Biomedical Engineering Reference
In-Depth Information
Table 5.2
General form of
governing equations for
incompressible flow in
Cartesian coordinates
S
1
0
0
1
ρ
∂p
∂x
+
S
u
u
ν
+
ν
T
−
1
ρ
∂p
∂y
+
S
v
v
ν
+
ν
T
−
1
ρ
∂p
∂z
+
S
w
w
ν
+
ν
T
−
ν
Pr
+
ν
T
Pr
T
T
S
T
k
ν
+
σ
k
ν
T
P
k
−
D
k
ω
ν
+
σ
ω
ν
T
P
k
−
D
k
to the time rate of change of momentum. It was shown that these equations conform
to a generic transport equation. Fundamentally the transport equation describes that
the
transient
and
convective transport
of a fluid variable is balanced by the
diffusive
transport
and the s
ources terms
.
The nature of turbulent flow and its characteristics were introduced to provide
insight into the inherent differences in comparison to laminar flows. The complex
geometry and the nature of the respiratory airways produce flows that are not only
laminar, and turbulent, but also in transitional regimes. The influence of flow charac-
teristics on quantities such as wall shear stress, pressure drop, and velocity profiles
were discussed. A summary of different turbulence models available in CFPD was
provided to introduce the reader to the difficulties associated with the task of sim-
ulating turbulent flows. Practical guidelines on turbulence model selection, mesh
creation, near-wall consideration, and boundary conditions are given to help the
users take the first steps in setting up the computational model for flow analysis
problems correctly.
5.6
Review Questions
∂ρ
∂t
∂
(
ρu
)
∂x
∂
(
ρv
)
∂y
∂
(
ρw
)
∂z
1. Simplify the general continuity equation
+
+
+
=
0, for a
two-dimensional constant density case.
2. In a converging nozzle, the flow accelerates due to the narrowing geometry.
Discuss the changes in the velocity gradients
∂u
∂x
∂v
∂y
during the flow (assume
a constant density). What is its corresponding pressure gradient?
3. Write a force balance equation for all the forces acting on a 2D differential
control volume. What is its equivalent 3D form?
4. For the momentum of a fluid property in the
x
direction discuss how the local
acceleration
and
∂u
∂t
and the convection terms
u
∂u
v
∂u
∂y
∂x
+
contribute to the overall
transport of the fluid.
Search WWH ::
Custom Search