Grid spaces

If the grid spacing is not uniform, the formulas must be revised. The notation is shown in Fig. 3-50. The finite-difference form of the equations is then... 

A priori estimates obtained in Chapter 6, Section 2 provides the sufficient background for a- uniform estimate of accuracy in the norm of the grid space C such as... 

Remark 1 Convergence in the norm of the grid space L2 occurs with the same rate if condition c) is relaxed ... 

Two last summands in the preceding inequality are some quantities of 0(r ), so there is some reason to make their contributions to the accurate account of the accuracy z. Thus, scheme (58) converges in the grid space... 

Pressure drop Grid spacing Field quantity Diffusivity... 

In LES, it is accepted that the flow is not fully resolved turbulent eddies smaller than the grid spacing A are not solved explicitly. These small eddies do contribute, however, to the redistribution of momentum within the flow field. The resulting set of NS equations then runs as... 

Even when the number of grid cells in a LB LES simulation of a stirred vessel 1.1 m3 in size amounts to some 36 x 106 grid cells, this implies a cell size, or grid spacing, of 5 mm only. Even a cell size of just a few millimeters makes clear that substantial parts of the transport of heat and species as well as all chemical reactions take place at scales smaller than those resolved by the flow simulation. In other words concentrations of species and temperature still vary and fluctuate within a cell size. The description of chemical reactions and the transport of heat and species therefore ask for subtle approaches to these SGS fluctuations. 

The second step in Ten Cate s two-step approach was to focus on crystal-crystal interaction by means of an explicit two-phase DNS of the turbulent suspension that completely resolves the translational and rotational motions and collisions of the spherical particles plus the turbulence of the liquid between the particles. The particle motions are driven by the turbulent flow and the particles affect the turbulent flow of the liquid in between. When particles approach one another down to a distance smaller than the grid spacing, lubrication theory is exploited to bridge the gap between them. 

The volume occupied by the host framework was calculated by subtracting the "available volume , after removal of the guest molecules, from the volume of the unit ceU. "Available volumes were calculated with Molecular Simulations Inc. Cerius2 (v. 35) software using a probe radius of 05A and "fine grid spacing. 

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. 

As described above, spatial transport in an Eulerian PDF code is simulated by random jumps of notional particles between grid cells. Even in the simplest case of one-dimensional purely convective flow with equal-sized grids, so-called numerical diffusion will be present. In order to show that this is the case, we can use the analysis presented in Mobus et al. (2001), simplified to one-dimensional flow in the domain [0, L (Mobus et al. 1999). Let X(rnAt) denote the random location of a notional particle at time step m. Since the location of the particle is discrete, we can denote it by a random integer i X(mAt) = iAx, where the grid spacing is related to the number of grid cells (M) by Ax = L/M. For purely convective flow, the time step is related to the mean velocity (U) by16... 

Although not denoted explicitly, we have seen in Section 6.8 that this estimate will depend on the grid spacing M and the number of particles Nv. In addition to the mean composition, the output data from the PDF code will usually be various composition statistics estimated at grid-cell centers. We will thus need accurate and efficient statistical estimators for determining particle fields given the ensemble of Nv notional particles. 

For most tests, if the drop height were too high (S3 m), no explosions resulted. Also, inhibition was obtained when a steel grid (spacing 2.5 cm) was placed between the crucible and water surface. 

Figure 3. a) Disclinations in an otherwise smooth nematic texture of PBT in MSA b) Ring images observed on warming a nematic PBT solution see text. The large grid spacing is 80/im. 

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