In this thesis, we present certain exact solutions of the mathematical equations governing the one-dimensional unsteady flow of a compressible fluid. In Chapter 2 we introduce the well-known simplification of the equations (1.1.10), (1.1.11) and (1.1.12) which occurs when the entropy is assumed to be constant, and conditions for parching solutions of the equations along characteristics are obtained. These results are used to generalise a problem solved by Mackie. In chapter 3 we meet the concept of a shook, and exact solutions are obtained for two problems in which shocks occur in non-uniform flows. In chapter 4 the case of waves in shallow water which has differential equations similar to those of gas flow is discussed. The results of the previous section are applied to this case and a problem attacked which permits a comparison to be made of the results obtained by this theory and a simpler linearized theory. Finally in chapter 5 we examine a method introduced by Martin for dealing with certain non-isentropic flows. Some new exact solutions of non-isentropic flows are thus obtained.

This work is concerned with the evaporation and condensation processes occurring when liquid helium II is in equilibrium with its saturated vapour. We define the condensation coefficient a as the fraction of atoms incident on the liquid vapour interface which cross it to form part of the liquid. Experiments to measure are described, and the results are discussed in terms of microscopic condensation processes. The measurements are made by reflecting second sound pulses from the liquid vapour surface at normal incidence and measuring the reflection coefficient. An account is given of the phenomenological theories of Osborne (1962a) and Chernikova (1964), which describe the reflection of second sound from the surface and the associated effect, its transformation into first sound in the gas. Neither of these agree with the experimental results, and Osborne's theory is modified by taking account of the conditions in the gas a small fraction of a mean free path above the surface (rather than many mean free paths above the surface, as in Osborne's original theory). Thus modified, the theory is shown to be in agreement with the measurements of the reflection coefficient. Also described are measurements made in second sound pulses generated at the interface by first sound pulses, themselves generated at the interface by second sound, propagated up the tube, and reflected from its closed and back to the surface. From the time intervals between these pulses the velocity of first sound in the vapour is deduced, and found to be in agreement with previous work. Measurements of pulse amplitude corroborate the reflection coefficient measurements, and taking the two sets of measurements together wo have concluded that a is probably 1 and not less than 0.8 between 1.0°K and 2.14°K. The microscopic processes by which condensation can take place are considered. Experiments due to beaker (unpublished, see Osborne, 1962a) and Osborne (1962b) are described, which indicate that the vapour exchanges momentum with normal fluid only. We have therefore supposed that processes in which a gas atom condenses to form excitations must conserve energy and momentum. Processes involving both bulk excitations and surface excitations are considered, but effects due to the finite lifetime of the excitations and the linewidth of the excitations spectrum are neglected. No attempt has been made to calculate the matrix elements for condensation processes, but plausible estimates have been made of their relative magnitudes. In particular, only processes involving one gas atom and one or two excitations have been considered. Using the requirements of conservation of energy and momentum, it is shown that as the temperature decreases, a decreasing fraction of the incident atom have enough energy to form two excitations, and condensation must take place by the collision of an atom with an existing excitation. A rough estimate of the collision probability for such a process leads to the conclusion that at 1°K, a should be about 0.2. This disagreement with experiment has not been resolved. Finally, some remarks are made about the implications for other work on liquid helium II, and some suggestions for future work.

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In many problems arising in the theory of compressible flow, the equations characterising the solution of the system are so intractable that recourse must be made to some approximate method which allows the essential features of the flow to be preserved, whilst to some degree, simplifying the mathematics. It is with certain methods of this type that this thesis is concerned. In the subsequent work, we shall assume that the effects due to viscosity and heat conduction are so small as to be negligible. These assumptions may be shown to be largely valid except in those domains of the flow-field where the modified system of equations predicts regions in which the solution is in general multivalued. In the modified system, however, such ‘regions' are avoided by the introduction of mathematical discontinuities and, assuming that the jump conditions across them can be determines, are sufficient to provide single-valued solutions valid everywhere, except at the discontinuity. The methods to be presented are formulated in the plane consisting of one space variable and one time variable.

