Prandtls Mixing Length Hypothesis for Turbulent Flow

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 

Let s consider the fully developed velocity profile in the middle of a wide open channel, with X-, y-, and z-components in the longitudinal, lateral, and vertical directions, respectively. It is fully developed because du/dx is close to zero. The fact that it is a wide channel means that du/dy is also very small in the middle. From equations (5.22) and (5.23), we can see that the turbulent transport of momentum in the x- and 

half of the w values will be positive, and the other half will be negative. We will use this criteria to divide them into two parts  

Equation (5.28) is a relation for a difference in velocity, which can be written as a velocity gradient times a distance  

Because turbulent eddies tend to approach spherical in shape  

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