2.1. Cauchy invariants for incompressible Euler and Navier–Stokes equations

Our study makes use of the Lagrangian formulation of vorticity dynamics by Cauchy (Reference Cauchy1815). This is mathematically equivalent to the circulation theorem of Kelvin (Reference Kelvin1868) but, rather than describing the evolution of surface integrals of vorticity, it applies to individual vorticity vectors and expresses their remarkable ‘frozen-in’ properties for ideal incompressible Euler flows. More recently, these invariants have been extended to viscous Navier–Stokes flows (Eyink et al. Reference Eyink, Gupta and Zaki2020a) based on a stochastic Lagrangian formulation of the incompressible Navier–Stokes equations (Constantin & Iyer Reference Constantin and Iyer2008, Reference Constantin and Iyer2011) and applied to analyse the viscous vorticity dynamics of near-wall sweep and ejection events in turbulent channel flows (Eyink et al. Reference Eyink, Gupta and Zaki2020b). We begin here with a basic introduction to these invariants for Euler equations and their stochastic extension to Navier–Stokes.

We start with the classical theory for an inviscid fluid described by the incompressible Euler equations and without boundaries. Consider the Lagrangian trajectory $\boldsymbol {X}(\boldsymbol {a},t)$ of a fluid particle whose initial position is $\boldsymbol {a}$ at time zero, which satisfies

(2.1)\begin{equation} \frac{{\rm d} \boldsymbol{X}(\boldsymbol{a},t)}{{\rm d} t}= \boldsymbol{u}(\boldsymbol{X}(\boldsymbol{a},t),t), \quad \boldsymbol{X}(\boldsymbol{a},0)=\boldsymbol{a}.\end{equation}

It is a straightforward consequence of the Helmholtz equation that vorticity satisfies

(2.2)\begin{equation} \frac{{\rm d} }{{\rm d} t} {\boldsymbol \omega}(\boldsymbol{X}(\boldsymbol{a},t),t)= {\boldsymbol \omega}(\boldsymbol{X}(\boldsymbol{a},t),t)\boldsymbol{\cdot}\nabla_{\boldsymbol{x}}\boldsymbol{u}(\boldsymbol{X}(\boldsymbol{a},t),t).\end{equation}

This result has the intuitive meaning that vorticity vectors are transported by the ideal flow exactly as infinitesimal material line vectors ${\boldsymbol l}(t)$ with initial vectors ${\boldsymbol l}_0$ located at point $\boldsymbol {a}$ at time zero,

(2.3)\begin{equation} {\boldsymbol l}(t)=\boldsymbol{X}(\boldsymbol{a}+{\boldsymbol l}_0,t) -\boldsymbol{X}(\boldsymbol{a},t)\simeq {\boldsymbol l}_0\boldsymbol{\cdot} \nabla_{\boldsymbol{a}}\boldsymbol{X}(\boldsymbol{a},t). \end{equation}

That ${\boldsymbol l}(t)$ satisfies the same evolution equation as ${\boldsymbol \omega }(\boldsymbol {X}(\boldsymbol {a},t),t)$ follows from the result

(2.4)\begin{equation} \frac{{\rm d} }{{\rm d} t}\nabla_{\boldsymbol{a}}\boldsymbol{X}(\boldsymbol{a},t)=\nabla_{\boldsymbol{a}}\boldsymbol{X}(\boldsymbol{a},t) \boldsymbol{\cdot} \nabla_{\boldsymbol x}\boldsymbol{u}(\boldsymbol{X}(\boldsymbol{a},t),t), \end{equation}

obtained by applying the gradient $\nabla _{a}$ to (2.1). This observation allows the Lagrangian evolution equation (2.2) to be exactly integrated as

(2.5)\begin{equation} {\boldsymbol \omega}(\boldsymbol{X}(\boldsymbol{a},t),t) ={\boldsymbol \omega}(\boldsymbol{a},0)\boldsymbol{\cdot} \nabla_{\boldsymbol{a}}\boldsymbol{X}(\boldsymbol{a},t),\end{equation}

which is the so-called Cauchy formula for vorticity, originally derived by Cauchy (Reference Cauchy1815) in the Lagrangian formulation of the Euler fluid without using the Helmholtz equation. It follows immediately that the initial vorticity ${\boldsymbol \omega }(\boldsymbol {a},0)$ can be written as

