1. Introduction
Most vortices observed in nature are curved and interact with nearby vortices. Although they can often be considered locally as axisymmetric (with possibly a jet component), their internal structure is actually azimuthally deformed by local effects (curvature, torsion) and non-local effects (remote vorticity). These azimuthal corrections are known to be the source of short-wavelength instabilities. The description of these corrections is therefore an important necessary step for the understanding and modelling of these instabilities. In this work, the first dipolar and quadrupolar corrections to a prescribed monopolar structure are computed for a single vortex ring, an array of rings, a helical vortex and an array of helices using two methods: an asymptotic analysis in the limit of small core size and dedicated numerical simulations.
Vortex rings are simple invariant vortical states that have been studied for more than a hundred years. In an inviscid framework, vortex rings are expected to propagate at a constant speed without changing their form. Many works have aimed at determining the ring propagation speed when the ring core size is small compared to its radius (see Saffman Reference Saffman1992). In this limit, the most recent work is by Fukumoto & Moffatt (Reference Fukumoto and Moffatt2000) who also includes viscous effects and computes the internal structure of a vortex ring up to third order. This analysis clearly shows that dipolar corrections are generated at first order, while quadrupolar corrections only appear at second order. The link between dipolar corrections and local curvature has been known for a long time (Ting & Tung Reference Ting and Tung1965; Widnall, Bliss & Zalay Reference Widnall, Bliss, Zalay, Olsen, Goldburg and Rogers1971; Moore & Saffman Reference Moore and Saffman1972). In a general setting, Callegari & Ting (Reference Callegari and Ting1978) showed how dipolar corrections depend on the local vortex curvature. These results were extended by Fukumoto & Miyazaki (Reference Fukumoto and Miyazaki1991) to account for an axial jet component within the vortex. In the present study, these asymptotic predictions are retrieved and compared for the first time to numerical simulations of finite core size vortices.
Helical vortices are more complicated than rings; in addition to their translation motion they also rotate. Many works are devoted to the rotation and translation speeds of such invariant structures (Moore & Saffman Reference Moore and Saffman1972; Widnall Reference Widnall1972; Ricca Reference Ricca1994; Kuibin & Okulov Reference Kuibin and Okulov1998; Boersma & Wood Reference Boersma and Wood1999). In particular Kuibin & Okulov (Reference Kuibin and Okulov1998) and Boersma & Wood (Reference Boersma and Wood1999) used the expression of the velocity field in terms of Kapteyn series derived by Hardin (Reference Hardin1982) for helical filaments. These results were further extended for multiple helices by Okulov (Reference Okulov2004) and to higher-order corrections by Fukumoto & Okulov (Reference Fukumoto and Okulov2005). The effect of torsion is not present in the rings. It was first characterized by Ricca (Reference Ricca1994) for helices. Torsion also generates a dipolar correction, as does curvature, but this effect is weaker since it is of second order (see Fukumoto & Okulov Reference Fukumoto and Okulov2005). Quadrupolar corrections responsible for the elliptic deformation of the inner core are also expected at second order (see for instance Fukumoto & Moffatt Reference Fukumoto and Moffatt2000). Such corrections are known to be generated when a vortex is subjected to an external strain field (Moffatt, Kida & Ohkitani Reference Moffatt, Kida and Ohkitani1994), or is exposed to the influence of other vortices (see Le Dizès & Verga Reference Le Dizès and Verga2002) or to distant parts of the same vortex, as for a ring. This quadrupolar correction has been fully computed for a single vortex ring (without jet) by Fukumoto & Moffatt (Reference Fukumoto and Moffatt2000). In the present work, this correction is also provided for an array of rings, a single helix and an array of helices. The effect of axial jet within the vortex core is also analysed.
These asymptotic results are then compared with numerical results obtained by DNS. For rings, the spectral DNS code developed by Bolnot (Reference Bolnot2012) is used, in which axisymmetry is enforced as well as axial periodicity, allowing short-wavelength instabilities (Widnall, Bliss & Tsai Reference Widnall, Bliss and Tsai1974; Hattori & Fukumoto Reference Hattori and Fukumoto2003) and the pairing instability (Levy & Forsdyke Reference Levy and Forsdyke1927; Bolnot, Le Dizès & Leweke Reference Bolnot, Le Dizès and Leweke2014) to be filtered out. For helical vortices, the DNS code developed by Delbende, Rossi & Daube (Reference Delbende, Rossi and Daube2012a ) and restricted to the simulation of helically symmetrical flows is used.
Figure 1. Vortex configurations studied in the present paper: (a) vortex ring, (b) array of rings, (c) helical vortex, (d)
$N$
helical vortices (here plotted for
$N=2$
).
The paper is organized as follows. In § 2, the framework of the analysis is presented. The configurations under study are introduced, as well as the different parameters and the local reference frame in which the internal structure of the vortex is analysed. The asymptotic analysis of the inner structure is performed in § 3. Both the leading-order dipolar correction and quadrupolar corrections are obtained. The quadrupolar contribution is shown to depend on a single constant, which varies from one configuration to another. This constant is related to the external strain field experienced locally by the vortex and is computed in § 4 for each configuration. In § 5, asymptotic results are compared to DNS results. Section 6 provides a summary of the main results and a discussion of their implications concerning short-wavelength instabilities.
2. Presentation of the framework
Four vortex configurations are considered, where the vorticity field (longitudinal and transverse) is confined in a region of radius
$a$
around the curve
$\mathscr{S}$
, as illustrated in figure 1. Depending on the configuration, the curve
$\mathscr{S}$
is
-
(i) a circle of radius
$R$
(figure 1
a); -
(ii) an infinite array of circles of radius
$R$
separated by a distance
$L_{z}$
(figure 1
b); -
(iii) an helix of radius
$R$
and pitch
$L$
(figure 1
c); -
(iv) a regular array of
$N$
helices of radius
$R$
and pitch
$L$
(figure 1
d).
Such a curve
$\mathscr{S}$
possesses a symmetry axis called the
$Oz$
axis and is parametrized by the arc length
$s$
. This means that in a global Cartesian frame, the point
$C(s)$
is given by
$C(s)=(R\cos {\it\phi},R\sin {\it\phi},L{\it\phi})$
with
${\it\phi}=s/\sqrt{R^{2}+L^{2}}$
(for rings,
$L$
is taken to be zero). In this Cartesian frame, the Serret–Frenet frame can be expressed as
$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{t}\equiv \frac{\displaystyle \frac{\text{d}C}{\text{d}s}}{\left|\displaystyle \frac{\text{d}C}{\text{d}s}\right|}=\frac{R(-\boldsymbol{e}_{\mathit{x}}\sin {\it\phi}+\boldsymbol{e}_{\mathit{y}}\cos {\it\phi})+L\boldsymbol{e}_{\mathit{z}}}{\sqrt{R^{2}+L^{2}}}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{n}\equiv \frac{\text{d}\boldsymbol{t}}{\text{d}s}=-(\boldsymbol{e}_{\mathit{x}}\cos {\it\phi}+\boldsymbol{e}_{\mathit{y}}\sin {\it\phi}), & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{b}\equiv \boldsymbol{t}\times \boldsymbol{n}=\frac{R\boldsymbol{e}_{\mathit{z}}+L(\boldsymbol{e}_{\mathit{x}}\sin {\it\phi}-\boldsymbol{e}_{\mathit{y}}\cos {\it\phi})}{\sqrt{R^{2}+L^{2}}}. & \displaystyle\end{eqnarray}$$
${\it\kappa}$
and torsion
${\it\tau}$
of this curve are given by 
$$\begin{eqnarray}\displaystyle {\it\kappa}=\frac{R}{R^{2}+L^{2}},\quad {\it\tau}=\frac{L}{R^{2}+L^{2}}. & & \displaystyle\end{eqnarray}$$
The vorticity field is assumed to be uniform along the curve
$\mathscr{S}$
, with a constant longitudinal circulation
${\it\Gamma}$
. The vorticity field is also assumed to possess a transverse component associated with a localized jet along
$\mathscr{S}$
. This will be further defined below. The radius
$a$
is assumed to be small compared to the other length scales i.e.
$R$
and
$L_{z}$
for rings, and
$R$
and
$L/N$
for helices. In this context, we are studying the internal structure within the region of radius
$a$
around
$\mathscr{S}$
. It is therefore useful to define a local frame centred on a point
$C(s)$
of
$\mathscr{S}$
. Following Callegari & Ting (Reference Callegari and Ting1978), we introduce the local polar frame
$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{e}_{r}=\boldsymbol{n}\cos {\it\varphi}+\boldsymbol{b}\sin {\it\varphi}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{e}_{{\it\varphi}}=-\boldsymbol{n}\sin {\it\varphi}+\boldsymbol{b}\cos {\it\varphi}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{e}_{s}=\boldsymbol{t}, & \displaystyle\end{eqnarray}$$
$r$
,
${\it\varphi}$
,
$s$
). The coordinate system is illustrated in figure 2 for ring and helix configurations. This non-orthogonal coordinate system is related to an orthogonal coordinate system (
$r$
,
${\it\theta}$
,
$s$
) where 
$$\begin{eqnarray}\displaystyle {\it\theta}={\it\varphi}+{\it\theta}_{0}(s),\quad \text{with }\frac{\partial {\it\theta}_{0}}{\partial s}={\it\tau}, & & \displaystyle\end{eqnarray}$$
which can be used to derive the governing equations in the local frame (see Callegari & Ting Reference Callegari and Ting1978, for details). In the following, the velocity vector field will be written
$\boldsymbol{V}=v_{{\it\rho}}\boldsymbol{e}_{{\it\rho}}+v_{{\it\phi}}\boldsymbol{e}_{{\it\phi}}+v_{z}\boldsymbol{e}_{z}$
in the global cylindrical frame and
$\boldsymbol{V}=u\boldsymbol{e}_{r}+v\boldsymbol{e}_{{\it\varphi}}+w\boldsymbol{e}_{s}$
in the local polar frame. The formulae that connect one representation to the other are provided in appendix A.
Figure 2. Illustration of the coordinate system for a ring (a) and a helix (b). The point M can be localized using the global cylindrical frame
$(\boldsymbol{e}_{{\it\rho}},\boldsymbol{e}_{{\it\phi}},\boldsymbol{e}_{z})$
, or the local frames: the Serret–Frenet frame
$(\boldsymbol{n},\boldsymbol{b},\boldsymbol{t})$
and the frame
$(\boldsymbol{e}_{r},\boldsymbol{e}_{{\it\varphi}},\boldsymbol{t})$
. Note that vectors
$\boldsymbol{e}_{r}$
and
$\boldsymbol{e}_{{\it\varphi}}$
are in the plane (
$\boldsymbol{n}$
,
$\boldsymbol{b}$
). Point
$C(s)$
is defined as the point on the curve
$\mathscr{S}$
which is the closest to point M.
In the local frame, the vortex structure is assumed, at leading order, to be a columnar axisymmetrical vortex, independent of
$s$
and
${\it\varphi}$
. Our goal is to determine the corrections to this axisymmetrical structure induced by the curvature and torsion of vortex lines, and the presence of distant vortices or vortical parts of the same vortex.
