Control volume Lagrangian

The control volume depicted in Figure 1.3 is for one fixed in position (i.e., fixed observation point) and of fixed size but allowing for variable mass within it this is often referred to as the Eulerian point of view. The alternative is the Lagrangian point of view, which focuses on a specified mass of fluid moving at the average velocity of the system the volume of this mass may change. 

In continuum dynamics the balance principles and conservation laws can be applied to the fluid contained in a control volume of arbitrary size, shape and state of motion. Conceptually, the simplest control volume is perhaps the Lagrangian or material one that moves at every point on its surface with... 

Fig. 1.1. (A) Finite Eulerian control volume fixed in space with the fluid moving through it. (B) Finite Lagrangian control volume moving with the fluid such that the same fluid particles are always in the same control volume (i.e., a material control volume). (C) Finite general Lagrangian control volume moving with an arbitrarily velocity not necessarily equal to the fluid velocity. The sohd line indicate the control volume surface (C5) at time t, while the dashed line indicate the same CS at time t + dt.

For a system the angular momentum conservation law is stated as follows The rate of change of the angular momentum of a material volume V(t) is equal to the sum of the torques. Let the vector Vj be the position of a point on the Lagrangian control volume surface with respect to a fixed origin. The relevant terms are formulated as follows [119] [134] [13] ... 

In an engineering view the ensemble of system points moving through phase space behaves much like a fluid in a multidimensional space, and there are numerous similarities between our imagination of the ensemble and the well known notions of fluid dynamics [35]. Then, the substantial derivative in fluid dynamics corresponds to a derivative of the density as we follow the motion of a particular differential volume of the ensemble in time. The material derivative is thus similar to the Lagrangian picture in fluid d3mamics in which individual particles are followed in time. The partial derivative is defined at fixed (q,p). It can be interpreted as if we consider a particular fixed control volume in phase space and measure the time variation of the density as the ensemble of system points flows by us. The partial derivative at a fixed point in phase space thus resembles the Eulerian viewpoint in fluid dynamics. 

The Lagrangian like control volume drdc in the six dimensional phase space may become distorted in shape as a result of the motion. But, in accordance with the Liouville s theorem, discussed in sect. 2.2.3, the new volume is simply related to the old one by the relation ... 

The model formulation used with moving meshes is of the arbitrary-Lagrangian-Eulerian (ALE) type. The integral form of the equations governing the incompressible Newtonian fluid in a time-varying control-volume V(t) is written as ... 

We begin with a general survey of atmospheric inverse methods. In all cases, the broad goal is to use concentration measurements in the air, together with information about atmospheric flow, to infer sources and sinks of entities at the earth s surface. Since the key concentration observations are remote from the surface sources and sinks, this entire class of methods relies explicitly or implicitly on an atmospheric mass or molar balance for the entity being measured, within a specified control volume. Such a balance can be either in an Eulerian framework, in which the control volume is fixed in space, or in a Lagrangian framework, in which the control volume moves with the flow. Considering the Eulerian framework first, the molar balance for a scalar entity can be written informally as... 

In the Lagrangian approach, the elemental control volume is considered to be moving with the fluid as a whole. In the Eulerian approach, in contrast, the control volume is assumed fixed in the space, the fluid is assumed to flow through and pass the control volume. The particle-phase equations are formulated in Lagrangian form, and the coupling between the two phases is introduced through particle sources in the Eulerian gas-phase equations. The standard k-e turbulence model, finite rate chemistry, and DTRM (discrete transfer radiation model) radiation model are used. 

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... 

It is remarked that in the standard literature on fluid dynamics and transport phenomena three different modeling frameworks, which are named in a physical notation rather than in mathematical terms, have been followed formulating the single phase balance equations [91]. These are (1) The infinitesimal particle approach [2, 3, 67, 91, 145]. In this case a differential cubical fluid particle is considered as it moves through space relative to some fixed coordinate system. By applying the balance principle to this Lagrangian control volume the conservation equations for... 

The fundamental conservation of mass law for a multi-component system can be formulated employing the material Lagrangian framework (e.g., [11, 13, 91, 170]). The mixture mass M in a macroscopic material control volume V (t), is given by ... 

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