4.1 Wandering amplitudes

Our wind and water tunnel experiments have provided several independent estimates of the standard deviations $\unicode[STIX]{x1D70E}_{y}$ and $\unicode[STIX]{x1D70E}_{z}$ (‘wandering amplitudes’) of the vortex axis coordinates in the $y$ and $z$ directions. Such wandering amplitudes for the wind tunnel tests were determined from the time series of the components of the vortex axis position vector $\boldsymbol{r}_{v}(t)$ (§ 3.1) and averaged over ensembles of measurements obtained with different probe separations $\unicode[STIX]{x0394}S$ at the same $X$ . These estimates were compared to independent estimates of the same properties obtained with the use of a single-probe signal analysis by Bailey & Tavoularis (Reference Bailey and Tavoularis2008), under the assumption that the vortex axis crossed the probe location when the measured velocity components in the $yz$ plane vanished simultaneously. That study obtained the joint probability density function of vortex position by traversing the measurement probe throughout the $yz$ plane and counting the instances at which the in-plane velocity was zero; its results were within 15 % of estimates obtained using the approach of Devenport et al. (Reference Devenport, Rife, Liapis and Follin1996). Vortex wandering amplitudes for the water tunnel results were estimated by three different methods: (i) from the SPIV measurements on transverse planes; (ii) from the polynomials fitted to the reconstructed vortex axis that was identified in the simultaneous pairs of LIF images on streamwise planes; and (iii) from the LIF images on transverse planes. The wandering amplitudes estimated with the use of these techniques were within 25 % of each other.

The wandering amplitudes determined from the different facilities and approaches are presented in figure 5(a) and it can be observed that the different free-stream conditions produced a fairly wide range of $\unicode[STIX]{x1D70E}_{y}$ and $\unicode[STIX]{x1D70E}_{z}$ at the same streamwise location. To compare and scale these results, it seems worthwhile to explore the approach of van Jaarsveld et al. (Reference van Jaarsveld, Holten, Elsenaar, Trieling and van Heijst2011), which was mentioned in § 1 and is presented in figure 5(b). This scaling method assumes that vortex wandering is analogous to the dispersion of a passive scalar injected from a point source in homogeneous turbulence. Taylor’s theory of passive scalar diffusion uses the Lagrangian time scale of the turbulence as an important parameter; van Jaarsveld et al. substituted this scale by the turnover time of the vortex, approximated as $T_{to}=2\unicode[STIX]{x03C0}r^{\ast }/U^{\ast }$ , as a way to account for the interaction of the vortex and the free-stream turbulence, albeit plainly admitting that ‘there is no sound physical explanation’ for this substitution. Van Jaarsveld et al. thus derived the following expressions for the local normalised wandering amplitudes:

(4.1) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{y}/r^{\ast },\unicode[STIX]{x1D70E}_{z}/r^{\ast }\propto u^{\prime }(T/\unicode[STIX]{x1D6E4}^{\ast })^{1/2},\end{eqnarray}$$

where $u^{\prime }$ is the standard deviation of turbulence fluctuations, $\unicode[STIX]{x1D6E4}^{\ast }=2\unicode[STIX]{x03C0}r^{\ast }|U^{\ast }|$ is the core circulation and $T$ is the residence time of the vortex in the turbulence; for the present configurations, we may estimate these properties as $u^{\prime }\approx (2/3k)^{1/2}$ ( $k$ is the local turbulent kinetic energy per unit mass) and $T\approx (X-3c/4)/U_{0}$ .

Figure 5.

(a) Streamwise evolution of normalised wandering amplitudes for different free-stream turbulence conditions and (b) the same results plotted using scaling proposed by van Jaarsveld et al. (Reference van Jaarsveld, Holten, Elsenaar, Trieling and van Heijst2011). The solid line is (4.1) using $0.07+4.21u^{\prime }(T/\unicode[STIX]{x1D6E4}^{\ast })^{1/2}$ as determined by van Jaarsveld et al. (Reference van Jaarsveld, Holten, Elsenaar, Trieling and van Heijst2011) from a fit of their results; ▵ denotes estimates from the analysis of § 3.1; ♢ denotes estimates by Bailey & Tavoularis (Reference Bailey and Tavoularis2008); ▫ denotes estimates from water tunnel LIF; ▿ are from water tunnel PIV. Uncertainty bars for the § 3.1 analysis were determined from variation between results for different $\unicode[STIX]{x0394}S$ . Uncertainty bars for the water tunnel results are based on per cent difference between the PIV and LIF estimates at overlapping conditions.

