2.1 Nonlinear parabolized stability equation

The NPSE are suitable for investigating both linear and weakly nonlinear evolutions of disturbances. The detailed formulation of the NPSE can be found in Bertolotti, Herbert & Spalart (Reference Bertolotti, Herbert and Spalart1992) and Chang & Malik (Reference Chang and Malik1994). Thus, only a brief introduction is given here. In the NPSE formulation, the disturbance is decomposed into a rapidly varying wave-like part and a slowly varying shape function as follows:

(2.1)$$\begin{eqnarray}\unicode[STIX]{x1D6F9}(x,y,\unicode[STIX]{x1D6FD}_{z},t)=\bar{\unicode[STIX]{x1D6F9}}(x,y)+\mathop{\sum }_{m,n}\hat{\unicode[STIX]{x1D6F9}}_{mn}(x,y)\exp \left(\text{i}\int _{0}^{x}\unicode[STIX]{x1D6FC}_{mn}\,\text{d}x-\text{i}\unicode[STIX]{x1D714}_{m}t+\text{i}n\unicode[STIX]{x1D6FD}_{z}\right),\end{eqnarray}$$

where $\unicode[STIX]{x1D6F9}\equiv (u,v,w,p)$; $\bar{\unicode[STIX]{x1D6F9}}$ denotes the basic state; $\hat{\unicode[STIX]{x1D6F9}}_{mn}\equiv (\hat{u} ,\hat{v},{\hat{w}},\hat{p})_{mn}$ are the shape functions for the Fourier mode ($m,n$); $\unicode[STIX]{x1D6FD}_{z}$ denotes the spanwise wave number; and $\unicode[STIX]{x1D6FC}_{mn}$ and $\unicode[STIX]{x1D714}_{m}$ are the associated streamwise (complex) wavenumber and frequency. The negative imaginary part of $\unicode[STIX]{x1D6FC}_{mn}$, $-\unicode[STIX]{x1D6FC}_{i}$, represents the growth rate of the mode, while the real part is related to the phase velocity as

(2.2)$$\begin{eqnarray}c_{mn}\equiv \frac{\unicode[STIX]{x1D714}_{m}}{\text{Re}(\unicode[STIX]{x1D6FC}_{mn})},\end{eqnarray}$$

where $\text{Re}$ denotes the real part. All the variables are non-dimensionalized using the free-stream velocity $U_{\infty }$ and the fixed characteristic thickness $\unicode[STIX]{x1D6FF}_{0}=\unicode[STIX]{x1D6FF}(x_{0})=(\unicode[STIX]{x1D708}\tilde{x}_{0}/U_{\infty })^{1/2}$, where $\tilde{x}_{0}$ is the initiation location where disturbances are forced, and $\unicode[STIX]{x1D708}$ is the kinematic viscosity of the fluid. Note that $x_{0}=\tilde{x}_{0}/\unicode[STIX]{x1D6FF}_{0}=U_{\infty }\unicode[STIX]{x1D6FF}_{0}/\unicode[STIX]{x1D708}=R_{0}$ and $R=(x/x_{0})^{1/2}R_{0}$, where $R_{0}$ and $R$ are Reynolds numbers based on $\unicode[STIX]{x1D6FF}_{0}$ and $\unicode[STIX]{x1D6FF}(x)$, respectively.

The mode shape, $\hat{\unicode[STIX]{x1D6F9}}_{mn}(x,y)$, is assumed to be a slowly varying function of the streamwise coordinate, $x$, so that its second derivative with respect to $x$ is negligible. Substituting (2.1) into the Navier–Stokes equation and subtracting the mean-flow equation yields the governing equations for the shape functions of each mode $(m,n)$ as

(2.3)$$\begin{eqnarray}({\mathcal{L}}_{0}+{\mathcal{L}}_{1})\hat{\unicode[STIX]{x1D6F9}}_{mn}+{\mathcal{L}}_{2}\frac{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D6F9}}_{mn}}{\unicode[STIX]{x2202}x}+\frac{\text{d}\unicode[STIX]{x1D6FC}_{mn}}{\text{d}x}{\mathcal{L}}_{3}\hat{\unicode[STIX]{x1D6F9}}_{mn}={\mathcal{F}}_{mn},\end{eqnarray}$$

where ${\mathcal{L}}_{0}$ contains the linear–parallel terms, ${\mathcal{L}}_{1}$ contains the non-parallel mean-flow terms, ${\mathcal{L}}_{2}$ and ${\mathcal{L}}_{3}$ arise from the non-parallel disturbance terms and ${\mathcal{F}}_{mn}$ is the nonlinear forcing term. For expression of the operators ${\mathcal{L}}_{0}$, ${\mathcal{L}}_{1}$, ${\mathcal{L}}_{2}$, ${\mathcal{L}}_{3}$ and ${\mathcal{F}}_{mn}$ refer to Guo & Tang (Reference Guo and Tang2010) and Zhu et al. (Reference Zhu, Lee, Chen, Wu, Chen and Gad-el Hak2018).

This system of equations is parabolic and thus requires boundary and initial conditions. The initial conditions are provided by the results from the linear stability theory (Chen, Zhu & Lee Reference Chen, Zhu and Lee2017). The boundary conditions are

(2.4)$$\begin{eqnarray}\displaystyle \hat{u} _{mn},\hat{v}_{mn},{\hat{w}}_{mn}=0,\quad \text{at}~y=0~\text{and}~\infty , & & \displaystyle\end{eqnarray}$$

which correspond to the no-slip boundary condition at the wall, and vanishing velocity disturbances as $y\rightarrow \infty$. However, for the mean-flow distortion, $\hat{\unicode[STIX]{x1D6F9}}_{00}$, the infinity condition for the wall-normal velocity is replaced by

(2.5)$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\hat{v}_{00}}{\unicode[STIX]{x2202}y}=0,\quad \text{at}~y=\infty ,\end{eqnarray}$$

which accounts for the boundary-layer growth. Provided initial and boundary conditions, equation (2.3) was solved by marching in the streamwise direction using an Euler backward scheme. At each marching step, the mode function as well as other stability characteristics of the mode (including growth rates, phase velocities, etc.) were obtained.

There is still ambiguity in the $x$-dependence between $\hat{\unicode[STIX]{x1D6F9}}_{mn}$ and $\unicode[STIX]{x1D6FC}_{mn}$. In this study, the wavenumber at each station was updated as

(2.6)$$\begin{eqnarray}\unicode[STIX]{x1D6FC}^{new}=\unicode[STIX]{x1D6FC}^{old}-\text{i}\frac{1}{E}\int _{0}^{\infty }\left(\hat{u} ^{\ast }\frac{\unicode[STIX]{x2202}\hat{u} }{\unicode[STIX]{x2202}x}+\hat{v}^{\ast }\frac{\unicode[STIX]{x2202}\hat{v}}{\unicode[STIX]{x2202}x}+{\hat{w}}^{\ast }\frac{\unicode[STIX]{x2202}{\hat{w}}}{\unicode[STIX]{x2202}x}\right)\text{d}y,\end{eqnarray}$$

where

(2.7)$$\begin{eqnarray}E\equiv \int _{0}^{\infty }(|\hat{u} |^{2}+|\hat{v}|^{2}+|{\hat{w}}|^{2})\,\text{d}y,\end{eqnarray}$$

with the asterisk denoting the complex conjugate. This updating process is continued until the largest change is less than $10^{-5}$.

