Potential Flow Through Cascades of Thin, Impermeable Aerofoils

Abstract

Potential flow past a periodic array of bodies is commonplace in a large range of fluid mechanical problems. For example, the flow through a rotor cascade in aerodynamics [20], the flow through structured porous materials [8], and the flow around large schools of fish [24]. Within these applications it is not merely the potential flow through the structure that must be calculated but also the complicated interactions between unsteady perturbations to the flow (such as turbulence) and the structures themselves.

Appendices

2.A Modified Plemelj Formulae

In this section we prove the Modified Plemelj formulae which is used in the solution of the Riemann–Hilbert problem in Sect. 2.3. The traditional Plemelj formulae must be adapted to be suitable for unbounded domains, as found in the cascade problem.

Theorem 1

If f(t) satisfies a Hölder condition on L as defined in Sect. 2.2, except possibly at the endpoints where it may have integrable singularities, and has period $\Delta \text {i} n$, then, for $z\notin L$,

$$\begin{aligned} \Phi (z):=\frac{1}{2 \pi \text {i}} \int _{L} \frac{f(\zeta )}{\zeta -z} \; \text {d} \zeta&=\frac{1}{2 \text {i} \Delta } \int _{-1}^1 f(\tau ) \coth \left( \frac{\pi (\tau -z)}{\Delta } \right) \; \text {d} \tau . \end{aligned}$$

(2.50)

Proof

By parametrising L, we may write

$$\begin{aligned} \frac{1}{2 \pi \text {i}} \int _{L} \frac{f(\zeta )}{\zeta -z} \; \text {d}\zeta&=\frac{1}{2 \pi \text {i}} \sum _{n=-\infty }^{\infty } \int _{-1}^1 \frac{f(\tau )}{(\tau +\text {i}n \Delta )-z} \; \text {d}\tau . \end{aligned}$$

We now use the dominated convergence theorem to interchange the orders of summation and integration for $z \notin L$. We write

$$\begin{aligned} h_N(\tau ,z)&:= f(\tau ) \sum _{n=-N}^{N} \frac{1}{(\tau +\text {i}n \Delta )-z} =f(\tau )\left( \frac{1}{\tau -z} +2 \sum _{n=1}^{N} \frac{(\tau - z)}{(\tau -z)^2 + n^2 \Delta ^2} \right) , \end{aligned}$$

and then

$$\begin{aligned} |h_N(\tau ,z)| \le g(\tau ,z) := |f(\tau )| \left( \frac{1}{|\tau -z|} +2 \sum _{n=1}^{\infty } \frac{|(\tau - z)|}{|(\tau -z)^2 + n^2 \Delta ^2|} \right) . \end{aligned}$$

(2.51)

To complete the proof, we must show that g is integrable. As f satisfies the Hölder condition and possibly has integrable singularities at the end points, all that remains is to show that the bracketed term is bounded. Since $z \notin L$, we may write $z-\tau = r \text {e}^{\text {i}\theta }$, where r and $\theta$ are functions of $\tau$. Moreover, $r \text {e}^{\text {i}\theta } \ne \pm \text {i}n \Delta$ and therefore,

$$\begin{aligned} g(\tau , z)&= \frac{|f(\tau )|}{r} \left( 1 +2 \frac{r^2}{\Delta ^2}\sum _{n=1}^{\infty } \frac{1}{\left| \frac{r^2}{\Delta ^2} \text {e}^{2\text {i}\theta }+ n^2\right| } \right) . \end{aligned}$$

By the comparison test with $\frac{1}{n^2}$, this sum converges for all $\tau \in [-1,1]$, so $g(\tau ,z)$ is bounded in the domain of integration and therefore integrable. By the dominated convergence theorem, we are free to interchange the order of limit and integral, so

$$\begin{aligned} \frac{1}{2 \pi \text {i}} \int _{L} \frac{f(\zeta )}{\zeta -z} \; \text {d}\zeta&=\frac{1}{2 \pi \text {i}} \int _{-1}^1 f(\tau )\sum _{n=-\infty }^{\infty } \frac{1}{(\tau +\text {i}n \Delta )-z} \; \text {d}\tau \\&=\frac{1}{2\text {i} \Delta } \int _{-1}^1 f(\tau ) \coth \left( \frac{\pi (\tau -z)}{\Delta } \right) \; \text {d}\tau , \end{aligned}$$

where the last identity is obtained from a classical formula [9, p. 296].

