Abstract
A new mathematical approach to kinematics and dynamics of planar uniform vortices in an incompressible inviscid fluid is presented. It is based on an integral relation between Schwarz function of the vortex boundary and induced velocity. This relation is firstly used for investigating the kinematics of a vortex having its Schwarz function with two simple poles in a transformed plane. The vortex boundary is the image of the unit circle through the conformal map obtained by conjugating its Schwarz function. The resulting analysis is based on geometric and algebraic properties of that map. Moreover, it is shown that the steady configurations of a uniform vortex, possibly in presence of point vortices, can be also investigated by means of the integral relation. The vortex equilibria are divided in two classes, depending on the behavior of the velocity on the boundary, measured in a reference system rotating with this curve. If it vanishes, the analysis is rather simple. However, vortices having nonvanishing relative velocity are also investigated, in presence of a polygonal symmetry. In order to study the vortex dynamics, the definition of Schwarz function is then extended to a Lagrangian framework. This Lagrangian Schwarz function solves a nonlinear integrodifferential Cauchy problem, that is transformed in a singular integral equation. Its analytical solution is here approached in terms of successive approximations. The self-induced dynamics, as well as the interactions with a point vortex, or between two uniform vortices are analyzed.
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Notes
1The asymptotic behaviour of G follows in terms of the moments M k (k = 0,1,2,…) of the curve ∂P as:
$$\boldsymbol{G}(\boldsymbol{x})= \sum\limits_{k = 0}^{\infty} \frac{\boldsymbol{M}_{k}}{\boldsymbol{x}^{k + 1}} \text{ with: } \boldsymbol{M}_{k} = \frac{1}{2\pi \boldsymbol{i}} {\int}_{\partial P} d \boldsymbol{y} \boldsymbol{y}^{k} \boldsymbol{\Phi}(\boldsymbol{y}). $$By naming |P| the area of the vortex, M0 is just |P|/π (see Riccardi and Durante (2008)).
In these flows other kinds of interactions were also found and classified, as the elastic one, the partial and the complete straining-out.
Note that the function x(ξ; t) has just the above geometrical meaning. It differs from the flow, introduced in Section 2 and indicated by \(\boldsymbol {x}^{{(\mathcal {L})}}(\boldsymbol {\xi };t)\).
In the present formulation, the search for relative equilibrium leads to the eigenvalue problem (12). In this framework, it is customary to indicate the eigenvalue with the symbol λ. It should not be confused with the same symbol used in Jin and Dubin (2001) and then in Section 1, because it has here a completely different meaning.
5The following result:
$$\boldsymbol{f}(\boldsymbol{z}) = \frac{1}{\pi\boldsymbol{i}} {\int-}_{\mathcal L} \frac{d\boldsymbol{\zeta}^{\prime}}{\boldsymbol{\zeta}^{\prime}-\boldsymbol{z}} \frac{1}{\pi\boldsymbol{i}} {\int{-}}_{\mathcal L} d\boldsymbol{\zeta}^{\prime\prime} \,\, \frac{\boldsymbol{f}(\boldsymbol{\zeta}^{\prime\prime})}{\boldsymbol{\zeta}^{\prime\prime}-\boldsymbol{\zeta}^{\prime}} \,\,, $$given in Muskhelishvili (2008) (Section 27, page 68 equation (B)) has been used. Here f satisfies the Hölder condition on the simple, smooth closed curve \({\mathcal {L}}\).
In this case the problem
$$\left\{ \begin{array}{l} (1-\lambda) \,\, \boldsymbol{\Psi}(\boldsymbol{\xi}) = - \boldsymbol{w}^{(i)}(\boldsymbol{\xi}) \hspace{5mm} \forall \boldsymbol{\xi} \in \partial P(0) \\ (1-\lambda) \,\, \overline{\boldsymbol{\xi}}_{j} = \boldsymbol{C}(\boldsymbol{\xi}_{j}) - \boldsymbol{w}^{(i)}_{j}(\boldsymbol{\xi}_{j}) \,\,, \hspace{3mm} j = 1, 2, \ldots, n \end{array} \right. $$is obtained. By using the Schwarz function on the vortex boundary given by the first equation inside the second one, it is easily shown that the Cauchy contributions C(ξ j ) vanish (for j = 1, 2, …, n). It follows the problem:
$$ \left\{ \begin{array}{l} (1-\lambda) \,\, \boldsymbol{\Psi}(\boldsymbol{\xi}) = - \boldsymbol{w}^{(i)}(\boldsymbol{\xi}) \hspace{5mm} \forall \boldsymbol{\xi} \in \partial P(0) \\ (1-\lambda) \,\, \overline{\boldsymbol{\xi}}_{j} = - \boldsymbol{w}^{(i)}_{j}(\boldsymbol{\xi}_{j}) \,\,, \hspace{3mm} j = 1, 2, \ldots, n \end{array} \right. $$(18)The problem (18) appears to be well-posed, giving also the correct behaviour of the velocity in a small neighbourhood of each point vortex. In order to prove this behaviour, consider an arbitrary point ξ′ inside P(0). By using the above value of the Schwarz function and the fact that all the point vortices lie inside P(0), it follows that C(ξ′) = 0 and the j th point vortex moves with the proper velocity:
$$\begin{array}{@{}rcl@{}} &&\lim_{\boldsymbol{\xi}} \to \boldsymbol{\xi}_{j} \left\{\overline{\boldsymbol{u}}[\boldsymbol{x}(\boldsymbol{\xi};t);t] + \boldsymbol{i} \,\, \frac{\omega}{2} \,\, e^{- \boldsymbol{i} {\Omega} t} \,\, \frac{{\Gamma}_{j}}{\boldsymbol{\xi}-\boldsymbol{\xi}_{j}}\right\}\\ && \qquad\qquad = \boldsymbol{i} \,\, \frac{\omega}{2} \,\, e^{- \boldsymbol{i} {\Omega} t} \,\, \left[ - \overline{\boldsymbol{\xi}}_{j} - \boldsymbol{w}_{j}(\boldsymbol{\xi}_{j}) \right] = - \boldsymbol{i} \,\, {\Omega} \,\, e^{- \boldsymbol{i} {\Omega} t} \,\, \overline{\boldsymbol{\xi}}_{j} \,\,. \end{array} $$The interesting solution of the problem (18) is under investigation at the present time.
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Acknowledgments
The author thanks Prof. Konstantin Koshel and Prof. Sergey Prants for their kind invitation and their valuable affection during and after the Conference “Vortices and Coherent Structures: from Ocean to Microfluids” held in Vladivostok at the end of August 2017.
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Responsible Editor: Sergey Prants
This article is part of the Topical Collection on the International Conference “Vortices and coherent structures: from ocean to microfluids”, Vladivostok, Russia, 28-31 August 2017
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Riccardi, G. A complex analysis approach to the motion of uniform vortices. Ocean Dynamics 68, 273–293 (2018). https://doi.org/10.1007/s10236-017-1129-1
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DOI: https://doi.org/10.1007/s10236-017-1129-1