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Non-normality and classification of amplification mechanisms in stability and resolvent analysis

Non-normality and classification of amplification mechanisms in stability and resolvent analysis
Non-normality and classification of amplification mechanisms in stability and resolvent analysis
Eigenspectra and pseudospectra of the mean-linearized Navier-Stokes operator are used to characterize amplification mechanisms in laminar and turbulent flows in which linear mechanisms are important. Success of mean flow (linear) stability analysis for a particular frequency is shown to depend on whether two scalar measures of non-normality agree: (1) the product between the resolvent norm and the distance from the imaginary axis to the closest eigenvalue and (2) the inverse of the inner product between the most amplified resolvent forcing and response modes. If they agree, the resolvent operator can be rewritten in its dyadic representation to reveal that the adjoint and forward stability modes are proportional to the forcing and response resolvent modes at that frequency. Hence the real parts of the eigenvalues are important since they are responsible for resonant amplification and the resolvent operator is low rank when the eigenvalues are sufficiently separated in the spectrum. If the amplification is pseudoresonant, then resolvent analysis is more suitable to understand the origin of observed flow structures. Two test cases are studied: low Reynolds number cylinder flow and turbulent channel flow. The first deals mainly with resonant mechanisms, hence the success of both classical and mean stability analysis with respect to predicting the critical Reynolds number and global frequency of the saturated flow. Both scalar measures of non-normality agree for the base and mean flows, and the region where the forcing and response modes overlap scales with the length of the recirculation bubble. In the case of turbulent channel flow, structures result from both resonant and pseudoresonant mechanisms, suggesting that both are necessary elements to sustain turbulence. Mean shear is exploited most efficiently by stationary disturbances while bounds on the pseudospectra illustrate how pseudoresonance is responsible for the most amplified disturbances at spatial wavenumbers and temporal frequencies corresponding to well-known turbulent structures. Some implications for flow control are discussed.
2469-990X
Symon, Sean
2e1580c3-ba27-46e8-9736-531099f3d850
Rosenberg, Kevin
505f63af-9b10-4649-8d31-ae6c060b8ffa
Dawson, Scott T. M.
e0bc000a-f03c-47b3-96ba-b97d0f47ff1b
McKeon, Beverley J.
4623066f-492f-4944-a541-151a6a130402
Symon, Sean
2e1580c3-ba27-46e8-9736-531099f3d850
Rosenberg, Kevin
505f63af-9b10-4649-8d31-ae6c060b8ffa
Dawson, Scott T. M.
e0bc000a-f03c-47b3-96ba-b97d0f47ff1b
McKeon, Beverley J.
4623066f-492f-4944-a541-151a6a130402

Symon, Sean, Rosenberg, Kevin, Dawson, Scott T. M. and McKeon, Beverley J. (2018) Non-normality and classification of amplification mechanisms in stability and resolvent analysis. Physical Review Fluids, 3 (5), [053902]. (doi:10.1103/PhysRevFluids.3.053902).

Record type: Article

Abstract

Eigenspectra and pseudospectra of the mean-linearized Navier-Stokes operator are used to characterize amplification mechanisms in laminar and turbulent flows in which linear mechanisms are important. Success of mean flow (linear) stability analysis for a particular frequency is shown to depend on whether two scalar measures of non-normality agree: (1) the product between the resolvent norm and the distance from the imaginary axis to the closest eigenvalue and (2) the inverse of the inner product between the most amplified resolvent forcing and response modes. If they agree, the resolvent operator can be rewritten in its dyadic representation to reveal that the adjoint and forward stability modes are proportional to the forcing and response resolvent modes at that frequency. Hence the real parts of the eigenvalues are important since they are responsible for resonant amplification and the resolvent operator is low rank when the eigenvalues are sufficiently separated in the spectrum. If the amplification is pseudoresonant, then resolvent analysis is more suitable to understand the origin of observed flow structures. Two test cases are studied: low Reynolds number cylinder flow and turbulent channel flow. The first deals mainly with resonant mechanisms, hence the success of both classical and mean stability analysis with respect to predicting the critical Reynolds number and global frequency of the saturated flow. Both scalar measures of non-normality agree for the base and mean flows, and the region where the forcing and response modes overlap scales with the length of the recirculation bubble. In the case of turbulent channel flow, structures result from both resonant and pseudoresonant mechanisms, suggesting that both are necessary elements to sustain turbulence. Mean shear is exploited most efficiently by stationary disturbances while bounds on the pseudospectra illustrate how pseudoresonance is responsible for the most amplified disturbances at spatial wavenumbers and temporal frequencies corresponding to well-known turbulent structures. Some implications for flow control are discussed.

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Published date: 16 May 2018

Identifiers

Local EPrints ID: 444778
URI: http://eprints.soton.ac.uk/id/eprint/444778
ISSN: 2469-990X
PURE UUID: f116eb78-81ed-4df9-bdf6-4b94b57efee8

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Date deposited: 04 Nov 2020 17:31
Last modified: 14 Sep 2021 18:47

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