Yano, Takeru

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Title共鳴管内の音響流の分岐について (非線形波動現象の数理とその応用)Author(s)Yano, TakeruCitation数理解析研究所講究録 (2003), 1311: 168-172Issue Date2003-04URL http://hdl.handle.net/2433/42924RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityAbstract The large amplitude standing wave excited in aresonator induces acousticstreaming ofRayleigh $\mathrm{t}\mathrm{y}\mu$ outside the acoustic boundary layer on the wall of the resonator.For the case that the resonator is atw0- imensional rectangular box, the streaming motionwith large Reynolds number is examined numerically. The tw0- imensional incompressibleNavier-Stokes equations with no external force are used as the governing equations forthe streaming velocity, which is defined by atime-averaged mass flux density vector. Thesteady velocity component at the outer edge of the acoustic boundary layer, which inducesRayleigh type streaming, is employed as the boundary condition for the Navier-Stokesequations. By using afinite-difference method, we shall show the existence of multiplesteady solutions.INTRODUCTIONFIGURE 1. The schematic of the model.Recently, one of the present authors has numerically studied the resonant gasoscillation with aperiodic shock wave in aclosed tube by solving the system ofcompressible Navier-Stokes equations [6]. The result has suggested the occurrenceof turbulent acoustic streaming when astreaming Reynolds number is sufficientlylarge. However, the direct numerical simulation of viscous compressible flow is anextraordinarily hard task if one tries to resolve all phenomena ffom an initial stat1311 2003 168-172168of uniform and at rest to an almost steady oscillation state throughout the flow fieldincluding the boundary layer. In the present paper, we shall adopt asimple model basedon the linear standing wave solution and boundary layer theory, and thereby executeanumber of numerical simulations to clarify how the classical symmetric streamingpattern changes into different flow patterns as the streaming Reynolds number increases.FORMULATIONWe shall consider the streaming motion induced by resonant gas oscillations in atw0-dimensional rectangular box filled with an ideal gas (see Fig. 1). The box, whoselength is $L$ and width is $W$ , is closed at one end by asolid plate and the other by apiston(sound source) oscilating harmonically with an amplitude $a$ and angular frequency $\omega$ .In the case that the excitation at the sound source is moderately weak and thedissipation effect is sufficiently small outside the boundary layer, the wave motion in thebody of the fluid is aplane standing wave accompanied with small correction terms dueto nonlnear and dissipation effects,u $=M \frac{\sin(x-b)}{\sin b}\sin t+\cdots$ , $\rho=1+M\frac{\cos(x-b)}{\sin b}\cos t+\cdots$ , (1)where $x=x^{*}\omega/c_{0}$ is anormalized axial coordinate, $t=\omega t^{*}$ is anormalized time, $u=$$u^{*}/c_{0}$ is the $x$ component of normalized fluid velocity, $\rho=\rho^{*}/\rho_{0}$ is normalized density,$M=a\omega/c_{0}(\ll 1)$ is the acoustic Mach number at the sound source, and $b=L\omega/\mathrm{q}_{1}$ is anormalized box length or anormalized angular frequency ($\mathrm{q}_{1}$ is the speed of sound in aninitial undisturbed state of density $\rho_{0}$).We shall assume that the oscillation is near the second mode resonance, i.e.,b $=2\pi+\sqrt{M}\Delta$, (2)where Ais nondimensional parameter which measures the detuning. $\mathrm{I}\mathrm{f}|\Delta|=O(\sqrt{M})$ ,then the oscillation includes two periodical shock waves as long as the dissipation effectis sufficiently small. The formation of shock waves may induce the turbulent streamingmotion as shown in [6]. In what foUows we assume $\Delta\approx 1$ . Equation (1) can then berewritten intou $= \sqrt{M}\frac{1}{\Delta}\sin x\sin t+\cdots$ , $\rho=1+\sqrt{M}\frac{1}{\Delta}\cos x\cos t+\cdots$ . (3)From the standard boundary layer analysis, we have the s0-called limiting velocityat the outer edge of the boundary layer,$\overline{u}=-U_{0}\sin 2x$ , $\overline{v}=0$ , $U_{0}= \frac{M}{\Delta^{2}}[\frac{3}{8}+\frac{\sqrt{\mathrm{P}\mathrm{r}}(\gamma-1)}{4(\mathrm{E}+1)}]$ , (4)where the bar denotes atime average, $\gamma$ the ratio of specific heats, and Pr the Prandt169Acoustic streaming velocity is usually defined as atime-averaged mass flux densityvector. We introduce astreaming velocity normalized by $U_{0}$$U= \frac{1}{2\pi}\int_{t}^{t+2\pi}\frac{pu}{U_{0}}\mathrm{d}t$ , $V= \frac{1}{2\pi}\int_{t}^{t+2\pi}\frac{\rho v}{U_{0}}\mathrm{d}t$. (5)Then, the governing equations for the streaming outside the boundary layer are$\frac{\partial U}{\partial x}+\frac{\partial V}{\partial\frac{\partial Vy}{\partial\tau}}=0,\frac{\partial U}{\partial\tau}+U\frac{\partial U}{\frac{\partial x\partial p}{\partial y}}+V\frac{\partial U}{,\mathrm{R}\mathrm{e}1\partial y},=(\frac{\partial^{2}U}{\partial x^{2}}+\frac{\partial^{2}U}{\partial y^{2}})+U\frac{\partial V}{\partial x}+V\frac{\partial V}{\partial y}+=\frac,(\frac{+\frac{\partial p}{x^{2}\partial xV}\partial^{2}}{\partial}+\frac{\partial^{2}\frac{1}{V{\rm Re}}}{\partial y^{2}}),’\}$ (6)where aslow scale time $\tau$ and the streaming Reynolds number ${\rm Re}$ are defined as$\tau=U_{0}t$ , ${\rm Re}= \frac{M}{\epsilon^{2}\Delta^{2}}[\frac{3}{8}+\frac{\sqrt{\mathrm{P}\mathrm{r}}(\gamma-1)}{4(\mathrm{R}+1)}]$ $( \epsilon=\frac{\sqrt{\nu_{0}\omega}}{c_{0}})$ . (7)In the definition of Re, $\epsilon$ represents the ratio of acoustic boundary layer thicknessto atypical wavelength. In the present study, we assume$\epsilon^{2}\ll M<<1$ and $\epsilon\ll w=\frac{W\omega}{c_{0}}$ , (8)where $W$ is the width of the box. Condition (8) supports Eq. (1) and enables largeReynolds number flows. We shall emphasize that the momentum equations in (6) haveno driving force terms because the sound wave is unattenuated; this is also supported bycondition (8).NUMERICAL ANALYSISWe shall numerically solve the incompressible Navier4tokes equations in atw0-dimensional box with the boundary condition (see Fig. 2),U $=-\sin 2x$ , V $=0$ at y $=0$ and y $=w$ , U $=0$, V $=0$ at x $=0$ and x $=2\pi$ . (9)The solution method is the standard projection (MAC) method with accuracy offirst order in time and second order in space. Aregular even-spaced $512\cross 128$ grid pointsare mainly used. The time step is changed from $5\cross 10^{-4}$ to $7\cross 10^{-5}$ . The normalizedbox width $w$ is fixed to $2\pi/5$ .FIGURE 2. The $2\mathrm{D}$ box for simulation170FIGURE 3. Streamlines for the classical symmetric flows, (a): ${\rm Re}=1\propto 1$ , (b): ${\rm Re}=230$.$. \frac{\frac{\triangleright}{.