Ansys Fluent predicts the trajectory of a discrete phase particle (or droplet or bubble) by integrating the force balance on the particle, which is written in a Lagrangian reference frame. This force balance equates the particle inertia with the forces acting on the particle, and can be written as

(12–1)

where is the particle mass, is the fluid phase velocity, is the particle velocity, is the fluid density, is the density of the particle, is an additional force, is the drag force, and is the droplet or particle relaxation time [215] calculated by:

(12–2)

Here, is the molecular viscosity of the fluid, is the particle diameter, and Re is the relative Reynolds number, which is defined as

(12–3)

Particle rotation is a natural part of particle motion and can have a significant influence on the trajectory of a particle moving in a fluid. The impact is even more pronounced for large and/or heavy particles with high moments of inertia. In this case, if particle rotation is disregarded in simulation studies, the resulting particle trajectories can significantly differ from the actual particle paths. To account for particle rotation, an additional ordinary differential equation (ODE) for the particle’s angular momentum is solved:

(12–4)

where is the moment of inertia, is the particle angular velocity, is the fluid density, is the particle diameter, is the rotational drag coefficient, is the torque applied to a particle in a fluid domain, and is the relative particle–fluid angular velocity calculated by:

(12–5)

For a spherical particle, the moment of inertia is calculated as:

(12–6)

From Equation 12–4, it is apparent that the torque results from equilibrium between the particle inertia and the drag.

For details on how to use the particle rotation capability, see Particle Rotation in the Fluent User's Guide.

While Equation 12–1 includes a force of gravity on the particle, it is important to note that in Ansys Fluent the default gravitational acceleration is zero. If you want to include the gravitational force, you must remember to define the magnitude and direction of the gravity vector in the Operating Conditions Dialog Box.

Equation 12–1 incorporates additional forces () in the particle force balance that can be important under special circumstances. The first of these is the "virtual mass" force, the force required to accelerate the fluid surrounding the particle. This force can be written as

(12–7)

where is the virtual mass factor with a default value of 0.5.

An additional force arises due to the pressure gradient in the fluid:

(12–8)

The virtual mass and pressure gradient forces are not important when the density of the fluid is much lower than the density of the particles as is the case for liquid/solid particles in gaseous flows (). For values of approaching unity, the Virtual Mass and Pressure Gradient forces become significant and it is recommended that they be included when the density ratio is greater than 0.1. See Including the Virtual Mass Force and Pressure Gradient Effects on Particles in Fluent User's Guide for details on how to include these forces in your model.

The additional force term, , in Equation 12–1 also includes forces on particles that arise due to rotation of the reference frame. These forces arise when you are modeling flows in moving frames of reference (see Flow in a Moving Reference Frame). For rotation defined about the axis, for example, the forces on the particles in the Cartesian and directions can be written as

(12–9)

where and are the particle and fluid velocities in the Cartesian direction, is the RPM, and

(12–10)

where and are the particle and fluid velocities in the Cartesian direction.

Small particles suspended in a gas that has a temperature gradient experience a force in the direction opposite to that of the gradient. This phenomenon is known as thermophoresis. Ansys Fluent can optionally include a thermophoretic effect on particles in the additional force in Equation 12–1:

(12–11)

where is the thermophoretic coefficient. You can define the coefficient to be constant, polynomial, or a user-defined function, or you can use the form suggested by Talbot [644]:

(12–12)

This expression assumes that the particle is a sphere and that the fluid is an ideal gas.

For sub-micron particles, the effects of Brownian motion can be optionally included in the additional force term. The components of the Brownian force are modeled as a Gaussian white noise process with spectral intensity given by [363]:

(12–13)

where is the Kronecker delta function, and

(12–14)

is the absolute temperature of the fluid, is the kinematic viscosity, is the Cunningham correction (defined in Equation 12–55), and is the Boltzmann constant. Amplitudes of the Brownian force components are of the form

(12–15)

where are zero-mean, unit-variance-independent Gaussian random numbers. The amplitudes of the Brownian force components are evaluated at each time step. The energy equation must be enabled in order for the Brownian force to take effect. Brownian force is intended only for laminar simulations.

The Saffman’s lift force, or lift due to shear, can also be included in the additional force term as an option. The lift force used is from Li and Ahmadi [363] and is a generalization of the expression provided by Saffman [562]:

(12–16)

where and is the deformation tensor. This form of the lift force is intended for small particle Reynolds numbers and is recommended only for sub-micron particles.

The Magnus or rotational lift force arises when particle is rotating in a fluid. The lift is caused by a pressure differential along a particle’s surface ([124]). For high Reynolds numbers, the Magnus force is scaled by a rotational lift coefficient :

(12–17)

For the rotational lift coefficient , different approaches are available in the literature. Ansys Fluent offers the following formulations:

For rotating particle simulations, you can include the Magnus lift force in the additional force term in the particle force balance equation (Equation 12–1). For details, see Particle Rotation in the Fluent User's Guide.