Abstract
When numerically integrating the equation describing the motion of a particle in a carrier fluid, the computation of the Basset (history) force becomes by far the most expensive and cumbersome, as opposed to forces such as drag, virtual mass, lift, buoyancy and Magnus. The expression representing the Basset force constitutes an integro-differential term whose standard integrand is singular when the upper integration limit is enforced. These shortcomings have led some researchers to either disregard or outright neglect the contribution of the Basset force to the total force, even in those cases where it may yield to important errors in the determination of particle trajectories in the computation of sediment transport and other environmental flows. This work is devoted to review four recent contributions associated with the computation of the Basset force, and to compare their proposals to diminish the inherent problems of the term integration. All papers, except one, use variants of a window-based approach; the most recent contribution, in turn, employs a specialized quadrature to increase the accuracy of the computation. An analysis was carried out to compare CPU computation times, rates of convergence and accuracy of the approximations versus a known analytical solution. All methods provide sound solutions to the issues associated with the computation of the Basset force; further, a road map to select the best solution for each given problem is provided. Finally, we discuss the implications of the techniques for the simulation of sediment transport processes and other environmental flows.
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References
Amador A, Sanchez-Juny M, Dolz J (2006) Characterization of the nonaerated flow region in a stepped spillway by PIV. J Fluids Eng 128(6):1266–1273
Armenio V, Fiorotto V (2001) The importance of the forces acting on particles in turbulent flows. Phys Fluids 13(8):2437–2440
Ali BA, Pushpavanam S (2011) Analysis of unsteady gas-liquid flows in a rectangular tank: comparison of Euler-Eulerian and Euler-Lagrangian simulations. Int J Multiph Flow 37:268–277
Bombardelli FA, González AE, Niño YI (2008) Computation of the particle Basset force with a fractional-derivative approach. J Hydraul Eng ASCE 134(10):1513–1520
Bombardelli FA, Jha S (2009) Hierarchical modeling of the dilute transport of suspended sediment in open channels. Environ Fluid Mech 9(2):207–235. doi:10.1007/s10652-008-9091-6
Bombardelli FA, Chanson H (2009) Progress in the observation and modeling of turbulent multi-phase flows. Environ Fluid Mech 9(2):121–123
Bombardelli FA, Meireles I, Matos J (2011) Laboratory measurements and multi-block numerical simulations of the mean flow and turbulence in the non-aerated skimming flow region of steep stepped spillways. Environ Fluid Mech 11:263–288
Bombardelli FA, Moreno PA (2012) Exchange at the bed sediments-water column interface. In: Gualtieri C, Mihailovic DT (eds) Fluid mechanics of environmental interfaces, Chapter 8, 2nd edn. Taylor & Francis Group, London, pp 221–253
Brunner H, Tang T (1989) Polynomial spline collocation methods for the nonlinear Basset equation. Comput Math Appl 18(5):449–457
Brush JS, Ho HW, Yen BC (1964) Acceleration motion of sphere in a viscous fluid. J Hydraul Div 90:149–160
Crowe C, Sommerfeld M, Tsuji M (2011) Multiphase flows with droplets and particles. CRC Press, Boca Raton
Daitche A (2013) Advection of inertial particles in the presence of the history force: higher order numerical schemes. J Comput Phys 254:93–106
Dorgan A, Loth E (2007) Efficient calculation of the history force at finite Reynolds numbers. Int J Multiph Flow 33:833–848
Drew DA, Passman SL (1999) Theory of multicomponent fluids. Springer, Berlin
García MH (2008) Sedimentation engineering: processes, measurements, modeling, and practice. ASCE Manuals and Reports on Engineering Practice N° 110, Reston
Gatignol R (1983) The Faxén formulae for a rigid particle in an unsteady non-uniform Stokes flow. J Méc Théor 1:143–160
González AE, Bombardelli FA, Nino YI (2006) Improving the prediction capability of numerical models for particle motion in water bodies. In: Proceedings of the 7th International Conference on HydroScience and Engineering ICHE 2006, Philadelphia, USA
González AE (2008) Coupled numerical modeling of sediment transport near the bed using a two-phase flow approach. Ph.D. Thesis, University of California, Davis, p 188
Kim I, Elghobashi S, Sirignana WB (1998) On the equation for spherical-particle motion: effect of Reynolds and acceleration numbers. J Fluid Mech 367:221–253
