本研究旨在建立一個不可壓縮光滑粒子法(SPH)來模擬跌水效應中的自由液面流動。這個SPH的方法可於GPU平行計算環境中執行。並且以自由液面流經渠道中一水下圓柱體造成的水位變化為例進行計算模擬。在本模式中，Rhie和Chow的內插法、自由液面顏色函數追蹤法以及修正的粒子轉移被引入來增加方法的精確性與效率。在參數研究中，雷諾數150的流場被考慮來檢視福祿數在0.2-0.6範圍、潛置比例(H/D)在0.5-1.0(H 為液面至圓柱表面的距離，D為圓柱直徑)以及間距比(G/D)在1-5( G 為圓柱下表面至底床的距離）對於自由液面流經圓柱的流場型態以及造成的水動力影響。研究中發現圓柱後方的渦旋逸出與這些參數息息相關。當福祿 數增加時，渦旋逸出會被壓抑。但如果 H/D 與 G/D 增加時，壓抑渦旋逸出的臨界福祿數也會增加。平均阻力係數則與 H/D 與 G/D 有關，而與福祿數無關。根據水動力與自由液面高度的頻譜分析結果發現自由液面的變形與所造成的升力在有渦旋逸出的情況下息息相關

An incompressible SPH (smoothed particle hydrodynamics) method capable simulating viscous free-surface flow with open boundaries is developed. The present SPH method is implemented for parallel computing in GPU and applied to model flow past a circular cylinder located between free-surface and wall boundaries. A number of enhancements including Rhie and Chow interpolation, non-uniform particle size, colour function-based free surface tracking, and modified particle shifting are introduced to increase the accuracy and efficiency of the method. A parameter study at the Reynolds number, Re =150, examines the influence of Froude number (Fr = 0.2-0.6$), submergence ratio (H/D=0.5-1.0 where H is the distance between the apex of the cylinder and undisturbed free surface and D is the cylinder diameter) and bottom gap ratio (G/D=1.0-5.0 where G is the distance between the base of the cylinder and the bed) on the ambient free-surface flow pattern and hydrodynamic force on the cylinder. It is found that the vortex shedding pattern depends on all three parameters. As the Froude number increases for fixed submergence ratio and bottom gap ratio, vortex shedding is eventually suppressed. As the submergence and bottom gap ratios increase, the threshold of vortex shedding suppression shifts to higher values of Froude number. The mean drag coefficient depends on the submergence ratio and bottom gap ratio but is independent of Froude number. Meanwhile, the lift coefficient depends on submergence ratio and Froude number but is independent of bottom gap ratio. Spectral analysis of force and free-surface elevation time signals shows that the free-surface deformation and lift force are closely related during the vortex shedding regime.

[1] NVIDIA Corporation (2009) NVIDIA’s Next Generation CUDA Compute Architecture:
Fermi. Tech. Rep., NVIDIA Corporation
[2] Williamson CHK (1996) Vortex dynamics in the cylinder wake. Annual Review of Fluid Mechanics 28:477–539
[3] Zdravkovich MM (1997) Flow Around Circular Cylinders Vol 1: Fundamentals. Oxford
University Press, New York
[4] Price SJ, Sumner D, Smith JG, Leong K, Pa¨ıdoussis MP (2002) Flow visualization
around a circular cylinder near to a plane wall. Journal of Fluids and Structures 16:175–191
[5] Sumer BM, Fredsøe J (2006) Hydrodynamics around cylindrical structures. World Scientific, Singapore
[6] Sheridan J, Lin JC, Rockwell D (1997) Flow past a cylinder close to a free surface. Journal of Fluid Mechanics 330:1–30
[7] Reichl P, Hourigan K, Thompson M (2005) Flow past a cylinder close to a free
surface. Journal of Fluid Mechanics 533:269–296
[8] Bearman PW, Zdravkovich MM (1978) Flow around a circular cylinder near a plane
boundary. Journal of Fluid Mechanics 89:33–47
[9] Sarkar S, Sarkar S (2009) Large-Eddy Simulation of Wake and Boundary Layer Interactions Behind a Circular Cylinder. Journal of Fluids Engineering 131:091201
[10] Dipankar A, Sengupta T (2005) Flow past a circular cylinder in the vicinity of a
plane wall. Journal of Fluids and Structures 20:403–423
[11] He GS, Wang JJ, Pan C, Feng LH, Gao Q, Rinoshika A (2017) Vortex dynamics
for flow over a circular cylinder in proximity to a wall. Journal of Fluid Mechanics 812:698–720
[12] Buresti G, Lanciotti A (1992) Mean and fluctuating forces on a circular cylinder in cross-flow near a plane surface. Journal of Wind Engineering and Industrial Aerodynamics 41:639–650
