2.1 Governing equations

Let ${\mathcal{D}}_{z}$ denote the domain exterior to $M+1$ thin plates, where $M=0$ corresponds to the single-plate case and $M=2$ to the Kasper wing case. A schematic of the configurations is shown in figure 1(a). To evaluate the velocity field $V(z)=u-\text{i}v$ induced at a point $z=x+\text{i}y$ in ${\mathcal{D}}_{z}$ , the complex potential of the system is first constructed in a conformally equivalent pre-image circular domain. This circular domain is labelled ${\mathcal{D}}_{\unicode[STIX]{x1D701}}$ and consists of the interior of the unit disk with $M$ excised non-overlapping smaller disks. The boundary of the unit circle will be labelled $C_{0}$ and those of the $M$ excised disks $C_{k}$ for $k=1,\ldots ,M$ . Complex coordinates in the transformed domain will be denoted $\unicode[STIX]{x1D701}$ (and thus $z=z(\unicode[STIX]{x1D701})$ ). Let the centres and radii of the $M$ excised disks be respectively $\unicode[STIX]{x1D6FF}_{k}\in \mathbb{C}$ and $q_{k}\in \mathbb{R}$ for $k=1,\ldots ,M$ . An example illustrating such a pre-image domain is shown in figure 1(b). In what follows, overlines will be used to denote complex conjugation and for some complex function $f(\unicode[STIX]{x1D701})$ , its conjugate function (Schwarz conjugate) is defined as $\bar{f}(\unicode[STIX]{x1D701}):=\overline{f(\bar{\unicode[STIX]{x1D701}})}$ (where ‘ $:=$ ’ means that the expression on the left-hand side is defined by the one on the right-hand side).

Figure 1.

(a) Schematic of the flow configuration. The thick dashed lines correspond to the auxiliary plates present in the Kasper wing system (with $M=2$ ) but not in the single-plate system (with $M=0$ ). (b) Schematic of a typical pre-image domain.

Let $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}\in {\mathcal{D}}_{\unicode[STIX]{x1D701}}$ , respectively, denote the location of a point vortex and the point that is mapped to infinity in ${\mathcal{D}}_{z}$ , i.e. $z(\unicode[STIX]{x1D6FD})=\infty$ . The flow under consideration consists of the following components (for each of which the complex potential at a point $\unicode[STIX]{x1D701}$ is stated):

1. (i) a potential flow which tends to a uniform flow inclined at angle $\unicode[STIX]{x1D712}_{0}$ as $|z|\rightarrow \infty$

   (2.1) $$\begin{eqnarray}W_{U}(\unicode[STIX]{x1D701},\unicode[STIX]{x1D6FD})=Ua\left[\text{e}^{\text{i}\unicode[STIX]{x1D712}_{0}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\bar{\unicode[STIX]{x1D6FD}}}-\text{e}^{-\text{i}\unicode[STIX]{x1D712}_{0}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FD}}\right]\log \left(\frac{\unicode[STIX]{x1D714}(\unicode[STIX]{x1D701},\unicode[STIX]{x1D6FD})}{|\unicode[STIX]{x1D6FD}|\unicode[STIX]{x1D714}(\unicode[STIX]{x1D701},\bar{\unicode[STIX]{x1D6FD}}^{-1})}\right),\end{eqnarray}$$
   where $a$ is a scaling constant such that $|\text{d}W/\text{d}z|\rightarrow U$ as $|z|\rightarrow \infty$ (see Nelson & Sakajo Reference Nelson and Sakajo2014);

2. (ii) flow due to a single point vortex

   (2.2) $$\begin{eqnarray}W_{V}(\unicode[STIX]{x1D701};\unicode[STIX]{x1D6FC})=-\frac{\text{i}\unicode[STIX]{x1D705}}{2\unicode[STIX]{x03C0}}\log \left(\frac{\unicode[STIX]{x1D714}(\unicode[STIX]{x1D701},\unicode[STIX]{x1D6FC})}{|\unicode[STIX]{x1D6FC}|\unicode[STIX]{x1D714}(\unicode[STIX]{x1D701},\bar{\unicode[STIX]{x1D6FC}}^{-1})}\right),\end{eqnarray}$$
   where $\unicode[STIX]{x1D705}$ denotes the circulation of the point vortex;

3. (iii) flow due to a point vortex with circulation $-\unicode[STIX]{x1D705}$ at the point $\unicode[STIX]{x1D6FD}$ (to remove the circulation around the disk $C_{0}$ owing to the point vortex at $\unicode[STIX]{x1D6FC}$ ):

   (2.3) $$\begin{eqnarray}W_{\infty }(\unicode[STIX]{x1D701},\unicode[STIX]{x1D6FD})=\frac{\text{i}\unicode[STIX]{x1D705}}{2\unicode[STIX]{x03C0}}\log \left(\frac{\unicode[STIX]{x1D714}(\unicode[STIX]{x1D701},\unicode[STIX]{x1D6FD})}{|\unicode[STIX]{x1D6FD}|\unicode[STIX]{x1D714}(\unicode[STIX]{x1D701},\bar{\unicode[STIX]{x1D6FD}}^{-1})}\right);\end{eqnarray}$$

