Effects of non-denumerable fixed points in finite dynamical systems

The motion of a spinning football brings forth the possible existence of a whole class of finite dynamical systems where there may be non-denumerably infinite number of fixed points. They defy the very traditional meaning of the fixed point that a point on the fixed point in the phase space should remain there forever, for, a fixed point can evolve as well! Under such considerations one can argue that a free-kicked football should be non-chaotic.Comment: This paper is a replaced version to modify the not-so-true claim, made unknowingly in the earlier version, of being first to propose the peculiar dynamical systems as described in the paper. With respect to the original workers, we present here our original finding

Transition to turbulence in particulate pipe flow

We investigate experimentally the influence of suspended particles on the transition to turbulence. The particles are monodisperse and neutrally-buoyant with the liquid. The role of the particles on the transition depends both upon the pipe to particle diameter ratios and the concentration. For large pipe-to-particle diameter ratios the transition is delayed while it is lowered for small ratios. A scaling is proposed to collapse the departure from the critical Reynolds number for pure fluid as a function of concentration into a single master curve.Comment: 4 pages, 4 figure

Eddy genesis and manipulation in plane laminar shear flow

Eddy formation and presence in a plane laminar shear flow configuration consisting of two infinitely long plates orientated parallel to each other is investigated theoretically. The upper plate, which is planar, drives the flow; the lower one has a sinusoidal profile and is fixed. The governing equations are solved via a full finite element formulation for the general case and semi-analytically at the Stokes flow limit. The effects of varying geometry (involving changes in the mean plate separation or the amplitude and wavelength of the lower plate) and inertia are explored separately. For Stokes flow and varying geometry, excellent agreement between the two methods of solution is found. Of particular interest with regard to the flow structure is the importance of the clearance that exists between the upper plate and the tops of the corrugations forming the lower one. When the clearance is large, an eddy is only present at sufficiently large amplitudes or small wavelengths. However, as the plate clearance is reduced, a critical value is found which triggers the formation of an eddy in an otherwise fully attached flow for any finite amplitude and arbitrarily large wavelength. This is a precursor to the primary eddy to be expected in the lid-driven cavity flow which is formed in the limit of zero clearance between the plates. The influence of the flow driving mechanism is assessed by comparison with corresponding solutions for the case of gravity-driven fluid films flowing over an undulating substrate. When inertia is present, the flow generally becomes asymmetrical. However, it is found that for large mean plate separations the flow local to the lower plate becomes effectively decoupled from the inertia dominated overlying flow if the wavelength of the lower plate is sufficiently small. In such cases the local flow retains its symmetry. A local Reynolds number based on the wavelength is shown to be useful in characterising these large-gap flows. As the mean plate separation is reduced, the form of the asymmetry caused by inertia changes, and becomes strongly dependent on the plate separation. For lower plate wavelengths which do not exhibit a cinematically induced secondary eddy, an inertially induced secondary eddy can be created if the mean plate separation is sufficiently small and the global Reynolds number sufficiently large

Numerical simulations of compressible Rayleigh-Taylor turbulence in stratified fluids

We present results from numerical simulations of Rayleigh-Taylor turbulence, performed using a recently proposed lattice Boltzmann method able to describe consistently a thermal compressible flow subject to an external forcing. The method allowed us to study the system both in the nearly-Boussinesq and strongly compressible regimes. Moreover, we show that when the stratification is important, the presence of the adiabatic gradient causes the arrest of the mixing process.Comment: 15 pages, 11 figures. Proceedings of II Conference on Turbulent Mixing and Beyond (TMB-2009

Utilization of granular solidification during terrestrial locomotion of hatchling sea turtles

Biological terrestrial locomotion occurs on substrate materials with a range of rheological behaviour, which can affect limb-ground interaction, locomotor mode and performance. Surfaces like sand, a granular medium, can display solid or fluid-like behaviour in response to stress. Based on our previous experiments and models of a robot moving on granular media, we hypothesize that solidification properties of granular media allow organisms to achieve performance on sand comparable to that on hard ground. We test this hypothesis by performing a field study examining locomotor performance (average speed) of an animal that can both swim aquatically and move on land, the hatchling Loggerhead sea turtle (Caretta caretta). Hatchlings were challenged to traverse a trackway with two surface treatments: hard ground (sandpaper) and loosely packed sand. On hard ground, the claw use enables no-slip locomotion. Comparable performance on sand was achieved by creation of a solid region behind the flipper that prevents slipping. Yielding forces measured in laboratory drag experiments were sufficient to support the inertial forces at each step, consistent with our solidification hypothesis

