Turbulent fluid flow models

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. 

The extension of generic CA systems to two dimensions is significant for two reasons first, the extension brings with it the appearance of many new phenomena involving behaviors of the boundaries of, and interfaces between, two-dimensional patterns that have no simple analogs in one-dimension. Secondly, two-dimensional dynamics permits easier (sometimes direct) comparison to real physical systems. As we shall see in later sections, models for dendritic crystal growth, chemical reaction-diffusion systems and a direct simulation of turbulent fluid flow patterns are in fact specific instances of 2D CA rules and lattices. 

In real life, the parcels or blobs are also subjected to the turbulent fluctuations not resolved in the simulation. Depending on the type of simulation (DNS, LES, or RANS), the wide range of eddies of the turbulent-fluid-flow field is not necessarily calculated completely. Parcels released in a LES flow field feel both the resolved part of the fluid motion and the unresolved SGS part that, at best, is known in statistical terms only. It is desirable that the forces exerted by the fluid flow on the particles are dominated by the known, resolved part of the flow field. This issue is discussed in greater detail in the next section in the context of tracking real particles. With a RANS simulation, the turbulent velocity fluctuations remaining unresolved completely, the effect of the turbulence on the tracks is to be mimicked by some stochastic model. As a result, particle tracking in a RANS context produces less realistic results than in an LES-based flow field. 

In studies that involve the CFD analysis of turbulent fluid flow, the k-t model is most frequently used because it offers the best compromise between width of application and computational economy (Launder, 1991). Despite its widespread popularity the k-e model, if used to generate an isotropic turbulent viscosity, is inappropriate for simulation of turbulent swirling flows as encountered in process equipment such as cyclones and hydrocyclones (Hargreaves and Silvester, 1990) and more advanced turbulence models such as the ASM or the RSM should be considered. Because these models are computationally much more demanding and involve an increased number of empirical parameters compared to the k-e model, other strategies have been worked out (Boysan et al, 1982 Hargreaves and Silvester, 1990) to avoid the isotropic nature of the classical k-e model. 

In the last decade very significant progress has been made in modeling turbulent fluid flow. There remain, however, very significant problems of which, in addition to the problems mentioned in the previous section, we would like to mention the following problem areas (I) availability of accurate turbulence models which can be used with confidence in complex geometries while at the same time the computational cost should be acceptable and (2) availability of DNS-generated databases to validate semiempirical turbulence models. 

The multifractal behavior of time series such as SRV, HRV, and BRV can be modeled using a number of different formalisms. For example, a random walk in which a multiplicative coefficient in the random walk is itself made random becomes a multifractal process [59,60], This approach was developed long before the identification of fractals and multifractals and may be found in Feller s book [61] under the heading of subordination processes. The multifractal random walks have been used to model various physiological phenomena. A third method, one that involves an integral kernel with a random parameter, was used to model turbulent fluid flow [62], Here we adopt a version of the integral kernel, but one adapted to time rather than space series. The latter procedure is developed in Section IV after the introduction and discussion of fractional derivatives and integrals. 

The simulation of convective effects on current distributions in the presence of turbulent fluid flow has not been treated extensively, even though turbulence is common in many practical applications. Wang et alJ provided a literature review of some of the previous work. They also presented simulation results for a two-equation kinetic energy-dissipation turbulence model. - " The model equations were solved numerically using the SIMPLE algorithm. 

Deacon [28] developed a boundary layer model based on turbulent fluid flow in the vicinity of a smooth rigid wall. By assuming that the wind stress is continuous across the air-water interface, producing a constant flux of momentum, the friction velocity on the water side can be determined as u = w a(9a/9w)° in which a and w refer to air and water, respectively. This approach has been found to provide a reasonable description of gas transfer in wind tunnels at low wind speeds [10]. Another boundary layer model [35] allows some surface divergence and predicts the -2/3 power of the Sc for low wind speeds and -1/2 power at higher wind speeds. 

In spite of the success of CFD simulations for the multiphase turbulent fluid flow in stirred-tank bioreactors (see Section 3.4), their application to coupled material balance equations in case of more complicated reaction networks is still limited by the required computing power. Even in case of successful approaches for model reduction, the number of compounds necessary for reliable portrayal of cellular dynamics in response to spatial variation of extracellular compounds may be still too large. An interesting method to overcome these numerical difficulties is the general hybrid multizonal/CFD [27-36], which gave momentum to the application of CFD modeling for bioreactors. 

C. The liquidus line corresponds closely wilh the locations where the liquid phase has been observed, thus validating the temperatures calculated using the turbulent heat transfer and fluid flow model. Adjacent to the liquid phase, a single- phase ferrite region is observed. This single... 

Instead of assigning different shear rates, he employed different breakage rate expressions for the two zones. The problem of coupling population balance models with fluid flow models has received some attention recently and coupled PB-CFD models have been developed for a wide variety of processes such as fluidization [70], gas-liquid reactions in bubble columns [71] and nanoparticle synthesis in flame aerosol reactors [72]. Complete description of aggregation in turbulent environments requires simultaneous solution of basic balance equations for mass, momentum, energy and concentration of species present along with population balances for particles/aggregates of different size classes. 

Patel, B. R. and Sheikoholeslami, Z., Numerieal modelling of turbulent flow through the orifiee meter. International Symposium on Fluid Flow Measurement, Washington, D.C., November 1986. 

Hanjalic, K. Adv.inced turbulence enclosure models A view of current status and future prospects. Int. ]. Heat Fluid Flow, vol. 15, pp. 178-203, 1994. 

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. 

Peng, S. H., Davidson, L., Holmberg, S. The two-equation turbulence k-o> model applied to recirculating ventilation flows. Report 96/13, Dept, of Thermo and Fluid Dynamics, Chalmers University of Technology, Gothenburg, 1996. 

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. 

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 availability of large and fast computers, in combination with numerical techniques to compute transient, turbulent flow, has made it possible to simulate the process of turbulent, premixed combustion in a gas explosion in more detail. Hjertager (1982) was the first to develop a code for the computation of transient, compressible, turbulent, reactive flow. Its basic concept can be described as follows A gas explosion is a reactive fluid which expands under the influence of energy addition. Energy is supplied by combustion, which is modeled as a one-step conversion process of reactants into combustion products. The conversion (combustion)... 

Glaser and Litt (G4) have proposed, in an extension of the above study, a model for gas-liquid flow through a b d of porous particles. The bed is assumed to consist of two basic structures which influence the fluid flow patterns (1) Void channels external to the packing, with which are associated dead-ended pockets that can hold stagnant pools of liquid and (2) pore channels and pockets, i.e., continuous and dead-ended pockets in the interior of the particles. On this basis, a theoretical model of liquid-phase dispersion in mixed-phase flow is developed. The model uses three bed parameters for the description of axial dispersion (1) Dispersion due to the mixing of streams from various channels of different residence times (2) dispersion from axial diffusion in the void channels and (3) dispersion from diffusion into the pores. The model is not applicable to turbulent flow nor to such low flow rates that molecular diffusion is comparable to Taylor diffusion. The latter region is unlikely to be of practical interest. The model predicts that the reciprocal Peclet number should be directly proportional to nominal liquid velocity, a prediction that has been confirmed by a few determinations of residence-time distribution for a wax desulfurization pilot reactor of 1-in. diameter packed with 10-14 mesh particles. 

By its random nature, turbulence does not lend itself easily to modelling starting from the differential equations for fluid flow (Navier-Stokes). However, a remarkably successful statistical model due to Kolmogorov has proven very useful for modelling the optical effects of the atmosphere. 

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