2.1 Experimental set-up

The trailing vortex was generated by a 304.8 mm chord NACA0012 airfoil with a half-aspect-ratio of 1.25, a rectangular wingtip, and positioned at an angle of attack of $5^{\circ }$ . Results reported herein are non-dimensionalized by the free-stream velocity $U_{\infty }$ and chord $c$ for a chord-based Reynolds number of $Re_{c}\equiv U_{\infty }c/{\it\nu}_{\infty }=530\,000$ , where ${\it\nu}_{\infty }$ is the free-stream kinematic viscosity. Experimental measurements were acquired in the University of Florida Anechoic Flow Facility (Mathew et al. Reference Mathew, Bahr, Sheplak, Carroll and Cattafesta2005), a $1.12~\text{m}\times 0.74~\text{m}$ open-jet wind tunnel installed inside a semi-anechoic chamber. Owing to the open-jet nature of the facility, the wandering effects may be further exacerbated by the shear layers on three of the four sides when compared with conventional wind tunnels with rigid tunnel walls. However, we expect minimal differences between open and closed-wall settings as the wind-tunnel blockage is less than 1.5 % with a thin streamlined body at a small angle of attack. We performed SPIV measurements perpendicular to the free stream from $x/c=0.67$ to 5 on eight measurement planes, as shown in figure 1. Here, $x$ denotes the streamwise direction and $c$ is the chord of the airfoil. These measurements focused on the vortex core, which contained the trailing-edge wake of the wing up to $x/c\approx 3$ , after which the wake exited the measurement plane due to downwash. Details of the experimental set-up are provided in Deem et al. (Reference Deem, Edstrand, Reger, Pascioni and Cattafesta2013). The instantaneous vortex centres, as shown in figure 2(a), and the fitted Gaussian joint probability density function (j.p.d.f.), shown in figure 2(b), demonstrate that wandering is present and significant. The j.p.d.f. contains a slight correlation between the $y$ - and $z$ -motion, corresponding to a correlation coefficient of ${\it\rho}=0.24$ . After collocating the instantaneous vortex cores, the mean flow field at $x/c=5$ is shown in figure 2(c).

2.2 Triple decomposition

A triple decomposition (Reynolds & Hussain Reference Reynolds and Hussain1972) is employed to extract the coherent wandering motion from the SPIV data. To this end, for each SPIV snapshot, the vortex centres are collocated by linearly shifting the instantaneous core positions, $\boldsymbol{x}_{c}(t_{k})$ , to the mean vortex axis, shown as a red $\times$ in figure 2(a). This results in the wandering-corrected instantaneous flow field, $\hat{\boldsymbol{u}}(\boldsymbol{x}-\boldsymbol{x}_{\boldsymbol{c}}(t_{k}),t_{k})$ , where the image is shifted by $\boldsymbol{x}_{\boldsymbol{c}}(t_{k})$ . The wandering-corrected mean flow field $\bar{\hat{\boldsymbol{u}}}$ , see figure 2(c), is then obtained by averaging the snapshots in time according to

(2.1) $$\begin{eqnarray}\bar{\hat{\boldsymbol{u}}}(\boldsymbol{x})=\frac{1}{N}\mathop{\sum }_{k=1}^{N}\hat{\boldsymbol{u}}(\boldsymbol{x}-\boldsymbol{x}_{\boldsymbol{ c}}(t_{k}),t_{k}).\end{eqnarray}$$

We obtain the velocity fluctuations devoid of wandering, $\boldsymbol{u}^{\prime }$ , for each snapshot by subtracting the wandering-corrected mean as

(2.2) $$\begin{eqnarray}\boldsymbol{u}^{\prime }(\boldsymbol{x},t_{k})=\hat{\boldsymbol{u}}(\boldsymbol{x}-\boldsymbol{x}_{\boldsymbol{c}}(t_{k}),t_{k})-\bar{\hat{\boldsymbol{u}}}(\boldsymbol{x}).\end{eqnarray}$$

We then assume vortex wandering to be the consequence of a coherent motion, and by using the triple decomposition of Reynolds & Hussain (Reference Reynolds and Hussain1972), the instantaneous flow field is decomposed into mean, coherent and random components, given by

