The local topology of stream- and vortex lines in turbulent flows

• Published in: Physics of fluids (1994) Vol. 26; no. 4
• Main Authors: Boschung, J., Schaefer, P., Peters, N., Meneveau, C.
• Format: Journal Article
• Language: English
• Published: Melville American Institute of Physics 01.04.2014
• Subjects: Computational fluid dynamics; Computer simulation; Convergence; Curvature; Divergence; Eigenvalues; Fields (mathematics); Isotropic turbulence; Magnetism; Probability density functions; Tensors; Topology; Tubes; Vortices
• ISSN: 1070-6631; 1089-7666
• DOI: 10.1063/1.4871097

Tangent lines to a given vector field, such as streamlines or vortex lines, define a local unit vector t that points everywhere in the line's direction. The local behavior of the lines is characterized by the eigenvalues of the tensor \documentclass[12pt]{minimal}\begin{document}$\bf T = {\bf \nabla } \cdot \bf t$\end{document}T=∇·t. In case of real eigenvalues, t can be interpreted as a normal vector to a surface element, whose shape is defined by the eigenvalues of T. These eigenvalues can be used to define the mean curvature −H and the Gaussian curvature K of the surface. The mean curvature −H describes the relative change of the area of the surface element along the field line and is a measure for the local relative convergence or divergence of the lines. Different values of (H, K) determine whether field lines converge or diverge (elliptic concave or elliptic convex surface element, stable/unstable nodes), converge in one principal direction and diverge in another (saddle) or spiral inwards or outwards (stable/unstable focus). In turbulent flows, a plethora of local field line topologies are expected to co-exist and it is of interest to find out whether certain topologies are more likely to occur than others. With this question in mind, the joint probability density function (JPDF) of H and K are evaluated for streamlines and vortex lines from several datasets obtained from direct numerical simulations of forced isotropic turbulence at four different Reynolds numbers. The JPDF for streamlines is asymmetric with a long tail towards negative H, implying that stream tubes tend to expand rapidly while shrinking more gently – a manifestation of negative skewness in turbulence. On the other hand, the JPDF for vortex lines is symmetrical with respect to H, indicating that the convergence and divergence of vortex lines is similar, different to streamlines.