Mixing length ($l$)

A mixing length is the characteristic transverse displacement over which a turbulent parcel retains its streamwise momentum before mixing with its surroundings. Taylor expansion of a mean velocity profile gives a fluctuation of magnitude $l|U_y|$. Combining comparable transverse fluctuations with down-gradient Reynolds stress transport gives the closure $\overline{u^{\prime }{v}^{\prime }}=-l^2|U_y|U_y$. This is a phenomenological approximation, not a consequence of the Navier-Stokes equation. In a free jet, a prescribed local mixing length must also be compatible with centreline regularity and vanishing outer momentum flux.

A mixing-length closure uses a prescribed mixing length $\ell$ to approximate turbulent transport. For mean shear ${\partial }_{z}U$, a standard model gives eddy viscosity ${\nu }_t={\ell }^2|{\partial }_{z}U|$ and down-gradient Reynolds stress. Scalar-diffusion versions replace momentum transport by an appropriate scalar flux. The choice of length and correlations is a modelling assumption rather than an exact closure of the Navier-Stokes equation.