数理解析研究所講究録
第 1874 巻 2014 年 135-138

135

A certain double series in aerodynamic interference

calculations(空力干渉計算における 2 重級数について)
Takumi Noda*\dagger(野田 工日本大学工学部)

Department of Mathematics, College of Engineering, Nihon University,
K\^oriyama, Fukushima 963-8642, Japan

1Abstract

In 1949, F. Olver [4] established a transformation formula which converts a certain slowly
convergent series into a rapidly convergent and easily computable form. The original (double) series occurs in aerodynamic interference calculations, and its numerical estimates havesome practical importance. In this report, we revisit this double series and show their transformation properties by using Mellin-Bames type integrals (cf. [3]).

2Classical results

We consider the correction interference of wind tunnel testing for the aerofoil in the lowspeed case, and its mathematical model. It is known that the constraint ofthe tunnel walls on
the flow due to the vorticity is equivalent to a distribution of up wash in the neighbourhood
ofthe wing and be evaluated from a system of images ofthe vorticity (cf. figure below).
Here we set the tunnel section is rectangular. Let be the span of the aerofoil and$\mu=h/b$ be the ratio oftunnel height to tunnel breadth
$s$

$h$

$b.$

$–1111$$–1^{+\fbox_{+}’+}-------.--------\tau_{I}^{1}--III1/-1-1-11$$–41—–1^{-}L-\lrcorner-----a_{1}--II11III1$$–J—–L_{----------4--}^{I}$*takumi@ge. ce. nihon-u. ac. jp
The author was supported by Grants-in-Aid for Scientific Research (No. 23540032), Japan Society for thePromotion of Science (JSPS) and the Ministry of Education, Culture, Sports, Science and Technology of Japan.
$\uparrow$

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According to H. Glauert [2], we introduce the double series for the effect of the wall tocause an upward inclination ofthe stream as follows:鞠$\sum_{m=-\infty}^{\infty}(-1)^{n}\frac{m^{2}-\mu^{2}n^{2}}{(m^{2}+\mu^{2}n^{2})^{2}}$$= \frac{s}{\pi b}.$$\sum_{n=\infty}^{\infty}$$(m,n)\neq(0,0)$$= \frac{s}{\pi b}\cdot\{\frac{\pi^{2}}{3}+8\pi^{2}\sum_{n=1}^{\infty}\frac{n}{1+\exp(2\mu\pi n)}\}.$The second equality was deduced by Glauert using the partial fraction decomposition of$\cot z.$

Remark. Above result can be interpreted in tems of arithmetic functions. Let, we define a certain kind of
be the complex upper halfplane. For
the holomorphic Eisenstein series by$\mathscr{H}=$$z\in \mathscr{H}$$\{z\in \mathbb{C}|{\rm Im} z>0\}$$\mathscr{T}_{k}(z)=(m,n)\neq(0,0)\sum_{n=\infty}^{\infty}\{\sum_{m=-\infty}^{\infty}\frac{(-1)^{n}}{(m+nz)^{k}}\},$

for$k\geqq 2$. Then the Fourier series expansion$\mathscr{T}_{k}(z)=2\zeta(k)+\frac{2(2\pi i)^{k}}{(k-1)!}\sum_{l=1}^{\infty}\overline{\sigma_{k-1}}(l)\exp(2\pi ilz)$$=2 \zeta(k)-\frac{2(2\pi i)^{k}}{(k-1)!}\sum_{n=1}^{\infty}\frac{n^{k-1}}{1+\exp(-2\pi inz)}$

holds.
Here we denote$\overline{\sigma_{k-1}}(l)=\Sigma_{n|l}(-1)^{l/n}n^{k-1}$

鞠. As a consequence, we obtain$= \frac{s}{\pi b}\cdot \mathscr{T}_{2}(\mu\sqrt{-1})$.

3 Olver’s result
Letand be a positive parameter. In the study ofthe upwash angle at the tailis approximately evaluated by L. W.
due to tunnel interference, the double series
Bryant and H. C. Gamer (cf. [1]), which is defined as follows:$(\alpha,\beta)\in \mathbb{R}^{2}$$\mu$$F(\alpha,\beta)$$F( \alpha,\beta)=\sum_{n=1}^{\infty}\sum_{m=-\infty}^{\infty}(-1)^{n}[f_{n}(\alpha-m)-f_{n}(\beta-m)]$with$f_{n}(y)= \frac{y\psi+2\mu^{2}n^{2})}{n^{2}(y^{2}+\mu^{2}n^{2})^{3/2}}.$

137

The elliptical shape in the figure above shows the tail, -axis extends forward in theflow direction, -axis is along the span of the aerofoil and -axis indicates the downwash
direction.
$x$

