Journal Article

A space–time variational approach to hydrodynamic stability theory

Patera, Anthony T., Yano, Masayuki

2013

Overview

We present a hydrodynamic stability theory for incompressible viscous fluid flows based on a space–time variational formulation and associated generalized singular value decomposition of the (linearized) Navier–Stokes equations. We first introduce a linear framework applicable to a wide variety of stationary- or time-dependent base flows: we consider arbitrary disturbances in both the initial condition and the dynamics measured in a ‘data’ space–time norm; the theory provides a rigorous, sharp (realizable) and efficiently computed bound for the velocity perturbation measured in a ‘solution’ space–time norm. We next present a generalization of the linear framework in which the disturbances and perturbation are now measured in respective selected space–time semi-norms; the semi-norm theory permits rigorous and sharp quantification of, for example, the growth of initial disturbances or functional outputs. We then develop a (Brezzi–Rappaz–Raviart) nonlinear theory which provides, for disturbances which satisfy a certain (rather stringent) amplitude condition, rigorous finite-amplitude bounds for the velocity and output perturbations. Finally, we demonstrate the application of our linear and nonlinear hydrodynamic stability theory to unsteady moderate Reynolds number flow in an eddy-promoter channel.

• Publisher: The Royal Society Publishing
• Subject: Brezzi–rappaz–raviart Theory / Computational fluid dynamics / Disturbances / Fluid flow / Functional Outputs / Hydrodynamic Stability / Hydrodynamics / Mathematical analysis / Navier-Stokes equations / Non-Modal Analysis / Perturbation methods / Singular-Value Decomposition / Space–time Variational Analysis / Stability