israel, Dow, Clark and Tachie2024) or in uniform flow (Arie & Rouse Reference Arie and Rouse1956; Kiya & Sasaki Reference Kiya and Sasaki1983; Moore et al. Reference Moore, Letchford and Amitay2019; Cimarelli et al. Reference Cimarelli, Corsini and Stalio2024).
Square and rectangular prisms with sharp leading edges in uniform flow have received special research attention owing to their geometric simplicity. The separated flow is exemplified by the presence of small-scale Kelvin–Helmholtz (KH) vortices, which originate from the roll-up of the separated shear layers formed at the leading edge, and von Kármán (VK) vortices shed alternately from either side of the prism and manifest as a sequence of alternating signed large-scale vortices convected downstream. The effects of Reynolds number,
$ \textit{Re}_H$
=
$U_{\infty }H$
/
$\nu$
(where
$U_{\infty }$
,
$H$
and
$\nu$
are the free-stream velocity, prism thickness and kinematic viscosity, respectively), chord-to-thickness ratio,
$ \textit{CR}$
=
$C$
/
$H$
(where
$C$
is the streamwise chord length of the rectangular prism) and blockage ratio,
$ \textit{BR}$
(defined as the ratio of the body height to the flow depth), on the KH instability and VK vortex shedding have been critically examined (Okajima Reference Okajima1982; Moore et al. Reference Moore, Letchford and Amitay2019; Abdul-Salam & Tachie Reference Abdul-Salam and Tachie2024; Kumahor & Tachie Reference Kumahor and Tachie2023; Cimarelli et al. Reference Cimarelli, Corsini and Stalio2024). These investigations demonstrated that the dimensionless vortex shedding frequency, or Strouhal number,
$St_{\textit{VS}}$
(=
$f_{\textit{VS}}H$
/
$U_{\infty }$
, where
$f_{\textit{VS}}$
is the vortex shedding frequency), depends on the chord-to-thickness ratio but is independent of Reynolds number at moderate and high values. Previous studies have also demonstrated that, as the chord-to-thickness ratio increases,
$f_{\textit{KH}}$
/
$f_{\textit{VS}}$
, where
$f_{\textit{KH}}$
is the frequency of KH instability, decreases due to the reduced interaction between the KH instability and wake-induced VK instability (Moore et al. Reference Moore, Letchford and Amitay2019; Kumahor & Tachie Reference Kumahor and Tachie2023). Abdul-Salam & Tachie (Reference Abdul-Salam and Tachie2024) also demonstrated that
$f_{\textit{KH}}$
/
$f_{\textit{VS}}$
increases with blockage ratio.
Kiya & Sasaki (Reference Kiya and Sasaki1983, Reference Kiya and Sasaki1985) and Cherry et al. (Reference Cherry, Hillier and Latour1984) investigated flow separation over a long rectangular plate in uniform flow and identified two distinct unsteady features: a quasi-periodic vortex shedding at a frequency of
$f_{\textit{VS}}$
= (0.5–0.6)
$U_{\infty }$
/
$L_P$
, where
$L_P$
is the streamwise length of the recirculation region, and quasi-periodic enlargement and shrinkage of the separation bubble, a phenomenon commonly referred to as ‘low-frequency flapping motion’, at a frequency of
$f_{\textit{FM}}$
= (0.1–0.2)
$U_{\infty }$
/
$L_P$
, where
$f_{\textit{FM}}$
is the frequency of flapping motion. This low-frequency flapping motion was observed in other separated flows, including those over rectangular prisms (Tafti & Vanka Reference Tafti and Vanka1991; Fang et al. Reference Fang, Tachie and Dow2022; Fang & Wang Reference Fang and Wang2024), backward-facing steps (Eaton & Johnston Reference Eaton and Johnston1981; Spazzini et al. Reference Spazzini, Iuso, Onorato, Zurlo and Di Cicca2001) and forward–backward-facing steps (Chalmers et al. Reference Chalmers, Fang and Tachie2021). While different mechanisms have been proposed to explain the dynamics of flapping motion in these studies, the characteristic frequencies are within the range of
$f_{\textit{FM}}$
= (0.08–0.2)
$U_{\infty }$
/
$L_P$
.
Despite extensive research on geometry-induced flow separation, knowledge of the spatio-temporal development of the separated shear layers is limited. Spazzini et al. (Reference Spazzini, Iuso, Onorato, Zurlo and Di Cicca2001) studied the separated shear layer induced by a backward-facing step using hot-wire anemometry (HWA) and particle image velocimetry (PIV). They reported that the evolution of the separated shear layer is governed by a cyclic interaction between the primary vortex shed from the step edge and a secondary vortex formed in the corner of the step and bottom wall, with the secondary vortex growing and decaying in phase with the development of the primary vortex, periodically altering the reattachment location and modulating the global unsteadiness of the separated shear layer. The spatio-temporal development of the separated shear layer over a rectangular prism with
$ \textit{CR}$
= 5 was investigated by Moore et al. (Reference Moore, Letchford and Amitay2019) using HWA and PIV, and by Cimarelli et al. (Reference Cimarelli, Corsini and Stalio2024) using direct numerical simulation. In both studies, the KH frequency continuously migrates towards the comparatively lower vortex shedding frequency along the separated shear layer. Fang et al. (Reference Fang, Tachie and Dow2022) evaluated the effects of surface roughness on the separated shear layer beneath submerged rectangular prisms with
$ \textit{CR}$
= 10 using time-resolved PIV. They observed that the KH instability was bypassed in the rough cases. For the smooth case, the KH frequency decreases discretely to lower frequencies along the separated shear layer due to successive vortex pairing events and the growth of larger vortical structures, and the frequency migration ceases around the middle of the recirculation bubble. Fang et al. (Reference Fang, Tachie and Dow2022) attributed the continuous frequency migration in Moore et al. (Reference Moore, Letchford and Amitay2019) and Cimarelli et al. (Reference Cimarelli, Corsini and Stalio2024) to the interruption of the development of the KH instability by the VK vortex shedding in the wake region.
In the present study, a vertical fence with a splitter plate in uniform flow is used to provide fundamental insight into the spatio-temporal dynamics of separated shear layers. This flow configuration is adopted because the recirculation bubble formed downstream of the fence may extend up to
$L_P$
= 17.2
$h$
–31.7
$h$
(where
$h$
is the fence height above the splitter plate), depending on the blockage ratio and Reynolds number (Smits Reference Smits1982; Ruderich & Fernholz Reference Ruderich and Fernholz1986; Hancock Reference Hancock1995, Reference Hancock2000), which is significantly longer than 2.5
$H$
–4.8
$H$
reported for rectangular prisms (Moore et al. Reference Moore, Letchford and Amitay2019; Smith et al. Reference Smith, Pisetta and Viola2021; Liu et al. Reference Liu, Kumahor and Tachie2024; Cimarelli et al. Reference Cimarelli, Corsini and Stalio2024). This enlarged recirculation region provides a sufficiently large domain over which the spatial and temporal development of vortical structures can be examined in detail. Additionally, the separation point is elevated above the splitter plate by the fence, therefore, the spatial development of the separated shear layer is less constrained by the splitter plate than in the case of rectangular prisms.
Separated flows over a fence with a splitter plate in uniform flow have been studied in the past, however, the velocity measurements were performed using Pitot tubes, hot-wire or pulsed-wire anemometry. Arie & Rouse (Reference Arie and Rouse1956) demonstrated that the splitter plate suppresses vortex shedding and reduces the turbulence levels in the separated shear layer compared with those of a fence without a splitter plate. Hancock (Reference Hancock1995, Reference Hancock2000) found that the mean flow characteristics, including the reattachment length, are independent of Reynolds number over the range 2000
$\leq$
$ \textit{Re}_h$
$\leq$
13 000. Smits (Reference Smits1982), who investigated blockage ratio effects over 1.4 %
$\leq$
$ \textit{BR}$
$\leq$
12.3 %, found that the reattachment length decreases with increasing blockage ratio due to enhanced flow entrainment at higher blockage ratios. Ruderich & Fernholz (Reference Ruderich and Fernholz1986) observed that the integral length scale increases along the shear layer. Castro & Haque (Reference Castro and Haque1987, Reference Castro and Haque1988) reported larger integral time scales in the first half of the separation bubble, corresponding to a frequency of 0.005
$U_{\infty }$
/
$h$
, compared with those measured near reattachment. The larger integral time scales in the first half of the separation bubble were associated with a low-frequency quasi-periodic ‘bursting’ of the separation bubble. Hudy et al. (Reference Hudy, Naguib and Humphreys2003) identified two distinct frequency ranges: a lower-frequency range, (0.004–0.008)
$U_{\infty }$
/
$h$
, and a higher-frequency range, (0.023–0.035)
$U_{\infty }$
/
$h$
, which were linked to the flapping motion of the shear layer and the vortex shedding motion, respectively. Despite the valuable insights gained from these studies, knowledge of the spatio-temporal dynamics of the separated shear layer over the fence-splitter-plate configuration is still limited. In particular, the dynamics of the main instabilities (e.g. the KH instability, vortex shedding and flapping motions), the spatial development of the coherent structures associated with these instabilities, as well as momentum transport and turbulence production, remain poorly understood.
The objectives of the present study are to investigate the spatio-temporal dynamics of the separated shear layer induced by a fence with a splitter plate in uniform flow, to characterise the dominant coherent structures and momentum transport, and to elucidate the dynamic role of low-frequency flapping motion in turbulence production. To achieve these objectives, a time-resolved particle image velocimetry (TR-PIV) system was employed to obtain whole-field velocity measurements at a sufficiently high sampling frequency to resolve the aforementioned instabilities. The remainder of this paper is organised as follows: the experimental set-up and measurement procedure are described in § 2, the results and discussion are presented in § 3, and the major conclusions of the present study are summarised in § 4.
2. Experimental procedure
The experiments were conducted in an open recirculating water channel located in the Turbulence and Hydraulic Engineering Laboratory (THEL) at the University of Manitoba. This facility was previously employed for studies on separated flows (Fang & Tachie Reference Fang and Tachie2019, Reference Fang and Tachie2020; Essel et al. Reference Essel, Tachie and Balachandar2021, Reference Essel, Balachandar and Tachie2023; Kumahor & Tachie Reference Kumahor and Tachie2023). Therefore, only the key features are summarised here for brevity. The length, width and height of the test section are 6.00 m, 0.60 m and 0.45 m, respectively. The side and bottom walls of the test section were manufactured using 31.8 mm thick Super Abrasion
$\textrm {Resistant}^{\circledR }$
transparent acrylic plates to facilitate optical access to the flow.
Figure 1(a) presents a schematic of the experimental set-up, the fields of view (
$\mathrm{FOV}_{1}$
and
$\mathrm{FOV}_{2}$
) of the cameras used to image the flow field and the Cartesian coordinate system employed in this study. The origin of the coordinate system is located at the corner of the upper half of the fence and the splitter plate, with
$x$
,
$y$
and
$z$
denoting the streamwise, vertical and spanwise directions, respectively. The fence was fabricated from aluminium and the splitter plate was machined from smooth acrylic. The fence has a thickness of
$t$
= 3.0 mm and a height of
$H$
= 27.0 mm, as shown in figure 1(b). A
$45^\circ$
angle is introduced at the leading corners of the fence, with the angle measured relative to the vertical front face of the fence. The splitter plate has a thickness of 6.00 mm and a length of
$L$
= 1000 mm. As shown in figure 1(c), the fence-splitter-plate assembly was mounted between two 6.0 mm thick acrylic plates that were clamped to the sidewalls of the test section. Following Hudy et al. (Reference Hudy, Naguib and Humphreys2003), the vertical distance from the upper surface of the splitter plate to the upper tip of the fence (
$h$
= 10.5 mm) was employed as the characteristic length scale for data analysis. The aspect ratio of the splitter plate was
$L$
/
$h$
= 95.2, which is sufficiently large to ensure permanent flow reattachment onto the splitter plate with no interaction between the separated shear layer from the leading edge of the fence and the wake behind the splitter plate (Smith et al. Reference Smith, Pisetta and Viola2021). The widths of the fence and the splitter plate were
$W = 600$
and
$588$
mm, resulting in aspect ratios of
$W/h = 57.1$
and
$56.0$
, respectively. Preliminary measurements in the
$x{-}z$
plane at
$y$
/
$h$
= 0.5 demonstrated that variations in the mean velocities and Reynolds stresses were within measurement uncertainties.