(2.6)\begin{equation} {\boldsymbol \omega}(\boldsymbol{a},0) =(\nabla_{\boldsymbol{a}}\boldsymbol{X}(\boldsymbol{a},t))^{-{\rm T}} \boldsymbol{\cdot} {\boldsymbol \omega}(\boldsymbol{X}(\boldsymbol{a},t),t), \end{equation}

where the superscript $(\bullet )^{-{\rm T}}$ represents inverse transpose of a matrix. The initial vorticity is thus expressed as a formally conserved quantity of the Lagrangian flow $\boldsymbol {X}(\boldsymbol {a},t),$ the so-called Cauchy invariant, satisfying $({\rm d}/{\rm d}t)\omega (\boldsymbol {a},0)=0$ for each position label $\boldsymbol {a}$. For the interested reader, we have included a brief discussion of the connection to geometric fluid mechanics in Appendix C.

Because of the time reversibility of the Euler fluid equations, the Cauchy formula (2.5) and the Cauchy invariants (2.6) are valid also backward in time, providing an exact reconstruction of the vorticity $\boldsymbol {\omega }(\boldsymbol {x},t)$ from its value $\boldsymbol {\omega }(\boldsymbol {a},s)$ at an earlier time $s< t.$ Here it is useful to generalize the prior discussion and introduce Lagrangian particle positions $\boldsymbol {X}^s_t(\boldsymbol {a})$ at time $t$ which are ‘labelled’ by positions $\boldsymbol {a}$ at time $s$ rather than at time 0. The superscript thus refers to the labelling time, and the subscript denotes the specific time when the particle location is considered. These particles positions evolve according to the same (2.1) but now satisfy $\boldsymbol {X}^s_s(\boldsymbol {a})=\boldsymbol {a}.$ By introducing the ‘back-to-labels’ map $\boldsymbol {x}\mapsto \boldsymbol {a}=\boldsymbol {A}^s_t(\boldsymbol {x})$, which is inverse to $\boldsymbol {a}\mapsto \boldsymbol {x}=\boldsymbol {X}^s_t(\boldsymbol {a})$, the Cauchy formula (2.5) can then be expressed as

(2.7a,b)\begin{equation} \boldsymbol \omega(\boldsymbol x,t) = \boldsymbol D_t^s(\boldsymbol x) \boldsymbol{\cdot} \boldsymbol \omega(\boldsymbol A_t^s(\boldsymbol x),s), \quad \boldsymbol D_t^s(\boldsymbol x) = (\nabla_{\boldsymbol a}\boldsymbol{X}^s_t)^{\rm T} = (\nabla_{\boldsymbol x} \boldsymbol A_t^s )^{-{\rm T}}, \end{equation}

where $\boldsymbol D_t^s$ is the so-called ‘deformation matrix’ quantifying the vorticity stretching and tilting from $s$ to $t.$ It is furthermore useful to observe that $\boldsymbol {A}^s_t=\boldsymbol {X}^t_s,$ since evolving the particle backward in time from $t$ to an earlier time $s$ via the flow $\boldsymbol {X}^t_s$ exactly recovers its label $\boldsymbol {a}$ at time $s.$ Thus, the back-to-labels map $\boldsymbol {A}^s_t$ satisfies the same advection equation (2.1) as does $\boldsymbol {X}^s_t(\boldsymbol {a}),$ but now in the time variable $s$,

(2.8)\begin{equation} \frac{{\rm d} \boldsymbol A_t^s(\boldsymbol x)}{{\rm d} s} = \boldsymbol u(\boldsymbol A_t^s(\boldsymbol{x}),s), \quad s < t; \quad \boldsymbol A_t^t(\boldsymbol x) = \boldsymbol x. \end{equation}

