3.1. Initial vortex array characteristics before interaction with cutting wall

The properties of the semi-infinite vortex array were first characterized in the centre plane of the tunnel without the cutting walls using planar MTV. The results of this study have been discussed in detail in Bohl & Koochesfahani (Reference Bohl and Koochesfahani2009). Results salient to the current work are summarized here, as they provide the reference initial conditions that we use for the post-interaction vortex characteristics. Results shown in figure 6 for $k= 5.2$ and 11.5, corresponding to the lowest and highest vortex strengths studies here, indicate the vortices were fully formed and isolated, with a Gaussian vorticity profile $\omega =\omega _{peak}\exp (-r^2/r_c^2)$ at $0.5C$ downstream of the oscillating airfoil trailing edge where the false walls would be placed. The measured azimuthal (swirl) velocity profiles showed equally good agreement with the swirl velocity expected from a Gaussian vorticity field

(3.1)\begin{equation} V_{sw}=\frac{\varGamma}{2{\rm \pi} r}(1-e^{-(r/r_c)^2}). \end{equation}

Selected properties of the vortex array at this location are shown in figure 7 for all the cases considered in the current study. The peak vorticity, $\langle \omega \rangle _{p,i}$, and circulation, $\varGamma _i$, showed a nominally linear dependence on the reduced frequency while the vortex core radius, $r_{c,i}$, was nominally constant at $r_{c,i} \approx 0.4$ cm. The maximum swirl speed, $V_{swm,i}$, which occurs at $r/r_c = 1.12$, was calculated using (3.1) and the measured $\varGamma _i$ and $r_{c,i}$. The data shown in figure 7 will be used as the reference normalization data for the remainder of this work.

Figure 6.

Horizontal profiles of (a) spanwise vorticity and (b) azimuthal velocity taken from a vortex at 0.5$C$ downstream of the oscillating airfoil trailing edge (Bohl & Koochesfahani Reference Bohl and Koochesfahani2009). The term ($x - x_c$) is the distance measured from the centre of the vortex. Lines indicate curve fits to vorticity and azimuthal velocity profiles for a Gaussian distribution of vorticity; see text.

Figure 7.

Selected vortex properties as a function of reduced frequency at 0.5$C$ downstream of the airfoil trailing edge (Bohl & Koochesfahani Reference Bohl and Koochesfahani2009). Maximum swirl velocity was computed from (3.1). Straight lines are least squares fits to data.

In the following sections, the interaction of the vortices with the cutting walls are discussed in detail for the two reduced frequencies, $k= 5.2$ and 11.5, which represent the limits of the vortex strengths considered here. The corresponding values of vortex circulation were found to be $\varGamma _i=21\,{\rm cm}^{2}\,{\rm s}^{-1}$ and $\varGamma _i =55\,{\rm cm}^{2}\,{\rm s}^{-1}$, respectively (see figure 7). These cases are referred to as the ‘low $\varGamma$’ and ‘high $\varGamma$’ cases in the remainder of the text. The corresponding Reynolds number of these vortices based on circulation is $Re=\varGamma _i/\nu =2100$ and 5500, respectively. The Reynolds number per unit length of the undisturbed laminar boundary layer on the cutting plate is approximately 1050.

3.2. Phase-averaged vorticity and spanwise velocity fields after interaction

The normalized phase-averaged vorticity fields, $\omega ^*=\langle \omega \rangle / \langle \omega \rangle _{p,i}$, at multiple span locations are shown in figures 8(a,c,e,g) and 9(a,c,e,g) to detail the downstream and spanwise development of the vorticity field for the low and high circulation cases, respectively. The data for four data planes, $z=r_{c,i}$, $2r_{c,i}$, $10r_{c,i}$, $20r_{c,i}$, will be discussed in terms of the ‘near-wall’ ($z=r_{c,i}$, $2r_{c,i}$) and ‘far-field’ ($z=10r_{c,i}$, $20r_{c,i}$) planes. These data are for a single phase and so can be considered instantaneous views of the flow field at that phase.

Figure 8.

(a,c,e,g) Normalized phase-averaged vorticity, $\omega ^*$, and (b,d, f,h) spanwise velocity, $w^*$ fields at multiple span locations for the low $\varGamma$ case.

Figure 9.

(a,c,e,g) Normalized phase-averaged vorticity, $\omega ^*$, and (b,d, f,h) spanwise velocity, $w^*$ fields at multiple span locations for the high $\varGamma$ case.

