Lagrangian coordinates

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 introduction of Lagrangian coordinates in the previous section allows a more natural treatment of a continuous flow in one dimension. The derivation of the jump conditions in Section 2.2 made use of a mathematical discontinuity as a simplifying assumption. While this simplification is very useful for many applications, shock waves in reality are not idealized mathematical... 

Figure 2.12. A flow tube used to derive one-dimensional flow equations in Lagrangian coordinates. Internal surfaces are massless, impermeable partitions to aid in visualizing elements of fluid in Lagrangian coordinates.

Equations (2.45), (2.46), and (2.49) express the conservation of mass, momentum, and energy in Lagrangian coordinates for continuous flow. 

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. 

Lagrangian coordinates The coordinate system in whieh the material position (h) and time (t) are the independent variables. The dependent variables are deseribed as funetions of a partiele position within the material which had coordinate x = hat time t = 0. Also known as material coordinates. 

The motion of a particle in the flow field can be described in the Lagrangian coordinate with the origin placed at the center of the moving particle. There are two modes of particle motion, translation and rotation. Interparticle collisions result in both the translational and the rotational movement, while the fluid hydrodynamic forces cause particle translation. Assuming that the force acting on a particle can be determined exclusively from its interaction with the surrounding liquid and gas, the motion of a single particle without collision with another particle can be described by Newton s second law as... 

These equations have been obtained by Bisnovatyi-Kogan (1966), using the expression of the energy with the prescribed distributions of the density (Em-den polytrope n = 3) and entropy (arbitrary) over the Lagrangian coordinate... 

Let us consider a particle of mass m located at a point p and subjected to a force T given by the gradient of potential T) in p. Denoting X/, where / = 1, 2 and 3, the Cartesian coordinates in p, and denoting qj the corresponding Lagrangian coordinates, the two systems of spatial coordinates are related by ... 

Ibid, pp 527-37 [A brief description of the following numerical methods for calculation a) "Finite Difference Scheme in Lagrangian Coordinates , previously described by Goad (Ref 5) b) Particle-in Cell Method, previously described by Evans Harlow (Ref 1)... 

An interesting class of exact self-similar solutions (H2) can be deduced for the case where the newly formed phase density is a function of temperature only. The method involves a transformation to Lagrangian coordinates, based upon the principle of conservation of mass within the new phase. A similarity variable akin to that employed by Zener (Z2) is then introduced which immobilizes the moving boundary in the transformed space. A particular case which has been studied in detail is that of a column of liquid, initially at the saturation temperature T , in contact with a flat, horizontal plate whose temperature is suddenly increased to a large value, Tw T . Suppose that the density of nucleation sites is so great that individual bubbles coalesce immediately upon formation into a continuous vapor film of uniform thickness, which increases with time. Eventually the liquid-vapor interface becomes severely distorted, in part due to Taylor instability but the vapor film growth, before such effects become important, can be treated as a one-dimensional problem. This problem is closely related to reactor safety problems associated with fast power transients. The assumptions made are ... 

The substantial derivative in a Lagrangian coordinate moving with the discrete particle can be expressed by... 

Trajectory models quantify the dynamic characteristics of particles in Lagrangian coordinates. The trajectory model is useful when the particle phase is so dilute that the description of particle behavior by continuum models may not be suitable. 

Mixing times are the times required for 6, or A, or Ig to fall from their initial value (before mixing), down to some prescribed small value (for instance 0.05 or 0.01). We shall see that in Lagrangian coordinates, - o /(da /dt) is also a mixing time. 

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. 

Now we see that the slope of the jump condition on the P-u plane is also a function of shock velocity. The interesting part to note is that U in this equation is the shock velocity in laboratory or Eulerian coordinates. The quantity (U -Uq) is the shock velocity in Lagrangian coordinates, or relative to the material. So we see that for the jump condition on the P-u plane... 

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. 

Here m is proportional to the volume of a spherical shell of liquid between the bubble wall and the radial position coordinate r, and hence, represents a Lagrangian coordinate, providing that a negligible volume of liquid is vaporized. U is therefore a measure of the heat content of the spherical shell, to within an arbitrary additive function of time, K(t) alternatively, it may be viewed as a temperature potential function. In terms of the new coordinates the diffusion equation becomes... 

Solving discrete phase particle track under particle effect balance equation of lagrangian coordinates. The form of particle effect balance equation under cartesian coordinate system (x direction) is (Morsi, S.A. Alexander, A.J. 1972) ... 

---