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.
N1 - Funding Information:
This research is supported by EPSRC grants GR/S96616/01, GR/S96616/02 and EP/D035635/1. The authors thank: the Department of Mathematics of the University of York for providing excellent working conditions, the Hull Institute of Mathematical Sciences and Applications (HIMSA) where this research was initiated, the Department of Computer Sciences of the University of Hull and Professor R.Phillips for the given access to the computer cluster in the HIVE Laboratory. The part of the team from the Southern Federal University is partially supported by CRDF RUMI-2842-RO-06, RFBR-08-01-00895-a, by the programme for development of high school research potential (Russia) nos. 2.1.1/554 and 2.1.1/6095, and by the research environment of the European group in Regular and Chaotic Hydrodynamics. The authors are grateful to Profsessors K. I. Ilin, H. K. Moffatt, T. J. Pedley, H. Aref and V. L. Berdichevsky for useful discussions.
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
AN - SCOPUS:77957124170
SN - 0022-1120
VL - 659
SP - 420
EP - 472
JO - Journal of Fluid Mechanics
JF - Journal of Fluid Mechanics
ER -