2.1 DNS of temporal transitional channel flow

A diagram of the computation domain of channel flow is shown in figure 1. The sides of the domain in the streamwise $x$ -, the wall-normal $y$ - and the spanwise $z$ -directions are $L_{x}$ , $L_{y}$ and $L_{z}$ , respectively. The three-dimensional incompressible Navier–Stokes equations for the velocity $\boldsymbol{u}=(u,v,w)$ are non-dimensionalized by the channel half-height ${\it\delta}$ and the bulk velocity $U_{b}$ as

where $p$ is the pressure, ${\it\nu}$ is the kinetic viscosity, $Re_{b}=U_{b}{\it\delta}/{\it\nu}$ denotes the Reynolds number with $U_{b}=\int _{0}^{2{\it\delta}}u\,\text{d}y/L_{y}$ . The channel half-height is set as ${\it\delta}=1$ , and the flow is driven by a time-dependent external force $\boldsymbol{f}(t)$ to maintain a constant flow flux with $U_{b}\approx 1$ in the streamwise direction.

Figure 1.

Diagram of the computational domain.

For DNS, equation (2.1) is solved by the Fourier–Chebyshev pseudo-spectral method (Kim et al. Reference Kim, Moin and Moser1987). The non-slip conditions are applied at the walls at $y=0$ and $y=2{\it\delta}$ , and the periodic boundary conditions are applied in the streamwise and spanwise directions. The nonlinear term is dealiased using the two-thirds truncation method (Canuto et al. Reference Canuto, Hussaini, Quateroni and Zang1988). The low-storage third-order semi-implicit Runge–Kutta method (Spalart, Moser & Rogers Reference Spalart, Moser and Rogers1991) is applied for the temporal integration. The number of grids $N_{x}$ , $N_{y}$ and $N_{z}$ in all of the directions and other parameters of the DNS are listed in table 1. The wall-friction Reynolds number $Re_{{\it\tau}}=u_{{\it\tau}}{\it\delta}/{\it\nu}$ in this table is calculated after the flow reaches the fully developed turbulent state, where $u_{{\it\tau}}=\sqrt{{\it\tau}_{w}/{\it\rho}}$ is the wall-friction velocity with the wall shear stress ${\it\tau}_{w}$ and the density  ${\it\rho}$ . The grid sizes ${\rm\Delta}x$ and ${\rm\Delta}z$ in the streamwise and the spanwise directions are scaled in wall units as ${\rm\Delta}x^{+}={\rm\Delta}x/{\it\delta}_{{\it\nu}}$ and ${\rm\Delta}z^{+}={\rm\Delta}z/{\it\delta}_{{\it\nu}}$ with the viscous length scale ${\it\delta}_{{\it\nu}}={\it\delta}u_{{\it\tau}}/{\it\nu}$ for the fully developed turbulent state. In the wall-normal direction, ${\rm\Delta}y_{w}^{+}$ denotes the distance of the first grid point from the wall scaled by ${\it\delta}_{{\it\nu}}$ .

Table 1.

Summary of DNS parameters.

The initial condition is set as the laminar Poiseuille flow with three Tollmien–Schlichting (TS) waves to trigger the K-type transition. The velocity profile of the Poiseuille flow is

where $U_{0}=1.5$ is the laminar velocity on the centreline of the channel. The initial disturbances (see Sandham & Kleiser Reference Sandham and Kleiser1992; Schlatter, Stolz & Kleiser Reference Schlatter, Stolz and Kleiser2004)

consist of a two-dimensional TS wave with the amplitude $A_{1}=3.5\,\%$ and a pair of superimposed oblique three-dimensional waves with the amplitude $A_{2}=0.09\,\%$ , which are similar to the simulations in Gilbert & Kleiser (Reference Gilbert, Kleiser, Kline and Afgan1990). In (2.3), $\boldsymbol{u}_{2D}(y)$ and $\boldsymbol{u}_{3D}(y)$ are the eigenfunctions of the two-dimensional TS wave and the three-dimensional TS waves, respectively. They are obtained by solving the Orr–Sommerfeld equations using a fourth-order compact difference method (Malik Reference Malik1990). With different temporal frequencies, the streamwise wavenumber is ${\it\alpha}=1.12$ and the spanwise wavenumber is ${\it\beta}=\pm 2.1$ for the TS waves. The numerical solver and initial disturbances used in this DNS were verified by Zhao et al. (Reference Zhao, Xia, Shi, Xiao and Chen2014).

The flow gradually evolves from the laminar state into the fully developed turbulent state with the imposed initial disturbances. The temporal evolution of the wall-friction Reynolds number $Re_{{\it\tau}}$ is shown in figure 2. The transition in this calculation can be roughly divided into three periods. First, the initial disturbances gradually amplify at $t<80$ , with the integral flow quantity such as $Re_{{\it\tau}}$ remains at the laminar value. After the slow amplification of the disturbances, the integral quantities change significantly in a short period, which implies the onset of the late stages of transition at $80\leqslant t\leqslant 108$ . The flow structures undergo the first significant topological changes around $t=110$ (shown in § 3) followed by a series of interactions of vortical structures, and finally the flow develops into the fully turbulent state around $t=200$ . The evolution of flow structures in the beginning of the late transitional stages for $80\leqslant t\leqslant 108$ will be studied in §§ 3–5 in detail.

Figure 2.

Temporal evolution of the wall-friction Reynolds number $Re_{{\it\tau}}$ in the K-type temporal transitional channel flow. The initial and the ending tracking times are marked by red circles.

2.2 Backward-particle-tracking method for the Lagrangian field

The transport equation for the Lagrangian scalar field ${\it\phi}(\boldsymbol{x},t)$ reads

and isosurfaces of the scalar field are material surfaces in the temporal evolution. A set of ordinary differential equations (ODEs) are converted from (2.4) to calculate trajectories of fluid particles as

where $\boldsymbol{X}(\boldsymbol{x}_{0},t_{0}\mid t)$ is the location at time $t$ of the particle which was located at $\boldsymbol{x}_{0}$ at the initial time $t_{0}$ and $\boldsymbol{V}(\boldsymbol{x}_{0},t_{0}\mid t)$ is the particle velocity at time  $t$ .

A backward-particle-tracking method is implemented to solve (2.5). More details about this numerical scheme can be found in Yang et al. (Reference Yang, Pullin and Bermejo-Moreno2010) and Yang & Pullin (Reference Yang and Pullin2011b ). First, the particles are placed on the uniform grids $N_{x}^{L}\times N_{y}^{L}\times N_{z}^{L}=768\times 768\times 768$ at a particular time $t$ . Here, the superscript $L$ denotes the grids used for the Lagrangian scalar field. Then, these particles are tracked backwards to the initial time $t_{0}$ with the Eulerian velocity field $\boldsymbol{u}$ saved on disk from DNS. After the backward tracking, the initial locations of the particles $\boldsymbol{x}_{0}$ can be obtained, and the corresponding flow map is

Then the Lagrangian field ${\it\phi}(\boldsymbol{x},t)$ at any given time $t$ can be obtained as

from (2.4) and (2.6) with the initial Lagrangian field ${\it\phi}_{0}\equiv {\it\phi}(\boldsymbol{x}_{0},t_{0})$ .

2.3 Determination of the initial Lagrangian field

Although the initial Lagrangian field ${\it\phi}_{0}=f(x,y,z)$ can be an arbitrary three-dimensional scalar field in a general case, several criteria are proposed to uniquely determine ${\it\phi}_{0}$ that is suitable for this particular laminar–turbulent transition problem.

