Stressed transport

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. 

As we shall see in Chapter 4, models for izf have much in common with those used for IZij in the Reynolds stress transport equation. Indeed, as shown using transported PDF methods in Chapter 6, the model for uniquely determines the model for 7zf. 

The Reynolds-stress transport equation then becomes... 

We shall see that transported PDF closures forthe velocity field are usually linear in V. Thus (/ D) will depend only on the first two moments of U. In general, non-linear velocity models could be formulated, in which case arbitrary moments of U would appear in the Reynolds-stress transport equation. 

The final form of the Reynolds-stress transport equation will then depend on the choice of Gij. For example, in the simplified Langevin model (SLM)... 

An alternative procedure is to simplify the transport equation for the kinematic momentum fluxes or Re molds stresses. The contraction of the Re3molds stress transport equation (1.394) (that is, when the 3 equations for the 3 normal stresses, i = k = 1,2,3) are summed up) gives an exact transport equation for the turbulent kinetic energy (e.g., [167] [131] [106]). 

It is noted that in recent papers several extended turbulence models, i.e., the standard k-e [50], RNG k-e [97, 98, 99, 82, 66], realizable k-e [81], Chen-Kim k-e [11], optimized Chen-Kim k-e [44], standard k-uj [96], k-uj shear-stress transport (SST) [56, 57, 58] and the standard Reynolds stress models, have been proposed and validated. However, little or no significant improvements have been achieved considering the predictivity of the turbulence models, although each of them may have minor advantages and disadvantages. A few... 

Reynolds-stress transport equation Models for conditional acceleration... 

The commercial software ANSYS FLUENT was used for CFD analysis. Unsteady flow was modeled to obtain the variation of granule temperature with time. The k-to with shear-stress transport turbulence model was used. The modeled flue gases included four gas species, referring to CO2, H2O, N2, and O2. The boundary condition was velocity-inlet at the inlet, pressure outlet at the outlet, and symmetry for the four sides. Second-order upwind scheme was used for the momentum, species, and energy equations. 

For turbulent expiratory conditions, avoiding the intensive computational efforts involved with a three-dimensional Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS), a Reynolds-Averaged Navier Stokes (RANS) equations coupled to a Shear Stress Transport (SST) fc- y turbulent model is used to model the fluid. The governing equations are essentially similar to (1) and (2) above, but with the inclusion of Reynolds stress... 

The two-fluid method using the volume of fluid (VOF) approach, coupled with the k-co SST (shear stress transport) turbulence model, is employed to simulate the formation and fragmentation of a swirling conical liquid sheet firom a typical swirl nozzle. In the VOF approach, the volume firaction of liquid (/) is defined in each computational cell. If the cell is completely filled with liquid then/= 1 and if it is filled with gas then its value is 0. At the gas-liquid interface, the value of / is between 0 and 1. In the OpenFOAM VOF formulation, the interface motion is governed by the convective transport equation of the volume fraction... 

Similarly, we have the wj equation. Let w- equation be multiplied by wj and wj equation be multiplied by u p then, the two resulting equations are added and averaged to yield the following Reynolds stress transport equation ... 

Reynolds stress model (RANS) in which the Reynolds stress transport equation is computed directly by modeling Eq. (1.6) to the form suitable for numerical computation ... 

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