[13] Lei C, Cheng L, Kavanagh K (1999) Re-examination of the effect of a plane boundary on force and vortex shedding of a circular cylinder. Journal of Wind Engineering and Industrial Aerodynamics 80:263–286
[14] Zdravkovich MM (1985) Forces on a circular cylinder near a plane wall. Applied
Ocean Research 7:197–201
[15] Chiew Y (1991) Flow Around Horizontal Circular Cylinder in Shallow Flows. Journal of Waterway, Port, Coastal, and Ocean Engineering 117:120–135
[16] Chu CR, Lin YA,Wu TR,Wang CY (2018) Hydrodynamic force of a circular cylinder
close to the water surface. Computers and Fluids 171:154–165
[17] Gingold RA, Monaghan JJ (1977) Smoothed particle hydrodynamics-theory and application to non-spherical stars. Monthly Notices of the Royal Astronomical Society
181:375–389
[18] Lucy LB (1977) A numerical approach to the testing of the fission hypothesis. The Astronomical Journal 82:1013–1024
[19] Cummins SJ, Rudman M (1999) An SPH Projection Method. Journal of Computational
Physics 152:584–607
[20] Chorin AJ (1968) Numerical Solution of the Navier-Stokes Equations. Mathematics
of Computation 22:745–762
[21] Shao S, Ji C, Graham DI, Reeve DE, James PW, Chadwick AJ (2006) Simulation of
wave overtopping by an incompressible SPH model. Coastal Engineering 53:723–735
[22] Ataie-Ashtiani B, Shobeyri G, Farhadi L (2008)Modified incompressible SPH method
for simulating free surface problems. Fluid Dynamics Research 40:637–661
[23] Gotoh H, Khayyer A, Ikari H, Arikawa T, Shimosako K (2014) On enhancement of
Incompressible SPH method for simulation of violent sloshing flows. Applied Ocean
Research 46:104–115
[24] Liang D, Jian W, Shao S, Chen R, Yang K (2017) Incompressible SPH simulation of solitary wave interaction with movable seawalls. Journal of Fluids and Structures
69:72–88
[25] Marrone S, Antuono M, Colagrossi a, Colicchio G, Le Touz´e D, Graziani G (2011) delta-SPH model for simulating violent impact flows. Computer Methods in Applied
Mechanics and Engineering 200:1526–1542
[26] Vacondio R, Rogers BD, Stansby PK, Mignosa P, Feldman J (2013) Variable resolution for SPH : A dynamic particle coalescing and splitting scheme. Computer
Methods in Applied Mechanics and Engineering 256:132–148
[27] Marrone S, Colagrossi a, Antuono M, Colicchio G, Graziani G (2013) An accurate
SPH modeling of viscous flows around bodies at low and moderate Reynolds numbers.
Journal of Computational Physics 245:456–475
[28] Sefid M, Fatehi R, Shamsoddini R (2016) A Modified Smoothed Particle Hydrodynamics Scheme to Model the Stationary and Moving Boundary Problems for Newtonian Fluid Flows. Journal of Fluids Engineering 137:031201
[29] Gholami M, Galindo-torres SA, Scheuermann A, Williams DJ (2017) Parametric study on smoothed particle hydrodynamics for accurate determination of drag coefficient
for a circular cylinder. Water Science and Engineering 10:143–153
[30] Bouscasse B, Colagrossi A, Marrone S, Souto-Iglesias A (2017) SPH modelling of
viscous flow past a circular cylinder interacting with a free surface. Computers &
Fluids 146:190–212
[31] Colagrossi A, Nikolov G, Durante D, Marrone S, Souto-Iglesias A (2019) Viscous
flow past a cylinder close to a free surface: Benchmarks with steady, periodic and
metastable responses, solved by meshfree and mesh-based schemes. Computers and
Fluids 181:345–363
[32] Shao S, Lo EYM (2003) Incompressible SPH method for simulating Newtonian and
non-Newtonian flows with a free surface. Advances in Water Resources 26:787–800
[33] Rhie CM, Chow WL (2008) Numerical study of the turbulent flow past an airfoil
with trailing edge separation. AIAA Journal 21:1525–1532
[34] Liu GR, Liu MB (2010) Smoothed Particle Hydrodynamics - A Meshfree Particle
Method. World Scientific Publishing Co. Pte. Ltd.
