2 Model derivation

Two fundamental assumptions form the basis of the LEV trajectory model: (i) the vortex core location is stationary relative to the body, and (ii) there exists uni-directional axial flow through the LEV core directed away from the CoR. Due to the second assumption, the model will only be valid in regions inboard of vortex breakdown, which generates flow reversal in the vortex core, e.g. Sarpkaya (Reference Sarpkaya1971). That said, when these two conditions are satisfied, an axial streamline aligned with the LEV core trajectory must exist (see figure 1). It is the trajectory of such a streamline that the presented model will predict.

Figure 2. The Navier–Stokes equations are cast in streamline coordinates on an axial streamline through the LEV core. The defined polar coordinate system, which lies in the $xy$ -plane, allows the predicted trajectory of the axial streamline to be expressed in the form ${\it\sigma}={\it\sigma}({\it\theta})$ .

The Navier–Stokes equations are cast in the rotating frame of reference onto this axial streamline through the LEV core, as done by Fay (Reference Fay1994) for the Euler equations. Figure 2 depicts the streamline-fixed coordinate system. Consider first the streamline-normal direction, denoted by unit vector $\hat{\boldsymbol{n}}$ :

(2.1) $$\begin{eqnarray}{\it\rho}\frac{\text{D}u_{n}}{\text{D}t}=-\frac{\partial p}{\partial n}+{\it\mu}({\rm\nabla}^{2}\boldsymbol{u})\boldsymbol{\cdot }\hat{\boldsymbol{n}}+2{\it\rho}{\it\Omega}u_{s}+{\it\rho}{\it\Omega}^{2}\boldsymbol{r}\boldsymbol{\cdot }\hat{\boldsymbol{n}},\end{eqnarray}$$

where ${\it\rho}$ is density, ${\it\mu}$ is viscosity, $p$ is pressure, ${\it\Omega}$ is the constant angular velocity of the plate, $\boldsymbol{u}$ is the velocity vector, and $u_{n}$ and $u_{s}$ are streamline-normal and -aligned components of velocity, respectively. By construction, $u_{n}$ is always zero, but its material derivative does not necessarily vanish. $\boldsymbol{r}$ is the radial vector from the CoR, which lies in a plane normal to the rotation vector. In accordance with the stated hypothesis, it is postulated that the pseudo-forces dominate this equation, and by comparison, the pressure gradient and viscous terms tend to zero. These simplifications can be justified physically, since the vortex core tends to be a local pressure minimum, and can be expected to lie at or near an inflection point of velocity (where the second spatial derivatives of velocity tend to zero). The $\hat{\boldsymbol{n}}$ -momentum equation thus reduces to

(2.2) $$\begin{eqnarray}\frac{\text{D}u_{n}}{\text{D}t}=2{\it\Omega}u_{s}+{\it\Omega}^{2}\boldsymbol{r}\boldsymbol{\cdot }\hat{\boldsymbol{n}}.\end{eqnarray}$$

Since the position of a stable LEV is stationary relative to the body, the material derivative can be related to the local curvature of the streamline, ${\it\kappa}$ , as given by

(2.3) $$\begin{eqnarray}\frac{\text{D}u_{n}}{\text{D}t}={\it\kappa}u_{s}^{2}.\end{eqnarray}$$

If one can designate a starting location of the LEV, i.e. the most inboard location of the stable LEV core, and the local curvature at all points along the axial streamline is known, then its trajectory is fully defined. This trajectory will be expressed in polar coordinates $({\it\sigma},{\it\theta})$ , as shown in figure 2. The origin of the polar coordinate system is offset from the CoR by ${\bf\sigma}_{0}$ to ensure that the radial coordinate ${\it\sigma}$ is a single-valued function of ${\it\theta}$ in the domain of interest. The curvature of the streamline can now be expressed in terms of ${\it\sigma}$ and ${\it\theta}$ (Gray, Abbena & Salamon Reference Gray, Abbena and Salamon2006):