2.2 Numerical setting and mode description

In the present study, in the language of frequency–wavenumber, the notation ($\unicode[STIX]{x1D714}$, $\unicode[STIX]{x1D6FD}$) is used, where $\unicode[STIX]{x1D714}$ and $\unicode[STIX]{x1D6FD}$ denote the streamwise frequency and spanwise wavenumber, respectively. For simplicity, $\unicode[STIX]{x1D714}$ and $\unicode[STIX]{x1D6FD}$ are normalized with the corresponding values of the primary waves. For K-regime transition, a 2-D T–S wave of normalized frequency $F=\unicode[STIX]{x1D714}/R_{0}=106\times 10^{-6}$, and a pair of oblique waves at the same frequency, are introduced at the initiation point ($R_{0}=400$), i.e. modes (1, 0) and (1, $\pm 1$). Therefore, the corresponding non-dimensional period is $T=2\unicode[STIX]{x03C0}/(R_{0}F)=148.2$, and the streamwise wavelength $\unicode[STIX]{x1D706}_{x}\approx 50$. The spanwise wavenumber $\unicode[STIX]{x1D6FD}$ is chosen as 0.05 to make the spanwise wavelength $\unicode[STIX]{x1D706}_{z}\sim O(\unicode[STIX]{x1D706}_{x})$, i.e. $\unicode[STIX]{x1D706}_{z}\approx 126$. The amplitude ${\mathcal{A}}$ for the primary 2-D wave is $0.2\,\%\,U_{\infty }$, and $0.14\,\%\,U_{\infty }$ for the oblique waves. For N-regime transition, the same 2-D T–S wave as employed for the K-regime transition is forced with a pair of oblique waves at half the frequency, i.e. (1, 0) and (0.5, $\pm 1$). For O-regime transition, only a pair of oblique waves are introduced, namely (1, $\pm 1$). The detailed information for the initial forcing wave components applied at the initiation location for the K, N and O transition scenarios is shown in table 1.

Table 1.

Numerical settings for all three simulations. A normalized notation ($\unicode[STIX]{x1D714},\unicode[STIX]{x1D6FD}$) is used in mode description. The first digit indicates multiples of the fundamental streamwise frequency and the second digit refers to multiples of the fundamental spanwise wavenumber.

2.3 Method of Lagrangian tracking

Three-dimensional data sets of the spatial-temporal transition process are obtained from the solution to the NPSE and also from the tomo-PIV measurements. These data sets are used to reconstruct flow structures by employing Lagrangian tracking of marked particles. So, it is helpful to revisit the concepts of the pathlines, streaklines and timelines.

A particle pathline shows the sequence of particle locations over time as a particle moves through a fluid medium. It is determined by integrating an ordinary differential equation over real time

(2.8)$$\begin{eqnarray}\boldsymbol{V}(\boldsymbol{X}(t),t)=\text{d}\boldsymbol{X}(t)/\text{d}t.\end{eqnarray}$$

Determining how a vector field changes with time from pathlines is possible, if the vector field is not too complicated. However, the information provided by a pathline is limited, for it only shows the positional history for a specific particle. On the other hand, streamlines provide no information on the history of a vector field, and only reflect the instantaneous velocity vector field. A compromise between particle pathlines and streamlines is a streakline, which provides a more comprehensive and intuitive physical view, like a dye stream in water. A streakline is comprised of a series of fluid elements that pass through, or are released from, a fixed location repeatedly and in succession. The position of each particle at a specific time step can be calculated from (2.8). Timelines are lines connecting a series of particles released from different locations at the same time. The periodic generation of a series of timelines, such as is done by pulsing the current through a hydrogen bubble wire at a constant frequency, creates a pattern of timelines, which yield what can be considered time-streak markers of the fluid behaviour.

In the present study, fluid trajectories are extracted from the three-dimensional data sets, based on the integration of (2.8). This is achieved using a MATLAB algorithm that employs an ordinary differential equation (ODE) solver and interpolation scheme. The Dormand–Prince method (ode45 in MATLAB) is chosen as the solver for trajectory advection, and the absolute and relative tolerances of the ODE solver were set at $10^{-6}$. Equation (2.8) is integrated over a time interval $[t_{0},t_{1}]$, for a specified initial condition. The initial condition can be either a specific point, straight line, curve, surface or a grid volume. In order to generate timelines similar to hydrogen bubble timelines, the initial condition is set as a fixed horizontal or vertical line, and the integration is implemented iteratively with an updating release time. Timeline patterns are formed by determining, and displaying, a sequence of timelines generated at a fixed frequency. Emulating the term of ‘hot-wire visualization’ introduced by Chernoray et al. (Reference Chernoray, Kozlov, Löfdahl and Chun2006), the patterns extracted from the NPSE and tomo-PIV data bases using this integration process are referred to as ‘NPSE visualization’ and ‘tomographic visualizations’, respectively.

Note that the result of flow visualizations should be interpreted with caution, as mentioned in Acarlar & Smith (Reference Acarlar and Smith1987a), ‘… the visible patterns are not the structures themselves, but their “footprints”’. Many of the previous studies of wall-bounded flow behaviours were based on technique of manual or automated image processing of pictures of hydrogen bubble timelines taken using a single-view or dual-view system (Kim, Kline & Reynolds Reference Kim, Kline and Reynolds1971; Cornelius, Takeuchi & Deutsch Reference Cornelius, Takeuchi, Deutsch, Patterson and Zakin1979; Smith & Paxson Reference Smith and Paxson1983; Lu & Smith Reference Lu and Smith1985; Acarlar & Smith Reference Acarlar and Smith1987a). These approaches proved effective and useful for assessing the spatial-temporal characteristics of boundary layers, and detection of coherent flow structures. However, acquisition of sequences of instantaneous, quantitative, three-dimensional velocity data is not feasible using hydrogen bubble visualization. The solution of NPSE and time-resolved tomo-PIV can obtain such types of spatial-temporal velocity field data, making the detection of three-dimensional, unsteady velocity vector fields (even vorticity fields) possible. Such spatial-temporal data can then be interrogated, and timelines can be generated for comparison with hydrogen bubble images, and for assessing and identifying three-dimensional flow.

3.1 Basic features of three transition scenarios

The downstream development of the averaged skin friction coefficient, $C_{f}$, for the three transition regimes is shown in figure 1(a). The skin friction coefficient is calculated as

(3.1)$$\begin{eqnarray}C_{f}=\frac{2\bar{\unicode[STIX]{x1D707}}(\unicode[STIX]{x2202}\bar{u}/\unicode[STIX]{x2202}y)|_{y=0}}{R}.\end{eqnarray}$$

All the skin friction curves end at $R=728$, where the NPSE becomes invalid. This position corresponds to $x=(R/R_{0})^{2}x_{0}=1325$, according to the normalization in § 2.1. The skin friction coefficient curves for the K- and N-regimes initially follow the laminar curve up to $R\approx 690$ (corresponding to $x\approx 1190$), then the curve for the K-regime increases slightly toward the end, while the curve for the N-regime rises steeply. The skin friction behaviour for the O-regime follows the laminar curve up to $R\approx 580$ ($x\approx 841$), then increases slightly up to $R\approx 660$ ($x\approx 1090$), where it begins to slowly decay. Beyond $R\approx 708$ ($x\approx 1253$), the curves rise again but more steeply than before. The initial departure of the skin friction coefficient from the laminar value, well before the final transition, is considered as the beginning of nonlinearity (Schmid & Henningson Reference Schmid and Henningson2001). These early nonlinear regions are what we focus in the preset study.

Figure 1.

Skin friction coefficient $C_{f}$ for the three transition regimes (a) and mode amplitude evolution for K-regime transition (b) O-regime transition (c) and N-regime transition (d). The arrows in (b,c) indicate a slight declination of the primary wave. The numbers in brackets denote normalized frequency–wavenumber as described in § 2.2 (table 1). MFD in (c) means mean-flow distortion.

