In the *Pre-Analysis & Start-Up* step, we'll review the following:

- Theory for Fluid Phase
- Theory for Particle Phase
- Choosing the Cases

A particle-laden flow is a multiphase flow where one phase is the fluid and the other is dispersed particles. Governing equations for both phases are implemented in Fluent. To run a meaningful simulation, a review of the theory is necessary.

In the simulations considered for this tutorial, the fluid flow is a 2D perturbed periodic double shear layer as described in the first section. The geometry is Lx = 59.15 m, Ly = 59.15 m. The mesh size is chosen as in order to resolve the smallest vortices. As a rule of thumb. One typically needs about 20 grid points across the shear layers, where the vortices are going to develop. The boundary conditions are periodic in the x and y directions. The fluid phase satisfies the Navier-Stokes Equations:

-Momentum Equations

where **u** is the fluid velocity, p the pressure, the fluid density and **f** a momentum exchange term due to the presence of particles. When the particle volume fraction and the particle mass loading are very small, it is legitimate to neglect the effects of the particles on the fluid: **f** can be set to zero. This type of coupling is called one-way. In these simulations, the fluid phase is air, while the dispersed phase is constituted of about 400 glass beads a few dozens of microns in diameter. This satisfies both conditions and

One way coupling is legitimate here. See Ansys documentation (16.2) for further details about the momentum exchange term.

The suspended particles are considered as rigid spheres of the same diameter d, and density . Newton’s second law written for the particle i stipulates:

where is the velocity of particle i, the forces exerted on it, and its mass.

In order to know accurately the hydrodynamic forces exerted on a particle, one needs to resolve the flow to a scale significantly smaller than the particle diameter. This is computationally prohibitive. Instead, the hydrodynamic forces can be approximated roughly to be proportional to the drift velocity.

where is known as the particle response time, the particle density, and D the particle diameter. This equation needs to be solved for all particles present in the domain. This is done in Fluent via the Discrete Phase Model (DPM).

The particle response time measures the speed at which the particle velocity adapts to the local flow speed. Non-inertial particles, or tracers, have a particle response time of zero: they follow the fluid streamlines. Inertial particles with might adapt quickly or slowly to the fluid speed variations depending on the relative variation of the flow and the particle response time.

This rate of adaptation is measured by a non-dimensional number called the Stokes number, given by the ratio of the particle response time to the flow characteristic time scale.

In these simulations, the characteristic flow time is the inverse of the growth rate of the vortices in the shear layers. This is also predicted by the Orr-Sommerfeld equation. For the particular geometry and configuration we used in this tutorial, the growth rate is . When St = 0 the particles are tracers. They follow the streamlines and, in particular, they will not be able to leave a vortex once caught inside.

When , particles have a ballistic motion and are not affected by the local flow conditions. They are able to shoot through the vortices without a strong trajectory deviation. In intermediate cases, where , there is a maximum coupling between the two phases: Particles are attracted to the vortices, but once they reach the highly swirling vortex cores they are ejected due to their non-zero inertia.

In this tutorial, we will consider a near tracer case St = 0.2, an intermediate case St = 1 and a nearly ballistic case St = 5.