4.1. Fundamental mode amplitude and phase

The development of the fundamental mode’s streamwise-velocity perturbation amplitude ( $A^{\tilde {u}}$ ) in step and baseline conditions is shown in figure 6(a), as obtained from HWA measurements and DNS. Similarly, the $u$ -perturbation profiles measured at six streamwise stations in the vicinity of the step are presented in figure 6(b–g). In figure 6(a), $A^{\tilde {u}}$ is retrieved at the wall-normal location where $|\tilde {u}|$ (shown in figure 6 b–g) becomes maximum. Excellent agreement is observed between DNS and HWA results in terms of TS amplification and $u$ -perturbation shape. Despite HWA measurements not being able to capture the secondary near-wall lobe emerging downstream of the step (see DNS solution in figure 6 d), they accurately capture the highly distorted TS perturbation profile at P15 (see figure 6 g). The excellent agreement between experiments and DNS is further confirmed when comparing the spatial distribution of streamwise-velocity perturbation amplitude ( $|\tilde {u}|$ , figure 7 a–c) and phase ( $\varphi ^{\tilde {u}}$ , figure 8 b,c,e) obtained from DNS and PIV measurements, e.g. compare figures 7(a,b) and 8(b,c).

Figure 6.

(a) The amplitude of the fundamental TS mode and zoomed view close to the step vicinity from HWA measurements ( $\circ$ ) and DNS (). Grey solid lines in the zoomed figure correspond to the amplitude of the near-wall peak emerging in the perturbation profile close to the step. (b–g) The TS wave perturbation profile upstream (b,c) and downstream (d–g) of the FFS. Refer to table 3 for the corresponding streamwise location ( $x^*$ ) of each profile. Results in FFS conditions are indicated in black () and in baseline conditions in orange (). Streamwise location of profiles shown in (b–g) are indicated by vertical grey dash-dotted lines () in (a).

Figure 7.

Contours of the perturbation velocity ( $\tilde {u}$ and $\tilde {v}$ ) from DNS (a,c,d) and PIV (b) in FFS conditions. Subfigures (c,d) constitute contours of streamwise (c) and wall-normal (d) perturbation velocity in a zoomed region delimited by a yellow box in (a) around the step corner. Solid lines denote the maxima () and minima () in the TS shape profile for $|\tilde {u}|$ (a–c) and $|\tilde {v}|$ (d). Dashed lines () indicate the inflection points in $U$ (a–c) and $V$ (d).

Figure 6 shows exponential growth of the fundamental mode’s amplitude upstream of the FFS, following the progressive pressure increase (note $C_p$ curve in figure 5 b). In this region upstream of the step ( $-15\leqslant x^* \leqslant 0$ ), the TS shape exhibits an increase in perturbation amplitude at a higher wall-normal location (note thickening of the TS wave shape in figure 7 c around $y/\delta ^*_0 \approx 1$ ), which coincides with a region where the mean shear is maximum (note the mean shear profiles in figure 5 d,e around $y/\delta ^*_0 \approx 1$ and black dashed line in figure 7, indicating the inflection points). This is a well-known behaviour observed in TS waves when entering regions of strong adverse pressure gradients, like a laminar separation bubble (see, e.g. figure 17 in Dovgal, Kozlov & Michalke Reference Dovgal, Kozlov and Michalke1994) or upstream of a 2-D hump (see, e.g. figure 10 in Park & Park Reference Park and Park2013).

Immediately upstream of the step ( $-1\leqslant x^* \leqslant 0$ ), the distortion induced by the FFS on the TS instability results in a substantial transfer of streamwise perturbation momentum in the wall-normal direction. This can be clearly distinguished by comparing the amplitude fields of $|\tilde {u}|$ (figure 7 c) and $|\tilde {v}|$ (figure 7 d), where $|\tilde {u}|$ decays upstream of the step, while $|\tilde {v}|$ is strongly amplified. This is also well observed from the growth rate results of $\tilde {u}$ and $\tilde {v}$ presented in figure 12(b) in § 5.2.

Figure 8.

Contours of the streamwise ( $\varphi ^{\tilde {u}}$ in b,c,e) and wall-normal ( $\varphi ^{\tilde {v}}$ in a and d) perturbation phase from DNS (a,b,d,e) and PIV (c) in FFS conditions. Subfigures (d,e) constitute contours of the streamwise (e) and wall-normal (d) perturbation phase in a zoomed region delimited by a yellow box in (a) and (b) around the step corner. Solid lines denote the maxima () and minima () in the TS shape profile for $|\tilde {u}|$ (b,c,e) and $|\tilde {v}|$ (a,d).

Immediately downstream of the FFS edge ( $0\leqslant x^*\leqslant 2$ ), a secondary peak develops in both $|\tilde {u}|$ and $|\tilde {v}|$ at the near-wall region (see black lines in figure 7 c,d and near-wall region of the TS profile in figure 6 d, i.e. $y/\delta ^*_0 \approx 0.2$ ). Similar observations are reported from the DNS results of Edelmann (Reference Edelmann2014) (see, e.g. figure 4.29 in Edelmann Reference Edelmann2014). Interestingly, this feature occurs at a wall-normal location that coincides with a peak in the mean shear profile. This can be observed from the $u$ -perturbation amplitude contours ( $|\tilde {u}|$ ) close to the FFS in figure 7(c) (note dashed lines and full lines overlapping at $0.5\leqslant x^* \leqslant 2$ and $y/\delta ^*_0 \approx 1$ ) and the near-wall peak emerging in the TS profiles, which also appears in the mean shear profile at the same streamwise station (see near-wall peaks in figures 5 f and 6 d). This peak in the mean shear was related in the previous section to the abrupt changeover from an adverse to favourable pressure gradient at the step. The amplitude of the near-wall $|\tilde {u}|$ peak decays in the streamwise direction (see grey lines in the inset of figure 6 a), indicating that the feature giving rise to this peak disappears or merges with the original TS wave from $x^* \geqslant 2$ . The perturbation phase ( $\varphi$ ) contours of $\tilde {u}$ and $\tilde {v}$ in figures 8(d) and 8(e) reveal that this near-wall feature exhibits a phase shift with respect to the phase contours of the TS structure above. In the next section it will be shown that this phase shift between the incoming TS structure and the near-wall feature is related to their opposite signs in the perturbation streamfunction ( $\textrm{Re}(\psi )$ , (4.1)).

Further downstream of the step ( $x^* \geqslant 5$ ), the TS wave experiences a gradual stabilisation up to $x^* \approx 40$ (see figure 6 a), followed by an inflection in growth and a prolonged destabilisation downstream up to $x^* \approx 100$ , where it decays again, returning to the decaying trend observed in baseline conditions. The latter can be observed by comparing baseline and FFS case growth rates in figure 12, presented later in § 5.1. The decay-growth-decay-growth behaviour shown in figure 6(a) resembles the TS amplitude evolution reported in previous works downstream of isolated roughness or geometrical elements (e.g. Wu & Dong Reference Wu and Dong2016 and Michelis et al. Reference Michelis, De Koning and Kotsonis2023a ).