As mentioned previously, Fourier mode decomposition is a basic method for investigating the transition behaviour. Thus, before presenting the structure-based results, it is helpful to examine the evolution of Fourier modes. Figure 1(c,d) shows the development of the $y$-maxima of the selected Fourier amplitudes in the streamwise direction, for the K-, O- and N-regimes, respectively. The same language of normalized frequency–wavenumber as described in § 2.2 (table 1) is used here. Since the flow is symmetric for ($\unicode[STIX]{x1D714},\unicode[STIX]{x1D6FD}$) and ($\unicode[STIX]{x1D714},-\unicode[STIX]{x1D6FD}$), only the positive part is shown. For K-regime transition, figure 1(b), we determined that the transition is initiated by the growth of the amplitude of oblique modes (1, 1) and streak modes (0, 1), which successively surpass the amplitude of the fundamental disturbance (1, 0) at approximately $R=680$. Note that this is the location where the skin friction begins to deviate from the laminar curve in figure 1(a). The fundamental resonance between modes (1, 0), (1, 1) and (0, 1) has been considered as the main cause for initiating K-regime breakdown (Herbert Reference Herbert1988).

Figure 1(c) illustrates the evolution of four primary modes for O-regime transition. The modes (0, 2), (0, 4) and mean-flow distortion (MFD) mode (0, 0) are excited nonlinearly due to the interaction of modes (1, $\pm 1$). After generation, the streak mode (0, 2) undergoes a transient growth stage due to a lift-up effect (Berlin, Lundbladh & Henningson Reference Berlin, Lundbladh and Henningson1994), and then becomes saturated for $R>650$. Mode (0, 4) also grows rapidly and becomes saturated driven by mode (0, 2) until $R\approx 670$, where it again begins to amplify, eventually surpassing the amplitude of mode (0, 2). The last growth of mode (0, 4) is probably due to a secondary instability of the streaks. The location where the mode (0, 4) surpasses the primary oblique mode is consistent with the location where the skin friction begins to deviate from its laminar value in figure 1(a). So, the first deviation of the skin friction curve is due to the growth of the streak mode, while the second deviation, which begins at $R\approx 708$, might be explained by the secondary instability of the streaks.

For N-regime transition in figure 1(d), during the amplification of the oblique mode, the primary wave undergoes a slight decrease, which is similar to the K-regime (see arrows in the figure). But the streak mode does not develop during the initial stage. Thus, the nonlinear amplification of the 3-D wave is mainly attributed to the growth of subharmonic oblique disturbances, which is consistent with previous results of experiments and DNS (Herbert Reference Herbert1997).

The development of Fourier amplitudes is closely related to the flow structure within the boundary layer. Figure 2(a–c) shows the contours of streamwise velocity (the grey contour) in the $x$–$z$ plane at $y=2.5$, superimposed on iso-surfaces of positive second invariant of the velocity gradient, i.e. $Q$ value (the red region), for the transition of the K-, O- and N-regimes, respectively. Here, $Q=[|\unicode[STIX]{x1D734}|^{2}-|\unicode[STIX]{x1D64E}|^{2}]/2=0.00002~\text{s}^{-2}$, with $\unicode[STIX]{x1D734}=[\unicode[STIX]{x1D735}\boldsymbol{v}-(\unicode[STIX]{x1D735}\boldsymbol{v})^{\text{T}}]/2$ is the vorticity tensor, and $\unicode[STIX]{x1D64E}=[\unicode[STIX]{x1D735}\boldsymbol{v}+(\unicode[STIX]{x1D735}\boldsymbol{v})^{\text{T}}]/2$ is the rate-of-strain tensor (Hunt, Wray & Moin Reference Hunt, Wray and Moin1988). As shown in figure 2(a), the K-regime is characterized by aligned low-momentum zones (LMZ), the darker regions as indicated by dashed circles in the figure. These regions have a spanwise spacing of $\unicode[STIX]{x1D706}_{z}=125.66$, which is the same as the spanwise wavelength of the corresponding oblique waves (i.e. $2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FD}=125.66$). The LMZs at far downstream locations are flanked by a pair of high-speed regions, labelled S1 and S2 in the figure. A 𝛬-vortex (‘ >’ in the figure) appears downstream of the location where the skin friction begins to deviate from the laminar curve ($x\approx 1190$).

Figure 2.

Contour of streamwise velocity $U$ (the grey shading) at $y=2.5$ and $t=0.8~\text{T}$, superimposed on iso-surfaces of $Q$-criterion detected structures (the red regions, $Q=0.00002~\text{s}^{-2}$). (a) K-regime transition; (b) O-regime transition; (c) N-regime transition. The flow is from left to right.

For the O-regime and N-regime, figure 2(b,c), the LMZs appear as staggered patterns. The staggered 3-D waves (dark regions within the dashed circles) for the O-regime develop nonlinearly from a pair of oblique waves (1, $\pm 1$), appearing farther upstream than for the N-regime, with a spanwise wavelength of $\unicode[STIX]{x1D706}_{z}=125.66$. The streaks (the dashed lines) shown for the O-regime are spaced by $\unicode[STIX]{x1D706}_{z}/2=62.83$, as a result of the growing streak mode (0, 2). The low-speed streak (LSS), labelled 1 in figure 2(b), consists of several 3-D waves, indicated by the dashed circles along the streak. The 𝛬-vortices are observed to develop in conjunction with LSS regions, in a staggered pattern.

For the N-regime, figure 2(c), shows that the LMZs and 𝛬-vortices that develop well downstream are larger than those for the K- and O-regimes, since the streamwise wavelength of the amplified 3-D wave is twice that of the primary 2-D wave. No streak is observed in figure 2(c), which agrees with the development of mode amplitudes in figure 1(d). Because the 3-D waves appear prior to a 𝛬-vortex, as indicated by the position of the LMZs in the figure, a 𝛬-vortex is most likely produced by the nonlinear 3-D wave. It will be shown later that the 3-D waves are very similar to the soliton-like wave structure mentioned previously.

3.2 K-regime transition

3.2.1 Evolution of timelines and streaklines

Figure 3.

K-regime timelines initiated at $x=1011$, $y=1.71$, with $\text{d}t=12$. (a) $t=5.6T$; (b) $t=6.1T$. WWF, warped wave front; K, kink structure; R, roll-up; LSS, low-speed streak; C, chevron structure.

Figure 3 shows two snapshots of timelines initiated at $y=1.71$ after development times of $t=5.6T$ and $6.1T$. Here, $T$ is the normalized time period of the primary wave defined in § 2.2. The timeline surface is displayed in greyscale, where the vertical location of the surface indicated by the scale to the right of the images (darker is nearer the wall, and lighter farther from the all). This represents a continuous timeline surface, similar to a timeline sheet formed by continuously releasing hydrogen bubbles. The translucency of the surface is set as 70 % to allow observation of the deformation of the timeline sheet. The volumetric rendering is sketched in the right schematic diagram. So, the shading in the figure shows the overlap of the surface, caused by the folding of the timeline surface. Figure 3(a) shows the deformation of the timeline sheet at $t=5.6T$, the behaviour of which is identified by a series of stages 1 to 5, showing the evolution from the initial T–S wave to a 𝛬-vortex. Figure 3(b) shows the spatial development of the timeline surface patterns at $t=6.1T$, a half-primary period later relative to figure 3(a). Figure 3 clearly shows how a 𝛬-vortex develops. Starting at stage 1, the 2-D T–S wave is distorted into a three-dimensional wave (3-D wave), with its wave front warped, indicated as a warped wave front – the wave front of a modulated two-dimensional wave that is strongly warped, as evidenced by the accumulation of times lines at stage 2. During stage 2, the WWF develops a fold in the timeline sheet and a low-speed streak appears, indicated by the retardation and downward motion of the timelines, labelled as LSS. By stage 3, the 3-D wave lifts up, with the WWF beginning to accelerate, leading to an initial roll-up of the wave front by stage 4 (see R1 and R2 in stage 4). This initial roll-up subsequently develops into an apparent 𝛬-vortex, at stage 5 in figure 3(a) and stage 4 in figure 3(b). The roll-up behaviour causes multiple folds of the surface, reflected by the darker shaded regions of the translucent sheet. During the folding process, streamwise depressions (V1 and V2) of the sheet toward the wall appear flanking the hairpin vortex, which subsequently develop into streamwise vortices, labelled SV in figure 3(a). Meanwhile, the kinked structure (labelled K), is observed to develop a chevron-shaped structure at a downstream position (labelled C). The details of the folding process are shown further in figures 4 and 5.