The water depth was maintained at
$D$
= 430 mm and the blockage ratio
$ \textit{BR}$
(=
$H$
/
$D$
) was 6.3 %. The free-stream velocities were
$U_{\infty }$
= 0.26, 0.54 and 0.73 m s−1, with corresponding Reynolds numbers
$ \textit{Re}_h$
(=
$U_{\infty }h$
/
$\nu$
) = 2700, 5500 and 7500, respectively. Preliminary velocity measurements performed in the absence of the bluff body showed that the boundary layer thicknesses on the bottom wall and sidewalls were less than 20 mm. Outside the boundary layers, the free-stream turbulence intensity
$T_u$
(=
$u'_{\textit{rms}}$
/
$U_{\infty }$
) was 1.3 %, where r.m.s. denotes root-mean-square. Based on these results, the fence and the splitter plate were placed 200 mm above the bottom wall to ensure that they were established in a uniform flow.
(a) Schematic (not to scale) of the test geometry, the coordinate system and the fields of view (
$\mathrm{FOV}_{1}$
and
$\mathrm{FOV}_{2}$
) of the two cameras, (b) detailed view of the fence geometry and (c) top view of the fence-splitter-plate assembly.
Time-resolved particle image velocimetry was used to perform two-component planar velocity measurements at the channel mid-span. The water flow was seeded with silver-coated hollow glass spheres with a nominal diameter of
$d_p$
= 10
$\unicode{x03BC}$
m and a specific gravity of 1.4. A diode-pumped, dual-cavity, high-speed Neodymium-doped yttrium lithium fluoride (Nd:YLF) laser with a wavelength of 527 nm and a maximum pulse energy of 30 mJ pulse−1 per cavity was positioned above the test section to illuminate the seeding particles. To mitigate laser sheet distortion caused by the free surface, an acrylic plate with a thickness of 9 mm, spanning the entire width of the test section and measuring 600 mm in length, was partially immersed in the water to a depth of 2 mm. The illuminated particle images were captured simultaneously by two side-by-side high-speed 12-bit Phantom VEO340L complementary metal-oxide-semiconductor (CMOS) cameras, each with a resolution of 2560 pixels
$\times$
1600 pixels. The cameras for
$\mathrm{FOV}_{1}$
and
$\mathrm{FOV}_{2}$
in figure 1(a) were fitted with a Sigma 105 mm lens and a Nikon 60 mm lens, respectively. The aperture was set to f/8 for both cameras. This arrangement was used to optimise spatial resolution near the fence, where the shear layer is relatively thin, while maximising the spatial extent of the total measurement domain. The dimensions of
$\mathrm{FOV}_{1}$
and
$\mathrm{FOV}_{2}$
were 125 mm
$\times$
80 mm and 231 mm
$\times$
148 mm, respectively. A 10 mm overlap was maintained in the streamwise direction to ensure continuous spatial coverage. Based on a convergence test, 60 000 samples were collected per field of view at a sampling frequency of 600 Hz.
Commercial software, DaVis version 10.0.5, supplied by LaVision Inc. (Göttingen, Germany), was used for data acquisition, image processing and velocity vector calculations. A multi-pass cross-correlation algorithm was used for velocity vector calculations (Willert & Gharib Reference Willert and Gharib1991; Westerweel Reference Westerweel1997; Scarano & Riethmüller Reference Scarano and Riethmüller2000). The initial interrogation area was 128 pixels
$\times$
128 pixels with 50 % overlap and the final interrogation area was 24 pixels
$\times$
24 pixels with 75 % overlap. The vector spacing was 0.29 mm (0.027
$h$
) for
$\mathrm{FOV}_{1}$
and 0.55 mm (0.052
$h$
) for
$\mathrm{FOV}_{2}$
. The window deformation technique (Scarano Reference Scarano2002) was applied in a multi-pass scheme, deforming and shifting interrogation windows based on predicted particle displacements to improve correlation accuracy by tracking identical particles across image pairs. A minimum correlation value of 0.5 for the primary correlation peak was imposed to ensure the reliability of the correlations. Spurious vectors were identified and removed using the universal outlier detection algorithm (Westerweel & Scarano Reference Westerweel and Scarano2005) with a 5
$\times$
5 neighbourhood median filter, and regions containing fewer than five vectors were discarded. The removed spurious vectors were subsequently replaced through interpolation. Post-processing of velocity vectors was performed with in-house
$\textrm {MATLAB}^{\circledR }$
scripts.
Statistical sampling uncertainty analysis was performed following the formulations of Benedict & Gould (Reference Benedict and Gould1996) and Bendat & Piersol (Reference Bendat and Piersol2010), as applied to PIV-derived statistical quantities by Sciacchitano & Wieneke (Reference Sciacchitano and Wieneke2016). The statistical sampling uncertainties at the 95 % confidence level for the mean velocity, Reynolds normal stresses and Reynolds shear stress were estimated to be
$\pm$
3 %,
$\pm$
7 % and
$\pm$
5 %, respectively.
3. Results and discussion
Since the results for all three Reynolds numbers (
$ \textit{Re}_h$
= 2700, 5500 and 7500) are qualitatively similar, most of the figures presented in this section are limited to the
$ \textit{Re}_h$
= 5500 case for brevity. Nonetheless, pertinent Reynolds number effects on the results are discussed.
3.1. Mean flow field
Figure 2(a) presents the contour of streamwise mean velocity (
$U$
), overlaid with mean streamlines. The incoming flow separates at the fence leading edge and reattaches on the splitter plate. Upon reattachment on the splitter plate, part of the flow is directed downstream to form a new boundary layer. The rest travels upstream, forming a large, clockwise-rotating primary vortex above the fence. The reverse flow beneath the primary vortex is redirected downward by the fence, forming a smaller counter-rotating secondary vortex in the corner of the fence and splitter plate (see figure 2
b). Isopleths of
$U$
= 0 and
$\phi (x,y=Y) = \int _{0}^{Y} U(x,y)\, {\rm d}y = 0$
, where
$\phi$
is the streamfunction for a two-dimensional incompressible mean flow, and
$Y$
represents the vertical location of
$\phi$
= 0, are overlaid to delineate the centre and upper bound of the primary and secondary vortices; the latter bounds the primary vortex and defines the mean separating streamline (Castro & Haque Reference Castro and Haque1987, Reference Castro and Haque1988; Mohammed-Taifour & Weiss Reference Mohammed-Taifour and Weiss2016; Fang & Tachie Reference Fang and Tachie2019; Li & Wang Reference Li and Wang2024). These isopleths originate from the leading edge of the fence and intersect with the splitter plate at the same location, marking the reattachment point of the primary vortex. Both the maximum streamwise mean velocity (
$U_{\textit{max}}$
= 1.22
$U_{\infty }$
, 1.27
$U_{\infty }$
and 1.29
$U_{\infty }$
for
$ \textit{Re}_h$
= 2700, 5500 and 7500, respectively) and backflow velocity (
$U_b$
$\approx$
−0.35
$U_{\infty }$
for all Reynolds numbers) occur in the mid-region of the primary vortex. The maximum backflow velocity agrees well with −0.3
$U_{\infty }$
reported for a similar bluff body configuration (Ruderich & Fernholz Reference Ruderich and Fernholz1986; Castro & Haque Reference Castro and Haque1987, Reference Castro and Haque1988; Hardman Reference Hardman1998). The maximum forward flow velocity of the secondary vortex is
$U_{\kern-1pt f}$
$\approx$
0.08
$U_{\infty }$
for all Reynolds numbers.
Contours of (a) streamwise mean velocity
$U$
, overlaid with mean streamlines (black solid lines), and (b) the mean velocity vector field near the corner of the fence and the splitter plate. The red solid and green dashed lines represent the isopleths of
$U=0$
and
$\phi$
= 0. Symbols
$+$
and
$\times$
in (a) mark the locations of maximum and minimum
$U$
, respectively.
The mean reattachment length of the primary vortex, defined as the streamwise distance from the origin of the coordinate system to the reattachment point, is
$L_P = 22.6h$
at
$ \textit{Re}_h = 2700$
and
$L_P = 21.7h$
at both
$ \textit{Re}_h = 5500$
and 7500. For the secondary vortex, the mean recirculation length, defined as the streamwise length from the intersection point of isopleths of
$\phi$
= 0 and
$U$
= 0 near the fence to the origin of the coordinate system, is
$L_S$
$\approx$
2.6
$h$
for all Reynolds numbers.
To examine the dynamics of shear layer development, figures 3(a,b) present contours of the streamwise and vertical mean velocity gradients,
$\partial U$
/
$\partial x$
and
$\partial U$
/
$\partial y$
, respectively. These gradients are also dynamically important in the transport of momentum and turbulence production, as discussed in §§ 3.4 and 3.6.2, respectively. Since
$\partial V$
/
$\partial x$
is orders of magnitude smaller than the other velocity gradients, and
$\partial V$
/
$\partial y$
= −
$\partial U$
/
$\partial x$
because of mass conservation and spanwise homogeneity,
$\partial V$
/
$\partial x$
and
$\partial V$
/
$\partial y$
are not shown. In figure 3(a), a region of positive
$\partial U$
/
$\partial x$
forms above the fence, indicative of mean flow acceleration induced by the sharp leading edge of the fence. Along the mean separating streamline,
$\partial U$
/
$\partial x$
is negative in the first half of the primary vortex, reflecting sustained mean flow deceleration as the shear layer evolves towards the vortex crest. Beyond the crest, the flow transitions to acceleration. Figure 3(b) reveals strong positive
$\partial U$
/
$\partial y$
along the mean separating streamline, which gradually weakens downstream. Such distributions are associated with localised velocity deficit (Fang & Wang Reference Fang and Wang2024) and momentum redistribution (Cimarelli et al. Reference Cimarelli, Corsini and Stalio2024). In the first half of the primary vortex, the coexistence of rapid streamwise mean flow deceleration and intense mean shear produces a velocity deficit and redistribution of momentum from the streamwise to the vertical direction, facilitating the upward deflection of the shear layer. Around the mid-bubble region, however, the opposing actions of strong free-stream and reverse flow result in reductions in both gradients. As the shear layer impinges on the splitter plate,
$\partial U$
/
$\partial y$
decays continuously due to enhanced mixing. Downstream of reattachment, both gradients decrease rapidly. In the corner of the fence and the splitter plate, the sign change in
$\partial U$
/
$\partial x$
(figure 3
a) and locally intensified
$\partial U$
/
$\partial y$
(figure 3
b) underscore the effect of geometric confinement in shaping the local flow dynamics.
Contours of mean velocity gradients, (a)
$\partial U$
/
$\partial x$
and (b)
$\partial U$
/
$\partial y$
. The green dashed and blue dash-dot-dotted lines represent the mean separating streamline and the isopleth of
$\partial U$
/
$\partial x_i$
= 0, respectively.
