Prandtl's mixing path hypothesis

The Prandtl mixing path hypothesis is one of the most important closure approaches in the context of the turbulence models in order to calculate the additional unknowns that appear in the Reynolds equations .

In fluid mechanics , according to the Prandtl mixing path hypothesis, the eddy viscosity can be represented as the product of a characteristic speed and a characteristic length, the so-called mixing path length. It was set up in 1925 by Ludwig Prandtl . The mixing path length can be at most as long as, for example, the thickness of the shear layer . It is also known as the coherence length, since it can also be interpreted as the path that a ball of turbulence travels on average before it loses its individuality.

The mixing path length can be represented as:

${\displaystyle l_{m}^{2}=l{\sqrt {(\Delta y)^{2}}}}$.

In this way one obtains the 1st Prandtl mixing path formula for the turbulent shear stress : ${\displaystyle \tau }$

${\displaystyle \tau =\rho l_{m}^{2}\left|{\frac {\partial {\overline {u}}}{\partial y}}\right|{\frac {\partial {\overline {u}}}{\partial y}}}$

Assuming that the mixing path length is constant, the Reynolds stress changes proportionally to the square of the mean flow velocity.

We finally get for the eddy toughness :

${\displaystyle \epsilon _{m}=l_{m}^{2}\left|{\frac {d{\overline {u}}}{dy}}\right|}$.

literature

• Julius C. Rotta: Turbulent currents . BG Teubner, Stuttgart 1972, pp. 171f.
• Erich Truckenbrodt : Fluid Mechanics . Springer, 1996, ISBN 3540585125