The transition from laminar to turbulent flow in boundary-layers occurs in three stages: onset of two-dimensional TS waves, onset of three-dimensional secondary disturbances of fundamental or subharmonic type, and onset of the turbulent regime. In free flight conditions, subharmonic disturbances are the most amplified. Recent modeling of the subharmonic disturbance as a parametric instability arising from the presence of a finite amplitude TS wave has given results in quantitative agreement with experiments conducted in a Blasius boundary-layer. The present work extends the analysis to the Falkner-Skan family of profiles, and develops a formulation for spatially growing disturbances to exactly match the experimental observations. Results show that subharmonic disturbances in Falkner-Skan flows behave similarly to those in a Blasius flow. The most noticeable effect of the pressure gradient is a decrease (favorable) or an increase (adverse) of the disturbance's growth rate. Due to the lack of experimental data, a comparison of subharmonic growth rates from theory and experiment is limited to the Blasius boundary-layer. A comparison of results from the spatial formulation with those previously obtained from a temporal formulation shows the difference to be small. A connection between disturbance growth in a separating boundary-layer profile and a free shear layer is presented. A modification of Caster's transformation from temporal to spatial growth rates for secondary disturbances is given. / M.S.

Potential theory is the name given to the broad field of analysis encompassing such topics as harmonic and subharmonic functions, the Dirichlet problem, Green’s functions, potentials and capacity. In this text, our main goal will be to gain a deeper understanding towards complex dynamics, the study of dynamical systems defined by the iteration of analytic functions, using the tools and techniques of potential theory. We will restrict ourselves to holomorphic polynomials in C. At first, we will discuss briefly about harmonic and subharmonic functions. In course, potential theory will repay its debt to complex analysis in the form of some beautiful applications regarding the Julia sets (defined in Chapter 8) of a certain family of polynomials, or a single one. We will be able to provide an explicit formula for computing the capacity of a Julia set, which in some sense, gives us a finer measurement of the set. In turn, this provides us with a sharp estimate for the diameter of the Julia set. Further if we pick any point w from the Julia set, then the inverse images q−n(w) span the whole Julia set. In fact, the point-mass measures with support at the discrete set consisting of roots of the polynomial, (qn-w) will eventually converge to the equilibrium measure of the Julia set, in the weak*-sense. This provides us with a very effective insight into the analytic structure of the set. Hausdorff dimension is one of the most effective notions of fractal dimension in use. With the help of potential theory and some ergodic theory, we can show that for a certain holomorphic family of polynomials varying over a simply connected domain D, one can gain nice control over how the Hausdorff dimensions of the respective Julia sets change with the parameter λ in D.