3.2. Vorticity thickness and dynamics along the shear layer centre
As noted by Hancock (Reference Hancock1995, Reference Hancock2000), the separated shear layer induced by a fence with a splitter plate behaves as a mixing layer subjected to an additional mean strain arising from mean streamline curvature and fluctuating strain imposed on the low-speed flow side by entrainment of the free-stream fluid near reattachment. Accordingly, the vorticity thickness,
$\delta _\omega$
, originally proposed for plane mixing layers (Brown & Roshko Reference Brown and Roshko1974) and adopted for separating flows (Ruderich & Fernholz Reference Ruderich and Fernholz1986; Castro & Haque Reference Castro and Haque1987), is used to quantify the growth rate of the separated shear layer:
\begin{align} \delta _{\omega} = \frac {\Delta U}{(\partial U/\partial y)_{\textit{max}}} \end{align}
Here,
$(\partial U/\partial y)_{\textit{max}}$
denotes the maximum vertical gradient of the streamwise mean velocity, and the velocity difference is defined as
$\Delta U = U_{\textit{max}} - U_{\textit{min}}$
, with
$U_{\textit{max}}$
and
$U_{\textit{min}}$
denoting the local maximum and minimum of streamwise mean velocity at each streamwise location. Within the recirculation bubble,
$U_{\textit{min}}$
corresponds to the local maximum backflow. Beyond the mean reattachment point, where a new boundary layer develops, some studies adopted
$U_{\textit{min}} = U_{min-w} (=0)$
(Ruderich & Fernholz Reference Ruderich and Fernholz1986; Castro & Haque Reference Castro and Haque1987, Reference Castro and Haque1988), while others adopted
$U_{\textit{min}} = U_{min-I}$
, which is the local minimum streamwise mean velocity outside the developing wall boundary layer (Hancock Reference Hancock2000; Essel et al. Reference Essel, Tachie and Balachandar2021).
Figure 4(a) shows streamwise variations of
$U_{\textit{max}}$
,
$U_{min-I}$
and
$U_{min-w}$
,
$\Delta U_I = U_{\textit{max}} - U_{min-I}$
and
$\Delta U_w = U_{\textit{max}} - U_{min-w}$
. Here,
$\Delta U_I$
represents the velocity difference across the separated shear layer, whereas
$\Delta U_w$
is referenced to the wall. In the figure, the profiles of
$\Delta U_I$
at
$ \textit{Re}_h$
= 2700, 5500 and 7500 are nearly identical. Starting from a value of 1.0 at the fence leading edge,
$\Delta U_I$
increases to a peak value of 1.6 in the mid-bubble region, where the backflow reaches its maximum magnitude, and subsequently decreases to 1.1 at reattachment. This peak value agrees well with 1.5 reported by Castro & Haque (Reference Castro and Haque1987) and Hancock (Reference Hancock2000). Upstream of the reattachment point, the evolution of
$\Delta U$
is governed by the initial acceleration of the reverse flow towards the mid-bubble region followed by its deceleration towards the fence and the formation of the secondary vortex. Downstream of reattachment,
$\Delta U_w$
is nearly constant and coincident with
$U_{\textit{max}}$
, whereas
$\Delta U_I$
continues to decay due to the growth of the new internal boundary layer.
Streamwise variations of (a) velocity differences,
$\Delta U_I$
and
$\Delta U_{w}$
, and local maximum and minimum of streamwise mean velocities,
$U_{\textit{max}}$
,
$U_{min-I}$
and
$U_{min-w}$
, (b) vorticity thickness,
$\delta _\omega$
, (c) loci of shear layer centre,
$Y_s$
, and the mean separating streamline and (d) deflection angle,
$\theta$
, of the shear layer centre. The vertical lines in figures (b, c, d) mark the boundaries of the defined regions (regions I, II, III and IV).
Figure 4(b) presents the streamwise variations of vorticity thickness,
$\delta _{\omega -I}$
and
$\delta _{\omega -w}$
, computed using
$\Delta U_I$
and
$\Delta U_w$
, respectively. Similar to
$\Delta U_I$
, the profiles of
$\delta _{\omega -I}$
show no appreciable variation over the investigated Reynolds number range. Within the separation bubble, three distinct linear growth rates are identified. The initial growth rate of
$k_\omega$
= 0.13 in the range of
$x$
/
$h$
$\lt$
7.5 aligns closely with previously reported values of
$k_\omega$
$\in [0.15,\,0.22]$
for turbulent separation bubbles and mixing layers (Brown & Roshko Reference Brown and Roshko1974; Cherry et al. Reference Cherry, Hillier and Latour1984; Ruderich & Fernholz Reference Ruderich and Fernholz1986; Hancock Reference Hancock2000; Cimarelli et al. Reference Cimarelli, Corsini and Stalio2024). This growth rate is attributed to the onset of the KH instability (Cimarelli et al. Reference Cimarelli, Corsini and Stalio2024). Over the range of
$x$
/
$h$
= 7.5–18.0, the growth rate decreases sharply to
$k_\omega$
= 0.06, owing to the opposing actions of the free-stream and reverse flow, which smooth the velocity gradients and stabilise the shear layer. In the downstream range of
$x$
/
$h$
= 18.0–21.7, the growth rate increases to
$k_\omega$
= 0.26. This change is explained by the impingement of the shear layer on the splitter plate, which induces large-scale vortex breakdown and enhances the shear layer growth (Li et al. Reference Li, Long and Wang2022). Beyond reattachment (
$x$
/
$h$
$\gt$
21.7),
$\delta _{\omega -I}$
increases with
$k_\omega$
= 0.10, whereas
$\delta _{\omega -w}$
decreases. The former trend is consistent with the observations of Hardman (Reference Hardman1998) and Li et al. (Reference Li, Long and Wang2022). Guided by the above analysis, the separated shear layer is divided into four regions: regions I, II, III and IV, marked by the locations where the growth rate of the shear layer changes.
The trajectory of the separated shear layer is characterised by tracking the streamwise evolution of its centre. Following Cimarelli et al. (Reference Cimarelli, Corsini and Stalio2024), the shear layer centre is defined as the location of the maximum mean spanwise vorticity,
$\omega _{z, max}$
. Figure 4(c) presents the streamwise variation of the shear layer centre,
$Y_s$
, and the mean separating streamline. Immediately downstream of the fence leading edge, the shear layer rapidly deflects upward and reaches its maximum in the mid-bubble region, closely following the mean separating streamline. Thus, the shear layer in this region is primarily governed by mean flow advection (Cimarelli et al. Reference Cimarelli, Corsini and Stalio2024). Farther downstream, both the shear layer and the mean separating streamline turn downward until the end of region II. While the mean separating streamline continues this downward trajectory and intersects with the splitter plate, the shear layer centre progressively departs from the mean separating streamline in regions III and IV.
To complement the trajectory analysis in figure 4(c), figure 4(d) presents the streamwise variations of the shear layer deflection angle, which quantifies the local orientation of the separated shear layer. The deflection angles are defined based on the local shear layer centre slope,
$\theta _g = \tan ^{-1}\! ( {\Delta Y_s}/{\Delta x} )$
(Kumahor & Tachie Reference Kumahor and Tachie2023) and the local mean velocities along the shear layer centre,
$\theta _v = \tan ^{-1}\! ( {V}/{U} )$
(Castro & Haque Reference Castro and Haque1987). All profiles exhibit a similar trend. Specifically, they are maximum at the separation point, decrease through regions I and II and transition from positive to negative values around the mid-bubble region. The reduction in deflection angles coincides with the near-flat vorticity thickness distribution (figure 4
b), reinforcing the stabilisation of the separated shear layer. In regions III and IV, the magnitude of the deflection angle continues to decrease, in agreement with the progressive flattening of the shear layer centre induced by the wall confinement.
Streamwise variations of (a) volumetric flow rate,
$Q_x$
, and (b) entrainment parameter,
$\beta$
. The vertical lines denote the boundaries of the defined regions (regions I, II, III and IV).
3.3. Entrainment
Li & Wang (Reference Li and Wang2024) demonstrated that the growth of the separated shear layer is strongly influenced by turbulent entrainment, which governs momentum exchange between the shear layer and the surrounding flow. This process, initiated at the leading edge, is linked to the continuous deformation and folding of the interface of the separated shear layer driven by turbulent motions (Cimarelli et al. Reference Cimarelli, Corsini and Stalio2024). To quantify this process, an entrainment velocity,
$V_e$
, is defined as follows (Cimarelli et al. Reference Cimarelli, Corsini and Stalio2024):
\begin{align} V_e = \frac {{\rm d}Q_x}{{\rm d}x} , \end{align}
where the volumetric flow rate,
$Q_x$
, is computed as
\begin{align} Q_x = \int _{0}^{y_{\omega _z}(x)} U(x,y)\,{\rm d}y . \end{align}
Here,
$y_{\omega _z}(x)$
denotes the streamwise evolving position of the shear layer interface, defined as the vertical location where the mean spanwise vorticity decays to
$|\omega _z(x,y_{\omega _z})| = 0.015\,|\omega _z|_{\textit{max}}(x)$
following a sensitivity analysis to distinguish the rotational flow region from the outer irrotational flow.
Figures 5(a,b) show the streamwise variations of
$Q_x$
and
$\beta = V_e / \Delta U_I$
, where
$\beta$
is the entrainment parameter that assesses the effectiveness of entrainment (Turner Reference Turner1986; Cimarelli et al. Reference Cimarelli, Corsini and Stalio2024). Downstream of the fence in region I,
$Q_x$
remains weak for
$x$
/
$h$
$\lt$
2.0, but increases rapidly to a local peak of
$\beta$
= 0.14 at
$x$
/
$h$
= 2.5 (figure 5
b). This value is twice the value (
$\beta$
= 0.07) reported in the vicinity of the leading edge of a rectangular prism (Cimarelli et al. Reference Cimarelli, Corsini and Stalio2024). This discrepancy likely arises from the stronger secondary vortex in the corner of the fence and splitter plate in the present configuration, which enhances the deformation and upward deflection of the shear layer interface and thereby promotes entrainment of high-momentum free-stream fluid into the shear layer. Farther downstream in regions I and II,
$Q_x$
continues to increase such that
$\beta$
remains approximately constant at
$\beta$
$\approx$
0.12 up to the mid-bubble region. This value is consistent with
$\beta$
$\approx$
0.12 reported by Cimarelli et al. (Reference Cimarelli, Corsini and Stalio2024), and comparable to 0.15 commonly reported in plane mixing layers (Sreenivasan et al. Reference Sreenivasan, Ramshankar and Meneveau1989). In region III, as the shear layer impinges on the splitter plate,
$Q_x$
increases rapidly, producing a second peak of
$\beta$
= 0.38 near reattachment. Downstream of reattachment (region IV),
$Q_x$
approaches a plateau, leading to a sharp reduction in
$\beta$
.
Contours of (a) streamwise and (b) vertical Reynolds normal stresses and (c) Reynolds shear stress. Symbols
$+$
and
$\times$
mark the peaks near the fence and the reattachment, respectively. The dashed and dash-dotted lines mark the mean separating streamline and the shear layer centre, respectively. Figure (d) shows the local maxima of the Reynolds stresses along the shear layer. The vertical lines denote the boundaries of the defined regions (regions I, II, III and IV).
3.4. Reynolds stresses and momentum balance
Figures 6(a,b,c) display contours of streamwise,
$\overline {u'u'}$
, and vertical,
$\overline {v'v'}$
, Reynolds normal stresses and Reynolds shear stress,
$\overline {u'v'}$
, respectively, for
$ \textit{Re}_h$
= 5500. Regions of elevated stresses originate from the leading edge of the fence and extend downstream along the shear layer. Near the fence, the
$\overline {u'v'}$
contour exhibits a region of positive values that transitions to negative values at
$x$
/
$h$
= 1.1. The positive region arises from misalignment between the mean streamline and the Cartesian coordinate system, consistent with observations for rectangular prisms mounted on the wall (Fang & Tachie Reference Fang and Tachie2019) and in uniform flow (Cimarelli et al. Reference Cimarelli, Leonforte, De Angelis, Crivellini and Angeli2019). Local peaks are observed near the fence (region I), signifying the transitional nature of the shear layer, and in region III. Similar patterns have been reported for rectangular prisms in wall-mounted (Essel et al. Reference Essel, Nematollahi, Thacher and Tachie2015), free-surface-mounted (Agyei-Agyemang et al. Reference Agyei-Agyemang, Essel and Tachie2024) and uniform flow (Cimarelli et al. Reference Cimarelli, Corsini and Stalio2024) configurations. Figure 6(d) examines the effects of Reynolds number on the local maxima of the Reynolds stresses along the shear layer. The local peaks in region I decrease in magnitude and shift toward the fence with increasing Reynolds number. For instance,
$\overline {u'u'}$
,
$\overline {v'v'}$
and −
$\overline {u'v'}$
decrease from 0.09
$U_\infty ^2$
to 0.06
$U_\infty ^2$
, 0.06
$U_\infty ^2$
to 0.02
$U_\infty ^2$
and 0.03
$U_\infty ^2$
to 0.01
$U_\infty ^2$
, respectively, as
$ \textit{Re}_h$
increases from 2700 to 7500. Beyond region I, the profiles are approximately unchanged across the three Reynolds numbers examined, except in region III, where the magnitudes of
$\overline {u'u'}$
and
$\overline {-u'v'}$
for
$ \textit{Re}_h$
= 2700 are lower than those for
$ \textit{Re}_h$
= 5500 and 7500.
