Periodic Schwarz–Christoffel mappings with multiple boundaries per period

Abstract

We present an extension to the theory of Schwarz–Christoffel (S–C) mappings by permitting the target domain to be a single period window of a periodic configuration having multiple polygonal (straight-line) boundaries per period. Taking the arrangements to be periodic in the x-direction in an (x, y)-plane, three cases are considered; these differ in whether the period window extends off to infinity as y →  ± ∞, or extends off to infinity in only one direction (y →  + ∞ or y →  − ∞), or is bounded. The preimage domain is taken to be a multiply connected circular domain. The new S–C mapping formulae are shown to be expressible in terms of the Schottky–Klein prime function associated with the circular preimage domains. As usual for an S–C map, the formulae are explicit but depend on a finite set of accessory parameters. The solution of this parameter problem is discussed in detail, and illustrative examples are presented to highlight the essentially constructive nature of the results.

1. Introduction

The classical Schwarz–Christoffel (S–C) formula gives the functional form of the conformal mapping from a canonical domain such as the unit disc or the upper-half plane to some given simply connected polygonal region. It is an important tool in a wide range of applications, not least because it is one of the few constructive techniques to effectivize the Riemann mapping theorem which guarantees the existence of such conformal mappings. The monograph by Driscoll & Trefethen [1] collects many of the important results in the theory of classical S–C mappings up to the early 2000s.

The classical formula is only relevant to the mapping of simply connected domains. The form of the mapping to doubly connected domains was derived, using various schemes, independently by Akhiezer [2] and Komatu [3]. Concerning S–C mappings of general multiply connected domains, two significant theoretical steps forward appeared in 2004 and 2005. DeLillo et al. [4] constructed a generalized S–C formula to the unbounded region exterior to a finite collection of polygonal holes using reflection arguments that led naturally to an infinite product representation of the required mapping which, in certain parameter regimes, were shown by those authors to be convergent. Adopting a different function theoretical approach, Crowdy [5] derived a formula for the S–C mapping to a bounded multiply connected polygonal domain in terms of the transcendental Schottky–Klein prime function, a function naturally associated with any multiply connected domain [6]. Crowdy later showed [7] how this approach generalizes to the unbounded case while DeLillo [8] elucidated how, for the bounded case, the two different approaches can be related. Other authors have since added to the general theory [9,10], and the topic remains a vibrant area of research.

The present paper is a natural sequel to [5,7]. We consider here the problem of finding an analytical expression for the S–C mapping from a canonical parametric region to a representative period window of a singly periodic geometrical arrangement where each period window has multiple polygonal boundaries. We categorize this class of periodic S–C mappings into three distinguished cases, as shown schematically in figure 1. Such domains are infinitely connected but it is nevertheless possible to uniformize the boundaries of a single period window using a finitely connected parametric preimage region. Indeed, following Crowdy [5,7], here we write the formulae relevant to each case in terms of the (Schottky–Klein) prime function of a conformally equivalent circular domain comprising the unit disc with multiple excised circular holes [6]. Our work is significant because, even in the case of a single polygonal boundary per period, this problem has received only limited attention in the literature and yet the approach advocated here captures the general case (of multiple polygonal boundaries per period) in a unified fashion.

Figure 1.

The preimage and target domains in cases 1, 2 and 3 when M = 2. The preimages of ± ∞, if they exist in the target domain, are at a_∞ ±. The branch cut is denoted by the light blue curve. (a) Case 1, (b) case 2 and (c) case 3. (Online version in colour.)

We categorize the class of singly periodic mappings into three cases depending on the nature of the period window. To motivate this, it is useful to consider two commonly used mappings f(ζ) and g(ζ) from a parametric ζ plane. The first is a mapping from the unit disc |ζ| < 1 given by

$$z=f(\zeta )=\frac{1}{2\pi \mathrm{i}}\log \left[\frac{\zeta -a_{{\mathrm{\infty }}^{+}}}{|a_{{\mathrm{\infty }}^{+}}|(\zeta -1/\overline{a_{{\mathrm{\infty }}^{+}}})}\right],$$ 1.1

where a_∞⁺ is some point inside the disc; the second is a mapping from the concentric annulus ρ < |ζ| < 1 given by

$$z=g(\zeta )=\frac{1}{2\pi \mathrm{i}}\log \zeta .$$ 1.2

It is easy to show that the function f(ζ) transplants the unit disc to a period window of unit width in a complex z = x + iy plane extending to y →  + ∞ (but not y →  − ∞); the unit circle |ζ| = 1 maps to a unit length straight line along the x-axis; a portion of the branch cut joining the logarithmic singularities at a_∞⁺ to $1/\overline{a_{{\mathrm{\infty }}^{+}}}$ lies inside the unit disc and its two ‘sides’ correspond to the two edges of the period window running to y →  + ∞. This is the simplest example of a case 2 mapping to be described later. The function g(ζ), on the other hand, transplants the concentric annulus to a bounded region of unit width with the two circles |ζ| = 1 and |ζ| = ρ being mapped to straight lines parallel to the x axis; a portion of the branch cut joining the logarithmic singularities at 0 and ∞ lies inside the annulus ρ < |ζ| < 1 and its two sides correspond to the two edges of the period window. This is the simplest example of what we call a case 3 periodic S–C mapping. Mappings in case 1 involve a period window that extends both to y →  + ∞ and y →  − ∞ and having a finite collection of polygonal holes.

Previous work has produced the formulae for the S–C mapping from a bounded multiply connected circular region to bounded [5] polygons in the form

$$z=f(\zeta )=A{\int }_1^{\zeta }{S}_B({\zeta }^{\mathrm{\prime }})\prod _{j=0}^M\prod _{k=1}^{n_j}{\left[\omega ({\zeta }^{\prime },\hspace{0.17em}{a}_k^{(j)})\right]}^{{\beta }_k^{(j)}}\hspace{0.17em}\mathrm{d}{\zeta }^{\mathrm{\prime }}+B,$$ 1.3

where $\omega \left(\cdot ,\cdot \right)$ is the Schottky–Klein prime function of a conformally equivalent preimage circular domain and

$$S_B(\zeta )=\frac{{\omega }_{\zeta }\left(\zeta ,\hspace{0.17em}a\right)\omega \left(\zeta ,1/\overline{a}\right)-{\omega }_{\zeta }\left(\zeta ,1/\overline{a}\right)\omega \left(\zeta ,a\right)}{\prod _{j=1}^M\omega \left(\zeta ,{\gamma }_1^{(j)}\right)\omega \left(\zeta ,{\gamma }_2^{(j)}\right)},$$ 1.4

where ω_ζ(ζ, a) is the derivative of the prime function with respect to its first argument. All the parameters appearing in (1.3) will be explained later. On the other hand, the S–C mapping to the unbounded region exterior to a finite set of polygonal holes was derived [7] to be