Some caution is necessary when applying the van Jaarsveld et al. scaling postulate to the present results, because, like Taylor’s diffusion theory, it applies to strictly homogeneous turbulence and not necessarily to grid turbulence, which decays with streamwise distance. Equation (4.1) leads to the paradoxical predictions that wandering would vanish when turbulence vanishes, even if it occurred at some upstream location, and that if turbulence were introduced at some intermediate location in the vortex path, finite-amplitude wandering would be enforced in a stepwise manner. These paradoxes could be resolved, at least in part, if local wandering were not scaled by local turbulence conditions but by scales that take into account the history of wandering motion. One may also postulate that, when turbulence and free-stream unsteadiness drop below a certain level, vortex wandering would persist as the result of intrinsic instabilities; equation (4.1) cannot be expected to hold under such conditions.

In any case, such problems do not apply to the present conditions, as we found good agreement between the values of $\unicode[STIX]{x1D70E}_{y}$ and $\unicode[STIX]{x1D70E}_{z}$ calculated from the components of $\boldsymbol{r}_{v}(t)$ , the same properties obtained with the use of a single-probe signal analysis by Bailey & Tavoularis (Reference Bailey and Tavoularis2008), the values determined from the water tunnel study, which were at a Reynolds number that was an order of magnitude lower than that in the wind tunnel tests, and the values determined from the measurements of van Jaarsveld et al. (which are summarised by the solid line in figure 5 b). Although the apparent scatter of the wandering amplitudes is slightly larger for the large-grid case, it is within the estimated uncertainty and in remarkably good agreement with the no-grid and small-grid cases. Such agreement provides support for van Jaarsveld et al.’s observation that the vortex turnover time is an important scaling parameter for the wandering amplitude.

Most importantly, the fair agreement between wandering amplitudes measured in two very different facilities, at two distant Reynolds numbers and six different free-stream conditions, and with the use of three different experimental techniques (hot-wire anemometry, LIF flow visualisation and SPIV) and five different data processing algorithms provides confidence in the analysis presented in § 3.1 and suggests that the wandering amplitude depends entirely on the kinetic energy of the turbulence and is insensitive to its integral length and time scales. The insensitivity to the relative size of dominant turbulence eddies (presumed to be proportional to the grid mesh size) can be clearly demonstrated by comparing the present wind tunnel results with the small and large grids and those by van Jaarsveld et al. Normalised by the grid mesh size, the corresponding vortex core sizes for these three cases were $r^{\ast }/M\approx 0.28$ , 0.14 and 0.05, nearly spanning an order of magnitude of values, but these differences do not appear to have any measurable effect on the normalised wandering amplitude.

4.2 Wandering frequency and wavelength

Most previous experimental investigations of vortex wandering induced by free-stream turbulence focused on the amplitude of wandering (Devenport, Zsolodos & Vogel Reference Devenport, Zsolodos and Vogel1997; Heyes et al. Reference Heyes, Jones and Smith2004; Bailey & Tavoularis Reference Bailey and Tavoularis2008; Beresh et al. Reference Beresh, Henfling and Spillers2010; van Jaarsveld et al. Reference van Jaarsveld, Holten, Elsenaar, Trieling and van Heijst2011), under the presumption that such wandering would be stochastic. There seems to be a general agreement that the meandering of the vortex has a relatively long wavelength, but little documentation of the actual wavelength or frequency of wandering. Our present reconstruction of the time series of vortex position in § 3.1, combined with analysis of the LIF imagery, provide us an opportunity to estimate both the wavelength and the frequency of wandering.

To examine the frequency $f$ of wandering, we computed the frequency spectra $\unicode[STIX]{x1D719}_{yy}(f)$ and $\unicode[STIX]{x1D719}_{zz}(f)$ of the $y$ and $z$ components of the $\boldsymbol{r}_{v}(t)$ time series; by definition of these spectra, $\unicode[STIX]{x1D70E}_{y}^{2}=\int _{0}^{\infty }\unicode[STIX]{x1D719}_{yy}\,\text{d}f$ and $\unicode[STIX]{x1D70E}_{z}^{2}=\int _{0}^{\infty }\unicode[STIX]{x1D719}_{zz}\,\text{d}f$ . The two types of spectra were virtually identical, so we will only discuss the $\unicode[STIX]{x1D719}_{yy}(f)$ estimates.

The ensemble-averaged frequency spectra determined for the no-grid, small-grid and large-grid cases are provided in figures 6(a) to 6(c) respectively, scaled by the core radius. The no-grid case is shown in figure 6(a) and is characterised by relatively constant $\unicode[STIX]{x1D719}_{yy}$ to $fr^{\ast }/U_{0}\approx 0.02$ , after which there is a decay to $fr^{\ast }/U_{0}\approx 0.2$ , at which frequency a shoulder appears. At higher frequencies $\unicode[STIX]{x1D719}_{yy}$ decays monotonically. Most of the increase in wandering amplitude with streamwise distance appears to come from the lower frequencies, $fr^{\ast }/U_{0}<0.2$ . Similar behaviour can be observed in the spectra measured from the small- and large-grid cases, although the shoulder is much less pronounced for the cases with free-stream turbulence, and there is a broadband hump in the spectra around $fr^{\ast }/U_{0}\approx 0.1$ to 0.2 which appears in the spectra measured closest to the wing in the small-grid case, and at all streamwise locations of the large-grid case.

