�
�
�
�
Therefore,
∂
∂x
(ρu) +
∂
∂y
(ρv) = 0
(10.2)
and
∂
∂x
(
ρ " u "
)
+
∂
∂y
(
ρ " v "
)
= 0
(10.3)
10.1.2 Momentum Equation: RANS
Therefore,
∂
∂
∂
∂
∂
∂
∂
ρu
2
+
ρuv +
ρu "2 +
ρu " v " +
ρ " u "2 +
ρ " u " v " +
2uρ " u "
∂x
∂y
∂x
∂y
∂x
∂y
∂x
(
)
∂
∂P
∂ 2 u ∂ 2 u
+
uρ " v " + vρ " u " = −
+ ρf x + ρ " f " + μ
+
x
∂y
∂x
∂x 2
∂y 2
Rearranging,
(
)
(
)
∂u
∂u
∂P
∂ 2 u ∂ 2 u
ρu + ρ " u "
+ ρv + ρ " v "
= −
+ ρf x + ρ " f " + μ
+
x
∂x
∂y
∂x
∂x 2
∂y 2
(10.5)
⎫
(
) ⎪ ⎪
+
∂ ρu "2 + ρ " u "2 + uρ " u " ⎬
∂x (
) 6 Reynolds Stress
∂ ρu " v " + ρ " u " v " + vρ " u " ⎪ ⎪
+
⎭
∂y
For incompressible flow,

ρ " = 0

X-Momentum:

198
Analytical Heat Transfer