$$z=f(\zeta )=A{\int }_1^{\zeta }{S}_{\mathrm{\infty }}({\zeta }^{\mathrm{\prime }})\prod _{j=0}^M\prod _{k=1}^{n_j}{\left[\omega ({\zeta }^{\prime },a_k^{(j)})\right]}^{{\beta }_k^{(j)}}\hspace{0.17em}\mathrm{d}{\zeta }^{\mathrm{\prime }}+B,$$ 1.5

where

$$S_{\mathrm{\infty }}(\zeta )=\frac{S_B(\zeta )}{\omega {\left(\zeta ,a_{\mathrm{\infty }}\right)}^2\omega {\left(\zeta ,1/\overline{a_{\mathrm{\infty }}}\right)}^2},$$ 1.6

and where some point a_∞ in the circular domain is the preimage of the point at infinity in the z-plane. Clearly the central difference between (1.3) and (1.5) is the form of the ‘prefactor’ functions S_B(ζ) and S_∞(ζ) premultiplying the product of powers of the prime function necessary to capture the turning of the corners at the vertices of the polygon boundaries. It will be shown here that the mapping formula for each of the cases 1–3 of the required S–C formulae to periodic domains is of the same form—see (6.3)—but that each of the three cases requires a different form of this prefactor function given, respectively, for cases 1, 2 and 3, in (6.4), (6.6) and (6.8), respectively.

Besides the elementary mappings (1.1) and (1.2), a few previous papers giving more complicated examples of periodic S–C mappings have appeared [11–13]. None of these studies, however, allow for a general collection of polygonal holes in each period window. It is in this respect that our results both complement and enhance this prior work. Moreover, in §8 we derive an alternative representation for these previous mappings as special cases of our new formulae.

Applications of the mapping formulae derived here are multifarious, indeed they are potentially relevant to any of the large number of application areas where S–C maps are already known to be useful. Conformal mapping theory for periodic domains certainly finds relevance in aerodynamics in the modelling of cascades of aerofoils [14–16], including stalled aerofoils [17] and cavitating aerofoils [18]. It is useful in solid mechanics where it can be used to study cascades of cracks [19], and in wave-type phenomena where understanding plasmonic gratings is the challenge [20]. Recent research has used a special class of periodic conformal mappings to study acoustic scattering by a rough surface [21]. It is also worth mentioning that, although they were not specifically concerned with mappings of S–C type, Crowdy & Green [22] have employed similar ideas to those developed in this paper to finding a class of analytical solutions for the so-called periodic von Kármán streets of hollow vortices which comprise a periodic array with two vortices (and hence two boundaries) per period. That problem is an example of a free boundary problem and the new formulae here are likely to find use in other free boundary problems solvable by a combination of classical free streamline theory with S–C mapping ideas.

2. Mathematical formulation

Our aim is to construct a conformal map from a multiply connected circular domain D_ζ to a target period window D_z of a singly periodic geometry having straight line boundaries. The situations we envisage are illustrated in figure 1 in the three cases of interest. The preimage domain D_ζ comprises the unit disc in a complex ζ plane with M smaller circular discs excised. The circular boundaries of these discs are denoted by C_j for j = 1, …, M; circle C_j is centred at δ_j and has radius q_j. We will refer to the geometrical data {q_j, δ_j|j = 1, …, M} as the conformal moduli.

The target period windows D_z for each of the three cases 1–3 are different but each has a branch cut inside the domain D_ζ corresponding to the preimage of two periodic ‘edges’ of the period window D_z. These are also shown in figure 1; the position of the branch points depends on the case being considered. It is important to point out that the shape of the edges of the period window are not required to be straight lines, although by appropriate choice of the branch cut this can be arranged if needed. We also note the case 3 only exists for M≥1 since we require two preimage circles to be transplanted to two ‘sides’ that span the period window.

The other boundaries in each period window are assumed to be a union of straight lines. Let us consider case 1. The period window D_z consists of the exterior of a singly periodic array of (M + 1) polygons whose boundaries we denote by {P_j|j = 0, 1, …, M}. The period of the arrangement is taken to be $\mathcal{P}$. Figure 1a shows the M = 2 situation. We assume that polygon P_j has n_j edges with vertices at {z^(j)_k|k = 1, …, n_j;j = 0, 1, …M}, where n_j≥2 are integers. At the vertex z^(j)_k, the interior angle is

$$\pi \left({\beta }_k^{(j)}+1\right).$$ 2.1

A necessary (but not sufficient) condition for P_j to be closed is

$$\sum _{k=1}^{n_j}{\beta }_k^{(j)}=+2,j=0,1,\dots ,M.$$ 2.2

The straight-line edges of P_j may be expressed as

$$\overline{z}={ϵ}_k^{(j)}z+{\kappa }_k^{(j)},$$ 2.3

where ϵ^(j)_k and κ^(j)_k are complex constants that can be readily computed from the given geometry of D_z and where |ϵ^(j)_k| = 1.

Cases 2 and 3 are a little different. For case 2, we have

$$\sum _{k=1}^{n_0}{\beta }_k^{(0)}=0,\sum _{k=1}^{n_j}{\beta }_k^{(j)}=+2,j=1,\dots ,M.$$ 2.4

For case 3, we have

$$\sum _{k=1}^{n_0}{\beta }_k^{(0)}=0,\sum _{k=1}^{n_1}{\beta }_k^{(1)}=0,\sum _{k=1}^{n_j}{\beta }_k^{(j)}=+2,j=2,\dots ,M.$$ 2.5

In cases 2 and 3, where either one or two edges span the length of the period window as seen in figure 1b,c, the total turning angle on traversing such an edge is zero.

3. The Schottky–Klein prime function

The main mathematical object to be used to construct the mapping formula is the same (Schottky–Klein) prime function—or ‘prime function’ for short—already used in previous work [5,7]. Indeed, that work was the first to demonstrate the relevance of this classical special function to the theory of S–C mappings. To avoid unnecessary repetition of the background theory, the reader is referred to [5,7] for more detailed information on this function; a recent review article [23] also gives further details and a dedicated monograph on the prime function is in preparation [6]; a classical treatment is given in Chapter 12 of the monograph by Baker [24] (although the scope there is much broader). The point to emphasize here is that there is a prime function ω(ζ, a) naturally associated with every circular domain D_ζ of the type just described above, and shown on the left in figure 1. We write it, for convenience, as a function of two variables ζ and a although, since it is naturally associated with a given domain D_ζ, it also depends on the conformal moduli {q_j, δ_j|j = 1, …, M} characterizing D_ζ. Its dependence on these geometrical parameters is suppressed for notational convenience.

