Edmund Chadwick

Diffraction of short waves modelled using new mapped wave envelope finite and infinite elements

We consider a two-dimensional wave diffraction problem from a closed body such that the complex progressive wave potential satisfies the Sommerfield condition and the Helmholtz equation. We are interested in the case where the wavelength is much smaller than any other length dimensions of the problem. We introduce new mapped wave envelope infinite elements to model the potential in the far field, and test them for some simple Dirichlet boundary condition problems. They are used in conjuction with wave envelope finite elements developed earlier [1] to model the potential in the near field. An iterative procedure is used in which an initial estimate of the phase is iteratively improved. The iteration scheme, by which the wave envelope and phase are recovered, is described in detail. Copyright

GMOS: the GEMINI Multiple Object Spectrographs

The two Gemini multiple object spectrographs (GMOS) are being designed and built for use with the Gemini telescopes on Mauna Kea and Cerro Pachon starting in 1999 and 2000 respectively. They have four operating modes: imaging, long slit spectroscopy, aperture plate multiple object spectroscopy and area (or integral field) spectroscopy. The spectrograph uses refracting optics for both the collimator and camera and uses grating dispersion. The image quality delivered to the spectrograph is anticipated to be excellent and the design is driven by the need to retain this acuity over a large wavelength range and the full 5.5 arcminute field of view. The spectrograph optics are required to perform from 0.36 to 1.8 microns although it is likely that the northern and southern versions of GMOS will use coatings optimized for the red and blue respectively. A stringent flexure specification is imposed by the scientific requirement to measure velocities to high precision (1 - 2 km/s). Here we present an overview of the design concentrating on the optical and mechanical aspects.

A slender–body theory in Oseen flow

Consider steady flow generated by a uniform velocity field past a fixed closed slender body whose major axis is aligned closely to the uniform stream direction. Let us assume Oseen flow with the slip boundary condition. A slender–body theory is presented. In the near field, let us assume that the second–derivative changes in the velocity and pressure fields are of lower order in the axial direction of the slender body than in the transverse plane. The hydrodynamic forces are then related to the body shape by using matched asymptotics.

The far–field Oseen velocity expansion

Consider uniform, steady, incompressible fluid flow in an unbounded domain past a fixed, closed body. In the far field the Oseen equations approximately hold. We give the Oseen velocity and pressure expansions in the far field for two– and three–dimensional flow by using two approaches: the first decomposes the velocity into a potential velocity and a wake velocity, and was introduced by Lamb and Goldstein; the second approach uses the Oseen representation of the velocity and pressure by a Greens integral distribution of singularities called ‘Oseenlets’. These are the force (drag and lift) singular solutions. We show that, in general, the Lamb–Goldstein approach will not model the Oseen velocity in the far field wake. In contrast, by using Oseens representation to expand each Oseenlet in a Taylor series, we obtain velocity and pressure expansions everywhere in the far field. Nevertheless, there are exceptions for which the Lamb–Goldstein approach can be used, notably: (i) the singular drag and lift solutions; (ii) axisymmetric flow, such that the axis of rotational symmetry is parallel to the uniform stream velocity direction; (iii) low Reynolds number flow, and (iv) two–dimensional flow. For these special cases, the two approaches are shown to be equivalent.

A slender-wing theory in potential flow

Consider uniform, steady potential flow past a slender wing. By considering a horseshoe vortex in the limit as γ/Us → ∞, where γ is the circulation, U is the uniform stream velocity and s is the span, a model representing a vortex sheet is obtained from which the lift on the slender wing can be determined. (This is in contrast to the textbook approach of Batchelor and Katz & Plotkin, who discretize the vortex sheet with horseshoe vortices in the limit as γ/Us → ∞, but then relate the vortex strength to lift by using the two–dimensional limit γ/Us → 0. We shall argue that using these different limits in the same analysis is inconsistent and leads to an incorrect result.) The resulting potential term is shown to be the same as the potential term of the lift Oseenlet in Oseen flow. In the limit of high–Reynolds–number flow, only half the contribution to the lift integral comes from the potential–velocity part of the lift Oseenlet. The other half comes from the vortex–wake–velocity part of the lift Oseenlet. We therefore assume potential flow everywhere except at the vortex sheet, along which we allow a singular vortex–wake–velocity term of the lift Oseenlet. From this, a slender–wing theory is presented together with integral expressions for the lift and change in lift over the wing surface. Applications to slender bodies and large–aspect–ratio wings, in particular, the Lanchester–Prandtl lifting line, are then considered.

