C. J. Chapman

Nonlinear Rayleigh–Bénard convection between poorly conducting boundaries

The convective instability of a layer of fluid heated from below is studied on the assumption that the flux of heat through the boundaries is unaffected by the motion in the layer. It is shown that when the heat flux is above the critical value for the onset of convection, motion takes place on a horizontal scale much greater than the layer depth. Following Childress & Spiegel (1980) the disparity of scales is exploited in an expansion scheme that results in a nonlinear evolution equation for the leading-order temperature perturbation. This equation which does not depend on the vertical co-ordinate, is solved analytically where possible and numerically where necessary; most attention is concentrated on solutions representing two-dimensional rolls. It is found that for any given heat flux a continuum of steady solutions is possible for all wave numbers smaller than a given cut off. Stability analysis reveals, however, that each mode is unstable to one of longer wavelength than itself, so that any long box will eventually contain a single roll, even though the most rapidly growing mode on linear theory has much shorter wavelength.

Sound radiation from a cylindrical duct. Part 1. Ray structure of the duct modes and of the external field

This paper determines the ray structure of a spinning acoustic mode propagating inside a semi-infinite circular cylindrical duct, and thereby determines the ray structure of the field radiated from the end of the duct. Inside the duct, but outside of a caustic cylindrical surface, the rays are piecewise linear helices; on striking the rim of the end-face of the duct, these rays produce ‘Keller cones’ of diffracted rays. The cones determine the structure of the radiated field: for example, no rays penetrate two cone-shaped far-field quiet zones centred on the duct axis; two rays pass through each point in a forward loud zone; and one ray passes through each point in a rearward loud zone. The two rays through each point in the forward loud zone interfere to produce an oscillatory directivity pattern. One quarter of the rays on each cone point back inside the duct and produce the reflected field. Thus the rim of the end-face of the duct acts as a ‘ring source’, in which the radiated and reflected fields have their origin. Every propagating duct mode determines a polar angle and an azimuthal angle; these are taken as parameters specifying the mode and are used to calculate the positions and angles of all the rays. The mathematical method on which the paper is based is Debyes approximation for the Bessel function which appears in the expression for the duct modes; the approximation shows also that the duct contains a region of smooth helical rays on which the field consists of inhomogeneous waves: this region is the inner cylinder, lying inside the annulus of piecewise linear helical rays. The results of the paper are very promising for the application of Kellers geometrical theory of diffraction to detailed calculations of the sound radiated from aeroengine ducts. An alternative description of the field, using Cargills meridional rays, is summarized.

Long wavelength thermal convection between non-conducting boundaries

Abstract Non-linear Rayleigh-Benard convection in a fluid layer is considered as a model of convection in the Earths upper mantle. Previous studies have shown that when the temperature is held fixed at one of the boundaries of the layer, convection takes place in cells of width of the order of the layer depth or less. We investigate the effects of a different thermal boundary condition, in which the flux of heat is held fixed on both layer boundaries; then if this flux is just greater than that required for the onset of convection, motion takes place on horizontal scales much greater than the layer depth. An analytical treatment of the equations, based on an expansion in the depth-to-width ratio of the cells, shows that cells of a definite horizontal scale are the fastest growing according to linearised theory, but that these cells are unstable to ones of larger wavelength than themselves. Thus the dominant wavelength lengthens with time. The results hold whether the heat flux is generated internally of comes from beneath the layer. These results produce flow patterns similar to those found when the heat flux is much greater than the critical value. The results have important consequences for the understanding of mantle convection.

