Prandtl mixing length turbulent

Prandtl mixing length, length scale of turbulence... 

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. 

A proper representation of the effective viscosity is often problematic. Based on the Prandtl mixing length model for turbulence, Bloor and Ingham-suggest that the variation in p, should be of the form... 

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. 

The distance (. is called the Prandtl mixing length. Prandtl also postulated that v would be of the same order of magnitude as u so that the turbulent-shear stress of Eq. (5-60) could be written... 

The constant A in Eq. 10.2.20 is the von Karman constant, equal to 0.4. Though the Prandtl mixing length hypothesis (Eq. 10.2.20) works for conditions in the turbulent core... 

By use of the eddy viscosity hypothesis (1.380) and the Prandtl mixing length model (1.356), a similar expression for the turbulent viscosity can be deduced, and given by... 

The existing turbulence models consist of approximate relations for the /ij-parameter in (5.246). The Prandtl mixing-length model (1.356) represents an early algebraic (zero-equation) model for the turbulent viscosity Ht in turbulent boundary layers. 

The Smagorinsky Model (cf. Ref. [51]) is an algebraic model in the same spirit as the Prandtl mixing length model discussed in section RANS Turbulence Modeling. In the Smagorinsky model, the SGS stresses are assumed to be proportional to the rate of strain, that is, = VtS, and the kinematic eddy viscosity is determined from the expression... 

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. 

The classical model of a normal turbulent jet was recently extended to a S3mthetic jet [6, 7]. This is summarized in this section. Axisymmetric free turbulent jets can be solved analytically by using the Prandtl mixing-length model. The model findings are summarized by... 

The molecular momentum diffusivity ffp in mVs is a function only of the fluid molecular properties. However, the turbulent momentum eddy diffusivity e, depends on the fluid motion. In Eq. (3.10-29) we related e, to the Prandtl mixing length L as follows ... 

B Prandtl Mixing Length and Turbulent Eddy Mass Diffusivity... 

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 Prandtl mixing length model, as well as the k-e model, coupled with (8) and (5), have given quite good results predicting turbulent diffusion fluxes is a number of cases however quite clear discrepancies of (5) have been emphasized in other important cases, and other models can now be proposed. For more details concerning turbulence models, a good basic book is the one of Tennekes and Lumley [1]. 

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