6.1. Barriers to linear momentum transport

Setting $\boldsymbol {f}:=\rho \boldsymbol {u}$, we can rewrite (2.1) as

(6.1)\begin{equation} \frac{\textrm{D}\boldsymbol{f}}{\textrm{D}t}=\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{T}_{vis}-\boldsymbol{\nabla}p+\boldsymbol{q}- \frac{\textrm{D}\rho}{\textrm{D}t}\boldsymbol{u},\end{equation}

and hence obtain

(6.2a,b)\begin{equation} \boldsymbol{h}_{vis}=\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{T}_{vis},\quad \boldsymbol{h}_{nonvis}=-\boldsymbol{\nabla}p+\boldsymbol{q}- \frac{\textrm{D}\rho}{\textrm{D}t}\boldsymbol{u}, \end{equation}

for the viscous and non-viscous terms in (2.5). The viscous stress tensor and its divergence are objective (Gurtin et al. Reference Gurtin, Fried and Anand2013), and hence the $\boldsymbol {h}_{vis}$ function in (6.2a,b) satisfies the objectivity condition (2.6). Accordingly, the barrier equations (4.1) and (5.1) for the diffusive transport of linear momentum become

(6.3)\begin{gather} \boldsymbol{x}_{0}^{\prime} =\overline{\det\boldsymbol{\nabla}\boldsymbol{F}_{t_{0}}^{t}(\boldsymbol{F}_{t_{0}}^{t} )^{*}[\boldsymbol{\boldsymbol{\nabla}}\boldsymbol{\cdot} \boldsymbol{T}_{vis}]}, \end{gather}

(6.4)\begin{gather}\boldsymbol{x}^{\prime} =\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{T}_{vis}. \end{gather}

Specifically, in the case of incompressible Navier–Stokes flows with kinematic viscosity $\nu$, we have the constitutive law $\boldsymbol {\nabla }\boldsymbol {\cdot } \boldsymbol {T}_{vis}=\nu \rho {\rm \Delta} \boldsymbol {u}$ in the general momentum equation (6.1); we also observe that $\det \boldsymbol {\nabla }\boldsymbol {F}_{t_{0}}^{t}\equiv 1$ holds by incompressibility. We then obtain the following result.

Theorem 6.1 For incompressible, uniform-density Navier–Stokes flows, the material and instantaneous barrier equations (6.3) and (6.4) for linear momentum take the specific forms

(6.5)\begin{gather} \boldsymbol{x}_{0}^{\prime} =\nu\rho \overline{(\boldsymbol{F}_{t_{0}}^{t})^{*} {\rm \Delta}\boldsymbol{u}}, \end{gather}

(6.6)\begin{gather}\boldsymbol{x}^{\prime} =\nu\rho\,{\rm \Delta}\boldsymbol{u}. \end{gather}

Each of (6.5) and (6.6) defines a 3-D, autonomous (or steady) dynamical system with respect to the time-like variable $s\in \mathbb {R}$, and hence can be analysed via tools developed for steady flows in the chaotic advection literature (Aref et al. Reference Aref, Blake, Budisic, Cardoso, Cartwright, Clercx, El Omari, Feudel, Golestanian and Gouillart2017). For the purpose of finding active transport barriers, all the relevant information about the unsteadiness of $\boldsymbol {u}(\boldsymbol {x},t)$ over the time interval $[t_{0},t_{1}]$ is encoded into (6.5) through the pullback and the temporal averaging operations. The instantaneous version (6.6) of these equations only contains the physical time $t$ as a parameter; it is, therefore, also a steady ordinary differential equation (ODE) with respect to the variable $s$ parametrizing its streamlines. Both dynamical systems in (6.5) and (6.6) are volume preserving because ${\rm \Delta} \boldsymbol {u}$ is divergence free for incompressible flows. Therefore, the three possible active barrier geometries arising from the analysis of these barrier equations are those shown in figure 3.

In order to solve for trajectories of (6.5) accurately over a domain $U$ with boundary $\partial U$, one must be aware of any special boundary condition that $\boldsymbol {b}_{t_{0}}^{t_{1}}(\boldsymbol {x}_{0})$ may have to satisfy along $\partial U$. We assume for simplicity that $\boldsymbol {u}(\boldsymbol {x},t)$ is incompressible and $\partial U$ is a no-slip boundary. Then, after projecting the Navier–Stokes equation (2.1) at a point $\boldsymbol {x}\in \partial U$ onto a local orthogonal basis $(\boldsymbol {e}_{1},\boldsymbol {e}_{2},\boldsymbol {e}_{3})$, with $\boldsymbol {e}_{3}$ normal to the wall, we obtain

(6.7)\begin{equation} \begin{pmatrix} 0\\ 0\\ 0\end{pmatrix}=\nu\begin{pmatrix} 0\\ 0\\ {\rm \Delta}\boldsymbol{u}\boldsymbol{\cdot} \boldsymbol{e}_{3} \end{pmatrix}+\begin{pmatrix} \left(\boldsymbol{q}-\dfrac{1}{\rho}\boldsymbol{\nabla}p\right)\boldsymbol{\cdot} \boldsymbol{e}_{1}\\ \left(\boldsymbol{q}-\dfrac{1}{\rho}\boldsymbol{\nabla}p\right)\boldsymbol{\cdot} \boldsymbol{e}_{2}\\ \left(\boldsymbol{q}-\dfrac{1}{\rho}\boldsymbol{\nabla}p\right)\boldsymbol{\cdot} \boldsymbol{e}_{3} \end{pmatrix}. \end{equation}

Therefore, if the wall-normal pressure gradient balances out the external body forces along $\partial U$ (as is often assumed in computational fluid dynamics (CFD) simulations), then ${\rm \Delta} \boldsymbol {u}$ satisfies a no-penetration boundary condition along the no-slip boundary $\partial U$, because ${\rm \Delta} \boldsymbol {u}\boldsymbol {\cdot } \boldsymbol {e}_{3}$ must vanish at each boundary point by (6.7). Given that such a boundary $\partial U$ is invariant under the flow map $\boldsymbol {F}_{t_{0}}^{t}$, we obtain that the pullback of ${\rm \Delta} \boldsymbol {u}$ under the flow map must also be tangent to the boundary. Consequently, any no-slip boundary $\partial U$ with a vanishing boundary-normal resultant force is an invariant manifold for the barrier equations (6.5) and (6.6).

6.2. Barriers to angular momentum transport

To analyse angular momentum barriers, we take the cross-product of eq. (2.1) with a vector $\boldsymbol {r}=\boldsymbol {x-\hat {x}}$, where $\boldsymbol {\hat {x}}\in U$ marks a fixed reference point. Setting then $\boldsymbol {f}:=\boldsymbol {r}\times \rho \boldsymbol {u},$ we obtain an evolution equation for $\boldsymbol {f}$ in the form

(6.8)\begin{equation} \frac{\textrm{D}\boldsymbol{f}}{\textrm{D}t}= (\boldsymbol{x-\hat{x}} ) \times\frac{\textrm{D}\rho}{\textrm{D}t}\boldsymbol{u}- (\boldsymbol{x-\hat{x}} ) \times\boldsymbol{\nabla}p+ (\boldsymbol{x-\hat{x}} ) \times\boldsymbol{{q}}+ (\boldsymbol{x-\hat{x}} ) \times\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{T}_{vis},\end{equation}

implying

(6.9a,b)\begin{equation} \boldsymbol{h}_{vis}=(\boldsymbol{x-\hat{x}})\times\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{T}_{vis},\quad \boldsymbol{h}_{nonvis}=(\boldsymbol{x-\hat{x}})\times \left[-\boldsymbol{\nabla}p+\boldsymbol{q}+\frac{\textrm{D}\rho}{\textrm{D}t} \boldsymbol{u}\right],\end{equation}

for the viscous and non-viscous terms in (2.5). Under a frame change of the form (2.3), this $\boldsymbol {h}_{vis}$ satisfies

(6.10)\begin{align} \boldsymbol{h}_{vis}=(\boldsymbol{x-\hat{x}})\times\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{T}_{vis}=\boldsymbol{Q}(t)(\boldsymbol{y-\hat{y}}) \times\boldsymbol{Q}(t)\tilde{\boldsymbol{\nabla}}\boldsymbol{\cdot} \tilde{\boldsymbol{T}}_{vis} =(\boldsymbol{y-\hat{y}})\times\tilde{\boldsymbol{\nabla}} \boldsymbol{\cdot} \tilde{\boldsymbol{T}}_{vis}= \tilde{\boldsymbol{h}}_{vis},\end{align}

where we have used the objectivity of $\boldsymbol {\nabla }\boldsymbol {\cdot } \boldsymbol {T}_{vis}$. We conclude from (6.10) that the objectivity condition (2.6) is satisfied for this choice of $\boldsymbol {f}$, and hence our formulation is applicable. We, therefore, obtain, as in the case of linear momentum, the following result.

Theorem 6.2 For incompressible, uniform-density Navier–Stokes flows, the material and instantaneous barrier equations (4.1) and (5.1) for angular momentum take the specific form

(6.11)\begin{gather} \boldsymbol{x}_{0}^{\prime} =\nu\rho\,\overline{(\boldsymbol{F}_{t_{0}}^{t})^{*} [(\boldsymbol{x-\hat{x}})\times{\rm \Delta}\boldsymbol{u}]}, \end{gather}

(6.12)\begin{gather}\boldsymbol{x}^{\prime} =\nu\rho\,(\boldsymbol{x-\hat{x}}) \times{\rm \Delta}\boldsymbol{u}. \end{gather}

These equations again define 3-D, steady, volume-preserving dynamical systems with respect to the time-like independent variable $s\in \mathbb {R}$. As in the case of barriers to the transport of linear momentum, we find that in the presence of zero boundary-normal resultant force, (6.8) implies any no-slip boundary $\partial U$ to be an invariant manifold for the two dynamical systems in (6.11) and (6.12).

6.3. Barriers to vorticity transport

To obtain the evolution equation for the active vector field $\boldsymbol {f}:=\boldsymbol {\omega }$, we divide (2.1) by $\rho$, take the curl of both sides and use the relation $\boldsymbol {\nabla }\times \boldsymbol {\nabla }p=\boldsymbol {0}$ to obtain the general vorticity-transport equation

(6.13)\begin{equation} \frac{\textrm{D}\boldsymbol{f}}{\textrm{D}t}=(\boldsymbol{\nabla} \boldsymbol{u})\boldsymbol{f}-(\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u})\boldsymbol{f}+\frac{1}{\rho^{2}} \boldsymbol{\nabla}\rho\times\boldsymbol{\nabla}p+\boldsymbol{\nabla}\times \left(\frac{1}{\rho}\boldsymbol{{q}}\right)+\nu\boldsymbol{\nabla} \times\left(\frac{1}{\rho}\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{T}_{vis}\right).\end{equation}

Consequently, our general formulation (2.5) applies with

(6.14a,b)\begin{align} \boldsymbol{h}_{vis}=\nu\boldsymbol{\nabla} \times\left(\frac{1}{\rho}\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{T}_{vis}\right),\enspace \boldsymbol{h}_{nonvis}= (\boldsymbol{\nabla}\boldsymbol{u}) \boldsymbol{f}-(\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{u}) \boldsymbol{f}+\frac{1}{\rho^{2}}\boldsymbol{\nabla}\rho\times \boldsymbol{\nabla}p+\boldsymbol{\nabla}\times \left(\frac{1}{\rho}\boldsymbol{{q}}\right). \end{align}

Following the derivation of the transformation formula for vorticity under an observer change (2.3) (see, e.g., Truesdell & Rajagopal Reference Truesdell and Rajagopal2009), we obtain that $\boldsymbol {h}_{vis}=\boldsymbol {Q}(t)\tilde {\boldsymbol {h}}_{vis}$. Therefore, the objectivity condition (2.6) is satisfied for this choice of $\boldsymbol {f}$, and hence our formulation is applicable. The barrier equations (4.1) and (5.1) for diffusive vorticity transport then become

(6.15)\begin{gather} \boldsymbol{x}_{0}^{\prime} =\nu\,\overline{\det\boldsymbol{\nabla}\boldsymbol{F}_{t_{0}}^{t} (\boldsymbol{F}_{t_{0}}^{t})^{*}\left[\boldsymbol{\nabla} \times\left(\frac{1}{\rho}\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{T}_{vis} \right)\right]}, \end{gather}

(6.16)\begin{gather}\boldsymbol{x}^{\prime} =\nu\,\boldsymbol{\nabla}\times\left(\frac{\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{T}_{vis}}{\rho}\right). \end{gather}

Specifically, as in the case of linear and angular momentum barriers, we obtain the following result.

