A survey of exact solutions of inviscid field equations in the theory of shear flow instabilty

This paper presents a survey of undisturbed flows that take one or another of the field equations of inviscid shear flow instability theory (e.g. theRayleigh equation,Taylor-Goldstein-Haurwitz equation or theKuo equation) to a differential equation satisfied by aknown transcendental function forarbitrary complex values of the parameters. Some mean velocity profiles having this feature are well known. Thus, piecewise linear mean velocity profiles take theRayleigh equations to a constant-coefficient differential equation and the exponential mean velocity profile takes theRayleigh equation to theGauss hypergeometric equation. Less well known is the fact that a variety of mean velocity profiles take theRayleigh equation to a differential equation due toKarl Heun. These profiles include: (i) the sinusoidal profile; (ii) the hyperbolic tangent profile (an example pointed out byMiles (1963)); (iii) the profile in the form of the square of a hyperbolic secant (theBickley jet); and, (iv) a skewed velocity profile in which each component has the form of a quadratic function of the variable exp(–z/l) (in whichz is the cross-stream coordinate andl is a length scale). In all of these cases, one or another author has previously identified aregular neutral mode solution of theRayleigh equation and has expressed that solution in the form of elementary functions. Such regular neutral modes apparently represent cases in which the solution ofHeun's equation (which is normally an infinite series) truncates to a single term. The survey concludes by noting that the parabolic mean velocity profile takes theRayleigh equation to thedifferential equation of the spheroidal wave function.Dedicated to Mårten T. Landahl on the occasion of his sixty-fifth birthday by a former apprentice as a token of his respect and gratitude.