Building on the preceding analysis of the shear layer dynamics, momentum transport is examined using the streamwise component of the Reynolds-averaged Navier–Stokes equations. Under the assumptions of spanwise homogeneity and statistically steady flow, the streamwise momentum equation simplifies to
\begin{align} \frac {\partial U^{2}}{\partial x} = -\frac {1}{\rho }\frac {\partial P}{\partial x} - \frac {\partial \big (\overline {u'u'}\big )}{\partial x} - \frac {\partial \big (\overline {u'v'}\big )}{\partial y} - \frac {\partial (UV)}{\partial y} + \nu \left (\frac {\partial ^{2} U}{\partial x^{2}} + \frac {\partial ^{2} U}{\partial y^{2}} \right ) \! . \end{align}
The viscous terms are generally two to three orders of magnitude smaller than the other terms and are therefore omitted. The pressure gradient term, which was not measured, is estimated as the imbalance,
$I = -({\partial (\overline {u'u'})}/{\partial x} )- ( {\partial (\overline {u'v'})}/{\partial y}) -({\partial (UV)}/ {\partial y}) -( {\partial U^{2}}/{\partial x})$
, where
$\rho$
denotes the fluid density and
$P$
denotes the mean pressure.
Vertical profiles of momentum budget terms at
$x$
/
$h$
= (a) 0.6, (b) 8.5, (c) 20.0 and (d) 29.0. The horizontal dash-dotted line marks the location of the shear layer centre.
Figure 7 presents vertical profiles of the terms in the streamwise momentum equation at selected locations,
$x$
/
$h$
= 0.6, 8.5, 20.0 and 29.0, representative of the dynamics in regions I, II, III and IV, respectively. To facilitate comparison across locations, the vertical coordinate is shifted by the local shear layer centre location,
$Y_s$
. The magnitudes of all terms decrease progressively with downstream distance. Among the four momentum budget terms,
$-\partial U^{2}/\partial x$
and
$-\partial (UV)/\partial y$
consistently dominate the streamwise momentum transport. In region I (figure 7
a), negative
$-\partial (UV)/\partial y$
is the dominant contributor near the fence and exceeds the contribution of positive
$-\partial U^{2}/\partial x$
, while the Reynolds stress terms,
$-\partial (\overline {u'u'})/\partial x$
and
$-\partial (\overline {u'v'})/\partial y$
, are of similar magnitude but opposite sign. This behaviour agrees with the negative
$\partial U/\partial x$
and positive
$\partial U/\partial y$
in figure 3, and underscores the redistribution of streamwise momentum to the vertical direction to facilitate the upward deflection of the shear layer. The imbalance indicates that the pressure term is dynamically important for momentum balance near the fence. In region II (figure 7
b), negative
$-\partial (UV)/\partial y$
and positive
$-\partial U^{2}/\partial x$
are of comparable magnitudes, consistent with the shear layer stabilisation identified in figure 4. In region III (figure 7
c), both
$-\partial (UV)/\partial y$
and
$-\partial U^{2}/\partial x$
change sign, spatially consistent with the sign reversal in figure 3(a). The dominance of these two terms suggests that large-scale mean flow motions, rather than turbulence fluctuations, govern momentum transport. In region IV (figure 7
d), although all terms weaken downstream of reattachment, the mean flow terms, particularly
$-\partial U^{2}/\partial x$
, remain larger than the Reynolds stress terms.
Contours of the autocorrelation coefficients of
$u'$
, (a)
$R_{uu}$
, and
$v'$
, (b)
$R_{vv}$
. The reference locations are
$x$
/
$h$
= 1.0, 2.0, 3.0, 5.0, 7.0, 10.9, 16.5, 21.7 and 27.0 along the shear layer centre (black dash-dotted line) and the mean separating streamline (green dashed line). The vertical lines mark the boundaries of regions I, II, III and IV. The contour levels range from 0.9 to 0.4 at intervals of 0.1.
3.5. Vortical structures
3.5.1. Spatial and temporal characteristics
The spatial coherence of the turbulent structures is quantified using the two-point autocorrelation function of the fluctuating velocity component,
$u'_i$
, expressed as
\begin{align} R_{u_i u_i}(X_{\textit{ref}}, \Delta X) = \frac {\overline {u_i'(X_{\textit{ref}})\,u_i'(X_{\textit{ref}}+\Delta X)}} {u'_{i,rms}(X_{\textit{ref}})\,u'_{i,rms}(X_{\textit{ref}}+\Delta X)} , \end{align}
where
$X_{\textit{ref}}$
and
$\Delta X$
are the reference location and relative displacement, respectively, and
$u'_{i,rms}$
is the root-mean-square of
$u'_i$
. Figure 8 presents contours of the two-point autocorrelation,
$R_{uu}$
and
$R_{vv}$
, along the shear layer centre and the mean separating streamline. Near the fence leading edge, small-scale vortices induced by the KH instability are observed. In regions I and II, both
$R_{uu}$
and
$R_{vv}$
contours exhibit spatial growth, elongating in the streamwise and vertical directions, respectively. However, no further expansion is observed along the shear layer centre in regions III and IV. Near the reattachment, both correlations show a reduction due to the breakdown of the large-scale vortex (Bradshaw & Wong Reference Bradshaw and Wong1972; Saathoff & Melbourne Reference Saathoff and Melbourne1997). Here,
$R_{uu}$
also exhibits a varying degree of inclination along the shear layer centre. The inclination angles were determined following the procedures outlined in Christensen & Adrian (Reference Christensen and Adrian2001) and Liu et al. (Reference Liu, Bo and Liang2017). The inclination angle is approximately
$30^\circ$
near the fence and gradually decreases to
$10^\circ$
through regions I and II, consistent with the initial steep upward deflection and downstream weakening of the vertical growth of the shear layer in figure 4(c). In regions III and IV, the inclination angle diminishes considerably, aligning with the trend of the shear layer deflection angle in figure 4(d).
Previous pointwise measurement studies have shown that the integral time scale varies along separated shear layers, reflecting the low-frequency shear layer flapping (Castro & Haque Reference Castro and Haque1987; Hudy et al. Reference Hudy, Naguib and Humphreys2003) and the coexistence of wake-induced unsteadiness and KH instabilities (Moore et al. Reference Moore, Letchford and Amitay2019). In the present study, the availability of whole-field time-resolved velocity data enables the evaluation of the integral time scale over the entire flow field. Here,
$\tau$
is computed as
\begin{align} \tau _{u_i} = \int _{0}^{S} R_{u_i u_i}(\Delta t)\,{\rm d}(\Delta t) , \end{align}
where
$\Delta t$
denotes the temporal displacement and
$S$
is defined such that
$R_{u_i u_i}(S)=0$
. Figures 9(a,b) present the integral time scales
$\tau _u$
and
$\tau _v$
, based on
$u'$
and
$v'$
, respectively. Along the shear layer centre, both time scales are small near the fence leading edge, consistent with the formation of small-scale KH-type vortices (Moore et al. Reference Moore, Letchford and Amitay2019). Through regions I and II, both time scales increase gradually as the shear layer thickens, whereas in regions III and IV, their magnitudes remain nearly constant in the vicinity of reattachment, in agreement with the trends for fence-splitter-plate configurations (Castro & Haque Reference Castro and Haque1988) and long rectangular plates in uniform flow (Kiya & Sasaki Reference Kiya and Sasaki1983). This behaviour aligns with the development of large-scale vortices whose growth is constrained by the splitter-plate confinement. In the near-wall region,
$\tau _u$
exhibits a peak in region I at
$x$
/
$h$
= 4.5 (
$x$
/
$L_P$
= 0.21). Hudy et al. (Reference Hudy, Naguib and Humphreys2003) reported comparable time scales at
$x$
= 0.2–0.25
$L_P$
and linked them to the low-frequency shear layer flapping, which will be discussed in § 3.6. In contrast,
$\tau _v$
exhibits a distinct peak in the corner region and weakens downstream. The strong temporal coherence of the vertical velocity fluctuations in the corner is likely associated with the secondary vortex induced by the fence. In regions II and III, multiple near-wall peaks appear in both
$\tau _u$
and
$\tau _v$
but decrease in magnitude, while in region IV, both integral time scales continue to diminish.
Contours of the integral time scale based on
$u'$
, (a)
$\tau _u$
, and
$v'$
, (b)
$\tau _v$
. The red solid, black dash-dotted and purple dashed lines denote the isopleth of
$U = 0$
, the shear layer centre, and the isopleth of
$U = U_{\infty }$
, respectively. The vertical lines mark the boundaries of regions I, II, III and IV. Symbol
$\times$
marks the locations of the local peak in each region.
(a) Contour of the space–time correlation at the peak location of
$\overline {u'u'}$
(
$x/h = 15.8$
,
$y/h = 2.6$
) and (b) streamwise variations of the convective velocity,
$U_c$
. The contour levels range from 0.9 to 0.4 at an interval of 0.1. In (a), the red dashed line marks the locus of the local space–time correlation peaks. In (b), the vertical lines mark the boundaries of regions I, II, III and IV. The horizontal black dashed line corresponds to
$U_c = 0$
.
3.5.2. Convective velocity
Following Cherry et al. (Reference Cherry, Hillier and Latour1984), Tafti & Vanka (Reference Tafti and Vanka1991) and Hudy et al. (Reference Hudy, Naguib and Humphreys2003), the convective velocity of vortical structures is evaluated using the space–time correlation coefficient of the streamwise fluctuating velocity,
$u'$
. Figure 10(a) presents the contour of space–time correlation evaluated at the peak location of
$\overline {u'u'}$
(i.e.
$x$
/
$h$
= 15.8,
$y$
/
$h$
= 2.6). The red dashed line denotes the locus of the peak correlation, corresponding to the time delay
$(\Delta \tau )$
associated with the maximum correlation at each downstream location. The convective velocity (
$U_c$
) is estimated from the inverse slope of this locus (i.e.
$U_c = ({\rm d}\Delta \tau /{\rm d}\Delta x )^{-1}$
). Figure 10(b) presents the streamwise variations of
$U_c$
along the shear layer interface (
$x$
/
$h$
,
$y_{\omega _z}$
/
$h$
), mean separating streamline, shear layer centre (
$x$
/
$h$
,
$Y_s$
/
$h$
) and the near-wall region at
$y$
/
$h$
= 0.09. Along the shear layer interface,
$U_c$
$\approx$
0.90
$U_{\infty }$
. Along the mean separating streamline,
$U_c$
remains approximately constant at
$U_c$
$\approx$
0.65
$U_{\infty }$
in the first half of the primary vortex but decreases in the second half. The relatively larger upstream values are associated with the initial shear layer roll-up, where high-momentum free-stream fluid is entrained into the shear layer. Along the shear layer centre,
$U_c$
approaches a constant value of
$U_c$
$\approx$
0.50
$U_{\infty }$
for
$x$
/
$h$
$\gt$
18.0. In the near-wall region,
$U_c$
is negative, and its magnitude increases as the reverse flow strengthens toward the mid-bubble region and subsequently decreases as the reverse flow is constrained by the fence.
