The calculation of the instantaneous fluid velocity, in Equation 6–6, depends on the flow regime and the type of particle tracking desired (mean or with turbulent dispersion). In laminar flows or in flows where mean particle tracking is calculated, is equal to the mean local fluid velocity, , surrounding the particle. The path of a particle is deterministic (that is, there is a unique path for a particle injected at a given location in the flow).

In turbulent tracking, the instantaneous fluid velocity is decomposed into mean, , and fluctuating, , components. Now particle trajectories are not deterministic and two identical particles, injected from a single point, at different times, may follow separate trajectories due to the random nature of the instantaneous fluid velocity. It is the fluctuating component of the fluid velocity that causes the dispersion of particles in a turbulent flow.

The model of turbulent dispersion of particles that is used, which is due to Gosman and Ioannides [147], assumes that a particle is always within a single turbulent eddy. Each eddy has a characteristic fluctuating velocity, , lifetime, , and length, l _e. When a particle enters the eddy, the fluctuating velocity for that eddy is added to the local mean fluid velocity to obtain the instantaneous fluid velocity used in Equation 6–19. The turbulent fluid velocity, , is assumed to prevail as long as the particle/eddy interaction time is less than the eddy lifetime and the displacement of the particle relative to the eddy is less than the eddy length. If either of these conditions is exceeded, the particle is assumed to be entering a new eddy with new characteristic , , and l _e.

The turbulent velocity, eddy length and lifetime are calculated based on the local turbulence properties of the flow:

(6–33)

(6–34)

where k and are the local turbulent kinetic energy and dissipation, respectively, and is a turbulence constant. The factor was chosen to relate the characteristic length scale to the eddy dissipation length [39]. The variable is a normally distributed random number which accounts for the randomness of turbulence about a mean value. Because of this randomness, each component of the fluctuating velocity may have a different value in each eddy.