Figure 6.

Frequency spectra of $y_{v}$ for (a) no-grid, (b) small-grid and (c) large-grid cases. Symbols indicate measurement location: ▫  $X/c=3.75$ ; ▵  $X/c=5.75$ ; ▿  $X/c=7.75$ ; ♢  $X/c=9.75$ .

Further insight into these spectra can be gained by viewing them in pre-multiplied form. Pre-multiplied, ensemble-averaged spectra scaled by the core radius $f\unicode[STIX]{x1D719}_{yy}(f)/r^{\ast 2}$ for the no-grid, small-grid and large-grid cases are plotted in figure 7 versus the logarithm of the dimensionless frequency. With this choice of axes, the ‘relative energies’ (namely, the portions of $\unicode[STIX]{x1D70E}_{y}^{2}$ ) within different frequency bands can be easily compared, as each is proportional to the area under the curve within that band. The frequency spectra also identify the most probable frequencies of wandering as the frequencies of the spectral peaks. It must be noted, however, that the probability that any specific frequency, including a most probable one, will ever occur is zero and probabilities can only be assigned to finite frequency bands. Each wandering frequency value corresponds to a wavelength of wandering motion of the vortex axis, under the assumption that the vortex is convected past the probe by the free-stream velocity $U_{0}$ ; thus, the axis position wavelength that corresponds to a certain frequency of wandering can be estimated as $\unicode[STIX]{x1D706}\approx U_{0}/f$ .

A general observation that applies to all plots in figure 7 is that wandering energy was spread over approximately three orders of magnitudes of frequency. This clearly illustrates that, in all cases, wandering was largely stochastic and was composed of motions with very different scales. Although all spectra had well-defined peaks at one or two most probable frequencies, the relative energy within a band that was within 50 % of the peak frequency was in all cases but a very small fraction of the total energy. This proves that motions with frequencies at or near the most probable values were rare.

Figure 7.

The same spectra as figure 6 shown in pre-multiplied form for (a) no-grid, (b) small-grid and (c) large-grid cases. Symbols indicate measurement location: ▫  $X/c=3.75$ ; ▵  $X/c=5.75$ ; ▿  $X/c=7.75$ ; ♢  $X/c=9.75$ . Note: vertical axis scales are not the same for panels (a–c).

The frequency spectra at all streamwise locations of the no-grid case, shown in figure 7(a), exhibit two distinct, and relatively broad, bands, each centred about a strong peak. One band corresponds to relatively long wavelengths ( $10r^{\ast }<\unicode[STIX]{x1D706}<1000r^{\ast }$ ) and another corresponds to relatively short wavelengths ( $r^{\ast }<\unicode[STIX]{x1D706}<10r^{\ast }$ ). Near the wing ( $X/c=3.75$ ), motions within each of the two bands accounted for roughly half of $\unicode[STIX]{x1D70E}_{y}^{2}$ , but, as the vortex was convected downstream, the energy of the short-wavelength band decayed, whereas the energy of the long-wavelength band increased; as a result, at the last measuring station ( $X/c=9.75$ ) the long-wavelength band contributed 80 % of the wandering motion. These observations indicate that short-wavelength wandering motions likely received some initial energy from an upstream source, possibly near the wing or even further upstream, but lacked a mechanism capable of maintaining their energy and were dissipated by viscous actions. In contrast, long-wavelength motions were not dissipated with travel along the test section.

One may argue that short-wavelength wandering could be an artefact of the trajectory reconstruction process and a residual of conditions during vortex formation, which were specific to our set-up. Nevertheless, short-wavelength modes have also been observed previously under different conditions. For example, transient growth analysis has predicted an increased amplification of disturbances for streamwise wavenumbers around $k=1.4$ , which corresponds to a wavelength of approximately $5r^{\ast }$ or $fr^{\ast }/U_{0}\approx 0.2$ (Antkowiak & Brancher Reference Antkowiak and Brancher2004). Moreover, our observation that the energy of short-wavelength motions decayed downstream is consistent with the observations of Hussain et al. (Reference Hussain, Pradeep and Stout2011). It seems plausible that short-wavelength wandering was induced by residual turbulence and secondary structures originating at the wing.

