D. G. Crighton

Basic principles of aerodynamic noise generation

Abstract This paper gives a simple, unified, analytical description of a wide range of mechanisms associated with the generation of sound by unsteady fluid motion. Topics treated include radiation from compact and non-compact multipole sources, Lighthills theory of sound emission from free turbulence, effects of source convection, sound generation from flow interaction with solid surfaces and inhomogeneities of the medium, and singular perturbation aspects of the aerodynamic sound problem. The concluding section discusses several areas of current interest and importance, including noise generation by supersonic shear layers, shallow water wave simulation of flow noise, the excess-noise problem, and the general issues bound up with shear layer and jet instability and the orderly structure of turbulent jets.

Acoustics as a branch of fluid mechanics

This article gives a review of six areas of current activity and importance in aero-acoustics, including (i) the generation of sound and vorticity by vorticity and sound, respectively, (ii) the basis for, and consequences of, the application of a Kutta condition in unsteady leading- and trailing-edge flows, and (iii) the suppression or amplification of broadband hydrodynamic and acoustic fields in a jet under the influence of weak discrete tone forcing. The intention is also to promote acceptance once again of acoustics as a serious branch of fluid mechanics.

Shear-layer pressure fluctuations and superdirective acoustic sources

We consider a sequence of boundary-value problems for the acoustic wave equation, with the pressure specified on the boundary as a function of space and time, and simulating features of the pressure field measured just outside a turbulent shear layer supporting large-scale coherent structures. The boundary pressure field has the form of a travelling subsonic plane wave, modulated by a large-scale envelope function. Three models for the envelope distribution are studied in detail, and the particular features which they exhibit are shown to be representative of large classes of amplitude functions. We start by looking at the hydrodynamic near field of the boundary pressure fluctuations, over spatial regions throughout which the motion can be taken as incompressible. Very close to the boundary, the pressure fluctuations decay exponentially with transverse distance, while at sufficiently large distances from the whole wave packet on the boundary, the pressure fluctuations have a dipole algebraic decay. We investigate the transition from exponential to algebraic decay, and find that it is effected through quite a complicated multilayer structure which depends crucially on the detailed form of the envelope. Acoustic fields are then determined both from exact solutions to the wave equation, and from matching arguments. In some cases, where the boundary source is compact, the distant acoustic fields have a simple compressible dipole type of behaviour. In other cases, however, when the boundary source is non-compact, the acoustic field has a superdirective character, the angular variation being described by exponentials of cosines of the angle with the streamwise direction. It is shown how the superdirective acoustic sources are completely compatible with the features of the inner incompressible field, and a criterion for the occurrence of the superdirective acoustic fields will be given. Superdirective fields of this kind have been observed in measurements by Laufer & Yen (1983) on a low-speed round jet of Mach number 0.1, and the general relation of our results to those experiments is explained.

Fluid loading with mean flow. I, Response of an elastic plate to localized excitation

The response to localized forcing of a fluid-loaded elastic plate is studied in the case when there is uniform incompressible flow over the plate. Absolute instability of the fluid-plate system is found when the dimensionless mean velocity U exceeds a threshold Uc which is found exactly. For U ωp(U), ω being the excitation frequency: here asymptotic expressions are found for the frequencies and for the wavenumbers and amplitudes of the waves found upstream and downstream of the excitation. A significant feature is that ReA0 < 0 throughout 0 < ω < ωp, A0 being the drive admittance (velocity at the point of application of the force); this means that throughout the convectively unstable and the anomalous neutral frequency ranges, the exciting force must absorb energy. An exact energy equation is derived, and shown to require the introduction of a new fluid-plate interaction flux UnOt, where O is the fluid potential and n the plate deflexion. The energy equation is used to illuminate properties of the convectively unstable and neutral waves, to verify the property ReA0 < 0 and to trace the waves responsible for this. Anomalous features in the frequency range ωs(U) < ω < ωp(U) are investigated further from the viewpoint of the theory of negative energy waves, and it is found that not only can some wave modes in this frequency range have negative energy, but also group velocity in an inward direction (towards the excitation). It is argued that this does not contradict the outward group velocity ‘radiation condition’ of M. J. Lighthill, because that condition refers expressly to circumstances in which the excitation is the sole source of all the wave energy, whereas here the excitation acts also as a scatterer, transferring energy from the mean flow to the wave field.

The jet edge-tone feedback cycle; linear theory for the operating stages

The paper presents a linear analytical model to predict the frequency characteristics of the discrete oscillations of the jet-edge feedback cycle. The jet is idealized as having top-hat profile with vortex-sheet shear layers, and the nozzle from which it issues is represented by a parallel plate duct. At a stand-off distance h, a flat plate is inserted along the centreline of the jet, and a sinuous instability wave with real frequency w is assumed to be created in the vicinity of the nozzle and to propagate towards the splitter plate. Its interaction with the splitter plate produces an irrotational feedback field which, near the nozzle exit, is a periodic transverse flow producing singularities at the nozzle lips. Vortex shedding is assumed to occur, alleviating the singularities and allowing a trailing-edge Kutta condition to be satisfied; this Kutta condition is claimed to be the phase-locking criterion. The shed vorticity develops into a sinuous spatial instability, and the cycle of events is repeated periodically.Problems corresponding to the various physical processes described are analysed, for in viscid flow with vortex-sheet shear layers and aligned flat-plate boundaries, and solved in an appropriate asymptotic sense by Wiener-Hopf methods. Calculation of the phase changes occurring in the constituents of the cycle gives an equation for the frequency w in the Nth ‘stage’ as a function of jet width 2b, jet velocity U0, standoff distance h, and stage label N: \[ \omega b/U_0 = (b/h)^{\frac{3}{2}}[4\pi(N-{\textstyle\frac{3}{8}})]^{\frac{3}{2}}. \] The variations with b, U0, h and N are in excellent agreement with edge-tone experiments; the principal disagreement lies in the overall numerical factor and explanations are given for this. Possible effects associated with the inclusion of displacement thickness fluctuations in the splitter-plate boundary layers, and the enforcement of a leading-edge Kutta condition, are also considered and shown not to affect the frequencies of the operating stages.

