3.1. Some special functions

We will perform our construction in terms of certain special functions, labelled here as $P(\zeta,q)$, $K(\zeta,q)$ and $L(\zeta,q)$. We define these and state their relevant properties in this section. We refer the reader to Crowdy (Reference Crowdy2020) for further discussion of these functions, including their connection to more traditionally used elliptic functions (as used, for example, in Gurevitch Reference Gurevitch1965; Semenov & Wu Reference Semenov and Wu2020). The advantage of the functions that we use here is that their singularity and periodicity structures present themselves clearly, although it should be possible to derive similar properties for elliptic functions.

To begin, we define the transformation $\theta _n(\zeta )=q^{2n}\zeta$ for all $n\in \mathbb {Z}$ (note that $\theta _0(\zeta )=\zeta$ is the identity transformation), and the set $\varTheta =\{\theta _n(\zeta )\mid n\in \mathbb {Z}\}$. Next, we introduce $D_{\zeta }^{-1}$ to denote the reflection of $D_{\zeta }$ in $C_0$, where by reflection in $C_0$ we mean the transformation $\zeta \mapsto 1/\bar {\zeta }$: $D_{\zeta }^{-1}$ is the annular domain bounded by the circles $C_0$ and $C_{-1}$, where the latter denotes the reflection of $C_1$ in $C_0$ and is centred on the origin and of radius $1/q$ (see figure 3). We define $F$ to be the region that consists of the union of $\overline {D_{\zeta }}$ and $D_{\zeta }^{-1}$, where we use the ‘overline’ notation with respect to a domain to denote the domain's closure, i.e.

(3.4)\begin{equation} F=\{\zeta:q\leq|\zeta|< q^{{-}1}\}, \end{equation}

so $F$ does not contain $C_{-1}$. The images of $F$ under all elements of $\varTheta$ are mutually disjoint and cover the whole of the $\zeta$-plane, except for the origin and the point at infinity. $\varTheta$ is in fact an example of a Schottky group (Ford Reference Ford1972; Crowdy Reference Crowdy2020). We refer to $F$ as a fundamental region of $\varTheta$. (The fundamental region of a Schottky group is not unique.)

Figure 3.

The annuli appearing in the analysis: $D_{\zeta }^{-1}$ (light grey) is the reflection of the pre-image domain $D_{\zeta }$ (turquoise) of figure 2 in the unit circle $C_0$ (dotted blue). The union of $\overline {D_{\zeta }}$ and $D_{\zeta }^{-1}$ forms the fundamental region $F$ of (3.4) for the group $\varTheta$. The union of $\bar {F}$ and the reflection of $F$ (dark grey) in the circle $C_1$ (dotted red) forms the fundamental region $\hat {F}$ of (3.13) for the group $\hat {\varTheta }$. The complex velocity $W'(\zeta )$ has double poles (blue crosses) at $\zeta =-\mathrm {i}$ and $-\mathrm {i}q^2$, and simple zeros (blue circles) at $\zeta =\zeta _2$, $-\overline {\zeta _2}$, $1/\overline {\zeta _2}$ and $-1/\zeta _2$. The mapped complex velocity $\varOmega (\zeta )$ has simple poles (red crosses) at $\zeta =\zeta _1$ and $-1/\zeta _2$ (coinciding with a zero of $W'(\zeta )$), and simple zeros (red discs) at $\zeta =1/\overline {\zeta _1}$ and $-\overline {\zeta _2}$ (also coinciding with a zero of $W'(\zeta )$).

Now, the function $P(\zeta,q)$ is defined for all complex $\zeta$ and (real) $q$ with $0< q<1$, by

(3.5)\begin{equation} P(\zeta,q)=(1-\zeta)\prod_{n=1}^{\infty}(1-q^{2n}\zeta)(1-q^{2n}\zeta^{{-}1}). \end{equation}

Up to a normalisation, $P(\zeta,q)$ is the Schottky–Klein prime function associated with $\varTheta$. One can check that $P(\zeta,q)$ is analytic everywhere in $F$, and is non-zero in $F$ except for a simple zero at $\zeta =1$. Furthermore, one can deduce directly from (3.5) that

(3.6a,b)\begin{equation} P(q^2\zeta,q)={-}\zeta^{{-}1}\,P(\zeta,q), \quad P(\zeta^{{-}1},q)={-}\zeta^{{-}1}\,P(\zeta,q). \end{equation}

Relation (3.6a) can be used to continue $P(\zeta,q)$ to points $\zeta$ outside of $F$. In particular, one can deduce from (3.6a), and the properties of $P(\zeta,q)$ for $\zeta \in F$ noted above, that $P(\zeta,q)$ is analytic everywhere in the $\zeta$-plane except for essential singularities at the origin and the point at infinity, and that it has simple zeros at $\zeta =q^{2n}$ for all $n\in \mathbb {Z}$. Of course, one could also deduce these properties directly from (3.5).