The frequency spectra for the small- and large-grid cases (figures 7 b and 7 c, respectively) had values that were, respectively, one and two orders of magnitude larger than corresponding no-grid spectra. The shapes of the small- and large-grid spectra were consistent with each other but quite different from the no-grid spectra. In both cases with grids, and at all streamwise stations, there was a single spectral peak, surrounded by a relatively wide long-wavelength band. Although no peak was visible at relatively high frequencies, the spectra were somewhat elevated in a high-frequency spectral band, which indicates that some energy of short-wavelength motions persisted through grid turbulence. As this energy was relatively small (it varied from $0.2\unicode[STIX]{x1D70E}_{y}^{2}$ near the wing to $0.05\unicode[STIX]{x1D70E}_{y}^{2}$ far downstream), we will not discuss it any further, but will accept that grid turbulence essentially obliterated any short-wavelength motions that were introduced at or upstream of the wing. Turning again to the long-wavelength motions, we can make two observations. First, unlike the no-grid case, the energy of such motions increased monotonically with streamwise distance, with the rate of increase being significantly higher for the large-grid than the small-grid case. Second, as the vortex travelled downstream, the spectral peak kept shifting towards lower frequencies, whereas for the no-grid case the spectral peak was at a constant frequency, which means that the most probable wavelength of wandering tended to increase.

Having established that free-stream turbulence has a strong influence on vortex wandering, it seems worthwhile to explore any possible relationship between the wandering wavelength and the scales of turbulence. The integral length scale of the turbulence, which is a measure of the size of the energy containing eddies, was $L_{x}\approx 5r^{\ast }$ (Bailey & Tavoularis Reference Bailey and Tavoularis2008); this is an order of magnitude smaller than the most probable wavelength of wandering ( ${\approx}50r^{\ast }$ ) and so $L_{x}$ can be excluded as a possible scale of the wandering motion. Another relevant length scale of the turbulence would be the travelling distance $U_{0}L_{x}/u^{\prime }$ of a dominant eddy during its typical lifetime (eddy turnover time) $L_{x}/u^{\prime }$ ; this distance was in the range from $100r^{\ast }$ to $200r^{\ast }$ for the cases shown in figure 7, which is rather too large compared to the most probable wandering wavelength. In conclusion, the wavelength of wandering does not seem to be connected to a length scale of the free-stream turbulence.

Previous authors suggested that bending waves can be induced into the vortex by vortical coherent structures in the free-stream turbulence through a transient growth of perturbations (Pradeep & Hussain Reference Pradeep and Hussain2006; Hussain et al. Reference Hussain, Pradeep and Stout2011), which tend to get wrapped around the vortex and form azimuthally aligned secondary vortices (Melander & Hussain Reference Melander and Hussain1993; Marshall & Beninati Reference Marshall and Beninati2005; Takahashi et al. Reference Takahashi, Ishii and Miyazaki2005). Such secondary vortices were observed in our flow visualisation images (see examples in figure 3); their streamwise separation was between $2r^{\ast }$ and $6r^{\ast }$ , which is consistent with past observations (Marshall & Beninati Reference Marshall and Beninati2005). As noted previously, the scale of these structures is consistent with short-wavelength wandering, but too small to trigger long-wavelength wandering. This conjecture is consistent with the vortex appearance in figure 3(c), which shows that the amplitude of short-wavelength bending was much smaller than that of long-wavelength bending, even though short-wavelength bending in figure 3(c) was more intense than in figure 3(a,b) that correspond to flows with lower turbulence. In conclusion, the action of vortices from the free stream can make at most a small contribution to the overall wandering motion of the main vortex.

One potential source of long-wavelength motions is the transient growth mechanism, which was previously connected to short-wavelength motions. Long-wavelength modes are also excited by this mechanism, with the energy growth increasing with decreasing wavelength (as observed by Fontane et al. Reference Fontane, Brancher and Fabre2008, for example). However, the optimal time for such motions to develop also increases with decreasing wavelength. Such motions would not likely have sufficient time to develop within the time frame examined in the current study, which encompassed fewer than 15 vortex rotation periods. This postulate is supported by the lack of energy amplification at long wavelengths in the observations of Hussain et al. (Reference Hussain, Pradeep and Stout2011), who studied a vortex in the presence of decaying turbulence; a possible explanation of this phenomenon is that the turbulent perturbations decayed faster than the time required for the long-wavelength motions to be excited and attain a detectable amplitude. In summary, considering that the turbulence in our tests was decaying and the measurable vortex was relatively young, it is unlikely that the observed long-wavelength wandering motions would be caused by transient growth.

Figure 8.

Pre-multiplied frequency spectra of $y_{v}$ for (a) no-grid, (b) small-grid and (c) large-grid cases with the frequency scaled by vortex turnover time and the vertical axis normalised by wandering amplitude. Symbols indicate measurement location: ▫  $X/c=3.75$ ; ▵  $X/c=5.75$ ; ▿  $X/c=7.75$ ; ♢  $X/c=9.75$ .

