Shallow flows are those whose width is significantly larger than their depth. In these types of flows, two dimensional coherent structures can be generated and can influence the flow greatly by the lateral transfer of mass and momentum. The development of coherent structures as a result of flow instabilities has been a topic of interest for environmental fluid mechanics for decades. Studies on the use of linear modal stability analysis is commonly found in literature. However, the relatively recent development in the field of hydrodynamic stability suggests that the traditional linear modal stability analysis does not describe the behaviour of the perturbations in finite time. The discrepancy between asymptotic behaviour and finite time behaviour is particularly large in shear driven flows and it is most likely to be the case for shallow flows. This study aims to provide a better understanding of finite time growth of perturbation energy in shallow flows. The three cases of shallow flows evaluated are the mixing layer, jet and wake. The critical cases are obtained through the linear modal analysis and nonmodal analysis was conducted to show the transient behaviour in finite time for what is so-called marginally stable. Finally, the thesis concludes by generalizing the finite time energy growth in the S-k space.

The main focus of this work is on the non-Newtonian effects on the inertial instabilities in shear flows. Both inelastic (Carreau) and elastic models (Oldroyd-B and FENE-P) have been employed to examine the main features of the non-Newtonian fluids; shear-thinning, shear-thickening and elasticity. Several classical configurations have been considered; flow past a circular cylinder, in a lid-driven cavity and in a channel. We have used a wide range of tools for linear stability analysis, modal, non-modal, energy and sensitivity analysis, to determine the instability mechanisms of the non-Newtonian flows and compare them with those of the Newtonian flows. Direct numerical simulations have been also used to prove the results obtained by the linear stability analysis. Significant modifications/alterations in the instability of the different flows have been observed under the action of the non-Newtonian effects. In general, shear-thinning/shear-thickening effects destabilize/stabilize the flow around the cylinder and in a lid driven cavity. Viscoelastic effects both stabilize and destabilize the channel flow depending on the ratio between the viscoelastic and flow time scales. The instability mechanism is just slightly modified in the cylinder flow whereas new instability mechanisms arise in the lid-driven cavity flow. We observe that the non-Newtonian effect can alter the inertial flow at both baseflow and perturbation level (e.g. Carreau fluid past a cylinder or in a lid driven cavity) or it may just affect the perturbations (e.g. Oldroyd-B fluid in channel). In all the flow cases studied, the modifications in the instability dynamics are shown to be strongly connected to the contribution of the different terms in the perturbation kinetic energy budget. / <p>QC 20140113</p>

L’objectif de cette thèse est d’analyser le développement des instabilités secondaires bidimensionnelles et tridimensionnelles dans les couches de mélange à densité variable, incompressibles et à nombre de Froude infini. Dans ces conditions, la présence d’inhomogénéités de masse volumique modifie sensiblement la dynamique rotationnelle de l’écoulement et celle des instabilités secondaires sous l’action du couple barocline. Une analyse de stabilité linéaire non-modale est mise en oeuvre pour identifier les mécanismes physiques de croissance transitoire. Cette analyse permet également de prendre en compte le caractère instationnaire de la couche de mélange, absent dans l’analyse modale quasi-statique de Fontane (2005). Après établissement des équations de Navier–Stokes linéarisées directes et adjointes à densité variable, celles-ci sont utilisées dans une méthode d’optimisation itérative qui permet de déterminer les perturbations à croissance énergétique maximale. La première partie consiste en la description des perturbations optimales pour une couche de mélange homogène. Aux temps courts, lorsque la couche de mélange est quasi-parallèle, les perturbations optimales présentent de fortes amplifications transitoires, dont l’origine physique est due à la synergie des mécanismes classiques de Orr et de lift-up. Puis lorsque la couche s’enroule pour former un tourbillon de Kelvin–Helmholtz, les perturbations évoluent vers les instabilités tridimensionnelles elliptiques ou hyperboliques, selon le nombre d’onde latéral. Dans la deuxième partie, l’analyse est étendue aux couches de mélange à densité variable. Pendant la phase initiale de développement des perturbations optimales, les inhomogénéïtés de masse volumique ont une influence minime sur la croissance des perturbations. Ce n’est qu’une fois la couche de mélange enroulée que les effets de densité deviennent actifs, entraînant un supplément d’amplification significatif par rapport à la situation homogène. En particulier, le couple barocline favorise le développement des perturbations du côté du fluide léger du rouleau de Kelvin–Helmholtz. Enfin, lorsque le temps d’injection des perturbations est suffisamment retardé, la vorticité produite par le couple barocline favorise le développement d’une instabilité bidimensionnelle du type Kelvin-Helmholtz identifiée par Reinaud et al. (2000). / The purpose of this thesis is to analyse the development of two-dimensional and three-dimensional secondary instabilities in incompressible variable-density mixing layers, in the limit of infinite Froude number. Under these conditions, mass inhomogeneities alter significantly the rotational dynamics of the flow under the action of the baroclinic torque. A nonmodal stability analysis is implemented to identify the physical mechanisms of transient growth. This analysis allows to take into account the unsteady natureof the flow, which was absent in the quasi-static modal analysis (Fontane, 2005). After establishing of the direct and adjoint linearised Navier-Stokes equations for variable-density flows, they are used in an iterative optimization method to determine the perturbations that maximize their energy. The optimal perturbations are first obtained for a homogeneous time-evolving mixing layer. For times short enough, when the time-evolving mixing layer is almost parallel, optimal perturbations exhibit the largest transient growth. These amplifications arise from the synergy between the well-known Orr and liftup mechanisms. Once the mixing layer rolls up into a Kelvin–Helmholtz billow, the disturbances trigger the three-dimensional elliptical and hyperbolic instabilities. The analysis is then extended to variable-density mixing layers. During the initial development of optimal perturbations, mass inhomogeneities have no influence over the perturbations growth. Once the mixing layer has rolled up, the variable-density effects contribute significantly to the increase of the perturbation energy. In particular, the baroclinic torque enhances the development of perturbations in the light side of the Kelvin–Helmholtz billow. Finally, when the injection time of perturbations is delayed long enough, the baroclinic vorticity generation on the light side of the Kelvin–Helmholtz billow triggers a two-dimensional secondary Kelvin–Helmholtz instability, which has been identified by Reinaud et al. (2000).