4. (iv) flows due to prescribed circulations around each of the disks

   (2.4) $$\begin{eqnarray}W_{\unicode[STIX]{x1D6E4}}(\unicode[STIX]{x1D701},\unicode[STIX]{x1D6FD})=\mathop{\sum }_{k=0}^{M}\frac{\text{i}\unicode[STIX]{x1D6E4}_{k}}{2\unicode[STIX]{x03C0}}\log \left(\frac{\unicode[STIX]{x1D714}(\unicode[STIX]{x1D701},\unicode[STIX]{x1D6FD})}{\unicode[STIX]{x1D714}(\unicode[STIX]{x1D701},\bar{\unicode[STIX]{x1D703}}_{k}(\unicode[STIX]{x1D6FD}^{-1}))}\right),\end{eqnarray}$$
   where $\unicode[STIX]{x1D6E4}_{k}$ , $k=0,\ldots ,M$ , is the desired circulation around the disk $C_{k}$ .

In relations (2.2)–(2.4), $\unicode[STIX]{x1D714}(\unicode[STIX]{x1D701},\cdot )$ is a special function known as the Schottky–Klein prime function associated with ${\mathcal{D}}_{\unicode[STIX]{x1D701}}$ and $\{\unicode[STIX]{x1D703}_{k}\mid k=0,\ldots ,M\}$ are the related Möbius maps. Both of these functions will be described below in § 2.2. The total complex potential in the pre-image domain is then the sum of these components and is thus given by

The velocity field in the physical domain is retrieved via (see, e.g., Saffman Reference Saffman1992)

where the subscript $\unicode[STIX]{x1D701}$ represents the derivative with respect to $\unicode[STIX]{x1D701}$ and $\widetilde{W}_{\unicode[STIX]{x1D701}}$ is the velocity at the location of the point vortex in the pre-image domain with the self-induction term removed (see, e.g., Saffman Reference Saffman1992). To compute (2.6) it is required to know the form of the conformal mapping $z=z(\unicode[STIX]{x1D701})$ from ${\mathcal{D}}_{\unicode[STIX]{x1D701}}$ to ${\mathcal{D}}_{z}$ . For the domains under consideration, when $M=0$ ,

and when $M=2$ ,

where, owing to a degree of freedom in the Riemann mapping theorem, $\unicode[STIX]{x1D6FD}$ can be chosen to lie anywhere in ${\mathcal{D}}_{\unicode[STIX]{x1D701}}$ , but it will be required to compute the values of $S$ and $\unicode[STIX]{x1D6FE}$ corresponding to the desired configuration in ${\mathcal{D}}_{z}$ .

The map given in (2.7) is the celebrated Joukowski map which maps the unit disk ( $|\unicode[STIX]{x1D701}|=1$ ) to a thin slit located on the real axis ( $\text{Im}(z)=0$ ) between $-1\leqslant \text{Re}(z)\leqslant 1$ and the interior of the disk to the region exterior to this slit. Note that, for this map, the point mapped to infinity in ${\mathcal{D}}_{z}$ corresponds to $\unicode[STIX]{x1D6FD}=0$ . The form of the map appearing in (2.8) is known as a radial slit map (a subset of the more general group of conformal slit maps). Such maps have a wide range of applications in applied mathematics (Crowdy Reference Crowdy2012). Here, the map (2.8) will again map the unit disk to a thin slit located on the real axis between $-1\leqslant z\leqslant 1$ . Additionally, the two excised disks will be mapped to sections of ‘radial rays’ emanating from the point $z=1$ . The region interior to the unit disk and exterior to the two excised disks will be mapped to the region exterior to the three radial thin plates. It should be noted that the appearance of the Schottky–Klein prime function in (2.8) is a consequence of the type of conformal map required here and is not directly related to its appearance in (2.2)–(2.4). Further details regarding the relation between these maps and the configurations desired will be given shortly.

