Trailing edge noise theory for rotating blades in uniform flow

This paper presents a new formulation for trailing edge noise radiation from rotating blades based on an analytical solution of the convective wave equation. It accounts for distributed loading and the effect of mean flow and spanwise wavenumber. A commonly used theory due to Schlinker and Amiet (1981) predicts trailing edge noise radiation from rotating blades. However, different versions of the theory exist; it is not known which version is the correct one and what the range of validity of the theory is. This paper addresses both questions by deriving Schlinker and Amiet's theory in a simple way and by comparing it to the new formulation, using model blade elements representative of a wind turbine, a cooling fan and an aircraft propeller. The correct form of Schlinker and Amiet's theory (1981) is identified. It is valid at high enough frequency, i.e. for a Helmholtz number relative to chord greater than one and a rotational frequency much smaller than the angular frequency of the noise sources.Comment: 28 pages, 10 figures, submitted to Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences (final revision

Fundamental Investigation into the directivity function of multi-mode sound fields from ducts

Multi-mode sound radiation from hard-walled semi-infinite ducts with uniform subsonic flow is investigated theoretically. An analytic expression, valid in the high frequency limit, is derived for the multi-mode directivity function in the forward arc of the duct for a general family of mode distribution function. The multi-mode directivity depends on the amplitude of each mode, and on the single mode directivity functions. The amplitude of each mode is expressed as a function of cut-off ratio for a uniform distribution of incoherent monopoles, a uniform distribution of incoherent axial dipoles and for equal power per mode. The single mode directivity functions are obtained analytically by applying a Lorentz Transformation to the zero flow solution. The analytic formula for the multi-mode directivity with flow is derived by assuming total transmission of power at the open-end of the duct. The high frequency formula is compared to exact numerical solutions from the Wiener Hopf technique and for a flanged duct. The agreement is shown to be excellent

On wavenumber spectra for sound within subsonic jets

This paper clarifies the nature of sound spectra within subsonic jets. Three problems, of increasing complexity, are presented. Firstly, a point source is placed in a two-dimensional plug flow and the sound field is obtained analytically. Secondly, a point source is embedded in a diverging axisymmetric jet and the sound field is obtained by solving the linearised Euler equations. Finally, an analysis of the acoustic waves propagating through a turbulent jet obtained by direct numerical simulation is presented. In each problem, the pressure or density field are analysed in the frequency-wavenumber domain. It is found that acoustic waves can be classified into three main frequency-dependent groups. A physical justification is provided for this classification. The main conclusion is that, at low Strouhal numbers, acoustic waves satisfy the d'Alembertian dispersion relation.Comment: 20 pages, 9 figure

An integral formulation for wave propagation on weakly non-uniform potential flows

An integral formulation for acoustic radiation in moving flows is presented. It is based on a potential formulation for acoustic radiation on weakly non-uniform subsonic mean flows. This work is motivated by the absence of suitable kernels for wave propagation on non-uniform flow. The integral solution is formulated using a Green's function obtained by combining the Taylor and Lorentz transformations. Although most conventional approaches based on either transform solve the Helmholtz problem in a transformed domain, the current Green's function and associated integral equation are derived in the physical space. A dimensional error analysis is developed to identify the limitations of the current formulation. Numerical applications are performed to assess the accuracy of the integral solution. It is tested as a means of extrapolating a numerical solution available on the outer boundary of a domain to the far field, and as a means of solving scattering problems by rigid surfaces in non-uniform flows. The results show that the error associated with the physical model deteriorates with increasing frequency and mean flow Mach number. However, the error is generated only in the domain where mean flow non-uniformities are significant and is constant in regions where the flow is uniform

Prediction of noise from serrated trailing edges

A new analytical model is developed for the prediction of noise from serrated trailing edges. The model generalizes Amiet’s trailing-edge noise theory to sawtooth trailing edges, resulting in a complicated partial differential equation. The equation is then solved by means of a Fourier expansion technique combined with an iterative procedure. The solution is validated through comparison with the finite element method for a variety of serrations at different Mach numbers. The results obtained using the new model predict noise reduction of up to 10 dB at 90$^{\circ }$ above the trailing edge, which is more realistic than predictions based on Howe’s model and also more consistent with experimental observations. A thorough analytical and numerical analysis of the physical mechanism is carried out and suggests that the noise reduction due to serration originates primarily from interference effects near the trailing edge. A closer inspection of the proposed mathematical model has led to the development of two criteria for the effectiveness of the trailing-edge serrations, consistent but more general than those proposed by Howe. While experimental investigations often focus on noise reduction at 90$^{\circ }$ above the trailing edge, the new analytical model shows that the destructive interference scattering effects due to the serrations cause significant noise reduction at large polar angles, near the leading edge. It has also been observed that serrations can significantly change the directivity characteristics of the aerofoil at high frequencies and even lead to noise increase at high Mach numbers.The first author (BL) wishes to gratefully acknowledge the financial support co-funded by the Cambridge Commonwealth European and International Trust and China Scholarship Council. The second author (MA) would like to acknowledge the financial support of the Royal Academy of Engineering. The third author (SS) wishes to gratefully acknowledge the support of the Royal Commission for the exhibition of 1851.This is the author accepted manuscript. The final version is available from Cambridge University Press via http://dx.doi.org/10.1017/jfm.2016.13