We introduce the following parameters
$$\begin{eqnarray}\displaystyle {\it\varepsilon}=\frac{Ra}{R^{2}+L^{2}}={\it\kappa}a,\quad {\it\alpha}=\frac{L}{R}=\frac{{\it\tau}}{{\it\kappa}},\quad {\it\lambda}=\frac{L_{z}}{R}. & & \displaystyle\end{eqnarray}$$
The asymptotic analysis is performed in the limit
$a\ll R$
and
$a\ll L_{z}$
for rings, and
$a\ll R$
and
$a\ll L$
for helical vortices. The parameter
${\it\varepsilon}$
is thus assumed to be small, whereas the parameters
${\it\alpha}$
and
${\it\lambda}$
are
$O(1)$
. In the present study, we do not consider compressible and buoyancy effects and assume that the density is uniform and constant. Viscous effects are also neglected by assuming that the Reynolds number
${\it\Gamma}/{\it\nu}$
(
${\it\nu}$
is the kinematic viscosity) is sufficiently large. This hypothesis means that we consider vortex structures on time scales which are short compared to the viscous diffusion time scale
$a^{2}/{\it\nu}$
. This allows us to consider vortical structures that translate and rotate steadily without changing their shapes. Here, we shall assume that there exists a frame translating at the velocity
$V_{frame}\boldsymbol{e}_{\mathit{z}}$
where the solution is steady.
The computation of the displacement speed of helical vortices or rings has been the subject of numerous works (see for instance Saffman (Reference Saffman1992) for rings and Alekseenko, Kuibin & Okulov (Reference Alekseenko, Kuibin and Okulov2007) for helices). In each case, there is a dominant local velocity contribution which is proportional to
$-{\it\varepsilon}\log ({\it\varepsilon})$
and oriented along the binormal vector
$\boldsymbol{b}$
, as predicted by the local induction approximation (Saffman Reference Saffman1992). This local contribution is corrected by
$O({\it\varepsilon})$
non-local effects associated with distant vortex parts or other vortices. In all cases, the vortex structure displacement speed remains asymptotically small. The frame velocity
$V_{frame}$
is related to the vortex structure displacement speed in the manner illustrated in figure 3. For rings, the binormal vector is aligned along with the vertical axis, so the frame velocity correspond to the displacement speed
$\boldsymbol{V}_{ring}=V_{ring}\boldsymbol{b}=V_{frame}\boldsymbol{e}_{z}$
. For helices, the displacement speed can be decomposed as
$\boldsymbol{V}_{helix}=V_{helix}\boldsymbol{b}=V_{frame}\boldsymbol{e}_{z}+W_{0_{0}}\boldsymbol{t}$
, where
$V_{frame}=\sqrt{1+{\it\alpha}^{2}}V_{helix}$
and
$W_{0_{0}}=-{\it\alpha}V_{helix}$
because
$\boldsymbol{e}_{z}\boldsymbol{\cdot }\boldsymbol{b}=1/\sqrt{1+{\it\alpha}^{2}}$
and
$\boldsymbol{e}_{z}\boldsymbol{\cdot }\boldsymbol{t}={\it\alpha}/\sqrt{1+{\it\alpha}^{2}}$
. The component
$W_{0_{0}}$
represents a uniform jet contribution along the vortex axis. It is only present in helices. Though small, we shall take this effect into account in our analysis.
Figure 3. Definition of
$V_{frame}$
for a ring (a) and of a helix (b). The velocities
$V_{frame}\boldsymbol{e}_{\mathit{z}}$
and
$R{\it\Omega}_{frame}\boldsymbol{e}_{{\it\phi}}$
are such that their projection on the binormal vector
$\boldsymbol{b}$
is equal to vortex structure displacement speed.
In the uniformly translating frame, the solution satisfies the steady Euler equations. If, in the local frame, the velocity field
$(u,v,w)$
and pressure
$p$
are assumed to be independent of
$s$
, the steady Euler equations become (Callegari & Ting Reference Callegari and Ting1978)
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{{\it\varepsilon}(-{\it\alpha}u_{{\it\varphi}}+w\cos {\it\varphi})w}{h}+uu_{r}+\frac{v(u_{{\it\varphi}}-v)}{r}+p_{r}=0, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{{\it\varepsilon}(-{\it\alpha}v_{{\it\varphi}}-w\sin {\it\varphi})w}{h}+uv_{r}+\frac{v(v_{{\it\varphi}}+u)}{r}+\frac{p_{{\it\varphi}}}{r}=0, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{{\it\varepsilon}[(-{\it\alpha}w_{{\it\varphi}}-(u\cos {\it\varphi}-v\sin {\it\varphi}))w-{\it\alpha}p_{{\it\varphi}}]}{h}+uw_{r}+\frac{vw_{{\it\varphi}}}{r}=0, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{{\it\varepsilon}[-{\it\alpha}w_{{\it\varphi}}-(u\cos {\it\varphi}-v\sin {\it\varphi})]}{h}+\left[u_{r}+\frac{u}{r}+\frac{v_{{\it\varphi}}}{r}\right]=0, & \displaystyle\end{eqnarray}$$
$h=1-{\it\varepsilon}r\cos {\it\varphi}$
. Note that, in the text,
$v_{{\it\rho}}$
,
$v_{{\it\phi}}$
and
$v_{z}$
indicate the components in the global cylindrical frame, whereas the subscript indices for other variables such as the local components
$u$
,
$v$
,
$w$
refer to derivatives (e.g.
$u_{{\it\varphi}}=\partial u/\partial {\it\varphi}$
).
3. Internal vortex structure
In this section, we calculate the main dipolar and quadrupolar corrections to the local columnar axisymmetrical vortex. The velocity and pressure fields are expanded in powers of
${\it\varepsilon}$
$$\begin{eqnarray}\displaystyle u & = & \displaystyle \phantom{v^{(0)}(r)}+{\it\varepsilon}u^{(1)}(r,{\it\varphi})+{\it\varepsilon}^{2}u^{(2)}(r,{\it\varphi})+\cdots \,,\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle v & = & \displaystyle v^{(0)}(r)+{\it\varepsilon}v^{(1)}(r,{\it\varphi})+{\it\varepsilon}^{2}v^{(2)}(r,{\it\varphi})+\cdots \,,\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle w & = & \displaystyle w^{(0)}(r)+{\it\varepsilon}w^{(1)}(r,{\it\varphi})+{\it\varepsilon}^{2}w^{(2)}(r,{\it\varphi})+\cdots \,,\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle p & = & \displaystyle p^{(0)}(r)+{\it\varepsilon}p^{(1)}(r,{\it\varphi})+{\it\varepsilon}^{2}p^{(2)}(r,{\it\varphi})+\cdots \,.\end{eqnarray}$$
3.1. Monopolar structure
At leading order, the solution is a straight vortex of azimuthal velocity
$v^{(0)}(r)$
and axial velocity
$w^{(0)}(r)$
. The pressure
$p^{(0)}$
is related to
$v^{(0)}$
by
$$\begin{eqnarray}\displaystyle p^{(0)}(r)=\int _{0}^{r}\frac{[v^{(0)}({\it\eta})]^{2}}{{\it\eta}}\,\text{d}{\it\eta}. & & \displaystyle\end{eqnarray}$$
The longitudinal (axial) and transverse (azimuthal) distributions of vorticity are defined respectively from
$v^{(0)}$
and
$w^{(0)}$
by the following expressions
$$\begin{eqnarray}\displaystyle & \displaystyle {\it\zeta}^{(0)}=\frac{1}{r}\frac{\partial }{\partial r}(rv^{(0)}), & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle {\it\Upsilon}^{(0)}=-\frac{\partial w^{(0)}}{\partial r}. & \displaystyle\end{eqnarray}$$
The asymptotic analysis can be performed for any axisymmetrical vortex with jet but for computations and numerical comparisons we use the Batchelor family of profiles
$$\begin{eqnarray}\displaystyle & \displaystyle v^{(0)}(r)=\frac{1}{r}(1-\text{e}^{-r^{2}}), & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle w^{(0)}(r)=W_{0}\text{e}^{-r^{2}}+W_{0_{0}}. & \displaystyle\end{eqnarray}$$
$a$
, as a characteristic length and
${\it\Gamma}/(2{\rm\pi}a)$
as characteristic velocity. The constant
$W_{0}$
characterizes the jet strength and the constant
$W_{0_{0}}$
corresponds to a uniform flow component along the vortex axis. As explained above, such a component is present in helices. It is created by the change of reference frame, because the frame velocity does not correspond to the direction of propagation of the helix (see figure 3). If we take into account only this effect, it is related to the speed
$V_{frame}$
of the comoving frame by 
$$\begin{eqnarray}\displaystyle W_{0_{0}}=-\frac{{\it\alpha}V_{frame}}{\sqrt{1+{\it\alpha}^{2}}}. & & \displaystyle\end{eqnarray}$$
This term is null for vortex rings (
${\it\alpha}=0$
), and is
$O({\it\varepsilon}\log {\it\varepsilon})$
for helices. Although this term is a higher-order term, it is convenient to introduce it here. We shall see below that it improves the predictions of the axial flow component.
3.2. Dipolar correction
The problem at first order provides the main dipolar correction to the axisymmetric vortex. This problem was first solved in a general setting by Callegari & Ting (Reference Callegari and Ting1978). They showed that the first order corrections satisfy the system
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{v^{(0)}}{r}(u_{{\it\varphi}}^{(1)}-2v^{(1)})+p_{r}^{(1)}=-[w^{(0)}]^{2}\cos {\it\varphi}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle ru^{(1)}v_{r}^{(0)}+v^{(0)}(v_{{\it\varphi}}^{(1)}+u^{(1)})+p_{{\it\varphi}}^{(1)}=r[w^{(0)}]^{2}\sin {\it\varphi}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle u^{(1)}w_{r}^{(0)}+\frac{v^{(0)}}{r}w_{{\it\varphi}}^{(1)}=-[w^{(0)}v^{(0)}]\sin {\it\varphi}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{1}{r}(ru^{(1)})_{r}+\frac{1}{r}v_{{\it\varphi}}^{(1)}=-v^{(0)}\sin {\it\varphi}. & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle u^{(1)}=\hat{u} ^{(1)}(r)\sin {\it\varphi}=-\frac{\hat{{\it\psi}}^{(1)}}{r}\sin {\it\varphi}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle v^{(1)}=\hat{v}^{(1)}(r)\cos {\it\varphi}=(-\hat{{\it\psi}}_{r}^{(1)}+rv^{(0)})\cos {\it\varphi}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle w^{(1)}={\hat{w}}^{(1)}(r)\cos {\it\varphi}=\left(-\frac{w_{r}^{(0)}}{v^{(0)}}\hat{{\it\psi}}^{(1)}+rw^{(0)}\right)\cos {\it\varphi}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle p^{(1)}=\hat{p}^{(1)}(r)\cos {\it\varphi}=(-{\it\zeta}^{(0)}\hat{{\it\psi}}^{(1)}+v^{(0)}\hat{{\it\psi}}_{r}^{(1)}-r[v^{(0)}]^{2}-r[w^{(0)}]^{2})\cos {\it\varphi}, & \displaystyle\end{eqnarray}$$
$\hat{{\it\psi}}^{(1)}(r)$
satisfies 
$$\begin{eqnarray}\displaystyle \mathscr{L}^{(1)}(\hat{{\it\psi}}^{(1)})=2r{\it\zeta}^{(0)}+v^{(0)}+2r\frac{w^{(0)}w_{r}^{(0)}}{v^{(0)}}, & & \displaystyle\end{eqnarray}$$
with the operator
$\mathscr{L}^{(k)}$
for
$k=1,2,\ldots$
defined as
$$\begin{eqnarray}\displaystyle \mathscr{L}^{(k)}\equiv \left[\frac{\partial ^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial }{\partial r}-\left(\frac{k^{2}}{r^{2}}+\frac{{\it\zeta}_{r}^{(0)}}{v^{(0)}}\right)\right]. & & \displaystyle\end{eqnarray}$$
Using the method of variation of constants, (3.8) can be integrated in closed form as it possesses
$v^{(0)}$
as an exact homogeneous solution. If the centre of the local frame is chosen such that the velocity in the (
$\boldsymbol{n},\boldsymbol{b}$
) plane vanishes at the origin (up to
$O({\it\varepsilon}^{2})$
),
$\hat{{\it\psi}}_{r}^{(1)}(0)=0$
and
$\hat{{\it\psi}}^{(1)}(0)=0$
must be imposed, hence
$$\begin{eqnarray}\hat{{\it\psi}}^{(1)}(r)=v^{(0)}(r)\int _{0}^{r}\frac{\displaystyle \int _{0}^{z}v^{(0)}({\it\eta})\left[2{\it\eta}{\it\zeta}^{(0)}({\it\eta})+v^{(0)}({\it\eta})+2{\it\eta}\frac{w^{(0)}({\it\eta})w_{r}^{(0)}({\it\eta})}{v^{(0)}({\it\eta})}\right]{\it\eta}\,\text{d}{\it\eta}}{z[v^{(0)}(z)]^{2}}\,\text{d}z.\end{eqnarray}$$
As explained in Fukumoto & Moffatt (Reference Fukumoto and Moffatt2000), adding a homogeneous solution
$c_{1}v^{(0)}$
to
$\hat{{\it\psi}}^{(1)}$
corresponds to a change of frame centre. In the present work, the frame centre has been selected as the stagnation point of the flow, and this corresponds to
$c_{1}=0$
. As soon as the frame centre is fixed, there are no free parameters: streamfunction, velocity and pressure are given at first order by (3.7a–d
) and (3.10).
