Civil Engineering Reference
In-Depth Information
2.5. Transport of the velocity gradient
The transport equat
io
n for the local velocity gradient
u
i
, will be discussed in detail in
this topic, particularly in Chapter 3. It is obtained by taking
the derivative in relation to
x
j
of the Navier-Stokes
equation, which then becomes
x
j
, where
U
i
=
a
ij
= ∂
U
i
∂
U
i
+
2
a
ij
∂
a
ij
∂
U
k
∂
a
ij
2
P
∂
1
ρ
∂
[2.30]
+
x
k
+
a
ik
a
kj
=−
x
j
+ ν
t
∂
∂
x
i
∂
∂
x
k
∂
x
k
By applying the condition of incompressibility
a
ii
=
0
to the
above equation, we obtain the pressure
2
P
1
ρ
∂
[2.31]
a
ik
a
ki
=−
∂
x
i
∂
x
i
By subtracting the two equations term by term, we find
[CAN 92] the following equation:
Da
ij
Dt
a
km
a
mk
δ
ij
3
H
ij
[2.32]
+
a
ik
a
kj
−
=
where
δ
ij
is the Kronecker delta as usual,
D
/
Dt
is the
material derivative and
H
ij
is the source term in the
transport equation where
⎛
⎝
⎞
⎠
2
a
ij
2
P
2
P
δ
ij
3
∂
1
ρ
∂
∂
[2.33]
H
ij
=−
⎜
x
j
−
⎟
+ ν
∂
x
i
∂
∂
x
k
∂
x
k
∂
x
k
∂
x
k
Let us now decompose the velocity gradient tensor into a
symmetrical part
(
)
2
S
≡
S
ij
= ∂
U
i
∂
x
j
+ ∂
U
j
∂
x
i
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