[35] Violeau D (2012) Fluid Mechanics and the SPH Method: Theory and Applications,
vol. 9780199655. Oxford University Press, 1–616
[36] Bonet J, Lok TS (1999) Variational and momentum preservation aspects of Smooth
Particle Hydrodynamic formulations. Computer Methods in Applied Mechanics and
Engineering 180:97–115
[37] Chen JK, Beraun JE, Carney TC (1999) A corrective smoothed particle method for
boundary value problems in heat conduction. International Journal for Numerical
Methods in Engineering 46:231–252
[38] Chen J, Beraun J (2000) A generalized smoothed particle hydrodynamics method
for nonlinear dynamic problems. Computer Methods in Applied Mechanics and Engineering
190:225–239
[39] Oger G, Doring M, Alessandrini B, Ferrant P (2007) An improved SPH method:
Towards higher order convergence. Journal of Computational Physics 225:1472–1492
[40] Schwaiger HF (2008) An implicit corrected SPH formulation for thermal diffusion with linear free surface boundary conditions. International Journal for Numerical
Methods in Engineering 75:647–671
[41] Zhang GM, Batra RC (2009) Symmetric smoothed particle hydrodynamics (SSPH)
method and its application to elastic problems. Comput Mech 43:321–340
[42] Stranex T, Wheaton S (2011) A new corrective scheme for SPH. Computer Methods
in Applied Mechanics and Engineering 200:392–402
[43] Lee ES, Moulinec C, Xu R, Violeau D (2008) Comparisons of weakly compressible
and truly incompressible algorithms for the SPH mesh free particle method. Journal
of Computational Physics 227:8417–8436
[44] Kazemi E, Nichols A, Tait S, Shao S (2017) SPH modelling of depth-limited turbulent open channel flows over rough boundaries. International Journal for Numerical Methods in Fluids 83:3–27
[45] Tan SK, Cheng NS, Xie Y, Shao S (2015) Incompressible SPH simulation of open
channel flow over smooth bed. Journal of Hydro-Environment Research 9:340–353
[46] Lastiwka M, Basa M, Quinlan NJ (2009) Permeable and non-reflecting boundary conditions in SPH. International Journal for Numerical Methods in Fluids 61:709–
724
[47] Fu L, Jin YC (2013) A mesh-free method boundary condition technique in open
channel flow simulation. Journal of Hydraulic Research 51:174–185
[48] Randles PW, Libersky LD (1996) Smoothed Particle Hydrodynamics: Some recent
improvements and applications. Computer Methods in Applied Mechanics and Engineering
139:375–408
[49] Dilts GA (2000) Moving least-squares particle hydrodynamics II: conservation and
boundaries. International Journal for Numerical Methods in Engineering 48:1503–
1524
[50] Marrone S, Colagrossi A, Le Touz´e D, Graziani G (2010) Fast free-surface detection and level-set function definition in SPH solvers. Journal of Computational Physics 229:3652–3663
[51] Mayer S, Garapon A, Sørensen LS (1998) A fractional step method for unsteady free-surface flow with applications to non-linear wave dynamics. International Journal for Numerical Methods in Fluids 28:293–315
[52] Fuhrman DR, Madsen PA, Bingham HB (2006) Numerical simulation of lowest-order
short-crested wave instabilities. Journal of Fluid Mechanics 563:415–441
[53] Federico I, Marrone S, Colagrossi A, Aristodemo F, Antuono M (2012) Simulating 2D open-channel flows through an SPH model. European Journal of Mechanics, B/Fluids
34:35–46
[54] Feldman J, Bonet J (2007) Dynamic refinement and boundary contact forces in SPH
with applications in fluid flow problems. International Journal of Numerical Methods
in Engineering 72:295–324
[55] Spreng F, Schnabel D, Mueller A, Eberhard P (2014) A local adaptive discretization algorithm for Smoothed Particle Hydrodynamics. Computational Particle Mechanics 1:131–145
[56] Vacondio R, Rogers BD, Stansby PK, Mignosa P (2013) Shallow water SPH for flooding with dynamic particle coalescing and splitting. Advances inWater Resources
58:10–23
