1. Introduction
Turbulent boundary-layer separation can have significant implications for engineering and environmental flows. In aerodynamics applications, for example, separation is typically associated with a drag increase and a loss of lift. Such detriments are exacerbated by the unsteady nature of turbulent flow separation, potentially yielding noise emission and severely fluctuating thermal and mechanical loads. Understanding and predicting the unsteadiness of turbulent boundary-layer separation is therefore essential for optimising the performance of aircraft, improving energy efficiency in industrial processes, and mitigating environmental effects in natural fluid systems.
A relatively clear picture can be drawn for the case of geometry-induced turbulent flow separation where fluid particles detach from a solid surface due to a sudden geometry change. Here, along with small-scale turbulent fluctuations and the development of vortical structures inside the separated shear-layer, the dynamics is governed by a low-frequency flapping mode. Eaton & Johnston (Reference Eaton and Johnston1982) showed that the latter resembles an approximately two-dimensional (2-D) vertical motion of the shear-layer. They demonstrated that this leads to an up- and downstream displacement of the reattachment point that terminates the turbulent separation bubble (TSB) induced by a backward-facing step in their study. Further investigations of the flapping motion are concerned with flows past blunt flat plates (Kiya & Sasaki Reference Kiya and Sasaki1983; Cherry, Hillier & Latour Reference Cherry, Hillier and Latour1984; Saathoff & Melbourne Reference Saathoff and Melbourne1997), forward-facing steps (Farabee & Casarella Reference Farabee and Casarella1986; Camussi et al. Reference Camussi, Felli, Pereira, Aloisio and Di Marco2008) and a fence-like obstacle (Hudy, Naguib & Humphreys Reference Hudy, Naguib and Humphreys2003). All of these studies demonstrate the presence of low-frequency unsteadiness consistent with the description by Eaton & Johnston (Reference Eaton and Johnston1982). Furthermore, the associated normalised frequencies
${\textit{St}}=fL_{{b}}/U_\infty$
are of the same order of magnitude,
$\mathcal{O}({\textit{St}})=0.1$
; here,
$f$
denotes the dimensional frequency,
$L_{b}$
is the mean length of the separated flow region and
$U_\infty$
is the inflow velocity.
An alternative to the triggering of flow separation through a sudden geometrical change in the bounding surface is the application of an adverse pressure gradient (APG). In contrast, the latter yields a time-dependent location of the separation point, drastically increasing the complexity of flow dynamics, as appreciated by Simpson (Reference Simpson1981). The salient low-frequency feature of APG-induced flow separation is a large-scale contraction and expansion of the reverse-flow region. Smith & Kline (Reference Smith and Kline1974) conducted flow visualisations in a 2-D diffuser, indicating non-periodic low-frequency changes in the size of the separated flow region. Building on previous studies of the same flow configuration by Fox & Kline (Reference Fox and Kline1960), they report that, depending on the flow deflection induced by the diffuser, a partial or complete intermittent wash-out of the recirculating flow can be observed. Similarly, a low-frequency alteration between stalled and unstalled flow on aerofoils was reported by Zaman, McKinzie & Rumsey (Reference Zaman, Mckinzie and Rumsey1989) and Broeren & Bragg (Reference Broeren and Bragg1998) for angles of attack near the condition of maximum lift. More recently, the unsteadiness of APG-induced TSBs, where both separation and reattachment occurs on a solid flat surface, was addressed by Mohammed-Taifour & Weiss (Reference Mohammed-Taifour and Weiss2016). Based on pressure and velocity field measurements inside the symmetry plane, they reported a low-frequency contraction and expansion of the TSB, named breathing, that takes place at Strouhal numbers of
$\mathcal{O}({\textit{St}})=0.01$
, which is an order of magnitude lower in frequency than the flapping motion found in geometry-induced TSBs.
