P.E. Doak

Fluid dynamics of a flow excited resonance, Part II: Flow acoustic interaction

Abstract This is the second of two companion papers in which the physics and detailed fluid dynamics of a flow excited resonance are examined. The approach is rather different from those previously used, in which stability theory has been applied to small wavelike disturbances in a linearly unstable shear layer, with an equivalent source driving the sound field which provides the feedback. In the approach used here, the physics of the flow acoustic interaction is explained in terms of the detailed momentum and energy exchanges occurring inside the fluid . Gross properties of the flow and resonance are described in terms of the parameters necessary to determine the behaviour of the feedback system. In this second paper it is shown that two relatively distinct momentum balances can be considered in the resonator neck region. One can be identified with the vortically induced pressure and velocity fluctuations and the other with the reciprocating potential flow. The fluctuating Coriolis force caused by the interaction of the potential and vortical flows is shown to be the only term in the linearized momentum equation which is not directly balanced by a fluctuating pressure gradient. This force provides the mechanism for the exchange of the mean energies associated with the mean and fluctuating momenta, respectively. A source and sink of energy are identified in which mean energy associated with fluctuating momentum is extracted from and returned to the mean flow, respectively. The imbalance between the source and sink is responsible for both the radiated acoustic power and the power carried away by the vortices as they convect downstream. This radiated acoustic power and vortically convected power, and the source and sink powers, are all of the same order of magnitude. With the vortex shedding and reciprocating potential flow “phase locked” the amplitude of the steady state oscillations is determined by the condition that the net power produced in the resonator neck (the source power less the sink power) is equal to the sum of the radiated acoustic power and that carried by the vortices.

Fluid dynamics of a flow excited resonance, part I: Experiment

This is the first of two companion papers concerned with the physics and detailed fluid dynamics of a flow excited resonance. The phenomenon has been examined by using a rather different approach from others to date, in which usually stability theory has been applied to small wave-like disturbances in an unstable shear layer with an equivalent source to describe the radiation of sound providing the feedback. The physics of the flow acoustic interaction is explained in terms of the detailed momentum and energy exchanges occurring in the fluid itself. Gross properties of the flow and resonance are described in terms of the parameters necessary to determine the behaviour of the self-oscillatory system. In this first paper a full experimental investigation of a flow excited Helmholtz resonator is described, in which the detailed fluid dynamical and acoustic data necessary to develop a mathematical model for the flow was obtained, and a new theory of the interaction process is presented in the companion paper (Part II). The investigation described involved the use of a two-component Laser-Doppler Velocimeter (L.D.V.) and probe microphones to specify completely the velocity and pressure fields and a flow visualization to give qualitative information of the vortex shedding process. The overall aim of the work described in the two papers was to increase fundamental understanding of flow/acoustic interactions.

A subjective rating scale for timbre

Abstract The factors governing timbre are discussed and a subjective rating scale for their quantitative assessment devised. This scale was used with success to differentiate between a limited number of sounds of varying harmonic content. By using this scale a quantitative measure of the suitability of selected words for timbre was obtained.

Analysis of internally generated sound in continuous materials: 2. A critical review of the conceptual adequacy and physical scope of existing theories of aerodynamic noise, with special reference to supersonic jet noise

Abstract Existing theories of aerodynamic noise generation are critically reviewed with special emphasis on conceptual adequacy and physical scope, and with special reference to supersonic jet noise. In this review the basic work of Stokes, Kirchoff and Rayleigh on fluctuating motions in fluids is recalled and developed to provide a firm basis for the critique. The advantages and disadvantages of acoustic analogy theories such as Lighthills are thoroughly discussed. A contribution is made towards removing published criticisms by Lighthill of Ribners “isotropic source tensor” theory. New developments, such as those by Crow, Lilley and Doak, are emphasized.

Excitation, transmission and radiation of sound from source distributions in hard-walled ducts of finite length (II): The effects of duct length

Exact methods of calculating the sound fields both inside and outside a duct of finite length, due to an acoustic source (or force) distribution inside the duct, are derived and discussed. Specific results are presented for the case of a hard-walled duct carrying no mean flow and having, effectively, baffled termination apertures. The basis of the calculation methods is such that it can be extended to ducts carrying uniform mean flows, and with axial discontinuities in mean temperature. The results show that the sound field inside the duct, and the total radiated acoustic power, can be determined in terms of certain modal aperture coefficients, without the necessity of first calculating the entire radiated sound field.