We now turn to examining the possible connection of long-wavelength motions to intrinsic scales of the vortex. As mentioned in § 4.1, an intrinsic time scale of the vortex is its turnover time $T_{to}=2\unicode[STIX]{x03C0}r^{\ast }/U^{\ast }=\unicode[STIX]{x1D6E4}^{\ast }/U^{\ast 2}$ ; in the present wind tunnel experiments, this varied from approximately 0.006 to 0.010 s. When we normalised the frequency by the turnover time and the spectra by $\unicode[STIX]{x1D70E}_{y}^{2}$ (figure 8), we observed that the spectral peaks in the two cases with grids essentially collapsed; in the no-grid case, the locations of the low-frequency peaks also collapsed, but the spectral shapes at different streamwise locations differed significantly because of the contributions of short-wavelength motions. The period of wandering $T_{p}=1/f_{P}$ , corresponding to the frequency $f_{p}$ of the spectral peak, was in all cases $T_{p}\approx 2T_{to}$ ; spectral bands with significant energy around the low-frequency peak corresponded to periods in the range from 1 to $10T_{to}$ .

An estimate of the dimensionless average wavelength of the wandering vortex axis, under the assumption that the vortex is approximately convected downstream by the free-stream velocity $U_{0}$ , can be obtained as $\unicode[STIX]{x1D706}_{p}/r^{\ast }\approx U_{0}T_{p}/r^{\ast }\approx 2U_{0}T_{to}/r^{\ast }=4\unicode[STIX]{x03C0}U_{0}/U^{\ast }$ . The value of $\unicode[STIX]{x1D706}_{p}/r^{\ast }$ varied roughly from 30 to 60, depending on the free-stream conditions and the streamwise location. In the absence of direct measurements of $\unicode[STIX]{x1D706}$ , we cannot test our hypothesis that this scaling is appropriate for the wind tunnel data. Such a test can, however, be performed for the water tunnel data, because in these cases the wavelength $\unicode[STIX]{x1D706}$ of the instantaneous vortex axis was determined directly by inspection of reconstructed three-dimensional flow visualisation images (§ 3.2). The relatively low resolution of our visual approach did not allow us to determine a precise value of $\unicode[STIX]{x1D706}$ for each pair of images; instead, we classified each roughly estimated value in one of five different bins, assigned to represent waveforms with quarter, half, three-quarter, one and one-and-a-quarter full waves. Histograms of the observed wavelengths, normalised by $r^{\ast }$ , for the three free-stream conditions are shown in figure 9(a). In all cases, irrespective of turbulence conditions and distance downstream of the wing, the average detected wavelength was comparable to the length of the images that were considered. Although it is theoretically possible that a consideration of longer images would have resulted in a longer average wavelength, the statistical distribution of the present results does not support this conjecture. Furthermore, all estimates of $\unicode[STIX]{x1D706}$ in the water tunnel studies were in the range $\unicode[STIX]{x1D706}/r^{\ast }\leqslant 50$ , which is within the range of the wind tunnel estimates of this parameter. A comparison of the results shown in figure 9(a) for the three free-stream cases revealed that the ranges of values of $\unicode[STIX]{x1D706}/r^{\ast }$ shifted towards the right of the axis as the turbulent intensity was increased. This indicates that $r^{\ast }$ is not an appropriate scale by which to normalise the measured $\unicode[STIX]{x1D706}$ . When, however, we normalised $\unicode[STIX]{x1D706}$ by $U_{0}T_{to}$ (figure 9 b), we found that the ranges of the three sets of data collapsed (at least within the relatively large uncertainty of these estimates). In summary, we can assert with some confidence that the wandering wavelength is proportional to the vortex turnover time.

Figure 9.

Histograms of wandering wavelengths in the water tunnel studies, determined from analysis of LIF images and scaled by core radius (a) and by vortex turnover time (b), for the no-grid (▫), small-grid (▵) and large-grid (♢) cases.

