User:Raph123456/sandbox

= Introduction =

Motivation
In real-life conditions, the unsteady flow past any solid surface, lifting or not, results in the detachment of the boundary layer which curls and forms a vortex that is shed and left in the wake of the body. These coherent structures that persist in time and space are a source of unsteady forces, both for the body generating it and further downstream as fluid-structure interactions with other objects. Moreover, the presence of such strong vortices reduces the energy efficiency of the system as the energy is dissipated in the wake in an inefficient manner which thus contributes to the overall drag force. For instance, ground-vortex ingestion in an aircraft engine or the wake interaction in wind farms are recurrent challenges for aerodynamicists. Understanding the formation process of vortices is therefore of high importance to engineers to provide innovative and energy efficient designs. Finally, vortices are said to be "the sinews and muscles of fluid motions" \citep{Kuchemann1965} and play an evident role in how turbulent flows form, evolve and dissipate. As such, studying the life-cycle of vortices, and more importantly how initial and boundary conditions influence the formation and the behaviour of vortices, is critical to the engineer who wants to either mitigate their influence or use them to their benefit.

FIGURE: VAN DYKE

The vortex ring, as shown in figure \ref{fig:van_dyke}, is an archetypal type of flow easily reproducible experimentally which gathers all the features of complex vortex flows; the entire life-cycle of the vortex ring can be studied, from the formation to the asymptotic decay, via instability growth and eventual transition to turbulence.

Historical note
FIGURE: Unstable vortex ring exhibiting azimuthal instabilities

Vortex rings have been studied for more than 150 years, starting with \cite{Rogers1858}, and remain today a popular case-study for investigating the formation, propagation and disintegration of vortices, as well as hydrodynamic instabilities and the transition to turbulence. Illustrious names, such as \cite{Stokes1845}, \cite{Helmholtz1858, Helmholtz1867} , \cite{Thomson1878, Thomson1867, Thomson1883} , \cite{Reynolds1876} , \cite{Hill1894} or \cite{Hicks1884, Hicks1899} pioneered the research of vortex flows and vortex rings, all of which are summarised in the reference textbook of \cite{Lamb1932}. For instance the famous expression of the translation speed of thin-core vortex rings found by Thomson (Lord Kelvin) was appended by Tait \citep{Helmholtz1867} to the English translation of the original paper by \cite{Helmholtz1858}.

With the advent of new measurement techniques, such as flow visualisation, laser Doppler anemometry or hot wire anemometry, the study of vortex rings gained new momentum in the 1970's with the works of \cite{Maxworthy1972, Maxworthy1974, Maxworthy1977}, \cite{Sullivan1973} , \cite{Widnall1973} , \cite{Widnall1974} , \cite{Widnall1977} , \cite{Sallet1975} , or \cite{Saffman1970, Saffman1975, Saffman1978} , to name just a few. Stress was then put on the theoretical study of azimuthal instabilities developing around the core of initially laminar vortex rings (figure \ref{fig:azimuthal_instabilities} ). The formation process of nozzle-generated vortex ring was thoroughly investigated by \cite{Didden1979}, who, on top of making sounding observations, also captured the process in clean pictures which feature prominently in \citeapos{VanDyke1982} album of fluid motion (plates 76, 112, 114), although the quality of the photographs of \cite{Krutzsch1939} should be acknowledged likewise.

Later, with the development of numerical tools and particle image velocimetry, along with laser induced fluorescence visualisation, the vortex ring could be studied more in-depth with better precision and quantifiable evidence. For instance, \cite{Gan2010t}, \cite{Gan2010}, \cite{Gan2011} and \cite{Gan2012} made use of planar and stereoscopic particle image velocimetry to study the propagation of initially turbulent vortex ring and further validate the similarity theory proposed by \cite{Glezer1988} and \cite{Glezer1990}. Most importantly, \cite{Gharib1998} used the newly-developed (digital) particle image velocimetry to visualise the formation process of nozzle-generated vortex ring. Not only did \cite{Gharib1998} shed light on the limiting process from which a vortex ring stops accumulating fluid from the generator, but they also gave a theoretical explanation to the phenomenon invoking an energy maximisation principle, originally asserted by \cite{Thomson1878}, later demonstrated by \cite{Benjamin1976}. The limiting time scale at which a vortex ring starts exhibiting a trailing jet was dubbed as \emph{formation number} and was claimed to be universal. This led to a series of investigations, many of which were conducted by Prof. Gharib's research group, aiming at testing the universality of this number in different experimental conditions \citep{Dabiri2004a, Dabiri2005b, Krueger2006, Krueger2003b} and giving theoretical ground to the value found experimentally (\citealt{Mohseni1998}; \citealt{Shusser1999}; \citealt{Shusser2000a, Shusser2000b}; \citealt{Linden2001}).