For rectangular prisms in uniform flow, Tafti & Vanka (Reference Tafti and Vanka1991) reported a lower value (
$U_c$
= 0.7
$U_{\infty }$
) along the shear layer interface. However, their shear layer interface was defined based on
$u_{\textit{rms}, \textit{max}}$
= 0.25
$U_{\infty }$
, which is likely closer to the mean separating streamline than that in the present study. The values of
$U_c$
along the shear layer interface are in closer agreement with
$U_c$
= 0.8
$U_{\infty }$
for rectangular prisms in uniform flow determined using phase averaging (Nakagawa et al. Reference Nakagawa, Nitta and Senda1999) and proper orthogonal decomposition (POD) (Abdul-Salam et al. Reference Abdul-Salam, Fang and Tachie2024). Along the mean separating streamline, the present
$U_c$
falls within the range of
$U_c$
/
$U_{\infty }$
$\in [0.5,\,0.7]$
reported for studies on wall-mounted bluff bodies (Heenan & Morrison Reference Heenan and Morrison1998; Lee & Sung Reference Lee and Sung2001), and those at free surface (Israel et al. Reference Israel, Dow, Clark and Tachie2024) and in uniform flow (Cherry et al. Reference Cherry, Hillier and Latour1984; Tafti & Vanka Reference Tafti and Vanka1991; Hudy et al. Reference Hudy, Naguib and Humphreys2003).
3.5.3. Spectral analysis
Figure 11 shows the frequency spectra of
$u'$
at the peak locations of
$\tau _u$
and
$v'$
at the peak locations of
$\tau _v$
in figure 9, using Welch’s overlapped window technique (Welch Reference Welch1967). The window length for the Fourier transform was 8.33 s (corresponding to 5000 snapshots), and consecutive windows were shifted by 5 snapshots (0.0083 s), corresponding to an overlap of 4995 snapshots. The frequency resolution is
$0.0023\,U_\infty /h$
(
$0.1183\,\mathrm{Hz}$
). The distinct frequency peaks identified in figure 11 are summarised in table 1. To facilitate comparison with previous studies, two Strouhal numbers are employed:
$St_h$
$(\equiv f h/U_\infty )$
and
$St_L$
$(\equiv f L_P/U_\infty )$
, based on the length scale (
$h$
) and the mean reattachment length (
$L_P$
), respectively. Across all regions, a distinct low-frequency peak at
$St_h$
= 0.007 (
$St_L$
= 0.15) is observed at the local peak locations of both
$\tau _u$
and
$\tau _v$
. This value lies within the frequency range (0.10–0.18)
$U_\infty /L_P$
reported for shear layer flapping in similar bluff body configurations (Castro & Haque Reference Castro and Haque1987, Reference Castro and Haque1988; Hudy et al. Reference Hudy, Naguib and Humphreys2003). From region I to region II, the energetic peaks shift from low frequencies (
$St_h$
= 0.007 and 0.014 or
$St_L$
= 0.15 and 0.30) to high frequencies (
$St_h$
= 0.030, 0.040 and 0.050 or
$St_L$
= 0.65, 0.87 and 1.09), and these high frequencies persist in region III.
Summary of significant Strouhal numbers (
$St_h$
) at local
$\tau _u$
and
$\tau _v$
peak locations in regions I, II and III.
Table 1. Long description
A table with three rows and three columns. The columns are labeled ‘Region’, ‘St_h at peak locations of τ_u’, and ‘St_h at peak locations of τ_v’. The rows are labeled with regions I, II, and III. Row 1: Region I, St_h at peak locations of τ_u 0.007, St_h at peak locations of τ_v 0.007, 0.014. Row 2: Region II, St_h at peak locations of τ_u 0.007, 0.012, 0.030, 0.040, 0.050, St_h at peak locations of τ_v 0.007, 0.012, 0.030, 0.040, 0.050. Row 3: Region III, St_h at peak locations of τ_u 0.007, 0.012, 0.030, 0.040, 0.050, St_h at peak locations of τ_v 0.009, 0.030, 0.040, 0.050.
Premultiplied frequency spectra of the streamwise fluctuating velocity
$u'$
at the peak locations of
$\tau _u$
and the vertical fluctuating velocity
$v'$
at the peak locations of
$\tau _v$
in region I, (
$x$
/
$h$
= 4.5,
$y$
/
$h$
= 0.2) and (
$x$
/
$h$
= 0.4,
$y$
/
$h$
= 0.4); region II, (
$x$
/
$h$
= 11.6,
$y$
/
$h$
= 1.6) and (
$x$
/
$h$
= 12.2,
$y$
/
$h$
= 1.3); and region III, (
$x$
/
$h$
= 19.3,
$y$
/
$h$
= 1.6) and (
$x$
/
$h$
= 18.8,
$y$
/
$h$
= 1.1), respectively. To facilitate visualisation, each spectrum plot is vertically shifted by the value indicated on the left-hand side. Characteristic
$St_{h}$
numbers are marked by the vertical lines.
Figure 12 shows the streamwise evolution of the frequency spectra of
$u'$
at selected streamwise locations along the shear layer centre, i.e.
$x$
/
$h$
= 0.9, 1.5, 3.0, 9.0, 15.0, 18.0, 20.0 and 24.0. Near the fence (
$x$
/
$h$
= 0.9 or
$x$
/
$L_P$
= 0.04), a peak is observed at
$St_h$
= 1.347 (
$St_L$
= 29.23). A sub-dominant peak also appears around
$St_h$
= 0.650 (
$St_L$
= 14.10). As
$x$
/
$h$
increases from 0.9 to 1.5, the magnitude of this sub-dominant peak surpasses that of
$St_h$
= 1.347. Meanwhile, another peak (
$St_h$
= 0.350 or
$St_L$
= 7.60) occurs and strengthens, while the dominance of
$St_h$
= 1.347 weakens. Further downstream, the dominance shifts from
$St_h$
= 0.350 to
$St_h$
= 0.100. This frequency migration, from high frequency to low frequency, is due to the repeated small-scale vortex pairing (Fang et al. Reference Fang, Tachie and Dow2022), driving the growth of the separated shear layer (Winant & Browand Reference Winant and Browand1974; Ho & Huerre Reference Ho and Huerre1984). This trend is consistent with the spatial and temporal development of vortical structures within the shear layer in figures 8 and 9, respectively. This progression continues across region II until
$x$
/
$h$
= 18.0 (
$x$
/
$L_P$
= 0.8), where a peak at
$St_h$
= 0.030 (
$St_L$
= 0.65) appears and persists downstream. Interestingly, the frequency migration ceases around
$x$
/
$L_P$
= 0.5 in Fang et al. (Reference Fang, Tachie and Dow2022) and
$x$
/
$L_P$
= 0.28 in Spazzini et al. (Reference Spazzini, Iuso, Onorato, Zurlo and Di Cicca2001). The earlier cessation of frequency migration in Fang et al. (Reference Fang, Tachie and Dow2022) and Spazzini et al. (Reference Spazzini, Iuso, Onorato, Zurlo and Di Cicca2001), compared with the present study, is likely due to the geometric differences. In the present study, the elevated separation location on the fence delays the occurrence of KH instability and hence the downstream transition to the vortex shedding frequency is also delayed. Furthermore, the dominance of multiple high frequencies is maintained after
$x$
/
$h$
= 18.0. Indeed, as the shear layer approaches the splitter plate, further growth in vortex size is constrained by the splitter plate. Consequently, the characteristic frequencies are maintained as the vortices are convected downstream. This constrained behaviour by the splitter plate spatially coincides with the asymptotic convective velocities in figure 10(b). The frequency of the shear layer flapping (
$St_h$
= 0.007) remains detectable at all the selected streamwise locations. Compared with
$ \textit{Re}_h$
= 5500, the dominant frequency (
$St_L$
= 27.25) near the fence at
$ \textit{Re}_h$
= 2700 is lower and occurs farther downstream (
$x/L_P$
= 0.06), whereas that (
$St_L$
= 36.41) at
$ \textit{Re}_h$
= 7500 is higher and occurs closer to the fence (
$x/L_P$
= 0.03). Nevertheless, these values are comparable to
$St_L$
$\approx$
30.3 reported by Kiya & Sasaki (Reference Kiya and Sasaki1983) at
$x$
/
$L_P$
= 0.02, and
$St_L$
= 33.1 reported by Fang et al. (Reference Fang, Tachie and Dow2022) at
$x$
/
$L_P$
= 0.01, and are associated with the KH instability. In contrast, the dominant frequency in regions III and IV is
$St_L \approx$
0.65 for all Reynolds numbers. This value is comparable to the vortex shedding frequencies (
$St_L$
= 0.46–0.60) reported for similar bluff body configurations (Hudy et al. Reference Hudy, Naguib and Humphreys2003), rectangular bluff bodies in uniform flow (Kiya & Sasaki Reference Kiya and Sasaki1983; Moore et al. Reference Moore, Letchford and Amitay2019) and at a free surface (Fang et al. Reference Fang, Tachie and Dow2022), as well as backward-facing-step flows (Driver et al. Reference Driver, Seegmiller and Marvin1987; Spazzini et al. Reference Spazzini, Iuso, Onorato, Zurlo and Di Cicca2001). The resulting frequency ratio,
$St_{L,KH}$
/
$St_{L,VS}$
, increases from 41.92 at
$ \textit{Re}_h$
= 2700 to 44.97 and 56.02 at
$ \textit{Re}_h$
= 5500 and 7500, respectively, aligning with the trend reported for rectangular prisms (Moore et al. Reference Moore, Letchford and Amitay2019; Cimarelli et al. Reference Cimarelli, Corsini and Stalio2024).
Premultiplied frequency spectra of
$u'$
at streamwise locations
$x$
/
$h$
= 0.9, 1.5, 3.0, 9.0, 15.0, 18.0, 20.0 and 24.0 along the shear layer centre. To facilitate visualisation, each spectrum plot is vertically shifted by the value indicated on the left side. Characteristic
$St_h$
numbers are marked by the vertical lines.
Figure 12. Long description
Panel: A line graph showing premultiplied frequency spectra at various streamwise locations along the shear layer center. The horizontal axis represents the Strouhal number (S t h) on a logarithmic scale ranging from 10^-3 to 10^1. The vertical axis represents the premultiplied frequency spectra (f E u / U infinity squared) ranging from 0 to 90 times 10^3. Each spectrum plot is vertically shifted by a value indicated on the left side. Characteristic numbers are marked by vertical dashed lines at specific Strouhal numbers. The data lines are color-coded, with red and black lines representing different streamwise locations (x/h) ranging from 0.9 to 24.0. The trends show variations in the frequency spectra at different streamwise locations, with notable peaks and shifts in the spectra as the streamwise location increases.
To investigate the near-wall unsteadiness, figure 13 shows the contour of the premultiplied frequency spectra of
$u'$
along the line of
$y$
/
$h$
= 0.09. The horizontal dash-dotted lines mark characteristic frequencies at
$St_h$
= 0.007, 0.009, 0.012, 0.026, 0.030 and 0.040 (from bottom to top), and correspond to key unsteady features of the flow in the near-wall region. The distribution exhibits a clear shift in dominant spectral content around the mid-bubble region. In the first half of the primary vortex, the near-wall unsteadiness is governed primarily by low frequencies (
$St_h$
= 0.007, 0.009 and 0.012 or
$St_L$
= 0.15, 0.20 and 0.26). These frequencies persist downstream beyond reattachment, indicating their association with large-scale, low-frequency motions of the separation bubble. In the second half of the primary vortex, high frequencies (
$St_h$
= 0.026, 0.030 and 0.040 or
$St_L$
= 0.56, 0.65 and 0.87) dominate. The distinct frequency
$St_h$
= 0.030 emerges near reattachment and corresponds to the vortex shedding frequency identified in figure 12. The frequency
$St_h$
= 0.040 (
$St_L$
= 0.87) remains influential farther downstream, consistent with values (
$St_L$
= 0.81) reported by Kiya & Sasaki (Reference Kiya and Sasaki1985), and (
$St_L$
= 0.82) by Fang et al. (Reference Fang, Tachie and Dow2022), and becomes the primary contributor to the near-wall unsteadiness. In region IV, the large-scale vortices formed through vortex pairing interact directly with the splitter plate. These results indicate that although the vortex shedding frequency originates from the shear layer centre, its influence extends into the near-wall region and contributes significantly to the near-wall unsteadiness.
