Lagrangian acceleration

This is expressed in terms of the particle acceleration immediately behind the shock front. Equation (A. 15) can be expressed in terms of the Lagrangian stress gradient (dff/dX), and the Lagrangian longitudinal sound speed Q =... 

We shall see that a conditional acceleration model in the form of (6.48) is equivalent to a stochastic Lagrangian model for the velocity fluctuations whose characteristic correlation time is proportional to e/k. As discussed below, this implies that the scalar flux (u,

joint velocity, composition PDF level, and thus that a consistent scalar-flux transport equation can be derived from the PDF transport equation. 

As in Section 6.5, the Lagrangian conditional acceleration can be decomposed into mean and fluctuating components. However, unlike for the Eulerian PDF, the mean fields must be replaced by their conditional counterparts (see (6.170)) ... 

The Lagrangian generalized Langevin model (LGLM) for the fluctuating acceleration follows from (6.50) ... 

Erratum A stochastic Lagrangian model for acceleration in turbulent flow [Phys. Fluids 14, 2360 (2002)]. Physics of Fluids 15, 269. 

One- and two-particle Lagrangian acceleration correlations in numerically simulated homogeneous turbulence. Physics of Fluids 9, 2981-2990. 

In terms of these parameters, the acceleration of each atom can be written in terms of the forces on it (or, the Lagrangian for the system can be constructed, and Lagrange s equation can be written for each component of each u ). These provide a set of simultaneous equations of the form... 

For mathematical convenience, boundary conditions and initial conditions must be prescribed. For the simple marine propeller problem, a Lagrangian viewpoint was adopted. The frame of reference was attached to the propeller so that the propeller was fixed but the vessel was rotating. The boundary condition was then a zero velocity on the impeller, while the vessel wall rotated at -Qimpdier- The free surface was considered to be fiat, therefore the normal velocity was zero and a shear-free condition was assumed. It should be noted that in the Lagrangian viewpoint, the frame of reference is in rotation. The fluid is therefore subjected to a constant acceleration and the momentum conservation equation [Eq. (6)] must be modified to account for centrifugal forces and Coriolis forces.An advantage is, however, that the flow can be solved numerically at steady state provided the flow is fully periodic, which limits the computational efforts significantly. 

The basis for any derivation of the momentum equation is the relation commonly known as Newton s second law of motion which in the material Lagrangian form (see Fig. I.IB) expresses a proportionality between the applied forces and the resulting acceleration of a fluid particle with momentum density, P, (e.g., [89]) ... 

Since the Lagrangian q>proach yields scalar equations, it is seen as an advantage over a Newtonian approach. Only the velocity vector v, not the acceleration, of each body is required and any coordinate system orientation desired may be chosen. This is a result of the kinetic energy expressed in terms of a scalar quantity as demonstrated in Eq. (7.4). 

The kinematics table, shown in Table 7.1, introduces a method (referred to within this chapter as the table method) for efficiently managing the mathematics involved in analyzing multibody and multiple coordinate system problems, and can be used in either the Lagrangian or the Newton-Euler approach. A schematic diagram, which defines an inertial or body-fixed coordinate system, must accompany every kinematics table. The purpose of the schematic is to identify the point on the body at which the absolute velocity and acceleration is to be determined. The corresponding schematic for Table 7.1 is shown in Fig. 7.7. 

Table 7.4 is the kinematics table for the single elastic body pendulum shown in Fig. 7.3. The absolute velocity of the center of mass G is required to complete the Lagrangian approach, and the Newtonian approach utilizes die absolute acceleration of point <3 Eq. (7.86), in F = mao, for problems of constant mass. 

The motion of a particle in a flow field can be described in Lagrangian coordinates with its origin attached to the center of the moving particle. The motion of a single particle can be described by its acceleration and rotation in a nonuniform flow field. The particle accelerating in the liquid is governed by Newton s second law of motion as... 

For the rigid continumn of volume A consisting of such particles in accelerated motion, the virtual work may be formulated as given by Eq. (3.58). This extension of the principle of virtual displacements is referred to as d Alembert s principle in the Lagrangian version ... 

D Alembert s principle in the Lagrangian version, as derived in Section 3.4.5, uses infinitesimal virtual displacements about the instantaneous system state. For this reason, it is referred to as a differential principle. When infinitesimal virtual deviations from the entire motion of a system between two instants in time are examined, then it is an integral principle like Hamilton s principle, see Goldstein [86], Sokolnikoff [167], Szabb [172] or Morgenstern and Szaho [126]. Here the derivation from the prior to the latter principle will be demonstrated, starting with conversion of the virtual work of the inertia loads included in Eq. (3.59). With Eq. (3.54) and acceleration as derivative of velocity, it... 

D Alembert s principle in the Lagrangian version has been obtained in Section 3.4.5 in terms of virtual displacements and actual accelerations. Since it needs to be accounted for a superimposed guided motion, the position p x,s,t) in the inertial frame of reference, as described by Eq. (7.65), has to be taken into consideration. With the density p s, n) in accordance with Remark 7.1, the virtual work of inertia forces originating from Eq. (3.59) then reads... 

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