次臨界條件下振動柱體引致尾流流場響應之研究

Abstract

本研究使用二維寬頻元素法的 Navier-Stokes 方程式解題器，研究次臨界流場振動柱體後方尾流的響應現象。本研究主要分為三部份： 第一部份是開放流場振動圓柱後方尾流的響應現象。Reynolds數範圍從10至45，Roshko數範圍從0.1至6。對於微小擾動的頻率響應，本研提出一個半經驗Landau模型配合非線性迴歸，可有效率地得到Landau常數、強制項、共振頻等資訊。結果顯示，次臨界尾流的共振頻與流速呈現良好的線性關係。而量測位置對於尾流共振頻的影響顯示，共振頻隨量測位置不同而改變，於較下游的位置可以得到較低的共振頻，因此本研究提出次臨界條件下，利用不同量測位置的共振頻以預測臨界條件的新方法。同時尾流壓力與速度有一致的響應現象，本研究因此提出一個新穎的利用壓力頻率響應的壓力型流速計設計概念。 第二部份是開放流場振動橢圓柱後方尾流的響應現象，研究柱體長寬比對後方尾流響應的影響。長寬比範圍從0.01至2 ，Reynolds數範圍從15至60，Roshko數範圍從0.5至8。結果顯示，長寬比對於尾流共振頻影響較小，從阻力與尾流結構，本研究提出等效柱體的概念解釋了外部流場與共振頻的相似性。然而長寬比對於尾流振幅影響較大，隨長寬比越小，尾流振幅越大，訊號差異可達5倍以上。而量測位置對於尾流共振頻的影響顯示，尾流存在一個最佳量測位置可獲得最大響應訊號，且速度與壓力響應的比值隨量測距離呈現良好的線性關係。對於不同形狀橢圓柱，本研究分別使用速度與壓力不同位置的響應預測臨界條件，結果顯示誤差皆在6%以內。本部份最後以縮簡數統一不同橢圓柱流場的結果，提出一條次臨界開放流場振動橢圓柱尾流的泛用共振關係式。 第三部份是渠道流場振動圓柱後方尾流的響應現象，研究渠道阻塞比對後方尾流響應的影響。阻塞比範圍從1/2至1/12 ，Reynolds數範圍從10至80，Roshko數範圍從0.5至40。結果顯示，阻塞比對於尾流頻率響應影響很大，隨阻塞比越大，尾流共振頻越高、頻寬越寬、振幅則越小，訊號差異可達5倍以上。本部份最後以縮簡數統一不同渠道流場的結果，提出一條次臨界渠道流場振動圓柱尾流的泛用共振關係式。 最後，本研究進一步將開放流場與渠道流場的結果統一，提出一條適用不同柱體形狀和渠道或開放流場的次臨界尾流泛用共振關係式。對於超臨界流，本研究亦以縮簡數的概念呈現柱體長寬比與渠道阻塞比對渦旋洩離頻率的影響。綜合次臨界流與超臨界流的結果，本研究提供了設計渦旋流速計或流體混合器時，應用不同振動柱體形狀與不同阻塞比渠流的參考。

The resonant phenomena of the flow passing a vibrating cylinder in the subcritical regime are numerically investigated by using a two-dimensional Navier-Stokes equation-solver, and implementing the spectral element method. This study mainly contains three parts: In the first part, the resonant phenomena of the unconfined flow passing a vibrating circular cylinder in the subcritical regime are investigated. Both velocity and pressure responses are computed for Reynolds numbers, Re, between 10 and 45, and exciting Roshko numbers, between 0.1 and 6. For small vibrating amplitudes, a simplified Stuart-Landau model with a forcing term is used to describe the resonant behavior, and a four-parameter best-fit model is proposed for the regression of the Landau’s constants. This demonstrates that the proposed model is suitable for describing the flow in the subcritical regime and the linearity of the resonant Roshko number, Ro, with the Reynolds number can be further extended to a Reynolds number as low as 10. As a consequence, the critical Reynolds number of vortex shedding for the flow around a stationary cylinder can also be determined accurately and efficiently from this linear relationship. A study of probe locations reveals that the lower the Reynolds number is, the larger the deviation of resonant frequencies at various locations. Since the effect of probe locations is significant, a novel method to determine the critical condition is proposed by finding the intersection of the linear Ro-Re relationships from any two different locations. Since pressure and velocity have almost the same Ro-Re relationship, a new device for measuring free stream velocity in the regime of a very low Reynolds number has been proposed. In the second part, the resonant phenomena of the unconfined flow passing a vibrating elliptical cylinder in the subcritical regime are investigated. The effects of changing axis ratio of the cylinder on the resonant phenomenon are studied. The axis ratios range from Ar = 0.01 to Ar = 2.0, Reynolds numbers range from 15 to 60, and the Roshko numbers range from 0.5 to 8. The investigations of the drag coefficients and the wake streamlines indicate that the axis ratio of the cylinder has a minor effect on the resonant frequency, Ro. However, the axis ratio of the cylinder is found to have a prominent effect on the resonant amplitude; namely, the smaller the axis ratio, the stronger the occurrence of resonant amplitude. The investigation of resonant responses and the probe locations shows that the ratio of velocity and pressure responses poses a great linear relationship against the probe distance behind the vibrating cylinder. Moreover, the resonant method to predict the critical conditions is examined and verified for various elliptical cylinder flows. Based on the critical values found, a reduced Reynolds number and a reduced Roshko number are proposed to unify the various linear relationships resulting from various elliptical cylinder flows. The result indicates that the effect of axis ratio can be stripped off in the reduced plane, which may be applied to a more generalized cylinder shape. In the third part, the resonant phenomena of the channel flow passing a vibrating cylinder with various blockage ratios Br from 1/2 to 1/12 are investigated. Reynolds numbers range from 10 to 80, and the Roshko numbers range from 0.5 to 40. Since both velocity and pressure responses are crucial to a flow meter design, it is a given that special emphasis is made on the velocity and pressure responses. The blockage ratio of the channel is found to have a considerable effect on the resonant amplitude, the response bandwidth, and the resonant frequency. The higher blockage ratio resulted in a higher resonant frequency and wider response bandwidth, but lower resonant amplitude. Furthermore, the resonant Ro-Re relationships for the flow with various blockage ratios are investigated. The concept of the reduced numbers is applied to unify these relationships. Finally, a universal resonant relationship in the subcritical regime is presented regardless of the axis ratio of the cylinder and the blockage ratio of the channel. The result may lead to applications, such as flow meters and flow mixers in low Reynolds number flows before the onset of vortex shedding.