Turbulent models

The criterion of maintaining equal power per unit volume has been commonly used for dupHcating dispersion qualities on the two scales of mixing. However, this criterion would be conservative if only dispersion homogeneity is desired. The scale-up criterion based on laminar shear mechanism (9) consists of constant > typical for suspension polymerization. The turbulence model gives constant tip speed %ND for scale-up. 

The difficulty with Eq, (26-58) is that it is impossible to determine the velocity at every point, since an adequate turbulence model does not currently exist, The solution is to rewrite the concentration and velocity in terms of an average and stochastic quantity C = (C) -t- C Uj = (uj) + Uj, where the brackets denote the average value and the prime denotes the stochastic, or deviation variable. It is also helpful to define an eddy diffusivity Kj (with units of area/time) as... 

For most applications, the engineer must instead resort to turbulence models along with time-averaged Navier-Stokes equations. Unformnately, most available turbulence models obscure physical phenomena that are present, such as eddies and high-vorticity regions. In some cases, this deficiency may partially offset the inherent attractiveness of CFD noted earlier. 

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. 

Nullaswamy, M., Turbulence models and their applications to the predictions of internal flows, Computers and Eluids, 15, 151, 1987. 

HOTM AC/RAPTAD contains individual codes HOTMAC (Higher Order Turbulence Model for Atmospheric Circulation), RAPTAD (Random Particle Transport and Diffusion), and computer modules HOTPLT, RAPLOT, and CONPLT for displaying the results of the ctdculalinns. HOTMAC uses 3-dimensional, time-dependent conservation equations to describe wind, lempcrature, moisture, turbulence length, and turbulent kinetic energy. 

Another detailed method of determining pressures is computational fluid dynamics (CFD), which uses a numerical solution of simplified equations of motion over a dense grid of points around the building. Murakami et al. and Zhoy and Stathopoulos found less agreement with computational fluid dynamics methods using the k-e turbulence model typically used in current commercial codes. More advanced turbulence models such as large eddy simulation were more successful but much more costly. ... 

Murakami, S., A. Mochida, R. Ooka, S. Kato, and S. lizuka. 1996. Numerical prediction of flow around a building with various turbulence models Comparison of fe-e, EVM, ASM, DSM, and LES with wind tunnel tests. ASHRAE Transactions, vol. 102, no. I. 

Hunt and Kulmala have solved the full turbulent fluid flow for the Aaberg system using the k-e turbulent model or a variation of it as described in Chapter 13— the solution algorithm SIMPLE, the QUICK scheme, etc. Both commercial software and in-house-developed codes have been employed, and all the investigators have produced very similar findings. 

In ventilation problems, it is often sufficient to use simpler turbulence models, such as eddy-viscosity models. and Ujt are then re... 

Commonly used eddy-viscosity turbulence models are the k-e model and the k-(ji) model. The eddy viscosities for these models have the form... 

In displacement ventilation, there are regions with very low turbulence, and the flow can even be laminar. Hence it is important to use a turbulence model which can handle these regions. The k-f model gives rise to large numerical problems in regions of low turbulence. The reason is thar as k goes to zero, the destruction term in the e equation goes to infinity. The c equation is... 

When using LES, the time-dependent three-dimensional momentum and continuity are solved for. A subgrid turbulence model is used to mode the turbulent scales that are smaller than the cells. Instead of the traditional time averaging, the equations for using LES are filtered in space, and is a function of space and time. 

With LES we get much more information than with traditional time-averaged turbulence models, since we are resolving most of the turbulence. In Fig. T1.15 the computed u velocity is shown as a function of time in two cells one cell is located in the wall jet (Fig.. 15a), and the other cell is in the middle of the room (Fig. ll.lSh). It is found the instantaneous fluctuations are very large. For example, in the region of the wall jet below the ceiling where the time-averaged velocity u)/l] ) is typically 0.5, the instantaneous velocity fluctuations are between 0.2 and 0.9. In the middle of the room, which is a low-velocity region, the variation of u is much slower, i.e., the frequency is lower. 

Davidson, L. An introduction to turbulence models. Report 97/2, Llept, of Thermo aiivt l-luid Dynamics, Chalmers University of Technology, Gothenburg, 1997. 

