Correspondence between Koopman mode decomposition, resolvent mode decomposition, and invariant solutions of the Navier-Stokes equations
The relationship between Koopman mode decomposition, resolvent mode decomposition, and exact invariant solutions of the Navier-Stokes equations is clarified. The correspondence rests upon the invariance of the system operators under symmetry operations such as spatial translation. The usual interpretation of the Koopman operator is generalized to permit combinations of such operations, in addition to translation in time. This invariance is related to the spectrum of a spatiotemporal Koopman operator, which has a traveling-wave interpretation. The relationship leads to a generalization of dynamic mode decomposition, in which symmetry operations are applied to restrict the dynamic modes to span a subspace subject to those symmetries. The resolvent is interpreted as the mapping between the Koopman modes of the Reynolds stress divergence and the velocity field. It is shown that the singular vectors of the resolvent (the resolvent modes) are the optimal basis in which to express the velocity field Koopman modes where the latter are not a priori known.
© 2016 American Physical Society. (Received 29 March 2016; published 18 July 2016) No significant data were generated for the purposes of EPSRC's data policy. I.M. was partially supported by ARO Contract No. W911NF-14-C-0102 and AFOSR Grant No. FA9550-12-1-0230. B.J.M. gratefully acknowledges the support of the ONR under Grant No. N000141310739.
Submitted - 1603.08378.pdf
Published - PhysRevFluids.1.032402.pdf