An optimal vortex ring?
FIGURE: bartol jellyfish

The maximum normalised thrust was measured to occur in conditions close to the formation number by \cite{Krueger2003a} suggesting the existence of an optimal way for generating vortex rings, at least in terms of mass and momentum transfer. This corroborates observations made in biological flows, ranging from aquatic animal locomotion and propulsion \citep[review by][]{Dabiri2009} to cardiac flows \citep{Gharib2006}. For instance, it was shown by \citeauthor{Dabiri2005c} (\citeyear{Dabiri2005c}, \citeyear{Dabiri2006}) that jellyfishes propel themselves by forming a vortex ring in their wake in an "optimal" manner, which testify again of the powerful optimisation process of millions of years of evolution (figure \ref{fig:bartol}). In addition, \cite{Dabiri2005a, Dabiri2005b} showed that temporal variations of the jet exit diameter alters the formation of the leading ring and the subsequent transfer of mass and momentum to the flow. The temporally-varying valve, or funnel, then consists of a soft-material exhaust for which the fully open configuration can be modelled as a straight tube and the partially closed one as an orifice geometry. The concept of optimal vortex is not only associated to aquatic life and can be extended to the unsteady vortex formation of any vortex flows, whether they are naturally occurring in water or air, or designed by enlightened engineers. \cite{Krieg2008, Krieg2010}, for example, proposed to apply this concept to the design of thrusters for autonomous underwater vehicles and \cite{Renard2000}, among others, showed the importance of vortex formation in combustion engines. Last, but not least, the use of pulsed jets and synthetic jets, as shown in figure \ref{fig:synthetic_jet}, were proved to be appealing technologies for flow control, unsteady heat and mass transport and thrust generation (\citeauthor{Smith1998}, \citeyear{Smith1998} or review by \citeauthor{Glezer2002}, \citeyear{Glezer2002}). More precisely, synthetic jets, also called zero-net mass flux jets, have the ability of transferring linear momentum to the flow without addition of mass. As such, for a cross-flow over a surface, synthetic jets can induce a virtual change in the shape of the surface, which is of considerable interest for flow control applications.

Orifice-generated vortex rings
FIGURE: synthetic jets of Glezer

Despite the wide range of applications of orifice jets and vortex rings \textit{e.g.} biological flows and synthetic jet actuators, the understanding of the formation process of such flows is lacking, thus limiting the ability of the engineer to optimize their design for practical purposes. For instance, the well-accepted design for generating synthetic jets consists of an orifice plate covering a sealed cavity from which the flow is expelled periodically by a diaphragm of a piston. Although the vortex ring is the obvious key structure formed during ejection, very few studies have focused on the formation of single orifice-generated vortex rings; most results have been obtained using straight or converging nozzle geometries in which a boundary layer develops and detaches at the exhaust to form the vortex ring. The formation process of orifice-generated vortex rings is expected to be different from the case of nozzles as the boundary layer remains minimal in the thickness of the orifice plate.

Furthermore, using orifice plates enables to easily change geometrical shapes and mimic more accurately the complex vortex generation observed in nature. For instance, a dynamical variable diameter opening %, like the one used by \cite{Krieg2021}, could be used to simulate the ejection of fluid through biological orifice. Moreover, the use of multi-scale geometries, such as regular polygons or fractals, could be used for studying the influence of initial boundary conditions on the production of turbulence and its final decay.

The present thesis enters into this context of understanding the optimal formation process of vortex rings emanating from orifice geometries. The objective is to investigate the influence of the specific boundary conditions of orifices on the unsteady vortex formation to pave the way for innovative energy efficient industrial designs.