Theorem 6.3 For incompressible, uniform-density Navier–Stokes flows, the material and instantaneous barrier equations (6.3) and (6.16) for vorticity take the specific form

(6.17)\begin{gather} \boldsymbol{x}_{0}^{\prime} =\nu\,\overline{(\boldsymbol{F}_{t_{0}}^{t})^{*} {\rm \Delta}\boldsymbol{\omega}}, \end{gather}

(6.18)\begin{gather}\boldsymbol{x}^{\prime} =\nu\,{\rm \Delta} \boldsymbol{\omega}. \end{gather}

As in the case of the linear and angular momenta, the active barrier equations (6.17) and (6.5) define 3-D, autonomous, volume-preserving dynamical systems with respect to the time-like, evolutionary variable $s\in \mathbb {R}$, and hence can be analysed by adopting tools available such equations (see § 8).

As for boundary conditions for trajectories of the (6.17) along a no-slip boundary $\partial U$ in the incompressible case with $\rho _{0}(\boldsymbol {x})\equiv 1$, the vorticity-transport equation along the wall $\partial U$ takes the form

(6.19)\begin{equation} \begin{pmatrix} \left(\dfrac{\textrm{D}}{\textrm{D}t}\boldsymbol{\omega}\right)\boldsymbol{\cdot} \boldsymbol{e}_{1}\\ \left(\dfrac{\textrm{D}}{\textrm{D}t}\boldsymbol{\omega}\right)\boldsymbol{\cdot} \boldsymbol{e}_{2}\\ 0 \end{pmatrix}=\nu\begin{pmatrix} \boldsymbol{{\rm \Delta}\omega}\boldsymbol{\cdot} \boldsymbol{e}_{1}\\ \boldsymbol{{\rm \Delta}\omega}\boldsymbol{\cdot} \boldsymbol{e}_{2}\\ \boldsymbol{{\rm \Delta}\omega}\boldsymbol{\cdot} \boldsymbol{e}_{3} \end{pmatrix}{\rm \Delta}\boldsymbol{\omega}+\begin{pmatrix} \boldsymbol{\nabla} \times \boldsymbol{q}\boldsymbol{\cdot} \boldsymbol{e}_{1}\\ \boldsymbol{\nabla} \times \boldsymbol{q}\boldsymbol{\cdot} \boldsymbol{e}_{2}\\ \boldsymbol{\nabla} \times \boldsymbol{q}\boldsymbol{\cdot} \boldsymbol{e}_{3} \end{pmatrix},\end{equation}

with the vectors $\boldsymbol {e}_{i}$ defined as in formula (6.7). Consequently, whenever the curl of non-potential body forces is normal to a no-slip boundary $\partial U$, the vector field ${\rm \Delta} \boldsymbol {\omega }$ satisfies a no-penetration boundary condition along $\partial U$, given that $\boldsymbol {{\rm \Delta} \omega }\boldsymbol {\cdot } \boldsymbol {e}_{3}$ must then vanish by (6.19). As we have already noted in relation to formula (6.7), this in turn implies that $\partial U$ is an invariant manifold for the two flows in (6.17) and (6.18).

7.1. Two-dimensional Navier–Stokes flows viewed as 3-D Navier–Stokes flows with symmetry

We define the planar variable $\boldsymbol {\hat {\boldsymbol {x}}}=(x_{1},x_{2})\in \mathbb {R}^{2}$ and assume that a solution of the 3-D incompressible Navier–Stokes equation is of the form

(7.1a,b)\begin{equation} \boldsymbol{u}(\boldsymbol{x},t)=(\hat{\boldsymbol{u}}(\hat{\boldsymbol{x}},t), w(\hat{\boldsymbol{x}},t)),\quad p(\boldsymbol{x},t)=p(\hat{\boldsymbol{x}},t),\quad \boldsymbol{x}=(\boldsymbol{\hat{\boldsymbol{x}}},x_{3})\in\mathbb{R}^{3}, \end{equation}

with the 2-D velocity field $\hat {\boldsymbol {u}}(\hat {\boldsymbol {x}},t)$ and the scalar functions $w(\hat {\boldsymbol {x}},t)$ and $p(\hat {\boldsymbol {x}},t)$ (see, e.g., Majda & Bertozzi Reference Majda and Bertozzi2002). Under this 2-D symmetry ansatz, substitution of $\boldsymbol {u}$ and $\boldsymbol {p}$ into the 3-D Navier–Stokes equation gives

(7.2)\begin{gather} \partial_{t}\boldsymbol{\hat{\boldsymbol{u}}}+ (\boldsymbol{\nabla}_{\boldsymbol{\hat{\boldsymbol{x}}}}\hat{\boldsymbol{u}}) \hat{\boldsymbol{u}} =-\frac{1}{\rho}\boldsymbol{\nabla}_{\boldsymbol{\hat{\boldsymbol{x}}}}p +\nu{\rm \Delta}_{\hat{\boldsymbol{x}}}\hat{\boldsymbol{u}}, \end{gather}

(7.3)\begin{gather}\partial_{t}w+\boldsymbol{\nabla}_{\boldsymbol{\hat{\boldsymbol{x}}}}w\boldsymbol{\cdot} \hat{\boldsymbol{u}} =\nu{\rm \Delta}_{\hat{\boldsymbol{x}}}w, \end{gather}

with the subscript $\hat {\boldsymbol {x}}$ referring to the 2-D version of the differential operators involved. Therefore, the symmetry ansatz (7.1a,b) for a 3-D Navier–Stokes solution is valid if $w(\hat {\boldsymbol {x}},t)$ is chosen as a solution of the advection–diffusion equation appearing in (7.3). This advection–diffusion equation, however, coincides with the 2-D vorticity-transport equation, which is solved by

(7.4)\begin{equation} w(\boldsymbol{\hat{\boldsymbol{x}}},t)= \hat{\omega}(\boldsymbol{\hat{\boldsymbol{x}}},t),\end{equation}

with $\hat {\omega }(\boldsymbol {\hat {\boldsymbol {x}}},t)$ denoting the scalar vorticity field of the 2-D Navier–Stokes solution $\hat {\boldsymbol {u}}(\boldsymbol {\hat {\boldsymbol {x}}},t)$. In the following, we will choose the third component of $\boldsymbol {u}$ as in (7.4) and use the notation

(7.5)\begin{equation} \boldsymbol{J}=\begin{pmatrix} 0 & 1\\ -1 & 0 \end{pmatrix}, \end{equation}

for the 2-D canonical symplectic matrix $\boldsymbol {J}$. With this notation, we obtain the following results on active barriers to momentum transport in (7.1a,b).

Theorem 7.1 For 2-D incompressible, uniform-density Navier–Stokes flows, the material and instantaneous barrier equations (6.5) and (6.6) for linear momentum are autonomous Hamiltonian systems of the form

(7.6)\begin{gather} \hat{\boldsymbol{x}}_{0}^{\prime} =\nu\rho \boldsymbol{J}\boldsymbol{\nabla}_{0} \overline{\hat{\omega} (\hat{\boldsymbol{F}}_{t_{0}}^{t} (\hat{\boldsymbol{x}}_{0} ),t )}, \end{gather}

(7.7)\begin{gather}\hat{\boldsymbol{x}}^{\prime} =\nu\rho \boldsymbol{J}\boldsymbol{\nabla} \hat{\omega} (\hat{\boldsymbol{x}},t ), \end{gather}

respectively. Therefore, time-$t_{0}$ positions of material active barriers to linear momentum transport in these flows are structurally stable level curves of the time-averaged Lagrangian vorticity $\overline {\hat {\omega }(\hat {\boldsymbol {F}}_{t_{0}}^{t}(\boldsymbol {\hat {x}}_{0}),t)}$ viewed as a Hamiltonian. Similarly, instantaneous active barriers to linear momentum transport at time $t$ are structurally stable level curves of the vorticity $\hat {\omega }(\boldsymbol {\hat {\boldsymbol {x}}},t)$.

While streamlines in general are not objective, the streamlines of the vorticity $\hat {\omega }(\boldsymbol {\hat {\boldsymbol {x}}},t)$ are Eulerian objective and streamlines of the time-averaged Lagrangian vorticity $\overline {\hat {\omega }(\hat {\boldsymbol {F}}_{t_{0}}^{t}(\boldsymbol {\hat {x}}_{0}),t)}$ are Lagrangian objective (see Ogden Reference Ogden1984). This is consistent with the more general result established in (3.12) for the objectivity of all active barriers.

Active barriers to vorticity transport in (7.1a,b) also turn out to be trajectories of autonomous Hamiltonian systems. To state this result, we will use the notation

(7.8)\begin{equation} \delta\hat{\omega}(\hat{\boldsymbol{x}}_{0},t_{0},t_{1}) :=\hat{\omega}(\hat{\boldsymbol{F}}_{t_{0}}^{t_{1}} (\hat{\boldsymbol{x}}_{0}),t_{1})-\hat{\omega} (\hat{\boldsymbol{x}}_{0},t_{0}),\end{equation}

for the Lagrangian vorticity-change function along trajectories over the time interval $[t_{0},t_{1}]$.

Theorem 7.2 For 2-D incompressible, uniform-density Navier–Stokes flows, the material and instantaneous barrier equations (6.17) and (6.18) for linear momentum are autonomous Hamiltonian systems of the form

(7.9)\begin{gather} \hat{\boldsymbol{x}}_{0}^{\prime} =\frac{\nu}{t_{1}-t_{0}}\,\boldsymbol{J} \boldsymbol{\nabla}_{0}\,\delta\hat{\omega}(\hat{\boldsymbol{x}}_{0}, t_{0},t_{1}), \end{gather}

(7.10)\begin{gather}\hat{\boldsymbol{x}}^{\prime} =\nu\,\boldsymbol{J}\boldsymbol{\nabla} \frac{\textrm{D}}{\textrm{D}t}\hat{\omega}(\hat{\boldsymbol{x}},t), \end{gather}

respectively. Therefore, time-$t_{0}$ positions of material active barriers to linear momentum transport in these flows are structurally stable level curves of the Lagrangian vorticity-change function $\delta \hat {\omega }(\hat {\boldsymbol {x}}_{0},t_{0},t_{1})$ viewed as a Hamiltonian. Similarly, instantaneous active barriers to linear momentum transport at time $t$ are structurally stable level curves of the material derivative $({\textrm {D}}/{\textrm {D}t})\hat {\omega }(\hat {\boldsymbol {x}},t)$, or equivalently, of the vorticity Laplacian ${\rm \Delta} \hat {\omega }(\hat {\boldsymbol {x}},t).$

While vorticity is not objective, the level curves of the Lagrangian vorticity change $\delta \hat {\omega }(\hat {\boldsymbol {x}}_{0},t_{0},t_{1})$ is objective. This follows directly from the objectivity of the barrier equations that we have generally established, but can also be verified directly using the definition of objectivity.

Example 7.4 We consider the spatially doubly periodic Navier–Stokes flow family described by Majda & Bertozzi (Reference Majda and Bertozzi2002) in the form

(7.11a,b)\begin{gather} \hat{\boldsymbol{u}}(\hat{\boldsymbol{x}},t) =\textrm{e}^{-4\pi^{2}\ell\nu t} \hat{\boldsymbol{u}}_{0}(\hat{\boldsymbol{x}}), \quad p(\hat{\boldsymbol{x}},t)=\textrm{e}^{-4\pi^{2}\ell\nu t} p_{0}(\hat{\boldsymbol{x}}), \end{gather}

(7.12a,b)\begin{gather}\hat{\boldsymbol{u}}_{0}(\hat{\boldsymbol{x}}) = \sum_{ |\boldsymbol{k} |^{2}=\ell} \begin{pmatrix} a_{\boldsymbol{k}}k_{2}\sin (2\pi\boldsymbol{k}\boldsymbol{\cdot} \hat{\boldsymbol{x}} )-b_{\boldsymbol{k}}k_{2}\cos (2\pi\boldsymbol{k}\boldsymbol{\cdot} \hat{\boldsymbol{x}} )\\ -a_{\boldsymbol{k}}k_{1}\sin (2\pi\boldsymbol{k}\boldsymbol{\cdot} \hat{\boldsymbol{x}} )+b_{\boldsymbol{k}}k_{1}\cos (2\pi\boldsymbol{k}\boldsymbol{\cdot} \hat{\boldsymbol{x}} ) \end{pmatrix}, \end{gather}

where $\hat {\boldsymbol {u}}_{0}(\hat {\boldsymbol {x}})$ and $p_{0}(\hat {\boldsymbol {x}})$ solve the steady planar Euler equation for some positive integer $\ell$ (this flow family contains our motivating example (A 1a,b) in appendix A with the choice $k_{1}=0$, $\ell =k_{2}=1$, $a_{(1,0)}=b_{(1,0)}=a_{(0,1)}=0$ and $b_{(0,1)}=a$ if we let $x_{2}\to -x_{2}$). In that case, we have

(7.13)\begin{equation} {\rm \Delta}\boldsymbol{u} =\begin{pmatrix} {\rm \Delta}_{\hat{\boldsymbol{x}}}\hat{\boldsymbol{u}}\\ {\rm \Delta}_{\hat{\boldsymbol{x}}}\hat{\omega}\end{pmatrix}=\begin{pmatrix} -4\pi^{2}\ell \, \textrm{e}^{-4\pi^{2}\ell\nu t}\hat{\boldsymbol{u}}_{0}(\hat{\boldsymbol{x}})\\ {\rm \Delta}_{\hat{\boldsymbol{x}}}\hat{\omega}(\hat{\boldsymbol{x}},t) \end{pmatrix}. \end{equation}

