5.1 Flow parameters

This section features a comprehensive physical study of the system of a temporally developing pair of helical vortices. The dynamics of the various flow regimes is studied considering a range of flow parameters. When the helical pitch $h$ and circulation $\unicode[STIX]{x1D6E4}$ are chosen as reference parameters, three non-dimensional quantities govern the dynamics of the present configuration. Namely, the ratio of pitch to helix radius $h/R$ , the ratio of vortex radius to pitch $r_{c}/h$ and the Reynolds number $Re_{\unicode[STIX]{x1D6E4}}$ . A complete parametric study is performed to assess the influence of each quantity on the growth rate of vortex-core displacements driven by mutual inductance. The Reynolds number values span from $Re_{\unicode[STIX]{x1D6E4}}=7000$ to $Re_{\unicode[STIX]{x1D6E4}}=70\,000$ , the ratio helical radius to helical pitch from $h/R=0.36$ to $h/R=0.61$ and the vortex-core radius from $r_{c}/h=0.166$ to $r_{c}/h=0.333$ . To give an idea about the size of the vortex-core radius compared to the helix radius, which is often employed in the literature, the present flow conditions span values of $r_{c}/R$ between 0.06 and 0.20. Table 2 summarizes the parameter space for the present study, yielding 18 computations for the combination of all possible values. A thorough evaluation of the resolution quality for each Reynolds number has been performed including computations considering various mesh sizes, and is found in appendix A. It has been determined that resolutions of $288^{3}$ , $384^{3}$ and $480^{3}$ grids points are adapted to the Reynolds numbers $Re_{\unicode[STIX]{x1D6E4}}=7000$ , 21 000 and $70\,000$ , respectively.

Figure 5.

Flow regimes described from the evolution of volume-averaged enstrophy. Case $r_{c}/h=0.166$ , $Re_{\unicode[STIX]{x1D6E4}}=21\,000$ and $h/R=0.36$ .

Table 2.

Parameter space and corresponding symbols for the computations of the double helical vortex configuration. These conditions are reproduced for two different vortex-core radius ratios: $r_{c}/h=0.166$ and $r_{c}/h=0.333$ .

5.2 Regimes of flow evolution

The flow can be decomposed into three different regimes, represented on the enstrophy evolution in figure 5, as a function of the non-dimensional time $t^{\star }=t\unicode[STIX]{x1D6E4}/h^{2}$ . From the beginning of the computation up to $t^{\star }\approx 5$ (regime (A)), the flow dynamics is essentially inviscid, and the helical structures undergo relative displacements of their vortex cores with respect to the initial radius, induced by the mutual interaction between vortex filaments. During this first period, the volume-averaged enstrophy remains approximately constant in the flow as the vortices do not experience significant modification of their properties (vortex-core shapes and circulation). From $t^{\star }=5$ and onwards (regime (B)), we can notice a strong increase of enstrophy levels in the flow. The relative displacement of the vortex cores becomes significant enough such that the vortex lines become closer to each other and start interacting strongly, ultimately leading to the reconnection of vortex filaments at multiple locations on the helices. These vortex reconnection events generate small scales in the flow and modify significantly the shape of the initial helical vortices. The peak of enstrophy is reached when all the vortex reconnection events have occurred and all scales have developed in the flow. The time corresponding to the peak of enstrophy may vary depending on the flow parameters (Reynolds, helical pitch, initial vortex-core radius). After the peak of enstrophy is reached, the fully developed turbulent regime (C) is established.

A number of considerations can be made by inspecting the evolution of the volume-averaged kinetic energy (figure 6). For all cases, the energy weakly decays up to $t^{\star }\approx 10$ , corresponding to the period during which the flow evolution is mostly inviscid. Lower Reynolds number cases have the viscous diffusion impact at the large scales from the beginning of the computations, yielding stronger energy decrease. Another interesting observation lies in the fact that even if the non-dimensional time scale $t^{\star }$ related to parameters $h$ and $\unicode[STIX]{x1D6E4}$ is fit to collapse the flow evolution for all cases up to $t^{\star }\approx 10$ , it does not collapse the turbulent decay rates for $t^{\star }>10$ . Given that the non-dimensional time $t^{\star }$ is scaled with the helical pitch $h$ , the dimensional time at $t^{\star }=30$ is higher for higher values of $h$ . It is therefore likely that by $t^{\star }=30$ , the wake is at a more advanced stage of development for higher helical pitches.

Figure 6.

Evolution of kinetic energy normalized by its initial value as a function of dimensionless time $t^{\star }$ , for vortex-core radius $r_{c}/h=0.166$ (a) and $r_{c}/h=0.333$ (b). Different colours correspond to different Reynolds numbers (see table 2).

5.3 Inviscid regime and vortex-core displacement growth rates

Figure 7.

Vortex-core displacement rates as a function of non-dimensional time $t^{\star }$ , for vortex-core radii $r_{c}/h=0.166$ (a), $r_{c}/h=0.333$ (b). Dashed line: exponential growth rate $\unicode[STIX]{x1D6FC}^{\star }=20.2$ .

In this section, the flow dynamics during the initial inviscid regime is studied. In particular, we are interested in quantifying the growth rates of vortex-core displacements with respect to their rigid translation due to self-induced axial advection. The displacements are caused by mutual inductance due to the proximity of vortex filaments.

