2.1.1. Facility

The experiment was conducted in the closed-loop wind tunnel at Shanghai Jiao Tong University. The dimensions of the test section were $1.2\times 0.9\ \textrm {m}$. The free-stream velocity was set to $U_\infty =15$, $30$, $45\ \textrm {m}\ \textrm {s}^{-1}$, and the turbulent intensity was below 0.5 %. The boundary layer thickness was estimated to be less than 3.6 % of the height of the test section. The isolated trailing vortex and the vortex pair were generated by a swept wing of NACA0015 profile without and with a split winglet. The wing adopted the standard M6 wing platform and the winglet was of the Whitcomb configuration (Whitcomb Reference Whitcomb1976). The geometry of the wing platform and the winglet is shown in figure 1(a). The combination of an M6 wing platform and a Whitcomb winglet served to capture the complexity of an aircraft wingtip while remaining a clean, canonical wing model. The wing root was attached to an endplate to avoid any influence from the wind-tunnel boundary layer, as in Birch et al. (Reference Birch, Lee, Mokhtarian and Kafyeke2004). The wing model was mounted on a rail to allow an adjustment of the streamwise locations. The wingtip chord was $c = 0.08\ \textrm {m}$, which gave a chord-based Reynolds number of $Re_c={U_\infty }c/\nu =0.8\times 10^{5}, 1.6\times 10^{5}$ and $2.4\times 10^{5}$ for the three measured free-stream velocities. The incident angle was tuned to $6^{\circ }$, $8^{\circ }$ and $10^{\circ }$ to generate sufficient swirl. A schematic diagram of the SPIV experimental set-up is shown in figure 1(b).

Figure 1.

Schematic illustration of (a) the M6 wing model and the winglet geometry, (b) the SPIV set-up.

2.1.2. Velocity measurement

The three-component velocity field at each slice perpendicular to the free stream from $x/c=3\sim 14$ was measured by a SPIV system. The flow was seeded by droplets of atomized glycol with a diameter of 1 to $5\ {\mathrm {\mu }}\textrm {m}$, which were allowed to recirculate in the tunnel until a uniform distribution was achieved. The particles were then illuminated by a double-pulsed Nd:YAG laser ($2\times 380$ mJ) of 532 nm wavelength. The beam was formed into a 2 mm-thick light sheet throughout the measured frame using a cylindrical lens. The images were captured by two 12-bit CCD cameras (PCO 2000, $2048\times 2048$ pixels) positioned on each side of the test section with a $30^{\circ }$ angle of view to the free stream. To capture both vortices in the vortex pair, the calibrated frame of view was adjusted to $292\times 269$ mm, which yielded a raw-image resolution of $0.13\ \textrm {mm}\ \textrm {pixel}^{-1}$. The cameras and the laser were synchronized by a digital delay generator (DG646), and the time interval between two consecutive shots was tuned to ${\Delta }t=15\ {\mathrm {\mu }}\textrm {s}$, $7.5\ {\mathrm {\mu }}\textrm {s}$, $4\ {\mathrm {\mu }}\textrm {s}$ for $U_\infty =15\ \textrm {m}\ \textrm {s}^{-1}$, $30\ \textrm {m}\ \textrm {s}^{-1}$, $45\ \textrm {m}\ \textrm {s}^{-1}$, respectively, to ensure that the particles in each image pair remained in the light sheet. For each measured flow condition, 300 consecutive images were captured at a sampling frequency of 1 Hz to obtain the time-averaged flow field. The velocity vectors were then acquired using the cross-correlation algorithm built in the TSI Insight 4G software. Double interrogation windows were used to improve the resolution of the velocity vectors, with a starting interrogation window of $72\times 72$ pixels and an ending window of $36\times 36$ pixels. The unresolved vectors were interpolated by the surrounding vectors with a filling size of $9\times 9$ pixels. The resulting percentage of good vectors was over 85 %. The relative uncertainties of the in-plane velocities and the derived streamwise vorticity were 1 % and 3 %, respectively.

