A series of experiments were conducted to understand the sources of local, high-amplitude velocity fluctuations produced at the late stages of boundary-layer flow transition to turbulence. The laboratory experiments considered the controlled injection of Tollmien–Schlichting (TS) waves into a nearly zero pressure gradient, laminar boundary layer, resulting in H-type transition to turbulence. Proper orthogonal decomposition (POD) was used to extract the energetic coherent structures within the transitional flow field obtained with particle image velocimetry. The first three modes were observed to feature spatial mode shapes consistent with a cross-section of a canonical hairpin vortex structure and were associated with time-dependent amplitudes having consistent peak frequencies with the fundamental TS wave frequency. Higher-order modes exhibited a combination of sub- and super-harmonics of the TS wave frequency and were attributed to flow interactions produced by a hairpin packet. A conditional averaging method was used to establish a reduced-order model for the overshoot phenomena in Reynolds shear stress and turbulence kinetic energy observed at the late transition stage. The lower portion of the large-scale hairpin vortex structure was observed to be primarily responsible for the overshoot mechanisms, which was well captured in a reduced-order model of the velocity field. The first four POD modes were used to create this reduced-order model, which, while only consisting of ≈15% of the total turbulence kinetic energy of the original velocity field, was able to capture ≈85% of the peak Reynolds stress amplitude across the overshoot region.

## I. INTRODUCTION

Laminar to turbulent boundary-layer transition has received considerable attention, both historically and during recent decades due to its fundamental relevance and implications in engineering processes, with example application areas ranging from heat transfer^{1,2} to skin-friction drag on airfoils.^{3,4} Studies of transitional flows have revealed the generation of distinct flow structures and their transport properties, which play an essential role in fully developed wall-bounded turbulence.^{5} Consequently, a quantitative assessment of transition to turbulence is central to further understanding fully developed wall turbulence phenomena.^{6}

The transition process of a laminar to turbulent flow can be triggered through a series of different mechanisms. For instance, the flow may undergo a so-called “bypass transition” when disturbed by a high-amplitude perturbation and becomes turbulent almost immediately. Conversely, the amplification of Tollmien–Schlichting (TS) instability waves in a laminar boundary layer exposed to a low level of freestream turbulence, *I*_{u} ≤ 1%, often results in the amplification of a secondary, spanwise instability, which is subsequently followed by a nonlinear breakdown process to produce a turbulent boundary layer.^{7} Ever since the importance of TS waves to the natural transition process was experimentally confirmed,^{8} the theory of receptivity in the transition process has gained popularity.^{9} Direct numerical simulations^{5,10} and experiments^{11,12} have contributed to revealing the mechanisms responsible for the laminar–turbulent transition process for various input disturbance types, frequencies, and amplitudes. See Reed and Saric,^{9} Saric *et al.*,^{13} and Reed *et al.*^{14} for reviews on receptivity and linear stability on wall-bounded shear flows. While the foundational successes in receptivity and linear stability theories have transformed modern understanding of transitional flow dynamics, these theories are applicable only to the early stage of the transition. Thus, the flow behavior associated with secondary breakdowns and its relation to fully developed turbulence remain unresolved. Due to the highly nonlinear nature of the late-stage process of transition to turbulence, the use of innovative approaches, including conditional statistical or coherent structure analysis, is required to obtain novel insight into the dynamics of transitional shear flows.

The classic statistical theory of turbulence^{15–17} tackles the late stage of transition toward turbulence by considering higher-order statistical quantities. Although these statistical models are valuable for many applications, success in describing the underlying physics that produce these models have not yet been achieved. On the other hand, the analysis of coherent structures, or fluid motions with temporal coherence,^{18} as elementary building blocks of turbulent flows quickly became popular after first proposed by Theodorsen,^{19} which was subsequently followed by the experimental work of Kline *et al*.^{20} Through the production and dissipation mechanism, these structures maintain turbulence within a boundary-layer flow. Experiments^{21–23} and numerical simulations^{5,10,24,25} have revealed that horseshoe or hairpin vortices are structures that play a crucial role in the transport of mass and momentum in wall-bounded transitional flows and significantly contribute to the generation of near-wall turbulence kinetic energy.^{26} For reviews on these topics refer to, e.g., Adrian^{18} and Dennis.^{27}