The growth rates are studied for a range of flow parameters including helical pitch, Reynolds number and vortex-core radius. The vortex-core displacement growth rates are expressed as a function of the non-dimensional time $t^{\star }=t\unicode[STIX]{x1D6E4}/h^{2}$ . This non-dimensional time is obtained using the helical pitch as the reference length and the reference velocity $h/\unicode[STIX]{x1D6E4}$ , which corresponds to the self-motion of the helical vortex induced by the proximity of vortex filaments. Figure 7 shows the evolution of vortex-core displacement as a function of the time $t^{\star }$ . The growth rates collapse well and the transitional regime following an exponential temporal evolution of the vortex-core displacement ends near $t^{\star }=4$ . This result confirms that the helical pitch and therefore the mutual inductance between successive turn of the helices govern the growth rate of vortex-core displacements. This collapse of the vortex-core evolution confirms that the growth rate made non-dimensional using the helical pitch and circulation is a constant value when the mutual inductance mode is predominant. Here, we choose the same definition of non-dimensional growth rate as Nemes et al. (Reference Nemes, Lo Jacono, Blackburn and Sheridan2015):

(5.1) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}^{\star }=2\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FC}(2h)^{2}/\unicode[STIX]{x1D6E4}.\end{eqnarray}$$

In their experiment, they found a value of $\unicode[STIX]{x1D6FC}^{\star }$ close to 20 for various helical pitches (with experimental uncertainty between 5 and 12 %). In the present calculations, the growth rate obtained by fitting the data between $t^{\star }=2$ and $t^{\star }=4$ is $\unicode[STIX]{x1D6FC}^{\star }=20.2$ , which is close to the value of 20 found by Nemes et al. (Reference Nemes, Lo Jacono, Blackburn and Sheridan2015). Gupta & Loewy (Reference Gupta and Loewy1974) also reported linear stability calculations of the growth rates for two helical pitches $2h/R=0.63$ and $2h/R=0.95$ (in the context of a system of two helical vortices) and reported a maximum value of the growth rate of $\unicode[STIX]{x1D6FC}^{\star }=20$ for both, with respect to various types of perturbations. We can also compare the growth rate to those obtained for a single helical filament, for which $h$ is defined by the distance separating two successive turns of the single helix. The linear stability computations of Widnall (Reference Widnall1972) report maximal growth rates $\unicode[STIX]{x1D6FC}^{\star }\approx 20;21;23;20$ for helical pitches $h/R=0.63;1.26;1.9;6.3$ respectively. An experimental study of the stability of a single vortex filament by Quaranta et al. (Reference Quaranta, Bolnot and Leweke2015) reports a maximal growth rate of $\unicode[STIX]{x1D6FC}^{\star }\approx 20$ for the helical pitch $h/R=0.5$ . These results emphasize that the growth rate $\unicode[STIX]{x1D6FC}$ made non-dimensional using the helical pitch (or in the case of a double filament, the shortest distance between two successive helices), shows similar values for a range of theoretical and experimental studies. The present results confirm this observation.

Figure 7 also reveals that variations in the Reynolds number do not impact the transient regime. It is also seen, after $t^{\star }=4$ , that the Reynolds number as well as the helical pitch have an influence on the growth rates. Namely, low Reynolds numbers and low helical pitches impair the growth rates of vortex-core displacement, while high Reynolds and high helical pitches yield a prolongation of the exponential growth past $t^{\star }=4$ . These effects can be attributed to viscous interactions that can occur earlier when lower Reynolds numbers are considered, interrupting prematurely the inviscid evolution of the vortices. An increase in the vortex-core size is also found to reduce the growth of the vortex-core displacement amplitude in time, while still showing an exponential evolution. This result is in line with the observation of Gupta & Loewy (Reference Gupta and Loewy1974) who determined analytically that an increase in the vortex-core radius led to a reduction of the growth rates for two helical vortices. Widnall (Reference Widnall1972) determined the same behaviour in the context of the stability of a single helical vortex filament.

Figure 8.

$Q$ -criterion iso-surfaces coloured by pressure plotted at the beginning of transient regime starting from $t^{\star }=4$ , for the helical pitch $h/R=0.36$ . The lowest core radius $r_{c}/h=0.166$ case is considered.