= Literature Review =

A great corpus of literature on the mathematical description of axisymmetric flows and isolated vortex rings is available. The following section \ref{sec:intro_mathematical} summarizes the main equations for axisymmetric vortical flows and their application to vortex rings. The section is greatly inspired by \cite{Lamb1932}, \cite{Batchelor1967}, \citet{Lim1995}, \cite{Saffman1992} and \cite{Akhmetov2009}. Then, theoretical models for isolated vortex rings are presented (section \ref{sec:intro_theoretical_models}) and the essentials of the formation process are introduced (section \ref{sec:intro_formation_process}). Finally, a critical discussion of the concept of optimal vortex ring and formation number is presented in section \ref{sec:intro_formation_number}. \ul{This latter section includes original contributions which build on previous investigations (section {\ref{ssec:intro_theoretical_models}}).}

Equations of motion
The incompressibility of the flow and the continuity equation $$\nabla \cdot \mathbf{v}=0$$ implies the existence of a vector potential $$\boldsymbol{A}$$ such that $$\mathbf{v}=\nabla\times\boldsymbol{A}$$ and satisfying the Poisson's equation $$\nabla^2\boldsymbol{A}=-\boldsymbol{\omega}$$, where $$\boldsymbol{\omega}=\nabla\times\mathbf{v}$$ is the vorticity. This equation can be solved by means of the Green's function $$G\left(\boldsymbol{r},\boldsymbol{r'}\right)$$

$$\begin{align} \boldsymbol{A}\left(\boldsymbol{r}\right) = -\iiint{ G\left(\boldsymbol{r},\boldsymbol{r'}\right)\boldsymbol{\omega}\left(\boldsymbol{r'}\right)\,\mathrm{d}V' } = \frac{1}{4\pi} \iiint{ \frac{\boldsymbol{\omega}\left(\boldsymbol{r'}\right)}{\left|\boldsymbol{r}-\boldsymbol{r'}\right|}\,\mathrm{d}V' } \end{align}$$ \label{eqn:poisson_green} where $$\boldsymbol{r}$$ is the vector position and the integration is taken over all positions $$\boldsymbol{r'}$$ in the volume $$V'$$.

$$ \boldsymbol{A}\left(\boldsymbol{r}\right) = -\iiint{ G\left(\boldsymbol{r},\boldsymbol{r'}\right)\boldsymbol{\omega}\left(\boldsymbol{r'}\right)\,\mathrm{d}V' } = \frac{1}{4\pi} \iiint{ \frac{\boldsymbol{\omega}\left(\boldsymbol{r'}\right)}{\left|\boldsymbol{r}-\boldsymbol{r'}\right|}\,\mathrm{d}V' } $$ \label{eqn:poisson_green} where $$\boldsymbol{r}$$ is the vector position and the integration is taken over all positions $$\boldsymbol{r'}$$ in the volume $$V'$$.

For an axisymmetric flow, the motion is better described in a cylindrical coordinate system $$\left(x,r,\theta\right)$$. The vorticity is only non-zero in the azimuthal direction, and so is the vector potential $$\boldsymbol{A}=A\mathbf{e_\theta})$$. In the absence of swirl, the velocity field $$\mathbf{v}=\left(u,v,0\right))$$ can be retrieved from the vector potential, and introducing the Stokes stream function as $$\psi=rA)$$, one can write $$\begin{align}	u=+\frac{1}{r} \frac{\partial\psi}{\partial r} &&	v=-\frac{1}{r} \frac{\partial\psi}{\partial x} &&	\omega = \frac{\partial u}{\partial r}-\frac{\partial v}{\partial x} \end{align}$$ where $$u$$ and $$v$$ are the axial and radial components of velocity, respectively, and $$\omega$$ is the azimuthal vorticity. Moreover, the Stokes stream function and the vorticity are related by the elliptic equation $$\begin{align}	\frac{1}{r} \left(\frac{\partial^2\psi}{\partial x^2} + \frac{\partial^2\psi}{\partial r^2} - \frac{1}{r}\frac{\partial \psi}{\partial r}\right) = -\omega \end{align}$$ \label{eqn:axi_psi_omega}