First, in order to investigate the evolution of vortical structures, the isosurfaces of ${\it\phi}_{0}$ should be vortex surfaces (Yang & Pullin Reference Yang and Pullin2010). The vortex surface is a surface composed of vortex lines, which implies that the local vorticity vector is tangent at every point on the surface. As shown in figure 3, the unit vector $\boldsymbol{n}=(n_{x},n_{y},n_{z})\equiv \boldsymbol{{\rm\nabla}}{\it\phi}/g$ is the normal of the material surface, where $g\equiv (\boldsymbol{{\rm\nabla}}{\it\phi}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}{\it\phi})^{1/2}$ is the magnitude of the scalar gradient. Thus, it requires that the vorticity ${\bf\omega}\equiv \boldsymbol{{\rm\nabla}}\times \boldsymbol{u}$ is perpendicular to $\boldsymbol{n}$ in the whole field. In the initial state, the criterion can be written as

where the subscript 0 denotes a quantity at the initial time. Considering the base flow without the initial disturbances, the laminar Poiseuille flow (2.2) or any other simple shear flow as $\boldsymbol{u}_{0}=(h(y),0,0)$ , where $h(y)$ is an arbitrary smooth function of $y$ , the vorticity only has one component in the spanwise direction as ${\bf\omega}_{0}=(0,0,{\it\omega}_{z})$ . Thus, ${\it\phi}_{0}$ is reduced to a two-component scalar field as ${\it\phi}_{0}=f(x,y)$ .

Figure 3.

Diagram of the geometry of a material surface and typical vortex lines attached on the surface. The dashed vector $\boldsymbol{n}$ is the normal of the surface, and solid vectors are the tangential vector $\boldsymbol{t}_{{\it\omega}}$ , the normal vector $\boldsymbol{n}_{{\it\omega}}$ and the binormal vector $\boldsymbol{b}_{{\it\omega}}$ of a vortex line.

Second, the growth of disturbances in the natural transition is slow in a long period when the flow state is nearly laminar. To focus on the evolution of material surfaces in the transition and avoid the deformation under the mean shear, ${\it\phi}_{0}$ is expected to stay invariant as $\partial {\it\phi}_{0}/\partial t=0$ in a laminar shear flow $\boldsymbol{u}_{0}=(h(y),0,0)$ . With the transport equation (2.4) of ${\it\phi}$ , we obtain

so ${\it\phi}_{0}$ can be reduced as ${\it\phi}_{0}=f(y)$ . The isosurfaces of ${\it\phi}_{0}$ are streamwise–spanwise planes at different wall distances in figure 1, and they are both vortex and stream surfaces in the laminar state. In contrast, an arbitrary initial surface with an inclination angle with the streamwise direction is not a stream surface, and it can be persistently stretched by the mean shear. Eventually it evolves to nearly streamwise–spanwise planes within a finite time.

Finally, ${\it\phi}_{0}$ is required to be non-constant and as smooth as possible. Based on a polynomial expansion of ${\it\phi}_{0}=f(y)$ , we can assume ${\it\phi}_{0}=C_{0}+C_{1}y$ where $C_{0}$ and $C_{1}$ are constants. Without loss of generality, the initial scalar field is set as ${\it\phi}_{0}=y$ with $C_{0}=0$ and $C_{1}=1$ .

2.4 Tracking interval

The imposed growing disturbances such as (2.3) can break the temporal invariance of the scalar field. The initial tracing time $t_{0}=80$ is selected when the amplitudes of the initial disturbances are small enough, so that every initial material surface can be approximated as a vortex sheet composed of vortex lines. The choice of $t_{0}=80$ , instead of the theoretical value $t_{0}=0$ , can avoid a very long particle tracking time for $t\leqslant 80$ and greatly reduce the computational cost with negligible deviations. We find that the statistical results are not sensitive to the value of $t_{0}$ for $t_{0}\leqslant 80$ from numerical experiments.

The evolution and dynamics of material surfaces can be studied from the evolution of material surfaces in the late transitional stages for $80\leqslant t\leqslant 108$ except for a few of qualitative visualizations at $t=112$ . It is noted that the particle-backward-tracking method has no numerical diffusion. Although it ensures the geometrical and topological conservation properties for material surfaces, the fragmentation for the under-resolved material surfaces can occur for long times (see Yang et al. Reference Yang, Pullin and Bermejo-Moreno2010). To avoid the effect of numerical artifacts caused by the fragmentation on statistical results, we select the upper limit of the tracking period as $t=108$ . During this period, satisfactorily resolved material surfaces can be obtained by using the high-resolution grids $N_{x}^{L}\times N_{y}^{L}\times N_{z}^{L}$ for the Lagrangian field. Furthermore, iteration strategies with local grid refinement can be considered in the further study to reduce the computational cost.

3.1 Evolution of material surfaces

In the transition process, all of the material surfaces evolve from the streamwise–spanwise planes at different distances from the wall at the initial tracing time $t_{0}$ , but their evolutions can be different. The typical material surfaces studied in the present work are listed in table 2. They are labelled by the Lagrangian scalar field scaled by ${\it\delta}$ or ${\it\delta}_{{\it\nu}}$ that corresponds to the laminar state or the fully developed turbulent state, respectively. The evolution of the material surfaces in the late transition at $80\leqslant t\leqslant 112$ will be discussed from ${\it\phi}=0.024$ to ${\it\phi}=0.722$ .

Table 2.

Typical material surfaces labelled by scaled ${\it\phi}_{0}=y$ .

First of all, the material surface initially very close to the wall with ${\it\phi}=0.024$ is only slightly wrinkled in the evolution. Since the non-slip condition is applied at the wall and the amplitudes of the initial disturbances in this region are very small, the geometry of the surface has no obvious change.

In contrast, the material surface of ${\it\phi}=0.217$ , which is more distant from the wall than that of ${\it\phi}=0.024$ at the initial time, shows significant deformation in the evolution in figure 4. The planar surface is first folded and then rolled up at $t=90$ in figure 4(a). A triangle-shaped bulge is generated around $t=100$ in figure 4(b), and then gradually evolves into a typical hairpin-shaped structure in figure 4(c). Finally, the ‘head’ of the hairpin-like structure is lifted and the two ‘legs’ are stretched in figure 4(d). This stretching process was also observed by tracking an ensemble of material lines initially parallel to the wall in DNS of a transitional boundary layer in Rist & Fasel (Reference Rist and Fasel1995), but the evolution of the material surface can display more details, such as the rolling up of tube-like structures, than the scattered lines, and provide quantitative results based on the conditional statistics on the surfaces. It is noted that there are some fragments near the head and legs of the hairpin-like structure in figure 4(c,d). Since these parts of the surface endure persistent stretching and rolling during the evolution, they are partially under-resolved for the isosurface visualization. Although the fragmentation can be resolved by using higher resolutions for the scalar field as discussed in § 2.4, the current resolution is enough to show the skeleton of the hairpin-like structures.

Figure 4.

Temporal evolution of the material surface with ${\it\phi}=0.217$ or ${\it\phi}^{+}=45$ : (a)  $t=90$ ; (b)  $t=100$ ; (c)  $t=106$ ; (d)  $t=112$ . The surface is colour-coded by the wall-normal coordinate $y$ . The surfaces at $t=106$ and $t=112$ are partially under-resolved.

The evolution of the surface of ${\it\phi}=0.433$ is shown in figure 5. This process is mainly influenced by the lift of the hairpin-like structure as that in figure 4. Before the lift, this surface is only slightly disturbed in figure 5(a,b). A thumb-shaped bulge in figure 5(c) at $t=106$ is above the hairpin-like structure in figure 4(c). The material surface of ${\it\phi}=0.433$ warps around the structure of ${\it\phi}=0.217$ and also rolls into a hairpin-shaped structure in figure 5(d). A similar rolling-up process was observed from the evolution of a vortex layer from an initially disturbed sheet in Moin, Leonard & Kim (Reference Moin, Leonard and Kim1986), but the tracking object represented by scattered vortex lines was initially set in the mean-velocity profile of a fully developed channel flow rather than a transitional flow. Finally, the secondary hairpin-shaped structure is generated behind the primary one in figures 4(d) and 5(d). Furthermore, the interactions between different hairpin-like structures will be addressed in § 3.2, and the dynamics in the evolution of the hairpin-like structures will be discussed in § 5 in detail.

Figure 5.

Temporal evolution of the material surface with ${\it\phi}=0.433$ or ${\it\phi}^{+}=90$ : (a)  $t=90$ ; (b)  $t=100$ ; (c)  $t=106$ ; (d)  $t=112$ . The surface is colour-coded by the wall-normal coordinate $y$ . The surface at $t=112$ is partially under-resolved.