Compressible plane Couette flow is studied with superposed small perturbations. The steady mean flow is characterized by a non-uniform shear-rate and a varying temperature across the wall-normal direction for an appropriate perfect gas model. The studies are broadly into four main categories as said briefly below. Nonmodal transient growth studies and estimation of optimal perturbations have been made. The maximum amplification of perturbation energy over time, G max, is found to increase with Reynolds number Re, but decreases with Mach number M. More specifically, the optimal energy amplification Gopt (the supremum of G max over both the streamwise and spanwise wavenumbers) is maximum in the incompressible limit and decreases monotonically as M increases. The corresponding optimal streamwise wavenumber, αopt, is non-zero at M = 0, increases with increasing M, reaching a maximum for some value of M and then decreases, eventually becoming zero at high Mach numbers. While the pure streamwise vortices are the optimal patterns at high Mach numbers (in contrast to incompressible Couette flow), the modulated streamwise vortices are the optimal patterns for low-to-moderate values of the Mach number. Unlike in incompressible shear flows, the streamwise-independent modes in the present flow do not follow the scaling law G(t/Re) ~ Re2, the reasons for which are shown to be tied to the dominance of some terms (related to density and temperature fluctuations) in the linear stability operator. Based on a detailed nonmodal energy anlaysis, we show that the transient energy growth occurs due to the transfer of energy from the mean flow to perturbations via an inviscid algebraic instability. The decrease of transient growth with increasing Mach number is also shown to be tied to the decrease in the energy transferred from the mean flow (E1) in the same limit. The sharp decay of the viscous eigenfunctions with increasing Mach number is responsible for the decrease of E1 for the present mean flow. Linear stability and the non-modal transient energy growth in compressible plane Couette flow are investigated for the uniform shear flow with constant viscosity. For a given M, the critical Reynolds number (Re), the dominant instability (over all stream-wise wavenumbers, α) of each mean flow belongs different modes for a range of supersonic M. An analysis of perturbation energy reveals that the instability is primarily caused by an excess transfer of energy from mean-flow to perturbations. It is shown that the energy-transfer from mean-flow occurs close to the moving top-wall for “mode I” instability, whereas it occurs in the bulk of the flow domain for “mode II”.For the Non-modal transient growth anlaysis, it is shown that the maximum temporal amplification of perturbation energy, G max,, and the corresponding time-scale are significantly larger for the uniform shear case compared to those for its non-uniform counterpart. For α = 0, the linear stability operator can be partitioned into L ~ L ¯ L +Re2Lp is shown to have a negligibly small contribution to perturbation energy which is responsible for the validity of the well-known quadratic-scaling law in uniform shear flow: G(t/Re) ~ Re2 . In contrast , the dominance of Lp is responsible for the invalidity of this scaling-law in non-uniform shear flow. An inviscid reduced model, based on Ellignsen-Palm-type solution, has been shown to capture all salient features of transient energy growth of full viscous problem. For both modal and non-modal instability, the viscosity-stratification of the underlying mean flow would lead to a delayed transition in compressible Couette flow. Modal and nonmodal spatial growths of perturbations in compressible plane Couette flow are studied. The modal instability at a chosen set of parameters is caused by the scond least-decaying mode in the otherwise stable parameter setting. The eigenfunction is accurately computed using a three-domain spectral collocation method, and an anlysis of the energy contained in the least-decaying mode reveals that the instability is due to the work by the pressure fluctuations and an increased transfer of energy from mean flow. In the case of oblique modes the stability at higher spanwise wave number is due to higher thermal diffusion rate. At high frequency range there are disjoint regions of instability at chosen Reynolds number and Mach number. The stability characteristics in the inviscid limit is also presented. The increase in Mach number and frequency is found to further destabilize the unstable modes for the case of two-dimensional(2D) perturbations. The behaviors of the non-inflexional neutral modes are found to be similar to that of compressible boundary layer. A leading order viscous correction to the inviscid solution reveals that the neutral and unstable modes are destabilized by the no-slip enforced by viscosity. The viscosity has a dual role on the stable inviscid mode. A spatial transient growth studies have been performed and it is found that the transient amplification is of the order of Reynolds number for a superposition of stationary modes. The optimal perturbations are similar to the streamwise invariant perturbations in the temporal setting. Ellignsen & Palm solution for the spatial algebraic growth of stationary inviscid perturbation has been derived and found to agree well with the transient growth of viscous counterpart. This inviscid solution captures the features of streamwise vortices and streaks, which are observed as optimal viscous perturbations. The temporal secondary instability of most-unstable primary wave is also studied. The secondary growth-rate is many fold higher when compared with that of primary wave and found to be phase-locked. The fundamental mode is more unstable than subharmonic or detuned modes. The secondary growth is studied by varying the parameters such as β, Re, M and the detuning parameter.