3.1 Stage 1 – vortex generation

The vortex sheets shed from the separation lines along the spheroid discharge vorticity to generate vortical structures in the near wake. At the high inclination angle considered here, the separation lines are also inclined with respect to the free stream. The newly generated vortices are therefore inclined almost similarly as the spheroid. Since the main vortex is generated at the right side of the spheroid, it has positive axial vorticity $\langle {\it\omega}_{ax}\rangle D/U_{0}$ as shown in figure 2(a). The axial vorticity is the dominant component of the vorticity vector and retains the same positive sign all along the vortex tube (to be shown in the following subsection).

Figure 2.

Vorticity in the cross-sectional plane perpendicular to the vortex axis at $x/D=+1$ . The vortex centre is at the origin $r/D=0$ and the bold arrow indicates the clockwise rotation associated with the helical motion. (a) Distribution of axial vorticity $\langle {\it\omega}_{ax}\rangle D/U_{0}$ ; (b) distribution of azimuthal vorticity $\langle {\it\omega}_{{\it\theta}}\rangle D/U_{0}$ . Solid and dashed contour lines represent positive and negative values, respectively. The shadowed region in the upper left corner is a rough representation of the bluff body.

One can immediately see from figure 2 that the axial vorticity component is far larger than the magnitude of the azimuthal vorticity component. However, it is interesting to see the azimuthal vorticity distribution in figure 2(b) which reflects a so-called vortex composition phenomenon, i.e. positive and negative azimuthal vorticity interacting and spiralling together. The vortex composition in the present case is complex with multiple alternations between right- and left-handed swirl in the vortex core. Alekseenko et al. (Reference Alekseenko, Kuibin, Okulov and Shtork1999) first described the vortex composition phenomenon in a swirling flow generated by a vortex generator, but with only one alteration, as seen in their figure 17. In the present case of the streamlined body, the negative azimuthal vorticity in figure 2(b) is generated at the front side of the spheroid and the positive azimuthal vorticity is generated at the leeward side of the spheroid. The composition phenomenon observed in figure 2(b) is thus a result of interactions between sheets of azimuthal vorticity of opposite signs originating from two different parts of the spheroidal body. It is noteworthy that vortex composition can be seen all through the vortex generation stage, but disappears in the near wake.

3.2 Stage 2 – alteration of helical symmetry in the vortex core

The newly formed vortex tube develops with roughly the same inclination as the bluff body until $x/D\approx 2$ . However, the vortex tube is unable to align with the spheroid all the way to the upper pole, as seen in figure 1. As soon as the vortex tube leaves the shadow of the spheroid, the tube is bent by the free stream so that the inclination of the vortex axis with respect to the $x$ -direction is reduced to approximately $10^{\circ }$ . The variations of the axial velocity $\langle u_{ax}\rangle$ and the pressure coefficient $\langle C_{p}\rangle$ in the lower part of figure 1 show that the pressure and the axial velocity attain local maximum and minimum values, respectively, at approximately $x/D\approx 2$ where the vortex tube is being bent. This is indeed a reflection of the continuous bending of the vortex tube caused by the free stream. Nevertheless, the vortex is able to retain its coherence after this stage.

Figure 3.

Profiles of axial vorticity (a–c), azimuthal velocity (d–f) and axial velocity (g–i) along a diameter of the vortex cross-section. (a,d,g) Results at $x/D=4$ and 5 before the helical symmetry alteration; (b,e,h) at $x/D=6$ during the alteration; and (c,f,i) at $x/D=7$ and 8 after the alteration.