To evaluate the prime function numerically, several options are available as surveyed in the review article [23] (see also [6]). Importantly, recent work [23] has led to a number of rapid numerical schemes to compute the prime function. This facilitates the ready computation of the prime function, and consequently the new S–C mapping formulae derived here. All the examples in §8 are computed using these new algorithms (which are freely available [23]).

4. Building blocks of the S–C mapping

Crowdy [5,7] made use of the following key results on conformal slit maps to construct the S–C mapping to (non-periodic) multiply connected polygons, and we will employ these same constructs here. Further details of the following conformal slit maps expressed in terms of the prime function can be found in [6,25].

Proposition 4.1 (Bounded circular slit map). —

Let a be a point inside D_ζ. The mapping

$$\eta =g(\zeta ;a)=\frac{\omega \left(\zeta ,a\right)}{|a|\omega \left(\zeta ,1/\overline{a}\right)}$$ 4.1

has constant absolute value on each of the circles {C_j|j = 0, …, M}. Under this mapping each C_j for j = 1, …, M is mapped to a finite slit that is the arc of a circle centred at the origin in the complex η plane while C₀ maps to the entire unit η circle as illustrated in figure 2.

Figure 2.

Schematic, for M = 3, of the building block conformal slit maps in propositions 4.1–4.3 used in the construction of the periodic S–C formula. (a) Bounded circular slit map. Each L_j is an arc of a circle centred at the origin, (b) Cayley-type map and (c) radial slit map. (Online version in colour.)

For j = 0, 1, …, M, let us label the image of the circle C_j as L_j as shown in figure 2. Since the prime function $\omega \left(\zeta ,a\right)$ possesses a simple zero at ζ = a [23], then g(ζ;a) also possesses a simple zero at a. Furthermore, the derivative of g(ζ;a) with respect to ζ possesses 2M zeros, which we label as γ^(j)₁ and γ^(j)₂ for j = 1, …, M, at the preimages of the endpoints of the circular slits L_j. Hence,

$$\frac{\mathrm{d}g}{\mathrm{d}\zeta }({\gamma }_1^{(j)};a)=\frac{\mathrm{d}g}{\mathrm{d}\zeta }({\gamma }_2^{(j)};a)=0,j=1,\dots ,M.$$ 4.2

The values of γ^(j)₁ and γ^(j)₂ for j = 1, …, M will depend on the choice of a, and the conformal moduli characterizing D_ζ.

Proposition 4.2 (Cayley-type map). —

If ζ₁ and ζ₂ are any two distinct points on the same circle C_j for some j = 0, …, M, then the function

$$F_1(\zeta ;{\zeta }_1,{\zeta }_2)=\frac{\omega \left(\zeta ,{\zeta }_1\right)}{\omega \left(\zeta ,{\zeta }_2\right)}$$

has constant argument on all of the circles {C_k|k = 0, …, M}. As a conformal mapping this is a ‘Cayley-type map’ taking the canonical circular domain to a half plane with slits, as depicted in figure 2 with the image of circle C_j passing through the origin and the point at infinity. The proof of these results can be found in [26] (5.2).

Crowdy [6] refers to these as Cayley-type maps since, for the case M = 0, they correspond to the traditional Cayley map (a Möbius map) transplanting the interior of the unit disc to the upper half plane. The properties of the prime function mean that F₁ has a simple zero at ζ₁ and a simple pole at ζ₂.

Proposition 4.3 (Radial slit map). —

If ζ₁ and ζ₂ are any two distinct points inside D_ζ, then the function

$$F_2(\zeta ;{\zeta }_1,{\zeta }_2)=\frac{\omega \left(\zeta ,{\zeta }_1\right)\omega \left(\zeta ,1/\overline{{\zeta }_1}\right)}{\omega \left(\zeta ,{\zeta }_2\right)\omega \left(\zeta ,1/\overline{{\zeta }_2}\right)}$$

has constant argument on each of the circles {C_j|j = 0, …, M}.

As a conformal mapping the function F₂(ζ;ζ₁, ζ₂) is a radial slit map because it transplants all the circular boundaries of D_ζ to finite-length slits having constant argument, as illustrated in figure 2. The proof for this map can be found in [25]. The properties of the prime function mean that, inside D_ζ, F₂ possesses a simple zero at ζ₁ and a simple pole at ζ₂.

5. Properties of the periodic S–C mapping

The aim is to find the functional form of the conformal mapping

$$z=f(\zeta ),$$ 5.1

from the circular domain D_ζ to a single period of the periodic polygonal domain with multiple boundaries per period. Let the n_j prevertices (that is, the preimages of the n_j vertices) on each circle C_j be denoted by

$$\{a_k^{(j)}|k=1,\dots ,n_j\hspace{0.17em}\mathrm{and}\hspace{0.17em}j=0,1,\dots ,M\},$$ 5.2

so that

$$z_k^{(j)}=f\left(a_k^{(j)}\right),k=1,\dots ,n_j\hspace{0.17em}\mathrm{and}\hspace{0.17em}j=0,1,\dots ,M.$$ 5.3

Let the parameters in (2.1) determining the turning angles at these vertices be denoted by

$$\{{\beta }_k^{(j)}|k=1,\dots ,n_j\hspace{0.17em}\mathrm{and}\hspace{0.17em}j=0,1,\dots ,M\}.$$ 5.4

For all three cases 1–3, we will employ the same device of an intermediate mapping of D_z to a bounded circular-slit domain D_η as introduced in [5,7]. We know from the previous section that the form of such a mapping is given in terms of the prime function of D_ζ in (4.1) for some arbitrary choice of a in D_ζ. We also suppose that the circular arc L_j—the image of C_j under the mapping (4.1)—is of radius r_j in the η plane. We set r₀ = 1.

Now consider the composition of analytic functions defined by

$$z=f(\zeta )\equiv F(g(\zeta ;a))=F(\eta ),$$ 5.5

so that F(η) transplants the image D_η of D_ζ under the map (4.1) to the target region D_z. Since the image of C_j is a polygon in the z-plane then, by (2.3), we must have

$$\overline{F(\eta )}={ϵ}_k^{(j)}F(\eta )+{\kappa }_k^{(j)},j=0,1,\dots ,M,$$ 5.6

for some set of constants {ϵ^(j)_k, κ^(j)_k|k = 1, …, n_j; j = 0, 1, …, M} dictated by the geometry of the straight line image. Equivalently, we can write

$$\overline{F}\left(\frac{r_j^2}{\eta }\right)={ϵ}_k^{(j)}F(\eta )+{\kappa }_k^{(j)},j=0,1,\dots ,M,$$ 5.7

where we have used the fact that $\overline{\eta }=r_j^2/\eta$ if η lies on L_j (recall that L_j is a circular arc of finite length centred at η = 0). On differentiation with respect to η, we find

$$-\frac{r_j^2}{{\eta }^2}{\overline{F}}^{\prime }\left(\frac{r_j^2}{\eta }\right)={ϵ}_k^{(j)}{F}^{\prime }(\eta ),j=0,1,\dots ,M,$$ 5.8

where we use primes to denote the derivative of F(η) with respect to η, or

$$\frac{\eta F^{\prime }(\eta )}{\overline{\eta F^{\prime }(\eta )}}=-\frac{1}{{ϵ}_k^{(j)}},j=0,1,\dots ,M.$$ 5.9

This means that the quantity

$$\eta \frac{\mathrm{d}F}{\mathrm{d}\eta },$$ 5.10

must have piecewise constant argument on all boundaries of D_η and, hence, on all boundaries of the original circular domain D_ζ.

