Artificial smearing

When the transport equation for c is solved with a discretization scheme such as upwind, artificial diffusive fluxes are induced, effecting a smearing of the interface. When these diffusive fluxes are significant on the time-scale of the simulation, the information on the location of different fluid volumes is lost. The use of higher order discretization schemes is usually not sufficient to reduce the artificial smearing of the interface to a tolerable level. Hence special methods are used to guarantee that a physically reasonable distribution of the volume fraction field is maintained. 

A drawback of the Lagrangean artificial-viscosity method is that, if sufficient artificial viscosity is added to produce an oscillation-free distribution, the solution becomes fairly inaccurate because wave amplitudes are damped, and sharp discontinuities are smeared over an increasing number of grid points during computation. To overcome these deficiencies a variety of new methods have been developed since 1970. Flux-corrected transport (FCT) is a popular exponent in this area of development in computational fluid dynamics. FCT is generally applicable to finite difference schemes to solve continuity equations, and, according to Boris and Book (1976), its principles may be represented as follows. 

The code reproduced shock-jump conditions well, but many details in the solution were lost because of the smearing effect of artificial viscosity. 

Figure 10, Red cell smears after the application of the add elution technique. A, artificial mixture of cells from newborn and adult. B, blood sample from a patient with Fanconts anemia (Hb-Fjto 14.6%).

The Stockholm papyrus (third or fourth century A.D.) gives the following recipe for preparing verdigris for making artificial emeralds Clean a well-made sheet of Cyprian copper by means of pumice stone and water, dry, and smear it very lightly with a very little oil. Spread it... 

Spreading is the main method for making artificial leather. It comprises material spreading onto a moving fabric belt with a spreading knife or smearing roller 4>. 

Spreading (smearing) is the most widely used and well-studied method of plastisot processing for making linoleum, artificial leathers, wire enameling, etc. Descriptions of the process and its quantitative analysis may be found elsewhere3-4 31>. 

Figure 5 Calculated absorption spectra for four different Si clusters (a) Sii4H2o, (b) Sii4H2o>C>2, (c) Sii4Hi8=0, (d)Sii3Hi6>0=0 (an artificial Gaussian smearing of 0.05 eV has been applied).

Table 4 shows the residuals from the fits to these artificially perturbed data. Least squares, as represented by DORTHO, has smeared lack of fit across the 27 correct values in a quite unacceptable way. Instead of three residuals larger than 100, we now have (skipping the five parenthesized residuals, which ought to be large) four residuals larger than 20000. 

The theory now proceeds as developed in Sections V and VI, essentially unchanged. For example, P v) will have the same bimodal structure as shown in Fig. 14, but will now be continuous. Similar smoothing of all artificially introduced discontinuities will not affect the theory in any essential way. The loss of a sharp distinction between liquid- and solidlike cells could vitiate use of the percolation theory. The nonanalyticity in S will certainly be lost, leading to a communal entropy for which 9S/9p is always less than infinity. However, the first-order phase transition should be preserved, just as it was for most of the parameter space even when )3> 1. The discontinuity in p and v would be reduced, as would be the latent heat. One important effect of this smearing will be the appearance of a critical end point for the liquid, a temperature below which the liquid phase is no longer even metastable. The second-order transition, which is only a small region of parameter space for /8> 1, is now wiped out completely by the restoration of analyticity. Our theory thus leads to a first-order phase transition or no transition at all. However, the entropy catastrophe can be resolved within our theory only if a transition occurs. 

Invert soaps do not appear in nature but are important synkinons in the preparation of artificial membrane structures. The most common application of such monolayers is a cosmetic one. For centuries people smeared fats on their hair to make it shiny, but the hair then stuck together. Invert soaps adsorb strongly to hair proteins and provide them with the elegant luster of a monolayer and the fullness of non-greasy and non-polar hair. Hair with a nonsticky hydrocarbon monolayer on the surface looks irresistibly shiny, fluffy, and clean. Combinations with polymers, such as silicones, proteins, and poly(vinyl pyrrolidone) then help to build even more body in leave-in conditioning products. 

Enough artificial or real viscosity must be used to smear the shock waves over at least three cells. The PIC form with a constant of 2.0 is useful for many problems because of its scaling as a function of particle velocity. The time step is often determined by the Courant condition however, for most problems the time step can be estimated in microseconds as... 

Thus, it is clear that the finite difference numerical solutions offered by some authors are not really necessary because problems without capillary pressure can be solved analytically. Actually, such computational solutions are more damaging than useful because the artificial viscosity and numerical diffusion introduced by truncation and round-off error smear certain singularities (or, infinities) that appear as exact consequences of Equation 21-17. Such numerical diffusion, we emphasize, appears as a result of finite difference and finite element schemes only, and can be completely avoided using the more labor-intensive method of characteristics. For a review of these ideas, refer to Chapter 13. As we will show later, capillary pressure effects become important when singularities appear modeling these correctly is crucial to correct strength and shock position prediction. 

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