## 6.14Turbulent dissipation rate

A complete model for is still required to solve the ensemble-averaged momentum equation, e.g. Eq. (6.26 ). The discussion in Sec. 6.11 indicates is the product of and , requiring two models to represent each scale. Since , can represent the speed scale, modelled by Eq. (6.28 ).

The turbulent dissipation rate matches the rate of transfer of kinetic energy down the energy cascade , as discussed in Sec. 6.6 . This applies to all turbulent scales including the larger mixing length scales, so . Substituting and expressions into yields turbulent viscosity

 (6.31)
where is a constant. From empirical data, , except within the viscous and buﬀer layers close to a wall, see Sec. 7.4 .

### Transport of turbulent dissipation rate

A model is now required for , both to calculate by Eq. (6.31 ) and to provide the remaining unknown in Eq. (6.28 ). The model can be provided by a transport equation for

 (6.32)
where is the combined molecular and turbulent diﬀusion with an adjustable coeﬃcient , usually set to 1.3. The remaining coeﬃcients and are tuned to capture the behaviour of a range of ﬂows.

The -equation, Eq. (6.32 ), can be derived in terms of statistical properties, replacing high-order terms in by models with coeﬃcients , .18

Alternatively, it can be obtained by multiplying the principal variable, or , in each term of Eq. (6.28 ) by and introducing coeﬃcients , and .

The term in Eq. (6.32) causes to increase with . This is logical since the generated turbulence moves down the energy cascade, so ultimately aﬀects the rate of dissipation.

The term is justiﬁed by considering the free decay of turbulence. If the ﬂuid stops moving () and turbulence is no longer generated (), then (assuming constant and ignoring diﬀusion) Eq. (6.28 ) and Eq. (6.32) reduce to

 (6.33)
respectively. Integrating the combined equations yields where the “0” subscript indicates initial values.

Further integration yields a decay in over time to the power and a timescale , which is a reasonable approximation to real behaviour.

18B. I. Davydov, Statistical dynamics of an incompressible turbulent ﬂuid, Dokl. Akad. Nauk SSSR 136, 1961.

Notes on CFD: General Principles - 6.14 Turbulent dissipation rate