7.1. Streamline geometry and Frenet–Serret triad

In the field description, the counterpart of a material line element $\delta \boldsymbol{r}=\delta {r}\boldsymbol{e}$ is an infinitesimal streamline segment ${\mathcal{C}}$ , denoted as $\delta \bar {\boldsymbol{r}}=\delta {s}\boldsymbol{t}$ , where $s$ is the arclength parameter measured from a reference point on ${\mathcal{C}}$ , and $\delta {s}$ is its infinitesimal increment. At the considered instant, $\delta \bar {\boldsymbol{r}}$ coincides with $\delta \boldsymbol{r}$ . Each point on ${\mathcal{C}}$ is associated with an orthonormal right-handed triad $(\boldsymbol{t},\boldsymbol{n},\boldsymbol{b})$ , consisting of the unit tangent vector $\boldsymbol{t}$ , the principal normal vector $\boldsymbol{n}$ (pointing towards the streamline’s centre of curvature) and the binormal vector $\boldsymbol{b}\equiv \boldsymbol{t}\times \boldsymbol{n}$ (figure 3). The local curvature of ${\mathcal{C}}$ is denoted by $\kappa$ . The intrinsic evolution of $(\boldsymbol{t},\boldsymbol{n},\boldsymbol{b})$ along ${\mathcal{C}}$ is governed by the Frenet–Serret formulae from classical differential geometry (Serrin Reference Serrin1959; Chen Reference Chen2017). The osculating plane (spanned by $\boldsymbol{t}$ and $\boldsymbol{n}$ ) and the normal plane (spanned by $\boldsymbol{n}$ and $\boldsymbol{b}$ ) define the local geometry of ${\mathcal{C}}$ . Note that $\boldsymbol{n}$ is generally different from the unit normal vector $\boldsymbol{n}_{\varSigma }$ of an oriented surface. The velocity field along ${\mathcal{C}}$ is purely tangential and given by $\boldsymbol{u}=q\boldsymbol{t}$ , where $q\equiv \boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{t}=\lVert \boldsymbol{u}\rVert$ is the velocity magnitude. $\partial _{s}\equiv \partial /\partial s$ represents the partial derivative with respect to $s$ for a physical quantity.

Figure 3.

Schematic of the Frenet–Serret triad $(\boldsymbol{t},\boldsymbol{n},\boldsymbol{b})$ intrinsic to a regular streamline segment ${\mathcal{C}}$ with the local curvature radius $r_{0}\equiv \kappa ^{-1}$ . The streamline-based vorticity modes are shown as $(\boldsymbol{R}_{L}(\boldsymbol{t}),\boldsymbol{s}_{L}(\boldsymbol{t}),\boldsymbol{g}_{L}(\boldsymbol{t}))$ .

7.2. Streamline-based orbital-rotation mode

For a frozen velocity field, we observe that an orbital rotation along a streamline (segment) ${\mathcal{C}}$ occurs if and only if $\boldsymbol{t}\times \partial _{s}\boldsymbol{u}\neq \boldsymbol{0}$ (the non-collinearity between the tangent vector and the velocity variation in the osculating plane). Indeed, using the Frenet–Serret formula $\partial _{s}\boldsymbol{t}=\kappa \boldsymbol{n}$ and noting that $\boldsymbol{t}\times \boldsymbol{u}=\boldsymbol{0}$ , we find

(7.1) \begin{eqnarray} \boldsymbol{t}\times \partial _{s}\boldsymbol{u}=\kappa {q}\boldsymbol{b}, \end{eqnarray}

which indicates a local circular motion in the osculating plane about the binormal axis $\boldsymbol{b}$ . Thus, orbital rotation arises from streamline curvature $\kappa \neq {0}$ and non-zero speed $q\neq {0}$ . The radius of curvature is $r_{0}\equiv \kappa ^{-1}$ .

Since the directional derivative of velocity along a streamline can be expressed as $\partial _{s}\boldsymbol{u}=\boldsymbol{t}\boldsymbol{\cdot }\boldsymbol{A}$ , (7.1) can be adjusted to the form

(7.2) \begin{eqnarray} \boldsymbol{\psi }_{L}(\boldsymbol{t})\equiv \boldsymbol{t}\times \left (\boldsymbol{t}\boldsymbol{\cdot }\boldsymbol{A}\right )=\kappa {q}\boldsymbol{b}. \end{eqnarray}

We emphasize that in any flow field, the binormal vector $\boldsymbol{b}$ serves as the unique physically identifiable rotation axis external to the streamline orbit. Substituting $\boldsymbol{e} = \boldsymbol{t}$ into (3.4) yields the identity $\boldsymbol{W}_{L}^{\textit{eff}}(\boldsymbol{t}) = \boldsymbol{\psi }_{L}(\boldsymbol{t})$ . This reveals that $\boldsymbol{\psi }_{L}(\boldsymbol{t})$ in (7.2) can be interpreted as the effective angular velocity of $\boldsymbol{e}$ for a material line element instantaneously aligned with ${\mathcal{C}}$ (i.e. $\boldsymbol{e} = \boldsymbol{t}$ ). Then, the vorticity in a reference rigid body (frozen with $\boldsymbol{t}$ ) rotating with the angular velocity $\boldsymbol{\psi }_{L}(\boldsymbol{t})$ must be (figure 3)

(7.3) \begin{eqnarray} \boldsymbol{R}_{L}(\boldsymbol{t})=2\boldsymbol{t}\times \left (\boldsymbol{t}\boldsymbol{\cdot }\boldsymbol{A}\right )=2\kappa {q}\boldsymbol{b}. \end{eqnarray}

Here, $\boldsymbol{R}_{L}(\boldsymbol{t})$ is termed the streamline-based orbital-rotation mode (formally obtained from (3.13a ) via the substitution $\boldsymbol{e} = \boldsymbol{t}$ ), analogous to a speed skater navigating a turn. Its binormal component is denoted by $R_{L}(\boldsymbol{t})\equiv \boldsymbol{R}_{L}(\boldsymbol{t})\boldsymbol{\cdot }\boldsymbol{b}=2\kappa {q}$ .

7.3. Streamline-based spin mode

Equation (7.3) immediately evokes a fundamental but often overlooked vorticity representation using the Frenet–Serret triad. This formulation appears in the classical literature (Serrin Reference Serrin1959, § 20) when expressed in the local rectangular coordinates aligned with $(\boldsymbol{t},\boldsymbol{n},\boldsymbol{b})$ ,

(7.4) \begin{eqnarray} \boldsymbol{\omega }=\xi {q}\boldsymbol{t}+\partial _{b}q\boldsymbol{n}+\left (\kappa {q}-\partial _{n}q\right )\boldsymbol{b}, \end{eqnarray}

where the scalar factor $\xi$ ,

(7.5) \begin{eqnarray} \xi \equiv \boldsymbol{t}\boldsymbol{\cdot }\left (\boldsymbol{\nabla }\times \boldsymbol{t}\right ) =\boldsymbol{b}\boldsymbol{\cdot }\partial _{n}\boldsymbol{t}-\boldsymbol{n}\boldsymbol{\cdot }\partial _{b}\boldsymbol{t}, \end{eqnarray}

was historically termed as the abnormality (also referred to as the torsion of the curve system or torsion of neighbouring vector lines). It was introduced as a measure of the departure of the velocity field from the property of having a normal congruence (Truesdell Reference Truesdell1954). The physical interpretation of this subtle parameter once perplexed prominent researchers, owing to the inherent limitation of the Frenet–Serret formalism: it applies to an individual curve but not to a family of neighbouring curves. However, the ambiguity is readily resolved by recognizing that $\xi {q}^{2}=\boldsymbol{\omega }\boldsymbol{\cdot }\boldsymbol{u}\equiv {h}$ represents the helicity density, implying $\xi =h/q^2$ (the ratio of helicity density to twice the kinetic energy density). In contrast to Serrin’s classical derivation, we can straightforwardly express the vorticity as $\boldsymbol{\omega }=\boldsymbol{\nabla }\times (q\boldsymbol{t} )=q\boldsymbol{\nabla }\times \boldsymbol{t}+\boldsymbol{\nabla }q\times \boldsymbol{t}$ . The first term expands further using the Frenet–Serret relations, $q\boldsymbol{\nabla }\times \boldsymbol{t}=\xi {q}\boldsymbol{t}+\kappa {q}\boldsymbol{b}$ , and the second term captures the variation of $q$ along the principal normal and binormal directions, $\boldsymbol{\nabla }q\times \boldsymbol{t}=\boldsymbol{n}\partial _{b}q-\boldsymbol{b}\partial _{n}q$ , as appeared in (7.4).

Consider now the spin mode associated with a streamline segment ${\mathcal{C}}$ . Owing to (7.3), (7.4) does not directly provide a vorticity decomposition into orbital and spin modes in the field description. However, a minor modification yields the desired splitting. Let $r$ denote the radial coordinate measured from the curvature centre of a point on ${\mathcal{C}}$ . Since $\partial _{n}q=-\partial _{r}q$ , the binormal vorticity component $\omega _{b}\equiv \boldsymbol{\omega }\boldsymbol{\cdot }\boldsymbol{b}$ becomes

(7.6) \begin{eqnarray} \omega _{b}=\kappa {q}-{\partial }_{n}q=\frac {1}{r_0}\frac {\partial }{\partial r}\left (rq\right ). \end{eqnarray}

A naïve identification of $\partial _{r}q$ as the spin component along the binormal direction $(s_{L}(\boldsymbol{t})\equiv \boldsymbol{s}_{L}(\boldsymbol{t})\boldsymbol{\cdot }\boldsymbol{b})$ leads to a paradox. Consider freezing the fluid near point $P$ into a rigid body rotating with angular velocity $W$ , such that $q=Wr$ . Although $\omega _{b}=2W$ correctly matches the rigid-body rotation, this approach would erroneously yield $s_{L}(\boldsymbol{t})=W$ instead of $s_{L}(\boldsymbol{t})=0$ . The resolution lies in recognizing that the principal normal $\boldsymbol{n}$ itself rotates, making $\partial _{n}q$ alone no longer characterize spin. The correct spin component must account for this rotation by subtracting the orbital contribution,

(7.7) \begin{eqnarray} s_{L}(\boldsymbol{t})=-\partial _{n}q-\kappa {q}=r_{0}\frac {\partial }{\partial r}\left (\frac {q}{r}\right ), \end{eqnarray}

analogous to a figure skater’s quadruple jump characterized by the radial velocity gradient. Furthermore, observing that $q\boldsymbol{\nabla }q=\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{u}$ (namely, $\boldsymbol{\nabla }q=\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{t}$ ) and (7.2), we need to define, in general, the streamline-based spin mode as (figure 3)

(7.8) \begin{eqnarray} \boldsymbol{s}_{L}(\boldsymbol{t})\equiv \boldsymbol{\nabla }q\times \boldsymbol{t}-\boldsymbol{\psi }_{L}(\boldsymbol{t})=-2\boldsymbol{t}\times \left (\boldsymbol{t}\boldsymbol{\cdot }\boldsymbol{D}\right ). \end{eqnarray}

Remarkably, $\boldsymbol{s}_{L}(\boldsymbol{t})$ in (7.8) exhibits identical mathematical structure to the spin vorticity term $\boldsymbol{s}_{L}(\boldsymbol{e})$ in (3.13b ) when the alignment condition $\boldsymbol{e}=\boldsymbol{t}$ is imposed. Expanding (7.8) gives $\boldsymbol{s}_{L}(\boldsymbol{t})=\partial _{b}q\boldsymbol{n}+ (-\partial _{n}q-\kappa {q} )\boldsymbol{b}$ .

7.4. Streamline-based DVD

Based on the above discussion, we obtain the streamline-based DVD,

(7.9) \begin{eqnarray} \boldsymbol{\omega }=\boldsymbol{R}_{L}(\boldsymbol{t})+\boldsymbol{s}_{L}(\boldsymbol{t})+\boldsymbol{g}_{L}(\boldsymbol{t}), \end{eqnarray}

where the streamline-line orbital-rotation mode $\boldsymbol{R}_{L}(\boldsymbol{t})$ and spin mode $\boldsymbol{s}_{L}(\boldsymbol{t})$ are given in (7.3) and (7.8), respectively. The streamline-based streaming mode $\boldsymbol{g}_{L}(\boldsymbol{t})$ is defined as the streamline-parallel component (figure 3),

(7.10) \begin{eqnarray} \boldsymbol{g}_{L}(\boldsymbol{t})\equiv \left (\boldsymbol{\omega }\boldsymbol{\cdot }\boldsymbol{t}\right )\boldsymbol{t}=\xi {q}\boldsymbol{t}. \end{eqnarray}

We remark that (7.9) can also be derived from the Lamb decomposition of the convective acceleration, $\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{u}=\boldsymbol{\omega }\times \boldsymbol{u}+\boldsymbol{\nabla }(q^{2}/2)$ , where $\boldsymbol{\omega }\times \boldsymbol{u}$ is the Lamb vector. Taking the cross product of $\boldsymbol{u}$ with both sides of this identity and employing the identity $\boldsymbol{\nabla }q = \boldsymbol{A} \boldsymbol{\cdot }\boldsymbol{t}$ yields the desired result.

8.1. Streamsurface-based DVD

Consider a material surface element $\delta \boldsymbol{\varSigma }\equiv \delta \varSigma \boldsymbol{n}_{\varSigma }$ instantaneously coinciding with an infinitesimal streamsurface element $\delta \overline {\boldsymbol{\varSigma }}\equiv \delta {S}\boldsymbol{n}_{S}$ , where $\delta \varSigma =\delta {S}$ and $\boldsymbol{n}_{\varSigma }=\boldsymbol{n}_{S}$ . A streamsurface $\mathcal{S}$ is formed by the set of streamlines passing through a reference curve in the flow field. Locally, the surface can be parameterized by two independent parameters $(x_{1},x_{2})$ , with $x_{3}$ denoting the normal coordinate. The associated orthonormal basis $(\overline {\boldsymbol{e}}_{1},\overline {\boldsymbol{e}}_{2},\overline {\boldsymbol{e}}_{3})$ defines a surface-attached reference frame, where $\overline {\boldsymbol{e}}_{3}=\overline {\boldsymbol{e}}_{1}\times \overline {\boldsymbol{e}}_{2}=\boldsymbol{n}_{S}$ . The surface curvature tensor is given by $\boldsymbol{K}\equiv -\boldsymbol{\nabla }_{\pi }\boldsymbol{n}_{S}=K_{\alpha \beta }\overline {\boldsymbol{e}}_{\alpha }\overline {\boldsymbol{e}}_{\beta }\ (\alpha ,\beta =1,2)$ , where $K_{\alpha \beta }$ are the tensor components. For conciseness, the partial derivatives with respective to the arclength parameters $(s_{1},s_{2})$ of the coordinate curves are denoted by

(8.1) \begin{eqnarray} \partial _{\alpha }\equiv \frac {\partial }{\partial {s}_{\alpha }}=\frac {1}{h_{\alpha }}\frac {\partial }{\partial {x}_{\alpha }},\ \alpha =1,2, \end{eqnarray}

where $h_{\alpha }\equiv \sqrt {g_{\alpha \alpha }}$ are Lamé coefficients, and $g_{\alpha \alpha }$ are the covariant metric components of the surface. When $(\overline {\boldsymbol{e}}_{1},\overline {\boldsymbol{e}}_{2})$ coincide with the two principal directions $(\overline {\boldsymbol{e}}_{\xi },\overline {\boldsymbol{e}}_{\eta })$ of $\delta \overline {\boldsymbol{\varSigma }}$ , with $\overline {\boldsymbol{e}}_{3}=\overline {\boldsymbol{e}}_{\zeta }=\boldsymbol{n}_{S}$ , the non-zero components of $\boldsymbol{K}$ are $(K_{\xi \xi },K_{\eta \eta })=(\lambda _{\xi },\lambda _{\eta })$ , where $\lambda _{\xi }$ and $\lambda _{\eta }$ denote the principal curvatures of $\mathcal{S}$ . In this principal frame $(\overline {\boldsymbol{e}}_{\xi },\overline {\boldsymbol{e}}_{\eta },\overline {\boldsymbol{e}}_{\zeta })$ , the streamsurface property $u_{\zeta }=0$ yields the following vorticity components (see Appendix C):

(8.2) \begin{align} \omega _{\xi } & =-\partial _{\zeta }\widehat {u}_{\eta }+\lambda _{\eta }u_{\eta },\ \omega _{\eta }=\partial _{\zeta }\widehat {u}_{\xi }-\lambda _{\xi }u_{\xi }, \\[-9pt] \nonumber \end{align}

(8.3) \begin{align} \omega _{\zeta } & =\partial _{\xi }u_{\eta }-\partial _{\eta }u_{\xi }+\kappa _{g,\xi }u_{\xi }-\kappa _{g,\eta }u_{\eta }, \\[9pt] \nonumber \end{align}

where $\kappa _{g,\xi }$ and $\kappa _{g,\eta }$ are the geodesic curvatures of the coordinate curves on the streamsurface, solely determined by the covariant metric components.