Simple Viscous Flows: from Boundary Layers to the Renormalization Group

The seemingly simple problem of determining the drag on a body moving through a very viscous fluid has, for over 150 years, been a source of theoretical confusion, mathematical paradoxes, and experimental artifacts, primarily arising from the complex boundary layer structure of the flow near the body and at infinity. We review the extensive experimental and theoretical literature on this problem, with special emphasis on the logical relationship between different approaches. The survey begins with the developments of matched asymptotic expansions, and concludes with a discussion of perturbative renormalization group techniques, adapted from quantum field theory to differential equations. The renormalization group calculations lead to a new prediction for the drag coefficient, one which can both reproduce and surpass the results of matched asymptotics

Wave Energy

Part of the immense solar energy input to the earth is converted by natural processes into energy associated with ocean waves. The geographical location of the United Kingdom renders it one of the world ' s most favoured countries with respect to the potential availability of wave energy. In principle. the waves reaching our coastal waters from the North Atlantic might satisfy a considerable fraction of our electricity demand provided that reasonably high overall conversion efficiencies can be achieved . Inventors have recognised the power of the sea for many decades. and there has been no lack of ideas on how it might be tapped . But none of the ideas was developed on a substantial scale. since ample and relatively cheap supplies of other resources were always available . In recent years. however. there has been a growing recognition that--on a world scale-the presently used forms of energy may become too expensive,. too scarce or otherwise unavailable to meet our energy needs by themselves . The Government's responsibility is to ensure that as wide a range as possible of energy supply options are available when they may be needed . Research and development can provide the necessary technical and econom i c data on which the ultimate choices can be made . Within this context. the Government announced in 197 6 the start of an R and D programme on wave energy for which the first phase was to be a feasibility study lasting for two years. The funding level has been increased twice since that time to maintain the momentum of the programme in the light of technical progress . The programme has had three main components: -exploratory deve lopment of several different engineering concepts of wave energy converter; -supporting research in relevant engineering and scientific areas : • the collection and analysis of wave data. • analysis of the structural response to wave-induced motions. • mooring. • energy conversion and transm issi on. • environmental aspects ; -working up preliminary reference designs of full scale stations for tech nical and economic appraisal. The purpose of this paper is to review the present state of knowledge of wave energy in the light of the achievements of the first two-year phase of the programme . Development of the converters Four potential designs of converter were adopted for initial study. since the sparse data available were insufficient to enable a single concept to be chosen with confidence. Proposals for alternative concepts are received on a continuing basis and are assessed against a number of criteria: two of them have so far been added to the programme in order to explore new principles Apart from the basic technical differences the six designs differ from each other in their degree of complexity and their state of development. as described in Chapter 3 . Work on two of the designs has been advanced from laboratory wave tanks to the testing of 1 / 1 0th scale models in natural open water at Loch Ness and in the Solent. For all the designs . a combination of theoretical studies. laboratory work and engineering appraisal has clarified the factors which will prove to be the most crucial in determining which of them could be chosen for more extensive development. The programme has progressed from establishing the scientific feasibility of wave energy converters to confirming the engineering feasibility of designing and building some of the designs . In very broad terms: -the early part of the programme placed considerable emphasis on optimising the efficiency of extraction of the wave energy and proving the scientific feasibility ; -the present stage is concerned with the technical viability and is identifying the main cost centres in the designs . which can then be tackled by further R and D; -the immediate future must also place emphasis not only on the problems of construction . operation and maintenance. and on ways in which unit costs may be reduced . but also on the ability to survive in the most severe wave conditions . Whilst the technical feasibility of some types of converter has been established, we are far from the stage of recommending that a full scale generating station should be built . Of the four original concepts. no single design has yet emerged which is outstandingly better than the other designs when all factors are taken into account . The designs have changed considerably in the course of the feasibility study. and a continuing process of evol ution can be expected as in the early stages of any technical development programme. The optimum design may emerge from further changes in one of the original concepts. from a synthesis of ideas or from an alternative concept . The wave energy is