5.4 Transient break-up and study of flow topology

5.4.1 Dynamics for low helical pitch

The flow evolution is studied by inspecting the history of $Q$ -criterion iso-surfaces at various times for all Reynolds numbers considered, comparing low and high helical pitches. Figure 8 presents these vorticity plots for the lowest helical pitch and smallest core radius. Between $t^{\star }=0$ and 4, the two helical vortices interact, yielding an acceleration and radius reduction of one helix, and deceleration and radius increase for the other helix. This phenomenon leads to a leap frogging of the two helices, as seen from the plot at $t^{\star }=6$ . The proximity of parallel vortices travelling at different velocities due to the leap frogging triggers sinusoidal oscillations of the vortex filaments as seen at $t^{\star }=8$ and $t^{\star }=10$ . These short-wave oscillations have been experimentally observed by Felli et al. (Reference Felli, Camussi and Di Felice2011) in the context of the flow past marine propellers. The nature of these oscillations depends greatly on the Reynolds number. The highest Reynolds case displays small amplitude/small wavelengths oscillations, whereas at lower Reynolds numbers, the amplitude of the oscillations and their wavelengths increase. This enhanced amplitude of oscillations for the lower Reynolds cases facilitates the reconnection between vortex filaments. The two helices merge where the oscillation amplitude is the highest, forming an X-shaped structure seen at $t^{\star }=14$ for the $Re_{\unicode[STIX]{x1D6E4}}=7000$ case and for the $Re_{\unicode[STIX]{x1D6E4}}=21\,000$ case. For the $Re_{\unicode[STIX]{x1D6E4}}=70\,000$ case, the merging of these vortices occurs at $t^{\star }=18$ , as seen from figure 8. This merging of vortex filaments is the source of the generation of small scales in the flow as well as the decomposition of the main helical structures. This phenomenon is further studied by focusing on one merging event (figure 11). The event involves two parallel vortex filaments advected in the axial direction, one faster than the other. The two display sinusoidal displacements of their core location. The two vortices start pairing at the location of maximal displacement amplitude. This pairing is initiated by the appearance of vortex hooks that are perpendicular to the initial vortices. These secondary vortices interact with the main ones, and their reduced circulation/strength make them wind around both main vortices, a process similar to the one described by Zabusky & Melander (Reference Zabusky and Melander1989) in the interaction of strong and weak orthogonal vortices. This creates an entanglement of the two initial vortices at the location of maximal oscillation amplitude. Structures of different scales are observed at this location. The initial low-speed vortex keeps its vertical shape, while the high-speed one is strongly distorted and is shifted horizontally away from the reconnection zone. This phenomenon shares strong similarities with the Crow instability (Crow Reference Crow1970), which occurs when parallel counter-rotating vortices are subject to large oscillations. The presently observed phenomenon is also consistent with the computations performed by Laporte & Corjon (Reference Laporte and Corjon2000), who have super-imposed short-wave oscillations on two infinite, parallel vortex lines in the study of the elliptical Crow instability. The result is an entanglement of vortices at the locations where the distance between vortices is minimal.

A similar reconnection process occurs also for the perpendicular vortex filaments that can be observed at the top and bottom extremities of the helices at $t^{\star }=8$ and 10. This reconnection event also generates small-scale structures, visible for the higher Reynolds number cases. However, the time of reconnection of these perpendicular vortex filaments is independent of the Reynolds number. Small-scale generation related to this particular event can be observed for all three Reynolds cases at $t^{\star }=10$ .

These two events yield a significant alteration of the initial system of helical vortices. The transition towards fully developed turbulence is accelerated for lower Reynolds number cases as the first reconnection event promotes the onset of small scales and yields a decomposition of the coherent vortices. At $t^{\star }=26$ , for the $Re_{\unicode[STIX]{x1D6E4}}=7000$ and $Re_{\unicode[STIX]{x1D6E4}}=21\,000$ cases, the initial shape of helical vortices is essentially lost, whereas it is still present for the $Re_{\unicode[STIX]{x1D6E4}}=70\,000$ case.

Figure 9.

$Q$ -criterion iso-surfaces coloured by pressure plotted at the beginning of transient regime starting from $t^{\star }=4$ , for the helical pitch $h/R=0.61$ . The lowest core radius $r_{c}/h=0.166$ case is considered.

Figure 10.

Evolution of enstrophy normalized by its peak value as a function of time $t^{\star }$ , for vortex-core radius $r_{c}/h=0.166$ (a) and $r_{c}/h=0.333$ (b).

Figure 11.

Description of the parallel reconnection event occurring for the low pitch case $h/R=0.36$ , small core radius $r_{c}/h=0.166$ and $Re_{\unicode[STIX]{x1D6E4}}=7000$ . $Q$ -criterion iso-surfaces plotted in the region of interest at different times during the reconnection.

The influence of the reconnection of parallel vortices can also be observed by plotting the temporal evolution of volume-averaged enstrophy for all cases considered, see figure 10. In particular, for the lowest helical pitch case $h/R=0.36$ , the enstrophy peak’s location is clearly a function of the Reynolds number, as it occurs later for the high-Reynolds-number case $Re_{\unicode[STIX]{x1D6E4}}=70\,000$ . This early occurrence of enstrophy peak is related to the small scales generated by the parallel vortices reconnection. This difference is not visible for the larger core radius $r_{c}/h=0.333$ cases, which suggests that the short-wave oscillations responsible for vortex pairing are not present when larger core sizes are considered.

5.4.2 Dynamics for high helical pitch

Figure 9 shows the $Q$ -criterion iso-surfaces at times $t^{\star }$ between 4 and 26, here for the highest helical pitch case $h/R=0.61$ considered in the study. As seen from the plots, the flow shares similarities with the lower helical pitch but also shows some differences. First, the leap-frogging phenomenon occurs as well, yielding the reconnection of vortex filaments, but only in the location where the vortex filaments are perpendicular to each other. A close-up of the $Q$ -criterion iso-surfaces detailing accurately this vortex reconnection event is displayed in figure 12. The proximity of orthogonal vortex filaments of equivalent strength generates vortex hooks or fingers (as described by Boratav, Pelz & Zabusky (Reference Boratav, Pelz and Zabusky1992) and Jaque & Fuentes (Reference Jaque and Fuentes2017)), which are secondary vortices that wrap around the orthogonal vortices. The two initial vortices become aligned and undergo a typical anti-parallel separation reconnection mechanism as described by Kida & Takaoka (Reference Kida and Takaoka1994). Bridges are formed at the extremity of the region for which the vortices are parallel, corresponding to two new vortex structures. In between these newly formed structures, a number of elongated threads appear which the vortex cores are smaller than the initial parallel vortices. The short-wave oscillations seen for the lower pitch case $h/R=0.36$ are not observed for the high pitch case $h/R=0.61$ . This yields the small-scale turbulent structures to be mainly concentrated at the locations where the perpendicular vortex reconnection occurs. What is remarkable is that the Reynolds number does not significantly influence the flow topology at large scales. The main features of the flow remain the same for the three Reynolds numbers considered, except for the small-scale features that develop around the main vortices, but without perturbing extensively their shapes. This similarity of flow patterns is also seen from the enstrophy evolution, see figure 10, for which the peak occurs at the same time ( $t^{\star }\approx 17$ ) for the three Reynolds numbers considered, with the helical pitch $h/R=0.61$ . An interesting feature of the flow is the presence of secondary vortices, elongated in the axial direction and wrapped around the helical vortices. The number and size of these secondary vortices is highly dependent of the Reynolds number, as seen from figure 9, especially at times $t^{\star }=8$ and $t^{\star }=10$ .