Analogous to the material description in (4.13) and (4.14), the streamsurface-based DVD is expressed as

(8.4) \begin{eqnarray} \boldsymbol{\omega }=\boldsymbol{R}_{\varSigma }(\boldsymbol{n}_{S})+\boldsymbol{s}_{\varSigma }(\boldsymbol{n}_{S})+\boldsymbol{g}_{\varSigma }(\boldsymbol{n}_{S}). \end{eqnarray}

The last term in (8.4) is equal to ${\omega _{3}}\boldsymbol{n}_{S}$ being equivalent to $2*{\mathscr{A}}[{ (\boldsymbol{\nabla }_{\pi }\boldsymbol{u} )_{\pi }}]$ , where $*$ is the Hodge-star operator (see Appendix C). By (4.5) and (4.14a ), the streamsurface-based orbital-rotation mode is determined as

(8.5) \begin{eqnarray} \boldsymbol{R}_{\varSigma }(\boldsymbol{n}_{S})\equiv -2\boldsymbol{n}_{S}\times \left (\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{n}_{S}\right )=2\lambda _{\eta }u_{\eta }\overline {\boldsymbol{e}}_{\xi }-2\lambda _{\xi }u_{\xi }\overline {\boldsymbol{e}}_{\eta }, \end{eqnarray}

representing a superposition of two regular circular motions: one with curvature radius $R_{\xi }=\lvert \lambda _{\xi }\rvert ^{-1}$ , and the other with $R_{\eta }=\lvert \lambda _{\eta }\rvert ^{-1}$ . From (4.14b ) and (6.3), the streamsurface-based spin mode is

(8.6) \begin{eqnarray} \boldsymbol{s}_{\varSigma }(\boldsymbol{n}_{S})=2\boldsymbol{n}_{S}\times \left (\boldsymbol{D}\boldsymbol{\cdot }\boldsymbol{n}_{S}\right )=(-\partial _{\zeta }\widehat {u}_{\eta }-\lambda _{\eta }u_{\eta })\overline {\boldsymbol{e}}_{\xi }+(\partial _{\zeta }\widehat {u}_{\xi }+\lambda _{\xi }u_{\xi })\overline {\boldsymbol{e}}_{\eta }, \end{eqnarray}

where each component exhibits the same form as its counterpart in planar flow. The resulting surface shear stress $\boldsymbol{\tau }=\tau _{\xi }\overline {\boldsymbol{e}}_{\xi }+\tau _{\eta }\overline {\boldsymbol{e}}_{\eta }$ on $\mathcal{S}$ is

(8.7) \begin{eqnarray} \tau _{\xi }=\mu (\partial _{\zeta }\widehat {u}_{\xi }+\lambda _{\xi }u_{\xi }),\ \tau _{\eta }=\mu (\partial _{\zeta }\widehat {u}_{\eta }+\lambda _{\eta }u_{\eta }). \end{eqnarray}

8.2. Consistency between different vorticity decompositions for a cylindrical streamsurface

The vorticity decompositions based on streamline ${\mathcal{C}}$ and streamsurface ${\mathcal{S}}$ are established in § 7.4 and § 8.1, respectively. In this subsection, we show the consistency between these two vorticity decompositions for a cylindrical streamsurface ${\mathcal{S}}$ with ${\mathcal{C}}$ as its directrix. Consider a pair of material line elements, $(\delta \boldsymbol{r},\delta \boldsymbol{r}^{\prime })=(\delta {r}\boldsymbol{e},\delta {r}^{\prime }\boldsymbol{e}^{\prime })$ , with $(\boldsymbol{e},\boldsymbol{e}^{\prime })=(\boldsymbol{t},\boldsymbol{n})$ (figure 4). This line-element pair can equivalently be viewed as a pair of material surface elements $(\delta \boldsymbol{\varSigma },\delta \boldsymbol{\varSigma }^{\prime })\equiv (\delta \varSigma \boldsymbol{n}_{\varSigma },\delta \varSigma ^{\prime }\boldsymbol{n}_{\varSigma }^{\prime })$ coinciding with ${\mathcal{S}}$ , where $(\boldsymbol{n}_{\varSigma },\boldsymbol{n}_{\varSigma }^{\prime })=(\boldsymbol{n},\boldsymbol{t})$ . The surface curvature tensor of ${\mathcal{S}}$ is $\boldsymbol{K}\equiv -\boldsymbol{\nabla }_{\pi }\boldsymbol{n}_{S}={\kappa }\boldsymbol{tt}$ so that the principal curvatures of ${\mathcal{S}}$ reduce to $(\lambda _{\xi }, \lambda _{\eta })=(\kappa ,0)$ , where $\kappa$ is the curvature of the streamline ${\mathcal{C}}$ . Since $\boldsymbol{g}_{\varSigma }(\boldsymbol{n})$ vanishes, the analysis focuses exclusively on the orbital-rotation and spin modes.

Figure 4.

Sketch of a cylindrical streamsurface $\mathcal{S}$ generated by extending a regular streamline ${\mathcal{C}}$ (lying in the $\overline {\boldsymbol{e}}_{\xi }$ - $\overline {\boldsymbol{e}}_{\zeta }$ plane) along its binormal direction. The Frenet–Serret triad $(\boldsymbol{t},\boldsymbol{n},\boldsymbol{b})$ is defined at each point $P$ on ${\mathcal{C}}$ . The streamsurface normal $\boldsymbol{n}_{S}$ coincides with the principal normal $\boldsymbol{n}$ of ${\mathcal{C}}$ , with associated curvature radius $\kappa ^{-1}$ . The surface-attached frame $(\overline {\boldsymbol{e}}_{\xi },\overline {\boldsymbol{e}}_{\eta },\overline {\boldsymbol{e}}_{\zeta })$ is oriented such that $(\overline {\boldsymbol{e}}_{\xi },\overline {\boldsymbol{e}}_{\eta })$ align with the principal directions of $\mathcal{S}$ . Here $(\delta \boldsymbol{r},\delta \boldsymbol{r}^{\prime })$ represents a pair of orthogonal material line elements, with $\delta \boldsymbol{r}$ instantaneously coinciding with ${\mathcal{C}}$ . They can be viewed as a surface-element pair $(\delta \boldsymbol{\varSigma },\delta \boldsymbol{\varSigma }^{\prime })$ with $\delta \boldsymbol{\varSigma }$ residing on $\mathcal{S}$ .

Given the velocity field $\boldsymbol{u}=q\boldsymbol{t}$ on ${\mathcal{C}}$ , the streamsurface-based orbital-rotation mode in (8.5) equals to the streamline-based orbital-rotation mode,

(8.8) \begin{eqnarray} \boldsymbol{R}_{\varSigma }(\boldsymbol{n})=-2\kappa {q}\overline {\boldsymbol{e}}_{\eta }=2\kappa {q}\boldsymbol{b}=\boldsymbol{R}_{L}(\boldsymbol{t}). \end{eqnarray}

Equation (8.8) implies that the binormal components of $\boldsymbol{W}_{L}(\boldsymbol{t})$ and $\boldsymbol{W}_{\varSigma }(\boldsymbol{n})$ satisfy

(8.9) \begin{eqnarray} {W}_{\varSigma }(\boldsymbol{n})={W}_{L}(\boldsymbol{t})=\boldsymbol{t}\boldsymbol{\cdot }\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{n}=\kappa {q}, \end{eqnarray}

demonstrating that $\delta \boldsymbol{r}$ and $\delta \boldsymbol{\varSigma }$ rotates with the identical angular velocity $\kappa {q}$ . As a counterpart to (8.9), a similar relation follows,

(8.10) \begin{eqnarray} {W}_{L}(\boldsymbol{n})={W}_{\varSigma }(\boldsymbol{t}) =-\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{t}=-\partial _{\zeta }{q}, \end{eqnarray}

showing that $\delta \boldsymbol{r}^{\prime }$ and $\delta \boldsymbol{\varSigma }^{\prime }$ also rotate with the identical angular velocity $-\partial _{\zeta }q$ , where the square of the velocity magnitude is ${q}^2=\widehat {u}_{\xi }^{2}+\widehat {u}_{\zeta }^{2}$ in a small neighbourhood of ${\mathcal{C}}$ . The streamsurface-based spin mode in (8.6) is equal to the streamline-based spin mode,

(8.11) \begin{eqnarray} \boldsymbol{s}_{\varSigma }(\boldsymbol{n}) =(\partial _{\zeta }\widehat {u}_{\xi }+\kappa {u}_{\xi })\overline {\boldsymbol{e}}_{\eta } =(-\partial _{\zeta }q-\kappa {q})\boldsymbol{b}=\boldsymbol{s}_{L}(\boldsymbol{t}). \end{eqnarray}

The second equality in (8.11) is due to the validity of the relations $\partial _{\zeta }\widehat {u}_{\xi } = \partial _{\zeta }{q}$ and $u_{\xi } = q$ on the considered streamline ${\mathcal{C}}$ .

By substituting (8.9) and (8.10) into (8.11), we find that the spin mode can be expressed in terms of the angular velocities describing the rigid-rotation modes of the orthogonal line-element pair $(\delta \boldsymbol{r},\delta \boldsymbol{r}^{\prime })$ or surface-element pair $(\delta \boldsymbol{\varSigma },\delta \boldsymbol{\varSigma }^{\prime })$ ,

(8.12a) \begin{align} {s}_{L}(\boldsymbol{t}) & ={W}_{L}(\boldsymbol{n})-{W}_{L}(\boldsymbol{t})={W}_{\varSigma }(\boldsymbol{t})-{W}_{\varSigma }(\boldsymbol{n}), \\[-9pt] \nonumber \end{align}

(8.12b) \begin{align} {s}_{L}(\boldsymbol{t}) & ={W}_{\varSigma }(\boldsymbol{t})-{W}_{L}(\boldsymbol{t}) ={W}_{L}(\boldsymbol{n})-{W}_{\varSigma }(\boldsymbol{n}). \\[9pt] \nonumber \end{align}

As time evolves, the angle $\varTheta$ between the two material line elements will not remain $\pi /2$ due to their unequal angular velocities. Equation (8.12a ) implies that the rate of change of $\varTheta$ at the instant is equal to $s_{L}(\boldsymbol{t})$ . Furthermore, ${W}_{L}(\boldsymbol{t})$ and ${W}_{L}(\boldsymbol{n})$ can be decomposed as

(8.13) \begin{eqnarray} {W}_{L}(\boldsymbol{t})=\frac {1}{2}{\omega }_{b}+{W}_{D}(\boldsymbol{t}),\ {W}_{L}(\boldsymbol{n})=\frac {1}{2}{\omega }_{b}+{W}_{D}(\boldsymbol{n}), \end{eqnarray}

where their specific contributions are given by

(8.14) \begin{eqnarray} {W}_{D}(\boldsymbol{t})=-{W}_{D}(\boldsymbol{n})=\boldsymbol{t}\boldsymbol{\cdot }\boldsymbol{D}\boldsymbol{\cdot }\boldsymbol{n}=\frac {1}{2}\left (\kappa {q}+\partial _{\zeta }q\right ). \end{eqnarray}

9.1. Normal-nilpotent decomposition and rotational invariants

In the region where the discriminant $\Delta _{\textrm{3D}}\gt 0$ , $\boldsymbol{A}\equiv \boldsymbol{\nabla }\boldsymbol{u}$ has one real eigenvalue $\lambda _{r}$ and a pair of complex conjugate eigenvalues $(\lambda _{1,2}=\lambda _{\textit{cr}}\pm {\textrm{i}}\lambda _{\textit{ci}})$ , where $\chi \equiv \lambda _{\textit{cr}}$ and $\lambda _{\textit{ci}}(\gt 0)$ represent the real and imaginary parts, respectively. At a given point $P$ , the rotation axis is specified by the real eigenvector $\boldsymbol{e}_{3}$ of $\boldsymbol{A}$ , which is uniquely determined by imposing a constraint for the axial vorticity component ${\omega }_{3}\equiv \boldsymbol{\omega }\boldsymbol{\cdot }\boldsymbol{e}_{3}\gt 0$ . The eigenpair $(\lambda _{r},\boldsymbol{e}_{3})$ satisfies the equation $\boldsymbol{e}_{3}\boldsymbol{\cdot }\boldsymbol{A}=\lambda _{r}\boldsymbol{e}_{3}$ . The rotation-axis-normal plane ${\mathcal{P}}$ is then identified as a 2-D subspace orthogonal to $\boldsymbol{e}_{3}$ . According to the classical critical-point theory (Chong et al. Reference Chong, Perry and Cantwell1990), a complex conjugate eigenvalue pair indicates a closed or spiral streamline pattern appearing in a small neighbourhood of $P$ on ${\mathcal{P}}$ . Under a proper orthonormal basis $(\boldsymbol{e}_{1},\boldsymbol{e}_{2},\boldsymbol{e}_{3})$ where $\boldsymbol{e}_{1}$ and $\boldsymbol{e}_{2}$ span the plane ${\mathcal{P}}$ , by employing the Schur theorem (Schur Reference Schur1909), $\boldsymbol{A}$ can be expressed in a real Schur form (that is, as a block upper triangular matrix in the field of real numbers), yielding the NND (Li Reference Li2010; Liu et al. Reference Liu, Gao, Tian and Dong2018; Li Reference Li2024),

(9.1a) \begin{align} \boldsymbol{A} &= \begin{bmatrix} \chi & \psi +\gamma & -\beta \\[4pt] -\psi & \chi &\alpha \\[4pt] 0& 0& \lambda _{r} \end{bmatrix} =\boldsymbol{N}+\boldsymbol{S}, \\[-9pt] \nonumber \end{align}

(9.1b) \begin{align} \boldsymbol{N} &= \begin{bmatrix} \chi & \psi & 0\\[4pt] -\psi & \chi &0\\[4pt] 0& 0& \lambda _{r} \end{bmatrix},\quad \boldsymbol{S}= \begin{bmatrix} 0 & \gamma & -\beta \\[4pt] 0& 0&\alpha \\[4pt] 0& 0& 0 \end{bmatrix}, \\[9pt] \nonumber \end{align}

where $\boldsymbol{N}$ is a normal tensor, and $\boldsymbol{S}$ is a nilpotent tensor. For notational simplicity, the same mathematical notation is employed for both a tensor and its corresponding matrix throughout the paper. The orthonormal basis $(\boldsymbol{e}_1, \boldsymbol{e}_2, \boldsymbol{e}_3)$ , which serves as the fundamental basis for the NND, is referred to as the NND basis or the NND triad (figure 5). The diagonal elements $(\chi , \lambda _r)$ in $\boldsymbol{A}$ and $\boldsymbol{N}$ describe the relative stretching rates of the material line elements along the NND axes. The off-diagonal element $\psi$ in $\boldsymbol{A}$ and $\boldsymbol{N}$ represents the characteristic angular velocity associated with the rigid rotation about $\boldsymbol{e}_3$ within ${\mathcal{P}}$ . The remaining invariants $(\alpha ,\beta ,\gamma )$ in $\boldsymbol{A}$ and $\boldsymbol{S}$ characterize the rates of angular deformation in the three orthogonal planes spanned by the NND triad. The dilatation is given by $\vartheta \equiv \textrm {tr}(\boldsymbol{A})=\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{u}=2\chi +\lambda _{r}=\textrm {tr}(\boldsymbol{N})$ , where $\textrm {tr}$ denotes the trace. For $\Delta _{\textrm{3D}}\leq 0$ , $\boldsymbol{A}$ retains a matrix form analogous to (9.1a ), though no longer being a real Schur form (Li Reference Li2024).

Figure 5.

Schematic of the right-handed NND triad $(\boldsymbol{e}_{1},\boldsymbol{e}_{2},\boldsymbol{e}_{3})$ on the rotation-axis-normal plane ${\mathcal{P}}$ , showing the rotation axis $\boldsymbol{e}_{3}$ (grey arrow, defined by $\omega _{3}\gt 0$ ) and a typical swirling streamline (blue curve) near a point $P$ .

