2.1. Experimental set-up

2.1.1. Experimental facility and model

Figure 2. Experiment set-up: ( $a$ ) schematic of experimental facility; ( $b$ ) a linear configuration of Tomo-PIV.

Tomo-PIV measurements were performed at the Peking University Water Tunnel (PWT), which has the dimensions of 6.0 m $\times$ 0.4 m $\times$ 0.4 m (length $\times$ width $\times$ height). The turbulence level is less than $1\, \%$ at velocities ranging from 0.1 to 1.3 m s $^{-1}$ . In this paper, the coordinate axes $x$ , $y$ and $z$ are the streamwise, wall-normal and spanwise directions, respectively. The coordinate origin is defined as the centre of the perturbation jet orifice. The time origin of each measurement domain is defined as the moment when the leading edge of the turbulent spot appears in the specific measurement domain. A flat plate, of 1.8 m length, 0.5 m span and 18 mm thickness, was placed vertically in the water tunnel at zero angle of attack. A flap attached at the trailing edge of the flat plate was used to adjust the pressure gradient on its working surface, ensuring a zero pressure gradient. The perturbations were generated by a 2 mm diameter orifice wall-normal jet located 400 mm downstream from the leading edge of the flat plate. In the present experiments, the free stream velocity $U_\infty$ is 0.275 m s $^{-1}$ , which is used as the reference velocity. As illustrated in figure 2( $a$ ), a pulse signal from a signal generator activated a loudspeaker to trigger a pulsed wall-normal jet with an average velocity of 0.32 $U_\infty$ . The duration of the wall-normal jet was 70 ms. The wall-normal jet can reach an approximate height of 5.5 $\delta ^*_0$ at its end. The reference displacement thickness $\delta ^*_0$ is 1.78 mm, which is consistent with the reference displacement thickness employed in the subsequent numerical simulation. Here, $t$ is the dimensionless time, which is scaled on $\delta ^*_0/U_\infty$ .

2.1.2. Tomo-PIV set-up

As shown in figure 2( $b$ ), a linear camera configuration was employed for the Tomo-PIV measurements. Four Phantom v2512 high-speed cameras, resolution of 1280 $\times$ 800 pixels, were used. The cameras used Nikon lenses with 200 mm focal length. To satisfy the Scheimpflug criterion (Prasad & Jensen Reference Prasad and Jensen1995), a Scheimpflug adapter was used to connect the cameras and the lens, which ensures particles within the measurement area are in focus. The aperture was set to $f/16$ , which is a compromise between the depth of field and the light intensity of the tracking particles. A single-cavity DPQ- 527–60PIV-DF high-speed Laser (Nd:YLF, 527 nm, 30 mJ pulse $^{-1}$ at 1 KHz), manufactured by Beijing ZK Laser, was used as the light source. The laser beam source was expanded using two cylindrical lenses to a laser light volume with a thickness of 15 mm, illuminating the measurement domain. The cameras and laser were synchronised by a LaVision Programmable Timing Unit (PTU). The control of the measurement system and subsequent Tomo-PIV calculations were both performed in Davis 10.1 software. Hollow glass spheres of mean diameter 11 $\unicode {x03BC}$ m, provided by LaVision, were seeded in the water tunnel as tracking particles. The particle seeding density was approximately 0.035 particles per pixel, which is acceptable for tomographic reconstruction (Elsinga et al. Reference Elsinga, Scarano, Wieneke and van Oudheusden2006). After seeding the tracking particles, the water tunnel was kept running for several hours to ensure uniform particle distribution.

The Tomo-PIV data was acquired at a sampling rate of 500 Hz, for a time resolution of $\Delta t=0.002$ s. To establish the mapping relation between the world coordinate system and pixel coordinate system, perspective calibration was carried out. The volume self-calibration (Wieneke Reference Wieneke2008) was then employed to refine the previous calibration. This process was repeated until the maximum calibration error was reduced to less than 0.1 voxels. The measurements were made from 410 to 650 mm downstream of the leading edge of the flat plate, covering the entire development process from the initial disturbance to a turbulent spot. By using the time-resolved recordings in the present experiment, the sequential motion tracking enhancement (SMTE) approach (Lynch & Scarano Reference Lynch and Scarano2015) was automatically employed to reconstruct the particles in the measurement domain. While the dimensions of the three measurement regions differ, the cross-correlation set-up and post-processing of vectors remained the same. The various parameters of the Tomo-PIV measurements in the three domains are summarised in table 1. The final interrogation window size for all measurement domains was set to 24 $\times$ 24 $\times$ 24 voxels with a $75\,\%$ overlap. To remove outliers, a 3-D vector post-processing technique was used. A Gaussian smoothing with 5 $\times$ 5 $\times$ 5 vectors was used to reduce the background noise in the data set.