Interestingly, figure 6(g) shows an unconventional TS shape profile at P15 in FFS conditions. Both HWA and DNS capture a similarly distorted shape function. Further evidence of the substantial distortion undergone by the TS wave in this region ( $20 \leqslant x^* \leqslant 60$ ) is found from the $u$ -perturbation phase contours ( $\varphi ^{\tilde {u}}$ ) from DNS in figure 8(b). Unfortunately, the PIV domain does not extend sufficiently downstream to encompass the region where the pronounced TS distortion occurs. Nevertheless, the agreement between the fundamental mode amplitude and phase contours from PIV and DNS elsewhere, coupled with the fact that the TS distortion is captured by HWA, points to the occurrence of this behaviour in the experiments. The topology of the TS wave in this region and the origin of this distortion are investigated in the following sections.

4.2. Topology of the perturbation field

The results presented in the previous subsection revealed two spatial regions where TS waves experience a strong distortion due to the presence of the step. First, a region in the vicinity of the FFS is found ( $0\leqslant x^*\leqslant 2$ ), where a near-wall velocity perturbation appears immediately downstream of the FFS edge. Second, a region further downstream of the FFS ( $20 \leqslant x^* \leqslant 60$ ), where TS waves undergo a strong distortion in shape and phase, coinciding with local amplitude decay.

Figure 9 shows the perturbation structure around the FFS at three distinct phases within a TS period. In this study, the perturbation structure is visualised using isolines of the real part of the perturbation’s streamfunction, $\psi$ , which is defined as

(4.1) \begin{equation} u = \frac {\partial \psi }{\partial y} \quad \text{and} \quad v = -\frac {\partial \psi }{\partial x}, \end{equation}

where $u$ and $v$ denote the perturbation velocity components in the streamwise and wall-normal directions, respectively. These components are expressed as

(4.2) \begin{align} \begin{bmatrix} u \\ v \end{bmatrix} &= \begin{bmatrix} \tilde {u} \\ \tilde {v} \end{bmatrix} e^{-i\omega _0 t} + \textrm{c.c.}, \end{align}

(4.3) \begin{align} \tilde {u} &= |\tilde {u}| e^{i\varphi ^{\tilde {u}}}, \end{align}

(4.4) \begin{align} \tilde {v} &= |\tilde {v}| e^{i\varphi ^{\tilde {v}}}, \\[9pt] \nonumber \end{align}

Figure 9.

Real part of the streamwise-velocity perturbation component (contours) and isolines of the real part of the perturbation streamfunction (contour lines of $\textrm{Re}(\psi )$ , $-5.5\times 10^{-6}\leqslant \textrm{Re}(\psi )/(U_0\delta ^*_0) \leqslant 5.5\times 10^{-6}$ ) across different TS phases: $\phi = \pi /3$ rad (a,b), $\phi = 2\pi /3$ rad (c,d) and $\phi = \pi$ rad (e,f). Zoomed-out FOV (a,c,e) and zoomed-in region at the step edge (b,d,f), bounded by a yellow box in the zoomed-out fields. Negative values of $\textrm{Re}(\psi )$ are indicated with solid lines (), whereas positive values of $\textrm{Re}(\psi )$ are indicated with dashed lines (). Grey solid contour lines in the zoomed-in region (b,d,f) constitute isolines of the perturbation streamfunction’s real part ( $\textrm{Re}(\psi )$ ), evaluated within the range $-2\times 10^{-7}\leqslant \textrm{Re}(\psi )/(U_0\delta ^*_0) \leqslant 0$ , and delineating the topology of the near-wall structure. Additionally, the spatial organisation of the perturbation velocity field is indicated by black quivers in (b,d,f). A video showing the evolution of the TS structure over the FFS for multiple phases (similar to a,c and e) is included as supplementary movie 1.

where c.c. denotes complex conjugate. The streamfunction $\psi$ (4.1) is computed by integrating the perturbation field across the DNS domain using the trapezoidal rule.

In the vicinity of the FFS apex (see figure 9 b,d,f), the perturbation exhibits a secondary structure featuring opposite sign in $\textrm{Re}{(\psi )}$ with respect to the incoming TS wave (note the solid grey contour lines in figure 9 b,d,f). According to the results presented in the previous section, this secondary feature manifests in the TS shape function as a near-wall peak (see figure 6 d). The spatial region in which this near-wall structure is evident spans $0\leqslant x^*\leqslant 2$ , corresponding to about $11 \,\%$ of the local TS wave streamwise wavelength ( $\lambda _{\textit{TS}}/\delta ^*_0 \approx 18$ ). The small dimension of this near-wall feature and its eventual decay (inset of figure 6 a) in the streamwise direction (vanishing downstream, as observed in figure 9 b,d,f) leads to a negligible effect on the incoming TS wave stability.

The flow features observed at the FFS edge for an incoming TS wave are notably similar to those in flows governed by a S-CFI. Casacuberta et al. (Reference Casacuberta, Hickel, Westerbeek and Kotsonis2022), studying the interaction between a FFS and S-CFI, report the presence of spanwise-distributed near-wall streaks emerging immediately downstream of the step edge and counter-rotating with respect to the incoming primary crossflow mode. The near-wall structures observed in Casacuberta et al. (Reference Casacuberta, Hickel, Westerbeek and Kotsonis2022) also arise at the FFS apex, at a near-wall region where the mean flow exhibits high wall-normal shears.

Casacuberta et al. (Reference Casacuberta, Westerbeek, Franco, Groot, Hickel, Hein and Kotsonis2025) attribute the onset of near-wall streaks to the lift-up effect, caused by the redistribution of streamwise momentum by the cross-stream perturbation components of the crossflow mode in regions of high wall-normal shear ( $\partial U/\partial y$ ). The flow mechanism outlined in Casacuberta et al. (Reference Casacuberta, Westerbeek, Franco, Groot, Hickel, Hein and Kotsonis2025) can also explain the onset of the near-wall structure emerging at the FFS edge within a 2-D, time-periodic instability like the one concerned herein. At the FFS apex, the strong wall-normal perturbation velocity component ( $\tilde {v}$ , note length of velocity vectors in figure 9 d), resulting from the incoming $\textrm{Re}{(\psi )}\gt 0$ TS structure, displaces low-momentum fluid upwards in a region of high wall-normal shear ( $\partial U/\partial y$ ), creating an instantaneous defect of streamwise perturbation velocity ( $\tilde {u}$ ) near the wall (note blue contour in figure 9 d, overlapping with $\psi$ isolines of the near-wall structure). This $u$ -defect region, elongated in the streamwise direction due to flow convection, eventually vanishes (see figure 9 b,d,f) downstream as the peak in wall-normal shear weakens (see peak in $\partial U/\partial y$ profiles in figure 5 h,i). When the incoming TS structure features negative $\textrm{Re}{(\psi )}$ values, the opposite occurs, reversing the $\textrm{Re}{(\psi )}$ sign of the near-wall structure. Interestingly, the mechanisms generating the near-wall feature in both TS–FFS interactions and in stationary CFI-FFS interactions involve the effective displacement of streamwise momentum by the cross-stream perturbation component (i.e. only $v$ for the TS–FFS interaction) in a region of strong mean flow shear. For TS waves, this displacement is time dependent and shifts sign within the TS cycle, whereas for S-CFI, it is stationary and shifts sign along the spanwise direction, synchronous with the S-CFI wavelength, as reported by Casacuberta et al. (Reference Casacuberta, Westerbeek, Franco, Groot, Hickel, Hein and Kotsonis2025)