Expanding equation \ref{eqn:poisson_green} in cylindrical coordinates, the Stokes stream function reads $$\begin{align} \psi\left(r,x\right) = \frac{1}{4\pi} \iint{ \omega\left(r',x'\right) \int_0^{2\pi}{ \frac{\cos\theta\,\mathrm{d}\theta}{\sqrt{\left(x-x'\right)^2+r^2+r'^2-2rr'\cos\theta}}} rr'\,\mathrm{d}r'\,\mathrm{d}x' } \end{align}$$

The above equation can be rewritten in terms of the complete elliptic integrals of the first and second kind $$K$$ and $$E$$ as $$\begin{align} && \psi(r,x)=\frac{1}{2\pi}\iint{\left(rr'\right)^{1/2}\left[\left(\frac{2}{k}-k\right)K(k)-\frac{2}{k}E(k)\right]\omega\left(r',x'\right)\,\mathrm{d}r'\,\mathrm{d}x'} && \\ &\text{where} & k^2 = \frac{4rr'}{\left(x-x'\right)^2+\left(r+r'\right)^2} \end{align}$$

A more elegant expression can be obtained by means of Landen's transformations: $$\begin{align} && \psi(r,x)=\frac{1}{2\pi}\iint{\left(r_1+r_2\right)\left[K(\lambda)-E(\lambda)\right]\omega\left(r',x'\right)\,\mathrm{d}r'\,\mathrm{d}x'} && \\ &\text{with} & r_1^2 = \left(x-x'\right)^2+\left(r-r'\right)^2 \qquad r_2^2 = \left(x-x'\right)^2+\left(r+r'\right)^2 \qquad \lambda = \frac{r_2-r_1}{r_2+r_1} \end{align}$$ \label{eqn:psi_Landen} where $$r_1$$ and $$r_2$$ are the least and greatest distances, respectively, of the observation point $$\left(x,r\right)$$ from the vortex.

Finally, the complete elliptic integrals of the first and second kind $$K$$ and $$E$$ are given by $$\begin{align} K(k) = \int_0^1{ \frac{\mathrm{d}t}{\sqrt{\left(1-t^2\right)\left(1-k^2 t^2\right)}} } && \text{and} && E(k) = \int_0^1{ \frac{\sqrt{1-k^2 t^2}}{\sqrt{1-t^2}}\,\mathrm{d}t } \end{align}$$

For the case of steady axisymmetric flows, the quantity $$\omega/r$$ is constant along any streamline, that is $$\omega/r=f(\psi)$$, and so equation \ref{eqn:axi_psi_omega} reduces to $$\begin{align} \frac{1}{r^2} \left(\frac{\partial^2\psi}{\partial z^2} + \frac{\partial^2\psi}{\partial r^2} - \frac{1}{r}\frac{\partial \psi}{\partial r}\right) = -f(\psi) \end{align}$$ \label{eqn:steady_axi_psi_omega} where $$f$$ is any arbitrary function. Equation \ref{eqn:steady_axi_psi_omega} can also be seen as a condition for steadiness.

Invariants of the motion
The integral characteristics of the flow are of prime interest as they provide further information about the flow. The Euler equations of unbounded inviscid incompressible flows have a Hamiltonian structure and possess a total of seven conserved (integral) quantities associated with the symmetries of the equations; the hydrodynamic impulse, the angular impulse and the Hamiltonian functional, \textit{i.e.} the kinetic energy. Additionally, the degeneracy of the Hamiltonian operator leads in three dimensions to the invariance of the helicity and in two dimensions to the invariance of the area integrals $$\iint{ \mathcal{A}(\omega)\,\mathrm{d}S}$$, where $$\mathcal{A}$$ is any arbitrary function of $$\omega$$ \citep{Olver1982}.

Circulation
The circulation is originally defined as the line integral of velocity along a closed contour $$ \Gamma = \oint{ \mathbf{v}\cdot\mathrm{d}\boldsymbol{l} } $$

The circulation is a measure of the strength of a vortex and relates to the vortex force applied to a body (Kutta-Joukowski lift). Moreover, Kelvin's theorem states that the circulation around any material closed curve in an inviscid incompressible fluid is an invariant of the motion.