In this study, we focus on the late stage of the transition to turbulence and explore the overshoot phenomenon^{28} observed in the natural flat-plate boundary-layer transition process via a controlled injection of a TS wave instability. Particular attention was given to studying the structures responsible for the skin-friction overshoot often observed in the late stages of H- and K-type transitions.^{5,29} Proper orthogonal decomposition (POD) was used as a means of extracting a subset of contributions that describes an optimal lower-dimensional approximation of the original transitional velocity field. This technique has been widely applied in the past to extract coherent structures within various flows of interest.^{30–32} Through POD, a hierarchy of fluid structures form the basis of a low-order reconstruction, ranked by their respective energy contribution to the reconstruction of the original flow field. Taking advantage of the high-frame-rate particle image velocimetry (PIV) experiments acquired in the current study, the connection between the identified structures to the rapid rise in skin-fiction and near-wall Reynolds shear stress (*RSS*) is assessed by a phase-averaged, reduced-order model reconstruction. The paper is organized as follows: the experimental setup is described in Sec. II; the selection of TS wave generation parameters, basic statistics, POD conditional averaging analysis, and reduced-order modeling of the overshoot phenomenon are discussed in Sec. III; and the main conclusions are provided in Sec. IV.

## II. EXPERIMENTAL SETUP

The TS wave amplification and transition process of a flat-plate boundary layer was experimentally studied in a 2.5 m long, 112.5 mm wide, and 112.5 mm high recirculating, refractive-index matching (RIM) channel located in the Renewable Energy and Turbulent Environment Laboratory at the University of Illinois (Fig. 1). The upstream channel duct has a contraction section with an area ratio of 4.375:1, which induces a flow with negligible turbulence intensity. The boundary-layer profile without external disturbance at *Re*_{x} = 3 × 10^{5} is given in Fig. 2(a), which is observed to follow a Blasius profile. Aqueous sodium iodide solution of ∼63% by weight was used as the working fluid; its refractive index matched that of the flume walls, which significantly reduced the intensity of associated reflections in the vicinity of the wall surface. This aspect allowed for high accuracy velocity data to be resolved across the very-near wall locations in the PIV field of view (FOV). The fluid has a density of *ρ*_{0} = 1800 kg m^{−3} and a kinematic viscosity of *ν* = 1.1 × 10^{−6} m^{2} s^{−1}. For additional details on the facility, see Blois *et al.*;^{33} additional information on the refractive-index-matching technique can be found in Bai and Katz^{34} and Hamed *et al.*^{35}

A vibrating ribbon was used to induce Tollmien–Schlichting (TS) instability waves as a mechanism to trigger the laminar–turbulent transition process. This method has been widely used to produce TS waves since the original experimental verification of TS waves decades ago.^{8} A 355 *μ*m-thick copper ribbon was laminated by a layer of 25 *μ*m Kapton tape on each side to insulate the electric current passing through the ribbon from the fluid; see a basic schematic of the setup in Fig. 1. A verification test showed that, without the supply of current, the stationary ribbon presence did not significantly influence the boundary-layer profile; Blasius profiles were obtained with and without the stationary ribbon (see Fig. 2).

The ribbon's periodic sinusoidal motion was induced by a Lorentz force. The ribbon forcing was created by passing an alternating current at a prescribed frequency through the ribbon, which interacted with two rows of permanent magnets mounting under the channel bottom wall. The current was produced from a function generator and amplified, allowing for controllable variation of the ribbon motion frequency and amplitude. For the experiments described in Sec. III, the circuit oscillation frequency was set to 1.6 Hz and the current output of ±5.3 A. This resulted in a ≈6 mm maximum peak-to-peak vibration amplitude at the ribbon center. Additional details on the vibrating ribbon technique can be found in Boiko *et al.*,^{7} Gilev and Kozlov,^{36} and Klebanoff *et al.*^{37}

The ribbon's periodic vibration was tracked with a customized particle tracking velocimetry (PTV) system using a Mikrotron EoSens 4CXP MC4082 high-speed camera at 4 MP resolution and a Nikon AF Micro-Nikkor lens of 50 mm focal length and a focal ratio of f/2.8. Fiducial marks were added every 15 mm along the ribbon's top surface, which was illuminated with a Stanley Lithium–Ion Halogen Spotlight and tracked at 300 Hz over a 30 s sampling period. The trajectory of the ribbon at a given instant was obtained by linking the fiducial points using a piecewise polynomial interpolation. Figure 3 demonstrates the ribbon vibration trajectory throughout the spanwise direction, with *L* = 0.1125 m serving as the width of the channel.