The vorticity fields for the near-wall data planes ($z=r_{c,i}$, $2r_{c,i}$) were characterized by rapid distortion and breakup of the initially Gaussian vortex cores due to the viscous interaction with the wall. For the low $\varGamma$ case, figure 8, the vortices skewed and stretched into elongated regions of vorticity at $z=r_{c,i}$ due to the strong near-wall shear. This distortion of the vorticity field resulted in a widening of the vertical distance between the primary positive and negative regions of vorticity. Smaller secondary paired regions of vorticity were formed near the primary vortical structures. Farther away from the wall at $z=2r_{c,i}$ the vortices also experienced significant distortion from their initial Gaussian shape, stretching into nearly curved regions of vorticity with near-zero vorticity or even opposite sign vorticity inside the region (observable starting at $x\approx 22 r_{c,i}$). The development into the nearly closed curved regions of vorticity was not as a result of the decrease in the central peak vorticity of the Gaussian profile, but instead via distortion and re-distribution of the vorticity. While the regions of vorticity were highly stretched at these near-wall measurement planes, the vorticity levels remained significant throughout the entire measurement domain.

The dynamics for the high $\varGamma$ case, figure 9, near the wall ($z=r_{c,i}$, 2$r_{c,i}$) were qualitatively different, however. In this case, the previously noted strongly sheared vorticity regions near the wall at $z=r_{c,i}$ were not observed and the dynamics was instead characterized by the breakup of the primary vortex into multiple smaller compact regions of vorticity that evolved due to vortex interactions, diffused out and/or combined with vorticity of the opposite sign. For example at the $z=2r_{c,i}$ measurement plane, the initial single negative vortex structure observed at $x\approx 5r_{c,i}$ began to form a second distinct region of vorticity by $x\approx 15r_{c,i}$, which orbited and moved away from the original vortex core farther downstream. A third distinct local peak in the vorticity was formed by $x\approx 34r_{c,i}$. Similar dynamics was observed for the positive vortices. The regions of positive and negative vorticity then appeared to merge, resulting in a decrease of measurable regions of vorticity after $x\approx 45r_{c,i}$. The near-wall primary regions of vorticity for the high $\varGamma$ case appeared to change their vertical orientation farther downstream. For example at the $z=r_{c,i}$ measurement plane, the vortex array that was initially oriented with the positive vortices above the negative vortices up to $x\approx 20r_{c,i}$ had changed its orientation to the negative vortices being above the positive after $x\approx 40r_{c,i}$ as the result of the formation of the secondary regions of vorticity and the subsequent vortex interactions. The breakup of the primary vortex into multiple smaller compact vortical regions as the wall was approached is reminiscent of the observations in the direct numerical simulation of a forced plane wall jet by Visbal, Gaitonde & Gogineni (Reference Visbal, Gaitonde and Gogineni1998). There the spanwise shear-layer vortices split into a double-helical structure near the sidewall. The current measurements are unfortunately limited to only two near-wall planes and do not have sufficient spanwise information to unambiguously connect our observation of vortex breakup near the wall to the helical vortex structure described by Visbal et al. (Reference Visbal, Gaitonde and Gogineni1998).

We should mention that the computations of Liu & Marshall (Reference Liu and Marshall2004) and Saunders & Marshall (Reference Saunders and Marshall2015) have shed light on many details of the initial vortex–wall interaction when a single vortex is cut by a blade. As the vortex approaches the blade and the blade tip starts to penetrate the vortex core, the non-uniform induced velocity due to the spanwise vorticity of the approaching vortex creates opposite sign spanwise (i.e. blade normal) vorticity in the blade boundary layer ahead of the vortex. As the vortex moves farther downstream along the blade, the change of direction of the vortex induced velocity behind the vortex leads to the creation of same sign spanwise vorticity in the blade boundary layer behind the vortex. Similarly, positive and negative streamwise vorticity is generated in the blade boundary layer in front and back of the vortex as it passes over the blade surface. As mentioned before, our experiments were not designed to study such an intricate dynamics of vortex interaction with the cutting wall boundary layer vorticity. For example, in the plane closest to the wall shown in figures 8 and 9 (i.e. $z=r_{c,i}$), the data over the first third of the downstream region shown fall outside of the undisturbed boundary layer thickness of the cutting plate. Another complicating factor is the fact that the current experiments do not involve the cutting of a ‘single vortex’ by a plate whose boundary layer is initially undisturbed, but the cutting of a ‘vortex array’ by a plate, where the plate boundary layer interacting with a passing vortex has been previously disturbed by the passage of previous vortices of the array.

The phase-averaged vorticity fields for the far-field planes $z=10r_{c,i}$, $20r_{c,i}$ showed that the vortices remained isolated compact structures with peak levels dropping and the size of the vortices expanding with downstream distance for both low and high circulation cases. Both the diffusion of the vortices and the associated decrease in the magnitude of the vorticity levels were more pronounced for the high $\varGamma$ case. These data were consistent with the results of Bohl & Koochesfahani (Reference Bohl and Koochesfahani2009) which showed that the formation of axial flow increases the rate of diffusion and decreases the vorticity levels.