Table 1. Parameters of three measurement regions: $x_{d}$ is the distance from the centre of the measurement region to the leading edge of the flat plate; $L_{x}$ , $L_{y}$ and $L_{z}$ are the dimensions of the region in the streamwise, wall-normal and spanwise directions, respectively; $\varDelta _{x}$ , $\varDelta _{y}$ and $\varDelta _{z}$ are the spatial resolution in the streamwise, wall-normal and spanwise directions, respectively.

2.1.3. PIV accuracy and resolution

Figure 3. Comparison between the measured velocity profiles and Blasius profile.

In figure 3, the time and spanwise-averaged velocity profiles at the centre location of each measurement are compared with the Blasius profile (White & Majdalani Reference White and Majdalani2006). The local boundary layer displacement thickness was determined and employed to non-dimensionalise the wall-normal distance; velocities were scaled on the free stream velocity $U_\infty$ . The velocity profiles measured by Tomo-PIV exhibit a good agreement with the Blasius profile. The boundary layer parameters of these velocity profiles are summarised in table 2, demonstrating that the initial boundary layer in the present experiment is a typical Blasius laminar boundary layer. The time resolution is $\Delta t=0.002$ s for all cases and the spatial resolution for each region is presented in table 1. Using perspective calibration and volume self-calibration operations, the calibration error was reduced to less than 0.1 pixels. The signal-to-noise ratio (SNR) measures the quality of the reconstructed volume, calculated by comparing the intensity of the illuminated volume to the intensity outside of it. The SNR of each measurement domain is presented in table 1. All measurements have an SNR larger than 2, which indicates a high-quality reconstruction (Scarano Reference Scarano2013).

Table 2. Boundary layer characteristics: $\textrm {Re}_{\delta ^*}$ is the Reynolds number based on local displacement thickness; $\delta _{\theta }$ is the momentum thickness; $\delta ^*$ is the displacement thickness; $H$ is the shape factor.

2.2. Numerical method

2.2.1. DNS set-up

DNS was performed using an open-source pseudo-spectral solver Simson (Chevalier et al. Reference Chevalier, Schlatter, Lundbladh and Henningson2007), which has been employed by many previous boundary layer simulation investigations (Brandt & Henningson Reference Brandt and Henningson2002; Brandt et al. Reference Brandt, Schlatter and Henningson2004). The incompressible Navier–Stokes equations were solved to obtain a data set for turbulent spots developing within a zero pressure gradient flat plate boundary layer.

The spatial discretisation in the streamwise and spanwise directions employs Fourier series expansions. Chebyshev polynomial series are used in wall-normal direction discretisation. For time discretisation, a second-order Crank–Nicolson method is used to handle the linear term, while a third-order four-stage Runge–Kutta method is employed for the nonlinear components. The coordinate system is defined to match that used in the experiment, allowing comparison between the simulation and experimental results.

Figure 4. Schematic diagram of the computational domain.

As illustrated in figure 4, the computational domain dimensions in the streamwise, wall-normal and spanwise directions are $400\delta ^*_0$ , $25\delta ^*_0$ and $90\delta ^*_0$ , respectively, where $\delta ^*_0$ is the displacement thickness at the inlet of the computational domain. Grid resolution was selected based on studies (Levin & Henningson Reference Levin and Henningson2007) at comparable Reynolds numbers. Verification was conducted on three different grid densities: Grid 1 (800, 181 and 360 nodes in the streamwise, spanwise and wall-normal directions, respectively); Grid 2 (1600, 241 and 480 nodes); and Grid 3 (2000, 301 and 600 nodes). A comparison of structures within the young turbulent spot showed no observable differences between Grids 2 and 3. Consequently, Grid 2 was employed for this numerical simulation, featuring uniformly distributed nodes in the $x$ and $z$ directions, with a mesh refinement near the wall in the $y$ direction.

Periodic boundary conditions are applied in all horizontal directions. To achieve spatial simulation, an additional fringe region of length 100 $\delta ^*_0$ is added following the physical domain to dampen disturbances and ensure that the flow returns to the inlet condition. A no-slip boundary condition is specified at the wall surface. The upper surface of the computational domain is a uniform free stream boundary condition.

In the present simulation, the Reynolds number is expressed as $Re=U_0\delta ^*/\nu$ , where $\delta ^*$ is the displacement thickness at the local position and $\nu$ is the kinematic viscosity. The Reynolds number at the inlet of the domain is prescribed as 490. The displacement thickness $\delta ^*_0$ at the inlet is used to scale the length in the present simulation. Here, $U_0$ is the reference velocity, where $U_0$ is free stream velocity. The reference time is $\delta ^*_0/U_0$ . Turbulent spots are initialised by localised blowing of duration $t=3$ , where $t$ is the dimensionless time. The blowing location is 59 $\delta ^*_0$ downstream from the inlet. The Reynolds number is 571 at the blowing location, which is greater than the flat plate boundary layer critical Reynolds number of $ {Re}_{crit}=520$ (Schlichting & Gersten Reference Schlichting and Gersten2016). The dimensions of the blowing disturbance in the $x$ and $z$ directions are 2.4 $\delta ^*_0$ and 2 $\delta ^*_0$ , respectively. The spatial and temporal distribution of the perturbation is given by