Velocity fields of the flow transition process were acquired across a streamwise plane beginning 1.25 m downstream of the flume inlet using a high-speed planar PIV system from TSI. Two cameras were used to interrogate two 120 × 40 mm^{2} field of views (FOVs) with a slight overlap. The FOVs were located at the center of the RIM flume, immediately above the bottom wall. The FOVs were illuminated with a 1 mm thick laser sheet generated by 50 mJ pulses from a high speed, dual cavity Amplitude Terra PIV Nd:YLF laser. The working fluid was seeded with 14 *μ*m, silver-coated hollow glass spheres. Three sets of 1000 image pairs (3000 vector fields) were collected at an acquisition frequency of 100 Hz, using a pair of 2560 × 1600 pixels CMOS Phantom M340 cameras with 12 GB on-board memory. The image pairs were interrogated using a recursive cross-correlation method via the TSI Insight 4G software. The final interrogation window had a size of 24 × 24 pixels with 50% overlap, resulting in a vector grid spacing Δ*x* = Δ*y* = 420 *μ*m.

## III. RESULTS

### A. Parameter selection

Based on the location of the ribbon, the frequency of TS waves corresponding to a neutral stability condition were determined from linear stability theory,^{38,39} as shown in Fig. 4. For the experiments conducted in the current study, the freestream velocity of the flume corresponded to *U*_{∞} = 0.175 m s^{−1}, and the vibrating ribbon was placed *x* = 0.76 m downstream of the inlet. The working fluid was temperature-controlled such that constant density and viscosity were maintained, and a repeatable Reynolds number was achieved given a consistent freestream velocity and streamwise position. The excitation frequency and installation location of the vibrating ribbon corresponded to $F=2\pi f\nu /U\u221e2=1.8\xd710\u22124$ and $R=1.72Rex=598$. Linear stability theory also indicates that the TS wave excitation would occur at the resonance condition *ω*_{1} = 2*ω*_{1/2} for an H-type transition,^{40,41} where *ω*_{1} and *ω*_{1/2} are the frequencies of the two-dimensional TS and the oblique waves. This condition was obtained by placing a semi-elliptical roughness element with a diameter of *D* = 4 mm and a height of *h* = 4.5 mm at the flume center. The roughness element dimension was selected to have a *Re*_{h} ≡ *U*_{local}*h*/*ν* = 520, below the typical Reynolds number required for exciting bypass transition. Given the freestream velocity, the distance between the roughness element and the side walls (also acted as disturbance) was consistent with the wavenumber required to produce the associated *ω*_{1/2} condition for amplification for associated oblique waves of H-type transition, resulting in the generation of inherent spanwise sub-harmonics and a staggered pattern of coherent structures in the spanwise direction. This approach of introducing spanwise perturbations^{42,43} allowed the location of secondary breakdown mechanisms to be more effectively controlled.

The TS waves were measured across the late stages of the transition process, where exponential growth and nonlinear breakdown of initial disturbances occurred. The selected interrogation window covered a spatial domain where the onset of hairpin vortices and secondary breakdown were visible. The coordinate system is defined such that *x*, *y*, and *z* denote the streamwise, wall-normal, and spanwise directions, with *x* = 0 located at the beginning of the FOV, i.e., at 1.25 m downstream of the inlet.

### B. Time-averaged statistical analysis of the flow field

Basic features of the perturbed boundary layer are illustrated in Fig. 5. The mean streamwise velocity field shows a typical developing boundary layer; however, higher-order statistics, such as turbulence kinetic energy [$TKE=12(u\u20322\xaf+v\u20322\xaf)$] and Reynolds shear stress ($RSS=\u2212u\u2032v\u2032\xaf$), exhibit a region of increased levels across the region from *x* = 0.03 to *x* = 0.085 [see Figs. 5(b) and 5(c)]. Selected vertical profiles of these quantities using inner units, i.e., *y*^{+} ≡ *y*/*u*_{τ} and *u′*^{+} ≡ *u*′/*u*_{τ}, where $u\tau =\tau w/\rho $ is the local friction velocity, and *τ*_{w} and *ρ* are the local wall shear stress and fluid density, are shown in Figs. 6(a), 6(c), and 6(d) to further illustrate their evolution across the laminar-turbulent transitional region. Due to the inherent trade-off between FOV size and spatial resolution used in the current experiment, it was not feasible to include the unperturbed laminar boundary layer in the spatial domain of the time-resolved vector fields. As a result, a slight departure from the canonical laminar flow velocity profile is already observed at *x* = 0.02 m (*Re*_{x} = 2.02 × 10^{5}). The peak of the *RSS* and *TKE* (*y*^{+} ≈ 25–30) increased with streamwise distance, reaching maximum values at *x* ≈ 0.08 m (*Re*_{x} = 2.1 × 10^{5}). Further downstream, a decrease in the peak *RSS* and *TKE* is observed, which converged to classical turbulent correlation profiles across the downstream end of the PIV field of view.