2.1.3. Parameters and models of trailing vortex

The trailing vortex is characterized by its core radius $\delta$, axial velocity deficit $\xi$ and circulation $\varGamma$. The Reynolds number for the vortex is $Re_\delta = \xi \delta /\nu$ as in Mayer & Powell (Reference Mayer and Powell1992) and Fabre & Jacquin (Reference Fabre and Jacquin2004). These parameters were extracted by fitting the measured vortex with the Batchelor vortex, which reads in cylindrical coordinates as

(2.1a–c)\begin{equation} U_\theta=\frac{\varGamma}{2 {\rm \pi}r}\left( 1-\exp\left(- \frac{r^{2}}{{\delta}^{2}}\right) \right),\quad U_r=0,\quad U_x=U_\infty-\xi\exp\left(- \frac{r^{2}}{{\delta}^{2}}\right). \end{equation}

The swirl number is thus $q = \varGamma /(2{\rm \pi} \delta \xi )$, and the axial vorticity can be written as

(2.2)\begin{equation} \omega_x=\frac{\varGamma}{{\rm \pi}\delta^{2}}\exp\left( - \frac{r^{2}}{{\delta}^{2}} \right). \end{equation}

2.2.1. Governing equations

Considering the range of free-stream velocities measured in the present study, the incompressible, linearized Navier–Stokes equations (LNSEs) were used in the stability analysis. For the isolated trailing vortex, owing to its axial symmetry, the governing equations were transformed onto non-dimensionalized cylindrical coordinates $(\tilde {r},\theta ,\tilde {x})$. The equations were non-dimensionalized by the chord length and the free-stream velocity. These parameters corresponded to a core-radius-based Reynolds number of $Re_c\tilde {\delta }=14\,000$ to $58\,000$. The instantaneous flow field $\boldsymbol {q}=[u_r,u_\theta ,u_x,p]^{\textrm {T}}$ was decomposed into a steady basic state $\bar {\boldsymbol {q}}$ and an infinitesimal perturbation term $\boldsymbol {q}'$. The perturbation $\boldsymbol {q}'$ was then spanned into a Fourier series in the axial, tangential and timewise directions where the flow can be considered homogeneous, written as

(2.3)\begin{equation} {\boldsymbol{q'}}(\tilde{r},\theta,\tilde{x},\tilde{t})={\hat{\boldsymbol{q}}}(\tilde{r})\exp[\textrm{i}\left( {\alpha}\tilde{x}+m\theta-{\omega}\tilde{t} \right))]+\boldsymbol{c.c.} \end{equation}

in which $\alpha$ is the axial wavenumber, $m$ is the azimuthal wavenumber, $\omega$ is the frequency, c.c. is the conjugate complex and ${\hat {\boldsymbol {q}}}=[\textrm {i} \hat {u}_r,\hat {u}_\theta ,\hat {u}_x,\hat {p}]^{\textrm {T}}$ is the eigenvector. Here, the temporal local analysis is performed by solving the generalized eigenvalue problem (EVP) $\boldsymbol{\mathsf{A}}\hat {\boldsymbol{q}}={\omega }{\boldsymbol{\mathsf{B}}\hat {\boldsymbol{q}}}$ rearranged from the LNSE, where $\omega =\omega _r+\textrm {i}\omega _i$ is the complex frequency. The real part $\omega _r$ is the frequency of the perturbation and the imaginary part $\omega _i$ corresponds to the growth rate.

Differing from the isolated trailing vortex, while the parallel assumption still holds in the streamwise direction for a fully-developed vortex pair, the axial symmetry is broken for the vortex pair. Therefore, bi-global linear stability analysis was performed in the vortex pair scenario, which was conducted in non-dimensionalized Cartesian coordinates $(\tilde {x}, \tilde {y}, \tilde {z})$ with ${\boldsymbol {q}}=[u,v,w,p]^{\textrm {T}}$. All parameters and governing equations were non-dimensionalized by the core radius $\delta$ and characteristic velocity $U_c=\varGamma /2{\rm \pi} \delta$ of the upper vortex in the vortex pair. The bi-global analysis then allowed for resolved gradients in the cross-wise directions $(\tilde {y}, \tilde {z})$, and treated the streamwise gradient as the span of the Fourier series. The infinitesimal perturbation in the bi-global framework is hence given by

(2.4)\begin{equation} \boldsymbol{q'}(\tilde{x},\tilde{y},\tilde{z},\tilde{t})={\hat{\boldsymbol{q}}}(\tilde{y}, \tilde{z})\exp[\textrm{i}\left( {\alpha}\tilde{x}-{\omega}\tilde{t}\right))]+\boldsymbol{c.c.}\end{equation}

