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  • 1
    Electronic Resource
    Electronic Resource
    An asymptotic theory for the nonlinear instability of antiparallel pairs of vortex filaments (1993)
    New York, NY : American Institute of Physics (AIP)
    Physics of Fluids 5 (1993), S. 369-379 
    Details
    ISSN: 1089-7666
    Source: AIP Digital Archive
    Topics: Physics
    Notes: Simplified asymptotic equations describing the nonlinear dynamics of perturbed pairs of parallel vortex filaments are derived and analyzed here. The derivations are general enough to allow for vortices of unequal strength, but emphasis here is on the antiparallel vortex pair. The simplified asymptotic equations account for both the internal effects of self-induction and self-stretching for each filament and also the external effects of mutual induction that lead to a nontrivial coupling of the perturbations of the two filaments. When these nonlinear equations are linearized at the unperturbed filament pair, the linearized stability theory of Crow [AIAA J. 8, 2172 (1970)] is recovered in a systematic fashion. The asymptotic equations are derived in a novel singular limit at high Reynolds numbers through assumptions similar to the authors' recent theories [Physica D 49, 323 (1991); ibid. 53, 267 (1991); Phys. Fluids A 4, 2271 (1992)] for the dynamics of a single perturbed vortex filament. Through the Hasimoto transform [J. Fluid Mech. 51, 477 (1972)], these equations become two coupled perturbed nonlinear Schrödinger equations for a pair of filament functions. A series of numerical solutions of the asymptotic equations exhibits several new phenomena in the nonlinear instability of pairs of antiparallel vortex filaments.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.858860
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  • 2
    Electronic Resource
    Electronic Resource
    Asymptotic equations for the stretching of vortex filaments in a background flow field (1992)
    New York, NY : American Institute of Physics (AIP)
    Physics of Fluids 4 (1992), S. 2271-2281 
    Details
    ISSN: 1089-7666
    Source: AIP Digital Archive
    Topics: Physics
    Notes: Recently, two of the authors have derived [Physica D 49, 323 (1991)] and analyzed [Physica D 53, 267 (1991)] a new asymptotic equation for the evolution of small-amplitude short-wavelength perturbations of slender vortex filaments in high Reynolds number flows. This asymptotic equation differs significantly from the familiar local self-induction equation in that it includes some of the nonlocal effects of self-stretching of the filament in a simple fashion. Here, through systematic asymptotic expansions, the authors derive a modification of this asymptotic equation that incorporates the important additional effects of strain and rotation from a general background flow field. The main requirement on the background flow is that it does not displace the unperturbed background filament. The new asymptotic equations exhibit in a simple fashion the direct competition in filament dynamics between internal effects such as self-induction and self-stretching and external effects of background flows involving strain and rotation. Solutions of these asymptotic equations revealing various aspects of this competition are analyzed in detail through both theory and numerical simulation. An application is also presented for the nonlinear stability of a columnar vortex to suitable perturbations in a straining, rotating, background environment.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.858467
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  • 3
    Electronic Resource
    Electronic Resource
    Representation of core dynamics in slender vortex filament simulations (1996)
    [S.l.] : American Institute of Physics (AIP)
    Physics of Fluids 8 (1996), S. 2415-2425 
    Details
    ISSN: 1089-7666
    Source: AIP Digital Archive
    Topics: Physics
    Notes: The numerical description of slender vortex motion faces several major obstacles: (i) The stiffness induced by the rapid rotatory motion in the vortex core, where peak velocities are an order of magnitude larger than the filament velocity. In a vorticity-velocity formulation, this stiffness is reflected by the singular behavior of the line-Biot-Savart integral as one approaches the vortex geometry. Regularization occurs physically by viscous smoothing of the vorticity. (ii) The vortex core vorticity distribution has a crucial influence on the vortex filament motion. Thus, an accurate description of the core structure evolution due to vortex stretching and vorticity diffusion is necessary. We propose a numerical scheme that allows an accurate description of the effects of axial flow in the core, viscosity and vortex stretching on slender vortex filament motion. The approach is based on incorporating the detailed asymptotic analyses of the vortex core structure evolution by Callegari and Ting [SIAM J. Appl. Math. 15, 148 (1978)] and Klein and Ting [Appl. Math. Lett. 8, 45 (1995)] for stretched viscous slender vortices into the improved thin-tube vortex element schemes of Klein and Knio (1995). The resulting schemes overcome the difficulties mentioned above except for the issue of temporal stiffness, which we leave for future work. © 1996 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.869025
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  • 4
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    Asymptotic vorticity structure and numerical simulation of slender vortex filaments (1995)
    Cambridge University Press
    Details
    Publication Date: 1995-02-10
    Description: A new asymptotic analysis of slender vortices in three dimensions, based solely on the vorticity transport equation and the non-local vorticity-velocity relation gives new insight into the structure of slender vortex filaments. The approach is quite different from earlier analyses using matched asymptotic solutions for the velocity field and it yields additional information. This insight is used to derive three different modifications of the thin-tube version of a numerical vortex element method. Our modifications remove an 0(1) error from the node velocities of the standard thin-tube model and allow us to properly account for any prescribed physical vortex core structure independent of the numerical vorticity smoothing function. We demonstrate the performance of the improved models by comparison with asymptotic solutions for slender vortex rings and for perturbed slender vortex filaments in the Klein-Majda regime, in which the filament geometry is characterized by small-amplitude-short-wavelength displacements from a straight line. These comparisons represent a stringent mutual test for both the proposed modified thin-tube schemes and for the Klein-Majda theory. Importantly, we find a convincing agreement of numerical and asymptotic predictions for values of the Klein-Majda expansion parameter e as large as f. Thus, our results support their findings in earlier publications for realistic physical vortex core sizes. © 1995, Cambridge University Press. All rights reserved.