$y$

$z$

In [4], Olver proved the following theorem after the collaboration of Gamer who supplied him the aerodynamic information.
Theorem (F. Olver 1949) The double series
attached to aerodynamic calculation
is able to transformed as follows:$F(\alpha,\beta)$$F(\alpha,\beta)=\chi(\beta)-\chi(\alpha)$

,

where
$\chi(u)=\frac{1}{6}\pi^{2}u+\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\sin 2\pi mu\{8\pi\mu^{2}m(-1)^{n-1}$$\cross K_{0}(2\pi\mu mn)+4\mu\frac{(-1)^{n-1}}{n}K_{1}(2\pi\mu mn)\},$

and$K_{v}(z)$denotes the modified Bessel function.

4 Main Theorem
Our main purpose is to interpret Olver’s result in tenns of some anithmetic functions. For, real parameterand complex parameter, we define$z\in \mathscr{H}$$s\in \mathbb{C}\backslash vith{\rm Res}>1$$u\in \mathbb{R}\backslash \mathbb{Z}$$f_{m,n}(u;s;z)=$$\frac{(-1)^{n}}{n^{2}}\cdot\frac{{\rm Re}(m-u+nz)}{|(m-u)+nz|}\{1+\frac{n^{2}|z|^{2}}{|(m-u)+nz|^{s}}\},$and define a certain kind of the non-holomorphic Eisenstein series (or the Epstein zetafunction) by$\mathscr{T}(u;s;z)=(m,n)\neq(0,0)\sum_{n=-\infty}^{\infty}\{\lim_{Marrow\infty}\sum_{m=-M}^{M}f_{m,n}(u;s;z)\}.$

Following the method of the Mellin-Bames integral transformation in the study of theasymptotic expansion of the Epstein zeta function developed by M. Katsurada [3], we areable to obtain the transfomation formula foras below.
$\mathscr{T}(u;s;z)$138First, let $U(s_{1},s_{2};z)$ be the confluent hypergeometric fimction ofthe second kind definedby$U(s_{1};s_{2};z)= \frac{1}{\Gamma(s_{1})}\int_{0^{e^{-zu_{l}}}}^{\infty}f^{1^{-1}}(1+u)^{S2^{-S\downarrow-1}}du,$for ${\rm Re}(s_{1})>0$ and $|\arg(z)|<\pi/2$ . Then we have.

Theorem 1 ( 2012) Let ${\rm Res}>1$ for. Then the transformation formula$N$$s\in \mathbb{C}$

andput $\alpha=u-nx$. Assume$\alpha\not\in \mathbb{Z}$for any$n\in \mathbb{Z}$$\mathscr{T}(u;s;z)=2\sum_{n=-\infty}^{\infty}\frac{(-1)^{n-1}\alpha}{n^{2}}$$+ \sum_{n=-\infty}^{\infty}\sum_{m=1}^{\infty}\frac{(-1)^{n-1}(4\pi y)^{2}}{\sqrt{\pi}}$$\cross m\cdot\sin(2\pi m\alpha)e^{-2Z}U(\frac{3}{2};3;4Z)$$+ \sum_{n=-\infty}^{\infty}\sum_{m=1}^{\infty}\frac{(-1)^{n-1}(2\pi)^{s+1}|z|^{2}}{(s-1)\pi\Gamma(\frac{s}{2}-\frac{1}{2})}$$\cross m^{s-3}\cdot\sin(2\pi m\alpha)e^{-2Z}U(\frac{s}{2}-\frac{1}{2};s-1;4Z)$

holds. Here we have used the notation $Z=\pi ym|n|.$Remark. From Theorem 1, Olver’s transfomation formula is proved as a corollary.
Furthermore, it is possible to obtain asymptotic expansion ofas$\mathscr{T}(u;s;z)$${\rm Im} zarrow+\infty.$

5 Acknowledgment
The author would like to express sincere gratitude to Professor Koji Chinen who organized
the conference‘Analytic Number Theory” successffilly and giving us a chance to talk inthis conference.

References[1] L. W. Bryant and H. C. Gamer Control Testing in Wind Tunnels Aeronautical Research Coun-cil Reports and Memoranda, 2881, Her Majesty’s Stationery Office, (1951)[2] H. Glauert The Elements ofAerofoil andAirscrew Theory Cambridge University Press, (1926)[3] M. Katsurada, Complete asymptotic expansions associated with Epstein zeta-functions, Ramanujan . (2007), 14, 249-275$J$[4] Frank W. J. Olver, Transformation of certain series occurring in aerodynamic interference
calculations, Quart. J. Mech. Appl. Math. 2 (1949), 452-457.