Single-Point Lagrangian Equations

TABLE 2 Choices of the Arbitrary Concentration a and Source Density (p for the Scalars c, cR, cS, R,andS 

Single-Point Kulerian Equation Balance of a ip Lagrangian Methods Applicable  

The last column indicates the applicability of the Lagrangian equation (15) and its canopy version, Eq. (17). 

The quantities a, p, and P are all ensemble-averaged, an operation which is the same as the more familiar time average in a stationary (statistically steady) flow, but not in a nonstationary flow. 

The transition probability P(x, f xn, tg) carries all information about the velocity field that is needed to deduce (x, l). In practice there are three main ways of determining P first, one may obtain 

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The description is based on the previously defined single-particle (Lagrangian) or one-point (Eulerian) joint velocity-composition (micro-)PDF, /(r,yr). As mentioned in Section 12.4.1, in the one-point description no information on the local velocity and scalar (species concentrations, temperature,. ..) gradients and on the frequency or length scale of the fluctuations is included and the related terms require closure models. The scalar dissipation rate model has to relate the micro-mixing time to the turbulence field (see (12.2-3)), either directly or via a transport equation for the turbulence dissipation rate e. A major advantage is that the reaction rate is a point value and its behavior and mean are described exactly by a one-point PDF, even for arbitrarily complex and nonlinear reaction kinetics. 

This transformation gives new system of equations for any field distribution in a FGM of any complex structure from the single phenomenological point of view [6,15], where fields in domains and between them are considered within the same equations. Any external field, applied to the material with a certain temperature distribution, will not change the form of equations, since they will be ttdcen into account within the Lagrangian. 

Immersed Boundary (IB) and Immersed Interface (II) Techniques The immersed boundary technique [28] is a combined Eulerian-Lagrangian method in the sense that the flow is solved using a Eulerian approach on a fixed Cartesian mesh, and the interface is represented by a discrete set of points and advected in a Lagrangian way. In this method, a single set of conservation equations valid for both phases is solved. The interface conditions are incorporated into the governing field equa-... 

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