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In this thesis, we investigate mixing mechanisms in a stably stratified turbulent Taylor-Couette flow, which is the flow in the annular region of two co-axial cylinders, both capable of rotating at different speeds independently, in the presence of vertical stable stratification. Oglethorpe (2014) found that, for an initially linearly stratified Taylor-Couette (STC) flow with fixed outer cylinder, the flow spontaneously forms well-mixed layers of constant height separated by sharp density gradient interfaces. She also observed a quasi-periodic mixing phenomenon across the interfaces. Through laser induced fluorescence and particle images velocimetery, we discover the structure of this mixing instability. We find that the mixing occurs as a result of a flow phenomenon generated by two in-phase boundary trapped waves, with azimuthal wavenumber $m=1$, riding on the interface. We further look into the flux across the interface resulting from this instability. We find that, for high stratification, the molecular diffusion plays a significant part in the overall observed flux across the interface, and the buoyancy flux does not tend to a constant as previously discussed by Oglethorpe (2014). As a result, we find that the entrainment coefficient, $E\sim Ri_B^{-3/2}$, where $Ri_B = g'\frac{R_2}{(\Omega R_1)^2}$ with $R_1$ and $R_2$ being the inner and the outer cylinder radii respectively and $\Omega$ being the angular velocity of the inner cylinder, which is consistent with the classical observations of Turner (1968). Overall, we observe that the buoyancy flux monotonically increases as the mixing occurs (i.e. with reducing $Ri_B$) to a maximum where the interface is overturned by the turbulent eddies present in each of the layers. Furthermore, we perform stability analysis of the STC flow using a base flow having a dependence in both the radial $r$ and the axial $z$ directions,using the mean turbulent flow profile varying in $r$ as the base velocity profile and a density profile with a sharp gradient in the $z$ direction as the base density profile. Using our model, we are able to consistently predict the period of the empirically observed instability, which suggests that this instability has its origins in a linear instability. Finally, we look at the implications of rotating the outer cylinder on the observed instability. Through qualitative experiments and stability analysis, we discover that the same instability exists even outside the domain of centrifugal instability prescribed by Rayleigh's criterion (Rayleigh, 1917).

Free shear and wall-bounded buoyancy-driven turbulent flows occur in both natural environments and industrial situations. In this thesis, to better understand the entrainment process within these flows, experiments and theory have been used to investigate point and distributed buoyancy sources and, in particular, the effect of a bounding vertical wall on these flows. A free shear flow was first investigated by performing velocity and scalar edge measurements on an axisymmetric plume created by a continuous point source release of buoyancy. By conditionally sampling the velocity measurements based on the presence of both eddies and plume fluid, engulfment, whereby large pockets of ambient are engulfed in to the plume, was shown to be a dominant turbulent entrainment process. To isolate the effect of a wall on a turbulent buoyancy-driven flow, a line plume distant from all vertical boundaries and a wall plume, adjacent to a vertical wall were also studied. Simultaneous velocity and buoyancy field measurements were performed and a reduction in the net entrainment, and entrainment coefficient, for a wall plume were found. This reduction was investigated by considering an energy decomposition of the entrainment coefficient where the relative contributions of turbulent production, buoyancy and viscous terms were calculated. The reduced entrainment was also investigated by considering the statistics of the turbulent interface. Finally, simultaneous velocity and buoyancy field measurements on a vertically distributed buoyant plume were performed by forcing relatively dense fluid through a very low porosity plate. A reduced entrainment coefficient, compared to that of a wall plume, was observed. In order to model the ventilation of a room with a heated or cooled wall the flow was then enclosed within a mechanically ventilated model room. The evolving and steady-state ambient stratification was measured using dye-attenuation with an LED-light bank for varying buoyancy fluxes and ventilation flow rates.