Downstream of the FFS edge ( $20 \leqslant x^* \lt 60$ ), figure 9(a,c,e) shows a second region where the TS perturbation structure exhibits notable topological changes. In particular, the TS structure appears tilted in the mean shear ( $\partial U/\partial y$ ) direction ( $x^* \approx 30$ ), gradually changing orientation towards a tilt against the mean shear ( $x^* \approx 50$ ), and recovering further downstream to an upright orientation ( $x^* \geqslant 60$ ), characteristic of the TS wave’s asymptotic development. Notably, this region of topological changes in the perturbation structure coincides with a region where TS waves also exhibit an inversion in their growth trend (see figure 6 a). The following section explores in more detail the instability mechanisms driving the observed tilting of the perturbation structure and consequent modifications in growth.

5.1. Linear, parallel and non-parallel stability characteristics

The experiments and DNS performed in this work appear to capture the complete dynamics of the TS wave as it interacts with the step. However, towards elucidating the underlying mechanisms, it becomes instructive to leverage specific assumptions of flow linearity, parallelism and non-parallelism to isolate pertinent effects and reveal the relative influence on the complete perturbation behaviour. These assumptions are key for simplifications of the complete Navier–Stokes equations towards classical stability analysis methods such as LST and LPSE. Details on the numerical set-up employed to compute LPSE and LST are provided in § 2.3.2.

In practice, LPSE is typically used to avoid computationally expensive DNS simulations. However, since LPSE cannot march over the step (Tocci et al. Reference Tocci, Franco, Hein, Chauvat and Hanifi2021), where the highly non-parallel base flow invalidates PSE assumptions, an inflow condition is required downstream of the step. In the absence of more accurate solutions, this inflow is often set as a solution to the LST equations. Consequently, one possible scenario for the LPSE solutions downstream of the FFS would involve using a perturbation shape from LST as an inflow condition. This section compares two scenarios: LPSE results downstream of the step after initialisation with the perturbation profile from (i) DNS and (ii) the LST solution. Comparing these results, where DNS captures non-parallel effects that are not accounted for by LST, elucidates the importance of the upstream perturbation history in describing the TS wave distortion observed further downstream of the step. Table 4 summarises the different stability solutions studied.

Table 4.

Stability solutions studied in § 5.1.

To establish the DNS solution as the inflow for the LPSE when marching downstream of the step, it is necessary to compute the amplitude function and streamwise wavenumber of the fundamental mode from the DNS data. The streamwise wavenumber ( $\alpha$ ) is determined from the DNS solution by tracking the amplitude and the phase of the streamwise-velocity perturbation component ( $u$ ) of the fundamental mode at the wall-normal location where its amplitude ( $|\tilde {u}|$ ) is maximum, following:

(5.1) \begin{equation} \alpha = \alpha _r + i\alpha _i \quad \text{with} \quad \alpha _r = -\frac {\partial \left(\varphi ^{A^{\tilde {u}}}\right)}{\partial x}\delta _0 \quad \text{and} \quad \alpha _i = -\frac {\partial \left(\ln {A^{\tilde {u}}}\right)}{\partial x}\delta _0. \end{equation}

In (5.1), $\varphi ^{A^{\tilde {u}}}$ denotes the phase of the streamwise perturbation component where $|\tilde {u}|$ is maximum, i.e. $\varphi ^{A^{\tilde {u}}} = \varphi ^{\tilde {u}}(y = \text {max}_y(|\tilde {u}|))$ .

Figure 10 shows the growth rate ( $\alpha _i$ ) and amplitude evolution predicted by the different stability solutions investigated. Upstream of the step, both LPSE and LST accurately capture the perturbation growth rate up to $x^* \approx -20$ (see figure 10 a). However, as non-parallel effects intensify near the step, the TS growth predicted by LST deviates from the DNS solution earlier than LPSE due to the local formulation of LST. At the step edge ( $-20 \leqslant x^* \leqslant 0$ in figure 10 a), both LST and LPSE fail to capture the strong destabilisation predicted by DNS due to the large base flow streamwise gradients, which invalidate both the LST parallel-flow assumption and the LPSE slowly varying base flow assumption.

Figure 10.

Growth rate (a) and N factor (b) as computed with DNS (), DNS-initialised LPSE (), LST () and LST-initialised LPSE () in FFS conditions. For the DNS solution, the growth rate and amplification of both $|\tilde {u}|$ () and $|\tilde {v}|$ () perturbation components are presented. The DNS solution in baseline conditions () is also indicated. The grey-shaded region spanning $0 \leqslant x^* \leqslant 6.4$ represents the range where LPSE and LST results are unavailable.

Downstream of the step, LPSE and LST are initialised at $x^* = 6.4$ . This represents the first streamwise location where the LPSE converges to a physically meaningful solution when the DNS and LST solutions are used as inflow conditions. The inflow perturbation profiles used are shown in figure 11(f). Both LST-initialised LPSE and DNS-initialised LPSE results are presented in figure 10. Unlike the aforementioned upstream behaviour, where both LST and LPSE captured similar growth rate trends, DNS-initialised LPSE clearly outperforms LST downstream of the step. Comparison of DNS-initialised LPSE and LST-initialised LPSE with DNS results shows that initialising LPSE with a low-fidelity solution significantly degrades the agreement between DNS and LPSE downstream of the step (compare dashed magenta and dash-dotted blue lines to black lines in figure 10). The DNS-initialised LPSE solution effectively captures the non-monotonic growth-decay-growth pattern downstream. In contrast, both LST and LST-initialised LPSE predict an exponential decay in the growth rate, following the adverse pressure gradient increase downstream (see figure 5 b), failing to accurately capture the amplitude evolution downstream of the FFS (figure 10 b). The remarkable agreement between the DNS-initialised LPSE and DNS results highlights the following two key points.

1. (i) The perturbation behaviour remains linear at the investigated TS wave amplitude.

2. (ii) Accurately capturing the distortion experienced by the perturbation at the step is essential for describing the downstream evolution of the TS wave.

Figure 11.

Real part of the streamwise perturbation component (contours) and isolines of the real part of the perturbation streamfunction (contour lines of $\textrm{Re}(\psi )$ , $-5.5\times 10^{-6}\leqslant \textrm{Re}(\psi )/(U_0\delta ^*_0) \leqslant 5.5\times 10^{-6}$ ) at one TS phase, $\phi = \pi$ rad. Results obtained from (a) DNS, (b) DNS-initialised LPSE and (c) LST. Negative values of $\textrm{Re}(\psi )$ are indicated with grey solid lines (), whereas positive values of $\textrm{Re}(\psi )$ are indicated with grey dashed lines (). Green dash-dotted () vertical lines indicate the streamwise locations where the perturbation profiles in (d–g) are extracted. The TS shape function at (d) $x^* = -40.8$ , (e) $x^* = -7.1$ , (f) $x^* = 6.4$ and (g) $x^* = 40.8$ as obtained from DNS (), DNS-initialised LPSE () and LST ().

