# Effects of oncoming flow conditions on the flow upstream of a forward-facing step subject to a thin boundary layer

Emil Larose, Franck Kerhervé, Yann Fraigneau, Bérengère Podvin, Chris

Morton, Robert J Martinuzzi

► To cite this version:

Emil Larose, Franck Kerhervé, Yann Fraigneau, Bérengère Podvin, Chris Morton, et al. Ef-fects of oncoming flow conditions on the flow upstream of a forward-facing step subject to a thin boundary layer. International Journal of Heat and Fluid Flow, 2025, 115, pp.109866.10.1016/j.ijheatfluidflow.2025.109866. hal-05366300

HAL Id: hal-05366300

https://hal.science/hal-05366300v1

Submitted on 14 Nov 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>
<figure><img src="image_2.png" /></figure>

# Effects of oncoming flow conditions on the flow upstream of a forward-facing step subject to a thin boundary layer

Emil Larose<sup>a,∗</sup>, Franck Kerhervé<sup>b,†</sup>, Yann Fraigneau<sup>c,‡</sup>, Bérengère Podvin<sup>c,‡</sup>, Chris Morton<sup>d,‡</sup>, Robert J. Martinuzzi<sup>a,‡</sup>

## ARTICLE INFO

### Keywords:

Turbulent

Laminar

Boundary layer

Separated flow

Unsteady dynamics

Recirculation regions

Forward-facing step

## ABSTRACT

The influence of the oncoming flow on intrinsic flow dynamics of a forward-facing step (FFS) is investigated using planar, two-component particle image velocimetry (PIV) and direct numerical simulations (DNS) for thin ($\delta/H < 1$) laminar and turbulent boundary layers (LBL & TBL). Transition of the upstream separation region between *steady* and *unsteady* behaviour is shown to be sensitive to weak freestream disturbances above a critical Reynolds number (Re). *Unsteady* behaviour of the upstream separation, for oncoming TBLs, is characterised by the intermittent large-scale ejection of fluid over the step. *Steady* behaviour is observed for oncoming LBLs below a critical Re or disturbance-free. It is characterised by a low-frequency oscillation of the separating shear layer below $f^* = fH/U_\infty \approx 0.03$ and the recirculation region exhibits a natural instability around $f^* \approx 0.15$. Above a critical Re, this instability can be amplified by very small freestream disturbances and the behaviour of the upstream recirculation region becomes *transitional-unsteady*. The dynamics in this state – between *steady* and *unsteady* – exhibit intermittent ejections along with enhanced convective dynamics due to the formation of larger-scale motions at the unsteady laminar separation point. Unlike immersed steps ($\delta/H > 2$), scales from the oncoming boundary layer are not imprinted on the separation region, requiring a more complex description relating the oncoming flow to the recirculation dynamics about the step.

## 1. Introduction

The flow over a forward-facing step (FFS) is a diagnostic configuration to investigate fundamental instability and separation mechanisms. This geometry is relevant to real-world applications such as natural terrain and low-rise buildings, which are of particular interest for wind turbine and heliport placement. Thus, this flow configuration is particularly interesting from both scientific and engineering perspectives.

A characteristic of FFS flow is the presence of two separated flow regions: one at the foot of the step due to separation of the oncoming boundary layer (BL) resulting from the step-induced adverse pressure gradient (APG), and another immediately downstream of the step due to the salient edge of the step corner. The resulting instabilities of the shear layer and BL separation are also observed in other flow configurations like the blunt flat plate, square ribs and APG BLs.

The complexity of FFS flow lies in the many interacting flow phenomena, including laminar and turbulent BLs in adverse pressure gradients, interacting recirculation bubbles, shear layer flapping, and vortex

shedding. The underlying flow instabilities may be excited by intrinsic (i.e., naturally occurring) and extrinsic (i.e., external forcing) means, a topic which is explored in the present work.

FFS flows are typically classified according to three parameters: the state of the oncoming flow (laminar or turbulent), the Reynolds number ($Re_H = U_\infty H/\nu$, with $U_\infty$, $H$ and $\nu$ the freestream velocity, the step height, and the kinematic viscosity), and the relative thickness of the oncoming BL. A *thin* BL refers to thickness ($\delta$) smaller than $H$, $\delta/H < 1$, (protruding step) while *thick* BL refers to $\delta/H > 1$ (immersed step) (Sherry et al., 2010). For very thick ($\delta/H > 2$) turbulent boundary layers (TBLs) approaching a FFS, the scales of the oncoming TBL are imprinted on the flow near the FFS (Camussi et al., 2008; Sherry et al., 2010; Awasthi et al., 2014; Fang and Tachie, 2020). In contrast, the dynamics of FFS flow with very thin oncoming TBLs ($\delta/H < 0.3$) is dominated by the intrinsic local scales of the flow at the step (Largeau and Moriniere, 2007; Sherry et al., 2010; Graziani et al., 2017). As such, both very thick and thin TBLs have shown an insensitivity to $Re_H$ due

<footnote><sup>a</sup> Department of Mechanical and Manufacturing Engineering, University of Calgary, Calgary, Alberta, Canada</footnote>

<footnote><sup>b</sup> Institut Pprime, CNRS, Université de Poitiers, Poitiers, France</footnote>

<footnote><sup>c</sup> Laboratoire EM2C & Laboratoire interdisciplinaires des sciences numériques (LISN), CNRS, Université Paris-Saclay, Orsay, France</footnote>

<footnote><sup>d</sup> Department of Mechanical Engineering, McMaster University, Hamilton, Ontario, Canada</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: louis.larose@ucalgary.ca (E. Larose).</footnote>
Nomenclature

Abbreviations

<table><tr><td>APG</td><td>Adverse pressure gradient</td></tr><tr><td>BL</td><td>Boundary layer</td></tr><tr><td>DNS</td><td>Direct numerical simulation</td></tr><tr><td>FFS</td><td>Forward-facing step</td></tr><tr><td>LBL</td><td>Laminar boundary layer</td></tr><tr><td>PDF</td><td>Probability density function</td></tr><tr><td>POD</td><td>Proper orthogonal decomposition</td></tr><tr><td>PSDF</td><td>Power spectral density function</td></tr><tr><td>TBL</td><td>Turbulent boundary layer</td></tr><tr><td>TKE</td><td>Turbulent kinetic energy</td></tr></table>

English Symbols

<table><tr><td>U = (U, V, W)</td><td>Mean streamwise, wall-normal and span-wise velocity</td></tr><tr><td>u = (u, v, w)</td><td>Instantaneous streamwise, wall-normal and spanwise velocity</td></tr><tr><td>x = (x, y, z)</td><td>Streamwise, wall-normal and spanwise velocity</td></tr><tr><td>H</td><td>Heaviside function</td></tr><tr><td>Ψ</td><td>POD spatial mode</td></tr><tr><td>Fr</td><td>Froude number</td></tr><tr><td>med</td><td>Median value</td></tr><tr><td>Re</td><td>Reynolds number</td></tr><tr><td>a</td><td>POD temporal coefficient</td></tr><tr><td>A<sub>u</sub></td><td>Upstream reverse flow area</td></tr><tr><td>D</td><td>Water channel depth</td></tr><tr><td>f</td><td>Frequency</td></tr><tr><td>f<sub>n</sub></td><td>Natural frequency</td></tr><tr><td>f<sub>s</sub></td><td>Sampling frequency</td></tr><tr><td>g</td><td>Gravitational acceleration</td></tr><tr><td>H</td><td>Step height</td></tr><tr><td>h<sub>d</sub></td><td>Downstream recirculation height</td></tr><tr><td>H<sub>in</sub>, H<sub>out</sub></td><td>Inlet and outlet height</td></tr><tr><td>h<sub>u,a</sub></td><td>Step wall reattachment height</td></tr><tr><td>l<sub>d,a</sub></td><td>Downstream reattachment length</td></tr><tr><td>L<sub>foot</sub></td><td>Length of the foot of the step</td></tr><tr><td>L<sub>step</sub></td><td>Length of the top of the step</td></tr><tr><td>l<sub>u,s</sub></td><td>Upstream separation location</td></tr><tr><td>N</td><td>Number of snapshots (time steps)</td></tr><tr><td>r</td><td>Covariance function</td></tr><tr><td>T</td><td>Acquisition time</td></tr><tr><td>t</td><td>Time</td></tr><tr><td>U<sub>∞</sub></td><td>Freestream velocity</td></tr></table>

Greek Symbols

<table><tr><td>δ</td><td>Boundary layer thickness</td></tr><tr><td>δ<sup>*</sup></td><td>Boundary layer displacement thickness</td></tr><tr><td>δ<sub>ij</sub></td><td>Kronecker delta</td></tr><tr><td>γ</td><td>Intermittency</td></tr><tr><td>λ</td><td>POD eigenvalue</td></tr></table>

to the predominance of the scales of the oncoming flow or the local scales at the step.

For $0.3 < δ/H < 2$, where the scales of the oncoming flow and the step are similar, coupling between intrinsic instabilities of the oncoming BL and the step appears to dictate the dynamics of FFS flow.

<table><tr><td>μ</td><td>Mean value</td></tr><tr><td>ν</td><td>Kinematic viscosity</td></tr><tr><td>ω</td><td>Vorticity</td></tr><tr><td>φ</td><td>Power spectral density</td></tr><tr><td>Ψ</td><td>POD spatial mode</td></tr><tr><td>σ</td><td>Standard deviation</td></tr><tr><td>σ<sup>2</sup></td><td>Variance</td></tr><tr><td>τ</td><td>Time delay</td></tr><tr><td>θ</td><td>Boundary layer momentum thickness</td></tr><tr><td>μ̃<sub>3</sub></td><td>Skewness</td></tr><tr><td>μ̃<sub>4</sub></td><td>Kurtosis</td></tr></table>

Subscripts/Superscripts/Operators

<table><tr><td>(·)<sup>l</sup></td><td>Fluctuating quantity</td></tr><tr><td>(·)<sup>*</sup></td><td>Non-dimensionalised quantity</td></tr><tr><td>(·)<sup>+</sup></td><td>Viscous units</td></tr><tr><td>(·)<sup>×</sup></td><td>Flow cases with perturbations</td></tr><tr><td>(·)</td><td>Time-average</td></tr></table>

There is thus a greater variability in the downstream recirculation size (in Fig. 1(a), $l_{d,a} \approx [1.5, 5]$) and Reynolds stress distributions in earlier studies (Sherry et al., 2010), suggesting that the downstream flow is dependent on the behaviour of the recirculation at the foot of the step. As such, the relationship between the intrinsic instabilities of the oncoming BL, the upstream recirculation region and their receptivity to freestream perturbations, which remains poorly understood, warrants further investigation.

