K-e model of turbulence

In the k-e model of turbulence, turbulent viscosity is related to k and e by the following equation ... 

Turbulent stresses are determined by standard and modified K-e models of turbulence. For determination of flow field near the wall the method of parietal functions was used. Rates fields and pressures were corrected in the course of calculation according with SIMPLE-C algorithm. Diffusion of indicator was modeled as transmission of scalar introduced in less space of time in comparison with its average residence time in apparatus at fixed rate fields and turbulence parameters. 

Comini G, Del Giudice S (1985) A k-e model of turbulent flow. Numer Heat Transf 8 299-316... 

The modeled transport equations for z differ mainly in the diffusion and secondary source term. Launder and Spalding (1972) and Chambers and Wilcox (1977) discuss the differences and similarities in more detail. The variable, z = e is generally preferred since it does not require a secondary source, and a simple gradient diffusion hypothesis is fairly good for the diffusion (Launder and Spalding, 1974 Rodi, 1984). The turbulent Prandtl number for s has a reasonable value of 1.3, which fits the experimental data for the spread of various quantities at locations far from the walls, without modification of any constants. Because of these factors, the k-s model of turbulence has been the most extensively studied and used and is recommended as a baseline model for typical internal flows encountered by reactor engineers. 

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. 

Using turbulenee models, this new system of equations ean be elosed. The most widely used turbulenee model is the k-e model, whieh is based on an analogy of viseous and Reynolds stresses. Two additional transport equations for the turbulent kinetie energy k and the turbulent energy dissipation e deseribe the influenee of turbulenee... 

In these model equations it is assumed that turbulence is isotropic, i.e. it has no favoured direction. The k-e model frequently offers a good compromise between computational economy and accuracy of the solution. It has been used successfully to model stirred tanks under turbulent conditions (Ranade, 1997). Manninen and Syrjanen (1998) modelled turbulent flow in stirred tanks and tested and compared different turbulence models. They found that the standard k-e model predicted the experimentally measured flow pattern best. 

An Eulerian-Eulerian (EE) approach was adopted to simulate the dispersed gas-liquid flow. The EE approach treats both the primary liquid phase and the dispersed gas phase as interpenetrating continua, and solves a set of Navier-Stokes equations for each phase. Velocity inlet and outlet boundary conditions were employed in the liquid phase, whilst the gas phase conditions consisted of a velocity inlet and pressure outlet. Turbulence within the system was account for with the Standard k-e model, implemented on a per-phase basis, similar to the recent work of Bertola et. al.[4]. A more detailed description of the computational setup of the EE method can be found in Pareek et. al.[5]. 

In its turn, the turbulent viscosity may be position dependent and generally may be modeled in terms of a model, very usually a k-e model ... 

These two transport equations for k and e form an inherent part of any k i model of RANS-simulations. As the result of closing the turbulence modeling such that no further unknown variables and equations are introduced, the e-equation does contain some terms that are still the result of modeling, albeit at the very small scales (e.g., Rodi, 1984). 

The (isotropic) eddy viscosity concept and the use of a k i model are known to be inappropriate in rotating and/or strongly 3-D flows (see, e.g., Wilcox, 1993). This issue will be addressed in more detail in Section IV. Some researchers prefer different models for the eddy viscosity, such as the k o> model (where o> denotes vorticity) that performs better in regions closer to walls. For this latter reason, the k-e model and the k-co model are often blended into the so-called Shear-Stress-Transport (SST) model (Menter, 1994) with the view of using these two models in those regions of the flow domain where they perform best. In spite of these objections, however, RANS simulations mostly exploit the eddy viscosity concept rather than the more delicate and time-consuming RSM turbulence model. They deliver simulation results of in many cases reasonable or sufficient accuracy in a cost-effective way. 

The number of equations to be solved is, among other things, related to the turbulence model chosen (in comparison with the k-e model, the RSM involves five more differential equations). The number of equations further depends on the character of the simulation whether it is 3-D, 21/2-D, or just 2-D (see below, under The domain and the grid ). In the case of two-phase flow simulations, the use of two-fluid models implies doubling the number of NS equations required for single-phase flow. All this may urge the development of more efficient solution algorithms. Recent developments in computer hardware (faster processors, parallel platforms) make this possible indeed. 

Most RANS-based simulations make use of the k-e model for taking into account the momentum transport due to the turbulent eddies. This model is an... 