By the chain rule we have

$$\eta \frac{\mathrm{d}F}{\mathrm{d}\eta }=g(\zeta ;a)\frac{\mathrm{d}f/\mathrm{d}\zeta }{\mathrm{d}g/\mathrm{d}\zeta },$$ 5.11

from which it is clear that, when viewed as a function of ζ, η dF/dη will inherit all the same singularities at the prevertices as df/dζ (which are generally branch points). It will also have a simple zero at ζ = a from the appearance of η = g(ζ;a) in formula (5.11) and 2M simple poles at {γ^(j)₁, γ^(j)₂|j = 1, …, M} corresponding to the simple zeros (4.2) of dg/dζ at those points.

We now consider the properties of F(η) for each case separately.

• Case 1: The function F(η(ζ)) has the following properties:

  • (1a)In order that the mapping is one-to-one dF/dη must not vanish inside D_ζ.
  • (1b)
    Near ζ = a^(j)_k the function ηdF/dη has the local behaviour

    $$\eta \frac{\mathrm{d}F}{\mathrm{d}\eta }={(\zeta -a_k^{(j)})}^{{\beta }_k^{(j)}}{g}_k^{(j)}(\zeta ),$$ 5.12

    where g^(j)_k(ζ) is analytic at ζ = a^(j)_k and where the parameters {β^(j)_k} satisfy (2.2).
  • (1c)ηdF/dη must have a simple zero at ζ = a and 2M simple poles at γ^(j)₁ and γ^(j)₂ for j = 1, …, M.
  • (1d)ηdF/dη must have piecewise-constant argument on {C_j|j = 0, 1, …, M}.
  • (1e)ηdF/dη must have simple poles at two points ζ = a_∞⁺ and ζ = a_∞⁻ corresponding to the preimages of the period window as y →  ± ∞, respectively.

• Case 2: The function F(η(ζ)) has the following properties:

  • (2a)In order that the mapping is one-to-one dF/dη must not vanish inside D_ζ.
  • (2b)
    Near ζ = a^(j)_k the function ηdF/dη has the local behaviour

    $$\eta \frac{\mathrm{d}F}{\mathrm{d}\eta }={(\zeta -a_k^{(j)})}^{{\beta }_k^{(j)}}{g}_k^{(j)}(\zeta ),$$ 5.13

    where g^(j)_k(ζ) is analytic at ζ = a^(j)_k and where the parameters {β^(j)_k} satisfy (2.4).
  • (2c)ηdF/dη must have a simple zero at ζ = a and simple poles at γ^(j)₁ and γ^(j)₂ for j = 1, …, M.
  • (2d)ηdF/dη must have piecewise-constant argument on {C_j|j = 0, 1, …, M}.
  • (2e)ηdF/dη must have a simple pole at ζ = a_∞⁺ corresponding to the preimage of the period window as y →  + ∞.

• Case 3: The function F(η(ζ)) has the following properties:

  • (3a)In order that the mapping is one-to-one dF/dη must not vanish inside D_ζ.
  • (3b)
    Near ζ = a^(j)_k the function ηdF/dμ has the local behaviour

    $$\eta \frac{\mathrm{d}F}{\mathrm{d}\eta }={(\zeta -a_k^{(j)})}^{{\beta }_k^{(j)}}{g}_k^{(j)}(\zeta ),$$ 5.14

    where g^(j)_k(ζ) is analytic at ζ = a^(j)_k and where the parameters {β^(j)_k} satisfy (2.5).
  • (3c)ηdF/dη must have a simple zero at ζ = a and simple poles at γ^(j)₁ and γ^(j)₂ for j = 1, …, M.
  • (3d)ηdF/dη must have piecewise-constant argument on {C_j|j = 0, 1, …, M}.
  • (3e)The function F(η) is analytic inside D_ζ.

6. Constructing the periodic S–C mapping

To construct the periodic S–C mappings for cases 1–3, following [5,7] we will use the Cayley-type mappings F₁(ζ;ζ₁, ζ₂) and radial slit mappings F₂(ζ;ζ₁, ζ₂) as basic building blocks to construct ηdF/dη. Once this has been accomplished, the required df/dζ follows on use of (5.11).

A natural starting point is to pose that

$$\eta \frac{\mathrm{d}F}{\mathrm{d}\eta }=\left\{\prod _{j=0}^M\prod _{k=1}^{n_j}{\left[F_1(\zeta ,a_k^{(j)},{\gamma }_0^{(j)})\right]}^{{\beta }_k^{(j)}}\right\}C(\zeta )=\left\{\prod _{j=0}^M\prod _{k=1}^{n_j}{\left[\frac{\omega (\zeta ,a_k^{(j)})}{\omega (\zeta ,{\gamma }_0^{(j)})}\right]}^{{\beta }_k^{(j)}}\right\}C(\zeta ),$$ 6.1

where we have picked an arbitrary point γ^(j)₀ on each circle C_j for j = 0, 1, …, M. Since the term in curly brackets is a product of powers of Cayley-type maps, the form (6.1) ensures that ηdF/dη has the required singularities (see condition (1b, 2b and 3b)) at all these prevertices with the term in curly brackets also having piecewise constant argument on all the boundary circles {C_j|j = 0, 1, …, M}. Unfortunately, by virtue of (2.2), the function in curly brackets also has unwanted second-order poles at all the arbitrarily chosen points {γ^(j)₀|j = 0, 1, …, M}. The function C(ζ) is a correction function still to be determined and it must be analytic at all the prevertices {a^(j)_k|k = 1, …, n_j, j = 0, 1, …, M}. The relevant function C(ζ) must add in any missing zeros or singularities of the function, and delete any unwanted zeros or singularities, all the while preserving the piecewise constant argument of the function. This can be done using strategic choices of the Cayley-type maps and radial slit maps, as will now be shown.