By substituting $\boldsymbol {q'}$ into the LNSE and transforming the equations into a generalized EVP $\boldsymbol{\mathsf{A}}_{\boldsymbol{g}}\hat {\boldsymbol {q}}=\omega \boldsymbol{\mathsf{B}}_{\boldsymbol {g}}\hat {\boldsymbol{q}}$, the complex frequency $\omega$ and the associated eigenmode ${\hat {\boldsymbol {q}}}=[ \hat {u},\hat {v},\hat {w},\hat {p}]^{\textrm {T}}$ were sorted to quantify the asymptotic instability of the flow. More details of the local and global stability matrices and their boundary conditions are presented in Appendices A1 and A2.

2.2.2. Description of the numerical method

In this work, the Chebyshev spectral collocation method was chosen for discretization in both the local and bi-global linear stability analyses. As the physical domain ranged from $\tilde {r}=0$ to the truncation length $\tilde {r}_{max}$ for the isolated trailing vortex and $(\tilde {y}_{min}, \tilde {z}_{min})$ to $(\tilde {y}_{max}, \tilde {z}_{max})$ for the vortex pair, different mapping functions were employed in the local and global analyses to map the Chebyshev collocation points to the physical domain of the problem. A fine-tuned two-dimensional mapping was constructed to resolve both vortices in the vortex pair scenario, whose details can be found in Appendix A3.

The discretized EVP was then solved by the QR method for the local analysis. For the bi-global analysis, the QR method was only used when a full eigenvalue spectrum was needed. The spurious eigenvalues were filtered by their dependence upon discretization according to Oberleithner et al. (Reference Oberleithner, Sieber, Nayeri and Paschereit2011). To save computational cost, the global eigenmodes were acquired by the Arnoldi algorithm.

The numerical methods and mapping functions have been validated by Khorrami, Malik & Ash (Reference Khorrami, Malik and Ash1989), Mayer & Powell (Reference Mayer and Powell1992), Theofilis, Duck & Owen (Reference Theofilis, Duck and Owen2004) and Hein & Theofilis (Reference Hein and Theofilis2004). For the local analysis, the temporal growth rates for the classical pipe Poiseuille flow with/without rotation and the $q$-vortex model were acquired by the present method, which agreed with the published data to at least four significant digits (see table 3 in Appendix B.1). The bi-global stability code was validated by the published data for the duct flow and vortex/vortex pair. The results showed excellent agreement with Theofilis et al. (Reference Theofilis, Duck and Owen2004), Hein & Theofilis (Reference Hein and Theofilis2004) and the local results of Mayer & Powell (Reference Mayer and Powell1992) (see figure 28 in Appendix B.1). A grid-convergence verification (table 4) resulted in a collocation point of $N = 200$ for the local analysis and $(N_Y, N_Z) = (72, 72)$ for the bi-global analysis.

4.1.1. Spectra and mode shape with different azimuthal wavenumbers

In the scenario of the isolated trailing vortex, local stability analysis was performed on the measured vortex profiles, which were fitted to the Batchelor vortex. The purpose of fitting the measured vortex to a Batchelor vortex is to obtain a base flow as an exact solution of the N–S equation (Theofilis Reference Theofilis2003). The measured velocity profiles were not recentred to correct for vortex wandering before fitting, as the wandering amplitude was sufficiently small in the present research to have negligible effect on the mean profile. Owing to an inversion of the axial direction in the Batchelor vortex used herein, the direction of the azimuthal wavenumber $m$ here was opposite to that in Lessen et al. (Reference Lessen, Singh and Paillet1974), Lessen & Paillet (Reference Lessen and Paillet1974), Khorrami (Reference Khorrami1991) and Fabre & Jacquin (Reference Fabre and Jacquin2004). In this research, modes with positive $m$ represent perturbations countering the vortex rotation, and vice versa for negative $m$.

