Theoretical and experimental supersonic lateral-directional stability characteristics

For abstract, see A81-37375

Experimental and theoretical supersonic lateral-directional stability characteristics of a simplified wing-body configuration with a series of vertical-tail arrangements

An experimental investigation was conducted to provide a systematic set of lateral-directional stability data for a simplified wing-body model with a series of vertical-tail arrangements. The study was made at Mach numbers from 1.60 to 2.86 at nominal angles of attack from -8 to 12 deg and Reynolds number of 8.2 million per meter. Comparisons at zero angle of attack were made with three existing theoretical methods (MISLIFT - a second-order shock expansion and panel method; APAS - a slender body and first order panel method; and PAN AIR - a higher order panel method) and comparisons at angle of attack were made with PAN AIR. The results show that PAN AIR generally provides accurate estimates of these characteristics at moderate angles of attack for complete configurations with either single or twin vertical tails. APAS provides estimates for complete configurations at zero angle of attack. However, MISLIFT only provides estimates for the simplest body-vertical-tail configurations at zero angle of attack

A Girsanov approach to slow parameterizing manifolds in the presence of noise

We consider a three-dimensional slow-fast system with quadratic nonlinearity and additive noise. The associated deterministic system of this stochastic differential equation (SDE) exhibits a periodic orbit and a slow manifold. The deterministic slow manifold can be viewed as an approximate parameterization of the fast variable of the SDE in terms of the slow variables. In other words the fast variable of the slow-fast system is approximately "slaved" to the slow variables via the slow manifold. We exploit this fact to obtain a two dimensional reduced model for the original stochastic system, which results in the Hopf-normal form with additive noise. Both, the original as well as the reduced system admit ergodic invariant measures describing their respective long-time behaviour. We will show that for a suitable metric on a subset of the space of all probability measures on phase space, the discrepancy between the marginals along the radial component of both invariant measures can be upper bounded by a constant and a quantity describing the quality of the parameterization. An important technical tool we use to arrive at this result is Girsanov's theorem, which allows us to modify the SDEs in question in a way that preserves transition probabilities. This approach is then also applied to reduced systems obtained through stochastic parameterizing manifolds, which can be viewed as generalized notions of deterministic slow manifolds.Comment: 54 pages, 6 figure

Bifurcations of periodic orbits with spatio-temporal symmetries

Motivated by recent analytical and numerical work on two- and three-dimensional convection with imposed spatial periodicity, we analyse three examples of bifurcations from a continuous group orbit of spatio-temporally symmetric periodic solutions of partial differential equations. Our approach is based on centre manifold reduction for maps, and is in the spirit of earlier work by Iooss (1986) on bifurcations of group orbits of spatially symmetric equilibria. Two examples, two-dimensional pulsating waves (PW) and three-dimensional alternating pulsating waves (APW), have discrete spatio-temporal symmetries characterized by the cyclic groups Z_n, n=2 (PW) and n=4 (APW). These symmetries force the Poincare' return map M to be the nth iterate of a map G: M=G^n. The group orbits of PW and APW are generated by translations in the horizontal directions and correspond to a circle and a two-torus, respectively. An instability of pulsating waves can lead to solutions that drift along the group orbit, while bifurcations with Floquet multiplier +1 of alternating pulsating waves do not lead to drifting solutions. The third example we consider, alternating rolls, has the spatio-temporal symmetry of alternating pulsating waves as well as being invariant under reflections in two vertical planes. This leads to the possibility of a doubling of the marginal Floquet multiplier and of bifurcation to two distinct types of drifting solutions. We conclude by proposing a systematic way of analysing steady-state bifurcations of periodic orbits with discrete spatio-temporal symmetries, based on applying the equivariant branching lemma to the irreducible representations of the spatio-temporal symmetry group of the periodic orbit, and on the normal form results of Lamb (1996). This general approach is relevant to other pattern formation problems, and contributes to our understanding of the transition from ordered to disordered behaviour in pattern-forming systems

On the zero set of G-equivariant maps

Let $G$ be a finite group acting on vector spaces $V$ and $W$ and consider a smooth $G$-equivariant mapping $f:V\to W$. This paper addresses the question of the zero set near a zero $x$ of $f$ with isotropy subgroup $G$. It is known from results of Bierstone and Field on $G$-transversality theory that the zero set in a neighborhood of $x$ is a stratified set. The purpose of this paper is to partially determine the structure of the stratified set near $x$ using only information from the representations $V$ and $W$. We define an index $s(\Sigma)$ for isotropy subgroups $\Sigma$ of $G$ which is the difference of the dimension of the fixed point subspace of $\Sigma$ in $V$ and $W$. Our main result states that if $V$ contains a subspace $G$-isomorphic to $W$, then for every maximal isotropy subgroup $\Sigma$ satisfying $s(\Sigma)>s(G)$, the zero set of $f$ near $x$ contains a smooth manifold of zeros with isotropy subgroup $\Sigma$ of dimension $s(\Sigma)$. We also present a systematic method to study the zero sets for group representations $V$ and $W$ which do not satisfy the conditions of our main theorem. The paper contains many examples and raises several questions concerning the computation of zero sets of equivariant maps. These results have application to the bifurcation theory of $G$-reversible equivariant vector fields

Approximate solutions of problems involving normal operators

AbstractThe spectral theory for unbounded normal operators is used to develop a systematic method of approximating functions of operators with other, more easily computable functions, leading to a priori error estimates in the operator norm. In particular, polynomial approximations are obtained for resolvents and semigroups in terms of inverses and resolvents, respectively

Sub-100 nanosecond temporally resolved imaging with the Medipix3 direct electron detector

Detector developments are currently enabling new capabilities in the field of transmission electron microscopy (TEM). We have investigated the limits of a hybrid pixel detector, Medipix3, to record dynamic, time varying, electron signals. Operating with an energy of 60keV, we have utilised electrostatic deflection to oscillate electron beam position on the detector. Adopting a pump-probe imaging strategy we have demonstrated that temporal resolutions three orders of magnitude smaller than are available for typically used TEM imaging detectors are possible. Our experiments have shown that energy deposition of the primary electrons in the hybrid pixel detector limits the overall temporal resolution. Through adjustment of user specifiable thresholds or the use of charge summing mode, we have obtained images composed from summing 10,000s frames containing single electron events to achieve temporal resolution less than 100ns. We propose that this capability can be directly applied to studying repeatable material dynamic processes but also to implement low-dose imaging schemes in scanning transmission electron microscopy.Comment: 11 pages, 6 figures; improve ref formatting + revise tex

Penetration of a vortex dipole across an interface of Bose-Einstein condensates

The dynamics of a vortex dipole in a quasi-two dimensional two-component Bose-Einstein condensate are investigated. A vortex dipole is shown to penetrate the interface between the two components when the incident velocity is sufficiently large. A vortex dipole can also disappear or disintegrate at the interface depending on its velocity and the interaction parameters.Comment: 7 pages, 9 figure