Waves which travel upstream in boundary layers

Upstream propagation and diffusion of vorticity in a boundary layer is described by a numerical solution of the Orr-Sommerfeld equation. This traveling wave grows very rapidly in the downstream direction. The growth rate is approximately exp(+ R (sub delta)x) where R sub delta is the Reynolds number based on the characteristic boundary layer thickness, and x is the streamwise coordinate nondimensionalized against delta. Far from the boundary layer, the solution oscillates neutrally in the Y-direction. Analyses reveal high frequency wave which oscillates and decays in the y-direction approximately as exp(-i R(sub delta) y - omega Y) where omega is the frequency. This high frequency wave can survive into the freestream. Numerical solutions of the Orr-Sommerfeld equation with a Blasius layer are obtained by a series expansion of Chebyshev polynomials. Since the y-wavenumber of the oscillations increases with increasing Reynolds number, the calculations have been restricted to low Reynolds numbers. In the boundary-value problem, this solution appears as a branch line in Laplace space. It is one of the possible solutions in a mathematically complete description of the spatial evolution of fluctuations. This traveling wave represents one of the upstream influences of a boundary in a calculational domain. Another mechanism of upstream influence is the growing standing wave.