The first-order correction is thus a pure dipolar correction which depends on the local curvature only. This correction is thus identical for a ring, an array of rings, a helix and an array of
$N$
helices. It is easy to show that
$\hat{{\it\psi}}^{(1)}$
expands for large
$r$
(Fukumoto & Miyazaki Reference Fukumoto and Miyazaki1991) as follows
$$\begin{eqnarray}\displaystyle \hat{{\it\psi}}^{(1)}\sim \frac{1}{2}r\log r+rA+O\left(\frac{1}{r}\right), & & \displaystyle\end{eqnarray}$$
with
$$\begin{eqnarray}\displaystyle A & = & \displaystyle \frac{1}{2}\lim _{r\rightarrow \infty }\left(\int _{0}^{r}{\it\eta}(v^{(0)}({\it\eta}))^{2}\,\text{d}{\it\eta}-\log r\right)+\frac{1}{4}-\int _{0}^{\infty }{\it\eta}((w^{(0)}({\it\eta}))^{2}-W_{0_{0}}^{2})\,\text{d}{\it\eta}\nonumber\\ \displaystyle & = & \displaystyle \frac{1}{4}(1-W_{0}(W_{0}+2W_{0_{0}})+{\it\gamma}-\log 2)\end{eqnarray}$$
and
${\it\gamma}\approx 0.577$
being Euler’s constant. As
$W_{0}W_{0_{0}}$
is always small, we shall use the approximation
$A\approx 0.22-W_{0}^{2}/4$
. In figure 4, the streamfunction,
$\hat{{\it\psi}}^{(1)}/r$
is plotted together with its asymptotic behaviour for the Batchelor vortex for two jet parameters (
$W_{0}=0$
,
$W_{0}=1$
) and
$W_{0_{0}}=0$
. When
$W_{0}=0$
, the dipolar component of the axial velocity follows the simple linear expression
${\hat{w}}^{(1)}(r)=rW_{0_{0}}$
.
Figure 4. Characteristics of the dipolar correction for the Batchelor vortex.
$\hat{{\it\psi}}^{(1)}/r$
versus
$r$
(solid line:
$W_{0}=0$
; dashed line:
$W_{0}=1$
.
$W_{0_{0}}=0$
for both cases) The dotted lines correspond to the asymptotic behaviour (3.11).
As shown by Fukumoto & Okulov (Reference Fukumoto and Okulov2005), dipolar corrections are modified by torsion at second order. Thus, rings and helices generate a different dipolar correction only if we consider higher-order terms.
3.3. Quadrupolar correction
Quadrupolar corrections do not appear at first order but they are generated at second order. At second order, dipolar and monopolar corrections are also generated but we focus here on quadrupolar corrections, as monopolar and dipolar fields were already obtained at lower order. The perturbation equations for the second-order corrections are given by
$$\begin{eqnarray}\displaystyle \frac{v^{(0)}}{r}(u_{{\it\varphi}}^{(2)}-2v^{(2)})+p_{r}^{(2)} & = & \displaystyle \left[\frac{v^{(1)}}{r}(v^{(1)}-u_{{\it\varphi}}^{(1)})-u^{(1)}u_{r}^{(1)}\right]\nonumber\\ \displaystyle & & \displaystyle -\,w^{(0)}(2w^{(1)}\cos {\it\varphi}+w^{(0)}r\cos ^{2}{\it\varphi}),\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle ru^{(2)}v_{r}^{(0)}+v^{(0)}(v_{{\it\varphi}}^{(2)}+u^{(2)})+p_{{\it\varphi}}^{(2)} & = & \displaystyle -[ru^{(1)}v_{r}^{(1)}+v^{(1)}(v_{{\it\varphi}}^{(1)}+u^{(1)})]\nonumber\\ \displaystyle & & \displaystyle +\,rw^{(0)}(2w^{(1)}\sin {\it\varphi}+w^{(0)}r\cos {\it\varphi}\sin {\it\varphi}),\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle u^{(2)}w_{r}^{(0)}+\frac{v^{(0)}}{r}w_{{\it\varphi}}^{(2)} & = & \displaystyle -\!\left[u^{(1)}w_{r}^{(1)}+\frac{v^{(1)}}{r}w_{{\it\varphi}}^{(1)}\right]-v^{(0)}w^{(1)}\sin {\it\varphi}\nonumber\\ \displaystyle & & \displaystyle +\,w^{(0)}(u^{(1)}\cos {\it\varphi}-v^{(1)}\sin {\it\varphi}-rv^{(0)}\cos {\it\varphi}\sin {\it\varphi}),\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \frac{1}{r}(ru^{(2)})_{r}+\frac{1}{r}v_{{\it\varphi}}^{(2)}=u^{(1)}\cos {\it\varphi}-v^{(1)}\sin {\it\varphi}-rv^{(0)}\cos {\it\varphi}\sin {\it\varphi}. & & \displaystyle\end{eqnarray}$$
The quadrupolar field satisfying these equations is found to be of the form
$$\begin{eqnarray}\displaystyle & \displaystyle u_{quad}^{(2)}=\hat{u} ^{(2)}(r)\sin 2{\it\varphi}=-\frac{2}{r}\hat{{\it\psi}}^{(2)}\sin 2{\it\varphi}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle v_{quad}^{(2)}=\hat{v}^{(2)}(r)\cos 2{\it\varphi}=(-\hat{{\it\psi}}_{r}^{(2)}+V_{F})\cos 2{\it\varphi}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle w_{quad}^{(2)}={\hat{w}}^{(2)}(r)\cos 2{\it\varphi}=\left(-\frac{w_{r}^{(0)}}{v^{(0)}}\hat{{\it\psi}}^{(2)}+W_{F}\right)\cos 2{\it\varphi}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle p_{quad}^{(2)}=\hat{p}^{(2)}(r)\cos 2{\it\varphi}=(-{\it\zeta}^{(0)}\hat{{\it\psi}}^{(2)}+v^{(0)}\hat{{\it\psi}}_{r}^{(2)}+P_{F})\cos 2{\it\varphi}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle V_{F}=\frac{r^{2}}{4}(\hat{u} _{r}^{(1)}+2v^{(0)}), & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle W_{F}=\frac{r}{4v^{(0)}}\left[-\frac{{\hat{w}}^{(1)}\hat{v}^{(1)}}{r}+{\hat{w}}_{r}^{(1)}\hat{u} ^{(1)}+v^{(0)}({\hat{w}}^{(1)}+rw^{(0)})+w^{(0)}(\hat{v}^{(1)}-\hat{u} ^{(1)})\right],\quad & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle P_{F}=\frac{r}{4}\hat{u} ^{(1)}\hat{v}_{r}^{(1)}-\frac{\hat{v}^{(1)}}{4}(\hat{v}^{(1)}-\hat{u} ^{(1)})-\frac{r}{2}w^{(0)}\left({\hat{w}}^{(1)}+\frac{r}{2}w^{(0)}\right)-v^{(0)}V_{F}. & \displaystyle\end{eqnarray}$$
$\hat{{\it\psi}}^{(2)}$
satisfies 
$$\begin{eqnarray}\displaystyle \mathscr{L}^{(2)}(\hat{{\it\psi}}^{(2)})=\mathscr{F}, & & \displaystyle\end{eqnarray}$$
with
$$\begin{eqnarray}\displaystyle \mathscr{F}=\frac{3rv^{(0)}}{4}-\frac{r\hat{{\it\psi}}^{(1)}{\it\zeta}_{r}^{(0)}}{v^{(0)}}-\frac{(\hat{{\it\psi}}^{(1)})^{2}}{4v^{(0)}}\left(\frac{{\it\zeta}_{r}^{(0)}}{v^{(0)}}\right)_{r}+\frac{r\hat{{\it\psi}}^{(1)}}{[v^{(0)}]^{2}}\left[\frac{w^{(0)}w_{r}^{(0)}v_{r}^{(0)}}{v^{(0)}}-(w^{(0)}w_{r}^{(0)})_{r}\right].\qquad & & \displaystyle\end{eqnarray}$$
The general solution to (3.16) which is finite at the origin, can be written as
$$\begin{eqnarray}\displaystyle \hat{{\it\psi}}^{(2)}=S{\it\Psi}^{(2)}(r)+{\it\Psi}_{NH}^{(2)}(r), & & \displaystyle\end{eqnarray}$$
where
${\it\Psi}^{(2)}$
denotes the homogeneous function such that
$$\begin{eqnarray}\displaystyle \mathscr{L}^{(2)}({\it\Psi}^{(2)})=0,\quad {\it\Psi}^{(2)}\begin{array}{@{}c@{}}{\sim}\\ r\rightarrow 0\end{array}s_{0}r^{2},\quad {\it\Psi}^{(2)}\begin{array}{@{}c@{}}{\sim}\\ r\rightarrow +\infty \end{array}r^{2}, & & \displaystyle\end{eqnarray}$$
and
${\it\Psi}_{NH}^{(2)}$
denotes a particular solution to the non-homogeneous problem (3.16). Finally,
$S$
is an arbitrary constant.
It is worth mentioning that another homogeneous solution of (3.13a–d
) corresponding to a radial velocity proportional to
$(2/r){\it\Psi}^{(2)}(r)\cos 2{\it\varphi}$
could have been added in principle in (3.14a
). This solution turns out not to be present for rings and helices, as will be seen in the expressions of the outer solution given in the next section.
The function
${\it\Psi}^{(2)}$
describes how a strain field is transmitted within a vortex. It was first introduced by Moffatt et al. (Reference Moffatt, Kida and Ohkitani1994) for the asymptotic analysis of a vortex in a strain field. Eloy & Le Dizès (Reference Eloy and Le Dizès1999) showed that
$s_{0}\approx 2.525$
for a Gaussian vorticity profile. Function
${\it\Psi}_{NH}^{(2)}$
(and consequently
$\hat{{\it\psi}}^{(2)}$
) is expected to behave as
$3/16r^{2}\log r$
for large
$r$
(see figure 5) because
$\mathscr{F}\sim 3/4$
as
$r\rightarrow +\infty$
. Using the method of variation of constants and the fact that
${\it\Psi}^{(2)}$
is an exact solution of the homogeneous problem,
${\it\Psi}_{NH}^{(2)}$
is given by
$$\begin{eqnarray}\displaystyle {\it\Psi}_{NH}^{(2)}={\it\Psi}^{(2)}(r)\left[\frac{3}{16}\log r-\int _{r}^{+\infty }\left(\frac{\displaystyle \int _{0}^{s}{\it\eta}\mathscr{F}({\it\eta}){\it\Psi}^{(2)}({\it\eta})\,\text{d}{\it\eta}}{s[{\it\Psi}^{(2)}]^{2}}-\frac{3}{16s}\right)\,\text{d}s\right]. & & \displaystyle\end{eqnarray}$$
The strain rate
$S_{NH}^{(2)}=\lim _{r\rightarrow 0}{\it\Psi}_{NH}^{(2)}/r^{2}$
, which is associated with local curvature effects (it is not present for straight vortices (see, for instance, Le Dizès & Verga Reference Le Dizès and Verga2002)), strongly varies with respect to the jet parameter
$W_{0}$
(see figure 6). Finally, the function
${\it\Psi}^{(2)}$
only depends on
$v^{(0)}$
and the function
${\it\Psi}_{NH}^{(2)}$
only on
$v^{(0)}$
and
$w^{(0)}$
, that is, they only depend on the local properties of the underlying axisymmetrical vortex. These two functions are plotted in figure 5 for the Batchelor vortex for three values of the jet parameter.
