Modeling and numerical simulation of particulate flows by the Eulerian–Lagrangian approach
Introduction
Numerical schemes based on mathematical models of separated particulate multiphase flow have used the continuum approach for all the phases or a continuum approach for the fluid phase and a Lagrangian approach for the particles. These simulation methods can be applied in various settings; e.g. sedimenting and fluidized suspensions, lubricated transport, hydraulic fracturing of reservoirs, slurries, sprays, etc.
Continuum–continuum (or Eulerian–Eulerian) approach considers the particulate phase to be a continuous fluid interpenetrating and interacting with the fluid phase (Gidaspow, 1994). In the Eulerian–Lagrangian formulation each computational particle (called parcel) is considered to represent a group of particles interacting with the fluid and possessing the same characteristics such as size, composition etc. Positions of these parcels are then calculated using Newton's equations of motion. It has been found that the required number of parcels to accurately represent the particle phase is not excessive (Dukowicz, 1980).
The Eulerian–Lagragian numerical simulation methods have been developed with different assumptions. In problems such as the dispersion of atmospheric pollutants, it may be assumed that the particles do not perturb the flow field. The solution then involves tracing the particle trajectories in a known velocity field i.e. the fluid phase equations are solved independent of the particles (Gauvin et al., 1975). In other problems the particles may carry sufficient momentum to set the surrounding fluid in motion. In this case it is necessary to include the fluid-particle momentum exchange term in the fluid phase equation. However, the volume occupied by the particles in a computational cell in comparison with the volume of the fluid may still be neglected (Crowe et al., 1977). When the particle volume is significant it is important to model the volume fraction in both the momentum and continuity equations (Dukowicz, 1980). At high particle volume fractions the effective viscosity of the suspension is high and this effect should be ideally modeled through the viscous stress term in the governing equations.
Andrews and O'Rourke (1996) and Snider et al. (1998) presented a multiphase particle-in-cell (MP-PIC) method for particulate flows that accounts for full coupling between the fluid and particle phase as well as the interparticle stress to prevent the particle volume from exceeding the close packing limit. The fluid phase is assumed to be inviscid where viscosity is significant on the scale of the particles and is used only in the particle drag formula. In this approach the particle phase is considered both as a continuum and as a discrete phase. Interparticle stresses are calculated by treating the particles as a continuum phase. Particle properties are mapped to and from an Eulerian grid. Continuum derivatives that treat the particle phase as a fluid are evaluated to model interparticle stress and then mapped back to the individual particles. This results in a computational method for multiphase flows that can handle particulate loading ranging from dense to dilute and for particles of different sizes and materials. Patankar and Joseph (2001) presented a new Eulerian–Lagrangian numerical simulation (LNS) scheme for particulate flows in three-dimensional geometries. They apply the MP-PIC approach of Andrews and O'Rourke (1996).
The hydrodynamic force acting on the particle surface represents the fluid-particle momentum exchange. Different models can be used for the hydrodynamic force on the particle in the Eulerian–Lagrangian approach. These models can be developed through experimental investigation. At the same time, development of direct numerical simulation (DNS) techniques for rigid particulate flows (e.g. Hu et al., 1992; Hu, 1996; Johnson and Tezduyar, 1997; Glowinski et al., 1999; Patankar et al., 2000) have provided an invaluable tool for modeling the hydrodynamic force in many applications. It would be straightforward to use these models for the hydrodynamic force from experiments or DNS results, when available, in the LNS technique.
The primary objective of this work is to develop a model-based numerical simulation scheme for particulate flows that has the following features: (a) A flexibility to use different models for the fluid-particle force (drag and lift) and the effective properties (such as the effective viscosity) of the suspension. Such models are expected to be developed from our effort on the direct numerical simulation (DNS) of particulate flows. (b) A capability to simulate the motion of particles of different sizes and material properties. (c) A capability to handle dense particulate flows without exceeding the close-packing limit. (d) An efficient numerical algorithm capable of quick computations at each time-step with modest memory requirements as compared to the direct numerical simulation methods. The other objective is to suggest a model for the viscous stress terms in the fluid phase equations. We choose an Eulerian–LNS method because the models developed from DNS can be most naturally introduced in this approach. Additional continuity and momentum equations for each particulate phase are not required for multimodal simulations. The Lagrangian tracking of particles naturally predict the unstable flow features without complexities in the numerical implementation.
The numerical algorithm used in this paper is a variation of the scheme by Patankar and Joseph (2001). We solve the fluid phase continuity and momentum equations using a Chorin-type (Chorin, 1968) fractional-step method, but unlike the approach of Patankar and Joseph (2001) we solve the particle momentum equation coupled with the fluid phase equations. As a result the pressure equation set up in this approach is different from the one used by them. A non-staggered grid for velocity and pressure (Rhie and Chow, 1982) is used.
In the MP-PIC and LNS formulation of Patankar and Joseph (2001) a particle stress term that acts like pressure is used to prevent the particle volume fraction from exceeding the close-packing limit. This stress model introduces a repelling force on particles moving into regions where the particle volume fraction is large. In this paper we use this model for some problems. We also use a different “collision” scheme that prevents the particle volume fraction from exceeding the close packing limit. We introduce a parcel–parcel and parcel–wall collision force to ensure that there is no overlap between any two parcel regions and between the parcel regions and the wall. Consequently the particle volume fraction does not exceed the close-packing limit in any part of the computational domain. We have used this scheme for the simulation of bubbling fluidization of gas–particle and liquid–particle mixtures.
In Section 2 we will present governing equations for the Eulerian–Lagrangian formulation. In Section 3 the numerical scheme will be explained. This computational scheme will then be verified in Section 4 by comparing results with test cases and experiments. Simulation results for bimodal sedimentation, inclined sedimentation and bubbling fluidization of gas–particle and liquid–particle mixtures are presented. Conclusions will be stated in Section 5.
Section snippets
Governing equations
We use averaged equations of motion for the fluid phase (see Joseph and Lundgren, 1990 and references therein). Continuity equation is given bywhere ρf is the density of the suspending fluid, θf is the fluid volume fraction and is the average velocity of the fluid phase. The average velocity field of the fluid phase does not satisfy the divergence-free condition even if we consider an incompressible suspending fluid. The momentum equation of the fluid phase can be obtained
Numerical scheme
We use a finite-volume method on a three-dimensional Eulerian grid to solve the fluid phase equations in Cartesian coordinate system. A non-staggered grid for velocity and pressure (Rhie and Chow, 1982) is used. The particle phase equations are solved by considering the motion of a finite number of computational particles which represent a sample of the total population of particles. Each computational particle, henceforth referred to as a parcel, is considered to represent a group of particles
Numerical results
We solve the problems of bimodal sedimentation, inclined sedimentation, gas–solid fluidization and liquid–solid fluidization to verify the numerical scheme.
Conclusions
In this paper we report a new numerical scheme for simulation of particulate flows with the following unique features: (a) The fluid–particle momentum equations are solved in an entirely coupled fashion at a given time-step. (b) This coupled system is solved by a pressure based fractional step scheme for fast computations at each time-step. Traditionally, the pressure based Chorin-type fractional step schemes are devised for constant coefficients in the fluid phase equations. The unique feature
Acknowledgements
We acknowledge the support from NSF under KDI/NCC grant NSF/CTS-9873236 and STIMLAB.
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