Along with the flat plate configuration addressed by Mohammed-Taifour & Weiss (Reference Mohammed-Taifour and Weiss2016), the breathing motion of APG-induced TSBs has also been observed for the cases of a 2-D wing (Wang & Ghaemi Reference Wang and Ghaemi2022) and a backward-facing ramp (Steinfurth, Cura & Weiss Reference Steinfurth, Cura and Weiss2022). Despite the 2-D character of these set-ups, it is well known from surface flow visualisations that there exists a significant secondary flow driven by the presence of sidewalls enclosing the test sections (Mohammed-Taifour, Weiss & Dufresne Reference Mohammed-Taifour, Weiss and Dufresne2021; Simmons et al. Reference Simmons, Thomas, Corke and Hussain2022). It is therefore reasonable to believe that the low-frequency unsteadiness also features a spanwise component. Indeed, by analysing the results of a direct numerical simulation conducted by Coleman, Rumsey & Spalart (Reference Coleman, Rumsey and Spalart2018), Cura et al. (Reference Cura, Hanifi, Cavalieri and Weiss2024) suggested recently that the low-frequency dynamics of APG-induced TSBs are associated with low spanwise wavenumbers. Yet, due to inherent limitations of the methods employed so far, the attention of experimental studies has typically been restricted to events unfolding on the symmetry plane. Notable exceptions are the recent planar velocity field measurements along the test section span presented by Steinfurth et al. (Reference Steinfurth, Cura and Weiss2022), Borgmann et al. (Reference Borgmann, Cura, Weiss and Little2024) and Fuchs et al. (Reference Fuchs, Steinfurth, von Saldern, Weiss and Oberleithner2025), which all hint at three-dimensional (3-D) flow dynamics.
Contrasting with the experimental studies mentioned earlier, numerical simulations are not per se restricted to planar flow field information. However, the computational cost associated with 3-D simulations spanning a number of time steps sufficiently large to resolve the low-frequency spectrum in a wide spanwise domain appears to be unaffordable to date (Na & Moin Reference Na and Moin1998; Abe Reference Abe2017; Coleman et al. Reference Coleman, Rumsey and Spalart2018; Wu, Meneveau & Mittal Reference Wu, Meneveau and Mittal2020).
In summary, previous studies of TSB low-frequency dynamics were restricted to planar measurements or, in the case of 3-D numerical simulations, could not resolve the low-frequency spectrum. Therefore, a full description of the breathing phenomenon has so far remained elusive, potentially misguiding interpretations of the contraction and expansion unfolding on the TSB symmetry plane. With the current effort, we intend to fill the gap flanked by the 2-D frequency-resolved results obtained in past experiments and the 3-D non-frequency-resolved numerical simulations. Specifically, we conduct volumetric particle tracking velocimetry (PTV) measurements to quantify the time-resolved velocity field for the flow past a smooth backward-facing ramp to reveal a more complete picture of the breathing motion.
2. Experimental approach and data processing
The flow under consideration is characterised in the following before details on the volumetric velocity measurement procedure and data post-processing are provided.
2.1. Flow configuration
Experiments were conducted in a low-speed wind tunnel with a
$0.6 \,{\textrm{m}}\times 0.6 \,{\textrm{m}}$
nozzle exit connected to a
$1.2\,{\textrm{m}}$
long square channel. A splitter plate separates the channel in two parts. The upper side features the smooth backward-facing ramp shown in figure 1(a) with dimensions of
$l \approx 0.3\ {\textrm{m}}$
in length,
$H = 0.08\,{\textrm{m}}$
in height and
$S = 0.6\,{\textrm{m}}$
in span. The splitter plate was given a small negative incidence angle (two degrees) to prevent leading-edge separation, resulting in a slightly favourable pressure gradient (FPG) of approximately
$\Delta c_p/\Delta x = -0.4\,{\textrm{m}}^{-1}$
upstream of the ramp (figure 1
b). Note that
$\Delta c_p$
is the difference in pressure coefficients between to locations separated by
$\Delta x$
. A local pressure drop near the wall is also expected at the ramp entrance at
$x/S = 0$
due to streamline curvature. Further downstream, an APG is observed, ultimately yielding flow separation, visualised with tufts (not shown here) before proceeding with quantitative measurements. The incoming boundary-layer, presented in figures 1(c) and 1(d), was characterised by displacement and momentum thicknesses of
$\delta ^* \approx 5.3\ \textrm{mm}$
and
$\vartheta \approx 4.0\ \textrm{mm}$
, respectively; the resulting shape factor
$\delta ^*/\vartheta \approx 1.31$
lies at the lower end of the typical range for fully developed turbulent boundary-layers, indicating a low separation tendency. Yet, the scaled profile is well aligned with the logarithmic law of the wall. Based on the momentum thickness and the inflow velocity
$U_\infty \approx 6.7\ {\textrm{ms}}^{-1}$
, the Reynolds number was
$\textit{Re}_{\vartheta}\approx 1800$
.