Figure 5. Characteristics of the quadrupolar correction for a Batchelor vortex. (a)
${\it\Psi}^{(2)}/r^{2}$
versus
$r$
; (b)
${\it\Psi}_{NH}^{(2)}/r^{2}$
versus
$r$
(solid line:
$W_{0}=0$
, dashed line:
$W_{0}=0.4$
, dash-dot line:
$W_{0}=1$
). For all cases,
$W_{0_{0}}=0$
.
Figure 6. Variation of
$S_{NH}^{(2)}$
with respect to
$W_{0}$
for
$W_{0_{0}}=0$
.
The constant
$S$
is not determined from the above inner analysis but from matching with an outer solution. It is hence expected to be different for each case;
$S$
is actually related to the external strain field experienced by the vortex (see the following section). Contrary to the dipolar correction, the quadrupolar correction is dependent on non-local effects. Yet this dependence comes about through a single constant! Near the origin, the function
$\hat{{\it\psi}}^{(2)}$
represents a pure strain field
$$\begin{eqnarray}\displaystyle \hat{{\it\psi}}^{(2)}\sim S^{(2)}r^{2}, & & \displaystyle\end{eqnarray}$$
where
$S^{(2)}=s_{0}S+S_{NH}^{(2)}$
. The principal directions of this strain field are
${\it\varphi}=\pm {\rm\pi}/4$
.
The choice of the frame centre selected at first order does not affect the constant
$S$
, but has an impact on the forcing term
$\mathscr{F}$
, and thus on the function
${\it\Psi}_{NH}^{(2)}$
. When
$W_{0}=0$
, the axial flow component is particularly simple
${\hat{w}}^{(2)}=r^{2}W_{0_{0}}/2$
and the other velocity components become independent of
$W_{0_{0}}$
.
4. External strain field
The constant
$S$
depends on the strain field induced by the background flow. It comes about through an asymptotic matching of the inner solution previously obtained with the outer solution obtained by the Biot–Savart law.
Let us consider the velocity
$\boldsymbol{u}^{(out)}$
induced by the vortex system in a reference frame where the system is steady. Such a frame is chosen here as the frame uniformly translating with a velocity
$V_{frame}\boldsymbol{e}_{\mathit{z}}$
.
To get the outer solution, we assume that the vorticity is a monopolar field concentrated on the curve
$\mathscr{S}$
corresponding to one of the configurations shown in figure 1. The velocity is then such that
$$\begin{eqnarray}\displaystyle \boldsymbol{u}^{(out)}(\boldsymbol{x})=\boldsymbol{u}^{(BS)}(\boldsymbol{x})-V_{frame}\boldsymbol{e}_{\mathit{z}},\quad \text{with }\boldsymbol{u}^{(BS)}(\boldsymbol{x})=-\frac{1}{4{\rm\pi}}\int _{\mathscr{S}}{\it\Gamma}^{\prime }\frac{(\boldsymbol{x}-\boldsymbol{x}^{\prime })\times \boldsymbol{t}^{\prime }}{|\boldsymbol{x}-\boldsymbol{x}^{\prime }|^{3}}\,\text{d}s^{\prime }. & & \displaystyle\end{eqnarray}$$
All vortices have a constant circulation
${\it\Gamma}^{\prime }=2{\rm\pi}$
. Note that we do not consider the transverse distribution of vorticity on
$\mathscr{S}$
associated with the axial flow within the vortex. Indeed, this vorticity distribution is not expected to contribute to the external flow, as proved by Fukumoto & Miyazaki (Reference Fukumoto and Miyazaki1991). We also disregard higher-order corrections to (4.1) corresponding to multipolar vorticity concentration on
$\mathscr{S}$
; these corrections would be needed to perform a full matching of inner and outer solutions up to
$O({\it\varepsilon}^{2})$
(see for instance Fukumoto & Moffatt Reference Fukumoto and Moffatt2000) but are not required for the computation of
$S$
. Here, we simply match the quadrupolar field of one component of the velocity field at leading-order only. More precisely, we impose that the local radial velocity of the outer solution possesses a quadrupolar part
$u_{quad}^{(out)}$
of the form
$$\begin{eqnarray}\displaystyle u_{quad}^{(out)}=u_{quad}^{(BS)}\sim {\it\varepsilon}^{2}[(-2Sr-{\textstyle \frac{3}{8}}r\log r)\sin 2{\it\varphi}]\quad \text{for }1\ll r\ll 1/{\it\varepsilon}. & & \displaystyle\end{eqnarray}$$
4.1. A single vortex ring
The asymptotic description of a single ring was first considered by Widnall & Tsai (Reference Widnall and Tsai1977). It was recently re-examined by Fukumoto & Moffatt (Reference Fukumoto and Moffatt2000) and Fukumoto (Reference Fukumoto2002). Fukumoto & Moffatt (Reference Fukumoto and Moffatt2000) obtained an asymptotic solution up to
${\it\varepsilon}^{3}$
which can be used for the present analysis. Their expression (3.7) of the outer solution in the regime (
$1\ll r\ll R$
) leads to the radial velocity
$$\begin{eqnarray}\displaystyle u^{(BS)}\sim \frac{1}{2R}\left(\log \left(\frac{8R}{r}\right)-1\right)\sin {\it\varphi}+\frac{3r}{8R^{2}}\left(\log \left(\frac{8R}{r}\right)-\frac{4}{3}\right)\sin 2{\it\varphi}. & & \displaystyle\end{eqnarray}$$
Since
${\it\varepsilon}=1/R$
for the ring, the matching of their quadrupolar field with the inner quadrupolar part (4.2) requires that
$$\begin{eqnarray}\displaystyle S^{Ri}=-\frac{3}{16}\left[\log \left(\frac{8}{{\it\varepsilon}}\right)-\frac{4}{3}\right]. & & \displaystyle\end{eqnarray}$$
Note that such a formula could have also been obtained from expression (4.21) of Widnall & Tsai (Reference Widnall and Tsai1977).
4.2. An array of vortex rings
The velocity field of an array of vortex rings is equal to the field of a single ring plus the field generated by distant rings. The first contribution has been calculated in the previous section. The contribution from distant rings can be obtained directly from the Biot–Savart integral. The velocity field of the ring placed at position
$z=2{\rm\pi}nL_{z}$
can be written using the global cylindrical coordinate system (see appendix A) as
$$\begin{eqnarray}\displaystyle & \displaystyle v_{n,{\it\rho}}^{(BS)}({\it\rho},z)=-\frac{1}{2}\int _{0}^{2{\rm\pi}}\frac{R(z-2{\rm\pi}nL_{z})\cos s}{({\it\rho}^{2}-2R{\it\rho}\cos s+R^{2}+(z-2{\rm\pi}nL_{z})^{2})^{3/2}}\,\text{d}s, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle v_{n,z}^{(BS)}({\it\rho},z)=-\frac{1}{2}\int _{0}^{2{\rm\pi}}\frac{R(R^{2}-{\it\rho}R\cos s+(z-2{\rm\pi}nL_{z})^{2})\cos s}{({\it\rho}^{2}-2R{\it\rho}\cos s+R^{2}+(z-2{\rm\pi}nL_{z})^{2})^{3/2}}\,\text{d}s. & \displaystyle\end{eqnarray}$$
$L={\it\alpha}=0$
) 
$$\begin{eqnarray}\displaystyle u(r,{\it\varphi})=-v_{{\it\rho}}\cos {\it\varphi}+v_{z}\sin {\it\varphi} & & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle {\it\rho}=R-r\cos {\it\varphi},\quad z=r\sin {\it\varphi} & & \displaystyle\end{eqnarray}$$
Finally, the dipolar and quadrupolar terms are for
$r\ll R$
$$\begin{eqnarray}\displaystyle u^{BS}\sim \mathop{\sum }_{n\neq 0}\left[v_{n,z}^{(BS)}(R,0)\sin {\it\varphi}-\frac{r}{2}(\partial _{z}v_{n,{\it\rho}}^{(BS)}(R,0)+\partial _{{\it\rho}}v_{n,z}^{(BS)}(R,0))\sin 2{\it\varphi}\right]. & & \displaystyle\end{eqnarray}$$
The quadrupolar velocity contribution from distant rings becomes (
${\it\varepsilon}=1/R$
)
$$\begin{eqnarray}\displaystyle u_{quad}^{BS}\sim -2S^{DRi}{\it\varepsilon}^{2}r,\quad \text{with }S^{DRi}=\frac{R^{2}}{2}\mathop{\sum }_{n=1}^{\infty }(\partial _{z}v_{n,{\it\rho}}^{(BS)}(R,0)+\partial _{{\it\rho}}v_{n,z}^{(BS)}(R,0)). & & \displaystyle\end{eqnarray}$$
As shown by Levy & Forsdyke (Reference Levy and Forsdyke1927),
$S^{DRi}$
can also be expressed in terms of the complete elliptic integrals of the first and second kind,
$\boldsymbol{K}(Z)$
and
$\boldsymbol{E}(Z)$
, (Abramowitz & Stegun Reference Abramowitz and Stegun1965, p. 590)
$$\begin{eqnarray}\displaystyle S^{DRi}=\frac{1}{4}\mathop{\sum }_{n=1}^{\infty }{\it\Lambda}_{n}^{3/2}\left(\left(1-\frac{4}{n^{2}{\it\lambda}^{2}}\right)\boldsymbol{E}({\it\Lambda}_{n})-\boldsymbol{K}({\it\Lambda}_{n})\right), & & \displaystyle\end{eqnarray}$$
with
$$\begin{eqnarray}\displaystyle {\it\Lambda}_{n}=\frac{1}{1+n^{2}{\it\lambda}^{2}/4}. & & \displaystyle\end{eqnarray}$$
The function
$S^{DRi}$
is plotted in figure 7 as a function of the parameter
${\it\lambda}=L_{z}/R$
. The constant
$S$
for an array of vortex rings is
$$\begin{eqnarray}\displaystyle S^{ARi}=S^{DRi}+S^{Ri}. & & \displaystyle\end{eqnarray}$$
Both contributions are negative, and tend to elongate the vortex core along the
$z$
direction. In figure 8, the total contribution is plotted as a function of
${\it\lambda}$
for a few values of
${\it\varepsilon}$
.
Figure 7. Strain rate parameter
$S^{DRi}$
generated by distant rings in an array of rings as a function of the aspect ratio
${\it\lambda}=L_{z}/R$
.
Note that because
$$\begin{eqnarray}\displaystyle S^{DRi}\sim -\frac{{\rm\pi}^{2}}{6{\it\lambda}^{2}}\quad \text{as }{\it\lambda}\rightarrow 0, & & \displaystyle\end{eqnarray}$$
the complete external strain rate is equal to
$S_{ext}\equiv 2S^{DRi}{\it\varepsilon}^{2}\sim -{\rm\pi}^{2}/(3L_{z}^{2})$
which is equal to the external strain rate generated by an array of point vortices of circulation
$2{\rm\pi}$
separated by a distance
$L_{z}$
(Lamb Reference Lamb1932).
