Turbulence, wall functions

Yuan, X., Moser, A., Surer, P. Wall functions for numerical simulation of turbulent natura] convection along vertical plates, /nt. J. Heat Mass Transfer, vol. 36, pp. 4477—448,5, 1993. 

To save computational effort, high-Reynolds number models, such as k s and its variants, are coupled with an approach in which the viscosity-affected inner region (viscous sublayer and buffer layer) are not resolved. Instead, semiempiri-cal formulas called wall functions are used to bridge the viscosity-affected region between the wall and the fully turbulent region. The two approaches to the sublayer problem are depicted schematically in Fig. 2 (Fluent, 2003). 

It is important to place the first near-wall grid node far enough away from the wall at yP to be in the fully turbulent inner region, where the log law-of-the-wall is valid. This usually means that we need y > 30-60 for the wall-adjacent cells, for the use of wall functions to be valid. If the first mesh point is unavoidably located in the viscous sublayer, then one simple approach (Fluent, 2003) is to extend the log-law region down to y — 11.225 and to apply the laminar stress-strain relationship U — y for y < 11.225. Results from near-wall meshes that are very fine using wall functions are not reliable. 

The standard wall function is of limited applicability, being restricted to cases of near-wall turbulence in local equilibrium. Especially the constant shear stress and the local equilibrium assumptions restrict the universality of the standard wall functions. The local equilibrium assumption states that the turbulence kinetic energy production and dissipation are equal in the wall-bounded control volumes. In cases where there is a strong pressure gradient near the wall (increased shear stress) or the flow does not satisfy the local equilibrium condition an alternate model, the nonequilibrium model, is recommended (Kim and Choudhury, 1995). In the nonequilibrium wall function the heat transfer procedure remains exactly the same, but the mean velocity is made more sensitive to pressure gradient effects. 

The research on the flow regimes in packed tubes suggests that laminar flow CFD simulations should be reasonable for Re <100 approximately, and turbulent simulations for Re >600, also approximately. Just as RANS models provide steady solutions that are regarded as time averages of the real time-dependent turbulent flow, it may be suggested that CFD simulations in the unsteady laminar inertial range 100 time-averaged picture of the flow field. As with wall functions, comparisons with experimental data and an improved assessment of what information is really needed from the simulations will inform us as to how to proceed in these areas. 

The finite volume methods have been used to discretised the partial differential equations of the model using the Simple method for pressure-velocity coupling and the second order upwind scheme to interpolate the variables on the surface of the control volume. The segregated solution algorithm was selected. The Reynolds stress turbulence model was used in this model due to the anisotropic nature of the turbulence in cyclones. Standard fluent wall functions were applied and high order discretisation schemes were also used. 

The form and complexity of the wall functions is closely related to the turbulence model closure used. 

There are two main approaches to modeling the near-wall region. In one approach, the so-called wall function approach, the viscosity-affected inner regions (viscous and buffer layers) are not modeled. Instead, semi-empirical formulae (wall functions) are used to bridge the viscosity-affected region between the wall and the fully turbulent region. In another approach, special, low Reynolds number turbulence models are developed to simulate the near-wall region flow. These two approaches are shown schematically in Fig. 3.5(b) and 3.5(c). 

In most high Reynolds number flows, the wall function approach gives reasonable results without excessive demands on computational resources. It is especially useful for modeling turbulent flows in complex industrial reactors. This approach is, however, inadequate in situations where low Reynolds number effects are pervasive and the hypotheses underlying the wall functions are not valid. Such situations require the application of a low Reynolds number model to resolve near-wall flows. For the low Reynolds number version of k-s models, the following boundary conditions are used at the walls ... 

The influence of a wall on the turbulent transport of scalar (species or enthalpy) at the wall can also be modeled using the wall function approach, similar to that described earlier for modeling momentum transport at the wall. It must be noted that the thermal or mass transfer boundary layer will, in general, be of different thickness than the momentum boundary layer and may change from fluid to fluid. For example, the thermal boundary layer of a high Prandtl number fluid (e.g. oil) is much less than its momentum boundary layer. The wall functions for the enthalpy equations in the form of temperature T can be written as ... 