Using (9.1a ), the matrices of $\boldsymbol{D}$ and $\boldsymbol{\varOmega }$ in (2.1) are obtained as

(9.2) \begin{eqnarray} \boldsymbol{D}= \begin{bmatrix} \chi &\dfrac {1}{2}\gamma & -\dfrac {1}{2}\beta \\[8pt] \dfrac {1}{2}\gamma & \chi &\dfrac {1}{2}\alpha \\[8pt] -\dfrac {1}{2}\beta & \dfrac {1}{2}\alpha & \lambda _{r} \end{bmatrix}, \quad \boldsymbol{\varOmega }= \begin{bmatrix} 0 &\psi +\dfrac {1}{2}\gamma & -\dfrac {1}{2}\beta \\[8pt] -\left (\psi +\dfrac {1}{2}\gamma \right ) & 0&\dfrac {1}{2}\alpha \\[8pt] \dfrac {1}{2}\beta & -\dfrac {1}{2}\alpha & 0 \end{bmatrix}. \end{eqnarray}

Then, the vorticity $\boldsymbol{\omega }$ is derived as the dual vector of $\boldsymbol{\varOmega }$ , which is decomposed as

(9.3a) \begin{align} \boldsymbol{\omega } & =\boldsymbol{R}_{N}+\boldsymbol{s}_{N}, \\[-9pt] \nonumber \end{align}

(9.3b) \begin{align} \boldsymbol{R}_{N}=2\psi \boldsymbol{e}_{3}, & \quad \boldsymbol{s}_{N}=\alpha \boldsymbol{e}_{1}+\beta \boldsymbol{e}_{2}+\gamma \boldsymbol{e}_{3}. \\[9pt] \nonumber \end{align}

Equations (9.3a ) and (9.3b ) are collectively referred to as the invariant vorticity decomposition (IVD). Here $\boldsymbol{R}_{N}$ is termed the characteristic rigid-rotation mode; $\boldsymbol{s}_{N}$ is termed the characteristic spin mode whose in-plane components satisfy $(\omega _{1},\omega _{2})=(\alpha ,\beta )$ . The invariant vorticity splitting occurs exclusively for $\omega _{3}$ ,

(9.4) \begin{eqnarray} \omega _{3}=2\psi +\gamma . \end{eqnarray}

From (9.1a ) and (9.1b ), we obtain the line-element-based representations of $(\boldsymbol{R}_{N},\boldsymbol{s}_{N})$ ,

(9.5) \begin{eqnarray} \boldsymbol{R}_{N}=3\langle \boldsymbol{e}\times \left (\boldsymbol{e}\boldsymbol{\cdot }\boldsymbol{N}\right )\rangle ,\quad \boldsymbol{s}_{N}=3\langle \boldsymbol{e}\times \left (\boldsymbol{e}\boldsymbol{\cdot }\boldsymbol{S}\right )\rangle . \end{eqnarray}

The orientation of a reference frame in three-dimensional (3-D) space requires three scalars, while the second-order tensor $\boldsymbol{A}$ possesses nine independent elements. Consequently, $\boldsymbol{A}$ can be fully characterized by six intrinsic, orientation-invariant scalars. A conventional choice for these invariants comprises the three eigenvalues of $\boldsymbol{D}$ and the three vorticity components projected onto the $\boldsymbol{D}$ eigensystem, or alternatively, through properly defined tensor contractions (Meneveau Reference Meneveau2011). The real Schur form in (9.1a ) introduces a new set of rotational invariants $(\chi , \lambda _r, \alpha , \beta , \gamma , \psi )$ for generic compressible flows. Since $(\gamma , \psi )$ are functionally dependent on $(\lambda _{\textit{ci}}, \omega _3)$ (§ 9.4), an equivalent invariant set is $(\chi ,\lambda _{r},\alpha ,\beta ,\lambda _{\textit{ci}},\omega _{3})$ . This set encapsulates the complete eigenvalue information of $\boldsymbol{A}$ via $(\chi , \lambda _{\textit{ci}}, \lambda _r)$ , and the three vorticity components $(\alpha ,\beta ,\omega _{3})$ under the NND triad.

9.2. Rotational invariants and the generalized Caswell formula

The generalized Caswell formula (6.2a ) serves as the foundation for identifying the contribution of the surface-element-based spin mode to the surface vorticity. We now examine three special material surface elements $\delta \boldsymbol{\varSigma }^{(i)}=\delta \varSigma ^{(i)}\boldsymbol{e}_{i}$ ( $i=1,2,3$ ), with the unit normal vectors successively aligned with each NND basis vector. The material line element along $\boldsymbol{e}_{i}$ at the instant is denoted by $\delta \boldsymbol{r}^{(i)}=\delta {r}^{(i)}\boldsymbol{e}_{i}$ . Under the NND triad, the intrinsic decompositions of $\boldsymbol{D}$ in (6.2a ) and $\boldsymbol{\varOmega }$ in (6.2b ) are expressed as follows:

for $\delta \boldsymbol{\varSigma }^{(1)}=\delta \varSigma ^{(1)}\boldsymbol{e}_{1}$ ,

(9.6a) \begin{align} \boldsymbol{D} &=\begin{bmatrix} \chi &0 &0\\[8pt] 0 & 0&0\\[8pt] 0& 0& 0 \end{bmatrix}+ \begin{bmatrix} 0 &\dfrac {1}{2}\gamma & -\dfrac {1}{2}\beta \\[8pt] \dfrac {1}{2}\gamma & 0&0\\[8pt] -\dfrac {1}{2}\beta &0&0 \end{bmatrix}+ \begin{bmatrix} 0 &0 &0\\[8pt] 0 & \chi &\dfrac {1}{2}\alpha \\[8pt] 0& \dfrac {1}{2}\alpha & \lambda _{r} \end{bmatrix}, \end{align}

(9.6b) \begin{align} \boldsymbol{\varOmega } &=\begin{bmatrix} 0 &\dfrac {1}{2}\omega _{3} &-\dfrac {1}{2}\beta \\[8pt] -\dfrac {1}{2}\omega _{3} & 0&0\\[8pt] \dfrac {1}{2}\beta & 0& 0 \end{bmatrix}+ \begin{bmatrix} 0 &0 &0\\[8pt] 0& 0&\dfrac {1}{2}\alpha \\[8pt] 0&-\dfrac {1}{2}\alpha &0 \end{bmatrix}; \end{align}

for $\delta \boldsymbol{\varSigma }^{(2)}=\delta \varSigma ^{(2)}\boldsymbol{e}_{2}$ ,

(9.6c) \begin{align} \boldsymbol{D} &=\begin{bmatrix} 0 &0 &0\\[8pt] 0 &\chi &0\\[8pt] 0& 0& 0 \end{bmatrix}+ \begin{bmatrix} 0 &\dfrac {1}{2}\gamma & 0\\[8pt] \dfrac {1}{2}\gamma & 0&\dfrac {1}{2}\alpha \\[8pt] 0&\dfrac {1}{2}\alpha &0 \end{bmatrix}+ \begin{bmatrix} \chi &0 &-\dfrac {1}{2}\beta \\[8pt] 0 &0&0\\[8pt] -\dfrac {1}{2}\beta & 0& \lambda _{r} \end{bmatrix}, \end{align}

(9.6d) \begin{align} \boldsymbol{\varOmega} & =\begin{bmatrix} 0 &\dfrac {1}{2}\omega _{3} &0\\[8pt] -\dfrac {1}{2}\omega _{3} & 0&\dfrac {1}{2}\alpha \\[8pt] 0& -\dfrac {1}{2}\alpha & 0 \end{bmatrix}+ \begin{bmatrix} 0 &0 &-\dfrac {1}{2}\beta \\[8pt] 0& 0&0\\[8pt] \dfrac {1}{2}\beta &0&0 \end{bmatrix}; \end{align}

for $\delta \boldsymbol{\varSigma }^{(3)}=\delta \varSigma ^{(3)}\boldsymbol{e}_{3}$ ,

(9.6e) \begin{align} \boldsymbol{D} &=\begin{bmatrix} 0 &0 &0\\[8pt] 0 &0&0\\[8pt] 0& 0&\lambda _{r} \end{bmatrix}+ \begin{bmatrix} 0 &0 & -\dfrac {1}{2}\beta \\[8pt] 0 & 0&\dfrac {1}{2}\alpha \\[8pt] -\dfrac {1}{2}\beta &\dfrac {1}{2}\alpha &0 \end{bmatrix}+ \begin{bmatrix} \chi &\dfrac {1}{2}\gamma &0\\[8pt] \dfrac {1}{2}\gamma & \chi &0\\[8pt] 0& 0& 0 \end{bmatrix}, \end{align}

(9.6f) \begin{align} \boldsymbol{\varOmega} &=\begin{bmatrix} 0 &0 &-\dfrac {1}{2}\beta \\[8pt] 0 & 0&\dfrac {1}{2}\alpha \\[8pt] \dfrac {1}{2}\beta & -\dfrac {1}{2}\alpha & 0 \end{bmatrix}+ \begin{bmatrix} 0 &\dfrac {1}{2}\omega _{3} &0\\[8pt] -\dfrac {1}{2}\omega _{3}& 0&0\\[8pt] 0&0&0 \end{bmatrix}. \end{align}

On the right-hand side of (9.6a ), (9.6c ) and (9.6e ), the matrices (from left to right) represent the normal stretching component (N–N), the symmetric shear component ${\mathscr{S}}[\boldsymbol{n}_{\varSigma }(\boldsymbol{\omega }_{r}\times \boldsymbol{n}_{\varSigma })]$ (T–N and N–T) and the in-plane component ${\mathscr{S}}[{ (\boldsymbol{\nabla }_{\pi }\boldsymbol{u} )_{\pi }}]$ (T–T). The surface-element-based spin mode $\boldsymbol{s}_{\varSigma }(\boldsymbol{e}_{i})$ is solely determined by two spin components

(9.7a) \begin{align} \boldsymbol{s}_{\varSigma }(\boldsymbol{e}_{1})&=\boldsymbol{\omega }_{r}^{(1)}=\beta \boldsymbol{e}_{2}+\gamma \boldsymbol{e}_{3}=-\boldsymbol{s}_{L}(\boldsymbol{e}_{1}), \\[-9pt] \nonumber \end{align}

(9.7b) \begin{align} \boldsymbol{s}_{\varSigma }(\boldsymbol{e}_{2})&=\boldsymbol{\omega }_{r}^{(2)}=\alpha \boldsymbol{e}_{1}-\gamma \boldsymbol{e}_{3}=-\boldsymbol{s}_{L}(\boldsymbol{e}_{2}), \\[-9pt] \nonumber \end{align}

(9.7c) \begin{align} \boldsymbol{s}_{\varSigma }(\boldsymbol{e}_{3})&=\boldsymbol{\omega }_{r}^{(3)}=-\alpha \boldsymbol{e}_{1}-\beta \boldsymbol{e}_{2}=-\boldsymbol{s}_{L}(\boldsymbol{e}_{3}). \\[9pt] \nonumber \end{align}

Notably, only one spin component enters the matrix of ${\mathscr{S}}[{ (\boldsymbol{\nabla }_{\pi }\boldsymbol{u} )_{\pi }}]$ which characterizes the surface flexibility of $\delta \boldsymbol{\varSigma }^{(i)}$ . While the tensor ${\mathscr{S}}[{ (\boldsymbol{\nabla }_{\pi }\boldsymbol{u} )_{\pi }}]$ must be incorporated in the Cauchy–Poisson constitutive relation for the viscous stress tensor, it plays no dynamic role in generating surface shear stress. Moreover, the right-hand side matrices in (9.6b ), (9.6d ) and (9.6f ) represent the skew-symmetric tensors, ${\mathscr{A}}[\boldsymbol{n}_{\varSigma }(\boldsymbol{\omega }\times \boldsymbol{n}_{\varSigma })]$ and ${\mathscr{A}}[{ (\boldsymbol{\nabla }_{\pi }\boldsymbol{u} )_{\pi }}]$ , which determine the tangential and normal vorticity components, respectively.

Furthermore, for each orthogonal pair $(\delta \boldsymbol{\varSigma }^{(i)},\delta \boldsymbol{r}^{(i)})$ , the corresponding streamline- and streamsurface-based rigid-rotation modes are derived as

(9.8a) \begin{align} \boldsymbol{R}_{L}(\boldsymbol{e}_{1})&=2\beta \boldsymbol{e}_{2}+2(\psi +\gamma )\boldsymbol{e}_{3},\ \boldsymbol{R}_{\varSigma }(\boldsymbol{e}_{1})=2\psi \boldsymbol{e}_{3}, \\[-9pt] \nonumber \end{align}

(9.8b) \begin{align} \boldsymbol{R}_{L}(\boldsymbol{e}_{2}) &=2\alpha \boldsymbol{e}_{1}+2\psi \boldsymbol{e}_{3},\ \boldsymbol{R}_{\varSigma }(\boldsymbol{e}_{2})=2(\psi +\gamma )\boldsymbol{e}_{3}, \\[-9pt] \nonumber \end{align}

(9.8c) \begin{align} \boldsymbol{R}_{L}(\boldsymbol{e}_{3}) &=\boldsymbol{0},\ \boldsymbol{R}_{\varSigma }(\boldsymbol{e}_{3})=2\alpha \boldsymbol{e}_{1}+2\beta \boldsymbol{e}_{2}. \\[9pt] \nonumber \end{align}

Evidently, (9.7) and (9.8) satisfy the relation in (5.3). These relations demonstrate that the rigid-rotation modes derived from the analyses of material line and surface elements fundamentally differ from those in the characteristic description. The characteristic spin components can modify the rigid-rotation modes of both material line and surface elements. The material line element aligned with $\boldsymbol{e}_{3}$ exhibits no rigid-rotation mode, while the rigid-rotation mode of $\boldsymbol{e}_{3}$ -oriented material surface element is exclusively determined by the two characteristic spin components $(\alpha ,\beta )$ .

9.3. Rotational invariants and Helmholtz decomposition

The NND triad $\{\boldsymbol{e}_{1},\boldsymbol{e}_{2},\boldsymbol{e}_{3}\}$ defines a Cartesian coordinate system $\boldsymbol{x}=(x_{1},x_{2},x_{3})$ in a small neighbourhood of $P$ . According to the Helmholtz–Hodge theorem, a smooth velocity field admits a global decomposition into longitudinal and transverse parts (Wu et al. Reference Wu, Ma and Zhou2015)

(9.9) \begin{align} \boldsymbol{u}=\boldsymbol{\nabla }\varPhi +\boldsymbol{v},\quad \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{v}=0,\quad \boldsymbol{\nabla }\times \boldsymbol{v}=\boldsymbol{\omega }, \end{align}

where $\varPhi$ is the velocity potential, and $\boldsymbol{v}$ is the transverse velocity. This decomposition is unique in either an unbounded domain, or a bounded domain where $\boldsymbol{v}$ is parallel to the boundary. Evaluating the spatial gradient of (9.9) gives

(9.10) \begin{align} \boldsymbol{A}\equiv \boldsymbol{\nabla }\boldsymbol{u}=\boldsymbol{\nabla \boldsymbol{\nabla }}\varPhi +\boldsymbol{\nabla }\boldsymbol{v}, \end{align}

where $\boldsymbol{\nabla \boldsymbol{\nabla }}\varPhi =\partial _{i}\partial _{j}\varPhi \boldsymbol{e}_{i}\boldsymbol{e}_{j}$ is the potential Hessian, and the $\boldsymbol{\nabla }\boldsymbol{v}=\partial _{i}v_{j}\boldsymbol{e}_{i}\boldsymbol{e}_{j}$ is the transverse velocity gradient. The dilatation and vorticity are $\vartheta =\boldsymbol{\nabla} ^{2}\varPhi =\textrm {tr}(\boldsymbol{\nabla \boldsymbol{\nabla }}\varPhi )$ and $\boldsymbol{\omega }=\boldsymbol{\nabla }\times \boldsymbol{v}$ , respectively. By comparing the matrix in (9.1a ) with (9.10), $(A_{31},A_{32})=\partial _{3}\boldsymbol{u}_{\pi }=\boldsymbol{0}$ implies two constraints

(9.11) \begin{align} \partial _{3}\partial _{1}\varPhi =-\partial _{3}v_{1},\quad \partial _{3}\partial _{2}\varPhi =-\partial _{3}v_{2}, \end{align}

where $\boldsymbol{u}_{\pi }=(u_{1},u_{2})$ is the in-plane velocity field. Actually, it can be shown that $\partial _{3}\boldsymbol{u}_{\pi }=[\boldsymbol{e}_{3}\times (\boldsymbol{e}_{3}\boldsymbol{\cdot }\boldsymbol{A})]\times \boldsymbol{e}_{3}=\boldsymbol{W}_{L}^{\textrm{eff}}(\boldsymbol{e}_{3})\times \boldsymbol{e}_{3}$ . Since the rotation axis $\boldsymbol{e}_{3}$ satisfies $\boldsymbol{e}_{3}\boldsymbol{\cdot }\boldsymbol{A}=\lambda _{r}\boldsymbol{e}_{3}$ , the material line element aligned with $\boldsymbol{e}_{3}$ exhibits no effective rigid rotation, which ensures the intrinsic real Schur form of $\boldsymbol{A}$ .