Figure 8. Strain rate parameter
$S^{ARi}$
for an array of vortex rings as a function of
${\it\lambda}=L_{z}/R$
for various
${\it\varepsilon}=a/R$
. Solid line:
${\it\varepsilon}=0.33$
; dashed line:
${\it\varepsilon}=0.1$
; dash-dot line:
${\it\varepsilon}=0.033$
; dotted line:
${\it\varepsilon}=0.01$
.
4.3. A single helix
As shown by Hardin (Reference Hardin1982), the velocity field induced by a helical vortex filament can be expressed in cylindrical coordinates for
${\it\rho}<R$
(with our normalization) as
$$\begin{eqnarray}\displaystyle & \displaystyle v_{{\it\rho}}^{(BS)}({\it\rho},{\it\chi})=\frac{2}{R{\it\alpha}^{2}}\,\text{Im}\left[H_{1}^{1,1}\left(\frac{{\it\rho}}{R{\it\alpha}},\frac{1}{{\it\alpha}},{\it\chi}\right)\right], & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle v_{{\it\phi}}^{(BS)}({\it\rho},{\it\chi})=\frac{2}{{\it\rho}{\it\alpha}}\,\text{Re}\left[H_{1}^{0,1}\left(\frac{{\it\rho}}{R{\it\alpha}},\frac{1}{{\it\alpha}},{\it\chi}\right)\right], & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle v_{z}^{(BS)}({\it\rho},{\it\chi})=\frac{1}{R{\it\alpha}}-\frac{2}{R{\it\alpha}^{2}}\,\text{Re}\left[H_{1}^{0,1}\left(\frac{{\it\rho}}{R{\it\alpha}},\frac{1}{{\it\alpha}},{\it\chi}\right)\right], & \displaystyle\end{eqnarray}$$
${\it\chi}={\it\phi}-z/L$
and
$\text{Re}[\,]$
(respectively
$\text{Im}[\,]$
) indicates the real (respectively imaginary) part of a complex expression, and 
$$\begin{eqnarray}\displaystyle H_{M}^{I,J}(x,y,{\it\chi})=\mathop{\sum }_{m=1}^{\infty }m^{M}I_{m}^{(I)}(mx)K_{m}^{(J)}(my)\exp (\text{i}m{\it\chi}), & & \displaystyle\end{eqnarray}$$
where we used the notation introduced by Okulov (Reference Okulov2004) for the Kapteyn series.
$I_{m}^{(I)}$
and
$K_{m}^{(J)}$
denote the
$I$
th and
$J$
th derivative of modified Bessel functions. Other expressions for the velocity field can be obtained for
${\it\rho}>R$
, but we only need the above expressions valid for
${\it\rho}<R$
to compute the value of
$S$
.
The value
${\it\chi}=0$
defines the angular position of the helix. Without restriction, we can consider a local frame centred on the point in cylindrical coordinates
$({\it\rho},{\it\phi},z)=(R,0,0)$
. The local radial velocity
$u$
can be deduced from (4.14a–c
) using (A 4a
) and (A 5a–c
) given in appendix A. We are interested in the expansion of
$u$
as
$r/R$
goes to zero. As shown by Okulov (Reference Okulov2004), it is convenient to isolate the singularity of the Kapteyn series to obtain such an expansion (see appendix B). The derivation, which should be carried up to
$O(r/R)^{2}$
terms, is tedious but straightforward. It can be facilitated by using a symbolic software. The final result is expression (B 12) for
$u$
with the following expression for
$S$
:
$$\begin{eqnarray}\displaystyle S^{He} & = & \displaystyle -\frac{3}{16}\log \left(\frac{{\it\alpha}}{({\it\alpha}^{2}+1)^{3/2}{\it\varepsilon}}\right)-\frac{({\it\alpha}^{2}+1)^{2}}{24{\it\alpha}^{2}}-\frac{1}{4}({\it\alpha}^{2}+1)^{2}+\frac{1}{4}{\it\alpha}({\it\alpha}^{2}+1)^{3/2}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{1}{96}(28{\it\alpha}^{2}+11)+\mathop{\sum }_{m=1}^{\infty }R_{m}({\it\alpha}),\end{eqnarray}$$
with
$$\begin{eqnarray}\displaystyle R_{m}({\it\alpha}) & = & \displaystyle \frac{{\it\alpha}({\it\alpha}^{2}+1)^{3/2}}{2}+\frac{3}{16m}-\frac{(1+{\it\alpha}^{2})^{3}m}{2{\it\alpha}^{2}}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{m^{2}({\it\alpha}^{2}+1)^{5/2}}{{\it\alpha}^{3}}\left(-I_{m-1}\left(\frac{m}{{\it\alpha}}\right)+{\it\alpha}I_{m}\left(\frac{m}{{\it\alpha}}\right)\right)\left(K_{m-1}\left(\frac{m}{{\it\alpha}}\right)+{\it\alpha}K_{m}\left(\frac{m}{{\it\alpha}}\right)\right)\nonumber\\ \displaystyle & & \displaystyle +\,m({\it\alpha}^{2}+1)^{3/2}I_{m}\left(\frac{m}{{\it\alpha}}\right)\left(K_{m-1}\left(\frac{m}{{\it\alpha}}\right)+{\it\alpha}K_{m}\left(\frac{m}{{\it\alpha}}\right)\right).\end{eqnarray}$$
The function
$S^{He}$
is plotted in figure 9 for various values of
${\it\varepsilon}$
.
Figure 9. Strain rate parameter
$S^{He}$
as a function of
${\it\alpha}=L/R$
for various
${\it\varepsilon}$
. Solid line:
${\it\varepsilon}=0.33$
; dashed line:
${\it\varepsilon}=0.1$
; dash-dot line:
${\it\varepsilon}=0.033$
; dotted line:
${\it\varepsilon}=0.01$
.
Note that when
${\it\alpha}\rightarrow 0$
, we have
$S^{He}\sim -1/(24{\it\alpha}^{2})$
, which corresponds to the value (4.13) for an array of rings for small
${\it\lambda}$
since we have the relation
${\it\lambda}=2{\rm\pi}{\it\alpha}$
. From figure 9, we can also guess that
$S^{He}\sim O({\it\alpha}^{2})\text{ as }{\it\alpha}\rightarrow \infty$
, which implies that the self-external strain of a helix does vanish when its pitch
$L/R$
goes to infinity with a fixed
$a/R$
.
4.4. Multiple helices
When there are several helices, we must add to the previous contribution of a single helix, the contribution from the other distant helices. Expressions (4.14a–c
) for the velocity field can still be used. For the helix located in the horizontal plane
$(0xy)$
at
${\it\phi}_{k}=2{\rm\pi}k/N$
, we have to change
${\it\chi}={\it\phi}-z/L$
into
${\it\chi}=({\it\phi}-{\it\phi}_{k})-z/L$
. Summing the contribution from each helix for
$k=1,\ldots ,N-1$
and expanding every quantity as
$r/R\rightarrow 0$
leads to expression (B 15) for the local radial velocity (see appendix B) where
$S$
is given by
$$\begin{eqnarray}\displaystyle S^{DHe} & = & \displaystyle \frac{3\log (N)}{16}-\frac{({\it\alpha}^{2}+1)^{3}(N^{2}-1)}{24{\it\alpha}^{2}}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{{\it\alpha}({\it\alpha}^{2}+1)^{3/2}(N-1)}{4}+\mathop{\sum }_{m=1}^{\infty }R_{m}({\it\alpha}){\it\varrho}\left(\frac{m}{N}\right),\end{eqnarray}$$
with
$R_{m}({\it\alpha})$
given by (4.17) and
$$\begin{eqnarray}\displaystyle {\it\varrho}\left(\frac{m}{N}\right)=\mathop{\sum }_{k=1}^{N-1}\cos \left(2{\rm\pi}k\frac{m}{N}\right)=\left\{\begin{array}{@{}ll@{}}-1 & \displaystyle \frac{m}{N}\not \in \mathbb{N},\\ N-1 & \displaystyle \frac{m}{N}\in \mathbb{N}.\end{array}\right. & & \displaystyle\end{eqnarray}$$
We obtain for multiple helices
$S^{MHe}=S^{He}+S^{DHe}$
, that is
$$\begin{eqnarray}\displaystyle S^{MHe} & = & \displaystyle -\frac{3}{16}\log \left(\frac{{\it\alpha}}{({\it\alpha}^{2}+1)^{3/2}N{\it\varepsilon}}\right)-\frac{4({\it\alpha}^{2}+1)^{3}N^{2}+{\it\alpha}^{2}(20{\it\alpha}^{4}+12{\it\alpha}^{2}+9)}{96{\it\alpha}^{2}}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{{\it\alpha}({\it\alpha}^{2}+1)^{3/2}N}{4}+N\mathop{\sum }_{n=1}^{\infty }R_{nN}({\it\alpha}).\end{eqnarray}$$
The function
$S^{MHe}$
is plotted in figure 10 for
$N=1,2,3,5$
and
${\it\varepsilon}=1$
. Plots for other values of
${\it\varepsilon}$
are obtained by adding
$(3/16)\log {\it\varepsilon}$
as
$S^{MHe}({\it\varepsilon})=S^{MHe}(1)+(3/16)\log {\it\varepsilon}$
.
Figure 10. Strain rate parameter
$S^{MHe}$
versus
${\it\alpha}=L/R$
for
$N$
helices with
$N=1,2,3,5$
(solid, dash, dash-dot, dotted lines) for
${\it\varepsilon}=1$
. (a)
$S^{MHe}$
is normalized by its expression at the origin
$S_{0}^{MHe}=-N^{2}/(24{\it\alpha}^{2})$
; (b)
$S^{MHe}$
is normalized by
${\it\alpha}^{4}$
.
Note that we have
$$\begin{eqnarray}\displaystyle & \displaystyle S^{MHe}\sim -\frac{N^{2}}{24{\it\alpha}^{2}}\quad \text{as }{\it\alpha}\rightarrow 0, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle S^{MHe}\sim -\frac{(N-1)(N-5)}{24}{\it\alpha}^{4}\quad \text{as }{\it\alpha}\rightarrow \infty . & \displaystyle\end{eqnarray}$$
${\it\alpha}$
, the problem becomes a configuration of 2-D straight vortices on a polygon. This can be used to check (4.22). For
$N=2$
, the calculation is straightforward as the strain field at
$(x,y)=(R,0)$
is due to a vortex of circulation
$2{\rm\pi}$
located at
$(-R,0)$
. We immediately obtain a strain rate
$S_{ext}=1/(4R^{2})=(1/4){\it\varepsilon}^{2}(1+{\it\alpha}^{2})^{2}=2S{\it\varepsilon}^{2}$
which gives
$S=(1/8){\it\alpha}^{4}$
for large
${\it\alpha}$
in accordance with (4.22) for
$N=2$
.
For large
$N$
, we obtain
$$\begin{eqnarray}\displaystyle S^{MHe}\sim -\frac{(1+{\it\alpha}^{2})^{3}N^{2}}{24{\it\alpha}^{2}}, & & \displaystyle\end{eqnarray}$$
which corresponds to the external strain rate generated by an array of straight vortices of circulation
$2{\rm\pi}$
and separated by the shortest distance between the helices:
$L_{z}=2{\rm\pi}L/(N\sqrt{1+{\it\alpha}^{2}})$
.