Contour of the streamwise variation of premultiplied frequency spectra of
$u'$
along the
$x$
-axis at
$y$
/
$h$
= 0.09. The horizontal dash-dotted lines mark characteristic frequencies at
$St_h$
= 0.007, 0.009, 0.012, 0.026, 0.030 and 0.040, from bottom to top. The vertical lines mark the boundaries of regions I, II, III and IV.
3.6. Unsteadiness of flow separation
3.6.1. Reverse flow
The results in § 3.5.3 provide evidence of low-frequency motions in the separated flow, reminiscent of the flapping motion commonly observed in turbulent separation bubbles. This behaviour is associated with large-scale breathing of the separation bubble, modulation of entrainment and variations in reattachment location (Eaton & Johnston Reference Eaton and Johnston1981; Kiya & Sasaki Reference Kiya and Sasaki1983; Cherry et al. Reference Cherry, Hillier and Latour1984; Castro & Haque Reference Castro and Haque1987, Reference Castro and Haque1988; Tafti & Vanka Reference Tafti and Vanka1991; Cimarelli et al. Reference Cimarelli, Leonforte and De Angelis2018; Cimarelli et al. Reference Cimarelli, Corsini and Stalio2024). Previous studies have identified the flapping motion through low-frequency spectral peaks and spatio-temporal analysis of shear layer displacement. A more direct quantification has been achieved by examining the instantaneous reverse flow area (Pearson et al. Reference Pearson, Goulart and Ganapathisubramani2013; Fang et al. Reference Fang, Tachie and Dow2022; Abdul-Salam et al. Reference Abdul-Salam, Fang and Tachie2024). Motivated by these studies, the low-frequency unsteadiness is quantified here by analysing the instantaneous reverse flow area of the primary vortex and the forward flow area of the secondary vortex. Figure 14 shows representative snapshots of the instantaneous reverse flow (
$u$
$\lt$
0) of the primary vortex and the instantaneous forward flow (
$u$
$\gt$
0) of the secondary vortex. In the streamwise direction, the instantaneous reverse flow region of the primary vortex extends from
$x$
/
$h$
$\approx$
13.0 (figure 14
a) to
$x$
/
$h$
$\approx$
27.0 (figure 14
b), while its vertical extent varies from
$y$
/
$h$
$\approx$
2.3 (figure 14
a) to
$y$
/
$h$
$\approx$
2.9 (figure 14
b). Similarly, the streamwise extent of the forward flow region of the secondary vortex is approximately 1.0
$h$
in figure 14(c), and it elongates to about 2.0
$h$
in figure 14(d). In the vertical direction, it extends from
$y$
/
$h$
$\approx$
0.4 in figure 14(c) to
$y$
/
$h$
$\approx$
1.2 in figure 14(d).
Representative snapshots of (a,b) instantaneous reverse flow (
$u \lt 0$
) of the primary vortex and (c,d) instantaneous forward flow (
$u \gt 0$
) of the secondary vortex. The red solid line marks the isopleth of
$U = 0$
. The vertical dashed line in (a,b) marks the location of the midpoint of the mean reattachment length.
Figure 14. Long description
Panel A: A scatter plot with a line graph overlaid. The x-axis is labeled ‘x/h’ ranging from 0 to 30, and the y-axis is labeled ‘y/h’ ranging from 0 to 6. The scatter plot shows a distribution of blue points, and the line graph is a red curve. Panel B: A scatter plot with a line graph overlaid. The x-axis is labeled ‘x/h’ ranging from 0 to 30, and the y-axis is labeled ‘y/h’ ranging from 0 to 6. The scatter plot shows a distribution of blue points, and the line graph is a red curve. Panel C: A scatter plot with a line graph overlaid. The x-axis is labeled ‘x/h’ ranging from 0 to 4, and the y-axis is labeled ‘y/h’ ranging from 0 to 1.2. The scatter plot shows a distribution of orange points, and the line graph is a red curve. Panel D: A scatter plot with a line graph overlaid. The x-axis is labeled ‘x/h’ ranging from 0 to 4, and the y-axis is labeled ‘y/h’ ranging from 0 to 1.2. The scatter plot shows a distribution of orange points, and the line graph is a red curve.
The observations in figure 14 reveal expansion and contraction of both the primary and secondary vortices. To quantify this behaviour, the instantaneous reverse flow areas in the first and second halves of the primary vortex,
$A_{\textit{FH}}$
and
$A_{\textit{SH}}$
, the total reverse flow area of the primary vortex,
$A_P$
, and the instantaneous forward flow area of the secondary vortex,
$A_S$
, are computed by integrating regions of negative instantaneous streamwise velocity,
$u$
(
$x$
,
$y$
,
$t$
)
$\lt$
0 (Pearson et al. Reference Pearson, Goulart and Ganapathisubramani2013). These areas are defined as
\begin{align} A_i(t) = \int _{F_i} \mathcal{H}[u(x,y,t)]\,{\rm d}x\,{\rm d}y , \end{align}
where
$\mathcal{H}[u(x,y,t)]$
is an auxiliary function and equals 1 if
$u$
(
$x$
,
$y$
,
$t$
)
$\lt$
0 and 0 if
$u$
(
$x$
,
$y$
,
$t$
)
$\gt$
0. The integration domains (
$F_i$
) are defined as:
$F_{\textit{FH}} = (x/h,\,y/h)\in [0.0,10.9]\times [0.0,7.0]$
,
$F_{\textit{SH}} = (x/h,\,y/h)\in [10.9,33.0]\times [0.0,7.0]$
and
$F_{P} = (x/h,\,y/h)\in [0.0,33.0]\times [0.0,7.0]$
, corresponding to
$A_{\textit{FH}}$
,
$A_{\textit{SH}}$
and
$A_{P}$
, respectively. Spazzini et al. (Reference Spazzini, Iuso, Onorato, Zurlo and Di Cicca2001), who investigated the separated flow induced by a backward-facing step, observed that the forward flow area of the secondary vortex is upstream of the midpoint of the mean reattachment length in the streamwise direction and lies below the step height in the vertical direction. Consistent with these observations and the forward flow region identified in figures 14(c,d), a subdomain
$F_T = x/h \times y/h \in [0.0,\,10.9] \times [0.0,\,1.2]$
is introduced to isolate the secondary vortex region. The instantaneous reverse flow area (
$A_T$
) in this subdomain was evaluated and the instantaneous forward flow area of the secondary vortex was calculated as:
$A_S = F_T - A_T$
.
Figure 15(a) presents the probability density functions (PDFs) of
$A'_{\textit{FH}}$
,
$A'_{\textit{SH}}$
,
$A'_{P}$
and
$A'_{S}$
. The PDF of
$A'_{\textit{SH}}$
resembles that of
$A'_{P}$
, suggesting that the flapping of the second half of the primary vortex dominates the modulation of the entire primary vortex. In contrast, the PDF of
$A'_{\textit{FH}}$
is narrower than
$A'_{P}$
but broader than
$A'_{S}$
, which indicates that the dynamics of the first half of the primary vortex is influenced by both the large-scale modulation of the primary vortex and localised interaction with the secondary vortex. This intermediate variability highlights the coupling between the first half of the primary vortex and the secondary vortex.
Figure 15(b) presents the premultiplied frequency spectra of
$A'_{\textit{FH}}$
,
$A'_{\textit{SH}}$
,
$A'_{P}$
and
$A'_{S}$
. Multiple distinct low frequencies (
$St_h$
= 0.005, 0.007 and 0.009 or
$St_L$
= 0.11, 0.15 and 0.20, respectively) are identified in the spectra, consistent with previously identified multiple low frequencies in the near-wall region in figure 13. These frequencies fall within the flapping motion frequency range, (0.08–0.20)
$U_{\infty }$
/
$L_P$
, introduced earlier. The dominance of
$St_h$
= 0.009 occurs across all spectra and its presence in the spectrum of
$A'_{S}$
implies that the secondary vortex responds to the same large-scale modulation. In addition, the spectra of both
$A'_{\textit{FH}}$
and
$A'_{S}$
exhibit peaks at
$St_h$
= 0.005 and 0.007, whereas
$A'_{\textit{SH}}$
shows a peak at
$St_h$
= 0.016, which is suggestive of a coupling between the secondary vortex and the first half of the primary vortex. The larger magnitude of the spectrum of
$A'_{\textit{SH}}$
than that of
$A'_{\textit{FH}}$
indicates that the unsteadiness in the second half of the primary vortex is more energetic and exerts a dominant influence on the entire primary vortex. The statistical similarity between
$A_{\textit{SH}}$
and
$A_{P}$
, compared with
$A_{\textit{FH}}$
, indicates that the unsteadiness in the second half of the primary vortex primarily dominates that of the entire primary vortex, reinforcing the observations in figure 15(a).
(a) Probability density functions and (b) premultiplied frequency spectra of the fluctuating components of the instantaneous reverse flow areas of the primary vortex and instantaneous forward flow area of the secondary vortex.
3.6.2. Conditional averaging analysis
Building on the preceding analysis, conditional averaging is performed to examine the effects of flapping motion (contraction and expansion events) on the dynamics of the separated shear layer. Figure 16 displays contours of conditionally averaged streamwise mean velocity,
$\langle U\rangle$
, overlaid with conditionally averaged mean streamlines, for expansion and contraction events of
$A_{\textit{FH}}$
,
$A_{\textit{SH}}$
and
$A_{S}$
. In figures 16(a,c,e), multiple small-scale vortices are observed downstream of the primary vortex during expansion events. The conditionally averaged reattachment length (
$\langle L_{P}\rangle$
) is more sensitive to the flapping of
$A_{\textit{SH}}$
, extending to 24.8
$h$
during expansion (figure 16
c) and shortening to 18.0
$h$
during contraction (figure 16
d). In contrast,
$\langle L_{P}\rangle$
remains relatively unchanged during expansion or contraction of
$A_{\textit{FH}}$
and
$A_S$
. Conversely, the recirculation length (
$\langle L_{S}\rangle$
) responds strongly to the flapping of
$A_{\textit{FH}}$
and
$A_S$
but is relatively insensitive to the flapping of
$A_{\textit{SH}}$
. For instance,
$\langle L_{S}\rangle$
extends to 3.0
$h$
during expansion of
$A_S$
(figure 16
e) and shortens to 1.8
$h$
during contraction of
$A_S$
(figure 16
f), while it increases to 5.5
$h$
during the contraction of
$A_{\textit{FH}}$
(figure 16
b). The magnitude of the conditionally averaged maximum streamwise mean velocity (
$\langle U_{\textit{max}}\rangle$
) remains unchanged (
$\langle U_{\textit{max}}\rangle$
$\approx$
1.27
$U_{\infty }$
) across all cases, consistently peaking at
$x$
/
$h$
$\approx$
10.5. In contrast, the maximum backflow velocity (
$\langle U_{b}\rangle$
) becomes weaker (−0.25
$U_{\infty }$
) during the expansion of
$A_{\textit{SH}}$
and stronger (−0.42
$U_{\infty }$
) during the contraction of
$A_{\textit{SH}}$
.
Contours of conditionally averaged streamwise mean velocity,
$\langle U\rangle$
, superimposed with conditionally averaged mean streamlines during expansion and contraction of
$A_{\textit{FH}}$
(a,b), expansion and contraction of
$A_{\textit{SH}}$
(c,d) and expansion and contraction of
$A_S$
(e, f). The red solid line denotes the isopleth of
$\langle U\rangle = 0$
. Symbols
$+$
and
$\times$
mark the locations of maximum and minimum
$\langle U\rangle$
, respectively.
