Lagrangian correspondence

Note that Lagrangian correspondence does not automatically imply Eulerian correspondence, since the latter also requires that /x (x t) must be a uniform distribution for all t. Nevertheless, Lagrangian correspondence is the preferable choice, and is obtained by working directly with the stochastic differential equations. 

The maximum strain energy corresponding to a maximum elastic strain that <5,53Since represents a reference level of the Lagrangian corresponding to an optimum state of strains, it can be assumed to be zero for convenience. is a constant indepen-... 

The above description refers to a Lagrangian frame of reference in which the movement of the particle is followed along its trajectory. Instead of having a steady flow, it is possible to modulate the flow, for example sinusoidally as a function of time. At sufficiently high frequency, the molecular coil deformation will be dephased from the strain rate and the flow becomes transient even with a stagnant flow geometry. Oscillatory flow birefringence has been measured in simple shear and corresponds to some kind of frequency analysis of the flow... 

The equation for S is recognized as a Hamilton-Jacobi equation for a mechanical action. The solution can be written in terms of a Lagrangian L, for the nuclear motions, introducing the path Q t, Qin,Q), starting initially at positions Qin and ending at Q at time t, and the corresponding generalized velocities Q. The result is... 

In Eq. (9-62) MjL is the mass of atom //. The mass mq does not correspond to any physical mass and is simply set to a value small enough such that the charges follow the atomic coordinates adiabatically. The Lagrangian also includes an Nmoiec number of constraints to ensure that each molecule remains electrostatically neutral. 

Note this result agrees with Equation (h) in that four variables are included in the Lagrangian, but with one constraint corresponding to 3 degrees of freedom. 

Figure 3.9. Lagrangian scalars +(t) as a function of t. The scalar fields were extracted from DNS of isotropic turbulence (Rx = 140, (U) = 0) with collinear uniform mean scalar gradients. Dashed line Sc = 1/8. Solid line Sc = 1. The corresponding Lagrangian velocity is shown in Fig. 2.3. (Courtesy of P. K. Yeung.)...

B B. By correctly choosing the coefficient matrices (a u, a 7, By, and B ), (6.159) can be made to correspond with the Eulerian velocity, composition PDF transport equation (6.19). However, it is important to note that /L / Thus it remains to determine how the Lagrangian notional-particle PDF /,( is related to the Eulerian velocity, composition PDF fn,0. This can be done by considering Lagrangian fluid particles. 

It is important to recognize that (6.164) is a direct result of choosing the uniformly distributed initial locations (6.163). In contrast, if one chooses to start all notional particles at the origin /V(y) = <5(y), then /x (x t) will be non-uniform, and the Lagrangian notional-particle PDF will not correspond to the Lagrangian fluid-particle PDF. 

The key theoretical concept that makes Lagrangian PDF methods useful is the correspondence between the Eulerian PDF of the flow and the Lagrangian notional-particle PDF. As noted above, in the Lagrangian notional-particle PDF X is a random variable... 

As noted in Chapter 1, the composition PDF description utilizes the concept of turbulent diffusivity (Tt) to model the scalar flux. Thus, it corresponds to closure at the level of the k-e and gradient-diffusion models, and should be used with caution for flows that require closure at the level of the RSM and scalar-flux equation. In general, the velocity, composition PDF codes described in Section 7.4 should be used for flows that require second-order closures. On the other hand, Lagrangian composition codes are well suited for use with an LES description of turbulence. 

Simply by substituting the corresponding fields, die Lagrangian PDF code described in this section could be coupled to an LES code (Jaberi et al. 1999). 

Remark 7.2. In the above argument, we assumed that the C -action on a holomorphic symplectic Kahler manifold X satisfies iploJc = tujc for t E C. This is possible only when X is non-compact. The reason is as follows. If X is compact there exists a critical manifold corresponding to the maximum of the Morse function for which the unstable manifold is open submanifold of X, but this contradicts the propostion which asserts that every unstable manifold is Lagrangian. 

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 ... 

The diagonal Lagrangian-multipliers in the case of water-dimer are depicted on Fig. 4. One can see that the difference between the diagonal Fock-matrix element and the corresponding diagonal Lagrangian-multipliers is considerably smaller for the... 

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