Mixing length theory

Equation (6-37) represents the friction factor for Newtonian fluids in smooth tubes quite well over a range of Reynolds numbers from about 5000 to 105. The Prandtl mixing length theory and the von Karman and Blasius equations are referred to as semiempirical models. That is, even though these models result from a process of logical reasoning, the results cannot be deduced solely from first principles, because they require the introduction of certain parameters that can be evaluated only experimentally. 

At sufficiently high densities (e.g. cores of upper main-sequence stars), the > sign virtually becomes an equality (adiabatic stratification), but at lower densities (e.g. envelopes of the Sun and cooler stars) an exact calculation is very difficult and in most models a crude approximation based on mixing-length theory is used. In a situation where the chemical composition changes with depth, Eq. (5.24) (known as the Schwarzschild criterion) needs to be replaced by more complicated considerations. 

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. 

Then, mixing length theory may be used with momentum transport to derive... 

The mixing length theory relates the convective (enthalpy) flux F to the envelope structure quite well (Chan and Sofia 1987). For... 

The mixing length theory fails to address the significance of the flux of kinetic energy F (see Figure 1). F is negative and has a KE KE... 

Convection will not drastically alter the results of our calculations. Smarr et al. (1981) found that with complete core overturn the luminosity increased at most by 30%. Mayle (1985) in a series of calculations with mixing length theory found about a 20% enhancement of the luminosity. Our models are unstable by the LeDoux and salt finger criteria but not the Schwartzschild criteria. Future calculations will include convection. 

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 the mixing length theory it is assumed basically that lumps of fluid are carried transversely across the fluid flow by the turbulent eddies and during this motion they preserve their initial momentum and enthalpy. The motion continues over a transverse distance, lm, after which the "lumps interact with other fluid "lumps ... 

Using Eqs. (5.33) and (5.39) it follows that according to the mixing length theory the turbulent Prandtl number is given by ... 

This completes the list of the needed equations. If one employs a theory (like the mixing length theory of convection) to compute Xc, as well as a model for the opacity kop, then eqs. (9) and (12) determine the vertical structure of the nebula, i.e. T, p and F vs. z, provided appropriate boundary conditions are imposed. These are usually taken to be... 

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,. 

In many applications the flow in mass transfer is turbulent and not laminar. The turbulent flow of a fluid is quite complex and the fluid undergoes a series of random eddy movements throughout the turbulent core. When mass transfer is occurring, we refer to this as eddy mass diffusion. In Sections 3.10 and 5.7 we derived equations for turbulent eddy thermal diffusivity and momentum diffusivity using the Prandtl mixing length theory. 

A more rigorous viscous turbulent model of single-phase flow, based on a Prandtl mixing length theory was published by Bloor and Ingham. Like Rietema, these authors obtained theoretical velocity profiles, but they used variable radial velocity profiles calculated from a simple mathematical theory. The turbulent viscosity was then related to the rate of strain in the main flow and the distribution of eddy viscosity with radial distance at various levels in the cyclone was derived. 

The well-known logarithmic velocity profile law follows from Prandtl mixing length theory. It applies well to the constant flux surface layer and has been verified numerous times from measurements taken both in the laboratory and in the field for neutral and near-neutral stability atmospheric surface layers. The result is the following relationship for the skin friction ... 

---