One can verify by direct substitution that $\textrm {e}^{-4\pi ^{2}\ell \nu t}\hat {\boldsymbol {u}}_{0}(\hat {\boldsymbol {F}}_{t_{0}}^{t}(\boldsymbol {\hat {\boldsymbol {x}}}_{0}))$ is a solution of the equation of variations $\dot {\boldsymbol {\xi }}=\textrm {e}^{-4\pi ^{2}\ell \nu t}\boldsymbol {\nabla }_{\boldsymbol {\hat {\boldsymbol {x}}}} \hat {\boldsymbol {u}}_{0}(\hat {\boldsymbol {x}}(t))\boldsymbol {\xi }$ (whose fundamental matrix solution is $\nabla _{\hat {\boldsymbol {x}}_{0}}\boldsymbol {\hat {F}}_{t_{0}}^{t}(\boldsymbol {x}_{0})$) for the differential equation $\dot {\boldsymbol {x}}=\textrm {e}^{-4\pi ^{2}\ell \nu t}\hat {\boldsymbol {u}}_{0}(\hat {\boldsymbol {x}})$. As a consequence, we have

(7.14)\begin{equation} [\boldsymbol{\nabla}_{\boldsymbol{\hat{\boldsymbol{x}}}_{0}} \hat{\boldsymbol{F}}_{t_{0}}^{t} (\boldsymbol{\hat{\boldsymbol{x}}}_{0} ) ]^{-1}\textrm{e}^{-4\pi^{2}\ell\nu t}\hat{\boldsymbol{u}}_{0} (\hat{\boldsymbol{F}}_{t_{0}}^{t} (\boldsymbol{\hat{\boldsymbol{x}}}_{0} ) )= \textrm{e}^{-4\pi^{2}\ell\nu t_{0}}\hat{\boldsymbol{u}}_{0} (\boldsymbol{\hat{\boldsymbol{x}}}_{0} ), \end{equation}

and hence, by Theorem 7.1, the material and instantaneous barrier equations for linear momentum take the specific form

(7.15)\begin{equation} \left.\begin{array}{c@{}} \hat{\boldsymbol{x}}_{0}^{\prime} =\nu\rho \,\textrm{e}^{-4\pi^{2}\ell\nu t_{0}} \hat{\boldsymbol{u}}_{0} (\boldsymbol{\hat{\boldsymbol{x}}}_{0} ),\\ x_{03}^{\prime} =\nu\rho A(\boldsymbol{\hat{\boldsymbol{x}}}_{0}, t_{1},t_{0}),\\ \hat{\boldsymbol{x}}^{\prime} =\nu\rho \,\textrm{e}^{-4\pi^{2}\ell\nu t} \hat{\boldsymbol{u}}_{0} (\boldsymbol{\hat{\boldsymbol{x}}} ),\\ x_{3}^{\prime} =\nu\rho A(\boldsymbol{\hat{\boldsymbol{x}}},t,t), \end{array}\right\} \end{equation}

for an appropriate function $A(\boldsymbol {\hat {\boldsymbol {x}}}_{0},t_{1},t_{0})$. Therefore, both material and instantaneous barriers to linear momentum transport in the 2-D Navier–Stokes flow family in (7.11a,b) are structurally stable streamlines of the steady velocity field $\hat {\boldsymbol {u}}_{0}(\boldsymbol {\hat {\boldsymbol {x}}}_{0})$.

As for vorticity barriers in this example, note that

(7.16)\begin{gather} \hat{\omega} =\partial_{x_{1}}u_{2}-\partial_{x_{2}}u_{1}=-2 \pi\ell \, \textrm{e}^{-4\pi^{2}\ell\nu t}\hat{\omega}_{0}, \end{gather}

(7.17)\begin{gather}\hat{\omega}_{0}(\boldsymbol{\hat{\boldsymbol{x}}}) = \sum_{|\boldsymbol{k}|^{2}=\ell}a_{\boldsymbol{k}}\cos (2\pi\boldsymbol{k}\boldsymbol{\cdot} \hat{\boldsymbol{x}})+ b_{\boldsymbol{k}}\sin(2\pi\boldsymbol{k}\boldsymbol{\cdot} \hat{\boldsymbol{x}}). \end{gather}

As the steady part of the vorticity field solves the steady planar Euler equation, trajectories of $\hat {\boldsymbol {u}}(\hat {\boldsymbol {x}},t)$ remain confined to the steady streamlines of $\hat {\boldsymbol {u}}_{0}(\boldsymbol {\hat {\boldsymbol {x}}}_{0})$. Since these trajectories also conserve the vorticity $\hat {\omega }_{0}$ of the inviscid limit of the flow, the change in vorticity $\hat {\omega }(\hat {\boldsymbol {x}},t)$ along trajectories of $\hat {\boldsymbol {u}}(\hat {\boldsymbol {x}},t)$ can be written as

(7.18)\begin{equation} \delta\hat{\omega} (\hat{\boldsymbol{x}}_{0},t_{0},t_{1} ) =-2\pi\ell (\textrm{e}^{-4\pi^{2}\ell\nu t_{1}}-\textrm{e}^{-4\pi^{2}\ell\nu t_{0}} )\hat{\omega}_{0} (\boldsymbol{\hat{\boldsymbol{x}}}_{0} ). \end{equation}

Therefore, level curves of the vorticity change along trajectories coincide with those of the inviscid vorticity $\hat {\omega }_{0}(\boldsymbol {\hat {\boldsymbol {x}}}_{0})$, which are in turn just the streamlines of $\hat {\boldsymbol {u}}_{0}(\boldsymbol {\hat {\boldsymbol {x}}}_{0})$. Finally, we have

(7.19)\begin{equation} {\rm \Delta}\hat{\omega}(\hat{\boldsymbol{x}},t)= 8\pi^{3}\ell^{2}\textrm{e}^{-4\pi^{2}\ell\nu t}\hat{\omega}_{0} (\boldsymbol{\hat{\boldsymbol{x}}}), \end{equation}

and hence the level curves of ${\rm \Delta} \hat {\omega }(\hat {\boldsymbol {x}},t)$ also coincide with those of $\hat {\boldsymbol {u}}_{0}(\boldsymbol {\hat {\boldsymbol {x}}})$.

We conclude that both material and instantaneous active barriers to vorticity and linear momentum transport coincide with the streamlines of $\hat {\boldsymbol {u}}_{0}(\boldsymbol {\hat {\boldsymbol {x}}}_{0})$. In particular, we obtain the correct active barrier distributions that we inferred for our motivational 2-D channel-flow example in figure 1 (see (A 1a,b) in appendix A), which is part of the solution family (7.11a,b). Importantly, we obtain the same frame-indifferent conclusion about active barriers from any finite-time (or even instantaneous) analysis of the velocity field (7.11a,b).

7.2. Directionally steady Beltrami flows

Virtually all explicitly known, unsteady solutions of the 3-D incompressible Navier–Stokes equations satisfy the strong Beltrami property

(7.20)\begin{equation} \boldsymbol{{\omega}}(\boldsymbol{x},t)= k(t)\boldsymbol{u}(\boldsymbol{x},t)\end{equation}

for some scalar function $k(t)$ (see Majda & Bertozzi Reference Majda and Bertozzi2002). By definition, for any such incompressible strong Beltrami flow, we obtain

(7.21)\begin{equation} \left.\begin{array}{c@{}} {\rm \Delta}\boldsymbol{\omega} =\boldsymbol{\nabla} (\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{\omega} )-\boldsymbol{\nabla}\times (\boldsymbol{\nabla} \times\boldsymbol{\omega} )=-k^{3}\boldsymbol{u},\\ {\rm \Delta}\boldsymbol{u} =\dfrac{1}{k}{\rm \Delta}\boldsymbol{\omega}=-k^{2}\boldsymbol{u}. \end{array}\right\} \end{equation}

Recall that if a steady Euler flow is non-Beltrami, then it is integrable (Arnold & Keshin Reference Arnold and Keshin1998). Therefore, only velocity fields satisfying the Beltrami property can generate complicated particle dynamics in steady, inviscid flows.

We call an unsteady strong Beltrami flow with velocity field $\boldsymbol {u}(\boldsymbol {x},t)$ a directionally steady Beltrami flow if

(7.22a,b)\begin{equation} \boldsymbol{u}(\boldsymbol{x},t)=\alpha(t)\boldsymbol{u}_{0}(\boldsymbol{x}), \quad\boldsymbol{{\boldsymbol{\omega}}}(\boldsymbol{x},t)= \boldsymbol{\nabla}\times\boldsymbol{u}(\boldsymbol{x},t)=k(t)\alpha(t) \boldsymbol{u}_{0}(\boldsymbol{x}), \end{equation}

hold for some continuously differentiable scalar function $\alpha (t)$. Note that any steady strong Beltrami flow $\boldsymbol {u}_{0}(\boldsymbol {x})$ (which necessarily admits $k(t)\equiv k=\textrm {const.})$ solves the steady Euler equation and generates a directionally steady Beltrami solution $\boldsymbol {u}(\boldsymbol {x},t)=\exp (-\nu k^{2}t)\boldsymbol {u}_{0}(\boldsymbol {x})$ for the unsteady Navier–Stokes equation under conservative forcing (Majda & Bertozzi Reference Majda and Bertozzi2002).

For all directionally steady Beltrami flows, we obtain the following simple result on active transport barriers.

Theorem 7.5 shows that invariant manifolds for the Lagrangian particle motion in directionally steady Beltrami flows coincide with material and instantaneous active barriers to linear momentum and vorticity transport. This agrees with one's intuition: observed mass-transport barriers in these flows are expected to coincide with barriers to vorticity and momentum transport, given that momentum and vorticity are scalar multiples of each other. Remarkably, as in the case of 2-D flows analysed in the previous section, the exact barriers emerge from our analysis independently of the choice of the finite-time interval $[t_{0},t_{1}]$, including the case of instantaneous extraction with $t_{0}=t_{1}$.

Example 7.7 Examples of directionally steady Beltrami flows include the Navier–Stokes flow family (Ethier & Steinman Reference Ethier and Steinman1994)

(7.23a,b)\begin{align} \boldsymbol{u}(\boldsymbol{x},t)=\textrm{e}^{-\nu d^{2}t}\boldsymbol{u}_{0}(\boldsymbol{x}), \quad \boldsymbol{u}_{0}(\boldsymbol{x})=-a\begin{pmatrix} \textrm{e}^{ax_{1}}\sin(ax_{2}\pm \, {\textrm{d}}x_{3})+\textrm{e}^{ax_{3}}\cos (ax_{1}\pm \, {\textrm{d}}x_{2})\\ \textrm{e}^{ax_{2}}\sin(ax_{3}\pm \, {\textrm{d}}x_{1})+\textrm{e}^{ax_{1}}\cos (ax_{2}\pm \, {\textrm{d}}x_{3})\\ \textrm{e}^{ax_{3}}\sin(ax_{1}\pm \, {\textrm{d}}x_{2})+\textrm{e}^{ax_{2}}\cos (ax_{3}\pm \, {\textrm{d}}x_{1}) \end{pmatrix}, \end{align}

and the viscous, unsteady version of the classic Arnold–Beltrami–Childress (ABC) flow $\boldsymbol {u}_{0}(\boldsymbol {x})$ (Dombre et al. Reference Dombre, Frisch, Greene, Hé non, Mehr and Soward1986), given by

(7.24a,b)\begin{equation} \boldsymbol{u}(\boldsymbol{x},t)=\textrm{e}^{-\nu t}\boldsymbol{u}_{0}(\boldsymbol{x}),\quad \boldsymbol{u}_{0}(\boldsymbol{x})=\begin{pmatrix} A\sin x_{3}+C\cos x_{2}\\ B\sin x_{1}+A\cos x_{3}\\ C\sin x_{2}+B\cos x_{1} \end{pmatrix}.\end{equation}

All lengths in these examples are non-dimensional. Further examples of 3-D, unsteady but directionally steady Beltrami solutions are derived by Barbato, Berselli & Grisanti (Reference Barbato, Berselli and Grisanti2007) and Antuono (Reference Antuono2020).

For all these flows, Theorem 7.5 guarantees that all material and instantaneous active barriers to diffusive momentum and vorticity transport coincide with structurally stable, 2-D invariant manifolds of the flow generated by the steady velocity field $\boldsymbol {u}_{0}(\boldsymbol {x})$. Such manifolds can be captured via their intersections with Poincaré sections, with these intersections appearing as invariant curves of the associated Poincaré map, as first illustrated by Dombre et al. (Reference Dombre, Frisch, Greene, Hé non, Mehr and Soward1986) for one cross-section of the ABC flow. A more complete set of Poincaré maps along three orthogonal planes is shown in the figure 4(d), which reveals several families of 2-D invariant tori, appearing as spatially periodic cylinders.