Figure 11(a–c) shows the streamwise perturbation field for the fundamental TS mode as reconstructed from DNS, DNS-initialised LPSE and LST. Additionally, the TS wave shape function obtained from the three different stability solutions is presented in figure 11(d–g) for selected streamwise stations (indicated by green vertical lines in figure 11 a–c).

Results in figure 11(a–c) show that the perturbation field from both LPSE and LST present good agreement with the DNS upstream ( $x^* \leqslant 0$ , see also TS profiles in figure 11 d–e) and sufficiently downstream of the step ( $x^* \gt 60$ ). However, in the immediate vicinity downstream of the step ( $ 0 \leqslant x^* \leqslant 60$ ), while LPSE can capture the tilting of the TS wave structure in the mean shear ( $\partial U/\partial y$ ) direction, LST fails to do so (see figure 11 a–c). It is important to note that when LPSE is initialised using LST, the resulting perturbation field is similar to LST (not shown here). Thus, the performance of LPSE is largely subject to the shape of the inflow perturbation profile, as already concluded from the growth and amplitude results in figure 10.

The relative fidelity of these methods in capturing the TS wave growth and shape provides further insight into the stability mechanisms dominant in the near region downstream of the step. Specifically, as outlined in the seminal work of Herbert (Reference Herbert1997), the LPSE formulation is able to capture pertinent effects of non-parallelism on the evolution of boundary-layer instabilities. In purely parallel mean flows, LPSE results are expected to largely overlap with the local modal solutions of LST. However, when perturbation fields are distorted, the parabolic character of LPSE allows for the upstream history of the flow to travel downstream. This is evident in the close agreement between DNS-initialised LPSE and DNS results, and vice versa, in the disagreement between the results of LST and LST-initialised LPSE with DNS, as shown in figure 10. This effectively points to a spatial conditioning of the TS wave at the step towards a new distorted shape that is not an eigensolution of the local flow stability, yet behaves and develops similar to classical TS waves.

5.2. Non-modal growth

Results in the previous section reveal that, in order to resolve the perturbation growth downstream of the step using LPSE, it was necessary to initialise the simulation with the DNS perturbation solution. In contrast, initialisation with an eigenvalue solution failed to reproduce the correct perturbation dynamics. This suggests that classical modal eigenvalue approaches are inadequate in the region downstream of the FFS. Therefore, this section investigates potential non-modal growth near the step, following the methodology outlined in Marxen et al. (Reference Marxen, Lang, Rist, Levin and Henningson2009).

As evident in figure 10(a) in the previous section, the growth of the wall-normal ( $\alpha _i^{\tilde {v}}$ ) and streamwise ( $\alpha _i^{\tilde {u}}$ ) perturbation maxima diverge in the step vicinity. This divergence is evident upstream of the FFS edge ( $-20 \leqslant x^* \leqslant 0$ ) and further downstream, up to $x^* \approx 60$ . The decoupling between $u$ - and $v$ -perturbation growth rates is often linked in previous works to the presence of non-modal growth (e.g. Marxen et al. Reference Marxen, Lang, Rist, Levin and Henningson2009; Casacuberta et al. Reference Casacuberta, Hickel, Westerbeek and Kotsonis2022, and Michelis et al. Reference Michelis, Putranto and Kotsonis2023b ). Nevertheless, in regions where the base flow streamlines experience significant deflection in wall-normal direction, employing a wall-oriented reference system to distinguish between wall-normal and streamwise perturbation components is no longer deterministic towards identifying regions of non-modal behaviour (Marxen et al. Reference Marxen, Lang, Rist, Levin and Henningson2009). To circumvent this, Marxen et al. (Reference Marxen, Lang, Rist, Levin and Henningson2009) decompose the perturbation field using a base flow-oriented coordinate system, as originally proposed by Albensoeder, Kuhlmann & Rath (Reference Albensoeder, Kuhlmann and Rath2001). This approach yields perturbation components that are tangential ( $\tilde {\upsilon }_t$ ) and normal ( $\tilde {\upsilon }_n$ ) to the local base flow streamlines. The methodology is outlined in the works of Albensoeder et al. (Reference Albensoeder, Kuhlmann and Rath2001), Marxen et al. (Reference Marxen, Lang, Rist, Levin and Henningson2009), Lanzerstorfer & Kuhlmann (Reference Lanzerstorfer and Kuhlmann2012) and Casacuberta et al. (Reference Casacuberta, Hickel, Westerbeek and Kotsonis2022), among others.

Figure 12.

Growth rates of the tangential- ( $\alpha _i^{\tilde {\upsilon }_t}$ , ) and normal- ( $\alpha _i^{\tilde {\upsilon }_n}$ , ) to-the-base-flow perturbation maxima in baseline (a) and FFS (b) conditions. For the tangential perturbation component ( $\alpha _i^{\tilde {\upsilon }_t}$ ), solid lines track the TS lower lobe ( $\alpha _i^{\tilde {\upsilon }_t, \;l}$ , ), whereas dashed lines track the TS upper lobe ( $\alpha _i^{\tilde {\upsilon }_t, \;u}$ , ). Red and grey areas indicate the streamwise regions with total (red) and partial (grey) decoupling between $\alpha _i^{\tilde {\upsilon }_t}$ and $\alpha _i^{\tilde {\upsilon }_n}$ . Note that the values of $\alpha _i$ for $-1 \leqslant x^* \leqslant 1$ are excluded from (b) for clarity, as significant jumps occur due to the FFS discontinuity.

Figure 12 illustrates the growth rates of the tangential ( $\tilde {\upsilon }_t$ ) and normal ( $\tilde {\upsilon }_n$ ) to the base flow perturbation components in baseline and FFS conditions. The corresponding maximum of the two perturbation components is tracked in the streamwise direction to calculate the corresponding growth rates:

(5.2) \begin{align} \alpha _i^{\tilde {\upsilon }_n} &= -\frac {1}{A^{\tilde {\upsilon }_n}}\frac {\text{d}A^{\tilde {\upsilon }_n}}{\text{d}x}, \end{align}

(5.3) \begin{align} \alpha _i^{\tilde {\upsilon }_t} &= -\frac {1}{A^{\tilde {\upsilon }_t}}\frac {\text{d}A^{\tilde {\upsilon }_t}}{\text{d}x}. \end{align}

Owing to the dual peak shape of the streamwise ( $\tilde {\upsilon }_t$ ) perturbation component in TS waves, two distinct local maxima are distinguished, namely the lower ( $y/\delta ^*_0 \approx 0.5$ ) and upper ( $y/\delta ^*_0 \approx 3$ ) lobe of the TS wave shape function (see, e.g. the different TS maxima in figure 11 d). This distinction between upper lobe and lower lobe growth rates will be denoted as $\alpha _i^{\tilde {\upsilon }_t, \;u}$ and $\alpha _i^{\tilde {\upsilon }_t, \;l}$ , respectively.

