Discrete numerical simulation

In this section, we discuss the role of numerical simulations in studying the response of materials and structures to large deformation or shock loading. The methods we consider here are based on solving discrete approximations to the continuum equations of mass, momentum, and energy balance. Such computational techniques have found widespread use for research and engineering applications in government, industry, and academia. 

Numerical simulations offer several potential advantages over experimental methods for studying dynamic material behavior. For example, simulations allow nonintrusive investigation of material response at interior points of the sample. No gauges, wires, or other instrumentation are required to extract the information on the state of the material. The response at any of the discrete points in a numerical simulation can be monitored throughout the calculation simply by recording the material state at each time step of the calculation. Arbitrarily fine resolution in space and time is possible, limited only by the availability of computer memory and time. 

Large-scale numerical simulation for samples that are many times os large as the critical wavelength is perhaps the only way to develop a quantitative understanding of the dynamics of solidification systems. Even for shallow cells, such calculations will be costly, because of the fine discretizations needed to be sure the dynamics associated with the small capillary length scales are adequately approximated. Such calculations may be feasible with the next generation of supercomputers. 

So far, some researchers have analyzed particle fluidization behaviors in a RFB, however, they have not well studied yet, since particle fluidization behaviors are very complicated. In this study, fundamental particle fluidization behaviors of Geldart s group B particle in a RFB were numerically analyzed by using a Discrete Element Method (DEM)- Computational Fluid Dynamics (CFD) coupling model [3]. First of all, visualization of particle fluidization behaviors in a RFB was conducted. Relationship between bed pressure drop and gas velocity was also investigated by the numerical simulation. In addition, fluctuations of bed pressure drop and particle mixing behaviors of radial direction were numerically analyzed. 

Let us return for the moment to Eq. (2.2). In atmospheric problems it is impossible to solve the equations of motion analytically. Under these conditions information about the instantaneous velocity field u is available only from direct measurements or from numerical simulations of the fluid flow. In either case we are confronted with the problem of reconstructing the complete, continuous velocity field from observations at discrete points in space, namely the measuring sites or the grid points of the numerical model. The sampling theorem tells us that from a set of discrete values, only those features of the field with scales larger than the discretization interval can be reproduced in their entirety (Papoulis, 1%5). Therefore, we decompose the wind velocity in the form... 

In this section, we consider the description of Brownian motion by Markov diffusion processes that are the solutions of corresponding stochastic differential equations (SDEs). This section contains self-contained discussions of each of several possible interpretations of a system of nonlinear SDEs, and the relationships between different interpretations. Because most of the subtleties of this subject are generic to models with coordinate-dependent diffusivities, with or without constraints, this analysis may be more broadly useful as a review of the use of nonlinear SDEs to describe Brownian motion. Because each of the various possible interpretations of an SDE may be defined as the limit of a discrete jump process, this subject also provides a useful starting point for the discussion of numerical simulation algorithms, which are considered in the following section. 

Rather than using a medium that has a continually varying angle cp, we consider the rotation of cp to take place in discrete steps. To describe these changes, it is useful to replace the torque form of the propagation equation, (5.18), with one better suited to use in numerical simulation. 

In Figs. 4.1 and 4.2, the broken lines do not represent the sample paths of the process X(t), but join the outcoming states of the system observed at a discrete set of times f, t2,.. . , tn. To understand the behavior of X(t), it is necessary to know the transition probability. In Fig. 4.3 are given numerical simulations of a Wiener process W(t) (Brownian motion) and a Cauchy process C(t), both supposed one dimensional, stationary, and homogeneous. Their transitions functions are defined... 

The most widely adopted method for the turbulent flow analysis is based on time-averaged equations using the Reynolds decomposition concept. In the following, we discuss the Reynolds decomposition and time-averaging method. There are other methods such as direct numerical simulation (DNS), large-eddy simulation (LES), and discrete-vortex simulation (DVS) that are being developed and are not included here. 

