## 6.12Mixing 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)
The mixing length Eq. (6.21 ) is derived from the kinetic view of viscosity in Sec. 6.10 . Turbulent ﬂuctuations in the -direction, , carry ﬂuid to the plane from a distance .

At , the turbulent ﬂuctuations in the -direction, , correspond to the range of velocities at such that

For a positive , a positive at corresponds to a negative , and vice versa. The Reynolds stress can then be constructed by deﬁning a mixing length to replace , absorbing the constant of proportionality, to give
 (6.22)
From , we arrive at in Eq. (6.21 ).

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)
where is Kármán’s constant.

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).

16Ludwig Prandtl, Bericht uber Untersuchungen zur ausgebildeten Turbulenz, 1925.
17Johann Nikuradse, Strömungsgesetze in rauhen Rohren, 1933.

Notes on CFD: General Principles - 6.12 Mixing length