In baseline conditions, both perturbation components exhibit equal growth rates across the entire domain (figure 12 a), except for a localised region near $x^* \approx 0$ , where mild suction is applied in DNS to replicate experimental conditions (see details in § 2.3.1). Conversely, in step conditions, tangential and normal perturbation growth rates present significant deviations.

The decoupling between $\alpha _i^{\tilde {\upsilon }_n}$ and $\alpha _i^{\tilde {\upsilon }_t}$ observed around $-20 \leqslant x^*\leqslant 60$ from figure 12(b) suggests the presence of non-modal growth. In particular, two regions can be identified based on the decoupling behaviour:

1. (i) In the step vicinity ( $-20 \leqslant x^* \leqslant 20$ , note red area in figure 12 b), both upper ( $\alpha _i^{\tilde {\upsilon }_t, \;u}$ ) and lower ( $\alpha _i^{\tilde {\upsilon }_t,\; l}$ ) lobe growth rates deviate from the normal perturbation growth rate ( $\alpha _i^{\tilde {\upsilon }_n}$ ). Hereafter this will be denoted as total decoupling.

2. (ii) Further downstream ( $20 \leqslant x^* \leqslant 60$ , note grey area in figure 12 b), the lower lobe growth rate ( $\alpha _i^{\tilde {\upsilon }_t, \;l}$ ) and the normal perturbation growth rate ( $\alpha _i^{\tilde {\upsilon }_n}$ ) deviate. However, the upper lobe growth rate ( $\alpha _i^{\tilde {\upsilon }_t, \;u}$ ) converges to $\alpha _i^{\tilde {\upsilon }_n}$ . Hereafter this behaviour will be denoted as partial decoupling.

In the step vicinity ( $-20 \leqslant x^* \leqslant 20$ ), the total decoupling between $\alpha _i^{\tilde {\upsilon }_n}$ and $\alpha _i^{\tilde {\upsilon }_t}$ occurs in a region with significant non-parallel effects (e.g. note the abrupt modifications in base flow integral parameters along $x^*$ close to the step in figure 5 c). Due to the smaller streamwise extent of the FFS-induced base flow distortion compared with the incoming TS wavelength, the latter is scattered (or distorted) by the geometrical discontinuity (Goldstein Reference Goldstein1985; Wu & Hogg Reference Wu and Hogg2006). The near-wall structure observed after the FFS edge (discussed in § 4.2) reflects the strong distortion experienced by the TS wave. Consistent with these findings, recent studies also report significant non-modal perturbation growth in regions of wave scattering (Michelis et al. Reference Michelis, De Koning and Kotsonis2023a ; Casacuberta et al. Reference Casacuberta, Hickel, Westerbeek and Kotsonis2022).

Downstream of the step ( $20 \leqslant x^* \leqslant 60$ ), the partial decoupling observed between $\alpha _i^{\tilde {\upsilon }_n}$ and $\alpha _i^{\tilde {\upsilon }_t}$ indicates that the TS structure exhibits a stratified growth behaviour in the $y$ direction. The upper part of the TS structure follows a modal evolution, while the lower part is influenced by non-modal growth. This dual behaviour is similar to findings by Marxen et al. (Reference Marxen, Lang, Rist, Levin and Henningson2009), where the authors observe non-modal evolution of near-wall streamwise streaks – generated by the lift-up effect – coexisting with a modal (Görtler) instability above. In contrast to Marxen et al. (Reference Marxen, Lang, Rist, Levin and Henningson2009), a secondary structure cannot be distinguished herein from the modal TS structure in the lower lobe region (see, e.g. the TS shape in figure 11 g).

The partial decoupling described in the region $20 \leqslant x^* \leqslant 60$ suggests the presence of non-modal behaviour (Marxen et al. Reference Marxen, Lang, Rist, Levin and Henningson2009). At first glance, this appears to contradict the previously observed agreement (§ 5.1) between DNS and LPSE, given that classical LPSE has been historically associated with the modelling of modal perturbations. However, the LPSE formalism does not solve an eigenvalue problem, but instead propagates a prescribed initial perturbation downstream under the assumption that the dominant streamwise oscillation and growth at each frequency can be represented by a single complex wavenumber (Herbert Reference Herbert1997). As such, LPSE does not necessarily evolve a single mode. It can track a superposition of modes, but only those whose streamwise wavenumbers lie sufficiently close to one another. This interpretation is supported by previous studies in which the LPSE method has also been employed to analyse flows susceptible to non-modal growth, which inherently involve a superposition of modes (Tempelmann, Hanifi & Henningson Reference Tempelmann, Hanifi and Henningson2010; Lucas Reference Lucas2014). The observed indications of non-modal behaviour may arise from the streamwise propagation, via the LPSE marching scheme, of a DNS-derived perturbation field composed of a superposition of modes, originating from the strongly non-modal region near the step. The following section investigates further the role of superimposed modes in the DNS solution.

5.3. Origin of TS wave distortions downstream of the FFS

This section aims to elucidate the existence and nature of superimposed modes, encoded in the perturbation evolution downstream of the step. To do so, the different eigensolutions from LST applied on the DNS flow in the vicinity of the step are analysed. Figure 13(a) shows the real ( $\textrm{Re}{(\alpha ^n)}$ ) and imaginary ( $\rm Im {(\alpha ^n)}$ ) part of the two least stable eigenvalues predicted by LST for $F=90$ in the region downstream of the FFS ( $x^* \gtrsim 6.4$ ). The arrows in figure 13(a) indicate the direction of variation along the streamwise axis. A second mode (red symbols in figure 13 a) appears less stable than the classical TS mode (blue circles) closely downstream of the step (see markers of the first streamwise stations in figure 13 a). Nevertheless, it is important to note that this second mode remains stable at all the streamwise stations studied. The TS mode growth rate ( $\alpha _i$ ) becomes less stable than the second mode’s $\alpha _i$ from $x^* \gt 11.7$ onwards. This second mode, coexisting with the original TS mode, travels in the streamwise direction at a higher phase speed than the original TS wave, owing to its lower $\textrm{Re}{(\alpha )}$ .

Figure 13.

(a) Eigenvalues from LST downstream of the step ( $6 \leqslant x^* \leqslant 80$ ). (b) Weights assigned to the TS mode (blue, $w^1$ ) and second mode (red, $w^2$ ) by the non-negative least squares fit. (c,d) Shape functions of the fundamental mode from DNS (, $|\tilde {u}|$ ), weighted TS mode (, $\boldsymbol{\hat {C}}^1 w^1$ ), weighted second mode (, $\boldsymbol{\hat {C}}^2 w^2$ ) and linear superposition of LST modes (, $\sum _{\!j}^2 \boldsymbol{\hat {C}}^j w^j$ ) at the streamwise locations indicated by black vertical dashed lines in figure 14, namely $x^* = 6.4$ (c) and $x^* = 30.5$ (d).

