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Role of parasitic modes in nonlinear closure via the resolvent feedback loop

Role of parasitic modes in nonlinear closure via the resolvent feedback loop
Role of parasitic modes in nonlinear closure via the resolvent feedback loop
We use the feedback formulation of McKeon and Sharma [J. Fluid Mech. 658, 336 (2010)], where the nonlinear term in the Navier-Stokes equations is treated as an intrinsic forcing of the linear resolvent operator, to educe the structure of fluctuations in the range of scales (wave numbers) where linear mechanisms are not active. In this region, the absence of dominant linear mechanisms is reflected in the lack of low-rank characteristics of the resolvent and in the disagreement between the structure of resolvent modes and actual flow features. To demonstrate the procedure, we choose low Reynolds number cylinder flow and the Couette equilibrium solution EQ1, which are representative of very low-rank flows dominated by one linear mechanism. The former is evolving in time, allowing us to compare resolvent modes with dynamic mode decomposition (DMD) modes at the first and second harmonics of the shedding frequency. There is a match between the modes at the first harmonic but not at the second harmonic where there is no separation of the resolvent operator's singular values. We compute the self-interaction of the resolvent mode at the shedding frequency and illustrate its similarity to the nonlinear forcing of the second harmonic. When it is run through the resolvent operator, the “forced” resolvent mode shows better agreement with the DMD mode. A similar phenomenon is observed for the fundamental streamwise wave number of the EQ1 solution and its second harmonic. The importance of parasitic modes, labeled as such since they are driven by the amplified frequencies, is their contribution to the nonlinear forcing of the main amplification mechanisms as shown for the shedding mode, which has subtle discrepancies with its DMD counterpart.
2469-990X
1-8
Rosenberg, Kevin
505f63af-9b10-4649-8d31-ae6c060b8ffa
Symon, Sean
2e1580c3-ba27-46e8-9736-531099f3d850
McKeon, Beverley J.
4623066f-492f-4944-a541-151a6a130402
Rosenberg, Kevin
505f63af-9b10-4649-8d31-ae6c060b8ffa
Symon, Sean
2e1580c3-ba27-46e8-9736-531099f3d850
McKeon, Beverley J.
4623066f-492f-4944-a541-151a6a130402

Rosenberg, Kevin, Symon, Sean and McKeon, Beverley J. (2019) Role of parasitic modes in nonlinear closure via the resolvent feedback loop. Physical Review Fluids, 4 (5), 1-8, [052601]. (doi:10.1103/PhysRevFluids.4.052601).

Record type: Article

Abstract

We use the feedback formulation of McKeon and Sharma [J. Fluid Mech. 658, 336 (2010)], where the nonlinear term in the Navier-Stokes equations is treated as an intrinsic forcing of the linear resolvent operator, to educe the structure of fluctuations in the range of scales (wave numbers) where linear mechanisms are not active. In this region, the absence of dominant linear mechanisms is reflected in the lack of low-rank characteristics of the resolvent and in the disagreement between the structure of resolvent modes and actual flow features. To demonstrate the procedure, we choose low Reynolds number cylinder flow and the Couette equilibrium solution EQ1, which are representative of very low-rank flows dominated by one linear mechanism. The former is evolving in time, allowing us to compare resolvent modes with dynamic mode decomposition (DMD) modes at the first and second harmonics of the shedding frequency. There is a match between the modes at the first harmonic but not at the second harmonic where there is no separation of the resolvent operator's singular values. We compute the self-interaction of the resolvent mode at the shedding frequency and illustrate its similarity to the nonlinear forcing of the second harmonic. When it is run through the resolvent operator, the “forced” resolvent mode shows better agreement with the DMD mode. A similar phenomenon is observed for the fundamental streamwise wave number of the EQ1 solution and its second harmonic. The importance of parasitic modes, labeled as such since they are driven by the amplified frequencies, is their contribution to the nonlinear forcing of the main amplification mechanisms as shown for the shedding mode, which has subtle discrepancies with its DMD counterpart.

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Published date: 1 May 2019

Identifiers

Local EPrints ID: 444781
URI: http://eprints.soton.ac.uk/id/eprint/444781
ISSN: 2469-990X
PURE UUID: 1969bcf4-268f-4349-a1e8-ab98e0beac19

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Date deposited: 04 Nov 2020 17:31
Last modified: 14 Sep 2021 18:15

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