For oncoming laminar flow, most investigations are for a FFS immersed in a fully-developed laminar channel flow, where the flow over the FFS is subject to an oncoming parabolic velocity profile (and thus a shear gradient), which is tantamount to a thick laminar boundary layer (LBL). The flow thus depends on $Re_H$ and the step-to-inlet height ratio, $H/H_{in}$ in Fig. 1(a), which for small values is analogous to $(2\delta/H)^{-1}$. As $H/H_{in} \to 1/2$, the channel contraction ratio can result in inertial effects, e.g. increased velocity gradients, which may amplify disturbances or instabilities differently than would occur for FFS exposed to boundary layers for unconfined freestreams. Therefore, the intrinsic dynamics of the recirculation regions are better studied using thin laminar and turbulent BLs approaching the step due to the limited extrinsic forcing by the oncoming flow or boundaries.

The stability and the sensitivity to oncoming disturbances of the separation regions have been studied for laminar channel flows (Stüer et al., 1999; Wilhem et al., 2003; Lanzerstorfer and Kuhlmann, 2012; Marino and Luchini, 2009). Experiments by Stüer et al. (1999) showed the three-dimensional structure of the upstream separation region along with the formation of streaks extending over the step from the front of the FFS in a spatially quasi-periodic spanwise pattern. The number of streaks and structure complexity decreased with $Re_H$, speculating that at sufficiently low $Re_H$ the upstream separation is quasi-two-dimensional and steady. Wilhem et al. (2003) conducted direct numerical simulations (DNS) and linear stability analysis to demonstrate that the transition from a steady, large, two-dimensional recirculation to an unsteady, three-dimensional state was not a result of an absolute instability, rather a reaction to weak oncoming disturbances. These findings were generally corroborated by Lanzerstorfer and Kuhlmann (2012), who investigated a range of flow parameters (e.g., $Re_H$ and $H/H_{in}$). Once more, there is a marked difference between laminar channel flow with large $H/H_{in}$ (varying from 0.25 to 0.95) and a LBL approaching a FFS. To the authors' knowledge, such stability analysis has not been published for thin $(δ/H < 1)$ LBL approaching a FFS, and little is known about the response of the upstream separation to oncoming disturbances in this configuration.
Unsteady upstream separation behaviour, characterised by a three-dimensional structure at the foot of the step, was also observed for oncoming TBLs of varying $\delta/H$. Camussi et al. (2008), Pearson et al. (2013), Graziani et al. (2018) and Fang and Tachie (2020) reported that spectral energy content of velocity or pressure fluctuations shifted to lower frequencies as the flow approached the FFS, indicative of the formation of larger structures at the foot of the step. For thin TBLs, Graziani et al. (2018) reported spectral energy centred about a reduced frequency of $f^* = fH/U_\infty = 0.027$ from pressure measurements at the foot of the step. For thick TBLs, Camussi et al. (2008) found a frequency of $f^* = 0.01$ also from pressure measurements, while Pearson et al. (2013) and Fang and Tachie (2020) reported frequencies of $f^* = 0.09$ and 0.047, respectively, from analysis of the reverse flow area. For thick oncoming TBLs, the low-frequency dynamics are broadly ascribed to the expansion and contraction of the upstream separation and intermittent “ejections” of separated flow from the recirculation at the foot of the step over the salient edge (Pearson et al., 2013). They found that “ejections” are preceded by low-velocity regions from the oncoming BL that convect over the step thereby providing a connection between the oncoming flow, the upstream separation and the downstream recirculation. Fang et al. (2017) corroborated this finding, also for a thick TBL. The dependence of these low-frequency dynamics on $\delta/H$ and $Re_H$ suggests that FFS flow is influenced by the oncoming flow and associated extrinsic forcing, a relationship that remains unclear for thin laminar and turbulent BLs.

New in the present work is the identification of a transitional-unsteady behaviour of the upstream separation which is considered to be between steady and “fully” unsteady (hence “transitional”). The phenomena in this range develop with an oncoming LBL above a Reynolds number (Re) threshold in the presence of weak freestream perturbations which excite a shear layer instability at the upstream separation to the point of unsteady flow dynamics (hence “unsteady”). It remains to be seen if the Reynolds number effect is related to the BL ($Re_\theta$, the momentum thickness Re) or the local scales at the step to the BL ($Re_H$, the step height Re). The dynamics are distinct and provide insights to the transition from steady to unsteady behaviour of the upstream recirculation region.

The purpose of this study is thus to characterise the intrinsic dynamics of the upstream separation region for FFS flow using a variety of oncoming flow conditions. It will be shown that an instability of the upstream separation may indeed be excited by freestream disturbances, broadly corroborating previous findings for laminar channel flow (Wilhem et al., 2003; Lanzerstorfer and Kuhlmann, 2012; Marino and Luchini, 2009). Mean characteristic dimensions of the separation regions are also reported for thin LBL FFS flow which, to the authors' knowledge, have not been previously documented.

The article is organised as follows. The experimental, numerical and analytical methodologies are described in Section 2. Then, the oncoming BL is characterised in Section 3 prior to the analysis of the mean and dynamical characteristics of the flow in Sections 4–6.

## 2. Methodologies

In this section, the nomenclature and different test conditions are described. Details of the experimental setup and methodology are then provided in Section 2.1 followed by those for the DNS implementation in Section 2.2. Implementation of the proper orthogonal decomposition (POD) and Fourier decomposition, along with the associated terminology are briefly summarised.

Fig. 1(a) presents a schematic of the experimental configuration and FFS nomenclature. The origin of the coordinate system is the red dot at the corner of the foot of the step (mid-span of the model). Coordinates $(x, y, z)$ refer to streamwise, wall-normal, and spanwise directions, respectively. The instantaneous velocity components along each axes are thus $(u, v, w)$. In the following, $U_\infty$, $V$ and $W$ denote the time-averaged velocity components (denoted as $\cdot$) for other quantities).

Fluctuating components are represented by $(\cdot)^t$. Finally, superscript $(\cdot)^*$ refers to reduced variables, scaled by the step height $H$ and the mean freestream velocity $U_\infty$.

The cases are summarised in Table 1. The labels refer to the state of the oncoming BL (LBL or TBL), the $Re_H$ ($= U_\infty H/\nu$) and $(\cdot)^\times$ denotes the experimental cases and the associated existence of freestream disturbances.

### 2.1. Experimental

Experiments have been conducted in the closed-loop water channel in the Laboratory for Turbulence Research in Aerodynamics and Flow Control (LTRAC), University of Calgary, shown in Fig. 1(b). The channel straight section consists of glass walls through which we observe the flow. Prior to the straight section, a chamber with screens is used to settle the flow. The flow is contracted through converging section with a contraction ratio of approximately 6 : 1. The straight section is 0.38 m wide $\times$ 0.44 m tall $\times$ 3 m long. The free surface depth is $D = 0.44$ m. A variable frequency drive is used to adjust the test section flow speed from 0.07 to 0.42 m/s.

A FFS model, consisting of a flat plate with an 8:1 elliptical leading edge with a protruding sharp 90° step was suspended in the flume from 4 elliptical struts, as illustrated in Fig. 1(a). The step height is $H = 23.6$ mm. The model spans the entire spanwise extent of the channel except a 3 mm gap between the model edges and the channel side walls on either side. No end plates were used.

For the laminar cases, the incoming boundary layer develops naturally from the elliptical leading edge of the flat plate. For the turbulent cases, the boundary layer is tripped using a 2 mm thick sandpaper strip (25 mm long in $x$ and full model span) located 25 mm from the leading edge of the flat plate. The streamwise dimensions of the model before and after the step are $L_{fout}/H = 22.2$ and $L_{step}/H = 12.9$, respectively. The spanwise width of the model is $W/H = 16.1$ to limit extrinsic three-dimensional effects at the mid-span (de Brederode and Bradshaw, 1974; Martinuzzi and Tropea, 1993; Largeau and Moriniere, 2007). The distance between the top of the step and the bottom wall of the channel is $H_{out}/H = 12.8$ resulting in a constriction ratio of $H/H_{in} = 7.25\%$. No blockage corrections were applied.

Planar, two-component PIV measurements were undertaken at the mid-span of the model. Two Phantom Miro M340 cameras with Sigma 65 mm $f/2.8$ DG macro lenses were used to acquire PIV images. The native resolution of the cameras were reduced to $2560 \times 856$ px$^2$ to maximise the number of snapshots per acquisition to 1910.

A Photonics Industries DM20-527 Nd:YLF dual-cavity laser was used to produce a thin laser sheet ($\approx 2$ mm), effectively illuminating the hollow glass spherical seed particles (specific gravity $\approx 1.1$). The images were processed using LaVision DaVis 10.2 software using multi-pass cross-correlation (Wieneke, 2017b) with a final interrogation window size of $24 \times 24$ pixels$^2$ with 50% overlap, resulting in vector spacing of 0.67 mm (0.028H). Side-by-side calibration and image processing, available within the LaVision software package, were used to obtain one vector field from simultaneous image acquisition from two cameras.

Uncertainty for the velocity magnitude at each spatial point was calculated using the correlation statistics method (Wieneke, 2015) and was on average approximately 1% & 3% of $U_\infty$ in the freestream and the wake, respectively. To remove measurement noise, anisotropic denoising based on uncertainty (Wieneke, 2017a) was applied to the velocity fields (kernel size of $5 \times 5$ and smoothing strength of 2). The data sets were acquired at a sampling frequency ($f_s$) of 15 Hz, sufficient to capture the dominant coherent motions and satisfying the convergence of the statistical quantities. Acquisitions at 100 Hz were also collected to verify that aliasing effects were negligible but are not shown for brevity. Each case has $N = 7640$ total snapshots.

