Fundamentals of Particle Dynamics - Computational Fluid and Particle Dynamics in the Human Respiratory System - page 162

Fig. 6.16 Random discrete
eddies assumption in a
turbulent flow field for the
Eddy Lifetime model

square fluctuations via the following Langevin equation:

2 u i

2

du i
dt =−

u i − U i
T L

+ ξ i ( t ) (6.56)

T L

Here T L is the Lagrangian integral time scale, and ξ i ( t ) is a vector Gaussian white-
noise random process with spectral density 1 /π . The Lagrangian integral time scale
T L is estimated using

C 1 k
ε

T L = (6.57)

where C 1 is a constant. The solution of Eq. (6.56) provides the fluctuation velocity
vector at every time step.
Eddy Interaction Model (EIM): The EIM assumes that the flow field is comprised
of turbulent eddies that interacts with the particles (Fig. 6.16 ). As a particle moves
through the flow domain it interacts with the discrete eddy that surrounds it. The
fluctuation eddy velocity is given as,
u i = G u i

2

(6.58)
where G is a zero mean, unit variance normally distributed random number, u i is
the root mean-square (RMS) local fluctuation velocity in the i th direction. The time
scale τ e associated with each eddy (eddy lifetime) is given as
τ e =

2 T L (6.59)

where T L is given by Eq. (6.57). In addition to the eddy lifetime, a particle eddy
crossing time t cross is defined as

τ ln 1

L e
τ u f

u p

t cross =− − (6.60)

−

Here τ is the particle relaxation time, L e is the eddy length scale, and is the

magnitude of the relative slip velocity. The frequency of the particle encountering
turbulence eddies is the reciprocal of the lesser of τ e and t cross . Therefore, Eq. (6.56)
is used to generate a new random fluctuation velocity when the minimum of the eddy
lifetime or eddy crossing is reached.

| u f − u p |