Eulerian and Lagrangian Coordinates

On the other hand, the thickness of the slab in the Lagrangian system is the constant Ah = Axq, so for a Lagrangian shock velocity of C, the transit time is 

The transit time must be independent of the coordinate system, so these expressions can be equated, yielding 

The relative shock velocity t/ = (7 — Uj is the Eulerian shock velocity often used because it is a material property and is independent of the motion of the 

The form of the jump conditions also depends on the coordinate system. Substituting (2.40) into the general Eulerian form of the momentum jump condition (Table 2.1) yields the Lagrangian jump condition 

---

Recent efforts to distinguish between the terms burning velocity and flame speed on the basis of Eulerian and Lagrangian coordinate systems appear to introduce confusion. Therefore, the terms are used interchangeably here, as synonyms for such terms as deflagration velocity, wave speed, and propagation velocity. They all refer to velocities measured with respect to the gas ahead of the wave. 

The model is established on a mixture of Eulerian and Lagrangian coordinate systems. The river is approximated as a series of completely mixed cells (typically 10-1,000 m in length) fixed in position, as shown in Fig. 20.1. The slick is approximated as a series of completely mixed cells that move across the water surface in a Lagrangian coordinate system. This treatment of the slick as a series of moving cells allows for spatial variation in the concentration of the slick. The application of the model, per se, is to situations where both flow and slick can be described as one-dimensional. This occurs when the slick is spread completely across the river, as with relatively narrow streams. The length of the river from the spill site before the one-dimensional assumption can be applied is approximately ... 

While the Eulerian system has intuitive appeal, it is the Lagrangian coordinate system that is more convenient mathematically and in many practical applications. In this system, the coordinate is fixed to the material and moves with it. It is sometimes called the material coordinate system. In Fig. 2.2, the boxcars can be numbered, so the position of a car in this system never changes. By convention, the Lagrangian coordinate (h) is chosen so that it is equal to the Eulerian coordinate (x) at some time t = 0. Figure 2.10(b) illustrates a Lagrangian h-t diagram of the same system as shown in Fig. 2.10(a) with the Eulerian system. Because the flow is independent of the coordinate system chosen to describe it, both systems must lead to the same results. 

Figure 2.11. The sequence described in the text for (a) Eulerian and (b) Lagrangian coordinates. Hashmarks indicate material particle positions.

The properties required of a material in order for it to support a stable shock wave were listed and discussed. Rarefaction, or release waves were defined and their behavior was described. The useful tool of plotting shocks, rarefactions, and boundaries in the time-distance plane (the x-t diagram) was introduced. The Lagrangian coordinate system was defined and contrasted to the more familiar Eulerian coordinate system. The Lagrangian system was then used to derive conservation equations for continuous flow in one dimension. 

Note that according to Truesdell and Toupin [170] the material coordinates were introduced by Euler in 1762, although they are now widely referred to as the Lagrangian coordinates, while the spatial coordinates, often called Eulerian coordinates, where introduced by Jean le Rond d Alembert (1717-1783) in 1752. 

In Eulerian coordinates, shrinking causes an advective mass flux, which is difficult to handle. By changing the coordinate system to Lagrangian, i.e., the one connected with dry mass basis, it is possible to eliminate this flux. This is the principle of a method proposed by Kechaou and Roques (1990). In Lagrangian coordinates Equation 3.91 for one-dimensional shrinkage of an infinite plate becomes ... 

Initial reinforcement layup, followed by compaction and resin injection, are phases common to RTM. The volume of literature devoted to RTM can thus be applied directly to these phases (e.g., references 14,15). Once the resin has been injected and the gates have been closed, wet compression is initiated. The first theoretical analyses on the compression of saturated fibrous materials were carried out by Dave et and Gutowski et in the context of autoclave processing and bleeder ply moulding. Therein it was noted the similarity between these processes and the classic geomechanics consolidation problem, that is, the compression of a water-saturated soil or clay. i The theory can be formulated in terms of Lagrangian coordinates, but is more conveniently presented for the present purposes, as in what follows, in terms of an Eulerian description. ... 

We distinguish between the coordinate system Ei of the Lagrangian description and the coordinate system of the Eulerian description in order to understand the... 

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... 

Consider the impact of a semi-infinite space on a plate of thickness dp, separated from an identical plate by a gap of width d. If the impactor and plates are all composed of the same materials, what is the subsequent behavior Plot in both Lagrangian and Eulerian coordinates. 

The shock-change equation is the relationship between derivatives of quantities in terms of x and t (or X and t) and derivatives of variables following the shock front, which moves with speed U into undisturbed material at rest. The planar shock front is assumed to be propagating in the x (Eulerian spatial coordinate) or X (Lagrangian spatial coordinate) direction, p dx = dX. 

In preparation for this, the equations of gas dynamics will reproduce the conservation laws of impuls, mass and energy that can be written in a number of different ways with respect to Eulerian (x,t) or Lagrangian (s,f) variables, where x is the coordinate of a particle and s is the initial coordinate of a particle or the quantity... 

Other regional transport models, such as the Regional Lagrangian Model of Air Pollution (RELMAP Eder et al., 1986), use a different computational scheme than Eulerian models. In a Lagrangian model, the coordinate system moves with a parcel of air and mass balance of pollutant concentrations is computed on a parcel as it moves through space. 

To complete the computation of the concentration field of the puff in Eulerian coordinates, the position of the puff centroid must be updated based on the wind field velocity at the puff centroid the entire (Lagrangian) puff is then assumed to translate affinely with the centroid. A puff-splitting algorithm may be used to overcome the inaccuracies that arise as the puff dimensions become sufficiently large that the approximation inherent in assuming constant wind velocities throughout the puff becomes invalid (Sykes and Henn 1995). 

---