^{1,a)}and Nobumasa Sugimoto

^{2}

### Abstract

This paper studies quantitatively a thermoacoustic field by the Taconis oscillations with finite amplitude in a helium-filled, quarter-wavelength tube. Numerical simulations are performed based on the one-dimensional theory in the boundary-layer approximation developed in a previous paper [N. Sugimoto and D. Shimizu, Phys. Fluids20, 104102 (2008)] by solving initial- and boundary-value problems for a smooth step temperature distribution. It is found that the variations in the density, temperature, and entropy are so significant that the mean values deviate from the respective values in quiescent state. The mean acoustic energy flux and mean heat flux are calculated not only in the main-flow region but also in the boundary layer. The mean convective heat flux appears locally in the main-flow region due to higher-order nonlinear effects. While the total heat flux into the gas vanishes per one period, the local heat flux flows into the gas over a middle part of the tube.

I. INTRODUCTION

II. SUMMARY OF THE MODEL

A. Physical model

B. Nonlinear theory for the main flow

C. Evaluation of physical variables in the boundary layer

D. Initial- and boundary-value problems

III. THERMOACOUSTIC FIELD IN THE MAIN-FLOW REGION

IV. FLOW OF MEAN ENERGY FLUXES

A. Analysis of Fourier spectra

B. Distribution of acoustic energy flux

C. Distributions of enthalpy and heat fluxes

V. SUMMARY AND DISCUSSIONS

VI. CONCLUSIONS

### Key Topics

- Enthalpy
- 20.0
- Thermoacoustics
- 19.0
- Entropy
- 14.0
- Boundary value problems
- 9.0
- Nonlinear field theories
- 8.0

## Figures

The physical model of the Taconis oscillations in a quarter-wavelength tube of length and radius . The thermoacoustic field in the tube is divided into the acoustic main-flow region and the boundary layer on the tube wall bounded by the tube wall and the broken line where , , and denote, respectively, the cross sections of the tube, main-flow region, and boundary layer. The velocity at the edge of the boundary layer directed into the main-flow region is indicated by or along the normal vector to the tube wall.

The physical model of the Taconis oscillations in a quarter-wavelength tube of length and radius . The thermoacoustic field in the tube is divided into the acoustic main-flow region and the boundary layer on the tube wall bounded by the tube wall and the broken line where , , and denote, respectively, the cross sections of the tube, main-flow region, and boundary layer. The velocity at the edge of the boundary layer directed into the main-flow region is indicated by or along the normal vector to the tube wall.

Distributions of the temperature relative to room temperature and of the axial gradient of the logarithmic temperature in the solid and broken curves, respectively, for the temperature ratio .

Distributions of the temperature relative to room temperature and of the axial gradient of the logarithmic temperature in the solid and broken curves, respectively, for the temperature ratio .

Temporal variations in the excess pressure relative to at the closed end for the value of the temperature ratio .

Temporal variations in the excess pressure relative to at the closed end for the value of the temperature ratio .

Spatial profiles of the excess pressure and the axial velocity relative to and , respectively, for the temperature ratio at every 1/16th period of the oscillations from to 257.84, where (a) shows the profiles of , while (b) and (c) show the ones of over the first half period and the other half, respectively. The arrows indicate the sense of time evolution in the order of the red solid, blue solid, blue broken, and red broken arrows, and the color of the curves corresponds to the one of the arrows in the respective quarter periods. The black curve represents the profiles just at 8/16th period.

Spatial profiles of the excess pressure and the axial velocity relative to and , respectively, for the temperature ratio at every 1/16th period of the oscillations from to 257.84, where (a) shows the profiles of , while (b) and (c) show the ones of over the first half period and the other half, respectively. The arrows indicate the sense of time evolution in the order of the red solid, blue solid, blue broken, and red broken arrows, and the color of the curves corresponds to the one of the arrows in the respective quarter periods. The black curve represents the profiles just at 8/16th period.

Spatial profiles of the disturbances of the density , temperature , and entropy relative to , , and , respectively, in (a), (b), and (c) for the temperature ratio at every 1/16th period of the oscillations from to 257.84. The arrows indicate the sense of time evolution in the order of the red solid, blue solid, blue broken, and red broken arrows, and the color of the curves corresponds to the one of the arrows in the respective quarter periods. The black curve represents the profiles just at 8/16th period.

Spatial profiles of the disturbances of the density , temperature , and entropy relative to , , and , respectively, in (a), (b), and (c) for the temperature ratio at every 1/16th period of the oscillations from to 257.84. The arrows indicate the sense of time evolution in the order of the red solid, blue solid, blue broken, and red broken arrows, and the color of the curves corresponds to the one of the arrows in the respective quarter periods. The black curve represents the profiles just at 8/16th period.

