Turbulence, eddy viscosity models

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

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. 

For another class of models, the mixing length models, a different approach is used. Here, the turbulent eddy viscosity is assumed to be of the form... 

A three dimensional turbulent flow field in unbaffled tank with turbine stirrer or 6-paddle stirrer was numerically simulated by the method of finite volume elements [80], whereas in the case of free surface the vortex profile was also determined using iterative techniques. The prediction of the velocity and turbulence fields in the whole tank and the stirrer power was compared with literature data and their own results. Of the two simulation techniques used, turbulent eddy-viscosity/zc-e turbulence model and the DS model (differential 2. order shear stress), only the latter produced satisfactory results. In particular it proved that fluctuating Coriolis forces have to be taken into account by source terms in the transport equation for the Reynolds shear stress. 

Shih T-H, Liou WW, Shabbir A, Zhu J (1995) A New k-e Eddy Viscosity Model for High Reynolds Number Turbulent Elows. Comp Eluids 24(3) 227-238 Smith LM, Reynolds WC (1992) On the Yakhot-Orszag renormalization group method for deriving turbulence statistics and models. Phys Fluids A 4(2) 364-390... 

The turbulence model in FEMLAB is in dimensional (SI) units. Turbulence is modeled using the fe-e model. In this model, the turbulent kinetic energy is represented by k, and the rate of dissipation of turbulent kinetic energy is represented by s. Furthermore, the viscosity is augmented by a turbulent eddy viscosity, which is a function of k and s. Special equations have been developed for both variables, and these must be solved along with the momentum equation which has the turbulent eddy viscosity in it as well. All these equations are included in FEMLAB. 

The classical turbulence models express the eddy viscosity algebraically in terms of a turbulence scale and intensity that are related, respectively, to the characteristic length dimensions of the flow field and the local mean velocity gradients. This implies an equilibrium between the local turbulence and the mean motion. This requirement of equilibrium has been relaxed in some eddy viscosity models where the intensity and scale of turbulence used to evaluate the eddy viscosity are expressed by partial differential equations for the turbulence kinetic energy and its dissipation rate. This latter class of models is presented in detail, for example, in Refs. 76 and 77. 

Shih, T.H. Uou, W.W. Shabbir, A Yang, Z G et al. 1995. A new k-e eddy viscosity model for high Reynolds number turbulent flows. Comput Fluid, 24 3) 227-238. 

The family of two-equation A -f models is the most widely used of the eddy viscosity models. Ak-e model consists of two transport equations, one for the turbulent kinetic energy k and one for the energy dissipation rate e. The turbulent eddy viscosity is calculated from ... 

The standard k-s model, as presented by Launder and Spalding [2], is by far the most widely-used two-equation eddy viscosity model, also for modeling turbulence in stirred tank reactors. The popularity of the model and its wide use and testing has thrown light on both its capabilities and its shortcomings, which are well documented in the literature [2-8]. For high turbulent Reynolds numbers, the model may be summarized as follows ... 

Shih, T.-H., W. W. Liou, A. Shabbir, and J. Zhu (1995). A new k-s eddy-viscosity model for high Reynolds number turbulent flows model development and validation, Comput. Eluids, 24, 227-238. 

The quantity k is related to the intensity of the turbulent fluctuations in the three directions, k = 0.5 u u. Equation 41 is derived from the Navier-Stokes equations and relates the rate of change of k to the advective transport by the mean motion, turbulent transport by diffusion, generation by interaction of turbulent stresses and mean velocity gradients, and destmction by the dissipation S. One-equation models retain an algebraic length scale, which is dependent only on local parameters. The Kohnogorov-Prandtl model (21) is a one-dimensional model in which the eddy viscosity is given by... 

Reynolds Stress Models. Eddy viscosity is a useful concept from a computational perspective, but it has questionable physical basis. Models employing eddy viscosity assume that the turbulence is isotropic, ie, u u = u u = and u[ u = u u = u[ = 0. Another limitation is that the... 

Closure Models Many closure models have been proposed. A few of the more important ones are introduced here. Many employ the Boussinesq approximation, simphfied here for incompressible flow, which treats the Reynolds stresses as analogous to viscous stresses, introducing a scalar quantity called the turbulent or eddy viscosity... 

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. 

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... 

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

More advanced models, for example the algebraic stress model (ASM) and the Reynolds stress model (RSM), are not based on the eddy-viscosity concept and can thus account for anisotropic turbulence thereby giving still better predictions of flows. In addition to the transport equations, however, the algebraic equations for the Reynolds stress tensor also have to be solved. These models are therefore computationally far more complex than simple closure models (Kuipers and van Swaaij, 1997). 

Ekambra etal. [21] compared the results from ID, 2D, and 3D simulations of a bubble column with experimental results. They obtained similar results for holdup and axial velocity, while eddy viscosity, Reynolds stresses, and energy dissipation were very different in the three simulations as shown in Figure 15.7. This example also illustrates the importance of selecting the right variables for model vahdation. A 2D model will yield good results for velocity but will predict all variables based on turbulent characteristics poorly. 

Usually, however, the stresses are modeled with the help of a single turbulent viscosity coefficient that presumes isotropic turbulent transport. In the RANS-approach, a turbulent or eddy viscosity coefficient, vt, covers the momentum transport by the full spectrum of turbulent scales (eddies). Frisch (1995) recollects that as early as 1870 Boussinesq stressed turbulence greatly increases viscosity and proposed an expression for the eddy viscosity. The eventual set of equations runs as... 

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. 

Durbin, P. A., N. N. Mansour, and Z. Yang (1994). Eddy viscosity transport model for turbulent flow. Physics of Fluids 6, 1007-1015. 

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]. 

On the other hand, when the self-gravity does not dominate in the accretion disk, the prescription for the effective viscosity should become the standard or disk prescription in which typical turbulent eddy scale X is nearly the hight of the disk. According to standard ordisk model, we get for viscosity, = aCgH... 

The "correlative" multi-scale CFD, here, refers to CFD with meso-scale models derived from DNS, which is the way that we normally follow when modeling turbulent single-phase flows. That is, to start from the Navier-Stokes equations and perform DNS to provide the closure relations of eddy viscosity for LES, and thereon, to obtain the larger scale stress for RANS simulations (Pope, 2000). There are a lot of reports about this correlative multi-scale CFD for single-phase turbulent flows. Normally, clear scale separation should first be distinguished for the correlative approach, since the finer scale simulation need clear specification of its boundary. In this regard, the correlative multi-scale CFD may be viewed as a "multilevel" approach, in the sense that each span of modeled scales is at comparatively independent level and the finer level output is interlinked with the coarser level input in succession. 

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