Making use of Stoke's theorem, the circulation of two-dimensional axisymmetric flows can be written as $$ \Gamma = \iint{\omega\,\mathrm{d}r\,\mathrm{d}x} $$ \label{eqn:def_axi_circulation} Note that the invariance of the circulation in two-dimension can be deduced from the general derivation stated above, where $$\mathcal{A}$$ is replaced by the identity function.

Kinetic energy
The kinetic energy is a conserved quantity of the flow and corresponds to the invariance of the Euler equations to time. It is defined as $$ E = \frac{1}{2} \rho \iiint{ \mathbf{v}^2\,\mathrm{d}V } $$ \label{eqn:def_energy}

In an axisymmetric flow with no swirl, the kinetic energy reduces to $$ E = \rho\pi \iint{ \left(u^2+v^2\right)r\,\mathrm{d}r\,\mathrm{d}x } $$ \label{eqn:def_axi_energy}

For an unbounded axisymmetric flow with no swirl, the kinetic energy can be reformulated in terms of the Stokes stream function $$\psi$$: $$ E = \rho \pi \iint{\omega \psi \,\mathrm{d}r\,\mathrm{d}x} $$ \label{eqn:def_axi_energy_psi} Note that the above expression should not be confused with the equation $$ E = \frac{1}{2}\rho \iint{\omega\vartheta\,\mathrm{d}S} $$ which gives the kinetic energy per unit length of a two-dimensional unbounded flow and which involves the Lagrange stream function $$\vartheta$$.

Hydrodynamic impulse
For three-dimensional unbounded flows, the rate of change of hydrodynamic impulse is equal to the non-conservative external forces applied to the domain. In the absence of non-conservative forces, the hydrodynamic impulse remains invariant, which corresponds to the invariance of the Euler equations to spatial displacement. The hydrodynamic impulse is a vector quantity defined as $$ \boldsymbol{I} = \frac{1}{2}\rho\iiint{ \boldsymbol{x}\times\boldsymbol{\omega}\,\mathrm{d}V } $$

In the case of axisymmetric flows, the hydrodynamic impulse reduces to its axial component $$ I = \rho\pi \iint{ \omega r^2\,\mathrm{d}r\,\mathrm{d}x } $$ \label{eqn:def_axi_impulse}

The hydrodynamic impulse and the (linear) momentum of a fluid $$\iiint{\rho\mathbf{v}\,\mathrm{d}V}$$ are closely related but not equal. For instance, for a sphere of fluid in an unbounded domain, two-third of the hydrodynamic impulse contributes to its total momentum and one third to counteract the pressure field applied to the boundary at infinity and opposing the motion \citep{Cantwell1986}. As such, the hydrodynamic impulse can also be interpreted as the total impulsive force required to instantaneously generate the observed motion from rest.

Angular impulse
The angular impulse is a vector quantity defined as $$ \boldsymbol{A} = \frac{1}{3}\rho\iiint{\boldsymbol{x}\times\left(\boldsymbol{x}\times\boldsymbol{\omega}\right)\,\mathrm{d}V } $$ For unbounded flows, the rate of change of angular impulse is equal to the moment of the external non-conservative forces applied to the domain.

Helicity
The helicity is a scalar quantity defined in three dimensions as $$ H = \iiint{\boldsymbol{x}\cdot\boldsymbol{\omega}\,\mathrm{d}V } $$ In the case where vorticity is confined in vortex filaments, \cite{Moffatt1969} interpreted the helicity in terms of the degree of knottedness of the configuration.

For axisymmetric flows with no swirl, the last two integrals of the motion are zero.

Theoretical models of isolated vortex rings
\label{sec:intro_theoretical_models}

Definition of a vortex ring
\label{ssec:intro_definition_vortex_ring}

A vortex ring is a class of axisymmetric flows in which the vorticity is bounded in a three-dimensional toroidal structure. This region, referred as the vortex core, is surrounded by irrotational fluid moving with the torus and is usually qualified as the entrainment region, the ring atmosphere or the ring bubble. This compact coherent structure is primarily defined by its core size, diameter and speed, and by the invariants of the motion it carries.