Because $\boldsymbol {A}^s_t=\boldsymbol {X}^t_s,$ the formula (2.7a,b) formally expresses the vorticity $\boldsymbol {\omega }(\boldsymbol {x},t)$ as a Cauchy invariant of the backward-in-time evolution, independent of the choice of $s< t.$ This interpretation of (2.7a,b) is shown schematically in figure 2(a), where $\boldsymbol \omega (\boldsymbol x,t)$ is invariant along the backward Lagrangian trajectory (blue curve): no matter the stretching or tilting along the path or the choice of initial time $s$, the vorticity $\boldsymbol \omega (\boldsymbol x,t)$ is always equal to $(\nabla _{\boldsymbol x} \boldsymbol A_t^s )^{-{\rm T}} \boldsymbol \omega (\boldsymbol A_t^s(\boldsymbol x) ,s)$.

Figure 2.

(a) Lagrangian trajectory in inviscid fluid. The vorticity $\boldsymbol \omega (\boldsymbol x,t)$ originates from an earlier vorticity vector $\boldsymbol \omega (\boldsymbol a,s)$ through stretching and tilting. (b) Stochastic Lagrangian trajectories in viscous fluid. Earlier vorticity vectors $\boldsymbol \omega (\tilde {\boldsymbol{A}}_t^s,s)$ sampled by the stochastic trajectories are transported to $\boldsymbol \omega _s (\boldsymbol x,t)$ and ensemble averaged to obtain the target vorticity $\boldsymbol \omega (\boldsymbol x,t)$.

Physical fluids always involve viscosity however, so that the remarkable Lagrangian properties enjoyed by vorticity in inviscid flows do not seemingly apply to most real-world flows, except approximately in flow regimes and space–time regions where viscous effects are negligible. Recently, it was pointed out by Eyink et al. (Reference Eyink, Gupta and Zaki2020a) that the Cauchy invariant (2.7a,b) holds in a probabilistic sense for viscous incompressible fluids, based on a stochastic representation of incompressible Navier–Stokes solutions derived by Constantin & Iyer (Reference Constantin and Iyer2008). In this formulation, viscous diffusion and nonlinear advection of vorticity are simultaneously represented by the stochastic Lagrangian particle trajectories $\tilde{\boldsymbol {A}}_t^s(\boldsymbol x)$, which satisfy backward in time a Langevin equation with a noise term proportional to the square root of viscosity,

(2.9)\begin{equation} \hat{\!\mathrm{d}}\tilde{\boldsymbol{A}}_{t}^{s}(\boldsymbol{x}) =\boldsymbol{u}(\tilde{\boldsymbol{A}}_{t}^{s}(\boldsymbol{x}), s) \,\mathrm{d} s+\sqrt{2 \nu}\,\, \hat{\!\mathrm{d}} \tilde{\boldsymbol{W}}(s), \quad s < t; \quad \tilde{\boldsymbol{A}}_{t}^{t}(\boldsymbol{x}) = \boldsymbol x. \end{equation}

Here $\boldsymbol u$ is any solution to the incompressible Navier–Stokes equations, $\nu$ is the kinematic viscosity, $\hat{\!\mathrm{d}}$ denotes the backward It$\bar {{o}}$ differential and $\tilde {\boldsymbol {W}}(s)$ is a vector Brownian motion. Note that the backward It$\bar {{o}}$ differential is just the time reverse of the more widely known forward It$\bar {{o}}$ differential (see Constantin & Iyer (Reference Constantin and Iyer2011), § 4). Samples of backward-in-time stochastic Lagrangian trajectories starting from $(\boldsymbol x,t)$ are shown in figure 2(b). Due to stochasticity, the Lagrangian particles significantly diverge once leaving the starting location $\boldsymbol x$, and the trajectories are not smooth, especially if compared against figure 2(a). Constantin & Iyer (Reference Constantin and Iyer2008) proved that an expectation $\mathbb {E}$ over the ensemble of Brownian motions yields the solution $\boldsymbol {\omega }(\boldsymbol {x},t)$ of the viscous Helmholtz equation as

