Eulerian displacement

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... 

Riding along with a fluid packet is a Lagrangian notion. However, in the limit of dt - 0, the distance traveled dx vanishes. In this limit, (i.e., at a point in time and space) the Eulerian viewpoint is achieved. The relationship between the Lagrangian and Eulerian representations is established in terms of Eq. 2.52, recognizing the equivalence of the displacement rate in the flow direction and the flow velocity. In the Eulerian framework the... 

FIGURE 7.13 Time steps used for Eulerian-Lagrangian simulations. Atfiow Time step used to resolve macroscopic liquid flowfield, Atbubbie- Time step used to estimate forces acting on bubbles (macroscopic liquid flowfield is assumed to be constant), tab Time step used to displace bubbles and to account for collisions. Forces acting on bubbles are assumed to be constant. 

Introduction of a neighboring tide-raising body will disturb this equilibrium. Following an Eulerian approach, each mass element will be displaced an amount u(x, y, z), so that the new location will be x + ux,y + uy, z + uz. Moreover, each cubic mass element will be distorted during the displacement, each corner of the cube is displaced by a slightly different it. The new equilibrium written in terms of the conditions at the original location of the mass element (0) is given by... 

By contrast, the Lagrangian techniques attempt to describe the concentration statistics in terms of the statistical properties of the displacements of groups of particles released in the fluid. The mathematics of this approach is more tractable than that of the Eulerian... 

Small strains and small strain rates are assumed for solid deformation. Advective terms due to solid displacement are neglected after the formulation is transformed in terms of material derivatives (in fact, material derivatives are approximated as eulerian time derivatives). In... 

It is of interest to note that the identical result (2.38) is obtained for the librations by a treatment using cartesian displacement coordinates for the atoms of a diatomic molecule (Suzuki, 1970). Although we do not face the problems of separability of coordinates in the kinetic energy in this case, the assiunption of harmonicity of the potential energy presumably causes the same inaccuracies. In other words, it is immaterial which coordinates are used but in terms of Eulerian angle displacement coordinates the harmonic approximation is not confined to the potential energy but also appears in the kinetic energy. 

On the other hand, in terms of the Eulerian librational displacement coordinates, using Table II, for a particular component of (0,1 kf), we similarly obtain... 

Schnepp and Ron (1969) carried out a complete lattice dynamical calculation for a-Na throughout the Brillouin zone, using the potential model of Kuan, Warshel, and Schnepp (1969). This model contains three parameters which were calibrated for the optical modes [librational assignments of Brith, Ron, and Schnepp (1969)] and the equilibrium properties of the solid. The lattice dynamics was formulated in terms of Eulerian angle librational displacements. Dispersion curves for the symmetry directions of the Brillouin zone and density of states functions were reported with and without translation-libration interactions. The results clearly demonstrated the importance of these interactions. Schnepp... 

If, on the other hand, we were to use Eulerian angle displacement coordinates (Schnepp and Ron, 1969), the equivalent of (4.11) would be... 

Eulerian angles, and also the vibrational coordinates, will be differently defined on the basis of the initial and inverted configurations. Nevertheless, any configuration can be described in terms of either set of coordinates, but then the displacements from equilibrium will not necessarily be small. 

Following the motion of a fluid, the increment of velocity dv in the Eulerian specification results not only from a local increase at point rbut also from the displacement of the elementary volume of the fluid during the interval of time At. Taking into... 

The optimal pulse separation between two images is influenced by a number of parameters. Two main types of error affect the choice of pulse separation (At), that is, random error and acceleration error. Random error arises from noise during recording of images and subsequent interrogation of the particle images. Acceleration error arises from approximation of Lagrangian motion of tracer particle to local Eulerian velocity based on small particle displacement ... 

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