The extent of a vortex core can be defined as the point in space where the tangential velocity, in a frame of reference following the vortex, is maximum. This definition allows to precisely define the extent of the core, rather than the vorticity which decays as the inverse of the radial distance from the core centroid. But, experimentally, finding the stagnation point of a moving vortex in a fixed frame of reference can be problematic and one usually defines the extent of the core as the point in space where the vorticity drops below some threshold. %For a vortex model where the vorticity distribution is uniform inside the core and zero elsewhere, the two approaches give analogous results.

The vortex core centroid can also be defined in a number of different ways. Strictly speaking, the core centre is the point in space where the velocity in a frame of reference following the vortex is zero, \textit{i.e.} the stagnation point. \citet[\S 162]{Lamb1932} defined the vortex centroid, or centre of vorticity, for an axisymmetric (circular) vortex as $$ \begin{align} x_c=\frac{\iint{x r^2\omega\,\mathrm{d}r\,\mathrm{d}x}}{\iint{r^2 \omega\,\mathrm{d}r\,\mathrm{d}x}} && r_c^2=\frac{\iint{r^2 \omega\,\mathrm{d}r\,\mathrm{d}x}}{\iint{\omega\,\mathrm{d}r\,\mathrm{d}x}} \end{align}$$ \label{eqn:def_centre_of_vorticity} The equivalent definition for a two-dimensional (rectilinear) vortex is provided by \citet[\S 154]{Lamb1932} and was used by \cite{Gan2010t} in an axisymmetric case.

The above definitions puts emphasis on the core vorticity distribution and is somewhat subjective experimentally as it depends on the vorticity threshold defining the bounds of the integral. However, this is the definition adopted in the present investigation and is shown as a cross in the following figures.

An alternative definition was proposed by \cite{Saffman1970, Saffman1992} to define the core centroid as $$ \boldsymbol{x_c} = \iiint{\frac{\boldsymbol{x}\times\boldsymbol{\omega}\cdot\boldsymbol{I}}{I^2}}\boldsymbol{x}\,\mathrm{d}V $$ where $$I$$ is the norm of the hydrodynamic impulse. For an axisymmetric distribution of vorticity, this definition places the centre of the vortex on the axis of symmetry.

Others, such as \cite{Hettel2007}, used the position of the maximum of the stream function and the location of minimum static pressure to define the centre of the vortex. The limited benefit of those unconventional methods does not justify their cumbersome implementation.

Circular vortex line
\label{ssec:intro_circular_vortex_line}

For a single zero-thickness vortex ring, the vorticity is represented by a Dirac delta-function as $$\omega\left(r,x\right)=\kappa\delta\left(r-r'\right)\delta\left(x-x'\right)$$ where $$\left(r',x'\right)$$ denotes the coordinates of the vortex filament of strength $$\kappa$$ in a constant $$\theta$$ half-plane. Making use of equation \ref{eqn:psi_Landen}, and after simplifications, the Stokes stream function reads $$ \begin{align} && \psi(r,x)=\frac{\kappa}{2\pi}\left(r_1+r_2\right)\left[K(\lambda)-E(\lambda)\right] && \\ &\text{with} & r_1^2 = \left(x-x'\right)^2+\left(r-r'\right)^2 \qquad r_2^2 = \left(x-x'\right)^2+\left(r+r'\right)^2 \qquad \lambda = \frac{r_2-r_1}{r_2+r_1} \end{align} $$

The streamlines are shown in figure \ref{fig:vortex_line}. A circular vortex line is the limiting case of a thin vortex ring. Because there is no core thickness, the speed of the ring is infinite, as well as the kinetic energy. The hydrodynamic impulse can be expressed in term of the strength, or "circulation", of the vortex as $$I=\rho\pi\kappa R^2$$.

\begin{figure} \centering \includegraphics[width=0.8\textwidth]{vortex_line} \caption[Streamlines of a circular vortex line.]{\label{fig:vortex_line}Streamlines of a circular vortex line for equidistant values of $$\psi$$ in a fluid at rest at infinity} \end{figure}

Thin-core vortex ring
The discontinuity introduced by the Dirac delta-function prevents the computation of the speed and the kinetic energy of a circular vortex line. It is however possible to approximate these quantities for a vortex ring having a finite small thickness.

