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.
© 2018 American Physical Society. Received 14 December 2017; published 16 May 2018. The work in this study has been financially supported by a National Science Foundation Graduate Fellowship, Air Force Office of Scientific Research (AFOSR), under Grant No. FA 9550-16-1-0361 and Army Research Office (ARO) under Grant No. W911NF-17-1-0306. The authors would like to thank Denis Sipp for providing the resolvent code and Andres Goza who assisted setting it up on a cluster. Finally, the authors would like to acknowledge Theresa Saxton-Fox and Ryan McMullen for useful feedback on the manuscript.
Submitted - 1712.05473.pdf
Published - PhysRevFluids.3.053902.pdf