6.14 Turbulent 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) |
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) |
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) |
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.
