## 33 Modeling Approaches

Although the structural and deterministic approaches to characterizing turbulence have demonstrated promising results, a deductive approach based on solution of the basic governing equations of the flow processes is the most widely used approach for engineering applications. The basic premise in modeling turbulence is that it can be understood within the continuum assumption of fluid dynamics. There are some experimental facts which might shed doubt on the validity of the assumption. For example, small amounts of long chain polymers have significant influence on turbulence properties (drag), even though the polymer molecules are well dispersed and have dimensions significantly smaller than the dissipation scales of turbulence (more about these scales later). Despite this, the continuum assumption has formed the basis for modeling turbulence over the last several decades and therefore will be accepted here for modeling turbulent flows relevant to reactor engineering.

Accepting the continuum assumption implies accepting the use of the same basic momentum conservation equations (discussed in Chapter 2) to describe turbulent flows. If this is the case, one may wonder why there is a need for any further modeling? One has only to solve the momentum conservation equations described earlier with appropriate boundary conditions to predict the desired flow characteristics at any value of Reynolds number. However, at large Reynolds numbers, the inherent non-linearity in these equations manifest in terms of turbulence, which is a three-dimensional, unsteady phenomenon as is previously described. No doubt, there exist a number of numerical methods and computer programs capable of solving three-dimensional, time dependent momentum equations. The main difficulty in solving the basic governing equations under turbulent conditions is the inability to resolve the wide range of spatial and temporal scales simultaneously. From the foregoing discussion about the scales, it can be shown that the distance between the large and small scales grows with increase in Reynolds number (as Re3/4). Therefore, as the Reynolds number increases and the flow becomes more turbulent, the requirements on resolution become more and more stringent. The number of grid points and the smallness of the time steps required to resolve all the relevant time and space scales of turbulent motion push the computation of turbulent flows in industrial equipment beyond the realms of present computing capabilities. Estimates from various sources differ on the required mesh spacing and on when computer technology will have advanced to the point where turbulent flow calculations can be made from first principles. It appears that most engineering computations involving turbulent flow processes will have to rely on models of turbulent flows, at least for the foreseeable future. This is especially true for chemical reactor engineering applications, where, in addition to turbulence, there are many other complexities such as chemical reactions, multiple phases, complex geometry and so on.

It is sometimes argued that if Navier-Stokes equations can completely describe turbulent flows, it is futile to search for models which are simpler to solve and also retain a complete description of turbulent flows. Many reviews on turbulence modeling conclude by saying that we cannot calculate all the flows of engineering interest to the desired engineering accuracy with available turbulence models, which is of course true. However, the best modern computational models (and numerical methods) allow almost all the flows to be calculated to higher accuracy than the

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