Single-Point Eulerian Equations

In the limit V 0 at a single fixed point x, Eqs. (3) and (4) reduce to conventional Eulerian differential conservation equations. Anticipating application in vegetation canopies, we consider that some of the region V contains plant leaves and stems. Provided that the canopy is sufficiently finely textured to be regarded as a porous continuum from the standpoint of the airflow, a scalar source density in Eq. (3) can be defined by 

It is shown in Appendix A that in the limit V — 0, Eqs. (2) to (4) yield the following conservation equations for scalar mole fraction (c), iso-concentration (c8), and isotopic composition (8) at a fixed point x  

Equations (7) and (8) (for c and cS) include the usual time derivative and advection terms (the bracketed terms on the left-hand side), flux divergence terms, and source terms. Under steady, horizontally homogeneous conditions, both equations reduce to 

At this point it is useful to identify the isotopic source terms more precisely. If the scalar is CO2, the source density for CO2 mole fraction (d ) has an assimilation component (t/y) and a respiration compsonent (0k), so (f = (b + 0r. The sign convention that 4 is positive into V (that is, into the air) is retained, so t/y 0 and (/),( 0. The source terms in Eqs. (8) and (9), for cS and 6, then become 

Neglecting second-order (Ad, A ) and higher-order terms, the last of Eq. (12) simplifies to d = d — A. This approximate expression, or the exact Eq. (12), specifies d in terms of A and the air isotopic composition d. The source terms in Eq. (10) now 

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Appendix A Single-Point Eulerian Molar Balance Equations.57... 

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. 

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-... 

The similarity in form between the two real equations implied by the single-body spin-0 Schrddinger equation in the position representation (wave mechanics) and the equations of fluid mechanics with potential flow in its Eulerian formulation was first pointed out by Madelung in 1926 [1]. In this analogy, the probability density is proportional to the fluid density, and the phase of the wave function is a velocity potential. A novel feature of the quantum fluid is the appearance of quantum stresses, which are usually represented through the quantum potential. To achieve mathematical equivalence of the models, the hydrodynamic variables have to satisfy... 

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