The transition to turbulence in shear flows such as pressure driven pipe flow or plane Couette flow presents an interesting theoretical problem: how do we understand the existence of chaos when the laminar flow is stable to infinitesimal perturbations? A number of approaches to the problem have been used in recent years and a great deal of progress has been made towards understanding the transition, often utilising low-dimensional models to generate hypotheses. In the first part of this thesis I study the behaviour of a system of partial differential equations based on the damped Kuramoto-Sivashinsky equation which exhibit a subcritical transition to turbulence as a control parameter is varied. Typical lifetimes of the system are measured and align with the scenario for shear flows; they have an exponential distribution for a given value of the control parameter, and the typical lifetime scale superexponentially with that parameter. Coherent structures are found numerically and the linear stability measured to create a bifurcation diagram which is reminiscent of the ones found in shear flows. In the second part of the thesis a link is drawn between the apparent dynamical role of the lower branch states of the extended KS equation and the current understanding of transitional turbulence as belonging to the universality class of Directed Percolation (DP). A novel DP model is introduced which has a third state which represents the behaviour of the lower branches and the critical exponents of the system are measured and found to agree with the expected exponents for 1+1 dimensional DP. A non-universal parameter is found which varies with the strength of the bouncing behaviour, although it is unclear if it is possible to measure this parameter in a meaningful way for a real flow. Finally, in the third part of the thesis the extended KS equation is studied in an extended spatial domain, to confirm the hypothesis that this system also belongs to the DP universality class. Critical exponents are measured and found to agree with 1+1 DP. This confirms that the system has a transition which reproduces many of the important features subcritical fluid flows like pressure driven pipe flow, or plane Couette flow.

In this thesis I investigate a number of problems in the nonlinear stability of density stratified plane Couette flow. I begin by describing the history of transient growth phenomena, and in particular the recent application of adjoint based optimisation to find nonlinear optimal perturbations and associated minimal seeds for turbulence, the smallest amplitude perturbations that are able to trigger transition to turbulence. I extend the work of Rabin et al. (2012) in unstratified plane Couette flow to find minimal seeds in both vertically and horizontally sheared stratified plane Couette flow. I find that the coherent states visited by such minimal seed trajectories are significantly altered by the stratification, and so proceed to investigate these states both with generalised Koopman mode analysis and by stratifying the self-sustaining process described by Waleffe (1997). I conclude with an introductory problem I considered that investigates the linear Taylor instability of layered stratified plane Couette flow, and show that the nonlinear evolution of the primary Taylor instability is not coupled to the form of the linearly unstable mode, in contrast to the Kelvin-Helmholtz instability, for example. I also include an appendix in which I describe joint work conducted with Professor Neil Balmforth of UBC during the 2015 WHOI Geophysical Fluid Dynamics summer programme, investigating stochastic homoclinic bifurcations.

The aim of this thesis is to study the effects of turbulent flow on a fixed pitch tidal current turbine from the perspective of turbine design and operation. A prototype turbine, Deltastream as it is known, is being developed by Tidal Energy Ltd for deployment in Ramsey Sound, Wales. It is well known that turbulence plays an important role in the fatigue life of marine turbines. Field measurements of tidal flow at the turbine site were analysed to establish the velocity spectra and turbulence intensity. This revealed a wide range of anisotropic turbulence which is dependent upon the tidal direction with intensities ranging from 5-20%. A numerical turbine model based on momentum theory was constructed in a time marching formulation that accounts for the effects of dynamic inflow and rotationally augmented airfoil stall delay properties. The turbine rotor design allows for load alleviation by regulation of the turbine tip speed ratio. At flow velocities above the rated velocity the tip speed ratio can be increased to reduce turbine loads. The model has been combined with a novel rotor speed control algorithm that estimates unsteady turbine inflow velocity from turbine loading without the requirement for external sensing of flow speed. When the turbine is subjected to three dimensional turbulent inflow the rotor speed controller has been shown to significantly reduce the fatigue effect of unsteady, turbulent flow. The turbine blade design has been developed using the model established. Experimental validation studies were carried out at 1/16th scale in turbulent conditions. Studies using the model have; identified the relationship between turbulence intensity and turbine fatigue load, established a controller schedule to significantly reduce fatigue loading and determined the blading fatigue life in realistic turbulent flows.

