6.12 Mixing length
By analogy with kinetic theory, the eddy
viscosity can be expressed in terms of a characteristic speed
and length
. Prandtl produced the following model for the mixing
length
:16
![]() |
(6.21) |




At , the turbulent fluctuations in the
-direction,
, correspond to the range of velocities at
such that
![]() |







![]() |
(6.22) |


The mixing length Eq. (6.21) is
only effective as a turbulence model to calculate when the flow is
simple enough that
can be chosen appropriately.
Such an example is high , fully-developed flow
through a pipe of radius
. The mixing length
, calculated from
measured velocity profiles, follows a polynomial function of
distance
from the wall, given by17
![]() |
(6.23) |
Notably, close to the wall, e.g. , the mixing length
increases linearly according to
![]() |
(6.24) |

At the centre of the pipe, where
is pipe diameter.
Estimates of
are commonly cited for other simple examples,
e.g. mixing layer, jet, flat
plate boundary layer, etc.,
where
is the characteristic length of the problem (radius in the
case of a jet).