Figure 12.

Description of the orthogonal reconnection event occurring for the case $h/R=0.61$ , $r_{c}/h=0.166$ and $Re_{\unicode[STIX]{x1D6E4}}=7000$ . $Q$ -criterion iso-surfaces plotted in the region of interest at different times during the reconnection.

5.4.3 Energy spectra in the fully developed regime

Energy spectra are evaluated at the end of computations ( $t^{\star }=30$ ) for various pitch values and Reynolds numbers, with the value of vortex-core radius set to $r_{c}/h=0.166$ . Given the strong flow anisotropy, the plane-averaged one-dimensional energy spectra are considered, with the following definitions:

(5.2) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}E_{u}(k_{x})={\displaystyle \frac{1}{L_{y}L_{z}}}\displaystyle \int _{0}^{L_{z}}\displaystyle \int _{0}^{L_{y}}|\widehat{u}(k_{x},y,z)|^{2}\,\text{d}y\,\text{d}z\\ E_{r}(k_{r})={\displaystyle \frac{1}{2\unicode[STIX]{x03C0}L_{x}}}\displaystyle \int _{0}^{2\unicode[STIX]{x03C0}}\displaystyle \int _{0}^{L_{x}}|\widehat{u}_{r}(k_{r},\unicode[STIX]{x1D703},z)|^{2}\,\text{d}\unicode[STIX]{x1D703}\,\text{d}z.\end{array}\right\}\end{eqnarray}$$

The modes $\widehat{u}$ and $\widehat{u}_{r}$ are computed via a discrete Fourier transform (DFT) in the axial $x$ and radial $r$ directions, respectively. The radial direction is not periodic nor statistically homogeneous, however, the velocity component $u_{r}$ reaches negligible magnitudes values when approaching the helix axis, $r\rightarrow 0$ , or the edges of the computational domain, $r\gg R$ ; $\hat{u} _{r}$ hence effectively benefits from a compact support in $r$ , which permits the use of the DFT, or any other variant (e.g. direct cosine transform of an appropriate type accounting for the non-periodic boundaries), yielding the same result.

Figure 13 presents the compensated energy spectra plots, obtained using the total volume-averaged dissipation in the flow defined in appendix A. The $Re_{\unicode[STIX]{x1D6E4}}=21\,000$ and $Re_{\unicode[STIX]{x1D6E4}}=70\,000$ cases display a constant energy plateau for a range of wavenumbers in both directions, which suggests the presence of an inertial range. The lowest Reynolds $Re_{\unicode[STIX]{x1D6E4}}=7000$ case shows a strong intermediate and high wavenumber energy decay, emphasizing that the flow dynamics is mainly concentrated in the large scales. We can notice a collapse of spectra for the highest helical pitch and Reynolds numbers $Re_{\unicode[STIX]{x1D6E4}}=21\,000$ and $Re_{\unicode[STIX]{x1D6E4}}=70\,000$ . In this case, an increase in the Reynolds number generates additional smaller scales while retaining mostly the same low wavenumber content. This behaviour is different from the smaller helical pitches $h/R=0.36$ and $h/R=0.45$ cases, which display significant differences in terms of energy spectra when considering different Reynolds numbers. This is consistent with the different flow dynamics observed in figure 8, i.e. different short-wave distortion of the helical filaments which modifies the topology of the flow depending on the Reynolds number. In particular, the $Re_{\unicode[STIX]{x1D6E4}}=70\,000$ case presents a higher large-scale energy compared to the $Re_{\unicode[STIX]{x1D6E4}}=21\,000$ in both directions, mainly because the parallel filaments’ reconnection occurs later and preserves the large scales for a longer time.

Figure 13.

Energy spectra computed at $t^{\star }=30$ for $r_{c}/h=0.166$ . (a) Axial energy spectra; (b) radial energy spectra. $\unicode[STIX]{x1D702}$ is the Kolmogorov length scale defined in appendix A.

Figure 14.

Evolution of the components of the anisotropic stress tensor as a function of time $t^{\star }$ , for vortex-core radius $r_{c}/h=0.166$ (a) and $r_{c}/h=0.333$ (b). Volume-averaged quantities.