4.3 Vortex wandering velocity

To gain further insight into long-wavelength wandering, we calculated the part of the Lagrangian velocity of the vortex axis that corresponds to this spectral band. This velocity was estimated from the low-pass filtered time series $\boldsymbol{r}_{vf}$ of vortex axis position as $\text{d}\boldsymbol{r}_{vf}/\text{d}t$ . A fourth-order Butterworth, low-pass filter having 3 dB cutoff setting at $fr^{\ast }/U_{0}=0.05$ was used to reduce effects of short-wavelength wandering and random noise generated by the time series reconstruction process. First, the Cartesian components of the Lagrangian velocity were calculated as $V_{y}=\text{d}y_{vf}/\text{d}t$ and $V_{z}=\text{d}z_{vf}/\text{d}t$ with the use of a second-order central differencing scheme and then the time series of radial and circumferential components of this velocity were determined, respectively, as

(4.2) $$\begin{eqnarray}\displaystyle V_{r}(r,\unicode[STIX]{x1D703},t)=V_{y}(r,\unicode[STIX]{x1D703},t)\cos \unicode[STIX]{x1D703}+V_{z}(r,\unicode[STIX]{x1D703},t)\sin \unicode[STIX]{x1D703} & & \displaystyle\end{eqnarray}$$

and

(4.3) $$\begin{eqnarray}\displaystyle V_{\unicode[STIX]{x1D703}}(r,\unicode[STIX]{x1D703},t)=V_{z}(r,\unicode[STIX]{x1D703},t)\cos \unicode[STIX]{x1D703}-V_{y}(r,\unicode[STIX]{x1D703},t)\sin \unicode[STIX]{x1D703}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D703}=\tan ^{-1}z_{vf}/y_{vf}$ and $r^{2}=z_{vf}^{2}+y_{vf}^{2}$ .

Figure 10.

Ensemble-averaged probability density functions of $V_{r}$ for the (a) no-grid, (b) small-grid and (c) large-grid cases. Also shown are the probability density functions of $V_{\unicode[STIX]{x1D703}}$ for the (d) no-grid, (e) small-grid and (f) large-grid cases. Symbols indicate measurement location: ▫  $X/c=3.75$ ; ▵  $X/c=5.75$ ; ▿  $X/c=7.75$ ; ♢  $X/c=9.75$ .

The ensemble-averaged probability density functions (p.d.f.s) of $V_{r}$ and $V_{\unicode[STIX]{x1D703}}$ for the no-grid, small-grid and large-grid cases are shown in figure 10. A scale for normalising these velocities was devised as $U^{\ast 2}\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D6E4}^{\ast }$ by assuming that, during a vortex turnover time $\unicode[STIX]{x1D6E4}^{\ast }/U^{\ast 2}$ , the vortex would shift transversely by a distance that would be proportional to the average wandering amplitude $\unicode[STIX]{x1D70E}=(\unicode[STIX]{x1D70E}_{y}^{2}+\unicode[STIX]{x1D70E}_{z}^{2})^{1/2}$ . All p.d.f.s were nearly Gaussian, having skewnesses in the range $\pm 0.1$ and flatnesses in the range from 2.9 to 3.4, except close to the wing, where wake effects were more significant. This Gaussian behaviour enforces an interpretation that the wandering was largely stochastic. There is, however, a noticeable difference between the p.d.f.s of the two velocity components. Whereas the mean $V_{r}$ was essentially zero (actually, in the range $\pm 0.001~\text{m}~\text{s}^{-1}$ ) in all cases, the mean $V_{\unicode[STIX]{x1D703}}$ was measurably positive (approximately $0.01~\text{m}~\text{s}^{-1}$ for the no-grid, $0.1~\text{m}~\text{s}^{-1}$ for the small-grid and $0.5~\text{m}~\text{s}^{-1}$ for the large-grid cases). Combined with the streamwise translation of the vortex by approximately the free-stream velocity, the positive circumferential velocity of the vortex axis indicates that, on average, the vortex was in clockwise helical motion along a path around the time-averaged vortex axis when facing in the streamwise direction. The increase of the average $V_{\unicode[STIX]{x1D703}}$ from the no-grid to the small-grid cases and further to the large-grid case also indicates that the angular velocity of this helical motion increased with increasing turbulence intensity.

It is very important to emphasise that the presence of a helical motion was only detectable as an average of very disparate types of wandering motions. This becomes apparent if one considers that the average $V_{\unicode[STIX]{x1D703}}$ was less than 25 % of its standard deviation at $X/c=3.75$ and decreased to less than 15 % at $X/c=9.75$ . The rareness of occurrence of helical motions is further obviated by the fact that the probability of $V_{\unicode[STIX]{x1D703}}$ values in a range within 50 % of the mean was but a very small per cent of the total probability for all three free-stream conditions (figure 10 d–f). In fact, the probability of any clockwise vortex rotation was only slightly higher than that of any anticlockwise rotation.

Consistent with stochastic wandering, figure 10 further shows that the p.d.f.s of the normalised radial velocity for each of the three free-stream conditions essentially collapsed at all streamwise locations; the same observation applies to the circumferential velocity. Based on these observations, it appears that the choice of $U^{\ast 2}\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D6E4}^{\ast }$ as velocity scale was appropriate for scaling the stochastic axis motion of a vortex subjected to specific free-stream turbulence. The rationale for this choice will be discussed in § 5.

