An initial value method for eigenvalue problems using compound matrices

Tonal noise, the self-induced discrete frequency noise generated by aerofoils, is investigated. It is heard from an aerofoil placed in streams at low Mach number flows when inclined at a small angle to the stream. The tones are heard as a piercing whistle, commonly up to 30 dB above the background noise level. The work is motivated by the occurrence of tonal noise from rotors, fans and recently wind-turbines. Previous authors have attributed tonal noise to a feedback loop consisting of a coupling between laminar boundary-layer instability waves and sound waves propagating in the free stream. The frequency has been predicted by use of various methods based on this model.In this thesis a review of wind-tunnel results obtained by Dr. E.C. Nash at the University of Bristol is presented. Boundary-layer measurements show the presence of tonal noise is closely related to the existence of a region of separated flow close to the trailing edge of the aerofoil. Highly amplified boundary-layer instability waves were observed close to the trailing edge of the aerofoil at the frequency of the tone.A comprehensive analysis of the linear stability of the boundary-layer flow over the aerofoil is presented. The growth of boundary-layer instability waves over the aerofoil is calculated. The growth rates of the waves were obtained by solving the OrrSommerfeld problem at several stations on the aerofoil. The Falkner-Skan boundary layers were found to be a suitable form of velocity profiles to incorporate the adverse pressure gradients experienced by the flow over an aerofoil. The amplification of the instability waves is shown to be controlled almost entirely by the region of separated flow close to the trailing edge. The calculated frequency of the linear modes with maximum amplification over the aerofoil is found to be close to the observed frequency of the acoustic tone.A weakly nonlinear stability analysis was also performed and this appears to be a suitable description of the boundary-layer instability waves. The results indicate that the frequency of the tones may commonly be predicted to within 10% by using weakly nonlinear stability theory.The generation of sound by diffraction of the boundary-layer instability waves at the trailing edge of the aerofoil is also discussed as well as the proposed feedback models. A modified feedback model is proposed, being based on the experimental and theoretical results.i

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“…The method of compound matrices followed is described by Ng & Reid [36], and is also summarized in Drazin & Reid [13] pp. 311-317.…”

Section: Calculation Of the Eigenvalue Cmentioning

confidence: 99%

On the Generation of Discrete Frequency Tones by the Flow around an Aerofoil

McAlpine

Nash

Lowson

1999

Journal of Sound and Vibration

126

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“…It is easy to see that the reformulation of (32) as a first-order system would coincide with (20) but for the two entries a 21 and a 43 in the matrix A which transform into a 21 → a 21 + k 2 and a 43 → a 43 + k 2 . The inclusion of these extra terms proportional to k 2 does not affect the spectrum of A ∞ so the same construction of the Evans function as before is sufficient, save for the fact that now it also depends on the transverse wavenumber k as an extra parameter, i.e.…”

Section: Transverse Stabilitymentioning

confidence: 99%

“…If these two subspaces have a non-trivial intersection for some value of µ, then this is an eigenvalue of the problem (20).…”

Section: Longitudinal Stabilitymentioning

confidence: 99%

“…The second problem is that starting eigenvectors will not in general be analytic for all values of the spectral parameter. Both these problems can be circumvented by using exterior algebra and the compound matrix method introduced by Ng and Reid [20] for hydrodynamic stability problems. When using the compound matrix method, the solutions do not lie in a linear space, but rather on a Grassmanian manifold, and the numerical integration scheme should be chosen is such a way that it preserves this manifold.…”

Section: Introductionmentioning

confidence: 99%

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Front propagation in a phase field model with phase-dependent heat absorption

Blyuss

Ashwin

Wright

et al. 2006

Physica D: Nonlinear Phenomena

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We present a model for the spatio-temporal behaviour of films exposed to radiative heating, where the film can change reversibly between amorphous (glassy) and crystalline states. Such phase change materials are used extensively in read-write optical disk technology.In cases where the heat absorption of the crystal phase is less than that in the amorphous state we find that there is a bi-stability of the phases. We investigate the spatial behaviours that are a consequence of this property and use a phase field model for the spatio-temporal dynamics in which the phase variable is coupled to a suitable temperature field. It is shown that travelling wave solutions of the system are possible and, depending on the precise system parameters, these waves can take a range of forms and velocities. Some examples of possible dynamical behaviours are discussed and we show, in particular, that the waves may collide and annihilate. The longitudinal and transverse stability of the travelling waves are examined using an Evans function method which suggests that the fronts are stable structures.

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“…More recent calculations include those of Ng and Reid [19], using shooting with compound matrices, and of Dongarra, Straughan and Walker [4], who used a high precision Chebychev tau-algorithm coupled with quadruple precision arithmetic. For a review see the books [23] 1].…”

Section: Plane Poiseuille Flowmentioning

confidence: 99%

Eigenvalue enclosures and exclosures for non-self-adjoint problems in hydrodynamics

Brown¹,

Langer²,

Marletta³

et al. 2010

LMS J. Comput. Math.

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In this paper we present computer-assisted proofs of a number of results in theoretical fluid dynamics and in quantum mechanics. An algorithm based on interval arithmetic yields provably correct eigenvalue enclosures and exclosures for non-self-adjoint boundary eigenvalue problems, the eigenvalues of which are highly sensitive to perturbations. We apply the algorithm to: the Orr-Sommerfeld equation with Poiseuille profile to prove the existence of an eigenvalue in the classically unstable region for Reynolds number R = 5772.221818; the Orr-Sommerfeld equation with Couette profile to prove upper bounds for the imaginary parts of all eigenvalues for fixed R and wave number α; the problem of natural oscillations of an incompressible inviscid fluid in the neighbourhood of an elliptical flow to obtain information about the unstable part of the spectrum off the imaginary axis; Squire's problem from hydrodynamics; and resonances of one-dimensional Schrödinger operators.

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An initial value method for eigenvalue problems using compound matrices

Cited by 125 publications

References 10 publications

On the Generation of Discrete Frequency Tones by the Flow around an Aerofoil

On the Generation of Discrete Frequency Tones by the Flow around an Aerofoil

Front propagation in a phase field model with phase-dependent heat absorption

Eigenvalue enclosures and exclosures for non-self-adjoint problems in hydrodynamics

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