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This thesis concerns the stability of plane Couette flow and pipe Poiseuille flow in three space dimensions. The mathematical model for both flows is the incompressible Navier--Stokes equations. Both analytical and numerical techniques are used. We present new results for the resolvent corresponding to both flows. For plane Couette flow, analytical bounds on the resolvent have previously been derived in parts of the unstable half-plane. In the remaining part, only bounds based on numerical computations in an infinite parameter domain are available. Due to the need for truncation of this infinite parameter domain, these results are mathematically insufficient. We obtain a new analytical bound on the resolvent at s=0 in all but a compact subset of the parameter domain. This is done by deriving approximate solutions of the Orr--Sommerfeld equation and bounding the errors made by the approximations. In the remaining compact set, we use standard numerical techniques to obtain a bound. Hence, this part of the proof is not rigorous in the mathematical sense. In the thesis, we present a way of making also the numerical part of the proof rigorous. By using analytical techniques, we reduce the remaining compact subset of the parameter domain to a finite set of parameter values. In this set, we need to compute bounds on the solution of a boundary value problem. By using a validated numerical method, such bounds can be obtained. In the thesis, we investigate a validated numerical method for enclosing the solutions of boundary value problems. For pipe Poiseuille flow, only numerical bounds on the resolvent have previously been derived. We present analytical bounds in parts of the unstable half-plane. Also, we derive a bound on the resolvent for certain perturbations. Especially, the bound is valid for the perturbation which numerical computations indicate to be the perturbation which exhibits largest transient growth. The bound is valid in the entire unstable half-plane. We also investigate the stability of pipe Poiseuille flow by direct numerical simulations. Especially, we consider a disturbance which experiments have shown is efficient in triggering turbulence. The disturbance is in the form of blowing and suction in two small holes. Our results show the formation of hairpin vortices shortly after the disturbance. Initially, the hairpins form a localized packet of hairpins as they are advected downstream. After approximately $10$ pipe diameters from the disturbance origin, the flow becomes severely disordered. Our results show good agreement with the experimental results. In order to perform direct numerical simulations of disturbances which are highly localized in space, parallel computers must be used. Also, direct numerical simulations require the use of numerical methods of high order of accuracy. Many such methods have a global data dependency, making parallelization difficult. In this thesis, we also present the process of parallelizing a code for direct numerical simulations of pipe Poiseuille flow for a distributed memory computer. / QC 20100825

This dissertation consists of two parts, i.e. dynamic approaches for subgrid-scale (SGS) stress modelling for large eddy simulation and advanced assessment of the resolved scale motions related to turbulence geometrical statistics and topologies. The numerical simulations are based on turbulent Couette flow. The first part of the dissertation presents four contributions to the development of dynamic SGS models. The conventional integral type dynamic localization SGS model is in the form of a Fredholm integral equation of the second kind. This model is mathematically consistent, but demanding in computational cost. An efficient solution scheme has been developed to solve the integral system for turbulence with homogeneous dimensions. Current approaches to the dynamic two-parameter mixed model (DMM2) are mathematically inconsistent. As a second contribution, the DMM2 has been optimized and a modelling system of two integral equations has been rigorously obtained. The third contribution relates to the development of a novel dynamic localization procedure for the Smagorinsky model using the functional variational method. A sufficient and necessary condition for localization is obtained and a Picard's integral equation for the model coefficient is deduced. Finally, a new dynamic nonlinear SGS stress model (DNM) based on Speziale's quadratic constitutive relation [J. Fluid Mech., 178, p.459, 1987] is proposed. The DNM allows for a nonlinear anisotropic representation of the SGS stress, and exhibits a significant local stability and flexibility in self-calibration. In the second part, the invariant properties of the resolved velocity gradient tensor are studied using recently developed methodologies, i.e. turbulence geometrical statistics and topology. The study is a posteriori based on the proposed DNM, which is different than most of the current a priori approaches based on experimental or DNS databases. The performance of the DNM is further validated in terms of its capability of simulating advanced geometrical and topological features of resolved scale motions. Phenomenological results include, e.g. the positively skewed resolved enstrophy generation, the alignment between the vorticity and vortex stretching vectors, and the pear-shape joint probability function contour in the tensorial invariant phase plane. The wall anisotropic effect on these results is also examined.