With the velocity field throughout ${\mathcal{D}}_{z}$ determined, the instantaneous position of the free point vortex, which we will denote $\boldsymbol{X}(t):=[x~y]^{\text{T}}=[\text{Re}(z)~\text{Im}(z)]^{\text{T}}$ , is governed by the nonlinear dynamical system

First, stationary solutions are sought such that, in addition to the point vortex being in equilibrium, the Kutta conditions prescribing the flow separation at the trailing edge of each plate are satisfied. This choice is motivated by the original study of Saffman & Sheffield (Reference Saffman and Sheffield1977) on the single-plate configuration and more discussion regarding this choice can be found in Nelson & Sakajo (Reference Nelson and Sakajo2014, Reference Nelson and Sakajo2016). Imposing Kutta conditions at each trailing edge means that altogether $M+3$ equations must be satisfied, i.e. the Kutta conditions at $(M+1)$ plates and two conditions for the equilibrium coordinates $\text{Re}(z_{\unicode[STIX]{x1D6FC}})$ and $\text{Im}(z_{\unicode[STIX]{x1D6FC}})$ of the point vortex. The unknowns in the problem thus are $\{\text{Re}(z_{\unicode[STIX]{x1D6FC}}),\text{Im}(z_{\unicode[STIX]{x1D6FC}}),\unicode[STIX]{x1D705},\unicode[STIX]{x1D6E4}_{0},\ldots ,\unicode[STIX]{x1D6E4}_{M}\}$ giving $M+4$ real numbers. Solutions are therefore expected to trace out one-parameter continuous loci. Labelling the position of the trailing edge of each plate $z_{k}$ for $k=0,\ldots ,M$ , stationary solutions with a point vortex at $z_{\unicode[STIX]{x1D6FC}}$ are characterized by

where the angle $\unicode[STIX]{x1D712}_{0}$ can be chosen freely and the unknowns $z_{\unicode[STIX]{x1D6FC}},\unicode[STIX]{x1D705},\unicode[STIX]{x1D6E4}_{0},\ldots ,\unicode[STIX]{x1D6E4}_{M}$ are solved for. For the single-plate case ( $M=0$ ), equations (2.10) can be solved exactly. However, closed-form solutions are not possible for the Kasper wing system ( $M=2$ ) and a numerical approach is required. As in Nelson & Sakajo (Reference Nelson and Sakajo2014, Reference Nelson and Sakajo2016), a Brownian ratchets scheme is used to solve equations (2.10) to within a prescribed numerical tolerance. Further details regarding such Brownian ratchets schemes can be found in Newton & Chamoun (Reference Newton and Chamoun2007).

2.2 The Schottky–Klein prime function

In this section the aforementioned Schottky–Klein prime function and the associated Möbius maps are briefly introduced. For further details regarding this function and its properties we refer the reader to, e.g., Crowdy & Marshall (Reference Crowdy and Marshall2005). One manner in which this special function can be evaluated is via a classic product formula (Baker Reference Baker1897). Within ${\mathcal{D}}_{\unicode[STIX]{x1D701}}$ , for each of the $M$ excised circles $C_{k}$ , $k=1,\ldots ,M$ , we construct the Möbius map

The Schottky–Klein prime function is then given by

where the product is taken over all mappings $\unicode[STIX]{x1D717}$ belonging to a special subset $\unicode[STIX]{x1D6E9}^{\prime \prime }$ of the full Schottky group $\unicode[STIX]{x1D6E9}$ . (The full, or classical, Schottky group $\unicode[STIX]{x1D6E9}$ is defined to be the infinite free group of mappings generated by compositions of the $M$ basic Möbius maps $\{\unicode[STIX]{x1D703}_{k}\mid k=1,\ldots ,M\}$ and their inverses $\{\unicode[STIX]{x1D703}_{k}^{-1}\mid k=1,\ldots ,M\}$ and including the identity map.) This subset contains all mappings, excluding the identity and all inverse mappings, thus, for example, if the map $\unicode[STIX]{x1D703}_{1}\unicode[STIX]{x1D703}_{2}^{-1}$ is included in the set, $\unicode[STIX]{x1D703}_{2}\unicode[STIX]{x1D703}_{1}^{-1}$ must be excluded. Two important properties of the Schottky–Klein prime function are:

1. (i) $\unicode[STIX]{x1D714}(\unicode[STIX]{x1D701},\unicode[STIX]{x1D6FE})$ has a simple zero at $\unicode[STIX]{x1D701}=\unicode[STIX]{x1D6FE}$ ;

2. (ii) $\unicode[STIX]{x1D714}(\cdot ,\cdot )$ is such that the complex potential has constant imaginary part on all circles $C_{k}$ , $k=0,1,\ldots ,M$ .

Formula (2.12) is useful for concisely stating $\unicode[STIX]{x1D714}(\cdot ,\cdot )$ and can be applied in some practical situations. In general, however, expression (2.12) fails to converge or requires an impractically long time to compute (for example, when two circles are close together or when the connectivity is high). A more robust method is to compute $\unicode[STIX]{x1D714}(\cdot ,\cdot )$ via a Fourier–Laurent expansion, details of which are presented in Crowdy & Marshall (Reference Crowdy and Marshall2007). Briefly, the algorithm works by introducing the function $X(\cdot ,\cdot )$ defined as

in which

where $\unicode[STIX]{x1D6FF}_{k}^{\prime }$ is the centre of the circle $C_{k}^{\prime }$ obtained from reflecting $C_{k}$ about $|\unicode[STIX]{x1D701}|=1$ , $Q_{k}$ is related to the radius of $C_{k}^{\prime }$ , the constants $\{c_{m}^{(k)},d_{m}^{(k)}\mid k=1,\ldots ,M,m=1,2,\ldots \}$ are determined numerically (for some truncated $m$ ) and $A$ is a normalization coefficient chosen such that

The Schottky–Klein prime function is then retrieved via

where the branch of the square root is taken such that $\unicode[STIX]{x1D714}(\unicode[STIX]{x1D701},\unicode[STIX]{x1D6FE})$ behaves like $(\unicode[STIX]{x1D701}-\unicode[STIX]{x1D6FE})$ as $\unicode[STIX]{x1D701}\rightarrow \unicode[STIX]{x1D6FE}$ .