Lawrence CJ, Mei R (1995) Long-time behavior of the drag on a body in impulsive motion. J Fluid Mech 283:307–327
Lee HY, Hsu IS (1994) Investigation of saltating particle motions. J Hydraul Eng ASCE 120:831–845
Loth E, Dorgan A (2009) An equation of motion for particles of finite Reynolds number and size. Environ Fluid Mech 9(2):187–206
Lovalenti PM, Brady JF (1993) The hydrodynamic force on a rigid particle undergoing arbitrary time-dependent motion at small Reynolds number. J Fluid Mech 256:561–605
Lukerchenko N, Chara Z, Vlasak P (2006) 2D Numerical model of particle-bed collision in fluid-particle flows over bed. J Hydraul Res IAHR 44(1):70–78
Lukerchenko N, Piatsevich S, Chara Z, Vlasak P (2009) 3D numerical model of spherical particle saltation in a channel with a rough fixed bed. J Hydrol Hydromech 57(2):100–112
Maxey MR, Riley JJ (1983) Equation of motion for a small rigid sphere in a nonuniform flow. Phys Fluids 26:883–889
Mei R, Adrian RJ (1992) Flow past a sphere with an oscillation in the free-stream velocity and unsteady drag at finite Reynolds number. J Fluid Mech 237:323–341
Michaelides EE (2006) Particles, bubbles and drops: their motion, heat and mass transfer. World Scientific Publishing Co, Hackensack
Moorman RW (1955) Motion of a spherical particle in the accelerated portion of free-fall. Doctor of Philosophy Dissertation, University of Iowa
Mordant N, Pintot JF (2000) Velocity measurement of a settling sphere. Eur Phys J B 18:343–353
Moreno PA, Bombardelli FA (2012) 3D numerical simulation of particle-particle collisions in saltation mode near stream beds. Acta Geophys 60:1661–1688
Niño Y, García M, Ayala L (1994) Gravel saltation. Experiments. Water Resour Res 30(6):1907–1914
Niño Y, García M (1994) Gravel saltation. 2. Modeling. Water Resour Res 30(6):1915–1924
Niño Y, García M (1998) Experiments on saltation of sand in water. J Hydraul Eng 124(10):1014–1025
Niño Y, García M (1998) Using Lagrangian particle saltation observations for bed-load sediment transport modelling. Hydrol Process 12:1197–1218
OldhamY Spanier M (1974) The fractional calculus. Academic, New York
Parmar M, Haselbacher A, Balachandar S (2011) Generalized Basset-Boussinesq-Oseen equation for unsteady forces on a sphere in a compressible flow. Phys Rev Lett 106:084501
Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical recipes. Cambridge University Press, New York
Prosperetti A, Tryggvason G (2007) Computational methods for multiphase flow. Cambridge University Press, Cambridge
Rechiman LM, Dellavalle D, Bonetto FJ (2013) Path suppression of strongly collapsing bubbles at finite and low Reynolds numbers. Phys Rev E 87:063004
Sangani AS, Zhang DZ, Prosperetti A (1991) The added mass, Basset, and viscous drag coefficients in nondilute bubbly liquids undergoing small-amplitude oscillatory motion. Phys Fluids 3(12):2955–2970
Schmeeckle MW, Nelson JM (2003) Direct numerical simulation of bed-load transport using a local, dynamic boundary condition. Sedimentology 50:279–301
Sridhar G, Katz J (1995) Drag and lift forces on microscopic bubbles entrained by a vortex. Phys Fluids 7(2):389–399
Sungkorn R, Derksen JJ, Khinast JG (2012) Euler-Lagrange modeling of a gas-liquid stirred reactor with consideration of bubble breakage and coalescence. AIChE J 58(5):1356–1370
Tatom FB (1988) The Basset term as a semiderivative. Appl Sci Res 45:283–285
van Hinsberg MAT, Boonkkamp JHMT, Clercx HJH (2011) An efficient, second order method for the approximation of the Basset history force. J Comput Phys 230(4):1465–1478
Wakaba L, Balachandar S (2005) History force on a sphere in a weak shear flow. Int J Multiph Flow 31:996–1014
Wiberg P, Smith JD (1985) A theoretical model for saltating grains in water. J Geophys Res 90:7341–7354
Wood IR, Jenkins BS (1973) A numerical study of the suspension of a non-buoyant particle in a turbulent stream. In: Proceedings of IAHR internatinal symposium on river mechanics, vol 1, pp 431–442
Yen BC (1992) Sediment fall velocity in oscillating flow. Water resources environmental engineering resources. Report 11. Dept. of Civil Eng, University of Virginia
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Moreno-Casas, P.A., Bombardelli, F.A. Computation of the Basset force: recent advances and environmental flow applications. Environ Fluid Mech 16, 193–208 (2016). https://doi.org/10.1007/s10652-015-9424-1
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DOI: https://doi.org/10.1007/s10652-015-9424-1