The normalized spanwise (i.e. axial) flow fields, $w^*= \langle w\rangle /V_{swm,i}$, are shown on figures 8(b,d, f,h) and 9(b,d, f,h). In these figures the red/yellow coloured contours indicate positive spanwise flow away from the wall, while the blue/cyan contours indicate negative spanwise flow toward the wall. The positive axial flow first appeared at $z=r_{c,i}$ and then progressively at the farther data planes for both cases, as expected. The axial flow expanded away from the wall faster for the high $\varGamma$ case due to the higher axial flow speeds, e.g. in the low $\varGamma$ case in figure 8, the first positive axial flow region at $x\approx 5r_{c,i}$ appears only at the $z=r_{c,i}$ plane, whereas in the case of high $\varGamma$ in figure 9 the same appears at both $z=r_{c,i}$ and $2r_{c,i}$ planes. For both the low and high $\varGamma$ cases, the positive spanwise flow was initially found to be spatially coincident with the vortex core of either sign with peak magnitudes of the order of the maximum swirl speed, consistent with previous results of Bohl & Koochesfahani (Reference Bohl and Koochesfahani2004). Interestingly, the negative spanwise flow toward the wall was also clear in the two near-wall planes, where they were always located on the outer (free-stream) edges of the vortices, with magnitudes that were lower (nominally 0.5$V_{swm,i}$) than the primary positive axial flow.

The structure of the spanwise flow fields of both cases were consistent with the vorticity fields previously discussed. Consider first the near-wall measurement planes ($z=r_{c,i}$ and $2r_{c,i}$). For the low $\varGamma$ case, the regions of spanwise flow stretched into oblong structures qualitatively similar in shape, and spatially coincident, to the vorticity structures at the same downstream distance. The axial flow, like the vortical structures, remained isolated and distinct throughout the measurement domain. However, the structure of the spanwise flow for the high $\varGamma$ case changed from isolated regions to a more continuous sheet-like region farther downstream in the regions where primary vortices had broken up into multiple smaller compact regions of vorticity. Distinct maxima in the axial flow were, however, observed and found to be spatially coincident with the locations of the vortical structures. The axial flow field at these planes showed a marked decrease in intensity at the downstream edge of the measurement planes consistent with the disappearance of the vortical structures for these measurement planes.

The spanwise flow fields for $z=10r_{c,i}$ and $20r_{c,i}$ were qualitatively different from the near-wall cases. For the low $\varGamma$ case, axial core flow was observed in isolated regions spatially coincident with the location of the vortex cores. No reverse spanwise flow was observed at these planes for this case. Given the relatively weak positive spanwise core flow at this measurement plane, it was likely that any negative axial flow was at a level below the measurement uncertainty of the current experiments. As expected, the high $\varGamma$ case showed notably earlier measurable axial flow when compared with the low $\varGamma$ case. The axial flow developed into the sheet-like structure of the near-wall measurement planes with measurable negative spanwise flow only near the downstream edge of the measurement domain at the $z=10r_{c,i}$ plane. Previous line tagging measurements of Bohl & Koochesfahani (Reference Bohl and Koochesfahani2004) had also observed the continuous sheet-like structure of axial flow just mentioned. At $z=20r_{c,i}$ the spanwise flow entered the flow field as single peaked, isolated structures, but changed to multi-lobed or hollow profiles for $x>30r_{c,i}$, see figure 9. This may be a result of the influence of the opposite sign axial flow in the vortices cut by the far cutting wall and the proximity to the midspan of this measurement plane where one would expect the axial flow from the far wall to meet to enforce the zero velocity boundary condition. The low $\varGamma$ case did not show the same reduction in the spanwise flow at this plane. This likely would have been observed if the measurement region were longer downstream as the weaker axial flow only begins to enter the flow field at the end of the measurement domain for the low $\varGamma$ case.

3.3. Mean vorticity and spanwise velocity fields after interaction

The unique characteristics of the instantaneous spanwise vorticity and axial flow fields just discussed for different spanwise planes lead to interesting consequences for their mean fields. Mean normalized vorticity fields are shown in figures 10(a,c,e,g) and 11(a,c,e,g) for the low and high $\varGamma$ cases, respectively. It is noted that the maximum contour levels are set below the maxima for both the vorticity and spanwise flow in these figures so that the details of the lower mean levels can be more clearly observed. In the low $\varGamma$ case, figure 10, the primary regions of mean vorticity were located symmetrically about the $y$-axis of the flow field. The mean vorticity fields confirmed that the region of primary vorticity spread vertically during the initial interaction and then remained nominally equally spaced for the near-wall planes. Closest to the wall ($z = r_{c,i}$), the regions of secondary vorticity that resulted from the breakup of the initially compact regions of vorticity led to a mean secondary vorticity field that appeared mostly outside of the regions of the primary vorticity, except for the far downstream portion of the measurement domain. The mean vorticity fields for the $z=10r_{c,i}$, $20r_{c,i}$ far-field planes did not show significant vertical widening in the streamwise direction that was observed at the near-wall planes. Instead, they were characterized by two well-defined peaks of opposite vorticity with their transverse location and sign consistent with the isolated vortex array of the traditional Kármán street (see figure 8), with only a slight widening of the array in going from $z=10r_{c,i}$ to $20r_{c,i}$. Interestingly, these characteristics persisted for the primary mean vorticity in the two near-wall planes as well, in spite of the near-wall interactions and appearance of secondary vorticity regions.