[57] Barcarolo DA, Le Touz´e D, Oger G, De Vuyst F (2014) Adaptive particle refinement and derefinement applied to the smoothed particle hydrodynamics method. Journal of Computational Physics 273:640–657
[58] Chiron L, Oger G, de Leffe M, Le Touz´e D (2018) Analysis and improvements of
Adaptive Particle Refinement (APR) through CPU time, accuracy and robustness
considerations. Journal of Computational Physics 354:552–575
[59] Reyes L´opez Y, Roose D, Recarey Morfa C (2013) Dynamic particle refinement in
SPH: Application to free surface flow and non-cohesive soil simulations. Computational Mechanics 51:731–741
[60] Martin DF, Colella P, Graves D (2008) A cell-centered adaptive projection method
for the incompressible Navier-Stokes equations in three dimensions. Journal of Computational Physics 227:1863–1886
[61] Hu XY, Adams NA (2007) An incompressible multi-phase SPH method. Journal of Computational Physics 227:264–278
[62] Lo EYM, Shao S (2002) Simulation of near-shore solitary wave mechanics by an
incompressible SPH method. Applied Ocean Research 24:275–286
[63] Pozorski J, Wawrenczuk A (2002) SPH computation of incompressible viscous flows.
Journal of Theoretical and Applied Mechanics 40:917–937
[64] Aly AM (2015) Modeling of multi-phase flows and natural convection in a square cavity using an incompressible smoothed particle hydrodynamics. International Journal
of Numerical Methods for Heat & Fluid Flow 25:513–533
[65] Gui Q, Dong P, Shao S (2015) Numerical study of PPE source term errors in the
incompressible SPH models. International Journal for Numerical Methods in Fluids
77:358–379
[66] Xu R, Stansby P, Laurence D (2009) Accuracy and stability in incompressible SPH
(ISPH) based on the projection method and a new approach. Journal of Computational
Physics 228:6703–6725
[67] Lind SJ, Xu R, Stansby PK, Rogers B (2012) Incompressible smoothed particle hydrodynamics for free-surface flows: A generalised diffusion-based algorithm for
stability and validations for impulsive flows and propagating waves. Journal of Computational Physics 231:1499–1523
[68] Skillen A, Lind S, Stansby PK, Rogers BD (2013) Incompressible smoothed particle
hydrodynamics (SPH) with reduced temporal noise and generalised Fickian
smoothing applied to body–water slam and efficient wave–body interaction. Computer
Methods in Applied Mechanics and Engineering 265:163–173
[69] Tsuruta N, Khayyer A, Gotoh H (2015) Space potential particles to enhance the stability of projection-based particle methods. International Journal of Computational Fluid Dynamics 29:100–119
[70] Sun PN, Colagrossi A, Marrone S, Zhang AM (2017) The plus-SPH model: Simple
procedures for a further improvement of the SPH scheme. Computer Methods in
Applied Mechanics and Engineering 315:25–49
[71] Chow AD, Rogers BD, Lind SJ, Stansby PK (2018) Incompressible SPH (ISPH) with
fast Poisson solver on a GPU. Computer Physics Communications 226:81–103
[72] Harris M, Sengupta S, Owens JD (2007) Parallel prefix sum(scan) with CUDA. In:
Nguyen H (ed.) GPU Gems3, Addison Wesley, chap. 39, 851–876
[73] Kovasznay LSG (1949) Hot-wire investigation of the wake behind cylinders at low
Reynolds numbers. Proceedings of the Royal Society of London Series A Mathematical
and Physical Sciences 198:174–190
[74] Nishioka M, Sato H (1974) Measurements of velocity distributions in the wake of a circular cylinder at low Reynolds numbers. Journal of Fluid Mechanics 65:97–112
[75] Grove AS, Shair FH, Petersen EE, Acrivos A (1964) An experimental investigation
of the steady separated flow past a circular cylinder. Journal of Fluid Mechanics
19:60–80
[76] Reichl P (2001) Flow past a cylinder close to a free surface. Ph.D. thesis, Monash University
[77] Williamson CHK, Roshko A (1988) Vortex formation in the wake of an oscillating
cylinder. Journal of Fluids and Structures 2:355–381