(2.3) $$\begin{eqnarray}\boldsymbol{u}(\boldsymbol{x},t_{k})=\bar{\boldsymbol{u}}(\boldsymbol{x})+\tilde{\boldsymbol{u}}(\boldsymbol{x},t_{k})+\boldsymbol{u}^{\prime }(\boldsymbol{x},t_{k}).\end{eqnarray}$$

For the SPIV data, $\boldsymbol{u}$ represents the instantaneous measured velocity field, $\bar{\boldsymbol{u}}$ denotes the time-averaged velocity field with the effects of wandering, $\tilde{\boldsymbol{u}}$ is the coherent wandering velocity field and $\boldsymbol{u}^{\prime }$ is the random perturbation obtained from (2.2). It is straightforward to solve for $\tilde{\boldsymbol{u}}$ using (2.3) and (2.2) to obtain

(2.4) $$\begin{eqnarray}\tilde{\boldsymbol{u}}(\boldsymbol{x},t_{k})=[\boldsymbol{u}(\boldsymbol{x},t_{k})-\bar{\boldsymbol{u}}(\boldsymbol{x})]-[\hat{\boldsymbol{u}}(\boldsymbol{x},t_{k})-\bar{\hat{\boldsymbol{u}}}(\boldsymbol{x})],\end{eqnarray}$$

where the first bracketed term contains the coherent wandering motion and background turbulence, while the second bracketed term removes the turbulence. The resulting velocity field, $\tilde{\boldsymbol{u}}(\boldsymbol{x},t_{k})$ , describes the coherent wandering motion. From these snapshots, the most energetic structures of the coherent portion are extracted via the POD (Sirovich Reference Sirovich1987).

2.3 Batchelor-vortex model

For comparison of the experimental results with a canonical vortex base flow, the Batchelor vortex (Batchelor Reference Batchelor1964) is fitted to the computed mean flow,

(2.5a-c ) $$\begin{eqnarray}V_{x}=1-{\it\gamma}\exp \left(-\frac{r^{2}}{{\it\delta}^{2}}\right),\quad V_{r}=0,\quad V_{{\it\theta}}=\frac{{\it\kappa}}{r}\left[1-\exp \left(-\frac{r^{2}}{{\it\delta}^{2}}\right)\right],\end{eqnarray}$$

originally developed as a model for trailing vortices. In the above expression, ${\it\kappa}$ represents the swirl parameter, ${\it\delta}$ is the core radius and ${\it\gamma}$ denotes the axial velocity deficit at $r=0$ . To fit (2.5) to the experimental data, the mean velocity field at $x/c=5$ , $\overline{\boldsymbol{u}}$ , is azimuthally averaged (see figure 3 c) and curve fitted using a nonlinear least-squares method. The resulting fit is shown in figure 3(a,b) with the non-dimensional parameters ${\it\kappa}=0.0152$ , ${\it\delta}={\it\delta}^{\ast }/c=0.0514$ and ${\it\gamma}=0.0671$ . All parameters and equations have been non-dimensionalized by the chord, $c=0.3048~\text{m}$ , and the free-stream velocity, $U_{\infty }=28.5~\text{m}~\text{s}^{-1}$ . These values correspond to the non-dimensional swirl parameter of $q={\it\kappa}/({\it\delta}{\it\gamma})=4.41$ of a $q$ -vortex. These parameters correspond to a Reynolds number based on the core radius of $Re_{{\it\delta}^{\ast }}=Re_{c}{\it\delta}=27\,200$ . Averaging over the azimuth, excluding the trailing-edge wake, shows that the fitting parameters are rather insensitive to the averaging region; the swirl ${\it\kappa}$ has been found to be the most sensitive parameter, varying by 11.5 %. These variations, however, have negligible impact on the stability results.

Figure 3.

