2.1. Flow facility

The flow facility employed for this study was the Miniature Large Cavitation Channel (MLCC), a recirculating water flow facility. The MLCC is a 1/14th scale model of the US Navy’s William B. Morgan Large Cavitation Channel in Memphis, TN (Etter et al. Reference Etter, Cutbirth, Ceccio, Dowling and Perlin2005) with a 22.5 by 22.5 cm square test section (with chamfered corners) with removable, clear acrylic windows on all four sides. The test section flow can operate at a range of speeds from 1 to 25 m s⁻¹, but in this work, only speeds between 1 and 10 m s⁻¹ were employed. Upstream of the test section is a 6.4–1 contraction. Separate rings of tubing around the inlet and outlet of this contraction each connect four pressure taps on the tunnel walls (top, bottom and two sidewalls). Given the known contraction ratio, the pressure differential between these pressure measurements is used to estimate the flow velocity in the channel using the Bernoulli pressure change. The ring immediately upstream of the test section is also used to measure the free stream pressure in the test section. An Omega PX409 differential pressure transducer with 0.08 % accuracy was used to measure the pressure difference across the contraction, giving an uncertainty in velocity of 0.06 %. An Omega PX419 transducer with 0.08 % accuracy was used to measure static pressure immediately upstream of the test section, giving an uncertainty of 0.08 % in static pressure.

Upstream of the contraction, a flow straightener and screen are used to break up large vortices generated by the impeller and turning vanes and leave low turbulence in the test section, and the flow is strained as it moves through the contraction, reducing the small-scale turbulent fluctuations. Here PIV was used to estimate the free stream turbulence in the test section. With an empty test section, the free stream turbulence level (root mean square of $u^{\prime }/U_0$ where $u^{\prime }$ is the fluctuating streamwise velocity component and $U_0$ is the mean free stream velocity) was measured with SPIV to be less than 0.25 %. For these free stream turbulence measurements, frame-straddled SPIV data were taken at a range of speeds, from 1 to 10 m s⁻¹, and this turbulence level was consistent across this range. Critically, that data was taken with particle displacements of approximately 80–100 pixels. It is necessary to use such large displacements to measure the fluctuations because using smaller displacements (8–12 pixels would be typical values in SPIV) will lead to a high relative uncertainty in the measurements, which artificially inflates the fluctuations measured (Scharnowski, Bross & Kähler Reference Scharnowski, Bross and Kähler2015; Sciacchitano et al. Reference Sciacchitano, Neal, Smith, Warner, Vlachos, Wieneke and Scarano2019). The free stream turbulence level comports with that of the geometrically similar water channel, the William B. Morgan Large Cavitation Channel where Park, Cutbirth & Brewer (Reference Park, Cutbirth and Brewer2003) used laser doppler anemometry to find levels of 0.2 % to 0.5 % depending on free stream velocity. The measured level of free stream turbulence in the MLCC tunnel is similar and is consistent with the level of turbulence expected for a tunnel with modest contraction ratios (Barlow, Rae & Pope Reference Barlow, Rae and Pope1999).

2.2. Test models and conditions

A pair of cambered hydrofoils with modified NACA66 section (Shen & Dimotakis Reference Shen and Dimotakis1989) and different planform were used to generate the pair of line vortices. The larger hydrofoil, generating the primary or stronger vortex, has a rectangular plan form, while the smaller hydrofoil, generating the secondary or weaker vortex has either a tapered plan form or a rectangular plan form of equal chord to the primary foil, depending on the case. The root chord, c, for both hydrofoils is 167 mm and a maximum thickness of 15 mm. The tapered secondary hydrofoil has a rounded tip to encourage roll-up of the tip vortex with the tapered end having a thickness, $d_s$ , of 6.25 mm. The rectangular planform hydrofoils for the primary vortex and matching secondary foil have squared-off tips. Several different angles of attack of the secondary foil were studied with the tapered foil, though only one angle of attack was studied for the matched hydrofoil. The complete, as built, geometry of the hydrofoils and the test section are provided in Knister (Reference Knister2024).

Figure 1. (a) A side view of the water channel with primary hydrofoil on the top and a secondary hydrofoil (non-matching) on the bottom of the water tunnel. The section in the middle represents the view through the side window. Downstream of the hydrofoils, high-speed video is taken in the regions outlined with blue boxes and used for visualization of developed cavitation. The volumetric velocimetry is done at three streamwise locations. The laser comes from below and is shown in green. (b) The layout of the water channel as set up for volumetric velocimetry measurements and viewed from above. The brass hydrofoils are in orange on the right. The green regions downstream of the foils are the locations of the illuminated volumes for velocimetry. The light purple structures outside the test section are water filled boxes to allow the cameras (with Scheimpflug adapters) to interrogate the flow from non-orthogonal views. (c) A cartoon of the development of the vortex instability studied here. An upstream measurement (0.7 c, where c is foil chord) is taken for the ‘initial conditions’ of the instability. A midstream measurement is taken in the linear regime of the instability (1.2 c), and a measurement is taken in the nonlinear regime (1.7 c downstream).