To get more detailed boundary layer statistics and spectra, additional PIV images were acquired in a small window near the wall far
<figure><img src="image_3.png" /><figcaption>**Fig. 1.** (a) Schematic of the experimental configuration with typical mean recirculation bubble topology of FFS flow with notations, adapted from Sherry et al. (2010). Note that this schematic is inverted with respect to the actual experimental configuration to align with the results presented in subsequent sections. (b) Rendering of the Eidetics Model 1520 Low-Turbulence Water Channel at LTRAC (University of Calgary).</figcaption></figure>

<caption>**Table 1**</caption>

Main oncoming flow properties at $x/H = -9.5$; freestream velocity ($U_∞$), Froude number ($Fr = U_∞/\sqrt{gD}$), step height Reynolds number ($Re_H = U_∞H/v$), BL thickness ($δ/H$), displacement thickness ($δ^*/H$), momentum thickness ($θ/H$), shape factor ($δ^*/θ$), displacement thickness Reynolds number ($Re_δ$), momentum thickness Reynolds number ($Re_θ$), scaled sampling frequency ($f_s^*$) and scaled facility natural frequency ($f_n^*$). For experimental cases, $f_s = 15$ Hz and $f_n = 0.086$ Hz.

<table><thead><tr><th>Case</th><th>Meth.</th><th>U<sub>∞</sub> (m/s)</th><th>Fr</th><th>Re<sub>H</sub></th><th>δ/H</th><th>δ*/H</th><th>θ/H</th><th>δ*/θ</th><th>Re<sub>δ</sub></th><th>Re<sub>θ</sub></th><th>f<sub>s</sub><sup>a</sup></th><th>f<sub>n</sub><sup>a</sup></th></tr></thead><tbody><tr><td>LBL<sub>1700</sub></td><td>PIV</td><td>0.07</td><td>0.12</td><td>1700</td><td>0.47</td><td>0.165</td><td>0.062</td><td>2.65</td><td>280</td><td>105</td><td>5.03</td><td>0.029</td></tr><tr><td>LBL<sub>4800</sub></td><td>PIV</td><td>0.20</td><td>0.22</td><td>4800</td><td>0.28</td><td>0.102</td><td>0.038</td><td>2.69</td><td>490</td><td>182</td><td>1.74</td><td>0.010</td></tr><tr><td>LBL<sub>7900</sub></td><td>PIV</td><td>0.33</td><td>0.28</td><td>7900</td><td>0.23</td><td>0.078</td><td>0.030</td><td>2.58</td><td>616</td><td>237</td><td>1.07</td><td>0.006</td></tr><tr><td>LBL<sub>8000</sub></td><td>DNS</td><td>1</td><td>–</td><td>8000</td><td>0.41</td><td>0.141</td><td>0.050</td><td>2.84</td><td>1128</td><td>400</td><td>3.33</td><td>–</td></tr><tr><td>TBL<sub>8100</sub></td><td>PIV</td><td>0.34</td><td>0.28</td><td>8100</td><td>0.43</td><td>0.078</td><td>0.045</td><td>1.75</td><td>632</td><td>365</td><td>1.04</td><td>0.006</td></tr><tr><td>TBL<sub>8300</sub></td><td>DNS</td><td>1</td><td>–</td><td>8300</td><td>0.48</td><td>0.092</td><td>0.056</td><td>1.64</td><td>764</td><td>465</td><td>10.0</td><td>–</td></tr></tbody></table>

upstream of the step ($x/H \approx [-10, -9]$ & $y/H \approx [0, 0.9]$) as depicted in Fig. 1(a). For these acquisitions, only one camera was used with increased sampling frequency to 100 Hz. The camera sensor resolution was reduced to 428 × 428 to increase the number of snapshots to 20 924. Images were processed using the same procedure with 24 × 24 pixels² with 75% overlap, resulting in a vector spacing of 0.31 mm.

## 2.2. Numerical

Direct Numerical Simulations (DNS) of a LBL and TBL approaching a FFS were performed at Laboratoire interdisciplinaires des sciences numériques (LISN), Université de Paris-Saclay (Fraigneau, 2024; Podvin and Fraigneau, 2024). A brief description of the numerical

methods used in the simulation follows. Full details on numerical schemes can be found in Faugaret et al. (2022).

The Navier–Stokes equations are discretised in space with a second-order centred scheme on a staggered cartesian grid (Harlow and Welch, 1965), following a finite volume approach. The time discretisation relies on the second-order backward differentiation formula (BDF2 scheme). A semi-implicit technique on the viscous terms is used to ensure the stability of the numerical method with respect to the time step, which is based on the Courant–Friedrichs–Lewy (CFL) criterion. The discretised Navier–Stokes equations are solved using an ADI method (Peaceman and Rachford, 1955; Beam and Warming, 1976). The divergence-free velocity field is obtained from the incremental projection method in rotational formulation (Timmermans et al., 1996). For each time step, a prediction of the velocity field is computed
from the discretised Navier-Stokes equations, in which the pressure gradient is defined from the previous time step. At the end of this stage, divergence-free is not ensured. The projection method involves solving a Poisson equation to obtain a potential scalar field which provide the updates for the pressure and velocity field as divergence-free. The incremental projection method in rotational formulation is known to enforce the consistency of the Neumann boundary condition of the Poisson equation, usually used to solve it, and the momentum equation. The accuracy of the method is also improved (Guermond et al., 2006; Hosseini and Feng, 2011). The Poisson equation is solved from the successive over-relaxation method (SOR), in a red-black form, coupled with a geometric multigrid method in order to improve the convergence performance (for an example, see Saad (2003)). The code is parallelised following a domain decomposition approach performed by the MPI implementation. The computational domain is divided into cartesian blocks that perfectly fit the step topology. Each subdomain is therefore discretised according to a body-fitted grid. The Navier-Stokes equations are integrated using the technique of Schur complement in addition to the ADI method while the SOR algorithm used for solving the Poisson's equation is completed by an usual overlapping technique (one common point of discretisation for adjoined subdomains).

The computational domain size is 34H (LBL) and 37H (TBL) × πH × 22H in the streamwise, spanwise and wall-normal directions, respectively. The constriction ratio is thus small ($H/H_{in} = 1/22 = 4.55\%$). The step is placed 14H (LBL) and 17H (TBL) downstream from the inlet. The boundary conditions are defined as the usual no-slip conditions at walls, tangential stress-free and zero mass flow rate conditions at the top of the domain and periodic conditions in the spanwise direction. For the LBL case, a Blasius profile with $\delta/H = 1/3$ is prescribed at the inlet. In contrast, for the TBL case, a synthetic model for turbulent fluctuations at the inlet (Deck and Laraufie, 2013) is used to mimic a turbulent boundary layer at $Re_{\theta} = U_{\infty}\theta/v = 300$ (Spalart, 1988).

The Cartesian grid is made up of 1920 × 384 × 768 cells for both LBL and TBL simulations. For the TBL case, the spatial discretisation is expressed in wall units, defined in relation to the boundary layer features at the inlet. The grid spacing in the streamwise direction varies between 4.5⁺ and 12.5⁺ wall units, with a minimum value in the vicinity of the step. In the spanwise direction, the spacing is uniform at approximately 3.7⁺. Finally, the grid spacing evolves between 0.3⁺ and 4.7⁺ in the wall-normal direction over one step height from the wall. For the LBL case, the grid spacing is quite similar. In the spanwise direction, the spacing is identical with $\Delta z/H = 8.18 \times 10^{-3}$. The spacing distribution yields $1.0 \times 10^{-2} \le \Delta x/H \le 2.50 \times 10^{-2}$ in the streamwise direction and $1.0 \times 10^{-3} \le \Delta x/H \le 10^{-2}$ in the normal direction (over one step height from the wall).

The time step is set to $4.5 \times 10^{-4}$ time units in order to satisfy a CFL value less than 0.4 over the range of the entire domain. Data have been recorded over a time range of 162 (540 snapshots at $f_s^* = 0.3$) and 400 (4000 snapshots at $f_s^* = 0.1$) time units for the LBL and TBL simulations, respectively.

### 2.3. Proper orthogonal decomposition

The instantaneous velocity vector, $\mathbf{u}(\mathbf{x}, t)$, is represented using a Reynolds' decomposition into time-averaged, $\mathbf{U}(\mathbf{x})$, and fluctuating, $\mathbf{u}'(\mathbf{x}, t)$, components as

$$ \mathbf{u}(\mathbf{x}, t) = \mathbf{U}(\mathbf{x}) + \mathbf{u}'(\mathbf{x}, t). \qquad (1) $$

Proper Orthogonal Decomposition (POD) of $\mathbf{u}'$ is used to identify coherent motions (Lumley, 1967, 1970; Berkooz et al., 1993). Following the method of snapshots of Sirovich (1987), the fluctuating velocity field is decomposed into a finite sum of time-invariant spatial eigenfunctions $\Psi_k(\mathbf{x})$ multiplied by a time-varying temporal coefficient $a_k(t)$,

$$ \mathbf{u}'(\mathbf{x}, t) = \sum_{k=1}^{N} a_k(t) \Psi_k(\mathbf{x}). \qquad (2) $$

Vectors $\Psi_k(\mathbf{x})$ satisfy the orthonormality condition, where the inner-product is defined in the Hilbert $L^2$ space as

$$ (\Psi_l(\mathbf{x}), \Psi_j(\mathbf{x})) = \iint_{\Omega} \Psi_l(\mathbf{x}) \cdot \Psi_j(\mathbf{x}) dA = \delta_{lj}, \qquad (3) $$

where $\Omega$ is the PIV domain, with

$$ a_k(t) = (\mathbf{u}'(\mathbf{x}, t), \Psi_k(\mathbf{x})), \qquad (4) $$

such that the mean total turbulent kinetic energy is,

$$ \mathrm{TKE} = \frac{1}{2} \sum_{k=1}^{N} a_k a_k = \frac{1}{2} \sum_{k=1}^{N} \lambda_k, \qquad (5) $$

where modes are ordered by decreasing energy ($\lambda_1 > \lambda_2 > \dots > \lambda_N$). A reduced-order estimate, hereafter $\tilde{u}(\mathbf{x}, t)$, can be obtained by retaining the first $N_c$ terms (with $N_c \ll N$) of the summation in Eq. (2) associated to the most energetic modes. Note that in the following, $\mathbf{u}'(\mathbf{x}, t)$ consists of $\mathbf{u}'(\mathbf{x}, t)$ and $\mathbf{v}'(\mathbf{x}, t)$.

### 2.4. Power spectral density function estimates

The estimation of the power spectral density function (PSDF) is used throughout this work for spectral analysis of time-varying signals. The autocovariance function of an arbitrary ergodic, statistically stationary signal, $g(t)$, is

$$ r_{gg}(\tau) = \lim_{T \to \infty} \frac{1}{T} \int_{-\infty}^{\infty} g^*(t - \tau)g(t) d\tau. \qquad (6) $$

The PSDF of $g(t)$ is then defined as the Fourier transform of $r_{gg}(\tau)$ as

$$ \phi(f) = \int_{-\infty}^{\infty} r_{gg}(\tau) \exp(-i2\pi f \tau) d\tau. \qquad (7) $$

To estimate $\phi(f)$ in the present work, Welch's averaging method is used with 50% overlapping segments and a Hamming window. The PSDFs in the present work are not scaled, thereby satisfying Parseval's identity.

