Turbulence models mixing-length

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 ... 

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 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. 

At the next level of complexity, a second transport equation is introduced, which effectively removes the need to fix the mixing length. The most widely used two-equation model is the k-e model wherein a transport equation for the turbulent dissipation rate is... 

Reynolds stresses generated by time averaging. Thus, additional equations are needed to correlate these terms with time-averaged quantities. These additional equations may come from turbulence models. The two most commonly used turbulence models, the mixing length model and the k-e model, are introduced. 

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]. 

The last term in the momentum equation, i.e., Eq. (9.78). represents the affect of the buoyancy forces on the mean momentum balance. However, these buoyancy forces also affect the variation of e and e in the flow. To illustrate how the buoyancy forces can effect e and e, consider again the simple mixing length model discussed in Chapter 5. Lumps or eddies of fluid are assumed to move across the flow through a transverse distance, lm, while retaining their initial velocity and temperature. They then interact with the local fluid layer giving rise to the fluctuations in velocity and temperature that occur in turbulent flow. 

A numerical procedure for calculating the heat transfer rate with turbulent boundary layer flow was discussed in Chapter 5. This procedure used a mixing length-based turbulence model. Discuss the modifications that must be made to this procedure to apply it to mixed convective flow over a vertical plate. 

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