(2.10)\begin{equation} \boldsymbol \omega(\boldsymbol x,t) = \mathbb{E}[ \tilde{\boldsymbol{D}}_t^s(\boldsymbol x) \boldsymbol{\cdot} \boldsymbol \omega(\tilde{\boldsymbol{A}}_t^s(\boldsymbol x),s) ] \,{:=}\, \mathbb{E}[\tilde{\boldsymbol \omega}_s(\boldsymbol{x},t) ], \end{equation}

where $\tilde {\boldsymbol {D}}_t^s(\boldsymbol x) = (\nabla _{\boldsymbol x} \tilde {\boldsymbol {A}}_t^s )^{-{\rm T}}$. The interpretation of (2.10) is provided in figure 2(b). Starting from $(\boldsymbol x,t)$, the stochastic Lagrangian trajectories (coloured curves) are integrated backward in time until $s$. The vorticity vectors sampled by the Lagrangian particles (coloured arrows annotated with $\boldsymbol \omega (\tilde{\boldsymbol {A}}_t^s(\boldsymbol x),s)$) constitute the origin of the target vorticity $\boldsymbol \omega (\boldsymbol {x},t)$ (black arrow). Quantitatively, the earlier vorticities are transported by the deformation matrix $\tilde {\boldsymbol {D}}_t^s$ to obtain the stochastic Cauchy invariant $\tilde {\boldsymbol \omega }_s(\boldsymbol {x},t)$, whose ensemble average is equivalent to the vorticity of interest $\boldsymbol \omega (\boldsymbol {x},t)$. Since the Navier–Stokes equations are time irreversible, it is natural that such stochastic invariants exist only backward in time and the formula (2.10) yields a causal representation of the vorticity $\boldsymbol {\omega }(\boldsymbol {x},t)$ in terms of its values $\boldsymbol {\omega }(\boldsymbol {a},s)$ at each earlier time $s< t.$ This formula thus represents exactly how vortex lines move, or more precisely evolve, in a viscous fluid, through the combination of both nonlinear advection and viscous diffusion.

In wall-bounded flows, the treatment of the stochastic trajectories when they reach the wall is equivalent to the choice of the boundary conditions for the viscous Helmholtz equations. Constantin & Iyer (Reference Constantin and Iyer2011) proved that stopping the trajectories at the wall is equivalent to the Dirichlet boundary condition of the vorticity. Unfortunately, this formulation is inadequate for the purposes of our current investigation, because the problem is precisely to understand the evolution of the wall stress $\boldsymbol {\tau }_W=\nu \boldsymbol {\omega }_W\times \hat {\boldsymbol {n}}$ and, thus, $\boldsymbol {\omega }_W$ cannot be taken as given. In this context it is appropriate instead to adopt Neumann boundary conditions, where the wall vorticity flux is prescribed. The derivation of a stochastic Lagrangian representation for the Navier–Stokes vorticity with such boundary conditions is one of the main results of the present paper and is accomplished in the following section.

2.2. Stochastic Cauchy invariant with Neumann boundary condition

Here we adopt a Neumann boundary condition determined by vorticity source density at the wall, which is equivalent to reflecting the stochastic Lagrangian trajectories every time when they reach the wall (Drivas & Eyink Reference Drivas and Eyink2017). At no-slip boundaries most terms in the momentum equation vanish, except the pressure gradient and the viscous diffusion terms. Therefore, in the case of a flat wall considered here, the diffusion of vorticity, or so-called Lighthill source (Lighthill Reference Lighthill1963; Panton Reference Panton1984), has tangential components balanced by the pressure gradient

(2.11)\begin{equation} \boldsymbol \sigma \,{:=}\, -\nu \hat{\boldsymbol n} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{\omega}|_{w}={-} \hat {\boldsymbol n} \times \boldsymbol{\nabla} p |_{w}, \end{equation}

where $\hat {\boldsymbol n}$ is the wall-normal unit vector pointing into the fluid. The stochastic trajectory (2.9) is augmented with a reflecting boundary condition,