-\dot{\mathrm{e}}}}{\triangleright v}*$$\overline{\mathrm{e}}$$\mathrm{e}\mathrm{g}\mathrm{h}v$$\frac{\underline{\mathrm{z}}\approx}{\infty}$$\underline{v}$FIGURE 4. Residual error and total vorticity $({\rm Re}=280)$.EF$\frac{v}{v}$$.\not\in\Leftrightarrow v$.$\underline{\Xi \mathrm{e}}$FIGURE 5. Total kinetic energy.Atypical example of convergence and accuracy check is shown in Fig. 4. Thetime evolution of aresidual error denoted by the bold curve has aminimum at about$\tau=120$, when the flow pattern is similar to that shown in Fig. $3(\mathrm{b})$ . However, theclassical symmetric flow is unstable at ${\rm Re}=280$, and hence the residual error growsuntil $\tau=3W$ and thereafter decreases again. Finally, at about $\tau=\mathit{6}tD$, the flow fieldconverges to anew steady state, where the flow pattern is similar to that shown in Fig.$6(\mathrm{b})$ . The dashed curve in Fig. represents the fluctuation of total vorticity, which shouldbe zero due to the restriction from boundary condition (9)171Figure 5shows the coexistence of various steady flows. For ${\rm Re}<130$ , theclassical flow is the unique stable steady solution. Note that the classical flow has twotypes of symmetry, i.e., amirror symmetry about $x=\pi$ and arotational symmetryaround $(x, y)=(\pi, w/2)$ . Beyond ${\rm Re}=130$, another steady solution emerges, theflow pattern of which loses the rotational symmetry [see Figs. $6(\mathrm{a})$ , $6(\mathrm{b})$ , and $6(\mathrm{e})$]. At${\rm Re}=240$, the classical flow becomes unstable but even after that it continues to be asteady (unstable) solution of the problem. At ${\rm Re}=180$, another new steady solutionappears, whose flow pattern no longer has symmetry [see Figs. $6(\mathrm{c})$ , $6(\mathrm{d})$ , and $6(\mathrm{e})$ ].At ${\rm Re}=7\mathfrak{X}$ , afurther new steady solution appears, which possesses the rotationalsymmetry but loses the mirror symmetry [see Figs. $6(\mathrm{e})$ and $6(\mathrm{D}$].FIGURE 6. Streamlines of various steady flows. (a): ${\rm Re}=140$ , (b): ${\rm Re}=230$, (c): ${\rm Re}=180$, (d):${\rm Re}=240$, (e): ${\rm Re}=7\propto$), (f): ${\rm Re}=1000$, (g): ${\rm Re}=1000$ , (b): ${\rm Re}=1000$ .The mirror image and the rotational image of asolution ofboundary value problem(6) and (9) are also solutions of the same problem. Accordingly, the number of stablesteady solutions changes as $1arrow 3arrow 7arrow 6arrow 8$ with the increase of ${\rm Re}(<2\propto)0)$ . Thetype of bifurcation seems to be subcritical, although we have not yet verified.REFERENCES1. Lord Rayleigh, The Theory ofSound (Dover, New York, 1945).2. Andrade, E. N. da C, “On the circulations caused by the vibration of air in a tube,” Proc. R. Soc. A130, 445A70 (1995).3. Swil G. W., “Thermoacoustic engines and refrigerators,” Phys. Today 48, 22-28 (1995).4. Kawahashi, M. and Hirahara, H., “Velocity and density field measurements by digital speckle metheA”Opt. Laser Technol. 32, 575-582 (2000).5. CampklL M., et al., “Review of LDA and PIV applied to the measurement of sound and acousticsstreaming, Opt. Laser Technol. 32, 629-639 (2000).6. Yano, T., ’Turbulent acoustic streaming excited by resonant gas oscillation with periodic shock wavesin closed tube,” J. Acoust. Soc. Am. 106, L7-L12 (1999)172

共鳴管内の音響流の分岐について (非線形波動現象の数理とその応用)

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共鳴管内の音響流の分岐について (非線形波動現象の数理とその応用)

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