6.1 Anisotropy of velocity fluctuations

In this section, the anisotropy of turbulent stresses is assessed as a function of time, for all cases considered. First, the anisotropy stress tensor is computed as follows:

(6.1) $$\begin{eqnarray}\unicode[STIX]{x1D623}_{ij}=\frac{\langle u_{i}^{\prime }\,u_{j}^{\prime }\rangle }{\langle u_{r}^{\prime 2}\rangle +\langle u_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}}^{\prime 2}\rangle +\langle u_{xx}^{\prime 2}\rangle }-\frac{\unicode[STIX]{x1D6FF}_{ij}}{3},\end{eqnarray}$$

where $\langle u_{i}^{\prime }u_{j}^{\prime }\rangle =\langle u_{i}u_{j}\rangle -\langle u_{i}\rangle \langle u_{j}\rangle$ are the velocity fluctuations in the axial (index 1), radial (index 2) and azimuthal (index 3) directions, and $\langle \cdot \rangle$ is a volume-averaging operator. The adoption of such an averaging operator over the whole computational domain does not imply that the flow is assumed to be homogeneous in all directions; quantities such as (6.1) are intended as flow diagnostics, rather than equivalent ensemble-averaged quantities (the statistical inhomogeneity of the turbulent fluctuations in the radial direction is taken into account in results presented in § 6.3).

All components of the tensor $\unicode[STIX]{x1D623}_{ij}$ are plotted in figure 14 as a function of time, for the span of flow conditions summarized in table 2. First, it is observed that for all conditions considered, all cross-components of $\unicode[STIX]{x1D623}_{ij}$ are zero, the radial and azimuthal components $b_{rr}$ and $b_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}}$ are similar and the turbulence is dominated by the axial component $b_{xx}$ which carries most of the kinetic energy, hence showing a strong and persistent anisotropy. Second, it is seen that the degree of flow anisotropy is sensitive to the Reynolds number, vortex-core radius and helical pitch. Higher vortex-core radii strengthen the anisotropy of turbulence, while increasing the helical pitch or the Reynolds number tend to lower the anisotropy.

A qualitative view of the flow anisotropy can be obtained by plotting the position of the invariants of the anisotropic stress tensor in the Lumley triangle. The two non-zero invariants, for an axisymmetric representation of the Reynolds tensor, read (see Simonsen & Krogstad Reference Simonsen and Krogstad2005):

(6.2) $$\begin{eqnarray}\displaystyle & \text{II}=-{\textstyle \frac{1}{2}}(b_{rr}^{2}+b_{xx}^{2}+b_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}}^{2}+2b_{rx}^{2}) & \displaystyle\end{eqnarray}$$

(6.3) $$\begin{eqnarray}\displaystyle & \text{III}=(b_{rr}b_{xx}-2b_{rx}^{2})b_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}}. & \displaystyle\end{eqnarray}$$

The invariants for all cases are plotted at the end of computations ( $t^{\star }=30$ ) in the Lumley triangle and presented in figure 15. The cases with $r_{c}/h=0.333$ yield a quasi one-component turbulence, characterized by a dominating axial component and vanishing radial and azimuthal components. In this case, the influence of the Reynolds number on the anisotropy is minimal. Increasing the helical pitch reduces slightly the anisotropy. For the lower vortex-core radius $r_{c}/h=0.166$ , an increase in the Reynolds number or helical pitch yields invariant positions moving along the delimiting line towards an isotropic turbulence state at $\text{II}=\text{III}=0$ . This line characterizes an axisymmetric turbulence with a dominating axial component and identical radial and azimuthal components. In particular, a high helical pitch coupled to high Reynolds yields a flow state closer to isotropy. In general, in terms of the geometrical representation of the stress tensor proposed by Simonsen & Krogstad (Reference Simonsen and Krogstad2005), the shape of stress tensor evolves from a line in the thick core radius and low Reynolds case, to an elongated prolate spheroid when the Reynolds is increased and core radius decreased. Finally, the elongation of the prolate spheroid is reduced as the Reynolds keeps increasing and the helical pitch is high, tending towards a spherical shape characterizing isotropic turbulence.

Figure 15.

Anisotropic Reynolds stress tensor invariants plotted in Lumley triangle at $t^{\star }=30$ for vortex-core radii $r_{c}/h=0.166$ (a) and $r_{c}/h=0.333$ (b).

6.2 Scale-by-scale anisotropy study

This section is devoted to the study of the relevant scales that develop during the turbulent regime, and their relation with the flow parameters. The integral scales of turbulence are defined as the integrals of the two-point correlation functions (see e.g. Pope Reference Pope2000):

(6.4) $$\begin{eqnarray}L_{ij}=\int _{0}^{x_{0}}\frac{\langle u_{i}(\boldsymbol{x})u_{i}(\boldsymbol{x}+x_{j}\text{}\underline{e}_{j})\rangle _{kl}}{\langle u_{i}(\boldsymbol{x})^{2}\rangle _{kl}}\,\text{d}x_{j},\end{eqnarray}$$

where $k\neq i$ , $l\neq i$ and $k\neq l$ , such that the correlations are averaged in the two directions different than $i$ . $\text{}\underline{e}_{j}$ is the unit vector in the $j$ th direction and $x_{0}$ is the coordinate corresponding to the correlation function having a zero value. In our case, the index 1 corresponds to the axial direction, while the index 2 corresponds to the radial direction, yielding four different integral scales. $L_{xx}$ and $L_{rr}$ are the axial scales related to the velocity components $u$ and $u_{r}$ , respectively. $L_{xr}$ and $L_{rx}$ are the transverse scales. We first plot the axial scales $L_{xx}$ as a function of $h/R$ in figure 16 to assess the dependency of the length of the streamwise vortices to the helical pitch. The values of $L_{xx}$ are computed at $t^{\star }=30$ which corresponds to the fully developed turbulent regime. Interestingly, the size of the streamwise structures is found to be directly proportional to the helical pitch, as the values $L_{xx}/h$ stay constant for different values of $h$ . This means that the initial value of helical pitch sets the size of streamwise turbulent structures even during the fully developed turbulent regime, and that the influence of the initial condition remains important for a long time after the breakdown of the helical structures. The Reynolds number and initial vortex-core radius are found to have little influence on the integral scale in the axial direction.