The vortex line in steady, incompressible Oseen flow

The horseshoe vortex is given in Oseen flow as a constant spanwise distribution of lift Oseenlets. From this, the vortex line is represented in steady, incompressible Oseen flow. The velocity near to the vortex line is determined, as well as near to and far from the far field wake. The velocity field in the transverse plane near to the vortex line is shown to approximate to the two-dimensional Lamb–Oseen vortex, and the velocity field in the streamwise direction is generated by the bound vortex line of the horseshoe vortex giving a streamwise decay much faster than that of the Batchelor vortex. The far field wake description is shown to be consistent with laminar wake theory.

Gravitational theoretical development supporting MOND

Conformal geometry is considered within a general relativistic framework. An invariant distant for proper time is defined and a parallel displacement is applied in the distorted space-time, modifying Einstein’s equation appropriately. A particular solution is introduced for the covariant acceleration potential that matches the observed velocity distribution at large distances from the Galactic Center, i.e. modified Newtonian dynamics. This explicit solution of a general framework that allows both curvature and explicit local expansion of space-time, thus reproduces the observed flattening of galaxys’ rotation curves without the need to assume the existence of dark matter. The large distance expansion rate is found to match the speed of a spherical shock wave.

LIFT ON SLENDER BODIES WITH ELLIPTICAL CROSS SECTION EVALUATED BY USING AN OSEEN FLOW MODEL

Consider uniform, incompressible flow past a slender body with an elliptical cross section such that the major axis of the body is inclined slightly to the flow direction. Assume that the flow is inviscid everywhere except in a thin boundary layer region and in the vortex core of trailing line vortices that emanate from the body into the vortex wake. Hence, the flow is quasi-inviscid, and so the slip (impermeability) boundary condition is applied. Further assume that outside the boundary layer the velocity is to first order the uniform stream velocity. Then the Oseen approximation can be applied. The resulting solution, up to the slender body approximation, is given, and the lift over the slender body is determined. This solution is then compared with the theoretical and experimental results for flow past a delta wing, the viscous cross-flow method and experimental results for flow past a body with a circular cross section, and Newtonian impact theory and experimental results for flow past a body with an ...

The physical interpretation of the lift discrepancy in Lanchester-Prandtl lifting wing theory for Euler flow, leading to the proposal of an alternative model in Oseen flow

Consider uniform, steady potential and incompressible flow past a fixed thin wing inclined at a small angle to the flow. An investigation is conducted into the physical interpretation and consequences of the revision by Chadwick (Chadwick 2005 Proc. R. Soc. A 461, 1–18) of the Lanchester–Prandtl lifting wing theory in Euler flow. In the present paper, the lift is evaluated from the pressure distribution over the top and bottom surfaces together with a contribution across the trailing edge of the wing. It is shown that this contribution across the trailing edge has previously been erroneously omitted in the standard approach but confirms and provides a physical explanation for the discrepancy in the lift calculation found by Chadwick. This results in a reduction of the lift by a half, but this reduction in lift from the additional calculation is not the right answer, and instead arises from a mathematical discrepancy with the physically observed lift. The discrepancy is due to the pressure becoming singular at the trailing edge in the Euler model. The physical explanation is that in real flow the pressure is regularized by the action of viscosity and so is not singular at the trailing edge. So this lift force at the trailing edge is present in the Euler model but not in a real flow. In a real flow, the viscous effects prevent the pressure becoming singular and so there is no lift force, and consequently no large torque, concentrated at the trailing edge. That the lift force at the trailing edge has been ignored in the Lanchester–Prandtl theory in Euler flow has led to fortuitous agreement with the experimental results on real flows. This shows that the Euler model does not properly predict forces for this problem in which there are singularities (vorticity) within the flow field. We propose a revision to the Euler model by allowing a counterbalancing singular viscous velocity term to reside on the trailing vortex sheet, which is derived from the lift oseenlet. This viscous term ensures that the pressure and velocity are not singular in the flow field. The consequences for the flow due to the inclusion of this term for extending triple-deck and similar asymptotic theories to the case for flow past wings rather than aerofoils are discussed, as well as for the (ideal) high Reynolds number limit and for slender body lift.

The evaluation of the far-field integral in the Green’s function representation for steady Oseen flow

Consider the Green’s function representation of an exterior problem in steady Oseen flow. The far-field integral in the formulation is shown to be zero.