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Sparse space–time resolvent analysis for statistically stationary and time-varying flows

Published online by Cambridge University Press:  21 November 2024

Barbara Lopez-Doriga Open the ORCID record for Barbara Lopez-Doriga [Opens in a new window] ,
Eric Ballouz Open the ORCID record for Eric Ballouz [Opens in a new window] ,
H. Jane Bae Open the ORCID record for H. Jane Bae [Opens in a new window]  and
Scott T.M. Dawson Open the ORCID record for Scott T.M. Dawson [Opens in a new window]
Barbara Lopez-Doriga*
Affiliation:
Mechanical, Materials, and Aerospace Engineering Department, Illinois Institute of Technology, Chicago, IL 60616, USA
Eric Ballouz
Affiliation:
Mechanical and Civil Engineering, California Institute of Technology, Pasadena, CA 91125, USA
H. Jane Bae
Affiliation:
Graduate Aerospace Laboratory, California Institute of Technology, Pasadena, CA 91125, USA
Scott T.M. Dawson
Affiliation:
Mechanical, Materials, and Aerospace Engineering Department, Illinois Institute of Technology, Chicago, IL 60616, USA
*
†Email address for correspondence: barbara.ldoriga@gmail.com
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Abstract

Resolvent analysis provides a framework to predict coherent spatio-temporal structures of the largest linear energy amplification, through a singular value decomposition (SVD) of the resolvent operator, obtained by linearising the Navier–Stokes equations about a known turbulent mean velocity profile. Resolvent analysis utilizes a Fourier decomposition in time, which has thus far limited its application to statistically stationary or time-periodic flows. This work develops a variant of resolvent analysis applicable to time-evolving flows, and proposes a variant that identifies spatio-temporally sparse structures, applicable to either stationary or time-varying mean velocity profiles. Spatio-temporal resolvent analysis is formulated through the incorporation of the temporal dimension to the numerical domain via a discrete time-differentiation operator. Sparsity (which manifests in localisation) is achieved through the addition of an $l_1$-norm penalisation term to the optimisation associated with the SVD. This modified optimisation problem can be formulated as a nonlinear eigenproblem and solved via an inverse power method. We first showcase the implementation of the sparse analysis on a statistically stationary turbulent channel flow, and demonstrate that the sparse variant can identify aspects of the physics not directly evident from standard resolvent analysis. This is followed by applying the sparse space–time formulation on systems that are time varying: a time-periodic turbulent Stokes boundary layer and then a turbulent channel flow with a sudden imposition of a lateral pressure gradient, with the original streamwise pressure gradient unchanged. We present results demonstrating how the sparsity-promoting variant can either change the quantitative structure of the leading space–time modes to increase their sparsity, or identify entirely different linear amplification mechanisms compared with non-sparse resolvent analysis.

JFM classification

Turbulent Flows: Turbulence theory Turbulent Flows: Turbulent boundary layers
Type
JFM Papers
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Journal of Fluid Mechanics , Volume 999 , 25 November 2024 , A87
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© The Author(s), 2024. Published by Cambridge University Press

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