Figure 16.

Axial integral scale $L_{xx}$ as a function of helical pitch $h/R$ plotted at $t^{\star }=30$ for vortex radii $r_{c}/h=0.166$ (a) and $r_{c}/h=0.333$ (b).

Figure 17.

Ratio of axial integral scales $\widetilde{L}_{x}/\widetilde{L}_{r}$ as a function of helical pitch $h/R$ plotted at $t^{\star }=30$ for vortex radii $r_{c}/h=0.166$ (a) and $r_{c}/h=0.333$ (b).

The anisotropy at large scales can be assessed by plotting the ratio of axial to radial integral scales, which should be equal to 1 for isotropic turbulence, see Pope (Reference Pope2000). In particular, we choose the definitions introduced by Carter & Coletti (Reference Carter and Coletti2017), $\widetilde{L}_{x}=(L_{xx}L_{xr})^{1/2}$ and $\widetilde{L}_{r}=(L_{rr}L_{rx})^{1/2}$ , that account for the transverse integral length scales. Figure 17 shows the ratio $\widetilde{L}_{x}/\widetilde{L}_{r}$ at $t^{\star }=30$ for the various flow parameters. All cases display large-scale anisotropy, with the size of vortices being larger in the axial than in the radial direction (by a factor 2 to 4). Regarding the smaller core radius $r_{c}/h=0.166$ cases, the ratio is highest for the intermediate helical pitch $h/R=0.45$ , and lower for the helical pitches $h/R=0.36$ and 0.61. This can be explained by two factors. First, the increase in helical pitch yields an increase of the axial integral scale, which explains the high values for the intermediate helical pitch. Second, the low values of integral scales ratio for the highest helical pitch can be explained by the fact that the turbulence is more isotropic for higher helical pitches, as seen from the Lumley triangle plots. This effect counteracts the increase of the axial-to-radial integral scale ratio, by an elongation of the turbulent structures in the radial direction.

Considering the larger vortex-core radius $r_{c}/h=0.333$ , a quasi-linear increase of the ratio as a function of the helical pitch is observed. From the observation that $L_{xx}$ scales with $h$ , this behaviour can be ascribed to radial scales the size of which is independent from the helical pitch. This fact is consistent with the observation from the Lumley triangle that the anisotropy is stronger for the larger vortex-core radius. The larger vortex core somewhat inhibits turbulent structure elongation in the radial direction, yielding increased large-scale anisotropy for increasing helical pitch.

Overall, it is quite remarkable that the initial parameters defining the helical vortices have a persisting influence on the dynamics of the fully developed wake. In particular, the axial integral scales are directly proportional to the helical pitch, and the initial large vortex-core radius strengthens the anisotropy of the flow, preventing the turbulent structures becoming isotropic by restricting their radial extent.

The small-scale anisotropy is also investigated. As stated by George & Hussein (Reference George and Hussein1991), the average of the squared velocity gradients depend considerably on the dynamics of the smallest scales of the flow. Therefore, they proposed to study the small-scale anisotropy by computing the following ratios:

(6.5) $$\begin{eqnarray}\displaystyle & \displaystyle K_{1}=2\left.\left\langle \left(\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x}\right)^{2}\right\rangle \right/\left\langle \left(\frac{\unicode[STIX]{x2202}u_{r}}{\unicode[STIX]{x2202}x}\right)^{2}\right\rangle & \displaystyle\end{eqnarray}$$

(6.6) $$\begin{eqnarray}\displaystyle & \displaystyle K_{2}=2\left.\left\langle \left(\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x}\right)^{2}\right\rangle \right/\left\langle \left(\frac{\unicode[STIX]{x2202}u_{\unicode[STIX]{x1D703}}}{\unicode[STIX]{x2202}x}\right)^{2}\right\rangle & \displaystyle\end{eqnarray}$$

(6.7) $$\begin{eqnarray}\displaystyle & \displaystyle K_{3}=2\left.\left\langle \left(\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x}\right)^{2}\right\rangle \right/\left\langle \left(\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}r}\right)^{2}\right\rangle & \displaystyle\end{eqnarray}$$

(6.8) $$\begin{eqnarray}\displaystyle & \displaystyle K_{4}=2\left.\left\langle \left(\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x}\right)^{2}\right\rangle \right/\left\langle \left(\frac{1}{r}\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}\right)^{2}\right\rangle , & \displaystyle\end{eqnarray}$$

where $\langle \cdot \rangle$ denotes the volume-averaging operator. All these ratios should be equal to 1 for isotropic flows, and deviation from unity indicates a level of small-scale anisotropy. Moreover, these ratios are useful to verify the axisymmetric isotropy of small-scale structures (in the radial and azimuthal directions), satisfied if the equalities $K_{1}=K_{2}$ and $K_{3}=K_{4}$ hold.