Spatial profiles of the velocity at the edge of the boundary layer and the conductive heat flux into the gas through the side wall relative to and , respectively, in (a) and (b) for the temperature ratio at every 1/16th period of the oscillations from to 257.84. The arrows indicate the sense of time evolution in the order of the red solid, blue solid, blue broken, and red broken arrows, and the color of the curves corresponds to the one of the arrows in the respective quarter periods shown in Figs. 4 and 5. The black curve represents the profiles just at 8/16th period.

Spatial profiles of the velocity at the edge of the boundary layer and the conductive heat flux into the gas through the side wall relative to and , respectively, in (a) and (b) for the temperature ratio at every 1/16th period of the oscillations from to 257.84. The arrows indicate the sense of time evolution in the order of the red solid, blue solid, blue broken, and red broken arrows, and the color of the curves corresponds to the one of the arrows in the respective quarter periods shown in Figs. 4 and 5. The black curve represents the profiles just at 8/16th period.

Fourier spectra of the excess pressure at the closed end and of the axial velocity at the open end drawn in the red and blue curves, respectively, where the highest peaks are located at the frequency , and the smaller peaks occur at their second- and third-harmonic frequencies.

Fourier spectra of the excess pressure at the closed end and of the axial velocity at the open end drawn in the red and blue curves, respectively, where the highest peaks are located at the frequency , and the smaller peaks occur at their second- and third-harmonic frequencies.

Spatial profiles of the amplitudes and relative to and , respectively, calculated by the Fourier spectra of the excess pressure and the axial velocity .

Spatial profiles of the amplitudes and relative to and , respectively, calculated by the Fourier spectra of the excess pressure and the axial velocity .

Profile of the function defined by Eq. (18) where , , and indicate the positions at which takes the values of , , and so that the right-hand sides of Eqs. (21), (26), and (27) may vanish.

Profile of the function defined by Eq. (18) where , , and indicate the positions at which takes the values of , , and so that the right-hand sides of Eqs. (21), (26), and (27) may vanish.

Distribution of the mean acoustic energy flux due to the action of the boundary layer where the solid and broken curves represent, respectively, calculated directly by the numerical solutions and calculated by the formula (21), being and designates a position at which given by Eq. (21) vanishes for .

Distribution of the mean acoustic energy flux due to the action of the boundary layer where the solid and broken curves represent, respectively, calculated directly by the numerical solutions and calculated by the formula (21), being and designates a position at which given by Eq. (21) vanishes for .

Distributions of the mean acoustic energy flux in the main-flow region , the defect in the boundary layer , and the total acoustic energy flux over the whole cross section per unit area, all referenced to , where the solid and broken curves represent, respectively, the fluxes calculated by the numerical solutions and calculated by the formula in terms of the pressure amplitude .

Distributions of the mean acoustic energy flux in the main-flow region , the defect in the boundary layer , and the total acoustic energy flux over the whole cross section per unit area, all referenced to , where the solid and broken curves represent, respectively, the fluxes calculated by the numerical solutions and calculated by the formula in terms of the pressure amplitude .

Distributions of the mean values of the excess enthalpy flux and of the convective heat flux in the main-flow region, and the mean values of the defects in the enthalpy flux and in the convective heat flux in the boundary layer, all fluxes referenced to . Here and designate the points of at which and vanish.

Distributions of the mean values of the excess enthalpy flux and of the convective heat flux in the main-flow region, and the mean values of the defects in the enthalpy flux and in the convective heat flux in the boundary layer, all fluxes referenced to . Here and designate the points of at which and vanish.

Comparison of the mean value of the full excess enthalpy flux in the main-flow region with its approximation and with the mean acoustic energy flux . The difference is also depicted with the mean heat flux , all fluxes referenced to .

Comparison of the mean value of the full excess enthalpy flux in the main-flow region with its approximation and with the mean acoustic energy flux . The difference is also depicted with the mean heat flux , all fluxes referenced to .

Distributions of the mean conductive heat flux from the side wall calculated by in Eq. (29) and by in Eq. (30), referenced to .

Distributions of the mean conductive heat flux from the side wall calculated by in Eq. (29) and by in Eq. (30), referenced to .

Distributions of the mean mass flux in the main-flow region, the mean value of the defect over the boundary layer, and the sum , all referenced to .

Distributions of the mean mass flux in the main-flow region, the mean value of the defect over the boundary layer, and the sum , all referenced to .

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