At the outlet, extrapolation of the velocity to the boundary (zero gradient at the outlet boundary) can usually be used. At impermeable walls, the normal velocity is set to zero. The wall shear stress is then included in the source terms. In the case of turbulent flows, wall functions are used near walls instead of resolving gradients near the wall (refer to the discussion in Chapter 3). Careful linearization of source terms arising due to these wall functions is necessary for efficient numerical implementation. Other boundary conditions such as symmetry, periodic or cyclic can be implemented by combining the formulations discussed in Chapter 2 with the ideas of finite volume method discussed here. More details on numerical implementation of boundary conditions may be found in Patankar (1980) and Versteeg and Malalasekara (1995). 

Turbulence models user-defined model (UD) Wall functions constraints on near wall cells... 

In general CFD models show a good applicability for risk assessments in urban areas however, their results can differ depending on turbulent closure models and other assumptions (Schatzmann, 2005 [562]). For practical applications these models contain a substantial amount of empirical knowledge, not only in the turbulent closure schemes but also in the use of wall functions and in other parameterisation schemes. To cast doubt on the results is perfectly justified, as it was shown by systematic studies in which applications of the same model by different modellers to a given problem (Hall, 1997 [244]) and applications of different models by either the same or different modellers to the same problem (Ketzel et al., 2002 [332]) revealed significant differences. 

In this paragraph the wall function concept is outlined. The wall functions are empirical parameterizations of the mean flow variable profiles within the inner part of the wall boundary layers, bridging the fully developed turbulent log-law flow quantities with the wall through the viscous and buffer sublayers where the two-equation turbulence model is strictly not valid. These empirical parameterizations thus allow the numerical flow simulation to be carried out with a finite resolution within the wall boundary layers, and one avoids accounting for viscous effects in the model equations. Therefore, in the numerical implementation of the k-e model one anticipates that the boundary layer flow is not fully resolved by the model resolution. The first grid point or node used at a wall boundary is thus placed within the fully turbulent log-law sub-layer, rather than on the wall itself [95]. In effect, the wall functions amount to a synthetic boundary condition for the k-e model. In addition, the limited boundary layer resolution required also provides savings on computer time and storage. 

Both equilibrium and non-equilibrium wall boundary implementations are considered. For equilibrium flows the local production rate of turbulence equals the dissipation rate in the near wall grid node. The first set of wall function boundary conditions reported was apparently used for equilibrium flows by Gosman et al. [59]. Denoting the dependent variables in the first point near the wall by a subscript P, an approximate sketch of their approach is given next. 

Step 4 Set the boundary conditions as follows. The centerline, inlet velocity, and exit velocity/pres sure are set as in the laminar case slip/symmetry, v = 2, Normal flow/ Pressure, p = 0. The wall boundary condition, though, is set to the Logarithmic wall function. This is an analytic formula for the velocity, turbulent kinetic energy, and rate of dissipation, as determined by experiment (Deen, 1998, pp. 527-528). 

Gas flow in flow field can be described by Navi-er-Stokes equations, and then close equations by revised turbulent fluctuation RNG k-e two-way model, dealing with wall flow by wall function method (Wang et al. 2004). The model reflect the impact of small-scale through viscosity which is got after large-scale motion and amendments, remove small-scale movement systematically from... 

The remainder of the chapter focuses on the actual spray modeling. The exposition is primarily done for the RANS method, but with the indicated modifications, the methodology also applies to LES. The liquid phase is described by means of a probability density function (PDF). The various submodels needed to determine this PDF are derived from drop-drop and drop-gas interactions. These submodels include drop collisions, drop deformation, and drop breakup, as well as drop drag, drop evaporation, and chemical reactions. Also, the interaction between gas phase, liquid phase, turbulence, and chemistry is examined in some detail. Further, a discussion of the boundary conditions is given, in particular, a description of the wall functions used for the simulations of the boundary layers and the heat transfer between the gas and its confining walls. 

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