In this study, we have considered the modal and non-modal stability of fluids with shear-dependent viscosity flowing in a rigid straight pipe. A second order finite-difference code is used for the simulation of pipe flow in the cylindrical coordinate system. The Carreau-Yasuda model where the rheological parameters vary in the range of 0.3 < n < 1.5 and 0.1 < λ < 100 is represents the viscosity of shear- thinning and shear thickening fluids. Variation of the periodic pulsatile forcing is obtained via the ratio Kω/Kο and set between 0.2 and 20. Zero and non-zero streamwise wavenumber have been considered separately in this study. For the axially invariant mode, energy growth maxima occur for unity azimuthal wave number, whereas for the axially non-invariant mode, maximum energy growth can be observed for azimuthal wave number of two for both Newtonian and non-Newtonian fluids. Modal and non-modal analysis for both Newtonian and non-Newtonian fluids show that the flow is asymptotically stable for any configuration and the pulsatile flow is slightly more stable than steady flow. Increasing the maximum velocity for shear-thinning fluids caused by reducing power-low index n is more evident than shear-thickening fluids. Moreover, rheological parameters of Carreau-Yasuda model have ignored the effect on the peak velocity of the oscillatory components. Increasing Reynolds number will enhance the maximum energy growth while a revers behavior is observed by increasing Womersley number.

Experimental findings show that transition from laminar toturbulent ow may occur also if the exponentially growingperturbations, eigensolutions to the linearised disturbanceequations, are damped. An alternative non-modal growthmechanism has been recently identi fied, also based on thelinear approximation. This consists of the transient growth ofstreamwise elongated disturbances, with regions of positive andnegative streamwise velocity alternating in the spanwisedirection, called streaks. These perturbation are seen toappear in boundary layers exposed to signi ficant levels offree-stream turbulence. The effect of the streaks on thestability and transition of the Blasius boundary layer isinvestigated in this thesis. The analysis considers the steadyspanwise-periodic streaks arising from the nonlinear evolutionof the initial disturbances leading to the maximum transientenergy growth. In the absence of streaks, the Blasius pro filesupports the viscous exponential growth of theTollmien-Schlichting waves. It is found that increasing thestreak amplitude these two-dimensional unstable waves evolveinto three-dimensional spanwiseperiodic waves which are lessunstable. The latter can be completely stabilised above athreshold amplitude. Further increasing the streak amplitude,the boundary layer is again unstable. The new instability is ofdifferent character, being driven by the inectional pro filesassociated with the spanwise modulated ow. In particular, it isshown that, for the particular class of steady streaksconsidered, the most ampli fied modes are antisymmetric andlead to spanwise oscillations of the low-speed streak (sinuousscenario). The transition of the streak is then characterisedby the appearance of quasi-streamwise vorticesfollowing themeandering of the streak. Simulations of a boundary layer subjected to high levels offree-stream turbulence have been performed. The receptivity ofthe boundary layer to the external perturbation is studied indetail. It is shown that two mechanisms are active, a linearand a nonlinear one, and their relative importance isdiscussed. The breakdown of the unsteady asymmetric streaksforming in the boundary layer under free-stream turbulence isshown to be characterised by structures similar to thoseobserved both in the sinuous breakdown of steady streaks and inthe varicose scenario, with the former being the mostfrequently observed. <b>Keywords:</b>Fluid mechanics, laminar-turbulent transition,boundary layer ow, transient growth, streamwise streaks,lift-up effect, receptivity, free-stream turbulence, secondaryinstability, Direct Numerical Simulation.