2.3 Linear stability analysis of point-vortex equilibria

Linear stability analysis of the single plate and Kasper wing systems is carried out by adding a small perturbation $z^{\prime }=x^{\prime }+\text{i}y^{\prime }$ to the point-vortex equilibrium $z_{\unicode[STIX]{x1D6FC}}$ and performing linearization. The evolution of this perturbation is governed by the system

where $\boldsymbol{X}^{\prime }(t)=[x^{\prime }(t)~y^{\prime }(t)]^{\text{T}}$ and $\unicode[STIX]{x1D63C}$ is given by (see Nelson & Sakajo Reference Nelson and Sakajo2016)

with

Eigenvalue analysis of the matrix $\unicode[STIX]{x1D63C}$ reveals that the system is:

1. (i) unstable when $a^{2}+bc>0$ (corresponding to purely real eigenvalues);

2. (ii) neutrally stable when $a^{2}+bc<0$ (corresponding to purely imaginary eigenvalues).

We emphasize that when the linearization has purely imaginary eigenvalues and there are no eigenvalues with positive real parts, as is the case here, then such linearization is inconclusive as regards the stability properties of the equilibrium and one must account for the effect of nonlinearity through suitable invariant manifold reductions (Protas Reference Protas2007). We also note that, since system (2.9) is Hamiltonian, the two eigenvalues may be either purely real and of opposite signs or form a conjugate imaginary pair (Newton Reference Newton2001). It should also be noted that when the system is perturbed away from the equilibrium, the Kutta conditions (2.10c ) are no longer satisfied. For these conditions to be satisfied throughout any time-dependent evolution, a method for shedding vorticity from the rear tip of each plate is required. This topic will be revisited in § 6.

2.4 Configurations and analysis

We now introduce the configurations which will be subject to further study and analyse their properties in the uncontrolled setting. In all configurations, the main plate (which $C_{0}$ is mapped to) will lie in the region defined by $-1\leqslant x\leqslant 1$ and $y=0$ and, when $M=2$ , the centres of the auxiliary plates (which $C_{1}$ and $C_{2}$ are mapped to) will lie at $z=1+0.4\text{e}^{\pm \text{i}\unicode[STIX]{x1D719}}$ . Three values of the angle $\unicode[STIX]{x1D719}$ will be used, these being $\unicode[STIX]{x03C0}/12$ , $\unicode[STIX]{x03C0}/6$ and $5\unicode[STIX]{x03C0}/12$ , and the length of the auxiliary plates will be set to $0.1$ . This means that $C_{1}$ and $C_{2}$ are mapped to straight segments with endpoints, respectively, at $z=1+(0.4\pm 0.05)\text{e}^{\text{i}\unicode[STIX]{x1D719}}$ and $z=1+(0.4\pm 0.05)\text{e}^{-\text{i}\unicode[STIX]{x1D719}}$ . The angle of attack of the oncoming flow is set to $\unicode[STIX]{x1D712}_{0}=0.1$ .

Owing to the configurations chosen in ${\mathcal{D}}_{z}$ , when $M=2$ , $\unicode[STIX]{x1D6FE}$ must be set unity. The constant $\unicode[STIX]{x1D6FD}$ can be chosen freely and will be set to $-0.4$ in all results that follow. In the construction of ${\mathcal{D}}_{\unicode[STIX]{x1D701}}$ and the conformal map, the remaining unknowns are $S,\unicode[STIX]{x1D6FF}_{1},q_{1},\unicode[STIX]{x1D706}_{1}$ and $\unicode[STIX]{x1D706}_{2}$ , where $\unicode[STIX]{x1D706}_{1}$ and $\unicode[STIX]{x1D706}_{2}$ are the angles (with respect to $\unicode[STIX]{x1D6FF}_{1}$ ) of the points on $C_{1}$ that map to the auxiliary plate edges in ${\mathcal{D}}_{z}$ , which leaves six real unknowns. We note that due to the symmetry of the configurations considered here, $\unicode[STIX]{x1D6FF}_{2}=\overline{\unicode[STIX]{x1D6FF}_{1}}$ and $q_{2}=q_{1}$ (see Nelson & Sakajo (Reference Nelson and Sakajo2016) for details regarding non-symmetric Kasper wing configurations). The six unknowns can be found from the six real-valued equations obtained from

