Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-28T06:49:18.436Z Has data issue: false hasContentIssue false

Linear and nonlinear interactions between a columnar vortex and external turbulence

Published online by Cambridge University Press:  10 January 2000

TAKESHI MIYAZAKI
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge, CB3 9EW, UK Permanent address: Department of Mechanical and Control Engineering, University of Electro-Communications, Chofu, Tokyo 182-8585, Japan.
JULIAN C. R. HUNT
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge, CB3 9EW, UK

Abstract

The structure of initially isotropic homogeneous turbulence interacting with a columnar vortex (with circulation Γ and radius σ), idealized both as a solid cylinder and a hollow core model is analysed using the inhomogeneous form of linear rapid distortion theory (RDT), for flows where the r.m.s. turbulence velocity u0 is small compared with Γ/σ. The turbulent eddies with scale Γ are distorted by the mean velocity gradient and also, over a distance Γ from the surface of the vortex, by their direct impingement onto it, whether it is solid or hollow. The distortion of the azimuthal component of turbulent vorticity by the differential rotation in the mean flow around the columnar vortex causes the mean-square radial velocity away from the cylinder to increase as (Γt/2πr2)2 (Γx/r)u20, when (r − σ) > Γx, but on the surface of the vortices ((r − σ) < Γx) where 〈u2r〉 is reduced, 〈u2z〉 increases to the same order, while the other components do not grow. Statistically, while the vorticity field remains asymmetric, the velocity field of small-scale eddies near the vortex core rapidly becomes axisymmetric, within a period of two or three revolutions of the columnar vortex. Calculation of the distortion of small-scale initially random velocity fields shows how the turbulent eddies, as they are wrapped around the columnar vortex, become like vortex rings, with similar properties to those computed by Melander & Hussain (1993) using a fully nonlinear direct numerical simulation. A mechanism is proposed for how interactions between the external turbulence and the columnar vortex can lead to non-axisymmetric vortex waves being excited on the vortex and damped fluctuations in its interior. If the columnar vortex is not significantly distorted by these linear effects, estimates are made of how nonlinear effects lead to the formation of axisymmetric turbulent vortices which move as result of their image vorticity (in addition to the self-induction velocity) at a velocity of order u02 parallel to the vortex. Even when the circulation (γ) of the turbulent vortices is a small fraction of Γ, they can excite self-destructive displacements through resonance on a time scale σ/u0.

Type
Research Article
Copyright
© 2000 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)