# Numerical realization of helical vortices: application to vortex instability

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## Abstract

The need to numerically represent a free vortex system arises frequently in fundamental and applied research. Many possible techniques for realizing this vortex system exist but most tend to prioritize accuracy either inside or outside of the vortex core, which therefore makes them unsuitable for a stability analysis considering the entire flow field. In this article, a simple method is presented that is shown to yield an accurate representation of the flow inside and outside of the vortex core. The method is readily implemented in any incompressible Navier–Stokes solver using primitive variables and Cartesian coordinates. It can potentially be used to model a wide range of vortices but is here applied to the case of two helices, which is of renewed interest due to its relevance for wind turbines and helicopters. Three-dimensional stability analysis is performed in both a rotating and a translating frame of reference, which yield eigenvalue spectra that feature both mutual inductance and elliptic instabilities. Comparison of these spectra with available theoretical predictions is used to validate the proposed baseflow model, and new insights into the elliptic instability of curved Batchelor vortices are presented. Furthermore, it is shown that the instabilities in the rotating and the translating reference frames have the same structure and growth rate, but different frequency. A relation between these frequencies is provided.

## Keywords

Helical vortices Vortex dynamics Mutual inductance instability Elliptic instability## 1 Introduction

The motion of an infinite helical vortex is one of the classical problems in applied mathematics and fluid mechanics that was perhaps first time addressed by Levy and Forsdyke [1]. Such a vortex moves through the fluid without changing its shape [2] and is an idealization of a common flow structure. Related to this problem is that of determining the corresponding velocity field that a helical vortex induces in a certain domain \({\mathscr {D}}\). This is of prime interest during numerical simulations and is a prerequisite for stability calculations.

*s*, \(\varvec{t}\) is the tangent vector to the line, \(\varGamma \) is the vortex strength and \({\varvec{x}}'\) denotes a point on \({\mathscr {C}}\) [3] (\(\Vert \cdot \Vert \) refers to the Euclidean norm). The analysis of a flow field induced by Eq. (1) is obstructed by the presence of the singularity in the integrand when evaluated for a point on the vortex line. One way of circumventing this issue is to introduce a cut-off length \(\delta \) and exclude a small region \(\Vert {\varvec{x}}-{\varvec{x}}'\Vert <\delta \) from the integration [4]. Another approach that was suggested by Rosenhead [5] and elaborated upon by Moore [6] is to desingularize the integral by modifying the denominator in (1) to \((\Vert {\varvec{x}}-{\varvec{x}}'\Vert ^2 + \mu ^2)^{3/2}\). The parameters \(\delta \) or \(\mu \) can be chosen such that the induced velocity at a point on the vortex matches that of a vortex ring for which an analytical expression is known [7]. Although these methods will give an accurate representation of the induced velocity far away from the vortex, they are

*ad hoc*and will provide an ill-defined flow field in the vicinity of the vortex core, which might be the most important region from a stability point of view. To overcome this issue, two approximate solutions valid inside and outside the core, respectively, of a perturbed straight vortex were derived by [8] using matched asymptotic expansions. Later, Hardin [9] presented analytical expressions for the velocity field in the interior and exterior of a helix, which were furthered by Okulov [10]. These expressions however provide no details of the flow inside the vortex core. Elaborate expressions for this region have instead been presented by Callegari and Ting [11] and by Blanco-Rodríguez et al. [12]. Among the more recent contributions, we also mention the wake model proposed in [13], where the velocity field is evaluated through an asymptotic expansion of the Biot–Savart law after the vortex line has been determined iteratively.

The main motivation of this work is the desire to accurately represent a system consisting of *M* helical vortices numerically in order to perform a global stability analysis of this flow. A method of generating helical vortices that have gained much popularity in the wind turbine community is the actuator line (ACL) method [14]. Despite the success of the ACL method, it requires measured blade data to be available, which limits its application and makes it difficult to implement for an arbitrary helix. It also generates a root vortex, which is an inevitable feature in most real-world flows, but can be undesirable in a theoretical study. Another issue is the requirement of a Gaussian kernel to distribute the forcing from the actuator lines onto the grid points of the mesh, which may greatly influence the results and thus must be judiciously tuned [15].

