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Y-Momentum:
∂v
∂v ∂v
1 ∂P μ ∂ 2 v ∂ 2 v
∂u " v "
∂v "2
u
+ v
+
= −
+
+
−
−
+ f y
∂x
∂y
∂t
ρ ∂y
ρ ∂x 2
∂y 2
∂x
∂y
'
-v
'
'
-v
'
Reynolds stress, Turbulent stress
viscous forces
(10.7)
From boundary-layer approximation, Y-Momentum equation is not important as compared with X-Momentum equation.
10.1.3 Enthalpy/Energy Equation
Therefore,
(
)
∂T
∂T
c p (ρu + ρ " u " )
+ ρv + ρ " v "
∂x
∂y
∂ ∂T
∂ ∂T
∂P
∂P
=
k
+
k
+ u
+ v
+ Φ + ρε
(10.9)
∂x ∂x
∂y ∂y
∂x
∂y
(
)
2
∂
∂u
+ c p
ρu " T " + ρ " u " T " + uρ " T " +
+ . . .
∂x
∂x
2
(
)
∂
∂u "
+ c p
ρv " T " + ρ " v " T " + vρ " T " +
+ . . .
∂y
∂x
'
-v
'
6 Reynolds Flux
where Φ is the viscous dissipation and ε is the turbulent dissipation.
199
Turbulent Flow Heat Transfer