For the evolution of the material surface that is initially remote from the wall, the surface of ${\it\phi}=0.722$ is almost undisturbed until $t=106$ (not shown). Then it is influenced by the rapid elevation of the hairpin-like structure as shown in figure 4, and the rolling-up process is similar to the surface of ${\it\phi}=0.433$ .

3.2 Evolution of Lagrangian fields on the streamwise and wall-normal plane cut

The evolution of the Lagrangian scalar field on the streamwise and wall-normal ( $x$ – $y$ ) plane at $z=L_{z}/2$ is used to study evolving material surfaces, because the most rapidly growing fluctuations, the tip of the triangular bulge and the head of the hairpin-like structures occur on this ‘peak plane’ (Klebanoff et al. Reference Klebanoff, Tidstrom and Sargent1962; Kleiser & Zang Reference Kleiser and Zang1991). The number of two-dimensional grids for the Lagrangian field on this plane is increased as $N_{x}^{L}\times N_{y}^{L}=1536\times 1536$ to achieve the visualization with high resolution.

The temporal evolution of the Lagrangian field on the $x$ – $y$ plane at the peak position $z=L_{z}/2$ in the late transitional flow is shown in figure 6. The scalar is colour-coded by $0\leqslant {\it\phi}\leqslant 2$ . Corresponding to the structures in figure 4(b), the heads of primary hairpin-like structures marked as $A$ and $A^{\prime }$ are stretched out from the triangular bulges at $t=106$ in figure 6(a), and they are then lifted in figure 6(b). The spiral patterns with large scalar gradients indicate swirling motions near the heads of hairpin-like structures.

Subsequently the secondary hairpin-like structures marked as $B$ and $B^{\prime }$ at $t=108$ form in the upstream of the primary ones. Both structures continue lifting, and at the mean time the tertiary hairpin-like structures marked as $C$ and $C^{\prime }$ are generated in figure 6(c). It is noted that the tertiary hairpin-like structures cannot fully develop in the transitional channel flow, because they are suppressed by structures $A$ and $A^{\prime }$ in figure 6(d). With the periodic boundary condition in the streamwise direction, the structure $A$ reenters the domain from the left boundary and then catch $C$ . Furthermore, since the induced velocity of the vortical structure $A$ at the location of $C$ is downward, the lift of $C$ is also suppressed. It is noted that the structures marked as $A^{\prime }$ , $B^{\prime }$ and $C^{\prime }$ can be considered as the images of $A$ , $B$ and $C$ with corresponding phase differences in the streamwise direction owing to the reflectional symmetry with the mid-plane of the channel flow.

Figure 6.

Temporal evolution of the Lagrangian scalar field on the spanwise and wall-normal plane cut at the peak position $z=L_{z}/2$ : (a)  $t=106$ ; (b)  $t=108$ ; (c)  $t=110$ ; (d)  $t=112$ . The heads of the hairpin-like structures from the lower half of the channel are marked by $A$ , $B$ and $C$ , and those from the upper half are marked by $A^{\prime }$ , $B^{\prime }$ and $C^{\prime }$ .

It can be inferred from figure 6(d) that the hairpin-like structures from upper and bottom halves of the channel have strong interactions, which were observed and called the ‘cross-channel interactions’ by Sandham & Kleiser (Reference Sandham and Kleiser1992). In particular, the structure $A$ appears to collide with $B^{\prime }$ at $110\leqslant t\leqslant 112$ and they are entangled with each other. This structure interaction or reconnection causes flow to break down into the turbulent state, because a vortex surface can have violent topological changes, and then evolve into small-scale, tube-like structures with stretching and twisting processes (Yang & Pullin Reference Yang and Pullin2011a ). At this point, the evolution of material surfaces and vortex surfaces with the same initial condition can have essential differences, which will be addressed in § 5.1.

4.1 Deformation of material surfaces

In order to identify influential material surfaces, we first analyse the geometric deformation of different material surfaces. The level of deformation of the material surfaces can be quantified by the statistics for the magnitude of the scalar gradient $g$ , which is critical in scalar transport and is straightforward to be obtained in numerical or experimental data set. The transport equation of $g$ can be derived from (2.1) and (2.4) as

where $\text{D}/\text{D}t\equiv \partial /\partial t+\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}$ is the material derivative, $\unicode[STIX]{x1D64E}=(\partial u_{i}/\partial x_{j}+\partial u_{j}/\partial x_{i})/2$ is the rate-of-strain tensor. From (4.1), the evolution of $g$ is mainly controlled by the alignment of the scalar gradient and the most compressive strain direction of the rate-of-strain tensor (see Brethouwer, Hunt & Nieuwstadt Reference Brethouwer, Hunt and Nieuwstadt2003).

For the initial field ${\it\phi}_{0}=y$ with $g=1$ in the whole field at the initial time, ${\it\phi}$ is time invariant in the laminar state (2.2). The imposed disturbances cause the evolution of ${\it\phi}$ and $g$ . In figure 7, the conditional averages $\langle g\mid {\it\phi}\rangle$ for all the surfaces monotonically increase in the late transition, where $\langle \cdot \rangle$ denotes the volume average. This growth owing to the local straining motion can be explained from the statistical nature of the transport equation (4.1), which was discussed in Yang et al. (Reference Yang, Pullin and Bermejo-Moreno2010).

Figure 7.

Averaged $\langle g\mid {\it\phi}\rangle$ conditioned on different material surfaces. The dashed line denotes the level of $0.5\langle g\mid {\it\phi}\rangle _{max}$ at $t=108$ .

Meanwhile, the growth of $\langle g\mid {\it\phi}\rangle$ corresponds to the evolution of the structures presented in § 3. In figure 7, we can see that the surfaces with $0.1<{\it\phi}<0.35$ have significant deformation with $\langle g\mid {\it\phi}\rangle >0.5\langle g\mid {\it\phi}\rangle _{max}$ at $t=108$ . In particular, the surface of ${\it\phi}=0.217$ (see also table 2) shown in figure 4 is very close to the peaks of $\langle g\mid {\it\phi}\rangle$ at $t=106$ and $t=108$ . It is interesting that this surface with the most intensive deformation shows the generation of the signature hairpin-like structure. Furthermore, the range of ${\it\phi}$ with significant deformations is broaden with time in figure 7, which implies that the elevation of the hairpin-like structures near ${\it\phi}=0.217$ can have an impact on the motion of neighbouring material surfaces.

As an additional note, the scalar gradient magnitude $g$ is essentially similar to the DLE field that was used to identify the hyperbolic LCS in Green et al. (Reference Green, Rowley and Haller2007), which is discussed in appendix A. Compared with the contour lines of the DLE field on two-dimensional plane cuts in Green et al. (Reference Green, Rowley and Haller2007) rather than the explicit identification of LCSs as parameterized material surfaces (Haller Reference Haller2015), figures 4 and 5 display the evolution of uniquely defined material surfaces in three-dimensional space in the transition.

The conditional averaged $\boldsymbol{n}$ is used to characterize the geometry of the evolving material surfaces. As shown in figure 8, the influential material surface of ${\it\phi}=0.217$ with the maximum deformation is selected to study the conditional averages of the absolute values of normal components $n_{x}$ , $n_{y}$ and $n_{z}$ , and other surfaces initially located at different wall distances with ${\it\phi}=0.024,0.433$ and 0.722 are also presented for comparison. With $\boldsymbol{n}=(0,1,0)$ for the initial planar surfaces, the averaged $|n_{y}|$ decreases from $|n_{y}(t_{0})|=1$ at $t=t_{0}$ , while both $|n_{x}|$ and $|n_{z}|$ grow from $|n_{x}(t_{0})|=|n_{z}(t_{0})|=0$ .

Figure 8.

Temporal evolution of the conditional average of normal components for material surfaces with different initial wall distances, and the volume-averaged normal component: (a) absolute value of the streamwise component $|n_{x}|$ ; (b) absolute value of the wall-normal component $|n_{y}|$ ; (c) absolute value of the spanwise component $|n_{z}|$ .