The structure of rotating sound fields

An account is given of the three-dimensional structure of the sound field produced by a spinning sinusoidal distribution of thickness or loading sources. Particular attention is paid to the creeping evanescent waves in the near field and their physical interpretation as trapped edge waves; an exponentially small amount of energy leaks or tunnels through them to emerge as far-field acoustic radiation (as it does through the inhomogeneous waves carried by a waveguide in a cylindrically layered medium). The dependence of the structure on three parameters is investigated in detail: the Mach number M at the outermost radius of the source; the harmonic number n, defined so that the source strength is a function of nθ, where θ denotes azimuthal angle; and the type of source, i. e. thickness or loading. Parameter values considered include those for subsonic, sonic and supersonic motion, and for high and low harmonics. The field is calculated by reducing a special case of Rayleigh’s double integral to a single integral containing a function related to the Chebyshev polynomials, then integrating numerically to give contour plots of pressure as a function of position on various plane and cylindrical sections. These show that the evanescent waves occupy a spherical or ellipsoidal region, and consist of crescents of alternating high and low pressure, shaped and arranged like the segments of an orange; its ‘peel’ marks the transition to the propagating spiral waves of the far field, i. e. the radiation zone. Contour plots on meridional sections are similar to those for the oscillating hertzian electric dipole, suggesting that the field is approximately that produced by a suitably phased arrangement of its acoustic counterpart. When M > 1, the source distribution straddles both the evanescent and the radiation zone; at high supersonic M, the meridional contour plots display an intense beaming pattern, with side-lobes between the main beam and source plane. The results of the paper agree with previous work on propeller acoustics, especially the asymptotic theory.

Shocks and singularities in the pressure field of a supersonically rotating propeller

When linear acoustic theory is applied to the thickness noise problem of a supersonic propeller, it can give rise to a surface on which the pressure is discontinuous or singular. A method is described for obtaining the equation of this surface (when it exists), and the pressure field nearby; jumps, logarithms and inverse square roots occur, and their coefficients may be calculated exactly. The special case of a blade with a straight radial edge gives a cusped cone, whose sheets, each with a different type of discontinuity or singularity in pressure, are separated by lines of cusps; the coefficients in formulae for the pressure near the surface tend to infinity as a cusp line is approached, in proportion to the inverse quarter power of distance from the line. These results determine regions of space where nonlinear effects are important, and they suggest a strong analogy with sonic boom.

The finite-product method in the theory of waves and stability

This paper presents a method of analysing the dispersion relation and field shape of any type of wave field for which the dispersion relation is transcendental. The method involves replacing each transcendental term in the dispersion relation by a finite-product polynomial. The finite products chosen must be consistent with the low-frequency, low-wavenumber limit; but the method is nevertheless accurate up to high frequencies and high wavenumbers. Full details of the method are presented for a non-trivial example, that of anti-symmetric elastic waves in a layer; the method gives a sequence of polynomial approximations to the dispersion relation of extraordinary accuracy over an enormous range of frequencies and wavenumbers. It is proved that the method is accurate because certain gamma-function expressions, which occur as ratios of transcendental terms to finite products, largely cancel out, nullifying Runge’s phenomenon. The polynomial approximations, which are unrelated to Taylor series, introduce no spurious branches into the dispersion relation, and are ideal for numerical computation. The method is potentially useful for a very wide range of problems in wave theory and stability theory.

Energy paths in edge waves

In this paper the energy streamlines, energy paths, and energy streak lines in a steady or unsteady inhomogeneous acoustic field next to an unstable oscillating boundary, such as a vortex sheet or shear layer, are determined. The theory in the paper applies also to an evanescent wave produced by total internal reflection, and to any other type of edge wave, e.g. a coastally or topographically trapped wave in geophysical fluid dynamics. The idea of the paper is that energy velocity, i.e. energy flux divided by energy density, is defined at every point in space and time, not merely when averaged over a cycle. Integration of the ordinary differential equation for energy velocity as a function of position and time gives the energy paths. These paths are calculated explicitly, and are found to have starting and finishing directions very different from those of cycle-averaged paths. The paper discusses the physical significance of averaged and non-averaged energy paths, especially in relation to causality. Many energy paths have cusps, at which the energy velocity is instantaneously zero. The domain of influence of an arbitrary point on the boundary of a steady acoustic edge wave is shown to lie within 45° of a certain direction, in agreement with a known result on shear-layer instability in compressible flow. The results are consistent with flow visualization photographs of near-field jet noise. The method of the paper determines domains of influence and causality in any wave problem with an explicit solution, for example as represented by a Fourier integral.