Figure 4.

Plan view of the evolution of a single structure based on K-regime timelines (a,c,e,g,i) and a timeline surface (b,d,f,h,j) initiated at $y=1.71$, with release interval $\text{d}t=0.2T$. (a,b) $t=5.1T$; (c,d) $t=5.6T$; (e,f) $t=6.1T$; (g,h) $t=6.6T$; (i,j) $t=7.1T$. 1st F, first folding; 2nd F, second folding; SV, streamwise vortex. (See the supplementary Movie 1 for the complete evolution.)

Figure 5.

End view of K-regime timelines (looking downstream) at the same time intervals as figure 4. (a) $t=5.1T$; (b) $t=5.6T$; (c) $t=6.1T$; (d) $t=6.6T$; (e) $t=7.1T$. L, lift-up.

The first and second rows of figure 4 show the plan-view evolution of timelines and a timeline surface from $t=5.1T$ to $7.1T$ at intervals of $0.5T$, corresponding to the structure shown initially as stage 3 in figure 3(a) (see the supplementary Movie 1, available at https://doi.org/10.1017/jfm.2020.107, for the complete evolution). The volumetric rendering for the surface is the same as figure 3. Figure 4(a,c) clearly shows a LSS with retarded timelines appearing below the WWF, which indicates that it is not the presence of a 𝛬-vortex that induces the low-speed streak pattern, but the lifting of the 3-D wave, since the 𝛬-vortex has not yet formed. The 3-D wave is initially reflected as a warping and accumulation of timelines, as shown in figure 4(a,c), or as a transverse fold in the timeline surface, as shown in figure 4(b,d). In figure 4(f), it is observed that the surface appears to develop a new fold (labelled 2nd F), which projects outward and above the portion of the surface marking the WWF. This second surface fold is more obvious at the later time of $t=6.6T$, and will progressively develop into a 𝛬-vortex, as shown in figure 4(h,j). When a 𝛬-vortex appears, the timeline surface comprising the WWF begins to squeeze together and lift up, as the sequence of figure 4(e,g,i) shows. Previous studies assumed that the first folding (1st F) leads to the development of a 𝛬-vortex (Hama & Nutant Reference Hama and Nutant1963). However, figure 4 clearly demonstrates that a 𝛬-vortex develops from the second folding (2nd F) of the wave front. The lifting of the 3-D wave deforms the 2-D timelines, and causes the first and second foldings. It should be noted that the chevron-like structure, labelled C in figure 4(f,h), is actually a reflection of the low-speed streak developing near the wall along the centre line (red line in the figure). The two high-speed valleys (V1, V2) that were previously identified in figure 3, which flank the chevron-like structure (C), are shown in figure 4(h,j) to develop into rotational streamwise vortices (SV1, SV2). Note that the above warping process is unobservable when the timelines are initiated at a higher position such as $y=3.83$. In that case, the evolution of three-dimensional wave is obvious, see the supplementary Movie 2 for the complete evolution.

Figure 5 shows the corresponding end views of the five timeline patterns shown in figure 4(a,c,e,g,i). These end-view patterns clearly demonstrate the lift-up (labelled L in the figures) of a 3-D wave. Note that figure 5(d,e) shows that the second folding of the timelines protrudes above the first folding, which develops into the head of a 𝛬-vortex. The development of the two streamwise vortices is also clearly shown in figure 5(c,d), labelled SV1 and SV2 flanking the sides of the lifted WWF.

Figure 6.

Side view of vertical timelines initiated at $z=126$ (near centre plane) for K-regime transition, with release intervals $\text{d}t=0.16T$. (a) $t=2T$; (b) $t=3T$; (c) $t=4T$; (d) $t=5T$. The wavelength $\unicode[STIX]{x1D706}\approx 52.7$. HSL, high-shear layer; W, 3-D wave; B, wave front leading edge; RT, rotation; S, downward sweep. See the supplementary Movie 7 for the complete evolution.

The breakdown of a wave motion into hairpin eddies has been demonstrated to be a consequence of the instability arising from three-dimensional wave behaviour (Klebanoff et al. Reference Klebanoff, Tidstrom and Sargent1962). Klebanoff suggested the instability was similar to a secondary inflectional instability. The results of the present investigation also indicate that the 𝛬-vortex develops from a 3-D wave-induced inflectional instability. Figure 6 shows the evolution of vertical timelines initiated at $z=126$ (near the centre plane shown as red line in figure 4), with release intervals of $\text{d}t=0.16T$. The streamwise extent shown for the $x$-direction is almost $5.5\unicode[STIX]{x1D706}$, where $\unicode[STIX]{x1D706}$ is the wavelength of the primary T–S wave. The instantaneous velocity profiles at $t=2T$, $3T$, $4T$, $5T$ are markedly similar to those shown by Hama & Nutant (Reference Hama and Nutant1963) using hydrogen bubble timelines. As shown in figure 6(a), an inflectional high-shear layer (HSL) develops as the 3-D wave lifts up. This is followed by the development of a warped wave front, as shown in figure 6(b). Figure 6(c) shows that the WWF undergoes further stretching and develops the first folding of timelines (1st F), with the kink structure (K) also developing during the folding process. The leading edge (B) of a WWF creates a strong downward sweep (S) as transition proceeds. Finally, as shown in figure 6(d), rotation (RT) of the WWF progressively evolves into the head of a 𝛬-vortex, which results in the entrainment of higher-speed outer flow into the 𝛬-vortex and toward the wall.

Figure 7 shows streaklines initiated at both $y=1.71$ and 3.83 near the centre plane ($z=126$), which illustrate the amplification and evolution of the wave behaviour. The streakline initiated at $y=1.71$ clearly shows the first folding process of the WWF. The wave front undergoes a rotational behaviour downstream similar to those shown by Kelvin–Helmholtz waves or Holmboe waves (Smyth & Peltier Reference Smyth and Peltier1991; Lefauve et al. Reference Lefauve, Partridge, Zhou, Dalziel, Caulfield and Linden2018). However, the upper streakline, initiated higher at $y=3.83$, only undergoes a modest wave amplification. Two factors are considered to be relevant to the different wave behaviour: the first is the shear strength, and the second is the amplitude of the 3-D disturbance. Note that the upper streakline seems to behave as a superposition of the primary wave and its subharmonic, indicated by the wavelength of $2\unicode[STIX]{x1D706}$ on the figure. As pointed out by Kachanov (Reference Kachanov1994), subharmonic resonances are also important in K-regime transition, and the streakline behaviour of figure 7 may be a reflection of the development of subharmonic resonance.

Figure 7.

Evolution of K-regime streaklines initiated at $y=1.71$ and 3.83 on the plane of $z=126$: red line for $t_{0}=4T$; black line for $t_{0}+0.72T$.

3.2.2 Evolution of material surfaces

Figure 8 shows the evolution of an initially flat material surface ($\unicode[STIX]{x0394}z=98$, $\unicode[STIX]{x0394}x=316$) initiated at $y=1.71$ and $x_{initial}=800\sim 1116$. The deformation of the material surface from $1T$ to $3T$ shown in figure 8(a–c) is similar to that of the timeline surface shown in figure 3. Figure 8(a) illustrates a stage where the 2-D wave has undergone a weak 3-D deformation, with its crest inclined slightly downstream. As the wave amplifies, figure 8(b) shows that the inclined 2-D wave undergoes strong spanwise deformation, developing a three-dimensional WWF. This WWF is the typical first folding stage. Figure 8(c) shows that the WWF continues to deform three-dimensionally, and gradually evolves into a 𝛬-vortex. A black material line along the centreline of the wave amplification process illustrates the same folding behaviour as illustrated by the timelines and streaklines of figures 4 and 6.