The other rotational invariants in (9.1a ) are expressed as

(9.12a) \begin{align} \chi &=\partial _{1}^{2}\varPhi +\partial _{1}v_{1}=\partial _{2}^{2}\varPhi +\partial _{2}v_{2},\quad \lambda _{r}=\partial _{3}^{2}\varPhi +\partial _{3}v_{3}, \\[-9pt] \nonumber \end{align}

(9.12b) \begin{align} \alpha &=\partial _{2}\partial _{3}\varPhi +\partial _{2}v_{3},\quad \beta =-\partial _{1}\partial _{3}\varPhi -\partial _{1}v_{3}, \\[-9pt] \nonumber \end{align}

(9.12c) \begin{align} 2\psi &=-2\partial _{2}\partial _{1}\varPhi -2\partial _{2}v_{1},\quad \gamma =2\partial _{1}\partial _{2}\varPhi +\partial _{1}v_{2}+\partial _{2}v_{1}. \\[9pt] \nonumber \end{align}

Applying (9.11) to (9.12b ) isolates the purely transverse contributions as

(9.13) \begin{align} \alpha =\partial _{2}v_{3}-\partial _{3}v_{2}=\omega _{1},\quad \beta =\partial _{3}v_{1}-\partial _{1}v_{3}=\omega _{2}. \end{align}

Note that both $2\psi$ and $\gamma$ in (9.12c ) incorporate the contributions from the potential and transverse velocities. However, the potential contributions (i.e. $\pm \partial _{1}\partial _{2}\varPhi$ ) show opposite signs in their expressions, which cancel upon summation to yield $\omega _{3}=\partial _{1}v_{2}-\partial _{2}v_{1}$ in (9.4). Moreover, applying the SAD (2.1a ) to (9.10) gives

(9.14) \begin{align} \boldsymbol{D}=\boldsymbol{\nabla \boldsymbol{\nabla }}\varPhi +{\mathscr{S}}[\boldsymbol{\nabla }\boldsymbol{v}],\quad \boldsymbol{\varOmega }={\mathscr{A}}[\boldsymbol{\nabla }\boldsymbol{v}]. \end{align}

Equation (9.14) shows that the potential Hessian $\boldsymbol{\nabla \boldsymbol{\nabla }}\varPhi$ only contributes to $\boldsymbol{D}$ , whose diagonal matrix elements $(\partial _{1}^{2}\varPhi ,\partial _{2}^{2}\varPhi ,\partial _{3}^{2}\varPhi )$ affect the relative stretching rates $(\chi ,\lambda _{r})$ in (9.12a ). $\boldsymbol{\varOmega }$ is solely determined by the transverse velocity $\boldsymbol{v}$ .

9.4. Two IVDs

At the considered point $P$ , by restricting $\boldsymbol{A}$ in (9.1a ) to the rotation-axis-normal plane ${\mathcal{P}}$ , we obtain the restricted 2-D VGT under the basis $(\boldsymbol{e}_{1},\boldsymbol{e}_{2})$ ,

(9.15) \begin{eqnarray} \kern1.5pt \widehat {\kern-1.5pt \boldsymbol{A}} \equiv \begin{bmatrix} \boldsymbol{e}_{1}\boldsymbol{\cdot }\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{e}_{1}&\boldsymbol{e}_{1}\boldsymbol{\cdot }\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{e}_{2}\\ \boldsymbol{e}_{2}\boldsymbol{\cdot }\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{e}_{1} & \boldsymbol{e}_{2}\boldsymbol{\cdot }\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{e}_{2} \end{bmatrix} =\begin{bmatrix} \chi &\psi +\gamma \\ -\psi & \chi \end{bmatrix}, \end{eqnarray}

whose characteristic equation is expressed as

(9.16) \begin{eqnarray} p(\lambda )\equiv \lambda ^{2}-2\chi \lambda +\chi ^{2}+\psi (\psi +\gamma )=0, \end{eqnarray}

where $p(\lambda )$ is a polynomial of the variable $\lambda$ . The discriminant of $p(\lambda )$ is

(9.17) \begin{eqnarray} \varDelta =4\lambda _{\textit{ci}}^{2}=4\psi (\psi +\gamma )=\omega _{3}^{2}-\gamma ^2. \end{eqnarray}

Obviously, in the region $\varDelta \gt 0$ (i.e. $\omega _{3}\gt \lvert \gamma \rvert$ ), the complex conjugate eigenvalue pair of $\kern1.5pt \widehat {\kern-1.5pt \boldsymbol{A}}$ is $\lambda _{1,2}=\lambda _{\textit{cr}}\pm \textrm {i}\lambda _{\textit{ci}}$ , being exactly the same as that of $\boldsymbol{A}$ . The dilatation of the parallel flow on ${\mathcal{P}}$ is given by $\widehat {\vartheta }\equiv \textrm {tr}(\kern1.5pt \widehat {\kern-1.5pt \boldsymbol{A}})=2\chi$ . From a mathematical perspective, either $\gamma \gt 0$ (denoted as $\gamma ^{+}$ ) or $\gamma \lt 0$ (denoted as $\gamma ^{-}$ ) may serve as an alternative uniqueness condition for determining the real Schur form of $\boldsymbol{A}$ . The degenerate case with $\gamma =0$ implies $\omega _{3}=2\psi$ , representing purely the characteristic rigid rotation on ${\mathcal{P}}$ . To the best of our knowledge, this fundamental problem has not been explicitly analysed in the existing literature. As discussed below, this will yield two distinct IVDs corresponding to Case I and Case II, respectively.

(Case I) When $\gamma \gt 0$ , the condition $\omega _3 \gt \gamma \gt 0$ implies $\psi \gt 0$ . Let $(\psi ^+, \gamma ^+)$ denote the corresponding invariants and $(\boldsymbol{e}_{1}^{+},\boldsymbol{e}_{2}^{+},\boldsymbol{e}_{3}^{+})$ the associated NND triad (with $\boldsymbol{e}_{3}^{+}=\boldsymbol{e}_{3}$ ). The first IVD is then established as

(9.18a) \begin{align} \omega _{3}&=2\psi ^{+}+\gamma ^{+}, \\[-9pt] \nonumber \end{align}

(9.18b) \begin{align} R_{N}^{+}\equiv 2\psi ^{+}&=\omega _{3}-\sqrt {\omega _{3}^{2}-4\lambda _{\textit{ci}}^{2}},\quad {s}_{N}^{+}\equiv \gamma ^{+}=\sqrt {\omega _{3}^{2}-4\lambda _{\textit{ci}}^{2}}. \\[9pt] \nonumber \end{align}

Equation (9.18a ) is essentially the Liutex-shear decomposition proposed by Liu et al. (Reference Liu, Gao, Tian and Dong2018). The characteristic rigid-rotation mode $\boldsymbol{R}_{N}^{+}\equiv 2\psi ^{+}\boldsymbol{e}_{3}$ is precisely the Liutex (or Rortex) vector introduced by Liu et al. (Reference Liu, Gao, Tian and Dong2018), which has been widely used to extract vortex in complex flows. The explicit expressions in (9.18b ) has previously been reported in studies such as Xu et al. (Reference Xu, Gao, Deng, Liu and Liu2019), Liu & Wang (Reference Liu and Wang2020) and more recently reviewed by Mao et al. (Reference Mao, Li, Yao and Wu2023). This case coincides with the KKB mechanism (§ 1): axial vortices form via the wrapping of shear layers, accompanied by a transfer from spin mode to rigid-rotation mode, ultimately resulting in vortex intensification.

(Case II) When $\gamma \lt 0$ , there is $2\psi \gt \omega _{3}\gt -\gamma \gt 0$ , which implies $\psi \gt -\gamma \gt 0$ . Let $(\psi ^{-},\gamma ^{-})$ denote the corresponding invariants and $(\boldsymbol{e}_{1}^{-},\boldsymbol{e}_{2}^{-},\boldsymbol{e}_{3}^{-})$ the associated NND triad (where $\boldsymbol{e}_{3}^{-}=\boldsymbol{e}_{3}$ ). The second IVD is then established as

(9.19a) \begin{align} \omega _{3}&=2\psi ^{-}+\gamma ^{-}, \\[-9pt] \nonumber \end{align}

(9.19b) \begin{align} R_{N}^{-}\equiv 2\psi ^{-}&=\omega _{3}+\sqrt {\omega _{3}^{2}-4\lambda _{\textit{ci}}^{2}},\quad s_{N}^{-}\equiv \gamma ^{-}=-\sqrt {\omega _{3}^{2}-4\lambda _{\textit{ci}}^{2}}. \\[9pt] \nonumber \end{align}

In this case, the characteristic rigid-rotation mode is $\boldsymbol{R}_{N}^{-} \equiv 2\psi ^{-}\boldsymbol{e}_{3}$ . Equation (9.19a ) has been overlooked in the existing literature, primarily because $\gamma \gt 0$ is usually introduced as a uniqueness condition for determining the real Schur form of $\boldsymbol{A}$ . When the axial vortex has sufficient swirling strength to maintain its coherence, such that opposing spin effects cannot significantly disrupt the primary vortex structure, which we refer to as the anti-KKB mechanism/phenomenon. The physical validity of this case will be demonstrated through the Burgers vortex (§ 10.3) and the hurricane-eye structure on Saturn (§ 10.6).

From (9.18b ) and (9.19b ), the following identities are derived:

(9.20) \begin{eqnarray} \psi ^{+}=\psi ^{-}+\gamma ^{-},\quad \psi ^{-}=\psi ^{+}+\gamma ^{+},\quad \gamma ^{-}=-\gamma ^{+},\quad \omega _{3}=\psi ^{+}+\psi ^{-}. \end{eqnarray}

This leads to the concise representations for the discriminant $\varDelta$ and the imaginary part $\lambda _{\textit{ci}}$ :

(9.21) \begin{eqnarray} \varDelta =4\psi ^{+}\psi ^{-},\quad \lambda _{\textit{ci}}=\sqrt {\psi ^{+}\psi ^{-}}. \end{eqnarray}

Under the NND bases $(\boldsymbol{e}_{1}^{+},\boldsymbol{e}_{2}^{+})$ (Case I) and $(\boldsymbol{e}_{1}^{-},\boldsymbol{e}_{2}^{-})$ (Case II) on ${\mathcal{P}}$ , by using (9.15) and (9.20), the matrices of $\kern1.5pt \widehat {\kern-1.5pt \boldsymbol{A}}$ are, respectively, obtained as

(9.22) \begin{eqnarray} \text{Case I,}\ \kern1.5pt \widehat {\kern-1.5pt \boldsymbol{A}} =\begin{bmatrix} \chi &\psi ^{-}\\ -\psi ^{+} & \chi \end{bmatrix}; \quad \text{Case II,}\ \kern1.5pt \widehat {\kern-1.5pt \boldsymbol{A}} =\begin{bmatrix} \chi &\psi ^{+}\\ -\psi ^{-} & \chi \end{bmatrix}. \end{eqnarray}

We remark that $(\boldsymbol{e}_{1}^{+},\boldsymbol{e}_{2}^{+})$ can be obtained by rotating a given right-handed basis $(\boldsymbol{e}_{1}^{-},\boldsymbol{e}_{2}^{-})$ anticlockwise by an angle $\pi /2$ or $3\pi /2$ . Interestingly, we find that $\psi ^{+}$ and $\psi ^{-}$ are precisely the two eigenvalues of the matrix $\boldsymbol{Q}_{\theta }$ introduced by Nakayama (Reference Nakayama2017), which govern the azimuthal component of the relative velocity near the point $P$ ,

(9.23) \begin{eqnarray} \widetilde {u}_{\theta }=(r\psi ^{+})\cos ^{2}\varTheta +(r\psi ^{-})\sin ^{2}\varTheta +O(r^2), \end{eqnarray}

where $r\gt 0$ is the radial distance from $P$ . To first order in $r$ , the minimum and maximum of $\widetilde {u}_{\theta }$ are $r\psi ^{+}$ and $r\psi ^{-}$ , respectively. Given that $\psi ^{+}\gt 0$ and $\psi ^{-}\gt 0$ , (9.23) demonstrates their combined role in establishing a unidirectional swirling motion around $P$ .

9.5. Relationship between characteristic and field descriptions

The streamline direction is a distinctive, physically significant direction. In the region where $\varDelta \gt 0$ , consider a regular streamline ${\mathcal{C}}$ passing through the point $P$ on the rotation-axis-normal plane ${\mathcal{P}}$ , equipped with the Frenet–Serret triad $(\boldsymbol{t},\boldsymbol{n},\boldsymbol{b})$ . The binormal vector aligns with the rotation axis $(\boldsymbol{b}=\boldsymbol{e}_{3})$ , giving $\omega _{b}=\omega _{3}\gt 0$ . Using (8.9) and (8.10), the matrix of $\kern1.5pt \widehat {\kern-1.5pt \boldsymbol{A}}$ in the streamline-attached basis $(\boldsymbol{t},\boldsymbol{n})$ is written as

(9.24) \begin{eqnarray} \kern1.5pt \widehat {\kern-1.5pt \boldsymbol{A}}_{\textrm{F-S}} \equiv \begin{bmatrix} \boldsymbol{t}\boldsymbol{\cdot }\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{t}&\boldsymbol{t}\boldsymbol{\cdot }\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{n}\\ \boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{t} & \boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{n} \end{bmatrix} =\begin{bmatrix} \chi (\boldsymbol{t}) &W_{L}(\boldsymbol{t})\\ -W_{L}(\boldsymbol{n}) & \chi (\boldsymbol{n}) \end{bmatrix}, \end{eqnarray}

where the relative stretching rates along $\boldsymbol{t}$ and $\boldsymbol{n}$ are

(9.25) \begin{eqnarray} \chi (\boldsymbol{t})\equiv \boldsymbol{t}\boldsymbol{\cdot }\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{t}=\boldsymbol{t}\boldsymbol{\cdot }\boldsymbol{D}\boldsymbol{\cdot }\boldsymbol{t},\quad \chi (\boldsymbol{n})\equiv \boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{n}=\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{D}\boldsymbol{\cdot }\boldsymbol{n}. \end{eqnarray}

Unlike in (9.15) and (9.22), the diagonal elements here are generally unequal: $\chi (\boldsymbol{t})\neq \chi (\boldsymbol{n})$ . The emergence of the irreducible real Schur form in (9.24) is particularly noteworthy, as it reveals the intrinsic structure of $\boldsymbol{A}$ through a streamline-based field description. Since (8.12a ) implies $W_{L}(\boldsymbol{n})=W_{L}(\boldsymbol{t})+s_{L}(\boldsymbol{t})$ , the matrix of $\kern1.5pt \widehat {\kern-1.5pt \boldsymbol{A}}$ under the basis $(-\boldsymbol{n},\boldsymbol{t})$ is given by

(9.26) \begin{eqnarray} \kern1.5pt \widehat {\kern-1.5pt \boldsymbol{A}}_{\textrm{F-S}} \equiv \begin{bmatrix} \boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{n}&-\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{t}\\ -\boldsymbol{t}\boldsymbol{\cdot }\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{n} & \boldsymbol{t}\boldsymbol{\cdot }\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{t} \end{bmatrix} =\begin{bmatrix} \chi (\boldsymbol{n}) &W_{L}(\boldsymbol{t})+s_{L}(\boldsymbol{t})\\ -W_{L}(\boldsymbol{t}) & \chi (\boldsymbol{t}) \end{bmatrix}. \end{eqnarray}

The NND basis $(\boldsymbol{e}_{1}^{-},\boldsymbol{e}_{2}^{-})$ and the streamline basis $(\boldsymbol{t},\boldsymbol{n})$ are related through the following orthogonal transformation:

(9.27) \begin{eqnarray} \begin{bmatrix} \boldsymbol{t} & \boldsymbol{n} \end{bmatrix} =\begin{bmatrix} \boldsymbol{e}_{1}^{-} &\boldsymbol{e}_{2}^{-} \end{bmatrix} \begin{bmatrix} \cos \varTheta &-\sin \varTheta \\ \sin \varTheta & \cos \varTheta \end{bmatrix}, \end{eqnarray}

where $\varTheta$ is the azimuthal angle from the $\boldsymbol{e}_{1}^{-}$ axis. Combining (9.15), (9.20), (9.24) and (9.27) yields the following relations:

(9.28a) \begin{align} & \chi (\boldsymbol{t})=\chi +\gamma ^{-}\sin \varTheta \cos \varTheta =\chi +\frac {1}{2}\gamma ^{-}\sin {2\varTheta }, \\[-9pt] \nonumber \end{align}

(9.28b) \begin{align} & \chi (\boldsymbol{n})=\chi +\gamma ^{+}\sin \varTheta \cos \varTheta =\chi +\frac {1}{2}\gamma ^{+}\sin {2\varTheta }, \\[-9pt] \nonumber \end{align}

(9.28c) \begin{align} & W_{L}(\boldsymbol{t})=\psi ^{-}+\gamma ^{-}\cos ^{2}\varTheta =\psi ^{+}\cos ^{2}\varTheta +\psi ^{-}\sin ^{2}\varTheta , \\[-9pt] \nonumber \end{align}

