Instability Patterns, Wakes and Topological Limit Sets

Many hydrodynamic instability patterns can be put into correspondence with a subset of characteristic surfaces of tangential discontinuities. These topological limits sets to systems of hyperbolic PDE's are locally unstable, but a certain subset associated with minimal surfaces are globally stabilized, persistent and non-dissipative. Sections of these surfaces are the spiral scrolls so often observed in hydrodynamic wakes. This method of wake production does not depend explicitly upon viscosity. Introduction From the topological point of view it is remarkable how often flow instabilities and wakes take on one or another of two basic scroll patterns. The first scroll pattern is epitomized by the Kelvin-Helmholtz instability (Figure 1a) and the second scroll pattern is epitomized by the Raleigh-Taylor instability (Figure 1b). The repeated occurrence of these two patterns (one similar to a Cornu Spiral and the other similar to a Mushroom Spiral) often deformed but still recognizable and persistent, even in dissipative media, suggests that a basic simple underlying topological principle is responsible for their creation. The mushroom pattern is of particular interest to this author, who long ago was fascinated by the long lived ionized ring that persists in the mushroom cloud of an atomic explosion. Although the mushroom pattern appears in Figure 1b. The pattern of the Rayleigh-Taylor Instability. Figure 1a. The pattern of the Kelvin-Helmholtz instability. many diverse physical systems (in the Frank-Reed source of crystal growth, in the scroll patterns generated in excitable systems, in the generation of the wake behind an aircraft, ...), no simple functional description of the mushroom pattern is given in the literature known to this author. Classical geometric analysis applied to the equations of hydrodynamics has failed to give a satisfactory description of these persistent structures, so often observed in many different situations. Viscosity vs. Compressibility Conventional analysis of wake production and boundary layer formation subsumes that the dominant physical effect is governed by viscous creation (and destruction) of vorticity. To quote Batchelor, "The term wake is commonly applied to the whole region of non-zero vorticity on the downstream side of a body in an otherwise uniform stream of fluid." The conventional perspective presumes that the wake is approximated as a tangential discontinuity, defined as a vortex sheet. The viscous creation of the vortex sheet is not precisely defined, but once the sheet is formed, its evolution is presumed to be governed by an integral form of the Biot-Savart law, known as the Rott-Birchoff equation. These assumptions have been shown to lead to the production of discontinuities in finite time. Asymptotic spiral type solutions in the vicinity of the singularity have been investigated both analytically and numerically [Kaden, Rott, Kambe, Pullin, Cayflisch, Kransky]. The viscous process essentially is one of diffusion, and is governed by a nonhyperbolic system of PDE's. It is most certainly applicable to the decay of wake phenomena. However, this point of view, which has its domain of applicability in the infinitely far field behind a body, is often at odds with experiments of the near field, which indicate that wake features persist with sharp definition for long periods of time, over many characteristic lengths, without diffusive blurring, and at high Reynolds numbers where viscosity effects are minimal. To quote Browand [Browand, 1986] "At the present time, there is no theoretical framework to describe the structural features of high Reynolds number shear flows." In this article an alternate approach to the creation of wakes is proposed. The basic physical mechanism for wake production is assumed to be associated with the fact that all real fluids have a finite speed of sound, hence a finite compressibility. Therefore, there can exist domains in every (perhaps slightly) compressible fluid where the system of PDE's describing the fluid evolution is hyperbolic, and not diffusively parabolic or elliptic. Hyperbolic domains have the feature that they can be associated with topological limit sets upon which the solutions to the PDE's are not unique. Therefore topological discontinuities are admissible in such dynamical systems. Such discontinuities are of two types: shock waves, where C0 differentiability is lost for the component of velocity normal to the discontinuity surface (but the tangential components of velocity are continuous) and tangential discontinuities, where C0 differentiability is lost for the tangential components of the velocity, but the normal component is continuous. It is this set of limit points associated with the tangential discontinuities that can be put into correspondence with the two basic instability patterns described above. Recall that propagating discontinuities are associated with singular solutions to wave equations. In fact, the very definition of a wave, according to Hadamard, is a propagating discontinuity. To recap, the point of view taken in this article is that the creation of a wake is to be associated with a discontinuous process in a hyperbolic domain, while the decay of a wake is associated with a diffusive process in an elliptic domain. This alternate topological approach to the problem of wake creation is independent from viscosity, and gains credence from the fact that not only is a mechanism offered for the creation of the tangential discontinuity, but also closed form solutions to the vector fields that describe the aforementioned instability patterns can be obtained. An equivalent result is not known in the literature familiar to this author. Not only does the topological point of view give new insight into how wakes may be generated, but also points out how such wakes possibly may be controlled. As described below, the resultant wake creation phenomena is closely related to the problems of diffraction and interference in electromagnetism. In fact, it may be said that from the topological point of view the near field hydrodynamic wake is a diffraction pattern caused by the physical obstruction. Hence, it is plausible that phase interference mechanisms may be used to control wakes. At high Reynolds numbers the effect of viscous diffusion is to smear out or thicken the tangential discontinuity.