Figure 1(a) shows a schematic of the configuration with the primary and secondary hydrofoils on the top and bottom window, respectively. Downstream of the hydrofoils, high-speed video is taken in the regions outlined with blue boxes and used for visualization of developed cavitation (to be discussed later). The volumetric velocimetry is done at three streamwise locations shown in green. Four secondary hydrofoil arrangements generating counter-rotating vortices were examined in detail with the primary hydrofoil fixed at $+6^\circ$ angle of attack. The arrangements are denoted by `M’ or `N’ for a matching or non-matching (tapered) secondary hydrofoil planform set-up. A digit represents the negative angle of attack of the secondary hydrofoil, and the four conditions are therefore denoted as N.0, N.2, N.4, M.4. Two of these, M.4 and N.4, which have the secondary at $-4^\circ$ angle of attack, were examined over a range flow speeds to examine the effect of chord-based Reynolds number ( $\textit{Re}_{c}$ , (2.1)) on the formation of the vortices, where $\nu$ is the kinematic viscosity of water at room temperature. The cavitation number is defined using (2.2), where $P_{0}$ is the test-section inlet pressure, $P_{v}$ is the vapour pressure of water and $\rho$ is the water density:

(2.1) \begin{equation} \textit{Re}_{c} = \frac {U_{0}c}{\nu }, \end{equation}

(2.2) \begin{equation} \sigma _{c} = \frac {P_{0}-P_{v}}{\frac {1}{2} \rho U_{0}^{2}}. \end{equation}

2.3. High-speed cinematography

Reduction in the inlet pressure caused hydrodynamic cavitation in the vortex cores as they interacted and experienced the long-wavelength instability. This enabled the visualization of the instability and its evolution using high-speed cinematography, based on which regions of volumetric velocimetry measurements were identified. High-speed video was acquired with two Phantom v1212 cameras with 1280 × 800 pixel resolution at 12 600 frames per second with 12 bit depth dynamic range. The cameras were used with 50 mm AF Nikkor or 105 mm AF Micro-Nikkor lenses. The camera length scales were found by calibration from a simple ruler, allowing some quantitative information to be derived from the high-speed videos. Higher frame rates were achieved by reducing the portion of the camera sensor recorded. The cameras were placed to view interaction region from the side, as shown in figure 1(a). In some cases, when a top-down view was needed, a mirror was placed on top of the tunnel. An Arrilux Pocketpar 200 backlit the tunnel while an Arrilux Pocketpar 400 illuminated the tunnel from the same side as the cameras. The backlight was necessary to minimize shadows appearing behind the cavitating vortices.

2.4. Volumetric velocimetry

Volumetric velocimetry was used to measure the non-cavitating vortex properties at several locations downstream of the hydrofoils generating the vortices. Four Phantom v1212 cameras with 1280 x 800 pixel resolution at 12 600 frames per second with 12 bit depth dynamic range were used, as shown in figures 1(a), 1(b) and 1(c). LaVision Scheimpflug adapters were used to view non-orthogonally into the test section, as shown in figure 1(b). Four 105 mm Micro-Nikkor macro lenses were used, sometimes in combination with Nikon TC-201 2X and Tele Plus 1.4X DGX teleconverters. A LaVision High Speed PTU X commanded the devices and controlled the system timing. A Photonics Dual Head DM-100 high-speed laser with a maximum power of 10 mJ per pulse at 10 kHz for each cavity was used to illuminate the measurement domains. The data collected were time series data, where both heads are fired simultaneously to approximately double the illumination. Acquisition rates varied from 2–20 kHz for the range of flow speeds covered in the upstream location. Acquisition rates were fixed at 10 kHz for the midstream and downstream measurement locations. Shake-the-box (STB) particle tracking velocimetry (PTV) measurements were employed to estimate particle tracks.

Table 1. Locations and sizes of volumetric velocimetry measurement volumes. The PIV performed at 0.7 c, 1.2 c and 1.7 c are referred as upstream, midstream and downstream, respectively. Here c is the chord of the hydrofoil.

Three measurements regions were examined as shown in figures 1(b) and 1(c) and defined in table 1. The upstream volume is approximately 0.7 c downstream of the trailing edge of the primary foil, the midstream volume is approximately 1.2 c and the downstream volume is approximately 1.7 c, where c is the root chord of the hydrofoils, discussed below. Due to the small extents required to adequately interrogate the velocity fields of the vortices, three different measurement volumes were used at each of these locations, one for each secondary foil angle of attack, as the vortices were in different locations when changing between the different angles of attack for the several conditions studied. The variations in extents between the three regions used at each downstream location are small (typically less than 5 %). The sizes of the different measurement volumes are reported in table 1. The typical spatial resolution of the upstream region was 0.42 mm, and the typical spatial resolution for the midstream and downstream measurements was 0.35 mm. In the upstream region, the initial vortex properties are measured as the vortices are finishing rolling up and while they are initially almost parallel. These properties serve as the initial conditions for the Crow instability that the vortices then undergo and will be discussed in § 5. The midstream region captures the linear stage of the instability, where the linear stability analysis for the Crow instability accurately describes the dynamics of the interaction and the instability gives the secondary a modest quasisinusoidal centreline displacement motion. The downstream region measures the nonlinear stage of the interaction. A cartoon of these regions is shown in figure 1(c) and the conditions of measurements in these regions is shown in table 2.

Table 2. The conditions for volumetric velocimetry measurement at the three measurement locations (scaled by foil chord, c). Note that at each streamwise location, three separate volumes had to be set up to adequately measure the vortices as they shifted positions between different cases, but the differences in extent of those different volumes are minimal. In all cases, the primary hydrofoil was fixed at $\alpha _{P}=6^\circ$ . Optical visualization of developed hydrodynamic cavitation occurring during the interaction of the vortices for all four cases are shown in the Supplementary movies are available at https://doi.org/10.1017/jfm.2026.11120.

