Citation 数理解析研究所講究録 (2004), 1368:

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176-179 Issue Date 2004-04 URL http://hdlhandlenet/2433/25422 Right

1 Title Numerical study of acoustic wave sc of cylinders (Nonlinear Wave Phenom Author(s) Yano, Takeru Citation 数理解析研究所講究録 (2004), 1368: Issue Date URL Right Type Departmental Bulletin Paper Textversion publisher Kyoto University

2 $\mathrm{t}$ an $\mathrm{t}\mathrm{r}\{$ $\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{m}\mathrm{l}\pi \mathrm{g}$ $\mathrm{e}$ Numerical study of acoustic wave scattering from assemblies of cylinders Takeru Yano Department of Mechanical Science, Hokkaido University, Sapporo , Japan 1 Introduction Wave motions in inhomogeneous media, such as bubbly liquids, biological tissue, etc, have been extensively studied in physics and engineering The multiple scattering of acoustic waves is one of typical examples Averaging techniques and continuum approximation may resolve these problems However, understanding of real phenomena will require the direct simulation of the problem In the present study, we shall consider the problem of acoustic wave scattering from assemblies of cylinders by the numerical method PHYSALIS, originally devised for incompressible potential flows by Prosperetti and Oguz1 2 Formulation of the problem We shall consider a two-dimensional acoustic wave propagation in an ideal gas The geometry of the problem considered is shown in Fig 1 gi al tte $\mathrm{m}$ $\mathrm{e}$ yv $\neg$ fle 1 $\mathrm{w}_{\mathrm{o}1}$ $oe$ $x_{\mathrm{o}}$ rigi wall $x$ $oe$ Fig 1 Schematic of the model $=e^{\mathrm{i}[\mathrm{x}(\mathrm{z}-t)+\mathrm{w}\mathrm{o}]}$ As shown in Fig 1, a plane acoustic wave 1 propagates from $\kappa$ the left-side of a two- imensional channel with rigid wals, where is a nondimensional wave number, is a nondimensional coordinate along the channel, $x$ $t$ is a $\varphi_{0}$ nondimensional time, and is an initial phase There are a number of circular cylinders with the same radius in the channel The incident plane acoustic wave

3 om are is 177 is scattered by the cylinders, and as a result, a part of which propagates forward as a transmitted wave and the other propagates backward as a reflected wave The acoustic wave motion is governed by the Helmholtz equation $\Delta\phi+\kappa^{2}\phi=0,$ $\phi=\phi_{r}+i\phi_{i}$ (1) $\phi$ where is a complex velocity potential The boundary condition on the sidewall of the channel is $\frac{\partial\phi}{\partial n}=0$ (2) and at the entrance of the channel $(x=x0)$ and at the exit $(x=x_{1})$, $\frac{\partial\phi}{\partial x}=-it$ $jt$ $+2i\kappa e^{:(\kappa x\mathrm{o}+\varphi 0)}$ $(x=x_{0})$ (3) $\frac{\partial\phi}{\partial x}=i\kappa\phi$ $(x=x_{1})$ (4) Equations (3) and (4) are a kind of non-reflecting boundary condition based on a local one-dimensional approximation The acoustic pressure $p$ and the component of fluid velocity can be retrieved $x$ $u$ $\phi$ from the velocity potential by the following relations: $p=- \frac{\partial}{\partial t}(e^{- \kappa t}\phi)=i\kappa e^{-:\kappa t}\phi$ (5) $u= \frac{\partial\phi}{\partial x}e^{-i\kappa t}$ (6) The acoustic energy flux (acoustic intensity) based on the local one-dimensional approximation can be given as $q=\{\begin{array}{l}-\frac{1}{2}\kappa^{2} \phi-\phi^{i} ^{2}\frac{1}{2}\kappa^{2} \phi-\phi^{i} ^{2}\end{array}$ $x=x_{0}x=x_{1}$ (7) 3 Numerical method Numerical approach is based on the method of PHYSALIS originally devised for incompressible potential flows by Prosperetti and Oguzl We here extend the method to the Helmholtz equation In the neighborhood of each cylinder, the method utilizes a local analytical representation of a general solution of the Helmholtz equation, $\phi=\sum_{n=0}^{\infty}[j_{n}(\kappa r)\mathrm{y}_{n}(\kappa a)-y_{n}(\kappa r)j_{n}(\kappa a)](a_{n}\cos n\theta+b_{n}\sin n\theta)$ (8) $\mathrm{y}_{n}$ $J_{n}$ where and the Bessel functions of the first and second kinds, is the $r$ $\theta$ $\mathrm{f}$ distance a distmce ffom acylinder, is the radius of the cylinder, the azimuthal angle