In the evolution, the conditional average for the surface of ${\it\phi}=0.217$ is much earlier and more disturbed than those from other initial wall distances, which is similar to $\langle g\rangle$ . In particular, $|n_{x}|$ amplifies much earlier than $|n_{z}|$ before the triangular bulge is generated in figure 4(b), so the Lagrangian field is essentially two-component in terms of $x$ and $y$ before $t=85$ only with a slight increase of $|n_{z}|$ .

In addition, the generation of the triangular bulge quantified by the increase of $|n_{x}|$ and $|n_{z}|$ corresponds to the crucial event usually mentioned as ‘peak–valley splitting’ (Klebanoff et al. Reference Klebanoff, Tidstrom and Sargent1962; Kleiser & Zang Reference Kleiser and Zang1991) in transitional wall flows. A qualitative analysis about the formation mechanism of the triangular bulge is presented in appendix B. It is noted that the amplitude of the initial two-dimensional TS wave is much larger than those of the three-dimensional TS waves in (2.3), and the two-dimensional TS wave amplifies earlier than the three-dimensional ones (Kleiser & Zang Reference Kleiser and Zang1991). Therefore, the flow is approximately quasi-two-component until three-dimensional disturbances are amplified by the secondary instability (Herbert Reference Herbert1988). This transition can be represented by the formation of the triangular bulge for the most deformed surface with the evolution of $|n_{x}|$ and $|n_{z}|$ . Thus, the peak–valley splitting can be illustrated in a quantitative Lagrangian approach.

4.2 Lagrangian elevation and descent

The analysis presented in § 4.1 is on the evolutionary geometry of the material surfaces themselves, which is also related to other underlying flow physics such as momentum transport. The contours presented in figure 6 show that the Lagrangian scalar is transported violently in the wall-normal direction. Since the streamwise flow velocity is increasing from the wall at the initial state, the scalar transport, indicating the blending of rapid and slow motions from different wall distances, can be related to momentum transport in wall turbulence.

We define the wall-normal Lagrangian displacement as

where $Y$ is the wall-normal location of a fluid particle on a material surface at time  $t$ . The displacement ${\rm\Delta}Y$ quantifies the scalar transport in the wall-normal direction within a time interval of interest, and we define Lagrangian events ‘elevation’ with ${\rm\Delta}Y>0$ and ‘descent’ with ${\rm\Delta}Y<0$ .

We can prove that the overall elevation/descent of a material surface is caused by the mean shear and vortical structures. A material surface ${\it\phi}={\it\phi}_{0}$ can be presented as a graph function $y_{{\it\phi}}=h(x,z)$ , where $y_{{\it\phi}}$ is the coordinate of the surface in the wall-normal direction. Since there is no mass flux through a material surface and the boundaries are periodic in the streamwise and spanwise directions, the volume of the fluid enclosed by two material surfaces, including $y_{{\it\phi}}$ and the rigid wall, and four periodic boundaries is conserved for constant-density flows. Hence, $\langle {\rm\Delta}Y\mid {\it\phi}\rangle$ is always vanishing in a channel flow from the integration of $h(x,z)$ over the $x$ – $z$ plane with a constant area, but only if $h(x,z)$ is a single-valued function. In other words, the variation of $\langle {\rm\Delta}Y\mid {\it\phi}\rangle$ implies that $h(x,z)$ becomes multivalued for the surface. It corresponds to folding or rolling up of material surfaces caused by the mean shear and vortical structures as shown in figures 4 and 5.

As shown in figure 9(a), generally the fluid from the near-wall region is elevated from the wall, and the fluid close to the mid-plane descends toward the wall. From the visualizations in figures 4 and 6, the elevation is mainly caused by the hairpin-like structures with the intensive folding and rolling up of material surfaces. Thus, the elevation events are strong, and a peak of $\langle {\rm\Delta}Y\mid {\it\phi}\rangle$ appears near ${\it\phi}=0.125$ . In contrast, few coherent patterns moving toward the wall are observed, and relatively weak descent events occur in a broad range of $0.3<{\it\phi}<0.9$ . We can also observe that the elevation of the region near ${\it\phi}=0.125$ is significantly intensified in the temporal evolution, which appears to be related to the elevation of the hairpin-like vortical structure in figure 4.

Figure 9.

(a) Lagrangian wall-normal displacement $\langle {\rm\Delta}Y\mid {\it\phi}\rangle$ conditioned on different material surfaces. (b) Lagrangian streamwise velocity difference $\langle {\rm\Delta}u\mid {\it\phi}\rangle$ averaged on material surfaces.

Figure 10.

Lagrangian wall-normal displacement ${\rm\Delta}Y$ on the spanwise and wall-normal plane cut at $t=108$ : (a) at the peak position $z=L_{z}/2$ ; (b)  $0.17{\it\delta}$ away from the peak position, the plane cut is on the ‘hairpin legs’. The solid lines represents contour lines of shear $\partial u/\partial y$ , and the dashed lines represents contour lines of $u^{\prime }v^{\prime }$ .

The elevation and descent events are closely related to ejections and sweeps. The ejection event and the sweep event indicate low-speed fluid moving away from the wall and high-speed fluid moving toward the wall, respectively (Carino & Brodkey Reference Carino and Brodkey1969; Willmarth & Lu Reference Willmarth and Lu1972). To investigate the significance of ejections and sweeps in turbulence production, these events are often studied by the Eulerian quadrant analysis of $u^{\prime }v^{\prime }$ , where $u^{\prime }$ and $v^{\prime }$ are fluctuating velocities in the streamwise and wall-normal directions. Both the ejection event in the second quadrant (Q2) and the sweep event in the fourth quadrant (Q4) are of importance for producing turbulent energy (Adrian Reference Adrian2007).

Furthermore, the elevation and descent events can be observed directly in experiments using Lagrangian-type tracers such as hydrogen bubbles (see e.g. Hama Reference Hama, Roshko, Sturtevant and Bartz1960; Hama & Nutant Reference Hama and Nutant1963; Lee & Li Reference Lee and Li2007; Lee & Wu Reference Lee and Wu2008; Guo et al. Reference Guo, Borodulin, Kachanov, Pan, Wang, Lian and Wang2010). Since the experimental visualizations are based on the concentration of hydrogen bubbles that are initially released in a series of lines parallel or perpendicular to the wall, the bubbles are similar to material lines and their concentration can be described by the Lagrangian field ${\it\phi}$ with a particular initial condition as banded Heaviside functions on a plane. Compared with experimental visualizations, the visualization of ${\it\phi}$ with a smooth initial condition can provide the evolution of the entire collection of material surfaces and lines, and corresponding quantifications can be easily obtained from the conditional statistics in the scalar field.

The distribution of ${\rm\Delta}Y$ and the contour lines of large negative $u^{\prime }v^{\prime }$ with $v^{\prime }>0$ on the peak plane and another plane slightly aside from the peak position are shown in figure 10, where ${\rm\Delta}Y>0$ indicates the elevation and $u^{\prime }v^{\prime }<0$ represents the Q2 event in the lower half of channel. In figure 10(a), the region with the intensive Q2 event is mainly beneath the strong elevation region, which demonstrates that the elevation of low-speed fluid is related to the Q2 event. The most intensive elevation is near the ‘head’ of the primary hairpin-like structure (the dark red region around $x=5.2$ ) and the second most intensive elevation is near the triangular bulge and the secondary hairpin-like structure (red region from $x=1.0$ to $x=2.8$ ). This quantitatively supports the observation that the ejection event caused by hairpin-like vortical structures is much stronger than that caused by the ${\rm\Lambda}$ -shaped structures in Guo et al. (Reference Guo, Borodulin, Kachanov, Pan, Wang, Lian and Wang2010).

It is interesting that the elevation near the ‘head’ of the hairpin-like structures is highly concentrated, and the nearby fluids tend to follow the motion of the ‘head’. In contrast, the descent events are scattered and they are not as strong as the localized elevations in figure 10, and they are quantified as the broad range of $\langle {\rm\Delta}Y\mid {\it\phi}\rangle <0$ in figure 9(a).

Figure 11.