(2.4) $$\begin{eqnarray}{\it\kappa}=\left[{\it\sigma}^{2}+2\left(\frac{\text{d}{\it\sigma}}{\text{d}{\it\theta}}\right)^{2}-{\it\sigma}\frac{\text{d}^{2}{\it\sigma}}{\text{d}{\it\theta}^{2}}\right]\left({\it\sigma}^{2}+\left(\frac{\text{d}{\it\sigma}}{\text{d}{\it\theta}}\right)^{2}\right)^{-3/2}.\end{eqnarray}$$

Finally, the dot product in the last term of (2.2) can be expressed in terms of ${\it\sigma}$ and its derivative ${\it\sigma}^{\prime }({\it\theta})$ . It can be readily shown that the angle ${\it\phi}$ between the vector ${\bf\sigma}$ and the unit vector $\hat{\boldsymbol{s}}$ is given by

(2.5) $$\begin{eqnarray}\tan {\it\phi}=\frac{{\it\sigma}}{{\it\sigma}^{\prime }({\it\theta})}.\end{eqnarray}$$

Combining the previous four equations and noting that $\boldsymbol{r}={\bf\sigma}+{\bf\sigma}_{0}$ , the following expression is obtained after algebraic manipulation:

(2.6) $$\begin{eqnarray}\displaystyle \frac{\text{d}^{2}{\it\sigma}}{\text{d}{\it\theta}^{2}} & = & \displaystyle {\it\sigma}+\frac{2}{{\it\sigma}}\left(\frac{\text{d}{\it\sigma}}{\text{d}{\it\theta}}\right)^{2}-\frac{2{\it\Omega}}{u_{s}{\it\sigma}}\left({\it\sigma}^{2}+\left(\frac{\text{d}{\it\sigma}}{\text{d}{\it\theta}}\right)^{2}\right)^{3/2}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{{\it\Omega}^{2}}{u_{s}^{2}}\left({\it\sigma}^{2}+\left(\frac{\text{d}{\it\sigma}}{\text{d}{\it\theta}}\right)^{2}\right)\left[{\it\sigma}-\frac{{\it\sigma}_{0}}{{\it\sigma}}\left(\frac{\text{d}{\it\sigma}}{\text{d}{\it\theta}}\cos ({\it\theta}-{\it\beta})-{\it\sigma}\sin ({\it\theta}-{\it\beta})\right)\right],\qquad\end{eqnarray}$$

where ${\it\beta}$ is the angle between the $y$ -axis and the vector ${\bf\sigma}_{0}$ . This is a second-order, nonlinear, ordinary differential equation (ODE) that can be solved to yield ${\it\sigma}={\it\sigma}({\it\theta})$ . The only remaining unknown in (2.6) is the velocity along the streamline, which can be solved simultaneously using the $\hat{\boldsymbol{s}}$ -momentum equation:

(2.7) $$\begin{eqnarray}\frac{\text{D}u_{s}}{\text{D}t}=u_{s}\frac{\partial u_{s}}{\partial s}=-\frac{1}{{\it\rho}}\frac{\partial p}{\partial s}+\frac{{\it\mu}}{{\it\rho}}({\rm\nabla}^{2}\boldsymbol{u})\boldsymbol{\cdot }\hat{\boldsymbol{s}}+{\it\Omega}^{2}\boldsymbol{r}\boldsymbol{\cdot }\hat{\boldsymbol{s}}.\end{eqnarray}$$

At sufficiently low Reynolds numbers, viscous effects can inhibit span-wise flow in the LEV core (Birch, Dickson & Dickinson Reference Birch, Dickson and Dickinson2004). However, at Reynolds number of the order $O(10^{3})$ or greater, significant span-wise flow is observed in the core (Birch et al. Reference Birch, Dickson and Dickinson2004; Harbig et al. Reference Harbig, Sheridan and Thompson2013; Wojcik & Buchholz Reference Wojcik and Buchholz2014), and thus the viscous term in (2.7) is assumed to be negligible. The axial pressure gradients, however, cannot be neglected. In fact, Garmann & Visbal (Reference Garmann and Visbal2014) showed the span-wise pressure gradients to be an order of magnitude greater than centrifugal force in the flow over a rotating inclined plate. Maxworthy (Reference Maxworthy2007) developed a simplified model of a conical LEV, and showed that, in such a flow, the span-wise pressure gradient would scale in the same manner as centrifugal force. Thus, the following proportionality is implemented:

