Stochastic forcing of the Lamb–Oseen vortex

Fontane, Jérôme; Brancher, Pierre; Fabre, David

doi:10.1017/s002211200800308x

Cited by 24 publications

(21 citation statements)

References 34 publications

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“…They found qualitative agreement between the two, further supported by recent results of HPS showing that the dominant core perturbation is a bending wave mode (at k ≈ 1.4, the peak k value predicted from linear analysis). This is further supported by Fontane, Brancher & Fabre (2008), who found that stochastic forcing of the Oseen vortex also excited large growth at m = 1, k = 1.4. Dominance of m = 1 is expected as turbulence could simultaneously excite multiple transiently growing perturbations, with m = 1 being the most unstable.…”

Section: Transient Growth In Turbulent Vorticessupporting

confidence: 63%

Self-limiting and regenerative dynamics of perturbation growth on a vortex column

Hussain

Stout

2013

J. Fluid Mech.

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We study the mechanisms of centrifugal instability and its eventual self-limitation, as well as regenerative instability on a vortex column with a circulation overshoot (potentially unstable) via direct numerical simulations of the incompressible Navier-Stokes equations. The perturbation vorticity (ω ) dynamics are analysed in cylindrical (r, θ, z) coordinates in the computationally accessible vortex Reynolds number, Re(≡circulation/viscosity), range of 500-12 500, mostly for the axisymmetric mode (azimuthal wavenumber m = 0). Mean strain generates azimuthally oriented vorticity filaments (i.e. filaments with azimuthal vorticity, ω θ ), producing positive Reynolds stress necessary for energy growth. This ω θ in turn tilts negative mean axial vorticity, −Ω z (associated with the overshoot), to amplify the filament, thus causing instability. (The initial energy growth rate (σ r ), peak energy (G max ) and time of peak energy (T p ) are found to vary algebraically with Re.) Limitation of vorticity growth, also energy production, occurs as the filament moves the overshoot outward, hence lessening and shifting |−Ω z |, while also transporting the core +Ω z , to the location of the filament. We discover that a basic change in overshoot decay behaviour from viscous to inviscid occurs at Re ∼ 5000. We also find that the overshoot decay time has an asymptotic limit of 45 turnover times with increasing Re. After the limitation, the filament generates negative Reynolds stress, concomitant energy decay and hence self-limitation of growth; these inviscid effects are enhanced further by viscosity. In addition, the filament transports angular momentum radially inward, which can produce a new circulation overshoot and renewed instability. Energy decays at the Re studied, but, at higher Re, regenerative growth of energy is likely due to the renewed mean shearing. New generation of overshoot and Reynolds stress is examined using a helical (m = 1) perturbation. Regenerative energy growth, possibly resulting in even vortex breakup, can be triggered by this new overshoot at practical Re (∼10 6 for trailing vortices), which are currently beyond the computational capability.

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Section: Transient Growth In Turbulent Vorticessupporting

confidence: 63%

Self-limiting and regenerative dynamics of perturbation growth on a vortex column

Hussain

Stout

2013

J. Fluid Mech.

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“…The same trend was observed by Devenport et al (1996), who used hot-wire anemometry to estimate the amplitude of the lateral motion of a single trailing vortex, the so-called vortex meandering (Roy & Leweke 2008). This phenomenon is currently not fully understood; a probable explanation is related to transient growth of perturbations triggered by random external perturbations (Antkowiak & Brancher 2004;Fontane, Brancher & Fabre 2008;Roy & Leweke 2008), which can either come from the turbulence of the free stream, or be generated in the wake of the wing, possibly through flow separation at the relatively low Reynolds numbers considered here. The development of the Crow instability is the second factor leading to an increase of a M and a m , as the amplitude of the long sinusoidal deformations of the vortices increases.…”

