Axially-symmetric eddies embedded in a rotational stream

Abstract

The structure of eddies embedded in a swirling stream of an incompressible, inviscid fluid is examined. Laminar motion is assumed, as is axial symmetry of the flow field.It is supposed that the eddies are formed either (i) as wakes behind solid obstacles placed in the stream, or (ii) as the products of vortex breakdown occurrences, in which case the eddies are completely surrounded by fluid of the outer stream. In either event, it is assumed that the presence of an eddy only slightly disturbs the motion of the outer stream, i.e. that the eddies are slender. In particular, this requires that r₀Ω_∞/W_∞ [Lt ]1, where r₀ is the maximum radial extent of the eddy, and Ω_∞ and W_∞ are defined below.The analysis is carried to completion when the undisturbed outer stream is a wheel flow. Principal results derived are summarized below. (1) Eddy lengths are related directly and simply to the internal vorticity, providing that there is swirl in the outer flow. (2) A corollary is that very slender eddies have a maximum possible length. In the case (ii) above, this length is 2πW_∞/Ω_∞, where W_∞ is the axial velocity and Ω_∞ the angular velocity of the undisturbed outer flow. (3) The shapes of the free streamlines bounding the eddy are calculated. A feature is the cusped nature of an eddy free end. (4) The motion inside the eddy and the disturbance to the outer stream are calculated.Irrotational base flows are exceptional. Results (1) and (2) above cannot be found for a potential outer flow, but (3) and (4) remain.