Weighting function density

The physics and modeling of turbulent flows are affected by combustion through the production of density variations, buoyancy effects, dilation due to heat release, molecular transport, and instabiUty (1,2,3,5,8). Consequently, the conservation equations need to be modified to take these effects into account. This modification is achieved by the use of statistical quantities in the conservation equations. For example, because of the variations and fluctuations in the density that occur in turbulent combustion flows, density weighted mean values, or Favre mean values, are used for velocity components, mass fractions, enthalpy, and temperature. The turbulent diffusion flame can also be treated in terms of a probabiUty distribution function (pdf), the shape of which is assumed to be known a priori (1). 

The contribution F is computed in a nonlocal manner by employing the concept of smoothed density [49], p(r), i.e., the density obtained by averaging the local density with a weight function W(r)... 

The density functional theories are also accurate for the density profiles of fused-sphere chains. Figures 4(a) and 4(b) compare the theory of Yethiraj [39] (which is a DFT with the Curtin-Ashcroft weighting function) to Monte Carlo simulations of fused-hard-sphere chains at hard walls for N = 4 and 16, respectively. For both chain lengths the theory is in quantitative agreement with the simulation results and appears to get more accurate as the chain length is increased. Similarly good results were also found by SCMC who compared... 

Not all the data points from the nt trajectories are used in the interpolation. The nsei new data points are selected using the h weight function [133] that balances the desire to place new points as far as possible from the existing nj data points with the need to have a higher data density in dynamically important regions. In particular, the relative importance of a candidate data point Zk (k denotes the trajectory) is given by... 

This so-called Hirshfeld scheme is particularly popular within the so-called conceptual density functional theory (DFT) [26,27], The weighting function, which identifies the AIM as one that is most similar to the isolated atom [28], has been shown to be directly derivable from information entropy [6,29-33]. Here again, the atoms do not... 

Fig. 12. Window shapes and weighting functions used in calculation of effective density from a given layout.

The final density calculation step is the stage at which the 2-D weighting density window function is first employed. The 2-D filter of the correct... 

Now, in the statistical limit for which we can treat our final manifold of closely spaced twisted triplets as an effective continuum, the sum in eq. (12-24) can be replaced by a definite integral whose weight function is taken to be the density of vibronic states, p(E). Thus we write... 

Here / = 1/7 in the standard notation. From our general statements in Section in. A, the spinodal criterion derived from the exact free energy (38) must be identical to this this is shown explicitly in Appendix C. Note that the spinodal condition depends only on the (first-order) moment densities p, and the second-order moment densities py of the distribution p(cr) [given by Eqs. (40) and (41)] it is independent of any other of its properties. This simplification, which has been pointed out by a number of authors [11, 12], is particularly useful for the case of power-law moments (defined by weight functions vt>f(excess free energy only depends on the moments of order 0, 1... K — 1 of the density distribution, the spinodal condition involves only 2K— moments [up to order 2(K — 1)]. 

If extra moment densities with weight functions Wi(o) are included, these simply add a term A,p,- each. 

We now turn to the second general question, regarding the choice of weight functions for the extra moment densities. Comparing Eq. (60) with the formally exact solution (59) of the coexistence problem tells us at least in principle what is required The log ratio ln/f(cr)/p(0)(cr) of the effective prior and the parent needs to be well approximated by a linear combination of the weight functions of the extra moment densities. However, the effective prior is unknown (otherwise we would already have the exact solution of the phase coexistence problem), and so this criterion is of little use [56]. 

Figure 12. The effect of the weight functions for extra moment densities on coexistence calculations, (a) The triangle and bell weight functions, for the case n— 1=3. (b) Coexistence curves for the triangle weight functions, for the same parent as in Fig. 9. Note that multiphase coexistence is generally detected for smaller n than in Fig. 9, where power-law weight functions were used for the extra moments. On the other hand, the predicted number of phases now no longer varies monotonically with n. (c) Dependence of log-error 5 on n for fixed y = 10 and three choices of weight functions for the extra moment densities Power law (solid line), triangle (dashed), bell (dot-dashed). The number of phases is marked next to the curves.

VI.3 EVALUATION OF THE DENSITY-DEPENDENT AVERAGING WEIGHT FUNCTION... 

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