In this thesis I study a variety of two-dimensional turbulent systems using a mixed analytical, phenomenological and numerical approach. The systems under consideration are governed by the two-dimensional Navier-Stokes (2DNS), surface quasigeostrophic (SQG), alpha-turbulence and magnetohydrodynamic (MHD) equations. The main analytical focus is on the number of degrees of freedom of a given system, defined as the least value $N$ such that all $n$-dimensional ($n$ ≥ $N$) volume elements along a given trajectory contract during the course of evolution. By equating $N$ with the number of active Fourier-space modes, that is the number of modes in the inertial range, and assuming power-law spectra in the inertial range, the scaling of $N$ with the Reynolds number $Re$ allows bounds to be put on the exponent of the spectrum. This allows the recovery of analytic results that have until now only been derived phenomenologically, such as the $k$[superscript(-5/3)] energy spectrum in the energy inertial range in SQG turbulence. Phenomenologically I study the modal interactions that control the transfer of various conserved quantities. Among other results I show that in MHD dynamo triads (those converting kinetic into magnetic energy) are associated with a direct magnetic energy flux while anti-dynamo triads (those converting magnetic into kinetic energy) are associated with an inverse magnetic energy flux. As both dynamo and anti-dynamo interacting triads are integral parts of the direct energy transfer, the anti-dynamo inverse flux partially neutralises the dynamo direct flux, arguably resulting in relatively weak direct energy transfer and giving rise to dynamo saturation. These theoretical results are backed up by high resolution numerical simulations, out of which have emerged some new results such as the suggestion that for alpha turbulence the generalised enstrophy spectra are not closely approximated by those that have been derived phenomenologically, and new theories may be needed in order to explain them.

Numerical simulation is becoming increasingly used to support theoretical effort into understanding the turbulence problem. We develop theoretical ideas related to the transfer and dissipation of energy, which clarify long-standing issues with the energy balance in isotropic turbulence. These ideas are supported by results from large scale numerical simulations. Due to the large number of degrees of freedom required to capture all the interacting scales of motion, the increase in computational power available has only recently allowed flows of interest to be realised. A parallel pseudo-spectral code for the direct numerical simulation (DNS) of isotropic turbulence has been developed. Some discussion is given on the challenges and choices involved. The DNS code has been extensively benchmarked by reproducing well established results from literature. The DNS code has been used to conduct a series of runs for freely-decaying turbulence. Decay was performed from a Gaussian random field as well as an evolved velocity field obtained from forced simulation. Since the initial condition does not describe developed turbulence, we are required to determine when the field can be considered to be evolved and measurements are characteristic of decaying turbulence. We explore the use of power-law decay of the total energy and compare with the use of dynamic quantities such as the peak dissipation rate, maximum transport power and velocity derivative skewness. We then show how this choice of evolved time affects the measurement of statistics. In doing so, it is found that the Taylor dissipation surrogate, u^3 / L, is a better surrogate for the maximum inertial flux than dissipation. Stationary turbulence has also been investigated, where we ensure that the energy input rate remains constant for all runs and variation is only introduced by modifying the fluid viscosity (and lattice size). We present results for Reynolds numbers up to Rλ = 335 on a 1024^3 lattice. Using different methods of vortex identification, the persistence of intermittent structure in an ensemble average is considered and shown to be reduced as the ensemble size increases. The longitudinal structure functions are computed for smaller lattices directly from an ensemble of realisations of the real-space velocity field. From these, we consider the generalised structure functions and investigate their scaling exponents using direct analysis and extended self-similarity (ESS), finding results consistent with the literature. An exploitation of the pseudo-spectral technique is used to calculate second- and third-order structure functions from the energy and transfer spectra, with a comparison presented to the real-space calculation. An alternative to ESS is discussed, with the second-order exponent found to approach 2/3. The dissipation anomaly is then considered for both forced and free-decay. Using different choices of the evolved time for a decaying simulation, we show how the behaviour of the dimensionless dissipation coefficient is affected. The Karman-Howarth equation (KHE) is studied and a derivation of a work term presented using a transformation of the Lin equation. The balance of energy represented by the KHE is then investigated using the pseudo-spectral method mentioned above. The consequences of this new input term for the structure functions are discussed. Based on the KHE, we develop a model for the behaviour of the dimensionless dissipation coefficient that predicts Cɛ= Cɛ(∞)+CL/RL. DNS data is used to fit the model. We find Cɛ(∞) = 0.47 and CL = 19.1 for forced turbulence, with excellent agreement to the data. Theoretical methods based on the renormalization group and statistical closures are still being developed to study turbulence. The dynamic RG procedure used by Forster, Nelson and Stephen (FNS) is considered in some detail and a disagreement in the literature over the method and results is resolved here. An additional constraint on the loop momentum is shown to cause a correction to the viscosity increment such that all methods of evaluation lead to the original result found by FNS. The application of statistical closure and renormalized perturbation theory is discussed and a new two-time model probability density functional presented. This has been shown to be self-consistent to second order and to reproduce the two-time covariance equation of the local energy transfer (LET) theory. Future direction of this work is discussed.

