Lagrangian computation, compared

Because this method avoids iterative calculations to attain the SCF condition, the extended Lagrangian method is a more efficient way of calculating the dipoles at every time step. However, polarizable point dipole methods are still more computationally intensive than nonpolarizable simulations. Evaluating the dipole-dipole interactions in Eqs. (9-7) and (9-20) is several times more expensive than evaluating the Coulombic interactions between point charges in Eq. (9-1). In addition, the requirement for a shorter integration timestep as compared to an additive model increases the computational cost. 

Although a direct comparison between the iterative and the extended Lagrangian methods has not been published, the two methods are inferred to have comparable computational speeds based on indirect evidence. The extended Lagrangian method was found to be approximately 20 times faster than the standard matrix inversion procedure [117] and according to the calculation of Bernardo et al. [208] using different polarizable water potentials, the iterative method is roughly 17 times faster than direct matrix inversion to achieve a convergence of 1.0 x 10-8 D in the induced dipole. 

First, to demonstrate the operational procedure in LES/FMDF, the instantaneous number density of the Monte-Carlo elements or particles is shown in Fig. 4.1a. This figure shows the total number of particles within the computational domain. However, the weighted distribution of the particles is not, and should not be, uniform. Rather, it must be proportional to the flow density. This is demonstrated in Fig. 4.16 in which the ensemble-mean values of the weighted Monte-Carlo particle number density are compared with the fluid density as obtained from finite difference solution of the filtered density field. The very good correlation attained in this way verifies the consistency of the stochastic procedure and its Lagrangian Monte-Carlo solver. 

The finite difference numerical simulation was carried out by solving the Euler equations by the Lagrangian approach. The DANE code was used for computation. The air bubble radius was 1.0 mm and the shock overpressure was 1 kbar. Computational grids were, in the axlsymmetric Cartesian coordinates, 150 x 300 and one grid size was 0.025 mm. Figure 5 shows the sequential isobars. It is clearly seen that the peak pressure appear on the side where the shock first impinged the bubble, and the bubble deformation starts which indicates the microjet initiation. However, the rebound shock is so weak that, if compared with the incident shock, the wave front could not be resolved in this numerical scheme. 

Discrete particle modeling (DPM) is an advanced computational technique for particulate systems (in this case, fluidized beds) that has already been presented in Chapter 7 of Volume 3, Modem Drying TechnoU. DPM combines continuous (Eulerian) CFD for the gas phase with a discrete (Lagrangian) consideration of the particle phase by means of a discrete element method (DEM), and is therefore often also denoted as DEM-CFD. Its appHcation enables the resolution of not only interactions between the gas and the particle phase, but also of particle-particle and particle-wall interactions, in the sense of a four-way coupling (compare also with Chapter 5 in Volume 1, Modem Drying Technology). 

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