A symmetry analysis of some classes of evolutionary nonlinear (2+1)-diffusion equations with variable diffusivity

In this paper the (2+1)-nonlinear diffusion equation u _t ?div(f(u)grad u)=0 with variable diffusivity is considered. Using the Lie method, a complete symmetry classification of the equation is presented. Reductions, via two-dimensional Lie subalgebras of the equation, to first- or second-order ordinary differential equations are given. In a few interesting cases exact solutions are presented.     

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.     

The structure of compressible starting vortices

The structures of highly two-dimensional shock-induced compressible starting vortices were investigated. Density distributions across the vortex were measured by dual-pulsed holographic interferometry. Pressure profiles of the vortex were measured by fast-response miniature Kulite transducers. From these two independent measurements the velocity profile was calculated using the radial momentum equation. The detailed vortex structure is similar to that from the tip of a wing and consists of four well-defined regions: a core region, a logarithmic region, a transition region, and an inviscid region. The distribution of circulation of the vortices can be expressed by a set of empirical formulae. The proportionality constants of the logarithmic region were found to be 0.43±0.01 for three different two-dimensional vortices.Presently at U.S. Army Aeroflightdynamics Directorate, ATCOM, NASA-Ames Research Center, Moffett Field, CA 94035Professor     

On the elastodynamics of stiffened plates

Summary A method of determining the response of a stiffened elastic plate to a dynamic excitation is presented here. Green's function for such a combined system has been systematically derived based on the eigenfunctions of the system obtained through the application of Galerkin's method. With that Green's function a wide range of dynamic forcing functions can be considered.

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Zur elastodynamik versteifter platten

Übersicht Vorgestellt wird eine Methode zur Ermittlung der Antwort versteifter Platten auf dynamische Erregungen. Mit Hilfe der Eigenfunktionen des Systems, die durch Anwendung der Galerkin-Methode gewonnen werden, wird die Greensche Funktion solcher kombinierten Tragwerke systematisch hergeleitet. Damit kann ein weiter Bereich dynamischer Anregungsfunktionen behandelt werden.

     

Stopping a Viscous Fluid by a Feedback Dissipative Field: I. The Stationary Stokes Problem

We consider a planar stationary flow of an incompressible viscous fluid in a semiinfinite strip governed by the Stokes system with a body forces field. We show how this fluid can be stopped at a finite distance of the entrance of the semi-infinite strip by means of a feedback field depending in a sub-linear way on the velocity field. This localization effect is proved reducing the problem to a non-linear bi-harmonic type one for which the localization of solutions is obtained by means of the application of a suitable energy method. Since the presence of the non-linear terms defined through the body forces field is not standard in the fluid mechanics literature, we establish also some results about the existence and uniqueness of weak solutions for this problem.     

Pressure field due to drag reducing outer layer devices in turbulent boundary layers

The wall static pressure in the vicinity of drag reducing outer layer devices in flat wall turbulent boundary layers has been measured and compared with an inviscid theory. Symmetric and cambered airfoil devices have been examined at small angles of attack and very low chord Reynolds numbers. Airfoil devices impose a sequence of strong favorable and adverse pressure gradients on the boundary layer whose drag is to be reduced. At very small angles of attack (± 2°), this pressure field extends up to about three chord lengths downstream of the trailing edge of an airfoil device. Also examined are the pressures on the upper and lower surfaces of a symmetric airfoil device in the freestream and near the wall. The freestream pressure distribution around an airfoil section is altered by the wall proximity. The relevance of lift enhancement caused by wall proximity to drag reduction has been discussed. The pressure distributions on the flat wall beneath the symmetric airfoil devices are predicted well by the inviscid theory. However, the remaining pressure distributions are predicted only qualitatively, presumably because of strong viscous effects.     

Effets d'entrée en écoulements pulsés non-newtoniens

Sommaire Le présent travail concerne l'étude des champs de vitesse dans la région d'entrée dans un écoulement pulsé, newtonien et pseudoplastique en conduites cylindriques circulaires. Trois méthodes d'étude ont été utilisées: une méthode analytique, une méthode aux différences finies et une méthode expérimentale basée sur la vélocimétrie à laser. Les résultats obtenus à l'aide de ces méthodes sont confrontés entre eux. Les influences des paramètres rhéologiques et hydrodynamiques sur ces écoulements sont également analysées.

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Summary The purpose of this work is the study of the entrance effects for pulsating flow of inelastic non-newtonian fluids in cylindrical tubes. Founding upon classic methods relating to steady flows, we consider this problem as development of hydrodynamic boundary layer. The evolutions of velocity profiles have been obtained on the one hand by laser velocimetry on aqueous solutions of high polymers and, on the other hand, by numerical resolution of non-linear equations relating to such flows. The relative influences of rheological and hydrodynamical parameters have been analysed and compared.

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Zusammenfassung Die vorliegende Veröffentlichung befaßt sich mit der Einlaufströmung in zylindrische Rohre für den Fall pulsierender Strömungen newtonscher und unelastischer nicht-newtonscher Flüssigkeiten. Dabei werden drei Untersuchungsmethoden verwendet: eine analytische Methode, eine numerische Methode auf der Basis der Finiten Differenzen und eine experimentelle Methode unter Einsatz eines Laser-Doppler-Anemometers. Die mit Hilfe dieser Methoden gewonnenen Ergebnisse werden miteinander verglichen und der Einfluß der verschiedenen rheologischen und hydrodynamischen Parameter auf die Strömungsform wird analysiert.

