Waves which travel upstream in boundary layers
Abstract
Upstream propagation and diffusion of vorticity in a boundary layer is described by a numerical solution of the OrrSommerfeld 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 Ydirection. Analyses reveal high frequency wave which oscillates and decays in the ydirection 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 OrrSommerfeld equation with a Blasius layer are obtained by a series expansion of Chebyshev polynomials. Since the ywavenumber of the oscillations increases with increasing Reynolds number, the calculations have been restricted to low Reynolds numbers. In the boundaryvalue 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.
 Publication:

Final Report
 Pub Date:
 July 1985
 Bibcode:
 1985urc..rept.....R
 Keywords:

 Boundary Layer Stability;
 Boundary Value Problems;
 Completeness;
 Diffusion Waves;
 Numerical Flow Visualization;
 Numerical Stability;
 Rates (Per Time);
 Reynolds Number;
 Stable Oscillations;
 Standing Waves;
 Traveling Waves;
 Unsteady Flow;
 Upstream;
 Vortices;
 Wave Propagation;
 Boundary Layers;
 Chebyshev Approximation;
 Coordinates;
 Growth;
 High Frequencies;
 Polynomials;
 Series Expansion;
 Solutions;
 Streams;
 Thickness;
 Fluid Mechanics and Heat Transfer