Next, the function $K(\zeta,q)$ is defined by

(3.7)\begin{equation} K(\zeta,q)=\zeta\,\frac{\mathrm{d}}{\mathrm{d}\zeta}\log P(\zeta,q). \end{equation}

It follows from (3.5) that

(3.8)\begin{equation} K(\zeta,q)= \frac{1}{\zeta-1}+1 +\sum_{n=1}^{\infty} q^{2n} \left( \frac{1}{\zeta-q^{2n}} -\frac{1}{\zeta^{{-}1}-q^{2n}} \right). \end{equation}

One may check that $K(\zeta,q)$ is analytic everywhere in $F$ except for a simple pole at $\zeta =1$ with residue $1$. Also, it follows from (3.6) that

(3.9a,b)\begin{equation} K(q^2\zeta,q)=K(\zeta,q)-1, \quad K(\zeta^{{-}1},q)=1-K(\zeta,q). \end{equation}

Finally, the function $L(\zeta,q)$ is defined by

(3.10)\begin{equation} L(\zeta,q)=\zeta\,\frac{\mathrm{d}}{\mathrm{d}\zeta}K(\zeta,q). \end{equation}

It follows from (3.8) that

(3.11)\begin{equation} L(\zeta,q)= \frac{-1}{(\zeta-1)^2} -\frac{1}{\zeta-1} -\zeta\sum_{n=1}^{\infty} q^{2n} \left( \frac{1}{(\zeta-q^{2n})^2} +\frac{1}{(1-q^{2n}\zeta)^2} \right). \end{equation}

One may check that $L(\zeta,q)$ is analytic everywhere in $F$ except for a double pole at $\zeta =1$ with residue $-1$. Also, it follows from (3.9) that

(3.12a,b)\begin{equation} L(q^2\zeta,q)=L(\zeta,q), \quad L(\zeta^{{-}1},q)=L(\zeta,q). \end{equation}

In addition to the above, we will also make use of the functions $P(\zeta,q^2)$, $K(\zeta,q^2)$ and $L(\zeta,q^2)$. Of course, with $0< q<1$, we also have $0< q^2<1$, so $P(\zeta,q^2)$ is defined by (3.5) simply with $q$ replaced by $q^2$. And $P(\zeta,q^2)$ is (up to a normalisation) the Schottky–Klein prime function associated with $\hat {\varTheta }=\{\theta _{2n}(\zeta )\mid n\in \mathbb {Z}\}$, which is a subgroup of $\varTheta$ and itself a Schottky group (Vasconcelos, Marshall & Crowdy Reference Vasconcelos, Marshall and Crowdy2015). A fundamental region of $\hat {\varTheta }$ is

(3.13)\begin{equation} \hat{F}=\{\zeta:q^3<|\zeta|\leq q^{{-}1}\}, \end{equation}

i.e. the region that consists of the union of $\bar {F}$ and the reflection of $F$ in the circle $C_1$, where reflection in $C_1$ is given by $\zeta \mapsto q^2/\bar {\zeta }$ (see figure 3).

It follows directly from (3.5) that

(3.14)\begin{equation} P(\zeta,q)=P(\zeta,q^2)\,P(q^2\zeta,q^2), \end{equation}

and hence from (3.7) and (3.10) that

(3.15a,b)\begin{equation} K(\zeta,q)=K(\zeta,q^2)+K(q^2\zeta,q^2), \quad L(\zeta,q)=L(\zeta,q^2)+L(q^2\zeta,q^2). \end{equation}

3.2. Constructing the complex potential $W(\zeta )=w(z)$

The construction of $W(\zeta )$ is straightforward as it is simply the complex potential for flow in the annulus $D_{\zeta }$, driven by a dipole at $-\mathrm {i}$ and having circulation $\varGamma$ around $C_1$ with stagnation points at $\zeta _2$ and $\zeta _3$. Since both $D_{\zeta }$ and the dipole flow are right–left symmetric, $\zeta _3=-\overline {\zeta _2}$, as shown in § 3.2.1 below. Figure 4(a) shows contours of $\mathrm {Im}\{W(\zeta )\}$ – i.e. flow streamlines – in $D_{\zeta }$ for a typical solution.

Figure 4.

Illustrations of the solution components for a typical solution (namely, that which is illustrated in figure 6(b) below). Here, $\alpha =-{\rm \pi} /4$ and $y_c=0.3$ (see (3.51)). (a) Contours of $\mathrm {Im}\{W(\zeta )\}$ as given by (3.19), giving the flow streamlines in the pre-image domain $D_{\zeta }$ with a dipole at $\zeta =\mathrm {-i}$ (which corresponds to the point at infinity in $D$), stagnation points symmetrically at $\zeta =\zeta _2$ (the trailing edge in $D$) and $-\overline {\zeta _2}$ (on the leading face in $D$), and tangential flow along $C_0$ (the free surface in $D$) and $C_1$ (the foil). (b) Isotachs, contours of $|\varOmega (\zeta )|$ as given by (3.34), the flow speed mapped to the pre-image domain $D_{\zeta }$, with infinite speed at $\zeta _1$ (the leading edge in $D$), a single stagnation point at $\zeta =-\overline {\zeta _2}$ (on the leading face in $D$), and constant speed along $C_0$ (the free surface in $D$), with no stagnation point at $\zeta =\zeta _2$ (the trailing edge in $D$) where the speed is finite.