Scatter plots for Lagrangian displacement ${\rm\Delta}Y/{\it\phi}_{0}$ and Lagrangian streamwise velocity difference ${\rm\Delta}u/U_{0}$ for the most elevated material surface with ${\it\phi}=0.125$ : (a)  $t=100$ , ${\it\rho}_{Yu}=0.896$ ; (b)  $t=108$ , ${\it\rho}_{Yu}=0.780$ .

In figure 10(a), the strong elevation region is covered by the high-shear layer represented by the contour lines of large $\partial u/\partial y$ . It is noted that the instability of the high-shear layer is crucial in the transition process (see e.g. Klebanoff et al. Reference Klebanoff, Tidstrom and Sargent1962; Gilbert & Kleiser Reference Gilbert, Kleiser, Kline and Afgan1990). The quantified visualization in figure 10(a) provides a direct proof for the relation between the high-shear layer and the elevation event. The high-shear layer can be generated between the elevated low-speed fluid and the surrounding high-speed fluid. In figure 10(b), there exists a high-shear layer close to the wall on the planes aside from the peak position, which accompanies the descent events near the ‘legs’ of the primary hairpin-like structure.

The Lagrangian elevation and descent events are relevant to momentum transport. Guo et al. (Reference Guo, Borodulin, Kachanov, Pan, Wang, Lian and Wang2010) found that the Lagrangian acceleration and deceleration are qualitatively related with the ejections and sweeps from the visualization of decelerating fluid streaks identified by the motion of hydrogen bubbles near vortical structures. We quantify the momentum transport using the Lagrangian streamwise velocity difference as

which represents the fluid acceleration/deceleration in the streamwise direction. In figure 9, the profile of $\langle {\rm\Delta}u\mid {\it\phi}\rangle$ is qualitatively similar to $\langle {\rm\Delta}Y\mid {\it\phi}\rangle$ , in particular, the most elevated material surface of ${\it\phi}=0.125$ is most accelerated. The correlations of ${\rm\Delta}Y/{\it\phi}_{0}$ and ${\rm\Delta}u/U_{0}$ for the material surface ${\it\phi}=0.125$ at $t=100$ and $t=108$ are shown as scatter plots in figure 11. We can see that ${\rm\Delta}Y$ and ${\rm\Delta}u$ are positively correlated. In the scatter plots, the elevated fluid or the descending fluid tend to be accelerated or decelerated, respectively. The correlation coefficient ${\it\rho}_{Yu}=0.896$ is high at $t=100$ , and decreases with time because the mean velocity is no longer monotonically increased with $y$ during the transition.

4.3 Vorticity growth and distribution

It is widely accepted that the vortical structures play key roles in the momentum and scalar transport (Adrian Reference Adrian2007; Wallace Reference Wallace2013), so the evolutionary geometry of material surfaces is affected by the mean shear and the vortices. On the other hand, the Lagrangian evolution of the surfaces can also characterize the evolution of vortical structures themselves (discussed in § 5.1) and it appears to more natural to capture the growth of vorticity than Eulerian methods.

The conditional averages of the change of the vorticity magnitude are calculated over both material surfaces with different initial wall distances and fixed Eulerian $x$ – $z$ planes at different $y$ -coordinates at $t=100$ and $t=108$ . The averaged Lagrangian difference ${\it\Delta}_{L}|{\bf\omega}|\equiv |{\bf\omega}(\boldsymbol{X}(\boldsymbol{x}_{0},t_{0}\mid t),t)|-|{\bf\omega}(\boldsymbol{x}_{0},t_{0})|$ conditioned on ${\it\phi}$ and Eulerian difference ${\it\Delta}_{E}|{\bf\omega}|\equiv |{\bf\omega}(\boldsymbol{x},t)|-|{\bf\omega}(\boldsymbol{x},t_{0})|$ conditioned on $y$ are shown in figure 12 for comparison. Since the material surfaces close to the wall are hardly disturbed, both $\langle {\it\Delta}_{L}|{\bf\omega}||{\it\phi}\rangle$ and $\langle {\it\Delta}_{E}|{\bf\omega}||y\rangle$ show the similar vorticity accumulation in the near-wall region. The growth of $|{\bf\omega}|$ corresponds to the increase of the wall shear stress ${\it\tau}_{w}={\it\rho}{\it\nu}\partial u/\partial y$ at $y=0$ , which is usually observed in the laminar–turbulent transition.

Figure 12.

Change of vorticity magnitude: (a) averaged on material surfaces $\langle {\it\Delta}_{L}|{\bf\omega}||{\it\phi}\rangle$ ; (b) averaged on Eulerian planes $\langle {\it\Delta}_{E}|{\bf\omega}||y\rangle$ . The location of the peak of Lagrangian vorticity accumulations is indicated by the dashed line.

Unlike the Eulerian result, the Lagrangian vorticity accumulation has an obvious peak near ${\it\phi}=0.325$ at $t=108$ . We analyse the terms in the governing equation of  $|{\bf\omega}|$

to investigate the vorticity transport, where $\boldsymbol{t}_{{\it\omega}}={\bf\omega}/|{\bf\omega}|$ is the unit vector for vorticity. In the right-hand side, the transport of $|{\bf\omega}|$ is governed by the stretching term $(\boldsymbol{t}_{{\it\omega}}\boldsymbol{\cdot }\unicode[STIX]{x1D64E}\boldsymbol{\cdot }\boldsymbol{t}_{{\it\omega}})|{\bf\omega}|$ and the viscous diffusion term ${\it\nu}\boldsymbol{t}_{{\it\omega}}\boldsymbol{\cdot }{\rm\nabla}^{2}{\bf\omega}$ . Although the two terms involve non-local effects on the trajectory of a fluid particle, the qualitative relation between the different mechanisms and the growth of $|{\bf\omega}|$ can be explained by the conditional statistics. In the near-wall region, the vorticity is generated through the boundary vorticity flux (Lighthill Reference Lighthill and Rosenhead1963; Wu, Ma & Zhou Reference Wu, Ma and Zhou2005), and is diffused away from the high-vorticity region by the viscous term.

The major mechanism for the vorticity intensification remote from the wall is through the vorticity stretching term $(\boldsymbol{t}_{{\it\omega}}\boldsymbol{\cdot }\unicode[STIX]{x1D64E}\boldsymbol{\cdot }\boldsymbol{t}_{{\it\omega}})|{\bf\omega}|$ . The distribution of $|{\bf\omega}|$ on the streamwise and wall-normal plane cut at the peak position $z=L_{z}/2$ at $t=108$ is presented in figure 13. We can see that the vorticity concentrates in ribbon-like regions, and the distribution appears to resemble the high-shear layers in figure 10(a), because the shear $\partial u/\partial y$ is the dominating component in ${\it\omega}_{z}$ . Several material surfaces around ${\it\phi}=0.325$ with the maximum Lagrangian vorticity growth ${\it\Delta}_{L}|{\bf\omega}|$ are warped in the high-vorticity region in figure 13, while their Eulerian counterpart, the plane at $y=0.325$ , does not capture this strong vorticity growth and the corresponding vortex stretching mechanism in figure 12.

Figure 13.

Vorticity distribution on the spanwise and wall-normal plane cut at the peak position $z=L_{z}/2$ at $t=108$ . The black solid lines represents the material surfaces around ${\it\phi}=0.325$ , and the yellow dashed line represents the Eulerian plane at $y=0.325$ .

