Planar inviscid flows in a channel of finite length

Washout, trapping and self-oscillations of vorticity

V. N. Govorukhin, A. B. Morgulis, V. A. Vladimirov

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

The paper addresses the nonlinear dynamics of planar inviscid incompressible flows in the straight channel of a finite length. Our attention is focused on the effects of boundary conditions on vorticity dynamics. The renowned Yudovich's boundary conditions (YBC) are the normal component of velocity given at all boundaries, while vorticity is prescribed at an inlet only. The YBC are fully justified mathematically: the well posedness of the problem is proven. In this paper we study general nonlinear properties of channel flows with YBC. There are 10 main results in this paper: (i) the trapping phenomenon of a point vortex has been discovered, explained and generalized to continuously distributed vorticity such as vortex patches and harmonic perturbations; (ii) the conditions sufficient for decreasing Arnold's and enstrophy functionals have been found, these conditions lead us to the washout property of channel flows; (iii) we have shown that only YBC provide the decrease of Arnold's functional; (iv) three criteria of nonlinear stability of steady channel flows have been formulated and proven; (v) the counterbalance between the washout and trapping has been recognized as the main factor in the dynamics of vorticity; (vi) a physical analogy between the properties of inviscid channel flows with YBC, viscous flows and dissipative dynamical systems has been proposed; (vii) this analogy allows us to formulate two major conjectures (C1 and C2) which are related to the relaxation of arbitrary initial data to C1: steady flows, and C2: steady, self-oscillating or chaotic flows; (viii) a sufficient condition for the complete washout of fluid particles has been established; (ix) the nonlinear asymptotic stability of selected steady flows is proven and the related thresholds have been evaluated; (x) computational solutions that clarify C1 and C2 and discover three qualitatively different scenarios of flow relaxation have been obtained.

Original languageEnglish
Pages (from-to)420-472
Number of pages53
JournalJournal of Fluid Mechanics
Volume659
DOIs
Publication statusPublished - Sep 2010

Fingerprint

inviscid flow
self oscillation
fallout
Vorticity
vorticity
trapping
Boundary conditions
channel flow
Channel flow
boundary conditions
steady flow
Steady flow
Vortex flow
vortices
oscillating flow
counterbalances
incompressible flow
Incompressible flow
viscous flow
Viscous flow

Keywords

  • application
  • general fluid mechanics
  • vortex dynamics

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering

Cite this

Planar inviscid flows in a channel of finite length : Washout, trapping and self-oscillations of vorticity. / Govorukhin, V. N.; Morgulis, A. B.; Vladimirov, V. A.

In: Journal of Fluid Mechanics, Vol. 659, 09.2010, p. 420-472.

Research output: Contribution to journalArticle

@article{17e6e9ab2ffd4c739c33eeb7ffe9b293,
title = "Planar inviscid flows in a channel of finite length: Washout, trapping and self-oscillations of vorticity",
abstract = "The paper addresses the nonlinear dynamics of planar inviscid incompressible flows in the straight channel of a finite length. Our attention is focused on the effects of boundary conditions on vorticity dynamics. The renowned Yudovich's boundary conditions (YBC) are the normal component of velocity given at all boundaries, while vorticity is prescribed at an inlet only. The YBC are fully justified mathematically: the well posedness of the problem is proven. In this paper we study general nonlinear properties of channel flows with YBC. There are 10 main results in this paper: (i) the trapping phenomenon of a point vortex has been discovered, explained and generalized to continuously distributed vorticity such as vortex patches and harmonic perturbations; (ii) the conditions sufficient for decreasing Arnold's and enstrophy functionals have been found, these conditions lead us to the washout property of channel flows; (iii) we have shown that only YBC provide the decrease of Arnold's functional; (iv) three criteria of nonlinear stability of steady channel flows have been formulated and proven; (v) the counterbalance between the washout and trapping has been recognized as the main factor in the dynamics of vorticity; (vi) a physical analogy between the properties of inviscid channel flows with YBC, viscous flows and dissipative dynamical systems has been proposed; (vii) this analogy allows us to formulate two major conjectures (C1 and C2) which are related to the relaxation of arbitrary initial data to C1: steady flows, and C2: steady, self-oscillating or chaotic flows; (viii) a sufficient condition for the complete washout of fluid particles has been established; (ix) the nonlinear asymptotic stability of selected steady flows is proven and the related thresholds have been evaluated; (x) computational solutions that clarify C1 and C2 and discover three qualitatively different scenarios of flow relaxation have been obtained.",
keywords = "application, general fluid mechanics, vortex dynamics",
author = "Govorukhin, {V. N.} and Morgulis, {A. B.} and Vladimirov, {V. A.}",
year = "2010",
month = "9",
doi = "10.1017/S0022112010002533",
language = "English",
volume = "659",
pages = "420--472",
journal = "Journal of Fluid Mechanics",
issn = "0022-1120",
publisher = "Cambridge University Press",