The presence of a local overshoot region in the mean velocity and turbulent statistics near the wall observed in Fig. 6 is consistent with the skin-friction overshoot (see Figs. 9 and 10 in Ref. 5) produced during the late stages of boundary transition observed in previous Direct Numerical Simulation (DNS) and experimental literature.^{10,44–46} According to Sayadi *et al.*,^{5} the initial departure from the laminar regime occurs with the formation of Λ vortices inherent to the natural boundary-layer transition process. Figures 5 and 6 suggest the presence of coherent structures that provide a dominant contribution to the turbulent fluctuations throughout the transition process, as observed through locally increased values of *RSS* and *TKE*.^{18,47} After the initial formation near the wall, these vortices moved away from the wall with increased streamwise distance, creating the characteristic Λ vortex shape around *x* = 0.015 m (*Re*_{x} = 2 × 10^{5}).

### C. Proper orthogonal decomposition

Proper orthogonal decomposition (POD) is a common modal analysis technique that is used to extract the dominant spatial features produced in a given flow field; it extracts modes that optimizes the mean square value of the flow field variable of interest by utilizing singular value decomposition (SVD).^{64} This technique has proven to be extremely useful in studying the flow characteristics effected by coherent structures in transitional and spatially evolving flows.^{6} For more information on POD and applications, see, e.g., Berkooz *et al.*^{52} and Holmes *et al.*^{32} In this paper, the snapshot POD as introduced by Sirovich^{48} is chosen among various POD algorithms. Here, the two-dimensional, two-component POD decomposes the stochastic velocity fluctuation matrix $u\u2032\xaf(x,y,t)=[u\u2032v\u2032]T$ into deterministic spatial-correlated patterns *ϕ*^{n}(*x*, *y*) (POD modes)^{49} and their time-dependent coefficients *a*^{n}(*t*), as follows:

where *N* = 3000 is the number of snapshots and *u*′ is the velocity fluctuation vector matrix. Also, POD returns modes ranked by their energy content, *E*_{n}, after decomposing the velocity fluctuations' auto-covariance matrix. The energy content of a mode is obtained by dividing the eigenvalue of a particular mode by the sum of all *N* eigenvalues,^{50} i.e., $En=\lambda n/\u2211m=1N\lambda m,$ and can be interpreted as the single-mode contribution to the total turbulence kinetic energy. More information regarding implementation details of POD can be found in Meyer *et al.*^{51} and Berkooz *et al.*^{52} The energy contribution of the first 40 modes obtained through POD analysis of the time-resolved velocity vector fields is shown in Fig. 7. The first mode contributes about 6% of the total turbulence kinetic energy, with relatively slow convergence observed in the mode energy spectrum. The first ≈35 modes contribute ≈50% of the total *TKE*, demonstrating a rather complex flow produced by the transition process. More details on the mode shape, amplitude, and their relationship are discussed below.

The spatial mode shapes associated with the first six streamwise POD modes *ϕ*_{u′} are shown in Fig. 8. These mode shapes demonstrate structural features consistent with those of horseshoe or hairpin vortices, which have been widely recognized in wall-bounded flows after first proposed by Theodorsen;^{19,53} however, further arguments relating these modes to the proposed flow features are given in Sec. III D.