Figure 9 shows the local eigenspectrum of an isolated trailing vortex for different $m$ at $\alpha = 2$. Each spectrum for $m = 0, \pm 1, \pm 2$ consisted of three families. These families shared a similar topology as the A, P and S families categorized by Mack (Reference Mack1976) for the boundary layer. Therefore, we kept the naming used in Mack (Reference Mack1976) for the trailing vortex herein. A convergence study shown in figure 9(c,d) reveals two types of discrete modes in the P and A families. These were named as Mode-P and Mode-A. The structure of these discrete modes, which is transformed from one-dimensional eigenfunctions, is presented in figure 10. The S family, which corresponds to the continuous modes in the free-stream (Mao & Sherwin Reference Mao and Sherwin2011), will not be discussed herein.

Figure 9.

Eigenspectra with $m = 0, \pm 1, \pm 2, \alpha = 2.0$, (a) non-negative $m$, (b) negative $m$, (c,d) zoomed-in view of the $|m| = 1$ spectra with different numbers of collocation points. The flow condition is $AoA = 8^{\circ }, Re_c = 1.6\times 10^5$. Mode-P and Mode-A are marked by $\square$ and ${\bigcirc}$, respectively.

Figure 10.

Axial vorticity distribution of Mode-P and Mode-A with $m = 0, \pm 1, \pm 2$ for $\alpha = 2.0$ as marked in figure 9. The contour range is uniformly $[-10, 10]$. The dashed circle denotes the vortex core.

From figure 9(a,b), it can be observed that Mode-P was exclusive for $|m|\leqslant 1$, and was otherwise missing for higher $|m|$. Such a mode was neutral given its closeness to $\omega _i=0$. Mode-P with $m = 0$ is an axisymmetric mode (figure 10a1); and Mode-P with $|m| = 1$ is of a dipole structure, which corresponds to a displacement wave (figure 10b1,b2). The latter gives rise to the coherent motion of vortex wandering (Edstrand et al. Reference Edstrand, Davis, Schmid, Taira and Cattafesta2016).

For Mode-A, those with non-positive $m$ ($m = 0, -1, -2$) completely fell into the stable half-plane of the eigenspectrum (figure 9a). Mode-A with $m = 0$ was of an axisymmetric structure (figure 10a2), while those with $m = -1, -2$ featured a short spiral structure (figure 10c1,d1). For Mode-A with positive $m$ ($m = 1, 2$), we observed a piercing into the unstable half-plane, which indicated that the measured trailing vortex was temporally unstable with these azimuthal wavenumbers. The observation hence distinguished the measured trailing vortex from the two-dimensional Lamb—Oseen vortex where all the modes were essentially damped (Fabre et al. Reference Fabre, Sipp and Jacquin2006). The structure of Mode-A with positive-$m$ featured distinct spiral structures winding around the vortex centre (figure 10c2,d2). This structure was similar to that of the purely viscous mode, as reported by Fabre et al. (Reference Fabre, Sipp and Jacquin2006), but extended further beyond the core radius.

4.1.2. Stability curves for different modes

To further quantify the instability of trailing vortex, the growth rates of Mode-P and Mode-A are presented in figure 11 as a function of axial wavenumber $\alpha$. The resulting curve is referred to as the stability curve in the present study. Figure 11 confirms that the trailing vortex was dominated by Mode-A with positive-m, whose growth rate was the highest amongst all the modes and increased with $m$ until $m = 4$. Such a mode grew in the asymptotic limit and overwhelmed the other modes. Mode-A with negative-$m$ was stabilized monotonously with the increase of $|m|$ and remained stable for the entire dispersion relation. Mode-P with $|m| \leqslant 1$ remained marginally stable until $\alpha = 5.0$, beyond which it was completely stabilized. A zoom-in image of the neutrally stable Mode-P revealed that the $m = 1$ mode decayed faster at large axial wavenumbers than the $m = -1$ mode and the axisymmetric mode ($m = 0$) was indeed marginally unstable.

Figure 11.

Growth rate $\omega _i$ as a function of the axial wavenumber $\alpha$ for (a) Mode-P with $m = -1$ and Mode-A with $m = [-8, 8]$, (b) Mode-P with $m = 0, \pm 1$ (zoomed-in view). Solid curves, Mode-P; dashed curves, Mode-A. The labels denote $m$ for each curve. The flow condition is the same as that in figure 9.