(2.1) \begin{equation} v_{wall}=A_0f(x)f(z)f(t), \end{equation}

where the function $f$ is defined as

(2.2) \begin{equation} f(x)=S\left (\frac {x-x_{start}}{\varDelta _{rise}}\right )-S\left (\frac {x-x_{end}}{\varDelta _{fall}}+1\right ), \end{equation}

(2.3) \begin{equation} S(x) = \left \{ \begin{array}{ll} 0, & x\leqslant 0, \\[6pt] 1/\left [1+\exp ({1/(x-1)+1/x})\right ], & 0\lt x\lt 1, \\[6pt] 1, & x\geqslant 1. \end{array} \right . \end{equation}

Here, $v_{wall}$ is the wall-normal velocity at the wall surface; $\varDelta _{rise}$ and $\varDelta _{fall}$ are the rise and fall distance of the function $f(x)$ . The amplitude $A_0$ is set to 0.8, which was determined to be sufficient to trigger a turbulent spot.

2.3. Lagrangian tracking approach

The 3-D velocity fields obtained from Tomo-PIV measurements and DNS are used for Lagrangian tracking of specific particles, enabling the observation of flow patterns created by a turbulent spot. Streaklines are constructed by connecting multiple particles initiated at different times, from a fixed position. Timelines are obtained by simultaneously tracking multiple particles initiated at different initial positions but at the same initial time. Tracking a group of particles comprising an initially flat surface allows the behaviour and evolution of an initially flat material surface to be assessed. The trajectory of a particle in a fluid medium is determined by numerically integrating the following ordinary differential equation (ODE):

(2.4) \begin{equation} \overrightarrow {V}(\overrightarrow {X}(t),t)= {\rm d}\overrightarrow {X}(t)/{\rm d}t. \end{equation}

The instantaneous velocity, $V(X,t)$ , is derived from the datasets obtained from the Tomo-PIV experiments or DNS. The integration time step, ${\rm d}t$ , is used to calculate the displacement vector, ${\rm d}X(t)$ , for each integration step. The integration time step ( ${\rm d}t$ ) for the experimental tracking is 0.002 s, consistent with the time resolution of the Tomo-PIV. The integration time step ( ${\rm d}t$ ) for the DNS tracking is one dimensionless time unit. The integration process is conducted from the initial time, $t_0$ , to the final time, $t_1$ , with an initial condition of $X_0$ . A MATLAB algorithm using the Dormand–Prince method is employed to numerically solve (2.4), with absolute and relative tolerances of the ODE solver set at $10^{-6}$ . The present study employs the Lagrangian tracking methods developed by Jiang et al. (Reference Jiang, Lee, Chen, Smith and Linden2020a ,Reference Jiang, Lee, Smith, Chen and Linden b , Reference Jiang, Gu, Lee, Smith and Linden2021), which have been used effectively for transitional and turbulent boundary layers.

4.1. Development of the turbulent spot

The simulation conditions are set to be the same as the experiment conditions. The Q criterion (Jeong & Hussain Reference Jeong and Hussain1995) is employed to illustrate the initial vortical structure development. Here, Q is the second invariant of the velocity-gradient tensor, which is defined as

(4.1) \begin{equation} Q=\frac {1}{2}\left ( \frac {\partial u_i}{\partial x_j}\frac {\partial u_i}{\partial x_j}-\frac {\partial u_i}{\partial x_j}\frac {\partial u_j}{\partial x_i}\right ). \end{equation}

Einstein summation is used here. Figure 16 shows the isosurfaces of the Q criterion at $t=$ 16, 36, 56 and 76. The colour of the $Q$ isosurfaces reflects the dimensionless distance $y/\delta ^\ast_0$ to the wall surface. As shown in figure 16( $a$ ), the near-wall fluid is ejected from the wall due to the initial localised blowing disturbances, resulting in the formation of a hairpin vortex referred to as the primary hairpin vortex A. Concurrently, the ejected low-speed fluid from the wall obstructs the surrounding flow, resulting in the formation of a U-shaped vortex labelled B, characterised by streamwise legs rotating opposite to the hairpin vortex legs. The vortex C in figure 16( $a$ ) is a secondary vortex loop near the wall, induced by the initial disturbance. In figure 16( $b$ ), the primary hairpin vortex A evolves into a hairpin vortex with elongated legs and several spanwise vortices form above its legs. Figure 16( $b$ ) shows that the U-shaped vortex B has evolved into two quasi-streamwise vortices by $t=36$ . The vortex D shown in figure 16(b) consists of a pair of counter-rotating streamwise vortices located beneath the elongated legs of the primary vortex A, rotating in the opposite direction to the hairpin vortex legs. The elongated legs of the primary vortex A induce vortex D. As shown in figure 16(c), the primary hairpin vortex A at t = 36 has by $t=56$ evolved into the primary vortex head A1, a secondary hairpin vortex A2 and a tertiary hairpin vortex A3. This evolution is similar to the hairpin vortex regeneration process observed by Sabatino & Rossmann (Reference Sabatino and Rossmann2016). As shown in figure 16(d), by $t=76$ , a large number of secondary hairpin vortices have developed from primary vortex A and are labelled A1 to A5 in the order of their formation. Vortex connections form between hairpin vortex A5 and the $\Lambda$ -shaped vortex C, sharing one vortex head labelled E in figure 16(d).