### D. Coherent structures

The POD modes shown in Fig. 8 were inspected to determine their relation to characteristics of known coherent structures, beginning with the inclination angle, *β*, of spatial features observed in the POD modes; this is defined as the angle between the structure center line and the horizontal plane.^{54} The velocity fluctuations attributed to mode 1 exhibited an inclination angle, *β*_{1} ≈ 7°, across nearly the entire spatial domain interrogated [Fig. 8(a)]. Similarly, the velocity fluctuations attributed to modes 2 and 3 exhibited a similar inclination angle *β*_{1} away from the wall, which upon reaching *y* = 0.01 m transitions into an upper section with an inclination angle, *β*_{2} ≈ 25°, between *y* = 0.01 m and *y* ≈ 0.02 m [Figs. 8(b) and 8(c)]. Distinct angles, *γ* ≈ 15° and ≈14°, for the vortex packet^{55} are noted in modes 5 and 6 [Figs. 8(e) and 8(f)].

In addition to the aforementioned spatial features, the time-dependent amplitudes, *a*^{n}(*t*), of the POD modes provide distinct characteristics consistent with known coherent structures; see Figs. 9(a) and 9(b). The first three modes are associated with a single sinusoidal-like cycle at a consistent frequency, suggesting a correlation to a common flow feature between these primary modes. This commonality is contrasted with the higher modes, where multi-scale flow structures are expected due to nonlinear interactions, such as sub-harmonic and super-harmonic excitation. This observation is further supported through a Fourier analysis of the mode amplitudes, shown in Figs. 9(c) and 9(d). These power spectra indicate a dominant peak related to the fundamental TS wave frequency for the first three modes at 1.6 Hz, with a much smaller peak at its sub-harmonic frequency. Conversely, modes 5 and 6 exhibit spectral peaks at the sub- and super-harmonics of the excitation frequency. These higher mode amplitudes cover a wider frequency range across the spectral domain; they may be associated with many vortical structures. For even higher modes (not shown here for brevity), the spectra were associated with even smaller scales and associated with smaller structures, likely due to nonlinear interactions associated with the breakdown process. More detail on the linkage of these POD modes with flow features is discussed later.

The observations indicate that the first three modes are associated with the development of a single primary hairpin vortex (PHV).^{47} A distinct pattern in mode 4 appears to follow a similar scale as the PHV development observed across the first three modes. Still, this mode indicates a reversal in the perturbation velocity direction of the mode shapes across the hairpin head region, which might be attributed to the difference of convection speed between the near wall and the outer region. However, a clear distinction of the flow scales observed in modes 5 and 6 to the first four modes can be seen in Fig. 8. Instead of a single hairpin vortex, the higher modes (≥5) are conjectured to be associated with a hairpin packet, with the nesting angle *γ* in agreement with literature (12° ≤ *γ* ≤ 20° for hairpin vortex packets^{27,55–57}). The hairpin vortex packet is known to form once the initial PHV amplitude exceeds a certain threshold.^{18} These hairpin packets consist of self-similar secondary and tertiary hairpin vortices that form downstream of the PHV as a result of nonlinear interactions produced across the later stage of transition.^{18,56}

To further correlate the underlying spatial characteristics of the modes produced by the current POD analysis with flow interactions produced by coherent structures, those produced by modes 1 and 2 are shown in Figs. 10(a) and 10(b) with a superimposed distribution of velocity vectors. It should be noted that the time-dependent mode amplitudes can be positive or negative, so the specific directionality of the mode influence on the dimensional velocity can be either in the direction indicated or in reverse, relative to the vectors shown. The vector field corresponding to the first mode presents a high-amplitude ejection event (*u*′ < 0 and *v*′ > 0), which is caused by the interaction between the near-wall, low-speed flow that is pumped away from the wall, and the relatively high-speed, freestream flow.^{58} Given that the mode amplitude was predominantly positive, as shown in Fig. 9, this ejection behavior was observed in mode 1 far more frequently than related sweep events. Conversely, the second POD mode takes the form of a coupled sweep and ejection events, showing the reversed streamwise directionality between the upstream and the downstream flows.

