Eulerian numerical scheme

Eulerian Numerical Scheme. Equation (7) assumes that the material parameters are differentiable with respect to the spatial coordinates. The scheme presented here will approximate discontinuites in media to second order accuracy in space. A somewhat similar scheme has been published for a wave equation able to describe the propagation of torsion waves ( ). That work considers a wave equation after the assumptions of material continuity have removed the material modulus from the second order spatial differentiation and was unable to successfully model the problem considered in the present paper. 

Sunderland and Grosh (Sll) use an explicit numerical scheme for solving the Landau problem. An Eulerian coordinate system is used with the origin at the melt interface. As noted by Landau, the numerical integration is simplified by appropriate choice of the ratio of the space and time intervals. Extension to time-dependent heat flux by either a numerical or a graphical technique is indicated. 

An important aspect of Eulerian reactor models is the truncation errors caused by the numerical approximation of the convection/advection terms [82], Very different numerical properties are built into the various schemes proposed for solving these operators. The numerical schemes chosen for a particular problem must be consistent with and reflect the actual physics represented by the model equations. 

The Eulerian (bottom-up) approach is to start with the convective-diffusion equation and through Reynolds averaging, obtain time-smoothed transport equations that describe micromixing effectively. Several schemes have been proposed to close the two terms in the time-smoothed equations, namely, scalar turbulent flux in reactive mixing, and the mean reaction rate (Bourne and Toor, 1977 Brodkey and Lewalle, 1985 Dutta and Tarbell, 1989 Fox, 1992 Li and Toor, 1986). However, numerical solution of the three-dimensional transport equations for reacting flows using CFD codes are prohibitive in terms of the numerical effort required, especially for the case of multiple reactions with... 

The Eulerian finite difference scheme aims to replace the wave equations which describe the acoustic response of anechoic structures with a numerical analogue. The response functions are typically approximated by series of parabolas. Material discontinuities are similarly treated unless special boundary conditions are considered. This will introduce some smearing of the solution ( ). Propagation of acoustic excitation across water-air, water-steel and elastomer-air have been computed to accuracies better than two percent error ( ). In two-dimensional calculations, errors below five percent are practicable. The position of the boundaries are in general considered to be fixed. These constraints limit the Eulerian scheme to the calculation of acoustic responses of anechoic structures without, simultaneously, considering non-acoustic pressure deformations. However, Eulerian schemes may lead to relatively simple algorithms, as evident from Equation (20), which enable multi-dimensional computations to be carried out in a reasonable time. 

In this approach, the finite volume methods discussed in the previous chapter can be applied to simulate the continuous fluid (in a Eulerian framework). Various algorithms for treating pressure-velocity coupling, and the discussion on other numerical issues like discretization schemes are applicable. The usual interpolation practices (discussed in the previous chapter) can be used. When solving equations of motion for a continuous fluid in the presence of the dispersed phase, the major differences will be (1) consideration of phase volume fraction in calculation of convective and diffusive terms, and (2) calculation of additional source terms due to the presence of dispersed phase particles. For the calculation of phase volume fraction and additional source terms due to dispersed phase particles, it is necessary to calculate trajectories of the dispersed phase particles, in addition to solving the equations of motion of the continuous phase. 

A practical limitation of the Eulerian schemes is that the convergence of the numerical integration is achieved only if the adopted timestep At < CAx/u, where the constant C is generally equal to unity. This criterion, called the Courant-Fredrichs-Lewy (CFL) condition, imposes timesteps that are generally much smaller than the physical timescales of interest. 

The MAC method, which allows arbitrary free surface flows to be simulated, is widely used and can be readily extended to three dimensions. Its drawback lies in the fact that it is computationally demanding to trace a large number of particles, especially in 3D simulation. In addition, it may result in some regions void of particles because the density of particles is finite. The impact of the MAC method is much beyond its interface capmring scheme. The staggered mesh layout and other features of MAC have become a standard model for many other Eulerian codes (even numerical techniques involving mono-phase flows). 

The basic idea behind the VOF method is to discretize the equations for conservation of volume in either conservative flux or equivalent form resulting in near-perfect volume conservation except for small overshoot and undershoot. The main disadvantage of the VOF method, however, is that it suffers from the numerical errors typical of Eulerian schemes such as the level set method. The imposition of a volume preservation constraint does not eliminate these errors, but instead changes their symptoms replacing mass loss with inaccurate mass motion leading to small pieces of fluid non-physically being ejected as flotsam or jetsam, artificial surface tension forces that cause parasitic currents, and an inability to calculate accurately geometric information such as normal vector and curvature. Due to this deficiency, most VOF methods are not well suited for surface tension-driven flows unless some improvements are made [19]. 

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