4 $\sim--\sim\sim--\sim\backslash \simarrow\cdot---\sim\backslash$, \backslash \backslash \backslash$ $;^{\vee-\prime}\sim\simarrow,arrow\sim\backslash \backslash \sim\prime\prime\cdot\backslash -\backslash -\sim\backslash \prime\prime\prime_{\backslash }$ $\backslash \backslash \backslash \backslash \backslash \backslash \backslash \backslash \backslash \backslash \backslash$ $\backslash$ $,$ $\backslash \backslash \backslash \backslash$ $\cdot$ $\backslash \backslash$ $\nearrow\backslash _{\backslash }\backslash \backslash \backslash \backslash$ $ -\sim\sim,\backslash \sim-arrowarrow\backslash -\sim\sim\sim\sim-\wedge\wedge\wedge$ $\wedge-\overline{\backslash \sim}\sim\sim-\sim-\sim\sim\wedge\wedge$ $-\vee\sim\sim\sim\wedge\sim\sim\sim\sim\sim\sim\sim\sim\sim$ \sim-\sim\sim\sim\sim\sim\sim\sim\backslash \sim-\sim\sim\sim\sim\sim\sim\sim$ 178 measured from the axis, and the prime denotes the differentiation with respect $x$ to the argument The coefficient of expansion, and, are determined in an iteration: $A_{n}$ $B_{n}$ (0) Start with an appropriate initial expansion coefficients (1) Prom the local analytical representation (truncated suitably), we evaluate the normal derivative of the velocity potential along a closed curve enclosing each cylinder (2) We solve the Helmholtz equation with the finite-difference method with the second-order central difference The finite-difference solution is directly obtained with the help of FFT, without any iterative procedure (3) The expansion coefficients are updated by the finite-difference solution near each cylinder $\sim--\sim$ $\sim\sim\sim-$ $\sim\#$ - $rightarrow\wedge--\wedge\cdot,\prime\prime\prime$, $,-$ $arrow-\sim\backslash \backslash \backslash \backslash \backslash \backslash \backslash \backslash$ 1t $\backslash $\backslash $\prime\prime,-$ $\backslash, $\sim\sim\sim\sim\sim\sim\simarrow\sim$ "\sim\sim\sim\sim\sim\simarrow\sim$, $\ldots\wedge-\sim\sim\cdot\sim\sim\sim\vee\sim\sim\sim\sim$ $\cdot$ \sim -- $\backslash \sim\sim^{-}$,, $\backslash \sim-\sim-$ $\dot{\mathrm{j}},\cdot-arrow,,--arrow--_{1}"\cdot\sim\dagger\sim\overline{\sim\sim^{-arrow-}}-$ $\sim\wedge$ $\sim$, $ \sim\sim\sim\sim\sim$ $\backslash$ $-arrow\wedge--\wedge-$ $\vee\wedge$ \prime\prime $\backslash \backslash \backslash \backslash \backslash \cdot\backslash \sim-\wedge$ $/$ $\sim,-\sim$ : $-\vee-$ $\mathrm{j}_{\prime\prime\backslash $\backslash -arrow--\sim\wedge\prime\prime}^{\backslash \sim-\sim-arrowarrow\prime\prime\prime \sim-\prime^{\iota}}\backslash$ 1^{\cdot} -\sim\sim\cdot-\backslash \simarrow--\backslash :-arrow\wedge$ Fig 2 Numerical example In Fig 2, we present an example as an illustration of the numerical method, where one can see nine cylinders in the channel, The boundary of each cylinder is denoted by a circle plotted by a thin solid curve Along a closed line like octagon in the circle, we evaluate the normal derivative of the velocity potential, with which the exterior boundary-value problem of the Helmholtz equation is solved At the 17 black spots around each cylinder, we update the values of expansion coefficients $A_{n}$ and with the help of the finite-difference solution In this example, we $B_{n}$ $\mathrm{x}2$ use 17 coefficients The small arrows are the fluid velocity obtained from the finite-difference solution 4 Results Figure 3 shows the pressure wave fields for the case that the incident wave is scattered by a column of cylinders and by three columns of cylinders One