Figure 16. Evolution of the turbulent spot illustrated by iso-surface $Q/(U_{\infty }/\delta ^\ast_0)^{2}=0.0025$ : ( $a$ ) $t= 16$ ; ( $b$ ) $t= 36$ ; ( $c$ ) $t= 56$ ; ( $d$ ) $t= 76$ .

Figure 17. Evolution of the turbulent spot illustrated by iso-surface $Q/(U_{\infty }/\delta ^\ast_0)^{2}=0.0025$ : ( $a$ ) $t= 96$ ; ( $b$ ) $t= 136$ ; ( $c$ ) $t= 196$ ; ( $d$ ) $t= 296$ .

Figure 18. Development of the isosurfaces of streamwise fluctuating velocity for DNS: ( $a$ ) $t= 76$ ; ( $b$ ) $t= 136$ ; ( $c$ ) $t= 196$ ; ( $d$ ) $t= 296$ . The blue and red isosurfaces correspond to the streamwise fluctuating velocity at $u'/U_{\infty }=-10\, \%$ and $u'/U_{\infty }=10\, \%$ , respectively, where $U_{\infty }$ is the free stream velocity.

Figure 17 shows the isosurfaces of the Q criterion at further developmental stages of the turbulent spot. In figure 17( $a$ ), secondary hairpin vortices develop inside the legs of vortex C, while the hairpin head E evolves into an $\Omega$ -shaped vortex head. Figure 17( $b$ ) shows the generation of hairpin vortices labelled G at the spanwise edge of the incipient turbulent spot and a hairpin vortex head, labelled F, forms on the legs of vortex C. Meanwhile, the $\Omega$ -shaped head E is essentially separated from the main body of vortex C, which closely resembles the experimental observations of hairpin vortex regeneration in a laminar boundary layer (Sabatino & Rossmann Reference Sabatino and Rossmann2016). Additionally, figure 17( $b$ ) shows that several hairpin vortices develop inside the leg of vortex C, sharing one leg with vortex C. The appearance of these hairpin vortices results in the deformation and oscillation of the legs of vortex C, which eventually break into small-scale vortices. This process adds complexity to the vortical structures within the incipient turbulent spot. An apparent breaking away of the vortex head of vortex D observed in figure 17( $c$ ) is speculated to be the result of the isosurface visualisation process using the Qcriterion. Comparing figures 17( $b$ ) and 17( $c$ ), downstream of vortex head E, a pair of streamwise vortices labelled D and several small hairpin vortices gradually weaken and eventually dissipate. These structures exhibit minimal influence on the further development of the turbulent spot. The dominant vortex structure developing into an incipient turbulent spot is the $\Lambda$ -shaped vortex labelled as C in figure 17( $a$ ). The vortical structures shown in figure 17( $c$ ) exhibit extreme complexity, as anticipated. The vortex head E has evolved into an apparent vortex loop. Due to the interaction with surrounding vortices, the vortex heads F do not develop into vortex loops like vortex head E. Figure 17( $c$ ) indicates that by $t=196$ , several vortices G develop in the streamwise direction, continually expanding in scale with streamwise distance. As the turbulent spot continues downstream, hairpin vortices continually develop at the spanwise edges of the spot, which is the reason for the continued spanwise growth of the spot, which is similar to the initial growth process shown in figure 17( $b$ ). As shown in figure 17( $d$ ), the incipient turbulent spot has matured into a characteristic arrowhead shape. The larger-scale vortices dominate the core region of the turbulent spot, while smaller-scale vortices develop and dominate in the spanwise and trailing edge regions.

The DNS fluctuating velocity field is obtained in the same way as for experimental figure 6, by subtracting the undisturbed laminar boundary layer velocity field from the instantaneous local velocity field, whose streamwise, wall-normal and spanwise components are represented as $u'$ , $v'$ and $w'$ , respectively. Figure 18 shows the isosurfaces of the streamwise fluctuating velocity at $u'/U_{\infty }=-0.1$ (blue) and $u'/U_{\infty }=0.1$ (red). Figure 18( $a$ ) shows the initial low- and high-speed regions at $t=76$ . By $t=136$ , figure 18( $b$ ) shows that a low-speed streak labelled L1 develops at the spanwise edge of the incipient turbulent spot. The vortices labelled G in figures 17( $b$ ) and 17( $c$ ) straddle the low-speed streak L1. By $t=196$ , figure 18( $c$ ), the low-speed streak L1 has grown larger. By $t=$ 196 and 296, because of the interaction between vortices and low-speed streaks, the low-speed streaks within the turbulent spot become more sinuous. In figure 18( $d$ ), the low-speed streaks within the turbulent spot have begun to amalgamate, which makes it challenging to differentiate the low-speed streaks L1. This amalgamation of low-speed streaks suggests intense interactions among the vortices within and comprising the turbulent spot, which is quite apparent in the previous figures 17( $c$ ) and 17( $d$ ). This continued development of low-speed streaks near the edges of turbulent spots is consistent with the experimental results.

