Reynolds Average Navier Stokes approach

The model species, total mass, momentum, and energy continuity equations are similar to those presented in Section 13.7 on fluidized bed reactors. Constant values of the gas and liquid phase densities, viscosities, and diffusivities were assumed, as well as constant values of the interphase mass transfer coefficient and the reaction rate coefficient. The interphase momentum transfer was modelled in terms of the Eotvos number as in Clift et al. [1978]. The Reynolds-Averaged Navier-Stokes approach was taken and a standard Computational Fluid Dynamics solver was used. In the continuous liquid phase, turbulence, that is, fluctuations in the flow field at the micro-scale, was accounted for using a standard single phase k-e model (see Chapter 12). Its applicability has been considered in detail by Sokolichin and Eigenberger [1999]. No turbulence model was used for the dispersed gas phase. Meso-scale fluctuations around the statistically stationary state occur and were explicitly calculated. This requires a transient simulation and sufficiently fine spatial and temporal grids. 

Consider a single-phase homogeneous stirred-tank reactor with a time-invariant velocity field U(x, y, z ) a single reaction of the form /) —> B. (This approach can be extended to the case of time-dependent velocity fields. If the flow in the tank is turbulent, then the velocity field is the solution of the Reynolds averaged Navier-Stokes equations). The tank is divided into a three-dimensional network of n spatially fixed volumetric elements, or n-interacting... 

The three main numerical approaches used in turbulence combustion modeling are Reynolds averaged Navier Stokes (RANS) where all turbulent scales are modeled, direct numerical simulations (DNS) where all scales are resolved and large eddy simulations (LES) where larger scales are explicitly computed whereas the effects of smaller ones are modeled ... 

By far, the most widely employed models for reactive flow processes are based on Reynolds-averaged Navier Stokes (RANS) equations. As discussed earlier in Chapter 3, Reynolds averaging decomposes the instantaneous value of any variable into a mean and fluctuating component. In addition to the closure equations described in Chapter 3, for reactive processes, closure of the time-averaged scalar field equations requires models for (1) scalar flux, (2) scalar variance, (3) dissipation of scalar variance, and (4) reaction rate. Details of these equations are described in the following section. Broadly, any closure approach can be classified either as a phenomenological, non-PDF (probability density function) or as a PDF-based approach. These are also discussed in detail in the following section. 

Reynolds averaged Navier-Stokes (RANS) equation Equation representing the conservation of momentum in a fluid flow, subjected to a temporal or spatial averaging process in line with the approach proposed by Osborne Reynolds. 

Three different theoretical approaches have been established describing turbulent flows in general, as outlined in sect 1.3. These methods are the direct numerical simulations (DNS), large eddy simulations (LES), and the Reynolds average Navier-Stokes (RANS) approach. 

CFD simulations at high Reynolds numbers for technical applications are nowadays mainly based on solutions of the Reynolds averaged Navier-Stokes (RANS) equations. The main reason are that they are simple to apply and computationally more efficient than other turbulence modelling approaches such as LES.It is known, however, that in many flow problems the condition of a turbulent equilibrium is not satisfied, i.e., when strong pressure gradients or flow separation occurs, which reduces the prediction accuracy of the results obtained by one-and two-equation turbulence models used to close the RANS equations [13,15]. 

As of this writing, the only practical approach to solving turbulent flow problems is to use statistically averaged equations governing mean flow quantities. These equations, which are usually referred to as the Reynolds equations of motion, are derived by Reynold s decomposition of the Navier-Stokes equations (18). The randomly changing variables are represented by a time mean and a fluctuating part ... 

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