Turbulence model Prandtl mixing length

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... 

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... 

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]. 

These models are usually categorized according to the number of supplementary partial differential transport equations which must be solved to supply the modeling parameters. The so-called zero-equation models do not use any differential equation to describe the turbulent quantities. The best known example is the Prandtl (19) mixing length hypothesis ... 

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]. 

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

Prandtl s model derivation can then be briefly sketched, introducing the Boussinesq [19] [20] approximation for the turbulent viscosity. Starting out with the simple kinetic theory relation that the molecular viscosity equals the molecular velocity times the mean free path, an analogous relation can be formulated for the turbulent viscosity in terms of the turbulent mixing length and a suitable velocity scale, Ut Iv. ... 

The first analytical study to predict the performance of tubes with straight inner fins for turbulent airflow was conducted by Patankar et al. [118]. The mixing length in the turbulence model was set up so that just one constant was required from experimental data. Expansion of analytical efforts to fluids of higher Prandtl number, tubes with practical contours, and tubes with spiraling fins is still desirable. It would be particularly significant if the analysis could predict with a reasonable expenditure of computer time the optimum fin parameters for a specified fluid, flow rate, etc. 

The first successful eddy viscosity turbulence model, referred to as an algebraic mixing length model, was introduced by Prandtl in the 1920s [43]. Prandtl postulated that... 

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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