3.1. Vorticity strands

Thin strands of vortical regions are seen to wrap around both vortices for all cases (see Supplementary material is available at https://doi.org/10.1017/jfm.2026.11120), even at locations very close to the secondary hydrofoil generating the larger vortices. These strands are not from the secondary vortex but should be considered tertiary vorticity. Because the diameters of these cavitating strands are small relative to the primary and secondary vortices, these tertiary vortices have a much weaker pressure drop than even the secondary vortex. Since the primary and secondary vortices are roughly antiparallel and relatively far apart in the upstream region, these tertiary structures cannot be vorticity bridging that can occur between a pair of corotating vortices when their separation, b, is close to the combined radii of the vortices. Instead, these structures are likely a consequence of the vortex roll-up process between the tips of the primary and secondary hydrofoils.

3.2. Vortex straining

Figure 3. Developed cavitation in the vortices in Case N.2 is used to visualize how the secondary vortex can lead to temporary suppression of cavitation in the primary vortex. The above is a time sequence of images, with the time noted in the lower left-hand corner of each image. The red arrows denote the structure described in § 3.2.

As mentioned earlier, for Case N.2 substantial straining of the primary vortex core by the secondary was observed. This resulted in regular suppression of cavitation as portions of the secondary vortex wrap around the primary, followed by breaking up of the secondary vortex into vortex rings, as illustrated in figure 3. Figure 3 is a time sequence of images, with the time noted in the lower left-hand corner of each image and the red arrows denote the structure described hereafter. Initially, a perturbed portion of the secondary vortex is drawn up by the primary vortex. While not visible from this view, the secondary votrex is also moving towards the vortex out of the plane of the page. At t =1.11 ms, the secondary vortex is near enough to the primary vortex that it causes significant straining in the primary vortex, resulting in elliptical streamlines within the primary vortex (to be discussed in detail later). Note that the portion of the primary vortex nearest the secondary at this point is drawn upwards. The strain in the vertical direction is enough that at t =1.67 ms, cavitation is suppressed in the primary vortex. Some vertically oriented strands of vorticity run between the secondary and the remnants of the primary vortices. A weakened portion at the base of the primary vortex draws together and cavitates at t =2.22 ms. As this happens, the secondary breaks apart and by t =3.33 ms, the relevant portion of the secondary is folding away from itself to eventually form vortex rings with neighbouring peaks in the secondary. Although the mechanism is not immediately clear from cavitation alone, it will be shown in the upcoming sections that the secondary is straining the primary vortex out of its initially circular cross-section based on volumetric velocimetry measurements. For Case M.4, there appears to be some amount of cross-axis staining in the primary as the secondary wraps around it, but not to the same extent as Case N.2. In Cases N.0 and N.4, the primary vortex does not undergo such cavitation suppression as in Case N.2, but it does take a somewhat elongated or stretched form as opposed to a strictly circular cross-section.

This straining out of the primary vortex in Case N.2 comports with what was noted by Marshall et al. (Reference Marshall, Brancher and Giovannini2001) in Case E of their computations. They considered pairs of unequal strength, counter-rotating vortices with varying circulation ratios. Case E was the only case they considered that also had a smaller core radius in the secondary vortex. In Case E, they noted a straining of the primary akin to what is reported in this study, which they attributed to the higher peak vorticity in the core of the secondary vortex leading to the weaker vortex not straining out as much as the stronger vortex. Due to the smaller core radii of the secondary vortices in our study, the peak vorticities of the secondary are approximately equal to or higher than the peak vorticities in the primary vortices in all cases (figure 2). During this straining out of the primary, the secondary vortex starts to break apart and will eventually form vortex rings, though these are not completely formed and thus not visible by cavitation in the viewing area.

4.1. Experimental measurement of vortex properties before the onset of instability

As mentioned earlier, volumetric PIV measurements were used to determine the properties of the two nearly parallel line vortices before the onset of their strong interaction. This region, located at approximately 0.7 chord lengths downstream of the trailing edge of the primary foil, is referred to as the upstream region in tables 1 and 2. Figure 4 shows the location of the upstream region with instantaneous vorticity measurements for Case N.2 as an example. The two vortices are close to completing their roll-up, and their axes are nearly parallel for all cases considered. Thus, a relatively thin volume can be used to cut through their axes and measure their properties to characterize the initial conditions of the vortex interaction. For the non-matching foil cases (N.4, N.2., N.0), measurements were acquired for a sweep over angles of attack of the secondary foil at a fixed speed and a sweep over speeds at fixed angle of attack. For the matched foil case, data were taken over a sweep of speeds for a fixed angle of attack. As noted above, the vortices take substantially different paths between the different conditions, so the different cases each required their own location of the measurement volumes, though the dimensions of the different measurement volumes are within a few per cent of each other. Accordingly, relevant parameters such as circulation ( $\varGamma$ ), core radius( $r_c$ ), axial velocity ( $U_a$ ), eccentricity ( $\epsilon$ ) and peak vorticity $\omega _{\textit{max}}$ ) were measured.

Figure 4. Location and an example instantaneous realization of the upstream STB vortex in cell (VIC) measurement of the vortices for Case N.2.Here $\omega _z$ is the streamwise vorticity scaled with $d_s$ , the thickness of the secondary foil tip and $U_0$ , the free stream velocity. All cases had different measurement volumes so that the vortices would be near the centre of the volume and away from the typically lower-quality measurement regions on the periphery. The vortices are visualized by $\lambda _2$ isosurfaces and coloured by streamwise vorticity. Note that the dominant structures are largely the two vortices, the primary and secondary. Occasionally, tertiary vortical structures (usually oriented vertically, not streamwise) will convect between the two vortices. The vector spacing of the velocimetry measurement is 0.42 mm, but here the vectors are spaced at 2 mm so as not to crowd the image.