In order to shed light on the vorticity growth and the evolutionary geometry of vortex lines, the vorticity components averaged on material surfaces are analysed. Both the streamwise and the wall-normal components of vorticity are strongly intensified in the evolution. The peak of $|{\it\omega}_{x}|$ and $|{\it\omega}_{y}|$ for the surfaces with ${\it\phi}\approx 0.2$ at $t=100$ , which are close to the most deformed material surface, suggests that the triangular bulge in the evolution of the material surface with ${\it\phi}=0.217$ is related to the inclined quasi-streamwise vortices. In addition, the migration of the peak of $|{\it\omega}_{y}|$ from ${\it\phi}=0.2$ at $t=100$ to ${\it\phi}=0.375$ at $t=108$ appears to be relevant to the elevation of the hairpin-like structures in figure 5. Further explanations on the mechanism of the intensification of $|{\it\omega}_{x}|$ and $|{\it\omega}_{y}|$ are discussed in § 5.3. Since the dominant term of ${\it\omega}_{z}$ is the shear $\partial u/\partial y$ , the change of the spanwise vorticity ${\it\Delta}_{L}|{\it\omega}_{z}|\equiv |{\it\omega}_{z}(\boldsymbol{X}(\boldsymbol{x}_{0},t_{0}|t),t)|-|{\it\omega}_{z}(\boldsymbol{x}_{0},t_{0})|$ averaged over material surfaces is related with the momentum transport as indicated in figure 9, and it decreases significantly around ${\it\phi}=0.2$ .

Figure 14.

Vorticity components and the curvature of vortex lines averaged on material surfaces: (a) the absolute value of streamwise vorticity $|{\it\omega}_{x}|$ ; (b) the absolute value of wall-normal vorticity $|{\it\omega}_{y}|$ ; (c) the Lagrangian difference of absolute value of spanwise vorticity ${\it\Delta}_{L}|{\it\omega}_{z}|$ ; (d) the local curvature of vortex lines  ${\it\kappa}$ .

Although the Lagrangian change of the vorticity magnitude ${\it\Delta}_{L}|{\bf\omega}|$ averaged over the material surfaces around ${\it\phi}=0.2$ is small in figure 12, all of the vorticity components have significant variations on the surfaces in figure 14. This implies that the deformation of the material surfaces around ${\it\phi}=0.2$ is accompanied by intensive stretching and folding of vortex lines with high curvatures. The local curvature of vortex lines can be calculated as

and the averaged curvatures $\langle {\it\kappa}\mid {\it\phi}\rangle$ conditioned over material surfaces are shown in figure 14(d). The curvatures of vortex lines increase for all of the surfaces, which indicates that the vortex lines evolve from nearly straight lines into curved lines. In particular, the peak of $\langle {\it\kappa}\mid {\it\phi}\rangle$ is near ${\it\phi}=0.23$ , suggesting that vortex lines are most curved on these labelled material surfaces during the transition.

4.4 Brief summary and discussion

From three different criteria and perspectives, the influential material surfaces are identified and characterized. First, the most-deformed material surface near ${\it\phi}=0.225$ shows the signature shapes as the triangular bulge and the hairpin-like structure, and its evolution provides a quantitative Lagrangian presentation of the peak–valley splitting. Second, the maximum elevation of the material surface near ${\it\phi}=0.125$ is related to ejections, momentum transport and the formation of high-shear layers. Finally, the material surface with the largest vorticity growth is around ${\it\phi}=0.325$ , but the largest curvature of vortex lines and variations for three vorticity components, in general, occur near ${\it\phi}=0.2$ , which implies that the vortex lines become highly curved on the surface with ${\it\phi}\approx 0.2$ .

Figure 15.

Temporal evolution of $\langle {\it\lambda}_{{\it\omega}}\rangle$ , where ${\it\lambda}_{{\it\omega}}$ is the cosine of the angle between ${\bf\omega}$ and $\boldsymbol{{\rm\nabla}}{\it\phi}$ . The dashed line shows the 15 % level.

Therefore, although the whole flow field is filled with an infinite number of material surfaces, the influential material surface of ${\it\phi}=0.217$ or ${\it\phi}^{+}=45$ with the maximum deformation as well as the maximum curvature of vortex lines and variation of vorticity components, is selected as a representative structure to study the structural evolution in the K-type transitional flow in the next section. It has been justified that the surfaces close to this influential surface show similar dynamical behaviours in this section, and the surfaces remote from the representative one, namely the surfaces that are close to the wall or near the central region, are only slightly disturbed in the evolution and provide little understanding on flow dynamics as discussed in § 3.1.

5.1 Approximation of vortex surfaces

As remarked in Yang & Pullin (Reference Yang and Pullin2010), the material surface can be considered as a good approximation for the vortex surface before significant topological changes in high-Reynolds-number flows. As mentioned in § 2.3, the initial material surface of ${\it\phi}_{0}=y_{0}$ is a good approximation of the vortex surface as sketched in figure 1.

The cosine ${\it\lambda}_{{\it\omega}}\equiv \boldsymbol{t}_{{\it\omega}}\boldsymbol{\cdot }\boldsymbol{n}$ of the angle between the vorticity and the Lagrangian scalar gradient can be used to quantify the local deviation between an isosurface of ${\it\phi}$ and a vortex surface (see Yang et al. Reference Yang, Pullin and Bermejo-Moreno2010; Yang & Pullin Reference Yang and Pullin2011a ). As shown in figure 3, if ${\bf\omega}$ is perpendicular to $\boldsymbol{n}$ in the whole field, the material surfaces completely coincide with the vortex surfaces with $\langle {\it\lambda}_{{\it\omega}}\rangle =0$ .

Temporal evolution of $\langle {\it\lambda}_{{\it\omega}}\rangle$ calculated from the present transitional channel flow is shown in figure 15. For strictly inviscid incompressible flow with conservative body forces, the Helmholtz vorticity theorem shows that the material surfaces that are vortex surfaces at $t=0$ remain so for $t>0$ . Thus, $\langle {\it\lambda}_{{\it\omega}}\rangle =0$ can be satisfied for $t>0$ with $\langle {\it\lambda}_{{\it\omega}}\rangle =0$ at $t=0$ in inviscid flow. In a viscous flow, $\langle {\it\lambda}_{{\it\omega}}\rangle$ slowly increases owing to the viscous effect and the breakdown of the Helmholtz theorem in figure 15. The averaged deviation $\langle {\it\lambda}_{{\it\omega}}\rangle$ , however, remains less than the 15 % level for $t<110$ in the present transitional flow. Therefore, we can reasonably assume that vortex surfaces are still well approximated by the material surfaces before significant topological changes of vortical structures in the transition.

It is noted that the quantitative results of material surfaces in the present study are restricted to the evolution during $80\leqslant t\leqslant 108$ with $\langle {\it\lambda}_{{\it\omega}}\rangle \leqslant 15\,\%$ to ensure the good approximation of vortex surfaces. The whole process in the evolution of the influential material surface will be roughly divided into three stages as follows.

5.2 Stage $1$ , triangular bulge

In the evolution of vortical structures in transitional wall flows, the so-called ‘ ${\rm\Lambda}$ -vortex’ is frequently mentioned and often considered as a prior stage of hairpin-like strictures (e.g. Sandham & Kleiser Reference Sandham and Kleiser1992; Bake et al. Reference Bake, Meyer and Rist2002; Borodulin et al. Reference Borodulin, Gaponenko, Kachanov, Meyer, Rist, Lian and Lee2002a ; Guo et al. Reference Guo, Borodulin, Kachanov, Pan, Wang, Lian and Wang2010; Sayadi, Hamman & Moin Reference Sayadi, Hamman and Moin2013). Controversies, however, still exist about the definition and existence of the ‘ ${\rm\Lambda}$ -vortex’. For example, Kleiser & Laurien (Reference Kleiser, Laurien and Kozlov1985) found that the ${\rm\Lambda}$ -like structure is represented by the dislocation of vortex lines within a ${\rm\Lambda}$ -shaped region from the evolution of material lines in transitional plane Poiseuille flow. Similar observations were found in the transitional boundary layer, and Yan et al. (Reference Yan, Chen, Fu and Liu2014) concluded that the ‘ ${\rm\Lambda}$ -vortex tubes’ do not exist because the vortex lines can penetrate the ${\rm\Lambda}$ -shaped tubes visualized by the Eulerian vortex identification criterion. Bernard (Reference Bernard2011, Reference Bernard2013) pointed out that it is not reasonable to consider the coherent structures simply as rotational regions. Alternatively, the ‘vortex furrows’ evolving from spanwise-aligned vortex filaments were proposed as the principal structural element in the K-type transition. Therefore, the disagreement on the ${\rm\Lambda}$ -shaped vortical structure is partially caused by the different methods used to characterize the coherent structures. In the present study, we seek the essence of the ${\rm\Lambda}$ -shaped structure using material surfaces along with almost attached vortex lines and the vorticity magnitude on the lines, which combines the strengths of different identification methods.