The characteristics of this dominating Mode-A with positive-m was examined for the rest of the flow conditions. Figure 12 shows that the trailing vortex at $AoA = 8^{\circ }, Re_c = 2.4\times 10^5$ was the most unstable amongst the measured flow conditions. The corresponding growth rate peaked at $m=6$. The envelopes of the stability curves for the trailing vortex at $AoA = 6^{\circ }$ and $8^{\circ }, Re_c = 1.6\times 10^5$ almost coincided with each other. Both of them peaked at $m = 4$ and manifested an unstable nature. For the trailing vortex at $AoA = 10^{\circ }$, Mode-A was stabilized, probably owing to an increased swirl, and fell in the stable half-plane.

Figure 12.

Envelopes of the Mode-A stability curve at different flow conditions, as shown by the solid lines. The envelope at $AoA = 8^{\circ }, Re_c = 1.6\times 10^{5}$ is substituted by its stability curves (dashed line) as shown in figure 11 for a better illustration. The labelled brackets denote $[AoA, Re_c]$, and the most unstable $m$ of each flow condition is marked by $\circ$.

It is noticeable from figure 12 that the degree of instability, as demonstrated by the maximum growth rate, matched qualitatively with the wandering amplitude in figure 7. The vortex with a high growth rate, e.g. at $AoA = 8^{\circ }, Re_c = 1.6\times 10^{5}$ with $\omega _{i,max}$ reaching up to 0.3, manifested a larger wandering amplitude than the less unstable vortices, e.g. at $AoA = 10^{\circ }, Re_c = 1.6\times 10^{5}$. Such a correlation between the wandering amplitude and the growth rate was also observed for the other flow conditions. In addition, the temporal frequencies of Mode-P ($m=-1$) were in the range of 5–15 Hz for the measured condition, which agreed well with the frequencies of vortex wandering reported by Devenport et al. (Reference Devenport, Rife, Liapis and Follin1996) and Bailey & Tavoularis (Reference Bailey and Tavoularis2008) at similar flow conditions.

6.2.1. Shear penetration for the isolated trailing vortex

The penetration depth of the dominating Mode-A with positive-$m$ was inspected as a function of $\alpha$ along the most amplified stability curve, as in figure 12. Figure 21 shows that the penetration depth enlarged significantly with increasing $\alpha$. For the unstable vortex at $[AoA, Re_c] = [6^{\circ }, 1.6\times 10^{5}; 8^{\circ }, 1.6\times 10^{5}; 8^{\circ }, 2.4\times 10^{5}]$, the penetration depth increased drastically after $\alpha$ exceeded a threshold. For the less unstable vortex of $AoA = 8^{\circ }, Re_c = 0.8\times 10^{5}$, $d$ increased with $\alpha$ but only mildly. For the vortex at $AoA = 10^{\circ }, Re_c = 1.6\times 10^{5}$, however, $d$ was maintained at approximately zero. A similar trend where the perturbation modes enlarge in size with increasing $\alpha$ was also observed by Fabre et al. (Reference Fabre, Sipp and Jacquin2006). However, Mode-A herein was found to transit in terms of its modal structure for large $\alpha$.

Figure 21.

Penetration depth along the most amplified stability curve of the isolated vortex. (a) Penetration depth versus axial wavenumber, (b) a zoom-in view of the framed zone in (a) to show the small variation of $d$ for $[AoA, Re_c] = [8^{\circ }, 0.8\times 10^{5}; 10^{\circ }, 1.6\times 10^{5}]$.

As the growth rate is a function of $\alpha$, the nearly monotonic dependence of penetration depth on $\alpha$ infers that there should be a limiting penetration depth at which the perturbation amplifies the most. Figure 22 demonstrates $\omega _i$ as a function of $\alpha$. Notably, the growth rate of the four asymptotically unstable vortices peaked at an approximately identical penetration depth, which fell into a narrow band of $d = 0.14\sim 0.24$. The penetration depth corresponding to the neutrally stable vortex at $AoA = 10^{\circ }, Re_c = 1.6\times 10^{5}$, however, was below 0.01. The coincidence of the limiting $d$ for the four unstable trailing vortices is interesting as $d$ is indeed a rather sensitive parameter given its fractional form. A unified mechanism accounting for asymptotically unstable perturbations is hence expected to be underlying this specific parameter.

Figure 22.

Variation of the growth rate with penetration depth for the isolated trailing vortex.

