On the Variational Principles of Linear and Nonlinear Resolvent Analysis

Citation

Barthel, Benedikt (2022) On the Variational Principles of Linear and Nonlinear Resolvent Analysis. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/sy44-d841. https://resolver.caltech.edu/CaltechTHESIS:03222022-135834919

Abstract

Despite decades of research, the accurate and efficient modeling of turbulent flows remains a challenge. However, one promising avenue of research has been the resolvent analysis framework pioneered by McKeon and Sharma (2010) which interprets the nonlinearity of the Navier-Stokes equations (NSE) as an intrinsic forcing to the linear dynamics. This thesis contributes to the advancement of both the linear and nonlinear aspects of resolvent analysis (RA) based modeling of wall bounded turbulent flows. On the linear front, we suggest an alternative definition of the resolvent basis based on the calculus of variations. The proposed formulation circumvents the reliance on the inversion of the linear operator and is inherently compatible with any arbitrary choice of norm. This definition, which defines resolvent modes as stationary points of an operator norm, allows for more tractable analytical manipulation and leads to a straightforward approach to approximate the resolvent (response) modes of complex flows as expansions in any arbitrary basis. The proposed method avoids matrix inversions and requires only the spectral decomposition of a matrix of significantly reduced size as compared to the original system, thus having the potential to open up RA to the investigation of larger domains and more complex flow configurations. These analytical and numerical advantages are illustrated through a series of examples in one and two dimensions. The nonlinear aspects of RA are addressed in the context of Taylor vortex flow. Highly truncated and fully nonlinear solutions are computed by treating the nonlinearity not as an inherent part of the governing equations but rather as a triadic constraint which must be satisfied by the model solution. Our results show that as the Reynolds number increases, the flow undergoes a fundamental transition from a classical weakly nonlinear regime, where the forcing cascade is strictly down scale, to a fully nonlinear regime characterized by the emergence of an inverse (up scale) forcing cascade. It is shown analytically that this is a direct consequence of the structure of the quadratic nonlinearity of the NSE formulated in Fourier space. Finally, we suggest an algorithm based on the energy conserving nature of the nonlinearity of the NSE to reconstruct the phase information, and thus higher order statistics, from knowledge of solely the velocity spectrum. We demonstrate the potential of the proposed algorithm through a series of examples and discuss the challenges and potential applications to the study and simulation of turbulent flows.

Thesis Files