The axisymmetry of small-scale structures is further checked by computing additional velocity gradient ratios as proposed by Ganapathisubramani, Lakshminarasimhan & Clemens (Reference Ganapathisubramani, Lakshminarasimhan and Clemens2008):

(6.9) $$\begin{eqnarray}\displaystyle & \displaystyle M_{1}=\left.\left\langle \left(\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}r}\right)^{2}\right\rangle \right/\left\langle \left(\frac{1}{r}\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}\right)^{2}\right\rangle & \displaystyle\end{eqnarray}$$

(6.10) $$\begin{eqnarray}\displaystyle & \displaystyle M_{2}=\left.\left\langle \left(\frac{\unicode[STIX]{x2202}u_{r}}{\unicode[STIX]{x2202}x}\right)^{2}\right\rangle \right/\left\langle \left(\frac{\unicode[STIX]{x2202}u_{\unicode[STIX]{x1D703}}}{\unicode[STIX]{x2202}x}\right)^{2}\right\rangle & \displaystyle\end{eqnarray}$$

(6.11) $$\begin{eqnarray}\displaystyle & \displaystyle M_{3}=\left.\left\langle \left(\frac{\unicode[STIX]{x2202}u_{\unicode[STIX]{x1D703}}}{\unicode[STIX]{x2202}r}\right)^{2}\right\rangle \right/\left\langle \left(\frac{1}{r}\frac{\unicode[STIX]{x2202}u_{\unicode[STIX]{x1D703}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}\right)^{2}\right\rangle & \displaystyle\end{eqnarray}$$

(6.12) $$\begin{eqnarray}\displaystyle & \displaystyle M_{4}=\left.\left\langle \left(\frac{1}{r}\frac{\unicode[STIX]{x2202}u_{r}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}\right)^{2}\right\rangle \right/\left\langle \left(\frac{1}{r}\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}\right)^{2}\right\rangle . & \displaystyle\end{eqnarray}$$

Axisymmetric isotropy is achieved when all these ratios are equal to 1. Figure 18 shows the values of ratios $K$ and $M$ for all cases considered, at $t^{\star }=30$ . First, the axisymmetric isotropy seems to be well verified for all Reynolds numbers. Indeed, these cases show values of $M_{1},M_{2},M_{3},M_{4}$ close to 1 and also verify the relations $K_{1}=K_{2}$ and $K_{3}=K_{4}$ . All computations exhibit a ratio $K_{1}$ that tends to a value around 1.5, emphasizing a strong small-scale anisotropy. Similarly to the large-scale anisotropy, the small-scale anisotropy is stronger when the Reynolds number or the helical pitch decreases. The anisotropy identified at large scales seems therefore to propagate through the smallest scales via energy transfers. The fact that the ratios $K_{1}$ and $K_{2}$ are greater than 1 also suggests an elongation of dissipative scales in the axial direction.

Figure 18.

Small-scale anisotropy factors $K$ and $M$ as a function of $h/R$ for an initial vortex-core radius $r_{c}/h=0.166$ .

Figure 19.

Radial velocity profiles plotted at $t^{\star }=30$ , cases $Re_{\unicode[STIX]{x1D6E4}}=21\,000$ and $r_{c}/R=0.166$ . The mean components are normalized by $\unicode[STIX]{x1D6E4}/h$ and the Reynolds stresses by $(\unicode[STIX]{x1D6E4}/h)^{2}$ . Quantities averaged in the axial and azimuthal directions.

6.3 Radial velocity profiles in the wake

In this section, the velocity profiles in the wake are studied using the classical Reynolds decomposition $u_{i}=U_{i}+u_{i}^{\prime }$ , where $U_{i}$ is the (statistical) mean of the $i$ th component of velocity, hence averaged in the axial and azimuthal directions, with the respective operator indicated as $\langle \cdot \rangle _{x\,\unicode[STIX]{x1D703}}$ or $\langle \cdot \rangle _{13}$ ; the corresponding fluctuating quantity is indicated with $u_{i}^{\prime }$ . As a reminder, the directions $(x,r,\unicode[STIX]{x1D703})$ are also herein referred to as $(1,2,3)$ . When the subscript is omitted for brevity, it is assumed that the symbol refers to the axial component, e.g. $U=U_{x}$ . The statistical quantities $U=\langle u\rangle _{13}$ and $\langle u_{i}^{\prime }\,u_{j}^{\prime }\rangle _{13}$ are therefore only a function of the radial direction  $r$ .

Figure 19 shows the mean velocity profiles as well as the Reynolds stresses at various pitches for $Re_{\unicode[STIX]{x1D6E4}}=21\,000$ and $r_{c}/R=0.166$ . The profiles are plotted at the time $t^{\star }=30$ at which the turbulent wake develops. Regarding the mean profiles, the axial component displays maximal values near the helix centreline, then decreases progressively when approaching the helix radius. This profile corresponds to the typical velocity defect observed in the wake of rotating devices. The mean radial and azimuthal velocities display values close to zero, which indicates that no average rotating motion is present in the flow. The axial fluctuations are dominant compared to the radial and azimuthal fluctuations which have the same levels. The peak of velocity fluctuations is located around $r=R$ , indicating that the region of turbulent production and small-scale generation is related to the initial helix radius. Overall, it is interesting to note that the wake development is more advanced when the helical pitch is increased, as seen from the mean centreline axial velocity, $U_{c}=U_{x}(r=0)$ , which is reduced for higher helical pitch, and the velocity fluctuations more spread out along the radial direction. This can be attributed to the time scale issue addressed in § 5.2 from the observation of the evolution of volume-averaged kinetic energy. The time scale $t^{\star }=t\unicode[STIX]{x1D6E4}/h^{2}$ is indeed suited to the collapse of the flow evolution for different helical pitches up to the occurrence of the reconnection events, but is not appropriate afterwards to describe the evolution of the turbulent wake. Hence, various wakes at the same dimensionless time $t^{\star }=30$ are in fact in more advanced stages of development for higher helical pitches.

