Large eddy simulation Direct numerical simulation

Increased computational cost per iteration

FIGURE 3.3 Modeling approaches for turbulent flows.

Increased computational cost per iteration

FIGURE 3.3 Modeling approaches for turbulent flows.

best-informed guess. This means that the methods are genuinely useful for engineering applications even though they cannot replace experiments. The major efforts in the area of turbulence modeling have been and are still directed towards developing tractable computational models of turbulent flows with reasonable demands on computational resources. Large numbers of models have been developed in the last three decades or so. These modeling approaches (which are summarized in Fig. 3.3) can be classified into three categories; direct numerical simulations (DNS), large eddy simulations (LES) and Reynolds-averaged Navier-Stokes equations (RANS). These three approaches are briefly reviewed. As one progresses from DNS to RANS, more and more of the turbulent motions are approximated and, therefore, require less computational resources.

3.3.1. Direct Numerical Simulation (DNS)

Direct numerical simulation, as the name implies, attempts to simulate all the dynamically important scales of turbulent flows, directly. It is based on the hypothesis that direct simulations may be carried out by artificially decreasing the Reynolds number to the point where important scales can be simulated accurately on existing computers. This is probably the most exact approach to turbulence simulation without requiring any additional modeling beyond accepting the Navier-Stokes equations to describe the turbulent flow processes. The result is equivalent to a single realization of a flow or a short duration laboratory experiment. It is also the simplest approach conceptually. In DNS, all the motions contained in the flow are resolved.

Specification of initial and boundary conditions is one of the most important and difficult steps in applying DNS. For example, it is necessary to provide inlet boundary conditions specifying the time variation of velocities at all the grid points lying on the inlet boundary, throughout the simulation. For DNS simulations, the usual application of the symmetry boundary condition will no longer be valid because small-scale motions in turbulent flows will not be symmetric. Since the approach aims to resolve all the spatial and temporal gradients, the application of DNS requires huge computational resources. It must be remembered that a large number of grid points are required to resolve small spatial scales and for each grid point, time history needs to be stored to make meaningful simulations. Thus, DNS generates a huge amount of data containing time history at each point, which may not be necessary for the engineering application under consideration. DNS can, however, provide valuable information about the interaction of small-scale and large-scale motions. Often, such information is very difficult or impossible to obtain from experiments. The information obtained from DNS can be helpful in evaluating and validating more approximate models and may lead to the development of better models. More details of the DNS approach and some applications can be found in Rogallo and Moin (1984), Kim et al. (1987), Reynolds (1990), Choi et al. (1994) and Leonard (1995) among others.

The DNS approach may also give useful information about the interaction of chemical reactions and turbulence. Such interactions are discussed in Chapter 5. In general, DNS generates a lot more information than that needed for typical reactor engineering applications. In order to use the information, a reactor engineer has to resort to some sort of averaging, and such averaging may introduce errors. When the accuracy of some of the input data to the model, such as reaction kinetics, is not very high, it may be worthwhile to explore alternative approaches, which are less exact than DNS but require much less computational resources. Lower demands on computational resources mean less turnover time (time required to complete the simulations) and more opportunities for engineering applications, provided the possible errors and their implications are fully recognized. Two such approaches are discussed below.

3.3.2. Large Eddy Simulations (LES)

Large eddy simulations are based on the hypothesis that the relevant scales in turbulent flows can be separated into large-scale and small-scale (sub-grid) components. It is assumed that such separation does not have a significant effect on the evolution of large-scale turbulent motions. The range of scales occurring in turbulent flows and their relationship with modeling approaches is shown schematically in Fig. 3.4. The large-scale motions are generally much more energetic than the small scale motions and are the main contributors to the transport of conserved quantities. LES attempts to simulate these large-scale motions more precisely than the small-scale motions. The small scales of turbulence are believed to be more universal in character than large scales, which facilitate their modeling. Therefore, in LES, large-scale motions are resolved rigorously and small-scale motions (large wave numbers) are modeled in lieu of being resolved. LES models are also three-dimensional and time-dependent but are much less costly (and more flexible) than DNS.

The maximum wave number resolved with the LES approach is chosen to lie in the inertial sub-range of the turbulence energy spectrum. The governing transport equations are derived either by filtering the Navier-Stokes equation or using volume u

FIGURE 3.4 Schematic representation of scales in turbulent flows and their relationship with modeling approaches (adapted from Ferziger and Peric, 1995).

averaging concepts. Both methods lead to similar sets of governing equations. Any flow variable, 0, in the fluid domain, D, is decomposed into a large scale part, <0> and a small scale part, 0" as:

0 = <0} + 0" v <0} = I G(x - x*, A)0(x*) d3x* (3.3)

Jd where A is the characteristic filter scale and G is a filter function obeying the following property:

0 0

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