Major changes in the vortex topology occur in the near wake from $x/D=4$ to $x/D=8$ , as seen in figure 3 where the velocity and vorticity variations across the vortex tube are shown at five different cross-sections. Let us first consider the jet-like profiles of the axial velocity $\langle u_{ax}\rangle$ at $x/D=4$ and 5 shown in figure 3(g). It is noteworthy that the maximum value of $\langle u_{ax}\rangle$ at the vortex centre even exceeds the free-stream velocity $U_{0}$ by some 20 %. Such jet-like profiles are unexpected in the near wake of the inclined spheroid since we are not aware of any earlier observations of jet-like velocity profiles in three-dimensional wakes from which wake-like profiles are normally reported. Jet-like profiles are important, however, because birds and fishes move very effectively leaving such wake profiles behind (Taylor et al. Reference Taylor, Liberzon, Gurka, Holzman, Reesbeck and Diez2013). To the authors’ best knowledge, jet-like profiles in a near wake have only been seen in conjunction with two-dimensional von Kármán streets (Taylor et al. Reference Taylor, Liberzon, Gurka, Holzman, Reesbeck and Diez2013). The jet-like profiles seen in figure 3(g) are unforeseen and probably caused by three-dimensional effects in the near wake.

Figure 4.

Azimuthal vorticity $\langle {\it\omega}_{{\it\theta}}\rangle D/U_{0}$ over three different vortex cross-sections. (a) At $x/D=5$ and before the helical symmetry alteration. (b) At $x/D=6$ and during the alteration. (c) At $x/D=7$ and after the alteration.

From the jet-like velocity profile at $x/D=5$ in figure 3(g) the flow field experiences a sudden change to the wake-like variation at $x/D=7$ in figure 3(i). This change is accompanied by a change of the sign of azimuthal vorticity shown in figure 4. At $x/D=5$ in figure 4(a), the azimuthal vorticity is positive all over the vortex core and the positive $\langle {\it\omega}_{{\it\theta}}\rangle$ corresponds with the jet-like $\langle u_{ax}\rangle$ in figure 3(g). At $x/D=7$ in figure 4(c), the azimuthal vorticity has become distinctly negative within the vortex core, corresponding to the wake-like axial velocity profile in figure 3(i). In between, i.e. at $x/D=6$ , we cannot observe any particular dominating sign of azimuthal vorticity in figure 4(b). Indeed, the axial velocity profile in figure 3(h) shows that $\langle u_{ax}\rangle$ is roughly uniformly distributed over this vortex cross-section. According to Alekseenko et al. (Reference Alekseenko, Kuibin, Okulov and Shtork1999), a helical vortex with positive azimuthal vorticity is a right-handed vortex while one with negative azimuthal vorticity is a left-handed vortex. The plots of figures 3 and 4 suggest a change in helical symmetry of the vortex core from right to left (Okulov Reference Okulov1996; Martemianov & Okulov Reference Martemianov and Okulov2004).

The switching from a right-handed to a left-handed helical vortex is usually accompanied by a change of axial velocity profile, just as we observed in the present study. Okulov (Reference Okulov1996) and Martemianov & Okulov (Reference Martemianov and Okulov2004) associated such a change of the vorticity distribution together with the formation of a flow reversal in the vortex centre as a ‘classical’ vortex breakdown (Leibovich Reference Leibovich1978; Alekseenko et al. Reference Alekseenko, Kuibin and Okulov2007). However, figure 3(g–i) shows no negative values of the axial velocity; i.e. no recirculation bubble with stagnation points separating flows in opposite directions exists. For low Reynolds number flows at relatively low swirl levels Sarpkaya (Reference Sarpkaya1971) and Faler & Leibovich (Reference Faler and Leibovich1977) observed a similar phenomenon. Nevertheless, the abrupt changes of the present vortex flow represents an original form of abrupt change of helical symmetry in which the switch from a right to left helix at $x/D\approx 6$ is not accompanied by a flow reversal.