The existence of the limiting $d$ was further illustrated in figure 23(a,c,e,g,i). As $\alpha$ increased, the perturbations corresponding to the four unstable cases (figure 23a,c,e,g) gradually spread over the core boundary where the vortex shear was located. The only exception was the case at $AoA=10^{\circ }, Re_c=1.6\times 10^{5}$, given the stable nature of its Mode-A. Notably, as the axial wavenumber continued to increase, perturbations in the four unstable cases convected away from the shear region, which is consistent with the dramatic increase of $d$ shown in figure 21. For the most-amplified modes in the four unstable cases, their relative locations with the core boundary were strikingly identical, which corresponded to the narrow band of limiting $d$ in figure 22. The limiting $d$ signified that there exists an extent of shear-penetration where the perturbations are most amplified. From the variation of perturbation with $\alpha$, as shown in figure 23, it was implied that there might be an optimal interaction between the perturbation and the vortex shear.

Figure 23.

Mode variation with the axial wavenumber for the isolated trailing vortex: radial distribution of $|\hat {u}_x|$ (a,c,e,g,i) and the corresponding $\gamma$ (b,d,f,h,j). The red curve corresponds to the most unstable perturbation. The critical point shifts continuously and the grey thin lines fill the gap in between, as shown in the right column for clearer illustration. The vertical line marks the core boundary. Flow conditions and azimuthal wavenumbers $[AoA, Re_c, m]$: (a,b) $[6^{\circ }, 1.6\times 10^{5}, 4]$, (c,d) $[8^{\circ }, 0.8\times 10^{5}, 2]$, (e,f) $[8^{\circ }, 1.6\times 10^{5}, 4]$, (g,h) $[8^{\circ }, 2.4\times 10^{5}, 6]$, (i,j) $[10^{\circ }, 1.6\times 10^{5}, 1]$.

The limiting $d$ was then investigated by introducing the critical point in the trailing vortices. According to Lessen et al. (Reference Lessen, Singh and Paillet1974), the growth rate should be sensitive to the mean flow properties at this point. The critical point corresponds to the radial location $r$, where the azimuthal frequency $f_\theta$ and axial frequency $f_x$ of the perturbation cancel each other (Lessen et al. Reference Lessen, Singh and Paillet1974). The definition of a critical point is thus given by

(6.2)\begin{equation} \gamma^*=\alpha^* \left( U_x(r)-c \right)+mU_\theta(r)/r=0 \end{equation}

in which $\gamma ^*$ is the local difference between $f_\theta$ and $f_x$, and $c=\omega ^*/\alpha ^*$ is the phase speed. The asterisk denotes a dimensional variable.

In the right column of figure 23, the $\gamma$-curves correspond to the varying perturbations with increasing $\alpha$ as shown in the left column. The critical point is denoted by the zeros of the $\gamma$-curve, where $\gamma$ is non-dimensionalized by the reference time. Looking into the case at $AoA=6^{\circ }, Re_c=1.6\times 10^{5}$ as an example, as $\alpha$ increased, the critical point shifted away from the vortex centre. Below the critical point, the perturbation was featured by an excess of $f_\theta$ over $f_x$, where the flow experienced more azimuthal disturbances than axial disturbances. Comparing the left and right columns of figure 23, the peak of each mode was found to fall in the margin of $r < r_{cr}$ that pertained to the region of $f_\theta$-excess. The perturbation extended together with the region of $f_\theta$-excess and eventually permeated the core boundary as $\alpha$ increased. Furthermore, as the critical point shifted outward, the perturbation attained the maximum growth rates where the critical point coincided with the core boundary. The same was observed for the other three unstable cases, where the growth rate reached its maximum as the critical point approached the core boundary. However, the tendency for $AoA=10^{\circ }, Re_c=1.6\times 10^{5}$ was not evident as the perturbation was completely restricted in the vortex core and the shift of critical points was quite marginal. This was consistent with the stable nature of these perturbations. The correlation between the critical point and the vortex shear therefore highlights an interaction between the two characteristic layers of the $O(\epsilon )$ and the $O(1)$ flow. This interaction selects the dominating mode by the proximity of the critical point to the vortex shear layer.

