Reduced‐order modelling of an adaptive mesh ocean model

A novel proper orthogonal decomposition (POD) model has been developed for use with an advanced unstructured mesh finite‐element ocean model, the Imperial College Ocean Model (ICOM, described in detail below), which includes many recent developments in ocean modelling and numerical analysis. The advantages of the POD model developed here over existing POD approaches are the ability:

• 1. To increase accuracy when representing geostrophic balance (the balance between the Coriolis terms and the pressure gradient). This is achieved through the use of two sets of geostrophic basis functions where each one is calculated by basis functions for velocities u and v.
• 2. To speed up the POD simulation. To achieve this a new numerical technique is introduced, whereby a time‐dependent matrix in the discretized equation is rapidly constructed from a series of time‐independent matrices. This development imparts considerable efficiency gains over the often‐used alternative of calculating each finite element over the computational domain at each time level.
• 3. To use dynamically adaptive meshes in the above POD model.

Copyright © 2008 John Wiley & Sons, Ltd.     

Thermoacoustically driven flame motion and heat release variation in a swirl-stabilized gas turbine burner investigated by LIF and chemiluminescence

Laser-induced fluorescence and chemiluminescence, both phase-locked to the dominant acoustic oscillation, are used to investigate phenomena related to thermoacoustic instability in a swirl-stabilized industrial scale gas turbine burner. The observed sinusoidal phase-averaged flame motion in axial (main flow) direction is analyzed using different schemes for defining the flame position. Qualitative agreement between experimental data and theoretical analysis of the observed flame motion is obtained, interpreted as originating primarily from variation of the burning velocity. The heat release variation during an acoustic cycle is determined from the sinusoidally varying total OH* chemiluminescence intensity.

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On the applicability of some approximate similitude laws in hypersonic aerodynamics

The range of applicability of some similitude laws for heat transfer, friction and drag coefficients is discussed on the basis of numerical solutions of the complete viscous shock layer equations describing hypersonic flow past blunt bodies. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 121–130, March–April, 1994.     

On the dynamic behaviour of a flexible beam carrying a moving mass

The behaviour of a system containing a mass traveling on a cantilever beam is considered. The mass is induced to move by an applied force as opposed to the case which has been considered in most literature where the position of the moving mass is assumed to be known and independent of the motion of the beam. Furthermore, the system to be discussed has the unique characteristic that the motions of the mass and the beam are coupled. The mathematical model of the system includes two coupled nonlinear integral/partial differential equations which are impossible to solve analytically and are difficult to solve numerically in their original form. As a remedy, the solution is discretized into space and time functions and the equations of motion are reduced to a set of ordinary differential equations. The shape function is chosen so that it satisfies the boundary conditions of the beam as well as the transient conditions imposed by the traveling mass. This choice of the shape function, which considers the mass-beam interaction, provides an improvement over the conventional method of using a simple cantilever beam mode shapes.The ordinary differential equations of motion using the improved shaped functions, are solved numerically to obtain the dynamic behaviour of the system. The results illustrate the validity of the model, and demonstrate the advantages of the improved model to the un-improved equations.     

A time-reversal method for an acoustical pulse propagating in randomly layered media

In the recent years a considerable amount of mathematical work has been devoted to the study of reflected signals obtained by the propagation of pulses in randomly layered media. We refer to [M. Asch, W. Kohler, G. Papanicolaou, M. Postel and B. White, “Frequency content of randomly scattered signals”, SIAM Review 33 (4), 519–625 (1991)] for an extensive survey and applications to inverse problems. The analysis is based on separation of scales between the correlation scale of the inhomogeneities present in the medium, the typical wavelengths of the pulse and the macroscopic variations of the medium. On the other hand, in the context of ultrasounds, time-reversal mirrors have been developed and their effects have been studied experimentally by Mathias Fink and his team at the Laboratoire Ondes et Acoustique (ESPCI-Paris). We refer to: [M. Fink, “Time reversal mirrors”, J. Phys. D: Appl. Phys. 26, 1333–1350 (1993)]. Our goal is to present a mathematical analysis of a time-reversal method for analyzing reflected signals in the model described in [M. Asch, W. Kohler, G. Papanicolaou, M. Postel and B. White, “Frequency content of randomly scattered signals”, SIAM Review 33(4), 519–625 (1991)]. We restrict our analysis to the one-dimensional case, the three-dimensional layered case being the content of a forthcoming paper. It is noticeable that we do not introduce new mathematics in the problem but simply put together an already existing mathematical theory and a new device, the time-reversal mirror.     