Measurements of the normal acoustic impedance of ground surfaces

This investigation was undertaken to provide some data on the sound absorption properties of ground surfaces and to develop a method for field measurement of the acoustic impedance of the ground surface. Initial tests with an impedance tube showed that although results were satisfactory for sand with no living vegetation present, the tube was quite ineffective when used for tests on living material, placing unacceptable restrictions on the environment of the material under test. In these tests it was found that there was a slight increase in the sound absorption of granular materials such as sand when a little moisture is present. When vegetable roots are present in the soil it appears that sound absorption is greater than without them, and the attenuation is a maximum at zero moisture content. For field measurements a free wave reflection method has been developed. The specific normal acoustic impedance ratio of the ground surface measured for a range of frequencies from 200 to 1000 Hz was found to vary considerably for different ground surfaces—for example, from 3−3 i for a soft grass patch to 7−17 i for an area of rock covered with chippings.

Attenuation of plane wave and higher order mode sound propagation in lined ducts

Abstract A perturbation method, for obtaining analytical expressions, for the attenuation of higher-order and plane-wave sound propagation in lined ducts, is presented. A detailed analysis is given for ducts with hollow circular cross-sections. Both the nearly soft wall and the nearly hard wall cases have been included. The results are then extended for other cross-sections: a very narrow annulus of a cylinder, the general case of a cylinder and a rectangular cross-section. Some generalizations regarding ducts with uniform but arbitrary cross-section have been made. The effects on attenuation of a uniform mean flow (but not shear) have been added.

Momentum potential theory of energy flux carried by momentum fluctuations

Abstract The momentum potential theory of time-stationary fluctuating flows is briefly reviewed and then extended to include energy flux carried by momentum fluctuations. It is shown that the mean (time-averaged) energy flux can be expressed as a linear superposition of mean, turbulent, acoustic and thermal components. A mean energy flux balance relating turbulent, acoustic and thermal energy fluxes only is obtained. For homentropic flows the Cantrell and Hart expression for acoustic intensity is obtained as a special case. Physical and analog interpretations of the mean energy flux balance are presented for the general case. A formal solution of the equation for mean flow of energy due to momentum fluctuations is presented, leading to a definition of local fluctuating dynamical equilibrium of a time-stationary flow and an identification of sources of far field radiated acoustic power. In the Appendix, the momentum potential theory version of Jenveys partitioning of the stagnation enthalpy into mean, turbulent, acoustic and thermal parts is presented.

Fundamentals of aerodynamic sound theory and flow duct acoustics

Some fundamental aspects of the theory of internally generated sound (or “sound generated aerodynamically”) are reviewed and discussed. Particular stress is laid on the functional relationships, between the radiated sound field and the equivalent source distribution of Lighthills “acoustic analogy”, model, as exposed by multipole analysis. Recent theoretical and experimental progress in both turbulent mixing region noise and flow duct acoustics is cited, and discussed in the context of its fundamental, implications for the future development of “aerodynamic noise” theory.

Analysis of internally generated sound in continuous materials: 3. The momentum potential field description of fluctuating fluid motion as a basis for a unified theory of internally generated sound†

The reasons for a need for a unified theory of aerodynamic noise generation are discussed and justified. In a previous paper, existing theories of aerodynamic noise generation have been critically reviewed with special emphasis on conceptual adequacy and physical scope, and with special reference to supersonic jet noise. On the basis of the evidence provided by this critical review a new unified theory for internally generated sound in general, and jet noise in particular, has been devised. It is shown that the new, generalized theory, in which the previous work by Doak on the identification of acoustic, thermal and turbulent motions plays a fundamental role, includes all valid previous theories as special cases and that it can be explicitly formulated for general, non-linear fluctuating motion. A quasi-linear version of this new, generalized theoretical formulation of the problem of aerodynamic noise generation and propagation is presented. A detailed discussion is given of the manner in which Lilleys theory of mixing noise in a unidirectional, transversely sheared turbulent mixing layer arises as a special case of the new, generalized theory. The new theory is shown to be of a form in which no restrictions are implied on either mean velocity gradients or mean temperature gradients, with respect to either their magnitudes or directions.