6.14 Turbulent dissipation rate

A complete model for eqn is still required to solve the ensemble-averaged momentum equation, e.g. Eq. (6.26 ). The discussion in Sec. 6.11 indicates eqn is the product of eqn and eqn, requiring two models to represent each scale. Since eqn, eqn 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 eqn, as discussed in Sec. 6.6 . This applies to all turbulent scales including the larger mixing length scales, so  3 . Substituting eqn and eqn expressions into eqn yields turbulent viscosity

 2 t = c k-;
where eqn is a constant. From empirical data, eqn, 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 eqn by Eq. (6.31 ) and to provide the remaining unknown in Eq. (6.28 ). The model can be provided by a transport equation for

|----------------------------------------| | D
where eqn is the combined molecular and turbulent diffusion with an adjustable coefficient , usually set to 1.3. The remaining coefficients eqn and eqn 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 eqn by models with coefficients eqn, eqn.18

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

The c1 G term in Eq. (6.32) causes to increase with eqn. This is logical since the generated turbulence moves down the energy cascade, so ultimately affects the rate of dissipation.

The c2 term is justified by considering the free decay of turbulence. If the fluid stops moving (eqn) and turbulence is no longer generated (eqn), then (assuming constant eqn and ignoring diffusion) Eq. (6.28 ) and Eq. (6.32) reduce to

dk=dt =
respectively. Integrating the combined equations yields where the “0” subscript indicates initial values.

PICT\relax \special {t4ht=

Further integration yields a decay in eqn over time to the power eqn and a timescale  ? t = nk0=, 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