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

• Main Authors: Hsin-Hung Lee, 李信宏
• Other Authors: 伍次寅
• Format: Others
• Language: zh-TW
• Published: 2005
• Online Access: http://ndltd.ncl.edu.tw/handle/45500314983360975650

博士 === 國立臺灣大學 === 機械工程學研究所 === 94 === 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.