Note that although
$S^{MHe}$
diverges for small and large
${\it\alpha}$
, the external strain rate defined by
$S_{ext}=2S^{MHe}{\it\varepsilon}^{2}=2S^{MHe}(a/R)^{2}/(1+{\it\alpha}^{2})^{2}$
remains always small when the core size,
$a$
, remains small compared to the shortest distance to the other parts of the vortex structure.
5. Comparison with numerical results
In this section, the asymptotic solution is compared to direct numerical simulations.
5.1. Simulation of a ring array
For an array of vortex rings, the pseudo-spectral code developed by Bolnot (Reference Bolnot2012) is used. This code has been validated and run to analyse the stability of vortex rings with respect to the pairing instability in Bolnot et al. (Reference Bolnot, Le Dizès and Leweke2014). The system is assumed axisymmetric and periodic in the axial direction. The numerical formulation is based on Chebyshev and Fourier decompositions in the radial and axial directions, and on an extrapolation Adams–Bashforth scheme for time evolution. The simulation is initialized by a normalized Gaussian profile for the azimuthal vorticity and velocity centred at
$({\it\rho},z)=(R_{0},0)$
in a box of axial length
$L_{z}$
with a prescribed jet parameter. At
$t=0$
, circulation and core radius of the vortex ring and Reynolds number are fixed to
${\it\Gamma}_{0}=2{\rm\pi}$
,
$a_{0}=1$
and
$Re={\it\Gamma}_{0}/{\it\nu}=2000$
.
After a short relaxation process, the solution reaches a quasi-steady state in the frame moving with the vortex ring (Bolnot et al.
Reference Bolnot, Le Dizès and Leweke2014). The core size and jet parameter have slightly evolved during the relaxation process but are adequately predicted using the viscous expansion factor
$\sqrt{1+4t/Re}$
(Bolnot et al.
Reference Bolnot, Le Dizès and Leweke2014). It is this quasi-equilibrium state (frozen at a time
$t_{r}$
) that is compared to the asymptotic solution. In order to agree with the definition used in the asymptotic analysis, the radius
$R$
of the ring should correspond to a radial and axial velocity stagnation point. As the plane of symmetry of the solution is fixed in the centre
$z=0$
of the box, the radius
$R$
is obtained from the condition
$u_{z}(R,0)=0$
which leads to a value slightly different from
$R_{0}$
.
As soon as the centre of the local frame is obtained, the local velocity field
$(u,v,w)$
can be computed using (A 4a–c
), (A 5a–i
) and (A 1a–c
). An azimuthal decomposition is then performed to get the monopolar, dipolar and quadrupolar contributions that we compare to the theory.
Figure 11 shows a comparison between asymptotic and DNS results for an array of vortex rings for the parameters
${\it\varepsilon}=0.11$
,
${\it\lambda}=L_{z}/R=3$
,
$W_{0}=0.1$
and
$Re=2000$
. In these figures, we plotted the monopolar, dipolar and quadrupolar contributions of each velocity component in the local frame, using the same normalisation as in the theory. The core size, circulation and jet parameter have been evaluated by comparing the monopolar part (a,d,g) of the numerical velocity fields with the Gaussian model used. It is found that the theoretical profile selected at zero order is appropriate (see comparison figure 11
a,d,g). A good agreement between numerical and theoretical curves is also found for all dipolar and quadrupolar components. Other configurations have been tested, and a similar agreement has always been observed. This constitutes a strong validation for both theory and code.
Figure 11. Array of swirling vortex rings with jet strength
$W_{0}=0.1$
, at
$L_{z}/R=3$
,
${\it\varepsilon}=0.11$
,
$Re=2000$
: comparison between asymptotic results (dashed lines) and DNS results (solid lines). (a,d,g) Monopolar component; (b,e,h): dipolar component; (c,f,i) quadrupolar component. (a–c) Radial velocity, (d–f) azimuthal velocity, (g–i) axial velocity.
5.2. Simulation of helices
The simulations of helix systems have been carried out using the helical code developed in Delbende et al. (Reference Delbende, Rossi and Daube2012a ), which implements a generalisation of the vorticity-streamfunction formulation in a circular domain, with finite differences in the radial direction and spectral decomposition along the azimuth. The helical symmetry is explicitly enforced in such a way that the 3-D Navier–Stokes equations are reduced to a 2-D unsteady problem. The code has also been validated and used in Delbende, Rossi & Piton (Reference Delbende, Rossi and Piton2012b ), Delbende, Piton & Rossi (Reference Delbende, Piton and Rossi2015).
Similarly to vortex ring simulations, we start the simulation with Gaussian profiles for the axial velocity and axial vorticity around one or several helical curves (see figure 1) with prescribed
$a_{0}$
,
$R_{0}$
,
$W_{0}$
,
${\it\Gamma}$
and
$L$
. The Reynolds number is always fixed at
$Re={\it\Gamma}/{\it\nu}=10\,000$
. After a relaxation process, the system reaches a quasi-equilibrium state in the frame rotating at the rotation rate
${\it\Omega}_{frame}$
of the helical structure. The rotation rate
${\it\Omega}_{frame}$
is related to the displacement speed of the helical structure as sketched in figure 3. It can then also be expressed as a function of
$V_{frame}$
using
${\it\Omega}_{frame}=-V_{frame}/({\it\alpha}R)$
.
The difficulty concerning the treatment of the numerical data is the prescription of the local frame. Such a frame is not naturally defined from the DNS, and is obtained here by an iterative procedure. Let us define the helical vorticity
${\it\omega}_{B}$
by
${\bf\omega}\boldsymbol{\cdot }\boldsymbol{e}_{B}$
where
$\boldsymbol{e}_{B}$
is a unit vector tangent to any helical line (for details see Delbende et al. (Reference Delbende, Rossi and Daube2012a
)). This quantity is directly provided by the helical code. First, one finds the position
$M_{0}$
where the helical vorticity
${\it\omega}_{B}$
reaches its maximum in a global horizontal plane. Second, one defines the plane
${\it\Pi}_{\bot }(M_{0})$
normal to the local helical vector
$\boldsymbol{e}_{B}(M_{0})$
passing through this point
$M_{0}$
. Third, one determines the stagnation point
$M_{1}$
of the velocity components normal to
$\boldsymbol{e}_{B}(M_{0})$
in
${\it\Pi}_{\bot }(M_{0})$
. At this stagnation point,
$M_{1}$
, the local helical vector
$\boldsymbol{e}_{B}(M_{1})$
is oriented in a slightly different direction than
$\boldsymbol{e}_{B}(M_{0})$
, so a new plane
${\it\Pi}_{\bot }(M_{1})$
can be defined in which we can obtain another stagnation point
$M_{2}$
of the velocity components normal to
$\boldsymbol{e}_{B}(M_{1})$
in
${\it\Pi}_{\bot }(M_{1})$
. This operation is repeated until it has converged to the point that will be the centre of the local frame. In practice, around
$100$
iterations are necessary. At the end of the process, the stagnation point
$C$
has been located, providing a helical line
$\mathscr{S}$
. The radial coordinate of
$C$
defines the value of
$R$
. The local frame is such that
$\boldsymbol{e}_{B}(C)$
corresponds to the vector
$\boldsymbol{e}_{\boldsymbol{s}}$
of the theory.
Velocity and vorticity fields are expressed in the cylindrical coordinates attached to this local frame and thereafter Fourier decomposed in the azimuthal direction to obtain the monopolar, dipolar and quadrupolar components. The monopolar components are used to correct the value of the parameters
$a$
,
${\it\Gamma}$
and
$W_{0}$
, by fitting axial vorticity and axial velocity with Gaussian profiles, after subtracting the uniform vorticity associated with the rotation of the frame. At the end of this procedure, all theoretical parameters have been obtained and each component can be adequately renormalized for comparison. Note that the frame used in the theory to obtain a steady vortex system is different; it is translating rather than rotating. To express the theoretical results in the rotating frame, one simply subtracts velocity
$-V_{frame}\boldsymbol{e}_{\mathit{z}}+{\it\Omega}_{frame}{\it\rho}\boldsymbol{e}_{{\it\phi}}$
from the theoretical solution. Using (A 6), this amounts to performing the following modifications on the theoretical fields:
$$\begin{eqnarray}\displaystyle & \displaystyle v^{(0)}\rightarrow v^{(0)}+{\it\varepsilon}\sqrt{1+{\it\alpha}^{2}}V_{frame}r, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle w^{(0)}\rightarrow w^{(0)}+\frac{\sqrt{1+{\it\alpha}^{2}}}{{\it\alpha}}V_{frame}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle w^{(1)}\rightarrow w^{(1)}-\frac{\sqrt{1+{\it\alpha}^{2}}}{{\it\alpha}}V_{frame}r. & \displaystyle\end{eqnarray}$$
$V_{frame}$
is
$O({\it\varepsilon}\log {\it\varepsilon})$
for helices, these changes are higher-order terms. Nevertheless, it is convenient to introduce them here because they improve the predictions of the axial flow component, as shown below.
Figure 12. Single helix with
$W_{0}=0.54$
,
$L/R=0.5$
,
${\it\varepsilon}=0.045$
,
$Re=10^{4}$
: comparison between asymptotic (dashed lines) and DNS (solid lines). (a,d,g) Monopolar component; (b,e,h) dipolar component; (c,f,i) quadrupolar component. (a–c) Radial velocity, (d–f) azimuthal velocity, (g–i) axial velocity. The dash-dotted lines correspond to the theory without taking into account the effect of the change of frame (
$W_{0_{0}}=0$
). For these parameters, the theory gives
${\it\Omega}_{frame}=-0.0120$
and
$S=-0.6341$
, which gives a strain rate in the vortex centre
$S^{(2)}=-1.699$
(defined in (3.21)). The numerical values are
${\it\Omega}_{frame}=-0.0110$
and
$S^{(2)}=-1.803$
.
Figure 13. System of two helical vortices with
$W_{0}=0$
,
$L/R=0.5$
,
${\it\varepsilon}=0.06$
,
$Re=10^{4}$
: comparison between asymptotic (dashed lines) and DNS (solid lines). (a,d,g) Monopolar component; (b,e,h) dipolar component; (c,f,i) quadrupolar component. (a–c) Radial velocity, (d–f) azimuthal velocity, (g–i) axial velocity. The dash-dotted lines correspond to the theory without taking into account the effect of the change of frame (
$W_{0_{0}}=0$
). For these parameters, the theory predicts
${\it\Omega}_{frame}=-0.0258$
and
$S=-1.2832$
, which gives a strain rate in the centre
$S^{(2)}=-3.441$
(defined in (3.21)). The numerical values are
${\it\Omega}_{frame}=-0.0261$
and
$S^{(2)}=-3.496$
.
In figure 12 (respectively figure 13), theoretical and numerical results are compared for a single helix with axial flow
$W_{0}=0.54$
(respectively for a system of two helices without axial jet
$W_{0}=0$
). For radial and azimuthal velocity components, a fair agreement is found between theory and DNS results. Despite a smaller value of
${\it\varepsilon}$
, the agreement is found to be less convincing than for an array of rings. It is also seen that the change (5.1a–c
) of reference frame has to be taken into account. However, a small mismatch between DNS and theory remains for the monopolar component. This is probably at the origin of the deteriorated agreement for the dipolar and quadrupolar components (see bottom figures in figures 12 and 13).
As for the ring arrays, other configurations have been tested for helical vortex systems and we always observed a good agreement with the asymptotic theory. The values obtained from the numerical simulations for the internal strain rate
$S^{(2)}$
and for the frame rotation rate
${\it\Omega}_{frame}$
have also been compared to the theory and we have systematically found a good agreement. This provides a validation of expressions (4.20) and (B 17) for the external strain field and translation speed of multiple helices.