Figure 8.

Evolution of a material surface initiated at $y=1.71$ for K-regime transition. (a) $t=1T$; (b) $t=2T$; (c) $t=3T$. Note that the wall-normal coordinate scale is 10 times magnified.

3.3 O-regime and N-regime transition

Figure 9.

The evolution of a single structure based on O-regime timelines initiated at $y=1.71$, with release interval $\text{d}t=0.3T$. (a,d,g,j,m) End view of timelines; (b,e,h,k,n) plan view of timelines; (c,f,i,l,o) plan view of a rendered timeline surface. (a–c) $t=4T$; (d–f) $t=5T$; (g–i) $t=6T$; (j–l) $t=7T$; (m–o) $t=8T$. (See the supplementary Movie 3 for the complete evolution.)

The evolution of a single structure based on timelines generated for the O-regime is shown in figure 9. (See the supplementary Movie 3 for the complete evolution of how a 3-D wave develops into a 𝛬-vortex in O-regime transition.) Figure 9(a,d,g,j,m) shows end views of five timeline patterns from $t=4T$ to $8T$. Similar to the patterns shown in figure 5 for the K-regime, the 3-D wave is initially very weak, figure 9(a), but then develops into a WWF and expands in both the normal and spanwise directions, moving progressively upward. Some instability occurs adjacent to the two sides of the WWF, as illustrated by the intertwining timelines in figure 9(m). figures 9(b,e,h,k,n) and 9(c,f,i,l,o) show the corresponding plan view of the development of a 𝛬-vortex in O-regime transition, both for individual timeline behaviour (b,e,h,k,n) and as a developing timeline surface (c,f,i,l,o). First, the interaction of oblique waves produces a 3-D wave, manifested as weakly retarded timelines (LSS) or a small bulge (red dashed circle), figure 9(b,c). Next, the lift-up of the 3-D wave produces a warped wave front, with rhombic creases appearing on the timeline surface, as indicated by the red dashed lines in figure 9(f). The first folding (1st F) of the surface strengthens the creases (red dashed lines) that flank the WWF, figure 9(i). During the strengthening process, regions of apparent depression (labelled V1, V2, SV1, SV2) form, similar to those shown in figure 3. Subsequently, a chevron-like structure (C) develops below the WWF, constituting a more confined LSS, as shown in figure 9(k,l). Finally, the second folding develops on the timeline surface to form a 𝛬-vortex, during which the previous rhombic creases begin to rotate, developing streamwise vortices from the initial regions of depression, figure 9(n,o). Note that the above multi-folding process is not discernible when the timelines are initiated at $y=3.83$. In that case, the evolution of three-dimensional wave is clearly observed, which seems to constitute a low-speed streak (see the supplementary Movie 4 for the complete evolution).

Figure 10.

The evolution of a single structure based on N-regime timelines initiated at $y=1.71$, with release interval $\text{d}t=0.3T$. (a,d,g,j,m) End view of timelines; (b,e,h,k,n) plan view of timelines; (c,f,i,l,o) plan view of rendered timeline surface. (a–c) $t=5.1T$; (d–f) $t=6.4T$; (g–i) $t=7T$; (j–l) $t=7.5T$; (m–o) $t=8T$.

Figure 10, similar to the presentation in figure 9, shows the evolution of a single structure developing from a 2-D T–S wave into a 𝛬-vortex for the N-regime (see the supplementary Movie 5 for the complete evolution). The end-view timelines (upper row) demonstrate lift-up behaviour similar to the K- and O-regimes, but no apparent streamwise vortex develops, unlike those apparent shown in figures 5 and 9 (labelled SV1 and SV2). As shown in figures 10(b,e,h,k,n) and 10(c,f,i,l,o), during the lift-up process, a two-dimensional wave front (2D-WF) gradually evolves into a three-dimensional warped wave front (WWF), from $t=5.1\sim 7T$, which subsequently develops into a 𝛬-vortex; this process is illustrated most clearly by figure 10(c,f,i,l,o). The development process looks markedly similar to those shown by Hama & Nutant (Reference Hama and Nutant1963). The wave front is initially two-dimensional (see 2D-WF), as shown by the 2-D timelines in figure 10(a), with the transverse folding of the timeline surface (1st F) developing in figure 10(c). As time progresses, the second folding develops, figure 10(d,f), which corresponds to a denser accumulation of timelines indicated by DA in figure 10(e), which differs from K- and O-regime transition. When the warping of the wave front takes place (regions labelled 3 and 4 in figure 10i), a WWF develops, accompanied by retarded regions (labelled 1 and 2) preceding the WWF. As the warping progresses, the timelines coalesce and deform to form the head of a 𝛬-vortex, with upward motion of the WWF intensifying, figure 10(j–l). The rotation and coalescence of the timelines along what appears as a swept-back region (regions labelled 1, 2, 3, 4 in figure 10l) indicate that vorticity is concentrated along the swept-back sides. Eventually, a 𝛬-vortex develops and is strengthened due to streamwise stretching, with red arrows indicating the rotation of the swept-back legs in figure 10(o). The retarded region indicated by K in figure 10(i), is shown in figure 10(o) to have evolved into a chevron-shaped structure (labelled C) beneath the 𝛬-vortex, and appears to be a localized LSS, as shown in figure 10(n). Note also that the WWF and its warping process is indiscernible when the timelines are initiated at $y=3.83$ (see the supplementary Movie 6 for the complete evolution).

Figure 11.

Vertical timelines of O-regime transition at (a) $t=3T$ and (b) $t=4.6T$, initiated near the centre plane $z=126$. (See the supplementary Movie 7 for the complete evolution.)

Figure 12.

Vertical timelines of N-regime transition at (a) $t=3T$ and (b) $t=4.6T$, initiated near the centre plane $z=126$. (See the supplementary Movie 7 for the complete evolution.)

The vertical timeline patterns for the O-type and N-type regimes are shown in figures 11 and 12. These profiles are initiated at $x=1011$ and $z=1256$ (the centre plane), with release intervals $\text{d}t=0.2T$. Figures 11 and 12 show patterns similar to figure 6 for the K-regime. In both these figures, an inflectional region (A) develops in the near-wall region of the profiles, which progressively develops a humplike pattern, implying an HSL, and created by a lifting 3-D wave. The leading edge of this humplike pattern subsequently precipitates a strong downward sweep (S), and evolves into a warped wave front, which will further develop into a transverse rotational pattern as the head of a 𝛬-vortex. The humplike structure observed here corresponds to what is described by Hama & Nutant (Reference Hama and Nutant1963) as a ‘kink’ structure, whose breakdown was considered to be responsible for the formation of discrete vortices. Note that this process of development of the hump-like kink pattern for the N-regime occurs closer to the wall than for the K- and O-regimes, as illustrated by a comparison of figure 12 with figures 6 and 11, with the size of the hump-like patterns for the N-regime appearing smaller (see the supplementary Movie 7 for the comparison between the three regimes).

Figures 13 and 14 show the evolution of streaklines for both O-regime and N-regime transitions. The streaklines are initiated at four different wall-normal positions, i.e. $y_{0}=1.71$, 2.54, 3.83 and 4.25, and show streakline patterns for two different development periods. For the O-regime, the streaklines at the near-wall position $y_{0}=1.71$ are stretched into a thin filament projecting strongly into the outer layer (labelled A in the figure), indicating a severe wave deformation. When compared with patterns shown in figure 9, the location of the thin, stretched filament is consistent with the location where a 𝛬-vortex develops. However, note that comparably strong wave deformation does not occur for the streaklines initiated at $y_{0}=3.83$ and 4.25, which behave essentially as a modest wave-like amplification. The streakline initiated at $y_{0}=2.54$ in figure 13(b) shows the development of the first folding of the wave front, which is striking in its similarity to the streakline behaviour for the K-regime transition, generated at $y_{0}=1.71$ (figure 7).