This dissertation consists of two parts, i.e. dynamic approaches for subgrid-scale (SGS) stress modelling for large eddy simulation and advanced assessment of the resolved scale motions related to turbulence geometrical statistics and topologies. The numerical simulations are based on turbulent Couette flow. The first part of the dissertation presents four contributions to the development of dynamic SGS models. The conventional integral type dynamic localization SGS model is in the form of a Fredholm integral equation of the second kind. This model is mathematically consistent, but demanding in computational cost. An efficient solution scheme has been developed to solve the integral system for turbulence with homogeneous dimensions. Current approaches to the dynamic two-parameter mixed model (DMM2) are mathematically inconsistent. As a second contribution, the DMM2 has been optimized and a modelling system of two integral equations has been rigorously obtained. The third contribution relates to the development of a novel dynamic localization procedure for the Smagorinsky model using the functional variational method. A sufficient and necessary condition for localization is obtained and a Picard's integral equation for the model coefficient is deduced. Finally, a new dynamic nonlinear SGS stress model (DNM) based on Speziale's quadratic constitutive relation [J. Fluid Mech., 178, p.459, 1987] is proposed. The DNM allows for a nonlinear anisotropic representation of the SGS stress, and exhibits a significant local stability and flexibility in self-calibration. In the second part, the invariant properties of the resolved velocity gradient tensor are studied using recently developed methodologies, i.e. turbulence geometrical statistics and topology. The study is a posteriori based on the proposed DNM, which is different than most of the current a priori approaches based on experimental or DNS databases. The performance of the DNM is further validated in terms of its capability of simulating advanced geometrical and topological features of resolved scale motions. Phenomenological results include, e.g. the positively skewed resolved enstrophy generation, the alignment between the vorticity and vortex stretching vectors, and the pear-shape joint probability function contour in the tensorial invariant phase plane. The wall anisotropic effect on these results is also examined.

Extraction processes employing emulsion liquid membranes (ELMs), water-in-oil emulsions dispersed in aqueous phase, have been shown to be highly efficient in removing a variety of organic and inorganic contaminants from industrial wastewaters. As a result, they have been considered as alternative technologies to other more common separation processes such as pressure-driven membrane processes. Unfortunately, a widespread use of the ELM process has been limited due to the instability of emulsion globules against fluid shear. Breakup of emulsions and subsequent release of the internal receptor phase to the external donor phase would nullify the extraction process. Numerous studies have been, therefore, made in the past to enhance the stability of ELMs. Examples include adding more surfactants into the membrane phase and increasing the membrane viscosity. However, increased stability has been unfortunately accompanied by loss in extraction efficiency and rate in most reported attempts. The primary objective of this research is to apply the ELMs in a unique contacting device, a Taylor-Couette column, which provides a relatively low and uniform fluid shear that helps maintaining the stability of emulsion without compromising the extraction efficiency of a target compound. The ELM used in this study is made of membrane phase converted into non-Newtonian fluid by polymer addition, which provides additional uncommon remedy for the problem. This innovative ELM process was optimized to treat various types of simulated industrial wastewaters containing selected phenolic compounds and heavy metals. Experiments performed in this study suggested that the newly developed ELM process achieved exceptionally high overall removal efficiencies for the removal of these target compounds in relatively short contact time. Mechanistic predictive models were further developed and verified with the experimental data. Combined with the experimental data and novel mathematical predictive models, this study is expected to have a high impact on immediate practices of emulsion liquid membrane technologies in relevant industries.

