On the linear receptivity of trailing vortices

Bölle, Tobias; Brion, Vincent; Robinet, Jean-Christophe; Sipp, Denis; Jacquin, Laurent

doi:10.1017/jfm.2020.898

Cited by 11 publications

(26 citation statements)

References 58 publications

(188 reference statements)

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“…According to Heaton (2007), for non-zero viscosity, σ 0 c is replaced by a large number of closely packed discrete eigenmodes, but a detailed explanation was not given. Numerical observations by Bölle et al (2021) identified randomly scattered eigenvalues in the shaded region in figure 4(b), suggesting that they are the viscous remnants of σ 0 c . Mao & Sherwin (2011), who earlier discovered this region, named it the potential spectrum, denoted σ ν p , and suggested that it could be continuous based on the shape of the surrounding pseudospectra.…”

Section: Spectrummentioning

confidence: 95%

“…Since our numerical method was specifically designed not to deal with such non-physical eigenmodes, we do not discuss them further in this paper and clarify that our method is not the tool for those who wish to investigate σ ν f . We remark that Bölle et al (2021) argued that the viscous free-stream spectrum is rather an 'artefact' of the mathematical model of an unbounded domain. With the exception of the viscous free-stream eigenmodes, our numerical method is capable of computing the families of eigenvalues and eigenmodes indicated in figure 4.…”

Section: Spectrummentioning

confidence: 96%

“…Radial domain truncation is not required in our numerical method as it is designed for an unbounded radial domain. Other methods that require a radially bounded domain typically use a large truncation point to mimic unboundedness and impose arbitrary boundary conditions (Khorrami 1991;Mayer & Powell 1992;Mao & Sherwin 2011;Bölle et al 2021). Moreover, our use of Galerkin basis functions eliminates the need for such explicit boundary conditions, reducing numerical error and eliminating further treatments for boundary conditions (see Zebib 1987;McFadden, Murray & Boisvert 1990;Hagan & Priede 2013).…”

Section: Research Goalsmentioning

confidence: 99%

“…Otherwise, all other families that we depicted in figure 4 are in the scope of the analysis. Mao & Sherwin (2011) and Bölle et al (2021) reported that the inviscid critical-layer spectrum σ 0 c changes with viscosity and spreads to an area on the left half-plane of the complex σ plane, which they called the potential spectrum σ ν p . In this section we demonstrate that our numerical method can produce randomly scattered eigenvalues, which represent σ ν p numerically as reported by previous authors, and investigate their spatial characteristics.…”

Section: Viscous Linear Analysismentioning

confidence: 99%

“…The critical-layer eigenmodes arise from the non-normality of the governing equations (i.e. non-commutativity with the Hermitian adjoint) and are associated with continuously distributed eigenvalues in the complex plane, which are known to be significant in transient growth (Mao & Sherwin 2011, 2012Bölle et al 2021). Throughout this paper, we refer to the region where critical-layer eigenvalues exist as a non-normal region.…”

Section: Introductionmentioning

confidence: 99%

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Linear stability analysis of wake vortices by a spectral method using mapped Legendre functions

Lee

Marcus

2023

J. Fluid Mech.

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A spectral method using associated Legendre functions with algebraic mapping is developed for a linear stability analysis of wake vortices. These functions serve as Galerkin basis functions, capturing correct analyticity and boundary conditions for vortices in an unbounded domain. The incompressible Euler or Navier–Stokes equations linearised on a swirling flow are transformed into a standard matrix eigenvalue problem of toroidal and poloidal streamfunctions, solving perturbation velocity eigenmodes with their complex growth rate as eigenvalues. This reduces the problem size for computation and distributes collocation points adjustably clustered around the vortex core. Based on this method, strong swirling $q$ vortices with linear perturbation wavenumbers of order unity are examined. Without viscosity, neutrally stable eigenmodes associated with the continuous eigenvalue spectrum having critical-layer singularities are successfully resolved. The inviscid critical-layer eigenmodes numerically tend to appear in pairs, implying their singular degeneracy. With viscosity, the spectra pertaining to physical regularisation of critical layers stretch out toward an area, referring to potential eigenmodes with wavepackets found by Mao & Sherwin (J. Fluid Mech., vol. 681, 2011, pp. 1–23). However, the potential eigenmodes exhibit no spatial similarity to the inviscid critical-layer eigenmodes, doubting that they truly represent the viscous remnants of the inviscid critical-layer eigenmodes. Instead, two distinct continuous curves in the numerical spectra are identified for the first time, named the viscous critical-layer spectrum, where the similarity is noticeable. Moreover, the viscous critical-layer eigenmodes are resolved in conformity with the $Re^{-1/3}$ scaling law. The onset of the two curves is believed to be caused by viscosity breaking the singular degeneracy.

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Section: Spectrummentioning

confidence: 95%

Section: Spectrummentioning

confidence: 96%

Section: Research Goalsmentioning

confidence: 99%

Section: Viscous Linear Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Linear stability analysis of wake vortices by a spectral method using mapped Legendre functions

Lee

Marcus

2023

J. Fluid Mech.

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Interaction of quasi-two-dimensional vortical gusts with airfoils, unswept and swept wings

2022

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A nearly two-dimensional vortex of small core size has been produced by the transient plunging motion of an upstream airfoil and it interacted with the downstream wings. Depending on the offset distance of the vortex and wing angle of attack, the incident vortex filament deforms, diffuses, and loses coherence, while inducing leading-edge vortex formation and shedding from the wing. No significant spanwise flow develops in the incident vortices during the interaction. The interaction with the swept wing at each spanwise plane appears to be unaffected by the other spanwise planes. The counter-clockwise vortex induces a positive lift peak as it approaches the wing, which can be predicted by the potential flow assumption. The peak lift force is proportional to the circulation of the incident vortex and has its maximum near the zero-offset distance. The minimum lift coefficient is reached after the vortex has just passed and caused flow separation on the lower surface. The maximum lift coefficients for the finite unswept and swept wings can be estimated by making a correction for the aspect ratio and using the independence principle. The only exception is observed for the swept wing at a post-stall angle of attack for which the leading-edge vortex shedding becomes parallel to the leading-edge and increases the peak lift force. Graphical abstract

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