Spectra of the Orr–Sommerfeld equation

Murdock, John W.; Stewartson, K.

doi:10.1063/1.862035

Cited by 32 publications

(15 citation statements)

References 8 publications

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“…Recent calculations of the discrete eigenmodes of the Orr-Sommerfeld equation (Jordinson 1971;Mack 1976;Corner, Houston & Ross 1976;Murdock & Stewartson 1977) have indicated that, for a given Reynolds number and wavenumber (frequency), the Orr-Sommerfeld equation for Blasius flow has only a finite number of discrete temporal (spatial) eigenfunctions. Since a finite set of functions cannot be complete, these calculations raised the question of how to expand the stream function of an arbitrary disturbance in terms of the normal modes.…”

Section: Introductionmentioning

confidence: 99%

The continuous spectrum of the Orr-Sommerfeld equation. Part 2. Eigenfunction expansions

1981

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The expansion of an arbitrary two-dimensional solution of the linearized stream-function equation in terms of the discrete and continuum eigenfunctions of the Orr-Sommerfeld equation is discussed for flows in the half-space, y ε [0, ∞). A recent result of Salwen is used to derive a biorthogonality relation between the solution of the linearized equation for the stream function and the solutions of the adjoint problem.For the case of temporal stability, the orthogonality relation obtained is equivalent to that of Schensted for bounded flows. This relationship is used to carry out the formal solution of the initial-value problem for temporal stability. It is found that the vorticity of the disturbance at t = 0 is the proper initial condition for the temporal stability problem. Finally, it is shown that the set consisting of the discrete eigen-modes and continuum eigenfunctions is complete.For the spatial stability problem, it is shown that the continuous spectrum of the Orr-Sommerfeld equation contains four branches. The biorthogonality relation is used to derive the formal solution to the boundary-value problem of spatial stability. It is shown that the boundary-value problem of spatial stability requires the stream function and its first three partial derivatives with respect to x to be specified at x = 0 for all t. To be applicable to practical problems, this solution will require modification to eliminate disturbances originating at x = ∞ and travelling upstream to x = 0.For the special case of a constant base flow, the method is used to calculate the evolution in time of a particular initial disturbance.

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Section: Introductionmentioning

confidence: 99%

The continuous spectrum of the Orr-Sommerfeld equation. Part 2. Eigenfunction expansions

1981

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“…The exact nature of all these modes is unknown for the Ekman problem. Mack (197G), Murdock & Stewartson (1977) and Gustavsson (1979) have examined the case of the Orr-Sommerfeld equation for the Blasius layer and have presented evidence that the eigenvalues consist of a continuous spectrum as well as the discrete set of eigenvalues. The present perturbation problem has many features in common with the Orr-Sommerfeld equation and therefore there is reason to suspect that a similar distribution of eigenvalues should exist for the Ekman layer.…”

Section: Discussion Of the Short-time Problemmentioning

confidence: 99%

The evolution of disturbances in an Ekman boundary layer

Spooner

Criminale

1982

J. Fluid Mech.

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A linear model of a wave packet in a laminar Ekman boundary layer is proposed for tracing the development of an initially localized pulsed perturbation at the boundary. The model disturbance was built up from a linear combination of growing modes summed numerically over all wavenumbers and frequencies. The input spectrum was assumed to be flat so that there was no biasing at any wavenumber or frequency and the evolution was calculated on the basis of linear-stability theory. The wave packet generated by the summation of modes developed uniformly downstream of the disturbance source location and, depending upon the choice of the Reynolds number, vividly displayed results representative of the different kinds of instabilities that are present in the Ekman layer. Outputs in the form of perspective plots are given in order to explain the evolution of the packet into either a single wave patch or a sum of individual wave patches.

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“…In order for vorticity to homogenize inside closed streamlines, as it is assumed to do in the BBD solution where the Prandtl-Batchelor theorem was invoked, some viscosity must be present. Stewartson argued that the cat's eyes would decay before the vorticity had become fully homogenized, and indeed it is known that such a situation occurs for the Blasius boundary layer [37,38]. However, since structures resembling the BBD solutions were reported in simulations such as those of [39], it was widely assumed that the BBD solutions must be correct.…”

Section: Mathematical Problems In Engineeringmentioning

confidence: 99%