Simulation of Particulate Flows With the Force-Coupling Method (Keynote Paper)
Authors: Maxey M. R. , Dong S. , Liu D. , Xu J. 
Published: February 2009

One of the challenges in the numerical simulation of a system of particles in a fluid flow is to balance the need for an accurate representation of the flow around individual particles with the feasibility of simulating the fully-coupled dynamics of large numbers of particles. Over the past few years, several techniques have been developed for the direct numerical simulation of dispersed two-phase flows. Examples include the ALE-FEM formulation described by Hu et al. [1] and the DLM method of Patankar et al. [2]. The former uses a finite element mesh that conforms to the shape and position of each particle and evolves dynamically as the particles move, while the latter employs a fixed mesh and constraints are imposed in the volume of fluid occupied by the particle to reproduce a corresponding rigid body motion. In both the aim is to fully resolve the flow dynamics for each particle and there is a corresponding demand for high resolution of the flow. A typical approach used for gas-solid flows has been the point-force method that combines a Lagrangian tracking of individual particles with an Eulerian formulation for force feedback on the fluid flow. The latter approach has worked well for very small particles in systems of negligible void fraction but significant mass loading. The resolution level is very low and often the particles are smaller than the spacing between grid points. Its success comes from the averaging effect of large numbers of small particles and the fact that the influence of an individual particle is weak. The approach though is inaccurate for liquid-solid or bubbly flows when the individual particles are of finite size and the void fractions may easily be larger than 1%. In tracking the individual particles an equation of motion is formulated that relates the particle acceleration to the fluid forces acting on the particle, and these forces such as drag and lift are parameterized in terms of the local fluid velocity, velocity gradients and history of the fluid motion. Once flow modification is included however, it is harder to specify the local flow. The parameterizations also become more complex as effects of finite Reynolds number or wall boundaries are included. As a numerical procedure, the force-coupling method (FCM) does not require the same level of resolution as the DLM or ALE-FEM schemes and avoids the limitations of the point-force method. It gives a self-consistent scheme for simulating the dynamics of a system of small particle using a fixed numerical mesh and resolves the flow except close to the surface of each particle. Distributed, finite force-multipoles are used to represent the particles, and FCM is able to predict quite well the motion of isolated particles in shear flows and the interaction between moving particles. The method also provides insights into how the two-phase flow may be described theoretically and modeled. The idea of the force-coupling method was first introduced by Maxey et al. [3]. The basic elements of the method are given by Maxey & Patel [4] and Lomholt & Maxey [5]. In the basic version of the method, fluid is assumed to fill the whole flow domain, including the volume occupied by the particles. The presence of each particle is represented by a finite force monopole that generates a body force distribution f(x,t) on the fluid, which transmits the resultant force of the particles on the flow to the fluid. The velocity field u(x,t) is incompressible and satisfies ∇·u=0(1)ρDuDt=−∇p+μ∇2u+f(x,t),(2) where μ is the fluid viscosity and p is the pressure. The body force due to the presence of NP bubbles is f(x,t)=∑n=1NpF(n)Δ(x−Y(n)(t)),(3)Y(n)(t) is the position of the nth spherical particle and F(n)(t) is the force this exerts on the fluid. The force monopole for each particle is determined by the function Δ(x), which is specified as a Gaussian envelope Δ(x)=(2πσ2)−3/2exp(−x2/2σ2)(4) and the length scale σ is set in terms of the particle radius a as a/σ = π. The velocity of each particle V(n)(t) is found by forming a local average of the fluid velocity over the region occupied by the particle as V(n)(t)=∫u(x,t)Δ(x−Y(n)(t))d3x.(5) If mP and mF denote the mass of a particle and the mass of displaced fluid, the force of the particle acting on the fluid is F(n)=(mP−mF)(g−dV(n)dt).(6) This force is the sum of the net external force due to buoyancy of the particle and the excess inertia of the particle over the corresponding volume of displaced fluid. In addition a short-range, conservative force barrier is imposed to represent collisions between particles and prevent overlap. A similar barrier force is imposed, normal to the wall, to represent collisions between a particle and a rigid wall. With this scheme the body forces induce a fluid motion equivalent to that of the particles. The dynamics of the particles and the fluid are considered as one system where fluid drag on the particles, added-mass effects and lift forces are internal to the system. The method does not resolve flow details near to the surface of a particle, and indeed the no-slip condition is not satisfied on surface. At distances of about half a particle radius from the surface the flow though is fairly well represented. While there is no explicit boundary condition on the particle surface, the condition (5) ensures that the bubble and the surrounding fluid move together. The method has been applied to a variety of flow problems. Lomholt et al. [6] compared experimental results for the buoyant rise of particles in a vertical channel filled with liquid with results from corresponding simulations with FCM. The particle Reynolds numbers were in the range of 0 to 5 and the results agreed well. The wake-capture and the drafting, kissing and tumbling of pairs of particles, or of a group of three particles were found to match. Comparisons have made too with full direct numerical simulations performed with a spectral element code [7]. Liu et al. [8] examined the motion of particles in a channel at both low and finite Reynolds numbers, up to Re = 10. There was in general good agreement between the FCM results and the DNS for the particle motion, and the flow details were consistent away from the particle surface. There has been extensive work in the past on the sedimentation of particles in a homogeneous suspension, mainly for conditions of Stokes flow. Climent & Maxey [9] have verified that the FCM scheme reproduces many of the standard features found for Stokes suspensions. The results for finite Reynolds numbers illustrate how the structure of the suspension changes as fluid inertia is introduced, in particular limiting the growth in velocity fluctuation levels with system size. Further work has been done by Dance [10] on sedimenting suspensions in bounded containers. Recently we have been studying the dynamics of drag reduction by injecting micro-bubbles into a turbulent channel flow. This has been proven through experiments over the past 30 years to be an effective means for drag reduction but the details of the mechanisms involved have not been determined. Numerical simulations by Xu et al. [11] have shown clear evidence of drag reduction for a range of bubble sizes. A key feature is the need to maintain a concentration of bubbles in the near-wall region. In the talk, the method will be described and example results given. Specific issues relevant to gas-solid flows will be discussed.