It follows from the properties of $w(z)$ and $z(\zeta )$ noted above that $W(\zeta )$ must be analytic for all $\zeta \in D_{\zeta }$ except that (as follows from (2.1) and (3.1)) for $\zeta$ local to $-\mathrm {i}$, to leading order,

(3.16)\begin{equation} W(\zeta)\sim\frac{Ua}{\zeta+\mathrm{i}}. \end{equation}

Also, it follows from (2.2) that

(3.17)\begin{equation} \mathrm{Im}\{W(\zeta)\}=\psi_j\quad\text{for }\zeta\in C_j,\ j=0,1. \end{equation}

Furthermore, it follows from (2.6) that

(3.18)\begin{equation} \oint_{\hat{\mathcal{C}}} \,\mathrm{d}W(\zeta)=\varGamma, \end{equation}

where we can take $\hat {\mathcal {C}}$ to be a circle centred on the origin, of some radius $\hat {q}$, where $q<\hat {q}<1$ (so that $\hat {\mathcal {C}}$ lies in the interior of $D_{\zeta }$), and we integrate around $\hat {\mathcal {C}}$ in the anticlockwise direction. Recall that $\varGamma$ is still to be determined. We propose that

(3.19)\begin{equation} W(\zeta) = Ua\mathrm{i}\,K(\mathrm{i}\zeta,q) -\frac{\mathrm{i}\varGamma}{2{\rm \pi}}\log\zeta. \end{equation}

One can verify that $W(\zeta )$ as stated by (3.19) possesses the properties stated above, as follows. First, it follows from the properties of $K(\zeta,q)$, that $W(\zeta )$ as given by (3.19) is analytic for all $\zeta \in D_{\zeta }$ except for a simple pole with residue $Ua$ at $\zeta =-\mathrm {i}$, as required by (3.16). It is also evident that this form for $W(\zeta )$ satisfies (3.18). Finally, to check the boundary conditions (3.17), it is helpful to first note that

(3.20) \begin{equation} \mathrm{Re}\{K(\mathrm{i}\zeta,q)\}=\tfrac{1}{2}\left(K(\mathrm{i}\zeta,q)+K(1/(\mathrm{i}\zeta),q)\right)=\tfrac{1}{2}\quad\text{for }\zeta\in C_0, \end{equation}

where the first equality follows from the fact that for $\zeta \in C_0$, $\bar {\zeta }=1/\zeta$, and the second follows from (3.9b). Similarly,

(3.21)\begin{equation} \mathrm{Re}\{K(\mathrm{i}\zeta,q)\}=\tfrac{1}{2}(K(\mathrm{i}\zeta,q)+K(q^2/(\mathrm{i}\zeta),q))=0\quad\text{for }\zeta\in C_1, \end{equation}

where now the first equality follows from the fact that for $\zeta \in C_1$, $\bar {\zeta }=q^2/\zeta$, and the second follows by using both equations in (3.9). Then one may deduce that (3.17) holds (with $\psi _0=Ua/2$ and $\psi _1=-(\varGamma /(2{\rm \pi} ))\ln q$). This completes our verification of (3.19). Similar arguments to those below for $\varOmega (\zeta )$ show that $W(\zeta )$ is unique.

Differentiating (3.19) gives

(3.22)\begin{equation} W'(\zeta) = \frac{1}{\zeta} \left( Ua\mathrm{i}\,L(\mathrm{i}\zeta,q)-\frac{\mathrm{i}\varGamma}{2{\rm \pi}} \right). \end{equation}

Now, in order to impose the Kutta condition at the trailing endpoint $z_2$ of the hydrofoil, we must choose $\varGamma$ such that $W'(\zeta _2)=0$, giving

(3.23)\begin{equation} \varGamma=2{\rm \pi} Ua\,L(\mathrm{i}\zeta_2,q). \end{equation}

(Note that it follows from (3.24) that $L(\mathrm {i}\zeta _2,q)$ is real.)

3.2.1. Properties of $W'(\zeta )$, the complex velocity in $D_\zeta$

We now note some properties of $W'(\zeta )$ that will be useful later. First, $W'(-\overline {\zeta _2})=0$. This follows from (3.22) using the fact that $W'(\zeta _2)=0$ and

(3.24)\begin{equation} L(-\mathrm{i}\overline{\zeta_2},q)=L(q^2/(\mathrm{i}\zeta_2),q)=L(\mathrm{i}\zeta_2,q), \end{equation}

where the second equality follows by using both equations in (3.12). We will henceforth assume that $-\overline {\zeta _2}\neq \zeta _2$, or equivalently, that $\zeta _2\neq \pm \mathrm {i}q$ (although, in § 5.2, we will consider the limit of our results as $\zeta _2\rightarrow \pm \mathrm {i}q$, whilst $\zeta _1\rightarrow \mp \mathrm {i}q$, respectively). We now claim that the zeros of $W'(\zeta )$ at $\zeta =\zeta _2$ and $-\overline {\zeta _2}$ are the only zeros of $W'(\zeta )$ in $F$, and are simple zeros. To demonstrate this, note that it follows from (3.22) and (3.12a) that

(3.25)\begin{equation} W'(q^2\zeta)=\frac{1}{q^2}\,W'(\zeta). \end{equation}

Thus

(3.26)\begin{equation} \mathrm{d}\log W'(q^2\zeta)=\mathrm{d}\log W'(\zeta). \end{equation}

It then follows from (3.26) and an application of the argument principle (Ahlfors Reference Ahlfors1979, § 5.2) that $W'(\zeta )$ has the same number of poles as zeros in $F$, where these are both counted according to their multiplicities. (Here, $\zeta _2$ and $-\overline {\zeta _2}$ lie on the boundary of $F$, but by standard arguments one can adapt the argument principle to take account of this.) But it follows from (3.22) and the properties of $L(\zeta,q)$ that $W'(\zeta )$ has a double pole at $\zeta =-\mathrm {i}$ and no other singularities in $F$. Thus, as claimed, the zeros of $W'(\zeta )$ at $\zeta =\zeta _2$ and $-\overline {\zeta _2}$ must be the only zeros of $W'(\zeta )$ in $F$, and must be simple zeros. It also then follows that

(3.27)\begin{equation} \zeta_3={-}\overline{\zeta_2}. \end{equation}

Finally, one may also deduce that $W'(\zeta )$ is analytic for all $\zeta \in \hat {F}$ except for double poles at $\zeta =-\mathrm {i}$ and $-\mathrm {i}q^2$, and that the only zeros of $W'(\zeta )$ in $\hat {F}$ are simple zeros at $\zeta =\zeta _2$, $-\overline {\zeta _2}$, $1/\overline {\zeta _2}$ and $-1/\zeta _2$ – see figure 3. ($W'(\zeta )$ also has zeros at $\zeta =q^2\zeta _2$ and $-q^2\overline {\zeta _2}$, but both of these points have modulus $q^3$ and so are not contained in $\hat {F}$.)