For a thin vortex ring, the core can be approximated by a disk of radius $$a$$ which is assumed to be infinitesimal compared to the radius of the ring $$R$$, \textit{i.e.} $$a/R\ll1$$. As a consequence, inside and in the vicinity of the core ring, one may write: $$r_1/r_2\ll1$$, $$r_2\approx2R$$ and $$1-\lambda^2\approx4r_1/R$$. Moreover, in the limit of $$\lambda\approx1$$, the elliptic integrals can be approximated by $$K(\lambda)=1/2~\ln({16}/(1-\lambda^2))$$ and $$E(\lambda)=1$$. $$ \begin{align} K(\lambda) = \frac{1}{2} \ln\left(\frac{16}{1-\lambda^2}\right) && \text{and} && E(\lambda) = 1 \end{align} $$

As a consequence, for a uniform vorticity distribution $$\omega\left(r,x\right)=\omega_0$$ in the disk, the stream function can be approximated from equation \ref{eqn:psi_Landen} as $$ \psi(r,x) = \frac{\omega_0}{2\pi} R \iint{ \left(\ln\frac{8R}{r_1}-2\right)\,\mathrm{d}r'\,\mathrm{d}x' } $$

After a suitable change of coordinates and making use of known definite integrals, the stream function in the proximity of the core ring is $$ \psi(r,x)=\frac{1}{2}\omega_0 R a^2\left(\ln\frac{8R}{a}-\frac{3}{2}-\frac{1}{2}\frac{s^2}{a^2}\right) $$ where $$s$$ is the radial coordinate of the observation point $$\left(r,x\right)$$ in a polar coordinate system centred on the vortex core.

The invariants of the motion, as defined in equations \ref{eqn:def_axi_circulation}, \ref{eqn:def_axi_energy_psi} and \ref{eqn:def_axi_impulse}, can be expressed in terms of the core radius $$a$$, the ring radius $$R$$ and the uniform vorticity distribution $$\omega_0$$ as $$ \begin{align} \Gamma &= \pi\omega_0a^2 \\ I &= \rho \pi R^2 \Gamma \\ E & = \frac{1}{2} \rho \Gamma^2R\left(\ln\frac{8R}{a}-\frac{7}{4}\right) \end{align} $$ \label{eqn:thin_ring_C} \label{eqn:thin_ring_I} \label{eqn:thin_ring_E}

The time derivative of the axial position of the core centroid, as defined in equation \ref{eqn:def_centre_of_vorticity}, is the translational speed of the vortex. Owing to the invariance of the circulation and the impulse, a general expression for the axial velocity of any axisymmetric vortex flow in terms of the invariants of the motion can be expressed as: $$	U = \frac{E}{2I} + \frac{3\pi\rho}{I} \iint{ \omega\left(r',x'\right)r'x'v\,\mathrm{d}r'\,\mathrm{d}x'} $$

For a finite small-core vortex ring, the trajectory of the fluid particles near the core can be assumed to be a circular path, and the translational speed may be written as $$ U = \frac{E}{2I} + \frac{3}{8\pi}\frac{\Gamma}{R} $$ \label{eqn:thin_ring_U} Interestingly, the above expression suggests that the speed of an isolated vortex ring is only a function of the size of the ring $$R$$, the integrals of the motion $$\Gamma$$, $$I$$ and $$E$$ being conserved quantities.

Making use of equations \ref{eqn:thin_ring_C}, \ref{eqn:thin_ring_I} and \ref{eqn:thin_ring_E}, the famous expression of the translational speed of a thin vortex ring is obtained: $$	U = \frac{\Gamma}{4\pi R} \left(\ln\frac{8R}{a}-\frac{1}{4}\right) $$ \label{eqn:Kelvin_U} This result was originally derived by Sir William Thomson, 1\textsuperscript{st} Baron Kelvin, and published as an appendix to the translation by Tait of \citeapos{Helmholtz1858} paper \citep{Helmholtz1867}.

