Wavelet-based resolvent analysis for statistically-stationary and temporally-evolving flows

Ballouz, Eric; Lopez-Doriga, Barbara; Dawson, Scott T. M.; Bae, H. Jane

doi:10.2514/6.2023-0676

Cited by 4 publications

(5 citation statements)

References 56 publications

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“…where (•) denotes a Fourier transform for a given streamwise and spanwise wavenumber pair (k x , k z ) composed with a wavelet transform in time, ρ is density, and p is the pressure. These transformed quantities are functions of wall-normal position y, and the wavelet scale and shift parameters α and β , which respectively determine the time interval and frequency range of the wavelet mode onto which we project our physical quantities [30]. The nonlinear term, denoted by f , is assumed to act as an exogenous forcing function on the velocity field.…”

Section: Spatio-temporal Resolvent Modesmentioning

confidence: 99%

“…We introduce the additional step of constraining the forcing along a wavelet-shaped pulse of any desired scale α and shift β using a temporal windowing matrix B [39,40,30]. We then take the singular value decomposition (SVD) of the combined operator…”

Section: And the Modified Time Derivative Operator Is Defined Asmentioning

confidence: 99%

“…These resolvent modes, which are Fourier modes in time, lack transient linear growth information. In light of the importance of transient linear growth to wall-bounded turbulence, the aim of this work is to reformulate resolvent analysis in a time-localized basis, such as one constructed with wavelets rather than Fourier modes [30], and to further examine the interactions of these transient resolvent modes with the fully nonlinear turbulent flow in the minimal flow unit.…”

Section: Introductionmentioning

confidence: 99%

“…This paper is organized as follows. In §2.1, we compute resolvent modes in a time-localized wavelet basis as in Ballouz et al [30], and constrain the forcing to a wavelet-shaped pulse. This formulation yields a forcing mode in the shape of streamwise rolls that is compactly supported in time, in addition to an optimal streak-like response that grows transiently before decaying.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Transient growth of wavelet-based resolvent modes in the buffer layer of wall-bounded turbulence

Ballouz,

Dawson,

Jane Bae

2024

J. Phys.: Conf. Ser.

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In this work, we study the transient growth of the principal resolvent modes in the minimal flow unit using a reformulation of resolvent analysis in a time-localized wavelet basis. We target the most energetic spatial wavenumbers for the minimal flow unit and obtain modes that are constant in the streamwise direction and once-periodic in the spanwise direction. The forcing modes are in the shape of streamwise rolls, though pulse-like in time, and the response modes are in the form of transiently growing streaks. We inject the principal transient forcing mode at different intensities into a simulation of the minimal flow unit and compare the resulting nonlinear response to the linear one. The peak energy amplification scales quadratically with the intensity of the injected mode, and this peak occurs roughly at the same time for all forcing intensities. However, the larger energy amplification intensifies the magnitude of the nonlinear terms, which play an important role in damping the energy growth and accelerating energy decay of the principal resolvent mode. We also observe that the damping effect of the nonlinearities is less prominent close to the wall. Finally, we find that the principal resolvent forcing mode is more effective than other structures at amplifying the streak energy in the turbulent minimal-flow unit. In addition to lending support to the claim that linear mechanisms are important to near-wall turbulence, this work identifies time scales for the nonlinear breakdown of linearly-generated streaks.