(2.12)\begin{equation} \hat{\!\mathrm{d}} \tilde{\boldsymbol{A}}_{t}^{s}(\boldsymbol{x})=\boldsymbol{u} (\tilde{\boldsymbol{A}}_{t}^{s}(\boldsymbol{x}), s) \, \mathrm{d} s+\sqrt{2 \nu}\,\, \hat{\!\mathrm{d}} \tilde{\boldsymbol{W}}(s) - \nu \hat{\boldsymbol n}(\tilde{\boldsymbol{A}}_{t}^{s}(\boldsymbol{x}), s) \hat{\mathrm d} \ell_{t}^{s}(\boldsymbol{x}), \quad s < t, \end{equation}

enforced by the backward boundary local time density $\ell _t^s(\boldsymbol x)$, formally defined as

(2.13)\begin{equation} \ell_{t}^{s}(\boldsymbol{x})=\int_{t}^{s} \, {\rm d} r \int_{w} \, {\rm d} S(\boldsymbol{z})\delta^{3} (\boldsymbol{z}-\tilde{\boldsymbol{A}}_{t}^{s}(\boldsymbol{x})), \quad s< t, \end{equation}

where $\delta (\bullet )$ is the Dirac delta function, and $\int _{w} \, {\rm d} S(z)$ represents surface integration over the wall (Lions & Sznitman Reference Lions and Sznitman1984; Burdzy, Chen & Sylvester Reference Burdzy, Chen and Sylvester2004; Drivas & Eyink Reference Drivas and Eyink2017). By definition, $\ell _t^s(\boldsymbol x)$ has the dimension of time divided by length. The physical interpretation of $\ell _t^s(\boldsymbol x)$ is provided in figure 3. A reflected stochastic Lagrangian trajectory governed by (2.12) is shown on the left. Given a distance from the wall $\epsilon$, the backward time that the particle spent within $y \in [0,\epsilon ]$ is $\mathcal {T}(\epsilon )$ (defined to be negative). Due to stochasticity of the trajectory, as $\epsilon \rightarrow 0$, the ratio $\mathcal {T}(\epsilon ) / \epsilon$ converges to a finite value, the boundary local time density $\ell _t^s(\boldsymbol x)$. Therefore, $\ell _t^s(\boldsymbol x)$ remains non-positive and unchanged when the particle does not reach the wall. If reflection occurs, $\ell _t^s(\boldsymbol x)$ decreases, and the last term in (2.12) quantifies the reflected distance from the wall.

Figure 3.

Schematic of the stochastic Lagrangian trajectory with Neumann boundary condition, and the physical interpretation of the boundary local time density. Given a distance from the wall $\epsilon$, $\mathcal {T}(\epsilon )$ denotes the time spent by the particle within $\epsilon$.

We prove in Appendix B that the expectation of the stochastic Cauchy invariant supplemented with a term from the Lighthill source is conserved in backward time $s< t$,

(2.14)\begin{equation} \boldsymbol \omega(\boldsymbol x,t) = \mathbb{E} [\tilde {\boldsymbol \omega}_s(\boldsymbol x,t) ] = \mathbb{E} [ \tilde{\boldsymbol{D}}_{t}^{s}(\boldsymbol{x}) \boldsymbol{\cdot} \boldsymbol{\omega} (\tilde{\boldsymbol{A}}_{t}^{s}(\boldsymbol{x}), s)+ \tilde {\boldsymbol L}_t^s(\boldsymbol x) ], \end{equation}

where matrix $\tilde{\boldsymbol {D}}_{t}^{s}(\boldsymbol {x})$ is given by the solution of the final-value problem

(2.15)\begin{equation} \frac{{\rm d} }{{\rm d} s} \tilde{\boldsymbol{D}}_{t}^{s}(\boldsymbol{x}) ={-}\tilde{\boldsymbol{D}}_{t}^{s}(\boldsymbol{x}) (\boldsymbol \nabla_x \boldsymbol{u}(\tilde{\boldsymbol{A}}_{t}^{s}(\boldsymbol{x}),s))^{\rm T} \qquad \tilde{\boldsymbol{D}}_{t}^{t}(\boldsymbol{x})={\boldsymbol I},\end{equation}

and where the source term is

(2.16)\begin{equation} \tilde{\boldsymbol L}_t^s(\boldsymbol x) = \int_{s}^{t} \tilde{\boldsymbol{D}}_{t}^{r}(\boldsymbol{x}) \boldsymbol{\cdot} \boldsymbol{\sigma}(\tilde{\boldsymbol{A}}_{t}^{r}(\boldsymbol{x}), r) \hat{\mathrm d} \ell_{t}^{r}(\boldsymbol{x}). \end{equation}