A highly accurate baseflow is required when performing a numerical stability analysis. For the case of an open flow with a prescribed (helical) vortex, this implies that a detailed flow representation is required both inside the vortex core and in the far-field. Somewhat surprisingly, the above literature review has led us to conclude that a general procedure for generating such a field is missing. In this article, we therefore propose a step-by-step algorithm to generate a baseflow that is simple enough to potentially be implemented in a wide range of solvers, yet yields sufficiently accurate results to enable a meaningful stability analysis to be carried out. Our method is in certain aspects reminiscent of the technique recently discussed in [16], although more general. Selçuk et al. [16] considers a specific formulation of the Navier–Stokes equations that imposes helical symmetry and thereby reduces the three-dimensional flow problem to a two-dimensional one. Our method is, however, derived in the context of primitive variables in a Cartesian coordinate system. As a result, any vortex geometry and core structure is in principle possible in our framework. For the sake of brevity we will however limit the discussion to the case of two interlaced helical vortices. In spite of a substantial interest for this geometry within applications (e.g. rotor wakes *etc.*), there are still numerous open questions related to the transition of this flow, such as the competition between different types of instabilities and the appearance of these instabilities in different frames of reference, which motivate its further analysis. It is also a flow for which theoretical predictions of the eigenvalues and eigenvectors are available, which provides a direct means of validating the modeling approach. A recent experimental study by Quaranta et al. [17] has provided detailed measurements for the configuration at hand. This ensures that the flow case investigated is physically meaningful.

## 2 Baseflow generation

The fluid motion may in general be described as a combination of an isotropic expansion, a volume preserving straining motion, and a rigid-body rotation [3]. In what follows, we will consider a flow field \({\varvec{U}}\) that is assumed to be solenoidal (i.e. have zero rate of volume expansion) and consist of a set of *M* helical vortices that are embedded in a specified background flow. To determine \({\varvec{U}}\), the flow will be decomposed into an irrotational part \({\varvec{U}}_\mathrm{s}\) and a rotational part \({\varvec{U}}_\mathrm{v}\) such that \({\varvec{U}}={\varvec{U}}_\mathrm{s}+{\varvec{U}}_\mathrm{v}\).

### 2.1 Vortex-induced flow component

*M*-multiple of helical vortices must be determined. The center line of the

*j*th individual vortex is a curve in \({\mathbb {R}}^3\) parameterized by

*s*and given by

*R*and

*L*is the radius and the reduced pitch of the helices, respectively. The reduced pitch is related to the helical pitch, i.e. the streamwise separation

*h*between two consecutive helix loops (of the same helix) as \(L=h/(2\pi )\), and the angle \(\phi \) is related to

*s*by

*s*is provided by the Frenet–Serret formulae [18]

*r*is a local radial coordinate (see Fig. 1). (The tilde-sign is used to denote variable quantities in the local coordinate systems.)

*j*th vortex, the appropriate \({{\mathscr {P}}}\)-plane containing this point must be determined. This is done by finding the shortest distance from \({\varvec{x}}\) to \({{\mathscr {C}}}_j\), which amounts to solving the nonlinear problem

*s*and \(\phi (s)\), the global Cartesian coordinates are mapped to the local Frenet–Serret coordinate system \(\varvec{\xi }(s,j) = (\xi (s,j),\eta (s,j),\beta (s,j))^\mathrm{T}\) by,

*a*is the radius of the vortex core. The corresponding velocity profiles \(\tilde{\varvec{U}}^{ r}=({\tilde{U}}^{r},{\tilde{U}}^\theta ,{\tilde{U}}^\xi )^\mathrm{T}\) are given as

### 2.2 Background flow component

*z*-direction and a slipping along the filament [2]. Hence, two equilibrium frames exist in which the vortices will be steady (or quasi-steady), namely a rotating and a translating frame of reference.