## 3. Oncoming flow characterisation

The dynamics of the separation at the foot of the step is sensitive to the oncoming flow conditions. In this section, the oncoming boundary layer and freestream are characterised for the experimental cases using the PIV measurements far upstream of the step ("cam 1 BL" in Fig. 1(a)).

### 3.1. Boundary layer

The time-averaged streamwise velocity ($U/U_{\infty}$) and Reynolds stress ($\overline{u'^2}/U_{\infty}^2$, $\overline{v'^2}/U_{\infty}^2$ and $-\overline{u'}\overline{v'}/U_{\infty}^2$) profiles at $x/H = -9.5$ for the experimental and numerical TBL cases (TBL$_{8100}^\times$ & TBL$_{8300}$) in Fig. 2 show some discrepancies. On one hand, a power law profile ($U/U_{\infty} = (y/\delta)^{1/6}$) agrees well with the $U/U_{\infty}$ profile of TBL$_{8100}^\times$ in Fig. 2(a). The power law coefficient $1/6$ is appropriate for lower Reynolds number flows as in the current case (Schlichting and Gersten, 1979). On the other hand, the development of the BL in TBL$_{8300}$ is surmised to not have reached an "equilibrium state" at $x/H = -9.5$ due to the inflow conditions, the small $Re_{\theta}$, and the lack of freestream perturbations. The magnitude of the Reynolds stresses are thus different, but the distributions have a satisfactory match with the experimental counterpart. The shape factors and $Re_{\theta}$ in Table 1 also suggest that the oncoming boundary layers are indeed turbulent.

For the simulated LBL case (LBL$_{8000}$), an inflection point in the $U/U_{\infty}$ profile is observed at $U/U_{\infty} \approx 0.54$ and $y/\delta \approx 0.33$ in Fig. 2(a). As a result, the shape factor, $\delta^*/\theta$, is slightly larger than that expected for a LBL. The inflection point in the $U/U_{\infty}$ profile, the large displacement thickness $Re(Re_{\delta^*} = 1128)$ and the APG indicate that the LBL is in the susceptible range for the excitation of an instability by disturbances (Schlichting and Gersten, 1979). However, there are
<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Profiles of $U/U_\infty$ (a), $\overline{u'^2/U_\infty^2}$ (b), $\overline{v'^2/U_\infty^2}$ (c) and $-\overline{u'v'/U_\infty^2}$ (d) at $x/H = -9.5$ for LBL$^{\times}_{1700}$ (○), LBL$_{8000}$ (■), LBL$^{\times}_{4800}$ (△), LBL$^{\times}_{7900}$ (△), TBL$_{8100}$ (▲) and TBL$_{8300}$ (▼). The Blasius solution (---) and power law $U/U_\infty = (y/\delta)^{1/6}$ (---) profiles are added for laminar and turbulent boundary layer reference, respectively.</figcaption></figure>

no perturbations to trigger laminar–turbulent BL transition and the Reynolds stresses in Fig. 2(b)–(d) are very small, indicating that an instability is not excited.

For the experimental LBL cases (LBL$^{\times}_{1700}$, LBL$^{\times}_{4800}$ & LBL$^{\times}_{7900}$), the $U/U_\infty$ profiles agree with Blasius' solution regardless of $Re_H$ in Fig. 2(a). The shape factor, $\delta^*/\theta$, and $Re_\theta$ are consistent with a LBL ($Re_\theta < 300$ and $\delta^*/\theta \approx 2.62$; Smits (2010)).

While the Reynolds stresses remain weak for LBL$^{\times}_{1700}$ and LBL$^{\times}_{4800}$ in Fig. 2(b)–(d), $\overline{u'^2/U_\infty^2}$ is elevated for LBL$^{\times}_{7900}$ with the same background fluctuation level (see Section 3.2) with a maximum at $y/\delta = 0.4$. This observation, along with the inflection in $U/U_\infty$ for LBL$_{8000}$ (comparable $Re_H$ to LBL$^{\times}_{7900}$), hints at the receptivity of the oncoming LBL to freestream disturbances. Therefore, further characterisation of the freestream and BL fluctuations are warranted for the experimental cases.

## 3.2. Freestream disturbances

Spectral analysis of $u'$ in the freestream ($x/H = -9.5$ & $y/\delta = 1.5$) for the experimental cases reveals the presence of a spectral energy concentration centred at $f_n = 0.086$ Hz in the PSDFs in Fig. 3(a). While

not shown here for brevity, consistent small-amplitude spectral peaks at $f_n$ are observed in the freestream throughout the PIV domain independently of the spatial location, $U_\infty$ or $Re_H$. There is no discernible peak at $f_n$ in the $v'$-spectra. It was also determined from spatial correlations, that over a field of view of 10H, the oscillations at $f_n$ remained in phase, suggesting that the wavelength is very long. These observations support the presence of a long wavelength, low-frequency streamwise disturbance intrinsic to the facility (i.e., facility natural frequency).

Therefore, coherent and random fluctuations constitute the disturbances in the flow. Each may still trigger a change in the flow behaviour, as Wilhem et al. (2003) showed for an FFS immersed in a laminar channel flow. A conventional wavelet filtering procedure (Lewalle et al., 2000) was used to estimate the relative contribution of the disturbance at $f_n$ to the mean velocity fluctuation intensity $\sqrt{u'^2}$. In the freestream, the amplitude of the natural disturbance is weak, approximately constant at 0.5% of $U_\infty$ for every flow speed within the limits of PIV. The total freestream velocity fluctuation intensity is estimated to be $\sqrt{u'^2 + v'^2} = 1.2\%$ of $U_\infty$ for all experimental cases.

Within the BL at $x/H = -9.5$ & $y/\delta = 0.35$ in Fig. 3(b), spectral energy content at $f_n$ is embedded in the turbulent fluctuations for TBL$^{\times}_{8100}$ while it remains discernible in the other cases. The results in Figs. 2 & 3 suggest that cases LBL$^{\times}_{1700}$, LBL$^{\times}_{4800}$, LBL$^{\times}_{7900}$, and LBL$_{8000}$ all exhibit characteristics of a laminar boundary layer. For the unique case of LBL$^{\times}_{7900}$, there are elevated Reynolds stresses (Fig. 2(b)) and the development of higher frequency scales within the PSDF of $u'$ in Fig. 3, indicating initiation to transition and the response of the LBL to freestream perturbations. This is further evidenced by an excerpt of the time series of $u'$ and $v'$ at $y/\delta = 0.35$ for LBL$^{\times}_{7900}$ which exhibit a spike at $t^* \approx 400$ in Fig. 4. Since the location is near the boundary (the foot of the step), we would expect any $v'$ fluctuations to be much lower as observed. Despite these occasional perturbations, the oncoming BL agrees with the Blasius profile closely, indicating that the BL has not yet transitioned. Over one measurement time for LBL$^{\times}_{7900}$ ($T^* = 1780$), too few of these events occur to change the global dynamics at the step.

As the flow approaches the step, it is exposed to an APG in which the BL instability will influence the separation upstream of the step and the resulting recirculation region topology. Thus, comparison of the mean flow characteristics over the FFS will elucidate the influence of the oncoming BLs and freestream disturbances on the intrinsic flow phenomena at the step, especially for cases which appear susceptible to instability (LBL$^{\times}_{4800}$ & LBL$^{\times}_{7900}$).

## 4. Mean flow characterisation

The mean flows are first compared to examine the change in recirculation topology and Reynolds stress distribution resulting from a change in the underlying dynamics of FFS flow due to oncoming flow conditions. Time-averaged streamlines overlaid on $-\overline{u'v'/U_\infty^2}$ contours along with profiles of $U/U_\infty$ and $-\overline{u'v'/U_\infty^2}$ characterise the mean flow structures in Figs. 5, 6 & 7. Table 2 compares estimates of the mean upstream separation length ($l_{u,s}$), step wall reattachment height ($h_{u,a}$), downstream recirculation height ($h_d$) and downstream reattachment length ($l_{d,a}$) based on the dividing streamline.

### 4.1. Turbulent boundary layer cases

For TBL$^{\times}_{8100}$ & TBL$_{8300}$ cases, the upstream recirculation region at the foot of the step is small in Figs. 5(a) & 5(b). In contrast, the downstream recirculation region is large. The characteristic length scales reported in Table 2 agree well with earlier observations for TBL FFS flow of similar $\delta/H$ (Largeau and Moriniere, 2007; Sherry et al., 2010). The rapid change of the sign of the shear stresses ($-\overline{u'v'/U_\infty^2}$) near the salient edge of the step is associated with the reorientation of the principle axes of the mean strain rate tensor due to the strong
<figure><img src="image_5.png" /><figcaption>Fig. 3. PDFs of $u'$ for experimental cases at $x/H = -9.5$ in the freestream (a) and the boundary layer (b) for LBL$^{\times}_{1700}$ (—), LBL$^{\times}_{4800}$ (—), LBL$^{\times}_{7900}$ (—) and TBL$^{\times}_{8100}$ (—). Vertical lines correspond to the facility natural frequency for each case ($f_n^*$).</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 4. Excerpt of the time series of $u'/U_\infty$ and $v'/U_\infty$ at $x/H = -9.50$ & $y/\delta = 0.35$ for LBL$^{\times}_{1700}$ (—), LBL$^{\times}_{4800}$ (—), LBL$^{\times}_{7900}$ (—) and TBL$^{\times}_{8100}$ (—). The signals were filtered with a 5 point sliding average to improve clarity.</figcaption></figure>

curvature of the separated streamlines as observed by Largeau and Moriniere (2007), Hattori and Nagano (2010) and Fang and Tachie

<caption>Table 2</caption>

Mean dimensions of the separation regions: upstream separation location ($l_{u,a}$), step wall reattachment height ($h_{u,a}$), height of the downstream recirculation region ($h_d$) and downstream reattachment length ($l_{d,a}$). See Fig. 1(a) for notations. Note that for LBL$^{\times}_{1700}$ the separation occurs upstream of the PIV field of view.

