arXiv subjects

Michael Carley

Publications and source records attributed to Michael Carley.

16 recordsLinked to original sources

arXiv2022

Shielding of rotor noise by plates and wings

A method of noise reduction proposed for the next generation of aircraft is to shield noise from the propulsion system, by positioning the noise source over a wing or another surface. In this paper, an approximate analysis is developed for the acoustic field far from a circular source placed near the edge of a semi-infinite plate, a model problem for shielding of noise by a wing and for scattering by a trailing edge. The approximation is developed for a source of small radius and is found to be accurate when compared to full numerical evaluation of the field.

physics.flu-dyn↗

arXiv2020

Sound from rotors in non-uniform flow

An analysis is presented for the evaluation of the acoustic field of a rotating source in a non-uniform potential flow. Other than the restriction to low flow Mach numbers, the method is exact and general. The variation in radiation properties with source position is handled by representation as a Fourier series in source angle, giving rise to an asymmetrically varying acoustic field evaluated by summation of the series. The method is used to develop an exact solution for the model problem of a rotor operating near a cylinder in cross-flow and sample calculations demonstrate the accuracy of the technique when compared to full numerical evaluation. The calculations show changes of greater than one decibel in the acoustic field due to flow non-uniformity at a flow Mach number of 0.15, a typical speed for propeller aircraft at take-off.

physics.flu-dyn↗

arXiv2019

Closed-form evaluation of potential integrals in the Boundary Element Method

A method is presented for the analytical evaluation of the singular and near-singular integrals arising in the Boundary Element Method solution of the Helmholtz equation. An error analysis is presented for the numerical evaluation of such integrals on a plane element, and used to develop a criterion for the selection of quadrature rules. The analytical approach is based on an optimized expansion of the Green's function for the problem, selected to limit the error to some required tolerance. Results are presented showing accuracy to tolerances comparable to machine precision.

math.NA↗

arXiv2013

Fourier transform of a Bessel function multiplied by a Gaussian

An analytical result is given for the exact evaluation of an integral which arises in the analysis of acoustic radiation from wave packet sources: $ I_{mn}(β,q) = \int_{-\infty}^{\infty} e^{-β^{2}x^{2}-i q x}x^{m+1/2}J_{n+1/2}(x) \,d x, $ where $m$ and $n$ are non-negative integers, and $J_{n+1/2}(\cdot)$ is a Bessel function of order $n+1/2$.

math-ph↗

arXiv2013

Quadrature for second-order triangles in the Boundary Element Method

A quadrature method for second-order, curved triangular elements in the Boundary Element Method (BEM) is presented, based on a polar coordinate transformation, combined with elementary geometric operations. The numerical performance of the method is presented using results from solution of the Laplace equation on a cat's eye geometry which show an error of order $P^{-1.6}$, where $P$ is the number of elements.

math.NA↗

arXiv2012

Potential integrals on triangles

The problem of evaluating potential integrals on planar triangular elements has been addressed using a polar coordinate decomposition. The resulting formulae are general, exact, easily implemented, and have only one special case, that of a field point lying in the plane of the element. Results are presented for the evaluation of the potential and its gradients, where the integrals must be treated as principal values or finite parts, for elements with constant and linearly varying source terms. These results are tested by application to a single triangular element to the evaluation of the potential gradient outside the unit cube. In both cases, the method is shown to be accurate and convergent.

math.NA↗

arXiv2011

Numerical Solution of the Modified Bessel Equation

A Green's function based solver for the modified Bessel equation has been developed with the primary motivation of solving the Poisson equation in cylindrical geometries. The method is implemented using a Discrete Hankel Transform and a Green's function based on the modified Bessel functions of the first and second kind. The computation of these Bessel functions has been implemented to avoid scaling problems due to their exponential and singular behavior, allowing the method to be used for large order problems, as would arise in solving the Poisson equation with a dense azimuthal grid. The method has been tested on monotonically decaying and oscillatory inputs, checking for errors due to interpolation and/or aliasing. The error has been found to reach machine precision and to have computational time linearly proportional to the number of nodes.

math.NA↗

arXiv2010

The radiating part of circular sources

An analysis is developed linking the form of the sound field from a circular source to the radial structure of the source, without recourse to far-field or other approximations. It is found that the information radiated into the field is limited, with the limit fixed by the wavenumber of source multiplied by the source radius (Helmholtz number). The acoustic field is found in terms of the elementary fields generated by a set of line sources whose form is given by Chebyshev polynomials of the second kind, and whose amplitude is found to be given by weighted integrals of the radial source term. The analysis is developed for tonal sources, such as rotors, and, for Helmholtz number less than two, for random disk sources. In this case, the analysis yields the cross-spectrum between two points in the acoustic field. The analysis is applied to the problems of tonal radiation, random source radiation as a model problem for jet noise, and to noise cancellation, as in active control of noise from rotors. It is found that the approach gives an accurate model for the radiation problem and explicitly identifies those parts of a source which radiate.