<p>This thesis deals with numerical simulation of turbulentflows in geometrically simple cases. Both plane and cylindricalgeometries are used. The simplicity of the geometry allows theuse of spectral methods which yield a very high accuracy usingrelatively few grid points. A spectral method for planegeometries is implemented on a parallel computer. Thetransitional Reynolds number for plane Couette flow is verifiedto be about 360, in accordance with earlier findings. TurbulentCouette flow at twice the transitional Reynolds number isstudied and the findings of large scale structures in earlierstudies of Couette flow are substantiated. These largestructures are shown to be of limited extent and give anintegral length scale of six half channel heights, or abouteight times larger than in pressure-driven channel flow.Despite this, they contain only about 10 \% of the turbulentenergy. This is demonstrated by applying a very smallstabilising rotation, which almost eliminates the largestructures. A comparison of the Reynolds stress budget is madewith a boundary layer flow, and it is shown that the near-wallvalues in Couette flow are comparable with high-Reynolds numberboundary layer flow. A new spectrally accurate algorithm isdeveloped and implemented for cylindrical geometries andverified by studying the evolution of eigenmodes for both pipeflow and annular pipe flow. This algorithm is a generalisationof the algorithm used in the plane channel geometry. It usesFourier transforms in two homogeneous directions and Chebyshevpolynomials in the third, wall-normal, direction. TheNavier--Stokes equations are solved with a velocity-vorticityformulation, thereby avoiding the difficulty of solving for thepressure. The time advancement scheme used is a mixedimplicit/explicit second order scheme. The coupling between twovelocity components, arising from the cylindrical coordinates,is treated by introducing two new components and solving forthem, instead of the original velocity components. TheChebyshev integration method and the Chebyshev tau method isboth implemented and compared for the pipe flow case.</p>

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Cette thèse porte sur les effets combinés de l'excentricité et du débit axial sur les propriétés de stabilité linéaire de l'écoulement de Couette circulaire avec cylindre extérieur fixe. Cet écoulement intervient, entre autres, lors du forage de puits de pétrole. Une méthode pseudospectrale est mise en oeuvre pour calculer l'écoulement de base, stationnaire et invariant suivant la direction axiale, ainsi que les modes normaux d'instabilité. L'écoulement est régi par quatre paramètres adimensionnels : rapport de rayons _ et excentricité e pour la géométrie, nombres de Reynolds azimuthal et axial, Re et Rez, pour la dynamique. La première partie de l'étude est consacrée aux propriétés de stabilité temporelle. Il apparaît que l'excentricité repousse le seuil d'instabilité convective vers de plus fortes valeurs de Re. L'effet de l'advection axiale sur le seuil est principalement stabilisant également. L'excentricité a pour conséquence de déformer la structure des modes par rapport au cas concentrique. Le mode au plus fort taux de croissance temporelle est ainsi constitué de tourbillons de Taylor " pseudo-toroïdaux " lorsque le débit axial est nul, et de structures " pseudo-hélicoïdales " d'ordre azimuthal croissant lorsque Rez augmente. Les résultats sont qualitativement similaires lorsque l'on change le rapport de rayons. Les prédictions théoriques sont en bon accord avec les quelques résultats expérimentaux disponibles. Dans une seconde partie, l'instabilité absolue est étudiée par application d'un critère de point selle à la relation de dispersion. Le débit axial a pour effet d'inhiber fortement l'instabilité absolue, d'origine centrifuge, et la valeur de Re au seuil est typiquement supérieure à celle de Rez d'un ordre de grandeur. L'effet de l'excentricité est plus complexe : légère stabilisation aux faibles valeurs de e, puis déstabilisation marquée aux excentricités modérées lorsque Rez est suffisament grand, et enfin stabilisation lorsque e croît davantage. Contrairement au cas de l'instabilité convective, le mode dominant l'instabilité absolue correspond à l'écoulement tourbillonnaire " pseudo-toroïdal " pour toute la gamme de paramètres considérée.