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

The longitudinal (axial) and transverse (azimuthal) distributions of vorticity are defined respectively from $v^{(0)}$ and $w^{(0)}$ by the following expressions

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

This vortex model is often used because it corresponds to profiles of fully viscous self-similar solutions; viscosity is then expected to modify the core size and the jet strength but not the profiles. In expressions (3.4a,b ), the velocity is made dimensionless using the vortex core size, $a$ , 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

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

This system possesses a solution of the form where the streamfunction amplitude $\hat{{\it\psi}}^{(1)}(r)$ satisfies

with the operator $\mathscr{L}^{(k)}$ for $k=1,2,\ldots$ defined as

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

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

with

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

The quadrupolar field satisfying these equations is found to be of the form

with The streamfunction amplitude $\hat{{\it\psi}}^{(2)}$ satisfies

with

The general solution to (3.16) which is finite at the origin, can be written as

where ${\it\Psi}^{(2)}$ denotes the homogeneous function such that

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

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

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 $W_{0_{0}}$ .

4.1. 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

Since ${\it\varepsilon}=1/R$ for the ring, the matching of their quadrupolar field with the inner quadrupolar part (4.2) requires that

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

Using the expression for the local radial velocity (see appendix A), one obtains ( $L={\it\alpha}=0$ )

Finally, the dipolar and quadrupolar terms are for $r\ll R$

The quadrupolar velocity contribution from distant rings becomes ( ${\it\varepsilon}=1/R$ )

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)

with

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

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

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

where ${\it\chi}={\it\phi}-z/L$ and $\text{Re}[\,]$ (respectively $\text{Im}[\,]$ ) indicates the real (respectively imaginary) part of a complex expression, and

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 $S$ :

with

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

with $R_{m}({\it\alpha})$ given by (4.17) and

We obtain for multiple helices $S^{MHe}=S^{He}+S^{DHe}$ , that is

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

For large ${\it\alpha}$ , 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

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.

5.1. 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:

Since $V_{frame}$ 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.