A coherence-matched linear source mechanism for subsonic jet noise

We investigate source mechanisms for subsonic jet noise using experimentally obtained datasets of high-Reynolds-number Mach 0.4 and 0.6 turbulent jets. The focus is on the axisymmetric mode which dominates downstream sound radiation for low polar angles and the frequency range at which peak noise occurs. A linearized Euler equation (LEE) solver with an inflow boundary condition is used to generate single-frequency hydrodynamic instability waves, and the resulting near-field fluctuations and far-field acoustics are compared with those from experiments and linear parabolized stability equation (LPSE) computations. It is found that the near-field velocity fluctuations closely agree with experiments and LPSE computations up to the end of the potential core, downstream of which deviations occur, but the LEE results match experiments better than the LPSE results. Both the near-field wavepackets and the sound field are observed directly from LEE computations, but the far-field sound pressure levels (SPLs) obtained are more than an order of magnitude lower than experimental values despite close statistical agreement of the near hydrodynamic field up to the potential core region. We explore the possibility that this discrepancy is due to the mismatch between the decay of two-point coherence with increasing distance in experimental flow fluctuations and the perfect coherence in linear models. To match the near-field coherence, experimentally obtained coherence profiles are imposed on the two-point cross-spectral density (CSD) at cylindrical and conical surfaces that enclose near-field structures generated with LEEs. The surface pressure is propagated to the far field using boundary value formulations based on the linear wave equation. Coherence matching yields far-field SPLs which show improved agreement with experimental results, indicating that coherence decay is the main missing component in linear models. The CSD on the enclosing surfaces reveals that the application of a decaying coherence profile spreads the hydrodynamic component of the linear wavepacket source on to acoustic wavenumbers, resulting in a more efficient acoustic source.This is the author accepted manuscript. The final version is available from Cambridge University Press via http://dx.doi.org/10.1017/jfm.2015.32

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.Comment: Submitted to Journal of the Acoustical Society of Americ

Geometric algebra and an acoustic space time for propagation in non-uniform flow

This study aims to make use of two concepts in the field of aeroacoustics; an analogy with relativity, and Geometric Algebra. The analogy with relativity has been investigated in physics and cosmology, but less has been done to use this work in the field of aeroacoustics. Despite being successfully applied to a variety of fields, Geometric Algebra has yet to be applied to acoustics. Our aim is to apply these concepts first to a simple problem in aeroacoustics, sound propagation in uniform flow, and the more general problem of acoustic propagation in non-uniform flows. By using Geometric Algebra we are able to provide a simple geometric interpretation to a transformation commonly used to solve for sound fields in uniform flow. We are then able to extend this concept to an acoustic spacetime applicable to irrotational, barotropic background flows. This geometrical framework is used to naturally derive the requirements that must be satisfied by the background flow in order for us to be able to solve for sound propagation in the non-uniform flow using the simple wave equation. We show that this is not possible in the most general situation, and provide an explicit expression that must be satisfied for the transformation to exist. We show that this requirement is automatically satisfied if the background flow is incompressible or uniform, and for both these cases derive an explicit transformation. In addition to a new physical interpretation for the transformation, we show that unlike previous investigations, our work is applicable to any frequency

An implementation of an aeroacoustic prediction model for broadband noise from a vertical axis wind turbine using a CFD informed methodology

This paper presents an enhanced method for predicting aerodynamically generated broadband noise produced by a Vertical Axis Wind Turbine (VAWT). The method improves on existing work for VAWT noise prediction and incorporates recently developed airfoil noise prediction models. Inflow-turbulence and airfoil self-noise mechanisms are both considered. Airfoil noise predictions are dependent on aerodynamic input data and time dependent Computational Fluid Dynamics (CFD) calculations are carried out to solve for the aerodynamic solution. Analytical ow methods are also benchmarked against the CFD informed noise prediction results to quantify errors in the former approach. Comparisons to experimental noise measurements for an existing turbine are encouraging. A parameter study is performed and shows the sensitivity of overall noise levels to changes in inflow velocity and inflow turbulence. Noise sources are characterised and the location and mechanism of the primary sources is determined, inflow-turbulence noise is seen to be the dominant source. The use of CFD calculations is seen to improve the accuracy of noise predictions when compared to the analytic ow solution as well as showing that, for inflow-turbulence noise sources, blade generated turbulence dominates the atmospheric inflow turbulence