(2.8) $$\begin{eqnarray}-\frac{1}{{\it\rho}}\frac{\partial p}{\partial s}\propto {\it\Omega}^{2}\boldsymbol{r}\boldsymbol{\cdot }\hat{\boldsymbol{s}}.\end{eqnarray}$$

Introducing a constant of proportionality, $k$ , equation (2.7) becomes

(2.9) $$\begin{eqnarray}\frac{\partial u_{s}}{\partial s}=\frac{\text{d}u_{s}}{\text{d}s}=(k+1)\frac{{\it\Omega}^{2}}{u_{s}}\boldsymbol{r}\boldsymbol{\cdot }\hat{\boldsymbol{s}}.\end{eqnarray}$$

$k$ is a dimensionless variable that describes the ratio of the axial pressure gradient to the centrifugal acceleration in the LEV core:

(2.10) $$\begin{eqnarray}k=-\frac{\text{axial pressure gradient}}{\text{centrifugal acceleration}}.\end{eqnarray}$$

When $k=0$ , centrifugal force drives the flow and pressure gradients are considered negligible. When $k\gg 1$ , the pressure gradient is dominant. $k$ will hereafter be referred to as the axial pressure gradient coefficient.

Expressing (2.9) in terms of the polar coordinate ${\it\theta}$ , the following expression is obtained:

(2.11) $$\begin{eqnarray}\frac{\text{d}u_{s}}{\text{d}{\it\theta}}=\frac{(k+1){\it\Omega}^{2}}{u_{s}}\left[{\it\sigma}\frac{\text{d}{\it\sigma}}{\text{d}{\it\theta}}+{\it\sigma}_{0}\left(\frac{\text{d}{\it\sigma}}{\text{d}{\it\theta}}\sin ({\it\theta}-{\it\beta})+{\it\sigma}\cos ({\it\theta}-{\it\beta})\right)\right].\end{eqnarray}$$

Given suitable boundary conditions, equations (2.6) and (2.11) can be solved to yield a unique solution for the trajectory of the axial streamline through an LEV core, and the velocity along that same streamline.

3 Boundary conditions and solution method

Equations (2.6) and (2.11) are coupled ODEs that can be solved simultaneously to yield the trajectory of a steady, axial streamline through a stable LEV core. To obtain a unique solution to the system of ODEs, three boundary conditions are required: (i) the initial location of the LEV, $\boldsymbol{r}_{0}$ , i.e. the most inboard location where a stable LEV core can be said to exist, (ii) the initial velocity along the streamline, $u_{s}({\it\theta}_{0})$ , and (iii) the initial trajectory of the LEV core, ${\it\sigma}^{\prime }({\it\theta}_{0})$ , which is taken to be parallel to the leading edge. A solution to the system of ODEs is obtained numerically using the built-in MATLAB^® function ode45, which is an adaptive step-length Runge–Kutta solver.

The results of Garmann & Visbal (Reference Garmann and Visbal2014) and Phillips et al. (Reference Phillips, Knowles and Bomphrey2015) are compared to two solutions of the model, denoted as solution 1 and solution 2, respectively. Boundary conditions (i) and (ii) for each solution are extracted directly from the corresponding study. In the case of Phillips et al. (Reference Phillips, Knowles and Bomphrey2015), this was made possible by the generous provision of raw data by the authors of that study. Boundary condition (iii) is obtained in both cases by assuming the axial streamline through the LEV core to be parallel to the leading edge at the starting location. The model parameters and boundary conditions for both solutions are summarized in table 1.

Note that the axis of rotation is aligned with the plate differently in these two studies; in the work of Garmann & Visbal (Reference Garmann and Visbal2014), it is aligned with the mid-chord, whereas it is aligned with the leading edge of the plate in the work of Phillips et al. (Reference Phillips, Knowles and Bomphrey2015). In both solutions, the origin of coordinate system $xyz$ lies on the axis of rotation, with $z$ parallel to the rotation vector.