(9.28d) \begin{align} & W_{L}(\boldsymbol{n})=\psi ^{-}+\gamma ^{-}\sin ^{2}\varTheta =\psi ^{-}\cos ^{2}\varTheta +\psi ^{+}\sin ^{2}\varTheta . \\[9pt] \nonumber \end{align}

Obviously, the upper and lower bounds of both $\chi (\boldsymbol{t})$ and $\chi (\boldsymbol{n})$ are $\chi + ({1}/{2})\gamma ^{+}$ and $\chi + ({1}/{2})\gamma ^{-}$ , respectively. The sum and difference of (9.28a ) and (9.28b ) give

(9.29) \begin{eqnarray} \chi (\boldsymbol{t})+\chi (\boldsymbol{n})=2\chi ,\quad \varphi (\boldsymbol{t})\equiv \chi (\boldsymbol{t})-\chi (\boldsymbol{n})=\gamma ^{-}\sin 2\varTheta . \end{eqnarray}

From (8.12a ), (9.28c ) and (9.28d ), it is implied that

(9.30) \begin{eqnarray} W_{L}(\boldsymbol{t})+W_{L}(\boldsymbol{n})=\omega _{3},\quad {s}_{L}(\boldsymbol{t})=\gamma ^{+}\cos 2\varTheta . \end{eqnarray}

Using (9.29) and (9.30), we obtain a concise relation for $(\gamma ^{\pm },s_{L}(\boldsymbol{t}),\varphi (\boldsymbol{t}))$ :

(9.31) \begin{eqnarray} \gamma ^{2}=s_{L}^{2}(\boldsymbol{t})+\varphi ^{2}(\boldsymbol{t}); \end{eqnarray}

and the angle $\varTheta$ must satisfy

(9.32) \begin{eqnarray} \tan 2\varTheta =-\frac {\varphi (\boldsymbol{t})}{s_{L}(\boldsymbol{t})}. \end{eqnarray}

We note that (9.31) reproduces the result reported by Chen & Liu (Reference Chen and Liu2025, (33a)). However, our derivation approach differs fundamentally from that used by Chen & Liu (Reference Chen and Liu2025) based on the frame-invariance of the discriminant $\varDelta$ for generic 2-D flows. The analysis in this subsection equally applies to a material line element $\delta \boldsymbol{r}\equiv \delta {r}\boldsymbol{e}$ on ${\mathcal{P}}$ , provided that the streamline-attached basis $(\boldsymbol{t},\boldsymbol{n})$ is replaced by $(\boldsymbol{e},\boldsymbol{e}_{\perp })$ where $\boldsymbol{e}_{\perp }\equiv \boldsymbol{e}_{3}\times \boldsymbol{e}$ .

9.6. Vorticity extrema

From (9.28c ) and (9.28d ), two fundamental equalities are derived for $({R}_{L}(\boldsymbol{t}),{s}_{L}(\boldsymbol{t}))$ :

(9.33) \begin{eqnarray} 2\psi ^{+}\leq {R}_{L}(\boldsymbol{t})\leq 2\psi ^{-},\quad \gamma ^{-}\leq {s}_{L}(\boldsymbol{t})\leq \gamma ^{+}. \end{eqnarray}

Equation (9.33) reveals that the streamline-based DVD vorticity modes $({R}_{L}(\boldsymbol{t}),{s}_{L}(\boldsymbol{t}))$ are rigorously bounded by the IVD vorticity modes $(2\psi ^{\pm },\gamma ^{\pm })$ . As illustrated in figure 6, $({R}_{L}(\boldsymbol{t}),{s}_{L}(\boldsymbol{t}))$ or $({R}_{L}(\boldsymbol{e}),{s}_{L}(\boldsymbol{e}))$ must reside along the principal diagonal of the square vorticity-mode space, bounded by the two critical configurations below.

1. (i) When $\boldsymbol{e}=\boldsymbol{e}_{2}^{+}$ or $\boldsymbol{e}_{1}^{-}$ , the DVD vorticity modes correspond to the minimum rigid rotation $2\psi ^{+}$ and the maximum spin $\gamma ^{+}$ in Case I:

   (9.34) \begin{align} R_{L}(\boldsymbol{e}_{2}^{+})=R_{L}(\boldsymbol{e}_{1}^{-})=2\psi ^{+},\quad s_{L}(\boldsymbol{e}_{2}^{+})=s_{L}(\boldsymbol{e}_{1}^{-})=\gamma ^{+}. \end{align}

2. (ii) When $\boldsymbol{e}=\boldsymbol{e}_{1}^{+}$ or $\boldsymbol{e}_{2}^{-}$ , the DVD vorticity modes correspond to the maximum rigid rotation $2\psi ^{-}$ and the minimum spin $\gamma ^{-}$ in Case II:

   (9.35) \begin{align} R_{L}(\boldsymbol{e}_{1}^{+})=R_{L}(\boldsymbol{e}_{2}^{-})=2\psi ^{-},\quad s_{L}(\boldsymbol{e}_{1}^{+})=s_{L}(\boldsymbol{e}_{2}^{-})=\gamma ^{-}. \end{align}

From this perspective, the Liutex $\boldsymbol{R}_{N}^{+}\equiv 2\psi ^{+}\boldsymbol{e}_{3}$ (Liu et al. Reference Liu, Gao, Tian and Dong2018; Liu & Wang Reference Liu and Wang2020), widely used for vortex extraction, implicitly incorporates a minimization principle in its selection of the characteristic rigid-rotation mode.

The IVD and DVD actually provide complementary descriptions of vorticity kinematics at different levels for a generic flow field. The IVD modes (i.e. the vorticity extrema) can be evaluated to extract the primary features of flow structures at the characteristic algebraic level, albeit with some loss of flow details. In contrast, the DVD vorticity modes offer a more detailed description based on material and field perspectives, delivering deeper physical insight into the flow details. Therefore, a coupled IVD–DVD approach could enhance the physical understanding of the vorticity kinematics and structures, as demonstrated through several representative examples in § 10.

Figure 6.

Illustration of the vorticity-mode diagram for the parallel flow in the rotation-axis-normal plane ${\mathcal{P}}$ . The DVD vorticity modes $(R_{L}(\boldsymbol{t}),s_{L}(\boldsymbol{t}))$ (represented by the cat symbol) are confined along the diagonal of the square domain of the vorticity-mode space, bounded by the IVD vorticity modes $(R_{N}^{\pm },s_{N}^{\pm })\equiv (2\psi ^{\pm },\gamma ^{\pm })$ (indicated by the yellow and red solid circles).

10.1. Simple shear flow and Joseph Bertrand’s puzzle

Consider a simple shear flow aligned with the Cartesian coordinate system $(x,y)$ , where $(\boldsymbol{e}_{x},\boldsymbol{e}_{y})$ denote the streamwise and vertical basis vectors, respectively. The velocity field $\boldsymbol{u}$ is prescribed as $(u_{x},u_{y})=(ky,0)\ (k\gt 0)$ . Hence, using the SAD in (2.1a ) and (2.1b ), the VGT $\boldsymbol{A}$ , the strain-rate tensor $\boldsymbol{D}$ , and the rotation-rate tensor $\boldsymbol{\varOmega }$ are evaluated as

(10.1) \begin{eqnarray} \boldsymbol{A}=k\boldsymbol{e}_{y}\boldsymbol{e}_{x},\quad \boldsymbol{D}=\frac {1}{2}(k\boldsymbol{e}_{x}\boldsymbol{e}_{y}+k\boldsymbol{e}_{y}\boldsymbol{e}_{x}),\quad \boldsymbol{\varOmega }=\frac {1}{2}(-k\boldsymbol{e}_{x}\boldsymbol{e}_{y}+k\boldsymbol{e}_{y}\boldsymbol{e}_{x}). \end{eqnarray}

The vorticity $\boldsymbol{\omega }$ is deduced as the dual vector of $\boldsymbol{\varOmega }$ , namely, $\boldsymbol{\omega }=-k\boldsymbol{e}_{z}$ . At the times of Helmholtz, Cauchy and Stokes, one could only apply the classical interpretation to postulate that a volume element of fluid must rotate with the angular velocity $\boldsymbol{W}_{V}=\boldsymbol{\omega }/2=-(k/2)\boldsymbol{e}_{z}$ . But then any material line element within a fluid volume element would obey the classical prediction, which however, fundamentally contradicts the truth that a material line element along $\boldsymbol{e}_{x}$ along the streamline direction shows no intuitive rotation. This precisely captures the essence of Joseph Bertrand’s famous puzzle in 1868: simple shear flow exhibits non-zero vorticity despite having entirely straight streamlines (Truesdell Reference Truesdell1954).

Figure 7.

The $\boldsymbol{e}_{z}$ -components of (a) $(\boldsymbol{W}_{\kern-1.5pt D}(\boldsymbol{e}),\boldsymbol{W}_{L}(\boldsymbol{e}))$ and (b) $(\boldsymbol{R}_{L}(\boldsymbol{e}),\boldsymbol{s}_{L}(\boldsymbol{e}))$ for a material line element $\delta \boldsymbol{r}=\delta {r}\boldsymbol{e}$ , with the unit orientation vector $\boldsymbol{e}=\cos \theta \boldsymbol{e}_{x}+\sin \theta \boldsymbol{e}_{y}$ . The angle is normalized as $\theta ^* \equiv \theta /\pi$ . The angular velocities and vorticity components are normalized by $k/2$ and $k$ , respectively.

This paradox naturally disappears once the specific angular velocity $\boldsymbol{W}_{\kern-1.5pt D}(\boldsymbol{e})$ is recognized. For a material line element with $\boldsymbol{e}=\cos \theta \boldsymbol{e}_{x}+\sin \theta \boldsymbol{e}_{y}$ ( $\theta$ is the angle from $\boldsymbol{e}_{x}$ ), by using (3.7) and (3.9), we derive the specific and total angular velocities as

(10.2) \begin{align} & \boldsymbol{W}_{\kern-1.5pt D}(\boldsymbol{e})=\boldsymbol{e}\times (\boldsymbol{e}\boldsymbol{\cdot }\boldsymbol{D})=\frac {1}{2}k\cos 2\theta \boldsymbol{e}_{z}, \\[-9pt] \nonumber \end{align}

(10.3) \begin{align} & \boldsymbol{W}_{L}(\boldsymbol{e})=\boldsymbol{W}_{\kern-1.5pt D}(\boldsymbol{e})+\frac {1}{2}\boldsymbol{\omega }=\frac {1}{2}k(\cos 2\theta -1)\boldsymbol{e}_{z}. \\[9pt] \nonumber\end{align}

Correspondingly, the DVD vorticity modes are determined from (3.13a ) and (3.13b ) as

(10.4) \begin{eqnarray} \boldsymbol{R}_{L}(\boldsymbol{e})=k(\cos 2\theta -1)\boldsymbol{e}_{z},\quad \boldsymbol{s}_{L}(\boldsymbol{e})=-k\cos 2\theta \boldsymbol{e}_{z}. \end{eqnarray}

When $\boldsymbol{e}=\boldsymbol{e}_{x}=\boldsymbol{t}$ ( $\theta =0$ ) aligns with the streamline, the specific angular velocity $\boldsymbol{W}_{\kern-1.5pt D}(\boldsymbol{e}_{x})=(k/2)\boldsymbol{e}_{z}$ exactly cancels out $\boldsymbol{W}_{V}=-(k/2)\boldsymbol{e}_{z}$ , yielding $\boldsymbol{W}_{L}(\boldsymbol{e}_{x})=\boldsymbol{0}$ . The DVD modes are evaluated as $\boldsymbol{R}_{L}(\boldsymbol{t})=\boldsymbol{0}$ and $\boldsymbol{s}_{L}(\boldsymbol{t})=-k\boldsymbol{e}_{z}=\boldsymbol{\omega }$ . Hence, vorticity arises solely from the streamline-based spin mode, with no contribution from the orbital-rotation mode. Vorticity is not an appropriate physical measure to represent vortex in the spin-dominant region, despite a potentially high vorticity magnitude. Figures 7(a) and 7(b) display the $\boldsymbol{e}_{z}$ -components of $(\boldsymbol{W}_{\kern-1.5pt D}(\boldsymbol{e}),\boldsymbol{W}_{L}(\boldsymbol{e}))$ and $(\boldsymbol{R}_{L}(\boldsymbol{e}),\boldsymbol{s}_{L}(\boldsymbol{e}))$ , respectively. When $\theta$ increases from 0 to values within $(0,\pi /4)$ , both angular velocity components change: $0\lt W_{Dz}\lt k/2$ and $-k/2\lt W_{Lz}\lt 0$ , causing the clockwise rotation of the line element. At $\theta =\pi /4$ (i.e. the principal stretching axis of $\boldsymbol{D}$ ), the specific angular velocity vanishes with $(W_{Dz},W_{Lz})=(0,-k/2)$ . Beyond this point $(\theta \in (\pi /4,3\pi /4))$ , the specific angular velocity component (dominated as $\cos 2\theta$ ) becomes negative $(-k/2\leq {W}_{Dz}\lt 0)$ . The rotation continues until $\theta =3\pi /4$ (the principal compression axis of $\boldsymbol{D}$ ), where $W_{Dz}$ again vanishes. Both components reach their minima at $\theta =\pi /2$ , with $(W_{Dz},W_{Lz})=(-k/2,-k)$ , while $W_{Lz}$ returns to zero at $\theta =\pi$ .

In the characteristic algebraic description, the condition $\omega _{3}\gt 0$ uniquely determines the rotation axis as $\boldsymbol{e}_{3}=-\boldsymbol{e}_{z}$ . From (10.4), the $\boldsymbol{e}_{3}$ -components of the line-element-based vorticity modes are $\overline {R}_{L}(\boldsymbol{e})=2k\sin ^{2}\theta$ and $\overline {s}_{L}(\boldsymbol{e})=k\cos 2\theta$ . For Case I ((9.18a ) and (9.18b )), under the NND basis $(\boldsymbol{e}_{1}^{+},\boldsymbol{e}_{2}^{+})=(\boldsymbol{e}_{y},\boldsymbol{e}_{x})\ \text{or}\ (-\boldsymbol{e}_{y},-\boldsymbol{e}_{x})$ , the rotational invariants of $\boldsymbol{A}=k\boldsymbol{e}_{1}^{+}\boldsymbol{e}_{2}^{+}$ are given by $(2\psi ^{+},\gamma ^{+},\omega _{3})=(0,k,k)$ . Here $(\overline {R}_{L})_{\textit{min}}=2\psi ^{+}$ and $(\overline {s}_{L})_{\textit{max}}=\gamma ^{+}$ occur at $\theta =0$ (i.e. $\boldsymbol{e}=\boldsymbol{e}_{x}=\boldsymbol{t}$ ) and $\pi$ (i.e. $\boldsymbol{e}=-\boldsymbol{e}_{x}$ ). For Case II ((9.19a ) and (9.19b )), under the NND basis $(\boldsymbol{e}_{1}^{-},\boldsymbol{e}_{2}^{-})=(\boldsymbol{e}_{x},-\boldsymbol{e}_{y})\ \text{or}\ (-\boldsymbol{e}_{x},\boldsymbol{e}_{y})$ , the rotational invariants of $\boldsymbol{A}=-k\boldsymbol{e}_{2}^{-}\boldsymbol{e}_{1}^{-}$ are given by $(2\psi ^{-},\gamma ^{-},\omega _{3})=(2k,-k,k)$ . $(\overline {R}_{L})_{\textit{max}}=2\psi ^{-}$ and $(\overline {s}_{L})_{\textit{min}}=\gamma ^{-}$ occur at $\theta =\pi /2$ (i.e. $\boldsymbol{e}=\boldsymbol{e}_{y}$ ) and $3\pi /2$ (i.e. $\boldsymbol{e}=-\boldsymbol{e}_{y}$ ). Physically, Case I corresponds to the rectilinear streamlines that exhibit no swirling motion, whereas Case II describes the behaviour of the material line elements oriented orthogonal to the streamline direction.

