Eulerian mean

The main approach for modelling multiphase flows has been through solving conservation equations described in terms of Eulerian phase-averaged mean quantities - the two-fluid approach [665], The Eulerian mean velocity in a control volume V (such as the volume within the perimeter S of the cloud of particles) is defined as the velocity ux (for each component) averaged over the volume occupied by the fluid (ie the fluid space between the bodies),... 

The definition (7.32) masks the local and non-local contributions from bodies to the flow. A more systematic approach to characterising the Eulerian mean velocity is to decompose the flow into (i) a far field flow contribution - far from each body but still within the group of bodies - and (ii) a near field flow contribution - local to each body. This concept, originally described qualitatively by Kowe et al. [353], is strictly valid for dilute arrays since it formally requires the bodies to be widely separated, so that there is a separation of lengthscales between the near and far field, scaling approximately as 0(a) and 0(LS) respectively. The decomposition is defined formally here for potential flows. The far field flow, u, is defined mathematically as the sum of the dipolar and source contributions from the bodies, by assuming the bodies shrink to zero, so that (from (7.31))... 

The Eulerian mean velocity is computed by averaging over the whole fluid region between the bodies, as in (7.31). It can be expressed, using Gauss s Theorem, as... 

Example (b) Circular array of bodies. For a circular array of bodies, the Eulerian mean velocity is... 

Figure 3.21. Mass streamfunction representing the Eulerian mean circulation in January calculated by Becker and Schmitz (2002). Values are expressed in 109kg/s.

Lagrangian Mean, Diabatic, and Transformed Eulerian Mean Circulations... 

It defines the transformed Eulerian mean (TEM) or residual circulation. When these definitions are applied to the simplified form (3.63) of the thermodynamic equation, the resulting circulation is called the diabatic circulation and the meridional and vertical velocities will be noted (vd,wd) rather than (v, w ). In this case, the simplified thermodynamic equation and the steady state approximation for the continuity equation become... 

It is important to note that the diabatic circulation can be estimated from the thermodynamic and continuity equations only if the temperature distribution is known a priori. However, a fully self-consistent mean meridional circulation can be obtained when the momentum budget is considered together with the thermodynamic and continuity equations. When transformations (3.64a, b) are applied to the zonal mean equations (3.53-3.60), the solution of the following momentum, continuity, and thermodynamic equations defines the transformed Eulerian mean (TEM) or residual circulation (here expressed in log-pressure coordinates) ... 

As in the case of the potential temperature, the Eulerian mean and eddy components of meridional transport for the mixing ratio of chemical species tend to cancel, and the continuity equation for species i, written as a function of the TEM winds becomes... 

The conceptual advantage of the diabatic or transformed Eulerian mean circulations is that the eddy-mean flow cancellation is avoided, and a first approximation to the net meridional and vertical transport needed for atmospheric studies is therefore readily obtained. The most important features of this transport description are that air generally enters the lower stratosphere in the tropics and exits at middle and high latitudes in both hemispheres. In the upper stratosphere and mesosphere, air flows from the summer hemisphere to the winter hemisphere, and downward near the winter pole. This transport description is qualitatively very similar to that proposed by Brewer based... 

The formalism of the transformed Eulerian mean circulation shows that meridional transport in the middle atmosphere is generated primarily by non-local momentum forcing associated with wave dissipation. This forcing, represented by the Eliassen-Palm flux divergence in equation (3.67), acts as an extratropical pump producing strong upward air motions in the tropics and downward return ... 

Figure 3.27. Schematic representation of the global diffuser model (upper panel) and tropical pipe model (lower panel). Gray arrows denote meridional transport by the transformed Eulerian mean circulation while the heavy solid arrows show quasi-horizontal mixing by large scale waves. The vertical lines in the lower panel represent dynamical barriers against meridional transport in the tropics. From Plumb and Ko (1992).

Juckes, M., A generalization of the transformed Eulerian-mean meridional circulation. 

Palmer, T. N., Diagnostic study of a wavenumber 2 stratospheric sudden warming in a transformed Eulerian mean formalism. J Atmos Sci 38, 544, 1981. 