This thesis is concerned with three-dimensional vortex dynamics, in particular the modelling of vortex structures in an inviscid context. We are motivated by the open problem of regularity of the inviscid equations, i.e. whether or not these equations possess solutions. This problem is manifest in small scales, where vortex filaments are stretched and intensify as they are drawn into increasingly thin tendrils. This creates great difficulty in the investigation of such flows. Our only means of experimentation is to perform numerical simulations, which require exceptionally high resolution to capture the small scale vortex structures. A new numerical method to solve the inviscid Euler equations for three-dimensional, incompressible fluids is presented, with special emphasis on spatial adaptivity to resolve as broad a range of scales as possible in a completely self-similar fashion. We present a hybrid vortex method whereby we discretise the vorticity in Lagrangian filaments and perform and inversion to compute velocity on an arbitrary unstructured finite-volume grid. This allows for a two-fold adaptivity strategy. First, although naturally spatially adaptive by definition, the vorticity filaments undergo ‘renoding'. We redistribute nodes along the filament to concentrate their density in regions of high curvature. Secondly the Eulerian mesh is adapted to follow high strain by increasing resolution based on local filament dimensions. These features allow vortex stretching and folding to be resolved in a completely automatic and self-similar way. The method is validated via well known vortex rings and newly discovered helical vortex equilibria are also used to test the method. We begin by presenting this new class of three-dimensional vortex equilibria which possess helical symmetry. Such vortices are observed in propeller and wind turbine wakes, and their equilibria shapes have until now been unknown. These vortices are described by contours bounding regions of uniform axial vorticity. Material conservation of axial vorticity enables equilibria to be calculated simply by a restriction on the helical stream function. The states are parameterised by their mean radius and centroid position. In the case of a single vortex, the parameter space cannot be fully filled by our numerical approach. We conjecture that multiply connected contours will characterise equilibria where the algorithm fails. We also consider multiple vortices, evenly azimuthally spaced about the origin. In such cases instabilities often lead to a single helical vortex.

Marine bodies have complex geometry and are subject to various physical phenomenon, such as the free surface and turbulence. These require a variety of cell topologies which, when combined, often restrict the geometrical discretisation of the domain. The approach taken has been to use control volumes of arbitrary definition to allow the benefits of mesh adaptivity as well as a mix of cell topologies Three dimensional unstructured meshes are constructed using a data structure that lowers memory requirements from previous methodology and allows efficient and complex mesh adaption. A three dimensional Euler flow solver has been developed that utilises this mesh definition with artificial compressibility. The generalised fluxes for arbitrarily shaped moving meshes have been mathematically derived for a flux vector splitting scheme. The solver operates upon a face structure using a first order upwinding spatial discretisation. Validation has shown that results are convergent. Case studies have included flow over an arc hump, a foil at an angle of attack and a truncated structure. Meshes have been assessed with respect to both geometrical and flow solution properties, and numerical quantification of the parameters relevant to the solution accuracy has been defined. Facial skew is the most influential geometrical property while the local pressure gradient has been used to assess local solution accuracy. Quality quantification parameters have been used to control a variety of mesh adaptation techniques. Control volume splitting, merging and vertex manipulation have been used to improve solution accuracy with respect to computational efficiency. In addition, the cellular splitting algorithm has been used to create a multigrid scheme that increases the computational efficiency still further.