10.2. Potential flow outside a point vortex

For the pure potential flow $(\varDelta =-\gamma ^{2}\lt {0}\ \text{when}\ \gamma \neq {0})$ , $\boldsymbol{\omega }=\boldsymbol{0}$ implies $\alpha =\beta =0$ and $\gamma =-2\psi =2\partial _{1}\partial _{2}\varPhi$ . The characteristic spin component $\gamma$ coexists with the characteristic rigid-rotation component $\psi$ , which generates no surface shear stress due to the absence of viscosity. Consequently, $\boldsymbol{A}$ is fully described by three rotational invariants $(\chi ,\lambda _{r},\psi )$ where $\chi =\partial _{1}^{2}\varPhi =\partial _{2}^{2}\varPhi$ and $\lambda _{r}=\partial _{3}^{2}\varPhi$ (§ 9.3). The matrix of $\boldsymbol{A}$ in (9.1a ) is simplified as

(10.5) \begin{align} \boldsymbol{A}=\boldsymbol{D}=\boldsymbol{\nabla \boldsymbol{\nabla }}\varPhi = \begin{bmatrix} \chi & -\psi & 0\\[8pt] -\psi & \chi &0\\[8pt] 0& 0& \lambda _{r} \end{bmatrix} = \begin{bmatrix} \chi & \frac {1}{2}\gamma & 0\\[8pt] \frac {1}{2}\gamma & \chi &0\\[8pt] 0& 0& \lambda _{r} \end{bmatrix}. \end{align}

For a material line element $\delta \boldsymbol{r}=\delta {r}\boldsymbol{e}$ , (4.14) gives $\boldsymbol{R}(\boldsymbol{e})=-\boldsymbol{s}(\boldsymbol{e})=2\boldsymbol{e}\times (\boldsymbol{e}\boldsymbol{\cdot }\boldsymbol{\nabla \boldsymbol{\nabla }}\varPhi )$ .

Consider the potential flow outside a 2-D point vortex with the constant circulation $\varGamma =2\pi {K}\ (K\gt 0)$ . We adopt the cylindrical coordinate system $(r,\theta ,z)$ with the basis $(\boldsymbol{e}_{r},\boldsymbol{e}_{\theta },\boldsymbol{e}_{z})$ . For $r\gt 0$ , the azimuthal velocity component is $u_{\theta }(r)=K/r$ , and the VGT $\boldsymbol{A}$ is expressed as $\boldsymbol{A}=(\partial _{r}u_{\theta })\boldsymbol{e}_{r}\boldsymbol{e}_{\theta }-(u_{\theta }/r)\boldsymbol{e}_{\theta }\boldsymbol{e}_{r}$ . Under the NND triad $(\boldsymbol{e}_{1},\boldsymbol{e}_{2},\boldsymbol{e}_{3})=(\boldsymbol{e}_{r},\boldsymbol{e}_{\theta },\boldsymbol{e}_{z})$ , the invariants $(\psi ,\gamma ,\omega _{3})$ are given by (note $R_{N}=2\psi$ and $s_{N}=\gamma$ )

(10.6) \begin{eqnarray} \psi =\frac {u_{\theta }}{r}=\frac {K}{r^{2}},\quad \gamma =\frac {\partial {u}_{\theta }}{\partial r}-\frac {u_{\theta }}{r}=-\frac {2K}{r^{2}},\quad \omega _{3}=2\psi +\gamma =0. \end{eqnarray}

In contrast, the Liutex-based analysis (Liu et al. Reference Liu, Gao, Tian and Dong2018) yields $(\psi ,\gamma ,\omega _{3})=(0,0,0)$ in the region where $\varDelta \lt 0$ , which does not include the solution like (10.6).

To reveal the flow details, we examine the line-element-based DVD (in (3.12) and (3.13)) and the streamline-based DVD (in (7.3) and (7.8)). In this example, the Frenet–Serret frame of a circular streamline is just $(\boldsymbol{t},\boldsymbol{n},\boldsymbol{b})=(\boldsymbol{e}_{\theta },-\boldsymbol{e}_{r},\boldsymbol{e}_{z})$ . When $\boldsymbol{e}=\boldsymbol{t}=\boldsymbol{e}_{\theta }$ holds, we obtain

(10.7) \begin{eqnarray} \boldsymbol{R}_{L}(\boldsymbol{t})=2\frac {u_{\theta }}{r}\boldsymbol{e}_{z}=2\psi \boldsymbol{e}_{z}=\boldsymbol{R}_{N}, \quad \boldsymbol{s}_{L}(\boldsymbol{t})=\left (\frac {\partial u_{\theta }}{\partial r}-\frac {u_{\theta }}{r}\right )\boldsymbol{e}_{z}=\gamma \boldsymbol{e}_{z}=\boldsymbol{s}_{N}, \end{eqnarray}

where $(\boldsymbol{R}_{N},\boldsymbol{s}_{N})$ are the IVD vorticity modes in (9.3). When $\boldsymbol{e}=\boldsymbol{n}=-\boldsymbol{e}_{r}$ , we obtain

(10.8) \begin{align} & \boldsymbol{R}_{L}(\boldsymbol{n})=2\frac {\partial u_{\theta }}{\partial r}\boldsymbol{e}_{z}=2(\psi +\gamma )\boldsymbol{e}_{z}=-2\psi \boldsymbol{e}_{z}=-\boldsymbol{R}_{N}, \\[-9pt] \nonumber \end{align}

(10.9) \begin{align} & \boldsymbol{s}_{L}(\boldsymbol{n})=-\left (\frac {\partial u_{\theta }}{\partial r}-\frac {u_{\theta }}{r}\right )\boldsymbol{e}_{z}=-\gamma \boldsymbol{e}_{z}=-\boldsymbol{s}_{N}. \\[9pt] \nonumber \end{align}

10.3. Burgers vortex

The celebrated Burgers vortex is a steady, axisymmetric exact solution of the Navier–Stokes equations, widely used to model fine-scale turbulence structures owing to its finite and viscosity-independent total dissipation rate (Burgers Reference Burgers1948; Wu et al. Reference Wu, Ma and Zhou2015). It consists of an inward radial flow that concentrates and rotates around the symmetry axis, while the fluid moves outward in opposite directions along the axial dimension. In the following analysis, we adopt the cylindrical coordinate system $(r,\theta ,z)$ with the orthonormal basis $\{\boldsymbol{e}_{r},\boldsymbol{e}_{\theta },\boldsymbol{e}_{z}\}$ . The velocity field of the Burgers vortex is then expressed as

(10.10a) \begin{align} & u_{r}=-\frac {1}{2}ar, \\[-9pt] \nonumber \end{align}

(10.10b) \begin{align} & u_{\theta }=\frac {\varGamma }{2\pi r}\left [1-\exp \left (-\frac {r^2a}{4\nu }\right )\right ], \\[-9pt] \nonumber \end{align}

(10.10c) \begin{align} & u_{z}=az. \\[9pt] \nonumber \end{align}

Here, $\varGamma$ denotes the total circulation that quantities the vortex strength; the positive constant $a$ represents the axial stretching rate, $\nu$ is the kinematic viscosity and $r$ is the radial distance measured from the $z$ -axis (i.e. the symmetry axis). For definiteness, we use the parameters ( $a=0.084$ , $\varGamma =1.45$ , $\nu =0.01$ ) reported in Shrestha et al. (Reference Shrestha, Nottage, Yu, Alvarez and Liu2021). Using (10.10), the matrices of $\boldsymbol{A}$ and $\boldsymbol{D}$ under $\{\boldsymbol{e}_{r},\boldsymbol{e}_{\theta },\boldsymbol{e}_{z}\}$ are given by

(10.11) \begin{eqnarray} \boldsymbol{A}= \begin{bmatrix} -\dfrac {1}{2}a &\dfrac {\partial u_{\theta }}{\partial r}& 0\\[8pt] -\dfrac {u_{\theta }}{r} &-\dfrac {1}{2}a&0\\[8pt] 0&0& a \end{bmatrix},\quad \boldsymbol{D}=\dfrac {1}{2} \begin{bmatrix} -a &\dfrac {\partial u_{\theta }}{\partial r}-\dfrac {u_{\theta }}{r}& 0\\[8pt] \dfrac {\partial u_{\theta }}{\partial r}-\dfrac {u_{\theta }}{r} &-a&0\\[8pt] 0&0& 2a \end{bmatrix}. \end{eqnarray}

Notably, the matrix $\boldsymbol{A}$ in (10.11) coincides precisely with the real Schur form in (9.1a ). Comparing (9.1a ) and $\boldsymbol{A}$ in (10.11) yields $(\chi ,\lambda _{r})=(-({1}/{2})a,a)$ and $(\alpha ,\beta )=(0,0)$ .

For Case II, the rotational invariants $(\psi ^{-},\gamma ^{-},\omega _{3})$ in (9.19a ) under the NND triad $\{\boldsymbol{e}_{1}^{-},\boldsymbol{e}_{2}^{-},\boldsymbol{e}_{3}^{-}\}=\{\boldsymbol{e}_{r},\boldsymbol{e}_{\theta },\boldsymbol{e}_{z}\}$ are given by

(10.12a) \begin{align} & \psi ^{-}=\frac {u_{\theta }}{r},\quad R_{N}^{-}=2\psi ^{-}, \\[-9pt] \nonumber\end{align}

(10.12b) \begin{align} & \gamma ^{-}=\frac {\partial {u}_{\theta }}{\partial r}-\frac {u_{\theta }}{r} =r\frac {\partial }{\partial r}\left (\frac {u_{\theta }}{r}\right ),\quad s_{N}^{-}=\gamma ^{-}, \\[-9pt] \nonumber\end{align}

(10.12c) \begin{align} & \omega _{3}=2\psi ^{-}+\gamma ^{-}=\frac {\partial {u}_{\theta }}{\partial r}+\frac {u_{\theta }}{r} =\frac {1}{r}\frac {\partial }{\partial r}\left (ru_{\theta }\right ). \\[9pt] \nonumber\end{align}

Intuitively, $\psi ^{-}$ describes a virtual circular motion with speed $u_{\theta }$ and curvature radius $r$ . The corresponding characteristic rigid-rotation and spin modes are

(10.13) \begin{eqnarray} \boldsymbol{R}_{N}^{-}\equiv {R}_{N}^{-}\boldsymbol{e}_{3}^{-}=2\frac {{u}_{\theta }}{r}\boldsymbol{e}_{z},\quad \boldsymbol{s}_{N}^{-}\equiv {s}_{N}^{-}\boldsymbol{e}_{3}^{-}=r\frac {\partial }{\partial r}\left (\frac {u_{\theta }}{r}\right )\boldsymbol{e}_{z}. \end{eqnarray}

Substituting (10.10b ) into (10.12), we obtain the concrete expressions as

(10.14) \begin{align} & \psi ^{-}=\frac {\varGamma }{2\pi r^2}\left [1-\exp \left (-\frac {r^2a}{4\nu }\right )\right ], \\[-9pt] \nonumber\end{align}

(10.15) \begin{align} & \gamma ^{-}=\frac {\varGamma a}{4\pi \nu }\exp \left (-\frac {r^2a}{4\nu }\right )-\frac {\varGamma }{\pi r^2}\left [1-\exp \left (-\frac {r^2a}{4\nu }\right )\right ], \\[-9pt] \nonumber\end{align}

(10.16) \begin{align} & \omega _{3}=\frac {\varGamma a}{4\pi \nu }\exp \left (-\frac {r^2a}{4\nu }\right ). \\[9pt] \nonumber\end{align}

For Case I, the rotational invariants $(\psi ^{+},\gamma ^{+},\omega _{3})$ in (9.18b ) under the NND triad $\{\boldsymbol{e}_{1}^{+},\boldsymbol{e}_{2}^{+},\boldsymbol{e}_{3}^{+}\}=\left \{\boldsymbol{e}_{\theta },-\boldsymbol{e}_{r},\boldsymbol{e}_{z}\right \}$ are identified as

(10.17a) \begin{align} \psi ^{+} & = \frac {\partial {u}_{\theta }}{\partial r},\quad R_{N}^{+}=2\psi ^{+}, \\[-9pt] \nonumber \end{align}

(10.17b) \begin{align} \gamma ^{+} & = \frac {u_{\theta }}{r}-\frac {\partial {u}_{\theta }}{\partial r}=-r\frac {\partial }{\partial r}\left (\frac {u_{\theta }}{r}\right ),\quad s_{N}^{+}=\gamma ^{+}, \\[-9pt] \nonumber \end{align}

(10.17c) \begin{align} \omega _{3} &= 2\psi ^{+}+\gamma ^{+}=\frac {\partial {u}_{\theta }}{\partial r}+\frac {u_{\theta }}{r}=\frac {1}{r}\frac {\partial }{\partial r}\left (ru_{\theta }\right ). \\[9pt] \nonumber \end{align}

Accordingly, the characteristic rigid-rotation and spin modes are given by

(10.18) \begin{eqnarray} \boldsymbol{R}_{N}^{+}\equiv {R}_{N}^{+}\boldsymbol{e}_{3}^{+}=2\frac {\partial {u}_{\theta }}{\partial r}\boldsymbol{e}_{z}, \quad \boldsymbol{s}_{N}^{+}\equiv {s}_{N}^{+}\boldsymbol{e}_{3}^{-}=-r\frac {\partial }{\partial r}\left (\frac {u_{\theta }}{r}\right )\boldsymbol{e}_{z}. \end{eqnarray}

For both cases, the discriminant $\varDelta$ in (9.17) admits a unified formulation:

(10.19) \begin{eqnarray} \varDelta =4\psi ^{+}\psi ^{-}=4\frac {u_{\theta }}{r}\frac {\partial {u}_{\theta }}{\partial r}. \end{eqnarray}

Unlike $\boldsymbol{R}_{N}^{-}$ in (10.13) governed by ${u_{\theta }}/{r}$ , $\boldsymbol{R}_{N}^{+}$ in (10.18) is just the Liutex vector associated with the radial velocity gradient $\partial _{r}u_{\theta }$ . This distinction renders $\boldsymbol{R}_{N}^{+}$ devoid of a kinematic interpretation via orbital rotation, due to its lack of a direct association with a specific orbit.

Figure 8.

(a) Contour map of the normalized azimuthal velocity component $u_{\theta }$ with superposed streamlines for the Burgers vortex. (b) Radial profile of the normalized $u_{\theta }$ . (c) Comparison of the radial profiles of the normalized $\partial _{r}u_{\theta }$ , $u_{\theta }/r$ and the scaled discriminant $4\Delta$ .

The Burgers vortex emerges from a dynamic equilibrium among vorticity convection, vortex stretching and viscous diffusion. The reference scales for length, velocity and vorticity are thus introduced as $L_{\textit{ref}}\equiv ({\nu }/{a})^{1/2}$ , $U_{\textit{ref}}\equiv {\varGamma }/{L_{\textit{ref}}}$ and $\omega _{\textit{ref}}\equiv {\varGamma }/{L_{\textit{ref}}^2}$ , with an asterisk denoting normalized quantities. Figure 8(a) displays the distribution of the azimuthal velocity component $u_{\theta }^{*}$ on the rotation-axis-normal plane ${\mathcal{P}}$ , and its radial profile (along $y=0$ ) is given in figure 8(b). The swirling streamlines indicate a 3-D vortex structure with axial stretching, because mass conservation requires outward axial flow to balance inward radial motion. We observe that $u_{\theta }$ exhibits a linear increase for small $r$ , following $u_{\theta }\sim {(\varGamma ar)}/{(8\pi \nu )}$ (or $u_{\theta }^{*}\sim {r^*}/{8\pi }$ in normalized form), matching the rigid-body rotation behaviour. It reaches a global maximum at the critical radius $r_{c}=r_{1}=2.242{L}_{\textit{ref}}$ (or $r_{c}^{*}=2.242$ ), satisfying $\partial _{r}u_{\theta }(r_{c})=0$ . For $r\gt r_{c}$ , the velocity decays asymptotically as $u_{\theta }\sim {\varGamma }/{(2\pi r)}$ (or $u_{\theta }^{*}\sim {1}/{(2\pi {r}^*)}$ ), showing characteristic of a potential vortex flow. Thus, $r_{c}$ naturally defines the vortex core radius. Figure 8(c) shows the radial distributions of $\partial _{r}u_{\theta }$ and ${u_{\theta }}/{r}$ , and the discriminant $\varDelta$ in (10.19), which govern the invariants in (10.12a – c ) and (10.17a – c ). As $r\rightarrow {0}$ , both $\partial _{r}u_{\theta }$ and ${u_{\theta }}/{r}$ approach ${\varGamma {a}}/{(8\pi \nu )}$ , confirming rigid-rotation characteristic within the vortex core. The quantity ${u_{\theta }}/{r}$ remains positive throughout and decays towards zero as $r\rightarrow {+\infty }$ , whereas $\partial _{r}u_{\theta }$ and $\varDelta$ reverse their signs at $r_c$ .

Figure 9.

Contour maps of the normalized IVD variables for the Burgers vortex: (a) $2\psi ^{-}$ , (b) $\gamma ^{-}$ and (c) $\omega _{3}$ , with superposed streamlines. (d) Radial profiles of the normalized $2\psi ^{-}$ , $\gamma ^{-}$ , $\omega _{3}$ and the scaled discriminant $10\Delta$ .

