Fundamentals of Particle Dynamics - Computational Fluid and Particle Dynamics in the Human Respiratory System - page 139

6.2.1 Lagrangian Method

Particles in the Lagrangian method are treated in the form of points in space that are
tracked individually using Newton ' s second law of motion . The Lagrangian approach
is widely used because the associated equation of motion is in the form of an ordinary
differential equation, which can be integrated to calculate the path of a discrete
particle as it traverses through a flow domain. The force balance equation of the
particle is given as

F n

ρ p V p du p

dt =

(6.1)

where u p is the instantaneous particle velocity and the mass of the particle is given
by the product of the particle density ρ p and the particle volume V p . On the right-
hand side is the sum of the forces that act on the particle, which include the drag,
lift, buoyancy, pressure gradient, thermophoretic, and diffusion forces (which are
discussed in Sect. 6.4). In most practical flows of engineering interest, the drag force
is the most important force exerted on the particle by the surrounding fluid.
Each particle is tracked from its release point until it exits the flow domain, is
terminated after it has impacted onto a surface, or the number of integration time steps
has been reached. The particles can exchange mass, momentum, and energy with
the continuous phase through source terms in the governing equations of fluid flow.
Typically, the shape of the particle is assumed to be spherical; however, non-spherical
particles can be considered through the drag correlation. Particle interactions with
the boundaries occur when the distance between the boundary and the particle centre
is equal or less than the particle radius. By tracking individual particles, we are
able to determine precisely the particle deposition sites in the respiratory airways.
Furthermore, we can introduce additional forces and relevant physics, including
thermal history and mass exchange due to phase change. This method is also called
the Eulerian-Lagrangian method because the fluid phase is modelled by the Eulerian
method, and the dispersed phase is modelled by the Lagrangian method.
The forces that are applied to each particle depend on its interaction with the
surrounding air (particle-fluid interaction) and with other particles (particle-particle
interaction). Let us first consider fluid-particle interaction, which is called coupling,
between phases. For one-way coupling, the fluid influences the particles trajectory
but the particles do not affect the fluid. This occurs when the disperse phase is dilute
and has a very low volume fraction in comparison to the continuous phase. The
volume fraction α is defined as the fraction of volume occupied by a phase within a
given volume. The sum of the volume fraction of all the phases in a volume is equal
to 1. Typically, one-way coupling can be used where the discrete phase volume
fraction is 10 − 3 or less. This means that the volume taken up by the particles is small
enough that its influences, including the forces exerted on the continuous phase, can
be neglected. At increased loading for higher volume fractions, the particles occupy
more volume and the influence of the forces they exert on the fluid phase increases
(Fig. 6.2 ).