Figure 19. Timelines of the turbulent spot at a sequence of stages: ( $a$ ) $t=61$ ; ( $b$ ) $t=76$ ; ( $c$ ) $t=96$ ; ( $d$ ) $t=116$ . The timelines in panels ( $a$ ) and ( $b$ ) are initiated at $t=16$ , $x/\delta ^\ast_0=16$ and $y/\delta ^\ast_0=1.2$ . The timelines in panels ( $c$ ) and ( $d$ ) are initiated at $t=21$ , $x/\delta ^\ast_0=41$ and $y/\delta ^\ast_0=1.2$ .

Figure 20. Timelines of the turbulent spot at a sequence of stages: ( $a$ ) $t=156$ ; ( $b$ ) $t=176$ ; ( $c$ ) $t=276$ . The timelines in panels ( $a$ ) and ( $b$ ) are initiated at $t=36$ , $x/\delta ^\ast_0=61$ and $y/\delta ^\ast_0=1.2$ . The timelines in panel ( $c$ ) are initiated at $t=86$ , $x/\delta ^\ast_0=121$ and $y/\delta ^\ast_0=1.2$ .

To better understand the manifold behaviour taking place within the developing spot, spanwise timelines (similar to hydrogen bubble visualisation) are generated at a series of locations and initial times to visualise the timeline patterns created by the turbulent spot during the stages of its development. The timelines are initiated at time intervals of $\Delta t=1$ and shown in plan view (viewed towards the wall) in figures 19 and 20. The colour of the tracking particles indicates the wall-normal distance. In figure 19( $a$ ), the timeline pattern of the initial structures at $t=61$ is shown. The intertwining timelines marked as A at $z/\delta ^\ast_0=0$ in figures 19( $a$ ) and 19( $b$ ) represent the elongated legs of the initial vortex A, as identified in figure 16. In figure 19( $a$ ), timelines at locations B and C begin to intertwine due to the presence of vortices. Figure 19( $b$ ) displays the timelines at $t=76$ , which reveals the initial structures that are elongated in the streamwise direction. The intertwining of timelines at locations B and C clearly shows the development of vortical motions. Quasi-streamwise vortices and a $\Lambda$ -vortex are observed in locations B and C, respectively. Figure 19( $c$ ), $t=96$ , shows a similar, but more developed, pattern than figure 19( $b$ ), displaying the quasi-streamwise vortices and $\Lambda$ -vortex. Meanwhile, three low-speed regions, labelled as D, are observed in the trailing area of the turbulent spot. Low-speed streaks, marked as I, are observed to develop to the outside of the $\Lambda$ -vortex. By figure 19( $c$ ), the timelines within the region I do not intertwine, indicating that vortices are very weak. The behaviour of these edge-region low-speed streaks I will be further discussed in § 4.2. The development of the low-speed streaks I is observed to contribute to both spanwise and streamwise growth of the turbulent spot. Figure 19( $d$ ) shows that the associated structures are further elongated downstream. The accumulation and rotation of timelines in the low-speed streaks I suggest vortical behaviour, which eventually is incorporated into the turbulent spot. The black dashed line in figure 19( $d$ ) tracks the skeleton of the $\Lambda$ -vortex C from figure 19( $b$ ). There is a tendency for the head of the $\Lambda$ -vortex C to pinch off and separate from the main body of the $\Lambda$ -vortex. Note that the timeline patterns shown in figure 19 qualitatively agree with the experimental timeline patterns shown in figures 7 and 8. Timeline patterns in figure 19 display the development process from the initial disturbance to an incipient turbulent spot, which closely resembles the hydrogen bubble visualisation results shown in figures 15–18 of Haidari & Smith (Reference Haidari and Smith1994).