Experimental set-up and inflow conditions: (a) schematical side-view of closed test section, camera viewing angles indicated, (b) mean wall pressure distribution in the test section centre line, (c) and (d) mean turbulent boundary-layer profile obtained at
$x/S\approx -0.4$
;
$y^+$
and
$u^+$
denote the dimensionless wall distance in viscous units and the dimensionless velocity, respectively.
2.2. Velocity measurements and data assimilation
Volumetric velocity field information was obtained by performing 3-D PTV. Helium-filled soap bubbles (HFSBs) with a mean diameter of
$d_{{p}}\approx 0.3\ \textrm{mm}$
(Faleiros et al. Reference Faleiros, Tuinstra, Sciacchitano and Scarano2019) were introduced in the settling chamber, with a concentration of approximately
$0.25\,{\textrm{cm}}^{-3}$
in the test section. The mean particle distance of
$\Delta x_{{p}}\approx 16\ \textrm{mm}$
is small compared with the expected length scales associated with the low-frequency unsteadiness. The tracers were illuminated by two arrays of pulsed LEDs (LaVision LED Flashlight 300) placed vis-à-vis on both sides of the test section. The illumination was restricted in height to the volume of interest (approximately 100 mm), as shown in figure 1(a). The light scattered by HFSB tracers was recorded with four complementary metal-oxide-semiconductor (CMOS) cameras (Photron FASTCAM Mini AX100, 1024
$\times$
1024 pixels at
$4\,000 \,\textrm{f.p.s.}$
, pixel size 20
$\unicode {x03BC}$
m
$\times$
20
$\unicode {x03BC}$
m) subtending an arc of approximately 60 degrees. Each camera was equipped with a 35 mm photographic lens set with an aperture of
$f_{\#}=8$
. The cameras and LEDs were synchronised with the acquisition computer using a LaVision programmable timing unit (PTU-X). The 3-D particle reconstruction benefited from volumetric self-calibration (Wieneke Reference Wieneke2008), and the availability of the Shake-the-Box (STB) algorithm (Schanz, Gesemann & Schröder Reference Schanz, Gesemann and Schröder2016) involving iterative particle reconstruction and tracking (Wieneke Reference Wieneke2013). The component in the
$x$
direction for a single time step is highlighted by coloured circles in figures 2(b) and 2(c).
As is shown in figure 2(a), a time-resolved short sequence of eight snapshots was obtained to allow for the application of the STB algorithm with bursts sampled at
$f_{{b}}=2\,{\textrm{kHz}}$
, returning a velocity field with high accuracy and robustness. However, to cover the relatively wide spectrum of the velocity fluctuations, the frequency of burst repetition was set to
$f_{{s}}=60\,\textrm{Hz}$
. The latter choice allowed velocity measurements over a long observation time (
$30\,\textrm{s}$
) for a full description of the low-frequency fluctuations in the separated flow. As a result, several events of the breathing motion, expected to occur at a frequency of the order of
$f=1\,\textrm{Hz}$
(based on pressure measurements), could be captured. The sample rate corresponding to
$St\gt 1$
was chosen such as to also resolve shedding events alongside the breathing.
(a) Illustration of burst data acquisition, signal defined by burst frequency
$f_{{b}}=2000\,\textrm{Hz}$
and sampling frequency
$f_{{s}}=60\,\textrm{Hz}$
, (b) instantaneous velocity field in the test section symmetry plane (
$z/S=0$
), PTV data with maximum distance of 10 mm from that plane highlighted by circles, assimilated velocity field shown as a contour plot, (c) the same as in (b) but for a cross-section located at
$x/S=0.5$
.
