Variational formulation of resolvent analysis

Abstract

The conceptual picture underlying resolvent analysis is that the nonlinear term in the Navier-Stokes equations acts as an intrinsic forcing to the linear dynamics, a description inspired by control theory. The inverse of the linear operator, defined as the resolvent, is interpreted as a transfer function between the forcing and the velocity response. From a theoretical point of view this is an attractive approach since it allows for the vast mathematical machinery of control theory to be brought to bear on the problem. However, from a practical point of view, this is not always advantageous. The inversion of the linear operator inherent in the control theoretic definition obscures the physical interpretation of the governing equations and is prohibitive to analytical manipulation, and for large systems it leads to significant computational cost and memory requirements. In this work we suggest an alternative, inverse-free, definition of the resolvent basis based on an extension of the Courant–Fischer–Weyl min-max principle in which resolvent modes are defined as stationary points of a constrained variational problem. This definition 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. To illustrate this method and the advantages of the variational formulation we present three examples. First, we consider streamwise constant fluctuations in turbulent channel flow where an asymptotic analysis allows us to derive closed form expressions for the optimal resolvent mode. Second, to illustrate the cost-saving potential and investigate the limits of the proposed method, we apply our method to both a two-dimensional, three-component equilibrium solution in Couette flow and, finally, to a streamwise developing turbulent boundary layer. For these larger systems we achieve a model reduction of up to two orders of magnitude. Such savings have the potential to open up RA to the investigation of larger domains and more complex flow configurations.

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