The influential material surface of ${\it\phi}=0.217$ or ${\it\phi}^{+}=45$ , which has the maximum deformation in a period of interest as discussed in § 4.1, shows a triangular bulge at $t=100$ in figure 16(a). The formation of this bulge can be qualitatively analysed from the evolution of material surfaces in appendix B instead of the wave interactions (see Herbert Reference Herbert1988; Kachanov Reference Kachanov1994). In addition, the vortex lines, colour-coded by the magnitude of vorticity, are integrated from the material surface, and they are almost on the surface. This demonstrates that the material surface can be considered as a surrogate of the vortex surface.

For comparison, the Eulerian structures are extracted by the iso-surfaces of the swirling strength ${\it\lambda}_{ci}$ that is the imaginary part of the complex eigenvalue of the velocity gradient tensor (Zhou et al. Reference Zhou, Adrian, Balachandar and Kendall1999). The ${\rm\Lambda}$ -like structure identified in the present transitional channel flow is shown in figure 16(b). At the mean time, vortex lines are also drawn near the isosurface of ${\it\lambda}_{ci}$ , but tube-like vortical structures cannot be identified from the vortex lines. Other Eulerian vortex identification criteria based on the velocity gradient, such as the $Q$ - and ${\it\lambda}_{2}$ -criteria, show the similar result (not shown). It is interesting to observe that the ridge of the Lagrangian triangular bulge covered with folding vortex lines spatially coincides with the Eulerian ${\rm\Lambda}$ -like structure, and a high-shear layer with large magnitudes of vorticity is generated on the top of the bulge (Rist & Fasel Reference Rist and Fasel1995).

Figure 16.

Comparison of Lagrangian and Eulerian vortical structures at $t=100$ . Vortex lines are integrated from the surfaces, and colour-coded by the magnitude of vorticity $|{\bf\omega}|$ . (a) Material surface of ${\it\phi}=0.217$ . (b) Isosurface of the swirling strength ${\it\lambda}_{ci}$ (50 % of the maximum value).

Therefore, the conceptual ${\rm\Lambda}$ -shaped vortex tubes composed of vortex lines do not exist at this stage, and the ${\rm\Lambda}$ -like structure from the Eulerian identification criterion can be misinterpreted without caution. Moreover, the generation mechanism of the ${\rm\Lambda}$ -like structure is not easy to be elucidated and traced from the initial laminar state using the Eulerian methods applied on the structures that are not uniquely defined at different time instants.

From the Lagrangian study, the triangular bulge, which consists of ${\rm\Lambda}$ -shaped swirling regions at two ridges and the triangular high-shear layer on the top, is identified as the representative structure in this stage. Furthermore, from $|{\bf\omega}|$ on the vortex lines drawn in figure 16, there is no obvious vorticity concentration region near the triangular bulge. This indicates that the mean shear, rather than the vortical structures with obvious swirling motions, is still dominating in the evolution of material surfaces at this stage.

5.3 Stage $2$ , vorticity intensification

In figure 4, the material surface subsequently evolves into a hairpin-like structure after the formation of the triangular bulge. In this stage, the vorticity begins to be intensified in local regions and the tube-like vortical structures are generated in figures 16 and 17, so the evolution of vorticity becomes important in this stage. The governing equation for the vorticity in an incompressible flow can be written as

Figure 17.

Comparison of Lagrangian and Eulerian vortex structures at $t=104$ . Vortex lines are integrated from the surfaces, and colour-coded by the magnitude of vorticity $|{\bf\omega}|$ . (a) Material surface of ${\it\phi}=0.217$ . (b) Isosurface of the swirling strength ${\it\lambda}_{ci}$ (50 % of the maximum value).

For the flow at $t=100$ in figure 16, $\partial u/\partial y$ is the largest component in the rate-of-strain tensor $\unicode[STIX]{x1D64E}$ owing to the domination of the mean shear at this time, so (5.1) can be simplified as

Here, the viscous term ${\it\nu}{\rm\nabla}^{2}{\bf\omega}$ in (5.1) is ignored before topological changes of the vorticity for high-Reynolds-number flows.

Assuming the mean shear $\partial u/\partial y=2S_{0}$ is a constant, the solution of (5.1) is

where the initial values for $x$ - and $y$ -components of vorticity are ${\it\omega}_{x0}$ and ${\it\omega}_{y0}$ , respectively. Because of the symmetry, only the lower half of the channel with $S_{0}\geqslant 0$ is considered. From the first term in the right-hand side of (5.3), the amplitudes of ${\it\omega}_{x}$ and ${\it\omega}_{y}$ increase exponentially if ${\it\omega}_{x0}$ and ${\it\omega}_{y0}$ are finite and have the same sign.

As shown in figure 16(a), the inclination angles of the vortex lines near the bulge ridges imply that ${\it\omega}_{y}$ and ${\it\omega}_{x}$ are finite with ${\it\omega}_{x}{\it\omega}_{y}>0$ . As indicated in (5.3), ${\it\omega}_{x}$ and ${\it\omega}_{y}$ can increase rapidly in this region, which is also shown in figure 14. This implies that the vortex lines are intensively stretched and the most intensified $|{\bf\omega}|$ occurs at the ridge of the triangular bulge, which is also observed in figure 18 at $t=104$ .

Figure 18.

Vortex lines extracted near the material surface with ${\it\phi}=0.217$ . The folding of vortex lines is highlighted in the blue rectangle.

The structural evolution appears to be led by the tip of the triangular bulge where is the most distant from the wall on the material surface. The vortex lines at the tip are advected by large mean shear and induced velocities, so they are highly curved as an ${\rm\Omega}$ -shape. In the Lagrangian view, the bulge is lifted and elongated in figure 16(a). Then the surface rolls up in wall-normal and spanwise planes, and tube-like structures are generated beneath the bulge tip in figure 17(a). The vortex lines are stretched in the streamwise direction because of the mean flow shear, which was inferred as a possible mechanism for the formation of the hairpin-like structures in Kim & Moin (Reference Kim and Moin1986), and then they are folded inward at the neck as shown in the blue rectangle in figure 18. The folding vortex lines are concentrated and they are rapidly pulled out with intensified $|{\bf\omega}|$ . The accumulation of the streamwise vorticity in transitional boundary layer was also observed from the stretch and folding of material lines and surfaces in the K-type transition in Bernard (Reference Bernard2013), but the material surface was discretized on a coarse mesh so that the possible rolling up of the tube-like structures was not observed. Finally, the arch (or head) of the hairpin-like structure are pulled out at this part of the material surface with two vortex tubes behind as ‘legs’. This procedure is generally consistent with the conceptional model qualitatively illustrating the generation of a hairpin-like vortex in Perry & Chong (Reference Perry and Chong1982).

We remark that the vorticity intensification stage is skipped in some simplified model for the generation of the hairpin-like or ring-like vortical structures (e.g. Hama & Nutant Reference Hama and Nutant1963; Moin et al. Reference Moin, Leonard and Kim1986). In these models, the initial vorticity is assumed to be concentrated in thin tube-like structures and is much higher than the background vorticity produced by the mean shear, which can be considered as the beginning of the third stage below.

5.4 Stage $3$ , hairpin-like structures

The ‘hairpin vortex’ is often described as the hairpin-shaped, tube-like structures. In the evolution, the Lagrangian hairpin-like structures are stretched out from the triangular bulge at $t=106$ in figure 19(a) after the vorticity intensification stage. At the stage after $t=106$ , the vorticity distribution could be considered as isolated, thin and elongated vortex tubes or filaments.

Figure 19.

Comparison of Lagrangian and Eulerian vortex structures at $t=106$ . Vortex lines are integrated from the surfaces, and colour-coded by the magnitude of vorticity $|{\bf\omega}|$ . (a) Material surface of ${\it\phi}=0.217$ . (b) Isosurface of the swirling strength ${\it\lambda}_{ci}$ (25 % of the maximum value).