Linear mode conversion processes are much studied in plasma physics because they determine the efficiency of any radio frequency heating scheme. Mode coupling model equations, extracted with varying degrees of rigour from the Maxwell-linearized kinetic equations, are usually fourth or higher order O.D.E's. These are solved by complicated methods to obtain transmission, conversion, reflection and absorption coefficients. Recently, Fuchs et al and Cairns and Lashmore-Davies (C.L-D.) have postulated second order O.D.E's to describe pairwise coupling events. The second order theories have reproduced results previously obtained by much more sophisticated treatments. In this thesis, we firstly examine the hybrid resonances in a cold plasma and show that they have a mode conversion interpretation in the framework of the C.L-D. model. The Budden tunnelling coefficients are recovered for this case. Next, mode conversion between the fast and slow electromagnetic waves in the lower hybrid frequency range is considered. This phenomenon determines the accessibility of the lower hybrid resonance to the slow wave, and is also of theoretical interest because the mode coupling differs in certain aspects from cases previously investigated by C.L-D. A second order approximation to the dispersion relation is used in the mode conversion region leading to Weber's equation from which transmission coefficients are then obtained in various cases. Finally, we provide justification for the use of Weber's equation. The exact fourth order system of O.D.E's for the problem is set down, and a linear transformation, which is an extension of that given by Heading, reveals the second order nature of the coupling process. Numerical solutions of the fourth order system yield transmission coefficients in excellent agreement with the second order theory, and also demonstrate that the electric field variation across the mode conversion region is well approximated, via the above transformation, by our second order theory.

This thesis is concerned with the unsteady expansion of an initially uniform, stationary gas into a low density, stationary atmosphere, studied from the viewpoint of inviscid gasdynamics. It is found that, there are two regions in the k-σ parameter space having distinct forms for the large time solution, when the atmospheric density is initially proportional to r⁻^k, r being the spatial coordinate, k being constant and σ, the geometry index, has its usual meaning. First of all a constant asymptotic shock velocity is assumed and matched expansions, for large r, are constructed. Inner expansions, valid near the shock, are matched to zeroth and first orders with the outer expansions which are valid near the contact front. Zeroth order matching, which, yields the constant asymptotic shock velocity, is possible only in a restricted region of the k-σ parameter space and this situation is clarified by appealing to the similarity solutions which are extended to cover cases which have not been dealt with previously. In the other region of the k-σ parameter space the asymptotic shock velocity is proportional to r^∈ where ∈, a positive constant, is found from the similarity solutions as a function of k, γ, σ. An attempt is made at constructing matched asymptotic expansions for large r. The inner solution can be obtained, apart from the evaluation of certain constants, to zeroth and first orders but the outer solution is inaccessible and can only be determined from the full inviscid solution. However it is shown that there exists a solution to the outer equations which matches with the inner solution up to first order. In both cases matching of the first order inner terms to the outer solution produces an eigenvalue problem, the solution of which is not attempted here. Finally full numerical solutions of the inviscid equations, one for each case, were produced using the method of backward drawn characteristics, devised by Hartree, and it will be seen that they compare most favourably with the asymptotic analysis.

Superfluid 4 He (He II) is a quantum liquid whose flow is strongly affected by quantum mechanical effects. This thesis presents experimental and numerical studies of turbulent flows in He II - quantum turbulence. Experimentally, quan- tum turbulence is investigated in thermal counterflow, pure superflow and coflow using second sound attenuation, precision local thermometry and by visualisa- tion of helium excimer molecules. The steady state and decay of the vortex line density in pure superflow and counterflow is studied and the universal quasi clas- sical decay is characterised by measurements of the effective kinematic viscosity. General dynamical behaviour is studied in detail in unsteady thermal counter- flow, with various theoretical models tested. A new model where the mean tangle curvature is dependent on the vortex line density is proposed. Temperature de- pendence and enhancement of intermittency in quasi-classical flow in the wake of a moving grid is found using visualisation. Numerically, the interaction of the tangle of quantized vortices with solid tracers is investigated, where a back reaction of the seeding particles on the tangle is identified and its relevance to visualisation experiments is discussed. Additionally, an interesting and as-yet overlooked spherical counterflow is...