Figure 4. Three non-objective diagnostics ((a) sectional streamline plots, (b) the vorticity norm and (c) the $Q$-parameter) for the unsteady ABC flow (7.24a,b) with $A=\sqrt {3}$, $B=\sqrt {2}$ and $C=1$, at time $t=0$. All three plots remain the same for all times, because the velocity field is directionally steady. Also shown in (d) are three objective, active Poincaré maps computed for the associated barrier equations. The Lagrangian and Eulerian barrier equations for this flow are given by $\boldsymbol {x}_{0}^{\prime }=\boldsymbol {u}_{0}(\boldsymbol {x}_{0})$ by Theorem 7.5, both for momentum and vorticity. Black dots on the active Poincaré sections indicate repeated return locations of barrier trajectories launched from the same section. Intersections of 2-D, toroidal transport barriers with the three Poincaré sections are visible as invariant curves of these Poincaré maps. Outermost members of these torus families define objective coherent vortex boundaries.

As discussed in Remark 7.6, these torus families form objectively defined coherent vortices, with each torus acting as an internal barriers to both momentum and vorticity transport within the vortex. Outermost members of these torus families provide objective, active-transport-based coherent vortex boundaries. By their invariance under the flow map, they remain perfectly coherent under advection. For comparison, we also show in figure 4 other common Eulerian diagnostics applied to this flow: sectional streamlines (computed from velocities projected onto the three faces of the cube at time $t=0$); vorticity levels for $\boldsymbol {u}(\boldsymbol {x},t)$ at $t=0$; and levels of the parameter $Q=|\boldsymbol {W}|^{2}-|\boldsymbol {S}|^{2}$ at $t=0$, with the spin tensor $\boldsymbol {W}$ and the rate-of-strain tensor $\boldsymbol {S}$ defined as

(7.25a,b)\begin{equation} \boldsymbol{W}=\tfrac{1}{2} [\boldsymbol{\nabla}\boldsymbol{u}- (\boldsymbol{\nabla}\boldsymbol{u} )^\textrm{T} ], \quad\boldsymbol{S}=\tfrac{1}{2} [\boldsymbol{\nabla}\boldsymbol{u}+ (\boldsymbol{\nabla}\boldsymbol{u} )^\textrm{T} ].\end{equation}

The $Q>0$ region is often used to define vortices, and hence the white level sets are considered vortex boundaries by the $\boldsymbol {Q}$-criterion of Hunt, Wray & Moin (Reference Hunt, Wray and Moin1988). The structures appearing in the latter three plots change under an observer change and do not remain invariant under advection by the flow map.

Further studies revealing the same invariant manifolds in the steady ABC flow using finite-time Lyapunov exponents (FTLE) and the polar rotation angle (PRA) were given by Haller (Reference Haller2001) and Farazmand & Haller (Reference Farazmand and Haller2016), each emphasizing different classes of barriers from the complete collection revealed in figure 4. The FTLE and the PRA are generally usable structure detection tools along any cross-section of an unsteady flow, whereas Poincaré maps are only defined for trajectories returning to the same cross-section of a steady or time-periodic flow. In the next section, we will also show the passive FTLE and PRA plots computed for the ABC flow (7.24a,b), as well as active versions of the FTLE and PRA applied to the barrier equations of the ABC flow over the same time interval.

8.1. Computation from highly resolved numerical data

All applications of our main results in Theorems 6.1–6.3 require the analysis of the associated 3-D autonomous, divergence-free dynamical systems that depend on the Laplacian of $\boldsymbol {u}(\boldsymbol {x},t)$ for momentum-transport barriers, or on the Laplacian of $\boldsymbol {\omega }(\boldsymbol {x},t)$ for vorticity-transport barriers. In direct numerical simulations (DNS) of the Navier–Stokes equation, the required Laplacians can be computed spectrally with high accuracy, as our numerical results in § 9.2 will illustrate. With these Laplacians at hand, one proceeds to find invariant manifolds of the barrier equations in Theorems 6.1–6.3, which invariably involves computing trajectories of these equations. In generating these trajectories numerically, it is usually helpful to omit the (small) viscosity $\nu$ from the right-hand sides of the barrier equations to speed up the simulation. This omission of $\nu$ is equivalent to a rescaling of the time-like variable $s$ in the barrier equations, which does not alter the trajectories of these autonomous differential equations.

For 2-D incompressible Navier–Stokes flows, Theorems 7.1 and 7.2 show the relevant barrier equations to be computed. The right-hand sides of these equations are autonomous Hamiltonian vector fields whose trajectories coincide with the level curves of the corresponding Hamiltonians. Strictly speaking, therefore, the numerical solution of these barrier equations can be avoided by simply plotting the level curves of their Hamiltonians, which can be computed by finite differencing the velocity field (but see also Remark 7.3 in § 7.1).

8.2. Computation from experimental or lower-resolved numerical data

Taking second and third spatial derivatives of a velocity field obtained from an already finalized numerical simulation or experiment is challenging. An alternative is to work with the original material derivatives arising in our definition of active transport, rather than with the Laplacians of the velocity and the vorticity. More specifically, if we let $\boldsymbol {a}(\boldsymbol {x},t)= ({\textrm {D}\boldsymbol {u}}/{\textrm {D}t})(\boldsymbol {x},t)$ denote the Lagrangian particle acceleration along fluid trajectories, then using the general momentum equation (2.1), the active barrier equations (6.3) and (6.4) for the linear momentum can be rewritten as

(8.1)\begin{gather} \boldsymbol{x}_{0}^{\prime} =\overline{\det\boldsymbol{\nabla}\boldsymbol{F}_{t_{0}}^{t} (\boldsymbol{F}_{t_{0}}^{t})^{*}[\rho\boldsymbol{a}+ \boldsymbol{\nabla}p-\boldsymbol{q}]}, \end{gather}

(8.2)\begin{gather}\boldsymbol{x}^{\prime} =\rho\boldsymbol{a}+\boldsymbol{\nabla}p-\boldsymbol{q}. \end{gather}

These equations involve the Lagrangian acceleration, $\boldsymbol {a}(\boldsymbol {x},t)$, which can be obtained from high-resolution numerical or experimental data via the temporal differentiation of the velocity vector along trajectories.

Similarly, the most general active barrier equations (6.15) and (6.16) for vorticity can be rewritten as

(8.3)\begin{gather} \boldsymbol{x}_{0}^{\prime} =\overline{\det\boldsymbol{\nabla}\boldsymbol{F}_{t_{0}}^{t} (\boldsymbol{F}_{t_{0}}^{t})^{*}\boldsymbol{\nabla}\times \left[\boldsymbol{a}+\frac{1}{\rho}(\boldsymbol{\nabla}p- \boldsymbol{q})\right]}, \end{gather}

(8.4)\begin{gather}\boldsymbol{x}^{\prime} =\boldsymbol{\nabla}\times \left[\boldsymbol{a}+\frac{1}{\rho}(\boldsymbol{\nabla}p- \boldsymbol{q})\right]. \end{gather}

In particular, for incompressible, constant-density, Newtonian fluids subject only to potential body forces, the material and instantaneous barrier equations for vorticity in (8.3) and (8.4) simplify to

(8.5)\begin{gather} \boldsymbol{x}_{0}^{\prime} =\overline{(\boldsymbol{F}_{t_{0}}^{t})^{*} \boldsymbol{\nabla}\times\boldsymbol{a}}, \end{gather}

(8.6)\begin{gather}\boldsymbol{x}^{\prime} =\boldsymbol{\nabla}\times \boldsymbol{a}, \end{gather}

given that $\boldsymbol {\nabla }\times [({1}/{\rho })(\boldsymbol {\nabla }p-\boldsymbol {q})]= ({1}/{\rho })\boldsymbol {\nabla }\times [\boldsymbol {\nabla }p-\boldsymbol {q}]\equiv \boldsymbol {0}$ holds for such flows.

8.3. Passive vs. active Poincaré maps

Passive Poincaré maps for 3-D steady flows map initial conditions of trajectories launched from a selected 2-D section to their first return to the section, if such a return exists. We refer to a Poincaré map computed for the 3-D steady barrier equations (4.1) or (5.1) as active Poincaré map (see figure 4 for an example). This 2-D mapping generally does not preserve the standard 2-D area, but preserves a general area form, which makes the active Poincaré map a 2-D symplectic map (Meiss Reference Meiss1992). One-dimensional invariant curves of 2-D symplectic maps satisfy the only available formal definition of advective transport barriers by MacKay et al. (Reference MacKay, Meiss and Percival1984), as we noted in the Introduction. Structurally stable invariant curves of 2-D symplectic maps include stable and unstable manifolds of hyperbolic fixed points and Kolmogorov–Arnold-Moser curves, i.e. nested families of closed curves satisfying non-resonance and twist conditions (Arnold Reference Arnold1978).

In contrast to active Poincaré maps, the mapping relating subsequent returns of trajectories to a selected section in the general unsteady velocity field $\boldsymbol {u}(\boldsymbol {x},t$) is not well defined as a single Poincaré map. Rather, this map will be different for different initial times $t_{0}$. Therefore, passive Poincaré maps are generally inapplicable to LCS detection in $\boldsymbol {u}(\boldsymbol {x},t)$, whereas active Poincaré maps are well defined on barrier-equation trajectories that return to a cross-section. In case they do not, the active versions of the FTLE and PRA fields introduced next provide alternative tools to uncover structurally stable invariant manifolds in the barrier equations.

8.4. Passive FTLE vs. active FTLE (aFTLE)

We fix a time interval $[t_{0},t_{1}]$ over which we would like to identify LCSs as coherent material surfaces in the advective transport induced by the unsteady velocity field $\boldsymbol {u}(\boldsymbol {x},t)$. With the notation of § 2, the right Cauchy–Green strain tensor $\boldsymbol {C}_{t_{0}}^{t_{1}}(\boldsymbol {x}_{0})$ is defined as

(8.7)\begin{equation} \boldsymbol{C}_{t_{0}}^{t_{1}}(\boldsymbol{x}_{0}):= [\boldsymbol{\nabla}\boldsymbol{F}_{t_{0}}^{t_{1}} (\boldsymbol{x}_{0})]^\textrm{T}\boldsymbol{\nabla}\boldsymbol{F}_{t_{0}}^{t_{1}} (\boldsymbol{x}_{0}),\end{equation}

with the superscript $T$ referring to the transpose. Then, if $\lambda _{{max}}(\boldsymbol {C}_{t_{0}}^{t_{1}})$ denotes the maximal eigenvalues of the symmetric, positive definite tensor $\boldsymbol {C}_{t_{0}}^{t_{1}}$, then the (passive) FTLE field of $\boldsymbol {u}(\boldsymbol {x},t)$ over the $[t_{0},t_{1}]$ time interval is defined as

(8.8)\begin{equation} \mathrm{FTLE}_{t_{0}}^{t_{1}}(\boldsymbol{x}_{0})= \frac{1}{2(t_{1}-t_{0})}\log\lambda_{{max}} (\boldsymbol{C}_{t_{0}}^{t_{1}}(\boldsymbol{x}_{0})).\end{equation}

Two-dimensional ridges of $\mathrm {FTLE}_{t_{0}}^{t_{1}}(\boldsymbol {x}_{0})$ are quick indicators of the time $t_{0}$ locations of hyperbolic LCS. They signal locally most repelling material surfaces when $t_{1}>t_{0}$ and locally most attracting material surfaces when $t_{1}<t_{0}$. Valleys of $\mathrm {FTLE}_{t_{0}}^{t_{1}}(\boldsymbol {x}_{0})$ tend to indicate elliptic (vortical) LCSs, whereas trenches of $\mathrm {FTLE}_{t_{0}}^{t_{1}}(\boldsymbol {x}_{0})$ signal parabolic (jet-type) LCSs. The minimal and maximal value of $t_{0}$ and $t_{1}$ are governed by the length of the available data and the scales relative to which we wish to determine the LCSs in the flow. The flow-map gradient involved in the definition of $\lambda _{{max}}(\boldsymbol {C}_{t_{0}}^{t_{1}}(\boldsymbol {x}_{0}))$ can be computed by finite differencing a set of trajectories, launched from a regular grid of initial conditions, with respect to those initial conditions. The $\mathrm {FTLE}_{t_{0}}^{t_{1}}(\boldsymbol {x}_{0})$ is a simple but objective LCS diagnostic, with its strengths and limitations reviewed in Haller (Reference Haller2015).