Baseflow parameters used in the computations

Parameter | Notation | Value |
---|---|---|

Helix radius |
| 1.0 |

Circulation | \(\varGamma \) | \(-\) 1.0 |

Number of helices |
| 2 |

Core size |
| 0.031 |

Effective core size | \(a_\mathrm{e}\) | 0.050 |

Axial center line velocity | \(U_\mathrm{c}\) | \(-\) 1.9 |

Helix pitch |
| 0.82 |

Reynolds number | \({\text {Re}}\) | 6900 |

### 2.3 Validation and implementation

The described approach has been implemented in the high-order spectral element solver Nek5000 [24], which does not enforce helical symmetry but solves the Navier–Stokes equations in a Cartesian coordinate system using primitive variables in a \({\mathbb {P}}_N-{\mathbb {P}}_{N-2}\) formulation [25]. Homogeneous Dirichlet boundary conditions are imposed on the outer rim and the flow is periodic in the *z*-direction (only one period of the helix is generated). A cross section of the computational mesh is shown in Fig. 2a. Equation (5) is here preconditioned with an additive Schwarz preconditioner [26] and solved with GMRES [27]. To solve (11), a standard bisection method with interval bracketing is used [28].

The numerical values chosen are listed in Table 1 and correspond to the experimental configuration recently investigated by Quaranta et al. [17]. The parameters are made dimensionless with the circulation measured in the experiment \(\varGamma ^* = 68.8\;\text {cm}^2/\text {s}\) and the measured terminal helix radius \(R^*=8.7\;\text {cm}\). The Reynolds number is defined as \({\text {Re}}=\varGamma ^*/\nu ^*\) where \(\nu ^*\) is the kinematic viscosity. In addition, the experiments had a free-stream velocity of \(U_\infty ^*=37\;\text {cm}/\text {s}\), and the tip-speed ratio for this case was \(\lambda =5.4\). The resulting two helices are visualized in Fig. 2b.

### 2.4 Vortex adaptation

The evolution of a counter- and a co-rotating vortex-pair under the influence of viscosity has been studied numerically by Sipp et al. [32] and by Le Dizès and Verga [33], respectively. Recently, this process has also been investigated for the case of a Gaussian helical vortex with different core sizes and reduced pitches [16]. As shown in these studies, the evolution of the vortex until merging may be divided into two different stages, namely an initial non-viscous adaptation followed by a viscous diffusion of the vortex. During the initial stage, every vortex undergoes a rapid adaptation to the strain field produced by the curvature and the neighboring vortex filaments. This is a non-viscous process characterized by strong oscillations. Once the oscillations have decayed, the vortex settles into a slowly varying mean state where it evolves on a viscous timescale. During this stage the vortex cores expand slowly until the merging stage is reached and neighboring vortex loops rapidly approach each other [34].

*a*about the centroid are obtained from

*a*. The former is estimated by interpolating the velocities in \({\mathscr {S}}\) to a polar grid centered at \(({\bar{\eta }},{\bar{\beta }})\) and averaging in the azimuthal direction, whereas the latter is estimated from (23c). As seen in Fig. 4c, the expansion of the vortex core closely follows the viscous diffusion law of two-dimensional vortices given by

## 3 Stability analysis

In this section, the stability of the generated baseflow will be studied. The stability of helical filaments has previously (among others) been considered analytically by Widnall [36] for a single helix, and by Gupta and Loewy [37] and Okulov [10] for a helix multiple.

In similarity to (21a), the force term \(\varvec{f}\) on the right-hand side of (26a) corresponds to a volume force that must be included into the governing equation when considering the rotating frame of reference. However, in contrast to the baseflow calculation, there is no centrifugal force present in the perturbation formulation and in this case \(\varvec{f} = \varvec{f}_{\text {cor}}= 2\Omega _{\text {fr}}(u^y, -u^x, 0)^\mathrm{T}\) (and correspondingly \(\varvec{f}^\dagger = 2\Omega _{\text {fr}}(-u^{\dagger y}, u^{\dagger x}, 0)^\mathrm{T}\)).