Figure 20 shows the further development of the turbulent spot using the same timeline visualisation as figure 19. The timelines in figure 20( $a$ , $b$ ) are initiated at $x/\delta ^\ast_0=61$ and $t=36$ . Figure 20( $a$ ) shows that the vortices within the turbulent spot have evolved into small-scale vortices, leading to a random distribution of tracking particles within the centre of the incipient turbulent spot. Furthermore, certain tracking particles are drawn into the cores of vortices, akin to the phenomenon observed in experiments where hydrogen bubbles are drawn into vortex cores due to lower pressure within the vortex cores. In figure 20( $a$ ), the label G highlights this concentration phenomenon within the low-speed streak I region identified in figure 19, indicating the formation of vortices in that area. In figure 20( $b$ ), the tracking particles drawn into vortex cores show the presence of hairpin vortices, which are labelled G. Comparison of the timeline patterns in figures 19( $c$ , $d$ ) and 20( $a$ , $b$ ) suggests that the low-speed streaks I preceded the formation of hairpin vortices. Figure 20( $c$ ) displays the timelines of a fully developed turbulent spot at $t=276$ , which were initiated at $x/\delta ^\ast_0=121$ and $t=86$ . The random arrangement of particles within the turbulent spot signifies a degeneration to small-scale turbulence and makes it challenging to discern the individual structures within the spot. The region marked H illustrates a similar process to that shown in figures 19( $c$ ) and 19( $d$ ), wherein a low-speed region develops, followed by the formation of vortices. The region labelled II also indicates the development of low-speed regions in the trailing region of the turbulent spot. The timelines between $x/\delta ^\ast_0=121\ \mathrm{and}\ 140$ show that high- and low-speed streaks trail the turbulent spot.

4.2. Characteristics of low-speed streaks at edge of turbulent spots

The isosurfaces of streamwise fluctuating velocity and timelines shown in § 4.1 both reveal that low-speed regions develop at the edges of a turbulent spot, followed by the subsequent formation of vortices. In this section, we closely examine the localised region where this process takes place. Here, we use simultaneously generated spanwise and vertical timelines to investigate the evolution of the low-speed streaks labelled I in figure 19. The development of spanwise timelines at $t=$ 81, 91 and 101 are shown in figure 21( $a$ – $c$ ). The colour of the timelines indicates the distance from the wall surface. The timeline pattern in figure 21( $a$ ) indicates the development of a low-speed streak between $x/\delta ^\ast_0=40\ \mathrm{and}\ 50$ and $z/\delta ^\ast_0\approx -5$ . Figure 21( $b$ ) shows two bulges marked as A and B in this region at $t=91$ , which are similar to the wave-like structures on timelines (figure 1 $b$ ) observed by Jiang et al. (Reference Jiang, Lee, Smith, Chen and Linden2020b ) for an O-type transition boundary layer. As the low-speed streak travels downstream, it lifts away from the surface. In figure 21( $c$ ), almost all particles within the low-speed streak are coloured red, which indicates they have moved upward. Meanwhile, the accumulation of timelines at $z/\delta ^\ast_0=-5$ in the low-speed region suggests the development of vortices. These vortices eventually become part of the turbulent spot, contributing to the expansion in a spanwise direction.

Figure 21. Plan-view and side-view timeline patterns displaying the temporal evolution of the low-speed streak labelled I in figure 19. The spanwise timelines at $t=81, 91$ and 101, are shown in panel ( $a{-}c$ ) and are initiated at $x/\delta ^\ast_0=31$ , $y/\delta ^\ast_0=1.2$ and $t=31{-}64$ . The vertical timelines patterns at $t=$ 81, 91 and 101 are shown in panel ( $d{-}f$ ) and are initiated at $x/\delta ^\ast_0=31$ , $z/\delta ^\ast_0=-5.4$ . The red, orange, blue and green particles in panel ( $d$ – $f$ ) correspond to streakline patterns at heights $y/\delta ^\ast_0=1, 1.5, 2, 2.5$ , respectively.

Figure 22. Temporal evolution of the peaks of the two bulges in figure 21( $b$ ) over the time $t=81{-}91$ with a time interval of d $t=2$ : ( $a$ ) A; ( $b$ ) B.

The vertical timelines are initiated at the centre of the low-speed streak ( $z/\delta ^\ast_0=-5.4$ ) at time intervals of d $t=2$ . Figure 21( $d$ – $f$ ) displays the vertical timelines at $t=$ 81, 91 and 101 (the same times as the horizontal timelines). The different coloured particles indicate the behaviour of streaklines originating at different wall-normal heights. Figure 21( $d$ ) shows the presence of an inflectional region in the vertical timelines within the low-speed region, indicating instability within this region. As the low-speed streak lifts up, the vertical timelines in figure 21( $e$ ) show an accumulation of timelines in the inflectional region. Subsequently, the vertical timelines in figure 21( $f$ ) begin to intertwine, indicating the development of vortices.

Figure 23. Development of a material surface initiated at $y/\delta ^\ast_0=$ 1.5 and $t=76$ in the low-speed region labelled I in figure 19: ( $a$ ) $t=76$ ; ( $b$ ) $t=82$ ; ( $c$ ) $t=88$ ; ( $d$ ) $t=94$ .

Figure 24. Development of a material surface initiated at $y/\delta ^\ast_0=1.0$ and $t=76$ in the low-speed region labelled I in figure 19: ( $a$ ) $t=76$ ; ( $b$ ) $t=82$ ; ( $c$ ) $t=88$ ; ( $d$ ) $t=94$ .