Table 1. Boundary conditions and other model parameters. Solution 1 was obtained for comparison to the results of Garmann & Visbal (Reference Garmann and Visbal2014), and solution 2 was obtained for comparison to the work of Phillips et al. (Reference Phillips, Knowles and Bomphrey2015). Boundary conditions (i) and (ii) are obtained from the studies against which each solution will be compared. The LEV is assumed to start parallel to the leading edge in both cases to yield boundary condition (iii). $\boldsymbol{r}_{0}$ and ${\bf\sigma}_{0}$ are expressed in $x$ - and $y$ -coordinates as $[x,y]$ .

4 Results and comparison to previous work

4.1 Comparison to the work of Garmann & Visbal (Reference Garmann and Visbal2014)

Garmann & Visbal (Reference Garmann and Visbal2014) conducted time-resolved numerical simulations of rotating, rectangular plates at an angle of incidence of $60^{\circ }$ relative to the swept plane. Since the presented model does not consider edge effects, such as the presence of the wing tip, the solution of the model is compared to the flow field around the plate of aspect ratio 4 in the work of Garmann & Visbal (Reference Garmann and Visbal2014), which is the largest aspect ratio they studied.

The estimated LEV trajectory in the work of Garmann & Visbal (Reference Garmann and Visbal2014) is taken as the locus of points within the primary LEV structure, as viewed on span-normal planes, where the span-wise vorticity is a maximum. These estimates are taken at an azimuthal angle of $67.5^{\circ }$ from the start of rotation. Beyond this azimuthal angle, evidence of vortex breakdown is apparent and complex substructures are superimposed onto the overall LEV structure, as viewed with isosurfaces of total pressure; nonetheless, by qualitative observation, the overall LEV core trajectory does not appear to change greatly (Garmann & Visbal Reference Garmann and Visbal2014). LEV trajectory data, as extracted from their published results in this way, are only available at locations up to the midspan.

Garmann & Visbal (Reference Garmann and Visbal2014) noted that span-wise pressure gradients in the LEV core are an order of magnitude greater in strength than centrifugal acceleration. As such, an axial pressure gradient coefficient of $k=10$ is used in the model.

Figure 3. Comparison of the LEV trajectory predicted by the presented model and the LEV core locations at various span-wise locations in the numerical simulation of Garmann & Visbal (Reference Garmann and Visbal2014): (a) isometric view, (b) projection onto the $x^{\prime }y$ -plane, i.e. the plane of the plate, (c) projection onto the $xz$ -plane. (d) The predicted velocity along the axial streamline versus span-wise position. An interactive three-dimensional plot of the LEV trajectory is provided in .fig format as supplementary material available at http://dx.doi.org/10.1017/jfm.2016.395.

Using the boundary conditions presented in table 1, solution 1 to the coupled system of ODEs predicts the observed LEV trajectory in the results of Garmann & Visbal (Reference Garmann and Visbal2014) quite well. Projections of the model solution and the numerical results onto the plate are very well aligned, as depicted in figure 3(b). However, as one moves outboard from the CoR, the observed LEV core locations tend to deviate from a $z$ -normal plane in a downwards direction, as shown in figure 3(c).

Inboard of the midspan, the predicted range of axial velocities in the LEV core, as plotted in figure 3(d), is in agreement with previous work, e.g. Birch et al. (Reference Birch, Dickson and Dickinson2004), Harbig et al. (Reference Harbig, Sheridan and Thompson2013), Wojcik & Buchholz (Reference Wojcik and Buchholz2014). Although the tip velocity has no effect on the solution of the system of ODEs, the velocity along the streamline is normalized as $u_{s}^{\ast \ast }=u_{s}/u_{tip}$ for consistency with previous literature, where $u_{tip}=4.5{\it\Omega}c$ in this case.

