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The Nonparallel Evolution of Nonlinear Short Waves in Buoyant Boundary Layers
Author(s) -
Denier James P.,
Bassom Andrew P.
Publication year - 2003
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/1467-9590.00234
Subject(s) - boundary layer , mechanics , wavelength , ordinary differential equation , boundary (topology) , turbulence , physics , overshoot (microwave communication) , nonlinear system , boundary value problem , blasius boundary layer , classical mechanics , flow (mathematics) , work (physics) , boundary layer thickness , mode (computer interface) , mathematical analysis , differential equation , mathematics , optics , quantum mechanics , electrical engineering , engineering , computer science , thermodynamics , operating system
Buoyant boundary‐layer flows, typified by the flow over a heated flat plate, have the curious property that they can exhibit regions of “overshoot” in which the streamwise velocity exceeds its free‐stream value. A consequence of this is the streamwise velocity develops a local maximum and is inflectional in nature. It is therefore inviscidly unstable, and the fastest growing wave mode is known to be one whose wavelength is short compared to the boundary‐layer thickness. In this work we consider the nonparallel evolution of these short waves and show that they can be described in terms of the solution of a system of ordinary differential equations. Numerical and asymptotic studies enable us to explain the ultimate fate of the wave and show, depending on a key parameter which is a function of the underlying boundary layer, that two possibilities can arise. Nonparallelism may be sufficiently stabilizing so as to extinguish the linearly unstable waves or, in other cases, the mode may intensify but concentrate itself in a very thin zone surrounding the maximum in the streamwise velocity. These findings enable us to give some indication of the part these modes play in the transition to turbulence in buoyant boundary layers.