Figure 13.

Evolution of streaklines for O-regime transition initiated at different wall-normal positions $y_{0}$. (a) $y_{0}=1.71$ and 3.83; (b) $y_{0}=2.54$ and 4.25. $t_{0}=4T$ (red), $t_{1}=4.72T$ (black).

Figure 14.

Evolution of streaklines for N-regime transition initiated at different wall-normal positions $y_{0}$. (a) $y_{0}=1.71$ and 3.83; (b) $y_{0}=2.54$ and 4.25. $t_{0}=4T$ (red), $t_{1}=4.72T$ (black).

For N-regime transition, shown in figure 14, the streaklines initiated at the upper locations also reflect a modest wave-like amplification similar to the O-regime, although they do not move outward and appear more amplified than for the O-regime of figure 13. The waveform shows a superposition of the primary wave (of wavelength of $\unicode[STIX]{x1D706}$) and its subharmonic (of wavelength of $2\unicode[STIX]{x1D706}$), similar to the K-regime behaviour in figure 7. As shown by the streaklines initiated at $y_{0}=3.83$ and 4.25 in figure 14, the subharmonic wavelength ($2\unicode[STIX]{x1D706}$) is strongly amplified downstream and becomes the dominant wavelength. This amplification may be a reflection of typical subharmonic resonance. Note that prior to the strong amplification to a 3-D wave, there appears to be a relatively quiet zone, labelled E in figure 14(b). The presence of this quiet zone is hypothesized to be the result of a nonlinear interaction between waves, i.e. the attenuation of the primary wave ($\unicode[STIX]{x1D706}$) and the growth of a low-frequency disturbance (at nearly $2\unicode[STIX]{x1D706}$). Indeed, the location of the quiet zone is identical to the position of the attenuation of the primary wave shown in figure 1(d); clearly, the mechanism for development of this quiet zone merits further study.

The streakline initiated at $y_{0}=1.71$, figure 14(a), displays a similar warping process (a 1st folding and 2nd folding) to the K- and O-regimes. The streakline initiated at $y_{0}=2.54$, figure 14(b), again displays a strongly stretched, thin filament (labelled A), which is also observed in the region where a 𝛬-vortex appears ($x=1250{-}1300$). A backward stretched filament (labelled D) at the bottom of the 3-D wave front is slightly lifted up away from the wall, which appears coincident with the chevron-like pattern (C) in figure 10. The projecting thin filament pattern, which develops at the downstream warped wave front, is only observed for streaklines initiated close to the wall for all regimes. The results of figures 13 and 14 illustrate that the amplitude of the 3-D wave decreases with distance from the wall, and that the lift-up of the 3-D wave is much stronger in the near-wall region. Note that the evolution of streaklines for N-regime transition is markedly similar to the simulation of streaklines in a 2-D shear layer perturbed by one or two travelling waves (Hama Reference Hama1962; Williams & Hama Reference Williams and Hama1980), although no second folding behaviour was observed in those studies.

Figure 15.

Material surfaces at $t=1T$, initiated at $x_{initial}=795\sim 1309$, $z_{initial}=77\sim 175$, $y_{0}=3.83$. (a) K-regime transition; (b) O-regime transition; (c) N-regime transition.

Figure 15 shows the deformation of an initially flat material surface ($x_{initial}=795{-}1309$, $z_{initial}=77{-}175$) initiated at $y=3.83$, for the three transition regimes at $t=1T$. The black contour lines illustrate respective changes in $y$ height, and show the development and growth of three-dimensional waves. Figure 15(a) shows that for the K-regime, an upstream 2-D wave progressively amplifies into an apparent 3-D wave, appearing at approximately $x\approx 1200$. This 3-D wave subsequently amplifies with its wave front (P) lifting up. At the front of the 3-D wave, there is a region of depression labelled S, which may be the result of a downward sweep event. As the 3-D wave develops downstream, this region of depression evolves into two regions (labelled V1, V2) flanking both sides of the rising 3-D wave. Note that the wave front at this position does not undergo as severe a warping as shown in figure 8, which was for a material sheet initiated much closer to the wall.

For O-regime transition, figure 15(b) shows that a rising 3-D wave (labelled P) develops much sooner than the K-regime, which is also accompanied by regions of depression (labelled S1 and S2). However, figure 15(b) shows the development of four longitudinal depressions or valleys (labelled V1, V2, V3, V4) flanking the sides of the 3-D wave fronts, which is markedly different from figure 15(a). These valleys span several wavelengths of the 3-D wave in the streamwise direction. For the K-regime of figure 15(a), the spacing ($\unicode[STIX]{x1D706}_{1}$) between the valleys V1 and V2 is approximately $1/2\unicode[STIX]{x1D706}_{z}$, where the $\unicode[STIX]{x1D706}_{z}$ is the spanwise wavelength of the introduced oblique waves. However, the spacing ($\unicode[STIX]{x1D706}_{2}$) between valleys V3 and V4 in figure 15(b) for the O-regime is approximately half that of V1 and V2, i.e. $1/4\unicode[STIX]{x1D706}_{z}$. Two longitudinal streaks (LSS1 and LSS2) are observed outside the valleys V3 and V4, which are also spaced at approximately $\unicode[STIX]{x1D706}_{1}$, similar to the spacing between V1 and V2. These streaks are identified as low-speed streaks (LSS), which are a result of low-speed fluid moving upwards due to some type of uplift event.

Figure 15(c) shows the deformation of the material surface for N-regime transition, which looks similar to the K-regime, except that the dimensions of the 3-D wave are larger, and the 3-D waves develop later in the transition process. Regions of depression labelled V1, V2 and S0, S1 surround the 3-D wave, forming a staggered pattern. No longitudinal streaks or valleys are observed at this point of development for the N-regime.

In the present section, ‘kernel’ studies of both the O- and N-regimes followed the evolution of a 3-D wave into a 𝛬-vortex. These studies were facilitated using spatial-temporal visualizations of synthetic multi-perspective timelines, streaklines and material surfaces initiated at different heights and amplification periods. The results of O- and N-regime transition are compared to those of K-regime shown in § 3.2, and the similarities of WWF and its warping process are pointed out. A mutual comparison of the deformation of material surfaces for all three regimes, illustrates a similar pattern of 3-D wave development in the transition process.

4.1 Description of the experiments

4.1.2 Experimental techniques and data acquisition

A series of hydrogen bubble timeline visualizations were done employing horizontal-wire configurations as described by Lee & Li (Reference Lee and Li2007), see figure 16(a). The hydrogen bubble wire was 0.03 mm diameter platinum wire. The distance of the wire from the plate surface was adjusted using insulated, calibrated spacers between the wire and the plate. By pulsing the current through the hydrogen bubble wire at a constant frequency, a pattern of timelines is created. If the current to the wire is kept constant, a continuous bubble sheet will be generated. The free-stream velocity for all studies was approximately $0.17~\text{m}~\text{s}^{-1}$, and the distance, $x_{d}$, from leading edge to the centre of the tomo-PIV measurement region was 376 mm.

Figure 16.

Experimental set-up of tomo-PIV: (a) plan view of set-up for hydrogen bubble visualization; (b) schematic layout of the arrangement of the tomo-PIV system.

