6.12 Mixing length

By analogy with kinetic theory, the eddy viscosity eqn can be expressed in terms of a characteristic speed eqn and length eqn. Prandtl produced the following model for the mixing length eqn:16

 2 @ux- t = lm @y : \relax \special {t4ht=
The mixing length Eq. (6.21 ) is derived from the kinetic view of viscosity in Sec. 6.10 . Turbulent fluctuations in the eqn-direction, eqn, carry fluid to the eqn plane from a distance eqn.

PICT\relax \special {t4ht=

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

 0 juxj lj@ux=@yj: \relax \special {t4ht=
For a positive eqn, a positive eqn at eqn corresponds to a negative eqn, and vice versa. The Reynolds stress eqn can then be constructed by defining a mixing length eqn to replace eqn, absorbing the constant of proportionality, to give
 ----- @ux @ux tyx = u0xu0y = l2m ---- ----: @y @y \relax \special {t4ht=
From eqn, we arrive at eqn in Eq. (6.21 ).

The mixing length Eq. (6.21) is only effective as a turbulence model to calculate eqn when the flow is simple enough that eqn can be chosen appropriately.

Such an example is high eqn, fully-developed flow through a pipe of radius eqn. The mixing length eqn, calculated from measured velocity profiles, follows a polynomial function of distance eqn from the wall, given by17

 2 4 lm-= 0:14 0:08 1 y- 0:06 1 y- : R R R \relax \special {t4ht=

PICT\relax \special {t4ht=

Notably, close to the wall, e.g. eqn, the mixing length increases linearly according to

lm = y; \relax \special {t4ht=
where eqn is Kármán’s constant.

At the centre of the pipe, eqn where eqn is pipe diameter. Estimates of eqn are commonly cited for other simple examples, e.g. mixing layer, jet, flat plate boundary layer, etc., where eqn 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