Figure 9 presents the contour plots of the IVD variables $(2\psi ^{-},\gamma ^{-},\omega _{3})$ , with radial profiles shown in figure 9(d). The rigid-rotation mode $(R_{N}^{-}=2\psi ^{-})$ (figure 9 a) remains strictly positive and vanishes at infinity. In contrast, the spin mode $(s_{N}^{-}=\gamma ^{-})$ (figure 9 b) is exclusively negative, featuring a null point at the vortex-axis position $r_{\gamma }^{(1)}=0$ and a pronounced minimum at $r_{\gamma }^{(2)}$ ( $r_{c}=r_{1}\lt r_{\gamma }^{(2)}\lt r_{2}$ ), where $r_{2}$ is the inflection point of $u_{\theta }$ ( $\partial _{r}^{2}u_{\theta }(r_2)=0$ ). This minimum corresponds to the twin negative peaks in figure 9(d), revealing a dual-cored spin structure. The existence of $r_{\gamma }^{(2)}$ is qualitatively analysed as follows. At $r=r_{1}$ where $\partial _{r}u_{\theta }(r_{1})=0$ , figure 8(c) shows the negative slopes for both $\partial _{r}u_{\theta }$ and ${u_{\theta }}/{r}$ , with $\partial _{r}^{2}u_{\theta }(r_1)\lt \partial _{r}({u_{\theta }}/{r})(r_{1})$ , implying $\partial _{r}\gamma (r_{1})\lt 0$ . At $r=r_{2}$ , the slope reversal gives $\partial _{r}\gamma (r_{2})=-\partial _{r}({u_{\theta }}/{r})(r_{2})\gt 0$ . By the intermediate value theorem, there must exist a critical point $r_{\gamma }^{(2)}\in (r_{1},r_{2})$ of $\partial _{r}\gamma$ where $\gamma$ actually attains a minimum. The superposition of $2\psi ^{-}$ and $\gamma ^{-}$ produces the vorticity $\omega _z (=\omega _3)$ , with a more localized distribution than $2\psi ^{-}$ , as seen in both figures 9(c) and 9(d).

Note that for any fixed $z$ , a local comoving frame transformation $(\boldsymbol{u}\rightarrow \boldsymbol{u}-u_{z}(z)\boldsymbol{e}_{z})$ reduces the analysis to the rotation-axis-normal plane ${\mathcal{P}}$ which preserves all essential vortex kinematics. Since the in-plane velocity $\boldsymbol{u}_{\pi }(r,\theta )=u_{r}(r)\boldsymbol{e}_{r}+u_{\theta }(r)\boldsymbol{e}_{\theta }$ is independent of $z$ , we only need to analyse $z=0$ . Substituting (10.11) into (7.3) and (7.8) yields the streamline-based vorticity modes,

(10.20a) \begin{align} & {R}_{L}(\boldsymbol{t})=2\psi ^{-}+2\frac {u_r^2}{q^2}\gamma ^{-}=2\psi ^{-}+2\gamma ^{-}\cos ^2{\varTheta }, \\[-12pt] \nonumber \end{align}

(10.20b) \begin{align} & {s}_{L}(\boldsymbol{t})=\frac {u_{\theta }^{2}-u_{r}^{2}}{q^2}\gamma ^{-}=\gamma ^{+}\cos 2\varTheta , \\[9pt] \nonumber \end{align}

where $(\cos \varTheta ,\sin \varTheta )=(u_{r}/q,u_{\theta }/q)$ with $q^{2}=u_{\theta }^{2}+u_{r}^{2}$ and $\varTheta$ being the azimuthal angle from $\boldsymbol{e}_{r}$ . Equation (10.20a ) and (10.20b ) are consistent with the general theory in § 9.5.

Figure 10.

Contour maps of the normalized streamline-based DVD vorticity modes on the rotation-axis-normal plane ${\mathcal{P}}$ for the Burgers vortex: (a) ${R}_{L}(\boldsymbol{t})$ and (b) ${s}_{L}(\boldsymbol{t})$ , with superposed streamlines. (c) Radial profiles of the normalized ${R}_{L}(\boldsymbol{t})$ , ${s}_{L}(\boldsymbol{t})$ , $\omega _{3}$ and the scaled discriminant $10\Delta$ .

Figure 10 presents the axial components of the DVD variables $({R}_{L}(\boldsymbol{t}),{s}_{L}(\boldsymbol{t}),{\omega }_{z})$ , with the radial distributions shown in figure 10(c). The DVD vorticity modes (figures 10 a and 10 b) resemble their IVD counterparts (figures 9 a and 9 b): ${R}_L(\boldsymbol{t})$ retains the monopole structure of $R_{N}^{-}$ , while ${s}_L(\boldsymbol{t})$ preserves the dipole character of $s_{N}^{-}$ (figures 9 d and 10 c). The radial distributions of $(R_{N}^{-},R_{N}^{+},{R}_{L}(\boldsymbol{t}))$ and $(s_{N}^{-},s_{N}^{+},s_L(\boldsymbol{t}))$ are compared in figures 11(a) and 11(b), respectively. Indeed, the DVD vorticity modes are always bounded by the IVD vorticity modes. In the vortex core $(r\lt {r_c},\Delta \gt 0)$ , since $u_{\theta }$ dominates the in-plane velocity field $(\lvert {u}_{\theta }\rvert \gg \lvert {u}_{r}\rvert )$ , (10.20a ) and (10.20b ) imply $R_{L}(\boldsymbol{t})\approx 2\psi ^{-}=R_{N}^{-}$ and $s_{L}(\boldsymbol{t})\approx \gamma ^{-}=s_{N}^{-}$ , which explain their close agreement. The deviation of $R_N^{+}$ from $R_L(\boldsymbol{t})$ is due to the term $2\gamma ^{+}\sin ^2\varTheta =2\gamma ^{+}(u_{\theta }/q)^2$ , as seen from the alternative expression based on the IVD vorticity modes $(R_{N}^{+},s_{N}^{+})$ in Case I:

(10.21) \begin{eqnarray} {R}_{L}(\boldsymbol{t})=2\psi ^{+}+2\frac {u_\theta ^2}{q^2}\gamma ^{+}=2\psi ^{+}+2\gamma ^{+}\sin ^2{\varTheta }. \end{eqnarray}

Both $R_{N}^{+}$ and $R_{N}^{-}$ converge to the same limit ${\varGamma a}/{(4\pi \nu )}$ as $r\rightarrow {0}$ . In the region $r^{*}\gt 5$ , the difference between $R_{N}^{-}$ and $R_{L}(\boldsymbol{t})$ is caused by $2\gamma ^{-}\cos ^{2}\varTheta =2\gamma ^{-}(u_{r}/q)^{2}$ in (10.20a ). According to (10.20b ), the sign reversal of $s_L(\boldsymbol{t})$ with increasing $r$ is attributed to the growing influence of the radial velocity magnitude $|u_r|$ .

Figure 11.

Comparison of (a) $(R_{N}^{-},R_{N}^{+},{R}_{L}(\boldsymbol{t}))$ and (b) $(s_{N}^{-},s_{N}^{+},{s}_{L}(\boldsymbol{t}))$ for the Burgers vortex where $(R_{N}^{\pm },s_{N}^{\pm })=(2\psi ^{\pm },\gamma ^{\pm })$ . The discriminant $\varDelta$ has been multiplied by 5 and 3 in (a) and (b), respectively.

Furthermore, for a material line element on ${\mathcal{P}}$ with orientation $\boldsymbol{e}=\cos \hat {\varTheta }\boldsymbol{e}_{r}+\sin \hat {\varTheta }\boldsymbol{e}_{\theta }$ , the associated DVD vorticity modes are given by

(10.22) \begin{align} R_{L}(\boldsymbol{e})=2\psi ^{+}\cos ^{2}\hat {\varTheta }+2\psi ^{-}\sin ^{2}\hat {\varTheta },\quad s_{L}(\boldsymbol{e})=\gamma ^{+}\cos 2\hat {\varTheta }. \end{align}

Their extrema confirm the IVD modes as rigorous bounds, attained at specific orientations. Here $(R_{L})_{\textit{max}}=2\psi ^{-}$ and $(s_{L})_{\textit{min}}=\gamma ^{-}$ are achieved at $\hat {\varTheta }=\pi /2$ (i.e. $\boldsymbol{e}=\boldsymbol{e}_{\theta }$ ) and $3\pi /2$ (i.e. $\boldsymbol{e}=-\boldsymbol{e}_{\theta }$ ), while $(R_{L})_{\textit{min}}=2\psi ^{+}$ and $(s_{L})_{\textit{max}}=\gamma ^{+}$ are achieved at $\hat {\varTheta }=0$ (i.e. $\boldsymbol{e}=\boldsymbol{e}_{r}$ ) and $\pi$ (i.e. $\boldsymbol{e}=-\boldsymbol{e}_{r}$ ). In particular, when $\boldsymbol{e}=\boldsymbol{t}$ (i.e. $\hat {\varTheta }=\varTheta$ ) holds, (10.22) is consistent with (10.20) and (10.21). Thus, the IVDs robustly capture vortical features at the characteristic algebraic level, whereas the DVDs reveal finer flow details from material and field perspectives. Together, they offer complementary insights that collectively build a more complete physical description of the flow.

10.4. Instability of a vortex sheet to small disturbance

10.4.1. Configuration and Batchelor’s classical interpretation

The study of dynamical instability has become a critical and coherent branch of fluid mechanics. It is well documented that at the interface between two approximately uniform streams with different velocities, undulations of increasing amplitude tend to develop, typically leading to turbulent flow (Batchelor Reference Batchelor1967). This can be modelled by an idealized vortex sheet $\boldsymbol{\varSigma }$ with infinitesimal thickness $(\delta \rightarrow {0})$ , a material surface forming the common boundary between two streams of the same fluid. The velocity fields $(\boldsymbol{u}_{1},\boldsymbol{u}_{2})$ on the upper and lower sides of $\boldsymbol{\varSigma }$ are derived from the velocity potentials $(\varPhi _{1},\varPhi _{2})$ satisfying $\boldsymbol{\nabla} ^{2}\varPhi _{i}=0\ (i=1,2)$ , with $\boldsymbol{u}_{i}=\boldsymbol{\nabla }{\varPhi }_{i}$ . The velocity jump across the sheet $ [\! [ \boldsymbol{u} ]\! ]\equiv \boldsymbol{u}_{1}-\boldsymbol{u}_{2}$ determines the strength density of the vortex sheet by integrating the vorticity across the thin transition layer at a given point,

(10.23) \begin{eqnarray} \boldsymbol{\varGamma }\equiv \lim _{\delta \rightarrow {0}}\int _{-\delta /2}^{\delta /2}\boldsymbol{\omega }\, \text{d}\zeta =\boldsymbol{n}_{\varSigma }\times \left [\!\left [ \boldsymbol{u} \right ]\!\right ],\quad \zeta \in \left [-\frac {\delta }{2},\frac {\delta }{2}\right ], \end{eqnarray}

where the unit normal vector $\boldsymbol{n}_{\varSigma }$ points from Region 2 to Region 1. Equation (10.23) shows that $\boldsymbol{\varGamma }$ is entirely determined by the tangential velocity discontinuity across $\varSigma$ .

Figure 12.

Schematic illustrating the instability of a plane vortex sheet subjected to a small-amplitude sinusoidal perturbation. The blue arrows indicate the self-induced rotation direction at the discrete points satisfying $(\eta =0,\ \partial _{x}\eta \gt 0)$ , while the red arrows depict the resulting vorticity accumulation at these locations.

The linearized normal-mode theory predicts that an initially straight vortex sheet $\boldsymbol{\varSigma }$ with uniform strength density $\boldsymbol{\varGamma }_{0}$ is inherently unstable to infinitesimal perturbations (Kelvin–Helmholtz instability). Consider a planar vortex sheet initially at $y=0$ separating two uniform streams: $\boldsymbol{u}_{1}=-{1}/{2}U\boldsymbol{e}_{x}$ for the upper region $y\gt 0$ and $\boldsymbol{u}_{2}= {1}/{2}U\boldsymbol{e}_{x}$ for the lower region $y\lt 0$ (figure 12). Note that a generic small disturbance is a superposition of distinct Fourier modes. For a sinusoidal mode with wavelength $\lambda$ and wavenumber $k=2\pi /\lambda \ (k\gt 0)$ , the sheet displacement is $\eta (x,t)=O(\epsilon )$ , with $\epsilon \ll \lambda$ ( $\epsilon$ is a small parameter). Assuming a constant sheet thickness $\delta \ll \epsilon$ (and $\lambda =O(1)$ for convenience so that $\delta \ll \epsilon \ll {1}$ ), the total velocity potentials are $\varPhi _{1}=-{1}/{2}Ux+\phi _{1}$ and $\varPhi _{2}={1}/{2}Ux+\phi _{2}$ , where $(\phi _{1},\phi _{2})$ represent the disturbance-induced potentials $(U=O(1))$ . Then, $\boldsymbol{\varGamma }$ admits the following decomposition:

(10.24) \begin{eqnarray} \boldsymbol{\varGamma }=\boldsymbol{\varGamma }_{0}+\boldsymbol{\varGamma }^{\prime }, \quad \boldsymbol{\varGamma }_{0}\equiv \lim \limits _{\delta \rightarrow {0}} \int _{-\delta /2}^{\delta /2}\boldsymbol{\omega }_{0}\, \text{d}\zeta =U\boldsymbol{e}_{z},\quad \boldsymbol{\varGamma }^{\prime }=\varGamma ^{\prime }_{z}\boldsymbol{e}_{z}= \boldsymbol{n}_{\varSigma }\times \left [\!\left [ \boldsymbol{\nabla }\phi \right ]\!\right ], \end{eqnarray}

where $\boldsymbol{\varGamma }_{0}=O(1)$ and $\boldsymbol{\varGamma }^{\prime }=O(\epsilon )$ is the perturbation-induced local strength density.

Batchelor (Reference Batchelor1967, pp. 511–517) derived the solutions of $\eta$ , $\phi _{1,2}$ and $\varGamma ^{\prime }_{z}\approx -\partial _{x}(\phi _{1}-\phi _{2})_{y=0}$ (only their real parts are physically meaningful):

(10.25a) \begin{align} & \eta (x,t)=C\exp (\sigma t)\exp (ikx), \\[-9pt] \nonumber\end{align}

(10.25b) \begin{align} & \varGamma _{z}^{\prime }={2i}\sigma \eta ={2}\sigma {C}\exp (\sigma t)\exp \left [i\left (kx+\frac {\pi }{2}\right )\right ]. \\[9pt] \nonumber\end{align}

Here, $C\gt 0$ represents the initial disturbance magnitude, and $\sigma =kU/2\gt 0$ denotes the temporal growth rate, characterizing the exponential amplification of perturbations. The solution reveals that $\varGamma _{z}^{\prime }$ leads the phase of $\eta$ by $\pi /2$ , indicating a quarter-period phase difference between vorticity modulation and sheet displacement. As vividly sketched in Batchelor (Reference Batchelor1967, figure 7.1.3) and figure 12, the phase disparity drives the instability through two coupled processes occurring simultaneously: vorticity accumulation near the nodal points satisfying $(\eta =0,\partial _{x}\eta \gt 0)$ , and induced rotation of adjacent sheet segments. These mutually reinforcing mechanisms constitute the essential dynamics of the Kelvin–Helmholtz instability, explaining both the exponential growth of disturbances and the subsequent formation of discrete vortices.

