Eulerian element

The conductive heat loss per unit volume from a plane flame in a circular tube of diameter D can be estimated by the following simple reasoning. Consider an Eulerian element of gas of length dx in the tube whose walls are maintained at the temperature Tq. The energy per second conducted to the walls from this element is the product of the thermal conductivity A, a mean temperature gradient (T 7 )/(D/2), and the wall area nDdx. The rate of heat loss per unit volume is then obtained through division by the volume of the element dxnD I4 ... 

Two answers can be given by considering the thermodynamics of the cube. In one approach we consider a thermodynamic system whose boundaries coincide with those of the cube and move with the material in the other we consider a thermodynamic system whose boundaries coincide with the cube at some instant but remain fixed in space so that material flows across them (i.e., a Lagrangian element and an Eulerian element, respectively)—see Figure 11.10. 

Lagrangian-Eulerian (ALE) method. In the ALE technique the finite element mesh used in the simulation is moved, in each time step, according to a predetermined pattern. In this procedure the element and node numbers and nodal connectivity remain constant but the shape and/or position of the elements change from one time step to the next. Therefore the solution mesh appears to move with a velocity which is different from the flow velocity. Components of the mesh velocity are time derivatives of nodal coordinate displacements expressed in a two-dimensional Cartesian system as... 

Donea, J., 1992. Arbitrary Lagrangian-Eulerian finite element methods. In Belytschko, T. and Hughes, T. J. R. (eds), Computational Methods for Transient Analysis, Elsevier Science, Amsterdam. 

Figure 5.4 The finite element mesh configurations in the Arbitrary Lagrangian-Eulerian scheme...

Since the Lagrangian walls are impermeable, the mass of the Lagrangian element is constant. At time t, when the walls of the element are separated by an Eulerian distance dx, the density of the fluid within it must be... 

Arbitrary-Lagrangian-Eulerian (ALE) codes dynamically position the mesh to optimize some feature of the solution. An ALE code has tremendous flexibility. It can treat part of the mesh in a Lagrangian fashion (mesh velocity equation to particle velocity), part of the mesh in an Eulerian fashion (mesh velocity equal to zero), and part in an intermediate fashion (arbitrary mesh velocity). All these techniques can be applied to different parts of the mesh at the same time as shown in Fig. 9.18. In particular, an element can be Lagrangian until the element distortion exceeds some criteria when the nodes are repositioned to minimize the distortion. 

J. Donea, Arbitrary Lagrangian-Eulerian Finite Element Methods, Computational Methods for Transient Analysis (edited by J.D. Achenbach), North-Holland, Amsterdam, 1983. 

We will now describe the basic hydrodynamic relationships applicable in the case of steady-state flow in which the Eulerian velocity field is time-independent and written as v(r). Here the rate of strain elements are given by [1]... 

In simulating physical operations carried out in stirred vessels, generally one has the choice between a Lagrangian approach and a Eulerian description. While the former approach is based on tracking the paths of many individual fluid elements or dispersed-phase particles, the latter exploits the continuum concept. The two approaches offer different vistas on the operations and require different computational capabilities. Which of the two approaches is most... 

An alternative method to RTD theory for treating non-ideal reactors is the use of zone models. In this approach, the reactor volume is broken down into well mixed zones (see the example in Fig. 1.5). Unlike RTD theory, zone models employ an Eulerian framework that ignores the age distribution of fluid elements inside each zone. Thus, zone models ignore micromixing, but provide a model for macromixing or large-scale inhomogeneity inside the reactor. 

The selection rules for pure-rotation transitions are found by evaluation of the nine integrals IXOa,/ZOc these involve the nine direction cosines6 cos(XOa),..., cos(ZOc). The volume element in Eulerian angles can be shown to be... 

The set of properly orthogonal transformations R1 forms the group SO(3), the reflexion Z1 at the origin of the LS likewise leaves A symmetric, since the eulerian angles remain unaffected by Z1. Therefore, H is symmetric w.r.t. the full rotation group 0(3/. However, in agreement with the usual conventions we will omit the elements Z R1 0(3). As a consequence we will consider hence -forward the group... 

Three types of theoretical approaches can be used for modeling the gas-particles flows in the pneumatic dryers, namely Two-Fluid Theory [1], Eulerian-Granular [2] and the Discrete Element Method [3]. Traditionally the Two-Fluid Theory was used to model dilute phase flow. In this theory, the solid phase is being considering as a pseudo-fluid. It is assumed that both phases are occupying every point of the computational domain with its own volume fraction. Thus, macroscopic balance equations of mass, momentum and energy for both the gas and the solid... 

Equation (2) is expressed in the Eulerian frame of reference, in which the volume element under consideration is fixed in space, and material is allowed to flow in and out of the element. An equivalent representation of very different appearance is the Lagrangian frame of reference, in which the volume element under consideration moves with the fluid and encapsulates a fixed mass of material so that no flow of mass in or out is permitted. In this frame of reference, Eq. (2) becomes... 

If the N solutes are in the infinite dilution condition, we can simplify the integral in the previous equation considering that there is no interaction among the solute molecules. Flence for the great majority of the solutes rototranslational configurations, the integral on the other coordinates and solvent momenta is a constant (note that in the solute mass tensor no elements depend on the center of mass position and the Eulerian angles). Therefore... 

Let us consider the electron-vibrational matrix element. As is usually done, we consider two coordinate systems, the origins of which are located at the center of mass of the molecule. The first coordinate system is fixed in space, while the second system (the rotational one) is fixed to the molecule. For describing the orientation of the rotational system with respect to the fixed frame we use the Eulerian angles 6 = a, / , y. In the Born-Oppenheimer approximation, the motion of nuclei may be decomposed into the vibrations of the nuclei about their equilibrium position and the rotation of the molecule as a whole. Accordingly, the wave function of the nuclei X (R) is presented as a product of the vibrational wave function A V(Q) and the rotational wave function... 

Acoustic response of viscoelastic fluids may be simulated by Lagrangian finite element or Eulerian finite difference schemes. Both techniques have been used to predict the behaviour of discontinuous viscoelastic layers which may contain inserts such as air cavities and which are backed by metal plates. 

The acoustic response of resonant viscoelastic fluid structures to a pressure wave may be simulated by a four-dimensional calculation, three dimensions in space and one in time. The Lagrangian, primitive finite element and Eulerian finite difference schemes form the basis for two models presented in this paper which are able to simulate a wide range of fluid structures containing inclusions of arbitrary spacing, shape and composition. 

Nakayama, T., and Mori, M., An Eulerian finite element method for time-dependent free surface problems in hydrodynamics. Int. J. Num. Methods Fluids 22, 175 (1996). 

Yeh G. T., Carpenter S. L., Hopkins P. L., and Siegel M. D. (1995) Users Manual for LEHGC A Lagrangian-Eulerian Finite-element Model of HydroGeoChemical Transport through Saturated-unsaturated Media-version 1.1. Sandia National Laboratories. 

---