The exponentially convergent trapezoidal rule

It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators

The application of the boundary integral equation method to foundation dynamics

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Radiation-induced instability of a finite-chord Nemtsov membrane

We consider a problem of stability of a membrane of an infinite span and a finite chord length that is submerged in a uniform flow of finite depth with free surface. In the shallow water approximation, Nemtsov (1985) has shown that an infinite-chord membrane is susceptible to flutter instability due to excitation of long gravity waves on the free surface if the velocity of the flow exceeds the phase velocity of the waves and placed this phenomenon into the general physical context of the anomalous Doppler effect. In the present work we derive a full nonlinear eigenvalue problem for an integro-differential equation in the case of the finite-chord Nemtsov membrane in the finite-depth flow. In the shallow- and deep water limits we develop a perturbation theory in the small added mass ratio parameter acting as an effective dissipation parameter in the system, to find explicit analytical expressions for the frequencies and the growth rates of the membrane modes coupled to the surface waves. This result reveals a new intricate pattern of instability pockets in the parameter space and allows for its analytical description. The case of an arbitrary depth flow with free surface requires numerical solution of a new non-polynomial nonlinear eigenvalue problem. We propose an original approach combining methods of complex analysis and residue calculus, Galerkin discretization, Newton method and parallelization techniques implemented in MATLAB to produce high-accuracy stability diagrams within an unprecedentedly wide range of system's parameters. We believe that the Nemtsov membrane appears to play the same paradigmatic role for understanding radiation-induced instabilities as the famous Lamb oscillator coupled to a string has played for understanding radiation damping

Effect of chordwise forces and deformations and deformations due to steady lift on wing flutter

This investigation explores the effects of chordwise forces and deformations and steady-state deformation due to lift on the static and dynamic aeroelastic stability of a uniform cantilever wing. Results of this analysis are believed to have practical applications for high-performance sailplanes and certain RPV's. The airfoil cross section is assumed to be symmetric and camber bending is neglected. Motions in vertical bending, fore-and-aft bending, and torsion are considered. A differential equation model is developed, which included the nonlinear elastic bending-torsion coupling that accompanies even moderate deflections. A linearized expansion in small time-dependent deflections is made about a steady flight condition. The stability determinant of the linearized system then contains coefficients that depend on steady displacements. Loads derived from two-dimensional incompressible aerodynamic theory are used to obtain the majority of the results, but cases using three-dimensional subsonic compressible theory are also studied. The stability analysis is carried out in terms of the dynamically uncoupled natural modes of vibration of the uniform cantilever

Numerical modelling of polydispersed flows using an adaptive-mesh finite element method with application to froth flotation

An efficient numerical framework for the macroscale simulation of three-phase polydispersed flows is presented in this thesis. The primary focus of this research is on modelling the polydispersity in multiphase flows ensuring the tractability of the solution framework. Fluidity, an open-source adaptive-mesh finite element code, has been used for solving the coupled equations efficiently. Froth flotation is one of the most widely used mineral processing operations. The multiphase, turbulent and polydispersed nature of flow in the pulp phase in froth flotation makes it all the more challenging to model this process. Considering that two of the three phases in froth flotation are polydispersed, modelling this polydispersity is particularly important for an accurate prediction of the overall process. The direct quadrature method of moments (DQMOM) is implemented in the Fluidity code to solve the population balance equation (PBE) for modelling the polydispersity of the gas bubbles. The PBE is coupled to the Eulerian--Eulerian flow equations for the liquid and gas phases. Polydispersed solids are modelled using separate transport equations for the free and attached mineral particles for each size class. The PBE has been solved using DQMOM in a finite element framework for the first time in this work. The behaviour of various finite element and control volume discretisation schemes in the solution of the PBE is analysed. Rigorous verification and benchmarking is presented along with model validation on turbulent gravity-driven flow in a bubble column. This research also establishes the importance of modelling the polydispersity of solids in flotation columns, which is undertaken for the first time, for an accurate prediction of the flotation rate. The application of fully-unstructured anisotropic mesh adaptivity to the polydispersed framework is also analysed for the first time. Significant improvement in the solution efficiency is reported through its use.Open Acces

Aerodynamic Analyses Requiring Advanced Computers, part 2

Papers given at the conference present the results of theoretical research on aerodynamic flow problems requiring the use of advanced computers. Topics discussed include two-dimensional configurations, three-dimensional configurations, transonic aircraft, and the space shuttle

Survey and Evaluation of Supersonic Base Flow Theories

Survey and evaluation of supersonic base flow theorie

Radiative and diffusive instabilities in moving fluids

Radiation and diffusion are inherent phenomena in Nature that anyone of us already witnessed in the form of, e.g., the propagation of gravity waves at the surface of water or the thermal conduction in solid structures. A certain emission of energy is evidently associated with the presence of damping in all sorts of dynamical systems, from the simplest spring-mass configurations to the more complex stellar structures. For instance, as historically predicted for the model of a rotating self-gravitating mass of fluid, presence of dissipation in the form of viscosity may lead to the onset of a secular instability, in the form of oscillatory motions. This discovery is nowadays well-accepted as the first illustration of the so-called dissipation-induced instability, a particular sort of instability arising in the presence of damping. A similar effect can be encountered in the context of the emission of waves carrying modes of opposite energy sign and yields the analogue radiation-induced instability. We propose hereafter along this thesis to present a selection of problems involving radiative or diffusive mechanisms and to carry out an exhaustive linear stability analysis on them. Respectively, the different chapters are ordered as referring to the stability of the Maclaurin spheroids, the lenticular vortices, the rotating magnetohydrodynamics flows and the fluid-structure interactions of an elastic membrane with a uniform potential flow. A particular attention will be addressed to systems subject to two simultaneous dissipative mechanisms, in order to estimate the predominance of the damped mode of importance. By means of various analytical treatments, as well as supporting numerical methods, we manage to solve the different boundary value problems associated with each system and thus establish new stability criteria in the spaces of parameters. The developed methods remain general in their formulation and are not solely designed to only solve the problems of interest. As a matter of fact, one can easily adapt our approach to a broad range of application