Shock-generated ignition: the induction zone

Solutions of the small disturbance equations for shock-induced thermal explosions at high activation energies are discussed in the limit when the adiabatic and isothermal sound speeds lie close together. Temperature and pressure perturbations are shown to possess logarithmic singularities at ignition. Although the density perturbation appears to have a similar behaviour, a local adjustment occurs exponentially close to the ignition point and the density at ignition is bounded. Excellent agreement with numerical calculations is found.

Nonlinear wave motion governed by the modified Burgers equation

The modified Burgers equation (MBE) ∂V∂X+V2∂V∂τ=ε∂2V∂τ2 has recently been shown by a number of authors to govern the evolution, with range X, of weakly nonlinear, weakly dissipative transverse waves in several distinct physical contexts. The only known solutions to the MBE correspond to the steady shock wave (analogous to the well-known Taylor shock wave in a thermoviscous fluid) or to a similarity form. It can, moreover, be proved that there can exist no Bäcklund transformation of the WBE onto itself or onto any other parabolic equation, and in particular, therefore, that no linearizing transformation of Cole-Hopf type can exist. Attempts to understand the physics underlying the MBE must then, for the moment, rest on asymptotic studies and direct numerical computation.

Instability of flows in spatially developing media

This paper studies specific forms of partial differential equations which represent the amplitude evolution of disturbances to flows in media whose properties vary slowly in space (over a length scale ∊-1). A method is described for finding series solutions in the small parameter ∊, and general results for the Green’s function and the time-periodic response are obtained. With the aid of a computer-based symbolic algebra manipulation package, the method permits results to be obtained to any desired accuracy for any specified spatial variation. The method is then applied to find the Green’s function for the linearized Ginzburg-Landau equation (which is a good simple model of instability growth in many marginally unstable fluid-dynamical systems) in the case where the instability parameter μ (normally a constant in the equation) varies with the spatial coordinate. Even though the method produces only power series solutions in ∊, it is shown that with its help exact analytic solutions, valid for all ∊ (however large), can be found in some cases. The exact solutions for the cases of linear and quadratic spatial variation of μ are obtained and discussed, and criteria are found for the onset of global instability. Comparisons are made between these criteria and the local absolute and convective instability criteria, and the results are found to be in qualitative agreement with other authors’ numerical and analytic results; in particular, it is shown that a small region of local absolute instability (up to a maximal size which is known analytically) may exist and yet that the flow may be globally stable. It is also shown that in the case of quadratic spatial variation of μ with an infinite region of local absolute instability, the Green’s function may (for some particular parameter ranges) become infinite at all spatial locations simultaneously at some finite time.

Bäcklund transformations for nonlinear parabolic equations: the general results

Bäcklund transformations (BTs) are considered for nonlinear parabolic equations of the form ut + uxx + H(ux, u, x, t) = 0. (*) The most general form of both the BT and the class (*) is adopted, and we show that the only nonlinear equations (*) possessing BTs are the slight generalizations of Burgers’ equation obtained by adding a forcing term, ut + uxx + 2uux + a(x, t) = 0.

The modes, resonances and forced response of elastic structures under heavy fluid loading

This paper reports analytical studies of problems that involve the motion of plane elastic structures under conditions of heavy fluid loading. The main aspect concerns the description of the vibration response of a thin elastic plate (or membrane), of finite extent in at least one dimension, when the structure is excited by concentrated mechanical drive along a line or at a point; and as part of this the possibility of resonant response is discussed, and the resonance conditions and free modes of oscillation are obtained. There is also some discussion of the acoustic fields radiated by the structures under localized mechanical excitation. The analysis makes extensive use of results for the reflection of a structural wave (subject to heavy fluid loading) at an edge, and the paper gives results for that reflection process covering waves incident normally on eight different edge configurations and waves incident obliquely on two edge configurations. These results include the reflection coefficient (whose magnitude is unity in the leading-order approximation of low-frequency heavy fluid loading), and the amplitude and directivity of the edge-scattered sound. By using the argument that edge reflection is a local process, the response is then calculated for a strip plate, under both line and point forcing, and the response is, for the first time, obtained for structures finite in both dimensions and subject to heavy fluid loading. Specifically, solutions are given here for a circular plate with eccentric drive, and for a membrane model of a rectangular panel, with central point drive. For some conditions and geometries expressions in simple form are found for the natural frequencies and mode shapes, and for the off-resonance forced response. Expressions for the drive admittances are found which display a variety of interesting features.