To elucidate whether the superposition of this second mode with the TS mode improves the description of the fundamental mode evolution as predicted by DNS, a methodology inspired by the work of Edelmann (Reference Edelmann2014) is applied. This involves performing a non-negative least squares fit, where the absolute values of the normalised eigenfunctions of the two modes, shown in figure 13(c,d), are weighted to minimise the following function:

(5.4) \begin{align} \min _{w^j} & \qquad \|\boldsymbol{\hat {C}}^j w^j - \boldsymbol{\hat {d}}\|_2^2, \quad \text{subject to} \quad w^j \geqslant 0, \quad j \in \{1,2\}, \end{align}

(5.5) \begin{align} \text{with} \quad \boldsymbol{\hat {C}}^j &= \frac {1}{\max (|\tilde {u}_i^j|)} \begin{bmatrix} |\tilde {u}_{0}|^j \\ |\tilde {u}_{1}|^j \\ \vdots \\ |\tilde {u}_{n_y}|^j \end{bmatrix} \Bigg |_{\text{LST}} \quad \text{and} \quad \boldsymbol{\hat {d}} = \frac {1}{\max (|\tilde {u}_i|)} \begin{bmatrix} |\tilde {u}_{0}| \\ |\tilde {u}_{1}| \\ \vdots \\ |\tilde {u}_{n_y}| \end{bmatrix} \Bigg |_{\text{DNS}}. \end{align}

Here the subscript $\|\boldsymbol{\cdot }\|_2$ denotes the Euclidean norm. The term $\boldsymbol{\hat {C}}^j$ represents a column vector containing the normalised eigenfunction of the $ j$ th LST mode at a specific $x^*$ station, while $w^j$ is the corresponding weight for the $j$ th mode. Similarly, $\boldsymbol{\hat {d}}$ is a column vector containing the normalised shape function obtained from DNS. Equation (5.4) is minimised independently at each streamwise station $x^*$ , yielding a one-dimensional vector of weights, $\boldsymbol{\hat {w}}^j = [w_0^j, w_1^j, \ldots , w_{n_x}^j ]$ .

The resulting weights from the optimisation problem are shown in figure 13(b). The contribution of the second mode in describing the DNS solution is limited to the region downstream of the step, i.e. $0 \leqslant x^* \leqslant 35$ . Similar observations are reported in Edelmann (Reference Edelmann2014), where the author identified three additional modes that, when superimposed onto the original TS mode, largely contribute to describing the fundamental mode at the region directly downstream of the step. The weighted eigenfunctions, together with the final profile from linear superposition of LST modes ( $\boldsymbol{\hat {C}}^1 w^1+ \boldsymbol{\hat {C}}^2 w^2$ ) are compared with the DNS perturbation profile in figure 13(c–d) at different $x^*$ stations.

Figure 14.

Real part of the streamwise perturbation component (contours) and isolines of the real part of the perturbation streamfunction (contour lines of $\textrm{Re}(\psi )$ , $-5.5\times 10^{-6}\leqslant \textrm{Re}(\psi )/(U_0\delta ^*_0) \leqslant 5.5\times 10^{-6}$ ) at one TS phase, $\phi = \pi$ rad. Results for the real part of (a) the LST second mode, $u^2$ ; (b) the LST TS mode, $u^1$ ; (c) the linear superposition of LST modes, $u^1$ + $u^2$ ; and (d) the DNS solution. Negative values of $\textrm{Re}(\psi )$ are indicated with grey solid lines (), whereas positive values of $\textrm{Re}(\psi )$ are indicated with grey dashed lines (). Black dashed vertical lines indicate streamwise locations for the normalised shape functions shown in figure 13(c–d).

The streamwise perturbation field and streamlines for the two LST modes, the reconstructed mode (linear superposition) and the DNS solution are presented in figure 14, where the perturbation velocity for the two LST modes is obtained as

(5.6) \begin{equation} q^j = w^j\left (A^{\tilde {q}}_{x^*_0}|_{\text{DNS}}\frac {|\tilde {q}|^j}{\text {max}(|\tilde {q}|^j)}\text{e}^{-\int _{x^*_0}^{x^*}\alpha _i^j dx^*}\text{e}^{i(\varphi ^{\tilde {q}^j} + \int _{x^*_0}^{x^*}\alpha _r^j dx^* - \phi )}\right )\!, \end{equation}

and evaluated at different streamwise stations. In (5.6), $A^{\tilde {q}}_{x^*_0}$ denotes the perturbation amplitude from DNS at the LST inflow, i.e. $x^*_0 = 6.4$ .

In figure 14(a) the second mode predicted by LST exhibits a pronounced tilting in the mean shear direction ( $\partial U/\partial y$ ) close to the wall. Interestingly, the topology of this mode resembles the downstream tilted ( $\partial y/\partial x|_{\textrm{Re}{(\psi )}} \gt 0$ – refer to (5.20) presented later in § 5.4) structures identified in Åkervik et al. (Reference Åkervik, Ehrenstein, Gallaire and Henningson2008) as optimal forcing structures initiating TS waves in a 2-D flat-plate boundary layer. Åkervik et al. (Reference Åkervik, Ehrenstein, Gallaire and Henningson2008) refer to this structure as the Orr mode, named after the Orr mechanism, which drives the growth of the perturbation field in this context until it converges to the classic modal growth with the emergence of TS waves (see figure 3(c) in Åkervik et al. Reference Åkervik, Ehrenstein, Gallaire and Henningson2008).

The downstream tilting of the second mode helps to describe the topological features observed in DNS immediately downstream of the step (e.g. compare figures 14 c and 14 d at $6 \leqslant x^* \leqslant 30$ ). However, the reconstructed LST field fails to capture the perturbation evolution in the region of highly parallel flow ( $x^* \gtrsim 35$ ), particularly the transition from downstream ( $\partial y/\partial x|_{\textrm{Re}{(\psi )}} \gt 0$ ) to upstream ( $\partial y/\partial x|_{\textrm{Re}{(\psi )}} \lt 0$ ) tilting around $30 \leqslant x^* \leqslant 50$ . This discrepancy may be attributed to the inherently local nature of LST, which cannot account for the streamwise history of the perturbation. Furthermore, the omission of phase information in the optimisation procedure, as outlined in (5.4), could also contribute to the discrepancy between reconstructed LST and DNS perturbation fields.

Up to this point, the above analysis provides the following insights.

1. (i) In the region immediately downstream of the step, LST predicts the emergence of a second eigenmode that appears less stable than the original TS mode under the same conditions (figure 13 a).

2. (ii) The topological form of the second eigenmode is notably tilted in the mean shear direction (figure 14 a).

3. (iii) When reconstructing the DNS perturbation field, the superposition of the spatial structure of the second eigenmode and the original TS wave appear to be encoding the tilted perturbation topology observed in the DNS, which cannot be captured by the original TS mode alone (figures 13 c and 14 c).

