Biomedical Engineering Reference
In-Depth Information
Rearranging the equation for
y
+
= 5, the thickness of the viscous sublayer
y
is
u
δ
u
τ
→
5
ν
u
τ
=
25
ν
u
δ
u
+
=
5
=
yu
τ
/ν
,
5
=
y
=
(5.39)
where
u
δ
is the flow velocity at the edge of the viscous sublayer. This shows an
inverse relationship between the thickness of the viscous sublayer and the mean
flow velocity while the thickness is directly proportional to the kinematic viscosity.
This means at higher velocities (and therefore a higher Reynolds number) a thinner
viscous sublayer will be produced. This is an important consideration when dealing
with the near-wall mesh requirement to resolve this flow feature. Furthermore, the
presence of a mucous layer creates a non-smooth surface, and since CFD modelling
of the respiratory airways normally treats the walls as a smooth surface, this should
be kept in mind.
The outer flow is the turbulent core where turbulence effects dominate over molec-
ular viscous effects. Closer to the wall where the viscous effects are still negligible,
but wall effects becomes important, we reach to the
transition sublayer
(5 <
y
+
< 30).
In this region both turbulence and viscous effects are equally important. Furthermore
the production of turbulence kinetic energy reaches its peak value due to large gra-
dients in the mean velocity field and the presence of finite turbulence shear stress.
Thereafter there is the
logarithmic sublayer
(30 <
y
+
< 300
400). In this region,
the flow is characterized by one velocity scale (shear velocity) and one length scale
(distance from the wall), and the mean flow velocity has a logarithm profile that can
be expressed as
∼
u
+
=
2
.
5ln(
y
+
)
for
30
<y
+
<
400
+
5
.
45
(5.40)
Figure
5.15
shows the different regions of the mean turbulent velocity profile near a
wall. The logarithmic profile approximates the velocity distribution well and provides
a method to compute the wall shear stress.
Resolving the near-wall region in turbulent flows is important for many applica-
tions such as particle deposition in airway passages and for estimating the pressure
drop that depends on the local wall shear stress. In addition, most turbulence models
such as the
k-ε
are only valid in the turbulent core region and the log layer, which
means that they cannot be applied down to the wall. Two approaches are generally
used to model the flow in the near wall region.
The first is to use a low Reynolds number (LRN) turbulence model which resolves
the wall region with very small mesh elements in the direction normal to the wall.
These elements are typically stretched hexahedron mesh elements in 3D or quad
elements in 2D and referred to as prism layers or inflation layers (Fig.
5.16
). The
LRN turbulence model equations based on the dimensionless wall distance
y
+
resolve
the velocity profile through the viscous sublayer and all the way down to the wall.
The computational demands are greater for the LRN models because of the increased
number of mesh elements needed to provide the resolution in the near wall region to
capture the sharp variations in the velocity profile.
The second approach is to use
wall functions
which are specialized empirical cor-
relations that serve as boundary conditions to connect the flow properties at the wall
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