Computation of the Particle Basset Force witha Fractional-Derivative Approach
Fabián A. Bombardelli, A.M.ASCE; Andrea E. González; and Yarko I. Niño.

Several numerical simulations of bed-load sediment transport in open channels have satisfactorily estimated the global amount of material put into motion, and have provided information on the characteristics of sediment particle trajectories (Wiberg and Smith 1985; Sekine and Kikkawa 1992; Niño and García 1994; Lee and Hsu 1994; Niño and García 1998; Schmeeckle and Nelson 2003; Lukerchenko et al. 2006; Lee et al. 2006). In these studies, the flow field has been specified via either the semilogarithmic laws for turbulent open-channel flows (Niño and García 1994, 1998), or through velocity measurements (Schmeeckle and Nelson 2003). In turn, the acceleration of the particles has been described using Lagrangian models that consider all forces associated with the motion. One of the forces that may act on a given particle is the Basset force (Niño and García 1994, 1998; Mordant and Pinton 2000; Lukerchenko et al. 2006), which addresses the temporal delay in the development of the boundary layer surrounding the particle as a consequence of changes in the relative velocity (Crowe et al. 1998). This force is usually called the “history” force (Crowe et al. 1998). Although several authors have disregarded the Basset force in their models of bed-load transport [see for instance, Wood and Jenkins (1973); Lee and Hsu (1994); Schmeeckle and Nelson (2003)], there is recent evidence that the history force becomes important for relatively small particle sizes, i.e., for relatively small explicit particle Reynolds numbers, R_p=(Rgd_p³)^0.5 /υ, where R=(ρ_s/ρ)-1, ρ and ρ_s= fluid and particle density, respectively; g = acceleration of gravity; d_p= particle diameter, and υ=kinematic viscosity of water. Comparisons of numerical results with laboratory observations have shown that while the Basset force is negligible for gravels moving as bedload [with Rp of the order of 20,000; Niño and García (1994)], it becomes extremely important for sands [with Rp of the order of 100; Niño and García (1998)]. When the Basset force is neglected in the case of sands, the length of a single particle jump can be underpredicted by about 40%, and the jump height can be underpredicted by about 15% [Fig. 2 in Niño and García (1998)]. These differences can accumulate for multiple jumps and lead to very large errors in computations of transport. Mordant and Pinton (2000) in turn performed laboratory tests of spheres settling in water, with diameters ranging from 0.5 to 6 mm, and fall (limit) velocities varying from 0.07 to 1.16 m/s. They found that the Basset force needs to be included in the Lagrangian models in order to correctly describe the particle acceleration when the particle Reynolds number is smaller than 4,000 [given by Re_p=w_sd_p /ν, where w_s=particle fall (limit) velocity]. This result is in accord with the findings of Niño and García (1994, 1998). In addition, consistent with the above results, Armenio and Fiorotto (2001) found that the Basset force is appreciable for Re_p of the order of, and smaller than, 1, for a large range of density ratios.

In Lagrangian models, the evaluation of the Basset force can be extremely time consuming and requires the storage of the relative acceleration of the particle, which can make the simulation very demanding in terms of computer memory. Michaelides (1992) recast the linear equations of particle motion using Laplace transforms. He also employed “canonical” velocity fields to simplify the analysis, and presented a novel procedure to reduce the computational cost of the Basset force. Obviously, his procedure does not apply to nonlinear equations, or to random fields, which prevail in flumes and rivers. In this technical note, we present and discuss a method to reduce the computation time and the memory requirements of the Basset force.