Two equation models

The two-equation model of Launder and co-workers (22,23) and the one most often used employs the S equation given by... 

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... 

Other two-equation models have been developed using various combinations of k and e to derive alternative transport equations to replace (4.47). The most popular is, perhaps, the k-co model (see Wilcox (1993) for a detailed discussion of its advantages), where the turbulence frequency co = e/k is used in place of s. The standard transport equation for... 

Consider estimation of the following two equation model n. A sample of 50 observations... 

Computational fluid dynamics were used to describe the flow which undergoes a fast transition from laminar (at the fluid outlets) to turbulent (in the large mixing chamber) [41]. Using the commercial tool FLUENT, the following different turbulence models were applied a ke model, an RNC-ki model and a Reynolds-stress model. For the last model, each stream is solved by a separate equation for the two first models, two-equation models are applied. To have the simulations at... 

An enormous amount of effort has been expended in attempting to discover new equations for pv v. Five different levels of approach have been pursued in the literature involving various degrees of mathematical complexity. We cannot discuss all of them here. We outline only two of the most fruitful (1) the mixing length or zero-equation models and (2) the k-s or two-equation models. 

Actually, all methods of closure involve some type of modeling with the introduction of adjustable parameters that must be fixed by comparison with data. The only question is where in the hierarchy of equations the empiricism should be introduced. Many different systems of modeling have been developed. The zero-equation models have already been introduced. In addition there are one-equation and two-equation models, stress-equation models, three-equation models, and large-eddy simulation models. Depending on the complexity of the model and the problem... 

Computational experience has revealed that the two-equation models, employing transport equations for the velocity and length scales of the fluctuating motion, often offer the best compromise between width of application and computational economy. There are, however, certain types of flows where the k-e model fails, such as complex swirling flows, and in such situations more advanced turbulence models (ASM or RSM) are required that do not involve the eddy-viscosity concept (Launder, 1991). According to the ASM and the RSM the six components of the Reynolds stress tensor are obtained from a complete set of algebraic equations and a complete set of transport equations. These models are conceptually superior with respect to the older turbulence models such as the k-e model but computationally they are also (much) more involved. 

The following two-equation model [202], provides a better adiabatic flame temperature and a reasonable estimate of the CO concentration at equilibrium ... 

The two-equation models (especially, the k-s model) discussed above have been used to simulate a wide range of complex turbulent flows with adequate accuracy, for many engineering applications. However, the k-s model employs an isotropic description of turbulence and therefore may not be well suited to flows in which the anisotropy of turbulence significantly affects the mean flow. It is possible to encounter a boundary layer flow in which shear stress may vanish where the mean velocity gradient is nonzero and vice versa. This phenomenon cannot be predicted by the turbulent viscosity concept employed by the k-s model. In order to rectify this and some other limitations of eddy viscosity models, several models have been proposed to predict the turbulent or Reynolds stresses directly from their governing equations, without using the eddy viscosity concept. 

Saffman, P.G. (1977), Results of a two-equation model for turbulent flows and development of relaxation stress model for application to staining and rotating flows. Proceedings of Project SQUID Workshop on Turbulence in Internal Flows, Hemisphere, New York, p. 191. 

Squire, K.D. and Eaton, J.K. (1994), Effect of selective modification of turbulence on two equation models for particle laden turbulent flows, ASME J Fluids Eng., 116, 778. 

In multiphase reactor simulations the zero- and two equation models are very popular, only a few attempts have been made to implement Re3molds Stress Models, and LES models are still really rare. 

Due to their robustness and reasonable accuracy, the first-order two-equation models, such as the k-e closure originally proposed by Harlow and Nakayama... 

Examination of several two-equation models reveals that there is only very small differences between the various models of this t3q>e [106]. This may be expected since all proposals for formulating the 2nd equation are closely related, though they differ in the forms of diffusion and near wall terms employed [95]. However, as mentioned above, the k-e model of Jones and Launder [78] has been predominant in the literature, and this model also determine the basis for most multi-phase turbulence models adopted in the more fundamental (CFD) reactor modeling approaches. 

Jones WP, Launder BE (1972) The Prediction of Laminarization with a Two-Equation Model of Turbulence. Int J Heat Mass Transfer 15 301-314 Jones WP (1980) Models for Turbulent Plows with Variable Density and Combustion. In Kollman W (ed) Prediction Methods for Turbulent Plow, Hemisphere Publishing Company... 

Quintard M. and Whitaker S. 1993b. One- and two-equation models for transient diffusion processes in two-phase systems. In, Advances in Heat Transfer, Harnett J.P., Irvine T.F. Jr and Cho Y.I. (Eds.), Vol. 23. Academic Press, New York, pp. 369-465. 

Jones, W.P. Launder, B.C. 1972. The Prediction of lam-inarization with a two-equation model of turbulence. bit. J. Heat and mass transfer, (15) 301-314. 

We pick up horizontal roadway of Zhangxiaolou mine to set up the physical model. And then we adopt the VOF method to form the mathematical model for the wall. We define materials as water, turbulent kinetic energy of 0.01, dissipation rate is 0.007 mVS. Calculation model select standard fc- two equations model. The opening size of the parallel wall is 4 m, length of 400 m. As shown in Figure 1. 

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