No inverse necessary: a variational formulation of resolvent analysis

Resolvent analysis interprets the nonlinear term in the Navier-Stokes equations (NSE) as an intrinsic forcing to the linear dynamics. The conceptual picture of this phenomenon is inspired by control theory (CT), where the inverse of the linear operator, defined as the resolvent, is interpreted as a transfer function between the forcing and the velocity response. However, the inversion of the linear operator inherent in the CT definition obscures the physical interpretation of the governing equations and is prohibitive to analytical manipulation. Additionally, for large systems this inversion leads to significant computational cost and memory requirements.  Here 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 response modes are defined as stationary points of a constrained variational problem.  This definition leads to a straightforward approach to approximate the resolvent modes of complex flows as expansions in any arbitrary basis. The proposed method avoids any matrix inversions and requires only the spectral decomposition of a matrix of significantly reduced size as compared to the original system.  This approach enables accurate reconstruction of the response modes regardless of the properties of the linear operator, however the non-normality and directional amplification of the NSE operator can lead to errors in the forcing modes and singular values. This variational framework is applied to a series of examples including a streamwise developing turbulent boundary layer to illustrate both the analytical and computational advantages of the proposed method.