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Section: Spatio-temporal Resolvent Modesmentioning

confidence: 99%

Section: And the Modified Time Derivative Operator Is Defined Asmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Transient growth of wavelet-based resolvent modes in the buffer layer of wall-bounded turbulence

Ballouz,

Dawson,

Jane Bae

2024

J. Phys.: Conf. Ser.

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“…Carr, McAlister & McCroskey (1981) and Chandrasekhara, Carr & Wilder (1994), to cite a few) but the small size of the LSB, especially at higher Reynolds numbers, make their experimental analysis difficult (Raffel et al 2006). Recently, high-fidelity numerical simulations of dynamic stall have been able to confirm that the bursting of the LSB at the leading edge plays a crucial role in the onset of the dynamic stall at low-to-moderate Reynolds numbers (2.0 × 10 5 ) endeavour that has led to the recent extension of resolvent analysis to non-stationary flow configurations via wavelet transforms by Ballouz et al (2023).…”

Section: Introductionmentioning

confidence: 99%

Direct numerical simulations of an airfoil undergoing dynamic stall at different background disturbance levels

Kern,

Blanco,

Cavalieri

et al. 2024

J. Fluid Mech.

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Thin airfoil dynamic stall at moderate Reynolds numbers is typically linked to the sudden bursting of a small laminar separation bubble close to the leading edge. Given the strong sensitivity of laminar separation bubbles to external disturbances, the onset of dynamic stall on a NACA0009 airfoil section subject to different levels of low-amplitude free stream disturbances is investigated using direct numerical simulations. The flow is practically indistinguishable from clean inflow simulations in the literature for turbulence intensities at the leading edge of ${Tu} = 0.02\,\%$ . At slightly higher turbulence intensities of ${Tu} = 0.05\,\%$ , the bursting process is found to be considerably less smooth and strong coherent vortex shedding from the laminar separation bubble is observed prior to the formation of the dynamic stall vortex (DSV). This phenomenon is considered in more detail by analysing its appearance in an ensemble of simulations comprising statistically independent realisations of the flow, thus proving its statistical relevance. In order to extract the transient dynamics of the vortex shedding, the classical proper orthogonal decomposition method is generalised to include time in the energy measure and applied to the time-resolved simulation data of incipient dynamic stall. Using this technique, the dominant transient spatiotemporally correlated features are distilled and the wave train of the vortex shedding prior to the emergence of the main DSV is reconstructed from the flow data exhibiting dynamics of large-scale coherent growth and decay within the turbulent boundary layer.

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An invitation to resolvent analysis

Rolandi,

Ribeiro,

Yeh

et al. 2024

Theor. Comput. Fluid Dyn.

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Resolvent analysis is a powerful tool that can reveal the linear amplification mechanisms between the forcing inputs and the response outputs about a base flow. These mechanisms can be revealed in terms of a pair of forcing and response modes and the associated energy gains (amplification magnitude) at a given frequency. The linear relationship that ties the forcing and the response is represented through the resolvent operator (transfer function), which is constructed through spatially discretizing the linearized Navier–Stokes operator. One of the unique strengths of resolvent analysis is its ability to analyze statistically stationary turbulent flows. In light of the increasing interest in using resolvent analysis to study a variety of flows, we offer this guide in hopes of removing the hurdle for students and researchers to initiate the development of a resolvent analysis code and its applications to their problems of interest. To achieve this goal, we discuss various aspects of resolvent analysis and its role in identifying dominant flow structures about the base flow. The discussion in this paper revolves around the compressible Navier–Stokes equations in the most general manner. We cover essential considerations ranging from selecting the base flow and appropriate energy norms to the intricacies of constructing the linear operator and performing eigenvalue and singular value decompositions. Throughout the paper, we offer details and know-how that may not be available to readers in a collective manner elsewhere. Towards the end of this paper, examples are offered to demonstrate the practical applicability of resolvent analysis, aiming to guide readers through its implementation and inspire further extensions. We invite readers to consider resolvent analysis as a companion for their research endeavors.

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Wavelet-based resolvent analysis for statistically-stationary and temporally-evolving flows

Cited by 4 publications

References 56 publications

Transient growth of wavelet-based resolvent modes in the buffer layer of wall-bounded turbulence

Transient growth of wavelet-based resolvent modes in the buffer layer of wall-bounded turbulence

Direct numerical simulations of an airfoil undergoing dynamic stall at different background disturbance levels

An invitation to resolvent analysis

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