4.1.1. Estimating vortex properties

Calculations of the initial vortex properties are performed on individual velocity fields on a plane oriented with its normal vector parallel to the local vortex axis. The computed properties of each field are then averaged. It is important to average the vortex properties after calculating them for individual fields because of vortex wandering as discussed in Devenport et al. (Reference Devenport, Rife, Liapis and Follin1996). The mean vortex properties were calculated from 1000 independent fields of velocity data for each case in the upstream and midstream locations. Circulation, defined in (4.1) is estimated from the vorticity field derived from the measured velocity field, and integrated by considering only vorticity greater than a given threshold. This integral was performed for each vortex since the vortices have opposite orientations with a threshold of 10 % of the peak vorticity. Halving this threshold increased the calculated circulation by no more than a few per cent. An additional constraint was that the region had to be contiguous and compact, as identified with MATLAB’s bwareafilt function. As a check, contour integrals of the velocity field around the vortices were also computed and agreed with the integral of the vorticity approach:

(4.1) \begin{equation} \varGamma = \iint \omega (\omega \gt \omega _{\textit{threshold}}) \,{\rm d}A. \end{equation}

With the circulation known, the vortex centres were then calculated as

(4.2) \begin{equation} l_{v\textit{ortex}} = \frac {1}{\varGamma }\int l\omega (\omega \gt \omega _{\textit{threshold}}) \,{\rm d}A. \end{equation}

In (4.2), we consider only the region where the vorticity exceeded the threshold from above. The variable $l$ is the y, z position coordinate and $l_{v\textit{ortex}}$ is the y, z coordinate of the centre of the vortex. With the two vortex centres known, the separation distance between them, b, is found. The vortex core radius, $r_c$ , is found using

(4.3) \begin{equation} r_{c} = \frac {1}{\varGamma }\int |l-l_{v\textit{ortex}}|^{2}\omega (\omega \gt \omega _{\textit{threshold}}) \,{\rm d}A. \end{equation}

In addition, the axial velocities in the vortex cores within the region of vorticity above the threshold were also determined, and the maximal axial velocity was recorded. Area moments of the identified vortex regions in the vorticity field were also calculated using the relation

(4.4) \begin{equation} \phi _{fg} = \int (y-\overline {y})^{f}(z-\overline {z})^{g}\omega (y,z) \,{\rm d}A. \end{equation}

In the above expression, $\overline {y}$ and $\overline {z}$ are the locations of the centre, and $f$ , $g$ vary from 0 to 2. The vortex centroids and radii noted above are specific cases of these area moment integrals, and the integrals were discretized because of the discrete nature of the velocimetry data. A covariance matrix of these moments is used to define the eccentricity $\epsilon$ , as

(4.5) \begin{equation} \epsilon = \sqrt {\left(1-\frac {\lambda _{2}}{\lambda _{1}}\right)}. \end{equation}

Here $\lambda _{1}$ and $\lambda _{2}$ are the eigenvalues of the covariance matrix $\textit{Co}v(\textit{Vortex})$ . All measurements in the upstream region were acquired in time series mode, and the particle tracks were calculated with STB PTV. Eulerian flow fields were estimated using VIC (VIC#), as implemented in LaVision’s DaVis 10.2 software. To ensure that the velocity fields used were independent samples, every 10th velocity field was considered for analysis,

(4.6) \begin{equation} \textit{Co}v(\textit{Vortex}) = \begin{bmatrix} \frac {\phi _{20}}{\phi _{00}} & \frac {\phi _{11}}{\phi _{00}}\\[11pt] \frac {\phi _{11}}{\phi _{00}} & \frac {\phi _{02}}{\phi _{00}} \end{bmatrix}\!. \end{equation}

4.1.3. Variation with secondary attack angle $\alpha _s$

Figure 5. Effect of attack angle of secondary hydrofoil on the vortex properties measured upstream at $x/c=0.7$ for the non-matching hydrofoils at 3 m s⁻¹.

Figure 6. Effect of Re on the vortex properties measured upstream at $x/c=0.7$ for the non-matching hydrofoils at attack angle of $\alpha _{s}=-4^{\circ }$ , corresponding to Case N.4.

Figure 5(a–d) shows the variation of vortex properties measured upstream (before the onset of the instability) with $\alpha _s$ , the attack angle of the secondary vortex for non-matching planforms. All measurements shown in figure 5 were performed at a constant free stream speed and at locations shown in table 1. In the following subsection, the effect of chord-based $\textit{Re}$ on the vortex properties will be discussed.

In figure 5, square symbols represent the primary vortex and circles represent the secondary vortex. The symbols are coloured in figure 5(a–c) as black or blue corresponding to the quantity represented in the $Y_1$ and $Y_2$ axes, respectively. This convention holds for the figures 6, 7 and 8 to be discussed soon.