3.3. Constructing the mapped complex velocity $\varOmega (\zeta )=w'(z)$

It follows from the properties of $w'(z)$ and $z(\zeta )$ stated above that $\varOmega (\zeta )$ must be analytic for all $\zeta \in D_{\zeta }$ except for a simple pole at $\zeta =\zeta _1$ (as follows from (2.5) and the fact that $z'(\zeta )$ has a simple zero at $\zeta _1$). Furthermore, $\varOmega (\zeta )$ must be non-zero for all $\zeta$ in the closure of $D_{\zeta }$ except for a simple zero at $\zeta =-\overline {\zeta _2}$ (as follows from (2.7), recalling (3.27)). Note that $\varOmega (\zeta )$ is non-zero at $\zeta =\zeta _2$ – i.e. $w'(z)$ is non-zero at the trailing endpoint $z_2$ of the hydrofoil – because $W'(\zeta )$ and $z'(\zeta )$ both have simple zeros at $\zeta =\zeta _2$ (and $\varOmega (\zeta )=W'(\zeta )/z'(\zeta )$). In addition, it follows from (2.3) and (2.4) that

(3.28a,b)\begin{equation} \left|\varOmega(\zeta)\right|=U\text{ for }\zeta\in C_0, \quad \mathrm{Im}\{\mathrm{e}^{\mathrm{i}\alpha}\,\varOmega(\zeta)\}=0\text{ for }\zeta\in C_1, \end{equation}

and from (2.1) that

(3.29)\begin{equation} \varOmega(-\mathrm{i}) = U. \end{equation}

Figure 4(b) illustrates the mapped flow speed $|\varOmega (\zeta )|$ in $D_{\zeta }$ for a typical solution.

It follows from (3.28a) that for $\zeta \in C_0$, since $\bar {\zeta }=1/\zeta$,

(3.30)\begin{equation} \varOmega(\zeta)=\frac{U^2}{\bar{\varOmega}(1/\zeta)}, \end{equation}

where we define $\bar {\varOmega }(\zeta )=\overline {\varOmega (\bar {\zeta })}$. However, it follows from the properties of $\varOmega (\zeta )$ that $1/\bar {\varOmega }(1/\zeta )$ is analytic for all $\zeta \in D_{\zeta }^{-1}$ except for a simple pole at $\zeta =-1/\zeta _2$, and also non-zero for all $\zeta$ in the closure of $D_{\zeta }^{-1}$ except for a simple zero at $\zeta =1/\overline {\zeta _1}$. It then follows by analytic continuation that (3.30) must in fact hold for all $\zeta$ in the closure of $F$, and that $\varOmega (\zeta )$ is analytic for all $\zeta \in F$ except for simple poles at $\zeta =\zeta _1$ and $-1/\zeta _2$. Furthermore, the only zeros of $\varOmega (\zeta )$ in $F$ are simple zeros at $\zeta =-\overline {\zeta _2}$ and $1/\overline {\zeta _1}$.

Next, one may deduce from (3.28b) that for $\zeta \in C_1$, since $\bar {\zeta }=q^2/\zeta$,

(3.31)\begin{equation} \varOmega(\zeta)=\mathrm{e}^{{-}2\mathrm{i}\alpha}\,\bar{\varOmega}(q^2/\zeta). \end{equation}

Then, by arguments similar to those just stated after (3.30), it follows that (3.31) must in fact hold for all $\zeta$ in the closure of $\hat {F}$. But furthermore, one may deduce that $\varOmega (\zeta )$ is analytic for all $\zeta \in \hat {F}$ except for simple poles at $\zeta =\zeta _1$ and $-1/\zeta _2$, and that the only zeros of $\varOmega (\zeta )$ in $\hat {F}$ are simple zeros at $\zeta =-\overline {\zeta _2}$ and $1/\overline {\zeta _1}$ – see figure 3.

Now note that by repeated analytic continuation, one may show that (3.30) and (3.31) in fact hold for all $\zeta$, except at $0$ and infinity. Combining these two relations gives