Physics-informed neural networks (PINNs) were employed to reduce the scattered velocity field vectors onto a Cartesian grid. Equalling the number of detected tracers, approximately 3500 vectors were processed per time step. The approach recently introduced by Steinfurth et al. (Reference Steinfurth, Hassanein, Doan and Scarano2024) was followed: fully connected networks consisting of four hidden layers with 256 neurons mapped the spatio–temporal coordinates
$(x, y, z, t)$
to the corresponding velocity and pressure values
$(u, v, w, p)$
. The networks were trained in a supervised fashion, constrained by the PTV data on the one hand and residuals of the 3-D incompressible Navier–Stokes equations on the other. The data and physics-informed losses were evaluated at approximately 3500 PTV measurement points and 3000 randomly sampled collocation points per time step. Furthermore, the no-slip condition was enforced at the solid wall by penalising non-vanishing velocities at 3000 wall points evenly distributed across the solid boundaries. Several PINNs were trained to capture the temporal evolution of the flow field, one for each burst. The target time step corresponded to the fourth sample in the burst sequence. The velocity fields at the preceding and following snapshots were used to determine the time derivatives needed to formulate the physics-informed loss term.
The PINN-based assimilation introduced by Steinfurth et al. (Reference Steinfurth, Hassanein, Doan and Scarano2024) was modified in the present study by adopting Fourier feature embeddings mapping the input layer to a higher-dimensional space using a sinusoidal transformation. The transformed input features higher-frequency components compared with the raw input data, which has been shown to address the bias towards learning (only) low-frequency components (Tancik et al. Reference Tancik, Srinivasan, Mildenhall, Fridovich-Keil, Raghavan, Singhal, Ramamoorthi, Barron and Ng2020). The mean absolute difference between PINN outputs and the PTV measurement data is below
$0.01 U_\infty$
for the streamwise velocity component. An example of the PINN prediction for a single time step is offered in the shape of heatmaps shown in figure 2, sharing a colour map with the highlighted scattered PTV data.
While the trained models can be inferred at arbitrary locations, we chose to evaluate them only inside the volume of interest (figure 1
a) with a resolution corresponding to
$\Delta x = 5.0\,\textrm{mm}\approx 0.008 S$
,
$\Delta y = 2.5\,\textrm{mm}\approx 0.004 S$
and
$\Delta z = 10.0\,\textrm{mm}\approx 0.017 S$
in the streamwise, vertical and spanwise directions, respectively. The results presented in the following section are based on the analysis of the streamwise velocity component evaluated on structured Cartesian grids with approximately 165 000 points at 1800 time steps (86 points in the
$x$
, 32 in the
$y$
and 60 in the
$z$
direction).
2.3. Spectral proper orthogonal decomposition
As opposed to space-only proper orthogonal decomposition (POD), SPOD makes use of temporal Fourier transforms to decompose the flow both in space and in time, providing frequency-dependent modes (Schmidt & Colonius Reference Schmidt and Colonius2020). This allows us to limit our focus to a specific frequency regime (
${\textit{St}}\approx 0.01$
) while neglecting spectral content associated with smaller-scale fluctuations. For this analysis, the time series was divided into blocks of 300 snapshots with 50 % overlap. Hamming filters were applied to each block before Fourier transformation. The Fourier coefficients were compiled in cross-spectral density matrices (one for each frequency), and eigendecompositions of these matrices returned the complex-valued SPOD modes ranked by fluctuating energy.
To afford a physical interpretation of the dominant SPOD modes
$\Psi^1_u$
found at low frequencies, the mean flow
$\bar{u}(x)$
is superposed by the real part
$\mathcal{R}$
of the evolving first mode, defined by
\begin{equation} u_{\varPsi ^1_u}(\boldsymbol{x}, t) = \bar {u}(\boldsymbol{x}) + |a^1| \mathcal{R}\big(\varPsi ^1_u (\boldsymbol{x}) \textrm{e}^{i \omega t}\big), \end{equation}
where
$\omega = 2\unicode{x03C0} f$
is the angular frequency of oscillation at the selected frequency
${\textit{St}}=0.02$
. Note that the mode is scaled with the related coefficient averaged throughout the time series
$|a^1|$
, hence the resulting model should be viewed as an idealised harmonic representation of the flow dynamics with a fixed magnitude.
Additionally, dominant SPOD modes were extracted and band-limited to the range
${\textit{St}}\lt 0.05$
following the low-order modelling approach suggested by Nekkanti & Schmidt (Reference Nekkanti and Schmidt2021). An inverse Fourier transform was then applied to recover the time-resolved spatial fluctuations. Finally, the fluctuations were superimposed onto the time-averaged flow field to yield the velocity field evolution over one block duration.
3. Results
Before turning our attention to the 3-D dynamics, it is necessary to characterise the mean field and the breathing signature to relate this study to current state-of-the-art research.
