管壁對圓管中落球所引生流場之效應研究

Boundary-wall effect on the flow generated by a sphere falling in a pipe

李信宏(Hsin-Hung Lee)

指導教授 ： 伍次寅

國立臺灣大學/工學院/機械工程學研究所/博士(2006年)

https://doi.org/10.6342/NTU.2006.00557

摘要

本文主要在於探討軸對稱管內不可壓縮定常流流場的不穩定性與流場物理現象。流場的形式為管中央的軸中心線上置有一圓球，並以軸向速度為1當作管入口與壁面的邊界條件。此一物理模型除了可探討圓球垂直落入管中受壁面效應所造成的影響外，在不考慮重力項的情形下，本文之模式即等同於一般數值計算中，以有限域來模擬無窮域均勻自由流流經圓球的情形。而改變管球徑比(D/d)的大小即可探討數值模擬中將無限域截斷成有限域所造成的誤差影響。在數值方法上，本文採用虛擬壓縮性法(pseudo-compressibility)並配合有限體積法(finite volume formulation)來計算不可壓縮流場之奈維爾-史托克(Navier-Stokes)方程式的數值解。在空間離散方面採用TVD雙曲型守恆率之高解析算則，時間離散方面則採用LUSSOR隱式解法以求得穩定的收斂值。
在流場穩定性的分析方面，本文採用線性穩定(linear stability analysis)的分析方法，並配合ARPACK程式庫以求解特徵方程組的領導特徵值，以藉此判別流場的穩定與否。計算結果顯示隨著雷諾數的增加，流場首次不穩定是發生在對稱破壞分歧點(symmetry-breaking bifurcation point)上，而在D/d=20（模擬無窮域）下臨界雷諾數為209。接著在管徑逐漸縮小的情況下，會造成臨界雷諾數(critical Reynolds number)逐次的增加並在管球徑比(D/d)等於3時達到最高，但當小於3時臨界雷諾數則會急速的降低。
另外在流場及圓球阻力的計算上，本文考慮了8種不同的管球徑比與4種不同的雷諾數情形下的流場，並採用 、 、 三種不同的網格數目，而後利用Richardson外插法得到一較高階準確的圓球阻力數值結果。此結果將與他人的實驗及無窮域模擬作比較，最後並利用計算所得之結果，經曲線擬合推導出一修正後的阻力、管球徑比及雷諾數的相關方程式。而此式不但可以分析在有限管壁下圓球的受力情形,甚至在考慮無窮域流場的情形時,亦具有相當程度的參考價值。

關鍵字

圓球 ； 管壁效應 ； 穩定性 ； 阻力

並列摘要

This work is to study the instability and the physical phenomenon for the falling sphere problem (a steady, axisymmetric, uniform flow of Newtonian fluid passing an axially-located sphere in a pipe with a moving wall). For the sake of convenience in the numerical simulation, the coordinate is set on the sphere and applies the uniform flow as the inlet velocity, the no-slip condition on the fixed sphere and moving wall with the same velocity as well as the inlet axial velocity to be the boundary conditions to simulate the problem. The boundary configuration has two objectives. The first is to study the drag on the sphere, wake length, wake width and the flow stability affected by the wall when a sphere falls into the pipe. The second is to simulate the infinite fluid passing a sphere without considering the gravity in the different numerical finite domain, and examined the truncation effect in the numerical calculation.
For the flow stability, the linear stability analysis is applied to determine the critical Reynolds number for each pipe-to-sphere diameter ratio (D/d). The finite volume method with the TVD strategy and the LUSSOR implicit scheme are adopted to solve the incompressible Navier-Stokes equations with artificial compressibility to calculate the base flow solutions and examine the flow instability to three-dimensional modal perturbations. Finally, the ARPACK package is utilized to obtain the leading eigenvalue of the resulting perturbation eigenvalue problem. The numerical results reveal that the critical Reynolds number increases gradually with the decreasing diameter ratio (D/d), but drops suddenly when D/d<3. This phenomenon is found to be related to the large pressure gradient behind the sphere and rapid pressure drop of a global minimum-pressure ring in the wake. Additionally, the bifurcation condition and the critical Reynolds number in large diameter ratios (D/d 10) are found to be consistent with the results of Natarajan and Acrivos (1993), who investigated the stability of the flow passing a sphere in an unbounded domain.
For the computation of the flow field, four different Reynolds number ranging from 50 to 200 and eight different diameter ratios (D/d=1.5~20) are selected. The results show that the wake length would vary from monotonically decreasing to asymptotically decreasing when Reynolds number exceeds 100 and diameter ratio (D/d) below 5, but the wake width still remains the tendency of monotonic decrease. Finally, a least squared regression technique is applied to collapse the calculated results into a single expression exhibiting the functional relationship between the drag coefficient, Reynolds number and the diameter ratio.

並列關鍵字

sphere ； wall effect ； stability ； drag

參考文獻

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