These equations are solved via Newton’s iteration and solutions for three different values of $\unicode[STIX]{x1D719}$ are given in table 1. The scaling constant $a$ (appearing in (2.1)) must be set to

when $M=0$ , and

when $M=2$ . Loci of solutions will be parametrized in terms of their vertical coordinate in the physical domain. Thus, moving along any given locus will correspond to an increasing $\text{Im}(z_{\unicode[STIX]{x1D6FC}})$ . Additionally, since they are more relevant from the physical point of view, only loci stemming from the rear tip of the main plate will be considered (see Nelson & Sakajo Reference Nelson and Sakajo2014).

Table 1.

Solutions of equations (2.20) for three different values of $\unicode[STIX]{x1D719}$ .

Figure 2.

Rear tip equilibria and their linear stability for (a) $M=0$ (b) $M=2$ , $\unicode[STIX]{x1D719}=\unicode[STIX]{x03C0}/12$ , (c) $M=2$ , $\unicode[STIX]{x1D719}=\unicode[STIX]{x03C0}/6$ and (d) $M=2$ , $\unicode[STIX]{x1D719}=5\unicode[STIX]{x03C0}/12$ . Thick solid lines represent plates, whereas thin solid and dashed curves represent, respectively, the neutrally stable and unstable parts of the loci. The dotted horizontal lines separate the sections of the equilibrium loci characterized by linear instability and neutral stability. Square and triangle symbols represent, respectively, the locations of neutrally stable and unstable equilibria that will be subject to further analysis (information regarding these equilibria is summarized below in table 2).

Figure 2(a)–(d) shows the equilibrium loci emanating from the rear tip of the main plate of the configurations just introduced and we also identify parts of the loci characterized by different stability properties. For the configuration with $M=0$ , it is seen that a linearly unstable region exists close to the plate (for $0<\text{Im}(z_{\unicode[STIX]{x1D6FC}})\lessapprox 0.3$ ). Small perturbations to equilibria with $z_{\unicode[STIX]{x1D6FC}}$ further away from the main plate exhibit neutral stability. When $M=2$ and $\unicode[STIX]{x1D719}=\unicode[STIX]{x03C0}/12$ , the shape of the locus and stability properties are relatively similar to the case with $M=0$ , but the unstable region has shrunk substantially so that now only equilibria lying in the region with $\text{Im}(z_{\unicode[STIX]{x1D6FC}})\lessapprox 0.075$ exhibit linear instability. Again, above this unstable region small perturbations exhibit neutral stability. For the configuration with $M=2$ and $\unicode[STIX]{x1D719}=\unicode[STIX]{x03C0}/6$ , the unstable region shrinks even further so that linear instability is only witnessed when $\text{Im}(z_{\unicode[STIX]{x1D6FC}})\lessapprox 0.003$ . However, when $M=2$ and $\unicode[STIX]{x1D719}=5\unicode[STIX]{x03C0}/12$ , the shape of the locus and the corresponding stability properties of equilibria change quite dramatically. Equilibria stemming from the rear tip of the main plate remain extremely close to the plate prior to a sharp rise away from it when $\text{Re}(z_{\unicode[STIX]{x1D6FC}})$ is roughly $0.2$ (given the finite graphical resolution, this latter region cannot be discerned in figure 2 d). The neutrally stable region now also extends down to the plate and unstable equilibria are present only in a small neighbourhood of the rear tip.

For each configuration, the lift force experienced by the main plate for increasing $\text{Im}(z_{\unicode[STIX]{x1D6FC}})$ is shown in figure 3. These forces are computed using the Blasius formula (Crowdy Reference Crowdy2006)

where $F_{x}$ and $F_{y}$ are the force components acting in the directions $x$ and $y$ with integration carried out numerically using a trapezoidal rule. The lift, $F_{N}$ , is then retrieved through taking the component of (2.23) in the direction normal to the oncoming flow. For small values of $\text{Im}(z_{\unicode[STIX]{x1D6FC}})$ the lift experienced by the main plate is similar for all flow configurations. As $\text{Im}(z_{\unicode[STIX]{x1D6FC}})$ increases, the lift in the configuration with $M=2$ and $\unicode[STIX]{x1D719}=\unicode[STIX]{x03C0}/12$ grows slowest, whereas for vortex equilibria furthest away from the plate the configuration with $M=2$ and $\unicode[STIX]{x1D719}=5\unicode[STIX]{x03C0}/12$ experiences the highest lifts. In regard to the data shown in figure 3, we can conclude that in all cases with auxiliary plates the lift $F_{N}$ increases continuously with the inclination angle $\unicode[STIX]{x1D719}$ from values lower than in the single-plate configuration ( $M=0$ ) to higher values. In particular, when the inclination angle is close to $\unicode[STIX]{x1D719}=3\unicode[STIX]{x03C0}/10$ , the lift is comparable to its value in the single-plate configuration. Although the configurations with small inclination angles $\unicode[STIX]{x1D719}$ have smaller lift than the single-plate configuration, they can still be useful since, as shown in figure 2, they are characterized by more favourable stability properties. In addition, differences in lift become more evident only for vortex equilibria located further away from the main plate, i.e. with larger $\text{Im}(z_{\unicode[STIX]{x1D6FC}})$ , which are less relevant from the application point of view.