<table><thead><tr><th>Case</th><th>l<sub>u,a</sub>/H</th><th>h<sub>u,a</sub>/H</th><th>l<sub>d,a</sub>/H</th><th>h<sub>d</sub>/H</th></tr></thead><tbody><tr><td>LBL<sup>×</sup><sub>1700</sub></td><td>&lt; -5</td><td>0.85</td><td>3.70</td><td>0.31</td></tr><tr><td>LBL<sub>8000</sub></td><td>-7.16</td><td>0.90</td><td>2.53</td><td>0.26</td></tr><tr><td>LBL<sup>×</sup><sub>4800</sub></td><td>-0.87</td><td>0.67</td><td>1.45</td><td>0.20</td></tr><tr><td>LBL<sup>×</sup><sub>7900</sub></td><td>-0.70</td><td>0.60</td><td>2.31</td><td>0.31</td></tr><tr><td>TBL<sup>×</sup><sub>8100</sub></td><td>-0.65</td><td>0.65</td><td>3.37</td><td>0.42</td></tr><tr><td>TBL<sub>8300</sub></td><td>-0.74</td><td>0.55</td><td>4.25</td><td>0.43</td></tr></tbody></table>

(2019, 2020). The DNS case captures the formation of another small recirculation near the step corner (opposite direction of rotation to the large one) in the zoomed inset of Fig. 5(b). The structure of the large recirculation in Fig. 5(a) indicates an area where such a characteristic may exist but the limitations of PIV do not capture it for TBL$^{\times}_{8100}$.

The velocity and Reynolds shear stress profiles, in Figs. 5(c)–(f), obtained by DNS and PIV match closely both upstream and downstream of the step, suggesting that weak freestream disturbances have negligible effect on the underlying physics of FFS flow.

## 4.2. Laminar boundary layer cases

As will be made evident, the following discussion of the LBL cases is grouped according the behaviour of the upstream recirculation resulting from the oncoming flow conditions. As such, LBL$^{\times}_{1700}$ & LBL$_{8000}$ are first discussed, then LBL$^{\times}_{4800}$ & LBL$^{\times}_{7900}$.

A large upstream recirculation region is observed for LBL$^{\times}_{1700}$ & LBL$_{8000}$ cases in Fig. 6(a) & 6(b). For the numerical case, the mean upstream separation point was estimated to be $l_{u,a}/H \approx -7.16$ (Table 2). For LBL$^{\times}_{1700}$, however, the boundary layer separates upstream of the PIV window. Comparison of the mean streamlines suggest that the recirculation region is of a similar size despite the discrepancy in the dividing streamline, $f_{d,sep}^{L_0} U dy = 0$ (the dashed blue lines), due to lower spatial resolution of PIV measurements compared to DNS. The mean separation streamline attaches on the front face of the step at $h_{u,a}/H \approx 0.9$. The magnitude of Reynolds stresses upstream of the step are negligible.

In contrast to the TBL cases, negligible shear stresses near the step corner in Fig. 6(a) indicate that laminar flow separation occurs at the salient edge. The concentration of the shear stresses are at the tail of the downstream recirculation in Fig. 6(f), rather than at the step corner in the TBL cases, suggesting delayed laminar-turbulent shear layer transition. Increased mixing from shear layer breakdown leads turbulent reattachment to the top of the step, closing the mean recirculation region. In Fig. 6(e) & (f), the profiles of LBL$^{\times}_{1700}$ & LBL$_{8000}$ match closely for $0 \le x/H < 1$. At $x/H \approx 1$, the profiles of $-u'v'/U_\infty^2$ for LBL$_{8000}$ indicate transition of the shear layer. In contrast, this occurs at $x/H \approx 2$ for LBL$^{\times}_{1700}$ suggesting that this process is slower owing to a lower $Re_H$. The reattachment length $l_{d,sep}$ is thus longer in the experimental case (see Table 2). For $x/H \ge 3$, the profiles of $-u'v'/U_\infty^2$ match once again while the profiles of $U/U_\infty$ begin to converge due to turbulent mixing.

Despite the similarities in the oncoming LBL, the structure of the recirculation regions upstream and downstream of the step are different for LBL$^{\times}_{4800}$ & LBL$^{\times}_{7900}$ in Fig. 7(a) & 7(b), respectively. Upstream of the step, the contours of intermittency ($\gamma$) – the fraction of the time the flow experiences reverse flow at each point ($u(x, t) < 0$) (Simpson, 1989) – of 0.01, 0.2 and 0.5 (the incipient, intermittent transitory and transitory detachment at the boundary, respectively) extend to $x/H \le -4$ for both LBL$^{\times}_{4800}$ & LBL$^{\times}_{7900}$ in Fig. 8(a) & 8(b) suggesting incipient detachment of the LBL. The separated flow then appears to
<figure><img src="image_7.png" /><figcaption>**Fig. 5.** Mean streamlines overlaid on Reynolds shear stress ($-\overline{u'v'/U_\infty^2}$) contours for the TBL cases in (a) & (b). The dashed blue and green lines are the dividing streamline ($j_0^\gamma U_\infty y = 0$) and the contour of reverse flow ($U < 0$), respectively. In (c)–(f), profiles of $U/U_\infty$ and $-\overline{u'v'/U_\infty^2}$ for upstream and downstream regions, respectively, for TBL$_{8100}^\times$ (◀) and TBL$_{8300}$ (—). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

reattach to the boundary intermittently – note the streamlines returning towards, elevated $-\overline{u'v'/U_\infty^2}$ close to and the contours of $\gamma = 0.2$ and $0.5$ closing at the boundary for $-3 \le x/H < -1$ – due to enhanced mixing. Another BL detachment occurs closer to the step ($x/H \approx -0.8$) and a small recirculation forms at the foot of the step. This is in sharp contrast to the TBL cases which only have one separation in Fig. 8(c) & 8(d).

Downstream of the step, a concentration of negatively-signed $-\overline{u'v'/U_\infty^2}$ near the salient edge is observed in Fig. 7(a) & 7(b) like the TBL cases. In contrast, we also observe a larger concentration of positively-signed $-\overline{u'v'/U_\infty^2}$ upstream of the salient edge – more than the TBL cases in Fig. 5(a) – indicating large-scale fluctuations over the FFS. Larger-scales impinging on the salient edge may influence the dynamics of the downstream shear layer resulting in the observed reduction of the mean recirculation length in Fig. 7(a) & 7(b). This is analogous to thick TBL cases, in which larger scales originating from a thick TBL impinge on the step, promoting the development of the downstream shear layer and flow reattachment (smaller $l_{d,a}$) (Sherry

et al., 2010; Fang and Tachie, 2020). The origin of the structures impinging on the salient edge is different however; for $LBL_{4800}^\times$ & $LBL_{7900}^\times$, the large-scale structures are being generated at the separation upstream of the step.

For $LBL_{4800}^\times$ & $LBL_{7900}^\times$, the profiles of $U/U_\infty$ for $x/H \le -3.5$ exhibit an inflection in Fig. 7(c), indicating laminar separation like in the other LBL cases. For $-3.5 < x/H \le -1.5$, the profiles of $U/U_\infty$ begin to converge towards the TBL profiles – more rapidly for $LBL_{7900}^\times$ than $LBL_{4800}^\times$ as expected – suggesting a trend towards intermittent reattachment of the BL. At the foot of the step ($-1.5 < x/H < 0$), the profiles of $U/U_\infty$ match the TBL cases well. However, elevated $-\overline{u'v'/U_\infty^2}$ is consistent with convection of larger structures through the domain which form at the incipient separation ($x/H \le -3.5$). This is a marked difference to the unsteady upstream separation dynamics of the TBL cases, which warrants further analysis.

In summary, despite similar oncoming flow conditions, the mean flow structure over the FFS for $LBL_{4800}^\times$ and $LBL_{7900}^\times$ differ significantly
<figure><img src="image_8.png" /><figcaption>**Fig. 6.** Mean streamlines overlaid on Reynolds shear stress ($-\overline{u'v'}/U_\infty^2$) contours with a zoomed inset near the step corner for LBL$_{1700}$ (a) & LBL$_{8000}$ (b). The dashed blue and green lines are the dividing streamline ($f_0^*Uy=0$) and the contour of reverse flow ($U<0$), respectively. In (c)–(f), profiles of $U/U_\infty$ and $-\overline{u'v'}/U_\infty^2$ for upstream and downstream regions, respectively, for LBL$_{1700} (^{\circ})$, LBL$_{8000}$ (—) and TBL$_{8000}$ (—). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

from that for LBL$_{8000}$ or LBL$_{1700}$. While the latter two cases have a similar mean flow structure upstream of the FFS, for LBL$_{1700}$ the freestream disturbances are similar to those for LBL$_{4800}^\times$ & LBL$_{900}^\times$, whereas no freestream perturbation is present in LBL$_{8000}$. It is thus reasonable to propose that, above a critical Re, weak freestream disturbances excite an intrinsic instability of the upstream separation that leads to unsteady dynamics. Whether the critical Re should be related to the BL ($Re_θ$) or step ($Re_H$) scales cannot be determined at this point from mean statistical fields. It is unclear if a change in the BL separation triggers a change in the upstream recirculation dynamics or, conversely, if the excitation of an instability of the upstream recirculation changes the APG causing the BL separation to change. Thus, spectral analysis of the flow as it approaches the step will elucidate the separation process and instabilities therein.

## 5. Oncoming boundary layer separation

Spectral analysis of velocity fluctuations is carried out at a multiple locations approaching the step within the boundary layer ($y/H = 0.1$) to investigate whether frequencies within the oncoming BL are amplified in the APG and elucidate the differences observed in the mean flow characterisation.

There are no distinctive spectral energy concentrations at a particular frequency in the PSD of $u'$ for the TBL cases in Fig. 9 as expected from a turbulent flow undergoing turbulent separation. This is consistent with the PSD within the oncoming TBL in Fig. 3(b). At the foot of the FFS, spectral energy shifts towards lower frequencies in Fig. 9(d) indicating the formation of larger structures as observed previously (Camussi et al., 2008).