Figure 16. Long description
Panel A: A heat map showing contours of conditionally averaged streamwise mean velocity during the expansion phase of the primary vortex. The x-axis represents the streamwise direction (x/h) ranging from 0 to 30, and the y-axis represents the vertical direction (y/h) ranging from 0 to 5. The color scale indicates the velocity magnitude, with blue representing lower values and red representing higher values. The mean streamlines are superimposed, showing the flow patterns. Panel B: A heat map similar to Panel A but during the contraction phase of the primary vortex. Panel C: A heat map showing contours of conditionally averaged streamwise mean velocity during the expansion phase of the secondary vortex. Panel D: A heat map similar to Panel C but during the contraction phase of the secondary vortex. Panel E: A heat map showing contours of conditionally averaged streamwise mean velocity during the expansion phase of the combined vortices. Panel F: A heat map similar to Panel E but during the contraction phase of the combined vortices. The red solid line in each panel denotes the isopleth of zero velocity.
Vertical profiles of conditionally averaged Reynolds shear stress (left),
$\langle \overline {u'v'} \rangle$
, and turbulence kinetic energy (right),
$\langle \textit{TKE} \rangle$
, at
$x$
/
$h$
= 0.6 (a,b), 8.5 (c,d), 20.0 (e,f) and 29.0 (g,h), in regions I, II, III and IV, respectively. The vertical coordinates are shifted by the shear layer centre vertical location,
$Y_s$
. The horizontal dash-dotted lines indicate the locations of the shear layer centre.
To assess the effects of the shear layer flapping on the fluctuating velocity field, figure 17 presents the profiles of the conditionally averaged Reynolds shear stress,
$\langle \overline {u'v'} \rangle$
, and turbulence kinetic energy,
$\langle \textit{TKE}\rangle$
, during expansion and contraction events of
$A_{\textit{FH}}$
,
$A_{\textit{SH}}$
and
$A_{S}$
. Since the spanwise fluctuating velocity (
$w'$
) was not measured,
$ \textit{TKE}$
= 0.5(
$\overline {u'u'} + \overline {v'v'}$
) was employed. The conditionally averaged streamwise and vertical Reynolds normal stresses,
$\langle \overline {u'u'} \rangle$
and
$\langle \overline {v'v'} \rangle$
, exhibit similar behaviours to
$\langle \textit{TKE}\rangle$
and are therefore not shown here. The selected locations are the same as those in figure 7, i.e.
$x$
/
$h$
= 0.6, 8.5, 20.0 and 29.0, corresponding to regions I, II, III and IV, respectively. Near the fence (figures 17
a,b), the conditional profiles nearly coincide with the unconditional profiles, indicating that the large-scale flapping motion has negligible influence on the local turbulence during the initial roll-up of the shear layer. In region II (figures 17
c,d),
$\langle \overline {u'v'} \rangle$
and
$\langle \textit{TKE}\rangle$
are noticeably amplified during expansion and suppressed during contraction. Thus, the shear layer flapping contributes significantly to the modulation of the Reynolds stresses in this region. During expansion, the separation bubble elongates and entrains more high-momentum free-stream fluid into the shear layer, promoting stronger Reynolds stresses. Conversely, the contraction event suppresses shear layer development, diminishing the levels of both
$\langle \overline {u'v'} \rangle$
and
$\langle \textit{TKE}\rangle$
. Near reattachment in region III (figures 17
e,f), the influence of shear layer flapping becomes more pronounced. In particular,
$\langle \overline {u'v'} \rangle$
and
$\langle \textit{TKE}\rangle$
are suppressed during the contraction of
$A_{\textit{SH}}$
and amplified during the expansion of
$A_{\textit{SH}}$
. Beyond reattachment in region IV (figures 17
g,h), the modulation of both
$\langle \overline {u'v'} \rangle$
and
$\langle \textit{TKE}\rangle$
by the shear layer flapping diminishes. These observations suggest that the large-scale shear layer flapping plays a critical role in the fluctuating velocity field throughout the separated shear layer.
The effects of shear layer flapping on turbulence production are examined through the production term (
$P_k$
)
\begin{align} P_k = - \overline {u'u'}\,\frac {\partial U}{\partial x} - \overline {v'v'}\,\frac {\partial V}{\partial y} - \overline {u'v'}\,\frac {\partial U}{\partial y} - \overline {u'v'}\,\frac {\partial V}{\partial x} . \end{align}
Figure 18 compares the profiles of the conditionally averaged total production term,
$\langle P_k \rangle$
, and its individual components,
$ \langle -\,\overline {u'u'}\,({\partial U}/{\partial x} ) \rangle$
,
$ \langle -\,\overline {v'v'}\,({\partial V}/{\partial y}) \rangle$
,
$ \langle -\,\overline {u'v'}\,({\partial U}/{\partial y}) \rangle$
and
$ \langle -\,\overline {u'v'}\,( {\partial V}/{\partial x}) \rangle$
, during the expansion and contraction events of
$A_{\textit{FH}}$
,
$A_{\textit{SH}}$
and
$A_{S}$
, together with their unconditional counterparts, at
$x$
/
$h$
= 0.6, 8.5 and 20.0 in regions I, II and III, respectively. Since all terms in region IV (
$x$
/
$h$
= 29.0) are weak in magnitude and exhibit negligible variations compared with the unconditional profiles, their profiles are not included. Near the fence in region I (figure 18
a),
$\langle P_k \rangle$
is negative, indicating a local reversal of energy transfer from the fluctuating velocity field to the mean flow. Compared with the unconditional profile,
$\langle P_k \rangle$
is enhanced during expansion and suppressed during contraction. In regions II and III (figures 18
b,c),
$\langle P_k \rangle$
becomes positive and acts as an energy source to the fluctuating velocity field. In region II (figure 18
b),
$\langle P_k \rangle$
remains amplified during expansion and suppressed during contraction. Near reattachment in region III (figure 18
c),
$\langle P_k \rangle$
is most pronounced during the expansion and contraction of
$A_{\textit{SH}}$
, highlighting the dominant role of the second half of the primary vortex in governing downstream turbulence production.
Among the four individual production terms,
$ \langle -\,\overline {u'u'}\,({\partial U}/{\partial x}) \rangle$
and
$ \langle -\,\overline {u'v'}\,({\partial U}/{\partial y}) \rangle$
constitute the dominant contributions to
$\langle P_k \rangle$
. Near the fence in region I (figures 18
d, g, j, m), the negative
$\langle P_k \rangle$
in figure 18(a) is primarily associated with negative
$ \langle -\,\overline {u'v'}\,({\partial U}/{\partial y} ) \rangle$
. In this region, all individual terms are amplified during expansion and suppressed during contraction. Together with the nearly unchanged conditionally averaged Reynolds stresses (figures 17
a,b), this indicates that the flapping events strongly modulate local mean velocity gradients, thereby modulating turbulence production in the vicinity of the fence leading edge. In region II at
$x$
/
$h$
= 8.5 (figures 18
e, h, k, n), the modulation of
$\langle P_k \rangle$
arises primarily from
$ \langle -\,\overline {u'u'}\,({\partial U}/{\partial x}) \rangle$
and
$ \langle -\,\overline {u'v'}\,({\partial U}/{\partial y}) \rangle$
. The concurrent modulation of Reynolds stresses (figures 17
c,d) suggests that the flapping events alter both the Reynolds stresses and streamwise mean velocity gradients (
$ {\partial U}/{\partial y}$
and
$ {\partial U}/{\partial x}$
), thereby modifying turbulence production in region II. Near reattachment in region III (figures 18
f,i,l, o),
$ \langle -\,\overline {u'u'}\,( {\partial U}/{\partial x}) \rangle$
and
$ \langle -\,\overline {v'v'}\,({\partial V}/{\partial y}) \rangle$
reverse sign, consistent with the sign change of
${\partial U}/{\partial x}$
shown in figure 3(a). In this region, all production terms are amplified during expansion of
$A_{\textit{SH}}$
and suppressed during contraction, aligning with the modulation of Reynolds stresses in figures 17(e, f).
Vertical profiles of conditionally averaged total production term,
$\langle P_k\rangle$
(a, b, c), and individual production terms
$\langle -\overline {u'u'}\,\partial U/\partial x\rangle$
(d, e, f),
$\langle -\overline {v'v'}\,\partial V/\partial y\rangle$
(g, h, i),
$\langle -\overline {u'v'}\,\partial U/\partial y\rangle$
(j, k, l) and
$\langle -\overline {u'v'}\,\partial V/\partial x\rangle$
(m, n, o). The reference locations are
$x/h=0.6$
, 8.5 and 20.0, corresponding to regions I, II and III, respectively. The vertical coordinates are shifted by the shear layer centre location,
$Y_s$
. The horizontal dash-dotted lines indicate the shear layer centre.
Figure 18. Long description
The image contains fifteen line graphs arranged in a 5x3 grid. Each graph shows vertical profiles of conditionally averaged production terms at different streamwise locations. The vertical axis in each graph represents the normalized vertical coordinate (y-Ys)/h, and the horizontal axis represents different normalized production terms. The graphs are labeled from (a) to (o). The reference locations are x/h = 8.5 and 20.0, corresponding to regions I, II, and III. The vertical dash-dotted lines indicate the shear layer center. Graphs (a), (b), and (c) show the total production term for different conditions. Graphs (d), (e), and (f) display the first individual production term. Graphs (g), (h), and (i) show the second individual production term. Graphs (j), (k), and (l) depict the third individual production term. Graphs (m), (n), and (o) illustrate the fourth individual production term. Each graph includes multiple lines representing different conditions, such as unconditional and various specific conditions indicated by different colors and line styles.
The importance of
$\left \langle {\partial U}/{\partial y} \right \rangle$
and
$\left \langle{\partial U}/{\partial x} \right \rangle$
in modulating turbulence production is consistent with the mechanism of shear layer flapping proposed by Fang & Wang (Reference Fang and Wang2024), in which mean shear tilts the streamwise vorticity to form a strong curvature in the vortex filament whose vertical and streamwise segments are preferentially stretched in the first and second halves of the separation bubble by positive
$ {\partial V}/{\partial y}$
(equivalently negative
$ {\partial U}/{\partial x}$
) and positive
${\partial U}/{\partial x}$
, respectively. In the present study, expansion in the first half of the primary vortex (
$x$
/
$h$
= 0.6 in region I and
$x$
/
$h$
= 8.5 in region II) increases the shear layer thickness and modifies its curvature, enhancing the local mean shear and streamwise deceleration. These changes amplify
$\left \langle {\partial U}/{\partial y} \right \rangle$
and
$\left \langle {\partial U}/{\partial x} \right \rangle$
, leading to increased contributions from
$ \langle -\,\overline {u'u'}\,( {\partial U}/{\partial x} ) \rangle$
and
$ \langle -\,\overline {u'v'}\,( {\partial U}/{\partial y}) \rangle$
. Conversely, contraction reduces these gradients and suppresses turbulence production. In the second half of the primary vortex (
$x$
/
$h$
= 20.0 in region III), although the flapping dynamics in this region is primarily associated with the streamwise stretching (positive
${\partial U}/{\partial x}$
) of the vortex segment (Fang & Wang Reference Fang and Wang2024), the flapping of
$A_{\textit{SH}}$
simultaneously modifies the Reynolds stresses and the mean velocity gradients, leading to the modulation of all individual production terms. This behaviour underscores the dominance of the second half of the primary vortex in modulating the downstream turbulence production.
