Eulerian Codes

Other cell variables such as sound speed and heat capacities can be calculated using similar techniques. Some codes allow a variety of multimaterial element thermodynamic treatments. For example, CTH allows all materials in an element to have the same or different pressures or temperatures [44], Material interfaces in multimaterial elements do not coincide with element boundaries, as shown in Fig. 9.14 [45]-[49]. The interfaces must be constructed using pattern matching or some other technique. 

Large Deformation Wave Codes quantity advected 

Remapping methods must also be able to move material into neighbor eells that share only a node. This is ealled eorner eoupling and is required to aeeurately move material diagonally aeross the mesh. For example, the flow shown in Fig. 9.15 is at a 30° angle and material from element zero will flow into elements one, two, and three. Elements one and two share a faee with element zero. But element three shares only a node with element zero. The multistep algorithm moves material from element zero to element one, then from element one to element three when the direetion is switehed. 

First-order sehemes use a uniform distribution aeross an element and seeond-order sehemes use a linear distribution aeross the element as shown in Fig. 9.16. Higher-order adveetion sehemes use more eomplex distributions aeross an element [29]. The distributions aeross the donor eell must be eon-strained to prevent numerieal oseillations. As an illustration, for seeond-order van Leer sehemes, the slope is limited using (9.15) and Fig. 9.17. The slope 

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Eulerian codes are often used to simulate high-velocity impact and penetration events, such as shown in Fig. 9.26. Here the problem involves the penetration of armor steel by a tungsten projectile at normal incidence. 

G. Luttwak and R.L. Rabie, The Multimaterial Arbitrary Lagrangian-Eulerian Code MMALE and its Application to Some Problems of Penetration and Impact, LA-UR-2311, Los Alamos National Laboratory, Los Alamos, NM, 1985. 

A much more pronounced vortex formation in expanding combustion products was found by Rosenblatt and Hassig (1986), who employed the DICE code to simulate deflagrative combustion of a large, cylindrical, natural gas-air cloud. DICE is a Eulerian code which solves the dynamic equations of motion using an implicit difference scheme. Its principles are analogous to the ICE code described by Harlow and Amsden (1971). 

Mader then reprogrammed his computations, for an Eulerian code and considered the interactions of 4 cylindrical voids rather than a single void (Ref 16). He showed that shock interactions with four holes lead to much greater faster computed nitromethane decomposition than the shock interaction with a single hole for the same initial conditions... 

A more detailed study of fuel cloud dispersion, though one lacking direct exptl verification, was made by Rosenblatt et al (Ref 23). The purpose of their study was to develop and use physically based numerical simulation models to examine the cloud dispersion and cloud detonation with fuel mass densities and particle size distributions as well as the induced air pressures and velocities as the principal parameters of interest. A finite difference 2-D Eulerian code was used. We quote The basic numerical code used for the FAE analysis was DICE, a 2-D implicit Eulerian finite difference technique which treats fluid-particle mixtures. DICE treats par-... 

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). 

At the Lehrstuhl fiir Warmeiibertragung und Klimatechnik der RWTH Aachen, Germany, such an Eulerian code is developed in cooperation with FLUENT Europe Ltd, Sheffield, U. K. At the present stage of development, the time dependent distribution of velocities, pressures and volume fractions of both phases can be calculated for isothermal fluidized beds. In this paper, the basic equations and the physical models are explained, and for a simple test case results are compared with experimental data. 

The resolved reaction zones of nitromethane, liquid TNT, and ideal gases have also been studied using the two-dimensional Lagrangian code 2DL, described in Appendix B and the two-dimensional Eulerian code 2DE described in Appendix C. 

The development of the three-dimensional Eulerian code, 3DE, described in Appendix D, allowed the Hydrodynamic Hot Spot Model of heterogeneous shock initiation to be used to investigate the shock interaction with a matrix of holes and the resulting formation of hot spots, their interaction and build up toward a propagating detonation. The Hydrodynamic Hot Spot Model has been used to evaluate the relative effect of explosive shock sensitivity as a function of composition, pressure, temperature, density, and particle size. It has also been used to understand the desensitization of explosives by preshocking. 

