7.10 Specific dissipation rate

The k model is one of a family of two-equation models for turbulence. With two equations, the models can represent each of the scales, eqn and eqn, which characterise eqn. Most often, eqn is used to represent eqn.

The other variable must represent eqn and so far we have used with SI units eqn. The specific dissipation rate eqn, with SI units of eqn, is a popular alternative for this variable in turbulence modelling.

While Kolmogorov first proposed a two-equation eqn model,15 the eqn models used in CFD originate from Wilcox.16 Here, “models” is plural since there are several versions of eqn model with modifications and additions from its original form.

The original eqn model is presented below (with some changes to the original variable names), assuming eqn = constant for direct comparison with k in Sec. 7.1 .

@k-+ r (uk) r (D rk) = @t k 2- G 3 k(r u) c !k \relax \special {t4ht=
@!- @t + r (u!) r (D!r!) = ! 2 G-- - -!(r u) !2 k 3 \relax \special {t4ht=

The standard model coefficients are

Dk = + k t; D! = + ! t c = 0:09; = 3=40; = 5=9; = 0:5; = 0:5: k ! \relax \special {t4ht=
Comparing dissipation terms in Eq. (7.1 ) and Eq. (7.36) gives the relation . Substituting in Eq. (6.31 ) leads to a simple relation for turbulent viscosity, given by
 k- t = ! : \relax \special {t4ht=
Inlet and initial estimates for eqn can be calculated by
 1 k1=2 !in = c =4---; lm \relax \special {t4ht=
using eqn, in a manner similar to in Eq. (7.4 ).

With wall functions, the boundary condition applied to eqn set a near-wall cell value according to

 (c 1=4k1=2= yP for y+ > y+; ! = P2 P+ tr+ 6 = yP for yP ytr: \relax \special {t4ht=
The expression for eqn (eqn) is a solution to the following equation for the viscous sub-layer where diffusion and dissipation terms dominate in Eq. (7.37 ):
 2 @-!- !2 = 0: @y2 \relax \special {t4ht=
The equivalent dissipation terms for eqn in Eq. (7.37) and in Eq. (7.2 ) are eqn and  2 c2 respectively. The former is more stable in a numerical solution since it is insensitive to variations in eqn.
15Andrey Nikolaevich Kolmogorov, Equations of turbulent motion in an incompressible fluid, first published in Russian in Izv. Akad. Nauk SSSR 6, 1941.
16David Wilcox, Reassessment of the scale-determining equation for advanced turbulence models, 1988.

Notes on CFD: General Principles - 7.10 Specific dissipation rate