math-ph↗

arXiv2010

Radial cancellation in spinning sound fields

The radiating part of a circular acoustic source is determined on the basis of an exact analysis of the radiation properties of a source with angular dependence $\exp \J nθ$ and arbitrary radial dependence. It is found that the number of degrees of freedom in the radiated field is no greater than $k-n$, where $k$ is the wavenumber. The radiating part of the source at low frequency is the wavenumber. The radiating part of the source at low frequency is explicitly stated and used to analyze noise cancellation. The results are applied to the identification of sources in jet noise and an explanation for the low order structure of jet noise fields is proposed.

physics.flu-dyn↗

arXiv2010

Information in spinning sound fields

The information content of a spinning sound field is analyzed using a combination of exact and asymptotic results, in order to set limits on how accurately source identification can be carried out. Using a transformation of the circular source to an exactly equivalent set of line source modes, given by Chebyshev polynomials, it is found that the line source modes of order greater than the source wavenumber generate exponentially small fields. Asymptotic analysis shows that the remaining, lower order, modes radiate efficiently only into a region around the source plane, with this region shrinking as the mode order is increased. The results explain the ill-conditioning of source identification methods; the successful use of low order models in active noise control; and the low radiation efficiency of subsonic jets.

physics.flu-dyn↗

arXiv2010

Series expansion for the sound field of a ring source

An exact series expansion for the field radiated by a monopole ring source with angular variation in source strength is derived from a previously developed expression for the field from a finite disk. The derived series can be used throughout the field, via the use of a reciprocity relation, and can be readily integrated to find the field radiated by arbitrary circular sources of finite extent, and differentiated to find the field due to higher order sources such as dipoles and quadrupoles.

physics.flu-dyn↗

arXiv2009

BEM3D: a free adaptive fast multipole boundary element library

The design, implementation and analysis of a free library for boundary element calculations is presented. The library is free in the sense of the GNU General Public Licence and is intended to allow users to solve a wide range of problems using the boundary element method. The library incorporates a fast multipole method which is tailored to boundary elements of order higher than zero, taking account of the finite extent of the elements in the generation of the domain tree. The method is tested on a sphere and on a cube, to test its ability to handle sharp edges, and is found to be accurate, efficient and convergent.

physics.comp-ph↗

arXiv2009

Moving least squares via orthogonal polynomials

A method for moving least squares interpolation and differentiation is presented in the framework of orthogonal polynomials on discrete points. This yields a robust and efficient method which can avoid singularities and breakdowns in the moving least squares method caused by particular configurations of nodes in the system. The method is tested by applying it to the estimation of first and second derivatives of test functions on random point distributions in two and three dimensions and by examining in detail the evaluation of second derivatives on one selected configuration. The accuracy and convergence of the method are examined with respect to length scale (point separation) and the number of points used. The method is found to be robust, accurate and convergent.

math.NA↗

arXiv2008

Inversion of spinning sound fields

A method is presented for the reconstruction of rotating monopole source distributions using acoustic pressures measured on a sideline parallel to the source axis. The method requires no \textit{a priori} assumptions about the source other than that its strength at the frequency of interest vary sinusoidally in azimuth on the source disc so that the radiated acoustic field is composed of a single circumferential mode. When multiple azimuthal modes are present, the acoustic field can be decomposed into azimuthal modes and the method applied to each mode in sequence. The method proceeds in two stages, first finding an intermediate line source derived from the source distribution and then inverting this line source to find the radial variation of source strength. A far-field form of the radiation integrals is derived, showing that the far field pressure is a band-limited Fourier transform of the line source, establishing a limit on the quality of source reconstruction which can be achieved using far-field measurements. The method is applied to simulated data representing wind-tunnel testing of a ducted rotor system (tip Mach number~0.74) and to control of noise from an automotive cooling fan (tip Mach number~0.14), studies which have appeared in the literature of source identification.

physics.flu-dyn↗

arXiv2007

Evaluation of Biot-Savart integrals on tetrahedral meshes

An arithmetically simple method has been developed for the evaluation of Biot--Savart integrals on tetrahedralized distributions of vorticity. In place of the usual approach of analytical formulae for the velocity induced by a linear distribution of vorticity on a tetrahedron, the integration is performed using Gaussian quadrature and a ray tracing technique from computer graphics. This eliminates completely the need for the evaluation of square roots, logarithms and arc tangents, and almost completely eliminates the requirement for trigonometric functions, with no operation more costly than a division required during the main calculation loop. An assessment of the algorithm's performance is presented, demonstrating its accuracy, second order convergence and near-linear speedup on parallel systems.

math.NA↗

arXiv2007

Numerical quadratures for near-singular and near-hypersingular integrals in boundary element methods

A method of deriving quadrature rules has been developed which gives nodes and weights for a Gaussian-type rule which integrates functions of the form: f(x,y,t) = a(x,y,t)/((x-t)^2+y^2) + b(x,y,t)/([(x-t)^2+y^2]^{1/2}) + c(x,y,t)\log[(x-t)^2+y^2]^{1/2} + d(x,y,t), without having to explicitly analyze the singularities of $f(x,y,t)$ or separate it into its components. The method extends previous work on a similar technique for the evaluation of Cauchy principal value or Hadamard finite part integrals, in the case when $y\equiv0$. The method is tested by evaluating standard reference integrals and its error is found to be comparable to machine precision in the best case.

math.NA↗