Martemianov & Okulov (Reference Martemianov and Okulov2004) derived several different axial velocity profiles (see figure 2 in their paper) that may occur in a continuous vortex tube with operating conditions under the same flow integrals. Abrupt changes of different velocity profiles may appear from the same initial vortex structures with right-handed helical symmetry, e.g. the ‘classical’ vortex breakdown takes place when the original jet-like profile switches to a wake-like variation, together with a negative axial velocity at the vortex axis. However, their theoretical analysis also hypothesized other possible scenarios in which transition between different profiles might occur without a flow reversal. A transition between two wake-like profiles was observed experimentally by Velte, Okulov & Hansen (Reference Velte, Okulov and Hansen2011) and referred to as a ‘slight’ alteration of helical symmetry from one left-handed to another left-handed helical symmetry, accompanied by an expansion of the vortex core but without changes in flow topology. This so-called ‘slight’ alteration is different from the ‘classical’ vortex breakdown with the symmetry changing from right to left. The present vortex tube, however, experiences a sudden change between jet- and wake-like profiles as the right-handed helix transforms to a left helix, while maintaining a positive axial velocity without any flow reversal (i.e. without any change of flow topology as in the ‘slight’ case, but with strong change of helical symmetry like the ‘classical’ one). This phenomenon might be similar to the one visualized by Sarpkaya (Reference Sarpkaya1971) and Faler & Leibovich (Reference Faler and Leibovich1977), although they did not report on any alteration of the helical symmetry. The present vortex alteration associated with the changing of helical symmetry is therefore qualitatively different from both classical and slight alterations in the vortex core. Although the possible existence of this type of symmetry alteration was postulated by theoretical considerations (Martemianov & Okulov Reference Martemianov and Okulov2004; Okulov & Sørensen Reference Okulov and Sørensen2010), it has never been documented and analysed in a real vortex core. Let us therefore proceed to explore this phenomenon in greater detail.

Figure 5.

Variation of the helical pitch $l$ along the axis of the vortex tube. The numerical values obtained with the two different methods are almost indistinguishable.

Okulov (Reference Okulov1996) first pointed out that a vortex breakdown is essentially a change of the helical symmetry in the vortex core, which implies a change of sign of the helical pitch. The pitch $l$ can be used for identification of vortex breakdown in combination with examinations of the velocity and vorticity distributions, as in figures 3 and 4. Alekseenko et al. (Reference Alekseenko, Kuibin, Okulov and Shtork1999) proposed several alternative approaches to estimate the vortex pitch in swirling flow devices. More recently Velte, Hansen & Okulov (Reference Velte, Hansen and Okulov2009) applied two of these methods to estimate the helical pitch for vortex structures in unbounded turbulent flows. The first method obtains $l$ by minimizing the residual of $u_{ax}+(r/l)u_{{\it\theta}}=u_{a}(x)$ , where $u_{a}$ is the vortex advection velocity, in a least-square sense. In the second method $l$ is calculated from $l=-F_{mm}/(F_{m}-u_{a}G)$ where $F_{mm}=\int _{{\it\Sigma}}{\it\rho}u_{{\it\theta}}u_{ax}r\,\text{d}{\it\Sigma}$ , with $F_{m}=\int _{{\it\Sigma}}{\it\rho}u_{ax}^{2}\,\text{d}{\it\Sigma}$ and where $G$ is the mass flow rate. Results obtained with both methods are seen to collapse in figure 5. It is of particular interest to observe that the helical pitch $l$ is positive before $x/D=6$ where the axial velocity distribution is jet-like and the azimuthal vorticity has positive values. On the contrary, $l$ is negative downstream of $x/D=7$ where the axial velocity profile has changed to wake-like and the azimuthal vorticity became negative. Okulov, Sørensen & Voigt (Reference Okulov, Sørensen and Voigt2005) studied the way of change of the helical pitch during a vortex breakdown. They predicted and discovered that, despite the traditional picture in which a vortex ring acquires zero pitch in the limit, the transition may also occur when it acquires infinite pitch $l$ . The variation of $l(x)$ in figure 5 suggests that the scenario outlined by Okulov & Sørensen (Reference Okulov and Sørensen2010) occurs in the present case.

The fact that the helical pitch $l$ exhibits a sudden change from positive to negative sign provides further evidence that the main vortex in this complex three-dimensional wake experiences a helical symmetry alteration process, although the flow topology is unchanged. As already mentioned, an atypical form of a ‘slight’ vortex alteration was reported from the experimental study by Velte et al. (Reference Velte, Okulov and Hansen2011) in which the flow topology remained unchanged. In their case also the wake-like form of the axial velocity profile remained the same although the helical pitch changed abruptly. In the present study, however, another kind of symmetry transformation in the vortex core has been identified. Changes of the vortex helical parameters appear with a transition from jet-like to wake-like axial velocity profile, but without a change in flow topology (no flow reversal at the vortex axis). To the authors’ best knowledge, this is the first time that this vortex alteration scenario is detected and analysed.

