Lagrangian time-distance

Figure 6.7 shows a Lagrangian time-distance diagram of a symmetric impact by a driver plate with the target backed by a spall plate. The symmetry... 

Figure 6.7. Lagrangian time-distance diagram of a symmetric impact shock.

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. 

Figure 2.10. (a) An Eulerian x-t diagram of a shock wave propagating into a material in motion. The fluid particle travels a distance ut, and the shock travels a distance Uti in time ti. (b) A Lagrangian h-t diagram of the same sequence. The shock travels a distance Cti in this system. 

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... 

Figure 24.2 Calculation of the time-dependent concentration along the river axis x of a reactive compound. The effect of the reaction is calculated in the Lagrangian framework the effect of advection is accounted for by the relation between flow time t and distance x, x(t).

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... 

Before we finish this subsection, we would like to discuss the practical limitations of the Poincare sections, which require Lagrangian particle tracking for extended times. In reality. Fig. 2 presents stroboscopic images of the same four particles passing through thousands of mixing block boundaries. This has two basic implications. First, numerical calculation of the Poincare sections requires either analytical solutions or high-order accurate discretizations of the velocity field. Otherwise, the results may suffer from numerical diffusion and dispersion errors, and the KAM boundaries may not be identified accurately. Second, it is experimentally difficult, if not impossible, to track particles (in three-dimensions) beyond a certain distance allowed by the field of view of the microscopy technique. Despite these... 

In this regime the typical distance from the origin of motion increases as the square root of time. Thus, the dispersion in turbulent flows at long times is analogous to molecular diffusion or random walks with independent increments and comparison of Eq. (2.24) with (2.16) relates the turbulent diffusion coefficient, Dt, to the integral of the Lagrangian correlation function, Tl, as... 

Ounis et al. ° performed a series of Lagrangian simulation studies for dispersion and deposition of particles emitted from a point source in the viscons snblayer of a turbulent near-wall flow. Figures 23 to 25 show time variation of particle trajectory statistics for different diameters, for the case that the point source is at a distance of 0.5 wall units away from the wall. In these simulations, it is assumed that when particles touch the wan they will stick to it. At every time step, the particle ordinates are statistically analyzed, and the mean, standard deviation, and the sample minimum and maximum were evaluated. The points at which the minimum curve touches the wall identify Ihe locations of a deposited particle. Figure 23 shows that O.OS-pm particles have a narrow distribution, and in the duration of 40 wall units, none of... 

Here, d is diameter, U is velocity, is particle number concentration (m ) in a control volume AV, ij is coUision efficiency, and At is a time step for a droplet to move from cme positirai to another. The subscripts p and d denote the physical quantities relative to particle and droplet, respectively. The particle-droplet relative velocity and the particle number concentration along the droplet moving path can be obtained based on a Lagrangian tracking. The collision efficiency i/ is defined as the ratio of the number of particles which collide with the spheroid to the number of particles which could collide wifli the spheroid if their trajectories were straight lines. In the case of laminar flow past a sphere, where the particles are uniformly distributed in the incident flow, the collision efficiency can be determined as t] = 2ycildd), where is the distance from the central symmetry axis of the flow, at which the particles only touch the sphere while flowing past it and is the diameter of the sphere (as shown in Fig. 18.30). The particles, whose coordinates in the incident flow are y > jcn will not collide with the sphere. In Schuch and Loffler [33], the collision efficiency rj is correlated with Stokes number (St) as... 

The first approach starts from the stationary flame equations in Eulerian coordinates. In their second-order form these equations are obtained by omitting the time derivative from the equations of type (4.19) (cf. Lagrangian representations where the distance derivative on the left-hand side is omitted). The further development, as illustrated by Wilde (1972), is essentially the same... 

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