Figure 10.

(a,c,e,g) Normalized mean vorticity, $\omega ^*$ , and (b,d, f,h) spanwise velocity, $w^*$ fields at multiple span locations for the low $\varGamma$ case.

Figure 11.

(a,c,e,g) Normalized mean vorticity, $\omega ^*$ , and (b,d, f,h) spanwise velocity, $w^*$ fields at multiple span locations for the high $\varGamma$ case.

The normalized mean vorticity fields for the high $\varGamma$ case, figure 11, exhibited a fundamentally different behaviour compared with the low $\varGamma$ case just discussed, as a consequence of the different vorticity dynamics previously described in connection with figure 9. In the plane closest to the wall ($z=r_{c,i}$), the mean vorticity was initially characterized by two well-defined peaks of opposite vorticity whose transverse location and sign were consistent with the isolated vortex array of the reverse Kármán street depicted in figure 9. By $x\approx 15r_{c,i}$, the breakup of each primary vortex of either sign into two (a larger primary and a smaller secondary) vortices of the same sign resulted in the bifurcation of each mean vorticity peak into two, with the weaker thinner branch forming on the outside of the primary branch. Meanwhile, after breakup the primary vortices of either sign changed their transverse orientation and became aligned with each other nearly perfectly along a straight line, resulting in zero mean vorticity of the primary branch in the region in the range $x\approx 24-30r_{c,i}$ . Beyond this region the primary non-zero mean vorticity reappeared, but with its transverse orientation switched, a consequence of the change in the vertical orientation of the primary vortices described earlier in the phase-averaged vorticity map of figure 9.

The flow field characteristics described above at the plane closest to the wall ($z = r_{c,i}$) also occurred at $z=2r_{c,i}$, but the locations of the mean vorticity peak bifurcation and reappearance of mean vorticity peaks with switched orientation moved slightly downstream. The first far-field plane ($z = 10 r_{c,i}$) showed evidence of the same features near the downstream edge of the measurement plane, but the plane farthest away from the wall ($z = 20 r_{c,i}$) did not reveal those features over the measurement domain and the mean vorticity field was, instead, characterized throughout by two well-defined peaks of opposite vorticity corresponding to the isolated vortex array of the reverse Kármán street (see figure 9).

The normalized mean spanwise flow fields, shown in figures 10(b,d, f,h) and 11(b,d, f,h) for the low and high $\varGamma$ cases, reveal a common set of features regardless of the differences in the underlying vortical structure and interactions. Whereas the mean vorticity fields showed zero mean vorticity in the centre of the vortex array due to the alternating sign of vortices, the mean spanwise velocity showed generally positive axial flow away from the wall in the centre of vortex arrays due to the positive axial flow in the vortex cores for either sign vortices. The magnitude of the mean positive spanwise flow in the vortex array centre is determined by the strength of the axial flow in the vortex core, the core size, and the transverse separation between alternating sign vortices. The low $\varGamma$ case at $z=r_{c,i}$ (figure 10), and the early portion of the high $\varGamma$ case at $z=r_{c,i}$ (figure 11), are examples of situations where the mentioned parameters led to low levels of positive axial flow in the centre (near zero in some areas), and the maximum was found in two distinct regions that were spatially coincident with the locations of the peak vorticity. On the other hand, the downstream evolution of maximum axial flow in the centre for the high $\varGamma$ case at $z = r_{c,i}$ , 2$r_{c,i}$, and 10$r_{c,i}$ (figure 11) showed the regions of the maximum mean axial flow were located at the same downstream region where the primary alternating sign vortices changed their transverse orientation and became nearly perfectly aligned with each other. Note that negative mean spanwise flow was found outside of the region of mean positive spanwise flow, which was consistent with the location of the negative flow shown in the phase-averaged results.

The phase-averaged and mean fields, discussed in §§ 3.2 and 3.3, provide information about the instantaneous structure and average characteristics of the vorticity and spanwise velocity distributions. In the next section we discuss the dynamic characteristics of the flow in term of the evolution of selected vortex properties.

