Turbulence shear-stress transport model

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. 

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... 

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... 

The SST model uses the standard k-e model in the bulk flow and incorporates the transport of turbulent shear stress while using the k-(0 model in the boundary layer. This model can therefore be used over a wider range of operating parameters than the standard k-(0 model, such as the flow around curved bodies and flows with separating boundary layers. Prieske et al. (2007) examined the relationship between circulation velocity using the SST model and aeration flow rate in a pilot-scale MBR. 

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. 

The modeling procedure can be sketched as follows. First an approximate description of the velocity distribution in the turbulent boundary layer is required. The universal velocity profile called the Law of the wall is normally used. The local shear stress in the boundary layer is expressed in terms of the shear stress at the wall. From this relation a dimensionless velocity profile is derived. Secondly, a similar strategy can be used for heat and species mass relating the local boundary layer fluxes to the corresponding wall fluxes. From these relations dimensionless profiles for temperature and species concentration are derived. At this point the concentration and temperature distributions are not known. Therefore, based on the similarity hypothesis we assume that the functional form of the dimensionless fluxes are similar, so the heat and species concentration fluxes can be expressed in terms of the momentum transport coefficients and velocity scales. Finally, a comparison of the resulting boundary layer fluxes with the definitions of the heat and mass transfer coefficients, indiates that parameterizations for the engineering transfer coefficients can be put up in terms of the appropriate dimensionless groups. 

As mentioned previously, even when the flow becomes turbulent in the boundary layer, there exists a thin sub-layer close to the surface in which the flow is laminar. This layer and the fully turbulent regions are separated by a buffer layer, as shown schematically in Figure 7.1. In the simplified treatments of flow within the turbulent boundary layer, however, the existence of the buffer layer is neglected. In the laminar sub-layer, momentum transfer occurs by molecular means, whereas in the turbulent region eddy transport dominates. In principle, the methods of calculating the local values of the boundary layer thickness and shear stress acting on an immersed surface are similar to those used above for laminar flow. However, the main difficulty stems from the fact that the viscosity models, such as equations (7.13) or (7.27),... 

In an effort to improve the description of the Reynolds stresses in the rapid distortion turbulence (RDT) limit, the velocity PDF description has been extended to include directional information in the form of a random wave vector by Van Slooten and Pope (1997). The added directional information results in a transported PDF model that corresponds to the directional spectrum of the velocity field in wavenumber space. The model thus represents a bridge between Reynolds-stress models and more detailed spectral turbulence models. Due to the exact representation of spatial transport terms in the PDF formulation, the extension to inhomogeneous flows is straightforward (Van Slooten et al. 1998), and maintains the exact solution in the RDT limit. The model has yet to be extensively tested in complex flows (see Van Slooten and Pope 1999) however, it has the potential to improve greatly the turbulence description for high-shear flows. More details on this modeling approach can be found in Pope (2000). 

Gosman AD, Launder BE, Reece GJ (1985) Computer-Aided Engineering Heat Transfer and Fluid Flow. John Wiley Sons, New York Hanjalic K, Launder BE (1972) A Reynolds Stress Model of Turbulence and its Application to Thin Shear Flows. J Fluid Mech Part 4 52 609-638 Harlow FH, Nakayama 1 (1967) Turbulence Transport Equations. The Physics of Fluids 10(ll) 2323-2332... 

The physical interpretation of the terms in the equation is not necessary obvious. The first term on the LHS denotes the rate of accumulation of the kinematic turbulent momentum flux within the control volume. The second term on the LHS denotes the advection of the kinematic turbulent momentum flux by the mean velocity. In other words, the left hand side of the equation constitutes the substantial time derivative of the Reynolds stress tensor v v. The first and second terms on the RHS denote the production of the kinematic turbulent momentum flux by the mean velocity shears. The third term on the RHS denotes the transport of the kinematic momentum flux by turbulent motions (turbulent diffusion). This latter term is unknown and constitutes the well known moment closure problem in turbulence modeling. The fourth and fifth terms on the RHS denote the turbulent transport by the velocity-pressure-gradient correlation terms (pressure diffusion). The sixth term on the RHS denotes the redistribution by the return to isotropy term. In the engineering literature this term is called the pressure-strain correlation, but is nevertheless characterized by its redistributive nature (e.g., [132]). The seventh term on the RHS denotes the molecular diffusion of the turbulent momentum flux. The eighth term on the RHS denotes the viscous dissipation term. This term is often abbreviated by the symbol... 

---