These findings, together with the outcomes of § 5.1 and § 5.2, indicate that the perturbation distortion observed farther downstream of the FFS ( $x^* \gtrsim 10$ ) is likely a manifestation of the spatial non-modal growth. More specifically, this region can be interpreted as a zone of modes’ dominance changeover. Within, the perturbation field is converging towards its modal (asymptotic) state while coexisting with other local eigenmodes generated upstream, near the step ( $x^*\lesssim 7$ ). As the base flow gradually returns to the baseline configuration, these additional modes strongly decay in the streamwise direction (figure 13 a). However, their influence is encoded in the perturbation field and manifested as the tilting of the perturbation structure. The following section explores how the tilting of the perturbation structure drives the evolution of the perturbation kinetic energy.

5.4. The role of the Orr mechanism

The observations presented in the previous sections suggest a region of modes’ dominance changeover downstream of the step ( $30 \leqslant x^* \leqslant 50$ ), where the base flow is largely parallel. Notably, in this region the perturbation structure exhibits topological deformations (e.g. figure 9 e), accompanied by significant changes in growth, in comparison to the baseline case (figure 6 a). To further clarify whether the perturbation tilting holds connection with the observed inflections in growth (e.g. the region of decay-growth-decay in figure 10 b), the present section will analyse the evolution of the perturbation kinetic energy around the FFS.

Tollmien–Schlichting waves extract energy from the base flow through a positive Reynolds stress, expressed as

(5.7) \begin{equation} \tau = -\langle u v\rangle , \end{equation}

which is generated by viscosity in the near-wall region (Prandtl Reference Prandtl1921). A simplified conceptualisation of the instability mechanics can be found in § 5.2 of Mack (Reference Mack1984), following the original derivation of Prandtl (Reference Prandtl1921). To better illustrate how the instability exchanges energy with the base flow, it is indicative to examine the production term in the Reynolds–Orr equation (see (1.11) in Schmid & Henningson Reference Schmid and Henningson2001), which for 2-D flow is given by

(5.8) \begin{equation} P = -\int _{\text{S}}u_i u_{\!j} \frac {\partial U_i}{\partial x_{\!j}}\text{d}S. \end{equation}

Figure 15.

(a) Streamwise evolution of the production terms in (5.11)–(5.14) integrated in the $y$ direction. (b) Perturbation kinetic energy integrated in the $y$ direction in step () and baseline () conditions. Here $P_{m}$ denotes the value of $\int _y \sum _n \varPi _n^{\omega _0}$ at $x^* = -5$ , the location where $|\int _y \sum _n \varPi _n^{\omega _0}|$ becomes maximum upstream of the step.

Since this study focuses exclusively on the stability of the fundamental TS mode, the subscript (1,0) has been omitted for simplicity. Substituting the perturbation velocity definition from (4.2) into (5.8) and integrating over one TS wave period, $T = 2\pi /\omega _0$ , yields the time-averaged production term:

(5.9) \begin{equation} P_{\omega _0} = -\frac {2\pi }{\omega _0}\int _{\text{S}}(\tilde {u}_i + \textrm{c.c.})(\tilde {u}_{\!j} + \textrm{c.c.}) \frac {\partial U_i}{\partial x_{\!j}}\text{d}x\text{d}y. \end{equation}

Expanding the velocity component multiplication results in

(5.10) \begin{align} P_{\omega _0} = \frac {2\pi }{\omega _0} \int _{\text{S}} \sum _{n = 1}^{4} \varPi _n^{\omega _0} \text{d}x\text{d}y, \end{align}

where the individual production terms $\varPi _n^{\omega _0}$ are given by

(5.11) \begin{align} \varPi _1^{\omega _0} &= -(\tilde {u} \tilde {u}^{\dagger } + \textrm{c.c.}) \frac {\partial U}{\partial x},\\[-12pt] \nonumber \end{align}

(5.12) \begin{align} \varPi _2^{\omega _0} &= -(\tilde {u} \tilde {v}^{\dagger } + \textrm{c.c.}) \frac {\partial U}{\partial y}, \\[-12pt] \nonumber \end{align}

(5.13) \begin{align} \varPi _3^{\omega _0} &= -(\tilde {v} \tilde {u}^{\dagger } + \textrm{c.c.}) \frac {\partial V}{\partial x}, \\[-12pt] \nonumber \end{align}

(5.14) \begin{align} \varPi _4^{\omega _0} &= -(\tilde {v} \tilde {v}^{\dagger } + \textrm{c.c.}) \frac {\partial V}{\partial y}. \end{align}

Figure 15(a) shows the different $\varPi _n^{\omega _0}$ terms in (5.11)–(5.14) integrated in the wall-normal direction. It is apparent from this figure that the term $\varPi _2^{\omega _0}$ (red line in figure 15 a) constitutes the main contribution to the total production (black line) across the entire domain. This term represents the extraction of energy from the base flow by the disturbance via the build-up of a positive Reynolds stress ( $\tau$ ) averaged over one TS period.

Figure 16.

(a) Phase difference between streamwise and wall-normal perturbation components, represented by $\cos {(\varphi ^{\tilde {u}} - \varphi ^{\tilde {v}})}$ . (b) Contours of $\xi$ , defined in (5.20). The black dash-dotted lines () indicate the slope of $\textrm{Re}(\psi )$ , represented as $(\partial y/\partial x )_{\textrm{Re}(\psi )}$ . This slope is measured at a fixed wall-normal position ( $y/\delta _0^* = 2.2$ ) but at different streamwise locations $(x^* = 35, 44)$ , as marked by the white dots. (c) Contours of production term, $\varPi _2^{\omega _0}$ , as defined in (5.12). Superimposed contour lines represent the real part of the perturbation streamfunction ( $-5.5\times 10^{-6}\leqslant \textrm{Re}(\psi )/(U_0\delta ^*_0) \leqslant 5.5\times 10^{-6}$ ) obtained at a specific TS phase, $\phi = 0$ . Negative values of $\textrm{Re}(\psi )$ are indicated with grey solid lines (), whereas positive values of $\textrm{Re}(\psi )$ are indicated with grey dashed lines (). All results are in step conditions and obtained from DNS.

Additionally, comparing figure 15(a) with figure 15(b) reveals that the evolution of the perturbation kinetic energy can be fully described by examining the production term, i.e. regions of negative production align with regions of perturbation energy decay and vice versa. This again reaffirms the linear behaviour of the perturbation field under the conditions concerned herein (i.e. sufficiently low TS wave amplitude).

Equation (5.12) can be alternatively expressed in terms of the perturbation phase and amplitude:

(5.15) \begin{equation} \varPi _2^{\omega _0} = -2|\tilde {u}||\tilde {v}|\cos {(\varphi ^{\tilde {u}} - \varphi ^{\tilde {v}})}\frac {\partial U}{\partial y}. \end{equation}

The previous equation provides insight into the role that the phase shift between $\tilde {u}$ and $\tilde {v}$ plays in regard to the change of sign of the Reynolds stress and thereby in production. The strong distortion observed from the phase contours of $\tilde {u}$ (refer to figure 8 b in § 4.1) in FFS conditions suggests the emergence of regions downstream of the step where $\tilde {u}$ and $\tilde {v}$ are out of quadrature (i.e. $\varphi ^{\tilde {u}} - \varphi ^{\tilde {v}} \neq \pm \pi /2$ ). Consequently, the term $\cos {(\varphi ^{\tilde {u}} - \varphi ^{\tilde {v}})}$ in (5.15) approaches values closer to $\pm 1$ , leading to either an enhancement or suppression of the Reynolds stresses and, consequently, of the production term $\varPi _2^{\omega _0}$ , as shown in figures 16(a) and 16(c).