Consider the choice

$$C(\zeta )=F_2(\zeta ;a,a_{{\mathrm{\infty }}^{+}})F_2(\zeta ;{\gamma }_0^{(0)},a_{{\mathrm{\infty }}^{-}})\prod _{j=1}^M{F}_1(\zeta ;{\gamma }_0^{(j)},{\gamma }_1^{(j)})F_1(\zeta ;{\gamma }_0^{(j)},{\gamma }_2^{(j)})$$ 6.2

which, by virtue of being a product of Cayley-type maps and radial slit maps, has constant argument on each circle {C_j|j = 0, 1, …, M}. Once C(ζ) is substituted into (6.1) the first radial slit map F₂(ζ;a, a_∞⁺) adds in the required zero at a and the required simple pole at a_∞⁺; see conditions (1c) and (1e). The second radial slit map F₂(ζ;γ⁽⁰⁾₀, a_∞⁻) removes the unwanted second-order pole at γ⁽⁰⁾₀ and adds in the required simple pole at a_∞⁻; see condition (1e). The product of Cayley-type maps in (6.2) has the effect of removing the unwanted second-order poles at {γ^(j)₀|j = 1, …, M} and adding in the required simple poles at {γ^(j)₁, γ^(j)₂|j = 1, …, M}; see condition (1c).

If we now substitute (4.1) and (6.2) into (6.1), simplify, and then integrate the result the required mapping can be written as

$$f(\zeta )=A{\int }_1^{\zeta }{S}_P({\zeta }^{\prime };a,a_{{\mathrm{\infty }}^{+}},a_{{\mathrm{\infty }}^{-}})\prod _{j=0}^M\prod _{k=1}^{n_j}{\left[\omega \left({\zeta }^{\prime },a_k^{(j)}\right)\right]}^{{\beta }_k^{(j)}}\hspace{0.17em}\mathrm{d}{\zeta }^{\prime }+B,$$ 6.3

where A and B are complex constants and

$$S_P(\zeta ;a,a_{{\mathrm{\infty }}^{+}},a_{{\mathrm{\infty }}^{-}})\equiv \frac{S_B(\zeta )}{\omega (\zeta ,a_{{\mathrm{\infty }}^{+}})\omega (\zeta ,a_{{\mathrm{\infty }}^{-}})\omega (\zeta ,1/\overline{a_{{\mathrm{\infty }}^{+}}})\omega (\zeta ,1/\overline{a_{{\mathrm{\infty }}^{-}}})},$$ 6.4

where S_B(ζ) is defined in (1.4) and is the same function introduced in [5,7]. Finally, by adopting a similar approach to [7] (appendix A), it can be shown that a ratio of the required mapping function and the function just constructed above is necessarily constant leading to the sought after representation of the mapping function.

Cases 2 and 3 differ only in the relevant choices of C(ζ). For case 2, it can be argued that the relevant choice of C(ζ) is

$$C(\zeta )=F_2(\zeta ;a,a_{{\mathrm{\infty }}^{+}})\prod _{j=1}^M{F}_1(\zeta ;{\gamma }_0^{(j)},{\gamma }_1^{(j)})F_1(\zeta ;{\gamma }_0^{(j)},{\gamma }_2^{(j)}),$$ 6.5

leading to formula (6.3) but with S_P (ζ;a, a_∞⁺, a_∞⁻) replaced by

$$S_P(\zeta ;a,a_{{\mathrm{\infty }}^{+}})\equiv \frac{S_B(\zeta )}{\omega (\zeta ,a_{{\mathrm{\infty }}^{+}})\omega (\zeta ,1/\overline{a_{{\mathrm{\infty }}^{+}}})}.$$ 6.6

For case 3, C(ζ) is

$$C(\zeta )=F_2(\zeta ;a,{\gamma }_1^{(1)})F_1(\zeta ;{\gamma }_1^{(1)},{\gamma }_2^{(1)})\prod _{j=2}^M{F}_1(\zeta ;{\gamma }_0^{(j)},{\gamma }_1^{(j)})F_1(\zeta ;{\gamma }_0^{(j)},{\gamma }_2^{(j)})$$ 6.7

leading to formula (6.3) but with S_P (ζ;a, a_∞⁺, a_∞⁻) replaced by

$$S_P(\zeta ;a)\equiv S_B(\zeta )\frac{\omega (\zeta ,{\gamma }_1^{(1)})}{\omega (\zeta ,1/\overline{{\gamma }_1^{(1)}})}.$$ 6.8

It can be shown, using the general theory of the prime function [5,6], that

$$\frac{\omega (\zeta ,{\gamma }_1^{(1)})}{\omega (\zeta ,1/\overline{{\gamma }_1^{(1)}})}=c({\gamma }_1^{(1)}){\mathrm{e}}^{2\pi \mathrm{i}{v}_1(\zeta )},$$ 6.9

where c(γ⁽¹⁾₁) is a function of γ⁽¹⁾₁ (and the conformal moduli of D_ζ) but not ζ, while v₁(ζ) depends only on the conformal moduli {q_j, δ_j} [6]. This means that different choices of the arbitrary parameter a, which in turn will change the value of γ⁽¹⁾₁, only affects the prefactor function S_P (ζ;a) by a multiplicative factor which is inconsequential since it can be absorbed into the constant A in formula (6.3).

In this way, we have constructed the general form of the S–C mapping functions to periodic geometries within cases 1,2 and 3 and for any M≥0. The above theoretical approach using the prime function is very natural and readily facilitates generalization of the multiply connected formulae derived by Crowdy [5,7] to this (infinitely connected) periodic case.

7. The parameter problem

It is useful to confirm that the number of free parameters appearing in the formulae matches with the number of additional constraints on the mapping.

First note that the parameter a is arbitrary and, for a given set of conformal moduli {q_j, δ_j|j = 1, …, M}, the parameters {γ^(j)₁, γ^(j)₂|j = 1, …, M} can be directly computed from the chosen value of a; the latter 2M values are slaved to the choice of a and the conformal moduli and should not therefore be considered to be independent free parameters. Furthermore, the final mapping formulae can be shown to be independent of the choice of a (an appendix of [27] gives details). Thus, we discount both a and the set {γ^(j)₁, γ^(j)₂|j = 1, …, M} when tallying the number of degrees of freedom in the mapping.

For case 1, the complex-valued parameters A, B, a_∞⁺ and a_∞⁻ represent a total of eight real degrees of freedom in the mapping formula; the conformal moduli {q_j, δ_j|j = 1, …, M} provide a further 3M real degrees of freedom; on any circle C_j for j = 0, 1, …, M there are n_j real degrees of freedom associated with the angular location of the n_j prevertices on that circle C_j giving a total of $N\equiv \sum _{j=0}^M{n}_j$ real degrees of freedom. At the same time, we must remove three real degrees of freedom associated with an automorphism of the unit disc (the usual three real degrees of freedom associated with the Riemann mapping theorem). This produces a total count of 8 + 3M + N − 3 = 3M + N + 5 real parameters.