3.4. Vortex parameters

The phase-averaged vorticity and spanwise velocity data, such as those in figure 9, allow us to interrogate the downstream evolution of select vortex parameters, presented in figure 12 for the high $\varGamma$ case. The dynamics that is discussed was more pronounced for this case, but the qualitative description was similar for the low $\varGamma$ case. The parameters considered are the core radius, peak vorticity and peak spanwise velocity. Their values were found by identifying a vortex as it entered the measurement region and tracking its location and properties at each phase as it convected downstream. In this way the characteristics of the vorticity and spanwise flow fields were quantified throughout the entire measurement domain. Both the positive and negative vortices were tracked and the measured values were consistent for both. Peak vorticity values are shown for streamwise locations were the vortices remained as well-defined structures with a clear primary core, as the interpretation of the ‘peak vorticity’ was less clear after the vortices became distorted or broke up. The peak vorticity and vortex core data in figure 12 also include baseline results at the centre-span location without the cutting walls present from Bohl & Koochesfahani (Reference Bohl and Koochesfahani2009) for reference. Note that these two vortex parameters, which for the current data are from the flow that was strongly three-dimensional, agreed well initially with the properties from the centre-span data.

Figure 12.

Vortex parameters ($r_c^*$, $w_p^*$, $\omega _ p^*$) at multiple span locations for the high $\varGamma$ case. Baseline data are from centre span without the cutting walls (Bohl & Koochesfahani Reference Bohl and Koochesfahani2009).

The vortex core radius, $r_c^*$, initially followed the expansion trend of the baseline case, with the near-wall planes showing rapid increases in the vortex radius as the vortices distorted and the axial flow was generated. This was also observed for the far-field planes, however, the increased rate of expansion was delayed to progressively farther downstream locations. The initiation of the rapid expansion of the core was spatially coincident to the spatial location where the axial flow was first measured for all measurement planes. This indicated that the rapid expansion of the vortex radius was, at least initially, associated with the development of the axial flow. Smaller scale local oscillations in the core radius, which will be discussed in subsequent discussion, were also observed.

The peak spanwise velocity, $w_p^*$, showed slow developing (i.e. long spatial wavelength) trends along with a more local (i.e. short spatial wavelength) dynamics. Consider first the global trends in the data. All measurement planes showed initiation of axial flow that increased to a peak level and then decreased. The downstream location at which the axial flow was initiated at each plane progressively increased with spanwise plane location, as expected given the spanwise flow was generated at and then convected away from the wall. The global maximum values decreased with increasing distance away from the wall. The location of the overall maximum value for each spanwise plane moved downstream as the distance from the wall increased with the exception of the $z=20r_{c,i}$ plane. Here, the global maxima occurred earlier than was observed for the $z=10r_{c,i}$ plane. This was most likely due to the $z=20r_{c,i}$ measurement plane being near the centre of the span where spanwise flow from the opposite wall will have some influence and likely inhibited a higher magnitude of axial flow from being developed.

Within these overall trends smaller, local maxima/minima were also observed in the profiles. Figure 13 shows $w_p^\ast$ for $z=10 r_{c,i}$ and 20$r_{c,i}$ on the same plot for comparative purposes. Several local maxima/minima were observed at both spanwise planes. The first local maxima (S1) at each measurement plane were labelled with solid lines and the second (S2) by dashed. Other local minima were observed but not labelled on the plot for clarity. These data indicated the local maxima/minima, i.e. ‘spikes’, were not isolated incidents, but disturbances that convected away from the wall through multiple measurement planes. Further, these local spikes, even in the near-wall region could not be solely due to the initial cutting of the vortices as they continue to form well after the vortex was cut and the axial flow is initiated. These effects will be discussed in detail in §§ 3.5 and 3.6.

Figure 13.

Peak axial flow, $w_p^*$, as a function of downstream distance for the high $\varGamma$ case.

The peak vorticity, figure 12, showed a rapid decline in magnitude and interestingly also local oscillations, or spikes, for the near-wall planes. These effects were also observed at the far-field planes, although they were again delayed depending on the spanwise location of the plane. There are two possible explanations for this phenomenon. First, re-organization of the vorticity and circulation created dynamic changes in the peak vorticity levels. Second, local variations in the vortex radius (i.e. a vortex stretching effect) changed the vorticity levels. The evolution of the vorticity in the near-wall planes was complicated by both the initiation of the axial flow and distortion of the vortex due to the proximity of the wall. In the far-field region the vorticity remained in isolated structures and the re-organization of the vorticity was a less likely explanation. Therefore, vortex stretching was more likely the source of these local oscillations, at least in the far field.