This thesis documents three years of work involved in the numerical solution of atmospheric wave models. Derivation of these models is established whilst introducing the basic physical laws governing fluid motion. Numerical techniques axe investigated with particular reference to the solution of parabolic and elliptic partial differential equations. Parallel computer systems are discussed and basic concepts introduced with the emphasis placed on distributed virtual parallelism. The role of inertio-gravity waves under the influence of cyclonic Rossby waves is investigated with respect to the production of atmospheric turbulence. Results from evolving numerical systems bound by various conditions are presented. It is discovered that the wave interaction is not the sole cause of atmospheric blocking as was previously thought. The use of a loosely coupled parallel environment is discussed in relation to potential increases in speed or size of the numerical model. A solution technique is modified to enable such an implementation. The full nonlinear Barre de Saint-Venant model of fluid motion is solved using a combination of finite difference and spectral methods. Preliminary results are presented and further avenues of investigation are discussed.

The present study is aimed at enhancing the effectiveness of the large eddy simulation (LES) approach to the computation of turbulent flows by these two methods: i) developing a superior subgrid scale (SGS) model and ii) improving the economy of LES. First of all, the various existing SGS models are extensively investigated, and their advangages and disadvantages are addressed to highlight the areas requiring improvements. This study leads to the construction of a modified SGS dynamic model. In addition, a detailed derivation of the second-order velocity structure function SGS model is made, correcting an error found in that model. A new multiple mesh method is also designed to accelerate LES. After the above theoretical studies, several low-Reynolds-number channel flow simulations have been performed. Firstly, simulations with varying model constants are carried out, and the results agree with those of Deardorff [14], showing that a model constant of about 0.1 is optimum for channel flows. Secondly, simulations with varying numerical resolutions have been carried out. They reveal that the refinement of the mesh in the direction normal to the wall improves all the turbulence statistics, both higher- and lower-order statistics, over the whole channel, while the refinement of the resolution in the streamwise and spanwise directions improves lower-order statistics over the whole channel, but only improves higher-order turbulence statistics in the central region of the channel. Thirdly, a dissipation-range SGS model (i.e. the Smagorinsky model with low- Reynolds-number modification [67]) is, for the first time, tested and compared with the standard Smagorinsky model. The results obtained show some promise for automatically adjusting the SGS model with Reynolds number. Fourthly, the performance of the modified dynamic SGS model is assessed through a comparison of length scales computed respectively by this modified model, the Germano- Lilly dynamic SGS model and two empirical wall damping functions in conjunction with an optimum model coefficient, which have been successfully used in many simulations of channel flows. Two values of the ratio of filter widths are set for each of the dynamic models. The results have confirmed that the modified dynamic SGS model can be successfully extended to simulate low-Reynolds-number channel flows. Of great promise is that the modified SGS dynamic model gives the correct behaviour of the subgrid eddy viscosity in the region of a plane wall to an accuracy that exceeds the best-tuned wall damping function, and almost collapses with the theoretical behaviour of the length scale near the wall without any tuning and adjustment. In addition, the impact of the choice of the ratio of filter widths on the modified dynamic SGS model is much less than that on the Germano-Lilly model. Finally, simulations using the new and old multiple mesh methods are performed. The instantaneous results just after the interpolation of the coarse mesh velocity field onto the fine mesh show that the fine mesh velocity field created by the new multiple mesh method contains the information of the residual field. In contrast, there is no difference between the fine mesh results obtained by the old method and those from a simulation on the coarse mesh.

In this thesis we consider turbulent fluid systems. We develop a closure scheme in which the mean velocity field of an incompressible fluid is driven by a turbulent velocity field possessing a non-zero mean helicity. We use this to investigate the formation of large scale vortices and the behaviour of the mean kinetic energy, enstrophy and helicity. The same technique is then applied to the equations of magneto-hydrodynamics, in order to explain the self-generation of mean magnetic fields, and the joint formation of current and vortex structures. We then discuss the convection of a passive scalar by the fluid and determine an equation for the mean temperature. Finally we present a theory to account for the behaviour of a two-dimensional electrically conducting fluid subject to a constant external magnetic field driven by external forces. We explain the peaks in the power spectrum, the saturation of the magnetic and kinetic energies, and the insensitiveness of their equilibrium value on the external field. All of these are observed in numerical experiments.

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.

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.