The two terms in (2.14) quantify the contributions to $\boldsymbol \omega (\boldsymbol x,t)$ of interior vorticity deformation and of the wall vorticity flux, respectively. Thus, the Lighthill source (2.11) is sampled every time the particle hits the boundary and is reflected.

Although we focus in this work on a developing boundary layer over a flat wall, our method of solving stochastic differential equations with reflecting boundary conditions applies to flow domains with curved walls. Details can be found in the paper of Lions & Sznitman (Reference Lions and Sznitman1984), which treated any domain $\varOmega$ whose boundary is a smooth manifold. In fact, their analysis covered a more general class of ‘admissible’ open domains $\varOmega$ which includes those whose boundary is piecewise smooth with components intersecting at convex interior angles. The construction of Lions & Sznitman (Reference Lions and Sznitman1984) established globally in time both the existence and uniqueness of the reflected diffusion process and the boundary local time density for all such domains. Their result already covers a large number of flows of physical interest, including those in which the wall is mathematically smooth but ‘hydraulically rough’, e.g. when the height of the wall is given by a sinusoidal profile. Our analysis thus carries over to a large class of flow domains.

The stochastic Cauchy invariant (2.14) can be numerically evaluated using a Monte-Carlo scheme and Euler–Maruyama method for time discretization. At discrete times $s = s_k \,{:=}\, t - k({\rm \Delta} s), k=1,2,3,\ldots$, the particle locations $\tilde{\boldsymbol {A}}_t^{s_k}$, the deformation matrix $\tilde{\boldsymbol {D}}_t^{s_k}$ and the wall contribution $\tilde {\boldsymbol L}_t^{s_k}$ are obtained through backward integration,

(2.17)\begin{gather} \tilde{\boldsymbol{A}}_{t}^{s_{k}}(\boldsymbol{x}) = \tilde{\boldsymbol{A}}_{t}^{s_{k-1}}(\boldsymbol{x})-\boldsymbol{u} (\tilde{\boldsymbol{A}}_{t}^{s_{k-1}}(\boldsymbol{x}), s_{k-1}) {\rm \Delta} s+\sqrt{2 \nu {\rm \Delta} s} \tilde{\boldsymbol {N}}_{k} - \nu {\rm \Delta} \ell_{k} \hat{\boldsymbol y}, \end{gather}

(2.18)\begin{gather}\tilde{\boldsymbol{D}}_{t}^{s_{k}} = \tilde{\boldsymbol{D}}_{t}^{s_{k-1}} \boldsymbol{\cdot} [\boldsymbol{I}+(\boldsymbol{\nabla}_{x} \boldsymbol{u})^{{\rm T}}|_{(\tilde{\boldsymbol{A}}_{t}^{s_{k-1}}, s_{k-1})} {\rm \Delta} s], \end{gather}

(2.19)\begin{gather}\tilde{\boldsymbol{L}}_{t}^{s_{k}} = \tilde{\boldsymbol{L}}_{t}^{s_{k-1}}-\tilde{\boldsymbol{D}}_{t}^{s_{k-1}} \boldsymbol{\cdot} \boldsymbol{\sigma}(\tilde{\boldsymbol{A}}_{t}^{s_{k}}, s_{k-1}) {\rm \Delta} \ell_{k}. \end{gather}

In (2.17), $\tilde {\boldsymbol N}_k$ is a three-dimensional normal random vector with mean zero and covariance matrix $\boldsymbol I$, independently sampled for each step $k=1,2,3\cdots$. The increment of the boundary local time density is denoted as ${\rm \Delta} \ell _k = \ell _t^{s_k} - \ell _t^{s_{k-1}}$ and details about its evaluation are provided in Appendix A. Given a fully resolved Navier–Stokes solution $\boldsymbol u(\boldsymbol x,t)$, ((2.17)–(2.19)) can be exploited to evaluate the stochastic Cauchy invariant (2.14) in backward time. The numerical procedures are summarized in algorithm 1. In all the examined cases, the number of particles employed for Monte-Carlo evaluation of the expectation is always $N_p=10^4$.