CAUSTICS IN CYLINDRICAL DUCTS

This paper determines the caustic structure of the acoustic field reflected from the end face of a circular cylindrical duct when the incident field from inside the duct is a spinning acoustic mode. The rays of the reflected field are piecewise linear helices, which begin at the circular rim of the end face of the duct as straight–line diffracted rays lying on one–quarter of a Keller cone of diffracted rays, and continue by multiple reflection of straight–line ray segments off the circular wall of the duct. The caustics of the reflected field, which are envelopes of the axisymmetric curved surfaces obtained by rotating the piecewise linear helices about the duct axis, are infinite in number and have cusps. The paper analyses the resulting longitudinal chain of infinitely many cusped caustics, and shows that, because each cusp is close to a smooth caustic of the same family of rays, the reflected field contains a hyperbolic umbilic singularity. In terms of the reflection numberN of a ray segment, the hyperbolic umbilic region near segmentN has length of orderN−1/3 and width of orderN−2/3. The paper also determines the orders, as powers ofN, of the length and width of a wall region and an ‘outer region’. The orders are used to construct asymptotic expansions for the positions of the caustics in all the regions. The expansions satisfy Van Dykes matching rule, and the leading terms are shown to be numerically accurate as early as the first reflection. The results are promising for the application of Kellers geometrical theory of diffraction and Weinsteins exact Wiener–Hopf theory to the study of acoustic fields and resonant interactions in aeroengine ducts. The mathematical techniques used in the paper, reflection–number asymptotics and a method of dominant triple balances, apply also to multiple reflections of acoustic fields and of internal waves in shear layers and in ocean waveguides.

High‐speed leading‐edge noise

This paper determines the sound generated at the leading edge of a turbofan blade in a subsonic flow when the blade is struck by a convected gust of arbitrary shape. The gust may be localized in the span direction, i.e. may strike only a limited section of the leading edge, and it may strike the edge for only a limited time. The parameter range considered is the non‐compact limit in which the wavelength of the sound is less than the chord of the blade, so that the leading edge may be considered in isolation. By means of the Wiener‐Hopf technique and the use of aeroacoustic similarity variables, a simple integral expression is found for the unsteady three‐dimensional sound field everywhere in space, including the near field and the leading‐edge region remote from the gust, where the Lloyds mirror effect reduces the field amplitude. The expression for the field is well suited to numerical evaluation, e.g. of transients, and its far‐field approximation is simple enough that for many gust shapes all integrations may be performed analytically. The results provide a transfer function between the gust and the source term in the acoustic part of a computational fluid dynamics code.

The asymptotic theory of rapidly rotating sound fields

A rapidly rotating sound field, such as that produced by a supersonic propeller, may contain several types of diffraction pattern, each in a different region of space. This paper determines these patterns and their locations for the field around a propeller of simple type ; the method used is stationary phase analysis of a certain double integral, and leads to asymptotic formulae valid when the number of blades is not too small or the high harmonics are being investigated. Physically, the results describe propagation along rays : each stationary phase point is a ‘ loud spot ’, producing a ray which points directly at the observer; most of the noise comes from these loud spots, because extensive cancellation takes place everywhere else. At most two interior and two boundary stationary points may be present: the number and type depend on the position of the observer in relation to a cusped torus and two hyperboloids of one sheet. As these surfaces are crossed, the acoustic field changes in character. For example, when two stationary points coalesce and annihilate each other, as they do at a caustic, an Airy function describes the transition from a loud zone of rapid oscillation to a quiet zone of exponential decay; and when an interior stationary point crosses the boundary of the disc the transition region is described either by a Fresnel integral or by a generalized Airy function. Separate analyses are given for regions close to and well away from the transition surfaces, and inner and outer limits are calculated for use in the method of matched asymptotic expansions. In all cases, an overlap region is found in which the leading term s agree. The results of the paper determine completely the geometry of the acoustic field, because the different regions have boundaries at known positions and cover the whole of space.