The tomo-PIV configuration is shown in figure 16(b). Four high-speed CMOS cameras (Photron FASTCAM SA4) with resolution of $1024\times 1024$ pixels at framing rates up to 3600 f.p.s. were used, in a cross-configuration arrangement. The cameras are fitted with Nikon lenses of 200 mm focal length at an aperture setting of $f/11$. Illumination was provided by a DCQ-30Q, high-speed laser generator from Beijing ZK Laser. The thickness of the light volume is 30 mm. The four cameras and laser system were controlled by Davis 8.4 software, and were synchronized using a LaVision PTU timing controller. Hollow glass particles, with mean diameter $11~\unicode[STIX]{x03BC}\text{m}$, were introduced into the water tunnel 6 m downstream of the test section to assure that the particles were fully mixed and uniformly distributed after circulating to the measurement site. Experiments were conducted at a particle image density of 0.098 particles per pixel, which corresponds to particle concentrations of approximately $2.3~\text{particles}~\text{mm}^{-3}$.

The tomo-PIV sampling frequency was 500 Hz, with a maximum sampling time of 20 s. The final residual calibration errors for all cameras was less than 0.1 pixel. The calibration field of view had a digital resolution of $74.80~\unicode[STIX]{x03BC}\text{m}$ per pixel. The measurement domain extended 78 mm in the streamwise direction and 78 mm in the spanwise direction. Volume self-calibration (Wieneke 2008) was used on the pre-processed particle images. Sequential motion tracking enhancement (SMTE, Lynch & Scarano Reference Lynch and Scarano2015) was applied to volume reconstruction. The reconstructed volume was $1041\times 1048\times 187$ voxels. A multi-pass approach of four steps was carried out for the volume correlation, and the interrogation volume size for the final pass was $16\times 16\times 16$ voxels ($1.2\times 1.2\times 1.2~\text{mm}^{3}$) with a 75 % overlap. As a final step, a Gaussian smoothing filter of $3\times 3\times 3$ subvolume was applied to the velocity data. Proper orthogonal decomposition (POD) analysis has shown to be effective for noise removal and extraction of high modes in turbulent flow (Moin & Moser Reference Moin and Moser1989; Hellström & Smits Reference Hellström and Smits2017). In the present paper, POD analysis was applied to the smoothed velocity data sets, and the first four leading-order POD modes are extracted, in accordance with the methods of Zhu et al. (Reference Zhu, Wang, Wang and Wang2017). The POD approximated vector fields are the data sets used in the following postprocessing and analyses.

4.1.3 Properties of the boundary layers and PIV accuracy

Figure 17.

Measurements of two boundary-layer profiles compared with the Blasius profile.

Table 2.

Characteristic boundary-layer parameters and resolution parameters of tomo-PIV measurements: $Re_{\unicode[STIX]{x1D703}}$ is the momentum-thickness-based Reynolds number; $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D703}}$ is the momentum thickness and $\unicode[STIX]{x1D6FF}^{\ast }$ is the displacement thickness; $H$ is the shape factor.

The spanwise averaged boundary-layer velocity profiles at two streamwise positions, obtained from the tomo-PIV data sets, are shown in figure 17 compared with a conventional Blasius profile (White Reference White2006). The wall-normal coordinate is scaled on the displacement thickness $\unicode[STIX]{x1D6FF}^{\ast }$ of the laminar boundary layer, and shows that the averaged velocities agree well with the Blasius profile. Table 2 summarizes the characteristic parameters for the initial laminar boundary layer. Table 2 also lists the space and time resolution of the PIV data. The tomo-PIV measurements in this study provided velocity information on a grid of $0.3\times 0.3\times 0.3~\text{mm}^{3}$ over the volume of $78\times 78\times 30~\text{mm}^{3}$; the sample time increments are $\unicode[STIX]{x0394}t=0.002~\text{s}$. The turbulence intensity of the free stream measured by tomo-PIV is $0.0077U_{\infty }$, which suggests that the root-mean-square (r.m.s.) background noise of the PIV was less than 0.77 % of the free-stream velocity, and the r.m.s. divergence error is $4.34~\text{s}^{-1}$. The calibration inaccuracies were further reduced to less than 0.01 pixel by applying volume self-calibration on the pre-processed images, a process employing background subtraction and Gaussian smoothing with a kernel size of $3\times 3$ pixels. The reconstruction accuracy of the tomo-PIV was determined by the reconstruction signal-to-noise ratio (NSR), defined as the intensity level encountered in the reconstructed region compared with that outside the illumination (Scarano Reference Scarano2013). The distribution of light intensity across the depth of the measurement volume was determined using a MART tomographic algorithm, which gave a $NSR>2.17$, which is considered an acceptable reconstruction level.

4.2 Comparison with flow visualization

Figure 18(a,b) shows plan-view hydrogen bubble patterns in a controlled transitional boundary layer. The timelines in figure 18(a) clearly indicate a low-speed streak type of pattern, and also a downstream 𝛬-vortex straddling the streak. Note that there are several kink-like patterns that appear near the LSS, labelled as K in the figure. The 𝛬-vortex creates a butterfly-like pattern, which displays a ‘peak–valley’ behaviour. The two wings or ‘valleys’, labelled V1 and V2, pass over the timelines of the LSS and quickly surround the kink-like structure to form a 𝛬-vortex (see the supplementary Movie 8 for the complete evolution). Figure 18(b) also shows the same behaviour for a sheet of continuously generated hydrogen bubbles. The bubble sheet initially represents a warped wave front, stage 1 in the figure. The 𝛬-vortex is clearly seen to develop from the three-dimensionalized WWF (stage 2 in the figure). Progressing downstream, the kink-like patterns evolves into a much narrower chevron-like patterns (labelled C in the figure). Collectively, the patterns shown for the sequential stages in figure 18(b) are all associated with the formation of a LSS (within the dashed box). As shown by figure 18(a,b), the LSS appears to consist of several kink-like (or chevron-like) structures, encompassed by the 𝛬-vortex downstream. These patterns are markedly similar to hydrogen bubble visualizations by Lee (Lee Reference Lee2000; Lee & Fu Reference Lee and Fu2001; Lee & Li Reference Lee and Li2007; Lee & Wu Reference Lee and Wu2008), Hama & Nutant (Reference Hama and Nutant1963) and Wortmann (Reference Wortmann and Krause1981), an example of which is shown in figure 18(c). Combined with hot-film measurements within the kink-like region at different wall-normal heights, Lee et al. hypothesized that the kink-like pattern is a three-dimensional wave structure, termed a soliton-like coherent structure (SCS). The SCS was hypothesized as a natural consequence of the nonlinear amplification of a pair of oblique disturbance waves in the transitional boundary layer. Figure 18(d) is a timeline pattern from our NPSE study of K-regime transition, similar to figure 3, but with closer timeline spacing to more closely emulate to the hydrogen bubble visualization of figure 18(a). The brighter regions indicate where the timelines comprising the sheet are overlapping due to the folding process. This pattern looks markedly similar to the structures shown in figure 18(a–c), especially the patterns of the kink/chevron-like structures and 𝛬-vortex. The implications of these comparisons, and a hypothesis of the process that results in the development of the WWF and the 𝛬-vortex, will be discussed later in § 5.

Figure 18.

Hydrogen bubble visualization of transitional boundary layer: (a) plan-view timelines at $y=0.35\unicode[STIX]{x1D6FF}$; (b) continuous bubble visualization at $y=0.35\unicode[STIX]{x1D6FF}$ (c) low-speed streak and 𝛬-vortex at $y=0.15\unicode[STIX]{x1D6FF}$ (Lee & Wu Reference Lee and Wu2008); (d) ‘NPSE visualization’ of timelines initiated at $y_{0}=1.71$ for K-regime transition. The flow is left to right.

4.3 Comparison with ‘tomo-PIV visualization’

Figure 19.

Low-speed streak development in a controlled transitional boundary layer at $t=0.24~\text{s}$ after initiation. (a) Timelines generated from tomo-PIV; $x=0$ is a relative coordinate representing $X=378.5~\text{mm}$, with A, B, C, D indicating the streamwise location of the four cross-sections shown in 19(b); (b) cross-sections of streamwise velocity contours at: A, $x=-22.5~\text{mm}$; B, $x=-7~\text{mm}$; C, $x=8~\text{mm}$; and D, $x=24~\text{mm}$.

