BSA Flow Software Installation and User's Guide - CSI

π
d
6
considering the instantaneous velocity V ≡ Up - Uf, of the particle relative to
the fluid. (From Durst, Melling & Whitelaw, 1981):
t
dU
p
π dUfπdV
3
dV
ρ =− 3πµ
dp V + dpρf
− d pρf − d p πµρf
dt
6 dt 12 dt 2 ∫ dξ
3 3 3 2
p p
142 3
Accelerating
force
44 142 34 1 24
1 24
1 2
Stokes
viscous
drag
Lorenz-Mie light scattering theory
4 3
Pressure
gradient
force on
fluid
4 3
Fluid
resistance to
accelerating
sphere
t
0
dξ
t − ξ
4444 44443
Drag force
associated with
unsteady motion
-where subscript p refer to the seeding particle, and subscript f refer to the
fluid.
(7-15)
The first term in this equation represent the force required to accelerate the
particle, and the second term describe the viscous drag as given by Stokes
law. Acceleration of the fluid produce a pressure gradient in the vicinity of
the particle, and hence additional force on the particle as described by the
third term. The fourth term is the resistance of an inviscid fluid to
acceleration of the sphere, and is predicted by potential flow theory. The last
term is the “Basset history integral” representing the drag force arising from
derivation of the flow pattern from that occurring in steady flow.
Note that when the first, third and fourth terms are combined, the
accelerating force is equivalent to that of a sphere whose mass is increased
by an additional “virtual mass” equal to half the mass of the displaced fluid.
The above equation is valid within the following assumptions:
• The turbulence is homogeneous and time-invariant.
• Particles are smaller than the turbulence microscale
• Stokes drag law applies (particles are spherical)
• Particles are always surrounded by the same fluid molecules
• There is no interaction between particles.
Furthermore external forces, such as gravitational, centrifugal and
electrostatic forces have been ignored.
Depending on the nature of the flow, seeding particles used for LDAmeasurements
usually have particle diameters ranging from 0.1 to 50 µm.
This is comparable to the wavelength of the light used, which for a He-Ne
laser is 632.8 nm.
With particle sizes comparable to the wavelength of light, the Lorenz-Mie
light scattering theory apply. This theory consider spherical particles, and
thus describe only the dependency on particle size, but in practice also the
shape and orientation of seeding particles play a major role in the scattering
of light.
In general large particles scatter more light than smaller ones, but particle
size also affect the spatial distribution of the scattered light as shown in
Figure 7-13. For large particles the ratio of forward to backward scattered
light can be in the order of 10 2 to 10 3 , while smaller particles scatter more
evenly.
7-18 BSA Flow Software: Reference guide