Acoustic resonance of radially symmetric waves in a thermoviscous gas

Ellermeier, W.

doi:10.1007/bf01262525

Cited by 7 publications

(10 citation statements)

References 3 publications

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“…This equation is given in Ellermeier (1997), is implicit in Chester (1991), and yields incommensurate eigenvalues and hence no shocks for sufficiently small 3. Plane wave.…”

Section: (I) Homogeneous Gasmentioning

confidence: 99%

Resonant oscillations of an inhomogeneous gas between concentric spheres

2011

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We investigate the effects of nonlinearity, geometry and stratification on the resonant motion of a gas contained between two concentric spheres. The emphasis is on whether the motion is continuous, and on how the inhomogeneity, geometry or nonlinearity can move the motion to a shocked state. Linear undamped theory yields a standing wave of arbitrary amplitude and an eigenvalue equation in which the higher eigenvalues are not integer multiples of the fundamental; the system is said to be dissonant. Higher modes, generated by the nonlinearity, are not resonant and consequently shocks may not form. When the output is shockless, the amplitude is two orders of magnitude greater than that of the input. When the eigenvalues for a homogeneous gas are not sufficiently dissonant and shocks form, the introduction of a stratification in the gas can restore dissonance and allow a continuous output. Similarly, the introduction of an inhomogeneity can change a continuous motion to a shocked one, as can an increase in the input Mach number, or an increase in the geometrical parameter. Various limits of the eigenvalue equation are considered and previous results for simpler geometries are recovered; e.g. a full sphere, a cone and a straight tube.

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“…This equation is given in Ellermeier (1997), is implicit in Chester (1991), and yields incommensurate eigenvalues and hence no shocks for sufficiently small 3. Plane wave.…”

Section: (I) Homogeneous Gasmentioning

confidence: 99%

Resonant oscillations of an inhomogeneous gas between concentric spheres

2011

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“…These three fixed points explains the frequency response of the steady state oscillation. [10][11][12] Even in the case of detuning, as long as the dissipation effect is negligible, the resonant oscillation starting from the initial quiescent gas experiences the periodic modulation along the separatrix passing through the origin.…”

Section: -7mentioning

confidence: 99%

“…The fixed point corresponds to a steady state oscillation, and it has been examined, [10][11][12] while other trajectories have been left untouched. As can be seen from Fig.…”

Section: ͑50͒mentioning

confidence: 99%

“…[10][11][12][13] This is because the excitation of higher harmonics required for shock formation is impeded by the fact that the resonance frequency of each nth mode ͑n =2,3, ...͒ is not equal to the n multiple of that of the fundamental mode in cylindrical and spherical standing waves, as sometimes called dissonant. 14 As a result, the maximum wave amplitude in the shock free resonances of cylindrical and spherical standing waves attains O͑M 1/3 ͒, [10][11][12][13] which can be considerably large for M Ӷ 1 compared to the case of plane waves. The dissonant effect has been utilized for realization of large amplitude and shock free resonant oscillations.…”

Section: Introductionmentioning

confidence: 99%

“…The dissonant effect has been utilized for realization of large amplitude and shock free resonant oscillations. 15,16 The steady state oscillations of shock free resonances in a spherical shell and in an infinitely long cylinder were studied by Chester 10 and Ellermeier, 11,12 respectively. However, since they started the analyses from assuming the steady state, the temporal evolutions of wave motions were not clarified.…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear analysis of periodic modulation in resonances of cylindrical and spherical acoustic standing waves

Kurihara

Yano

2006

Physics of Fluids

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The nonlinear resonance of cylindrical acoustic standing waves of an ideal gas contained between two coaxial cylinders is theoretically investigated by the method of multiple scales. The wave motion concerned is excited by a small-amplitude harmonic oscillation of the radius of the outer cylinder, and the formulation of the problem includes the wave phenomenon in a hollow cylinder without the inner one as a limiting case. The spherical standing wave in two concentric spheres is also studied in parallel. The resonance occurs if the driving frequency falls in a narrow band around the linear resonance frequency, and in the weakly nonlinear regime, no shock wave is formed in contrast to the plane wave resonance. A cubic nonlinear equation for complex wave amplitude can then be derived by the method of multiple scales. Using a first integral of the cubic nonlinear equation, we shall demonstrate that the resonant oscillation is accompanied by a periodic modulation of amplitude and phase when the dissipation effect due to viscosity and thermal conductivity is negligible. The period of the modulation varies as the minus two-thirds power of the acoustic Mach number defined at the outer cylinder or sphere and decreases with an increase in the radius ratio of the inner and outer cylinders or spheres. When the dissipation effect is small but not negligible, the modulation is slowly weakened and the resonant oscillation approaches a steady state oscillation, which corresponds to the steady solution examined in earlier works.

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Nonlinear resonant oscillations and shock waves generated between two coaxial cylinders

Kurihara¹,

Inoue²,

Yano³

2005

Fluid Dyn. Res.

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Nonlinear evolution of a gas oscillation of large amplitude generated between two coaxial cylinders is numerically studied by solving the Euler equations for an inviscid ideal gas. The resonant oscillation is excited uniformly along the axis by harmonic oscillation of the radius of the outer cylinder with the fundamental resonance frequency. Nonlinear and geometrical effects are investigated by varying two nondimensional parameters, i.e., the acoustic Mach number M at the surface of the outer cylinder and the ratio of the radius of the inner cylinder to that of the outer cylinder. The result shows that no shock wave is formed if M is smaller than a critical value for a given , and then the amplitude of the oscillation reaches O(M 1/3 ). In this case the amplitude of gas oscillation is modulated periodically with a period of O(M −2/3 ). When is small ( ≈ 0.1) and M is larger than the critical value, the waveform including a shock wave is quite different from that in plane wave resonance because of the geometrical effect. Furthermore, the soaring of temperature takes place in the vicinity of the inner cylinder owing to the shock heating.

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Acoustic resonance of radially symmetric waves in a thermoviscous gas

Cited by 7 publications

References 3 publications

Resonant oscillations of an inhomogeneous gas between concentric spheres

Resonant oscillations of an inhomogeneous gas between concentric spheres

Nonlinear analysis of periodic modulation in resonances of cylindrical and spherical acoustic standing waves

Nonlinear resonant oscillations and shock waves generated between two coaxial cylinders

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