Figure 3.

Lift $F_{N}$ against $\text{Im}(z_{\unicode[STIX]{x1D6FC}})$ . The solid curve represents the single-plate configuration (with $M=0$ ), the dotted curve the configuration with $M=2$ and $\unicode[STIX]{x1D719}=\unicode[STIX]{x03C0}/12$ , the dashed curve the configuration with $M=2$ and $\unicode[STIX]{x1D719}=\unicode[STIX]{x03C0}/6$ and the dot-dash curve the configuration with $M=2$ and $\unicode[STIX]{x1D719}=5\unicode[STIX]{x03C0}/12$ .

Figure 4.

Trajectories (represented by the solid curves) of the solutions of system (2.9) in (a) case #3 and (b) case #4 (see table 2) in which the equilibria $z_{\unicode[STIX]{x1D6FC}}$ are perturbed to $z_{\unicode[STIX]{x1D6FC}}+\unicode[STIX]{x1D6FF}$ , where $\unicode[STIX]{x1D6FF}=0.005\text{i}$ , corresponding to the time interval $0\leqslant t\leqslant 50$ . The dotted rectangles represent the regions magnified in the insets. In the magnified regions the solid circle marks the unperturbed equilibrium.

Finally in this section, uncontrolled point-vortex trajectories are presented for configurations with $M=2$ and $\unicode[STIX]{x1D719}=\unicode[STIX]{x03C0}/6$ . Evolutions are computed through numerical integration of system (2.9) using Euler’s explicit method with a time step of $\text{d}t=0.001$ (the choice of this method is motivated by its straightforward extension to the stochastic case considered in § 6.1). Figure 4(a) and (b) show the trajectories of the vortex when it is perturbed by $0.005\text{i}$ away from the equilibria, respectively, at $\text{Im}(z_{\unicode[STIX]{x1D6FC}})\approx 0.2$ and $\text{Im}(z_{\unicode[STIX]{x1D6FC}})\approx 0.6$ . Responses to these perturbations show good agreement with the linear theory for $\text{Im}(z_{\unicode[STIX]{x1D6FC}})\approx 0.6$ ; that is, the vortex follows a closed orbit over a long period of time. However, when $\text{Im}(z_{\unicode[STIX]{x1D6FC}})\approx 0.2$ , the vortex quickly escapes showing that nonlinear effects become important even for very small perturbations. Again, it should be noted that during these evolutions the Kutta conditions are not enforced. We emphasize that, from the practical point of view, neutrally stable equilibria are not desirable, because neutral stability in the linearized setting does not imply the stability of the nonlinear system, an effect already observed in figure 4(a). In addition, vortices moving along closed trajectories circumscribing the equilibrium give rise to fluctuating loads on the airfoils which degrade their performance. To close this section, in table 2 we collect the information about the different cases which were discussed here and which will be further studied in the controlled setting below. These cases have been selected to be representative of the different behaviours in the controlled setting (cf. §§ 5 and 6). Note that due to the proximity of the vortex to the main plate in case #7, this case it is not necessarily as physically important as the others considered. It is however included in this study for completeness so that a Kasper wing configuration demonstrating linear instability is represented as well. It is also noted that this case will not be analysed in as much detail as the others considered. To complete the physical picture, the streamline patterns corresponding to the seven equilibrium configurations from table 2 are presented in figure 5. We see that the cases corresponding to equilibrium locations further away from the main plate feature larger recirculation regions.

Table 2.

Summary information about the different flow cases which will be analysed in the subsequent sections.

Figure 5.

(a–g) The streamline patterns $\unicode[STIX]{x1D713}=\text{Im}[W]$ corresponding to the vortex equilibria from table 2.

6.1 Control in the presence of random disturbances of the angle of attack

To check how the control performs in a fluctuating background flow, the flow angle of attack $\unicode[STIX]{x1D712}_{0}$ is periodically augmented with a Gaussian random variable, so that we obtain

where $\unicode[STIX]{x0394}t=0.01$ , $\lfloor \cdot \rfloor$ denotes the integer part and $\unicode[STIX]{x0394}\unicode[STIX]{x1D712}(l)$ is the $l$ th sample of the random variable with distribution ${\mathcal{N}}(\unicode[STIX]{x1D707},\unicode[STIX]{x1D70E}^{2})$ for some $\unicode[STIX]{x1D70E},\unicode[STIX]{x1D707}\in \mathbb{R}$ . In other words, the stochastic disturbance is frozen over the time window of length $\unicode[STIX]{x0394}t$ before a new sample is drawn. We are interested here in a single realization of the stochastic process, rather than in any statistic quantities, so once the random variable has been sampled, the closed-loop system (3.1) with (6.1) can be integrated as a deterministic system. We do so with the approach introduced in § 5, i.e. Euler’s explicit method with the time step $\text{d}t=0.001$ .

