Turbulent mixing length

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. ... 

This relation has been used successfully for many boundary layer flows. However, it is today generally accepted that the physical equivalence between the turbulent mixing length and the molecular mean free path is completely erroneous. The turbulent eddies are not small compared to the width of the mean flow, and they interact continuously rather than collide instantaneously. This model limitation was already recognized by Taylor [157]. 

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 ... 

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. 

Turbulence modeling capability (range of models). Eddy viscosity k-1, k-e, and Reynolds stress. k-e and Algebraic stress. Reynolds stress and renormalization group theory (RNG) V. 4.2 k-e. low Reynolds No.. Algebraic stress. Reynolds stress and Reynolds flux. k- Mixing length (user subroutine) and k-e. 

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... 

The ratio of Ihe mixing length to the distance from the pipe wall has a constant value of 0.4 for the turbulent flow of a fluid in a pipe. What is the value of the pipe friction factor if the ratio of the mean velocity to the... 

It may be assumed that the fully turbulent portion of the boundary layer starts at y+ — 30, that the ratio of the mixing length to the distance y from the surface, Af/v = 0.4, and that for a smooth surface u+ = I4 at v 30. 

With injection mixers (Figures 10.52b,c), in which the one fluid is introduced into the flowing stream of the other through a concentric pipe or an annular array of jets, mixing will take place by entrainment and turbulent diffusion. Such devices should be used where one flow is much lower than the other, and will give a satisfactory blend in about 80 pipe diameters. The inclusion of baffles or other flow restrictions will reduce the mixing length required. 

The difference in the Li abundances in the G-stars of the Pleiades and the Sun, combined with the probable similarities in their overall chemical composition tell us that PMS Li depletion cannot be the whole story. Another mechanism, additional to convective mixing, must be responsible for Li depletion whilst solar-type stars are on the main-sequence. Recent PMS models that have their convective treatments tuned to match the structure of the Sun reproduce the mass dependence of Li depletion, but deplete too much Li compared with the Pleiades, and can even explain the solar A (Li) in the case of full spectrum turbulence models [9]. The over-depletion with respect to the Pleiades gets worse at lower masses. Better fits to the Pleiades data are achieved with PMS models that feature relatively inefficient convection with smaller mixing lengths. 

The three models were calculated with the same chemical and physical inputs with the only exception of convection, for which we adopted the Full Spectrum of Turbulence convective model (FST, Canuto Mazzitelli 1991), and the MLT model (Vitense 1953) with two values of the free parameter connected to the mixing length a = 1.7 (the standard value, used to reproduce the evolution of the Sun) and a = 2.1. 

In the discussion above, we have considered only the velocity field in a turbulent flow. What about the length and time scales for turbulent mixing of a scalar field The general answer to this question is discussed in detail in Fox (2003). Here, we will only consider the simplest case where the scalar field 4> is inert and initially nonpremixed with a scalar integral length scale that is approximately equal to Lu. If we denote the molecular diffusivity of the scalar by T, we can use the kinematic viscosity to define a dimensionless number in the following way ... 

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