Velocity field filtered

The results of computations of T o for an isolated fiber are dhistrated in Figs. 17-62 and 17-63. The target efficiency T t of an individual fiber in a filter differs from T o for two main reasons (Pich, op. cit.) (1) the average gas velocity is higher in the filter, and (2) the velocity field around the individual fibers is influenced by the proximity of neighboring fibers. The interference effect is difficult to determine on a purely theoretical basis and is usually evaluated experimentally. Chen (op. cit.) expressed the effecd with an empirical equation ... 

The filtering process used in LES results in a loss of information about the SGS velocity field. For homogeneous turbulence and the sharp-spectral filter, the residual velocity field5... 

Note that we have made use of the fact that for a homogeneous flow with an isotropic filter (u U ) = 0. More generally, the conditional expected value of the residual velocity field will depend on the filter choice. 

The residual velocity covariance should not be confused with die Reynolds stresses. Indeed, most of the contribution to the Reynolds stresses comes from the filtered velocity field. Thus, in general, u[u j U ) < C UjUj). 

Note that hv operates on the random field U(r, f) and (for fixed parameters V, x, and t) produces a real number. Thus, unlike the LES velocity PDF described above, the FDF is in fact a random variable (i.e., its value is different for each realization of the random field) defined on the ensemble of all realizations of the turbulent flow. In contrast, the LES velocity PDF is a true conditional PDF defined on the sub-ensemble of all realizations of the turbulent flow that have the same filtered velocity field. Hence, the filtering function enters into the definition of /u u(V U ) only through the specification of the members of the sub-ensemble. 

Fig. 8. (a) Velocity fields at different sections through the reconstructed filter wall (gray denotes the solid material, violet denotes lowest and red denotes highest velocity) and visualization of flow paths in the reconstructed filter wall and (b) comparison of experimental and simulated filter permeabilities (see Plate 8 in Color Plate Section at the end of this book). 

Figure 7.6 Relaxation lengthscales of the unfiltered conditional average velocity field normalised by the bubble diameter d as a function of void fraction. (The difference from estimates based on the filtered conditionally averaged velocity field, are negligible). L and L d correspond to decay distance for up and downstream velocity. The theoretical prediction, from (7.29), of the relaxation lengthscale is Ld/d = 2d/aCd, for Ret = 10.

The equations for the evolution of the filtered velocity field are derived from the Navier-Stokes equations. 

Leonard [97] was apparently the first to use the term Large Eddy Simulation. He also introduced the idea of filtering as a formal convolution operation on the velocity field and gave the first general formulation of the method. Since Leonard s approach form the basis for application of LES to chemical reactor modeling, we discuss this approach in further details. 

At this end, to demonstrate superiority of our classical algorithms, we show some sample results of our most recent FDF simulation of the Sandia/Sydney swirl burner [35]. This configuration is selected as it is one of the most challenging turbulent flames for prediction. Figure 3 shows the contours of the azimuthal velocity field as predicted by our FDF. The simulated results agree with experimental data better than any other classical methods currently available [36]. But the computational time requirements are excessive. As another example. Fig. 4 shows the contour of filtered temperature field for the symbolic Taylor-Green vortex flow as obtained via FDF coupled with a discontinuous Galerkin flow solver [37]. Quantum computation may potentially provide a much more efficient means for such simulations. 

The fluid velocity field v is quite important in convective diffusion of particles occurring in filters and scruhhers used for gas cleaning. Knowledge of v in the separator is essential in predicting particle separation. 

Any solution of equation (6.3.36) or (6.3.37) requires detailed information about the flow field around the filter bed coUector/fiber. The flow field may be available via the three velocities Vx, Vy, (or v, Vg, Vz, etc.) or via the stream function ip, if it can be assumed that the particle motion does not affect the flow field. The solution of such a problem generally requires a numerical solution of the governing equations (e.g. equation (6.2.6b)) for the chosen velocity field around the fiber in the depth filter. 

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