}

TY - JOUR

T1 - Planar inviscid flows in a channel of finite length

T2 - Washout, trapping and self-oscillations of vorticity

AU - Govorukhin, V. N.

AU - Morgulis, A. B.

AU - Vladimirov, V. A.

PY - 2010/9

Y1 - 2010/9

N2 - The paper addresses the nonlinear dynamics of planar inviscid incompressible flows in the straight channel of a finite length. Our attention is focused on the effects of boundary conditions on vorticity dynamics. The renowned Yudovich's boundary conditions (YBC) are the normal component of velocity given at all boundaries, while vorticity is prescribed at an inlet only. The YBC are fully justified mathematically: the well posedness of the problem is proven. In this paper we study general nonlinear properties of channel flows with YBC. There are 10 main results in this paper: (i) the trapping phenomenon of a point vortex has been discovered, explained and generalized to continuously distributed vorticity such as vortex patches and harmonic perturbations; (ii) the conditions sufficient for decreasing Arnold's and enstrophy functionals have been found, these conditions lead us to the washout property of channel flows; (iii) we have shown that only YBC provide the decrease of Arnold's functional; (iv) three criteria of nonlinear stability of steady channel flows have been formulated and proven; (v) the counterbalance between the washout and trapping has been recognized as the main factor in the dynamics of vorticity; (vi) a physical analogy between the properties of inviscid channel flows with YBC, viscous flows and dissipative dynamical systems has been proposed; (vii) this analogy allows us to formulate two major conjectures (C1 and C2) which are related to the relaxation of arbitrary initial data to C1: steady flows, and C2: steady, self-oscillating or chaotic flows; (viii) a sufficient condition for the complete washout of fluid particles has been established; (ix) the nonlinear asymptotic stability of selected steady flows is proven and the related thresholds have been evaluated; (x) computational solutions that clarify C1 and C2 and discover three qualitatively different scenarios of flow relaxation have been obtained.

AB - The paper addresses the nonlinear dynamics of planar inviscid incompressible flows in the straight channel of a finite length. Our attention is focused on the effects of boundary conditions on vorticity dynamics. The renowned Yudovich's boundary conditions (YBC) are the normal component of velocity given at all boundaries, while vorticity is prescribed at an inlet only. The YBC are fully justified mathematically: the well posedness of the problem is proven. In this paper we study general nonlinear properties of channel flows with YBC. There are 10 main results in this paper: (i) the trapping phenomenon of a point vortex has been discovered, explained and generalized to continuously distributed vorticity such as vortex patches and harmonic perturbations; (ii) the conditions sufficient for decreasing Arnold's and enstrophy functionals have been found, these conditions lead us to the washout property of channel flows; (iii) we have shown that only YBC provide the decrease of Arnold's functional; (iv) three criteria of nonlinear stability of steady channel flows have been formulated and proven; (v) the counterbalance between the washout and trapping has been recognized as the main factor in the dynamics of vorticity; (vi) a physical analogy between the properties of inviscid channel flows with YBC, viscous flows and dissipative dynamical systems has been proposed; (vii) this analogy allows us to formulate two major conjectures (C1 and C2) which are related to the relaxation of arbitrary initial data to C1: steady flows, and C2: steady, self-oscillating or chaotic flows; (viii) a sufficient condition for the complete washout of fluid particles has been established; (ix) the nonlinear asymptotic stability of selected steady flows is proven and the related thresholds have been evaluated; (x) computational solutions that clarify C1 and C2 and discover three qualitatively different scenarios of flow relaxation have been obtained.

KW - application

KW - general fluid mechanics

KW - vortex dynamics

UR - http://www.scopus.com/inward/record.url?scp=77957124170&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77957124170&partnerID=8YFLogxK

U2 - 10.1017/S0022112010002533

DO - 10.1017/S0022112010002533

M3 - Article

VL - 659

SP - 420

EP - 472

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

SN - 0022-1120

ER -