6.4 Far-wake self-similarity

In this section, the self-similarity of the flow is assessed in the far wake, and the self-similar properties compared to those of known results related to the spatial development of helical vortices past wind turbines. A simulation with the parameters $Re_{\unicode[STIX]{x1D6E4}}=21\,000$ , $h/R=0.36$ and $r_{c}/h=0.166$ is performed up to $t^{\star }=240$ , in order to assess the self-similarity in the turbulent far wake, with a box size extended to $L_{y}=L_{z}=6.5R$ in the helix cross-section to accommodate the larger spreading of the wake. The number of grid points is increased accordingly to match the spatial resolution of the original simulation with a smaller box size. The region of self-similarity can be identified by plotting the evolution of the centreline velocity and displacement thickness which read:

(6.13) $$\begin{eqnarray}\displaystyle & \displaystyle U_{c}=\frac{1}{L_{\text{x}}}\int _{0}^{L_{\text{x}}}u(x,r=0)\,\text{d}x & \displaystyle\end{eqnarray}$$

(6.14) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FF}^{\star }=\frac{1}{U_{c}}\int _{0}^{\infty }U(r)r\,\text{d}r, & \displaystyle\end{eqnarray}$$

where $U(r)$ is the axial velocity component averaged in the axial and azimuthal directions. According to Dufresne & Wosnik (Reference Dufresne and Wosnik2013) or Okulov et al. (Reference Okulov, Naumov, Mikkelsen and Sørensen2015), the self-similar region, for a spatially developing wind turbine wake, is identified when the centreline velocity and displacement thickness verify $U_{c}\propto (x-x_{0})^{-2/3}$ and $\unicode[STIX]{x1D6FF}^{\star }\propto (x-x_{0})^{1/3}$ , respectively. For the present temporally developing wake configuration, the time replaces the longitudinal spatial coordinate such that these laws translate to $U_{c}\propto (t-t_{0})^{-2/3}$ and $\unicode[STIX]{x1D6FF}^{\star }\propto (t-t_{0})^{1/3}$ . The evolutions of $U_{c}$ and $\unicode[STIX]{x1D6FF}^{\star }$ are plotted in figure 20. The centreline velocity and displacement thickness remain constant up to approximately $t^{\star }=50$ , then start to respectively decrease and increase. Starting from $t^{\star }\approx 100$ , the agreement with the $-2/3$ and $1/3$ power laws for $U_{c}$ and $\unicode[STIX]{x1D6FF}^{\star }$ , respectively, is reasonable, indicating the onset of the self-similar region. To further assess the self-similarity in the wake, the axial velocity and turbulent kinetic energy profiles are plotted in figure 21 at various times corresponding to the self-similar region. The profiles are made non-dimensional using either the self-similarity variables $U_{c}$ and $\unicode[STIX]{x1D6FF}^{\star }$ (figure 21 b,d) or the non-self-similar variables $h,R$ and $\unicode[STIX]{x1D6E4}$ (figure 21 a,c). The mean axial velocity collapses well for the different times considered when made non-dimensional using the self-similarity variables, as well as the turbulent kinetic energy. The turbulent kinetic energy profiles show variations near the centre of the helix, which can be attributed to the relatively low statistical sampling depending directly on the axial extent of the box (in a spatially developing wake, this would correspond to the temporal sampling of the statistics). Considering larger boxes in the axial direction would probably yield a better statistical convergence allowing for a more thorough assessment of the wake self-similarity. As this can be a topic for future studies, the present results hint at the collapse of the velocity fluctuations in the self-similar region of the wake. Considering larger boxes in the cross-plane direction could also allow assessment for longer times of the spread of the wake and the extent of the self-similar regime. The results in the present section indicate that temporally developing helical vortices can relate to more applied rotor–wake configurations. An additional comparison against experiments in proposed in appendix C, in which the present methodology is assessed for the representation of large-scale instabilities that occur in the developed wake past rotor blades.

Figure 20.

Temporal evolution of centreline velocity and displacement thickness for the simulation with parameters $Re_{\unicode[STIX]{x1D6E4}}=21\,000$ , $h/R=0.36$ and $r_{c}/h=0.166$ .

Figure 21.

Mean velocity and turbulent kinetic energy radial profiles in the self-similar region of the wake. (a,c) Variables made non-dimensional using $h,R$ and $\unicode[STIX]{x1D6E4}$ . (b,d) Variables made non-dimensional using the self-similar quantities $U_{c}$ and $\unicode[STIX]{x1D6FF}^{\star }$ . The quantity $E=(1/2)\langle u^{\prime 2}+u_{r}^{\prime 2}+u_{\unicode[STIX]{x1D703}}^{\prime 2}\rangle _{13}$ is the turbulent kinetic energy.