Boundaryvalue problem for twodimensional fluctuations in boundary layers
Abstract
The streamwise evolution of disturbances in a boundary layer is described as an asymptotic solution of the forced OrrSommerfeld equation. The velocity fluctuations and their derivations are specified along the yaxis. With these boundary conditions, the effects are included of vortical and irrotational free stream disturbances, fluctuations originating from leading edges, and discrete eigenmodes. A Fourier transform in time and a Laplace transform in the streamwise direction are used. Complementary and particular integrals are found and the inverse transforms are taken. Five families of 2D fluctuations can exist in a parallelflow, incompressible boundary layer. Three families have exponentially growing fluctuations, one of which is the Tollmien stability wave. Another is an exponentiallygrowth standing wave that oscillates in time and does not travel. This fluctuation appears as a mathematical pole in transform space, like the stability waves, but does not vanish far away from the boundary layer. A third growing fluctuation appears in Laplace space as a branch line. This continuous spectrum diffuses and travels upstream. The last two of these three growing fluctuations are excluded in our quarterplane problem that extends forever downstream. Besides the Tollmien wave that can grow or decay in the streamwise direction, the other discrete modes appearing as mathematical poles are damped. Two other decaying fluctuations appear.
 Publication:

Final Report
 Pub Date:
 July 1985
 Bibcode:
 1985aedc.rept.....T
 Keywords:

 Asymptotic Series;
 Boundary Layer Stability;
 Boundary Value Problems;
 Eigenvectors;
 Fourier Transformation;
 Incompressible Flow;
 Integrals;
 Inversions;
 Laplace Transformation;
 Leading Edges;
 OrrSommerfeld Equations;
 Standing Waves;
 TollmienSchlichting Waves;
 Two Dimensional Flow;
 Velocity;
 Boundary Conditions;
 Continuous Spectra;
 Diffusion Waves;
 Free Flow;
 Growth;
 Incompressibility;
 Perturbation;
 Stable Oscillations;
 Two Dimensional Boundary Layer;
 Variations;
 Vortices;
 Fluid Mechanics and Heat Transfer