6.2.2. Shear penetration for the co-rotating vortex pair

The interlink between the modal feature of perturbations and asymptotic instability was further examined for the co-rotating vortex pair. Figure 24 shows a similar increase of penetration depth versus axial wavenumber. As $\alpha$ increased, $d$ uniformly increased beyond the scope of the core region. It should be noted that the variation of $d$ was very much complicated by the global nature of the vortex pair in which both $\omega$ and $m$ changed with $\alpha$. However, a general trend of $d$ increasing with $\alpha$ was still observed.

Figure 24.

Penetration depth versus the streamwise wavenumber for the co-rotating vortex pair, (a) with different $AoA$, (b) with different $Re_c$.

Figure 25 demonstrates that a limiting $d$ also existed for each measured vortex pair at which the perturbation was most amplified. However, the limiting $d$ of the co-rotating vortex pair was larger than that of the isolated vortex, which reached up to $d > 1$. The left column of figure 26 illustrates the radial distribution of associated perturbation modes. Except for the vortex pair at $AoA=8^{\circ }, Re_c=2.4\times 10^{5}$, which pertained to a smaller limiting $d$, the perturbations permeated the core boundary completely as $\alpha$ increased. The perturbed regions for the co-rotating vortex pair were again associated with the shifting of the critical points, as shown in the right column of figure 26. The indication of the $f_\theta$-excess region favoured by the perturbations also held for the global scenario of the vortex pairs, as the peak of each global perturbation still resided in this region. As the $f_\theta$-excess region moved outward with the critical layer, the perturbations were shifted away accordingly.

Figure 25.

Variation of the amplification rate with the penetration depth on a logarithmic scale. Symbols, data points from stability analysis; curves, polynomial fittings to show the trend of the data points. Here, ${\bigcirc} , AoA=6^{\circ }, Re_c=1.6\times 10^{5}$; ${\blacktriangledown }$, red, $AoA=8^{\circ }, Re_c=0.8\times 10^{5}$; $\blacktriangle$, $AoA=8^{\circ }, Re_c=1.6\times 10^{5}$; ${\blacklozenge }$, grey, $AoA=8^{\circ }, Re_c=2.4\times 10^{5}$; ${\blacksquare }$, blue, $AoA=10^{\circ }, Re_c=1.6\times 10^{5}$.

Figure 26.

Mode variation with the axial wavenumber for the co-rotating vortex pair. Dotted vertical line, the core boundary; dashed vertical line, the separatrix between the inner and outer recirculation zone as illustrated in Appendix C. Flow conditions $[AoA, Re_c]$: (a,b) $[6^{\circ }, 1.6\times 10^{5}]$; (c,d) $[8^{\circ }, 0.8\times 10^{5}]$; (e,f) $[8^{\circ }, 1.6\times 10^{5}]$; (g,h) $[8^{\circ }, 2.4\times 10^{5}]$; (i,j) $[10^{\circ }, 1.6\times 10^{5}]$.

The relatively large value of limiting $d$ for the co-rotating vortex pair is intriguing. In the right column of figure 26, the critical points were observed to also shift to larger radial locations than those of the isolated trailing vortex. Notably, for the isolated trailing vortex, only one shear layer existed which was located at the core boundary, while for co-rotating vortex pairs, an outer recirculation zone surrounded the two vortices where the fluid circulated as a whole. The recirculation zones for the co-rotating vortex pairs consisted of two core regions: two inner recirculation zones, where the fluid circulates the individual vortex, and an outer recirculation zone (see Appendix C). We found that the critical point of the most unstable perturbation in the vortex-pair scenario aligned with the radial position of the separatrix dividing the inner and outer recirculation zone. The radial position of the separatrix was determined by its outermost point (see figure 31). This particular layer of the co-rotating vortex pair separated the fluid rotating about each vortex centre from that circulating both the vortices. This layer therefore formed a mutual shear layer of the vortex pair in addition to the shear layer at the individual core boundary. Analogous to the isolated vortex scenario, the alignment between the critical point of the most unstable perturbation and the separatrix pointed to an interaction between the two characteristic layers of the $O(\epsilon )$ perturbation field and the $O(1)$ base flow. The perturbation was amplified in the asymptotic limit as its critical point moved toward the mutual shear of the vortex pair. The results from the vortex pair scenario further substantiated that the asymptotic growth of instability is associated with an interplay between the critical points of the perturbations and the corresponding characteristic layer of the base flow.