We emphasize that the baseflow \({\varvec{U}}\) does not correspond to a stationary solution of (21) as is commonly assumed. However, as was noted in the previous section, it is indeed an approximate solution to the steady Euler equations and can be considered to evolve on a larger timescale than the instabilities. We therefore extract the solution after the relaxation process in Sect. 2.4 is completed (i.e. at \(t=0.4\)) and treat it as being frozen in time. Such a procedure has also been employed in previous studies e.g. [42, 43, 44, 45]. It should be noted that since no stabilization or restriction (aside from streamwise periodicity) is imposed on the flow, the baseflow may contain a low amplitude component of possibly unstable eigenmodes at the end of the adaptation stage. However, given that the relaxation stage is short and the level of the numerical noise is low, the deformation of the baseflow due to any such modes will be negligible.

*k*is the azimuthal wavenumber in the \(\phi \)-direction annotated in Fig. 6. The subscripts \(\hbox {r}\) and \(\hbox {t}\) are here used to denote the rotating and the translating reference frames, respectively. See “Appendix B” for a derivation of (28).

The flow is unstable to both long- and short-wavelength instabilities, which correspond to deformations of the vortex filaments and of the vortex cores, respectively. As seen, the short-wavelength instabilities are weaker than the long-wavelength instabilities, which have up to twice as large growth rate. The differences in frequencies between the two types of instabilities are several orders of magnitude. They are further discussed in the following sections.

To ensure that all the relevant flow structures are resolved, the eigenvalue computations are performed using three different polynomial orders, \(N=\{11,14,17\}\). By comparing the different spectra, \(N=14\) is found to be sufficient (see the second quadrant of the mesh in Fig. 2a for a visualization of the mesh density). In addition to the resolution study, the eigenvalue computations are repeated using different temporal separation between the Krylov vectors [41] in order to certify that no aliasing is present in the spectra [39].

### 3.1 Mutual inductance instabilities

The obtained long-wavelength instabilities correspond to the mutual inductance instability [36], which arise from the self- and mutually induced velocities of the perturbed vortex filaments. By developing a single helical vortex in the \((R\phi )z\)-plane and locally approximating the filaments as a single row of co-rotating infinitely long slanted vortices, an analogy between the vortex pairing due to the mutual inductance instability and an array of point vortices was established by Quaranta et al. [20]. This analogy was furthered in [17], where the growth rates for a two-helix configuration computed using the Biot–Savart law (1) following [37] were successfully compared against the growth rates of three-dimensional perturbations in a single row of infinitely long vortices derived by Robinson and Saffman [46].

Following [17], we here compare the numerical growth rates of the mutual inductance instability with the predictions by Robinson and Saffman [46]. (Details on adapting the theoretical results to the helix geometry are omitted here and the reader is referred to [17].) To enable the comparison, the numerical growth rates in Fig. 6 have been scaled with \(2h^2/\Gamma \), whereby the maximum growth rate becomes close to \(\pi /2\) (i.e. the maximum growth rate for point vortices). As seen in Fig. 7, there is a close agreement between the numerical and the theoretical data sets. Note, however, that the present numerical setup only enables integer azimuthal wavenumbers in the \(\phi \)-direction to be recovered due to the periodic boundary conditions in the streamwise direction. If the length of the domain would be extended such that more periods of the helix are taken into account, the eigenvalue spectrum would become richer.

