An efficient, second order method for the approximation of the Basset history force
- a Department of Physics, Eindhoven University of Technology, P.O. Box 513, 5600MB Eindhoven, The Netherlands
- b Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600MB Eindhoven, The Netherlands
- c Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
- Received 22 July 2010
- Revised 9 November 2010
- Accepted 9 November 2010
- Available online 14 November 2010
Abstract
The hydrodynamic force exerted by a fluid on small isolated rigid spherical particles are usually well described by the Maxey–Riley (MR) equation. The most time-consuming contribution in the MR equation is the Basset history force which is a well-known problem for many-particle simulations in turbulence. In this paper a novel numerical approach is proposed for the computation of the Basset history force based on the use of exponential functions to approximate the tail of the Basset force kernel. Typically, this approach not only decreases the cpu time and memory requirements for the Basset force computation by more than an order of magnitude, but also increases the accuracy by an order of magnitude. The method has a temporal accuracy of O(Δt2) which is a substantial improvement compared to methods available in the literature. Furthermore, the method is partially implicit in order to increase stability of the computation. Traditional methods for the calculation of the Basset history force can influence statistical properties of the particles in isotropic turbulence, which is due to the error made by approximating the Basset force and the limited number of particles that can be tracked with classical methods. The new method turns out to provide more reliable statistical data.
Keywords
- Basset history force;
- Numerical approximation;
- Particle laden flow;
- Maxey–Riley equation;
- Isotropic turbulence
1. Introduction
The turbulent dispersion of small inertial particles plays an important role in environmental flows, and in this work we focus on small particles with densities of the same order as that of the surrounding fluid. Examples of such particles that may be present in well-mixed or in density stratified estuaries are plankton, algae, aggregates (all with densities similar to the fluid density) or resuspended sand from the sea bottom (particle densities in this case several times that of the fluid). Particle collisions and the formation of aggregates of marine particles or sediment depend on the details of the small-scale trajectories of the particles in locally homogeneous and isotropic turbulence. At these scales the details of the hydrodynamic force acting on (light) inertial particles are relevant.
Maxey and Riley [1] introduced the equation of motion for small (dp ≪ η, with dp the particle diameter and η the Kolmogorov length scale) isolated rigid spherical particles in a nonuniform velocity field u(x, t). An important assumption is that the particle Reynolds number Rep = dp∣u − up∣/ν ≪ 1, with up the velocity of the particle and ν the kinematic viscosity of the fluid. As we consider small particle diameter and small volume fraction of particles we ignore the effect of two-way and four-way coupling. The relative importance of the terms in the hydrodynamic force depends on the ratio of particle-to-fluid density and the particle diameter. The computation of all the different forces in the Maxey–Riley equation is an expensive time- and memory consuming job. Therefore, assumptions are often made regarding the forces that can be neglected in the study of particle dispersion. The number of studies underpinning these assumptions, however, is rather limited due, for example, to the lack of efficient algorithms to take into account the effects of the Basset history force with sufficient numerical accuracy. This term was first discovered by Boussinesq in 1885. An elaborate overview of the work on the different terms in the Maxey–Riley equation and their numerical implementation can be found in the paper by Loth [2] and a historical account of the equation of motion was given in a review article by Michaelides [3].
The term most often neglected is the Basset history force because of its numerical complexity. Many recent studies underline the importance of the Basset force compared to the other forces contributions in the Maxey–Riley equation for particle transport in turbulent flows, see Refs. [4], [5], [6] and [7]. Moreover, it can affect the motion of a sedimenting particle [8] or bed-load sediment transport in open channels, where the Basset force becomes extremely important for sand particles [9] and [10]. It also might alter the particle velocity in an oscillating flow field [11] or modify the trapping of particles in vortices [12].
Fast and accurate computation of the Basset force is far from trivial. Although several attempts have been made [13], [14] and [15], the computation of the Basset force is still far more time consuming and less accurate than the computation of the other forces in the MR equation. Therefore we present a new method that saves time, memory costs and is more accurate.
The MR equation and the subtlities with regard to the computation of the Basset history force are introduced in Section 2. Next, in Sections 3 and 4, the new method is introduced, where Section 3 focuses on the approximation of the tail of the Basset history force and Section 4 on the numerical integration of the Basset history force. Thereafter, validation of the method using analytical solutions is discussed in Section 5. A simulation of isotropic turbulence, with light inertial particles embedded in the flow, has been performed. In Section 6 we compare the results from this simulation with the new implementation of the full MR equation with the old version used by van Aartrijk and Clercx [6]. Finally, concluding remarks are given in Section 7.