(3.32)\begin{equation} \varOmega(q^2\zeta)=\frac{\mathrm{e}^{{-}2\mathrm{i}\alpha}\,U^2}{\varOmega(\zeta)}, \end{equation}

hence

(3.33)\begin{equation} \varOmega(q^4\zeta)=\varOmega(\zeta), \end{equation}

from which one may deduce that $\varOmega (\zeta )$ is automorphic with respect to the group $\hat {\varTheta }$. The property (3.33), together with the properties of $\varOmega (\zeta )$ for $\zeta$ in the fundamental region $\hat {F}$ of $\hat {\varTheta }$ that are stated just after (3.31), along with the normalisation (3.29), are enough to identify $\varOmega (\zeta )$ uniquely. To check this, consider the ratio of $\varOmega (\zeta )$ and any other function that has these properties. This ratio must be analytic everywhere in $\hat {F}$ (all poles of the numerator are cancelled by the same poles of the denominator; likewise all zeros of the denominator are cancelled by the same zeros of the numerator), and also automorphic with respect to $\hat {\varTheta }$. It then follows from the extended form of Liouville's theorem for automorphic functions (e.g. see Ford Reference Ford1972), that this ratio must equal a constant. But it follows from the normalisation on $\varOmega (\zeta )$ that is imposed by (3.29) that this constant must equal $1$. We thus seek to construct a function with these properties. One could construct this as a ratio of products of $P$ functions. However, it will be more convenient later – in particular, we will wish to differentiate $\varOmega (\zeta )$ (with respect to $\zeta$) – to instead construct it as a sum of the form

(3.34)\begin{equation} \varOmega(\zeta) = \alpha_1\,K(\zeta/\zeta_1,q^2) +\alpha_2\,K(-\zeta_2\zeta,q^2) +\alpha_0, \end{equation}

for some (unique) constants $\alpha _0$, $\alpha _1$ and $\alpha _2$, which we will determine in due course. We highlight the fact that the second argument of the $K$ functions that appear in (3.34) is $q^2$, not $q$.

One may verify (3.34) as follows. First, one may check from the properties of $K(\zeta,q)$ that for all values of $\alpha _0$, $\alpha _1$ and $\alpha _2$, the function on the right-hand side of (3.34) is analytic for all $\zeta \in \hat {F}$ except for simple poles at $\zeta =\zeta _1$ and $-1/\zeta _2$, as required. However, in order for it to satisfy (3.33), it follows from (3.9a) that

(3.35)\begin{equation} \alpha_1+\alpha_2=0. \end{equation}

Next, in order that $\varOmega (-\overline {\zeta _2})=0$, it follows – using the fact that $K(q^2,q^2)=0$ (which one may deduce by using both equations in (3.9) to show that $K(q^2,q^2)=-K(q^2,q^2)$) – that also

(3.36)\begin{equation} \alpha_1\,K(-\overline{\zeta_2}/\zeta_1,q^2)+\alpha_0=0. \end{equation}

Note that we will assume that $\zeta _1\neq -\overline {\zeta _2}$, as otherwise $K(-\overline {\zeta _2}/\zeta _1,q^2)$ is unbounded and – as will be shown in § 5.1 – the hydrofoil is horizontal.

Next, one could also write down a relation between $\alpha _0$, $\alpha _1$ and $\alpha _2$ by imposing on (3.34) the condition that $\varOmega (1/\overline {\zeta _1})=0$. However, one can show that this relation can be retrieved from (3.35) and (3.36) (using the fact that $\mathrm {Re}\{K(\zeta,q^2)\}=1/2$ for all $\zeta \in C_0$ – cf. (3.20)). Finally, then, it follows from (3.29) that

(3.37)\begin{equation} \alpha_1\,K(-\mathrm{i}/\zeta_1,q^2) +\alpha_2\,K(\mathrm{i}\zeta_2,q^2) +\alpha_0=U. \end{equation}

Combining (3.35)–(3.37), and also making use of (3.9b), one arrives at

(3.38a–c) \begin{align} \alpha_1=\frac{-U}{K(\mathrm{i}\zeta_1,q^2)+K(\mathrm{i}\zeta_2,q^2)-K(-\zeta_1/\overline{\zeta_2},q^2)}, \enspace \alpha_2={-}\alpha_1, \enspace \alpha_0={-}K(-\overline{\zeta_2}/\zeta_1,q^2)\,\alpha_1. \end{align}

This also completes our check on (3.34).

3.4. Completing the solution: constructing the mapping $z(\zeta )$

We now introduce the function

(3.39)\begin{equation} H(\zeta)=\zeta\,z'(\zeta), \end{equation}

which may be determined as follows. First, it follows from (3.3) that

(3.40)\begin{equation} H(\zeta)=\zeta\,\frac{W'(\zeta)}{\varOmega(\zeta)}. \end{equation}

Then it follows from (3.25) and (3.33) that

(3.41)\begin{equation} H(q^4\zeta)=H(\zeta). \end{equation}

Hence $H(\zeta )$ is automorphic with respect to $\hat {\varTheta }$. Furthermore, one can check from the properties of $W'(\zeta )$ and $\varOmega (\zeta )$ identified in §§ 3.2 and 3.3 that $H(\zeta )$ is analytic everywhere in $\hat {F}$ except for a simple pole at $\zeta =1/\overline {\zeta _1}$ and double poles at $\zeta =-\mathrm {i}$ and $-\mathrm {i}q^2$. These properties of $H(\zeta )$, along with its residues at the aforementioned poles and the coefficients of $(\zeta +\mathrm {i})^{-2}$ and $(\zeta +\mathrm {i}q^2)^{-2}$ in the Laurent series expansions of it about $\zeta =-\mathrm {i}$ and $-\mathrm {i}q^2$, respectively, identify $H(\zeta )$ uniquely, up to an additive constant. This follows by arguments similar to those that we have already applied to $\varOmega (\zeta )$ (see the paragraph just after (3.33)), although one should now consider the difference of $H(\zeta )$ and any other function with these properties; this difference must be analytic everywhere in $\hat {F}$, and automorphic with respect to $\hat {\varTheta }$, and so must equal a constant. It then follows from the properties of $K(\zeta,q)$ and $L(\zeta,q)$ that we can write

(3.42)\begin{align} H(\zeta)={}&\beta_1\,K(\overline{\zeta_1}\zeta,q^2) +\beta_2\,K(\mathrm{i}\zeta,q^2) +\beta_3\,K(\mathrm{i}q^2\zeta,q^2)\nonumber\\ &{}+\beta_4\,L(\mathrm{i}\zeta,q^2) +\beta_5\,L(\mathrm{i}q^2\zeta,q^2) +\beta_0, \end{align}

for some unique constants $\beta _0,\ldots,\beta _5$. We determine these constants in Appendix A.