On the other hand, the constraints on the mapping for case 1 are as follows. For each of the M + 1 polygons per period window, we must specify the location of one of the vertices and the direction from that vertex of one of the straight line edges, making a total of 3(M + 1) real conditions. Once the direction of an edge is given, the mapping formula automatically ensures the mapping ‘turns’ through the correct angle at each vertex but it does not guarantee that the length of each side is as required; the n_j real parameters associated with the angular locations of prevertices a^(j)_k on C_j must be chosen to enforce that the side lengths are correct. This provides N real constraints. Finally, we must specify two real degrees of freedom associated with the period $\mathcal{P}$ of the arrangement giving two additional conditions. The period is uniquely defined if the sum of the residues of df/dζ at a_∞⁺ and a_∞⁻ vanishes. However, this condition is already guaranteed by prior constraints on the side lengths:

$$\mathrm{Res}\left(\frac{\mathrm{d}f}{\mathrm{d}\zeta };a_{{\mathrm{\infty }}^{+}}\right)+\mathrm{Res}\left(\frac{\mathrm{d}f}{\mathrm{d}\zeta };a_{{\mathrm{\infty }}^{-}}\right)=\frac{1}{2\pi \mathrm{i}}{\oint }_{\mathrm{\partial }{D}_z}\frac{\mathrm{d}f}{\mathrm{d}\zeta }\mathrm{d}\zeta =\frac{1}{2\pi \mathrm{i}}{\oint }_{\mathrm{\partial }{D}_z}\hspace{0.17em}\mathrm{d}z=0,$$ 7.1

where the first equality is due to Cauchy's theorem and the final equality is due to the fact that every polygon is closed. In total, we therefore have 3(M + 1) + N + 2 = 3M + N + 5 real constraints which is consistent with the number of free parameters.

For case 2, all of the arguments above carry over except that the formula loses two real degrees of freedom since a_∞⁻ has dropped out of it. However, we are no longer free to independently specify the two real degrees of freedom in the period $\mathcal{P}$ since it is now forced by the geometry of the image of circle C₀.

For case 3, the mapping formula loses two more real degrees of freedom since a_∞⁺ has also now dropped out leaving a total of 3M + N + 1 real parameters. However in this case the image of C₀ and C₁ together with the images of two sides of a branch cut joining them are mapped to a single polygon having n₀ + n₁ + 2 sides but two of those sides must have equal length (by the periodicity of the arrangement) meaning that this polygon has n₀ + n₁ + 1 sides of specifiable length. The total number of real constraints therefore reduces to 3M + N + 1, which is again consistent with the parameter count.

8. Illustrative examples

We now present several examples of the new periodic S–C mapping for different values of M and in the various cases 1, 2 and 3. We focus on examples for which the parameter problem is relatively simple.

It was shown by Crowdy [5] that when M = 0 we can take S_B(ζ) = 1, and when M = 1 (and the preimage domain is taken to be a concentric annulus) it turns out that we can take S_B(ζ) = 1/ζ² without loss of generality.

(a). Examples with M = 0

First note that there is no case 3 when M = 0 because one needs M≥1. For case 1 with M = 0, where the prime function is simply ω(ζ, a) = ζ − a and S_B(ζ) = 1, then S_P (ζ) for case 1 can be written in the form

$$S_P(\zeta ;a,a_{{\mathrm{\infty }}^{+}},a_{{\mathrm{\infty }}^{-}})=\frac{1}{(\zeta -a_{{\mathrm{\infty }}^{+}})(\zeta -a_{{\mathrm{\infty }}^{-}})(\zeta -1/\overline{a_{{\mathrm{\infty }}^{+}}})(\zeta -1/\overline{a_{{\mathrm{\infty }}^{-}}})}.$$ 8.1

Consequently, we obtain the S–C map from the unit disc to a periodic domain with a single polygon per period as

$$z(\zeta )=A{\int }^{\zeta }\frac{\prod _{k=1}^{n_0}{\left[{\zeta }^{\mathrm{\prime }}-a_k^{(0)}\right]}^{{\beta }_k^{(0)}}}{\left({\zeta }^{\mathrm{\prime }}-a_{{\mathrm{\infty }}^{+}}\right)\left({\zeta }^{\mathrm{\prime }}\overline{a_{{\mathrm{\infty }}^{+}}}-1\right)\left({\zeta }^{\mathrm{\prime }}-a_{{\mathrm{\infty }}^{-}}\right)\left({\zeta }^{\mathrm{\prime }}\overline{a_{{\mathrm{\infty }}^{-}}}-1\right)}\hspace{0.17em}\mathrm{d}{\zeta }^{\mathrm{\prime }}+B.$$ 8.2

As an example consider a periodic array of slits (figure 3). We take the prevertices a⁽⁰⁾_1,2 = 1,   − 1, the turning angles β⁽⁰⁾_1,2 = 1, and the preimages of infinity a_∞ ± =  ± a_∞. We then obtain the mapping as

$$z=f(\zeta )=\frac{\mathcal{P}}{2\pi \mathrm{i}}\left(\log \left(\frac{\zeta -a_{\mathrm{\infty }}}{\zeta +a_{\mathrm{\infty }}}\right)+{\mathrm{e}}^{\mathrm{i}\sigma }\log \left(\frac{\zeta -1/\overline{a_{\mathrm{\infty }}}}{\zeta +1/\overline{a_{\mathrm{\infty }}}}\right)\right),\sigma =\arg \left(\frac{a_{\mathrm{\infty }}-\overline{a_{\mathrm{\infty }}}{\left|a_{\mathrm{\infty }}\right|}^2}{\overline{a_{\mathrm{\infty }}}-a_{\mathrm{\infty }}{\left|a_{\mathrm{\infty }}\right|}^2}\right).$$ 8.3

This mapping arises in turbomachinery studies [14–18], although it has not previously been identified as a periodic S–C mapping in the sense considered here. As one might expect, as we send $a_{{\mathrm{\infty }}^{\pm }}\to a_{\mathrm{\infty }}=0$ so that the branch points coalesce, we recover the familiar Joukowski map

$$z(\zeta )=C\left(\frac{1}{\zeta }+\zeta \right),$$

where C is a constant. A second example of a periodic array of squares is illustrated in figure 4.

Figure 3.

The periodic S–C mapping for a periodic array of slits with M = 0 in case 1. (a) The canonical circular domain in ζ-space. (b) The polygonal target domain in z-space. The light blue dashed line indicates the branch cut and its image. (Online version in colour.)

Figure 4.