For $t_{1}=t_{0}\equiv t$, the instantaneous of limit of the FTLE field is the maximal rate-of-strain eigenvalue

(8.9)\begin{equation} \mathrm{FTLE}_{t}^{t}(\boldsymbol{x})=\lambda_{{max}} (\boldsymbol{S}(\boldsymbol{x},t)),\end{equation}

with the rate-of-strain tensor $\boldsymbol {S}(\boldsymbol {x},t)$ defined in (7.25a,b), as noted by Serra & Haller (Reference Serra and Haller2016) and Nolan, Serra & Ross (Reference Nolan, Serra and Ross2020). This eigenvalue field can, in principle, be used to detect objective Eulerian coherent structures (OECS) as instantaneous limits of LCS. In practice, the field $\mathrm {FTLE}_{t}^{t}(\boldsymbol {x})$ often provides insufficient spatial detail, but the eigenvector field of $\boldsymbol {S}(\boldsymbol {x},t)$ can be used to define and extract OECS (see Serra & Haller Reference Serra and Haller2016).

In contrast to passive FTLE, by active FTLE (aFTLE) we mean here the implementation of the FTLE diagnostic on the steady material barrier equation (4.1), including its steady instantaneous version (5.1). We again select a physical time interval $[t_{0},t_{1}]$ over which we would like to locate barriers to the active transport of the vector field $\boldsymbol {f}(\boldsymbol {x},t)$ in the velocity field $\boldsymbol {u}(\boldsymbol {x},t)$. Let $\tilde {\boldsymbol {x}}_{0}(s;0,\boldsymbol {x}_{0})$ denote the trajectory of the barrier ODE (4.1) starting at the dummy time $s=0$ from the initial location $\boldsymbol {x}_{0}$. The corresponding autonomous flow map for this barrier ODE will be denoted by the active flow map $\boldsymbol {\mathcal {F}}_{t_{0},t_{1}}^{s}\colon \boldsymbol {x}_{0} \mapsto \tilde {\boldsymbol {x}}_{0}(s;0,\boldsymbol {x}_{0}).$ The associated active Cauchy–Green strain tensor for the barrier equation (4.1) can then be defined as

(8.10)\begin{equation} \boldsymbol{\mathcal{C}}_{t_{0},t_{1}}^{s} (\boldsymbol{x}_{0}):=[\boldsymbol{\nabla}\boldsymbol{\mathcal{F}}_{t_{0},t_{1}}^{s} (\boldsymbol{x}_{0})]^\textrm{T}\boldsymbol{\nabla}\boldsymbol{\mathcal{F}}_{t_{0},t_{1}}^{s} (\boldsymbol{x}_{0}).\end{equation}

Again, if $\lambda _{{max}}(\boldsymbol {\mathcal {C}}_{t_{0},t_{1}}^{s})$ denotes the maximal eigenvalue of the symmetric, positive definite tensor $\boldsymbol {\mathcal {C}}_{t_{0},t_{1}}^{s}$, then the aFTLE field of $\boldsymbol {u}(\boldsymbol {x},t)$ over the $[t_{0},t_{1}]$ time interval, with respect to the vector field $\boldsymbol {f}(\boldsymbol {x},t)$, is defined as

(8.11)\begin{equation} \mathrm{aFTLE}_{t_{0},t_{1}}^{s}(\boldsymbol{x}_{0}; \boldsymbol{f})=\frac{1}{2s}\log\lambda_{{max}} (\boldsymbol{\mathcal{C}}_{t_{0},t_{1}}^{s} (\boldsymbol{x}_{0})).\end{equation}

Here the time-like parameter $s$ governs the level of accuracy and spatial resolution in the visualization of active transport barriers. The only limitation to the choice of $s$ is that the trajectories of the barrier equation (4.1) may ultimately leave the spatial domain $U$ over which the barrier equation is known. This is, however, unrelated to the physical time that the trajectories of $\boldsymbol {u}(\boldsymbol {x},t)$ spend in the domain $U$.

For instance, in our 2-D turbulence simulation to be analysed in § 9.1, the maximal possible spatial detail for LCS from $\mathrm {FTLE}_{t_{0}}^{t_{1}}(\boldsymbol {x}_{0})$ is limited by the length of the time interval $[t_{0},t_{1}]=[0,50],$ given that this is the temporal length of the available data set. In contrast, on the same data set, $\mathrm {aFTLE}_{t_{0},t_{1}}^{s}(\boldsymbol {x}_{0};\boldsymbol {f})$ can be computed for arbitrarily large $|s|,$ because the barrier vector field is known globally for $U=\mathbb {R}^{2}$. Similarly, in our 3-D turbulent channel-flow example in § 9.2, trajectories of the barrier equation tend to stay in the finite channel domain for much longer (non-dimensional) dummy times than the non-dimensional residence time of fluid trajectories in the same channel.

As a consequence, aFTLE has the potential to provide much finer spatial detail for active barriers than one is able to obtain for LCSs in the same data set from the passive FTLE. Figure 5 shows this substantial refinement obtained from the vorticity-based aFTLE relative to the passive FTLE computed over the same time interval $[t_{0},t_{1}]=[0,5]$ for the unsteady ABC flow (7.24a,b). (For this particular flow, the linear-momentum-based aFTLE and aPRA would give identical results by Theorem 7.5.) In addition, aFTLE is always guaranteed to converge under increasing $s$, as illustrated in figure 5, while the convergence of $\mathrm {FTLE}_{t_{0}}^{t_{1}}(\boldsymbol {x}_{0})$ is generally not guaranteed in an unsteady flow with time-varying structures.

Figure 5. Passive and vorticity-based active versions of the FTLE and PRA diagnostics for the unsteady ABC flow (7.24a,b), computed over the same time interval ($[t_{0},t_{1}]=[0,5]$) and with the same spatial resolution ($300^{3}$ grid points in the spatial domain $[0,2\pi ]^{3}$): (a) $\mathrm {FTLE}_0^{5} (\boldsymbol {x}_0)$, (b) $\mathrm {aFTLE}_{0,5}^{10} (\boldsymbol {x}_0; \boldsymbol {\omega })$, (c) $\mathrm {aFTLE}_{0,5}^{15} (\boldsymbol {x}_0; \boldsymbol {\omega })$, (d) $\mathrm {PRA}_0^{5} (\boldsymbol {x}_0)$, (e) $\mathrm {aPRA}_{0,5}^{15} (\boldsymbol {x}_0; \boldsymbol {\omega })$ and (f) $\mathrm {aPRA}_{0,5}^{50} (\boldsymbol {x}_0; \boldsymbol {\omega })$. Two values for the barrier time $s$ were selected to illustrate the increasing spatial resolution and convergence of hyperbolic barriers by the aFTLE and of elliptic barriers by the aPRA under increasing $s$-times. With the exception of the passive PRA, all diagnostics shown here are objective.

The $t_{1}=t_{0}\equiv t$ limit of the aFTLE field in (8.11) is

(8.12)\begin{equation} \mathrm{aFTLE}_{t,t}^{s}(\boldsymbol{x};\boldsymbol{f})=\frac{1}{2s}\log\lambda_{{max}} (\boldsymbol{\mathcal{C}}_{t,t}^{s}(\boldsymbol{x})).\end{equation}

Here, $\boldsymbol {\mathcal {C}}_{t,t}^{s}(\boldsymbol {x})$ is simply computed from the autonomous flow map $\boldsymbol {\mathcal {F}}_{t,t}^{s}(\boldsymbol {x})$ of the instantaneous barrier equation (5.1), with the instantaneous time $t$ playing the role of a constant parameter in the computation. Again, the time-like evolutionary variable $s$ in this computation can be arbitrarily large in norm, as long as the trajectories generated by the barrier flow map $\boldsymbol {\mathcal {F}}_{t,t}^{s}(\boldsymbol {x})$ stay in the domain $U$ over which the barrier vector field $\boldsymbol {b}_{t}^{t}(\boldsymbol {x})$ is known. This guarantees convergence and higher resolution in the detection of instantaneous objective barriers from $\mathrm {aFTLE}_{t,t}^{s}(\boldsymbol {x};\boldsymbol {f})$ when compared with $\mathrm {FTLE}_{t}^{t}(\boldsymbol {x})$. The only practical limitation to resolving the details of active barriers via aFTLE is the spatial resolution of the available data.

8.5. Passive PRA vs. active PRA (aPRA)

By the polar decomposition theorem Gurtin et al. (Reference Gurtin, Fried and Anand2013), the deformation gradient $\boldsymbol {\nabla }\boldsymbol {F}_{t_{0}}^{t_{1}}(\boldsymbol {x}_{0})$ can be uniquely decomposed as

(8.13)\begin{equation} \boldsymbol{\nabla}\boldsymbol{F}_{t_{0}}^{t_{1}}=\boldsymbol{R}_{t_{0}}^{t_{1}} \boldsymbol{U}_{t_{0}}^{t_{1}},\end{equation}

with the proper orthogonal rotation tensor $\boldsymbol {R}_{t_{0}}^{t_{1}},$ the symmetric and the positive definite right stretch tensor $\boldsymbol {U}_{t_{0}}^{t_{1}}$. The decomposition (8.13) means that a general deformation can locally always be viewed as triaxial stretching and compression followed by a rigid-body rotation. One can verify by direct substitution into (8.13) that $\boldsymbol {R}_{t_{0}}^{t_{1}}$ and $\boldsymbol {U}_{t_{0}}^{t_{1}}$ must be of the form

(8.14a,b)\begin{equation} \boldsymbol{U}_{t_{0}}^{t_{1}}=[\boldsymbol{C}_{t_{0}}^{t_{1}}]^{1/2},\quad \boldsymbol{R}_{t_{0}}^{t_{1}}=\boldsymbol{\nabla}\boldsymbol{F}_{t_{0}}^{t_{1}} [\boldsymbol{U}_{t_{0}}^{t_{1}}]^{-1},\end{equation}

with $\boldsymbol {C}_{t_{0}}^{t}$ defined in (8.7). The first equation in (8.14a,b) shows that $\boldsymbol {U}_{t_{0}}^{t}$ can be computed using the singular-value-decomposition of $\boldsymbol {C}_{t_{0}}^{t}$. With $\boldsymbol {U}_{t_{0}}^{t_{1}}$ at hand, one can compute the rotation tensor $\boldsymbol {R}_{t_{0}}^{t_{1}}$ from the second equation of (8.14a,b).

Farazmand & Haller (Reference Farazmand and Haller2016) show that $\boldsymbol {R}_{t_{0}}^{t_{1}}(\boldsymbol {x}_{0})$ rotates material elements around an axis of rotation by the polar rotation angle satisfying

(8.15)\begin{align} \mathrm{PRA}_{t_{0}}^{t_{1}}(\boldsymbol{x}_{0})=\cos^{-1} \left[\tfrac{1}{2}(\mathrm{{tr}\,}\boldsymbol{R}_{t_{0}}^{t_{1}}(\boldsymbol{x}_{0})-1)\right] =\cos^{-1}\left[\tfrac{1}{2}\left(\sum_{i=1}^{3}\langle \boldsymbol{\xi}_{i}(\boldsymbol{x}_{0}),\boldsymbol{\eta}_{i}(\boldsymbol{x}_{0}) \rangle -1\right)\right],\end{align}

with $\boldsymbol {\xi }_{i}(\boldsymbol {x}_{0})$ and $\boldsymbol {\eta }_{i}(\boldsymbol {x}_{0})$ denoting the right and left singular vectors of $\boldsymbol {\nabla }\boldsymbol {F}_{t_{0}}^{t}(\boldsymbol {x}_{0})$. For 2-D flows viewed as 3-D flows with a symmetry, the intermediate eigenvalue of $\boldsymbol {C}_{t_{0}}^{t}$ is always one, which simplifies $\mathrm {PRA}_{t_{0}}^{t_{1}}(\boldsymbol {x}_{0})$ to

(8.16)\begin{equation} \mathrm{PRA}_{t_{0}}^{t_{1}}(\boldsymbol{x}_{0})=\cos^{-1} \langle \boldsymbol{\xi}_{1}(\boldsymbol{x}_{0}), \boldsymbol{\eta}_{1}(\boldsymbol{x}_{0})\rangle =\cos^{-1} \langle \boldsymbol{\xi}_{2}(\boldsymbol{x}_{0}),\boldsymbol{\eta}_{2}(\boldsymbol{x}_{0})\rangle , \quad\boldsymbol{x}_{0}\in\mathbb{R}^{2}.\end{equation}

Farazmand & Haller (Reference Farazmand and Haller2016) propose $\mathrm {PRA}_{t_{0}}^{t_{1}}(\boldsymbol {x}_{0})$ as a diagnostic tool for elliptic (rotational) LCS. They find that nested circular or toroidal level sets of $\mathrm {PRA}_{t_{0}}^{t_{1}}(\boldsymbol {x}_{0})$ indeed highlight elliptic LCS significantly sharper than FTLE valleys do. They also show, however, that these level sets are only objective for 2-D flows. Similarly to FTLE calculations for $\boldsymbol {u}(\boldsymbol {x},t)$, the spatial scales resolved by the passive PRA in 2-D flows are limited by the length of the time interval $[t_{0},t_{1}].$ For 3-D flows, an additional limitation of the PRA is the non-objectivity of its level surfaces. The instantaneous limit $t_{0}=t_{1}\equiv t$ of the PRA gives $\mathrm {PRA}_{t}^{t}(\boldsymbol {x})\equiv 0$, and hence this diagnostic is unable to detect instantaneous limits of elliptic OECS.