Now note that, evidently, as follows from (3.39), dividing the right-hand side of (3.42) by $\zeta$ provides an expression for $z'(\zeta )$. Recalling (3.7) and (3.10), it is straightforward to integrate this expression to find

(3.43)\begin{align} z(\zeta)={}&\beta_1 \log P(\overline{\zeta_1}\zeta,q^2) +\beta_2 \log P(\mathrm{i}\zeta,q^2) +\beta_3 \log P(\mathrm{i}q^2\zeta,q^2)\nonumber\\ &{}+\beta_4\,K(\mathrm{i}\zeta,q^2) +\beta_5\,K(\mathrm{i}q^2\zeta,q^2) +c, \end{align}

where $c$ is an additional constant. One might expect to see the term $\beta _0\log \zeta$ on the right-hand side of (3.43). However, we can omit this for the following reason. Of course, our map $z(\zeta )$ must be single-valued in $D_{\zeta }$; one can check that the form on the right-hand side of (3.43) is indeed so, as follows. First, note that it is straightforward to rewrite (3.5) as

(3.44)\begin{equation} P(\zeta,q)= \kappa(q)\,(\zeta-1)\prod_{n=1}^{\infty}(\zeta^{{-}1}(\zeta-q^{2n})(\zeta-q^{{-}2n})), \end{equation}

where $\kappa (q)$ is independent of $\zeta$ (in fact, $\kappa (q)=-\prod _{n=1}^{\infty }(-q^{2n})$). Then it is evident that the change in $\log P(\zeta,q)$ after $\zeta$ completes a circuit around a circle that is centred on the origin and of radius $\rho$, say, in the anticlockwise direction, is equal to $0$ if $q^2<\rho <1$ (since for each $n\geq 1$, the change due to the logarithmic singularity at $\zeta =q^{2n}$ is cancelled by that due to a logarithmic singularity of opposite strength at the origin), but equal to $2{\rm \pi} \mathrm {i}$ if $1<\rho < q^{-2}$, and so on. For $\zeta \in D_{\zeta }$, we have $q<|\zeta |<1$, and hence $q^4<|\overline {\zeta _1}\zeta |, |\mathrm {i}\zeta |, |\mathrm {i}q^2\zeta |<1$. It then follows that the terms $\log P(\overline {\zeta _1}\zeta,q^2)$, $\log P(\mathrm {i}\zeta,q^2)$ and $\log P(\mathrm {i}q^2\zeta,q^2)$ that appear in (3.43) are all single-valued in $D_{\zeta }$. And $K(\mathrm {i}\zeta,q^2)$ and $K(\mathrm {i}q^2\zeta,q^2)$ are also both single-valued in $D_{\zeta }$. However, $\beta _0\log \zeta$ is not single-valued in $D_{\zeta }$. Hence

(3.45)\begin{equation} \beta_0=0, \end{equation}

so the integrated form for our map $z(\zeta )$ is given by (3.43) and is single-valued in $D_{\zeta }$. Condition (3.45) places a constraint on our mapping parameters, as we discuss further in the next subsection.

3.5. Specifying parameter values

Our parametrisation depends on the following parameters: $q$, $\arg \{\zeta _1\}$, $\arg \{\zeta _2\}$, $a$ (which, recall, is real and positive) and $c$ (which is complex). Trivially, for any values of $q$, $\arg \{\zeta _1\}$ and $\arg \{\zeta _2\}$, we can fix $c$ and $a$ to ensure that $z_1=0$ and the length of the hydrofoil is $1$; more specifically, it follows from (3.43) that we should take

(3.46) \begin{align} c&={-}(\beta_1 \log P(q^2,q^2) +\beta_2 \log P(\mathrm{i}\zeta_1,q^2) +\beta_3 \log P(\mathrm{i}q^2\zeta_1,q^2)\nonumber\\&\qquad\ +\beta_4\,K(\mathrm{i}\zeta_1,q^2) +\beta_5\,K(\mathrm{i}q^2\zeta_1,q^2)) \end{align}

with

(3.47) \begin{align} a= \Bigl|&\hat{\beta}_1 \log P(\overline{\zeta_1}\zeta_2,q^2) +\hat{\beta}_2 \log P(\mathrm{i}\zeta_2,q^2) +\hat{\beta}_3 \log P(\mathrm{i}q^2\zeta_2,q^2)\nonumber\\ &{}+\hat{\beta}_4\,K(\mathrm{i}\zeta_2,q^2) +\hat{\beta}_5\,K(\mathrm{i}q^2\zeta_2,q^2) +\hat{c} \Bigr|^{{-}1}, \end{align}

where here we use $\hat {\beta }_j$ to denote $\beta _j/a$ for $j=1,\ldots,5$, and $\hat {c}$ to denote $c/a$. (It is evident that one may factor out $a$ from our expressions (A11), (A7), (A9), (A1) and (A5) for $\beta _1,\ldots,\beta _5$, and hence also from our expression (3.46) for $c$.) Next, in order to fix $\arg \{z_2\}=\alpha$, one could use the condition that one obtains by simply setting $\zeta =\zeta _2$ on the right-hand side of (3.43) and then requiring that the argument of this equals $\alpha$. However, a simpler condition is given by (3.49) below. One may deduce this by noting that it follows from (3.31) and (3.34) (along with both equations in (3.9)) that

