A space-time variational approach to hydrodynamic stability theory

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” spacetime norm. We next present a generalization of the linear framework in which the disturbances and perturbation are now measured in respective selected space-time seminorms; the semi-norm theory permits rigorous and sharp quantification of, for example, the growth of initial disturbances or functional outputs. We then develop a (BrezziRappaz-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.