Temporal Eigenvalues Research Articles

Spectra of the Orr–Sommerfeld equation

It is shown that the continuous spectra of both the temporal and spatial eigenvalues of the Orr–Sommerfeld equation for a Blasius flow can be re-interpreted as discrete spectra of the heat-conduction and diffraction types, respectively. Most of the general properties of the discrete spectrum of the Orr–Sommerfeld equation discovered by Mack can be understood on the basis of a simple model equation.

A numerical study of the temporal eigenvalue spectrum of the Blasius boundary layer

A numerical study is made of the temporal eigenvalue spectrum of the Orr-Sommerfeld equation for the Blasius boundary layer. Unlike channel flows, there is no mathematical proof that this flow has an infinite spectrum of discrete eigenvalues. The Orr-Sommerfeld equation is integrated numerically, and the eigenvalues located by tracing out the contour lines in the complex wave velocity (c = cr + ici) plane on which the real and imaginary parts of the secular determinant are zero. This method gives only a finite and small number of discrete eigenvalues for a wide range of Reynolds numbers and wavenumbers. The spectrum of plane Poiseuille flow is used as a guide to study the spectrum of an artificial two wall flow which consists of two Blasius boundary layers. As the upper boundary of this flow moves to infinity, it is found that the portion of the spectrum with an infinite number of eigenvalues moves towards cr = 1 and the spacing between eigenvalues goes to zero. It is concluded, on the basis of this result and the contour method, that the original few eigenvalues found are the only discrete eigenvalues that exist for Blasius flow over a wide portion of the c plane for cr < 1 and cr > 1. It is suggested that the discrete spectrum is supplemented by a continuous spectrum which lies along the cr = 1 axis for ci < −α/R.

Classical Noise. VI. Noise in Self-Sustained Oscillators near Threshold

Because of the relative narrowness of the threshold region, a general model for spectrally pure self-sustained oscillators (both classical and quantum, including gas lasers) can be reduced, in the threshold region, to a rotating-wave Van der Pol (RWVP) oscillator. By a scaling transformation, we reduce to the normalized RWVP oscillator which contains only one dimensionless parameter, a net pump rate $p$, which determines the operating point. The power spectra of phase and amplitude fluctuations and of amplitude (intensity) fluctuations in the normalized RWVP oscillator near threshold are calculated "exactly" by numerical Fokker-Planck methods. Using the appropriate scaling transformation, our results yield these power spectra for any oscillator of this general type. In particular, for gas lasers our results yield the one-sided Fourier transform of $〈{b}^{\ifmmode\dagger\else\textdagger\fi{}}(t)b(0)〉$ (the spectrum) and of $〈{b}^{\ifmmode\dagger\else\textdagger\fi{}}(0){b}^{\ifmmode\dagger\else\textdagger\fi{}}(t)b(t)b(0)〉\ensuremath{-}{〈{b}^{\ifmmode\dagger\else\textdagger\fi{}}b〉}^{2}$ (the intensity spectrum), where ${b}^{\ifmmode\dagger\else\textdagger\fi{}}$ and $b$ are the creation and destruction operators for the radiation field. Except for intensity fluctuations just above threshold, the power spectra were found to be nearly Lorentzian, with half-widths at half power approximately equal to the lowest nonzero temporal eigenvalue of the Fokker-Planck equation. For intensity fluctuations above threshold, the second-lowest nonzero eigenvalue was found to yield a significant contribution to the power spectrum as well as the lowest nonzero eigenvalue. These two eigenvalues become nearly degenerate for operation well above threshold. Thus the intensity fluctuation spectrum is Lorentzian below and well above threshold, but more complex in the threshold region.