In contrast, using the active rotation tensor

(8.17)\begin{equation} \boldsymbol{\mathcal{R}}_{t_{0,}t_{1}}^{s}=\boldsymbol{\nabla} \boldsymbol{\mathcal{F}}_{t_{0},t_{1}}^{s} [\boldsymbol{\mathcal{C}}_{t_{0},t_{1}}^{s}]^{-1/2},\end{equation}

the corresponding active PRA (aPRA) is obtained in three dimensions as

(8.18)\begin{equation} \mathrm{aPRA}_{t_{0,}t_{1}}^{s}(\boldsymbol{x}_{0};\boldsymbol{f})=\cos^{-1} \left[\tfrac{1}{2}(\mathrm{{tr}\,}\boldsymbol{\mathcal{R}}_{t_{0,} t_{1}}^{s}(\boldsymbol{x}_{0})-1)\right]\!=\!\cos^{-1}\left[\tfrac{1}{2} \left(\sum_{i=1}^{3}\langle \boldsymbol{\xi}_{i}^{a}(\boldsymbol{x}_{0}), \boldsymbol{\eta}_{i}^{a}(\boldsymbol{x}_{0})\rangle -1\right)\right],\end{equation}

with $\boldsymbol {\xi }_{i}^{a}(\boldsymbol {x}_{0})$ and $\boldsymbol {\eta }_{i}^{a}(\boldsymbol {x}_{0})$ denoting the right and left singular vectors of the active deformation gradient $\boldsymbol {\nabla }\boldsymbol {\mathcal {F}}_{t_{0},t_{1}}^{s}$. For 2-D flows, the corresponding formula is

(8.19)\begin{equation} \mathrm{aPRA}_{t_{0,}t_{1}}^{s}(\boldsymbol{x}_{0};\boldsymbol{f})=\cos^{-1} \langle \boldsymbol{\xi}_{1}^{a}(\boldsymbol{x}_{0}), \boldsymbol{\eta}_{1}^{a}(\boldsymbol{x}_{0})\rangle =\cos^{-1} \langle \boldsymbol{\xi}_{2}^{a}(\boldsymbol{x}_{0}), \boldsymbol{\eta}_{2}^{a}(\boldsymbol{x}_{0})\rangle , \quad\boldsymbol{x}_{0}\in\mathbb{R}^{2}.\end{equation}

Unlike for the passive PRA defined in (8.15), the spatial scales resolved by the aPRA can be gradually refined by increasing the time-like parameter $s$ in $\mathrm {aPRA}_{t_{0,}t_{1}}^{s}$. As in the case of the aFTLE, this increase is possible as long as the underlying trajectories $\tilde {\boldsymbol {x}}_{0}(s;0,\boldsymbol {x}_{0})$ of the barrier equation for $\boldsymbol {f}$ stay in the spatial domain $U$ where $\boldsymbol {u}(\boldsymbol {x},t)$ is known. As for aFTLE, the spatial resolution of the active barriers discoverable by aPRA is only limited by the resolution of the available velocity data. Figure 5 illustrates the substantial refinement and convergence for increasing $s$-values obtained from aPRA relative to the passive PRA computed over the same time interval $[t_{0},t_{1}]=[0,5]$ for the unsteady ABC flow (7.24a,b).

Another major advantage of $\mathrm {aPRA}_{t_{0,}t_{1}}^{s}$ over $\mathrm {PRA}_{t_{0}}^{t_{1}}$ is the objectivity of $\mathrm {aPRA}_{t_{0,}t_{1}}^{s}$, which follows from the objectivity of the barrier vector field $\boldsymbol {b}_{t_{0}}^{t_{1}}(\boldsymbol {x}_{0})$. Additionally, structures revealed by $\mathrm {aPRA}_{t_{0,}t_{1}}^{s}$ always converge as $s$ is increased, because $\mathrm {aPRA}_{t_{0,}t_{1}}^{s}$ operates on a steady flow, even though $\boldsymbol {u}(\boldsymbol {x},t)$ is unsteady. Finally, unlike $\mathrm {PRA}_{t_{0}}^{t_{1}}(\boldsymbol {x}_{0})$, its active version, $\mathrm {aPRA}_{t_{0,}t_{1}}^{s}$, has a non-degenerate instantaneous limit, $\mathrm {aPRA}_{t_{,}t}^{s}(\boldsymbol {x};\boldsymbol {f})$, which is just the PRA computed for the Eulerian barrier equation (5.1) over the barrier time interval $[0,s]$. This limit enables the detection of instantaneous limits of active elliptic LCSs as active elliptic OECSs.

8.6. Relationship between active and passive LCS diagnostics

Active LCS diagnostics are applied to barrier vector fields, whereas passive LCS diagnostics are applied to the underlying velocity field. As a consequence, active barriers highlighted by active LCS methods generally differ from passive barriers (coherent structures) detected by passive LCS methods. This is no surprise, given that these two types of barriers are constructed from different principles.

As an extreme case, all Eulerian and Lagrangian active barriers coincide with their passive counterparts in directionally steady Beltrami flows (see § 7.2). Therefore, the closer a generic flow is to a Beltrami flow in a given region, the closer its active and passive barriers will be to each other in that region. More broadly, the more correlated the velocity field is with its Laplacian (i.e. with the diffusive force field), the closer the Lagrangian and Eulerian momentum barriers are expected to be to their passive counterparts. Similarly, the more correlated the velocity field is with the vorticity Laplacian (i.e. with the curl of the diffusive force field), the closer the Lagrangian and Eulerian vorticity barriers will be to their passive counterparts.

As another extreme case, inviscid flows have barrier equations with identically vanishing right-hand sides. This is because there is no viscous transport in such flows and hence active barriers are not well defined. As a consequence, aFTLE and aPRA will identically vanish for such flows, while passive FTLE and PRA will only vanish if the inviscid flow has a spatially independent velocity field. Therefore, the more inviscid the flow is in a region, the more its active and passive barriers will differ from each other in that region.

A notable case between Beltrami and inviscid flows is a Lamb–Oseen velocity field modelling a vortex decaying due to viscosity (Saffman et al. Reference Saffman, Ablowitz, Hinch, Ockendon and Olver1992). Along each cylindrical streamsurface surrounding the origin in this flow, the viscous force is a constant negative multiple of the velocity at any given time. This immediately implies that all Eulerian active and passive barriers to momentum transport coincide with the cylindrical streamsurfaces of Lamb–Oseen vortices, even though their velocity field is not Beltrami. Indeed, we do find in our upcoming 2-D and 3-D turbulence examples that strong enough vortices have very similar overall signatures in the active and the passive LCS diagnostic fields, with the former field providing more detail. This stands in contrast to hyperbolic mixing regions outside those vortices, in which active and passive LCS diagnostics may differ substantially.

9.1. Two-dimensional homogeneous, isotropic turbulence

Here we evaluate our 2-D results from § 7.1 on active barriers in a 2-D turbulence simulation over a spatially periodic domain $U=[0,2\pi ]\times [0,2\pi ]$. Since all computations will be 2-D in this section, we drop the hat from the notation we used in § 7.1 for 2-D variables.

Obtained from a pseudo-spectral code applied to the 2-D, incompressible Navier–Stokes equation (see Farazmand, Kevlahan & Protas Reference Farazmand, Kevlahan and Protas2011), the spatial coordinates are resolved using $1024^{2}$ Fourier modes with $2/3$ dealiasing. The viscosity is $\nu =2\times 10^{-5}$. This data set comprises $251$ equally spaced velocity field snapshots spanning the time interval $[0,50]$. Whenever a numerical integration scheme is required, i.e. advection of particles and integration of the barrier fields, the Runge–Kutta 4 algorithm is employed. The same data set was already analysed by Katsanoulis et al. (Reference Katsanoulis, Farazmand, Serra and Haller2020), who located vortex boundaries as barriers to the diffusive transport of vorticity using the theory of constrained diffusion barriers from Haller et al. (Reference Haller, Karrasch and Kogelbauer2019). In contrast, here we use appropriate 2-D, steady barrier equations (7.6) and (7.7) and (7.9) and (7.10) (see also Remark 7.3) to visualize Lagrangian and objective Eulerian coherent vortices as regions bounded by maximal barriers to active transport.

9.1.1. Eulerian active barriers

For the instantaneous barrier calculations, we use the first snapshot of the data set at time $t=0$ and we compute the right-hand sides of (7.7) and (7.10) using a grid of $1024\times 1024$ points. In our experience, this grid spacing is much smaller than the size of the coherent vortices in this flow. As a consequence, the results do not change appreciably under further grid refinements, as long as one targets structurally stable objects in the Lagrangian particle dynamics, as we do (see Definition 4.1). For vorticity barriers, we use the 2-D version of (8.6) to illustrate the computational procedure for barriers in lower-resolved data. We then proceed to compute the aFTLE and aPRA for both the momentum and vorticity barrier fields from (8.12) and (8.19) using a central finite-differencing scheme for the active flow-map gradient required in (8.10).

We focus on the region $[2.8,4.9]\times [1,3]$ of the full computational domain to illustrate the level of spatial detail we obtain from instantaneous velocity data (see figure 6). We note the striking differences in the quality of the delineated structures between the instantaneous limit of the passive FTLE and the momentum-based aFTLE of figure 6. Advective LCSs tend to have relatively weak signatures in the instantaneous limit of the FTLE field (see formula (8.9)) which is given by the dominant rate-of-strain eigenvalue field. In contrast, active barriers remain sharply defined in the aFTLE fields, which offer increasing refinement of the flow features under increasing $s$-times. The only limitation to this refinement is the resolution of the available data. This is apparent in the vorticity-based aFTLE in figure 6(c), where the improvement is more modest, given that higher-order spatial derivatives need to be computed from the same data set.

Figure 6. Comparison of the $t=0$ instantaneous limits of the passive FTLE, the aFTLE with respect to $\rho \boldsymbol {u}$ with $s=0.05$ and the aFTLE with respect to $\boldsymbol {{\omega }}$ with $s=0.15$ in our 2-D turbulence example: (a) $\mathrm {FTLE}_0^{0} (\boldsymbol {x})$, (b) $\mathrm {aFTLE}_{0,0}^{0.05} (\boldsymbol {x}; \rho \boldsymbol {u})$ and (c) $\mathrm {aFTLE}_{0,0}^{0.15} (\boldsymbol {x}; \boldsymbol {\omega })$.

Figure 7 focuses on momentum-based active barriers in one of the vortical regions revealed by figure 6. The aFTLE provides a clear demarcation of the main vortex, which becomes even more pronounced for longer $s$-times, revealing secondary vortices around its neighbourhood. In contrast, none of these vortices are present in the passive FTLE in figure 7. A similar result emerges when the same region is analysed using the aPRA field in the same figure. Specifically, the effect of progressive refinement with increasing $s$-times is more prominent here as a number of elliptic structures become visible in the main vortical region. In contrast, the instantaneous limit of the passive PRA returns identically zero values, as the instantaneous limit of all polar rotation angles is zero by definition.

Figure 7. Comparison between the instantaneous limit of the passive FTLE (PRA) and the instantaneous limit of the momentum-based aFTLE (PRA) fields for $s=0.05$ and for $s=0.15$ in one of the vortical regions of our 2-D turbulence example at $t=0$: (a) $\mathrm {FTLE}_0^{0} (\boldsymbol {x})$, (b) $\mathrm {aFTLE}_{0,0}^{0.05} (\boldsymbol {x}; \rho \boldsymbol {u})$, (c) $\mathrm {aFTLE}_{0,0}^{0.15} (\boldsymbol {x}; \rho \boldsymbol {u})$, (d) $\mathrm {PRA}_{0}^{0} (\boldsymbol {x})$, (e) $\mathrm {aPRA}_{0,0}^{0.1} (\boldsymbol {x}; \rho \boldsymbol {u})$ and (f) $\mathrm {aPRA}_{0,0}^{0.15} (\boldsymbol {x}; \rho \boldsymbol {u})$.

9.1.2. Lagrangian active barriers

For the Lagrangian computations in this example, we use the same, slightly oversampled grid of Katsanoulis et al. (Reference Katsanoulis, Farazmand, Serra and Haller2020) with $1100\times 1100$ equally spaced initial conditions and we advect them over the time interval $[0,25]$ using all the available velocity snapshots. To compute the required Lagrangian averages along trajectories, we use $25$ snapshots of the appropriate quantities as using more snapshots does not bring any noticeable changes to the resulting barrier fields. Based on that, we compute the expressions for the active barrier fields from (7.6) and (7.9), which we then use for the evaluation of the aFTLE and aPRA.

Comparisons between these scalar diagnostic fields and the passive FTLE and PRA are shown in figure 8. We observe that the momentum-based aFTLE and aPRA reveal structures inside the vortical regions in much finer detail, as they do not rely on substantial fluid particle separation. In agreement with our arguments in § 8.6, aFTLE and aPRA consistently refine the same coherent vortices indicated by FTLE and PRA. In the mixing regions surrounding those vortices, however, active and passive LCS diagnostics tend to identify different barriers. As in the case of our Eulerian barrier calculations in § 9.1.1, the vorticity-based aFTLE and aPRA provide a more moderate enhancement, because they rely on second derivatives of the velocity data.

