Two-dimensional vortex motions of fluid in harbor-like basins at large Reynolds numbers

Abstract

This article considers the topology of the vortex regimes generated in harbor-like basins by the external potential long shore current at large Reynolds numbers. The proposed theory discusses the issues of what solution compatible with the Prandtl-Batchelor theorem for inviscid fluids, and under what conditions, may be realized as an asymptotic state of the open hydrodynamical system. The analysis developed is based on the variational principle. We formulated a validity criterion according to which stationary regimes of dissipative systems may be considered as extremals of a variational inviscid problem. In particular, such a situation is possible when the dissipative functionals represent some functions of motion invariants of the same problem. It is shown that the steady state corresponds to the circulational regime in which the system has minimal energy with the fixed enstrophy. This state is fixed by the Reynolds number. The approach that is formulated is applied to the model of a rectangular harbor-like basin in order to obtain the relation between the Reynolds number, the geometry factor and the topological number characterizing the number of vortex cells.

Hot Spots in Sgr A* Accretion Disk: Hydrodynamic Insights

“…Within this framework-called the contour dynamics method (CDM) (see Refs. [18,[22][23][24][25][26][27])-the distribution of vorticity q for a vortex structure may be written as q = q 0 Θ(x, y). Function Θ(x, y) is a two-dimensional Heaviside step-function equal to one inside the domain in consideration and zero outside.…”

“…In the simplest model-in which the vortex structures are realized in a "thin" flat sheet of an incompressible inviscid fluid (i.e., when h L, V 2 s 2 c 2 , Re 1)-the z-component of velocity, w, vanishes and may be dropped in all formulas (see, for example, [18]). Note also that 3jk ∂ j τ s ∂ k Φ ≡ 0 because of an axial symmetry of both τ s and Φ.…”

“…[21,22]), as well as Refs. [18,23]); (iii) For a strongly localized vortex, quantity q(x, t) can be parameterized by the function in form F(x − x 0 (t)), i.e., the one with the center at coordinate x = x 0 (t), which satisfies the equation of transport ẋ0i (t) + ik ∂ i ψ = 0, i.e., when the center of vortex moves according to ẋ0 (t) − vs.[x 0 (t)] = 0. (Here, the symbol "dot" signifies the derivative with respect to time.)…”

Some estimates on the space scales of vortex pairs emitted from river mouths

“…Consequently, various approximations are commonly used (Lavrentev and Shabat, 1965;Goncharov and Pavlov, 1998). Some of these essentially consist of the replacement of the real boundary by an open n-gon which is characterized in the z-plane by vertices A k and corners πα k (0 < πα k ≤ 2, k = 1, 2, ..., n) at the vertices.…”

Cyclostrophic vortices in polar regions of rotating planets

“…We consider below quasi two-dimensional flows in a thin sheet of an ideal fluid, an assumption which allows one to make some additional simplifications (see Goncharov and Pavlov, 1997, 1998, 2000, and references therein).…”