<p>Experimental findings show that transition from laminar toturbulent ow may occur also if the exponentially growingperturbations, eigensolutions to the linearised disturbanceequations, are damped. An alternative non-modal growthmechanism has been recently identi fied, also based on thelinear approximation. This consists of the transient growth ofstreamwise elongated disturbances, with regions of positive andnegative streamwise velocity alternating in the spanwisedirection, called streaks. These perturbation are seen toappear in boundary layers exposed to signi ficant levels offree-stream turbulence. The effect of the streaks on thestability and transition of the Blasius boundary layer isinvestigated in this thesis. The analysis considers the steadyspanwise-periodic streaks arising from the nonlinear evolutionof the initial disturbances leading to the maximum transientenergy growth. In the absence of streaks, the Blasius pro filesupports the viscous exponential growth of theTollmien-Schlichting waves. It is found that increasing thestreak amplitude these two-dimensional unstable waves evolveinto three-dimensional spanwiseperiodic waves which are lessunstable. The latter can be completely stabilised above athreshold amplitude. Further increasing the streak amplitude,the boundary layer is again unstable. The new instability is ofdifferent character, being driven by the inectional pro filesassociated with the spanwise modulated ow. In particular, it isshown that, for the particular class of steady streaksconsidered, the most ampli fied modes are antisymmetric andlead to spanwise oscillations of the low-speed streak (sinuousscenario). The transition of the streak is then characterisedby the appearance of quasi-streamwise vorticesfollowing themeandering of the streak.</p><p>Simulations of a boundary layer subjected to high levels offree-stream turbulence have been performed. The receptivity ofthe boundary layer to the external perturbation is studied indetail. It is shown that two mechanisms are active, a linearand a nonlinear one, and their relative importance isdiscussed. The breakdown of the unsteady asymmetric streaksforming in the boundary layer under free-stream turbulence isshown to be characterised by structures similar to thoseobserved both in the sinuous breakdown of steady streaks and inthe varicose scenario, with the former being the mostfrequently observed.</p><p><b>Keywords:</b>Fluid mechanics, laminar-turbulent transition,boundary layer ow, transient growth, streamwise streaks,lift-up effect, receptivity, free-stream turbulence, secondaryinstability, Direct Numerical Simulation.</p>

<p>Methods in hydrodynamic stability, systems and control theory are applied to spatially developing flows, where the flow is not required to vary slowly in the streamwise direction. A substantial part of the thesis presents a theoretical framework for the stability analysis, input-output behavior, model reduction and control design for fluid dynamical systems using examples on the linear complex Ginzburg-Landau equation. The framework is then applied to high dimensional systems arising from the discretized Navier–Stokes equations. In particular, global stability analysis of the three-dimensional jet in cross flow and control design of two-dimensional disturbances in the flat-plate boundary layer are performed. Finally, a parametric study of the passive control of two-dimensional disturbances in a flat-plate boundary layer using streamwise streaks is presented.</p>

La stabilité d'un écoulement de couche limite sur une paroi verticale et d'un écoulement de canal entre deux parois verticales est étudiée en présence d'une stratification en densité. Des analyses de stabilité modale et non-modale sont conduites.Pour l'écoulement de couche limite sur une paroi verticale, l'analyse de stabilité temporelle est réalisé pour un profil de vitesse en tanh. Les caractéristiques sont décrites en fonction du nombre de Reynolds (Re) et du nombre de Froude (F). Je montre que l'écoulement de couche limite est sujet à l'instabilité visqueuse et à l'instabilité radiative qui conduisent respectivement à la formation d'ondes de Tollmien-Schlichting (TS) et à la génération spontanée d'ondes internes. Je montre que l'instabilité radiative diminue le nombre de Reynolds critique et domine l'instabilité visqueuse pour des grands nombres de Reynolds. L'instabilité radiative devrait donc être observable dans les expériences et les écoulements géophysiques atmosphériques ou océaniques. Pour l'écoulement de canal, je réalise une étude de stabilité temporelle ainsi qu'une étude des perturbations optimales en utilisant le profil de vitesse de Poiseuille. Comme pour l'écoulement de couche limite, je montre que l'instabilité visqueuse est dominée par une instabilité 3D associée à la stratification. Cette dernière affecte également la croissance transitoire des perturbations. Les deux mécanismes fondamentaux de croissance transitoire que sont les mécanismes de Orr et de ``lift-up'' sont toujours présents mais le mécanisme de lift-up est fortement atténuée par la stratification et rapidement dominée par la présence des instabilités 3D. / The stability of a boundary layer on a vertical wall and a channel flow between two vertical walls is studied in the presence of density stratification. Both modal and non-modal analysis are conducted in these studies. For the boundary layer on a vertical wall, a temporal stability analysis is performed for a tanh velocity profile. The characteristics are analysed as functions of the Reynolds number (Re) and the Froude number (F). The boundary layer is found to be unstable to viscous instability and radiative instability. The viscous instability can lead to Tollmien–Schlichting (TS) waves, and the radiative instability may generate internal gravity waves spontaneously. The radiative instability reduces the critical Re for instability. And for large Reynolds numbers, it dominates the viscous instability. Consequently, radiative instability may develop in experiments and various geophysical situations in the ocean and atmosphere.For the channel flow, we choose plane Poiseuille flow as a prototype. Both the exponential instability and transient growth are analysed. There are also two kinds of exponential instabilities, viscous instability and a 3D instability. The 3D instability influences the behaviour of the transient growth. The fundamental mechanisms in transient growth are the inclination of upstream tilting waves and the growth of streamwise vortices, which are referred to as Orr mechanism and lift-up mechanism. In the presence of stratification, the Orr mechanism is not affected while the lift-up mechanism is weakened. The combination of these two mechanisms is amplified by the influence of the 3D instabilities.