Figure 8. Comparison between the passive FTLE (PRA) and the momentum- and vorticity-based aFTLE (aPRA): (a) $\mathrm {FTLE}_0^{25} (\boldsymbol {x}_0)$, (b) $\mathrm {aFTLE}_{0,25}^{0.35} (\boldsymbol {x}_0; \rho \boldsymbol {u})$, (c) $\mathrm {aFTLE}_{0,25}^{0.05} (\boldsymbol {x}_0; \boldsymbol {\omega })$, (d) $\mathrm {PRA}_0^{25} (\boldsymbol {x}_0)$, (e) $\mathrm {aPRA}_{0,25}^{0.35} (\boldsymbol {x}_0; \rho \boldsymbol {u})$ and (f) $\mathrm {aPRA}_{0,25}^{0.05} (\boldsymbol {x}_0; \boldsymbol {\omega })$. All computations were performed over the time interval $[t_{0},t_{1}]=[0,25]$ on the domain $[2.8,4.9]\times [1,3]$.

Next, we illustrate the extraction of active barriers to the transport of momentum and vorticity as parametric curves. This is possible in 2-D incompressible flows because the active barrier equations are Hamiltonian, and hence the barriers are level curves of a scalar function (see Theorems 7.1 and 7.2, as well as Remark 7.3). To perform this extraction, we follow the method presented in Haller et al. (Reference Haller, Hadjighasem, Farazamand and Huhn2016) for the extraction of coherent Lagrangian vortex boundaries as outermost level sets of the Lagrangian-averaged vorticity deviation. We will use the notation $H_{t_{0}}^{t_{1}}(\boldsymbol {x}_0)$ to denote the relevant Hamiltonian from § 7.1. The algorithm is the same for all those Hamiltonians, but we will restrict our computations here to the Hamiltonian governing Lagrangian momentum barriers in two dimensions, given by $H_{t_0}^{t_1}(\boldsymbol {x}_{0})=\nu \rho \, \overline {\omega (\boldsymbol {F}_{t_{0}}^{t}(\boldsymbol {x}_{0}),t)}$ (see (7.6)).

In all our computations, we focus on finding almost convex structurally stable level sets of $H_{t_{0}}^{t_{1}}(\boldsymbol {x}_0)$ that encircle a single local maximum of $|H_{t_{0}}^{t_{1}}(\boldsymbol {x}_0)|$. The need for relaxation of the strict convexity requirement in discrete data sets is discussed extensively in Haller et al. (Reference Haller, Hadjighasem, Farazamand and Huhn2016), so we will skip it here. Along these lines, we introduce the convexity deficiency of a closed curve in the plane as the ratio of the area between the curve and its convex hull to the area enclosed by the curve, which we denote with $d_{max}$. The maximum $d_{max}$ we used for the different extracted barriers was $5\times 10^{-2}$.

Small-scale local maxima of $|H_{t_{0}}^{t_{1}}(\boldsymbol {x}_0)|$ may appear either due to non-accurate resolution of these scales or because of computational noise. To address this issue, we only considered boundaries with arclength larger than a threshold $l_{min}$. This threshold was set to $0.4$ for all our computations because below this limit, boundaries contain too few grid points to be considered well resolved.

The main steps of the extraction procedure are delineated in Algorithm 1. All the MATLAB codes used for the extraction of the barriers of this section can be found at https://github.com/LCSETH?tab=repositories.

Algorithm 1 Coherent Lagrangian and Eulerian vortex boundaries for two-dimensional flows

We apply this algorithm to extract an active material barrier to the transport of momentum with high precision as a parametrized curve. This closed active barrier is shown in red in figure 9. We also show the impact of this barrier on the momentum landscape in Eulerian and Lagrangian coordinates, respectively, for the initial and final times in $[0,25]$. Furthermore, for reference, we show an elliptic LCS (black) extracted as a closed level curve of the passive PRA through a selected point of the active barrier. As expected from our discussion in § 8.6, these active and passive elliptic barriers are very close to each other at the initial time and remain equally close during their material evolution. In the Eulerian frame, we observe that the extracted active and passive barriers show no sign of filamentation throughout their whole extraction time. This is in agreement with the general expectation we stated earlier for diffusion-minimizing material curves. Furthermore, when viewed in the Lagrangian frame, we note the organizing role of the extracted barrier in the momentum landscape. Indeed, the barrier keeps encapsulating small values of the momentum norm for the entire extraction time.

Figure 9. Evolution of an extracted active material barrier (red) to the diffusive transport of momentum in (a) the Eulerian and (b) the Lagrangian frame, superimposed on the distribution of the norm of the linear momentum. This barrier was identified as a level curve of the Hamiltonian $H_{t_0}^{t_1}(\boldsymbol {x}_{0})=\nu \rho \, \overline {\omega (\boldsymbol {F}_{t_{0}}^{t}(\boldsymbol {x}_{0}),t)}$. Black curve indicates an elliptic LCS extracted as a level curve of the passive PRA, launched from the highlighted point of the active barrier. (c) The instantaneous viscous forces (normalized by $\rho \nu$) acting on the evolving barrier.

Figure 9 also shows the instantaneous viscous force (normalized by $\rho \nu$) along the extracted active momentum barrier. Note that this force remains almost tangent to the barrier for the most part. There are, however, some notable exceptions, illustrating that these barriers are not constructed to be tangent to the viscous forces at every time instance. Rather, the viscous forces are tangent to the barriers in a time-averaged sense after being pulled back under the flow map to the initial configuration.

9.2. Three-dimensional turbulent channel flow

We consider now the 3-D incompressible, turbulent flow of a Newtonian fluid in a doubly periodic channel, a well-studied physical setting for 3-D coherent structure studies.

Our analysis relies on velocity snapshots from a mixed-discretization parallel solver of the incompressible Navier–Stokes equations in the wall-normal velocity and vorticity formulation, developed by Luchini & Quadrio (Reference Luchini and Quadrio2006). The equations of motion are discretized via a Fourier–Galerkin approach along the two statistically homogeneous streamwise $(x)$ and spanwise $(z)$ directions. Fourth-order compact finite differences (Lele Reference Lele1992) based on a five-point computational stencil are adopted in the wall-normal direction $(y)$.

The governing equations are integrated forward in time at constant power input (Hasegawa, Quadrio & Frohnapfel Reference Hasegawa, Quadrio and Frohnapfel2014) with a partially implicit approach, combining the three-step, low-storage Runge–Kutta scheme with the implicit Crank–Nicolson scheme for the viscous terms. The friction Reynolds number is $Re_{\tau }=u_{\tau }h/\nu =200$, based on the friction velocity $u_{\tau }$, the channel half-height $h$ and the kinematic viscosity $\nu$, which corresponds to a bulk Reynolds number $Re_{b}=U_{b}h/\nu =3177$, where $U_{b}$ is the bulk velocity. The computational domain is $L_{x}=4\pi h$ long and $L_{z}=2\pi h$ wide. The number of Fourier modes is 256 both in the streamwise and spanwise direction; the number of points in the wall-normal direction is 256, unevenly spaced in order to decrease the grid size near the walls. The corresponding spatial resolution in the homogeneous directions is ${\rm \Delta} x^{+}=9.8$ and ${\rm \Delta} z^{+}=4.9$ (without accounting for the additional modes required for dealiasing according to the 3/2 rule); the wall-normal resolution increases from ${\rm \Delta} y^{+}=0.4$ near the walls to ${\rm \Delta} y^{+}=2.6$ at the centreline, while the temporal resolution is kept constant at ${\rm \Delta} t=0.005 h/U_b$, corresponding to ${\rm \Delta} t^{+}=0.063$. The superscript $+$ denotes non-dimensionalization in viscous units, i.e. with $u_{\tau }$ and $\nu$. At each DNS time step and thus with the same temporal resolution, a 3-D flow snapshot is stored for a total of 1500 snapshots. The 750$\mathrm {th}$ snapshot in the series is stored at time $t=0$. This is the instant at which we compute the Eulerian barriers to active transport. The last 750 snapshots are utilized for the computation of the active barriers and passive forward FTLE, while the first 750 ones are used for calculating the passive backward FTLE. The integration time for the Lagrangian calculations has been chosen based on pair-dispersion statistics of Lagrangian tracers (see, for instance, Pitton et al. Reference Pitton, Marchioli, Lavezzo, Soldati and Toschi2012). The averaging time for the bulk flow statistics is 8100 $U_b / h$. In the following, all quantities are non-dimensionalized using $U_{b}$ and $h$ unless stated otherwise.

The active barriers are computed in a two-step procedure. First, the active barrier field $\boldsymbol {b}_{t_0}^{t_1}(\boldsymbol {x}_0)$, appearing at the right-hand side of the barrier equation (4.1), is computed. Then, the barrier ODE is solved and visualized via the FTLE and PRA diagnostics. For Eulerian barriers, the barrier vector field appearing in the instantaneous (or Eulerian) barrier equation (5.1) is readily computed from the velocity field data as $\boldsymbol {b}_{t}^{t}=\boldsymbol {h}_{vis}$. Differentiation of the velocity field is performed with the same discrete operators used during DNS. For material barriers, $\boldsymbol {b}_{t_0}^{t_1}(\boldsymbol {x}_0)$ is simply obtained as the temporal average of $(\boldsymbol {F}_{t_0}^{t_1})^{\ast } \boldsymbol {h}_{vis}$, because $\det \boldsymbol {\nabla } \boldsymbol {F}_{t_0}^{t_1}(\boldsymbol {x}_0)\equiv 1$ by incompressibility. In this case, the vector field $\boldsymbol {b}_{t_0}^{t_1}(\boldsymbol {x}_0)$ is discretized on a Cartesian grid similar to the one used for the velocity field; the only difference is that the number of collocation points along the $x$- and $z$-directions is increased to 384 via Fourier interpolation.

At time $t_0$, a set of tracers is seeded in the neighbourhood of each point $\boldsymbol {x}_0$ at which $\boldsymbol {b}_{t_0}^{t_1}(\boldsymbol {x}_0)$ needs to be computed. Each set (see figure 10) is composed by 7 tracers. The central tracer is exactly located at $\boldsymbol {x}_0$ and is the only one along which the vector field $\boldsymbol {h}_{vis}$ is also computed. The other tracers are shifted by $\epsilon _i$ along the positive and negative $i$th spatial direction and are utilized to compute $\boldsymbol {\nabla } \boldsymbol {F}_{t_0}^{t_1}(\boldsymbol {x}_0)$ with second-order central finite differences. The shift $\epsilon _i$ is defined as $1/100$ of the minimum grid spacing along the $i$th spatial direction. A total of $2.64\times 10^{8}$ particles are seeded into the flow. The evolving positions of these tracers, which are images of their initial positions under the flow map $\boldsymbol {F}_{t_0}^{t_1}$, are advanced in time by integrating the $\boldsymbol {u}$ field with a third-order, four-stage Runge–Kutta algorithm. The vector fields $\boldsymbol {u}$ and $\boldsymbol {h}_{vis}$ required at the intermediate stages are obtained via linear interpolation of two consecutive flow snapshots and are evaluated at the particle position through a sixth-order, 3-D Lagrangian interpolation (van Hinsberg et al. Reference van Hinsberg, Ten Thije Boonkamp, Toschi and Clercx2012; Pitton et al. Reference Pitton, Marchioli, Lavezzo, Soldati and Toschi2012) of the underlying vector field, which reduces to fourth order only between the wall and the first grid point above it.

Figure 10. Sketch of the computational molecule utilized for the computation of the active barrier field $\boldsymbol {b}_{t_0}^{t_1}$. The large circle denotes the Lagrangian tracer at the centre of the molecule, where the vector field $\boldsymbol {h}_{vis}$ is computed. The cross denotes the further six tracers utilized to compute $\boldsymbol {\nabla }\boldsymbol {F}_{t_0}^{t_1}$.

Once the (Lagrangian or Eulerian) barrier equation is available, its active flow map $\boldsymbol {\mathcal {F}}_{t_0,t_1}^{s}$ is computed by solving the steady barrier ODE up to $s =31.0$ and $s =0.62$ for the momentum and vorticity barriers, respectively. We have chosen these $s$-times large enough for the computed barrier trajectories to reveal enough detail in the underlying barrier vector field but small enough to avoid accumulation of the integration error. The effect of changing the parameter $s_{{max}}$ is shown in the additional material Movie 1.mp4 and Movie 2.mp4 for Eulerian momentum and vorticity barriers, respectively. The seeds for the flow map are arranged in a Cartesian grid identical to the one of the active barrier field for 3-D computations of aFTLE/aPRA diagnostics, while the spatial resolution is increased by a factor 6 when only 2-D slices are computed. The comparison between the two resolutions has been utilized to verify the grid independence of the results. The aFTLE and aPRA diagnostics are then computed according to (8.11) and (8.18), respectively.

