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
At , the turbulent ﬂuctuations in the -direction, , correspond to the range of velocities at such that
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
Notably, close to the wall, e.g. , the mixing length increases linearly according to
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).