Since, as we have shown above, we may split up the integral into its contributions from each chord, we have

and the analogous result for the Plemelj formulae holds:

(2.52)

2.B Asymptotic Results at Endpoints

In this section, we consider the asymptotic behaviour of Cauchy-type integrals with $\coth$ kernels, which is necessary for the analysis of endpoint behaviour in Sect. 2.3.3. We restrict our attention to the endpoints , since the behaviour at will be identical by the periodicity of the kernel. We define

$$\begin{aligned} \Phi (z)=\frac{1}{2 \text {i}\Delta } \int _{-1}^1 f(\tau )\coth \left( \frac{\pi (\tau -z)}{ \Delta } \right) \; \text {d}\tau = \Phi _1(z) + \frac{1}{2 \pi \text {i}} \int _{-1}^1 \frac{f(\tau )}{\tau - z}\; \text {d}\tau , \end{aligned}$$

(2.53)

where f(t) satisfies a Hölder condition on $(-1,1)$, except possibly at the ends where it satisfies

$$\begin{aligned} f(t)&=\frac{\tilde{f}(t)}{(t-c)^{\beta }}, \end{aligned}$$

where c is an endpoint of L, $\beta$ is a real constant, and $\tilde{f}(t)$ satisfies a Hölder condition near and at c. In our case the relevant parameters are $\beta = 0, 1/2$. We have removed the principal value part in (2.53) so that $\Phi _1$ is bounded and takes a definite value as $z \rightarrow c$ along any path. When $z=t$ on the contour, the remaining integral is considered in the principal value sense. In the following formulae, ± correspond to taking $c=\mp 1$. The branch of $\log \frac{1}{z-c}$ is chosen to pass through the contour. Then the following limits, which can be deduced from the non-periodic analysis in [25, Sect. 29], are valid:

• 1. $\underline{\beta = 0}$

  We have the asymptotic behaviours:

  • (a) as $z\rightarrow c$, with z not on the contour,

    $$\begin{aligned} \Phi (z) \sim \pm \frac{f(c)}{2 \pi \text {i}} \cdot \log \left( \frac{1}{z-c} \right) + \Phi _1(z)+ \Phi _0(z), \qquad {(2.53\mathrm{a})}\end{aligned}$$

  • (b) as $t \rightarrow c$, with t on the contour,

    $$\begin{aligned} \Phi (t) \sim \pm \frac{f(c)}{2 \pi \text {i}} \cdot \log \left( \frac{1}{t-c} \right) +\Phi _1(t)+ \Psi _0(t), \qquad {(2.53\mathrm{b})}\end{aligned}$$

  where $\Phi _1$ is a function that is analytic at $z=c$, $\Psi _0$ satisfies a Hölder condition near and at $t=c$, and $\Phi _0$ is a bounded function tending to a definite limit as $z \rightarrow c$.

• 2. $\underline{\beta \ne 0}$

  We have the asymptotic behaviours:

  • (a) as $z \rightarrow c$, with z not on the contour,

    $$\begin{aligned} \Phi (z) \sim \pm \frac{\text {e}^{\pm \beta \pi \text {i}}}{2 \text {i}\sin (\beta \pi )} \cdot \frac{\tilde{f}(c)}{(z-c)^{\beta }}+\Phi _1(z) + \Phi _0(z), \qquad{(2.53\mathrm{c})}\end{aligned}$$

  • (b) as $t\rightarrow c$, with t on the contour,

    $$\begin{aligned} \Phi (t) \sim \pm \frac{\cot (\beta \pi )}{2 \text {i}} \cdot \frac{\tilde{f}(c)}{(t-c)^{-\beta }} + \Phi _1(t)+ \Psi _0(t), \qquad {(2.53\mathrm{d})}\end{aligned}$$

  where $\Phi _1$ is a function that is analytic at c, $\Psi _0 = o\left( (t - c)^{-\beta } \right)$ as $t\rightarrow c$, and $\Phi _0= o \left( (z - c)^\beta \right)$.