Algorithm 1:

Stochastic Cauchy analysis with Neumann boundary condition.

2.3. Computational set-up and events of interest

Our study is performed using the transitional boundary-layer dataset of the Johns Hopkins Turbulence Databases (JHTDB) (Wu et al. Reference Wu, Lee, Meneveau and Zaki2019). The computational domain and flow configuration are shown in figure 4. The dataset was produced from DNS of incompressible flow over a flat plate with an elliptical leading edge. The streamwise, wall-normal and spanwise coordinates are denoted by $x$, $y$ and $z$, and corresponding velocity components are $u$, $v$ and $w$. The reference length scale is half-thickness of the plate, $L$, and the reference velocity is the incoming free-stream speed $U_{\infty }$. At the upstream curved boundary of the domain, the inflow is a superposition of the uniform velocity $U_{\infty }$ and homogeneous isotropic turbulence. The free-stream turbulence decays as it is advected towards the leading edge where its intensity reaches $Tu = 3\,\%$. The free-stream forcing of the boundary layer leads to the formation of amplifying streaks and sporadic breakdown into turbulent spots (figure 4). As spots spread towards the wall and laterally, the wall friction in the footprint of the turbulence increases appreciably (solid line in the middle panel of figure 4 and the contours in the bottom panel). Accompanying the intermittent generation of turbulent spots, mean skin friction (dashed line) also increases to a turbulent level.

Figure 4.

(Top) Computational grid for DNS of transitional boundary layer. (Middle) Visualization of flow structure: high-speed ($u^{\prime } = 0.1$, white) and low-speed streaks ($u^{\prime } = -0.1$, dark grey); vortical structures identified using $\lambda _2$ criterion (red to green, $\lambda _2=-0.02$). Only half of the domain is shown in the spanwise direction. Mean (dashed line) and sample instantaneous (solid line) skin friction are shown on the side. (Bottom) Turbulent spot and the associated wall vorticity.

To generate the database flow, the incompressible Navier–Stokes equations were solved on a curvilinear grid (see figure 4) using a fractional-step method (Rosenfeld, Kwak & Vinokur Reference Rosenfeld, Kwak and Vinokur1991). A second-order volume-flux formulation was adopted for the spatial discretization. The advection terms were treated explicitly by the Adams–Bashforth scheme, and the Crank–Nicolson scheme was adopted for the diffusion terms. The pressure Poisson equation was solved using Fourier transform in the spanwise direction and multi-grid inversion for every spanwise wavenumber. The algorithm has been applied in numerous studies of transitional and turbulent flows (Zaki Reference Zaki2013; Lee, Sung & Zaki Reference Lee, Sung and Zaki2017). The domain size and the number of grid points are summarized in table 1, and more details about the numerical method and flow statistics can be found at JHTDB.

Table 1.

Domain size and the number of grid points for DNS within the curved domain. Grid resolution and flow statistics are reported for the transitional region $200 < x < 500$. Here $Re_{\theta } = U_{\infty } \theta /\nu$, where $\theta$ is the momentum thickness.

In order to evaluate the stochastic Cauchy invariant ((2.14), (2.17)–(2.19)) along stochastic Lagrangian trajectories, we adopt the JHTDB web service interface to obtain the velocity and its derivatives at instantaneous particle locations. The piecewise cubic Hermite interpolation (PCHIPInt) is adopted in time. Since the DNS was performed using second-order methods, we choose the fourth-order Lagrangian interpolation (Lag4) for the getVelocity subroutine, and the fourth-order finite-difference scheme (FD4Lag4) for getVelocityGradient and getVelocityHessian subroutines.

