Derek William Moore

The spontaneous appearance of a singularity in the shape of an evolving vortex sheet

The evolution of a small amplitude initial disturbance to a straight uniform vortex sheet is described by the Fourier coefficients of the disturbance. An approximation to the exact evolution equation for these coefficients is proposed and it is shown, by an asymptotic analysis valid at large times, that the solution of the approximate equations develops a singularity at a critical time. The critical time is proportional to In (ε-1), where ε is the initial amplitude of the disturbance and the singularity itself is such that the nth. Fourier coefficient decays like n-2.5 instead of exponentially. Evidence - not conclusive, however - is present to show that the approximation used is adequate. It is concluded that the class of vortex layer motions correctly modelled by replacing the vortex layer by a vortex sheet is very restricted; the vortex sheet is an inadequate approximation unless it is everywhere undergoing rapid stretching.

The motion of a vortex filament with axial flow

Infinitesimal waves on a uniform vortex with axial flow are studied. The equation for the frequency of helical waves is obtained, and solved for the case of long waves which leave the internal structure almost unaltered. A method is developed to obtain results for vortices of non-uniform structure and for displacements which are not necessarily small compared with the core radius. The approach consists of balancing the Kutta—Joukowski lift force, the momentum flux due to the axial motion, and the ‘tension’ of the vortex lines. A general equation for the motion of a vortex filament is obtained, valid for arbitrary shape and internal structure, and in the presence of an external irrotational velocity field. When the axial flow vanishes, the method is equivalent to using the Biot—Savart law for the self-induced velocity, with a suitable cutoff. The impulse of a vortex filament is discussed and its rate of change is given.

The instability of a straight vortex filament in a strain field

A straight infinite vortex of finite cross section is deformed by the action of weak irrotational plane strain. The deformed vortex is shown, in the absence of axial flow, to be unstable to disturbances whose axial wavelengths lie in a narrow band, whose width is proportional to the imposed strain. The band is centred on the wavelength of the helical wave which does not propagate on the unstrained circular vortex. Thus support is given to the instability mechanism proposed recently by Widnall, Bliss & Tsai (1974). The argument depends, however, on the mirror image of the helical wave also being a possible non-propagating disturbance on the unstrained vortex.

Axial flow in laminar trailing vortices

The structure of laminar trailing vortices behind a lifting wing is considered. The inviscid roll up of the trailing vortex sheet is examined, and the nature of the singularity at the centre of the spiral is determined. It is shown that viscosity removes the singularity and the structure of the viscous core is obtained. The pressure in the viscous core is found and used to calculate the axial velocities produced by streamwise pressure gradients. It is found that the perturbation of axial velocity can be either away from the wing or towards the wing depending on the distribution of tip loading on the wing. For elliptic loading, the perturbation is towards the wing. The axial flow deficit in the core due to the boundary layers on the wing is also estimated. A comparison with experiment is made and reasonable agreement is found.

ACOUSTIC DESTABILIZATION OF VORTICES

A line vortex which has uniform vorticity 2Ω0 in its core is subjected to a small two-dimensional disturbance whose dependence on polar angle is eimθ. The stability is examined according to the equations of compressible, inviscid flow in a homentropic medium. The boundary condition at infinity is that of outgoing acoustic waves, and it is found that this capacity to radiate leads to a slow instability by comparison with the corresponding incompressible vortex which is stable. Numerical eigenvalues are computed as functions of the mode number m and the Mach number M based on the circumferential speed of the vortex. These are compared with an asymptotic analysis for the m = 2 mode at low Mach number in which it is found that the growth rate is (π/ 32) M4Ω0 in good agreement with the numerical results.

The effect of compressibility on the speed of propagation of a vortex ring

A circular vortex filament of radius R, cross sectional area πa2 and circulation Г propagates steadily in an inviscid, calorically perfect gas. The flow outside the filament is assumed to be irrotational and isentropic. If it is further assumed that a/R ≪ 1, the cross section is approximately circular and the speed of propagation of the filament is shown to depend on the distribution of circulatory velocity v0 and entropy s0 within the core. If s0 is constant and equal to its value in the isentropic exterior of the filament, the vortex ring is slowed down by compressibility effects, whatever the distribution of circulatory velocity. If the circulatory velocity corresponds to rigid rotation in the core cross section, the speed, U, of propagation is given by U = Γ/4πR[ln 8R/a - ¼ - 5/12M2 + O(M4)], where M is the Mach number Γ/2πac∞ and c∞ is the sound speed far from the vortex ring. Numerical results for finite M are also given in this case. These results enable the cut-off theory of filament motion to be extended to compressible fluids.

The Rolling up of a Semi-Infinite Vortex Sheet

The rolling up of a semi-infinite initially straight vortex sheet is studied analytically. In its initial state the circulation in the sheet increases as the square root of the distance from its edge. Previous investigations have asserted that the asymptotic form for the equation of the rolled up portion given by Kaden could be improved on by finding higher terms in a locally determined asymptotic expansion. This assertion is contested and it is suggested that the correction to Kaden can not be found unless the shape of the whole vortex sheet is known. The correction proposed renders the turns of the spiral slightly elliptical, the precise magnitude involving an integral over the entire vortex sheet. While a useful analytical solution cannot be found this way, it is suggested that the result would be useful in a numerical study.

The velocity of a vortex ring with a thin core of elliptical cross section

The motion of a circular vortex ring with a thin elliptical core is considered. The core is untwisted so that the vortex ring is axisymmetric and the vorticity in the core is proportional to distance from the axis of symmetry. The core rotates with a constant angular velocity comparable to the circulation frequency, as in Kirchoff’s two-dimensional solution. The velocity of the ring, suitably defined, is periodic and the average velocity is Γ/4πR[ln(16R/a+b)-¼], where Γ is the circulation around the core, a and b are the semi-major and semi-minor axes of the core cross section and R is the radius of the ring. This mean velocity is smaller than the velocity of translation of a ring of the same radius and circulation but with a circular core of the same-cross-sectional area.

The Interaction of a Diffusing Line Vortex and an Aligned Shear Flow

An exact solution of the Navier─Stokes equations of incompressible flow, which represents the interaction of a diffusing line vortex and a linear shear flow aligned so that initially the streamlines in the shear flow are parallel to the line vortex, is presented. If Γ is the circulation of the line vortex and v the kinematic viscosity then, when Re ═ Γ/2πv is large, the vorticity of the shear flow is expelled from the circular cylinder 0 < r ≪ (vt)1/2 Re1/3, where r is the distance from the axis of the diffusing line vortex and t the time since initiation of the flow. At larger radii a peak vorticity 0.903Ω Re1/3 is found at a radial distance 1.26(vt)1/2 Re1/3, where Ω is the initial uniform vorticity in the shear flow. The vortex filament is embedded in a growing cylinder from which vorticity has been expelled, the cylinder itself being bounded by an annular region of thickness of order Re1/3 (vt)1/2 in which the vorticity is of order Ω Re1/3.

The interaction of a vortex ring and a coaxial supersonic jet

Previous experimental work by Baird has established that when a supersonic jet is first started it forms a vortex ring at the mouth of the jet pipe which subsequently travels downstream and carries with it a normal shock. This flowfield is here analysed on the basis of simplifying assumptions, in particular that when the vortex ring is sufficiently far from the mouth of the jet pipe the flow is steady in coordinates moving with the ring, and that time-dependent terms may be neglected. It is shown that the jet will be subjected to an adverse pressure gradient roughly where the shock forms (although the shock itself is excluded from the simplified mathematical model) and also that the vortex ring must grow in size at a slow rate. These results are in qualitative agreement with experiment.