Der ivation of the "-Equation
14.2 Der ivation of the "-Equation
The Navier Stokes equation
D C (14.16)
@u 0 is differentiated and multiplied with the derivative i
(14.17) @x k @t
and then it is averaged. The following identities are used in transformations: @
Cu i /
@x k @t @x k
@t
@x k
@x k
@t @x k @x k
1 @ @u 0 i @u 0 i
2 @t @x k @x k
@t
@ @u
@N u
@N
@x k @x j @x k
@x k
@x j @x
@x
@x j @x k
@u 0 0 i 0 i @
@u 0
@x k
@x j @x k
@x k
@x j @x k
@ @u 0 i 0 i
@N u j @u 0 0 i 0 @u 0 2
@u i
CN u
@x k @x j @x k
@x k @x j @x k
@x j @x k @x k
@N u
i 0 @u
i @u 0 1 @u 0 i @u D 0 i @ C i
D (14.20)
@x k @x j @x k
2 @x j @x k @x k
@N u i @u 0 @u 0 1 @
@x k @x j @x k
@x j
Equations of the k – ε Model
(14.23) % @x k @x j
(14.24) @x j
(14.25) This gives:
D " D u 0 j " 0 s 2 (14.27)
s D (14.29)
@x j @x k @x j @x k
@x k @x k
The terms on the r.h.s. were approximated according to the following for- mula:
Equations of the k – ε Model
C "1 " @N u i
C " "2 2
(14.30) @x j
Constants are taken from planar jet and mixing layer:
(14.31) Hereby the full closed system of the k " model reads:
C "1
D 1:44; C "2
Under assumption that the generation of the turbulent energy equals to the its dissipation (the turbulence is in equilibrium, turbulent scales are in the inertial range) Kolmogorov and Prandtl derived the relation between the kinetic energy, the dissipation rate and the integral lengths L:
k 3=2
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Equations of the k – ε Model
From the dimension analysis
t is com- puted from (14.35) and Reynolds stresses can be calculated from the Boussi-
nesq hypothesis and then substituted into the Reynolds averaged Navier Stokes equations. The problem is mathematically closed.
The k " model is the classical approach, which is very accurate at large Re numbers. At small Re number, for instance close to the wall, the approxima- tions used in derivation of k " model equations are not valid. To overcome this disadvantage various low Reynolds k " models were proposed.