(3.48)\begin{equation} \frac{\overline{\alpha_1}}{\alpha_1}={-}\mathrm{e}^{2\mathrm{i}\alpha}, \end{equation}

and hence from (3.38a) (and both equations in (3.9) again) that

(3.49)\begin{equation} \frac{K(\mathrm{i}\zeta_1,q^2)+K(\mathrm{i}\zeta_2,q^2)-K(-\zeta_1/\overline{\zeta_2},q^2)} {K(\mathrm{i}q^2\zeta_1,q^2)+K(\mathrm{i}q^2\zeta_2,q^2)+K(-\overline{\zeta_2}/\zeta_1,q^2)} =\mathrm{e}^{2\mathrm{i}\alpha}. \end{equation}

Finally, in addition to the above, we must also impose the single-valuedness condition (3.45) with $\beta _0$ as given by (A12), which one can show is equivalent to

(3.50)\begin{align} L(\mathrm{i}\zeta_1,q^2)+\mathrm{e}^{2\mathrm{i}\alpha}\,L(\mathrm{i}q^2\zeta_1,q^2) - \frac{\left(L(\mathrm{i}\zeta_1,q^2)-L(\mathrm{i}\zeta_2,q^2)\right)\left( K(\mathrm{i}\zeta_1,q^2)-\mathrm{e}^{2\mathrm{i}\alpha}\,K(\mathrm{i}q^2\zeta_1,q^2) \right)} {K(\mathrm{i}\zeta_1,q^2)+K(\mathrm{i}\zeta_2,q^2)-K(-\zeta_1/\overline{\zeta_2},q^2)} =0. \end{align}

Equations (3.49) and (3.50) fix two of the remaining three parameters $\arg \{\zeta _1\}$, $\arg \{\zeta _2\}$ and $q$. This leaves one remaining free parameter. As is borne out by the examples presented in § 6, one could interpret this remaining parameter as corresponding in some sense to the depth of the hydrofoil below the free surface. As a measure of this depth, we will adopt the following. Recall that we fix the leading and trailing endpoints of the hydrofoil to be at the origin and $\mathrm {e}^{\mathrm {i}\alpha }$, respectively. We now seek some convenient reference level for the $y$-coordinate of points on the free surface. As we show next in § 3.6, in general, $y$ does not tend to a constant along the free surface as $|x|\rightarrow \infty$. However (as is the case for the examples that we present in § 6), it appears that there is a single (finite) point, say $z_c$, on the free surface at which $\mathrm {d} y/\mathrm {d} x=0$; there is a peak of the free surface at $z_c$ when the leading endpoint of the hydrofoil is above its trailing endpoint – i.e. when $\alpha <0$ – and a trough at $z_c$ when $\alpha >0$. Fixing $\mathrm {Im}\{z_c\}=y_c$, say, then essentially fixes the depth of the hydrofoil. This places the following additional condition on our mapping parameters. Of course, $z_c=z(\zeta _c)$ for some $\zeta _c$ on $C_0$ at which $\mathrm {Im}\{\mathrm {d}z(\zeta )/\mathrm {d}\theta \}=0$, where here we use $\theta$ to denote the angular polar coordinate of $\zeta$. Hence (recalling (3.39)) one may deduce that

(3.51)\begin{equation} \mathrm{Im}\{z(\zeta_c)\}=y_c \end{equation}

and

(3.52)\begin{equation} \mathrm{Re}\{H(\zeta_c)\}=0. \end{equation}

3.6. Limit as $|x|\rightarrow \infty$

To conclude this section, we determine the limiting shape of the free surface as $|x|\rightarrow \infty$. To do so, we consider the limit of $z(\zeta )$ as given by (3.43) as $\zeta \rightarrow -\mathrm {i}$ along $C_0$. Equivalently, we may take $\zeta =\mathrm {e}^{\mathrm {i}((-{\rm \pi} /2)+\epsilon )}$ and consider the limit of $z(\zeta )$ as $\epsilon \rightarrow 0^{\pm }$. In this limit, the only terms in (3.43) that become singular are $\log P(\mathrm {i}\zeta,q^2)$ and $K(\mathrm {i}\zeta,q^2)$; recalling (3.5) and (3.8) (and using series expansions for $\exp$ and $\log$), one finds that

(3.53a,b)\begin{equation} \log P(\mathrm{i}\zeta,q^2)\sim\ln|\epsilon|+O(1),\quad K(\mathrm{i}\zeta,q^2)\sim\frac{\mp\mathrm{i}}{|\epsilon|}+O(1). \end{equation}

Hence, recalling (A1), it follows from (3.43) that in this limit, assuming that $\beta _2$ and $a$ are of the order of $1$,

(3.54)\begin{equation} z(\zeta)\sim\beta_2\ln|\epsilon|\pm\frac{a}{|\epsilon|}+O(1). \end{equation}

But, as shown in Appendix A (see its final paragraph), $\beta _2$ is purely imaginary. It then follows from (3.54) that as $|x|\rightarrow \infty$, the free surface tends to the curve given by

(3.55)\begin{equation} y(x)={-}\mathrm{Im}\{\beta_2\}\ln|x|. \end{equation}

Thus the $y$-coordinate of points on the free surface tends to infinity as $|x|\rightarrow \infty$. SW20 observe the same limiting behaviour for the free surface for flow past a submerged circular cylinder without gravity – see their equation (3.8). The sole exception to this appears to be for $\alpha <0$ in the limit when the depth of the hydrofoil below the free surface tends to zero, as in figure 5. Equally, (4.9) and (4.10) show that the lift coefficient $C_L$ is simply a multiple of $\mathrm {Im}\{\beta _2\}$, and figure 7(a) shows that $C_L\to 0$ as $y_c\to 0$ for $\alpha =-{\rm \pi} /4$.

