Eulerian equation of motion

Then the Eulerian equation of motion (2.104) can be written as follows  

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Primitive equations Eulerian equations of motion under the effect of Coriolis force and modified by the... 

The Eulerian equations of motion are more useful for numerical solution of highly distorted fluid flow than are Lagrangian equations of motion. Multicomponent Eulerian calculations require equations of state for mixed cells and methods for moving mass and its associated state values into and out of mixed cells. These complications are avoided by Lagrangian calculations. Harlow s particle-in-cell (PIC) method uses particles for the mass movement. The first reactive Eulerian hydrodynamic code EIC (Explosive-in-cell) used the PIC method, and it is described in reference 2. The discrete nature of the mass movement introduced pressure and temperature variations from cycle to cycle of the calculation that were unacceptable for many reactive fluid dynamic problems. A one-component continuous mass transport Eulerian code developed in 1966 proved useful for solving many one-component problems of interest in reactive fluid dynamics. The need for a multicomponent Eulerian code resulted in a second 2DE code, described in reference 4. Elastic-plastic flow and real viscosity were added in 1976. The technique was extended to three dimensions in the 1970 s and the resulting 3DE code is described in Appendix D. 

The starting point for obtaining quantitative descriptions of flow phenomena is Newton s second law, which states that the vector sum of forces acting on a body equals the rate of change of momentum of the body. This force balance can be made in many different ways. It may be appHed over a body of finite size or over each infinitesimal portion of the body. It may be utilized in a coordinate system moving with the body (the so-called Lagrangian viewpoint) or in a fixed coordinate system (the Eulerian viewpoint). Described herein is derivation of the equations of motion from the Eulerian viewpoint using the Cartesian coordinate system. The equations in other coordinate systems are described in standard references (1,2). 

A much more pronounced vortex formation in expanding combustion products was found by Rosenblatt and Hassig (1986), who employed the DICE code to simulate deflagrative combustion of a large, cylindrical, natural gas-air cloud. DICE is a Eulerian code which solves the dynamic equations of motion using an implicit difference scheme. Its principles are analogous to the ICE code described by Harlow and Amsden (1971). 

In this approach, the finite volume methods discussed in the previous chapter can be applied to simulate the continuous fluid (in a Eulerian framework). Various algorithms for treating pressure-velocity coupling, and the discussion on other numerical issues like discretization schemes are applicable. The usual interpolation practices (discussed in the previous chapter) can be used. When solving equations of motion for a continuous fluid in the presence of the dispersed phase, the major differences will be (1) consideration of phase volume fraction in calculation of convective and diffusive terms, and (2) calculation of additional source terms due to the presence of dispersed phase particles. For the calculation of phase volume fraction and additional source terms due to dispersed phase particles, it is necessary to calculate trajectories of the dispersed phase particles, in addition to solving the equations of motion of the continuous phase. 

The four equations of motion are obtained by applying the principle of the conservation of angular momentum. In the first place, the total angular momentum must remain constant in magnitude, and in a direction fixed in space this gives the Eulerian equations... 

The Eulerian equations of fluid motion in which the primary dependent variables are the velocity components of the fluid. In meteorology, they can be specialized to apply directly to the cylonic-scale motions, proxy climate indicators... 

For steady-state flows (here we consider the steady-state in the Eulerian viewpoint. In a steady Eulerian velocity field (which is not necessarily Lagrangian steady), 0M,/0f = 0. For the analysis of polymer melt flow occurring at a very low Reynolds number, the inertia term puj dujldxj will usually be neglected. The body force effect is usually negligible too. This reduces the equations of motion to... 

Then we can obtain the following Eulerian form of the equation of motion defined in the current deformed body ... 

Substituting Stokes law (2.234) into the equation of motion (2.104) under the Eulerian description yields the following equation of motion for the unknown velocity v ... 

We derive the conservation law of mechanical energy, referred to as Stokes power formula, starting with the following equation of motion in an Eulerian framework ... 

In the Eulerian approach, the equation of motion of an individual particle does not appear explicitly but is hidden in the formulation to express the dispersion tensor. We used the Tchen equation, although it is actually controversial. The use of another equation would require a modification of the details of the Eulerian approach. Such a modification is certainly not possible for any equation of motion, especially if it is not linear. 

