Prandtl’s mixing length theory

Prandtl s mixing length theory, the basis of which is outlined in Section 2.9, predicts that the three eddy diffusivities are equal. It is important to appreciate that these eddy diffusivities are not genuine physical properties of the fluid their values vary with position in the flow, as illustrated in Example 1.10. 

Another method of trying to describe the turbulence terms in the above equations is by means of Prandtl s mixing length theory. The mixing length concept will be introduced in this section and some simple turbulence models based on this concept will be discussed [1],[2],[3],[6],[7]. 

In order to calculate wx/wT by solving the differential equation (3.149), the Reynolds stress w xw has to be known. The hypothesis introduced by Boussinesq (3.140) is unsuitable for this, as according to it, the Reynolds stress does not disappear at the wall. However, the condition w xw y = 0 at the wall is satisfied by Prandtl s mixing length theory, which will now be explained. In order to do this we will consider a fluid element in a turbulent boundary layer, at a distance y from the wall, Fig. 3.16. It has, at a distance y, the mean velocity wx(y) and... 

Similarities among momentum, heat, and mass transport. Equation (5.7-26) is similar to Eq. (5.7-20) for total momentum transport. The eddy thermal diffusivity a, and the eddy momentum diffusivity s, have been assumed equal in the derivations. Experimental data show that this equality is only approximate. An eddy mass diffusivity for mass transfer has also been defined in a similar manner using the Prandtl mixing length theory and is assumed equal to a, and e,. 

Prandtl s model derivation can then be briefly sketched, introducing the Boussinesq [19] [20] approximation for the turbulent viscosity. Starting out with the simple kinetic theory relation that the molecular viscosity equals the molecular velocity times the mean free path, an analogous relation can be formulated for the turbulent viscosity in terms of the turbulent mixing length and a suitable velocity scale, Ut Iv. ... 

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