Prandtl mixing length

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... 

Equation (6-37) represents the friction factor for Newtonian fluids in smooth tubes quite well over a range of Reynolds numbers from about 5000 to 105. The Prandtl mixing length theory and the von Karman and Blasius equations are referred to as semiempirical models. That is, even though these models result from a process of logical reasoning, the results cannot be deduced solely from first principles, because they require the introduction of certain parameters that can be evaluated only experimentally. 

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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 equation set (5-73) is called the universal velocity profile and matches very well with experimental data however, we should note once again that the constants in the equations must be determined from experimental velocity measurements. The satisfying point is that the simple Prandtl mixing-length model yields an equation form which fits the data so well. 

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 Prandtl mixing-length model and the relation I a ny [123], yields ... 

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... 

This simple model for bubble induced viscosity is a Prandtl mixing length model with the bubble radius as the mixing length and 1.2ag v — v j as... 

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. 

If we combine this relationship with the Prandtl mixing length expression (1.356) in which the mixing length is approximated by / = Ky, we get the following dimensionless velocity gradient ... 

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... 

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