早期封閉管駐波共振不管是實驗還是數值計算都是研究低頻共振系統。主要是一般
機械傳動所能產生的頻率極限在200Hz 左右。這些機械系統所占空間大, 因此轉向使用
高頻振動系統來縮小系統空間, 且封閉管駐波共振裝置在實驗上, 對於要量測管內物理
量的變化, 以目前的技術來說還無法達成, 因此最佳的選擇便是使用數值模擬來監測管
內所有的物理量的變化。藉由數值模擬來探討非線性帶給內部流場之行為, 因此本文研
究驅動的活塞頻率為4000 Hz 、管長為0.0435 m , 內徑為0.026 m 的半波駐波共振裝
置, 假設三維軸對稱並考慮有黏性之空氣, 以及流場內的能量因壁面絕熱條件無法傳至
大氣, 分別觀察近似共振頻率對非線性流場造成的影響以及倍數成長的共振頻率對非線
性流場造成的影響。

當操作操作頻率接近共振頻率時, 震波的產生越明顯, 因此非線性效應及黏滯效應將
導致波形呈現上下不對稱, 而流場內的壓力和速度將會有相位差存在, 但壓力與溫度則
為同相。流體黏滯性的影響導致邊界上產生速度梯度, 因此邊界層內將有剪應力發生, 將
動能消耗成熱能, 而這些熱量將累積在邊界層內, 導致邊界層上的溫度和速度變化。溫
度邊界層的發展因壁面假設絕熱的關係, 使得溫度發生最大的地方在壁面上。由於模擬
結果得的現像, 將可對於往後熱聲裝置內的疊平板擺放的位置提供一個良好的參考依據
去設計。

Traditionally, the study of standing wave resonance inside a closed tube is
focussed on the low frequency resonance system (generally below 200 Hz), which is limited by the mechanical transmission mechanisms used. In order to reduce the space taken by the mechanical transmission mechanisms, high frequency resonance system is favored. In addition, it is very difficult to measure the variations of physical quantities inside the resonance tube with current technologies available.
However, this presents no difficulties in using numerical simulation to monitor
the variations of physical quantities inside the tube. In this work, the resonance behaviors inside a closed tube (0.0435 m in length, 0.026 m inside diameter) driven by a reciprocating piston at a frequency of 4000 Hz are studied. The tube is assumed to be axis-symmetric and the walls are assumed to be adiabatic. The effects of detuning and frequency multiplication on the nonlinear resonance are investigated.

Closed to the exact resonance, the pressure waveforms are significantly distorted and become asymmetric. Formation of shock waves can be clearly observed. Due to the steep velocity gradient inside the velocity boundary layer, significant shear stresses occurs near the velocity boundary layer and kinetic energy is dissipated and converted into heat. This heat will accumulate inside the velocity boundary layer due to the adiabatic boundary conditions. This, in turn, will lead to the growth of thermal boundary layer and result in a very thick thermal boundary layer.

The results of the simulations can provide the necessary guidelines for the selection of the proper locations of the stack plates inside a thermoacoustic resonance tube.

[1] R. Betchov ”Nonlinear oscillations of a column of gas”, Physics of Fluids ,1,pp.205−212,1958
[2] W. Chester ”Resonant oscillations in closed tubes”, Journal of Fluid Mechan-
ics ,18, pp.44−64,1964
[3] Alan B. Coppens and James V. Sanders”Finite Amplitude Standing Waves
in Rigid Walled Tubes”, Journal Society of America ,43, pp.516−529,1968
[4] A. L. Van Buren ”Mathematical model for non-linear standing waves in a
tube”, Journal of Sound and Vibration 42(3), pp. 273-280,1975
[5] Yurii A. Ilinskii ”Nonlinear standing waves in an acoustical resonator”, Journal of the Acoustic Society of America 104(5),pp. 2664-2674,1998
[6] Y.Chun,Y.Kim ”Numerical analysis for nonlinear resonant oscillations in
closed tubes”, Journal of the Acoustic Society of America 104(5), pp. 2664-
2674,1998
[7] Elvira-Sequra ”A finite element algorithm for the study of nonlinear standing waves”, Journal of the Acoustic Society of America 108(6), pp. 2765-2447,2000
[8] Christian Vanhille ”Numerical model for nonlinear standing waves and weak
shocks in thermoviscous fluids”, Journal of the Acoustic Society of America
109(6), pp. 2660-2677,1998
[9] Ning Fangli ”Two-dimensional Model and Numerical Method for Nonlinear
Standing Waves in Axisymmetric Resonators1”, CMCE, pp. 68-71,2010
[10] Rayleigh, Lord, ”The Theory of Sound”, London : Macmillan,II,2nd end, pp.301, 1901
[11] Pierce, Allan D., ”Acoustics : an introduction to its physical principles and applications ”, New York : McGraw-Hill Book Co, pp. 524, 1981
[12] Pierce, Allan D., ”Acoustics : an introduction to its physical principles and applications ”, New York : McGraw-Hill Book Co, pp. 553, 1981
[13] George Emanuel, ”Bulk viscosity in the Navier Stokes equations”, Interna-
tional Journal of Engineering Science,36, pp. 1313-1323,1998
[14] D.E. Weston, ”The theory of the propagation of plane sound waves in tubes”,
Proceedings of the Physical Society London,66, pp. 695-709,1953