本研究應用具有虛擬力的直接施力沈浸邊界法(Direct Forcing Immersed Boundary Method, DFIB)來模擬均勻流場中的圓球在一定雷諾數範圍內受渦漩引致振動(Vortex Induced Vibration, VIV)的現象。因渦漩引致振動所產生的水動力,在流場中對固體結 構物所造成的擾動力量可能會造成結構損害,但從另一方面來看,也可能成為再生流 動能源轉換的一個方式。本研究主要分析一可在均勻流的橫向方向運動的圓球前有一個固定的小圓球的振動行為。在模擬結果中,小球的直徑及兩球間的間距為本文對水 動力係數、渦漩振動以及能源效率為主要的研究因素。除此之外,此研究也將描述如 何預測小圓球的影響。本研究中發現,固定小圓球會提高的球下游處的升力係數及振幅比且阻力係數會逐漸減弱。在過程中所建立的數值模式以靜止下的球在雷諾數50的流場以及雷諾數300下球的振動現象的已發表的期刊文章進行比較與驗證。這個研究也顯示了直接施力沈浸邊界法對於此圓球在流場中的振動行為有良好的分析預測能力

A direct-forcing immersed boundary (DFIB) method with virtual force is used to investigate the vortex-induced vibration (VIV) of an elastically mounted sphere in uniform flow at a moderate Reynolds number. Fluctuating hydrodynamic forces may cause damage when a solid structure interacts with fluid flow. Conversely, such forces may be exploited as a renewable energy resource. In this study, a smaller fixed sphere placed in front of an elastically mounted sphere to investigate the effect that is given. Variation in the diameters of the small sphere and the gap between both spheres set as a parameter of the smaller fixed sphere. The effects of the small fixed sphere on the hydrodynamic force coefficients, sphere responses, vortex shedding modes and energy efficiency are discussed. In addition, an explanation of how to estimate the effects of the small sphere is also described in this study. The existence of smaller fixed spheres in front of the mounted sphere increases value of lift coefficient and amplitude ratio of the downstream sphere. However, the value of drag coefficient in downstream sphere becomes smaller. The method is validated by comparisons with previously published results for flow past a sphere at Re = 50 for stationary sphere and Re = 300 for VIV phenomena. This study proves the capability of the DFIB model for investigation of VIV on a structure and offers an analysis of the effects of a smaller upstream sphere on VIV phenomena of an elastically mounted sphere.

[1] Sarpkaya, T., 1979. Vortex-induced oscillations a selective review. Journal of Applied Mechanics. 46, 241-258.
[2] Bearman, P. W., 1984. Vortex shedding from oscillating bluff bodies. Annual Review of Fluid Mechanics. 16, 195-222.
[3] Parkinson, G., 1989. Phenomena and modeling of flow-induced vibrations of bluff
bodies. Progress in Aerospace Sciences. 26, 169-224.
[4] Williamson, C.H.K., Govardhan, R., 2004. Vortex induced vibrations. Annual Review of Fluid Mechanics. 36, 413-455.
[5] Govardhan, R.N., Williamson, C.H.K., 2005. Vortex-induced vibrations of a sphere. Journal of Fluid Mechanics. 531, 11-47.
[6] Willden, R.H.J., Graham, J.M.R., 2001. Numerical predictions of VIV on long flexible circular cylinders. Journal of Fluids and Structures. 15, 659-669.
[7] Yamamoto, C.T., Meneghini, J.R., Saltara, F., Fregonesi, R.A., Ferrari Jr, J.A., 2004. Numerical simulations of vortex-induced vibration on flexible cylinders. Journal of Fluids and Structures. 19, 467-489.
[8] Violette, R., de Langre, E., Szydlowski, J., 2007. Computation of vortex-induced
vibrations of long structures using a sake oscillator model: comparison with DNS
and experiments. Computational Structure. 85, 1134-1141.
[9] Borazjani, I., Sotiropoulos, F., 2009. Vortex-induced vibrations of two cylinders to tandem arrangement in the proximity-wake interface region. Journal of Fluid Mechanics. 621, 321-364.
[10] Bourguet, R., Karniadakis, G.E., Traintafyllou, M.S., 2011. Vortex-induced vibrations of a long flexible cylinder in shear flow. Journal of Fluid Mechanics. 677, 342-382.
[11] Sarpkaya, T., 2004. A critical review of the intrinsic nature of vortex-induced vibrations. Journal of Fluids and Structures. 19, 389-447.