(a) Mean streamwise velocity component, the blue isosurface indicates
$\bar {u}/U_\infty = 0$
, (b) normalised root-mean-square of the fluctuating streamwise velocity component
$u'_\mathrm{rms}/U_\infty$
shown in different slices,
$\bar {u}/U_\infty = 0$
now indicated by purple isosurface.
3.1. Mean field and breathing signature
Figure 3 shows the time-averaged streamwise velocity component. The mean flow resembles that past a straight backward-facing ramp, addressed by Steinfurth & Weiss (Reference Steinfurth and Weiss2024): a more distinct inflow velocity deficit is found near the sidewalls compared with the region around the centre line, resulting from the boundary-layers on the sidewalls. Hence, mean reverse flow is observed further upstream in this region. In contrast with the U-shaped mean separation line reported by Steinfurth & Weiss (Reference Steinfurth and Weiss2024), however, the separation line of the TSB assessed in the present study exhibits a W-shape (i.e. there exists a region close to the centre plane where separation occurs almost as far upstream as near the sidewalls). This difference in mean flow topology may be caused by minor inflow inhomogeneities. The length of the reverse-flow region inside the centre plane is
$L_{{b}} \approx S/3$
, equalling the aspect ratio found by Steinfurth & Weiss (Reference Steinfurth and Weiss2024).
The root-mean-square value of streamwise velocity fluctuations
$u'_{rms}$
is presented on the right-hand side of figure 3. Strong fluctuations are found throughout the separated shear-layer enclosing the mean TSB, most prominently in the regions close to the sidewalls where values close to 20 % of the inflow velocity are observed. These fluctuations can be expected to originate from two phenomena unfolding in distinct frequency regimes, namely shedding and breathing.
Space-only snapshot POD is applied to streamwise velocity fluctuations inside the centre plane, which has been proven to effectively elucidate the TSB dynamics in previous studies (e.g. Mohammed-Taifour & Weiss Reference Mohammed-Taifour and Weiss2016). Here, the dominant POD mode, shown in figure 4(a), encapsulates more than 70 % of the total kinetic energy and reveals a large region of correlated fluctuations spanning the downstream part of the mean TSB. The physical significance of this observation lies in a large-scale contraction and expansion of the TSB as the modal structure is modulated by the corresponding temporal coefficient
$a_1$
. The frequency content of the first POD mode is presented in figure 4(b), clearly falling inside the low-frequency regime around
$\mathcal{O}({\textit{St}})=0.01$
. The same is true for the wall pressure obtained upstream of the mean separation line in the region of the highest APG that is known to feature the low-frequency signature of the contraction/expansion motion.
(a) Normalised first spatial POD mode
$\Phi^1_u/U_\infty$
inside the centre plane, dashed line indicates
$\bar {u}/U_\infty = 0$
, (b) normalised power spectral densities
$PSD_{norm}$
of fluctuating parts of the first POD coefficient
$a'_1$
and wall pressure coefficient
$c'_p$
measured at
$x/S\approx 0.2$
.
In summary, despite some differences in the mean flow topology, the characteristics pertaining to the low-frequency unsteadiness observed inside the symmetry plane are consistent with findings reported in previous studies that were restricted to planar information.
3.2. Three-dimensional dynamics
Next, we turn our attention to the 3-D dynamics revealed by SPOD.
The SPOD spectrum is presented in figure 5(a). It shows low-rank behaviour in the range
${\textit{St}}\lt 0.05$
where a large proportion of the fluctuation energy is concentrated in the dominant modes. A minor peak in the leading and sub-leading eigenvalues is observed at
${\textit{St}}\approx 0.04$
, which is explained by the spectral estimation uncertainty rather than being of physical significance. As a representative of the entire low-frequency regime, we select the dominant SPOD mode at
${\textit{St}}\approx 0.02$
(
$f=0.8\,\textrm{Hz}$
) for further analysis. It contains approximately 50 % of the fluctuation energy at this frequency (figure 5
b), which is indicative of a dominant physical phenomenon.
Results of SPOD analysis: (a) spectrum of eigenvalues
$\mu$
, (b) fluctuation energy of modes at
$St\approx 0.02$
, represented by
$\mu$
relative to sum of all eigenvalues
$\sum \mu$
, (c) and (d) isosurfaces highlighting the real and imaginary part of the dominant mode at
$St\approx 0.02$
, respectively.