To further expose the relationship between the build-up of Reynolds stresses and the topology of the TS wave structure, it is instructive to write the production term $\varPi _2^{\omega _0}$ in terms of the streamfunction, $\psi$ , previously introduced in (4.1). Following the work of Butler & Farrell (Reference Butler and Farrell1992), it can be shown that by introducing (4.1) into (5.12), $\varPi _2^{\omega _0}$ can be re-expressed as

(5.16) \begin{align} \varPi _2^{\omega _0} &= \left (\frac {\partial \tilde {\psi }}{\partial y} \left (\frac {\partial \tilde {\psi }}{\partial x}\right )^{\dagger } + \left (\frac {\partial \tilde {\psi }}{\partial y}\right )^{\dagger } \frac {\partial \tilde {\psi }}{\partial x}\right )\frac {\partial U}{\partial y}, \end{align}

(5.17) \begin{align} & = 2\textrm{Re}\left [\left (\frac {\partial \tilde {\psi }}{\partial y}\right )^{\dagger } \frac {\partial \tilde {\psi }}{\partial x}\right ]\frac {\partial U}{\partial y}, \end{align}

(5.18) \begin{align} & = 2\textrm{Re}\left [ \left (\frac {\partial \tilde {\psi }}{\partial y}\right )^{\dagger }\left (\frac {\partial \tilde {\psi }}{\partial y}\right )\left (\frac {\partial \tilde {\psi }/\partial x}{\partial \tilde {\psi }/\partial y}\right )\right ]\frac {\partial U}{\partial y}, \end{align}

(5.19) \begin{align} & = -2\textrm{Re}\left [|\tilde {u}|^2\left (\frac {\partial y}{\partial x}\right )_{\tilde {\psi }}\right ]\frac {\partial U}{\partial y}. \end{align}

Equation (5.19) indicates that, in a boundary layer where the mean flow shear is everywhere positive (i.e. $\partial U/\partial y\gt 0$ , like the case concerned herein), the tilting of the perturbation streamlines ( $(\partial y/\partial x)_{\tilde {\psi }}$ ) uniquely governs the shift of sign of the production term, and therefore, the growth or decay of the instability.

Figure 16(b) presents the contours of the product between the mean shear $\partial U/\partial y$ and the slope of the real part of the perturbation streamfunction, $ (\partial y/\partial x )_{\textrm{Re}(\tilde {\psi })}$ . This product is denoted as

(5.20) \begin{equation} \xi = \frac {\partial U}{\partial y}\left (\frac {\partial y}{\partial x}\right )_{\textrm{Re}(\tilde {\psi })}. \end{equation}

According to Butler & Farrell (Reference Butler and Farrell1992), a perturbation structure tilted in the mean flow shear direction corresponds to $\xi \gt 0$ , whereas a tilt against the mean shear direction is indicated by $\xi \lt 0$ . In this work, upstream and downstream tilting is also employed to refer to regions where $\xi \lt 0$ and $\xi \gt 0$ , respectively.

To illustrate the relationship between $\xi$ and the topology of the TS wave structure, the real part of the perturbation streamfunction $\textrm{Re}(\tilde {\psi })$ at a specific TS phase ( $\phi = 0$ ) is also displayed in figure 16(b). These results highlight how changes in the sign of $\xi$ correlate with variations in the tilting of the TS structures, represented by isolines of $\textrm{Re}(\tilde {\psi })$ . Similarly, the contours of the production component $\varPi ^{\omega _0}_2$ are shown in figure 16(c) superimposed with $\textrm{Re}(\tilde {\psi })$ . In this case, this visualisation helps to establish a link between the tilting of the TS structures and the shift in the sign of production, indicating either growth or decay.

Upstream of the step, $\xi$ becomes negative, indicating a tilting of the perturbation streamlines against the mean shear direction as a result of the strong base flow distortion exerted by the adverse pressure gradient. This upstream tilting ( $\xi \lt 0$ ) creates a positive production ( $\varPi _2^{\omega _0} \gt 0$ , (5.19)) that enhances the growth of the instability, as observed from figure 15(a). Upon reaching the step corner, the perturbation streamlines are strongly bent when they encounter the step edge. The distortion imposed by the step on the perturbation induces a locally negative production ( $\varPi _2^{\omega _0} \lt 0$ ) that yields a stabilising effect; see the contours of $\varPi _2^{\omega _0}$ in figure 16(c).

While the increase and decrease of production (i.e. build-up of positive and negative Reynolds stresses, respectively) in the step neighbourhood can be attributed to the strong distortion experienced by the perturbation streamlines due to the abrupt base flow modification, observations downstream of the step do not obey the same reasoning since the base flow is largely parallel there. Comparison between contours of $\xi$ in figure 16(b) and contours of $\varPi _2^{\omega _0}$ in figure 16(c) in the region between $20 \leqslant x^* \leqslant 60$ reveal a good correlation between the shift from negative to positive production values and the shift from downstream ( $\xi \gt 0$ ) to upstream ( $\xi \lt 0$ ) tilting. Based on previous works (Orr Reference Orr1907; Butler & Farrell Reference Butler and Farrell1992), the growth of disturbances as a result of their tilting against the shear can be associated to the Orr mechanism. This mechanism is known to be most effective for disturbances featuring finite streamwise wavenumbers, like TS waves (Hwang, Moin & Hack Reference Hwang, Moin and Hack2021).

The Orr mechanism consists of the amplification of both streamwise and wall-normal perturbation components due to an increase in the perturbation’s circulation as it changes orientation relative to the mean shear (Hwang et al. Reference Hwang, Moin and Hack2021). However in contrast to exponential growth, the Orr mechanism amplifies both normal and tangential perturbation components in an algebraic manner, i.e. in a non-modal fashion (Orr Reference Orr1907; Butler & Farrell Reference Butler and Farrell1992). This is in line with the decoupling observed between the growth rates of tangential- ( $\alpha _i^{\tilde {\upsilon }_t}$ ) and normal- ( $\alpha _i^{\tilde {\upsilon }_n}$ ) to-the-base-flow perturbation components in figure 12(b). Finally, a comment can be made on the role of the Orr mechanism in the specific conditions of FFS flows. Traditionally, the Orr mechanism is usually associated with the destabilisation of the perturbation field. An example of this is the work of Åkervik et al. (Reference Åkervik, Ehrenstein, Gallaire and Henningson2008), where the evolution of a wavepacket tilted against the mean shear leads to the linear amplification of the perturbation via the Orr mechanism, eventually developing into a TS wave. However, the present results unveil an unconventional role for the Orr mechanism, as it is observed to be responsible also for the local decay of the perturbation.