Velocity profiles from experimental data for (a) the streamwise velocity and (b) the azimuthal velocity. The azimuthal average of the velocities is shown by the solid line, with the shaded area representing the maximum azimuthal variation. Using this mean flow field, the Batchelor vortex is fitted using a least-squares method, shown by the dashed line. The contours in (c) show the mean streamwise velocity, $\overline{u}$ , without wandering correction used for the Batchelor fit, whose parameters are ${\it\kappa}=0.0152$ , ${\it\delta}=0.0514$ and ${\it\gamma}=0.0671$ , which correspond to a swirl parameter of $q=4.39$ .

2.4 Linear stability analysis

The governing equations for infinitesimal disturbances are given by

(2.6) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\tilde{\boldsymbol{u}}=0,\\ \displaystyle \frac{\partial \tilde{\boldsymbol{u}}}{\partial t}+\overline{\boldsymbol{u}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\tilde{\boldsymbol{u}}+\tilde{\boldsymbol{u}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\overline{\boldsymbol{u}}=-\boldsymbol{{\rm\nabla}}\tilde{p}+\frac{1}{Re_{c}}{\rm\nabla}^{2}\tilde{\boldsymbol{u}},\end{array}\right\}\end{eqnarray}$$

where $\tilde{\boldsymbol{u}}=[\tilde{v}_{x},\tilde{v}_{r},\tilde{v}_{{\it\theta}}]^{\text{T}}$ denotes the disturbance velocity field, $\overline{\boldsymbol{u}}=[V_{x},V_{r},V_{{\it\theta}}]^{\text{T}}$ is the base velocity field and $\tilde{p}$ represents the disturbance pressure. The experimentally measured vortex as well as the Batchelor vortex vary slowly with downstream progression, making streamwise variations higher order in nature. As such, we employ the parallel approximation and assume wave-like disturbances of the form $\tilde{\boldsymbol{q}}(x,r,{\it\theta},t)=\boldsymbol{q}(r)\exp (\text{i}{\it\alpha}x+\text{i}m{\it\theta}-\text{i}{\it\omega}t)$ , while neglecting streamwise gradients in the base flow. Here, ${\it\alpha}\in \mathbb{C}$ is the streamwise wavenumber, $m$ is the azimuthal wavenumber and ${\it\omega}\in \mathbb{R}$ is the radian frequency. We perform a spatial stability analysis, which yields a nonlinear eigenvalue problem which is linearized by employing the companion matrix method (Schmid & Henningson Reference Schmid and Henningson2001). This results in the generalized eigenvalue problem of the form

(2.7) $$\begin{eqnarray}\unicode[STIX]{x1D63C}\boldsymbol{q}_{\boldsymbol{\ast }}={\it\alpha}\unicode[STIX]{x1D63D}\boldsymbol{q}_{\boldsymbol{\ast }},\end{eqnarray}$$

where $\boldsymbol{q}_{\boldsymbol{\ast }}=[v_{x},v_{r},v_{{\it\theta}},p,{\it\alpha}v_{r},{\it\alpha}v_{{\it\theta}}]^{\text{T}}$ is the shape function of the perturbation, while $\unicode[STIX]{x1D63C}$ and $\unicode[STIX]{x1D63D}$ are the stability matrices, omitted for brevity (see, e.g. Paredes Reference Paredes2014). The coefficient matrices $\unicode[STIX]{x1D63C}$ and $\unicode[STIX]{x1D63D}$ are discretized using a Chebyshev spectral collocation method, and treatment of the centreline singularity arising from the cylindrical coordinates is handled through the method of Mohseni & Colonius (Reference Mohseni and Colonius2000). The disturbance is evaluated on a mapped Gauss–Lobatto grid for $r/{\it\delta}\in (0,100]$ (Hein & Theofilis Reference Hein and Theofilis2004), where Dirichlet boundary conditions are enforced at the computational boundary, forcing the disturbances to become zero in the far field. The current formulation has been validated with the Batchelor-vortex results of Paredes (Reference Paredes2014).

Figure 4.

Leading POD mode for $\tilde{u}$ , $\tilde{v}$ and $\tilde{w}$ shown in (a,d,g,j), (b,e,h,k) and (c,f,i,l) respectively; (a,b,c), (d,e,f), (g,h,i) and (j,k,l) correspond to the downstream distances $x/c=1,2,3$ and 4 respectively.