Wakes as Topological Limit Sets

About 1990 it was established that the persistent scroll-like features of visible wakes can be captured by a special subset of surfaces of tangential discontinuities see Kiehn 1992 poitier.pdf. The patterns, or surface structures, so formed are recognized by their boundaries, which are interpreted as topological limit sets of instabilities induced by fluctuations.

 Spiral Wake past and Edge

 Frenet Curvature Analysis

### The Repere Mobile and Frenet Curvatures

The morphology of the flow pattern on the left still illudes the most powerful Cray computers. However, the pattern on the right was generated by a PC using an analysis based on the Cartan-Frenet-Serret concept of the moving basis frame or Repere Mobile. In the Frenet analysis the matrix of vectors representing the basis frame at a moving point on a space curve is restricted to be an element of the orthogonal group. Cartan ultimately extended this idea to basis frames which belong to the projective group.

The key idea is that if a basis frame [F] has a global inverse, [G], then the differential of any basis vector is closed with respect to the original basis set. That is, the differential of any basis vector leads to no surprises, and may be constructed in terms of linear combinations [C] of the original basis frame.
In short

d[F] = [F][C]
The matrix [C] = -d [G] [F] of differential coefficients used to express these linear combinations is defined as the matrix of connection 1-forms. The closure of the exterior derivatives of the matrix [C] leads to the notions of Curvature in Cartan's version of HREF="carfre50.htm">differential geometry.

 Kelvin - Helmholtz Instability

 Rayliegh - Taylor Instability

When the tangential discontinuity is dominated by either Frenet curvature or Frenet torsion, the space curve is described by the complex phase equation

By choosing various formats for the phase function, Q(s), the morphology of most wakes can be reproduced. The space curve is imbedded in a surface approximated by a Minimal Surface.

## Lanchester Tip "Vortices"

When the wake is torsion dominated the appearance replicates the features of the Lanchester theory of flight. The characteristic spirals of the phase function form the tip vortices, which are the basis of drag on a wing of finite aspect ratio. The spanwise spirals are mushrooms implying that the torsion is an even function of spanwise arclength. The spirals dominated by curvature start from the trailing edge and generate the Prandtl wake (not shown). The curvature is an odd function of the arclength.

## Lagrangian Wakes

Another aspect of wake phenomena is to be observed by comparing surfaces of equal phase. In the translatory flow around a cylinder with circulation, but without vorticity, the phase function of interest is generated by surfaces of equal "time". In the figure at the right, these equal time surfaces connect sequences of fluid elements with the same initial conditions. Note that the circulation component of flow produces a defect in the equal time phase pattern. Without circulation the defect disappears, and there is NO lift. The system is described by a Cartan 1-form, A, which is closed, dA= 0. For any smooth integration cycle around the cylinder, the circulation integral has a finite value.

This result is in apparent contradiction to Stokes theorem, which, as the value of vorticity is zero, would indicate that the circulation integral around a closed integration path must be zero as well. However, Stokes theorem requires that the integration chain be a boundary, and not just a cycle. For the flow field with a harmonic component superposed upon a translational component, a smooth curve around the origin is not a boundary in the topological sense. A true boundary curve, consisting of two equal time segments above and below the wing, and two flow line segments, does not cross the trailing edge defect line transversly.

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