Reduced-complexity modeling for control and nonlinear analysis of transitional flows

Abstract

The predominant causes for transition to turbulence in wall-bounded shear flows are transient energy growth and nonlinear interactions resulting from the governing linear and nonlinear dynamics. This work addresses both the problem of (i) minimizing transient energy growth (TEG) arising from non-modal dynamics and (ii) accounting for nonlinear interactions while performing stability analysis. This work aims to delay the transition and perform analysis to predict stability boundaries for the onset of turbulence---both with the help of reduced-complexity modeling. In part I, the first problem is addressed by developing control-oriented reduced-order models (COROMs) for designing controllers that minimize maximum transient energy growth (MTEG). The COROMs are developed on the linearized channel flow system using proper orthogonal decomposition and balanced truncation in conjunction. The COROMs enable the development of MTEG minimizing controllers, which otherwise were intractable due to the large dimension of the channel flow system. The performance of the designed controllers is investigated in response to optimal perturbations. In part II, the second problem is addressed by accounting for the nonlinear interactions for stability analysis. Typically in fluid flows, modeling the behaviour of nonlinear interactions exactly is notoriously difficult. Here, instead of using the exact nonlinear dynamics, the input-output properties of the nonlinearity are written as quadratic constraints. The stability analysis of only the linear dynamics is subject to the aforementioned quadratic constraints resulting in linear matrix inequalities~(LMIs). These LMIs enable stability analysis by estimating the region of attraction and the largest permissible perturbation in the nonlinear setting. The utility of the proposed framework is demonstrated on nonlinear low-order models of transitional flows such as the Couette flow. Further, two algorithms are developed, both these algorithms enable solving LMIs with quadratic constraints such that the predicted stability margins can be further improved. Finally, the application of the proposed framework in analyzing the stability of reduced-order viscous Burgers equations, is demonstrated. The viscous Burgers equation is a canonical yet simple model to demonstrate the applicability of the proposed framework. Since the viscous Burgers system has a quadratic convective nonlinearity which introduces transition and change in stability, similar to the Navier-Stokes. The stability of the Burgers equation is studied using the proposed framework and its results are benchmarked against methods like the sum-of-squares method.