The aforementioned observations suggest that mode 1 captures the quasi-streamwise vortex from the PHV (or the upstream portion of the hairpin leg) as both its inclination angle, *β*, and strong ejection behavior agree well with the literature. Zhou *et al.*^{47} reported hairpin angles ≈8° and ≈25° for the upstream and downstream sections of the PHV from DNS, and Clar and Markland^{59} experimentally identified counter-rotating streamwise vortices inclined at an angle of ≈5° to 7° in the wall region 7 ≤ *y*^{+} ≤ 70. Using dye visualizations of hairpin vortex structures, Haidari and Smith^{60} reported 6° ≤ *β* ≤ 15° for the upstream portion of the legs and 14° ≤ *β* ≤ 32° for the downstream portion of hairpin legs. Further details and descriptions of spatial characteristics of these PHV structures can be found in Smith,^{61} Miyake *et al.*,^{62} and Bernard and Wallace.^{63} In comparison to mode 1, modes 2 and 3 resemble a complete single primary hairpin vortex, possessing both upstream and downstream portions. Additionally, a “vortex tongue” is highlighted inside the dashed box in Fig. 8(c). Dennis^{27} pointed out that vortices located close to wall surfaces often feature an extra element, such as the hairpin vortex tongue observed here, which is also clearly captured in Fig. 8 of Haidari and Smith.^{60}

Phase diagrams showing the related variation of distributed amplitudes across pairs of modes are shown in Fig. 11 for the first four modes. Figure 11(a) demonstrates a lack of correlation between modes 1 and 2, with a bias toward positive values of *a*^{1}. In comparison, the mode amplitude pairs formed by modes 2 and 3 exhibit a clear *π*/2 phase offset between them as the modal amplitudes trace the circumference of a unit circle, as indicated by Fig. 11(b). The inspection of mode 4 suggests a relationship to modes 1 and mode 2, which is consistent with the similarity in flow scales previously observed in Fig. 8; however, the relation is not very clear from the planar velocity measurements of the current study.

### E. Reduced-order modeling

The similarity in the spatial topology of the first few mode shapes observed in Fig. 8 with the *TKE* distribution in Figs. 5(b) and 5(c) suggests the possibility of using the POD modes to generate a reduced-order model of the transitional velocity field. The generation of this model may reveal the dominant dynamics associated with the skin-friction overshoot phenomena observed during the boundary-layer transition. To provide evidence supporting this concept, Fig. 12 shows a subset of instantaneous snapshots corresponding to the increase, maximum, and the decrease in the *RSS* profile computed from the raw data that contain all POD modes during an overshoot event at a fixed location *x* = 0.08 m, alongside the instantaneous mode coefficients. Modes 1 and 3 provided the dominant energy contribution when these snapshots were extracted, suggesting that these two modes may serve as the main contributor to the overshoot.

A conditional sampling method is used to provide an averaged representation of the velocity fluctuations associated with the instantaneous, large-amplitude excursions in the *RSS*. The detection criterion to identify *RSS* excursions was developed based on an identification of the local maximum of a time series, given by

obtained at *x* = 0.08 m [*Re*_{x} = 2.1 × 10^{5}, see Fig. 6(c)]. An average advection velocity of 0.4*U*_{∞} (0.07 m s^{−1}) was also assumed for the near-wall coherent structures,^{65} in which, coupled to the time-resolved acquisition rate of the PIV acquisition system of with Δ*t* = 10 ms, the coherent structures are then expected to advect about a distance equivalent to ≈6% of the total FOV size between two consecutive vector field samples. To observe the influence of the average flow dynamics associated with the local excursions in *RSS*, the vector fields were conditionally sampled to include time intervals of −6Δ*t* to +6Δ*t*, allowing the coherent structure to travel ≈80% distance of our FOV size, centered about the identified maximum in *RS*(*t*_{0}). This conditional averaging was applied onto the acquired velocity fluctuation fields as well as POD mode amplitudes ($atic$), as a function of non-dimensional time $T=t\u2212t0/\Delta t$ in Figs. 13 and 14.

Figure 13 shows a local low-speed region, similar to the shape of first three modes, passing through *x* = 0.08 m around *T* = 0. This observation further suggests the fundamental role of these primary POD mode shapes in the overshot mechanism. As further support of this argument, the contributions of each mode to the local excursion in *RSS* are indicated in Fig. 14. The coefficients of the first and second modes, i.e., $a1c$ and $a2c$, undergo a rapid increase in magnitude and reach a maximum at *T* = 0. A similar process occurs in $a3c$ and $a4c$, though with a local maximum in magnitude occurring at *T* = 1 and *T* = −1 for these two modes. Note that the variation in the amplitudes of modes 2 and 3 exhibit a similar trend but are associated with opposing sign since they are related to the same structure with a *π*/2 phase shift, as discussed previously in relation to Fig. 11. The conditionally averaged amplitudes for all higher modes are small across the conditional sampling period of −6Δ*t* to +6Δ*t*, with much smaller standard deviations than modes 1 through 4. The conditional POD amplitude is only shown up to the sixth mode in Fig. 14 for brevity. Figure 14 provides further evidence that the first four modes, which were observed to be associated with a lower frequency and are related to a single hairpin vortex created by the fundamental TS wave excitation, indeed dominate the overshoot mechanism and are mostly responsible for the *RSS* and *TKE* generation near the wall. Conversely, other coherent structures created by nonlinear interaction, or the generation of a hairpin packet, did not have a significant role in dictating the local excursion observed in *RSS*. A similar conclusion had been shown in Sayadi *et al.*,^{10} where dynamic mode decomposition (DMD) was applied on DNS data of a transitional boundary-layer and showed that inclusion of the first three lowest frequency DMD modes was enough to reproduce the correct level of *RSS* gradient.