3.7. Spectral proper orthogonal decomposition analysis
Following Towne et al. (Reference Towne, Schmidt and Colonius2018), the spectral proper orthogonal decomposition (SPOD) method is employed to examine the spatial and temporal dynamics of the coherent structures associated with characteristic frequencies identified in the separated shear layer. Using singular value decomposition, a method commonly used for POD (Riches et al. Reference Riches, Martinuzzi and Morton2018), the Fourier-transformed velocity field is decomposed into POD modes. The fluctuating velocity field,
$u_i'(x,y,t)$
, is decomposed into different modes at various frequencies as follows:
\begin{align} u_i'(x,y,t) = \sum _{j=1}^{\infty }\sum _{k=1}^{M} a_j^{k}\,\phi _j^{k}(x,y)\,e^{i2\pi t f_j} + c.c. , \end{align}
where
$\phi _j^{k}(x,y)$
is the
$k$
th POD mode at the
$j$
th frequency and
$a_j^{k}$
is the corresponding modal coefficient. Here,
$M$
is the number of modes and c.c. is the complex conjugate. The SPOD analysis is performed over the entire measurement domain, excluding the region covered by the fence and splitter plate.
The first SPOD mode captures approximately 14 % of the total turbulence kinetic energy (
$ \textit{TKE}$
), nearly three times the 5 % captured by the second mode. The mode energy decays monotonically thereafter, with modes 3–6 contributing 3 %, 2 %, 1 % and 1 %, respectively. Figure 19 displays the premultiplied frequency spectra of the first two SPOD modes. Both modes exhibit similar peaks. The peaks at
$St_h$
= 0.007, 0.009 and 0.012 are consistent with those identified in the frequency spectra of the fluctuating velocities (figures 11, 12, 13) and reverse flow area (figure 15
b). Interestingly, the lower frequency
$St_h$
= 0.005, identified in the spectra of reverse flow area, is not prominent in either SPOD mode. This suggests that
$St_h$
= 0.005 is likely associated with spatially intermittent or localised flow motions that contribute relatively low energy to the flow field. The more energetic peaks at
$St_h$
= 0.030, 0.040 and 0.050 align with the vortex shedding frequencies along the shear layer centre (figure 12) and energetic frequencies in the near-wall region (figure 13), indicating that the vortex shedding motion and the energetic near-wall flow structures constitute the strong spatially coherent motions in the separated flow.
Premultiplied frequency spectra of the first and second SPOD modes. The vertical lines mark characteristic frequencies. The vertical axis uses arbitrary scales.
Contours of the streamwise component of the first SPOD mode at (a)
$St_h = 0.007$
, (b)
$St_h = 0.009$
, (c)
$St_h = 0.012$
, (d)
$St_h = 0.030$
, (e)
$St_h = 0.040$
and (f)
$St_h = 0.050$
. The dash-dotted and solid lines denote the isopleths of
$U = U_{\infty }$
and
$U = 0$
. The symbols ‘
$+$
’ mark the core of the dominant vortical structure.
Figure 20 presents the first SPOD modes at an arbitrary phase angle,
$\alpha \equiv 2\pi f_j t$
, at the dominant and sub-dominant frequencies identified in figure 19, to gain insight into their spatial structures. The cores of energetic vortical structures are marked by the symbol ‘
$+$
’. Figures 20(a,b,c) present the first SPOD modes at frequencies
$St_h$
= 0.007, 0.009 and 0.012, respectively, which are associated with the flapping motion. At
$St_h$
= 0.009, a vortical structure spanning the entire separation region is observed, highlighting the global nature of the flapping motion. A distinct vortex core is identified along the isopleth of
$U$
=
$U_{\infty }$
, far away from the fence at (
$x$
/
$h$
= 27.6,
$y$
/
$h$
= 5.5). At
$St_h$
= 0.007 and 0.012, a pair of distinct positive and negative regions is observed, encompassing the entire flow domain. The distinct vortex cores are primarily located along the isopleth of
$U$
= 0. Figures 20(d,e,f) show the first SPOD modes at frequencies
$St_h$
= 0.030, 0.040 and 0.050, respectively, corresponding to the vortex shedding motion in the shear layer and the most energetic near-wall flow structures. A train of alternating positive and negative structures is observed, illustrating the spatial periodicity of these coherent structures. As the frequency increases from
$St_h$
= 0.030 to 0.050, the number of positive and negative structures increases and the characteristic size of these vortices decreases. In figures 20(e) and 20(f), in the second half of the primary vortex, distinct counter-rotating vortex pairs are formed. Each pair contains one vortex core aligned with the isopleth of
$U$
=
$U_{\infty }$
and another in the near-wall region, indicating a strong vortical coupling across the separated shear layer. This coupling suggests that the vortex shedding dynamics, which primarily originates from the shear layer centre, extends into the near-wall region, reinforcing the observations in figures 12 and 13.
4. Summary and conclusions
Turbulent flow separation induced by a fence with a splitter plate in uniform flow at moderate Reynolds numbers was investigated using TR-PIV. The Reynolds numbers based on the free-stream velocity and the fence height above the splitter plate are 2700, 5500 and 7500. The mean separated flow is characterised by a large primary clockwise recirculating vortex above the splitter plate and a smaller counter-rotating secondary vortex in the corner of the fence and splitter plate. The mean reattachment length of the primary vortex decreases from
$L_P = 22.6h$
at
$ \textit{Re}_h = 2700$
to
$L_P = 21.7h$
at both
$ \textit{Re}_h = 5500$
and 7500. However, the mean recirculation length and maximum forward flow velocity of the secondary vortex, as well as the vorticity thickness, remain approximately unchanged across the three Reynolds numbers examined. Near the fence, the magnitudes of the Reynolds stresses decrease and their peak locations move closer to the fence as the Reynolds number increases. Farther downstream (
$7.5 \lt x/h$
), the Reynolds stresses show no appreciable variation between
$ \textit{Re}_h$
= 5500 and 7500. Guided by the vorticity thickness growth rates, the shear layer development is divided into four regions associated with the initial roll-up (region I: 0.0
$\lt x/h \lt$
7.5), shear layer stabilisation (region II: 7.5
$\lt x/h \lt$
18.0), wall-induced large-scale vortex breakdown (region III: 18.0
$\lt x/h \lt$
21.7) and post-reattachment recovery (region IV:
$21.7 \lt x/h$
). The shear layer trajectory demonstrates that the development in regions I and II is driven by mean flow advection, whereas downstream evolution in regions III and IV is constrained by wall confinement. Flow entrainment intensifies near the fence in region I and near reattachment in region III. The former is associated with the strong counter-rotating secondary vortex, while the latter results from the shear layer impingement on the splitter plate.
Streamwise momentum balance analysis demonstrated that mean flow dominates the local momentum transport throughout the separated shear layer. In region I, the streamwise momentum is redistributed vertically to facilitate upward shear layer deflection. The broad region of reduced momentum transport in region II coincides with shear layer stabilisation. Near reattachment in region III, sign changes occur in both the mean and Reynolds stress terms.
The spatio-temporal characteristics of the vortical structures were examined using two-point correlations, integral time scales and convective velocities. The two-point correlations indicate that the coherent structures grow spatially through regions I and II as the separated shear layer develops from the initial KH instability, but this growth diminishes downstream in regions III and IV due to confinement by the splitter plate. Similarly, small integral time scales are observed near the fence, indicative of the presence of small-scale KH vortices, but the time scales increase through regions I and II as the shear layer thickens and coherent structures grow and remain approximately constant in regions III and IV. The convective velocity (0.65
$U_{\infty }$
) in regions I and II is high due to the entrainment of high-momentum free-stream fluid during the initial roll-up of the shear layer and decreases downstream as reattachment is approached. In the near-wall region, the convective velocity is negative with its magnitude increasing toward the mid-bubble region and decreasing as the reverse flow is constrained by the fence.
Spectral analysis reveals a distinct frequency migration along the shear layer. The KH frequency (
$St_{L, KH}$
) near the fence increases from 27.25 at
$ \textit{Re}_h$
= 2700 to 29.23 and 36.41, respectively, at
$ \textit{Re}_h$
= 5500 and 7500. Successive vortex pairing events then drive a shift towards lower frequencies in regions I and II, followed by the persistence of the vortex shedding frequency,
$St_{L, VS}$
$\approx$
0.65 across the three Reynolds numbers examined, in regions III and IV. Thus, the frequency ratio,
$St_{L,KH}$
/
$St_{L,VS}$
, which is a measure of the relative influence of the vortex shedding on the shear layer instability, increases from 41.92 at
$ \textit{Re}_h=2700$
to 44.97 and 56.02 at
$ \textit{Re}_h=5500$
and 7500, respectively. The low-frequency unsteadiness identified from the spectral analysis is also evaluated using the instantaneous reverse flow area of the primary vortex and the forward flow area of the secondary vortex. The statistical similarity between the second half of the primary vortex and the entire primary vortex demonstrates that the second half dominates global unsteadiness, whereas the first half exhibits intermediate variability arising from its coupling with both the large-scale modulation of the primary vortex and localised interaction with the secondary vortex.
Conditional averaging analysis based on contraction and expansion events of the separation bubble demonstrates that expansion events weaken the maximum backflow velocity, whereas contraction events intensify the reverse flow. The Reynolds stresses are strongly modulated in regions II and III. Turbulence production exhibits strong regional dependence. The negative production near the fence (region I) is modulated primarily through variations in the mean velocity gradients during the flapping events. In region II, the production becomes positive and is amplified or suppressed through the combined modulation of Reynolds stresses and streamwise mean velocity gradients. In region III, the positive production is modulated mainly through the flapping of the second half of the primary vortex. Examination of the individual production terms highlights the importance of both
$\left \langle {\partial U}/{\partial y} \right \rangle$
and
$\left \langle {\partial U}/{\partial x} \right \rangle$
, consistent with the shear layer flapping mechanism described in Fang & Wang (Reference Fang and Wang2024).
Spectral proper orthogonal decomposition analysis shows that the first SPOD mode reveals the large-scale flapping motion, particularly at
$St_h$
= 0.009, with coherent vortices spanning the entire separation bubble. At higher frequencies (
$St_h$
= 0.030, 0.040 and 0.050), the mode contours exhibit spatially periodic counter-rotating vortex pairs, especially beyond the middle of the primary vortex, consistent with the vortex shedding motions in the separated shear layer.
Acknowledgements
The authors gratefully acknowledge Professor X. Fang for his help with the experiments and for many insightful discussions throughout this work.
Funding
The authors gratefully acknowledge the financial support provided by the Natural Sciences and Engineering Research Council (NSERC) of Canada. Support from the Canada Foundation for Innovation (CFI) for the experimental facility is also gratefully acknowledged.
Declaration of interests
The authors report no conflict of interest.
Data availability statement
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Author contributions
Conceptualisation: J. Kang; data collection: J. Kang, X. Fang; investigation: J. Kang; methodology: J. Kang; visualisation: J. Kang; writing the original draft: J. Kang. Conceptualisation (supporting): M.F. Tachie; investigation (supporting): M.F. Tachie; methodology (supporting): M.F. Tachie; review and editing: M.F. Tachie.
FOV1
FOV2
U
U=0
ϕ
+
×
U
∂U
∂x
∂U
∂y
∂U
∂xi
ΔUI
ΔUw
Umax
Umin−I
Umin−w
δω
Ys
θ
Qx
β
+
×
x
h
u′
Ruu
v′
Rvv
x
h
u′
τu
v′
τv
U=0
U=U∞
×
u′u′¯
x/h=15.8
y/h=2.6
Uc
Uc=0
Sth
τu
τv
u′
τu
v′
τv
x
h
y
h
x
h
y
h
x
h
y
h
x
h
y
h
x
h
y
h
x
h
y
h
Sth
u′
x
h
Sth
u′
x
y
h
Sth
u<0
u>0
U=0
⟨U⟩
AFH
ASH
AS
⟨U⟩=0
+
×
⟨U⟩
⟨u′v′¯⟩
⟨TKE⟩
x
h
Ys
⟨Pk⟩
⟨−u′u′¯∂U/∂x⟩
⟨−v′v′¯∂V/∂y⟩
⟨−u′v′¯∂U/∂y⟩
⟨−u′v′¯∂V/∂x⟩
x/h=0.6
Ys
Sth=0.007
Sth=0.009
Sth=0.012
Sth=0.030
Sth=0.040
Sth=0.050
U=U∞
U=0
+
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