6. Conclusions
We analysed the internal structure of vortex rings and helical vortices showing that they can be described using an asymptotic theory where the core size is considered small compared to the other scales of the vortex structure. At leading order, the vortex structure was assumed locally an axisymmetric vortex with axial flow. The asymptotic theory was used to determine dipolar and quadrupolar corrections associated with this structure. It was shown that dipolar corrections which appear at first order depend in a dominant way on the local curvature of the vortex. As a consequence, we expect rings, arrays of rings, helices and multiples helices to all exhibit the same dipolar correction if they possess the same curvature. We also computed the quadrupolar corrections which appear at second order. Contrary to the dipolar corrections, such corrections depend on the global geometry. To be more precise, quadrupolar corrections are composed of a contribution which depends on the local curvature and a non-local contribution associated with the strain field generated by distant part of the vortex structure. The non-local contribution has been calculated for a vortex ring, an array of rings, a helix and multiple helices by considering the flow field obtained from Biot–Savart law for vortex filaments.
The asymptotic results have been compared to numerical solutions obtained by DNS for both rings and helices. A good agreement has been observed for rings. For helices, a fair agreement has also been demonstrated. In practice, this case is more involved because of the necessity to define a plane orthogonal to the vortex core structure.
The present results are important to model the short-wavelength instabilities that can develop in rings or helices. Both dipolar and quadrupolar correction terms are expected to couple modes of the underlying vortex and be a source of instability. The dipolar correction term is known to be responsible of the so-called curvature instability. This instability has been theoretically predicted for rings (Hattori & Fukumoto Reference Hattori and Fukumoto2003; Fukumoto & Hattori Reference Fukumoto and Hattori2005) and helices (Hattori & Fukumoto Reference Hattori and Fukumoto2009, Reference Hattori and Fukumoto2014). However, the analysis has only been performed for particular vortices with uniform vorticity. It would be interesting to extend the analysis to more realistic vortices such as those considered in the present study. The quadrupolar correction is associated with the so-called elliptic instability (see for instance Kerswell Reference Kerswell2002). Widnall & Tsai (Reference Widnall and Tsai1977) provided a growth rate estimate for the elliptic instability in a single ring with a Rankine vorticity profile. No person however has so far provided any prediction for helices or rings with a realistic vorticity profile. Yet, we expect this instability to be present as the quadrupolar field contained a strain field part which is also present in straight vortices. An interesting study would then be to compute the effects of the additional quadrupolar field which is generated by curvature.
Both curvature and elliptic instabilities are also expected to be strongly affected by the presence of axial flow, as this parameter modifies the form of vortex modes involved in the instabilities (Lacaze, Ryan & Le Dizès Reference Lacaze, Ryan and Le Dizès2007; Roy et al. Reference Roy, Leweke, Thompson and Hourigan2011). We suspect that this parameter is crucial to an understanding of the competition between both instabilities and associate recent observations in helices (Leweke et al. Reference Leweke, Quaranta, Bolnot, Blanco-Rodriguez and Le Dizès2014) with one instability or another.
Acknowledgements
This work received support from the French Agence Nationale de la Recherche under the A*MIDEX grant ANR-11-IDEX-0001-02, the LABEX MEC project ANR-11-LABX-0092 and the ANR HELIX project ANR-12-BS09-0023-01. HPC resources from GENCI–IDRIS (grant 2015-2a1386) are also acknowledged.
Appendix A. Change of coordinate systems
In this section, we provide the formulae which can be used to obtain the velocity field in the local frame from its expression in the global cylindrical frame. Consider a point of coordinates
$({\it\rho},{\it\phi},z)$
in the global cylindrical coordinate system and assume that this point has the coordinates
$(r,{\it\varphi},s)$
in the local frame defined by (2.1) and (2.3) centred on the point
$C(s)$
of cylindrical coordinates
$(R,{\it\phi}_{0},L{\it\phi}_{0})$
(see figure 2). We obtain the following relation between the coordinates:
$$\begin{eqnarray}\displaystyle & \displaystyle {\it\rho}^{2}=(R-r\cos {\it\varphi})^{2}+\frac{{\it\alpha}^{2}}{1+{\it\alpha}^{2}}r^{2}\sin ^{2}{\it\varphi}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle {\it\rho}\sin {\it\phi}=(R-r\cos {\it\varphi})\sin {\it\phi}_{0}-\frac{{\it\alpha}}{\sqrt{1+{\it\alpha}^{2}}}r\sin {\it\varphi}\cos {\it\phi}_{0}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle z=L{\it\phi}_{0}+\frac{1}{\sqrt{1+{\it\alpha}^{2}}}r\sin {\it\varphi}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle s={\it\phi}_{0}\sqrt{R^{2}+L^{2}}. & \displaystyle\end{eqnarray}$$
If the velocity field can be written in the global cylindrical frame as
$$\begin{eqnarray}\displaystyle \boldsymbol{V}=\boldsymbol{V}({\it\rho},{\it\phi},z)=v_{{\it\rho}}\boldsymbol{e}_{{\it\rho}}+v_{{\it\phi}}\boldsymbol{e}_{{\it\phi}}+v_{z}\boldsymbol{e}_{\mathit{z}} & & \displaystyle\end{eqnarray}$$
and
$$\begin{eqnarray}\displaystyle \boldsymbol{V}=u\boldsymbol{e}_{\boldsymbol{r}}+v\boldsymbol{e}_{{\it\varphi}}+w\boldsymbol{e}_{\boldsymbol{s}} & & \displaystyle\end{eqnarray}$$
in the local frame, we derive the relation between
$(u,v,w)$
and
$(v_{{\it\rho}},v_{{\it\phi}},v_{z})$
:
$$\begin{eqnarray}\displaystyle & \displaystyle u(r,{\it\varphi},s)=v_{{\it\rho}}({\it\rho},{\it\phi},z)\boldsymbol{e}_{\boldsymbol{r}}\boldsymbol{\cdot }\boldsymbol{e}_{{\it\rho}}+v_{{\it\phi}}({\it\rho},{\it\phi},z)\boldsymbol{e}_{\boldsymbol{r}}\boldsymbol{\cdot }\boldsymbol{e}_{{\it\phi}}+v_{z}({\it\rho},{\it\phi},z)\boldsymbol{e}_{\boldsymbol{r}}\boldsymbol{\cdot }\boldsymbol{e}_{\mathit{z}}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle v(r,{\it\varphi},s)=v_{{\it\rho}}({\it\rho},{\it\phi},z)\boldsymbol{e}_{{\it\varphi}}\boldsymbol{\cdot }\boldsymbol{e}_{{\it\rho}}+v_{{\it\phi}}({\it\rho},{\it\phi},z)\boldsymbol{e}_{{\it\varphi}}\boldsymbol{\cdot }\boldsymbol{e}_{{\it\phi}}+v_{z}({\it\rho},{\it\phi},z)\boldsymbol{e}_{{\it\varphi}}\boldsymbol{\cdot }\boldsymbol{e}_{\mathit{z}}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle w(r,{\it\varphi},s)=v_{{\it\rho}}({\it\rho},{\it\phi},z)\boldsymbol{e}_{\boldsymbol{s}}\boldsymbol{\cdot }\boldsymbol{e}_{{\it\rho}}+v_{{\it\phi}}({\it\rho},{\it\phi},z)\boldsymbol{e}_{\boldsymbol{s}}\boldsymbol{\cdot }\boldsymbol{e}_{{\it\phi}}+v_{z}({\it\rho},{\it\phi},z)\boldsymbol{e}_{\boldsymbol{s}}\boldsymbol{\cdot }\boldsymbol{e}_{\mathit{z}}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{e}_{\boldsymbol{r}}\boldsymbol{\cdot }\boldsymbol{e}_{{\it\rho}}=-\left(\frac{R}{{\it\rho}}-\frac{r}{{\it\rho}}\cos {\it\varphi}\right)\cos {\it\varphi}+\frac{{\it\alpha}^{2}}{(1+{\it\alpha}^{2})}\frac{r}{{\it\rho}}\sin ^{2}{\it\varphi}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{e}_{\boldsymbol{r}}\boldsymbol{\cdot }\boldsymbol{e}_{{\it\phi}}=-\frac{{\it\alpha}}{\sqrt{1+{\it\alpha}^{2}}}\frac{R}{{\it\rho}}\sin {\it\varphi}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{e}_{\boldsymbol{r}}\boldsymbol{\cdot }\boldsymbol{e}_{\mathit{z}}=\frac{1}{\sqrt{1+{\it\alpha}^{2}}}\sin {\it\varphi}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{e}_{{\it\varphi}}\boldsymbol{\cdot }\boldsymbol{e}_{{\it\rho}}=\left(\frac{R}{{\it\rho}}-\frac{r}{{\it\rho}}\cos {\it\varphi}\right)\sin {\it\varphi}+\frac{{\it\alpha}^{2}}{(1+{\it\alpha}^{2})}\frac{r}{{\it\rho}}\sin {\it\varphi}\cos {\it\varphi}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{e}_{{\it\varphi}}\boldsymbol{\cdot }\boldsymbol{e}_{{\it\phi}}=\frac{{\it\alpha}}{\sqrt{1+{\it\alpha}^{2}}}\left(\frac{r}{{\it\rho}}-\frac{R}{{\it\rho}}\cos {\it\varphi}\right), & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{e}_{{\it\varphi}}\boldsymbol{\cdot }\boldsymbol{e}_{\mathit{z}}=\frac{1}{\sqrt{1+{\it\alpha}^{2}}}\cos {\it\varphi}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{e}_{\boldsymbol{s}}\boldsymbol{\cdot }\boldsymbol{e}_{{\it\rho}}=-\frac{{\it\alpha}}{(1+{\it\alpha}^{2})}\frac{r}{{\it\rho}}\sin {\it\varphi}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{e}_{\boldsymbol{s}}\boldsymbol{\cdot }\boldsymbol{e}_{{\it\phi}}=\frac{1}{\sqrt{1+{\it\alpha}^{2}}}\left(\frac{R}{{\it\rho}}-\frac{r}{{\it\rho}}\cos {\it\varphi}\right), & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{e}_{\boldsymbol{s}}\boldsymbol{\cdot }\boldsymbol{e}_{\mathit{z}}=\frac{{\it\alpha}}{\sqrt{1+{\it\alpha}^{2}}}. & \displaystyle\end{eqnarray}$$
These expressions can be used to express in the local frame with the local coordinates the vector