Given the importance of the first four POD modes in dictating the *RSS* and *TKE* overshoot during the transition process, a reconstruction of the velocity field using this reduced-order model, where we reconstruct the flow field using only the first few orthogonal basis from POD modes to capture the overshoot phenomena, was produced using this subset of modes. Such a reconstruction in the boundary-layer profile utilizing the original mode shapes and the conditionally averaged coefficients from Fig. 14 is given by

where *n* ranges from −6 to +6. Several conditionally averaged velocity fields from the reconstruction and the corresponding velocity profiles produced at *x* = 0.08 m are illustrated in Fig. 15. The convection of a single hairpin is captured successfully, such that high-frequency structures produced across *y* ≥ 0.015 m (*y*^{+} ≥ 35) are effectively filtered out of the vector field reconstructions. Figure 15(d) illustrates how the ejection events pump the low-speed fluid upward and decelerate the fluid near *y* = 0.01 m (*y*^{+} ≈ 25 – 30) significantly as the lower portion of the hairpin vortex (*y* ≤ 0.01 m) passed by. A similar process had been observed in Kline *et al.*,^{20} where the ejection phase of the low-speed streak is associated with a violent burst and breakup of the flow into small-scale structures. Figure 15(d) shows the velocity profile generated from the first four modes, i.e., the lower portion of the hairpin vortex, which is the main source of the burst; the variations observed in this velocity profile thus provide a representation of the influence these low-speed streaks have in relation to the skin-friction overshoot and the increase in the near-wall *RSS*.

The time-averaged *RSS* profile of the reduced-order model is given in Fig. 16(a); it exhibits a peak at *y*^{+} ≈ 25, proving the model's capability of capturing the overshoot phenomenon near the wall. It should be noted that this model-based prediction captures the peak Reynolds stress with ≈83% accuracy, though the first four modes only contained 16% of the total *TKE* of the vector field. Another *RSS* profile produced using the remaining higher modes is shown in Fig. 16(b) for comparison; the vanishing near-wall peak suggests that the first four low-frequency modes related to the large-scale coherent structures are the main contributors of near-wall Reynolds stress, while the high-frequency modes are linked to smaller-scale structures created at *y*^{+} ≈ 60 as shown in Fig. 16(b).

## IV. DISCUSSION AND CONCLUSIONS

Specially designed experiments were conducted on a canonical H-type boundary-layer transition velocity field, where TS waves were induced with a vibrating ribbon. Velocity field measurements of the TS wave growth and initial stages of the nonlinear breakdown were acquired using high-frame-rate PIV. Proper orthogonal decomposition successfully identified the main coherent structures produced across the wall-bounded transitional flow. The first four energetic modes were related to the leg portion of a hairpin vortex, whereas higher modes were likely created due to the formation of hairpin vortex packets due to the nonlinear growth and breakdown of the primary hairpin vortex. A conditional averaging method was used to identify large overshoots in the amplitudes of the first four POD modes around the instance of local excursions in the Reynolds shear stress; this led to the suggested use of a reduced-order model to characterize the overshoot phenomenon. The reconstruction produced by the reduced-order model shows the first four POD modes with ≈15% of the total flow field *TKE* predict the peak value of the Reynolds shear stress profile (≈85% of the overall amplitude), demonstrating the reduced-order model's ability to reproduce relevant flow dynamics of the overshoot process.

## ACKNOWLEDGMENTS

This work was supported by the Army Research Office, Contract No. W911NF-18-1-0030, under the Fluid Dynamics Program monitored by Dr. Matthew Munson.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.