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Correspondence between Koopman mode decomposition, resolvent mode decomposition, and invariant solutions of the Navier-Stokes equations

Correspondence between Koopman mode decomposition, resolvent mode decomposition, and invariant solutions of the Navier-Stokes equations
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.
2469-990X
Sharma, Ati
cdd9deae-6f3a-40d9-864c-76baf85d8718
Mezic, Igor
14a4e679-bdf1-493d-b88c-88527c41455e
McKeon, Beverley J.
4623066f-492f-4944-a541-151a6a130402
Sharma, Ati
cdd9deae-6f3a-40d9-864c-76baf85d8718
Mezic, Igor
14a4e679-bdf1-493d-b88c-88527c41455e
McKeon, Beverley J.
4623066f-492f-4944-a541-151a6a130402

Sharma, Ati, Mezic, Igor and McKeon, Beverley J. (2016) Correspondence between Koopman mode decomposition, resolvent mode decomposition, and invariant solutions of the Navier-Stokes equations. Physical Review Fluids, 1 (3), [032402(R)]. (doi:10.1103/PhysRevFluids.1.032402).

Record type: Article

Abstract

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.

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On the correspondence between Koopman mode decomposition, resolvent mode.pdf - Accepted Manuscript
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Accepted/In Press date: 23 June 2016
e-pub ahead of print date: 18 July 2016
Published date: 18 July 2016
Organisations: Aerodynamics & Flight Mechanics Group

Identifiers

Local EPrints ID: 398122
URI: http://eprints.soton.ac.uk/id/eprint/398122
ISSN: 2469-990X
PURE UUID: 35777297-b456-447b-b6d6-2dcb42a261c6
ORCID for Ati Sharma: ORCID iD orcid.org/0000-0002-7170-1627

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Date deposited: 19 Jul 2016 15:29
Last modified: 15 Mar 2024 03:46

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Contributors

Author: Ati Sharma ORCID iD
Author: Igor Mezic
Author: Beverley J. McKeon

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