圧縮性LESを用いたエアリード楽器の発音機構の数値解析 (数値解析と数値計算アルゴリズムの最近の展開)

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1 LES Numerical study on sounding mechanism of air-reed instruments (Kin ya Takahashi) * (Masataka Miyamoto) * (Yasunori Ito) * (Toshiya Takami), (Taizo Kobayashi), (Akira Nishida), (Mutsumi Aoyagi ) ** * Physics Laboratories, Kyushu Institute of Technology ** Research Institute for Information Technology, Kyushu University 1 ( )) [1,2,3]. ( 1)[2,3,4]. $\sim$ 140 $160dB$ [1,2,3] [1,3,5,6]. [7], 1:

2 27 [8, 9, 10, 11], LES(Large-Eddy Simulation) 2 LES [7]. OpenCFD OpenFOAM (oneeqeddy) [12]. LES 2: ( mm, ) OpenFOAM (LES, (RANS)) LES ( (DNS), ) OpenFOAM 2 [3,13], [3], [3,14]. [15]. LES 3 lom 10 Ocm (a)locm, $(b)2$.ocm 80 $\sim$ 90%

$28 (a) 0.0 0.5 1.0 Coordinate (m) 1.5 0.0 (b) 0.5 $.0$ Coordinate (m) $($ 3: $:\Delta t=1.$

3 28 (a) Coordinate (m) (b) 0.5 $.0$ Coordinate (m) $($ 3: $:\Delta t=1.0\cross 10^{-7}s)$. (a) locm. (b) 2Ocm. 3 $m/s$ $340m/s$ 1 Hz $34mm$ 2 $z$ $9mm$ 913.lHz 2 $5^{0}$ 4: ( ) LES SGS( ) [7,16].

4 29 1: ) ( Ols $\triangle t=1.0\cross 10^{-7_{S}}$ OpenCFD OpenFOAM LES (oneeqeddy) [12]. 4 5, 6 1 ( ) 2 A $10mm$ 4 a $)$ $V=12m/s$ 5(a), (b) A 5(a) Pa 5(b) $818Hz$ 913.lHz (a) Time $(s)$ 5: A (a) (b) 6(a) 6(a), (b) 6(b)

$[4]. $V$ $\nu=0.466j(100v-40)(1/(100l)-0.$

5 30 (a) (b) 6: (a) (b) 2/3 b $)$ $V=12m/s$ 7 3 Brown [4]. $V$ $\nu=0.466j(100v-40)(1/(100l)-0.07)$, (1) $l$ $j$

6 31 7: $i=1.0,2.3,3.8,5.4$ $l=5mm,$ $j=1$ 7 $(2\leq V\leq 40m/s)$ $2\leq V\leq 8m/s$ Brown ((1) ) $V\geq 10m/s$ (913.lHz) $V\geq 14m/s$ $V\geq 18m/s$ $18\leq V\leq 28m/s$ $V\geq 30m/s$ $3Hz$ $200Hz$ $V\geq 24m/s$ 3 7 [1,2,9,10]. LES C $)$ OpenFOAM oneeqeddy LES LES RANS LES oneeqeddy $F$ 8 4 LES (oneeqeddy, Smagorinsky, dynoneeqeddy, lowreoneeqeddy) $V=12m/s$ 0. $02s$ 9 3 (Smagorinsky, $dyno$neeqeddy, lowreoneeqeddy) 7 LES

7 $u\vee\cdot$ $\overline{*\neg n}$ $ _{\neg}^{-}::\backslash \backslash$ $\overline{-}$ $-\neg$ $-:_{\backslash }$ $\underline{\underline{\overline{\overline{\overline{ru}}}\backslash }}$ $-:^{\tau}$ $arrow$ $\overline{x}$ $n$ $\overline{r}$ $\urcorner$ $\simarrow$ $-$ $-$ $n$ $u*$ $n\cdot\cdot$ $rightarrow$ 32 $\prime 7--1$. ; (a). $arrow-$ $-$ $-$. $ ^{-}*$ $1_{-}^{3}\backslash$ un (c).-. $m$ - $\sim$ 8: LES (a) oneeqeddy. (b) Smagorinsky. (c) dynoneeddy. (d) lowreoneeqeddy. OpenFOAM RANS $k-\epsilon$ $k-\omega SST$ $k-\epsilon$ $k-\omega SST$ $k-\omega$ $k-\epsilon$ $k-\epsilon$ LES $k-\epsilon$ $k-\omega SST$ LES $k-\omega SST$ 5 Lighthill [3,5,6,17]. ( ) Lighthill [17]. Lighthill Navier-Stokes $( \frac{\partial^{2}}{\partial t^{2}}-c^{2}\nabla^{2})(\rho-\rho_{0})=\frac{\partial^{2}t_{ij}}{\partial x_{i}\partial x_{j}}$, (2) $T_{ij}$ Lighthill $T_{ij}=\rho v_{i}v_{j}+$ $((p- Po)-c^{2}(\rho-\rho 0))\delta_{ij}+\sigma_{ij}$. (3)

$$\rho$ 33 9: LES $c$ $p_{0}$ $\rho_{0}$ $p$ $\sigma_{ij}$ Lighthill $+$ ( $+Navier-S$tokes ) (2) (2) $\rho$ Lighthill $T_{ij}$ Lighthill Lighthill $(p-p_{0})-c^{2}(\rho-\rho_{0})=0$ (2) Lighthill 2$

