The time evolution of the mother body of a planar uniform vortex moving in an inviscid fluid

Abstract

The time evolution of the mother body of a planar, uniform vortex that moves in an incompressible, inviscid fluid is investigated. The vortex is isolated, so that its motion is just due to self-induced velocities. Its mother body is defined as the part internal to the vortex of the singular set of the Schwarz function of its boundary. In the present analysis, it is an arc of curve (branch cut), starting and ending in the two internal branch points of this function, across any point of which the Schwarz function experiences a finite jump. By looking at the mother body from the outside of the vortex, it behaves as a vortex sheet having density of circulation given by the jump of the Schwarz function. Its name (mother body) is taken from Geophysics, and it is here used due to its property of generating, outside the vortex and on its boundary, the same velocity as the vortex itself. The shape of the branch cut and the jump of the Schwarz function across any point of it change in time, by following the motion of the vortex boundary. As it happens for a physical vortex sheet, the mother body is not a material line, so that it does not move according to the velocities induced by the vortex. In the present paper, the cut shape, the above jumps, as well as the cut velocities are deduced from the time evolution equation of the Schwarz function. Numerical experiments, carried out by building the branch cut and calculating the limit values of the Schwarz function on its sides during the vortex motion, confirm the analytical calculations. Some global quantities (circulation, first and second order moments) are here rewritten as integrals on the cut, and their conservation during the vortex motion is analytically and numerically verified. Indeed, the numerical simulations show that they behave in the same way as their classical contour dynamics forms, written in terms of integrals on the vortex boundary. This proves that the shape of the cut, as well as the limit values of the Schwarz function on its sides, are correctly calculated during the motion.

Keywords:

• Mother body
• dynamics of uniform vortices
• Schwarz function approach

Acknowledgments

The author thanks prof. D.G. Dritschel for many helpful discussions about the topics of the present research, as well as for his invaluable contribution to the building of the present mathematical approach.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The Schwarz function is here denoted $\mathit{S}$, as in the paper (Akinyemi et al. 2017), for simplifying its reading. The Reader is kindly requested to keep in mind that the same function will be named $\mathit{\Phi }$ in the present analysis, as in the previous works of the author.

2 Indeed, assume that P rotates by α about the origin. The positions in the original ($\mathit{y}$) and in the rotated frame (${\mathit{y}}^{\prime }$) of the same point are related by ${\mathit{y}}^{\prime }={\mathrm{e}}^{-\mathbf{i}\alpha }\mathit{y}$, so that ${\mathit{\tau }}^{\prime }={\mathrm{e}}^{-\mathbf{i}\alpha }\mathit{\tau }$, ${\mathit{\Phi }}^{\prime }={\mathrm{e}}^{+\mathbf{i}\alpha }\mathit{\Phi }$, respectively, and the rotation does not affect their product, i.e. $\mathit{\tau }\left[\mathit{\Phi }\right]={\mathit{\tau }}^{\prime }\left[{\mathit{\Phi }}^{\prime }\right]$.

3 Direct calculation shows that the mth ($m\ge 0$) derivative of the integral is given by $\frac{{\mathrm{d}}^m}{\mathrm{d}{\mathit{x}}^m}{\int }_{\mathrm{\partial }P}\mathrm{d}{\mathit{x}}^{\prime }\frac{\mathit{\Phi }\left({\mathit{x}}^{\prime }\right)-\mathit{\Phi }\left(\mathit{x}\right)}{{\mathit{x}}^{\prime }-\mathit{x}}={\int }_{\mathrm{\partial }P}\mathrm{d}{\mathit{x}}^{\prime }\frac{{\mathit{\Phi }}^{\left(m\right)}\left({\mathit{x}}^{\prime }\right)-{\mathit{\Phi }}^{\left(m\right)}\left(\mathit{x}\right)}{{\mathit{x}}^{\prime }-\mathit{x}},$ where ${\mathit{\Phi }}^{\left(m\right)}$ is the mth derivative of $\mathit{\Phi }$ (see also section 4.4 on page 28 of Gakhov 1990). The above derivatives are continuous across $\mathrm{\partial P}$ due to the properties of $\mathit{\Phi }$.

4 Indeed, for any $j\ge 1$, the derivatives of the even and odd functions are ${\mathit{w}}_{2j}^{\prime }=2j(1-1/{\mathit{\omega }}^2)({\mathit{w}}_{2j-1}+{\mathit{w}}_{2j-3}+\dots +{\mathit{w}}_3+{\mathit{w}}_1)$ and ${\mathit{w}}_{2j+1}^{\prime }=(2j+1)(1-1/{\mathit{\omega }}^2)({\mathit{w}}_{2j}+{\mathit{w}}_{2j-2}+\dots +{\mathit{w}}_2+1)$.

5 The building of the external cuts is somewhat simpler than the one of $\mathcal{B}{C}_i$. The preimage of such a cut (in the $\mathit{z}$-plane) is drawn by assigning half preimage (typically, a straight half-line) an the other half is obtained as image of the first through the proper mirror function.

