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1 

The Ultimate Image Singularities for External Spheroidal and Ellipsoidal Harmonics
The image system of singularities of an arbitrary exterior potential field within a triaxial ellipsoid is derived. It is found that the image system consists of a source and doublet distribution over the fundamental ellipsoid. The present contribution is a generalization of previous theories on the image system of an exterior potential field with in a sphere and spheroid. A proof of Havelock's spheroid theorem which apparently is not available in the literature is also given. The knowledge of the image system is required, for example, when hydrodynamical forces and moments acting on an ellipsoid immersed in a potential flow are computed by the Lagally theorem.
The two examples given consider the image system of singularities of an ellipsoid in a uniform translatory motion and in pure rotation.

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2 

Development of a TwoScale Turbulence Model and Its Applications
The use of second order closure turbulence model in predicting turbulent flows is known to be more successful than the classical mixing length model. However, it is found that if the turbulence constants are not altered or modified, the second order closure turbulence model is unable to predict satisfactorily f or some flows such as round jet and wake flows. In order to improve the predictability of the second order closure model, the present work proposes to consider two turbulent scales in the modelling of turbulent flows. One of these scales is based on using the turbulent kinetic energy, k, and its dissipation rate, epsilon, to characterize the large energy containing eddies. The other scale is based on the dissipation rate and the kinematic viscosity, nie, to characterize the small energy dissipating eddies. The second scale is based on the well known Kolmogorov hypothesis that dissipation of turbulent kinetic energy occurs primarily at small eddies. The turbulence model derived based on the concept of two different scales is called the twoscale turbulence model. The existing turbulence model which is modelled based on the onescale concept of k and epsilon is called the onescale turbulence model.
The twoscale turbulence model is then applied to predict turbulent free shear flows and recirculating flows. The calculations were done in three parts. The first test case was nonbuoyant free shear flows which included round and plane jets in stagnant and moving streams, plane wakes and mixing layer. In the second part, the model was tested for plane and round buoyant jets having different Froude numbers. Finally, some results were obtained for recirculating flows, namely, backward facing step and flow past an obstruction.
It is shown in the present study that the twoscale turbulence model performs significantly better than the onescale turbulence model in all the cases concerned. The prediction capability of the twoscale turbulence model is shown since one does not need to alter or modify the turbulence constants as in the case of the onescale turbulence model.

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3 

An analysis of relationships between flow conditions and statistical measures of bed configurations in straight and curved alluvial channels
The reliability of predictors for friction factors and rates of sediment transport in alluvial channels is open to question when they are applied to sinuous channels. Bedform geometry in a curved channel and a straight flume which are subject to the same nominal flow conditions is investigated by statistical analysis of records of streamed profiles. Autocorrelation, spectral density and probability density functions of a process defined by the bed elevation as a function of the distance along the channel, or as a function of elapsed time at a fixed point of the channel are computed by digital computer. Comparison of the statistical descriptors obtained from the curved channel and from the straight flume permits a quantitative evaluation of the marked differences between bed geometry in curved and straight channels.
The total rate of sediment transport in the curved channel is approximately 15 times as much as that of a straight flume which is subject to nominally identical flow conditions. This difference increases with increase in Froude number. At the same time the overall mean water surface slope in the curved channel is comparable to the water surface slope in the straight flume.
It is shown that bedfriction factors in alluvial beds can be
determined either in terms of flow conditions or in terms of the size of the bed forms. The statistical approach described in the text permits practical and relatively simple methods to be used for obtaining characteristic heights and lengths of the bed forms in terms of the moments of the spectral density function. These characteristic bed form dimensions are used in turn to evaluate bed friction factors in a straight flume.
It is demonstrated that characteristic dimensions of the bed forms can be obtained from stationary as well as from nonstationary sample records.
Comparison between time and space spectra permits evaluation of ripple celerity. The resulting relationship shows that small ripples move faster than large ones and that the celerity of ripples increases with increasing flow velocity. These results are confirmed by results obtained from timelapse photography and are suggested for use in relating time and space domains.
It is shown that the theoretical second order linear Markov model used by other investigators, as well as other simple exponential, sine or consine spectral density functions do not fit the observed phenomenon. Suggestions for future study are listed.

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