Turbulence, eddy viscosity

There are two distinct modes of flow, laminar and turbulent. Fluid inertia tends to allow fluctuations to grow and give rise to turbulent eddies. Viscosity on the other hand, tends to damp out these fluctuations. A ratio of forces, inertial to viscous, is used to characterise the nature of the flow and is called the Reynolds Number, Re. For pipe flow this takes the form ... 

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. 

Balzer G, Simonin O (1996) Turbulent eddy viscosity derivation in dilute gas-solid turbulent flows. 8th Workshop on Two-Phase Plow Predictions, 26-29 March, Merseburg, Germany... 

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. 

Where is the turbulent eddy viscosity. The transport of momentum, which is related to turbulence, is thought of as turbulent eddies, which, like molecules, collide and exchange momentiun. 

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 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 universal turbulent velocity profile near the pipe wall presented in the preceding subsection Tncompressible Flow in Pipes and Channels may be developed using the Prandtl mixing length approximation for the 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. 

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... 

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). 

Fine suspensions are reasonably homogeneous and segregation of solid and liquid phases does not occur to any significant extent during flow. The settling velocities of the particles are low in comparison with the liquid velocity and the turbulent eddies within the fluid are responsible for the suspension of the particles. In practice, turbulent flow will always be used, except when the liquid has a very high viscosity or exhibits non-Newtonian characteristics. The particles may be individually dispersed in the liquid or they may be present as floes. 

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. 

When the concentration profile is fully developed, the mass-transfer rate becomes independent of the transfer length. Spalding (S20a) has given a theory of turbulent convective transfer based on the hypothesis that profiles of velocity, total (molecular plus eddy) viscosity, and total diffusivity possess a universal character. In that case the transfer rate k + can be written in terms of a single universal function of the transfer length L and fluid properties (expressed as a molecular and a turbulent Schmidt number) ... 

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. 

Note that the Eqs. (1), (2), and (8) are really and essentially different due to the absence or presence of different turbulent transport terms. Only by incorporating dedicated formulations for the SGS eddy viscosity can one attain that LES yield the same flow field as DNS. RANS-based simulations with their turbulent viscosity coefficient, however, essentially deliver steady flow fields and as such are never capable of delivering the same velocity fields as the inherently transient LES or DNS, irrespectively of the refinement of the computational grid ... 

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