For LBL$_{1700}^\times$ & LBL$_{8000}$, weak low-frequency fluctuations are dominant as spectral energy is found at $f^* = fH/U_\infty \approx 0.029$ & 0.015,
<figure><img src="image_9.png" /><figcaption>**Fig. 7.** Mean streamlines overlaid on Reynolds shear stress ($-\overline{u'v'}/U_\infty^2$) contours with a zoomed inset near the step corner for LBL$_{4800}^\times$ (a) & LBL$_{7900}^\times$ (b). The dashed blue and green lines are the dividing streamline ($\oint_0^\gamma U dy = 0$) and the contour of reverse flow ($U < 0$), respectively. In (c)–(f), profiles of $U/U_\infty$ and $-\overline{u'v'}/U_\infty^2$ for upstream and downstream regions, respectively, for LBL$_{4800}^\times$ (c), LBL$_{7900}^\times$ (d) and TBL$_{8300}$ (e). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

respectively, in Fig. 9. For LBL$_{1700}^\times$, the low-frequency peak coincides with $f_n^*$ and is not amplified at the foot of the step. In this case, a higher-frequency band ($f^* \approx 0.15$), however, is amplified as the flow progresses towards the step. The energy contribution at this frequency is negligible in the oncoming BL in Fig. 3(b), indicating that it may be related to an intrinsic instability of the upstream recirculation region. Although the spectral energy level for LBL$_{8000}$ is lower than LBL$_{1700}^\times$, it tends to increase overall in the vicinity of the step with a small amplification appearing around ($f^* \approx 0.15$), corroborating the presence of an instability.

In contrast to the other cases, larger-scales appear to form farther upstream for LBL$_{4800}^\times$ & LBL$_{7900}^\times$. For $x/H \le -2.3$ in Fig. 9(a)–(d), spectral energy is concentrated on a low-frequency band centred at $f^* \approx 0.02$ and a higher-frequency band centred at $f^* \approx 0.25$. Closer to

the step, in Fig. 9(e) & (f), the PSDFs of LBL$_{4800}^\times$ & LBL$_{7900}^\times$ shift towards a low-frequency broadband energy distribution resembling the TBL cases (with higher energy) which is consistent with the profiles near the step in Figs. 6(c) & (d).

Thus, there appears to be three distinct behaviours at the foot of the step: (i) *unsteady* dynamics related to TBL separation (TBL cases), (ii) *steady* dynamics related to laminar separation (LBL$_{1700}^\times$ & LBL$_{8000}^\times$), and (iii) *transitional-unsteady* dynamics related to LBL separation and the formation of larger structures due to excitation of an instability (LBL$_{4800}^\times$ & LBL$_{7900}^\times$). To the authors' knowledge, the latter has not been previously characterised in the available literature. Therefore, spatio-temporal evolution of the coherent motions for these behaviours is investigated in the following section to uncover the underlying phenomena.
<figure><img src="image_10.png" /><figcaption>**Fig. 8.** Mean streamlines overlaid on intermittency contours ($\gamma$: 0.01, 0.2 and 0.5 are the incipient, intermittent transitory and transitory detachment locations at the boundary, respectively (Simpson, 1989). The dashed blue and green lines are the dividing streamline and the contour of reverse flow, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

# 6. Upstream flow dynamics

The global motion of the recirculation region is first characterised using the analysis of the reverse flow area. Then, analysis of POD results distinguishes between high- and low-frequency coherent motions in the flow at the foot of the step defined as the region comprised of $x/H = [-5,0]$ & $y/H = [0,2]$. Representative snapshots sequences are shown to corroborate the findings of these analyses.

## 6.1. Reverse flow area

The reverse flow area, $A(t)$, can be used to estimate the time-varying size of a separated flow region. Upstream of the FFS, this estimate has been used to characterise the states of the recirculation region (Pearson et al., 2013). “Ejection events” for unsteady separation behaviour (thick oncoming TBL) which occur at peaks of $A(t)$ were identified by the same authors. In other words, the separated flow region grows ($A(t)$ increases) followed by an ejection of separated flow over the step

<table><caption>**Table 3**
Statistical moments of the scaled reverse flow area ($A_u(t)/H^2$): average ($\mu$), median (med), variance ($\sigma^2$), skewness ($\bar{\mu}_3$) and kurtosis ($\bar{\mu}_4$).</caption><thead><tr><th>Case</th><th>μ</th><th>med</th><th>σ²</th><th>μ̄₃</th><th>μ̄₄</th></tr></thead><tbody><tr><td>LBL<sup>×</sup><sub>1700</sub></td><td>0.81</td><td>0.83</td><td>0.011</td><td>-0.97</td><td>2.1</td></tr><tr><td>LBL<sup>×</sup><sub>4800</sub></td><td>0.48</td><td>0.45</td><td>0.051</td><td>+0.64</td><td>0.3</td></tr><tr><td>LBL<sup>×</sup><sub>7900</sub></td><td>0.27</td><td>0.23</td><td>0.027</td><td>+1.16</td><td>1.5</td></tr><tr><td>TBL<sup>×</sup><sub>8100</sub></td><td>0.18</td><td>0.15</td><td>0.013</td><td>+1.06</td><td>1.0</td></tr><tr><td>TBL<sub>8300</sub></td><td>0.20</td><td>0.17</td><td>0.015</td><td>+0.95</td><td>0.4</td></tr></tbody></table>

corner ($A(t)$ peaks and decreases). Thus, the time signature of $A(t)$ is intermittent for this dynamic. Until now, reverse flow area results have not been published for *steady* or *transitional-unsteady* upstream recirculation behaviour as in $LBL_{1700}^×$ and $LBL_{4800}^×$, respectively.

For a two-dimensional domain $\Omega$, the reverse flow area can be defined as

$$A(t) = \int_{\Omega} H(u(x, t)) dxdy \qquad (8)$$

with,

$$H(u(x, t)) = \begin{cases} 0, & u(x, t) > 0 \\ 1, & u(x, t) \le 0. \end{cases} \qquad (9)$$

Unsteady and steady global behaviours of the upstream recirculation region are discernible in Fig. 10. An intermittent time signature is observed for $LBL_{4800}^×$, $LBL_{7900}^×$ and the TBL cases in Fig. 10(a), consistent with the presence of ejections from the upstream recirculation region. The low-frequency expansion and contraction of the upstream separation related to the ejection motion is exhibited as broad spectral energy distribution around $f^* \approx 0.025$ and 0.05 in the PSDFs of $A_u(t)/H^2$ in Fig. 10(b) for the LBL and TBL cases, respectively. The lower frequency for the LBL cases suggests an increase in large-scale motions as observed in Fig. 9.

For unsteady upstream behaviour ($LBL_{4800}^×$, $LBL_{7900}^×$ and TBL cases), the trend is that as $Re_H$ increases, the recirculation region becomes smaller and large-scale changes in its size are reduced as evidenced by a reduction of the average $A_u(t)/H^2$ in Table 3 and the length of the tails of the probability density functions (PDF) in Fig. 10(c), respectively. These findings are consistent with the observed increase of the upstream recirculation size and the Reynolds stresses for the lower $Re_H$ cases in Section 4. Both the PSDFs and the PDFs in Fig. 10(b) & (c) indicate that the upstream separation behaviour of the *transitional-unsteady* cases tends towards *unsteady* behaviour of the TBL cases, but are clearly distinct.

To further investigate the evolution of the upstream separation behaviour with $Re_H$, the probability density function (PDF) and the main statistical moments of $A_u(t)/H^2$ are examined in Fig. 10(c) and Table 3. The PDF of the TBL cases exhibit comparable right-tail distributions such as those reported by different authors (such as Pearson et al., 2013; Graziani et al., 2018; Fang et al., 2017; Fang and Tachie, 2020) for oncoming TBL of varying $\delta/H$. Such a distribution is representative of a small upstream separation region with intermittent growth resulting in massive fluid ejections over the step as described by Pearson et al. (2013).

In contrast, steady, periodic behaviour of the upstream recirculation region is observed for $LBL_{1700}^×$ & $LBL_{8000}^×$ rather than intermittent ejections. A primary dynamic related to the low-frequency expansion and contraction of the separation is observed at $f^* \approx 0.03$ and 0.012 (period of 83) for $LBL_{1700}^×$ and $LBL_{8000}^×$, respectively. The low-frequency coincides with $f^*_{H}$ for $LBL_{1700}^×$. A weaker dynamic related to the intrinsic instability of the upstream separation centred at $f^* \approx 0.15$ is observed in Fig. 10(b) for these cases, whose PSDFs are consistent with the spectral energy content in Fig. 9.

While analysis of $A_u(t)$ reveals that the global behaviour of the upstream recirculation region is related primarily to its low-frequency
<figure><img src="image_11.png" /><figcaption>Fig. 9. PSDF of $u'$ for a series of points approaching the step at $y/H = 0.1$ for $LBL_{1700}^\times$ (—), $LBL_{8000}$ (—), $LBL_{4800}^\times$ (—), $LBL_{7900}^\times$ (—), $TBL_{8100}^\times$ (—) and $TBL_{8300}$ (—). Vertical lines correspond to the facility natural frequency for each case ($f_u^*$).</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 10. Excerpt of the time series of the reverse flow area upstream of the step ($A_u(t)/H^2$) (a), along with the premultiplied PDFs (b) and probability density functions (c) for $LBL_{700}^\times$ (—), $LBL_{8000}$ (—), $LBL_{4800}^\times$ (—), $LBL_{7900}^\times$ (—), $TBL_{8100}^\times$ (—) and $TBL_{8300}$ (—).</figcaption></figure>

expansion and contraction, other relevant dynamics are filtered out due to the spatial averaging arising from the integration over the area. Traces of the important high-frequency dynamics are evidenced in Sections 5 and 6.1. Therefore, we introduce the POD in the following to separate and identify dominant coherent motions at the foot of the step to further characterise the distinct behaviours of the separation regions and elucidate the intrinsic instabilities at play.

## 6.2. Coherent motions of the upstream flow

The POD eigenvalue distributions ($\lambda$) – the contribution to the turbulent kinetic energy (TKE) of each POD mode – highlight that the leading five modes are representative of the dominant dynamics of the flow for each case in Fig. 11(a). For higher-order POD modes, the TKE reduces slowly for unsteady and transitional-unsteady recirculation region behaviour cases, indicative of high-dimensional flow (i.e., many POD modes are needed to represent the flow with high fidelity). Conversely, the modal TKE reduces more rapidly for the steady recirculation cases ($LBL_{1700}^\times$ & $LBL_{8000}$), suggesting lower-dimensionality of the flow (i.e., few POD modes are needed to represent the flow with high fidelity). For direct comparison, the DNS modes also correspond to the decomposition of the $u'$, $v'$ correlation matrix.