10.4.2. A new interpretation of vortex-sheet instability

Building on (7.3) and (7.8), we analyse the streamline-based DVD vorticity modes. For both the upper and lower regions, the streamline curvature is $\kappa =\lvert \partial _{xx}\eta \rvert +O(\epsilon ^3)=O(\epsilon )$ , the velocity magnitude is $q=U/2+O(\epsilon )$ , and the binormal vector is $\boldsymbol{b}=\boldsymbol{e}_{z}$ (here, terms of $O(\delta )$ are absorbed into $O(\epsilon )$ since $\delta \ll \epsilon$ ). Under these approximations, the $\boldsymbol{e}_{z}$ -component of the orbital-rotation mode in (7.3) is

(10.26) \begin{eqnarray} {R}_{L}(\boldsymbol{t})=k^{2}C\exp (\sigma t)\lvert \cos (kx)\rvert {U}+O(\epsilon ^2)=k^{2}\lvert \eta \rvert +O(\epsilon ^2), \end{eqnarray}

which maintains phase synchronization with the absolute displacement $\lvert \eta \rvert$ . The maxima of ${R}_{L}(\boldsymbol{t})$ occur at the discrete locations where $\lvert \cos (kx)\rvert =1$ . Since $\boldsymbol{R}_{L}(\boldsymbol{t})=O(\epsilon )$ , its integral across the vortex sheet scales as

(10.27) \begin{eqnarray} \int _{-\delta /2}^{\delta /2}\boldsymbol{R}_{L}(\boldsymbol{t})\, \text{d}\zeta =O(\epsilon \delta )\ll O(\epsilon ^2). \end{eqnarray}

In the region where $\eta \gt 0$ (the case $\eta \lt 0$ follows analogously), given that $\boldsymbol{t}=-\boldsymbol{e}_{x}+O(\epsilon )$ and $\boldsymbol{n}_{\varSigma }=\boldsymbol{e}_{y}+O(\epsilon )$ , the spin mode $\boldsymbol{s}_{L}(\boldsymbol{t})$ in (7.8) is estimated as

(10.28) \begin{eqnarray} \boldsymbol{s}_{L}(\boldsymbol{t})\approx -\boldsymbol{t}\times \left (\boldsymbol{n}_{\varSigma }\partial _{\zeta }\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{t}\right )+O(\epsilon )=-\partial _{\zeta }u_{x}\boldsymbol{e}_{z}+O(\epsilon ), \end{eqnarray}

which scales as $O(\delta ^{-1})$ , thereby dominating the vortex-sheet dynamics. Although the detailed variation of $u_{x}$ within the sheet is not available, its values at the transition-layer boundaries $\zeta =\pm ({1}/{2})\delta$ are known. The sheet centreline is parameterized as $\boldsymbol{r}(x,t)={x}\boldsymbol{e}_{x}+\eta (x,t)\boldsymbol{e}_{y}$ . Denoting $\boldsymbol{r}^{\pm }\equiv \boldsymbol{r}\pm ({\delta}/{2})\boldsymbol{n}_{\varSigma }$ , $\boldsymbol{\varGamma }^{\prime }$ in (10.24) is estimated as

(10.29) \begin{eqnarray} \boldsymbol{\varGamma }^{\prime }=\boldsymbol{n}_{\varSigma }\times \left [\!\left [ \boldsymbol{\nabla }\phi \right ]\!\right ]=\left [(\partial _{x}\phi _{2})(\boldsymbol{r}^{+})-(\partial _{x}\phi _{1})(\boldsymbol{r}^{-})\right ]\boldsymbol{e}_{z}+O\big(\epsilon ^2\big). \end{eqnarray}

Then, integrating (10.28) across the vortex sheet gives

(10.30) \begin{eqnarray} \int _{-\delta /2}^{\delta /2}\boldsymbol{s}_{L}(\boldsymbol{t})\, \text{d}\zeta =\boldsymbol{\varGamma }_{0}+\boldsymbol{\varGamma }^{\prime }+O\big(\epsilon ^2\big)=\boldsymbol{\varGamma }+O\big(\epsilon ^2\big). \end{eqnarray}

Equation (10.30) shows that under small disturbances, $\boldsymbol{\varGamma }$ can be approximated by the integral of $\boldsymbol{s}_{L}(\boldsymbol{t})$ across the vortex sheet. Since the vorticity in the unperturbed sheet arises solely from the spin mode, we define the disturbance spin as $\boldsymbol{s}_{L}^{\prime }(\boldsymbol{t})\equiv \boldsymbol{s}_{L}(\boldsymbol{t})-\boldsymbol{\omega }_{0}$ . It follows that (10.30) is equivalent to

(10.31) \begin{eqnarray} \int _{-\delta /2}^{\delta /2}\boldsymbol{s}_{L}^{\prime }(\boldsymbol{t})\, \text{d}\zeta =\boldsymbol{\varGamma }^{\prime }+O\big(\epsilon ^2\big). \end{eqnarray}

Equations (10.27) and (10.31) collectively imply that $\boldsymbol{s}_{L}^{\prime }(\boldsymbol{t})$ is the dominant contributor to $\boldsymbol{\varGamma }^{\prime }$ . Hence, the phase difference of ${1}/{2}\pi$ between $\boldsymbol{\varGamma }^{\prime }$ and $\eta$ is primarily governed by the disturbance spin mode, subject to minor modulation by the orbital-rotation mode. This result provides a novel vorticity-dynamical interpretation of vortex-sheet instability, complementing the classical theory of Batchelor (Reference Batchelor1967).

10.5. Moffatt–Kida–Ohkitani asymptotic vortex solution

10.5.1. Background: stretched vortices in triaxial strain field

Moffatt, Kida & Ohkitani (Reference Moffatt, Kida and Ohkitani1994) (MKO94) proposed that stretched vortices constitute the sinews of turbulence. To substantiate this claim, they developed a theory for a vortex embedded in a generic, uniform, non-axisymmetric, irrotational strain field. The background straining flow is given by

(10.32) \begin{align} & \boldsymbol{U}=(\bar {\alpha } x,\bar {\beta } y,\bar {\gamma }z)=\boldsymbol{\nabla }\varPhi , \\[-9pt] \nonumber\end{align}

(10.33) \begin{align} & \bar {\alpha }+\bar {\beta }+\bar {\gamma }=0,\quad \bar {\alpha }\lt 0\lt \bar {\gamma },\quad \bar {\beta }\geq \bar {\alpha }, \\[9pt] \nonumber\end{align}

where $\bar {\gamma }$ is the positive stretching rate along $\boldsymbol{e}_{z}$ , and the velocity potential is $\varPhi = ({1}/{2})(\bar {\alpha }x^2+\bar {\beta }y^2+\bar {\gamma }z^2)$ . Note that $(\bar {\alpha },\bar {\beta },\bar {\gamma })$ in (10.33) are distinct from the spin components $(\alpha ,\beta ,\gamma )$ in (9.3b ). Different strain configurations are characterized by the dimensionless parameter $\lambda =(\bar {\alpha }-\bar {\beta })/(\bar {\alpha }+\bar {\beta })\geq {0}$ , which yields $\bar {\alpha }=-{1}/{2}\bar {\gamma }(1+\lambda )$ and $\bar {\beta }=-{1}/{2}\bar {\gamma }(1-\lambda )$ . When $\bar {\alpha }\leq \bar {\beta } \lt 0$ ( $0 \leq \lambda \lt 1$ ), the system has a single positive principal strain rate ( $\bar {\gamma }$ ), promoting the formation of vortex tubes aligned with the $z$ -axis. This case is known as axial strain. The case where $\bar {\alpha } = \bar {\beta } \lt 0$ ( $\lambda = 0$ ) corresponds to axisymmetric axial strain, while $\lambda =1$ and $\lambda \gt 0$ correspond to plane strain and biaxial strain, respectively.

In an incompressible fluid, the velocity field is described by a stream function $f(x,y)$ , yielding $u_{x}=\partial _{y}f$ , $u_{y}=-\partial _{x}f$ and vorticity $\omega =-\boldsymbol{\nabla} ^{2}f$ . For a vortex of finite circulation $\varGamma =\iint \omega (x,y)\,\text{d}x\,\text{d}y$ at high Reynolds number $R_{\varGamma }\equiv \varGamma /\nu \gg {1}$ ( $\nu$ is the kinematic viscosity), the small parameter $\epsilon \equiv {1}/R_{\varGamma }\ll {1}$ is defined. The characteristic radial length scale is $\delta =(\nu /\bar {\gamma })^{1/2}$ , and the maximum velocity in the vortex region is of the order $O(\varGamma /\delta )$ . After normalizing $(x,y)$ , $(\bar {\alpha },\bar {\beta })$ , and $f$ by $\delta$ , $\bar {\gamma }$ and $\varGamma$ , the dimensionless steady vorticity equation admits a series solution $f=f_{0}+\epsilon {f}_{1}+\epsilon ^{2}{f}_{2}+ {\cdots}$ . To first order in $\epsilon$ , the vorticity in polar coordinates $(r,\theta )$ is $\omega (r,\theta )=\omega _{0}(r)+\epsilon \lambda \varOmega (r)\sin (2\theta )$ (see figure 13 a), where $\varOmega (r)=(({1}/{4})r^2-f(r))\eta (r)$ is a well-behaved function. For small $r$ , the Taylor-expansion solutions of $f(r)$ and $\eta (r)$ are given in Moffatt et al. (Reference Moffatt, Kida and Ohkitani1994, (2.27) and (2.28)). The leading-order term $\omega _{0}(r)$ corresponds to the Burgers vortex solution (§ 10.3) (normalized by $\varGamma /\delta ^2=\bar {\gamma }/\epsilon$ ) where $\bar {\alpha }=\bar {\beta }=-\bar {\gamma }/2=-a/2\ (\lambda =0)$ .

Figure 13.

(a) Contour plot of the normalized vorticity field $\omega (r,\theta )$ for the MKO94 solution with $\lambda =0.725$ , where the green dashed lines mark the critical polar angles $\theta _c = \pi /4, 3\pi /4$ . Typical modes of the specific angular velocity $\boldsymbol{W}_{\kern-1.5pt D}(\boldsymbol{e})$ in (10.38), (10.39), and (10.40): (b) $\sin (2\phi )\sin \theta$ , (c) $\sin (2\phi )\cos \theta$ and (d) $(1-\cos (2\phi ))\sin (2\theta )$ .

10.5.2. Rotation of a material line element in a background strain field

We now analyse the rotation of a material line element ${\delta }\boldsymbol{r}=\delta {r}\boldsymbol{e}$ , retaining only the leading-order term from the MKO94 vortex solution. Here, the background strain field incorporates the non-axisymmetric effects $(\lambda \gt 0)$ arising from higher-order corrections. Let $\phi$ be the angle between $\boldsymbol{e}$ and the axial direction $\boldsymbol{e}_{z}$ , and $\theta$ the angle between the projection of $\boldsymbol{e}$ onto the $(x,y)$ -plane and $\boldsymbol{e}_{x}$ , so that $\boldsymbol{e}=\sin \phi (\cos \theta \boldsymbol{e}_{x}+\sin \theta \boldsymbol{e}_{y})+\cos \phi \boldsymbol{e}_{z}=\sin \phi \boldsymbol{e}_{r}+\cos \phi \boldsymbol{e}_{z}$ . From (10.33), the background VGT $\boldsymbol{A}_{\varPhi }$ and the strain-rate tensor $\boldsymbol{D}_{\varPhi }$ are given by

(10.34) \begin{eqnarray} \boldsymbol{A}_{\varPhi }=\boldsymbol{D}_{\varPhi }=\frac {1}{2}\bar {\gamma }\left [-(1+\lambda )\boldsymbol{e}_{x}\boldsymbol{e}_{x}-(1-\lambda )\boldsymbol{e}_{y}\boldsymbol{e}_{y}+2\boldsymbol{e}_{z}\boldsymbol{e}_{z}\right ]. \end{eqnarray}

It follows from (10.34) that the specific angular velocity due to $\boldsymbol{D}_{\varPhi }$ is

(10.35) \begin{eqnarray} \boldsymbol{W}_{D_{\varPhi }}(\boldsymbol{e}) &=& \boldsymbol{e}\times (\boldsymbol{e}\boldsymbol{\cdot }\boldsymbol{D}_{\varPhi })\nonumber \\ &=& \frac {1}{4}\bar {\gamma }\sin (2\phi )\left [(3-\lambda )\sin \theta \boldsymbol{e}_{x}-(3+\lambda )\cos \theta \boldsymbol{e}_{y}\right ]\nonumber \\ && +\, \frac {1}{4}\bar {\gamma }\lambda (1-\cos (2\phi ))\sin (2\theta )\boldsymbol{e}_{z}. \end{eqnarray}

Using (10.11) and (10.12), the disturbance VGT $\boldsymbol{A}_{\omega _{0}}$ and strain-rate tensor $\boldsymbol{D}_{\omega _{0}}$ arising from the leading-order solution are given by

(10.36) \begin{eqnarray} \boldsymbol{A}_{\omega _{0}}=(\psi ^{-}+\gamma ^{-})\boldsymbol{e}_{r}\boldsymbol{e}_{\theta }-\psi ^{-}\boldsymbol{e}_{\theta }\boldsymbol{e}_{r},\quad \boldsymbol{D}_{\omega _{0}}=\frac {1}{2}\gamma ^{-}(\boldsymbol{e}_{r}\boldsymbol{e}_{\theta }+\boldsymbol{e}_{\theta }\boldsymbol{e}_{r}). \end{eqnarray}

By (10.36), the specific angular velocity caused by $\boldsymbol{D}_{\omega _{0}}$ is evaluated as

(10.37) \begin{eqnarray} \boldsymbol{W}_{D_{\omega _{0}}}(\boldsymbol{e})&=&\boldsymbol{e}\times (\boldsymbol{e}\boldsymbol{\cdot }\boldsymbol{D}_{\omega _{0}})\nonumber \\ &=&-\frac {1}{4}\gamma ^{-}\sin (2\phi )\boldsymbol{e}_{r}+\frac {1}{4}\gamma ^{-}(1-\cos (2\phi ))\boldsymbol{e}_{z}. \end{eqnarray}

Defining $\gamma ^{*}\equiv \gamma ^{-}/\bar {\gamma }$ , the superposition of (10.35) and (10.37) yields

(10.38) \begin{align} \boldsymbol{W}_{\kern-1.5pt D}(\boldsymbol{e})& = \boldsymbol{e}\times (\boldsymbol{e}\boldsymbol{\cdot }\boldsymbol{D})\simeq \boldsymbol{W}_{{D}\perp }(\boldsymbol{e})+\boldsymbol{W}_{Dz}(\boldsymbol{e}), \\[-9pt] \nonumber\end{align}

(10.39) \begin{align} \boldsymbol{W}_{{D}\perp }(\boldsymbol{e}) & = \frac {1}{4}\bar {\gamma }\sin (2\phi )\left [(3-\lambda )\sin \theta -\gamma ^{*}\cos \theta \right ]\boldsymbol{e}_{x}\nonumber \\&\qquad -\frac {1}{4}\bar {\gamma }\sin (2\phi )\left [(3+\lambda )\cos \theta +\gamma ^{*}\sin \theta \right ]\boldsymbol{e}_{y}, \\[-9pt] \nonumber\end{align}

(10.40) \begin{align} & \boldsymbol{W}_{Dz}(\boldsymbol{e})=\frac {1}{4}\bar {\gamma }(1-\cos (2\phi ))(\gamma ^{*}+\lambda \sin (2\theta ))\boldsymbol{e}_{z}. \\[9pt] \nonumber\end{align}

Therefore, the total angular velocity $\boldsymbol{W}_{L}(\boldsymbol{e})$ is obtained as

(10.41) \begin{align} & \boldsymbol{W}_{L}(\boldsymbol{e})=\boldsymbol{W}_{{D}\perp }(\boldsymbol{e})+\boldsymbol{W}_{Lz}(\boldsymbol{e}),\quad W_{Lz}=W_{Dz}+\frac {1}{2}\omega _{z}, \\[-9pt] \nonumber\end{align}

(10.42) \begin{align} & W_{Lz}\simeq \frac {1}{2}\omega _{0}(r)+\frac {1}{4}\gamma ^{-}(1-\cos (2\phi ))+\frac {1}{4}\bar {\gamma }\lambda (1-\cos (2\phi ))\sin (2\theta ). \\[9pt] \nonumber\end{align}

Both $\boldsymbol{W}_{\kern-1.5pt D}(\boldsymbol{e})$ and $\boldsymbol{W}_{L}(\boldsymbol{e})$ exhibit non-trivial dependence on $(\phi ,\theta ,\lambda )$ . Typical modes of $\boldsymbol{W}_{\kern-1.5pt D}(\boldsymbol{e})$ in the $(\theta ,\phi )$ -space display periodic roll patterns, as shown in figure 13(b,d). We analyse two special orientations of $\boldsymbol{e}$ . When $\boldsymbol{e}=\boldsymbol{e}_{z}$ (i.e. $\phi =0$ , axial alignment), $\boldsymbol{W}_{\kern-1.5pt D}(\boldsymbol{e})=\boldsymbol{0}$ indicates the disappearance of specific rotation. When $\boldsymbol{e}\perp \boldsymbol{e}_{z}$ (i.e. $\phi =\pi /2$ , planar alignment), $\delta \boldsymbol{r}$ is confined to the $(x,y)$ -plane so that $\boldsymbol{W}_{{D}\perp }(\boldsymbol{e})=\boldsymbol{0}$ , and $\boldsymbol{W}_{\kern-1.5pt D}(\boldsymbol{e})=\boldsymbol{W}_{Dz}(\boldsymbol{e})={1}/{2}(\gamma ^{-}+\bar {\gamma }\lambda \sin (2\theta ))\boldsymbol{e}_{z}$ , demonstrating explicit dependence on $(\theta ,\lambda )$ . For a given $\lambda$ , $W_{Dz}$ reaches its maximum at $\theta =\pi /4$ (i.e. $y=x$ ) and the minimum at $\theta =3\pi /4$ (i.e. $y=-x$ ). Notably, these angles coincide with the principal axes of the isovorticity lines in figure 13(a). In addition, the viscous dissipation rate $2\nu \boldsymbol{D}\boldsymbol{:}\boldsymbol{D}$ , despite evolving in a rather complex manner, always exhibits two maxima near $\theta =3\pi /4$ (i.e. $y=-x$ ) in contour maps across varying $R_{\varGamma }$ and symmetrically displaced from the vortex centre (Kida & Ohkitani Reference Kida and Ohkitani1992; Moffatt et al. Reference Moffatt, Kida and Ohkitani1994).