Bohl & Koochesfahani (Reference Bohl and Koochesfahani2009) investigated the downstream evolution of the peak vorticity for the 2-D baseline case also presented in this work. They showed that the rate of decrease in the peak vorticity level showed a distinct change as the vortex convected downstream. The initial decrease in the peak vorticity was found to be due to diffusion of the vortex core. The rate of decrease in the peak vorticity then showed a distinct change and became faster than if diffusion alone were considered. They were able to show that the presence of even a small velocity gradient in the spanwise direction axial flow (${\partial w}/{\partial z}$) within the vortex core caused the additional decrease in the peak vorticity. In that work velocity gradients of the order of ${\partial w}/{\partial z}=0.1\,{\rm s}^{-1}$ compared with dimensional vorticity values 2–3 orders of magnitude larger were sufficient to produce the observed changes. The trends for the peak vorticity for the $z=10r_{c,i}$ and $20r_{c,i}$ planes of the current data were consistent with this model. Of note is that the current data, which measured vorticity and axial flow simultaneously, showed that the location where the rate of decrease changed was upstream of the point at which the axial flow itself was observable, supporting the conclusion that very small spanwise velocity gradients have a significant effect on the vortex dynamics.

3.5. Axial flow development

Axial flow data discussed so far were at only 4 selected spanwise locations and their purpose was to investigate in detail the spatial development in the streamwise, $x$, direction. Now we focus on the spanwise development of axial flow with finer resolution in $z$, but at specific downstream, $x$, locations. Line tagging MTV was applied here as it allowed finer spatial resolution in the spanwise direction ($z$) and a more detailed mapping of the axial flow than the sMTV experiments. Measurements were taken at multiple (16) span locations from 0.6$r_{c,i} \le z \le 25 r_{c,i}$ with an approximate spacing of $1.6r_{c,i}$ and from $2.5r_{c,i} \le x \le 67.5 r_{c,i}$ with steps of $2.5r_{c,i}$. Contour maps of the maximum positive and negative spanwise flows over the entire spatial extent for the two cases being discussed are shown in figure 14. Both cases show that the positive spanwise core flow began near the wall and moved towards the centre of the tunnel with increasing downstream distance, as expected. The axial flow ‘front’ (i.e. the boundary between the spanwise flow/no spanwise flow) advanced more rapidly for the higher circulation case. Both cases showed nominally the same normalized axial flow levels with the highest levels measured of the order of $V_{swm,i}$, indicating that the spanwise flow scaled with the swirl speed, and by extension, strength of the vortex. The region where the axial flow was nominally equivalent to the swirl speed was confined to the region near the wall for the low $\varGamma$ case. The high $\varGamma$ case showed that the region of high positive spanwise flow moved away from the wall with increasing downstream distance with a pronounced region of low spanwise flow near the wall. These data were qualitatively and quantitatively consistent with the sMTV results which showed the loss of the vortical regions and spanwise flow near the wall. Variation in the axial flow speed was observed along the axial flow front for both cases, although it was much more pronounced for the high circulation case.

Figure 14.

Contour plot of peak positive (a,c) and negative (b,d) normalized axial flow, $w _p^*$, over the measurement domain.

The peak negative spanwise flow fields were found to be aligned with regions of the highest axial flow, near the cutting walls for the low $\varGamma$ case and initially along the wall then angling into the flow field for the low $\varGamma$ case. The magnitude of the peak negative spanwise flow was higher for the low $\varGamma$ case than for the high $\varGamma$ case. This is again consistent with the phase-averaged spanwise flow fields in figure 8 and 9 where the negative spanwise flow was limited to isolated structures.

The peak positive spanwise flow data in figure 14 were used to plot spanwise profiles along the axial flow front ($2.5< x/r_{c,i}<10$) for the high $\varGamma$ case to investigate the initial dynamics of the positive axial flow within the vortex cores, figure 15. It is again important to recall that these data are from phase-averaged results that were taken at multiple locations and therefore the features they display are repeatable in both time and space. These profiles show that as the axial flow front propagated away from the wall a clear local overshoot or ‘spike’ in the axial flow velocity was observed. These initial local overshoots are identified in figure 15 by the solid arrows for each downstream location. The velocity spike followed the front boundary as it progressed away from the cutting wall. Spikes at the axial flow front boundary were observed for all circulation cases studied, although the magnitude of the spike was progressively weaker for lower $\varGamma$ cases. The profile at $x=10r_{c,i}$ also showed evidence of other velocity spikes behind the axial flow front indicating that these features continued to be formed as the vortex interacted with the cutting walls and were not limited to the initial cutting of the vortex.

Figure 15.

Spanwise profiles of peak axial flow vs downstream distance for the high $\varGamma$ case.

The results thus far show that the initiation and development of the axial flow within the vortex cores had significant effects on the vorticity distribution and core size of the vortex array. These results were suggestive that disturbances were generated near the wall and propagated away from the wall along the vortex. This dynamics is explored and compared with the analytic area-varying wave model of Lundgren & Ashurst (Reference Lundgren and Ashurst1989) in the following section.

