Topology of helical fluid flow

Abstract

Considering a coordinate-free formulation of helical symmetry rather than more traditional definitions based on coordinates, we discuss basic properties of helical vector fields and compare results from the literature obtained with other approaches. In particular, we discuss the role of the stream function for the topology of the streamline pattern in incompressible flows. On this basis, we perform a comprehensive study of the topology of the flow field generated by a helical vortex filament in an ideal fluid. The classical expression for the stream function obtained by Hardin (Hardin, J. C. 1982 Phys. Fluids 25, 1949-1952 contains an infinite sum of modified Bessel functions. Using the approach by Okulov (Okulov, V. L. 1995 Russ. J. Eng. Thermophys. 5, 63-75) we obtain a closed-form approximation which is considerably easier to analyse. Critical points of the stream function can be found from the zeroes of a single real function of one variable, and we show that three different flow topologies can occur, depending on a single dimensionless parameter. By including the selfinduced velocity on the vortex filament by a localised induction approximation, the stream function is slightly modified and an extra parameter is introduced. In this setting two new flow topologies arise, but not more than two critical points occur for any combination of parameters.

Flow topology of helical vortices

“…As shown by Velasco Fuentes (2018) all calculations that do not include the tangential component give erroneous values of U and Ω, particularly for relatively small values of the vortex pitch: the ratio U t /U b is approximately 0.3 for τ ≈ 0.4 when α = 0.1, whereas it is approximately 0.1 for τ ≈ 0.2 when α = 10 −5 . Consequently, Mezić et al (1998) and Andersen & Brøns (2014) found regime boundaries that were shifted with respect to the ones shown in figure 7.…”

“…The flow topology of a single helical vortex was first studied by Mezić et al (1998) and Andersen & Brøns (2014). They, however, only took into account the binormal component of the vortex motion: Mezić et al (1998) computed U b using both local and non-local effects whereas Andersen & Brøns (2014) used only local effects.…”

“…The flow topology of a single helical vortex was first studied by Mezić et al (1998) and Andersen & Brøns (2014). They, however, only took into account the binormal component of the vortex motion: Mezić et al (1998) computed U b using both local and non-local effects whereas Andersen & Brøns (2014) used only local effects. As shown by Velasco Fuentes (2018) all calculations that do not include the tangential component give erroneous values of U and Ω, particularly for relatively small values of the vortex pitch: the ratio U t /U b is approximately 0.3 for τ ≈ 0.4 when α = 0.1, whereas it is approximately 0.1 for τ ≈ 0.2 when α = 10 −5 .…”

“…This result has been widely used to compute the velocity of the vortex itself (see, e.g., Ricca 1994;Boersma & Wood 1999;Okulov 2004;Okulov & Sørensen 2007;Velasco Fuentes 2018). Mezić et al (1998) and Andersen & Brøns (2014) studied the topology of Hardin's velocity field in a system translating and rotating with the vortex, taking into account the binormal component of the vortex motion only.…”

Μελέτη Ισορροπίας Και Ευστάθειας Ελικοειδώς Συμμετρικού Μαγνητισμένου Πλάσματος

“…Therefore, a helical scalar function is a function which has invariance along a helix, or equivalently: a scalar function f : R 3 → R is helically symmetric if and only if it is independent of ζ: ∂f /∂ζ = 0 ⇒ f = f (r, u). In this respect, a vector field is helically symmetric if and only if all its components in basis of the helical coordinates defined in Appendix B.1 are helically symmetric scalar functions [61]. The characteristic coordinate surfaces of the helical system are illustrated in figure 1.2.…”