The correlation coefficient between $A_u(t)$ and the reverse flow area from reduced-order velocity fields reconstructed from the first $m$ POD modes ($\hat{A}_{u,m}(t)$) as function of the number of leading modes included in the reconstruction (m) in Fig. 11(b) supports the previous assertion that the leading five modes are representative of the dominant dynamics of the flow despite the high-dimensionality of the flow. Therefore, only the leading five POD spatial modes and premultiplied PSDF of the associated temporal coefficients are shown in Fig. 12.

For cases which exhibit unsteady and transitional-unsteady recirculation region behaviour, strong similarities are observed among the leading spatial POD modes ($\Psi_U, \Psi_V$) in Fig. 12 (columns 3–6). The first $\Psi_U$ appears as a streamwise elongated oscillatory motion immediately at the foot of the step which extends over the salient edge. For the TBL cases, the PSDFs of this mode exhibits a broad spectral energy distribution centred at $f^* \approx 0.05$ like the PSDFs of $A_u(t)$ in Fig. 10(b), suggesting that this mode is related to the motions of low-frequency expansion and contraction of the separation region and ejections. For the transitional-unsteady cases $LBL_{4800}^\times$ & $LBL_{7900}^\times$, the spectral energy concentration is at $f^* \approx 0.1$ & 0.07, respectively, which does not coincide with the PSDFs of $A_u(t)$ in Fig. 10(b). However, in viewing animations, it appears that this POD mode is related to the motion of the upstream recirculation region rather than a change in its size.

The spatial POD modes 2 to 5 for $LBL_{4800}^\times$, $LBL_{7900}^\times$ and the TBL cases exhibit spatial periodicity in both $\Psi_U$ & $\Psi_V$ in Fig. 12 (columns 3–6), indicating convective dynamics in the streamwise direction related to the global motion of the upstream separation. The motions exhibited by these modes are aligned with the streamlines immediately above the mean upstream recirculation and extend to salient edge of the step. The similarity across the peak frequencies ($f^* \approx 0.15$ for modes 2 & 3, and $f^* \approx 0.25$ for modes 4 & 5) and ($\Psi_U, \Psi_V$) for these cases would
<figure><img src="image_13.png" /><figcaption>**Fig. 11.** (a) POD modal energy distribution for the upstream domain. Inset: cumulative energy distribution as a percentage of the total TKE in the domain. (b) Correlation coefficient between $A_u(t)$ and reverse flow area of reconstructed velocity fields from the first *m* POD modes ($\bar{A}_{u,m}(t)$) for $L_{BL}^\times_{1700}$ (—), $L_{BL}^\times_{8000}$ (—), $L_{BL}^\times_{4800}$ (—), $L_{BL}^\times_{7900}$ (—), $TBL^\times_{8100}$ (—) and $TBL^\times_{8300}$ (—).</figcaption></figure>

indicate some common underlying dynamics.

In contrast, the concentration of spectral energy in the peak frequencies is more pronounced and the relative contribution of the POD modes 2–5 to the total TKE is greater per mode in $L_{BL}^\times_{4800}$ & $L_{BL}^\times_{7900}$ than in the TBL cases, suggesting that the convective motions associated with the unsteadiness of the upstream recirculation region are more significant. This is consistent with the notion of the formation of larger structures at the incipient LBL separation farther upstream which convect over the step.

The PSDFs of leading five POD modes of $L_{BL}^\times_{1700}$ and $L_{BL}^\times_{8000}$ in Fig. 12 (columns 1–2) contain low- and high-frequency spectral energy content which are consistent with the PSDFs of $A_u(t)$ in Fig. 10(b). The low-frequency dynamics ($f^* \approx 0.03$ & 0.012, respectively) may thus be ascribed to the global expansion and contraction of the recirculation region. The high-frequency dynamics spectral energy concentration at $f^* \approx 0.15$ is discernible in every mode, it features more prominently in the PSDFs of POD modes 3–5 of $L_{BL}^\times_{1700}$. The corresponding spatial modes exhibit enhanced spatial periodicity in the streamwise direction similar to the spatial POD modes 2–5 of $L_{BL}^\times_{4800}$ (column 3) where a high-frequency band and convective dynamics feature prominently. The evidence thus suggests that the high-frequency dynamics for a steady separation is related to an intrinsic instability which is excited by freestream disturbances.

For the LBL cases, the instability is not sufficiently excited in $L_{BL}^\times_{1700}$ (Re < Re\_crit) and $L_{BL}^\times_{8000}$ (perturbation-free) to lead to unsteady recirculation dynamics. For $L_{BL}^\times_{4800}$ & $L_{BL}^\times_{7900}$, on the other hand, the instability is sufficiently excited (freestream disturbances and Re >

$Re_{crit}$) to transition towards unsteady behaviour, in which we observe increased convective dynamics from the formation of structures. The dynamics of TBL cases, on the other hand, are primarily related to the low-frequency expansion and contraction of the recirculation region, along with ejections.

### 6.3. Representative snapshot sequences

Sequences of vorticity snapshots are presented in Figs. 13–15 to clearly display the dynamics discussed throughout the article. Vorticity fields from reduced-order velocity fields (leading POD modes representing 80% of the total TKE in the domain) with the contour of the reverse flow (green line) highlight the motion of the upstream shear layer and separation region. The top subplot shows excerpts of $A_u(t)$ and the first four POD temporal coefficients, $a_1(t)-a_4(t)$, preceding and following the central time point with vertical black lines on the plot corresponding to the displayed snapshots.

The snapshot sequence of $TBL^\times_{8100}$ displays the expansion of the upstream separation region (area of $u < 0$) immediately followed with an ejection over the step as observed in snapshots (f)–(i) in Fig. 13 with temporal spacing of $\Delta t^* \approx 1$ between snapshots. Upstream of the mean separation point, the maximum of vorticity remains concentrated near the boundary and smaller-scale structures from the oncoming TBL convect along the upstream shear layer and over the step continuously.

For cases of steady recirculation, the snapshot sequence in Fig. 14 for $L_{BL}^\times_{1700}$ highlights the behaviour of the laminar shear layer for $L_{BL}^\times_{1700}$. Two separate recirculations form occasionally beneath the continuous stream of vorticity connecting the separation and the step corner as the shear layer oscillates in snapshots (c), (d), (f), (i), (j) and (k) in Fig. 14. This is consistent with spatial periodicity in the spatial POD modes and an important spectral energy concentration at $f^* \approx 0.15$ in the PSDF in Fig. 12 (columns 1 & 7), supporting the hypothesis that the laminar shear layer instability is excited by freestream disturbances. Note that the excitation frequency from freestream perturbations ($f = 0.086$ Hz for all $Re_H$) is different from the shear layer response, indicating a nonlinear process.

The snapshot sequence of $L_{BL}^\times_{4800}$ in Fig. 15, representative of the transitional-unsteady dynamics of the upstream separation, demonstrates the formation and convection of structures as result of laminar-turbulent shear layer transition brought upon by weak freestream disturbances. In all snapshots, LBL separation occurs far upstream ($x/H < -4.5$) – indicated by the reverse flow contour – and maximum spanwise vorticity is found along the shear layer for $x/H < -4.5$ like $L_{BL}^\times_{1700}$ in Fig. 14. For $x/H > -4$, the shear layer instability grows and sheds vorticity concentrations towards the step (snapshots (b)–(e) in Fig. 15). These structures impinge on the main recirculation at the foot of the step in snapshot (f) and flow over the step in an ejection event in snapshots (g)–(i). These large-scale motions are consistent with the POD findings for $L_{BL}^\times_{4800}$ & $L_{BL}^\times_{7900}$ in Fig. 12 (columns 3–4), which suggested increased relevance of convective dynamics along with the underlying ejection motion.

## 7. Conclusions

Two-dimensional velocity fields of flow over a forward-facing step from experimental and numerical databases were examined to elucidate the effect of the oncoming flow conditions on the dynamics of the flow upstream of the step. The experimental cases are subjected to weak freestream disturbances, while the numerical simulations are freestream disturbance-free. The existence of three distinct behaviours of the upstream recirculation region are revealed based on the oncoming BL state (laminar or turbulent), the presence of freestream disturbances and the Reynolds number.

Unsteady upstream separation behaviour, ascribed to the TBL cases, is characterised by unsteady TBL separation and broadband dynamics. Low-frequency expansion and contraction of the separation (broadly
<figure><img src="image_14.png" /><figcaption>**Fig. 12.** Leading POD spatial modes of the upstream domain with the associated premultiplied PSDF of the temporal coefficients. Columns indicate the cases and every 2-row combination corresponds to a mode number. Every row therein corresponds to a velocity component $\Psi_U$ & $\Psi_V$. The colour of the curves in the PSDFs corresponds to the coloured box around the title of each column: $LBL_{1700}^\times$ (—), $LBL_{8000}^\times$ (—), $LBL_{4800}^\times$ (—), $LBL_{7900}^\times$ (—), $TBL_{8100}^\times$ (—) and $TBL_{8300}^\times$ (—). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

centred at $f^* \approx 0.05$), related to the intermittent ejection of separated flow over the step, is dominant. Secondary high-frequency dynamics are associated to the convection of smaller-scale structures from the oncoming TBL over the upstream recirculation and the salient edge continuously.

**Steady** upstream behaviour, observed for $LBL_{1700}^\times$ & $LBL_{8000}^\times$, is characterised by LBL separation and discernible low- and high-frequency dynamics. The upstream separation was found to oscillate weakly about a steady state at low frequency ($f^* \approx 0.03$ & 0.012, respectively). A higher frequency dynamic (at $f^* \approx 0.15$) is surmised to be an intrinsic instability of the separation region. If the flow is subjected to weak freestream disturbances as in $LBL_{1700}^\times$, this instability is particularly excited without a transition to unsteady dynamics.

**Transitional-unsteady** behaviour of the upstream separation, exhibited by $LBL_{4800}^\times$ & $LBL_{7900}^\times$, is particularly complex. LBL separation

occurs far upstream followed by the growth of an instability of the upstream shear layer as in $LBL_{1700}^\times$. The instability is sufficiently excited to breakdown and shed structures. Dynamically, the low-frequency expansion and contraction of the separation region related to the ejection motion is dominant as in the TBL cases. However, the convective dynamics are particularly enhanced – compared to the TBL cases – due to growth of the instability of the laminar shear layer. These structures, impacting the recirculation at the foot of the step, occasionally lead to an ejection from the upstream separation over the step.