It is noted that the skeleton of the hairpin-like structure can be also captured by the Eulerian ${\it\lambda}_{ci}$ -criterion at $t=106$ in figure 19(b), because the Eulerian criteria have the capability to identify the ‘vortex core’ at a time instant (Zhou et al. Reference Zhou, Adrian, Balachandar and Kendall1999). The tracking of the hairpin-like structures, however, appears to be ambiguous from the Eulerian criterion in figures 16(b), 17(b) and 19(b), because the isocontour value is not uniquely defined and usually it has to be varied ad hoc with time to keep the characteristic size of extracted structures similar at different times.

The vortex-induced velocity $\boldsymbol{u}_{i}$ at a point $\boldsymbol{r}$ in space can be calculated by the Biot–Savart law

in a Lagrangian viewpoint as discussed in Moin et al. (Reference Moin, Leonard and Kim1986), where $s^{\prime }$ is the arclength of the vortex filament, $\boldsymbol{r}^{\prime }(s^{\prime },t)$ is a curve representing the tube of concentrated vorticity, ${\it\Gamma}$ is the circulation of the vortex tube, $\boldsymbol{t}_{{\it\omega}}$ is the tangential vector of the vortex filament, and the function $G(|\boldsymbol{r}-\boldsymbol{r}^{\prime }|)$ models the effect of distributed vorticity within the core of effective radius ${\it\sigma}$ . This equation is valid in the thin-filament approximation, i.e.  ${\it\sigma}$ is very small and the structure of the vortex core does not change.

The self-induced velocity $\boldsymbol{u}_{si}$ of the vortex filament can be obtained by assuming the point $\boldsymbol{r}$ is also on the filament $\boldsymbol{r}^{\prime }(s^{\prime },t)$ . By only considering the local effect (Batchelor Reference Batchelor1967), Zhou et al. (Reference Zhou, Adrian, Balachandar and Kendall1999) showed that $\boldsymbol{u}_{si}$ can be approximated by the leading term of (5.4) as

where $K_{c}$ is a constant depending on the distribution of vorticity within the vortex core, and $\boldsymbol{b}_{{\it\omega}}$ is the local binormal vector of the vortex filament $\boldsymbol{r}^{\prime }(s^{\prime },t)$ .

As the vortex lines are concentrated within tube-like regions with growing vorticity, the geometry of the vortex lines plays an increasingly important role in the evolution of material surfaces that are mostly composed of vortex lines. From (5.5), the self-induced velocity $\boldsymbol{u}_{si}$ of the vortex filament is aligned with $\boldsymbol{b}_{{\it\omega}}$ of the vortex line and proportional to  ${\it\kappa}$ .

In this stage, the geometry of vortex lines is important in dynamics because the motions of vortex lines and material surfaces are affected by the self-induced velocity $\boldsymbol{u}_{si}$ . The distribution of the binormal vector $\boldsymbol{b}_{{\it\omega}}$ on the $x$ – $y$ plane at the peak position $z=L_{z}/2$ is shown in figure 20, where $\boldsymbol{b}_{{\it\omega}}$ can be calculated as

The vectors are colour-coded by the spanwise vorticity ${\it\omega}_{z}$ and overlayed by two typical contour lines of the Lagrangian field with ${\it\phi}=0.217$ and ${\it\phi}=0.433$ in table 2. It is shown that the vorticity is most concentrated at the arch of the hairpin-like structure near $x=3$ . From (5.5), $\boldsymbol{u}_{si}$ at the tip of the hairpin-like structure is upper left, so the hairpin is lifted under its self-induction. At the mean time, the hairpin-like structure is elongated in the streamwise direction controlled by the mean shear.

Figure 20.

The binormal vector $\boldsymbol{b}_{{\it\omega}}$ on the spanwise and wall-normal plane cut at the peak position $z=L_{z}/2$ at $t=106$ . The vectors are colour-coded by the spanwise vorticity ${\bf\omega}_{z}$ . The solid lines are the contour lines of the Lagrangian field with ${\it\phi}=0.217$ and ${\it\phi}=0.433$ .

Figure 21.

Comparison of Lagrangian and Eulerian vortex structures at $t=108$ . Vortex lines are integrated from the surfaces, and colour-coded by the magnitude of vorticity $|{\bf\omega}|$ . (a) Material surface of ${\it\phi}=0.217$ . (b) Isosurface of the swirling strength ${\it\lambda}_{ci}$ (25 % of the maximum value).

In figure 20, the secondary vorticity intensification region occurs near $x=1.1$ where $\boldsymbol{u}_{si}$ has a finite upward component at $t=106$ . After the stretching of the ‘legs’ of the primary hairpin-like structure, vortex lines start to concentrate behind the stretched structure, and the secondary hairpin-like structure is generated in this region at $t=108$ . This formation is shown in figure 21, and its mechanism is similar to that for the primary one.

Since the present study focuses on the period during the beginning of the late transitional stages, the ring-like vortical structures perhaps generated at later times have not been observed. The ring-like vortex in the very late stage of transition is usually considered to be generated by the reconnections of the ‘legs’ of ${\rm\Lambda}$ -like or hairpin-like structures (e.g. Bake et al. Reference Bake, Meyer and Rist2002; Borodulin et al. Reference Borodulin, Gaponenko, Kachanov, Meyer, Rist, Lian and Lee2002a ) with the Crow instability (Crow Reference Crow1970). In this stage, the deviation between the material surfaces and vortex surfaces can be greatly amplified, so a correction method must be introduced to faithfully describe the evolution of the vortex surfaces with significant topological changes (Yang & Pullin Reference Yang and Pullin2010).

5.5 Brief summary and discussion

In this section, a Lagrangian perspective on the formation and evolution of the signature structures is provided by tracking the influential material surface. This material surface with the maximum deformation during the evolution is uniquely defined to avoid the ambiguity, and is a reasonable surrogate of a vortex surface before significant topological changes of vortex surfaces. Therefore, the evolution of this material surface can shed light on the dynamics of vortical structures prior to possible vortex reconnections around $t=110$ .

A schematic diagram for the evolution of the Lagrangian hairpin-like structures, represented by typical vortex lines, in the late transitional channel flow is summarized in figure 22, which is also combined with previous findings focusing on the vortical structures in late transitional stages (see e.g. Kleiser & Zang Reference Kleiser and Zang1991; Kachanov Reference Kachanov1994; Bake et al. Reference Bake, Meyer and Rist2002; Borodulin et al. Reference Borodulin, Gaponenko, Kachanov, Meyer, Rist, Lian and Lee2002a ) and the generation of hairpin-like structure packets in turbulent channel flows (Zhou et al. Reference Zhou, Adrian, Balachandar and Kendall1999; Green et al. Reference Green, Rowley and Haller2007). Since the amplitudes of the initial disturbances are small, the flow is nearly laminar and the initial parallel vortex lines are attached on a planar material surface. A triangular bulge is generated because the disturbed surface is stretched by the mean shear, and the vortex lines also bend at two ridges of the bulge. The vorticity near the ridges begins to be intensified, and the vortex lines are folded and elongated in this region. The most intensified vorticity is at the bulge tip, and the vortex lines at this part are pulled out as the arch of the hairpin-like structure. The hairpin arch is lifted and legs are stretched by the combination of the mean shear and the self-induced velocity, then the signature hairpin-like structure develops into a self-sustained state. After the detachment of the primary hairpin-like structure from the triangular bulge, the secondary hairpin arch is generated at the tip of the original triangular bulge under the similar mechanism for the generation of the primary one.

Figure 22.

A schematic diagram of the evolution of the influential material surface in transitional channel flow. The skeleton of the evolving material surface is presented by four typical vortex lines marked by different colours.

It is noted that the influential material surface is only used as a representative surface and its evolution is not isolated. This surface can interact with, or ‘influence’ in some sense, the motion of neighbouring material surfaces. The surfaces below the influential one are elevated near the ‘head’ of the hairpin-like structure, and the surfaces above the influential surface warp around the hairpin-like structure with high-shear layers generated between the ‘legs’.