RANS model

Thus, the reactor will be perfectly mixed if and only if = at every spatial location in the reactor. As noted earlier, unless we conduct a DNS, we will not compute the instantaneous mixture fraction in the CFD simulation. Instead, if we use a RANS model, we will compute the ensemble- or Reynolds-average mixture fraction, denoted by ( ). Thus, the first state variable needed to describe macromixing in this system is ( ). If the system is perfectly macromixed, ( ) = < at every point in the reactor. The second state variable will be used to describe the degree of local micromixing, and is the mixture-fraction variance (maximum value of the variance at any point in the reactor is ( )(1 — ( )), and varies from zero in the feed streams to a maximum of 1/4 when ( ) = 1/2. 

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. 

RANS turbulence models are the workhorse of CFD applications for complex flow geometries. Moreover, due to the relatively high cost of LES, this situation is not expected to change in the near future. For turbulent reacting flows, the additional cost of dealing with complex chemistry will extend the life of RANS models even further. For this reason, the chemical-source-term closures discussed in Chapter 5 have all been formulated with RANS turbulence models in mind. The focus of this section will thus be on RANS turbulence models based on the turbulent viscosity hypothesis and on second-order models for the Reynolds stresses. 

The presentation in this section has been intentionally kept short, as our primary objective is to present the standard form of each model so that the reader can refer to them in later chapters. Nevertheless, it is extremely important for the reader to realize that the quality of a reacting-flow simulation will in no small part depend on the performance of the turbulence model. The latter will depend on a number of issues15 that are outside the scope of this text, but that should not be neglected when applying RANS models to complex flows. 

Owing to these unique characteristics, additional wall-reflection terms are required in the pressure redistribution model in order to obtain satisfactory agreement with data for the impinging jet flow. A detailed discussion of RANS models that employ a more physically realistic description of the pressure fluctuations in the near-wall region can be found in Pope (2000). However, one obvious shortcoming of current wall models is that they typically depend explicitly on the unit normal to the wall, which makes it very difficult to apply them to complex geometries. 

At present, there exists no completely general RANS model for differential diffusion. Note, however, that because it solves (4.37) directly, the linear-eddy model discussed in Section 4.3 can describe differential diffusion (Kerstein 1990 Kerstein et al. 1995). Likewise, the laminar flamelet model discussed in Section 5.7 can be applied to describe differential diffusion in flames (Pitsch and Peters 1998). Here, in order to understand the underlying physics, we will restrict our attention to a multi-variate version of the SR model for inert scalars (Fox 1999). 

Thus, solutions to the transported PDF equation will provide more information than is available from second-order RANS models without the problem of closing the chemical source term. 

Relative to velocity, composition PDF codes, the turbulence and scalar transport models have a limited range of applicability. This can be partially overcome by using an LES description of the turbulence. However, consistent closure at the level of second-order RANS models requires the use of a velocity, composition PDF code. 

While some of these disadvantages can be overcome by devising improved algorithms, the problem of level of description of the RANS turbulence model remains as the principal shortcoming of composition PDF code. One thus has the option of resorting to an LES description of the flow combined with a composition PDF code, or a less-expensive second-order RANS model using a velocity, composition PDF code. 

Time dependence The inherent time dependence of fires sets strong requirements for computational efficiency. In RANS models, the radiation field must be updated within the internal iterations of the time step, but the computational cost can be relaxed by solving RTE only every Mh iteration. In SOFIE, for example, it is typical to use N = 10. In FDS, the time accuracy of the radiation field has been relaxed by solving the FVM equations typically every third time step and only part of the directions at a time. 

In RANS models, the solid wall boundary conditions have traditionally been modeled using wall functions. Wall functions use empirical profiles to replace the actual boundary conditions, such as no-slip (zero velocity) condition at solid surfaces. An example of an empirical law is the logarithmic velocity profile ... 

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