5 $\vee--i-\cdot ^{-}_{}-,\cdot-\sqrt j\grave{\overline{\tau}^{j}\underline{\prime}}$ $_{i_{\overline{\prime}}}^{-}\cdot\cdot$ $\ovalbox{\tt\small REJECT}_{i}\cdotarrowarrow P\tilde{x_{j}}--\hat{p}--*\cdot$ $\overline{\omega}$ $\mathrm{f}_{\phi}^{\overline{\mathrm{q}}}\mathrm{f}\mathrm{i}$ $-\hat{p}*$ $_{\tilde{\dot{\sqrt}}}\neg\nearrow"\acute{"}\acute{,}\dot{\ovalbox{\tt\small REJECT}}x_{\ovalbox{\tt\small REJECT}}^{rightarrow,-}\nearrow i^{-}\prime\prime y_{\nearrow\nearrow}^{arrow}\acute{\mathrm{a}}_{\vee}\prime r\varphi\phi ff^{*}\cdot\acute{\hat{}}$ $_{\tilde{\dot{\sqrt}}}\neg\nearrow"\acute{"}\acute{,}\dot{\ovalbox{\tt\small REJECT}}x^{rightarrow-}\nearrow i^{-}\prime\prime y_{\nearrow\nearrow}^{arrow}\acute{\mathrm{a}}_{\mathit{5}^{*}}\prime r\phi\acute{}\hat{}$ $\mathrm{o}\mathrm{o}$ $\mathrm{s}\mathrm{o}$ $\mathrm{o}$ $\overline{\mathrm{q}}$ $\mathrm{f}^{\phi}\mathrm{f}$ $\not\in\omega$ $\vee\cdot\wedge\leftrightarrow \mathrm{c}\cdot\cdot$ J\hat{\mathrm{e}_{\mathrm{R}\prec}^{<}}\backslash \dot{\overline{\varphi}}d\delta r\cdot\dot{p}_{\gamma}^{-}r\tilde{j}\acute{\theta}_{\check{\acute{\mathrm{r}}}}\tilde{\hat{\wedge\cdot}}j$ $\sigmaj\dot{\overline{\varphi}}r_{\wedge,\dot{p}_{\gamma}^{-}}\cdot\acute{\theta}_{\check{\acute{\mathrm{r}}}}\tilde{\hat{}}$ $\cdot\cdot<\cdot\dot{\acute{p}}s^{-}\neq_{\simeq}^{k}\dot{\mathrm{f}}fi_{\dot{\tilde{\mathrm{g}}}_{\wedge\wedge}}\dot{\hat{\epsilon}}_{\mathrm{r}}^{\gamma}\ddot{\theta}^{-}\overline{\theta} i^{\nearrow}r\check{j}\wedge\acute{\sim}\theta_{\check{\acute{\mathrm{r}}}}<,\sim\acute{r}\sim\wedge \mathrm{r}\hat{\prec j}",\cdot$ $\dot{\triangleleft}<\acute{d}\neq^{\dot{\tilde{\mathrm{g}}}_{\wedge\wedge}}\hat{\ll,\mathrm{a}}\prec\rightarrow\acute{r}\sim\grave{\mathit{3}^{\mathrm{r}}\rightarrow}\simeq\delta\rightarrow-\tau_{k}\vee\cdot<f\dot{\acute{\phi}}",\cdot$ $\dot{\triangleleft}<\acute{d}\neq^{k}\sqrt\rightarrow\acute{r},\sim \mathit{3}^{\mathrm{r}}\rightarrow\simeq^{\dot{\tilde{\epsilon_{\vee}}}}\sim\acute{\mu}^{\wedge}\rightarrow n\tau_{\vee}\neg\yen_{\gamma}^{-\nearrow}f\dot{\acute{\theta}}\cdot$ $\mathrm{r}_{\wedge}\cdot \mathrm{x}_{i}-rt-\backslash \cdot$ $A-F\#\mathrm{r}_{-}\dot{\ovalbox{\tt\small REJECT}}^{\kappa}\ulcorner\check{j}$ $*i_{d}^{\leftrightarrow\sim}\dot{_{\vee}}^{arrow}$ $ arrow\cdot\cdot-\cdotarrow-\#-\mathrm{r},\cdot F\ \ovalbox{\tt\small REJECT}^{\check{arrow}}x_{\sqrt}^{x\dot{\wedge}}\dot{\swarrow}i_{-,*}\acute{\dot{d}}\dot{\overline{*}}i_{arrow}^{\Leftrightarrow\sim}\dot{\ovalbox{\tt\small REJECT}}_{\vee}^{\kappa_{\vee}}\ulcorner\check{j}\overline{\dot{\#}}\sum_{\approx}arrow I-\cdot$ $\underline{\sim}-\lrcorner*\cdot-\succ_{\dot{j}}^{\check{4}}\simj\sim,\backslash ^{\check{d}}\tilde{\mathrm{g}\cdot}\simeq\cdot$ $\wedge\vee\}0\cdot!\cdot 1^{\cdot}\backslash \cdot\backslash!^{1\mathfrak{l}^{\wedge}}\sim\cdot\ldots\cdot$ $-\cdot-$ 173 can easily see that the amplitude of transmitted wave depends on the number of $\mathrm{n}\mathrm{a}$ columns and the parameter $\sigma\cdot $\vee\rightarrow\prime\prime\cdot$ $-\cdot\yen_{\gamma}^{-\nearrow}\acute{\mu}^{\wedge}\vee$ $\cdot y\sim$ $x^{p_{x\dot{\wedge}}}\ \ovalbox{\tt\small REJECT}^{\Rightarrow}-*$ Fig 3 Scattering from columns of cylinders 10 $\cdot!\cdot 1^{\cdot}!:\cdot\cdot\ldots\cdot!$ 0 0 $\overline{\gamma}$ $\}$ $\mathrm{i}\mathrm{j}$ - columns $\mathrm{j}\mathrm{q}$ $\sim$ wide pacing $\kappa$ Fig 4 Reflection coefficient References $\mathrm{h}\mathrm{n}$ 1 A Prosperetti and Oguz PHYSALIS: A new $0\{\mathrm{N}$) method numerical simulation of disperse flows of spheres Part $\mathrm{i}$: Potential Comput Phys 167: , 2001 for the flow $J$

Citation 数理解析研究所講究録 (2001), 1191:

Traffic Model and a Solvable Differ Title (Interfaces, Pulses and Waves in No Systems : RIMS Project 2000 "Reacti theory and applications") Author(s) Nakanishi, Kenichi Citation 数理解析研究所講究録 (2001), 1191: