On the Local Axisymmetry of a Vortex

Abstract

The local 3D cylindrical axisymmetry of a vortex is examined. Simple scalar quantities with a clear geometrical interpretation are introduced as suitable analytical tools for visualization of the local axisymmetry and skewness of the flow in the domain characterized by complex eigenvalues of the velocity-gradient tensor. Both complex and real eigenvectors are employed, including the real-valued dual-eigenvectors representing the orthogonal elongated directions in the swirl plane. The proposed quantities are applied to an impulsively started incompressible flow around an inclined flat plate.

Appendix

The determination of dual-eigenvectors is briefly summarized according to [16]. Assume a real matrix with a pair of complex conjugate eigenvalues as in (5). The complex eigenvectors are not unique since multiplying either of them with a complex number yields another set of eigenvectors. However, if $\left( {{\varvec{a}} + i{\varvec{b}}} \right)$ is an eigenvector corresponding to the eigenvalue $\left( {\lambda_{cr} + i\lambda_{ci} } \right)$ then $\left( {{\varvec{a}} - i{\varvec{b}}} \right)$ is an eigenvector corresponding to $\left( {\lambda_{cr} - i\lambda_{ci} } \right)$. The multiplication by a complex number reads.

$$\left( {{\cos \phi } + i{\sin \phi }} \right)\left( {{\varvec{a}} + i{\varvec{b}}} \right) = \left( {{\varvec{a}}{\cos \phi } - {\varvec{b}}{\sin \phi }} \right) + i\left( {{\varvec{a}}{\sin \phi } + {\varvec{b}}{\cos \phi }} \right) = \left( {{\varvec{a}}^{\user2{*}} + i{\varvec{b}}^{\user2{*}} } \right).$$

(8)

The need to define a unique pair of complex eigenvectors leads to the natural choice of a pair of real-valued vectors $\left( {{\varvec{a}}^{\user2{*}} ,{\varvec{b}}^{\user2{*}} } \right)$, the so-called dual-eigenvectors, that are mutually perpendicular. To satisfy the orthogonality condition ${\varvec{a}}^{\user2{*}} \cdot{\varvec{b}}^{\user2{*}} = 0$, it requires leading to a quadratic equation for ${\tan \phi }$ of the form

$$\left( {{\varvec{a}}{\cos \phi } - {\varvec{b}}{\sin \phi }} \right)\cdot\left( {{\varvec{a}}{\sin \phi } + {\varvec{b}}{\cos \phi }} \right) = 0,$$

(9)

$${\varvec{a}}\cdot{\varvec{b}}\left( {\tan \phi } \right)^{2} + \left( {\left| {\varvec{b}} \right|^{2} - \left| {\varvec{a}} \right|^{2} } \right){\tan \phi }\user2{ } - {\varvec{a}}\cdot{\varvec{b}} = 0.$$

(10)

This quadratic equation has usually two distinct solutions which are mutually perpendicular. To cope with the problem of two different ways to align them we choose the solution $\left( {{\varvec{a}}^{\user2{*}} ,{\varvec{b}}^{\user2{*}} } \right)$ for which $\left| {{\varvec{a}}^{*} } \right| \ge \left| {{\varvec{b}}^{*} } \right|$. Then ${\varvec{a}}^{\user2{*}}$ denotes the major dual-eigenvector and ${\varvec{b}}^{\user2{*}}$ denotes the minor dual-eigenvector.