Prandtl s mixing length

Explain the importance of the universal velocity profile and derive the relation between the dimensionless derivative of velocity trf, and the dimensionless derivative of distance from the surface y+, using the concept of Prandtl s mixing length kE. 

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. 

C. Prandtl s Mixing Length Hypothesis for Turbuient Fiow... 

Prandtl s mixing length hypothesis (Prandtl, 1925) was developed for momentum transport, instead of mass transport. The end result was a turbulent viscosity, instead of a turbulent diffusivity. However, because both turbulent viscosity and turbulent diffusion coefficient are properties of the flow field, they are related. Turbulent viscosity describes the transport of momentum by turbulence, and turbulent diffusivity describes the transport of mass by the same turbulence. Thus, turbulent viscosity is often related to turbulent diffusivity as... 

C. PRANDTL S MIXING LENGTH HYPOTHESIS FOR TURBULENT FLOW... 

Equation (5.36) is Prandtl s mixing length hypothesis, and it works well, considering that the basis for the equation is so empirical. However, equation (5.36) does present a challenge for us that mixing length, L, stiU needs to be specified. Measurements have shown us the following ... 

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... 

Application of the Governing Equations to Turbulent Flow 123 Prandtl s Mixing-Length Model... 

Supported by experimental observations, v y is assumed to be of the same order of magnitude as v in the constant stress layer near a solid boundary. Hence, we may define Prandtl s mixing length model as ... 

For power law fluids, Tomlta (1959) extended his laminar flow model (discussed in Section 5.2.1.3) to turbulent flows in smooth pipes by applying Prandtl s mixing length concept, and developed a different implicit equation ... 

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