Modeling and Analysis of Stochastic Base Flow Uncertainties in Wall-bounded Shear Flows

Spatially distributed dynamical systems arise in a variety of science and engineering problems and are typically described by Partial Integro-Differential (P(I)DEs) equations. Important examples of such systems include the wave equations, Maxwell equations, Burgers equations, Schrodinger equations, and the Navier-Stokes equations. An appropriate way to study and control such systems often involves the spatio-temporal analysis of linearized forms of these equations around base profiles, which either describe a steady-state solution or a long-time averaged mean of a simulation- or experiment-based field. In addition, deterministic or stochastic forcing is commonly used to compensate for the neglected nonlinear terms and evaluate the input-output features of the linearized dynamics. However, uncertainty in both the base profile and nature of the inputs challenge the effectiveness of linearized models for analysis and control design. Motivated by applications in the analysis and control of complex fluid flows, this thesis demonstrates how modeling sources of stochastic base flow uncertainty can enable physical discovery and statistical modeling of quantities of interest. We provide an input-output framework to analyze the effect of base flow perturbations on the stability and receptivity properties of transitional and turbulent channel flows. Such base flow variations are modeled as persistent white-in-time stochastic excitations that enter the linearized dynamics as multiplicative sources of uncertainty that can alter the stability of the linearized dynamics and their receptivity to exogenous excitation. We provide verifiable conditions for mean-square stability and study the frequency response of the flow subject to additive and multiplicative sources of uncertainty using the solution to the generalized Lyapunov equation. Our approach does not rely on costly stochastic simulations or adjointbased sensitivity analyses. We use our framework to uncover the Reynolds number scaling of critically destabilizing variance levels of the base flow uncertainty, study the reliability of numerically estimated mean velocity profiles in turbulent channel flows, and the robust performance of a typical boundary control strategy for turbulence suppression in the wake of parametric uncertainties. For small-amplitude base flow perturbations, we adopt a perturbation analysis to provide a computationally efficient method for computing the variance amplification of velocity fluctuations around the uncertain base. Moreover, we study the flow structures that are extracted from a modal decomposition of the resulting velocity covariance matrix at energetically dominant locations of wall-parallel wavenumbers. In the final part of this thesis, we use the developed input-output framework to evaluate the robust performance transverse lower-wall oscillations as a flow control strategy when oscillations are subject to imperfections in amplitude and phase. These imperfections, cause the nominally harmonic flow control strategy to resemble a random oscillatory pattern.