The pattern of the dominant SPOD mode (normalised real part) is illustrated in figure 5(c). It features four elongated coherent structures with correlated fluctuations. The two smaller ones coincide with the regions of maximum velocity fluctuations near the sidewalls while the larger ones span the flow near the centre plane. Whereas the mode is relatively homogeneous in the streamwise direction, a cellular pattern of fluctuations is noted along the span. Specifically, a mitigation of the reverse flow near the left sidewall (positive velocity fluctuations) is accompanied by an increase in reverse flow between the centre line and the left sidewall (negative velocity fluctuations), while this pattern is reversed on the other side (i.e. more intense reverse flow close to the right sidewall and reduced TSB between the symmetry plane and the right sidewall).
The imaginary part of the dominant mode is aligned with the real part (figure 5
d). This in-phase behaviour indicates a standing oscillatory pattern rather than a travelling phenomenon. In other words, the mode is locked in space and oscillates in time. The relative homogeneity in the streamwise direction also suggests that the mode is not of a convective nature. Similar features are found for the dominant modes throughout the low-frequency regime (
${\textit{St}}\lt 0.05$
) whereas smaller-scale, convecting structures occur at higher frequencies (not shown here). Both for the real and imaginary part, a spanwise wavelength of
$\lambda \approx S/2$
can be estimated visually, corresponding to a wavenumber of two cycles per span. Allowing for a quantitative validation, the complex SPOD mode at
$St\approx 0.02$
is decomposed into spanwise wavenumbers
$k_z$
by applying a Fast Fourier Transform
$\displaystyle \hat {\varPsi }^1_u(x, y, k_z)=\mathcal{F}_z [\varPsi ^1_u(x, y, z) ]$
in the cross-section at
$x=S/2$
(centre slice in figure 5). The dominant SPOD mode at this location is displayed in figures 6(a) and 6(b), confirming the high alignment of the real and imaginary parts. In the two-sided wavenumber spectra in figure 6(c) for different wall-normal distances, it is apparent that low spanwise wavenumbers carry the majority of the fluctuation energy
$E$
. Specifically, a dominant wavenumber of
$k_{z, n}=2$
is found to arise from the low-frequency breathing motion, thereby confirming the visual analysis of 3-D mode structures (
$\lambda \approx S/2$
). Furthermore, the wavenumber spectra are nearly symmetric about the zero wavenumber (i.e.
$E(k_z)\approx E(-k_z)$
), indicating that the mode does not contain dominant directionality in the spanwise direction.
Four selected phases of the dynamical process captured by this mode are presented in figure 7(a–d). The first time step features an almost symmetric distribution of the cellular reverse-flow regions that occur close to the sidewalls and on both sides of the centre plane, respectively. Next, transitioning from phase
$\varphi = \unicode{x03C0} /2$
to
$\varphi = \unicode{x03C0}$
, the TSB expands on one side of the backward-facing ramp, namely that which corresponds to the positive (red) streaky structure in figure 5. At the same time, the reverse flow near the respective sidewall is diminished. On the other side of the ramp, the picture is reversed (i.e. the reverse-flow region near the sidewall increases while the TSB close to the centre plane is reduced in size). Finally, at
$\varphi = 3\unicode{x03C0} / 2$
(panel d), The maxima/minima associated with the localised reverse-flow region are shifted by
$S/3$
in the spanwise direction.
(a,b) Normalised real and imaginary parts of
$St\approx 0.02$
SPOD mode at
$x/S=0.5$
, (c) wavenumber spectra for different wall-normal distances for these SPOD modes;
$E$
and
$E_{max}$
denote the raw spectral energy at wavenumber
$k_{z,n}$
and the maximum energy at that height, respectively.
Dynamics represented by leading SPOD mode at
$St\approx 0.02$
: (a) streamwise and crosswise slices indicating the main velocity component along with isosurface defined by
$u_{\varPsi ^1_u}/U_\infty = 0$
, shown at phase
$\varphi = 0$
, (b)
$\varphi = \unicode{x03C0} /2$
, (c)
$\varphi = \unicode{x03C0}$
, (d)
$\varphi = 3 \unicode{x03C0} /2$
, (e) normalised length of TSB for different spanwise stations throughout one oscillation period, phase angles presented in (a–d) highlighted by circles.
