Computational Examples

The possibilities of CFD are demonstrated in this section on the basis of two examples. The focus is exclusively on post-processing in other words, the analysis of the computation results. 

The first example shows a simple screw element (Fig. 8.4). The geometry of the screw element is uniquely defined by the following six parameters  

Because the geometry of a double-flighted screw element repeats after half the pitch, it is sufficient to observe a length of 60 mm. Assuming a fully developed, isothermal flow, the upper and lower cross-section surfaces are linked together by periodic boundary conditions. 

The polymer melt used in this example has a density of 1000 kg/m3. The following initially assumes a Newtonian flow behavior with a viscosity of 1000 Pa-s. In later computations, a more realistic shear thinning flow behavior is assumed, which can be described using the power law equation. The flow exponent n ranges between 0.4 and 0.9 and the consistency 

Two cases are observed in more detail as specific operating points for a Newtonian flow 

Numerical studies allow us to explore aspects of these models for a number df molecular continua and pulse configurations. Consider first the effect of the puUe intensity on transition probabilities to a slowly varying continuum by considering continuum composed of single broad Lorentzian [Eq. (10.21)] of wid r, = 2000 cm-1, excited by a 120 cm-1 wide pulse (i.e., a pulse of 80fs durp tion). The central frequency of the pulse is tuned to the center of the continuutfi) (A, = 0) and the pulse peaks at t = 0. 

The situation is quite different for a structured continuum. Consider Fig where the strong-pulse-induced transition to a narrow continuum (Ts = 50 bfi displayed. The results show behavior that is intermediate between a flat , 1 nuum and a discrete set of levels. We see that center-line , caE l — coj 5= 0, ( nuum levels display recurrences, or Rabi oscillations, similar, though not idet 

Time dependence of continuum populations for a bound-continuum spectrum sed of two (r = 1,2) overlapping resonances, (a) The weak-field absorption spectrum. 

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We describe a simple computational example to demonstrate two key features of the new protocol Stability with respect to a large time step and filtering of high frequency modes. In the present manuscript we do not discuss examples of rate calculations. These calculations will be described in future publications. 

Gilli and coworkers57 have recognized many other examples of the RAHB phenomenon from the Cambridge Structural Database, documenting the structural correlations that strongly support the hypothesis of the covalent nature of these H-bonds. The computational examples presented in this section are fully consistent with their RAHB model, and similar NBO/NRT patterns would be expected to characterize the many interesting classes of compounds that were considered by these workers, but are beyond the scope of the present work. 

The primary piece of information obtained from most theoretical calculations is the molecular structure. If homoaromatic interactions are important in a molecule, they may cause the molecule to adopt an unusual geometry. In suitable radicals, ESR evidence has been taken to indicate systems of high symmetry which in turn has been interpreted in terms of homoaromatic interactions (Dai et al., 1990). A computational example of this effect is shown in the semiempirical calculations of Williams and Kurtz (1988) on the bisannelated semibullvalene [108]. Here simple configuration interaction... 

Figure 12.3 shows some computational examples of nonreactive and reactive turbulent flows in a combustor with the bluff-body flame holder. The size of the combustor in Fig. 12.3 is 35 x 8 cm. The characteristic height and length of the bluff body is H = 2 cm. The left boundary is set as inlet, right boundary as outlet, and the upper and lower boundaries as rigid walls. 

Numerical studies of combustion control in simple combustors with flame holders have been made. The criterion of flame stabilization, based on the unambiguously defined characteristic residence and reaction times, is suggested and validated against numerous computational examples. The results of calculations were compared with available experimental findings. A good qualitative and reasonable quantitative agreement between the predictions and observations were attained. Futher studies are planned to include mixing between fuel jets with oxidizer and to extend the analysis to transonic and supersonic flow conditions. 

Several computational examples are shown in Figures 14 and 15 and Tables 2 to 6. Three minimization codes are examined CONMIN (nonlinear CG and full-memory BFGS), LM-BFGS, and TNPACK (TN method). Preconditioning options, the number of stored updates (for LM-BFGS), starting points, and problem dimensions are varied for analysis. 

Figure 42 displays a computational example of P i) (solid line) and the survival probability C(f)p (dashed line) as a function of the total number of states N and the number of the prompt states K. The initial state is taken to have uniform weights for either the K prompt states or the N — K delayed states. It is seen that with fixed K and increasing N the decay of the initial delayed state is shifted to much longer time, whereas the decay of the initial prompt state changes little. 

Figure 43. A computational example oi P(t), Pj(t), and C(t), for three different initial states taken to be uniformly weighted in three different sub-subspaces (denoted by gi, Q2, and Q3) of the bound-state subspace. Note that there is coupling between the three sub-subspaces. [From F. Remade and R. D. Levine, J. Phys. Chem. 100, 7962 (1996).]...

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