In all cases measured with velocimetry, the secondary foil has a negative or zero angle of attack. The secondary hydrofoil at an attack angle of $\alpha _s= -4^{\circ }$ is expected not to generate substantial lift if isolated in the flow. However, in Cases N.4 and M.4 (shown in figure 7), a pair of ‘well-defined’ counter-rotating vortices are produced. This may be due to the interaction of the flows around the foils with one another as the vortices roll up, resulting in a coherent secondary vortex. Note that as the angle of attack of the secondary foil becomes more negative, its core radius ( $r_{c,s}$ ) is smaller, and the vortex circulation ( $\varGamma _{c,s}$ ) is weaker, as seen in figure 5(a). The circulation ( $\varGamma _{c,p}$ ) and the core radius ( $r_{c,p}$ ) of the primary vortex are modestly affected by the secondary attack angle. For both primary and secondary vortices, the axial velocity ( $u_{\textit{axial}}/U_{0}$ ) decreased with higher negative angle of the secondary ( $\alpha _{p}$ ), and the maximum vorticity ( $\omega _{\textit{max}}/U_{0}d$ ) also decreased with higher negative angle of the secondary hydrofoil. The observations in figures 5(a) and 5(b) suggest that the roll-up and formation processes of the two line vortices are influenced by the tip flow occurring between the two hydrofoils. This change in core radius and circulation is likely also due to the complexities of the flow immediately between the two hydrofoils. If the test section were longer, the foils could be set farther apart while still observing the instability in the test section, and the resulting rolling up of the vortices would likely show the effects of this interaction less.

Figure 7. Effect of Re on the vortex properties measured upstream at $x/c=0.7$ for matching hydrofoil planforms at attack angle of $\alpha _{s}=-4^{\circ }$ , corresponding to Case M.4.

Figure 8. Effect of $\alpha _s$ on the vortex properties measured in the midstream at $x/c=1.2$ for different configurations.

Figures 5(c) and 5(d) show other quantities of interest such as the eccentricity ( $\epsilon$ ), circulation ratio ( $\varGamma _{p}/\varGamma _{s}$ ), inferred separation distance ( $b$ ) and vortex strain-rate ( $\beta$ ). The uncertainty in these quantities is higher because they are inferred from measured velocity fields using numerical integration and differentiation. However, the observed trends show that the eccentricity of the primary vortex decreases with an increase in the negative attack angle of the secondary hydrofoil ( $\alpha _{p}$ ). The highest circulation ratio occurred at $\alpha _{s}=-4^{\circ }$ where the lowest measured was at $\alpha _{s}=0^{\circ }$ . The separation distance between the vortices changed modestly as seen in figure 5(d). Measured shear for both primary and secondary vortices remained unchanged within measurement uncertainty.

5.1. Linear stability analysis

Based on the upstream measurements in the prior section as an initial condition, perturbation growth rates and their wavelengths can be estimated using the generalized linear Crow theory. Such an analysis provides information on the validity of linear dynamics in the midstream region. The generalized approach discussed here follows the notation of Leweke et al. (Reference Leweke, Le Dizès and Williamson2016). It is assumed that the wavelength of the instability, $\lambda$ , is long relative to the size of the vortex core radius $r_{c}$ . The instability combines three elements: the self-induced rotation of a single vortex; the strain field induced by a given vortex on the other; the mutual induction of the perturbations of both vortices on each other. Given the unequal strength of the counter-rotating vortices considered here, they both also rotate about their combined centroid.

The growth rate, $\gamma$ , of the instability is estimated by the largest eigenvalue of the instability equation matrix, scaled by the larger vortex strength $\varGamma _{p}$ and the separation distance $b$ as follows:

(5.1) \begin{equation} \gamma = \frac {\varGamma _{p}}{2\pi b^{2}} {\textit{Re}}[{\gamma _{\textit{max}}}]. \end{equation}

From the above relationship, the growth rate as a function of wavelength can be estimated from the upstream measurements. We assume that the maximum growth rate corresponds to the wavelength observed in the experiments. As the Crow instability is a displacement instability, the wavelength and amplitude can be determined by tracking the vortex centreline displacement over time. Experimental observation of the growth rate is obtained by comparing the displacement amplitude values ( $\delta$ ) in the midstream to those in the upstream. From the estimates of the amplitudes and assuming linear interaction, the growth rate $\gamma$ can be written as

(5.2) \begin{equation} \delta (t) = \delta _{0}e^{\gamma t}. \end{equation}

Moving in the frame of reference of the vortex, $U_{0}$ and the known downstream location of interest from the initial condition ( $\Delta x$ ) can be used to estimate the growth rate $\gamma$ . The amplitudes $\delta _{0}$ and $\delta$ are estimated from the velocimetry measurements performed in the upstream and midstream locations, respectively.

5.2. Vortex kinematics comparison with linear theory

Figure 9 shows the comparison between observed (scatter points) and predicted growth rate from theory as a function of perturbation wavelength(solid lines). The magnitudes of the observed growth rate compare well to the predicted maximum growth rate, although the uncertainty in the observed growth rate measurements is considerable. Here, the large uncertainties result from the considerable physical variability of the measured phenomena. However, the wavelength of the predicted maximum growth rate is greater than the wavelength observed in the experiments. There is no deliberate forcing of the instability, so no wavelength is a priori assumed to cause the instability.

In the vortex configurations considered in the present study, the separation distance $b$ is not large compared with the vortex core radius $r_{c}$ , which violates an important assumption of the linear stability theory. This results in the predicted wavelength being more sensitive to the vortex core properties such as $r_{c}$ , as is evident by the large variation from the expected maximum wavelength for the Case N.0. However, since the growth rate depends more on the circulation ratio and the separation distance, the predicted and measured wavelengths match reasonably well. This suggests that in the midstream region, the vortex kinematics is driven by mechanisms that are well captured by linear stability theory.

Figure 9. Comparison of measured and linear-stability-theory-based instability growth rates shows reasonable agreement between the observed and predicted growth rates and wavelengths.