[12] Assi, G.R.S., Meneghini, J.R., Aranha, J.A.P., Bearman, P.W., Casaprima, E., 2006. Experimental investigation of flow-induced vibration interference between two circular cylinders. Journal of Fluids and Structures. 22, 819-827.
[13] Alam, Md.M., Moriya, M., Takai, K., Sakamoto H., 2003. Fluctuating fluid forces
acting on two circular cylinders in a tandem arrangement at a sub-critical Reynolds
number. Journal of Wind Engineering and Industrial Aerodynamics. 91, 139-154.
[14] Mysa, R.C., Kaboudian, A., Jaiman, R.K., 2016. On the origin of wake-induced
vibration in two tandemcircular cylinders at low Reynolds number. Journal of Fluids
and Structures. 61, 76-98.
[15] Behara, S., Borazjani, I., Sotiropoulos, F., 2011. Vortex-induced vibration of an elastically mounted sphere with three degrees of freedom at Re=300:hysteresis and
vortex shedding modes. Journal of Fluid Mechanics. 686, 426-450.
[16] Behara, S., Sotiropoulos, F., 2016. Vortex-induced vibration of an elastically mounted sphere the effects of reynolds number and reduced velocity. Journal of Fluid Mechanics. 66, 54-68.
[17] Rajamuni, M.M., Thompson, M.C., Hourigan, K., 2018. Transverse flow-induced
vibrations of a sphere. Journal of Fluid Mechanic. 837, 931-966.
[18] Rajamuni, M.M., Thompson, M.C., Hourigan, K., 2019. Vortex-induced vibration
of elastically-mounted spheres: A comparison of the response of three degrees of
freedom and one degree of freedom systems. Journal of Fluid and Structures.
[19] Peskin, C.S., 1972. Flow patterns around heart valve: a numerical method. Journal of Computational Physics. 10, pp. 252-271.
[20] Yusof, J.M., 1996. Interaction of massive particles with turbulence. PhD. Dissertation, Dept. of Mechanical and Aerospace Engineering, Cornell Univ.
[21] Noor, D. Z., Chern, M. J. and Horng, T. L., 2009. An immersed boundary method
to solve fluid-solid interaction problems. Computational Mechanic. 44, 447-453.
[22] Chern, M. J., Noor, D. Z., Liao, C. B., and Horng, T. L. Direct-Forcing Immersed Boundary Method for Mixed Heat Transfer. Commun. Comput. Phys., 2015. 18,
1072-1094.
[23] Chern, M. J., Hsu, W. C., Horng, T. L., 2012. Numerical prediction of hydrodynamic loading on circular cylinder array in oscillatory flow using direct-forcing immersed boundary method. Journal of Applied Mathematics. 531, 11-47.
[24] Leonard, B.P., 1979. Astable and accurate convective modelling procedure based
on quadratic upstream interpolation. Computer Method in Applied Mechanic and
Engineering 19, 59-98.
[25] Tomboulides, A. G., Orszag, S. A., 1993. Karniadakis, G. E. Direct and large-eddy simulation of axisymmetric wakes. AIAA Journal. 93-0546.
[26] Taneda, S., 1956. Experimental investigation of the wake behind a sphere at low
Reynolds numbers. Journal of Physics Society Japan. 11, 1104-1108.
[27] Roos, F. W., Willmarth, W. W., 1971. Some experimental results on sphere and disk drag. AIAA Journal. 9, 285-291.
[28] Johnson, T.A and Patel, V. C., 1999. Flow past a sphere up to a Reynolds number
of 300. Journal of Fluid Mechanics. 378, 19-70.
[29] Hunt, J.C.R., Wray, A.A., Moin, P., 1988. Eddies, streams, and convergence zones in turbulent flows. In: Proceedings of the 1988 Summer Program on Studying Turbulence Using Numerical Simulation Databases (SEE N89-24538 18-34), vol. 2, pp. 193-208.
[30] Feng, C. C., 1968. The measurement of vortex-induced effects in flow past stationary and oscillating circular and D-section cylinders. Master’s thesis, University of British Columbia, Vancouver, BC, Canada
[31] Assi, G.R.S., Bearman, P.W., Meneghini, J., 2010. On the wake-induced vibration
of tandem circular cylinders: the vortex interaction excitation mechanism. Journal
of Fluid Mechanics. 661, 365-401.
[32] Assi, G.R.S., 2014. Wake-induced vibration of tandem cylinders of different diameters. Journal of Fluids and Structures. 50, 329-339.