In figure 17 we show the vortex trajectories together with the corresponding histories of the measurements and control intensities obtained for disturbances with $\unicode[STIX]{x1D707}=0$ and two different values of $\unicode[STIX]{x1D70E}^{2}$ (0.2 and 0.4). In both these examples the initial perturbation has been chosen to lie close to the edge of the basin of attraction and in the absence of control the corresponding vortex trajectories are swept to infinity. For these relatively large perturbations it is seen that the control still performs well in the presence of a fluctuating background flow, even for large values of $\unicode[STIX]{x1D70E}^{2}$ . Figure 18 then presents two examples corresponding to cases #1 and #5 from table 2 with disturbances for which $\unicode[STIX]{x1D707}>0$ . In both these cases, $\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\sim {\mathcal{N}}(0.1,0.1)$ in (6.1) and the same initial perturbation with $\unicode[STIX]{x1D6FF}=-0.25$ was used, chosen to lie within each of the respective basins of attraction. In the single-plate configuration (figure 18 a,b) the control fails and the vortex quickly escapes. However, in the Kasper wing configuration (figure 18 c,d) the control is robust and the vortex gradually moves to, and then undergoes a random motion about, a point close to the uncontrolled equilibrium. We note that in case #5 in the absence of control the vortex trajectory also remained bounded, although in comparison with the controlled case, the departures from the equilibrium position were much larger (to avoid cluttering figures, these results are not shown here).

Figures 17 and 18 illustrate representative types of behaviour seen in the presence of the stochastic forcing in the form (6.1). The general behaviour can be summarized as follows. When $\unicode[STIX]{x1D707}$ was set to zero, the closed-loop control system was robust with respect to stochastic disturbances (6.1) in all cases considered, even for large values of $\unicode[STIX]{x1D70E}^{2}$ . Perturbed trajectories would eventually perform a ‘random walk’ in a region near the uncontrolled equilibrium whose extent depends on $\unicode[STIX]{x1D70E}^{2}$ and in this regime the control magnitude oscillates around zero. On the other hand, for values of $\unicode[STIX]{x1D707}>0$ the closed-loop control was unable to stabilize the equilibrium in cases #1 and #2 even for small values of $\unicode[STIX]{x1D707}$ . However, in cases #3–6 it remained robust up to larger values of $\unicode[STIX]{x1D707}$ .

Figure 17.

(a) Vortex trajectories (solid curve) and estimator trajectories (dotted curve) for initial perturbation $\unicode[STIX]{x1D6FF}=-0.1\text{i}$ and disturbances with $\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\sim {\mathcal{N}}(0,0.2)$ in a case #1. The square symbol indicates the initial position of the perturbed vortex and the solid circle the equilibrium position in the absence of any stochastic forcing. (b) The corresponding time histories of the measurement (solid curve) and control magnitude (dashed curve). (c,d) Same as (a,b), but for a case #4 configuration with $\unicode[STIX]{x1D6FF}=0.075\text{i}$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\sim {\mathcal{N}}(0,0.4)$ ; in (c) the inset represents a magnification of the neighbourhood of the equilibrium.

Figure 18.

(a) Vortex trajectories (solid curve) and estimator trajectories (dotted curve) for initial perturbation $\unicode[STIX]{x1D6FF}=-0.25$ and disturbances with $\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\sim {\mathcal{N}}(0.1,0.1)$ in a case #1. The square symbol indicates the initial position of the perturbed vortex and the solid circle the equilibrium position in the absence of any stochastic forcing. (b) The corresponding time histories of the measurement (solid curve) and control magnitude (dashed curve). (c,d) Same as (a,b), but for a case #5. The inset in (c) shows a magnification of the region close the uncontrolled vortex equilibrium.

