Abstract:
We develop a method for computing the optimal disturbance based on the Orr--Sommerfeld--Squire matrices. The method is similar to the one employed elsewhere in the literature. The basic method compares well when compared to a benchmark case in single-phase flow. In contrast, for two-phase flows, the basic method needs to be modified in a substantial manner before agreement can be obtained with known test cases. These modifications are discussed and derived below and eventually, good agreement between the present case and the two-phase test cases is obtained.

Abstract:
The present thesis deals with the non-modal linear analysis of 3D perturbations in wall flows. In the first part,a solution to the Orr-Sommerfeld and Squire IVP, in the form of orthogonal functions expansion, is researched. The Galerkin method is successfully implemented to numerically compute approximate solutions for bounded flows. The Chandrasekhar functions revealed to ensure a fifth order of accuracy. The focus of the subsequent analysis is on the transient behavior of the perturbation frequency and phase velocity. The results confirm recent observations about a jump in the temporal evolution of the frequency of the wall-normal velocity signal, considered as the end of an Early Transient. After this jump, the wave frequency for Plane Couette flow experiences a periodic modulation about the asymptotic value, which is motivated and investigated in detail. A new result is the presence of a second frequency jump for the wall-normal vorticity. This fact, together with the possibility for different values of the signals asymptotic frequency, shows the existence of an Intermediate Transient. Moreover, a connection between the frequency jumps and the establishing of a self-similarity condition in time for both the velocity and vorticity profiles is found and investigated for both Plane Couette flow and Plane Poiseuille flow. Eventually, through superposition of waves with limited wavenumber range, a wave packet is reconstructed for Plane Couette flow and Blasius boundary-layer flow . The linear spot evolution revealed to have many common features with the early stages of a turbulent spot, particularly the streaky structure and the spot shape.

Abstract:
We introduce a minimization formulation for the determination of a finite-dimensional, time-dependent, orthonormal basis that captures directions of the phase space associated with transient instabilities. While these instabilities have finite lifetime they can play a crucial role by either altering the system dynamics through the activation of other instabilities, or by creating sudden nonlinear energy transfers that lead to extreme responses. However, their essentially transient character makes their description a particularly challenging task. We develop a minimization framework that focuses on the optimal approximation of the system dynamics in the neighborhood of the system state. This minimization formulation results in differential equations that evolve a time-dependent basis so that it optimally approximates the most unstable directions. We demonstrate the capability of the method for two families of problems: i) linear systems including the advection-diffusion operator in a strongly non-normal regime as well as the Orr-Sommerfeld/Squire operator, and ii) nonlinear problems including a low-dimensional system with transient instabilities and the vertical jet in crossflow. We demonstrate that the time-dependent subspace captures the strongly transient non-normal energy growth (in the short time regime), while for longer times the modes capture the expected asymptotic behavior.

Abstract:
The model for Orr--Sommerfeld equation with quadratic profile on the finite interval is considered. The behavior of the spectrum of this problem is completely investigated for large Reynolds numbers. The limit curves are found to which the eigenvalues concentrate and the counting eigenvalue functions along these curves are obtained.

Abstract:
The Orr-Sommerfeld equation is a spectral problem which is known to play an important role in hydrodynamic stability. For an appropriate operator theoretical realization of the equation, we will determine the essential spectrum, and calculate an enclosure of the set of all eigenvalues by elementary analytical means.

Abstract:
The Orr-Sommerfeld equation with linear profile on the finite interval is considered. The behavior of the spectrum of this problem is completely investigated for large Reynolds numbers. The limit curves are found to which the eigenvalues concentrate and the counting eigenvalue functions along these curves are obtained.

Abstract:
By varying the wave number over a large interval of values and by keeping the Reynolds number constant, we analyze the phase and group velocity of linear three-dimensional traveling waves in two sheared flows, the plane channel and wake flows. Evidences are given about the possible compresence of dispersive and nondispersive effects which are associated to the long and short ranges of wavelength, respectively. We solve the Orr-Sommerfeld and Squire eigenvalue problem and observe the least stable mode. At low wave numbers, a dispersive behavior amenable to the least stable eigenmodes belonging to the left branch of the eigenvalue spectrum is observed. By rising the wave number, in both flows, we observe a sharp dispersive to nondispersive transition which is located at a nondimensional wave number of the order of the unity. Beyond the transition, the asymptotically dominant mode belongs to the right branch of the spectrum. The Reynolds number was varied in the ranges 20-100, 1000-8000 for the wake and channel flow, respectively. We also consider the transient behavior of the phase velocity of small three dimensional traveling waves. Given an arbitrary initial condition, we show that the shape of the transient highly depends on the wavelength value with respect to that of the dispersion/nondispersion transition. Furthermore, we show that the phase velocity can oscillate with a frequency which is equal to the width of the real part of the eigenvalue spectrum.

Abstract:
For applications regarding transition prediction, wing design and control of boundary layers, the fundamental understanding of disturbance growth in the flat plate boundary layer is an important issue. In the present work we investigate the stability of boundary layer in Poiseuille flow. We normalize pressure and time by inertial and viscous effects. The disturbances are taken to be periodic in the spanwise direction and time. We present a set of linear governing equations for the parabolic evolution of wavelike disturbances. Then, we derive the so-called modified Orr-Sommerfeld equation that can be applied in the layer. Contrary to what one might think of, we find that Squire’s theorem is not applicable for the boundary layer. We find also that normalization by inertial or viscous effects leads to the same order of stability or instability. For the 2-D disturbances flow, we find the same critical Reynolds number for our two normalizations. This value coincides with the one we know for neutral stability of the known Orr-Sommerfeld equation. We notice also that for all over values of k in the case , correspond the same values of at whatever the normalization. We therefore conclude that in the boundary layer with 2-D disturbances, we have the same neutral stability curve whatever the normalization. We find also that for a flow with high hydrodynamic Reynolds number, the neutral disturbances in the boundary layer are two dimensional. At last, we find that transition from stability to instability or the opposite can occur according to the Reynolds number and the wave number.

Abstract:
A model problem of the form -i\epsilon y''+q(x)y=\lambda y, y(-1)=y(1)=0, is associated with well-known in hydrodynamics Orr--Sommerfeld operator. Here (\lambda) is the spectral parameter, (\epsilon) is the small parameter which is proportional to the viscocity of the liquid and to the reciprocal of the Reynolds number, and (q(x)) is the velocity of the stationary flow of the liquid in the channel (|x|\leqslant 1). We study the behaviour of the spectrum of the corresponding model operator as (\epsilon\to 0) with linear, quadratic and monotonous analytic functions. We show that the sets of the accumulation points of the spectra (the limit spectral graphs) of the model and the corresponding Orr--Sommerfeld operators coincide as well as the main terms of the counting eigenvalue functions along the curves of the graphs.