Second, due to the difficulty of accessing multiphase flows with laser-based flow diagnostics, there is very little experimental data available for validating multiphase turbulence models to the same degree as done in single-phase turbulent flows. For example, thanks to detailed experimental measurements of turbulence statistics, there are many cases for which the single-phase k- model is known to yield poor predictions. Nevertheless, in many CFD codes a multiphase k-e model is used to supply multiphase turbulence statistics that cannot be measured experimentally. Thus, even if a particular multiphase turbulent flow could be adequately described using an effective viscosity, in most cases it is impossible to know whether the multiphase turbulence model predicts reasonable values for... 

The RNG model provides its own energy balance, which is based on the energy balance of the standard k-e model with similar changes as for the k and e balances. The RNG k-e model energy balance is defined as a transport equation for enthalpy. There are four contributions to the total change in enthalpy the temperature gradient, the total pressure differential, the internal stress, and the source term, including contributions from reaction, etc. In the traditional turbulent heat transfer model, the Prandtl number is fixed and user-defined the RNG model treats it as a variable dependent on the turbulent viscosity. It was found experimentally that the turbulent Prandtl number is indeed a function of the molecular Prandtl number and the viscosity (Kays, 1994). 

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... 

A variety of statistical models are available for predictions of multiphase turbulent flows [85]. A large number of the application oriented investigations are based on the Eulerian description utilizing turbulence closures for both the dispersed and the carrier phases. The closure schemes for the carrier phase are mostly limited to Boussinesq type approximations in conjunction with modified forms of the conventional k-e model [87]. The models for the dispersed phase are typically via the Hinze-Tchen algebraic relation [88] which relates the eddy viscosity of the dispersed phase to that of the carrier phase. While the simplicity of this model has promoted its use, its nonuniversality has been widely recognized [88]. 

Theoretical investigations of the problem were carried out on the base of the mathematical model, combining both deterministic and stochastic approaches to turbulent combustion of organic dust-air mixtures modeling. To simulate the gas-phase flow, the k-e model is used with account of mass, momentum, and energy fluxes from the particles phase. The equations of motion for particles take into account random turbulent pulsations in the gas flow. The mean characteristics of those pulsations and the probability distribution functions are determined with the help of solutions obtained within the frame of the k-e model. 

Kuhl, A. L., R. E. Ferguson, K. Y. Chien, J. P. Collins, and A. K. Oppenheim. 1995. Gasdynamic model of turbulent combustion in an explosion. Combustion, Detonation, Shock Waves. Zel davich Memorial Proceedings. Eds. A. G. Merzhanov and... 

The Fluent code with the RSM turbulence model, predict very well the pressure drop in cyclones and can be used in cyclone design for any operational conditions (Figs. 3, 5, 7 and 8). In the CFD numerical calculations a very small pressme drop deviation were observed, with less than 3% of deviation at different inlet velocity which probably in the same magnitude of the experimental error. The CFD simulations with RNG k-e turbulence model still yield a reasonably good prediction (Figs. 3, 5, 7 and 8) with the deviation about 14-20% of an experimental data. It considerably tolerable since the RNG k-e model is much less on computational time required compared to the complicated RSM tmbulence model. In all cases of the simulation the RNG k-< model considerably underestimates the cyclone pressme drop as revealed by Griffiths and Boysan [8], However under extreme temperature (>850 K) there is no significant difference between RNG k-< and RSM model prediction. 

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 k-e 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 k2/e. 

In these equations summations over repeated indices are implied. The values for the empirical constants Cu = 1.44, C2e = 1.92, Gi = 1.0, and ce = 1.3 are widely accepted (Launder and Spaulding, The Numerical Computation of Turbulent Flows, Imperial Coll. Sci. Tech. London, NTIS N74-12066 [1973]). The k-e. model has proved reasonably accurate for many flows without highly curved streamlines or significant swirl. It usually underestimates flow separation and overestimates turbulence production by normal straining. The k-e model is suitable for high Reynolds number flows. See Virendra, Patel, Rodi, and Scheuerer (A1AA J., 23, 1308-1319 [1984]) for a review of low Reynolds number k-e. models. 

The standard k-e model simulates the turbulence in the reactor. For flow within the porous catalyst bed, however, we suppress the turbulence. We enter the appropriate physical properties of the system, and employ standard boundary conditions at the impermeable walls and the reactor outlet. To represent the turbulence of the feed stream at the inlet, we treat it as pipe-flow turbulence. These model equations can then be solved for instance, via the well-known Simple algorithm [3]. To facilitate fast convergence, it is useful to make a reasonable initial guess of the pressure drop across the catalyst bed. 

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