The origin and subsequent development in space of Tollmien-Schlichting waves in a boundary layer

Two kinds of small-parameter expansions are used to derive a series of ordinary differential equations governing spatially growing or decaying disturbances of the travelling-wave type in non-parallel laminar boundary layers. Eigensolutions of the lowest-order equation are shown to tend to resemble eigensolutions of the unsteady boundary-layer equation as the Reynolds number Re decreases, and to the Tollmien-Schlichting wave solutions of the Orr-Sommerfeld equation as Re increases. Examination of the next-order approximation reveals that the unsteady boundary-layer equation has eigensolutions only if Re exceeds a critical value depending on the frequency of disturbances, while it remains a partial differential equation for lower Reynolds numbers. Thus the Tollmien-Schlichting wave may be considered to originate at the above critical value of Re. The subsequent development of the wave with fixed frequency can be followed by integrating the spatial growth rate of the eigensolutions with respect to the downstream distance. The amplitude is at first damped but grows even higher than its original value farther downstream, provided the frequency is less than a certain value.