6.1. Identifying secondary vortices and their properties from velocimetry measurements

Unlike in the midstream and upstream regions, in the downstream region, the secondary vortex is seldom oriented in the streamwise direction. Since the legs of the secondary loops are generally oriented in the vertical and spanwise direction, a singular laser sheet orientation to measure core properties is not practically feasible. While a SPIV plane or thin STB volume can readily be oriented perpendicular to the axes of the vortices in the upstream sections, this is not possible in the downstream region. Additionally, the secondary vortex wanders through a much larger region in the spanwise and vertical direction as the various portions of the secondary (peaks, troughs and legs) move past this streamwise location. As a result, a larger measurement domain in the vertical and spanwise directions is required. Having a large vertical and spanwise extent imposes a constraint on the thickness in the streamwise direction to measure reliable particle tracks. A thicker streamwise direction would lead to a higher particle number density and inhibit STB from tracking particles reliably. To address the challenging constraints imposed by the flow and the velocimetry measurement technique, only a portion of a particular secondary loop can be measured in a given instant.

To measure vortex properties in the downstream location, the locations of the primary and secondary are found first at the streamwise midplane of the measurement domain. Then a small plane orthogonal to each vortex at that location is interpolated. The location of these planes is recorded, and the vortex properties are calculated on these planes with the same method as used in the preceding section. Figure 10 shows an example section of the secondary vortex and the locally orthogonal plane. Here STB-VIC# is used for determining the velocity fields in the vortex flows considered here. While STB offers the highest spatial resolution of PTV methods by allowing high particle concentrations, the choice of method to calculate Eulerian velocity and pressure (VIC# in our case) is an evolving area of study. For each case considered, three runs of 15 000 images were taken.

Figure 10. Estimation of vortex properties from STB-based velocimetry measurements. (a) An isometric view of a leg of the secondary vortex (nearly vertically oriented $\lambda _{2}$ isosurface along the blue arrow, coloured by streamwise vorticity, $\omega _{z}$ , which is scaled by foil tip thickness $d_s$ and free stream velocity $U_0$ ) as it transits through the measurement domain near the primary vortex (horizontally oriented isosurface near the red arrow). The plane bisecting the secondary vortex is normal to the secondary vortex at that location, and the data on that plane are extracted and examined independently as in (b). On this plane, the vortex circulation, radius, velocity, strain rate, pressure and other properties are calculated and recorded. The boundary for calculations is indicated by the red line. The black arrow indicates the vortex centre and radius, and the black lines indicate the axes for calculation of eccentricity.

6.2. Vortex motion and topology identification

As the vortices move in the downstream location, their positions are tracked over time when finding the locally orthogonal planes. To illustrate this, figure 11(a) shows the separation distance between the primary and secondary in the downstream region scaled by separation distance in the upstream region, $b/b_0$ , for Case N.2 in an arbitrary time range. A slightly periodic behaviour is observed, exhibiting the characteristics of the loop behaviour discussed above. The secondary vortex is at a peak when the separation is low and a trough when the separation is large. A middling distance would typically indicate a `leg’ in the secondary, a classification made in conjunction with the vertical component of vorticity at that point.

Figure 11. (a) The distance ( $b$ , scaled by upstream vortex separation, $b_0$ ) between the secondary vortex and primary vortex in the midplane of the downstream measurement domain over time (scaled by the circulation of the primary vortex, $\varGamma$ , and the (upstream) separation distance $b_0$ for Case N.2). (b) The vertical component of the vorticity unit vector for the secondary vortex in Case N.2 over an arbitrary timespan. A vertical component of nearly 1 indicates that that portion of the secondary is a ‘leg’ of the secondary. This can be used in combination with vortex separation to determine whether the secondary is in a peak, trough or leg.

Another indication of the topology of the secondary loops is the vertical component of vorticity (in the global coordinate system), shown in figure 11(b) for an arbitrary range of time. When the absolute value of the vertical component of vorticity is large, the portion of the secondary in the measurement domain is a leg region. This is also the case if the spanwise component of vorticity is large, although the secondary legs tend to be oriented more vertically than in the spanwise direction. Together, the separation distance between the vortices and the relative local orientation of the secondary are used to define whether the secondary is at that instant in a peak, trough or leg. In other words, from time-resolved PTV measurements, vortex properties are found at every time step and used to assign that instance to a category of peak, trough or leg.

6.3. Axial flow, core size reduction and straining in the secondary vortex

Having defined a method for categorizing and measuring the vortices in the downstream, the secondary vortex properties can now be analysed. Given that the legs of the secondary vortex are generally orthogonal to the axis of the primary vortex, they are in line with the strain field induced by the primary. This strain field of the primary is then assumed to cause significant axial straining in the secondary vortex. With the stretching of the secondary vortex, it is possible that significant axial jetting may develop; however, our measurements suggest that the magnitude of any axial jetting in the secondary is small. Figure 12(a) shows a histogram of axial flow velocities for the legs of the secondary in Case N.2. This axial velocity is defined by considering the velocity in the core of the vortex, projected in the direction of the vortex at that point, and comparing it with the velocity of the neighbouring area outside the vortex that has also been projected in the direction of the vortex. As can be seen, no substantial axial velocity is noted. Similarly, the local strain rate tensor can be projected in the direction of the local vortex, as shown in figure 12(b) for Case N.2. No substantial axial straining is found. Although the magnitude of axial straining and jetting in the secondary is not uniquely zero at all times, it is essentially in the noise floor for our measurements of velocity gradients. As a comparison, the vorticity magnitude in the secondary, also in figure 12(b), is several times larger than even the outliers of the axial straining data. The other cases (N.0, N.4 and M.4) also show minimal axial jetting and straining.

