Resonant radial oscillations of an inhomogeneous gas in the frustum of a cone

Amundsen, David E.; Mortell, Michael P.; Seymour, Brian

doi:10.1007/s00033-015-0510-5

Cited by 9 publications

(6 citation statements)

References 26 publications

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“…When the slope of the frustum of a cone is then, using the superposition of oppositely travelling modulated simple waves, it can be shown that the resonant output at for an input of contains shocks, see Amundsen et al. (2015). In contrast, when the slope is the resonant output is for an input of and the signal is continuous, see Lawrenson et al.…”

Section: Discussionmentioning

confidence: 99%

“…For resonant flows in tubes with varying cross-section there are several analytical and numerical investigations for both small and finite , see Keller (1977), Chester (1991), Ockendon et al. (1993), Chun & Kim (2000), Ellermeier (1994) and Amundsen, Mortell & Seymour (2015). Ellermeier (1993) concluded that shocks must appear when the cross-section is sufficiently slowly varying and the system is close to resonance.…”

Section: Introductionmentioning

confidence: 99%

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On the transition between continuous and shocked solutions for resonant gas oscillations in the frustum of a closed cone

2022

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Resonant gas oscillations in a closed straight tube contain shocks for frequencies near the linear resonant frequency. As the tube geometry changes from straight to a cone with slope $a,$ the shock strength decreases until, for large enough $a,$ the motion is continuous. The analytical result for small $a$ follows from a nonlinearization of the linear resonant response, while the result for large $a$ is a dominant single mode approximation. The connection between these two forms is analysed numerically. The shocked solutions change to multimode continuous solutions as $a$ increases to cross the curve $a_{s}\doteqdot 12.8M^{1/3}$ , where $M\ll 1$ is the Mach number of the input. Then the amplitudes of the higher modes decrease as $a$ continues to increase until, having crossed $a_{m}\doteqdot 30M^{1/3}$ , the single mode solution emerges.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the transition between continuous and shocked solutions for resonant gas oscillations in the frustum of a closed cone

2022

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“…To provide a simple yet nontrivial example, consider in the case where

N false(u false) = \frac{\partial}{\partial x} (\frac{u^{2}}{2}),

and

p false(x false) = r false(x false) = {(1 + k x)}^{2} = s false(x false)

for parameter

k \geq 0

, with domain

L = π

ν = 0.25

M = 0 . 03^{3}

, and boundary conditions specified by

β_{1} = 0, β_{2} = 1

. This example is based on the case of acoustic waves in conical geometries where

s (x)

corresponds to the cross‐sectional area and k the opening angle of the cone, . It follows that

φ false(x, λ false) = \frac{sin \sqrt{λ} x}{1 + k x}

ψ false(x false) = \frac{cos \sqrt{λ} x}{1 + k x}

and the linear eigenvalue relation is

tan \sqrt{λ} π = \frac{\sqrt{λ}}{k} false(1 + k π false),

We note that in the limit

k \to 0

we see that

λ_{1} = \frac{1}{4}

and

λ_{n} = {(2 n - 1)}^{2} λ_{1}

, whereas for

k > 0

this commensurate structure is lost, as illustrated by Fig.…”

Section: Examplementioning

confidence: 99%

“…and p(x) = r (x) = (1 + kx) 2 = s(x) for parameter k ≥ 0, with domain L = π , ν = 0.25, M = 0.03 3 , and boundary conditions specified by β 1 = 0, β 2 = 1. This example is based on the case of acoustic waves in conical geometries where s(x) corresponds to the cross-sectional area and k the opening angle of the cone, [17]. It follows that φ(x, λ) = sin √ λx 1+kx , ψ(x) = cos √ λx 1+kx and the linear eigenvalue relation is…”

Section: Examplementioning

confidence: 99%

Resonances in Bounded Media

Amundsen

2017

Stud Appl Math

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Resonant behavior in bounded domains and under the influence of weakly periodic forcing with magnitude M≪1 is studied for a general class of one‐dimensional nonlinear wave systems. The model encompasses and provides analogy to numerous physically motivated cases such as acoustic resonators. Through a generalized weakly nonlinear analysis the linear response and associated resonant spectrum may be determined. In the case where the spectrum is sufficiently noncommensurate a single mode response emerges with amplitude O(M13). Dependence upon detuning and dissipative effects follow immediately from the subsequent nonlinear balance. In the case where the spectrum is commensurate a multimodal response arises, leading to a coupled system of solvability conditions. The amplitude of the response depends in detail on the commensurate structure, O(M12) in the case where it is fully commensurate. This is in keeping with the well‐known dichotomy between responses for acoustic waves in open and closed tubes. Through continuous variation of the model system, the nature of the transition between these distinct regimes is then studied and through an appropriate modal truncation the connection is achieved. A numerical example is presented to further illustrate and corroborate the general analysis.

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“…Recently Amundsen et al [15] investigated the effect of the density parameter in the resonant oscillation of a gas contained in the frustum of a cone. In order to study the continuous resonant solution, they constructed a variable density profile depending on an arbitrary parameter to approximate a desired density, then used it to reflect the eigenvalue equation corresponding to the linear theory.…”

Section: Rms Was Investigated Numerically Bymentioning

confidence: 99%

Resonant Response in Bounded Nonlinear Wave Systems

Shatnawi¹

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Resonant radial oscillations of an inhomogeneous gas in the frustum of a cone

Cited by 9 publications

References 26 publications

On the transition between continuous and shocked solutions for resonant gas oscillations in the frustum of a closed cone

On the transition between continuous and shocked solutions for resonant gas oscillations in the frustum of a closed cone

Resonances in Bounded Media

Resonant Response in Bounded Nonlinear Wave Systems

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