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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. Keywords: Mathematical completeness; Numerical boundary conditions; Growing solutions, Stability, Transition; Upstream-traveling waves; Upstream diffusion; Unsteady boundary layers; Orr-Sommerfeld equation.

A spectral numerical method is used to study the two-dimensional unsteady flow over a flat plate in the presence of a plane sound wave propagating parallel to the flow. For s = omega x/u at infinity 0(1), no observable interaction or energy interchange between the sound wave and the Tollmien-Schlichting wave is present. In the region s or = 0(1), near the leading edge, an interaction occurs; the magnitude of this interaction is computed. A plane sound wave with dimensionless frequency omega nu/U-square at infinity = .000056 generates a Tollmien-Schlichting wave of the same frequency and an amplitude at the first neutral point .0001 times the sound-wave amplitude. Computations at a second, lower frequency result in a lower neutral point amplitude. The response of the boundary layer to large unsteady disturbances is discussed and compared with published behavior based on the boundary layer equations. Solutions of the Orr-Sommerfeld equation are examined and found to have a different character for s or 0(1) consistent with the more complete equations. (Author)

A fourth-order Orr-Sommerfeld eigenvalue solver for heated boundary layers has been implemented on the VAX-11/785 computer. The solver is based on the compound matrix method and the effects of heating are limited to mean viscosity variation across the boundary layer. Some test computations have been carried out on flat plates with uniform wall temperature. For small overheat, the results agree with those previously published using other methods, but for large overheat, there are some minor differences. The calculation procedure requires a large number of integration steps across the boundary layer to obtain accurate eigenvalues, but is relatively insensitive to other physical and numerical parameters. A few computations have been carried out using a full sixth order model, which takes into account fluctuations in all the fluid properties. It is concluded that the fourth order method can be incorporated into an axisymmetric body calculation procedure.