Figure 12. (a) Probability density function of the axial velocity, $U_{\textit{axial}}$ , in the secondary vortex legs for Case N.2. There is generally minimal axial flow relative to the surrounding fluid, indicating that axial jetting is not present here. (b) The axial strain rate projected along the axis of the secondary vortex legs ( $\hat {\omega } \boldsymbol{\cdot }S_{ij} \boldsymbol{\cdot }\hat {\omega } d_s/U_0$ , the hat indicates a unit vector in the direction of the vortex) for Case N.2 is relatively small. The vorticity in the centre of the vortex ( $\omega$ ) is much greater than the strain rate along the axis, indicating that the swirling velocity gradients are more important than axial straining gradients for pressure drop.

The findings reported in this section thus indicate that the effects of axial jetting and straining in the secondary vortex legs are small. As a check of this conclusion, it is useful to estimate the strength of the strain field generated by the primary vortex in the region of the secondary vortex. This straining should scale as

(6.1) \begin{equation} \beta \propto \frac {\varGamma _{p}}{b^2}, \end{equation}

where the separation distance, $b$ , is the local, not the upstream, initial separation distance. On the other hand, the strength of the vorticity in the region of the secondary vortex should behave as

(6.2) \begin{equation} \omega \propto \frac {\varGamma _{s}}{r_{c,s}^2}. \end{equation}

Using typical values for the vortex circulations, sizes and separation distance of the primary from the legs, the magnitude of axial straining is less than a tenth of the vorticity magnitude. This indicates that the local swirling motion dominates the local axial straining in the legs of the secondary vortex. The scaling of the axial straining above also comports with the growth in amplitude of the secondary vortex sinusoidal instability structures between the midstream and downstream measurements.

Although axial strain and jetting have small effects in the secondary vortex, the reduction in core size in the secondary results has a significant effect. Figure 13(a) shows the typical core size of the secondary vortex for the various cases at the upstream, midstream and downstream locations. Although the difference between the upstream and midstream radii is relatively small, the radius is significantly reduced downstream. Figure 13(b) presents the radius of the secondary in the downstream as it varies depending on the location of the measurement (e.g. in the legs or troughs) compared with the overall average. The legs of the secondary have the smallest radius as they are stretched the most. Figure 13(c) presents the measured circulation of the secondary vortex across the conditions at the various locations. The change in circulation across the locations is unlikely due to viscous dissipation, as discussed in Lundgren & Koumoutsakos (Reference Lundgren and Koumoutsakos1999) and Terrington, Hourigan & Thompson (Reference Terrington, Hourigan and Thompson2022). At some time instances, the secondary vortex peaks and primary vortex are near enough to lead to vorticity annihilation. However, viscosity can result in vorticity diffusion, which may affect the interaction of the vortices. In addition, the method of circulation calculation using vorticity thresholding and insufficient spatial resolution of the measurement can result in the observed trends.

Figure 13. Comparison of secondary vortex radius for different configurations across different locations. Panel (a) shows the variation of vortex radius at different locations for the configurations considered, and (b) shows the vortex core radius measured in the legs, troughs and overall. Panel (c) shows the change in vortex circulation measured at different locations for different configurations considered. No appreciable change in circulation is measured, however, vortex core radius reduces suggesting an increase in peak vorticity.

6.4. Vortex strain-out leading to cavitation suppression in the primary vortex

As noted in § 3.2, under certain conditions, the secondary vortex suppresses cavitation in the primary vortex when the free stream pressure is low. This happens only for the N.2 case as the secondary vortex peaks draw near to the primary and wrap around it. Here we use the vorticity fields from the measurements to show quantitatively what was previously observed only in videos of developed cavitation.

Figure 14. A cross-section (streamwise normal) of the time evolution of the secondary and primary vortices as the peak of the secondary vortex is drawn near the primary vortex and coloured by streamwise vorticity. Shown here for Case N.2, the compact core of the secondary allows it to strain out the primary vortex. Initially, at t =0, the secondary is in a trough and far away from the primary at this location. As it is drawn near, at t =3.2 ms and after, the previously near circular cross-section of the primary vortex is then deformed to take an elongated shape. By t =4.8 ms, the primary is strained to the point of almost breaking into two distinct vortex patches. The upper patch is effectively peeled off and apparently merges with the secondary vortex as the secondary vortex breaks apart and orients itself spanwise, although this cannot be seen in this view. This straining out of the primary leads into a reduction in pressure drop in the primary vortex and thus a rise in core pressure, suppressing cavitation. After this interaction, the secondary and primary have a configuration similar to their initial state (as shown at t =12.0 ms). A movie of this interaction can be found in the Supplementary material.

Figure 14 shows a time series of the vortical interaction as the secondary vortex wraps around the primary vortex in a plane cutting through both vortices (streamwise normal), displayed with streamwise vorticity. Initially, the secondary vortex is in a trough state (t =0 ms). It is then drawn up so that it is at a peak (t =3.2 ms). The previously circular cross-section of the primary vortex now takes an elliptical form as it is strained out by the secondary vortex (t =4.0 ms). As the interaction continues, this elliptically shaped primary separates out into two vortical patches with near circular cross-sections (t =4.8 ms), and these patches have a connecting region of weaker vorticity. While not apparent from this view, the upper portion of the primary vortex is subsequently peeled off and joins with the secondary vortex as that portion of the secondary vortex breaks into vortex rings, since those portions of both vortices have spanwise oriented vorticity with the same sign.