3.6. Area-varying waves

Lundgren & Ashurst (Reference Lundgren and Ashurst1989) examined the development and propagation of area-varying waves on vortices using an analytic model. The current data allow examination of the propagation along the vortex core of the disturbance caused by the impulsive vortex cutting by a thin plate, and its connection to travelling area-varying waves predicted by Lundgren & Ashurst (Reference Lundgren and Ashurst1989). We focus on the first disturbance launched during the initial cutting of the vortex, where the vortex retains its Gaussian nature in the areas interrogated and its characteristics can be quantified without much complication. Lundgren & Ashurst (Reference Lundgren and Ashurst1989) used the gas dynamics equations as an analogous description of the behaviour of area-varying waves on a vortex tube. They defined an equivalent ‘speed of sound’ in their equation (4.7) as

(3.2)\begin{equation} c^2=\frac{\varGamma ^2}{8{\rm \pi} A}. \end{equation}

Here, $\varGamma$ is the circulation of the vortex and $A = {\rm \pi}r_c^2$ is it's area. The authors considered a sharp edged vortex with uniform vorticity within the core (i.e. ‘flat top vortex’), where the maximum swirl speed, $V_{swm}$, is defined by

(3.3)\begin{equation} V_{swm}=\frac{\varGamma}{2{\rm \pi} r_c}. \end{equation}

When (3.2) and (3.3) are combined, the non-dimensional ‘sound speed’, normalized by the maximum swirl speed, for a flat top vortex model simplifies to

(3.4)\begin{equation} \frac{c}{V_{swm}}=V^*_{FT}=\frac{1}{\sqrt{2}}=0.707. \end{equation}

The vortices in this work are well described by a Gaussian vortex model, as previously discussed. The maximum swirl speed for this vortex model using (3.1) is found when $r/r_c=1.12$. Using the maximum swirl speed of a Gaussian vortex for normalization in combination with (3.2) leads to the non-dimensional wave speed for the Gaussian vortex model being given by

(3.5)\begin{equation} \frac{c}{V_{swm}}=V^*_{G}=\frac{1}{\sqrt{2}}\frac{1}{0.6382}=1.11. \end{equation}

The data in figures 13 and 15 show evidence of disturbances that propagate along the vortex core. Two estimates of the propagation speed of this disturbance speed were calculated using the line tagging data, which were available for the entire circulation range studied, for comparison with the analytic model. The spatial location of the axial flow ‘front’ was defined by the span location where $\langle w\rangle _{peak} = 0.5$ cm s$^{-1}$. This level, while arbitrary, was chosen to be slightly above the uncertainty level in the measurements allowing for the determination of the location of the initial axial flow front to be made. We note that this choice of threshold level did not significantly affect the results that follow. Figure 15 shows the front threshold as the vertical dashed line and the spanwise, $z$, location of this position is indicated by the green arrows in that figure. The convection time was calculated using the pre-interaction convection speed, $U_{c,i}$, multiplied by the streamwise distance between measurement locations. The front speed, $V_f$, was determined by differentiation of a linear least squares fit of the spanwise location of the front with time. The intent of the calculation of the front speed with this definition was to compare the propagation of the leading edge of the axial flow with the propagation speed of first disturbance observed on the vortex cores. The propagation speed of the first disturbance, $V_d$, was determined in a similar way. However, in this case the spanwise location of the first local velocity maximum, shown with the solid arrows in figure 15, was used to define the spanwise location of the first disturbance. Data were again fit using a linear least squares curve and differentiated to determine the disturbance speed, $V_d$.

The stereoscopic data provided an independent method for calculating the propagation speed of the initial disturbance, $V_{ds}$, for the low and high $\varGamma$ cases. The tracking data for the peak spanwise flow, shown in figures 12 and 13 for the high $\varGamma$ case, were replotted as a function of time instead of space. We note that for these data time was known directly instead of via calculation using the initial convection speed. The time of the occurrence of the first spanwise flow spike was determined for each of the measurement planes, which defined the spanwise location. These data were also fitted using a linear least squares and then differentiated with respect to time. All disturbance speeds were then normalized using the maximum pre-interaction swirl speed, $V_{swm,i}$, which was taken as analogous to the maximum swirl speed, $V_{swm}$, for the analytic model.

The results of the preceding analysis are plotted in figure 16. The dimensional speeds, figure 16(a), showed a linear increase with vortex circulation as well as good agreement between the line tagging and stereoscopic data. The normalized speeds are shown in figure 16(b). The dashed line is the theoretical normalized sound speed for the Gaussian vortex model from (3.5). The solid line is the theoretical normalized sound speed for the flat top vortex model from (3.4) and is provided for reference. Experimental results are shown with symbols. The results shows that the normalized disturbance speeds, $V^*_f$ and $V^*_d$, were nominally constant and agreed well with the results from the stereoscopic data, $V^*_{ds}$. More significantly, these results indicated that the propagation speed of the first disturbance matched the analytic sound speed proposed by Lundgren & Ashurst (Reference Lundgren and Ashurst1989).

