Turbulent flow Prandtl mixing length

The universal turbulent velocity profile near the pipe wall presented in the preceding subsection Tncompressible Flow in Pipes and Channels may be developed using the Prandtl mixing length approximation for the eddy viscosity,... 

The Prandtl mixing length concept is useful for shear flows parallel to walls, but is inadequate for more general three-dimensional flows. A more complicated semiempirical model commonly used in numerical computations, and found in most commercial software for computational fluid dynamics (CFD see the following subsection), is the A — model described by Launder and Spaulding (Lectures in Mathematical Models of Turbulence, Academic, London, 1972). In this model the eddy viscosity is assumed proportional to the ratio /cVe. 

Obtain a dimensionless relation for the velocity profile in the neighbourhood of a surface for the turbulent flow of a liquid, using Prandtl s concept of a Mixing Length (Universal Velocity Profile). Neglect the existence of the buffer layer and assume that, outside the laminar sub-layer, eddy transport mechanisms dominate. Assume that in the turbulent fluid the mixing length Xe is equal to 0.4 times the distance y from the surface and that the dimensionless velocity u1 is equal to 5.5 when the dimensionless distance y+ is unity. 

We have already likened the macroscopic transport of heat and momentum in turbulent flow to their molecular counterparts in laminar flow, so the definition in Eq. (5-60) is a natural consequence of this analogy. To analyze molecular-transport problems (see, for example. Ref. 7, p. 369) one normally introduces the concept of mean free path, or the average distance a particle travels between collisions. Prandtl introduced a similar concept for describing turbulent-flow phenomena. The Prandtl mixing length is the distance traveled, on the average, by the turbulent lumps of fluid in a direction normal to the mean flow. 

Despite the fact that equation (3.37) is applicable to all kinds of time-independent fluids, numerous workers have presented expressions for turbulent flow friction factors for specific fluid models. For instance, Tomita [1959] applied the concept of the Prandtl mixing length and put forward modified definitions of the friction factor and Reynolds number for the turbulent flow of Bingham Plastic fluids in smooth pipes so that the Nikuradse equation, i.e. equation (3.37) with n = 1, could be used. Though he tested the applicability of his method using his own data in the range 2000 < Reg(l — 4>f 3 — )< 10, the validity of this approach has not been established using independent experimental data. 

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. 

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... 

This hypothesis works surprisingly well for many boundary layer flows. Prandtl suggested the estimation of characteristic length (mixing length) of turbulence (/) by postulating it to be proportional to the distance from the nearest wall. Several... 

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