The periodic S–C mapping for a periodic array of squares with M = 0 in case 1. (a) Preimage circular domain in ζ-space, (b) The polygonal target domain in z-space. The light blue dashed line indicates the branch cut and its image. (Online version in colour.)

For case 2 with M = 0, we can take

$$S_P(\zeta ;a,a_{{\mathrm{\infty }}^{+}})=\frac{1}{(\zeta -a_{{\mathrm{\infty }}^{+}})(\zeta -1/\overline{a_{{\mathrm{\infty }}^{+}}})}.$$ 8.4

If, in addition, we choose n₀ = 0 so that there are no vertices on the image of C₀ it is easy to show that on substitution of (8.4) for S_P (ζ;a, a_∞⁺, a_∞⁻) into (6.3) then we arrive, after integration and on taking a suitable choice of A and B, at the mapping (1.1) given in the Introduction. In the general case where n₀ > 0, the case 2 mapping can be shown to be equivalent to that of Floryan [13], where the upper half plane is mapped to an upper half plane with periodic polgyonal boundaries.

(b). Examples with M = 1

With M = 1 we can take, without loss of generality, the preimage circular domain to be a concentric annulus ρ < |ζ| < 1. In this case, it is known [5,7] that the prime function may be expressed as

$$\omega \left(\zeta ,\alpha \right)=-\frac{\alpha }{D^2}P\left(\frac{\zeta }{\alpha },\rho \right),$$ 8.5

where P(ζ, ρ) can be defined by the convergent infinite product

$$P(\zeta ,\rho )\equiv (1-\zeta )\prod _{k=1}^{\mathrm{\infty }}\left(1-{\rho }^{2k}\zeta \right)\left(1-{\rho }^{2k}{\zeta }^{-1}\right),\mathrm{and}D\equiv \prod _{k=1}^{\mathrm{\infty }}\left(1-{\rho }^{2k}\right).$$ 8.6

This function can be shown from this definition to satisfy the functional relation [5,7]

$$P({\rho }^2\zeta ,\rho )=-{\zeta }^{-1}P(\zeta ,\rho ).$$ 8.7

(There are, incidentally, infinite sum representations of the same function that are very rapidly convergent and are preferable for numerical evaluation [6].) For case 1 we find

$$S_P(\zeta ;a,a_{{\mathrm{\infty }}^{+}},a_{{\mathrm{\infty }}^{-}})=\frac{C}{{\zeta }^{2}P(\zeta /a_{{\mathrm{\infty }}^{+}},\rho )P(\zeta /a_{{\mathrm{\infty }}^{-}},\rho )P(\zeta \overline{a_{{\mathrm{\infty }}^{+}}},\rho )P(\zeta \overline{a_{{\mathrm{\infty }}^{-}}},\rho )},$$ 8.8

where C is a constant that can be absorbed into the constant A in the S–C formula (6.3). Equation (6.3) can be used to construct the periodic array of slits with two slits per period shown in figure 5 and the periodic array of rectangles with two rectangles per period shown in figure 6.

Figure 5.

Example with M = 1 (two slits per period), case 1. (a) Preimage circular domain in ζ-space. (b) The polygonal target domain in z-space. The light blue dashed line indicates the branch cut and its image. (Online version in colour.)

Figure 6.

Example with M = 1 (two rectangles per period), case 1. (a) Preimage circular domain in ζ-space. (b) The polygonal target domain in z-space. The light blue dashed line indicates the branch cut and its image. (Online version in colour.)

The new mapping formulae may be combined with other tools to conduct novel fluid dynamical investigations. For example, the mapping can be used to calculate the trajectories of point vortices through periodic domains by making use of known analytical formulae (in terms of the same prime functions used here) for the so-called Kirchhoff–Routh path function [28], as illustrated in figure 7. Such a geometry is an example of case 2, and could be used to model the aeroacoustic interaction of a point vortex with bioinspired noise reduction mechanisms [29] using the compact Green's function for multiple bodies [30]. In a similar vein, we may use a calculus for two-dimensional vortex dynamics [31] to calculate the uniform potential flow through a ‘finned channel’ as illustrated in figure 8. This geometry is an example of case 3. Both the conformal mapping and the associated complex potential describing this flow can be written explicitly in terms of the prime function providing a compact and convenient representation of the solution.

Figure 7.

Example with M = 1, case 2. (a) Preimage circular domain in ζ-space; (b) point vortex trajectories, calculated by using the periodic S–C map in the Kirchhoff–Routh theory of [28], in a half-plane with a periodic array of fins. The light blue dashed line indicates the branch cut and its image. (Online version in colour.)

Figure 8.

Example with M = 1, case 3 showing the streamlines for uniform potential flow through a finned channel. The light blue dashed line indicates the branch cut and its image. (Online version in colour.)

Finally, we note that, for case 3, S_B(ζ) = 1/ζ² and (6.8) becomes

$$S_P(\zeta ;a)=\frac{C}{\zeta },$$ 8.9

where C is a constant that can be absorbed into the constant A in the S–C formula (6.3). If we choose n₀ = n₁ = 0 so that there are no vertices on the images of either C₀ or C₁ then, on integration and taking an appropriate value of A, we arrive at the mapping (1.2) given in the Introduction.

Floryan [11] derived the functional form of the mapping

$$w=f(z),$$ 8.10

from an infinite strip in a complex z-plane

$$0<\mathrm{Im}[z]<1,$$ 8.11

to another infinite strip in a complex w plane whose top and bottom walls are no longer parallel but are made up of a periodic array of straight line segments. He found

$$\frac{\mathrm{d}f}{\mathrm{d}z}=C\prod _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\prod _{k=1}^N{\left[\sinh \frac{\pi }{2}(z-z_k-mT)\right]}^{{\beta }_k},$$ 8.12

where C is some constant, N is the total number of prevertices, T is the period of the prevertices, and β_k is the turning angle at prevertex z_k. We can retrieve this result by combining two instances of the new case 3 formula with M = 1 and eliminating the intermediate variable ζ. The ζ, z and w complex planes are illustrated in figure 9. The mapping

$$z=F(\zeta )=-\frac{\mathrm{i}T}{2\pi }\log \zeta ,\rho ={\mathrm{e}}^{-2\pi /T},$$ 8.13

is the simplest case 3 mapping (no vertices, M = 1), as just discussed above and which is just a rescaled form of the mapping (1.2). It transplants the annulus ρ < |ζ| < 1 to the rectangle of unit height and width T as shown in figure 9. The most general form of the case 3 mapping with M = 1 derived here can be written as

$$w=G(\zeta )=A{\int }_1^{\zeta }\prod _{j=0}^1\prod _{k=1}^{n_j}{\left[P\left(\frac{{\zeta }^{\mathrm{\prime }}}{a_k^{(j)}},\rho \right)\right]}^{{\beta }_k^{(j)}}\frac{\mathrm{d}{\zeta }^{\mathrm{\prime }}}{{\zeta }^{\mathrm{\prime }}}+B,$$ 8.14

where P( · , · ) was introduced in (8.6). Dropping the j index, setting N = n₀ + n₁, and allowing the prevertices to be on either C₀ or C₁, produces the modified formula

$$w=G(\zeta )=A{\int }_1^{\zeta }\prod _{k=1}^N{\left[P\left(\frac{{\zeta }^{\mathrm{\prime }}}{a_k},\rho \right)\right]}^{{\beta }_k}\frac{\mathrm{d}{\zeta }^{\mathrm{\prime }}}{{\zeta }^{\mathrm{\prime }}}+B,$$ 8.15

where, by (2.5), we have

$$\sum _{k=1}^N{\beta }_k=\sum _{k=1}^{n_0}{\beta }_k^{(0)}+\sum _{k=1}^{n_1}{\beta }_k^{(1)}=0.$$ 8.16