    Print ISSN: 0022-1120
    Electronic ISSN: 1469-7645
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
    Published by Cambridge University Press
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  • 5
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    Simplified equations for the interaction of nearly parallel vortex filaments (1995)
    Cambridge University Press
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    Publication Date: 1995-04-10
    Description: New simplified asymptotic equations for the interaction of nearly parallel vortex filaments are derived and analysed here. The simplified equations retain the important physical effects of linearized local self-induction and nonlinear potential vortex interaction among different vortices but neglect other non-local effects of self-stretching and mutual induction. These equations are derived systematically in a suitable distinguished asymptotic limit from the Navier-Stokes equations. The general Hamiltonian formalism and conserved quantities for the simplified equations are developed here. Properties of these asymptotic equations for the important special case involving nearly parallel pairs of interacting filaments are developed in detail. In particular, strong evidence is presented that for any filament pair with a negative circulation ratio, there is finite-time collapse in a self-similar fashion independent of the perturbation but with a structure depending on the circulation ratio. On the other hand, strong evidence is presented that no finite-time collapse is possible for perturbations of a filament pair with a positive circulation ratio. The present theory is also compared and contrasted with earlier linear and nonlinear stability analyses for pairs of filaments. © 1995, Cambridge University Press. All rights reserved.
    Print ISSN: 0022-1120
    Electronic ISSN: 1469-7645
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
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  • 6
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    Energy dissipation caused by boundary layer instability at vanishing viscosity – ERRATUM (2018)
    Cambridge University Press
    Details
    Publication Date: 2018-10-31
    Description: In the original version of this article (Nguyen van yen et al. 2018), the copyright statement incorrectly indicated that it had been transferred to Cambridge University Press. The authors retain the copyright of this work. The copyright line has been updated in the original to the following: © The Authors 2018 The publisher apologises to the authors for this mistake. © The Authors 2018.
    Print ISSN: 0022-1120
    Electronic ISSN: 1469-7645
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
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  • 7
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    Motion and structure of atmospheric mesoscale baroclinic vortices: dry air and weak environmental shear (2012)
    Cambridge University Press
    Details
    Publication Date: 2012-05-14
    Description: A strongly tilted, nearly axisymmetric vortex in dry air with asymmetric diabatic heating is analysed here by matched asymptotic expansions. The vortex is in gradient wind balance, with vortex Rossby numbers of order unity, and embedded in a quasi-geostrophic (QG) background wind with weak vertical shear. With wind speeds of 60-120 km h -1, such vortices correspond to tropical storms or nascent hurricanes according to the Saffir-Simpson scale. For asymmetric heating, nonlinear coupling of the evolution equations for the vortex tilt, its core structure, and its influence on the QG background is found. The theory compares well with the established linear theory of precessing quasi-modes of atmospheric vortices, and it corroborates the relationship between vortex tilt and asymmetric potential temperature and vertical velocity patterns as found by Jones (Q. J. R. Meteorol. Soc., vol. 121, 1995, pp. 821-851) and Frank & Ritchie (Mon. Weath. Rev., vol. 127, 1999, pp. 2044-2061) in simulations of adiabatic tropical cyclones. A relation between the present theory and the local induction approximation for three-dimensional slender vortex filaments is established. © 2012 Cambridge University Press.
    Print ISSN: 0022-1120
    Electronic ISSN: 1469-7645
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
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  • 8
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    Energy dissipation caused by boundary layer instability at vanishing viscosity (2018)
    Cambridge University Press
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    Publication Date: 2018-06-26
    Description: A qualitative explanation for the scaling of energy dissipation by high-Reynolds-number fluid flows in contact with solid obstacles is proposed in the light of recent mathematical and numerical results. Asymptotic analysis suggests that it is governed by a fast, small-scale Rayleigh-Tollmien-Schlichting instability with an unstable range whose lower and upper bounds scale as and , respectively. By linear superposition, the unstable modes induce a boundary vorticity flux of order , a key ingredient in detachment and drag generation according to a theorem of Kato. These predictions are confirmed by numerically solving the Navier-Stokes equations in a two-dimensional periodic channel discretized using compact finite differences in the wall-normal direction, and a spectral scheme in the wall-parallel direction. © © 2018 Cambridge University Press This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited..
    Print ISSN: 0022-1120
    Electronic ISSN: 1469-7645
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
    Published by Cambridge University Press
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