For the injection–suction controlled transition, the position of the low-speed streak is nearly stable, as shown in figure 19. Figure 19(a) is a plan view timeline visualization of a low-speed streak; figure 19(b) shows end-view cross-sections of streamwise velocity contours for the lifted low-speed streak. Figure 19(b) shows $U$ contours at a sequence of four different streamwise locations in the $z$–$y$ plane ($x=-22.5~\text{mm}$, $-7~\text{mm}$, 8.5 mm, 24 mm) that illustrate the streamwise development of the LSS. As shown by the modest ‘hump’ region in these $U$ contours in plane A, the low-speed streak is initially very weak near the wall. As the streak develops downstream (planes B $\rightarrow$ C $\rightarrow$ D), the low velocity region expands in both the normal and spanwise directions, moving progressively upward. The presence and expansion of the low-velocity region is also apparent in figure 19(a), as revealed by the continued retardation of the simulated timelines. Note that these patterns of streak development are consistent with the behaviour of timeline patterns shown in end view in figure 5.

Figure 20 shows the spatial and temporal evolution of a LSS in a controlled transitional boundary layer. Figure 20(a) is superposed timeline patterns for timelines initiated at two different wall-normal heights. These patterns clearly show that the low-speed region (the peak region labelled P in the figure) is more pronounced near the bounding wall ($y=0.4\unicode[STIX]{x1D6FF}$, red timelines), with significantly less retardation when the distance from the wall is doubled ($y=0.8\unicode[STIX]{x1D6FF}$, black time lines).

Figure 20.

Spatial-temporal evolution of a low-speed streak reconstructed from the tomo-PIV data of a controlled transitional boundary layer: (a) plan view of superposed spanwise timelines at $y=0.4\unicode[STIX]{x1D6FF}$ (red line) and $y=0.8\unicode[STIX]{x1D6FF}$ (black solid line); (b) vertical timelines released at $z=-13.7~\text{mm}$; markers indicate particles released at $y=0.2\unicode[STIX]{x1D6FF}$, $y=0.4\unicode[STIX]{x1D6FF}$, $y=0.8\unicode[STIX]{x1D6FF}$; A and B respectively indicate locations of lift-up and downward sweep events; (c) oblique view of the spanwise timeline pattern for timelines released at $y=0.8\unicode[STIX]{x1D6FF}$; (d) temporal evolution ($t_{0}\rightarrow t_{1}$) of streaklines initiated at a wall-normal position of $y=0.8\unicode[STIX]{x1D6FF}$, with time period $t=0.08T$ between realizations.

Figure 20(b) is a sequence of vertical timelines generated at a frequency of 10 Hz, and initiated at $x=-45.8~\text{mm}$, $z=-13.7~\text{mm}$. Three streaklines are also shown in the figure, which initiate at $y=0.2\unicode[STIX]{x1D6FF}$, $y=0.4\unicode[STIX]{x1D6FF}$ and $y=0.8\unicode[STIX]{x1D6FF}$. The arrows labelled A and B in the figure indicate local organized vertical movements of the fluid. The two events labelled $\text{A}_{1}$ and $\text{A}_{2}$ are outward-directed lift-up events, and the two events labelled $\text{B}_{1}$ and $\text{B}_{2}$ are inward-directed sweep events. The inflectional region of the timelines is similar to the vertical hydrogen bubble visualization of a transitional boundary layer by Hama & Nutant (Reference Hama and Nutant1963). They considered the strong inflection to have its origin in a strongly ‘warped wave front’. Haidari & Smith (Reference Haidari and Smith1994) also observed this type of inflectional behaviour in a laminar boundary layer disturbed by injection through a streamwise slot. The ejection simulated a lift up of near-wall fluid, which was balanced by a downward sweep of outer fluid. The result was the creation of a high-shear layer between the ejection and the sweep, which was subsequently observed to break down into a hairpin vortex. These vertical timeline patterns, with the associated lift-up and downward sweep events are markedly similar to the K-regime simulation patterns shown previously in figure 6.

Figure 20(c) is an oblique view of the black timelines of figure 20(a), initiated at $y=0.8\unicode[STIX]{x1D6FF}$. This figure clearly shows the development of a three-dimensional bulge, created by a preceding three-dimensional downward sweep (B), and a following three-dimensional lift up (A). Figure 20(d) shows the temporal evolution of streaklines initiated at $y=0.8\unicode[STIX]{x1D6FF}$ and $z=-13.7~\text{mm}$, the general location for the development of the peak position P in figure 20(a). It reveals a wave-like amplification of the streaklines in the upper boundary layer. Figure 20(c,d) suggests that the development of a ‘peak’ (LSS) propagates and amplifies as a wave-like structure.

Figure 21.

Evolution of material surfaces reconstructed from the tomo-PIV data. The two surfaces are initiated at $y=0.46\unicode[STIX]{x1D6FF}$ and $0.61\unicode[STIX]{x1D6FF}$ (initial flat surfaces shown at far right): (a) $t=0.16T$; (b) $t=0.32T$; (c) $t=0.48T$; (d) $t=0.64T$.

As pointed out in § 2.3, if the initial condition for Lagrangian tracking is a planar surface, the subsequent Lagrangian behaviour is that of an evolving material surface. Figure 21 shows the evolution of such a material surface for the present transitional boundary layer. The inset at the lower right of figure 21 shows two planar $z$–$x$ material surfaces initiated at $y=0.46\unicode[STIX]{x1D6FF}$ and $0.61\unicode[STIX]{x1D6FF}$ (initially flat surfaces shown at the far right of figure 21). As these surfaces evolve in time, figure 21(a–d), they undergo a transverse warping, developing a peak/valley behaviour commented on previously that is construed as the development of a streamwise fluid lift up, flanked by apparent fluid sweeps toward the surface. The lower surface in figure 21(b), obtained at $t=0.32T$, illustrates the initial transverse warping of the material surface, with a peak (labelled P) developing at $z\approx -20~\text{mm}$, the region where a LSS is observed. Downstream of the peak P, the material surface moves toward the centre, and develops a chevron-like pattern (labelled C). By $t=0.48T$, figure 21(c), the hump-like peak P is flanked by two material valleys, V1 and V2, due to the downward sweep motions. By $t=0.64T$, figure 21(d), a pronounced peak, P, has developed for both surfaces, which reflects the outward penetration of the LSS. The ejection/sweep motions create an HSL around the hump-like peak (dashed line in the figure), where an initial roll-up develops. The two distinct material valleys, V1 and V2, begin to subsequently roll up to form the legs of a 𝛬-vortex. The chevron-like pattern looks markedly similar to the hydrogen bubble visualization in figure 18(a), suggesting that a LSS is located beneath the humplike peak region. Note that the upper material surface does not appear as warped as the lower surface, i.e. the hump-like peak and the chevron-like pattern are more distinct on the lower surface, indicating that the HSL is also more pronounced at lower levels.

The evolutions of the material surfaces shown in figure 21 are supplements for the timeline patterns in figures 18 and 20. The bubble sheets appear much more convoluted than the material surfaces. But the evolution of the material surfaces caused by the developing flow field provides a more complete picture of the flow behaviour than the timelines. Both the evolution of the material surfaces (figure 21) and that of the hydrogen bubble sheet (figure 18) show that the LSS is flanked by developing vortices. And the LSS in figure 21 appears to lift up as a 3-D wave, the waveform of which is similar to that shown in figures 15(a) and 20(c,d). This suggests that the LSS revealed by the hydrogen bubble timelines of figure 18 is a 3-D wave structure, with the formation of vortices initiated by the lift up and breakdown of this 3-D wave.