Section: Three-dimensional Base Flowmentioning

confidence: 95%

Experiments on the elliptic instability in vortex pairs with axial core flow

et al. 2011

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International audienceResults are presented from an experimental study on the dynamics of pairs of vortices, in which the axial velocity within each core differs from that of the surrounding fluid. Co- and counter-rotating vortex pairs at moderate Reynolds numbers were generated in a water channel from the tips of two rectangular wings. Measurement of the three-dimensional velocity field was accomplished using stereoscopic particle image velocimetry, revealing significant axial velocity deficits in the cores. For counter-rotating pairs, the long-wavelength Crow instability, involving symmetric wavy displacements of the vortices, could be clearly observed using dye visualisation. Measurements of both the axial wavelength and the growth rate of the unstable perturbation field were found to be in good agreement with theoretical predictions based on the full experimentally measured velocity profile of the vortices, including the axial flow. The dye visualisations further revealed the existence of a short-wavelength core instability. Proper orthogonal decomposition of the time series of images from high-speed video recordings allowed a precise characterisation of the instability mode, which involves an interaction of waves with azimuthal wavenumbers m = 2 and m = 0. This combination of waves fulfils the resonance condition for the elliptic instability mechanism acting in strained vortical flows. A numerical three-dimensional stability analysis of the experimental vortex pair revealed the same unstable mode, and a comparison of the wavelength and growth rate with the values obtained experimentally from dye visualisations shows good agreement. Pairs of co-rotating vortices evolve in the form of a double helix in the water channel. For flow configurations that do not lead to merging of the two vortices over the length of the test section, the same type of short-wave perturbations were observed. As for the counter-rotating case, quantitative measurements of the wavelength and growth rate, and comparison with previous theoretical predictions, again identify the instability as due to the elliptic mechanism. Importantly, the spatial character of the short-wave instability for vortex pairs with axial flow is different from that previously found in pairs without axial flow, which exhibit an azimuthal variation with wavenumber m = 1

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“…We assume that F is a unitary matrix (F † F = I) so that its columns are orthogonal; the forcing is defined as the sum of an orthonormal set of uncorrelated processes. As mentioned by Fontane et al (2008), a spatial covariance matrix representing a true experimental noise environment may also be used in order to represent more specific perturbation fields. In our case, we assume that no information on the noise is available and keep the above-mentioned formulation so as to mimic the most generic unknown disturbances in an unbiased manner.…”

Section: Empirical Orthogonal Functionsmentioning

confidence: 99%

“…These are referred to as stochastic optimals (SOs), see Farrell & Ioannou (1993a,b, 2001. The stochastic forcing approach has been successfully applied in systems with a small number of degrees of freedom by Fontane, Brancher & Fabre (2008). Our goal is to show how these quantities may be computed for high dimensional fluid systems and how these structures (EOFs and SOs) may be related to the instabilities displayed by the system (optimal perturbations or optimal forcing responses).…”

Section: Introductionmentioning

confidence: 99%

Stochastic dynamics and model reduction of amplifier flows: the backward facing step flow

2013

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Methods for investigating and approximating the linear dynamics of amplifier flows are examined in this paper. The procedures are derived for incompressible flow over a two-dimensional backward-facing step. First, the singular value decomposition of the resolvent is performed over a frequency range in order to identify the optimal and suboptimal harmonic forcing and responses of the flow. These forcing/responses are shown to be organized into two categories: the first accounting for the Orr and Kelvin-Helmholtz instabilities in the shear layer and the second for the advection and diffusion of perturbations in the free stream. Next, we investigate the dynamics of the flow when excited by a white in space and time noise. We compute the predominant patterns of the random flow which optimally account for the sustained variance, the empirical orthogonal functions (EOFs), as well as the predominant forcing structures which optimally contribute to the sustained variance, the stochastic optimals (SOs). The leading EOFs and SOs are expressed as a linear combination of the suboptimal forcing and responses of the flow and are related to particular instability mechanisms and/or frequency intervals. Finally, we use the leading EOFs, SOs and balanced modes (obtained from balanced truncation) to build low-order models of the flow dynamics. These models are shown to accurately recover the time propagator and resolvent of the original dynamical system. In other words, such models capture the entire flow response from any forcing and may be used in the design of efficient closed-loop controllers for amplifier flows.

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Stochastic forcing of the Lamb–Oseen vortex

Cited by 24 publications

References 34 publications

Self-limiting and regenerative dynamics of perturbation growth on a vortex column

Self-limiting and regenerative dynamics of perturbation growth on a vortex column

Experiments on the elliptic instability in vortex pairs with axial core flow

Stochastic dynamics and model reduction of amplifier flows: the backward facing step flow

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