In order to extract events that represent the enhanced skin friction, we search for local maxima of streamwise wall shear stress $\tau _{xy} = \nu \partial _y u = -\nu \omega _z$ within the transitional region $200 < x < 500$. The grid resolution and flow statistics for the analysis subdomain are shown in table 1. For each local maximum of wall stress, we keep track of its temporal evolution $\omega _m(t)$ and select the time when

(2.20a,b)\begin{align} |\omega_{m}(t)| \leq \max_x \{ | \langle \omega_{z}\rangle |+ \omega^{\prime}_{z,{rms}} \} , \quad \text{and} \quad |\omega_{m}(t+{\rm \Delta} t)| > \max_x \{ | \langle \omega_{z}\rangle |+ \omega^{\prime}_{z,{rms}} \}. \end{align}

Here $\langle \omega _{z} \rangle$ and $\omega ^{\prime }_{z,{rms}}$ are the mean and root-mean-square fluctuation of the wall vorticity, averaged over time and the spanwise direction. The maximum value in (2.20a,b) is 2.79 at $x=429$, and the criterion ensures that the instantaneous wall stress at the selected location is much higher than the turbulent mean stress. After excluding events that are already inside a developed turbulent spot, we obtain 48 events in total representing the instant when spots just hit the wall. At later times, as a larger portion of the spot spreads along the wall, an increasing area of wall stress around each selected maximum reaches the turbulent level (bottom panel in figure 4) and finally merges into the fully turbulent boundary layer downstream. Detailed information about the location and time of these events are available in table 1 of the supplementary material (SM) which are available at https://doi.org/10.1017/jfm.2022.296.

The flow field around a sample event is visualized in figure 5. The stress maximum is located at $(x,y,z)=(203.5,0,220.0)$ and $t=916.25$, and the corresponding vorticity vector is $(\omega _x,\omega _y,\omega _z)=(0.78,0,-2.8)$. The isosurfaces of $\omega _z$ in figure 5(a) resemble the ‘inclined shear layers’ commonly observed in fully turbulent flows (Jimenez et al. Reference Jimenez, Moin, Moser and Keefe1988). As speculated by Thomas & Bull (Reference Thomas and Bull1983), ‘this shear layer appears to be responsible for the characteristic variations of wall shear.’ The vortex lines in figure 5(a) are lifted above the stress minimum, and depressed towards the stress maximum at the wall (panel (b)); the pattern is reminiscent of ejection and sweep events in fully developed wall turbulence (Sheng et al. Reference Sheng, Malkiel and Katz2009), which have been attributed to streamwise vortical structures in the buffer layer (Kravchenko, Choi & Moin Reference Kravchenko, Choi and Moin1993; Orlandi & Jiménez Reference Orlandi and Jiménez1994). In addition, fluctuations in the wall pressure gradient are appreciable during transition (figure 6ii), and, hence, the Lighthill source at early times may be important. In fact, the contribution of the Lighthill source to the extreme events in turbulence has been controversial: although the stress maximum is mostly associated with a strong pressure gradient at the wall (Ghaemi & Scarano Reference Ghaemi and Scarano2013), Thomas & Bull (Reference Thomas and Bull1983) claimed that the pressure pattern is not the direct cause of the bursting process, after analysing the phase relation between the pressure and wall shear stress. In transitional flows, whether the Lighthill source leads to the instantaneous high wall stress remains unexplored. In the next section we quantify the contribution from each of these mechanisms using a stochastic Cauchy analysis.

Figure 5.

(a) Three-dimensional view and (b) end view of the flow field around the stress maximum (marked by red asterisk in (b)). Black and red curves: vortex lines initiated at $y=0.5$; grey isosurface: $\omega _z = -1.5$; bottom plane: contour plot of spanwise wall vorticity.

Figure 6.

Instantaneous particle locations at (a–c) $\delta s = s - t = 0$, $-10$, $-20$. The bottom plane shows (i) spanwise wall vorticity and (ii) spanwise vorticity source. Grey isosurface: $\omega _z = -1.5$. Particles in (c) are separated into (magenta) near-wall cluster and (yellow) outer cluster.