A computational investigation was undertaken to examine techniques for predicting supersonic dynamic flows, involving unsteadiness over fixed and moving surfaces. The fixed geometries examined were cylinder-flares and compression ramps, and the moving body geometries a pitching aerofoil and a rapidly deployed flap. Investigation into the characteristics of incipient separation of a supersonic cylinder-flare flow revealed that the separated length varied with a power of the flare angle and that the variation in height of the separated region varies in a bi-modal manner with flare angle. For small-scale separations (flare angles less than those which would traditionally have been expected to induce separation) the height of the separated region was seen to vary slowly with flare angle. For larger flare angles, the separation bubble was found to grow rapidly in height and length with increasing flare angle and produce significant deflection of the external flow. Computations of a Mach 5, compression ramp induced unsteady shock boundary layer interaction exhibited self-sustained oscillations at frequencies and amplitudes consistent with experimental data. Large dynamic structures (up to 1.7 boundary layer thicknesses in extent) were observed, and their production, propagation and deformation illustrated. By modifying the turbulent viscosities produced by a non-dimensional implementation of the Baldwin-Lomax turbulence model (using under- relaxation) a turbulence model was produced which accurately predicted separation lengths for a series of Mach 6.85 compression ramp flows encompassing laminar, transitional and turbulent flow regimes (dependent on ramp angle). A technique was developed to enable efficient computation of dynamically moving and/or deforming body flows. This technique was based on hierarchical, adaptive mesh refinement coupled with automatic generation of body surfaces, in which mesh adaption was used to capture the body geometry to within a specified accuracy. This, in conjunction with automatic cell creation and destruction, enabled the derivation of steady and unsteady, time accurate, conservative boundary conditions. This algorithm was used to compute a quasi-steady laminar supersonic pitching aerofoil flow, and an unsteady turbulent supersonic flap deployment. In both cases agreement with experiment was found to be good.

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M.Ing. (Mechanical Engineering) / Shell and tube heat exchangers are commonly used in industrial processes. In the heat exchangers the tubes incur flow induced vibrations. These vibrations lead to fretting on the perimeter of the tube at the baffle plates and to fatigue failure at the tube sheets. Various studies were and are done to qualify and quantify these vibrations. This is then used to develop design criteria and methods for lowering the vibration amplitudes of the tubes. In the thesis the response of a tube bundle in cross flow is shown. The effect of increasing cross flow velocity on the vibration amplitude of the tubes is determined in a low speed wind tunnel. The results show similarity to the work done by Gorman [27]. By raising the natural frequency of the tubes in the first row upstream it is shown that the vibration peaks exhibited by the unmodified model.at cross flow velocities lower than the critical flow speed do not materialize in the modified tube bundle. This modification can be implemented by decreasing the unsupported length of the tubes in the first row. The inclusion of fluid damping when calculating the total damping of the tubes according to Blevins [24] is shown in the experiment. From experimental results it is shown that the prediction of the critical velocity according to Blevins [24] results in a better prediction of this velocity. Numerical solutions of two dimensional flow over a single cylinder and a pair of cylinders are obtained by simulating the flow on the STAR CD flow simulation package. The effect of neighbouring tubes on one another is determined by qualifying and quantifying the flow over and the forces on the tubes. The vortex shedding frequency determined from the simulation coincides with experimental results. This shows great potential in future simulation of flow over a tube bundle. A computerised design model is also developed. Themodel determines the vibration potential of the tube bundle in a shell and tube heat exchanger. If needed, modifications to new designs and built heat exchangers can be done and the vibration potential is easily determined with the model. The results of the experimental work are used to further develop the design criteria preventing detrimental vibrations in heat exchangers. Promising results to decrease vibration amplitudes of tubes in tube bundles are obtained in the experiments done. This and the results obtained in the simulation of the vortex shedding frequencies that coincide with the results in the experimental work, show great potential for flow simulation and development of methods to decrease flow induced vibrations of tubes in shell and tube heat exchangers.