As discussed in [20], the largest growth rate in the case of an array of point vortices is obtained for perturbations that are out-of-phase on neighboring vortices. Considering three-dimensional disturbances on two helical vortices, Fig. 7 shows that successive peaks in the growth rate curve correspond to perturbations that are either in- or out-of-phase (i.e. have phase difference equal to 0 or \(\pi \) radians, respectively). A sample of eigenvectors corresponding to the three most unstable modes is visualized in Fig. 8. The most unstable mode is shown in Fig. 8a. It corresponds to the eigenvalue \(\omega =4.742i\) with wavenumber \(k=0\). Since it has a zero frequency, the eigenvalue is not visible in Fig. 6 (due to the logarithmic scaling of the horizontal axis). Moreover, it is invariant with respect to the frame of reference. This mode corresponds to a relative displacement of the two helices relative to each other, whereby one helix contracts and the other expands. This induces a uniform pairing of the helices. The effect of the core size, the Reynolds number and the reduced pitch on the growth rate of this mode has recently been investigated in [44]. In Fig. 8b, c the eigenvectors corresponding to \(k=1\) and \(k=2\) are visualized. These modes induce a local pairing of the neighboring vortex loops at 2*k* locations in the azimuthal direction, and correspond to perturbations that are in- and out-of-phase on the two helices, respectively.

### 3.2 Elliptic instabilities

*l*, so called

*principal modes*[49]. The label

*l*represents the number of zero-crossing in the radial direction of \({\check{{\varvec{u}}}}\) [50]. For straight Lamb–Oseen vortices, the most unstable mode corresponds to a combination of Kelvin modes with azimuthal wavenumbers \(m=1\) and \(m=-1\). However, in the case of Batchelor vortices, this configuration has been shown to stabilize with increased axial flow, and be replaced by other unstable resonance configurations [42, 43]. Recently, the elliptic instability of curved Batchelor vortices has been investigated by Blanco-Rodríguez and Le Dizès [51]. These authors computed the growth rates for certain combinations of Kelvin modes using the following equation

*S*. Blanco-Rodríguez et al. [12] gave an expression for this parameter in the case of

*M*helical vortices as

*corrigendum*to some of these relations appended to [47]). The Reynolds number used in (31) is defined as \({\text {Re}}'={\text {Re}}/(2\pi )\).

Since the elliptically deformed vortex cores are incorporated into the baseflow, the unstable resonance conditions discussed above will appear as unstable eigenmodes in the present stability calculations. A close-up of the numerically determined eigenvalues associated with the elliptic instability is shown in the inset of Fig. 6. In Fig. 10a, the eigenvector corresponding to the most unstable elliptic instability with the global wavenumber \(k=45\) is visualized. This mode assumes the shape of a pair of secondary helices that winds around the vortex filament.

*c.f.*figure 3c of [51] as well as figure 15a and 16 of [42]). All other modes that appear in the spectrum of Fig. 6 and correspond to the elliptic instability are of the same type. In Fig. 12, the growth rates of these eigenmodes are plotted together with the largest root of (31) for different axial wavenumbers \(\left. a\right| _{t=0.4}\alpha \) (where \(\alpha =k/\gamma \)). This figure shows that the flow is unstable around the wavenumber for ideal resonance [52] \(\alpha _\mathrm{c}\approx 1.49\) in the interval \(1.286 \le \left. a\right| _{t=0.4}\alpha \le 1.788\). The agreement between the numerics and the theory is overall good, although the theory slightly underestimates the growth rates of the instabilities. This small discrepancy might be associated with the approximate expressions used to evaluate the coefficients in (31), possible changes in the helix radius

*R*(which is assumed constant during the relaxation time), the estimation of the vortex core size

*a*, or the estimation of the axial center line velocity \(U_\mathrm{c}\). The theoretical results are noted to be sensitive with respect to these parameters. For instance, a minor increase in the estimated core radius about 2% yields a nearly perfect match.

*vice versa*when \(\chi =\pi \). Correspondingly, eigenvectors with \(k=\{42, 44, 45, 48\}\) that satisfy property (ii) will enhance each other on both helices when \(\chi =\pm \pi /2\). Such constructive and destructive interference between the modes suggests that the breakdown to turbulence due to the elliptic instability is likely to appear different on the two vortices.