To study this system in a more realistic geometry, the Eulerian code, 2DE, described in Appendix C, was used because it can describe large distortion problems such as an explosive-air interface. The results calculated using the Forest Fire burn are shown in 4.19. Again, the results depend upon the detonation wave profile before it reaches the corner. If the wave started out flat, the explosive region near the explosive-air interface remained partially decomposed and the detonation wave never completely burned across the front until the wave became sufficiently curved at the front and near the interface. The failure process of a heterogeneous explosive is a complicated interaction of the effective reaction zone thickness which determines how flat the wave should be and the curvature required for decomposition to occur near the surface of the charge. 

P. Donguy and N. Legard, Numerical Simulations of Non-Ideal Detonations of a Heterogeneous Explosive with the Two Dimensional Eulerian Code C.E.E. , Seventh Symposium (International) on Detonation, 695-702 (1981). 

The most useful and simplest experiment that can be performed to obtain a good estimate of the detonation C-J pressure is the plate dent test described by Smith and performed by M. Urizer for almost 50 years at Los Alamos. Of the usual experiments used to study detonation performance, this experiment is also one of the most difficult to simulate numerically. Lagrangian codes, such as TDL, cannot describe the highly distorted flow around the surface of the dent. The Eulerian code, 2DE, described in Appendix C has realistic calibrated equations of state and elastic-plastic treatments for solid materials, so it has been used to model the plate dent test. 

The revolution in numerical modeling has been made possible by the development of the multi-material adaptive grid Eulerian codes SAGE/NOBEL/RAGE with models to describe the build-up to detonation using the Forest Fire heterogeneous shock initiation burn model and models to describe the build-up of detonation which results in factors of two variations in the explosion energy with time. The build-up to and of detonation was first modeled in 2002 by the NOBEL code. 

M. L. Gittings, 1992 SAIC s Adaptive Grid Eulerian Code , Defense Nuclear Agency Numerical Methods Symposium, 28-30 (1992). 

The Eulerian equations of motion are more useful for numerical solution of highly distorted fluid flow than are Lagrangian equations of motion. Multicomponent Eulerian calculations require equations of state for mixed cells and methods for moving mass and its associated state values into and out of mixed cells. These complications are avoided by Lagrangian calculations. Harlow s particle-in-cell (PIC) method uses particles for the mass movement. The first reactive Eulerian hydrodynamic code EIC (Explosive-in-cell) used the PIC method, and it is described in reference 2. The discrete nature of the mass movement introduced pressure and temperature variations from cycle to cycle of the calculation that were unacceptable for many reactive fluid dynamic problems. A one-component continuous mass transport Eulerian code developed in 1966 proved useful for solving many one-component problems of interest in reactive fluid dynamics. The need for a multicomponent Eulerian code resulted in a second 2DE code, described in reference 4. Elastic-plastic flow and real viscosity were added in 1976. The technique was extended to three dimensions in the 1970 s and the resulting 3DE code is described in Appendix D. 

The elastic-plastic treatment in 2DE was compared to one-dimensional, Lagrangian SIN calculations of a 0.3175 cm thick Aluminum plate initially traveling 0.0412 cm/nsec driving a 32.5 kbar shock into 1.27 cm of Aluminum. The yield was 2.5 kbar, the shear modulus was 0.25, and FLAP (see Appendix A) was 0.05 for both calculations. The shock profiles at 1 and 2 nsec are shown in Figure C.l using the same number of cells (100) in the SIN and 2DE calculations. The Eulerian calculations are more smeared, but the magnitude of the elastic component of the rarefaction is reproduced. The elastic-plastic treatment in the Eulerian code gives realistic, if more smeared, results. 

The three-dimensional Eulerian code is called 3DE and is a three-dimensional version of the 2DE code described in Appendix C. 

Boemer A, Qi H, Renz U, Vasquez S, Boysan F (1995) Eulerian computation of fluidized bed hydrodynamics—a comparison of physical models. Fluidized bed combustion, vol 2. ASME Campbell CS (1990) Rapid granular flows. Atmu Rev Fluid Mech 22 57-92 Carlo AD, Bocci E, Zuccari F, DeU Era A (2010) Numerical investigation of sorption enhanced steam methane reforming process using computational fluid dynamics Eulerian-Eulerian code. Ind Eng Chem Res 49 1561-1576... 

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