Fluid Dynamics: “Vorticity” 1961 Ascher Shapiro, MIT; PSSC; Vortex Physics
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“Ascher H. Shapiro presents experimental demonstrations of phenomena relating to vorticity, to circulation, and to the theorems of Crocco, Kelvin, and Helmholtz.”
Public domain film, slightly cropped to remove uneven edges, with the aspect ratio corrected, and one-pass brightness-contrast-color correction & mild video noise reduction applied.
The soundtrack was also processed with volume normalization, noise reduction, clipping reduction, and/or equalization (the resulting sound, though not perfect, is far less noisy than the original).
Wikipedia license:
In continuum mechanics, the vorticity is a pseudovector field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along with the flow.
Conceptually, vorticity could be determined by marking the part of continuum in a small neighborhood of the point in question, and watching their relative displacements as they move along the flow. The vorticity vector would be twice the mean angular velocity vector of those particles relative to their center of mass, oriented according to the right-hand rule. This quantity must not be confused with the angular velocity of the particles relative to some other point.
More precisely, the vorticity is a pseudovector field ω→, defined as the curl (rotational) of the flow velocity u→ vector. The definition can be expressed by the vector analysis formula:
ω → ≡ ∇ × u → , {displaystyle {vec {omega }}equiv nabla times {vec {u}},,} {vec {omega }}equiv nabla times {vec {u}},,
where ∇ is the del operator. The vorticity of a two-dimensional flow is always perpendicular to the plane of the flow, and therefore can be considered a scalar field.
The vorticity is related to the flow’s circulation (line integral of the velocity) along a closed path by the (classical) Stokes’ theorem. Namely, for any infinitesimal surface element C with normal direction n→ and area dA, the circulation dΓ along the perimeter of C is the dot product ω→ ∙ (dA n→) where ω→ is the vorticity at the center of C.
Many phenomena, such as the blowing out of a candle by a puff of air, are more readily explained in terms of vorticity rather than the basic concepts of pressure and velocity. This applies, in particular, to the formation and motion of vortex rings…
In a mass of continuum that is rotating like a rigid body, the vorticity is twice the angular velocity vector of that rotation. This is the case, for example, of water in a tank that has been spinning for a while around its vertical axis, at a constant rate.
The vorticity may be nonzero even when all particles are flowing along straight and parallel pathlines, if there is shear (that is, if the flow speed varies across streamlines). For example, in the laminar flow within a pipe with constant cross section, all particles travel parallel to the axis of the pipe; but faster near that axis, and practically stationary next to the walls. The vorticity will be zero on the axis, and maximum near the walls, where the shear is largest.
Conversely, a flow may have zero vorticity even though its particles travel along curved trajectories. An example is the ideal irrotational vortex, where most particles rotate about some straight axis, with speed inversely proportional to their distances to that axis. A small parcel of continuum that does not straddle the axis will be rotated in one sense but sheared in the opposite sense, in such a way that their mean angular velocity about their center of mass is zero…