Figure 7(e) shows the streamwise distance
$L_{{b}}$
between the separation and reattachment point observed in the centre plane (black curve) and two further spanwise stations. The separation/reattachment points are defined by the most upstream/downstream locations where negative streamwise velocity is found. It is immediately apparent that the TSB undergoes significant changes in length during the oscillation (up to
$0.5L_{{b}}$
). This observation is fully consistent with previous studies of the breathing motion limited to planar measurements (e.g. Mohammed-Taifour & Weiss Reference Mohammed-Taifour and Weiss2016). Here, we demonstrate that this motion is caused by spanwise modulations of the flow field rather than being a 2-D phenomenon.
Similar conclusions can be drawn from the low-order model presented in supplementary movie 1 available at https://doi.org/10.1017/jfm.2025.10630. Here, instead of only one specific frequency, a number of selected frequency bins were utilised to form a low-order representation of the original velocity field dynamics, as explained in § 2.3. The movie clearly indicates that the behaviour observed at
$St\approx 0.02$
in figure 7 is in fact representative of a broad low-frequency phenomenon.
4. Concluding remarks
The objective of this study was to elucidate the 3-D low-frequency dynamics of an APG-induced TSB. Although previous works have already reported a large-scale contraction and expansion of the reverse-flow region, they were restricted to planar observations of an intrinsically 3-D dynamical system. The research gap was closed by conducting time-resolved 3-D PTV measurements that quantified the flow in the entire domain of interest and covered the spatio–temporal dynamics of flow separation.
The mean flow, featuring a TSB that spans the entire channel width, is characterised by an inhomogeneous, W-shaped separation line. Specifically, flow reversal is observed the furthest upstream near both sidewalls and close to the centre plane. This observation somewhat differs from the U-shaped topology found in previous experiments (Steinfurth & Weiss Reference Steinfurth and Weiss2024) despite the very similar experimental conditions. Nevertheless, the low-frequency signature revealed by space-only POD and wall pressure measurements is consistent with previous experiments.
Spectral proper orthogonal decomposition, applied to fluctuations of the streamwise velocity component, suggests low-rank behaviour at frequencies corresponding to
${\textit{St}}\lt 0.05$
, which has been associated with the breathing motion in previous studies. The related dominant modes exhibit inhomogeneities in the spanwise direction at wavelengths of the order of
$S/2$
. Specifically, one elongated structure is found near each sidewall while two further streaks with larger spanwise extent enclose the mean region of reverse flow near the centre plane. Interestingly, this mirror-inverted pattern is not of a convective nature. Neither does it travel in the spanwise direction but instead represents a standing oscillatory pattern that is locked in space while oscillating in time. The centre node of this standing wave is slightly to the side of the centre plane due to a minor asymmetry ascribed to real-world experimental imperfection. As a consequence, the flow observed in the centre plane experiences significant changes in the extent of the reverse-flow region, which has been observed in previous planar measurements. A similar standing-wave pattern was recently observed by Fuchs et al. (Reference Fuchs, Steinfurth, von Saldern, Weiss and Oberleithner2025) for a straight backward-facing ramp. Based on a resolvent analysis, they suggest that this pattern is a consequence of interfering left- and right-travelling modes reflected by the wind tunnel confinement. Furthermore, they associate the formation of the low-frequency streaks with the lift-up mechanism (Brandt Reference Brandt2014).
The cellular standing-wave pattern found in the present study is reminiscent of stall cells occurring on aerofoils and wings (Liu & Nishino Reference Liu and Nishino2018). Ma, Gibeau & Ghaemi (Reference Ma, Gibeau and Ghaemi2020) recently showed that these localised regions of separated flow undergo low-frequency fluctuations consistent with the breathing phenomenon found in flow configurations where reattachment occurs on a solid surface. Considering that the number of cells strongly depends on the aspect ratio (Weihs & Katz Reference Weihs and Katz1983), this parameter and its relation to the resulting low-frequency dynamics may be investigated in future studies.
Supplementary movie
Supplementary movie is available at https://doi.org/10.1017/jfm.2025.10630.
Acknowledgements
The authors gratefully acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – project number 504349109. We would also like to express our gratitude to C. Cura, L. Fuchs, J. von Saldern and K. Oberleithner for stimulating discussions on TSB dynamics that contributed to the development of this work.
Declaration of interests
The authors report no conflict of interest.