For these cases where $\delta$ is on the order of $H$, the evidence suggests that $Re_H$ is not a sufficient parameter to characterise the dynamics of about the step. This is hypothesised to be due to an inherent instability of the upstream recirculation region which is receptive to freestream disturbances. The excitation of the instability may then change the APG, affecting the response of the upstream BL which forms a feedback.
<figure><img src="image_15.png" /><figcaption>**Fig. 13.** (a) Time series of $A_u/H^2$ and the first four POD temporal coefficients, $a_1(t)-a_4(t)$, scaled by $\sigma$, respectively, and offset. (b)–(k) Sequence of streamlines overlaid on vorticity snapshots for TBL$_{700}^*$ centred about a peak in $A_u(t)/H^2$. Reduced-order velocity fields using leading POD modes containing 80% of the TKE. The green contour delimits the area of reverse flow. The black box highlights the snapshot corresponding to the peak in $A_u/H^2$. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>**Fig. 14.** (a) Time series of $A_u/H^2$ and the first four POD temporal coefficients, $a_1(t)-a_4(t)$, scaled by $\sigma$, respectively, and offset. (b)–(k) Sequence of streamlines overlaid on vorticity snapshots for LBL$_{700}^*$ centred about a peak in $A_u(t)/H^2$. Reduced-order velocity fields using leading POD modes containing 80% of the TKE. The green contour delimits the area of reverse flow. The blue arrows highlight the secondary recirculation forming upstream of the main recirculation at the foot of the step. The black box highlights the snapshot corresponding to the peak in $A_u/H^2$. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

Complex interactions between upstream-downstream recirculation regions are also anticipated, since it is observed from the mean flow characterisation that the upstream recirculation behaviour affects the flow downstream of the step. The coupling between the recirculation regions will be subject of a forthcoming publication.

CRediT authorship contribution statement

**Emil Larose:** Writing – original draft, Visualization, Validation, Software, Project administration, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Franck Kerhervé:** Writing – review & editing, Supervision, Software, Conceptualization. **Yann Fraigneau:** Writing – review & editing, Software, Methodology, Funding acquisition, Conceptualization. **Bérengère Podvin:** Writing – review & editing, Funding acquisition, Conceptualization. **Chris Morton:** Writing – review & editing, Supervision, Conceptualization. **Robert J. Martinuzzi:** Writing – review & editing, Resources, Project administration, Funding acquisition, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors gratefully acknowledge the support of Natural Sciences and Engineering Research Council of Canada (NSERC) through the Discovery and Research Tools & Instruments grants and the Mitacs Globalink program. This work was also partially supported by the France 2030 program: LABEX INTERACTIFS, reference ANR-11-LABX-0017-01. Numerical results in the study were obtained using HPC resources from GENCI-IDRIS (grant 2023-AD012A02062R2).
<figure><img src="image_17.png" /><figcaption>**Fig. 15.** (a) Time series of $A_u/H^2$ and the first four POD temporal coefficients, $a_1(t)-a_4(t)$, scaled by $\sigma$, respectively, and offset. (b)–(k) Sequence of streamlines overlaid on vorticity snapshots for LBL centred about a peak in $A_u(t)/H^2$. Reduced-order velocity fields using leading POD modes accounting for 80% of the TKE. The green contour delimits the area of reverse flow. The black box highlights the snapshot corresponding to the peak in $A_u/H^2$. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

## Data availability

Data will be made available on request.

## References

Awasthi, M., Devenport, W.J., Glegg, S.A.L., Forest, J.B., 2014. Pressure fluctuations produced by forward steps immersed in a turbulent boundary layer. J. Fluid Mech. 756, 384–421. http://dx.doi.org/10.1017/jfm.2014.405.

Beam, R.M., Warming, R.F., 1976. An implicit finite-difference algorithm for hyperbolic conservation-law form. J. Comput. Phys. 22, 87–110.

Berkooz, G., Holmes, P., Lumley, J.L., 1993. The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech. 25, 539–575.

Camussi, R., Felli, M., Pereira, F., Aloisio, G., Di Marco, A., 2008. Statistical properties of wall pressure fluctuations over a forward-facing step. Phys. Fluids 20 (7), 075113.

de Brederode, V.A.S.L., Bradshaw, P., 1974. Three-Dimensional Flow in Nominally Two-Dimensional Separated Bubbles. Chapter 4: Flow Behind a Rear-Ward Facing Step (Ph.D. thesis). Imperial College of Science and Technology.

Deck, S., Laraufie, R., 2013. Numerical investigation of the flow dynamics past a three-element aerofoil. J. Fluid Mech. 732, 401–444.

Fang, X., Tachie, M.F., 2019. On the unsteady characteristics of turbulent separations over a forward-backward-facing step. J. Fluid Mech. 863, 994–1030.

Fang, X., Tachie, M.F., 2020. Spatio-temporal dynamics of flow separation induced by a forward-facing step submerged in a thick turbulent boundary layer. J. Fluid Mech. 892, A40.

Fang, X., Yang, Z., Wang, B.-C., Bergstrom, D.J., 2017. Direct numerical simulation of turbulent flow in a spanwise rotating square duct at high rotation numbers. Int. J. Heat Fluid Flow 63, 88–98. http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.011.

Faugaret, A., Duguet, Y., Fraigneau, Y., Martin-Witkowski, L., 2022. A simple model for arbitrary pollution effects on rotating free-surface flows. J. Fluid Mech. 935.

Fraigneau, Y., 2024. DATABASE-FLMEC: a CFD database on unsteady and turbulent flows. http://dx.doi.org/10.57745/IE8RPF.

Graziani, A., Kerhervé, F., Martinuzzi, R.J., Keirsbulck, L., 2018. Dynamics of the recirculating areas of a forward-facing step. Exp. Fluids 59 (154), 1–18.

Graziani, A., Lippert, M., Uystepruyst, D., Keirsbulck, L., 2017. Scaling and flow dependencies over forward-facing steps. Int. J. Heat Fluid Flow 67, 220–229.

Guermond, J.L., Minev, P.D., Shen, J., 2006. An overview of projection methods for incompressible flows. CMAME 195, 6011–6045.

Harlow, F.H., Welch, J.E., 1965. Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids 8.

Hattori, H., Nagano, Y., 2010. Investigation of turbulent boundary layer over forward-facing step via direct numerical simulation. Int. J. Heat Fluid Flow 31 (3), 284–294.

Hosseini, S.M., Feng, J.J., 2011. Pressure boundary conditions for computing incompressible flows with SPH. J. Comput. Phys. 230 (19), 7473–7487. http://dx.doi.org/10.1016/j.jcp.2011.06.013.

Lanzerstorfer, D., Kuhlmann, H.C., 2012. Three-dimensional instability of the flow over a forward-facing step. J. Fluid Mech. 695, 390–404.

Largeau, J.F., Moriniere, V., 2007. Wall pressure fluctuations and topology in separated flows over a forward-facing step. Exp. Fluids 42 (1), 21–40.

Lewalle, J., Delville, J., Bonnet, J.-P., 2000. Decomposition of mixing layer turbulence into coherent structures and background fluctuations. Flow Turbul. Combust. 64, 301–328.

Lumley, J.L., 1967. The structure of inhomogeneous turbulent flows. Atmospheric Turbul. Radio Wave Propag. 166–178.

Lumley, J.L., 1970. Stochastic Tools in Turbulence. Academic Press, New York, USA.

Marino, L., Luchini, P., 2009. Adjoint analysis of the flow over a forward-facing step. Theor. Comput. Fluid Dyn. 23 (1), 37–54.

Martinuzzi, R., Tropea, C., 1993. The flow around surface-mounted, prismatic obstacles placed in a fully developed channel flow (data bank contribution). J. Fluids Eng. 115 (1), 85–92.

Peaceman, D.W., Rachford, H.H., 1955. The numerical solution of parabolic and elliptic differential equations. Int. J. Ind. Math. 3, 28–41.

Pearson, D.S., Goulart, P.J., Ganapathisubramani, B., 2013. Turbulent separation upstream of a forward-facing step. J. Fluid Mech. 724, 284–3042.

Podvin, B., Fraigneau, Y., 2024. Low-dimensional analysis and modelling of the flow over a forward-facing step. J. Fluid Mech. 1000, A69. http://dx.doi.org/10.1017/jfm.2024.1051.

Saad, Y., 2003. Iterative methods for sparse linear systems. Other Titles in Applied Mathematics, Society for Industrial and Applied Mathematics.

Schlichting, H., Gersten, K., 1979. Boundary-Layer Theory, seventh ed. McGraw-Hill, pp. 596–600.

Sherry, M., Jacono, D.L., Sheridan, J., 2010. An experimental investigation of the recirculation zone formed downstream of a forward facing step. J. Wind. Eng. Ind. Aerodyn. 98 (12), 888–894.

Simpson, R.L., 1989. Turbulent boundary-layer separation. Annu. Rev. Fluid Mech. 21 (Volume 21, 1989), 205–232. http://dx.doi.org/10.1146/annurev.fl.21.010189.001225.

Sirovich, L., 1987. Turbulence and the dynamics of coherent structures. I. Coherent structures. Quart. Appl. Math. 45 (3), 561–571.

Smits, A.J., 2010. Viscous flows and turbulence. In: Lectures in Fluid Mechanics. Princeton University, pp. 331–339.

Spalart, P.R., 1988. Direct simulation of a turbulent boundary layer up to $R\theta=1410$. J. Fluid Mech. 187, 61–98.

Stüer, H., Gyr, A., Kinzelbach, W., 1999. Laminar separation on a forward facing step. Eur. J. Mech. B Fluids 18 (4), 675–692.

Timmermans, L.J.P., Minev, P.D., Van De Vosse, F.N., 1996. An approximate projection scheme for incompressible flow using spectral elements. Internat. J. Numer. Methods Fluids 22 (7), 673–688. http://dx.doi.org/10.1002/(SICI)1097-0363(19960415)22:7<673::AID-FLD373>3.0.CO;2-O.

Wieneke, B., 2015. PIV uncertainty quantification from correlation statistics. Meas. Sci. Technol. 26 (7), 074002.

Wieneke, B., 2017a. PIV anisotropic denoising using uncertainty quantification. Exp. Fluids 58.

Wieneke, B., 2017b. PIV Uncertainty Quantification and Beyond (Ph.D. thesis). Delft University of Technology, http://dx.doi.org/10.13140/RG.2.2.26244.42886.

Wilhem, D., Härtel, C., Kleiser, L., 2003. Computational analysis of the two-dimensional-three-dimensional transition in forward-facing step flow. J. Fluid Mech. 489, 1–27.