Figure 5.

Free surface profiles (blue) for flow past a hydrofoil (red) for various angles of attack $-\alpha$ and leading-edge submergence $y_c$: (a) $\alpha =-{\rm \pi} /4$, $y_c=0.01, 0.05$ and $0.1, 0.2, 0.3,\dots$; and (b) $\alpha ={\rm \pi} /3$, $y_c=0.28, 0.8$ and $1.2, 1.4, 1.6, \ldots$. Lengths here and in subsequent figures are normalised on the length of the foil.

5.1. A horizontal hydrofoil

In deriving our parametrisation, we have assumed that $\zeta _1\neq -\overline {\zeta _2}$ (see just after (3.36)), i.e. that $\zeta _1$ and $\zeta _2$ are not reflections of one another in the imaginary $\zeta$-axis. However, let us now consider the limit of our results as $-\zeta _1/\overline {\zeta _2}\rightarrow 1$. First, in this limit, it is evident from (3.38a,b) and (3.37) that (since $K(-\zeta _1/\overline {\zeta _2},q^2)$ becomes unbounded) $\alpha _1=\alpha _2=0$ and $\alpha _0=U$, and hence, as follows from (3.34), $\varOmega (\zeta )=U$ for all $\zeta$. Thus $w'(z)=U$ for all $z$. It follows that in this case the hydrofoil must be horizontal. Indeed, it follows that in this case our expressions for the constants $\beta _1,\ldots,\beta _5$ – see Appendix A – reduce to

(5.1a,b)\begin{equation} \beta_1=\beta_2=\beta_3=0,\quad\beta_4=\beta_5=a\mathrm{i}, \end{equation}

and hence our expression (3.43) for $z(\zeta )$ reduces to

(5.2)\begin{equation} z(\zeta)=a\mathrm{i}\,K(\mathrm{i}\zeta,q)+c, \end{equation}

where we have made use of (3.15a). And one can check by using the properties of $K(\zeta,q)$ that $z(\zeta )$ as given by (5.2) maps $C_1$ onto a horizontal line segment of finite length. Furthermore, one can also check that it maps $C_0$ onto a horizontal line that extends to infinity in both directions – this is the free surface in this case. We also mention that in this case, it follows from (5.1a,b) (and (3.15b)) that $\beta _0$ – as given by (A12) – reduces to

(5.3)\begin{equation} \beta_0 ={-}a\mathrm{i}\,L(\mathrm{i}\zeta_1,q). \end{equation}

And since $z'(\zeta _1)=0$, it follows from (5.2) (and (3.10)) that $L(\mathrm {i}\zeta _1,q)=0$, and hence from (5.3) that $\beta _0=0$. Thus in this case, the single-valuedness condition (3.45) is satisfied automatically. For a similar reason, it follows from (3.23) that the circulation $\varGamma$ around the hydrofoil in this case is also $0$. Furthermore, it follows from (4.9) (and the fact that now $\beta _2=0$) that in this case, the fluid exerts no force on the hydrofoil.

5.2. Near-vertical hydrofoils

We have also thus far assumed that $-\overline {\zeta _2}\neq \zeta _2$, or equivalently, that $\zeta _2\neq \pm \mathrm {i}q$ (see just after (3.24)). Let us now consider the limits of our results as $\zeta _2\rightarrow \pm \mathrm {i}q$ whilst $\zeta _1\rightarrow \mp \mathrm {i}q$, respectively. First, in both of these limits, it is evident from (3.38a) that $\alpha _1$ must be real, and hence, as follows from (3.48), that $\alpha$ must tend to either of $\pm {\rm \pi}/2$. On geometrical grounds, one may deduce that $\alpha \rightarrow \pm {\rm \pi}/2$ as $\zeta _2\rightarrow \pm \mathrm {i}q$, respectively. Indeed, it then follows from (A1), (A5), (A9), (A11), and the fact (already stated) that $\beta _2$ is purely imaginary, that in both of these limits, $\beta _1$, $\beta _3$, $\beta _4$ and $\beta _5$ are also all purely imaginary, and

(5.4a,b)\begin{equation} \beta_3=\beta_2,\quad\beta_5={-}\beta_4. \end{equation}

One can then check by using the properties of $P(\zeta,q)$ and $K(\zeta,q)$ that $z(\zeta )$ as given by (3.43) maps $C_1$ onto a vertical line segment of finite length. We also mention that in this case, it follows (using the facts that $K(\pm q,q^2)=-K(\pm q^3,q^2)$ and $L(\pm q,q^2)=L(\pm q^3,q^2)$, which follow from (3.9) and (3.12), respectively) that $\beta _0$ (as given by (A12)) is automatically equal to $0$, so the single-valuedness condition (3.45) is satisfied automatically.