Biomedical Engineering Reference
In-Depth Information
Table 5.1
Governing equations for incompressible flow in Cartesian coordinates
Mass conservation
∂u
∂x
+
∂v
∂y
+
∂w
∂z
=
0
u
-
momentum
∂u
∂t
+
(
ν
+
ν
T
)
∂u
∂x
∂
(
uu
)
∂x
∂
(
vu
)
∂y
∂
(
wu
)
∂z
1
ρ
∂p
∂x
+
∂
∂x
+
+
=−
(
ν
+
ν
T
)
∂u
∂y
(
ν
+
ν
T
)
∂u
∂z
∂
∂y
∂
∂z
+
+
+
S
u
v
-
momentum
∂v
∂t
+
(
ν
∂
(
uv
)
∂x
∂
(
vv
)
∂y
∂
(
wv
)
∂z
1
ρ
∂p
∂y
+
∂
∂x
ν
T
)
∂v
∂x
+
+
=−
+
(
ν
(
ν
∂
∂y
ν
T
)
∂v
∂y
∂
∂z
ν
T
)
∂v
∂z
+
+
+
+
+
S
v
w
-
momentum
∂w
∂t
+
(
ν
+
ν
T
)
∂w
∂x
∂
(
uw
)
∂x
∂
(
vw
)
∂y
∂
(
ww
)
∂z
1
ρ
∂p
∂z
+
∂
∂x
+
+
=−
(
ν
+
ν
T
)
∂w
∂y
(
ν
+
ν
T
)
∂w
∂z
∂
∂y
∂
∂z
+
+
+
S
w
Energy equation
∂T
∂t
+
ν
Pr
+
∂T
∂x
∂
(
uT
)
∂x
∂
(
vT
)
∂y
∂
(
wT
)
∂z
∂
∂x
ν
T
Pr
T
+
+
=
ν
Pr
+
∂T
∂y
ν
Pr
+
∂T
∂z
∂
∂y
ν
T
Pr
T
∂
∂z
ν
T
Pr
T
+
+
+
S
T
k equation
∂k
∂t
+
(
ν
+
σ
k
ν
T
)
∂k
∂x
(
ν
+
σ
k
ν
T
)
∂k
∂y
u
∂k
v
∂k
∂
∂x
∂
∂y
∂x
+
∂y
=
+
+
P
k
−
D
k
ω equation
∂ω
(
ν
+
σ
ω
ν
T
)
∂ω
∂x
(
ν
+
σ
ω
ν
T
)
∂ω
∂y
∂t
+
u
∂ω
∂x
+
v
∂ω
∂
∂x
∂
∂y
∂y
=
+
+
P
ω
−
D
ω
terms not shared between the equations inside the source terms. It is noted that the
additional source terms for the momentum equations
S
φ
,
u
,
S
φ
,
v
and
S
φ
,
w
comprise the
pressure and non-pressure gradient terms and other possible sources such as gravity
that influence the fluid motion, while the additional source term
S
T
in the energy
equation may contain heat sources or sinks within the flow domain. By setting the
transport property
φ
equal to 1,
u
,
v
,
w
,
T, k, ω
and selecting appropriate values for the
diffusion coefficient
and source terms
S
φ
, we obtain the special forms presented
in Table
5.2
for each equation for the conservation of mass, momentum, energy, or
turbulent properties.
5.5
Summary
Derivation of the governing equations of fluid flow was presented in detail: from basic
conservation principles based on an infinitesimal control volume to conserve mass
and energy to the statement that the net force acting on the control volume is equal
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