a Mathematical Analysis of the Effect of Freestream Turbulence on the Blasius Boundary Layer

This dissertation is an analysis of the boundary layer adjacent to an infinitely smooth, semi-infinite flat plate, situated in a free stream composed of a uniform flow upon which a field of random free stream turbulence, of small amplitude, is superimposed. An expansion of the solution in a power series, with respect to the amplitude of the free stream turbulence, results in a zeroth order set of equations; the Blasius problem, a first order set of equations, as well as problem sets of higher order. As the amplitude is taken to be small, only the zeroth and first order problems are retained. The governing partial differential operators for the first order problem are studied in some detail to determine their characteristic properties. They form a set of nonlinear, in the parameter, eigenvalue problems. New methods of solution for this type of problem are developed. The characteristics of the operators are then used to form a set of transfer functions which will mathematically represent the "receptivity" of the boundary layer. A multi-dimensional autocorrelation function, for the transverse velocity in the free stream, formed by close observation of what occurs in experiments of this type, is used to specify the statistical properties of the free stream turbulence. Other statistical properties, of the free stream turbulence, are obtained with the aid of the governing equations for the free stream. The semi-infinite domain, in the direction perpendicular to the plate, is mapped to a finite Chebyshev domain, and the set of variables for the first order solution is obtained by expansion in a finite Chebyshev series. Since the forcing function is a random field, the excited response in the boundary layer will also be random. This results in auto- and cross-correlations, of the excited response, which can be expressed as double Chebyshev series. The correlations for the excited velocity, pressure, and vorticity fields are obtained. Results for the root mean square velocity field, are compared, with favorable agreement, to natural transition experiments. These, as well as other, included results show that the free stream turbulence excites a patently three-dimensional velocity and vorticity field in the boundary layer.