Figure 15 shows a top-down view which illustrates this. Initially, at t = 0, the secondary is in a trough and far away from the primary at this location. The secondary is drawn up near the primary by its self-induced motion (t =4.0 ms). By t =4.8 ms, the primary is strained to the point of almost breaking into two distinct vortex patches, though this appearance is somewhat influenced by the isosurface level. At t =5.2 ms, the primary upper patch has separated and is now oriented as two spanwise-oriented sections. This continues into t =5.6 ms, and by t =6.0 ms and after, the secondary has also broken apart as it will begin to form vortex rings. By t =8.0 ms, it continues to appear that the upper section of the primary has been stripped off and joined to the secondary in spanwise-oriented structures. As the vortices convect downstream, they return to their prior compact shapes (t =12.0 ms).

Re-examining figure 14, the streamwise vorticity of both vortices is reduced, and spanwise oriented vorticity is present (t =6.4 ms). Meanwhile, the lower portion of the primary then stays connected to the rest of the primary as the secondary leg passes through. Finally, the secondary is back to a trough state and the primary is back to a streamwise-oriented vortex without a noticeably elliptical cross-section (t =12.0 ms).

Figure 15. A top-down view of the time evolution of the secondary and primary vortices as the peak of the secondary vortex is drawn near the primary vortex. Shown here with $\lambda _2$ isosurfaces, coloured by streamwise vorticity, $\omega _z$ , for Case N.2, the compact core of the secondary allows it to strain out the primary vortex. A movie of this effect can be found in the Supplementary material.

As this process occurs, the elliptical cross-section of the primary shows increasing pressure in the core region (to be discussed in detail in a companion paper), leading to suppression of cavitation in the primary when the vortices undergo developed cavitation. The dynamics suggested in the prior paragraph are also indicated by the high-speed video in that cavitation in the primary vortex often reinitiates after the suppression in the lower portion. In other words, the lower of the two patches of vorticity noted above has a drop in pressure as it regains a more circular cross-section. From the velocimetry, we see that the existing lower patch of vorticity is no longer straining cross-axially so its pressure drops correspondingly. Figure 16 shows the eccentricity of the vortices as the secondary is in a peak state. Both vortices are strained into more elliptical shapes and thus have higher eccentricity than when the secondary is in a trough state. However, despite the primary having a greater strength, the primary also has a higher eccentricity, presumably due to the compact nature of the secondary, which allows the secondary to better resist straining deformation than the primary.

Figure 16. The eccentricity of the secondary and primary vortices depends on how close they are to each other, as shown here with Case N.2. The eccentricity of both is increased as they draw near to each other (when the secondary is in the peaks (a) and (c)) relative to when they are far from each other (when the secondary is in troughs (b) and (d)). Notably, the primary (c) is made more eccentric than the secondary (a) when the secondary is in a peak even though the secondary is weaker. This is due to the smaller core radius of the secondary.

The interaction process described here was noticed first by Marshall et al. (Reference Marshall, Brancher and Giovannini2001). They used computations to examine the interaction of pairs of counter-rotating vortices of varying circulation ratios, albeit at lower Reynolds numbers than those considered here. In one case, their Case E, they prescribed a weaker vortex ( $\varGamma _S/\varGamma _S$ = −0.25 in our nomenclature) with a smaller radius than the primary $r_{c,s}/r_{c,p}$ = 0.5 while other cases had equal core radii sizes but different vortex strengths. They noted that with a smaller radius, the secondary would have higher peak vorticity in its core than the primary vortex. Due to the higher vorticity, the secondary would more readily withstand straining and was more able to strip vorticity from the primary. In their case, they found substantial cross-axial straining of the primary vortex. However, the secondary vortex eventually weakened due to the diffusion of vorticity between the two vortices as they near. This allowed the primary to regain its originally circular cross-section and subject the secondary to significant cross-axis straining. The case noted here has a relatively stronger secondary ( $\varGamma _S/\varGamma _P$ = −0.4) and a relatively smaller secondary radius. Accordingly, the same dynamics as Marshall et al. (Reference Marshall, Brancher and Giovannini2001) are not observed. Instead, the secondary strains out the primary and evidently strips vorticity from it during the secondary’s breakup into rings. Another effect that could lead to the differing results between the present study and Marshall et al. (Reference Marshall, Brancher and Giovannini2001) is that at the lower Reynolds numbers they studied, the viscous diffusion between the two vortices would have more of an effect. This would weaken the secondary vortex peak vorticity more than in our case, so the secondary they consider would be more apt to strain out under the influence of the primary vortex.

Case N.4 considered in the present study has a circulation ratio almost exactly that considered in Case E of Marshall et al. (Reference Marshall, Brancher and Giovannini2001), but the secondary core radius is smaller. Correspondingly, the peak vorticity in the secondary is generally noticeably higher than that in the primary in the N.4 case. As suggested by the physics discussed in Marshall et al. (Reference Marshall, Brancher and Giovannini2001), the secondary in the N.4 case resists cross-axial straining out better than the secondary in Case E of Marshall et al. (Reference Marshall, Brancher and Giovannini2001). The secondary does strain the primary into an elliptical cross-section in both Case E and Case N.4. However, the secondary in Case N.4 is not strong enough to strain the primary to the point of suppressing cavitation in it as in Case N.2. This likely results from the weaker strength of the secondary in Case N.4 as opposed to Case N.2.