Resolved scale

The filtering process removes all SGS fluctuations so that U(x, t) is considerably smoother than U(x, t). As a general rule, the bandwidth should be at least twice the smallest resolved scale in an LES calculation. However, it is not unusual to see LES simulations reported where the bandwidth is nearly equal to the grid spacing. 

Chapter 3 will be employed. Thus, in lieu of (x, t), only the mixture-fraction means ( ) and covariances ( , F) (/, j e 1,..., Nm() will be available. Given this information, we would then like to compute the reacting-scalar means and covariances (require additional information about the mixture-fraction PDF. A similar problem arises when a large-eddy simulation (LES) of the mixture-fraction vector is employed. In this case, the resolved-scale mixture-fraction vector (x, t) is known, but the sub-grid-scale (SGS) fluctuations are not resolved. Instead, a transport equation for the SGS mixture-fraction covariance can be solved, but information about the SGS mixture-fraction PDF is still required to compute the resolved-scale reacting-scalar fields. 

In the equilibrium-chemistry limit, the turbulent-reacting-flow problem thus reduces to solving the Reynolds-averaged transport equations for the mixture-fraction mean and variance. Furthermore, if the mixture-fraction field is found from LES, the same chemical lookup tables can be employed to find the SGS reacting-scalar means and covariances simply by setting x equal to the resolved-scale mixture fraction and x2 equal to the SGS mixture-fraction variance.88... 

The SGS mixture-fraction mean will be equal to the resolved-scale (or filtered) mixture fraction (f )sgs = (x> 0-... 

Physically, this closure attempts to model the effect of SGS fluctuations on the filtered reaction rate by assuming that the largest of the SGS are dynamically similar to the smallest of the resolved scales. The influence of the smallest eddies, in particular Kolmogorov eddies, on the filtered reaction rate is assumed negligible because the large viscosity in the flame tends to rapidly dissipate these scales [19]. Further discussion and assessment of this model with comparisons to other SGS combustion closures and DNS results can be found in [17]. 

Figure 4.5 Vertical profile of normalized turbulent kinetic energy (solid curve) and its components from a large-eddy simulation resolved-scale flow (long dash line) SGS energy (dash-dot line) wake kinetic energy (short dash line). Calculations were performed for an element dimension of 0.1 m and a canopy height of 10 m. (From Shaw and Patton, 2003 [578]).

Fig. 1.8. The energy spectrum divided into resolved scales (fc < kc) and subgrid scales k > kc), for the purpose of large-eddy simulation.

It follows from (1.382), (1.476), and (1.482) that the resolved scales satisfy the continuity equation given by ... 

The first engineering LES was Deardorff s [27] simulation of plane channel flow. Deardorff used Reynolds (spatial) averaging, applied to a unit cell of the finite difference mesh, to define the larger (or resolved) scales, and introduced the terminology filtered variables. Although only 6720 grid points were used, the comparison with literature laboratory experiments was sufficient favorable for the feasibility of the method to have been established. 

The quantity Vi denotes the resolvable-scale filtered velocity. 

In other cases the application of this concept has been further extended simulating faster turbulent fluctuations that are within the turbulence spectrum. For such dynamic simulations, using Reynolds averaged models, the Ic-quantity represents the turbulent kinetic energy accumulated on the fraction of the spectrum that is represented by the modeled scales. Therefore, to compare the simulated results obtained with this type of models with experimental data, that is averaged over a sufficient time period to give steady-state data (representing the whole spectrum of turbulence), both the modeled and the resolved scales have to be considered [68]. 

Most of the work done in this area has focused on the exploration of the potential of micromixers, which is seen in attaining narrower size distributions, resolving scale-up problems by a numbering-up approach, better predictability of attainable drop size and a higher energy efficiency of the emulsification process compared with conventional approaches [3]. Exemplary investigations - not necessarily vhth a background... 

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