distributed over a wide frequency and energy bandwidth and no des ign has yet been optimised to operate at or near peak efficiency over the whole spectrum. However, one of the new concepts introduced into the programme recently may offer significant advantages in this respect. Supporting research The extent of the available data on waves in the sea areas of primary interest is inadequate as yet for the full assessment of the resource . A start has been made in collecting and analysing new data. which will take several years to reach a satisfactory level. The results so far confirm the general point that the locations around the United Kingdom with the most abundant wave energy lie to the west of the Outer Hebrides, where several hundred km of searoom are available with average annual power levels in the range 35-60 kW / m of wavefront. The majority of the designs un der consideration are free-floating and the converters would operate on or near the water surface-- one of the most hostile environments for engineering structures . The ult imate feasibility. technical and economic . of all designs of floating converter will depend upon extensive further work on mooring and anchoring . Whilst over-designed mooring systems based on present knowledge have allowed the open water trials to proceed with the objective of gaining experience . the existing knowledge is not adequate to design cost-effective mooring systems which will ensure survival at full scale under storm conditions . Considerable progress has been made in assessing and under3tanding suitable energy conversion and transmission systems for the various designs of converter , but much more remains to be done to arrive at the most cost-effective solutions . The general engineering difficulties are quite basic and are related to the properties of the natural wave spectrum : -the conversion system must be able to handle large short-term variations in the instantaneous power level ; -the peak power level in the sea (of the order of 10,OOO kW /m) can be many times greater than the average power level (a few tens of kW / m) ; -the primary output is not in a form which can be handled conventionally (it is . of course . variable with time in a complex way). Moreover. apart from the randomness. other general problems arise from the low energy density of the input and the relatively low speeds and frequency of movement induced by the waves . Engineering devices to transmit large amounts of energy under such conditions must themselves be large. heavy and expensive . The efficient generation of electricity requires machinery operating at relatively high and preferably constant speed . The transition from the one regime to the other appears to be more straightforward for systems involving air turbines than for those which do not: some designs of converter may prove to be intractable in this respect. Many possible forms of energy transmission to the mainland have been reviewed . as summarised in Chapter 6 . Whilst it has been recommended that several options should be kept open in the continuing studies it is likely that most attention will be given to electricity. The overall flow of energy from the waves to a final user connected to the electricity grid involves many separate steps . each of which can involve loss of some of the energy . This can have a considerable influence on the system economics and further work in this area will need to concentrate on both reducing the number of steps and increasing the efficiency (including the directional efficiency of t he converters themselves ) of those which must remain . Unless this can be achieved the usable resource will be only a small proportion of our needs : some pointers to substantial improvements are beginning to emerge. Environmental studies have not revealed any major detrimental effects of the converters provided they are well offshore. More information is needed on the behaviour of salmon and herring off the Outer Hebrides to confirm that the fisheries would not be affected significantly by the widespread installation of converters. Concluding remarks: The costing studies of the reference designs which have been evolved so far indicate that wave-produced electricity is likely to be expensive compared with either nuclear or fossil fuels unless some major breakthrough in the engineering can be achieved . However . this does not imply that the possibility of wave energy should be abandoned at this stage . It must be emphasised that the subject is still at a very ear ly state of development and many unknown factors remain to be resolved . Under these circumstances, wave energy is best regarded at present as a possible insurance technology- the consequences of failure of one of our existing major energy supplies are so severe that it is worth paying an insurance premium to explore fully the alternatives . Nevertheless. the evidence from the feasibility study so far does not allow a recommendation for a full-scale development programme to be made at this time . Much more can be achieved to explore and then to narrow the design choices by continuing work at about the 1 / 1 0th scale coupled with . on the one hand . limited trials of some critical components at larger scale and . on the other hand . f urther creative engineering on the drawing board and laboratory work in a new generation of wave tanks (of which the forerunner has been successfully commissioned at Edinburgh University). The programme has generated a broad basis of knowledge of all aspects of wave energy which did not exist before. so that we can now identify clearly the critical problems to be tackled by further work