It has been well established in the literature that wavelengths with an azimuthal wavenumber of unity ( $|m|=1$ ) are subject to transient amplification (Antkowiak & Brancher Reference Antkowiak and Brancher2004) and can contribute to wandering (Edstrand et al. Reference Edstrand, Davis, Schmid, Taira and Cattafesta2016). One of the more dominant long-wavelength $|m|=1$ modes is Kelvin bending waves, also referred to as displacement waves (Fabre et al. Reference Fabre, Sipp and Jacquin2006), which may introduce a helical motion via Biot–Savart induction (Kelvin Reference Kelvin1880; Widnall Reference Widnall1975; Saffman Reference Saffman1995). This self-induced rotation can be explained qualitatively with the help of the sketch in figure 11. Assume that, as the result of interactions with free-stream turbulence, the vortex axis has a planar, quasi-sinusoidal shape. By the law of Biot–Savart, each segment of the vortex would be subjected to the velocity field induced by any other segment, but it is the closest segments that have the strongest influence. Therefore, segment A, i.e. the upper half of the left side of the wave in figure 11, would be mainly under the influence of the field induced by the neighbouring segment B, as segment C (the lower half of the left side of the wave) is nearly coaxial with segment A and has negligible effect on it. It is easy to see that segment A would tend to move out of the plane of the illustration under the influence of segment B. Similarly, segment B would also tend to move out of the plane under the influence of segment A. Conversely, segment C would tend to move into the plane under the influence of segment D. In summary, the entire vortex would be set in self-induced clockwise rotation in a preferential direction, which combined with streamwise transport by the free stream, would generate the observed clockwise helical motion of the axis.

Figure 11.

Sketch illustrating the self-induced rotation of the vortex under the influence of a bending wave.

We now proceed to devise a scale for the angular velocity of the helical motion of the vortex under the assumption that this motion is self-induced. As reviewed in Alekseenko, Kuibin & Okulov (Reference Alekseenko, Kuibin and Okulov2007), a helical vortex filament with a pitch that is much larger than its amplitude would be set in solid-body rotation with the same angular velocity as a plain sinusoidal vortex filament with the same wavelength (pitch) and amplitude as the helical filament. We approximate the influence of a perturbation on a filament of circulation $\unicode[STIX]{x1D6E4}^{\ast }$ by assuming that it introduces approximately sinusoidal deformation with amplitude $\unicode[STIX]{x1D70E}$ and a wavelength $\unicode[STIX]{x1D706}_{P}\approx U_{0}T_{p}$ . Then, following Alekseenko et al. (Reference Alekseenko, Kuibin and Okulov2007), one may calculate that the total induced velocity at the apices of this filament would be proportional to $\unicode[STIX]{x1D6E4}^{\ast }\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D706}_{P}^{2}$ , which means that the angular velocity of this filament would be proportional to $\unicode[STIX]{x1D6E4}^{\ast }/\unicode[STIX]{x1D706}_{P}^{2}\approx \unicode[STIX]{x1D6E4}^{\ast }/(U_{0}^{2}T_{p}^{2})$ . The latter parameter would then be a plausible scale for the observed helical vortex rotation.

To test the appropriateness of this scale, we plot in figure 12 radial profiles of scaled, conditionally averaged $V_{\unicode[STIX]{x1D703}}$ . These profiles of $V_{\unicode[STIX]{x1D703}}$ were calculated by conditionally averaging $V_{\unicode[STIX]{x1D703}}(r,\unicode[STIX]{x1D703},t)$ by $r$ . To compare the effectiveness of scaling by the approximate angular velocity, the results are plotted normalised by $U_{0}$ in figure 12(a) and $\unicode[STIX]{x1D6E4}^{\ast }/(U_{0}^{2}T_{p}^{2})$ in figure 12(b). Conditional averaging of $V_{\unicode[STIX]{x1D703}}$ was done by classifying each instance of $V_{\unicode[STIX]{x1D703}}(r,\unicode[STIX]{x1D703},t)$ to one among 50 radially spaced bins in the range $0\leqslant r\leqslant 6\unicode[STIX]{x1D70E}$ , and then averaging all values within each bin for all $\unicode[STIX]{x0394}S$ . Because the scatter of these results increased significantly as the sample population decreased, we only considered results at radial bins for which 2000 or more values were averaged; unfortunately, this restricted the radial ranges of the results, particularly for the no-grid case, and for this reason we only present values for the small- and large-grid cases. Calculations similar to those for $V_{\unicode[STIX]{x1D703}}$ were also performed for the radial velocity, but will not be presented, as the average of this velocity was approximately zero at all radial positions. The radial variation of $V_{\unicode[STIX]{x1D703}}$ in figure 12 was roughly linear, which indicates that, in a frame convected with the free-stream velocity, the vortex axis was on average in nearly rigid-body rotation about its average position. Note that the scaling used in figure 12(b) assumes that the wavelength of the bending wave was constant, whereas, as illustrated in figure 7, motions with wavelengths within a broad range were present, which is considered to be the reason for the radial dependence of the induced angular velocity.