The Eulerian approach is limited to restricted situations due to numerous underlying assumptions. For instance, only the Tchen equation of motion is implanted in the formulation. Furthermore, the CTE cannot be accounted for in an elegant way. Although we expect that some assumptions could be relaxed, it would lead to a difficult formal work to generalize the approach. Also, complex phenomena such as coalescence or break-ups, particle vaporization and combustion, would certainly be very difficult to study. Consequently, we also developped a Lagrangian approach which is more flexible. 

The Eulerian continuum approach, based on a continuum assumption of phases, provides a field description of the dynamics of each phase. The Lagrangian trajectory approach, from the study of motions of individual particles, is able to yield historical trajectories of the particles. The kinetic theory modeling for interparticle collisions, extended from the kinetic theory of gases, can be applied to dense suspension systems where the transport in the particle phase is dominated by interparticle collisions. The Ergun equation provides important flow relationships, which are useful not only for packed bed systems, but also for some situations in fluidized bed systems. 

Eulerian equations for the dispersed phase may be derived by several means. A popular and simple way consists in volume filtering of the separate, local, instantaneous phase equations accounting for the inter-facial jump conditions [274]. Such an averaging approach may be restrictive, because particle sizes and particle distances have to be smaller than the smallest length scale of the turbulence. Besides, it does not account for the Random Uncorrelated Motion (RUM), which measures the deviation of particle velocities compared to the local mean velocity of the dispersed phase [280] (see section 10.1). In the present study, a statistical approach analogous to kinetic theory [265] is used to construct a probability density function (pdf) fp cp,Cp, which gives the local instantaneous probable num-... 

The averaged Eulerian-Eulerian multi-fluid model denotes the averaged mass and momentum conservation equations as formulated in an Eulerian frame of reference for both the dispersed and continuous phases describing the time-dependent motion. For multiphase isothermal systems involving laminar flow, the averaged conservation equations for mass and momentum are given by ... 

Greenspan [31] outline the transformation of the Eulerian equations governing the motion of an incompressible viscous fluid from an inertial to a rotational frame. The transformation of the Navier-Stokes equations simply results in adding the artificial forces in the momentum balance. The additional equations are apparently not changed as the substantial derivative of scalar functions are Galilean invariant so the form of the terms do not change. 

The sections of this chapter deal with the following elements of atmospheric dynamics vertical structure of the atmosphere (Section 3.2), fundamental equations of atmospheric motions (Section 3.3), transport of chemical constituents and the relative importance of dynamical and chemical effects on photochemical species (Section 3.4), atmospheric waves (Section 3.5), the mean meridional circulation and the use of the transformed Eulerian formalism to illustrate the roles of mean meridional and eddy transports (Section 3.6), the important role of wave transience and dissipation (Section 3.7), vertical transport by molecular diffusion in the thermosphere (Section 3.8), and finally, models of the middle atmosphere (Section 3.9). 

The single-fluid approach has been undertaken utilizing Eulerian methods such as VOF, LS, and the mixed Eulerian-Lagrangian methods [11, 12], For most interface capturing schemes, which use single-fluid formulation, an additional equation is solved to obtain the interface evolution and topology. This equation governs the advection of a variable that can be attributed to the interface. The equation of interface motion is... 

Secondly, the authors give an Eulerian description of the motion of N independent and identical Brownian particles in terms of the phase space density /(r,u f) obeying the Fokker-Planck equation (FPE)... 

Analysis of the motion of the gas bubbles can be conducted either in the Eulerian or Lagrangian frame of reference. In the Eulerian frame of reference, the problem is formulated in terms of partial differential equations which describe the balances of mass and momentum, while in the latter approach, the trajectories of individual bubbles are tracked by solving ordinary differential equations in time. The Lagrangian method has distinct advantages over the Eulerian method in terms of simplicity of formulation, ability to accommodate complicated exchange process, computer memory requirements, and computational efforts. 

In this model, two level-set functions (d, p) are defined to represent the droplet interface (d) and the moving particle surface (p), respectively. The free surface of the droplet is taken as the zero in the droplet level-set function 0> and the advection equation (Eq. (3)) of the droplet level-set function (droplet surface. The particle level-set function (4>p) is defined as the signed distance from any given point x in the Eulerian system to the particle surface ... 

The motions of the individual fluid parcels may be overlooked in favor of a more global, or Eulerian, description. In the case of single-phase systems, convective transport equations for scalar quantities are widely used for calculating the spatial distributions in species concentrations and/or temperature. Chemical reactions may be taken into account in these scalar transport equations by means of source or sink terms comprising chemical rate expressions. The pertinent transport equations run as... 

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