6.2 Control in the presence of vortex shedding

We now augment our model with a vortex-shedding mechanism in which point vortices are ‘injected’ into the flow at locations close to the rear tips of each plate at discrete instances of time. The circulations of these injected vortices are chosen such that the Kutta condition (2.10c ) at the rear tip of each plate is satisfied at the time of injection. Integration is again carried out using Euler’s explicit method with a time step of $\text{d}t=0.001$ . Vortices are injected into the flow at $t=0$ and then at intervals of $\unicode[STIX]{x0394}t=0.1$ . The injection location is chosen to be at a distance of $0.1$ from each plate tip in the direction tangent to the plate. That is, in cases #1–6 a single vortex is injected at the point

Additionally, in cases #3–6 two further vortices are injected at

These injection points were chosen as they lie close to the rear tips of each plate and can therefore be considered a fair approximation to a vortex sheet representing a separating boundary layer. They also provide numerical stability over long integration times. Circulations of the injected vortices are calculated as follows. Prior to injection, the total velocity field, including that owing to any previously added vortices (see below), is calculated at each plate tip. Vortices are then injected at the locations given in (6.2)–(6.3) with circulations determined such that the total velocity field, including the contributions from both the vortices already present in the flow field and the newly created ones, vanishes at each plate tip. This corresponds to solving a single linear equation for the circulation of the one added vortex in single-plate cases and to solving a set of three coupled linear equations for the circulations of each of the three added vortices in Kasper wing cases. Following this, the circulations of any newly added vortices are kept constant as the vortices are allowed to move freely, i.e. their positions evolve in time owing to the velocity induced by other vortices plus that of the background flow. In the results that follow, after a vortex has been injected, it is then dealt with explicitly for the remainder of the simulation, i.e. no shed vortices are removed and no far-field averaging is employed.

Figure 19.

(a) Case #1 configuration with initial perturbation $\unicode[STIX]{x1D6FF}=0.3$ . The dashed curve represents the controlled vortex trajectory starting from the outside of the basin of attraction in the absence of shedding, the thin solid curve the controlled trajectory in the presence of shedding and the dotted curve the estimator trajectory in the presence of shedding. The solid circular symbol represents the unperturbed equilibrium position and the square the initial perturbed position. (b) The corresponding time history of the measurements $Y(t)$ (solid curve) and control intensity $m(t)$ (dashed curve) in the presence of shedding. (c) The corresponding circulation of the most recently shed vortex at $z_{i1}$ . (d–f) Same as (a–c) but for a case #5 configuration with $\unicode[STIX]{x1D6FF}=0.52\text{i}$ . Additionally, in (d) the thicker solid lines represent the plate boundaries and in (f) the solid, dotted and dashed curves represents the circulations of the most recently shed vortices at $z_{i1}$ , $z_{i2}$ and $z_{i3}$ respectively.

Examples of the various behaviours observed with vortex shedding added are presented in figures 19 and 20. These behaviours can be summarized as follows. In general, the addition of shed vortices has a stabilizing effect. In almost all configurations tested, perturbations larger than those in the absence of shedding could be stabilized (exceptions to this rule will be mentioned below and are discussed in a little further detail in the following section). In figure 19, two examples are shown in which the vortex is initially perturbed to outside the basins of attraction determined in § 5. Despite this, the vortex trajectories are rapidly stabilized and, after an initial increase, the circulations of the most recently shed vortices decay to zero as demonstrated in figure 19(c,f).

Figure 20(a–c) demonstrates a slightly different behaviour. Now, owing to the larger circulation of the point vortex, instead of returning to the unperturbed equilibrium position, the vortex is held at a location close to the equilibrium and the circulations of the shed vortices tend to constant values. That is, a new equilibrium state emerges. Finally, figure 20(d–f) shows an example of a ‘chaotic’ case. In situations where the perturbation $\unicode[STIX]{x1D6FF}$ is too large for the vortex to be stabilized, but not large enough for the vortex to escape, the vortex can undergo a complicated motion in the vicinity of the equilibrium location over a long period of time.

Although the presence of vortex shedding has, in general, a stabilizing effect in the majority of cases, two scenarios were identified in which the effect was in fact destabilizing. Firstly, in the configurations corresponding to case #2 equilibria would not be stabilized for any initial perturbation $\unicode[STIX]{x1D6FF}$ . Additionally, in cases #3–6 when an initial perturbation with a large positive real component is taken, the extent to which the corresponding evolutions can be controlled is reduced. Physical arguments for these scenarios are discussed in the following section.

Figure 20.

(a) Case #4 configuration with initial perturbation of $\unicode[STIX]{x1D6FF}=0.1\text{i}$ . The dashed curve represents the controlled vortex trajectory in the absence of shedding, the thin solid curve the controlled trajectory in the presence of shedding and the dotted curve the estimator trajectory in the presence of shedding. The thicker solid lines represent the plate boundaries, the solid circular symbol the unperturbed equilibrium position and the square the initial perturbed position. The inset shows a magnification of the region close the vortex equilibrium. (b) The corresponding time histories of the measurements $Y(t)$ (solid curve) and control intensity $m(t)$ (dashed curve) in the presence of shedding. (c) The corresponding circulations of the most recently shed vortices at $z_{i1}$ (solid curve), $z_{i2}$ (dotted curve) and $z_{i3}$ (dashed curve). (d–f) Same as (a–c) but for a case #1 configuration with $\unicode[STIX]{x1D6FF}=0.375\exp (-0.13\text{i}\unicode[STIX]{x03C0})$ . The inset shows the region close to the unperturbed equilibrium.