6 These properties are proved by verifying that a rotation of an angle β about the origin reduces the image of $\mathcal{C}$ through the map (B.1(B.1) $\mathit{x}\left(\mathit{z}\right)=\frac{{\mathit{a}}_{-1}}{\mathit{z}}+{\mathit{a}}_0+{\mathit{a}}_{+1}\mathit{z}+{\mathit{a}}_{+2}{\mathit{z}}^2$(B.1) ) (with ${\mathit{a}}_0={\mathit{a}}_{+2}=0$) to an ellipse centred on the origin and having semiaxes A (along β) and B ($<A$). By taking $\mathit{z}={\mathrm{e}}^{\mathbf{i}\theta }$ and combining real and imaginary parts of the position ${\mathit{x}}^{\prime }=\mathit{x}\left(\mathit{z}\right){\mathrm{e}}^{-\mathbf{i}\text{β}}$ in the rotated frame some algebra leads to the identity ($c_{\pm }:=\cos (\text{β}-{\theta }_{\pm 1})$, $s_{\pm }:=\sin (\text{β}-{\theta }_{\pm 1})$): (B.4) $\begin{array}{rl}& [{\left(\frac{a_{+1}{c}_{+}+a_{-1}{c}_{-}}{A}\right)}^2+{\left(\frac{a_{+1}{s}_{+}+a_{-1}{s}_{-}}{B}\right)}^2]{\cos }^2\theta +\\ & +2[\frac{(a_{+1}{c}_{+}+a_{-1}{c}_{-})(a_{+1}{s}_{+}-a_{-1}{s}_{-})}{A^2}-\frac{(a_{+1}{s}_{+}+a_{-1}{s}_{-})(a_{+1}{c}_{+}-a_{-1}{c}_{-})}{B^2}]\cos \mathrm{\theta sin}\theta +\\ & +[{\left(\frac{a_{+1}{s}_{+}-a_{-1}{s}_{-}}{A}\right)}^2+{\left(\frac{a_{+1}{c}_{+}-a_{-1}{c}_{-}}{B}\right)}^2]{\sin }^2\theta \equiv 1.\end{array}$(B.4) By naming as $q:=a_{+1}{c}_{+}+a_{-1}{c}_{-}$, $p:=a_{+1}{s}_{+}+a_{-1}{s}_{-}$, $r:=a_{+1}{c}_{+}-a_{-1}{c}_{-}$ and $t:=a_{+1}{s}_{+}-a_{-1}{s}_{-}$, and enforcing that equation (B.4(B.4) $\begin{array}{rl}& [{\left(\frac{a_{+1}{c}_{+}+a_{-1}{c}_{-}}{A}\right)}^2+{\left(\frac{a_{+1}{s}_{+}+a_{-1}{s}_{-}}{B}\right)}^2]{\cos }^2\theta +\\ & +2[\frac{(a_{+1}{c}_{+}+a_{-1}{c}_{-})(a_{+1}{s}_{+}-a_{-1}{s}_{-})}{A^2}-\frac{(a_{+1}{s}_{+}+a_{-1}{s}_{-})(a_{+1}{c}_{+}-a_{-1}{c}_{-})}{B^2}]\cos \mathrm{\theta sin}\theta +\\ & +[{\left(\frac{a_{+1}{s}_{+}-a_{-1}{s}_{-}}{A}\right)}^2+{\left(\frac{a_{+1}{c}_{+}-a_{-1}{c}_{-}}{B}\right)}^2]{\sin }^2\theta \equiv 1.\end{array}$(B.4) ) is verified for any θ, the nonlinear system $\{\begin{array}{r}{p}^2{A}^2+q^2{B}^2=A^2{B}^2\\ \mathrm{pr}{A}^2=\mathrm{qt}{B}^2\\ r^2{A}^2+t^2{B}^2=A^2{B}^2\end{array}$ follows. Its solution is $A^2=\frac{q}{r}(a_{+1}^2-a_{-1}^2),B^2=\frac{p}{t}(a_{+1}^2-a_{-1}^2),\mathrm{pq}=\mathrm{rt}.$ The orientation β is obtained from the last relation, and then A and B follow from the first two ones: $\text{β}=\frac{{\theta }_{-}+{\theta }_{+}}{2},A=a_{-1}+a_{+1},B=|a_{-1}-a_{+1}|.$ Furthermore, the vector $\mathit{l}$ joining the origin with a focus (that is a branch point of the corresponding Schwarz function) results to be $\mathit{l}=(A^2-B^2{)}^{1/2}{\mathrm{e}}^{\mathbf{i}\text{β}}=2(a_{-1}{a}_{+1}{)}^{1/2}{\mathrm{e}}^{\mathbf{i}({\theta }_{-}+{\theta }_{+})/2}=2\sqrt{{\mathit{a}}_{-1}{\mathit{a}}_{+1}}.$