Methods in hydrodynamic stability, systems and control theory are applied to spatially developing flows, where the flow is not required to vary slowly in the streamwise direction. A substantial part of the thesis presents a theoretical framework for the stability analysis, input-output behavior, model reduction and control design for fluid dynamical systems using examples on the linear complex Ginzburg-Landau equation. The framework is then applied to high dimensional systems arising from the discretized Navier–Stokes equations. In particular, global stability analysis of the three-dimensional jet in cross flow and control design of two-dimensional disturbances in the flat-plate boundary layer are performed. Finally, a parametric study of the passive control of two-dimensional disturbances in a flat-plate boundary layer using streamwise streaks is presented. / QC 20101103

Whether it is designing and controlling super-efficient high speed transport systems or understanding environmental fluid flows, a key question that arises is: what state does the fluid take and why? An answer to this question lies in understanding the hydrodynamic stability properties of the flow as a function of parameters. While much work has been done in this area in the past, there are many open questions that need to be addressed. Here we study the effect of spatial domain size, number of modes, non-hermitianness and non-normality on the finite time and infinite time stability properties of a standing, viscous shock flow problem. It has been shown that the above problems are not only non-normal but also non-hermitian, when the base flow has shear. The eigenvalue problems corresponding to infinite spatial domain, finite spatial domain, Forward and L2 adjoint problems are solved exactly by converting the linear partial differential equations into nonlinear Riccati equations. In the finite domain case, the full time dependent solutions are obtained analytically using bi-orthogonal basis functions. In the infinite domain case, the point spectrum of the forward operator is shown to be unbounded and that of the adjoint operator to be empty. In the unbounded case, the spectrum fills the entire area on one side of a parabola in the complex plane and is connected. As the fluid viscosity decreases the width of the parabola increases and in the limit of zero viscosity covers almost entire left half plane(LHP). On the other hand, as the fluid viscosity increases the width of parabola decreases and in the limit of infinite viscosity becomes negative real axis, which is the spectrum of heat equation. The spectrum of adjoint problem is empty for all values of the viscosity and prescribed velocity. In the finite spatial domain case, the point spectrum lies in the open left half plane for all Reynolds numbers and hence asymptotically stable. The results obtained showed that perturbations grow substantially large for finite time before they decay at large times. It is also found that retainig right number of modes is crucial for observing transient growth phenomena. Finally, the linear results are compared with the nonlinear finite amplitude simulation results. The relevance of current results to other fluid flows is presented.

No description available.

Power networks are sophisticated dynamical systems whose stable operation is essential to modern society. We study the swing equation for networks and its linearization (LSEN) as a tool for modeling power systems. Nowadays, phasor measurement units (PMUs) are used across power networks to measure the magnitude and phase angle of electric signals. Given the abundant data that PMUs can produce, we study applications of the dynamic mode decomposition (DMD) and Loewner framework to power systems. The matrices that define the LSEN model have a particular structure that is not recovered by DMD. We thus propose a novel variant of DMD, called structure-preserving DMD (SPDMD), that imposes the LSEN structure upon the recovered system. Since the solution of the LSEN can potentially exhibit interesting transient dynamics, we study the transient growth for the exponential matrix related to the LSEN. We follow Godunov's approach to get upper bounds for the transient growth and also analyze the relationship of such bounds with classical bounds based on the spectrum, numerical range, and pseudospectra. We show how Godunov's bounds can be optimized to bound the solution operator at a given time. The Loewner framework provides a tool for identifying a dynamical system from tangential measurements. The singular values of Loewner matrices guide the discovery of the true order of the underlying system. However, these singular values can exhibit rapid decay when the interpolation points are far from the poles of the system. We establish a range of bounds for this decay of singular values and apply this analysis to power systems. / Doctor of Philosophy / Power networks are sophisticated dynamical systems whose stable operation is essential to modern society. We study a mathematical model called the LSEN to understand and recover the dynamics of power networks. The LSEN model defines some matrices that have special structures dictated by the application. We propose a novel method to recover matrices with this desired structure from data. We also study some properties of the solution of the LSEN model related to the exponential of a matrix, connecting classical results with the particular approach that we follow. In the system identification context, we also study bounds on the singular values of Loewner matrices to understand the interplay between the data (measurements of the system) and mathematical artifacts (poles of the system).

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.