Viscosity, eddy

Eddy Viscosity Models. A large number of closure models are based on the Boussinesq concept of eddy viscosity ... 

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... 

Dukler Theory The preceding expressions for condensation are based on the classical Nusselt theoiy. It is generally known and conceded that the film coefficients for steam and organic vapors calculated by the Nusselt theory are conservatively low. Dukler [Chem. Eng. Prog., 55, 62 (1959)] developed equations for velocity and temperature distribution in thin films on vertical walls based on expressions of Deissler (NACA Tech. Notes 2129, 1950 2138, 1952 3145, 1959) for the eddy viscosity and thermal conductivity near the solid boundaiy. According to the Dukler theoiy, three fixed factors must be known to estabhsh the value of the average film coefficient the terminal Reynolds number, the Prandtl number of the condensed phase, and a dimensionless group defined as follows ... 

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). 

If the laminar sub-layer extends from y3 = 0 to y1 = 5, obtain the equation for the relation between and y+ in the buffer zone, and show that the ratio of the eddy viscosity to the molecular viscosity increases linearly from 0 to 5 through this buffer zone. 

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. 

Figure 15.7 Measured and simulated holdup (a), axial velocity (b), eddy viscosity (c), and Reynolds stresses (d), using a ID, 2D, and 3D simulations (From [21]).

Cheremisinoff and Davis (1979) relaxed these two assumptions by using a correlation developed by Cohen and Hanratty (1968) for the interfacial shear stress, using von Karman s and Deissler s eddy viscosity expressions for solving the liquid-phase momentum equations while still using the hydraulic diameter concept for the gas phase. They assumed, however, that the velocity profile is a function only of the radius, r, or the normal distance from the wall, y, and that the shear stress is constant, t = tw. ... 

For the liquid phase, Cheremisinoff and Davis (1979) solved the momentum equation using von Karman s and Deissler s eddy viscosity expressions. 

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) ... 

The most widely used model for the SGS stresses is due to Smagorinsky (1963) and involves a SGS eddy viscosity, ve, which is related to the local resolved deformation rate S ... 

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 ... 

Eddy-current separation, 75 435 of nonferrous metallics, 75 455-457 Eddy-current separator, 27 447-448 Eddy-current technique, in nondestructive evaluation, 77 420 Eddy diffusion, 9 658 Eddy viscosity, 77 779 Eddy-viscosity-based models, 77 780 Edecrin, 5 169... 

It has been assumed that the density is constant in writing these equations, which are therefore strictly valid only for incompressible flow. ed is called the eddy diffusivity and eh the eddy thermal diffusivity. Although s can be interpreted as the eddy diffusivity of momentum, it is usually called the eddy viscosity and sometimes by the better name eddy kinematic viscosity. 

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

Germano, M., U. Piomelli, P. Moin, and W. H. Cabot (1991). A dynamic subgrid-scale eddy viscosity model. Physics of Fluids 7, 1760-1765. 

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]. 

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