On the self-induced motion of a helical vortex

The velocity field in the immediate vicinity of a curved vortex comprises a circulation around the vortex, a component due to the vortex curvature, and a ‘remainder’ due to the more distant parts of the vortex. The first two components are relatively well understood but the remainder is known only for a few specific vortex geometries, most notably, the vortex ring. In this paper we derive a closed form for the remainder that is valid for all values of the pitch of an infinite helical vortex. The remainder is obtained firstly from Hardin's (1982) solution for the flow induced by a helical line vortex (of zero thickness). We then use Ricca's (1994) implementation of the Moore & Saffman (1972) formulation to obtain the remainder for a helical vortex with a finite circular core over which the circulation is distributed uniformly. It is shown analytically that the two remainders differ by 1/4 for all values of the pitch. This generalizes the results of Kuibin & Okulov (1998) who obtained the remainders and their difference asymptotically for small and large pitch. An asymptotic analysis of the new closed-form remainders using Mellin transforms provides a complete representation by a residue series and reveals a minor correction to the asymptotic expression of Kuibin & Okulov (1998) for the remainder at small pitch.

An interpretative model of carbon and nitrogentransformations applied to a residue incubation experiment

A simple model of 3 equations was devised to simulate the rates through time of gross mineralisation of nitrogen, nitrogen immobilisation, and microbial respiration relating to individual inputs into soil of plant residues of any age or type. Using published data from an incubation experiment carried out in Iowa, we applied the model to a residue newly added to soil, to the original soil organic matter (SOM), and to a mixture of these. Manipulation of the model allowed the derivation from the Iowa data of a net mineralisation index which seemed to summarise the nitrogen release characteristics of the residue in all treatments of the experiment. The equations and parameter values developed for the added residue were applied to SOM using results from unamended soil. The balance between respiration and mineralisation was found not to correspond to that expected for old organic matter near an equilibrium carbon/nitrogen ratio. Rate constants of mineralisation and respiration for SOM were adapted to overcome this apparent anomaly. To model the dynamics SOM and added residue simultaneously, the 2 sets of 3 equations were applied in parallel to 4 extreme treatments in the with-residue series (lowest and highest nitrate levels with low and high residue additions). To achieve the fits presented, only 2 of the 12 parameters required in each set of equations needed to differ between the set for SOM and that for added residue. The model reproduces well most of the primary Iowa data and also some derived results. Use of the model helped to interpret divergences between simulations and data.

7.—High-frequency Scattering in a Certain Stratified Medium

SynopsisThe propagation of scalar waves in a certain stratified medium is studied; the field is due to a line source situated on an opaque plane boundary. The exact field can be expressed in terms of a Fourier integral involving Airy functions. By deforming the real axis, which is the contour of integration of the Fourier integral, only in the neighbourhood of the real axis, it is possible to give a simple but rigorous derivation of the asymptotic nature of the field. Two separate cases are considered: the point of observation lying in the illuminated region, when a steepest descents analysis is appropriate, or lying in the shadow region, when the asymptotic field is given by a residue series.

XVI.—Creeping Waves in an Inhomogeneous Medium

SynopsisThis paper is concerned with high-frequency scattering in a medium, the square of whose refractive index varies linearly with height from a plane boundary. Two asymptotic methods are examined, namely the method of stationary phase and evaluation by residue series. The first of these corresponds to geometric optics and gives the high-frequency field in the illuminated region, while the second complements the first in the sense that if thsre is no point of stationary phase, the residue series is an asymptotic expansion of the field. The Airy functions in the residue series can be replaced by their asymptotic developments in terms of exponentials, and when this is done only the first term or first creeping wave is of genuine significance.

Supersonic wing-body interference

We consider here the supersonic inviscid flow past a wing-body combination consisting of a semi-infinite plane wing symmetrically placed about an infinite convex cylinder. When the cylinder is circular in cross-section a formal solution for the Laplace transform of the velocity potential exists (Neilson(l), and Stewartson, 1951, unpublished). This solution is given in the form of a series which is only slowly convergent near the boundary surface separating the disturbed and the undisturbed regions. However, this slow convergence has been overcome by Waechter(4) using the Poisson Summation formula on the series solution. This enables the solution to be expressed in terms of integrals which can be evaluated as a convergent residue series, taking on different forms in the different regions of the interaction.