8 $\rho$ 33 9: LES $c$ $p_{0}$ $\rho_{0}$ $p$ $\sigma_{ij}$ Lighthill $+$ ( $+Navier-S$tokes ) (2) (2) $\rho$ Lighthill $T_{ij}$ Lighthill Lighthill $(p-p_{0})-c^{2}(\rho-\rho_{0})=0$ (2) Lighthill 2 Reynolds Lighthill (3) $\rho v_{i}$ Lighthill ((2) ) [16] $\nabla^{2}p=-\rho_{0}\frac{\partial^{2}v_{i}v_{j}}{\partial x_{i}\partial x_{j}}$ (4) Lighthill 1 ( ) ((2) ) Lighthill $\rho=\rho 0$, $divv=0$ $\frac{\partial^{2}t_{ij}}{\partial x_{i}\partial x_{j}}\sim-2\rho_{0}(\frac{\partial v_{1}}{\partial x_{1}}\frac{\partial v_{2}}{\partial x_{2}}-\frac{\partial v_{2}}{\partial x_{1}}\frac{\partial v_{1}}{\partial x_{2}})$. (5)

9 34 6 Lighthill 10 6(b) Powell-Howe [5, 6, 18]. 10: Lighthill ( ) 6 4 LES RANS [7, 16]. LES RANS DNS LES RANS Kolmogorov [7,16]. LES RANS

$35 LES RANS [1]. [1]. [19,20]. LES RANS LES RANS $\overline{v}$ $v=\overline{v}+\check{v}$ $R_{ij}=-\rho_{0}\overline{\check{v}_{i}\check{v}j}$ [16].$

10 35 LES RANS [1]. [1]. [19,20]. LES RANS LES RANS $\overline{v}$ $v=\overline{v}+\check{v}$ $R_{ij}=-\rho_{0}\overline{\check{v}_{i}\check{v}j}$ [16]. Reynolds Reynolds Lighthill ((3) ) Reynolds LES RANS (No ) [1] N.H.Fletcher and T.D.Rossing, The Physics of Musical Instruments, 2nd Edition (Springer- Verlag, New York 1998). [2] J.W.Coltman, Sounding mechanism of the flute and organ pipe, J. Acoust. Soc. Am. 44 (1968) ; Acoustics of the flute, Physics Today 21 (1968) 25-32; Jet driven mechanisms in edge tones and organ pipes, J. Acoust. Soc. Am. 60 (1976) ; Momentum transfer in jet excitation of flute-like instruments, J.Acoust. Soc. Am. 69 (1981) [3] 10 ( 2007) [4] G.B. Brown, The vortex motion causing edge tones, Proc. Phys. Soc., London XLIX (1937)

11 36 [5] M.S. Howe, Contributions to the theory of aerodynamic sound with appliction to excess jet noise an the theory of the flute, J. Fluid Mech. 71 (1975) [6] M.S. Howe, Acoustics of Fluid-Structure Interactions, (Cambridge Univ. Press, 1998). [7] C. Wagner, T. H\"uttl, and P. Sagaut, eds. Large-Eddy Simulation for Acoustics, (Cambridge, 2007). [8] LES, 28, No.7 (2009) pp [9] :, $-$ : 150 [10] M. Miyamoto, Y. Ito, K. Takahashi, T. Takami, T. Kobayashi, A. Nishida, M. Aoyagi, Numerical study on sound vibration of an air-reed instrument with compressible LES, in preparation. [11] T. Kobayashi, T. Takami, M. Miyamoto, K. Takahashi, A. Nishida, M.Aoyagi, $3D$ Calculation with Compressible LES for Sound Vibration of Ocarina, Open Source CFD International Conference 2009(CD-ROM), November 12-13th, Barcelona, Spain $($http: $//arxiv.org/abs/ vl)$. [12] http: $//www$.openfoam. $com/$ [13] J.Tsuchida, T.Fujisawa, G.Yagawa, Direct numerical simulation of aerodynamic sounds by a compressible CFD scheme with node-by-node finite elements, Computer Methods in Applied Mechanics and Engineering (2006). [14] H.K\"uhnelt, Simulating and sound generation in flutes and flue pipes with the Lattice- Boltzmann-Method, Proc. of ISMA 2004, pp (Nara, Japan, 2004) [15] A. R. da Silva and G. P. Scavone, Lattice Boltzmann simulations of the acoustic radiation from waveguides, J.Phys.A 40 (2007) pp [16],( 1999) [17] M.J. Lighthill, On sound generated aerodynamically. Part I: General theory, Proc. Roy. Soc. London A211 (1952) [18] A. Powell, Theory of vortex sound, J.Acoust. Soc. Am. 33 (1964) [19] T.Idogawa,T.Kobata,K.Komuro, and M.Iwaki, Nonlinear vibrations in the air column of a clarinet artificially blown J.Acoust.Soc.Am. 98 (1993) [20] K.Takahashi,H.Kodama,A.Nakajima and T. Tachibana, Numerical study on multi-stable oscillations of woodwind single-reed instruments, Acta Acustica united with Acustica 95 (2009)

IPSJ SIG Technical Report Vol.2013-ARC-207 No.23 Vol.2013-HPC-142 No /12/17 1,a) 1,b) 1,c) 1,d) OpenFOAM OpenFOAM A Bottleneck and Cooperation

IPSJ SIG Technical Report Vol.2013-ARC-207 No.23 Vol.2013-HPC-142 No /12/17 1,a) 1,b) 1,c) 1,d) OpenFOAM OpenFOAM A Bottleneck and Cooperation 1,a) 1,b) 1,c) 1,d) OpenFOAM OpenFOAM A Bottleneck and Cooperation with the Post Processes in Numerical Calculation of Transient Phenomena Taizo Kobayashi 1,a) Yoshiyuki Morie 1,b) Toshiya Takami 1,c)