\subsection{Hill's spherical vortex}

An example of steady axisymmetric vortex flow solving equation \ref{eqn:steady_axi_psi_omega} is provided by \citeapos{Hill1894} spherical vortex. The arbitrary function $$f$$ is chosen to be constant equal to $$A$$. The Stokes stream function inside the spherical vortex is then given by $$ \psi(r,x) = -\frac{1}{10}Ar^2\left(a^2-r^2-x^2\right) $$ where $$a$$ is the radius of the sphere.

\begin{figure} \centering \begin{subfigure}[b]{0.8\textwidth} \includegraphics[width=\textwidth]{Hill_vorticity} \caption{Vorticity distribution} \end{subfigure} \begin{subfigure}[b]{0.8\textwidth} \includegraphics[width=\textwidth]{Hill_following} \caption{in a frame of reference following the vortex} \end{subfigure} \begin{subfigure}[b]{0.8\textwidth} \includegraphics[width=\textwidth]{Hill_fixed} \caption{in a fixed frame of reference} \end{subfigure} \caption[Streamlines of Hill's spherical vortex.]{\label{fig:Hill_vortex}\sampleline{} Streamlines of Hill's spherical vortex for equidistant values of $$\psi$$ \sampleline{dash pattern=on .7em off .2em on .2em off .2em} Extent of the spherical vortex \sampleline{dashed} Separating streamline, \textit{i.e.} $$\psi=0$$.} \end{figure}

The flow outside the sphere is provided by the classical stream function of a flow of axial velocity $$U$$ over a sphere of radius $$a$$: $$ \psi(r,x) = \frac{1}{2}Ur^2\left[1-\frac{a^3}{\left(x^2+r^2\right)^{3/2}}\right] $$

Although the condition $$\psi=0$$ at the interface for both expressions ensures the spherical interface to be a streamline for both the inner and the outer flows, thus satisfying the continuity of the normal velocity, the tangential velocity has to be made continuous by enforcing $$A=\frac{15}{2}\frac{U}{a^2}$$ and, from equation \ref{eqn:axi_psi_omega}, the vorticity inside the vortex becomes $$ \frac{\omega}{r} = \frac{15}{2}\frac{U}{a^2} $$ \label{eqn:Hill_omega}

To summarize, the stream function of Hill's spherical vortex flow is $$ \begin{align} & \psi(r,x) = -\frac{3}{4}\frac{U}{a^2}r^2\left(a^2-r^2-x^2\right) && \text{inside the vortex} \\ & \psi(r,x) = \frac{1}{2}Ur^2\left[1-\frac{a^3}{\left(x^2+r^2\right)^{3/2}}\right] && \text{outside the vortex} \end{align} $$

The above expressions of the stream function describe a steady flow, or equivalently a steadily propagating vortex in a frame of reference following the ring. In a fixed frame of reference, the vortex is propagating at a constant speed $$U$$ and so the stream function of the bulk flow must be added, such that $$ \begin{align} & \psi(r,x) = -\frac{3}{4}\frac{U}{a^2}r^2\left(a^2-r^2-x^2\right)-\frac{1}{2}Ur^2 && \text{inside the vortex} \\ & \psi(r,x) = \frac{1}{2}Ur^2\left[1-\frac{a^3}{\left(x^2+r^2\right)^{3/2}}\right]-\frac{1}{2}Ur^2 && \text{outside the vortex} \end{align} $$ \label{eqn:Hill_psi_inside_moving} \label{eqn:Hill_psi_outside_moving}

The conserved quantities as defined in equations \ref{eqn:def_axi_circulation}, \ref{eqn:def_axi_energy_psi} and \ref{eqn:def_axi_impulse} can easily be computed from equation \ref{eqn:Hill_omega} and equations \ref{eqn:Hill_psi_inside_moving} and \ref{eqn:Hill_psi_outside_moving}: $$ \begin{align} \Gamma_H &= 5Ua \\ I_H &= 2\pi Ua^3 \\ E_H & = \frac{10\pi}{7}U^2a^3 \end{align} $$ \label{eqn:Hill_C} \label{eqn:Hill_I} \label{eqn:Hill_E}

Note that the kinetic energy inside the vortex is $$(23\pi/21)U^2a^3$$ and the kinetic energy outside the vortex is $$(\pi/3)U^2a^3$$. The centre of vorticity, as defined in equation \ref{eqn:def_centre_of_vorticity}, is found to be at a radial position of $$r_c=a\sqrt{2/5}$$.

The streamlines are shown in figure \ref{fig:Hill_vortex}. The centre of vorticity is shown as a cross and the extent of the vorticity is shown as a dashed line.