Imposing Physical Structure within Input-Output Analysis of Fluid Flows: Methods and Applications

Abstract

Input-output (I/O) methods have recently been proposed as simulation-free methods for identifying and quantifying fluid flow instabilities. Recent developments in I/O methods have focused on imposing additional physical structure within the I/O framework by (i) accounting for the structure of the nonlinear terms (i.e., structured I/O), or (ii) promoting sparsity in the identified instability mechanisms (i.e., spatially-localized modal analysis). This dissertation contributes to the state of the art by formulating and applying I/O analysis algorithms that are both computationally efficient and impose this physical structure in a mathematically consistent manner. First, we propose algorithms for performing the structured I/O analysis, which involves computing structured singular value (SSV) bounds (worst-case gains) and mode shapes by exploiting the underlying mathematical structure of the convective nonlinearity in the incompressible Navier-Stokes equations (NSE). The analysis yields physical insights of the global flow mechanisms, which are useful for identifying flow instabilities. We demonstrate the analysis on a laminar channel flow, and a turbulent channel flow over riblets. For both models, we identify various relevant flow mechanisms that are consistent with the ones predicted in high-fidelity numerical simulations, e.g., Kelvin-Helmholtz (KH) vortices, lift-up effects and Near-Wall (NW) cycles. Second, we propose computationally efficient algorithms for spatially-localized modal analysis. Unlike state of the art methods that promote sparsity, the methods proposed here work to solve a cardinality-constrained optimization problem. The solution to the optimization are sparse modes that highlight most pertinent flow quantities for triggering instabilities. We demonstrate the analysis on a laminar channel flow, where the sparse modes identify various spatially-localized flow mechanisms that contribute to the kinetic energy growth of the flow, e.g., the lift-up effect and Tollmien-Schlichting instabilities.