Polymer contribution to the thermal conductivity and viscosity in a dilute solution (Fraenkel Dumbbell model)

The phase-space kinetic theory for polymeric liquid mixtures is used to obtain an expression for the polymer contribution to the thermal conductivity of a nonflowing, dilute solution of polymers, where the polymer molecules are modeled as Fraenkel dumbbells. This theory takes into account three mechanisms for the energy transport: diffusion of kinetic energy (including the Öttinger-Petrillo term), diffusion of intramolecular energy, and the work done against the intramolecular forces. This paper is an extension of previous developments for the Hookean dumbbell model and the finitely-extensible dumbbell model. A comparison among the dumbbell results suggests that the thermal conductivity increases with chain stiffness. In addition, the zero-shear-rate viscosity and first normal-stress coefficient are also given for the Fraenkel dumbbell model.Dedicated to Prof. John D. Ferry on the occasion of his 85th birthday.     

Compatibility conditions for the Cauchy-Green strain fields: Solutions for the plane case

This paper considers the issues related to uniqueness and existence of a finite deformation generated by prescribed right or left Cauchy-Green strain tensor field in the plane. First, the questions of uniqueness and existence to a pre-assigned right strain field C are discussed. It is shown that the existence condition, in the context of continuum mechanics, are naturally posed using the field corresponding to the square root of C instead of C, the latter a classical approach. Then, the corresponding questions for the left strain field are considered, which is more involved. The analysis of uniqueness gives rise to an appropriate classification of the deformation fields. The question of existence is discussed and a complete solution is presented. In both the right and left cases, we stress the techniques for obtaining the corresponding deformation fields.     

Finite amplitude,free vibrations of an axisymmetric load supported by a highly elastic tubular shear spring

The finite amplitude, free vibrational characteristics of a simple mechanical system consisting of an axisymmetric rigid body supported by a highly elastic tubular shear spring subjected to axial, rotational, and coupled shearing motions are studied. Two classes of elastic tube materials are considered: a compressible material whose shear response is constant, and an incompressible material whose shear response is a quadratic function of the total amount of shear. The class of materials with constant shear response includes the incompressible Mooney-Rivlin material and certain compressible Blatz-Ko, Hadamard, and other general kinds of models. For each material class, the quasi-static elasticity problem is solved to determine the telescopic and gyratory shearing deformation functions needed to evaluate the elastic tube restoring force and torque exerted on the body. For all materials with constant shear response, the differential equations of motion are uncoupled equations typical of simple harmonic oscillators. Hence, exact solutions for the forced vibration of the system can be readily obtained; and for this class, engineering design formulae for the load-deflection relations are discussed and compared with experimental results of others'. For the quadratic material, however, the general motion of the body is characterized by a formidable, coupled system of nonlinear equations. The free, coupled shearing motion for which either the axial or the azimuthal shear deformation may be small is governed by a pair of equations of the Duffing and Hill types. On the other hand, the finite amplitude, pure axial and pure rotational motions of the load are described by the classical, nonlinear Duffing equation alone. A variety of problems are solved exactly for these separate free vibrational modes, and a number of physical results are presented throughout.     

Natural oscillations of a jet membrane under hydrostatic pressure

The natural oscillations of a submerged elastic jet membrane modeling an axisymmetric hydrodynamic direct-flow radiator are analyzed. The fundamental frequency is calculated as a function of the geometric and hydrodynamic characteristics of the jet. The principle possibility of adjusting the frequency by changing the elasticity of the fluid, which is a function of hydrostatic pressure, is demonstrated. Numerical and experimental data are compared.__________Translated from Prikladnaya Mekhanika, Vol. 40, No. 12, pp. 92–98, December 2004.     

The discrete Babuška-Brezzi condition

Summary In the treatment of saddle point problems with finite elements a discrete Babuka-Brezzi condition is encountered. We demonstrate what this discrete Babuka-Brezzi condition entails and explain the mechanical background of this somewhat abstract inf sup condition.

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Die diskrete Babuka-Brezzi-Bedingung

Übersicht Behandelt man Sattelpunktprobleme mit Finiten Elementen, dann stößt man auf eine diskrete Babuka-Brezzi Bedingung. Wir zeigen, welche Rolle diese diskrete Babuka-Brezzi-Bedingung bei der numerischen Behandlung spielt und verdeutlichen den mechanischen Hintergrund dieser etwas abstrakten inf sup Bedingung.