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From Kuo’s book
From Turns book
Turbulence Closure ProblemLet the instantaneous velocity be broken up into a mean and a
fluctuating component:
The momentum equation can then be written as:
where body forces have been neglected, and where
Substituting and then averaging the entire equation over time gives the following equation:
The turbulence closure problem is how to model .
( ) ( )( )[ ] ( ) ( )[ ]ijijiji
jjjji
jj PPxuuuu
xuu
tττδ ′++′+−
∂∂
=′+′+∂∂
+′+∂∂
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂∂
≡i
j
j
iij x
uxuντ
( ) ( ) [ ]ijiji
jijii
j Pxuuuu
xu
tτδ +−
∂∂
=′′+∂∂
+∂∂
uuu ′+=
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
′∂+
∂′∂
≡′i
j
j
iij x
uxuντ
jiuu ′′
Boussinesq Hypothesis
Reynolds Stress Model
Boxes represent terms that must be modeled
Idea: Write transportequations for eachof the 6 Reynolds stress terms
Algebraic Stress Model
Idea:Write algebraic equationsfor each of the Reynoldsstress terms
Note:No terms with ( )uux ′′∂
∂
Standardk-ε
Non-reactingswirling flow
Negative centerlinevelocity region
Non-reacting
Non-reacting
Non-reacting
From Sloan et al., PECS, 1984
Reacting