P 61] The numerical simulations were based on the solution of the incompressible Navier-Stokes equation and a convection-diffusion equation for a concentration field by means of the finite-volume method [152], The Einstein convention of summation over repeated indices was used. For pressure-velocity coupling, the SIMPLEC algorithm and for discretization of the species concentration equation the QUICK differencing scheme were applied. Hybrid and the central differencing schemes referred to velocities and pressure, respectively (commercial flow solvers CFX4 and CFX5). 

The first approach is the discretization of the convection and the diffusion operators of the PDEs, which gives rise to a large (or very large) system of effective low-dimensional models. The order of these low-dimensional models depend on the minimum mesh size (or discretization interval) required to avoid spurious solutions. For example, the minimum number of mesh points (Nxyz) necessary to perform a direct numerical simulation (DNS) of convective-diffusion equation for non-reacting turbulent flow is given by (Baldyga and Bourne, 1999)... 

In shock-wave-chemistiy research, compacts of solid powders are usually used as samples. Because of porosity, the samples are inhomogeneous and not a continuous medium, so models based on continuum physics are not suitable for tlie numerical simulation of these problems. A discrete meso-dynamic mctliod (denoted as DM ) developed by Tang et al. in recent years is a better metliod for tliis purpose [20]. 

Xu, B.H. and Yu, A.B. (1997), Numerical simulation of gas solid flow in a fluidized bed by combining discrete particle method with computational fluid dynamics, Chem. Eng. Sci., 52, 2785. 

The position of the interface between phases is inferred from the values of marker function. The discretized form of this equation relates the value of F at the new time level with its previous time-level value and to the fluxes of F through the cell faces. Approximations used in calculating these cell fluxes may lead to smearing of the fluid-fluid interface, which has to be minimized during numerical simulations. Several... 

Syamlal, M. (1998), Higher order discretization methods for the numerical simulation of fluidized beds, AIChE Symposium Series, No. 318, 94, 53-57. 

The metapopulation modeling clearly demonstrates that a relationship exists between patch arrangement, initial population size, and carrying capacity in determining the number and frequencies of discrete outcomes. Changes can lead to new outcomes and alter the probabilities of occurrence. Are these results artifacts of the numerical simulations or can they be expressed in simulated populations and ecological systems ... 

There are two main approaches for the numerical simulation of the gas-solid flow 1) Eulerian framework for the gas phase and Lagrangian framework for the dispersed phase (E-L) and 2) Eulerian framework for all phases (E-E). In the E-L approach, trajectories of dispersed phase particles are calculated by solving Newton s second law of motion for each dispersed particle, and the motion of the continuous phase (gas phase) is modeled using an Eulerian framework with the coupling of the particle-gas interaction force. This approach is also referred to as the distinct element method or discrete particle method when applied to a granular system. The fluid forces acting upon particles would include the drag force, lift force, virtual mass force, and Basset history force.Moreover, particle-wall and particle-particle collision models (such as hard sphere model, soft sphere model, or Monte Carlo techniques) are commonly employed for this approach. In the E-E approach, the particle cloud is treated as a continuum. Local mean... 

To fully understand this concept of filtering, note that the values of flow properties at discrete points in a numerical simulation represent averaged values. To illustrate this explicitly, Rogallo and Moin [133] considered the central difference approximation for the first derivative of a continuous variable, v x), in a grid with points spaced a distance h apart. We can write this as follows (e.g., [133], p. 103 [186], p 323) ... 

There are several reasons for observing differences between the computed results and experimental data. Errors arise from the modeling, discretization and simulation sub-tasks performed to produce numerical solutions. First, approximations are made formulating the governing differential equations. Secondly, approximations are made in the discretization process. Thirdly, the discretized non-linear equations are solved by iterative methods. Fourthly, the limiting machine accuracy and the approximate convergence criteria employed to stop the iterative process also introduce errors in the solution. The solution obtained in a numerical simulation is thus never exact. Hence, in order to validate the models, we have to rely on experimental data. The experimental data used for model validation is representing the reality, but the measurements... 

In this equation, A represents the number of crystals, p is the length of the three-phase junction (i.e., the perimeter of the electrode/crystal interface), and Ax, Az denote the size of the discrete boxes in which the crystal is divided for numerical simulation procedures. 

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