In consideration of the facts that the linear variation of the conditionally averaged $V_{\unicode[STIX]{x1D703}}$ extended well beyond $r^{\ast }$ and that $\unicode[STIX]{x1D70E}/r^{\ast }<1.5$ for all current free-stream turbulence conditions, we may assert that the (on average) rigid-body rotation of the axis covered the entire domain of its wandering. It is worthwhile to note that the angular velocity of this rigid-body rotation appeared to be sensitive to the free-stream conditions.

Figure 12.

Circumferential velocity of the vortex axis, conditionally averaged based on radial position relative to the time-averaged vortex axis and normalised by (a) the free-stream velocity and (b) the ratio of circulation to wavelength; ▫ denotes small-grid cases and ♢ denotes large-grid cases.

The values in figure 12(b) for $r>r^{\ast }$ exhibited variations of approximately 10–20 %, which is approximately half the per cent variation of the unscaled $V_{\unicode[STIX]{x1D703}}/r$ . These results are subjected to considerable uncertainty, introduced by stochastic wandering motions, the roughness of assumptions and the small populations used for conditional averaging purposes. Differences in the scaling of the small- and large-grid cases may possibly be attributed to a weak dependence of the displacement wave instability on the properties of the vortex core (Fabre et al. Reference Fabre, Sipp and Jacquin2006). Additional sources of uncertainty are the broadening of the range of perturbation amplitudes with an increase in turbulence intensity, which entails a broadening of the range of $V_{\unicode[STIX]{x1D703}}$ , and deviations of the wandering waveform from the sinusoidal shape, which would affect the distributions of $V_{r}$ and $V_{\unicode[STIX]{x1D703}}$ by self-induced motion. All things considered, however, we may view figure 12 as evidence that the entire vortex was, on average, in self-induced rotation, at least for the two cases with grids. It is equally important to re-iterate that the average $V_{\unicode[STIX]{x1D703}}$ was in all cases very small by comparison to the width of the range of observed instantaneous values of $V_{\unicode[STIX]{x1D703}}$ and so rigid-body rotation of any part of the vortex would be a rare event.

Helical motion should also produce correlation between the $y_{v}$ and $z_{v}$ components of vortex motion. However, we found this correlation to be relatively weak for the cases with free-stream turbulence with $\langle y_{v}z_{v}\rangle /(\unicode[STIX]{x1D70E}_{y}\unicode[STIX]{x1D70E}_{z})<0.1$ for these cases and closer to 0.05 for most of the measurements. In addition, if we assume that the hump observed in figure 6 for the grid cases can be attributed to this helical motion, we found that this hump contributed less than 10 % of the total variance for the two measurement locations nearest the wing for the small-grid case, and could not be observed for the two downstream locations. For the large-grid case, the hump in the spectra accounted for 20 % of the variance closest to the wing, but this contribution decreased to less than 15 % by the furthest downstream location. All these observations prove conclusively that the organised helical motion was only a small component of the overall wandering, which was largely stochastic.

In closing this section, it seems worthwhile to comment on the work of Edstrand et al. (Reference Edstrand, Davis, Schmid, Taira and Cattafesta2016), which also addressed vortex wandering. These authors examined experimentally one location of a vortex in a very low turbulence stream, conducted a linear stability analysis of a Batchelor vortex and concluded that vortex wandering is the result of an instability. In the absence of multiple data from which to attempt constructing scaling laws for the wandering frequency, they normalised it with the wing chord length and the free-stream velocity. A rough estimate of the Strouhal number $fc/U_{0}$ for the present experiments gives a value of about 0.45, which is of the same order as the value of about 0.1 in the work of Edstrand et al. (Reference Edstrand, Davis, Schmid, Taira and Cattafesta2016). Nevertheless, this is hardly evidence for the validity of this scaling postulate, but rather evidence of reproducibility of the organised motion, as the conditions in our wind tunnel experiments were not vastly different from those used by Edstrand et al. Both wings had the same profile and angle of attack, the chord of our wing was only 40 % shorter and our free-stream speed was only 23 % lower. It seems evident to us that this Strouhal number would be sensitive to the strength of the vortex, which would depend, among other factors, on the angle of attack. The opposite would imply that a symmetric airfoil at near-zero angle of attack would produce a vortex with the same wandering frequency as the one observed here! Finally, our observations and scaling of stochastic vortex wandering cannot be tested against the Edstrand et al.’s measurements, which were obtained in the absence of significant free-stream turbulence.