Figure 16.

(a) Dimensional and (b) normalized axial flow disturbance speeds. Here, V_G ${*}$: analytic equivalent sound speed for a Gaussian vortex; ${V^*}_{FT}$: analytic equivalent sound speed for a flat top vortex from Lundgren & Ashurst (Reference Lundgren and Ashurst1989).

Lundgren & Ashurst (Reference Lundgren and Ashurst1989) showed in their equation (4.14) that the speed at which disturbances propagate along the vortex core depends on the equivalent sound speed, the area ratio up and down stream of the disturbance and the speed of the axial flow into which the disturbance is propagating. For cases with small variations in the area of the vortex core and no axial flow ahead of the disturbance, the wave speed of a disturbance simplifies to the equivalent sound speed, $c$. These conditions will be shown to be met for the initial disturbance caused by the initiation of the axial flow. The data in figure 16 were, therefore, suggestive that small amplitude area-varying waves were generated on the vortex cores as the axial flow was first initiated and then propagated through the vortex cores.

Figure 17 shows $r_c^*$, $w^*$ and $\omega ^*$ plotted on the same figure at $z=10r_{c,i}$ and $20r_{c,i}$ for the high $\varGamma$ case. These span locations were chosen for analysis because the vortices remained relatively undistorted and well defined over the measurement domain. Note that the vertical axis of the curves are offset to allow the curves to be displayed on the same figure without significant overlap. The intent of these plots is to show spatial coincidence of the features in each profile, not relative scales of the plotted values. Changes in the vortex core radius were difficult to distinguish in figure 17 because of the naturally increasing core radius as the vortex convected downstream and because these changes were small. The variable $\Delta r^*$ was calculated to highlight the local changes in the vortex core radius in the following way. The core radius was first smoothed using a $\pm$30 point moving spatial average. This value was subtracted from the local vortex core radius and divided by $r_{c,i}$ to provide a local normalized difference, $\Delta r^*$ at each streamwise location.

Figure 17.

Correlation of spatial features of vortex parameters for the high $\varGamma$ case.

At $z=10r_{c,i}$ a series of local maxima and minima were observed in the peak vorticity as a function of the downstream distance. The first local minimum in the peak vorticity (solid line) corresponded to a small but measurable increase in $r_c^*$, seen as a local maximum in $\Delta r^*$. There was also an associated local maximum in $w^*$. This was followed by a local maxima (dashed line) in the peak vorticity which was correlated to a minimum in the core radius ($\Delta r^*$) and axial flow. The vortex core radius showed the least pronounced local maximums and minimums. For example, the variation in $r_c^*$ for the first local peak was relatively small, at approximately $\Delta r^*\approx 1.3\,\%$. This level was significant, but near to the noise level in the radius estimation. The subsequent vorticity peaks were less intense, and the corresponding area changes were also smaller. At $z=20r_{c,i}$ the correlation between the vorticity and spanwise velocity was clear for the first three vorticity spikes. Changes in the vortex core radius were again small ($\Delta r^*<1\,\%$) but visible at this span location. These trends were also observable for the low $\varGamma$ case (not shown here). However, for this case the weaker initial vortex along with the distortion/reorganization of the vorticity and made this correlation less clear.

These results reveal important aspects of the dynamics of the development and progression of the axial flow within the vortex. The axial flow developed and then strengthened as it progressed away from the cutting walls. This led to a surge in the axial flow that created an increase in the vortex area which in turn resulted in a local decrease in the peak vorticity. The vortex radius decreased and the peak vorticity increased after the disturbance on the vortex core propagated past the spanwise measurement location. This view is of course simplistic as other dynamics (e.g. diffusion/expansion of the vortex) are occurring simultaneously, creating more complicated flow dynamics.

The data do, however, support the existence of travelling small amplitude, area-varying waves on the vortex core. Recall that the analytic wave speed simplified to the analytic sound speed, $c$, assuming small area variation and an initial axial flow velocity of $w=0$. These simplifications were met for the first disturbance on the vortex cores where the variation in the vortex core area was small (${<}1.3\,\%$) and there was no axial flow ahead of the disturbance. The measured disturbance speeds shown in figure 16 were shown to be nominally equal to the associated sound speed, confirming the analytic model proposed by Lundgren & Ashurst (Reference Lundgren and Ashurst1989). Moreover, the characteristics of the peak vorticity and axial flow, which are more pronounced and easier to measure than the vortex core area for small disturbances, appear to also be indicators of the existence and spatial/temporal locations of area-varying waves.