In this article, the zonal-mean equations are written in the Transformed Eulerian Mean (TEM) formalism, whose derivation can be found in several of the works listed in the bibliography. The vertical coordinate adopted is log-pressure altitude, an isobaric (constant-pressure) coordinate defined by... 

The quantity p—po) in Eq. (17) is the mean excess pressure and in general, it is nonzero for finite amplitude waves. It is also an Eulerian quantity since the timeaveraging is being performed at a fixed position in space. Wang and Lee [1] provide a succinct derivation for the value of this Eulerian mean excess pressure, correct to second order but expressed in terms of first-order quantities, in the form ... 

This flow field can be maintained in a steady state, at least in the Eulerian sense, either by use of a four-roll mill [18] as in Figure 2.8.4(a) or by means of opposed jet flow as in Figure 2.8.4(b). However, it is important to note that the flow is still transient in the Lagrangian sense. That is, pure planar extension is confined to the central stagnation... 

In summary, we have commented briefly on the microscopic applications of NMR velocity imaging in complex polymer flows in complex geometries, where these applications have been termed Rheo-NMR [23]. As some of these complex geometries can be easily established in small scales, NMR velocimetry and visc-ometry at microscopic resolution can provide an effective means to image the entire Eulerian velocity field experimentally and to measure extensional properties in elastic liquids non-invasively. 

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... 

As in Section 6.5, the Lagrangian conditional acceleration can be decomposed into mean and fluctuating components. However, unlike for the Eulerian PDF, the mean fields must be replaced by their conditional counterparts (see (6.170)) ... 

By definition, the mean scalar field ( (x, t) can be found from the Eulerian composition PDF ... 

As described above, spatial transport in an Eulerian PDF code is simulated by random jumps of notional particles between grid cells. Even in the simplest case of one-dimensional purely convective flow with equal-sized grids, so-called numerical diffusion will be present. In order to show that this is the case, we can use the analysis presented in Mobus et al. (2001), simplified to one-dimensional flow in the domain [0, L (Mobus et al. 1999). Let X(rnAt) denote the random location of a notional particle at time step m. Since the location of the particle is discrete, we can denote it by a random integer i X(mAt) = iAx, where the grid spacing is related to the number of grid cells (M) by Ax = L/M. For purely convective flow, the time step is related to the mean velocity (U) by16... 

For simple flows where the mean velocity and/or turbulent diffusivity depend only weakly on the spatial location, the Eulerian PDF algorithm described above will perform adequately. However, in many flows of practical interest, there will be strong spatial gradients in turbulence statistics. In order to resolve such gradients, it will be necessary to use local grid refinement. This will result in widely varying values for the cell time scales found from (7.13). The simulation time step found from (7.15) will then be much smaller than the characteristic cell time scales for many of the cells. When the simulation time step is applied in (7.16), one will find that Ni must be made unrealistically large in order to satisfy the constraint that Nf > 1 for all k. 

Although it seems natural to formulate the dynamic equations of a chemical in a river in terms of the Langrangian picture, the field data are usually made in the Eulerian reference system. In this system we consider the changes at a fixed point in space, for instance, at a fixed river cross section located atxQ. In Eq. 22-6 we adopted the Eulerian system and found that this representation combines the influence from in-situ reactions (the Langrangian picture) with the influence from transport. The latter appears in the additional advective transport term -udCJdx, where the mean flow velocity ... 

Overall our objective is to cast the conservation equations in the form of partial differential equations in an Eulerian framework with the spatial coordinates and time as the independent variables. The approach combines the notions of conservation laws on systems with the behavior of control volumes fixed in space, through which fluid flows. For a system, meaning an identified mass of fluid, one can apply well-known conservation laws. Examples are conservation of mass, momentum (F = ma), and energy (first law of thermodynamics). As a practical matter, however, it is impossible to keep track of all the systems that represent the flow and interaction of countless packets of fluid. Fortunately, as discussed in Section 2.3, it is possible to use a construct called the substantial derivative that quantitatively relates conservation laws on systems to fixed control volumes. 

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