Chien, K. Y. Predictions of channel and boundary layer flows with a low-Reynolds-nuraber turbulence model. AIAA J., vol, 20, pp. 33-18, 1982. 

Chen, H. C, Patel, V. C. Near-wall turbulence models for complex flows including separation, AIAA J., vol. 26, pp. 641-648, 1988. 

Abe, K., Kondoh, T., Nagano, Y. A new turbulence model for predicting fluid flow and heat transfer in separating and reattaching flows 1. Flow field calculations. Int. ]. Heat Mass Trans, fer, vol. 37, pp. 139-151, 1994. 

Muller, D., Renz, U, Measurements and predictions of room airflow patterns using different turbulence models. In Mundt, E., Malmstrdm, T. G., eds., Roomvent 98 6th Int. Conf. on Air Distributions in Rooms, vol. 1, pp. 109-116, Stockholm, 1998. 

Davidson, L. Turbulence modelling and calculation of ventilation parameters in ventilated rooms. Lie. thesis. Report 86/10, Dept, of Thermo and Fluid Dynamics, Chalmers Universirv of Technology, Gothenburg, 1986. 

Menter, F. R. Two-equation eddy-viscosity turbulence models for engineering applications. AIAA ., vol. 32, pp. 1598-1605, 1994. 

An appropriate model of the Reynolds stress tensor is vital for an accurate prediction of the fluid flow in cyclones, and this also affects the particle flow simulations. This is because the highly rotating fluid flow produces a. strong nonisotropy in the turbulent structure that causes some of the most popular turbulence models, such as the standard k-e turbulence model, to produce inaccurate predictions of the fluid flow. The Reynolds stress models (RSMs) perform much better, but one of the major drawbacks of these methods is their very complex formulation, which often makes it difficult to both implement the method and obtain convergence. The renormalization group (RNG) turbulence model has been employed by some researchers for the fluid flow in cyclones, and some reasonably good predictions have been obtained for the fluid flow. 

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. 

Computational fluid dynamics (CFD) is the numerical analysis of systems involving transport processes and solution by computer simulation. An early application of CFD (FLUENT) to predict flow within cooling crystallizers was made by Brown and Boysan (1987). Elementary equations that describe the conservation of mass, momentum and energy for fluid flow or heat transfer are solved for a number of sub regions of the flow field (Versteeg and Malalase-kera, 1995). Various commercial concerns provide ready-to-use CFD codes to perform this task and usually offer a choice of solution methods, model equations (for example turbulence models of turbulent flow) and visualization tools, as reviewed by Zauner (1999) below. 

The major mechanism of a vapor cloud explosion, the feedback in the interaction of combustion, flow, and turbulence, can be readily found in this mathematical model. The combustion rate, which is primarily determined by the turbulence properties, is a source term in the conservation equation for the fuel-mass fraction. The attendant energy release results in a distribution of internal energy which is described by the equation for conservation of energy. This internal energy distribution is translated into a pressure field which drives the flow field through momentum equations. The flow field acts as source term in the turbulence model, which results in a turbulent-flow structure. Finally, the turbulence properties, together with the composition, determine the rate of combustion. This completes the circle, the feedback in the process of turbulent, premixed combustion in gas explosions. The set of equations has been solved with various numerical methods e.g., SIMPLE (Patankar 1980) SOLA-ICE (Cloutman et al. 1976). 

Figure 7. Piston and tilt removed phase variance at a 4 m telescope versus altitude. Wavelength O.Spm. Solid line single LGS. Dashed line at zenith. Dotted line array of 4 LGSs 45°. Hufnagel-Valley21 turbulence model.

There are, however, additional issues when one wants to perform the LES of an industrial device, and these are to be solved if the LES methodology is ever to be a useful tool. First, the geometric intricacy of most industrial combustion chambers cannot be represented with a Cartesian mesh high-order methods on unstructured grids must be developed. Moreover, many issues about the boundary conditions are raised, such as boundary movement for blades or pistons, turbulence injection, and acoustic properties. Such challenges are not to be underestimated, as their impact on the structure of the flow might sometimes be greater than that of the turbulence model. 

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