3.3 Stage 3 – self-similarity in the far wake

The essential features of the coherent vortex structure are preserved through the far wake, i.e. downstream of $x/D\approx 8$ and all the way to the outlet of the computational domain. This is so for the positive axial vorticity, the negative azimuthal vorticity and the wake-like axial velocity profiles. Moreover, the vortex tube exhibits an almost axisymmetric shape, as demonstrated by the pressure coefficient contours at $x/D=21.5$ and 26 in figure 6. These plots are cross-sections through the main coherent vortex structure in which a significant fraction of the wake energy is concentrated. Although these data are taken far downstream of the part of the wake shown in figure 1, a distinct pressure minimum is observed at the vortex centre.

Figure 6.

Contours of the pressure coefficients $\langle C_{p}\rangle$ in the far wake. Both plots are centred at the local vortex centre. (a) At $x/D=21.5$ ; (b) at $x/D=26$ .

In this subsection, the validity of the self-similarity laws mentioned in the Introduction will be assessed. This is motivated by the hypothesized universality of the laws also for abnormal axisymmetric cases (Okulov et al. Reference Okulov, Naumov, Mikkelsen and Sørensen2015). According to the theory of Johansson et al. (Reference Johansson, George and Gourlay2003) the evolution of the maximum velocity deficit $U_{def}=(U_{0}-u_{ax})_{max}$ along the vortex axis and the transverse length scale ${\it\delta}_{\ast }$ of the axisymmetric vortex tube should follow the self-similarity laws:

where ${\it\theta}$ (the momentum thickness) and ${\it\delta}_{\ast }$ (the transverse length scale) are defined as:

Data for ${\it\delta}_{\ast }/{\it\theta}$ versus $(x-x_{0})/{\it\theta}$ and $U_{def}/U_{0}$ versus $(x-x_{0})/{\it\theta}$ are deduced from the simulated flow field in the range between $x/D=9$ to 26 and plotted as black dots in figures 7(a) and 7(b), respectively. The broken lines are the best-fit curves:

The results in figure 7 follow the self-similarity laws with the particular power-law variation in (3.1) exceptionally well. It is furthermore interesting to note that the best-fit coefficients $a=0.515$ , $b=4.1$ in (3.3) and (3.4) give $ab=1.087$ . This value is only 8.7 % above the constraint $ab=1$ proposed by Johansson & George (Reference Johansson and George2006). They also mentioned that the virtual origin $x_{0}$ should be the same in both (3.1). The present results are different in this respect since $x_{0}$ in (3.4) is almost exactly twice the value of $x_{0}$ in (3.3), i.e.  $5.135/2.512\approx 2.04$ . In the wake behind a single disk, Johansson & George (Reference Johansson and George2006) considered the entire wake. The present study is, on the other hand, focusing only on the main vortex structure. Although this vortex structure is the most dominant feature of the present wake flow, it is nevertheless only a part of the entire wake behind the spheroid. In fact, the vortex tube under consideration here stemmed from half of the vorticity shed from the inclined spheroid. This might explain the difference between our results and those of Johansson & George (Reference Johansson and George2006). Another explanation may be associated with the more complex vortex composition reported in § 3.1. Discussions by Alekseenko et al. (Reference Alekseenko, Kuibin, Okulov and Shtork1999) suggested that vortex composition can either reduce the velocity in the vortex core or change the real vortex size.

Figure 7.

(a) ${\it\delta}_{\ast }/{\it\theta}$ versus $(x-x_{01})/{\it\theta}$ , where $x_{01}=2.512$ ; (b) $U_{def}/U_{0}$ versus $(x-x_{02})/{\it\theta}$ , where $x_{02}=5.135$ . The black dots are calculated values based on the simulation data and dashed lines are best-fit curves obeying the power-law variations (3.1) proposed by George (Reference George, George and Arndt1989).