Figure 22 shows the temporal evolution of the peaks of the two bulges marked as A and B in figure 21( $b$ ). Each curve in figure 22 is obtained by connecting the particles that pass through the peak of the bulge at specific moments. Figure 22 shows that the bulge peaks move downstream and lift away from the wall surface. The bulge peaks travel downstream at 55 % of the free stream velocity and move upward at 2 %–3 % of the free stream velocity. These findings agree with the observation by Jiang et al. (Reference Jiang, Lee, Smith, Chen and Linden2020b ) for low-speed streak behaviour in a turbulent boundary layer.

To further assess the lift-up behaviour of low-speed streak I, a material surface is initiated parallel to the $x{-}z$ plane at $y/\delta ^\ast_0=$ 1.5 within the region of the low-speed streak I. The colour of the material surface indicates the distance from the wall. Figure 23( $a$ ) shows the initial material surface. At a subsequent time of $t=82$ , the material surface displays two 3-D bulges, revealing a lift-up behaviour. Figure 23( $c{,}d$ ) shows that as the two 3-D bulges progressively develop, the middle portion of the material surface exhibits wave-like peaks, which closely resemble the wave pattern on the material surface (figure 1 $a$ ) observed by Jiang et al. (Reference Jiang, Lee, Chen, Smith and Linden2020a ) in the late stage of O-type transition. Simultaneously, the lateral sides of the material surface undergo a downflow, indicating a significant downward movement of fluid. The deformation of the material surface indicates that the low-speed streak lifts up and exhibits wave-like behaviour.

To further illustrate the near-wall behaviour of the low-speed streak region I, the evolution of a material surface even closer to the wall at $y/\delta ^\ast_0 =1$ is examined in figure 24. Contour lines and colouration are again used to illustrate height variations of the material surface. The development of the contour lines in figure 24 again shows two wave-like lift-up regions marked as W1 and W2, which were also observed in the timeline patterns of figure 21. As shown in figure 24( $d$ ), two wave-like lift-up regions W1 and W2 form. The blue-coloured region at the right edge of the material sheet again indicates fluid moving towards the wall. This observation indicates that as the low-speed streaks lift up from the wall, higher-speed fluid moves down towards the wall, creating high-shear layers.

Figure 25. Cross-sections of streamwise fluctuating velocity and perturbation vorticity for the 3-D wave W1. ( $a$ – $c$ ) Colour maps and contour lines of $u'/U_{\infty }$ . In panel ( $d$ – $l$ ), the dashed (negative) and solid (positive) contour lines of the streamwise fluctuating velocity are overlaid on the colour maps of the perturbation vorticity. ( $d$ – $f$ ) Colour maps of $\omega '_x$ and contour lines of $u'/U_{\infty }$ ; ( $g$ – $i$ ) colour maps of $\omega '_y$ and contour lines of $u'/U_{\infty }$ ; ( $j$ – $l$ ) colour maps of $\omega '_z$ and contour lines of $u'/U_{\infty }$ . ( $a$ , $d$ ,g, $j$ ) At $t=82$ and $x/\delta ^\ast_0=43$ ; ( $b$ , $e$ , $h$ , $k$ ) at $t=88$ and $x/\delta ^\ast_0=46$ ; ( $c$ , $f$ , $i$ , $l$ ) at $t=94$ and $x/\delta ^\ast_0=49$ .

As observed in this paper, and in the works of Jiang et al. (Reference Jiang, Lee, Smith, Chen and Linden2020b , Reference Jiang, Gu, Lee, Smith and Linden2021, Reference Jiang, Lefauve, Dalziel and Linden2022), the lift-up of a low-speed streak appears to develop as the amplification of 3-D waves. The lift-up of near-wall low-speed fluid is accompanied by a downward sweep of higher-speed fluid at the flanks of the streak; this interaction of high- and low-speed fluid results in the development of local high-shear layers. The perturbation vorticity vector $\omega '_i$ is the curl of the fluctuating velocity vector $u'_i$ . To demonstrate the development of vorticity around a 3-D wave-like structure, the temporal evolution of the perturbation vorticity in the $y{-}z$ plane at the wave-like peak W1 identified in figures 23 and 24 is examined.

Figure 26. Plan-view and side-view timeline patterns showing the temporal evolution of the low-speed streak labelled II in figure 20. Spanwise timelines at $t=266, 271$ and 276 are shown in panels ( $a$ )–( $c$ ), respectively, and were initiated at $x/\delta ^\ast_0=121$ , $y/\delta ^\ast_0=1.2$ and $t=236-266$ . Vertical timelines at $t=$ 266, 271 and 276 are shown in panels ( $d$ )–( $f$ ) and were initiated at $x/\delta ^\ast_0=121$ , $z/\delta ^\ast_0=-17.3$ . The red, orange, blue and green particles in panels ( $d$ )–( $f$ ) correspond to streakline patterns at $y/\delta ^\ast_0=1.0, 1.5, 2.0, 2.5$ , respectively.

Figure 27. Development of material surfaces at $y/\delta ^\ast_0=1.2$ in the low-speed region labelled II in figure 20: ( $a$ ) $t=261$ ; ( $b$ ) $t=268$ ; ( $c$ ) $t=275$ ; ( $d$ ) $t=282$ .

