A space-time variational approach to hydrodynamic stability theory
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Patera_A space-time.pdf
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Author(s)
Yano, Masayuki
Patera, Anthony T.
Date Issued
May 2013
Journal
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Publisher
Royal Society
Citation
Yano, M., and A. T. Patera. “A Space-Time Variational Approach to Hydrodynamic Stability Theory.” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 469, no. 2155 (April 24, 2013): 20130036–20130036.
Version
Original manuscript
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Department of Mechanical Engineering
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DOI of Published Version
https://doi.org/10.1098/rspa.2013.0036
