A multi-physics computational model of fuel sloshing effects on aeroelastic behaviour

AbstractA multi-physics computational method is presented to model the effect of internally and externally-carried fuel on aeroelastic behaviour of a pitch–plunge aerofoil model through the transonic regime. The model comprises three strongly coupled solvers: a compressible finite-volume Euler code for the external flow, a two-degree of freedom spring model and a smoothed particle hydrodynamics solver for the fuel. The smoothed particle hydrodynamics technique was selected as this brings the benefit that nonlinear behaviour such as wave breaking and tank wall impacts may be included. Coupling is accomplished using an iterative method with subcycling of the fuel solver to resolve the differing timestep requirements. Results from the fuel-structural system are validated experimentally, and internally and externally-carried fuel is considered using time marching analysis. Results show that the influence of the fuel, ignoring the added mass effect, is to raise the flutter boundary at transonic speeds, but that this effect is less pronounced at lower Mach numbers. The stability boundary crossing is also found to be less abrupt when the effect of fuel is included and limit cycles often appear. An external fuel tank is seen to exhibit a lower stability boundary, while the response shows a beating effect symptomatic of two similar frequency components, potentially due to interaction between vertical and horizontal fuel motion

Future geodesic completeness of some spatially homogeneous solutions of the vacuum Einstein equations in higher dimensions

It is known that all spatially homogeneous solutions of the vacuum Einstein equations in four dimensions which exist for an infinite proper time towards the future are future geodesically complete. This paper investigates whether the analogous statement holds in higher dimensions. A positive answer to this question is obtained for a large class of models which can be studied with the help of Kaluza-Klein reduction to solutions of the Einstein-scalar field equations in four dimensions. The proof of this result makes use of a criterion for geodesic completeness which is applicable to more general spatially homogeneous models.Comment: 18 page

Fuchsian methods and spacetime singularities

Fuchsian methods and their applications to the study of the structure of spacetime singularities are surveyed. The existence question for spacetimes with compact Cauchy horizons is discussed. After some basic facts concerning Fuchsian equations have been recalled, various ways in which these equations have been applied in general relativity are described. Possible future applications are indicated

Existence of families of spacetimes with a Newtonian limit

J\"urgen Ehlers developed \emph{frame theory} to better understand the relationship between general relativity and Newtonian gravity. Frame theory contains a parameter $\lambda$, which can be thought of as $1/c^2$, where $c$ is the speed of light. By construction, frame theory is equivalent to general relativity for $\lambda >0$, and reduces to Newtonian gravity for $\lambda =0$. Moreover, by setting \ep=\sqrt{\lambda}, frame theory provides a framework to study the Newtonian limit \ep \searrow 0 (i.e. $c\to \infty$). A number of ideas relating to frame theory that were introduced by J\"urgen have subsequently found important applications to the rigorous study of both the Newtonian limit and post-Newtonian expansions. In this article, we review frame theory and discuss, in a non-technical fashion, some of the rigorous results on the Newtonian limit and post-Newtonian expansions that have followed from J\"urgen's work

Monotonic functions in Bianchi models: Why they exist and how to find them

All rigorous and detailed dynamical results in Bianchi cosmology rest upon the existence of a hierarchical structure of conserved quantities and monotonic functions. In this paper we uncover the underlying general mechanism and derive this hierarchical structure from the scale-automorphism group for an illustrative example, vacuum and diagonal class A perfect fluid models. First, kinematically, the scale-automorphism group leads to a reduced dynamical system that consists of a hierarchy of scale-automorphism invariant sets. Second, we show that, dynamically, the scale-automorphism group results in scale-automorphism invariant monotone functions and conserved quantities that restrict the flow of the reduced dynamical system.Comment: 26 pages, replaced to match published versio

The Einstein-Vlasov sytem/Kinetic theory

The main purpose of this article is to guide the reader to theorems on global properties of solutions to the Einstein-Vlasov system. This system couples Einstein's equations to a kinetic matter model. Kinetic theory has been an important field of research during several decades where the main focus has been on nonrelativistic- and special relativistic physics, e.g. to model the dynamics of neutral gases, plasmas and Newtonian self-gravitating systems. In 1990 Rendall and Rein initiated a mathematical study of the Einstein-Vlasov system. Since then many theorems on global properties of solutions to this system have been established. The Vlasov equation describes matter phenomenologically and it should be stressed that most of the theorems presented in this article are not presently known for other such matter models (e.g. fluid models). The first part of this paper gives an introduction to kinetic theory in non-curved spacetimes and then the Einstein-Vlasov system is introduced. We believe that a good understanding of kinetic theory in non-curved spacetimes is fundamental in order to get a good comprehension of kinetic theory in general relativity.Comment: 31 pages. This article has been submitted to Living Rev. Relativity (http://www.livingreviews.org

Cosmological post-Newtonian expansions to arbitrary order

We prove the existence of a large class of one parameter families of solutions to the Einstein-Euler equations that depend on the singular parameter \ep=v_T/c (0<\ep < \ep_0), where $c$ is the speed of light, and $v_T$ is a typical speed of the gravitating fluid. These solutions are shown to exist on a common spacetime slab M\cong [0,T)\times \Tbb^3, and converge as \ep \searrow 0 to a solution of the cosmological Poisson-Euler equations of Newtonian gravity. Moreover, we establish that these solutions can be expanded in the parameter \ep to any specified order with expansion coefficients that satisfy \ep-independent (nonlocal) symmetric hyperbolic equations

Computing Gowdy spacetimes via spectral evolution in future and past directions

We consider a system of nonlinear wave equations with constraints that arises from the Einstein equations of general relativity and describes the geometry of the so-called Gowdy symmetric spacetimes on T3. We introduce two numerical methods, which are based on pseudo-spectral approximation. The first approach relies on marching in the future time-like direction and toward the coordinate singularity t=0. The second approach is designed from asymptotic formulas that are available near this singularity; it evolves the solutions in the past timelike direction from "final" data given at t=0. This backward method relies a novel nonlinear transformation, which allows us to reduce the nonlinear source terms to simple quadratic products of the unknown variables. Numerical experiments are presented in various regimes, including cases where "spiky" structures are observed as the coordinate singularity is approached. The proposed backward strategy leads to a robust numerical method which allows us to accurately simulate the long-time behavior of a large class of Gowdy spacetimes.Comment: 19 pages, 12 figure