## 4 Discussion

An accurate baseflow is the starting point for every stability analysis. In most cases, the basic state will be completely determined by the geometry of the problem and the boundary conditions, and can in many situations be obtained by merely integrating the governing equations in time. Simple as it might seem, this task turns out not to be entirely trivial for a flow that features an infinitely long vortex multiple. In this situation, the obvious boundary conditions would be to impose a constant streamwise velocity or a solid body rotation in the far-field, and have periodic boundary conditions in the streamwise direction. This implies that the vortex necessarily must be positioned in the flow through the initial condition. The question thus becomes how to determine an initial field that will enable an accurate representation of the flow inside the vortex core, and still capture the external strain field due to the surrounding vortex filaments. In this respect, all the techniques that are known to us appear to fall short in one way or the other, and therefore disqualify for a study of the instability mechanisms in the flow.

In this article, a straightforward step-by-step algorithm to generate a suitable baseflow is proposed. The method has been implemented in a general purpose Navier–Stokes solver and is discussed in the context of a pair of helical vortices whose characteristics recently have been determined in an experiment [17]. The resulting flow field is seen to satisfy all the specified properties, and a stability analysis is observed to capture both long-wavelength mutual inductance instabilities and short-wavelength elliptic instabilities. Both the translating and the rotating frame of reference have been discussed, and the stability properties determined in these different frames been compared.

Although both the long- and the short-wavelength instabilities have been analyzed before, they are for the first time obtained through a single computation, which is remarkable given that the difference in frequency is several orders of magnitude. To the best of the authors’ knowledge, the elliptic instability is for the first time captured and analyzed numerically in a helical vortex multiple. Subsequently, the first validation of the theoretical growth rate expressions derived by Blanco-Rodríguez and Le Dizès [51] is presented. Good agreement between the numerical and the theoretical results is reported. The geometric multiplicity of the corresponding eigenvalues suggests that this instability will develop differently in the different vortices of the helix multiple. Regarding the mutual inductance instabilities, good agreement is also observed between the numerical eigenvalues and the theoretical predictions by Robinson and Saffman [46] adapted to the helix geometry [17]. The fact that both of these instabilities are captured simultaneously, enables their relative importance to be studied. It is shown that the mutual inductance instability for the present configuration has twice as large growth rate as the elliptic instability, and hence can be expected to dominate the transition process. This finding agrees well with the experimental observations by Quaranta et al. [17], where the mutual inductance instability always was observed for the parameters listed in Table 1. In addition, the receptivity of the two instability types have been studied and adjoint eigenvectors been reported. It is shown that the mutual inductance instability is most sensitive to disturbances that enhance vortex pairing by altering the distance between neighboring filaments. By comparing the spatial support of the adjoint eigenvectors for the elliptic instability with the distribution of the individual components of the strain rate tensor for the baseflow, it is found that the \((-\,2,0,1)\)-mode obtained for the present flow configuration is most sensitive to perturbations outside of the vortex core that affects the straining and shearing of the vortex in a plane orthogonal to the vortex center line.

Taken together, the presented way of imposing a vortex inside the numerical domain appears rather appealing and capable of increasing our understanding about instabilities in vortex systems. Since only periodicity in the streamwise direction is assumed, the present approach can readily be used to investigate many other configurations of higher complexity. Examples involve curved helices that are inscribed in a torus, which may be used to investigate the instabilities and breakdown of horizontal axis wind turbine wakes during yawed inflow conditions [53]. Another interesting configuration that might be considered is that of twin helices arising in rotor applications where vane tips are implemented [54]. A direct means of studying the elliptic instability in a geometry with curvature and torsion is also provided. For studies where an inflow/outflow arrangement is more suitable than a periodic configuration, the derived flow field may be used to extract an accurate inflow condition. This would in turn enable the optimal forcing of the flow to be computed, as well as transient analysis involving impulse response and optimal perturbation to be carried out.

## Notes

### Acknowledgements

Open access funding provided by KTH Royal Institute of Technology. We wish to thank Dr. Thomas Leweke for valuable discussions, and Dr. Elektra Kleusberg for help with the mesh generation. We also thank the anonymous referees for their helpful comments. Simulations have been performed at the PDC Center for High Performance Computing in Sweden with computer time granted by the Swedish National Infrastructure for Computing (SNIC). The work has been financed by VR—The Swedish Research Council (Vetenskapsrådet).

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