By the chain rule, it follows that we can write:

$$\frac{\mathrm{d}f}{\mathrm{d}z}=\frac{\mathrm{d}G/\mathrm{d}\zeta }{\mathrm{d}F/\mathrm{d}\zeta }=C\prod _{k=1}^N{\left[P\left(\frac{\zeta }{a_k},\rho \right)\right]}^{{\beta }_k},$$ 8.17

where C is some constant. Formula (8.17) can be shown to be equivalent to (8.12) by combining the following observations. On use of (8.13), it follows that

$${\left(\frac{\zeta }{a_k}\right)}^{1/2}{\rho }^m={\mathrm{e}}^{\pi \mathrm{i}(z-z_k+2m\mathrm{i})/T},z_k=-\frac{\mathrm{i}T}{2\pi }\log a_k.$$ 8.18

From these formulae, and the infinite product expression (8.6), it is readily demonstrated that

$$P\left(\frac{\zeta }{a_k},\rho \right)=-{\mathrm{e}}^{\pi \mathrm{i}(z-z_k)/T}2\mathrm{i}\sin \left(\frac{\pi (z-z_k)}{T}\right)\prod _{m\ne 0}\mathrm{sgn}(m){\rho }^{m\times \mathrm{sgn}(m)}2\mathrm{i}\sin \left(\frac{\pi (z-z_k+2m\mathrm{i})}{T}\right),$$ 8.19

where sgn(m) denotes the sign of m. The final step involves substituting (8.19) into (8.17) and taking a ratio of that result and Floryan's formula (8.12). Assuming the latter is a convergent representation it is easy to check, on use of condition (8.16), that this ratio is doubly periodic with periods 2i and T and all its singularities are removable; note that both (8.12) and (8.17) can be shown to have singularities of the same type at points z = z_k + 2mi + lT for integers m and l. Use of Liouville's theorem for doubly periodic analytic functions with no singularities in a representative period window establishes the ratio to be a constant, meaning that we have derived an alternative representation of Floryan's mapping formula [11]. It is clearly a special case of our more general formula to a much wider class of geometries.

Figure 9.

Floryan [11] found the functional form of the mapping from a single period window of an infinite strip in a z-plane to a single period window of a more complicated polygonal channel in a w plane. The arrows indicate how Floryan's map (8.12) can be constructed from the two case 3 mappings (8.13) and (8.15) from the intermediate ζ-plane. (Online version in colour.)

(c). Examples with M≥2

We show examples with M = 2 and M = 3 in figures 10 and 11, respectively. In order to avoid the parameter problem, the prevertices are chosen in a similar way to [7, section 11].

Figure 10.

Example with M = 2, case 1. (a) Preimage circular domain in ζ-space. (b) The polygonal target domain in z-space. The light blue dashed line indicates the branch cut and its image. (Online version in colour.)

Figure 11.

Example with M = 3, case 1. (a) Preimage circular domain in ζ-space. (b) The polygonal target domain in z-space. The light blue dashed line indicates the branch cut. (Online version in colour.)

9. Discussion

We have presented a significant extension to the theory of S–C mappings by allowing the target domain to be singly periodic with multiple straight-line boundaries in each period window. Written in terms of the prime function associated with a preimage circular domain, the S–C mapping formulae are applicable to three different cases 1, 2 and 3 shown in figure 1. It has also been demonstrated explicitly how our results generalize prior results of Floryan [11]. We have shown how to apply the new S–C mapping formulae to a range of simple geometries as might arise in applications.

As is true for S–C mappings of non-periodic multiply connected domains, much work remains to be done to find effective numerical schemes to tackle the parameter problem for this periodic case with multiple boundaries. It would be particularly useful in applications to dedicate effort towards establishing a general-purpose program that can swiftly calculate the accessory parameters as has been done for the classical (simply connected) S–C formula [1,32] and in other contexts [33,34].

The mapping functions presented here are relevant to the many situations where a conformal mapping description of a domain geometry is useful; indeed, they are potentially useful in any of the many application areas in which the theory of S–C mappings is applicable. They can be used, for example, in fluid dynamics to study a wide class of two-dimensional ideal irrotational flows via a new calculus involving the prime function recently expounded by one of the authors [31]. Of particular interest could be the configurations of stationary vortices [35–37], which have been shown to enhance the lift on an aerofoil. An intriguing application is to periodic arrays of slits, aerofoils or blades relevant in aviation and turbomachinery applications. As mentioned in §8a, a periodic slit map involving just a single blade per period has been used to analyse turbomachinery aerodynamics and aeroacoustics. The extension to multiple blades per period permits analysis of multiple rotor/stator rows, for example, the problem of the interaction between a moving rotor row with a stator row. This could be achieved via the calculus of vortex dynamics [31] and deploying an integral solution of the so-called modified Schwarz problem [38] to satisfy the no-penetration condition on the rotor blades. The mappings here provide new tools for modelling the effects of realistic blade geometry in turbomachinery and will facilitate studies of arbitrary two-dimensional blade shapes including analyses of steady potential flows, unsteady vortex flows and three-dimensional mesh generation. Recent research on the potential flow past a cascade of thin aerofoils [39]—an investigation that partly motivated the present study—could provide a useful benchmark for future conformal mapping studies.

Data accessibility

This article presents a mathematical prescription deriving general formulae to construct user-specified conformal mappings and contains no external data.

Authors' contributions

Both authors contributed equally to the contents of this paper.

Competing interests

We declare we have no competing interests.

Funding

Both authors acknowledge financial support from the CBMS-NSF Regional Conference workshop ‘Solving problems in multiply connected domains’ that took place at the University of California, Irvine, USA between 18 and 22 June 2018 and where this research was initiated. P.J.B. further acknowledges support from Engineering and Physical Sciences Research Council (EPSRC) award 1625902.