ANSYS FLUENT 12.0 Theory Guide - 15.2.1 Equations of Motion for Particles

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 (for the

direction in Cartesian coordinates) as

$\frac{d u_p}{dt} = F_D(u - u_p) + \frac{g_x (\rho_p - \rho)}{\rho_p} + F_x$

(15.2-1)

where

is an additional acceleration (force/unit particle mass) term,

is the drag force per unit particle mass and

$F_D = \frac{1 8 \mu}{\rho_p d^{2}_{p}} \;\; \frac{C_D {\rm Re}}{24}$

(15.2-2)

Here,

is the fluid phase velocity,

is the particle velocity, $\mu$ is the molecular viscosity of the fluid, $\rho$ is the fluid density, $\rho_p$ is the density of the particle, and

is the particle diameter. Re is the relative Reynolds number, which is defined as

${\rm Re} \equiv \frac{\rho d_p \left \vert u_{p} - u \right \vert}{\mu}$

(15.2-3)

While Equation 15.2-1 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 15.2-1 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

$F_x = \frac{1}{2}\frac{\rho}{\rho_p}\frac{d}{dt} \left(u - u_{p} \right)$

(15.2-4)

and is important when $\rho > \rho_p$ . An additional force arises due to the pressure gradient in the fluid:

$F_x = \left (\frac{\rho}{\rho_p} \right ) {u_p}_i \frac{\partial u}{\partial x_i}$

(15.2-5)

The additional force term,

, in Equation 15.2-1 also includes forces on particles that arise due to rotation of the reference frame. These forces arise when you are modeling flows in rotating frames of reference (see Section 2.2). For rotation defined about the

axis, for example, the forces on the particles in the Cartesian

and

directions can be written as

$\left(1 - \frac{\rho}{\rho_p} \right) \Omega^2x + 2 \Omega \left(u_{y,p} - \frac{\rho}{\rho_p} u_y \right)$

(15.2-6)

where $u_{y,p}$ and

are the particle and fluid velocities in the Cartesian

direction, and

$\left(1 - \frac{\rho}{\rho_p} \right) \Omega^2 y - 2 \Omega \left(u_{x,p}- \frac{\rho}{\rho_p} u_x \right)$

(15.2-7)

where $u_{x,p}$ 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 acceleration (force/unit mass) term,

, in Equation 15.2-1:

$F_x = -D_{T,p} \frac{1}{m_pT} \frac{\partial T}{\partial x}$

(15.2-8)

where $D_{T,p}$ 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 [ 345]:

$F_x = -\frac{6 \pi d_p \mu^2 C_s (K + C_t {\rm Kn})}{\rho (1... ...+ 2C_t {\rm Kn})} \frac{1}{m_pT} \frac{\partial T}{\partial x}$

(15.2-9)

where:	${\rm Kn}$	= Knudsen number = 2 $\lambda/d_p$
	$\lambda$	= mean free path of the fluid
		=
		= fluid thermal conductivity based on translational
		$\; \; \;$ energy only = (15/4) $\mu R$
		= particle thermal conductivity
		= 1.17
		= 2.18
		= 1.14
		= particle mass
		= local fluid temperature
	$\mu$	= fluid viscosity

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 $S_{n, ij}$ given by [ 191]

$S_0 = \frac{216 \nu k_B T}{\pi^2 \rho d_p^5 \left(\frac{\rho_p}{\rho}\right)^2 C_c}$

(15.2-11)

is the absolute temperature of the fluid, $\nu$ is the kinematic viscosity, and

is the Boltzmann constant. Amplitudes of the Brownian force components are of the form

where $\zeta_i$ 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 [ 191] and is a generalization of the expression provided by Saffman [ 298]:

${\vec F} = \frac{2 K \nu^{1/2} \rho d_{ij}}{\rho_p d_p (d_{lk} d_{kl})^{1/4}} \left({\vec v} - {\vec v}_p \right)$

(15.2-13)

where

and $d_{ij}$ is the deformation tensor. This form of the lift force is intended for small particle Reynolds numbers. Also, the particle Reynolds number based on the particle-fluid velocity difference must be smaller than the square root of the particle Reynolds number based on the shear field. This option is recommended only for submicron particles.