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An important aspect of the Orr Sommerfeld problem, which governs the linear stability of parallel shear flows, is concerned with the study of the temporal and spatial spectra for large but finite values of the Reynolds number R . By using only outer (WKB) approximations which are valid in the “complete” sense, we are able to derive approximations to the eigenvalue relation for channel flows, pipe flow, and boundary layer flows which are all remarkably simple and which have a relative error of order ( αR) -1/2 . In this paper, we discuss briefly the basic ideas involved in the derivation of these approximations for boundary layer flows. We then present some results to illustrate the effectiveness of these new approximations. For example, we are even able to compute eigenvalues which lie arbitrarily close to the continuous spectra where all previous numerical treatments have failed.
An important aspect of the Orr Sommerfeld problem, which governs the linear stability of parallel shear flows, is concerned with the study of the temporal and spatial spectra for large but finite values of the Reynolds number R. By using only outer (WKB) approximations which are valid in the “complete ” sense, we are able to derive approximations to the eigenvalue relation for channel flows, pipe flow, and boundary layer flows which are all remarkably simple and which have a relative error of order (αR) -1 / 2. In this paper, we discuss briefly the basic ideas involved in the derivation of these approximations for boundary layer flows. We then present some results to illustrate the effectiveness of these new approximations. For example, we are even able to compute eigenvalues which lie arbitrarily close to the continuous spectra where all previous numerical treatments have failed.