Figure 25 shows the cross-sections of streamwise fluctuating velocity $u'/U_{\infty }$ and perturbation vorticity $\omega '_i$ for the 3-D wave W1. Figure 25( $a{-}c$ ) shows that as the 3-D wave W1 amplifies, the position of the velocity deficit region undergoes a slight lift-up, accompanied by an expansion of its area and an increase in intensity. To display the positional relationship between the velocity deficit region and regions of vorticity concentration, figure 25( $d{-}f$ ) overlaps the dashed (negative) and solid (positive) contour lines of the streamwise fluctuating velocity on colour maps of perturbation vorticity $\omega '_i$ . Figure 25( $d$ – $f$ ) shows that as the 3-D wave W1 lifts up, a concentration of streamwise perturbation vorticity begins to develop at the interface between the low-speed and high-speed regions. In figure 25( $f$ ), the strong concentration of streamwise perturbation vorticity $\omega '_x$ at the right side of the velocity deficit region is attributed to a strong downward sweep motion of high-speed fluid on that side, as observed from the deformation of the material surface shown in figures 24( $c$ ) and 24( $d$ ). Figure 25( $g$ – $i$ ) shows that positive and negative wall-normal perturbation vorticity $\omega '_y$ flank the velocity deficit region. Compared with the wall-normal perturbation vorticity $\omega '_y$ on the left side of the velocity deficit region, a stronger concentration of wall-normal perturbation vorticity $\omega '_y$ is observed on the right side of the velocity deficit region, which is attributed to the presence of strong high-speed fluid on the right side. Figure 25( $j$ – $l$ ) shows the colour maps of spanwise perturbation vorticity $\omega _z'$ with the overlaid contour lines of $u'/U_{\infty }$ . As low-speed fluid lifts away from the wall, negative spanwise perturbation vorticity $\omega '_z$ appears at the top of the velocity deficit region. These results show that as the 3-D wave W1 develops, vorticity develops along its boundaries, serving as precursors to vortices. The difference in vorticity strength on either side of the velocity deficit region implies that the resulting vortices may exhibit asymmetry.

A low-speed region at the trailing edge of the spot, labelled II in figure 20( $c$ ), was also examined. The formation of vortices at the trailing edge of the mature turbulent spot is closely associated with the development of low-speed region II. In figure 26, spanwise and vertical timelines are used to illustrate the behaviour of the low-speed region marked II. Figures 26( $a$ )–26( $c$ ) show the spanwise timeline patterns for the low-speed region II at $t=266,271$ and 276. Figures 26( $a$ )–26( $c$ ) show the presence of a low-speed region in the vicinity of $z/\delta ^\ast_0=17.3$ , with an adjacent high-speed region located between $z/\delta ^\ast_0=16$ and $z/\delta ^\ast_0=17$ . As the timelines move downstream, the timelines within the low-speed region II begin to lift up. The timelines on both sides of the low-speed region do not exhibit any rotational behaviour, indicating that vortices have not yet developed. Figures 26( $d$ )–26( $f$ ) show the vertical timelines at $t=266,271$ and 276, which are initiated at the centre of the low-speed region ( $z/\delta ^\ast_0=-17.3$ ) at time intervals of d $t=2$ . Due to the sight uplift of the low-speed region, the vertical timelines only exhibit a mild inflection profile. The low-speed region II shown in figure 26 is at an earlier development stage than the low-speed streak I shown in figure 21. The results suggest that the formation of vortices is correlated with the degree of lift-up of the low-speed streaks.

In figure 27, the spatial-temporal evolution of the low-speed region II shown in figure 26( $a{-}c$ ) is further examined using a material surface initiated within this region at $t=261$ and $y/\delta ^\ast_0=1.2$ . Figure 27( $b$ ) shows that the material surface displays a slight lift-up of the low-speed region. By figure 27( $c$ ), a peak–valley pattern has developed, which is similar to the material surface pattern of wave structure in an O-type transitional boundary layer (Jiang et al. Reference Jiang, Lee, Chen, Smith and Linden2020 a ). Consistent with the material surface behaviour of the low-speed region I shown in figures 23 and 24, the material surface pattern of figures 27( $c$ ) and 27( $d$ ) suggest a wave-like behaviour of the low-speed region II. Figure 27( $d$ ) shows a more prominent downward movement of the material surface, coloured blue, on the right side of the surface, suggesting a strong downward movement of high-speed fluid.

This section examined the DNS results for two low-speed streaks developing at the edges of a turbulent spot. The timeline patterns and material surface behaviour closely resemble the wave structure patterns observed by Jiang et al. (Reference Jiang, Lee, Chen, Smith and Linden2020a ,Reference Jiang, Lee, Smith, Chen and Linden b ), which suggests that these low-speed streaks behave like 3-D waves. The vorticity concentrated around the 3-D wave W1, as shown in figure 25, is closely related to the formation of the hairpin vortices at the edge of a turbulent spot. These findings are consistent with the experimental results illustrated in § 3.