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Avec 9 figures et 3 tableaux     

On the elastic-viscoplastic proportional combined twist and stretch of a circular bar of rate-sensitive material

The elastic-plastic behaviour of a solid circular bar made of a homogeneous, isotropic and nonhardening material but of a rate-dependent type, subjected to different proportional deformation programmes of twist and stretch, is analysed. In this investigation the material is assumed to yield according to the von Mises criterion and then to follow a rate-dependent post-yield constitutive law of the Perzyna type. Four radial paths in the angte-of-twist and axial-stretch plane were investigated. For each deformation path the numerical solution of the governing system of quasi-linear partial differential equations gave the corresponding loading path and the radial variation of the stress field at selected times. This study complements that of S. A. Meguid and J.D. Campbell (1979) which considered mainly nonproportional deformation paths. In this work residual stress profiles were also calculated, based upon instantaneous unloading after various amounts of deformation. Comparison with predictions of a rateindependent theory for a limiting state, where elastic strains are negligible, is made. Over a considerable range, the loading paths beyond the initial yield were nearly straight, at the elastic slope. The load trajectory then bends over and approaches a point on the limit-state locus. The results showed that the approximate procedure of increasing the flow stress by an empirically determined factor corresponding to a mean plastic strain-rate gave a good estimate for the limit-state stresses.     

Wärmeübergang bei erzwungener turbulenter Strömung in Ringspalten und örtlich beliebig veränderlichen Randbedingungen zweiter Art

Zusammenfassung Das Problem des Wärmeübergangs bei turbulenter Strömung in konzentrischen Ringspalten wird für den dreidimensionalen Fall theoretisch gelöst, wobei die Wandwärmestromdichte sowohl in azimutaler als auch in axialer Richtung beliebig variiert. Die Lösung der Energiegleichung erfolgt mit der klassischen Methode der Superposition und Trennung der Variablen, wobei das dabei auftretende Sturm-Liouvillesche Eigenwertproblem numerisch gelöst wird. Zur Lösung werden Verteilungen für die Geschwindigkeit und anisotropen turbulenten Austauschgrößen verwendet, die mit dem phänomenlogischen Turbulenzmodell von Ramm berechnet wurden. Ergebnisse werden über einen weiten Bereich der Reynolds-Zahl (10⁴ Re 10⁶), der Prandtl-Zahl (0 Pr 100) und für verschiedene Radienverhältnisse diskutiert.

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Turbulent forced convection heat transfer in annuli with arbitrarily varying boundary conditions of second kind

The problem of turbulent flow heat transfer in concentric annuli is analysed for the general threedimensional case in which the wall heat flux varies arbitrarily in both the circumferential and axial directions. The energy equation is solved using the classical method of superposition and separating variables, where the resulting Sturm-Liouville problem are evaluated numerically. The solution is based on velocity profiles and anisotropic thermal turbulent transport properties evaluated by Ramm's phenomenological turbulence model. Results are discussed over a wide range of Reynolds number (10⁴ Re 10⁶), Prandtl number (0 Pr 100) and radius ratio.

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Bezeichnungen a,b Fourierkoeffizienten - B geometrische Funktion, [s(1-r) + r]/(1–s) - C Koeffizienten - D hydraulischer Durchmesser, 2(r₂ – r₁) - E Energietransportfunktion - f axiale Wärmestromdichteverteilung - F azimutale Wärmestromdichteverteilung - g radiale Temperaturfunktion - l Kanallänge - L dimensionslose Kanallänge, 1/D - M axialer Temperaturgradient im thermisch ausgebildeten Bereich - n harmonischer Parameter - Nu Nusselt-Zahl - Pe Péclet-Zahl - Pr Prandtl-Zahl - q Wärmestromdichte - Q dimensionslose Wärmestromdichte, q/q₀ - r dimensionslose radiale Koordinate, (R-r₁)/(r₂-r₁) - r₁,r₂ innerer und äußerer Ringspaltradius - R radiale Koordinate - Re Reynolds-Zahl - s Ringspaltverhältnis, r₁/r₂ - T dimensionslose Temperatur, 2· · (-_E/(D· q₀ - u dimensionslose Geschwindigkeit, U/U_m - U Geschwindigkeit - x dimensionslose axiale Koordinate, X/D - X axiale Koordinate - Wärmeübergangskoeffizient - _un modifizierter Eigenwert - halber Segmentwinkel - turbulente Austauschgröe - Temperatur - dimensionslose Temperaturdifferenz, T - T_m - Wärmeleitfähigkeit - _un Eigenwerte - kinematische Viskosität - azimutale Koordinate - Eigenfunktionen Indizes e thermischer Einlauf - E Eintritt bei x=0 - H Wärme - i Bedingung an der i-ten benetzten Oberfläche (i=1 – Innenrohr, i=2 - Außenrohr) - j Bedingung, wenn nur an der j-ten Oberfläche des Ringspaltes die Wärme übertragen wird (j=1,2) - ij Bedingung an der i-ten Oberfläche, wenn nur an der j-ten Oberfläche des Ringspaltes die Wärme übertragen wird (ij=11, 12, 22, 21) - m mittel - n Ordnung der Harmonischen - r radiale Richtung - u Ordnung des Eigenwertproblems - azimutale Richtung - 0 umfangskonstant - thermisch ausgebildet     

Point force in a MHB rotating fluid

We consider the steady flow of an inviscid, rotating fluid confined in a cylinder under the influence of a uniform axial magnetic field, in the presence of an isolated point force. It is found that, the wave-like terms, infinite in number, occurring on the downstream side are not persistent and decay at far distances, in contrast to the non-magnetic case. Also the presence of a uniform magnetic field induces an additional finite number of decaying modes on the upstream side and an infinite number of similar modes on the downstream side, which reduce to the geostrophic terms, in the corresponding non-magnetic case. The downstream solution reveals that discontinuities occur in the velocity gradients inside the fluid region.