9.2.1. Eulerian active barriers

Instantaneous aFTLE and aPRA are presented for $t=0$ in figures 11 and 12, respectively, and compared against their passive variants. Even though the results are computed for the complete 3-D field, only 2-D cross-sections are presented in the following. In figures 11 and 12, a $(y-z)$ cross-section located at $x=2 \pi h$ is shown. The 3-D visualization of the FTLE and PRA fields poses challenges in its own, which are subjects of ongoing research in computer visualization (see, e.g., Sadlo & Pikert Reference Sadlo and Pikert2009; Schindler et al. Reference Schindler, Peikert, Ruchs and Theisel2012) and are outside the scope of the present study. The 2-D visualization in different cross-sections results in different local flow structures which are, nevertheless, all reminiscent of classically known structures in channel flows. These include low-speed streaks, quasi-streamwise and hairpin vortices and packets thereof (Robinson Reference Robinson1991). The supplementary movies 3 and 4 show how figure 11(b,c) change throughout the channel length.

Figure 11. Comparison between the instantaneous limit of (a,d) the passive FTLE, (b,e) the aFTLE with respect to $\rho \boldsymbol {u}$ and (c,f) the aFTLE with respect to $\boldsymbol {\omega }$ at $t=0$ in a cross-sectional plane at $x/h = 2\pi$. The panels (d–f) magnify the region denoted with a rectangle in panels (a–c). All computations in the figure were performed on the same snapshot of the velocity field at $t=0$.

Figure 12. Comparison between the instantaneous limit of (a,d) the passive PRA (which is identically zero), (b,e) the aPRA with respect to $\rho \boldsymbol {u}$ and (c,f) the aPRA with respect to $\boldsymbol {\omega }$ at $t=0$ in a cross-sectional plane at $x/h = 2\pi$. The panels (d–f) magnify the region denoted with a rectangle in panels (a–c). All computations in the figure were performed on the same snapshot of the velocity field at $t=0$.

As already seen for 2-D turbulence in § 9.1, the aFTLE and aPRA highlight a broader range of structures in more detail from the same velocity data when compared to their passive variants. (Recall that the instantaneous limit of the passive PRA, in fact, vanishes identically, and hence reveals no elliptic coherent structures from a single velocity snapshot.) The Eulerian active barriers revealed by aFTLE and aPRA appear in figures 11 and 12 as an abundance of intersections of 2-D surfaces with the selected cross-section. Limiting to visual inspection, we recognize several open (or hyperbolic) barriers as ridges of the aFTLE fields. Given the quasi-streamwise nature of turbulent structures in wall-bounded flows, vortical (or elliptic) barriers to transport are often observed in cross-sectional planes as aFTLE ridges wrapping around closed regions, which are also captured as level sets of the corresponding aPRA fields. Example of such regions are shown in the magnifications of panels (d–f) in figures 11 and 12.

The results also reveal that large prominent aFTLE ridges penetrate into and span the bulk flow region, sometimes connecting the channel halves, as visible in 11(b) between $3 \leq z/h \leq 3.5$ and $0.3 \leq y/h \leq 1.5$. Other regions, such as between $2 \leq z/h \leq 3$ and $0.5 \leq y/h \leq 1.5$ in the same figure, display practically no discernible barriers and are bounded by the envelopes of filamented open (hyperbolic) transport barriers, which are finite-time generalizations of infinite-time classic stable and unstable manifolds. Unlike in previous approaches, however, these finite-time invariant manifolds are constructed here as perfect material barriers to active transport, rather than as LCS acting as backbones of advected fluid-mass patterns (Haller Reference Haller2015).

Figure 13 shows the Eulerian active barrier vector field of (a) $\rho \boldsymbol {u}$ and (b) $\boldsymbol {\omega }$ superimposed to the respective aFTLEs already shown in figure 11(e,f). Level-set curves of the $\lambda _{2}(\boldsymbol {x},t)=-0.015$ field (Jeong & Hussain Reference Jeong and Hussain1995), a common visualization tool for coherent vortical structures in wall-bounded turbulence, are also shown. The scalar field $\lambda _{2}(\boldsymbol {x},t)$ is defined as the instantaneous intermediate eigenvalue of the tensor field $\boldsymbol {S}^{2}(\boldsymbol {x},t)+\boldsymbol {W}^{2}(\boldsymbol {x},t)$, with $\boldsymbol {S}$ and $\boldsymbol {W}$ defined in (7.25a,b). This choice follows the heuristic convention to select a $\lambda _{2}^{+}$ value slightly below the negative of the root-mean-square peak of $\lambda _{2}(\boldsymbol {x},t)$ across the channel (Jeong et al. Reference Jeong, Hussain, Schoppa and Kim1997), which is approximately $0.0125$ in our case.

Figure 13. Eulerian active barriers of (a) $\rho \boldsymbol {u}$ and (b) $\boldsymbol {\omega }$ at $t=0$ in a cross-sectional plane located at $x/h = 2\pi$. The colour map shows the respective aFTLE fields, the vectors show the cross-sectional components of the underlying active barrier field, while the red lines are level-set curves $\lambda _{2}^{+}(\boldsymbol {x},t)=-0.015$ of the Eulerian vortex identification criterion proposed by Jeong et al. (Reference Jeong, Hussain, Schoppa and Kim1997).

Compared to the passive material barriers shown in figure 11(d), we observe that the active barriers not only yield a remarkably more complex flow structure but also carry a completely different physical meaning. Since the active barriers minimize the diffusive transport of, in this case, linear momentum or vorticity, we find that the cross-sectional components of the active barrier vector field $\boldsymbol {b}_t^{t}$ are parallel to the aFTLE ridges. This indicates that the resultant force of the viscous stresses is tangential to Eulerian active barriers of momentum transport. As noted previously, momentum barriers in $(y-z)$ cross-sections can roll-up into spiral patterns or form closed surfaces. In regions where this occurs, figure 13(a) shows that closed level-set curves of $\lambda _2$, typically used as indicators for the presence of quasi-streamwise vortices, tend to be found. This suggests that boundaries of quasi-streamwise vortices act as Eulerian active barriers to the transport of linear momentum. Interestingly, we find that the circulation of the active momentum barrier field in such areas is of opposite sign than the one of the velocity field. This indicates that viscous forces oppose the vortical motion that is observed in the analysed snapshots. In addition, Eulerian active barriers of vorticity tend to enter regions of closed momentum barriers or level-set curves of $\lambda _2$, thus highlighting regions in which vorticity diffuses into the vortex or is dissipated by viscosity.

Despite some similarities, it is important to note the practical and fundamental differences between the Eulerian momentum barriers and level-set surfaces of $\lambda _2$. On the fundamental side, we mention that the active barriers are objective, they have clear implications for the viscous transport of the active vector field and, most importantly, that they are extensible by definition to material barriers, thus accounting for the Lagrangian coherence of the barriers themselves. On the practical side, Eulerian active barriers do not require the convenient but arbitrary choice of a threshold and deliver information on the full active transport geometry, rather than just providing a few isolated curves.

9.2.2. Lagrangian active barriers

Figure 14 shows attracting material surfaces as passive backward $\mathrm {FTLE}_0^{-3.75}(\boldsymbol {x}_0 )$ at the same $(y-z)$ cross-section located at $x=2 \pi h$ discussed in § 9.2.1. These attracting material surfaces, forming the cores of experimentally observed fluid trajectory patterns at time $t=0$, show a striking resemblance to the Eulerian active barriers to linear momentum indicated in figure 11(b,e) by the $\mathrm {aFTLE}_{0,0}^{31}(\boldsymbol {x}_0, \rho \boldsymbol {u} )$ field. The close similarity between the two is not fully surprising. At the present low value of Reynolds number viscous effects dominate throughout a significant portion of the channel, and thus determine both the characteristics of the Eulerian momentum barriers and the finite-time dynamics of particle motion. The temporal horizon, over which the analogy between $\mathrm {FTLE}_0^{-3.75}(\boldsymbol {x}_0 )$ and $\mathrm {aFTLE}_{0,0}^{31}(\boldsymbol {x}_0, \rho \boldsymbol {u} )$ is observed, is expected to decrease with increasing Reynolds number, as the viscosity-dominated inner layer shrinks compared to the channel height. Whether the observed similarity holds at higher values of $Re$ is to be verified in later studies with high-$Re$ data. However, it is remarkable that the Eulerian momentum barriers, which are computed utilizing a single flow snapshot, reproduce the same features of material surfaces obtained from a Lagrangian computation, which requires storing the temporal evolution of the flow. Figures 15 and 16 show aFTLE and aPRA computed for momentum- and vorticity-based material barriers in a $(y-z)$ cross-section located at $x=2 \pi h$ and compare them against their passive variants. The integration interval is for all cases between $t_0=0$ and $t_1=3.75$ which corresponds to a time interval of 750 viscous units. The figures clearly show that some features of the Eulerian active barriers discussed in § 9.2.1, such as the spiralling or closed patterns of $\mathrm {aFTLE}_{0,0}^{31}(\boldsymbol {x}, \rho \boldsymbol {u} )$, do have a material character, since they persist almost unchanged over the temporal interval which we have considered. Examples are shown in the magnification of figures 15(e) and 16(e), showing promise for active LCS diagnostics in studying the lifetime of vortical structures in wall-bounded turbulence (Quadrio & Luchini Reference Quadrio and Luchini2003). In the vicinity of the wall, characterized by the strong intermittent turbulent events rapidly evolving with the viscous time scale, less detail is visible in the barriers, due to the lack of material coherence for the considered time frame. Consistent with the general principle discussed in § 8.6, passive and active LCS diagnostics tend to highlight the same vortical regions, but tend to differ in the mixing regions surrounding the vortices. As in our 2-D turbulence example, while the vorticity-based aFTLE and aPRA plots show a major enhancement over passive FTLE and PRA, some of their details are less clearly defined in comparison to their momentum-based counterparts. Again, this is due to the additional spatial differentiation involved in computing active LCS diagnostics for the vorticity compared to the same computation for the linear momentum. Figure 17 shows the material active barrier vector field of (a) $\rho \boldsymbol {u}$ and (b) $\boldsymbol {\omega }$ superimposed to the respective aFTLEs already shown in figure 15(e,f). Level-set curves of the $\lambda _{2}(\boldsymbol {x},t)=-0.015$ field at the temporal instant $t=t_0=0$ are also shown. It is confirmed that with the present definition of active barriers, the active vector field is tangent to the detected barriers visualized here as aFTLE ridges, in a temporally averaged sense. Figure 17(a) shows that closed $\mathrm {aFTLE}_{t_0,t_1}^{s}(\boldsymbol {x}_0, \rho \boldsymbol {u} )$ ridges can be in some instances close to level-set curves of the $\lambda _{2}$ criterion, as for instance at $(y/h, z/h) \approx (0.15,2.9)$ and $(0.55,3.3)$. In this sense, the material momentum barriers can be utilized as means to objectively identify vortical structures which play a role in inhibiting momentum transport and preserve material coherence over the considered time frame, without resorting to arbitrary choices of level-sets of $\lambda _{2}$. In the present example, we find streamwise vortices that are bounded by active momentum barriers for a time period of 750 viscous units.

Figure 14. Passive backward $\mathrm {FTLE}_0^{-3.75}(\boldsymbol {x}_0 )$ in a cross-sectional plane at $x/h = 2\pi$. The right panel magnifies the region denoted with a rectangle in the left panel.

Figure 15. Comparison between (a,d) the passive FTLE, (b,e) the aFTLE with respect to $\rho \boldsymbol {u}$ and (c,f) the aFTLE with respect to $\boldsymbol {\omega }$ in a cross-sectional plane at $x/h = 2\pi$. The integration interval is for all cases between $t_0=0$ and $t_1=3.75$. The panels (d–f) magnify the region denoted with a rectangle in panels (a–c).

Figure 16. Comparison between (a,d) the passive PRA, (b,e) the aPRA with respect to $\rho \boldsymbol {u}$ and (c,f) the aPRA with respect to $\boldsymbol {\omega }$ in a cross-sectional plane at $x/h = 2\pi$. The integration interval is for all cases between $t_0=0$ and $t_1=3.75$. The panels (d–f) magnify the region denoted with a rectangle in panels (a–c).

Figure 17. Material active barriers of (a) $\rho \boldsymbol {u}$ and (b) $\boldsymbol {\omega }$ at $t=0$ in a cross-sectional plane located at $x/h = 2\pi$ and for an integration interval between $t_0=0$ and $t_1=3.75$. The colormap shows the respective aFTLE fields, the vectors show the cross-sectional components of the underlying active barrier field, while the red lines are level-set curves $\lambda _{2}^{+}(\boldsymbol {x},t)=-0.015$ of the Eulerian vortex identification criterion proposed by Jeong et al. (Reference Jeong, Hussain, Schoppa and Kim1997) at the instant $t_0=0$.