$\boldsymbol{U}=-V_{frame}\boldsymbol{e}_{\mathit{z}}+{\it\Omega}_{frame}{\it\rho}\boldsymbol{e}_{{\it\phi}}$
associated with the change of frame in § 5.2. Using
${\it\Omega}_{frame}=-V_{frame}/({\it\alpha}R)$
, we obtain
$$\begin{eqnarray}\displaystyle \boldsymbol{U}=V_{frame}\left(-\frac{r}{R\sqrt{1+{\it\alpha}^{2}}}\boldsymbol{e}_{{\it\varphi}}+\left(-\frac{\sqrt{1+{\it\alpha}^{2}}}{{\it\alpha}}+\frac{r\cos {\it\varphi}}{R{\it\alpha}\sqrt{1+{\it\alpha}^{2}}}\right)\boldsymbol{e}_{\boldsymbol{s}}\right). & & \displaystyle\end{eqnarray}$$
Appendix B. Inner expansion of Hardin solution for helices
To derive the inner expansion of Hardin solution, it is convenient to isolate the singularity of the Kapteyn series using the technique introduced by Okulov (Reference Okulov2004). We have in particular used the expressions (valid for
${\it\rho}<R$
) given by Fukumoto & Okulov (Reference Fukumoto and Okulov2005):
$$\begin{eqnarray}\displaystyle & \displaystyle v_{{\it\rho}}^{(BS)}=\frac{2}{R{\it\alpha}^{2}}\text{Im}\left[{\it\lambda}^{1,1}\left(\frac{\text{e}^{{\it\xi}+\text{i}{\it\chi}}}{1-\text{e}^{{\it\xi}+\text{i}{\it\chi}}}+{\it\alpha}^{1,1}\log (1-\text{e}^{{\it\xi}+\text{i}{\it\chi}})+{\it\beta}^{1,1}\text{Li}_{2}(\text{e}^{{\it\xi}+\text{i}{\it\chi}})\right)+R_{1}^{1,1}\right],\qquad & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle v_{{\it\phi}}^{(BS)}=\frac{2}{{\it\rho}{\it\alpha}}\text{Re}\left[{\it\lambda}^{0,1}\left(\frac{\text{e}^{{\it\xi}+\text{i}{\it\chi}}}{1-\text{e}^{{\it\xi}+\text{i}{\it\chi}}}+{\it\alpha}^{0,1}\log (1-\text{e}^{{\it\xi}+\text{i}{\it\chi}})+{\it\beta}^{0,1}\text{Li}_{2}(\text{e}^{{\it\xi}+\text{i}{\it\chi}})\right)+R_{1}^{0,1}\right],\qquad & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle v_{z}^{(BS)}=\frac{1}{2{\it\alpha}}-\frac{2}{R{\it\alpha}^{2}}\text{Re}\!\left[{\it\lambda}^{0,1}\!\left(\frac{\text{e}^{{\it\xi}+\text{i}{\it\chi}}}{1-\text{e}^{{\it\xi}+\text{i}{\it\chi}}}+{\it\alpha}^{0,1}\log (1-\text{e}^{{\it\xi}+\text{i}{\it\chi}})+{\it\beta}^{0,1}\text{Li}_{2}(\text{e}^{{\it\xi}+\text{i}{\it\chi}})\!\right)\!+R_{1}^{0,1}\right], & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle {\it\chi}={\it\phi}-z/(R{\it\alpha}),\quad \text{e}^{{\it\xi}}=\frac{x(1+\sqrt{1+y^{2}})\exp (\sqrt{1+x^{2}})}{y(1+\sqrt{1+x^{2}})\exp (\sqrt{1+y^{2}})}, & & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle {\it\lambda}^{1,1}=-\frac{((1+x^{2})(1+y^{2}))^{1/4}}{2xy},\quad {\it\lambda}^{0,1}=-\frac{1}{2y}\left(\frac{1+y^{2}}{1+x^{2}}\right)^{1/4}, & & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle {\it\alpha}^{1,1}={\it\upsilon}_{1}(t_{x})-{\it\upsilon}_{1}(t_{y}),\quad {\it\beta}^{1,1}={\it\upsilon}_{2}(t_{x})+{\it\upsilon}_{2}(t_{y})-{\it\upsilon}_{1}(t_{x}){\it\upsilon}_{1}(t_{y}), & & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle {\it\alpha}^{0,1}={\it\vartheta}_{1}(t_{x})-{\it\upsilon}_{1}(t_{y}),\quad {\it\beta}^{0,1}={\it\vartheta}_{2}(t_{x})+{\it\upsilon}_{2}(t_{y})-{\it\vartheta}_{1}(t_{x}){\it\upsilon}_{1}(t_{y}), & & \displaystyle\end{eqnarray}$$
with
$$\begin{eqnarray}\displaystyle x=\frac{{\it\rho}}{R{\it\alpha}},\quad y=\frac{1}{{\it\alpha}},\quad t_{x}=\frac{1}{\sqrt{1+x^{2}}},\quad t_{y}=\frac{1}{\sqrt{1+y^{2}}}, & & \displaystyle\end{eqnarray}$$
and
$$\begin{eqnarray}\displaystyle & \displaystyle {\it\upsilon}_{1}(t)=(-9t+7t^{3})/24, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle {\it\vartheta}_{1}(t)=(3t-5t^{3})/24, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle {\it\upsilon}_{2}(t)=(-135t^{2}+594t^{4}-455t^{6})/1152, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle {\it\vartheta}_{2}(t)=(81t^{2}-462t^{4}+385t^{6})/1152. & \displaystyle\end{eqnarray}$$
The functions
$R_{1}^{1,1}$
and
$R_{1}^{0,1}$
are defined by
$$\begin{eqnarray}\displaystyle R_{1}^{1,1}=\mathop{\sum }_{m=1}^{\infty }ri_{m}^{1,1}(x,y)\text{e}^{\text{i}m{\it\chi}},\quad R_{1}^{0,1}=\mathop{\sum }_{m=1}^{\infty }ri_{m}^{0,1}(x,y)\text{e}^{\text{i}m{\it\chi}}, & & \displaystyle\end{eqnarray}$$
with
$$\begin{eqnarray}\displaystyle & \displaystyle ri_{m}^{1,1}(x,y)=mI_{m}^{\prime }(mx)K_{m}^{\prime }(my)-{\it\lambda}^{1,1}\text{e}^{m{\it\xi}}\left(1+\frac{{\it\alpha}^{1,1}}{m}+\frac{{\it\beta}^{1,1}}{m^{2}}\right), & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle ri_{m}^{0,1}(x,y)=mI_{m}(mx)K_{m}^{\prime }(my)-{\it\lambda}^{0,1}\text{e}^{m{\it\xi}}\left(1+\frac{{\it\alpha}^{0,1}}{m}+\frac{{\it\beta}^{0,1}}{m^{2}}\right). & \displaystyle\end{eqnarray}$$
An expression for
$u$
in terms of the local coordinates is obtained using (A 4a
) and (A 1a–c
) with
${\it\phi}_{0}=0$
. Expanding all the quantities as
$r/R\rightarrow 0$
, we find an expression for a single helix of the form
$$\begin{eqnarray}\displaystyle u^{(BS)}\sim [C^{He}-{\textstyle \frac{1}{2}}\log r]{\it\varepsilon}\sin {\it\varphi}+[-2S^{He}-{\textstyle \frac{3}{8}}\log r]{\it\varepsilon}^{2}r\sin 2{\it\varphi}, & & \displaystyle\end{eqnarray}$$
where
$S^{He}$
is given by (4.16) and
$$\begin{eqnarray}\displaystyle C^{He}=\frac{1-{\it\alpha}^{4}+{\it\alpha}^{3}\sqrt{1+{\it\alpha}^{2}}}{2{\it\alpha}\sqrt{1+{\it\alpha}^{2}}}+\frac{1}{2}\log \left(\frac{{\it\alpha}}{{\it\varepsilon}(1+{\it\alpha}^{2})^{3/2}}\right)+\mathop{\sum }_{m=1}^{\infty }c_{m}({\it\alpha}), & & \displaystyle\end{eqnarray}$$
with
$$\begin{eqnarray}\displaystyle c_{m}({\it\alpha})=-\frac{(1+{\it\alpha}^{2})^{3/2}}{{\it\alpha}}\left(\frac{2m}{{\it\alpha}}I_{m}\left(\frac{m}{{\it\alpha}}\right)K_{m}^{\prime }\left(\frac{m}{{\it\alpha}}\right)+1\right)-\frac{1}{2m}. & & \displaystyle\end{eqnarray}$$
Both
$R_{m}({\it\alpha})$
and
$c_{m}({\it\alpha})$
are
$O(1/m^{3})$
for large
$m$
, which guarantees that the sums in (4.16) and (B 10) are absolutely convergent.
If we now add the contribution from the displacement of the frame (which is
$-V_{frame}^{He}\boldsymbol{e}_{\mathit{z}}$
) we obtain from (4.1) for a single helix
$$\begin{eqnarray}\displaystyle u^{(out)}\sim \left[-\frac{V_{frame}^{He}}{\sqrt{1+{\it\alpha}^{2}}}+C^{He}-\frac{1}{2}\log r\right]{\it\varepsilon}\sin {\it\varphi}+\left[-2S^{He}-\frac{3}{8}\log r\right]{\it\varepsilon}^{2}r\sin 2{\it\varphi}.\quad & & \displaystyle\end{eqnarray}$$
The matching of this expression with the inner expansions provides the strain rate
$S^{He}$
and the frame velocity
$V_{frame}^{He}$
:
$$\begin{eqnarray}\displaystyle V_{frame}^{He}=\sqrt{1+{\it\alpha}^{2}}(C^{He}+A){\it\varepsilon}, & & \displaystyle\end{eqnarray}$$
where
$A$
is defined by (3.12). This expression is analogue to an expression already given in Alekseenko et al. (Reference Alekseenko, Kuibin and Okulov2007), p. 250.
The velocity field induced by the distant helices is obtained by summing the
$N-1$
contributions taken at
${\it\phi}_{k}=2{\rm\pi}k/N,k=1,\ldots ,N-1$
. There are many simplifications thanks to the relations
$$\begin{eqnarray}\displaystyle & \displaystyle \mathop{\sum }_{k=1}^{N-1}\frac{\exp (\text{i}{\it\phi}_{k})}{1-\exp (\text{i}{\it\phi}_{k})}=\frac{1-N}{2}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \mathop{\sum }_{k=1}^{N-1}\log (1-\exp (\text{i}{\it\phi}_{k}))=\log N, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \mathop{\sum }_{k=1}^{N-1}\text{Li}_{2}(\exp (\text{i}{\it\phi}_{k}))=\frac{{\rm\pi}^{2}}{6}\frac{1-N}{N}. & \displaystyle\end{eqnarray}$$
After a long but straightforward calculation, we obtain for the contribution from distant helices
$$\begin{eqnarray}\displaystyle u^{(BS)}\sim C^{DHe}{\it\varepsilon}\sin {\it\varphi}-2S^{DHe}{\it\varepsilon}^{2}r\sin 2{\it\varphi}, & & \displaystyle\end{eqnarray}$$
where
$S^{DHe}$
is given by (4.18) and
$$\begin{eqnarray}\displaystyle C^{DHe}=-\frac{\sqrt{1+{\it\alpha}^{2}}({\it\alpha}^{2}-1)(N-1)}{2{\it\alpha}}-\frac{1}{2}\log N+\mathop{\sum }_{m=1}^{\infty }c_{m}({\it\alpha}){\it\varrho}\left(\frac{m}{N}\right), & & \displaystyle\end{eqnarray}$$
with
${\it\varrho}(x)$
defined in (4.19).
The velocity field obtained by multiple helices is the sum of the contributions from a single helix and distant helices. From the complete matching of the dipolar component, we obtain using (B 12) and (B 15), an expression for the velocity
$V_{frame}^{MHe}$
of the frame for multiple helices:
$$\begin{eqnarray}\displaystyle V_{frame}^{MHe}=\sqrt{1+{\it\alpha}^{2}}(C^{MHe}+A){\it\varepsilon}, & & \displaystyle\end{eqnarray}$$
where
$C^{MHe}=C^{He}+C^{DHe}$
reduces to
$$\begin{eqnarray}\displaystyle C^{MHe}=\frac{{\it\alpha}^{2}}{2}-\frac{\sqrt{1+{\it\alpha}^{2}}({\it\alpha}^{2}-1)N}{2{\it\alpha}}+\frac{1}{2}\log \left(\frac{{\it\alpha}}{{\it\varepsilon}(1+{\it\alpha}^{2})^{3/2}N}\right)+N\mathop{\sum }_{k=1}^{\infty }c_{Nk}({\it\alpha}). & & \displaystyle\end{eqnarray}$$
It is this expression of
$V_{frame}$
, with
$A\approx 0.22-W_{0}^{2}/4$
which has been used in expression (3.5) for
$W_{0_{0}}$
and in (5.1a–c
).
Note finally that the expressions (B 9) and (B 15), and (4.3) and (4.8) for rings, contain no
$\cos 2{\it\phi}$
terms. This property validates the hypothesis made in § 3.3 that there is no homogeneous solution proportional to
$\cos 2{\it\phi}$
in the inner expression (3.14a
) of the quadrupolar contribution to the radial velocity.