4.2 Comparison to the work of Phillips et al. (Reference Phillips, Knowles and Bomphrey2015)

Phillips et al. (Reference Phillips, Knowles and Bomphrey2015) studied rectangular plates undergoing a simplified manoeuvre representative of the flapping motion of an insect wing. The plate was rotated about a vertical axis one chord length from the root of the plate, undergoing a rapid pitch motion from $90^{\circ }$ to $45^{\circ }$ angle of attack at the start of the manoeuvre. The angle of attack was then held constant for a period of time before the pitch and rotation were reversed to complete a symmetrical return stroke. For comparison to the steady-state model presented herein, their results for a plate of aspect ratio 4.5 at the time step closest to the end of the constant-pitch portion of the manoeuvre were considered. This time step corresponds to a dimensionless time of 0.4 (normalized by flapping period), and a traversed azimuthal angle of $61.5^{\circ }$ . The instantaneous LEV trajectory at this time is expected to approximate the steady-state trajectory, in accordance with the observations of Garmann & Visbal (Reference Garmann and Visbal2014) and Carr, DeVoria & Ringuette (Reference Carr, Devoria and Ringuette2015) for a plate of aspect ratio 4. As with the comparison to the work of Garmann & Visbal (Reference Garmann and Visbal2014), an axial pressure gradient coefficient of $k=10$ was used in the solution to the model.

Using stereo-PIV, Phillips et al. (Reference Phillips, Knowles and Bomphrey2015) quantified the velocity field over the suction side of the plate on span-normal planes separated by 1 mm, or $0.033c$ . At each instant, they identified saddle points and foci on each span-normal plane, and the axis of the LEV was identified as the curve connecting the primary focus on each plane. Given the close spacing of the investigated planes, the LEV trajectory was identified with high spatial resolution. The model-predicted trajectory is compared to their experimental results over the most inboard $5/8$ ths of the span, outboard of which the vortex lifts dramatically from the plate surface due to interaction with the tip vortex. This arching of the LEV in the vicinity of the tip is also observed in the results of Garmann & Visbal (Reference Garmann and Visbal2014).

Figure 4. Comparison of the model-predicted LEV trajectory and the trajectory of the LEV axis and nearby streamlines in the experimental results of Phillips et al. (Reference Phillips, Knowles and Bomphrey2015): (a) isometric view, (b) projection onto the $x^{\prime }y$ -plane, i.e. the plane of the plate, (c) projection onto the $xz$ -plane. (d) Comparison of the predicted velocity along the axial streamline and the measured velocity along streamlines in the vicinity of the LEV core in Phillips et al. (Reference Phillips, Knowles and Bomphrey2015).

The LEV trajectory of solution 2 is in good agreement with the results of Phillips et al. (Reference Phillips, Knowles and Bomphrey2015) for an aspect ratio of 4.5, as demonstrated in figure 4. Similar to figure 3, the projection of the results onto the plane of the plate shows excellent agreement (figure 4 b), whereas the greatest deviation between the experimentally identified LEV axis and the model is in the plate-normal direction (figure 4 c).

Phillips et al. (Reference Phillips, Knowles and Bomphrey2015) also identified 69 individual streamlines in the vicinity of the vortex core at $t^{\ast }=0.4$ , which have also been plotted in figure 4. The velocity magnitude along these streamlines, as normalized by tip velocity, is plotted in figure 4(d). The axial streamline velocity predicted by the model matches these data well in the region inboard of the midspan, where the streamlines are closely clustered around the vortex core. Near the midspan, the streamlines begin to disperse from one another, while the velocity along the streamlines is decelerated; these observations are suggestive of vortex breakdown, which invalidates the model in this region.

A longer plate of aspect ratio 7.5 was also studied by Phillips et al. (Reference Phillips, Knowles and Bomphrey2015), and the LEV over this plate would be less influenced by the tip vortex over much of the span. However, the LEV trajectory over this plate does not appear to have reached steady state before the onset of the return-stroke manoeuvre, making it a poor candidate for comparison to the presented low-order model. This was concluded by noting that the LEV persists for much more than four chord lengths from the CoR for the plate of aspect ratio 7.5, in contradiction to the steady-state observations of Kruyt et al. (Reference Kruyt, van Heijst, Altshuler and Lentink2015) for aspect ratios of 6.5, 8 and 10.