Eulerian measures

At this point it is useful to define the velocity of a sound (/ or b) wave propagating in a moving medium, which may also be compressed. The velocity with respect to stations moving with the medium is termed Lagrangian, C. The position of a station (for the purpose of calculating the velocity) is taken to be specified by its initial position. Sound velocity with respect to distances, measured with respect to the laboratory, is termed Eulerian, C . 

The Eulerian approach requires a measurement of the temperature or the progress variable at many sample points at a given normal distance from the ignition plane, at a given time elapsed since ignition. The progress variable introduced here can be for instance a normalized temperature or concentration that varies from... 

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. 

Measurements of 7l in fhe atmosphere are extremely difficult to perform and it is difficult to establish when t > Ti holds for urban scale flows. Csanady (1973) indicates that a typical eddy which is generated by shear flow near the ground has a Lagrangian time scale of the order of 100 sec. Lamb and Neiburger (1971), in a series of measurements in the Los Angeles Basin, estimated the Eulerian time scale to be 50 sec. In a discussion of some field experiments, Lumley and Panofsky (1964) suggested that Tl < 4Te If the averaging interval is selected to be equal to the travel time, then an approximate value for Kyy can be deduced from the measurements of Willis and Deardorff (1976). Their data indicate that for unstable conditions (L > 0) and a travel time t = 3zi/w,... 

The mathematical models used to infer rates of water motion from the conservative properties and biogeochemical rates from nonconservative ones were flrst developed in the 1960s. Although they require acceptance of several assumptions, these models represent an elegant approach to obtaining rate information from easily measured constituents in seawater, such as salinity and the concentrations of the nonconservative chemical of interest. These models use an Eulerian approach. That is, they look at how a conservative property, such as the concentration of a conservative solute C, varies over time in an infinitesimally small volume of the ocean. Since C is conservative, its concentrations can only be altered by water transport, either via advection and/or turbulent mixing. Both processes can move water through any or all of the three dimensions... 

For example, Kumar and Russell (1996) examined the effect on predicted ozone levels in the Los Angeles area of two different approaches now used for incorporating meteorology into a grid-based Eulerian model. The diagnostic approach is based on field measurements of the needed meteorological variables and in-... 

Figure 16.26, for example, shows ozone isopleths predicted for the Los Angeles area using the CIT Eulerian model under two different assumptions for the boundary and initial conditions (Winner et al., 1995). In Fig. 16.26a, the pollutant concentrations measured in the past were used to set the boundary and initial conditions. In Fig. 16.26b, concentrations measured in clean air over the Pacific Ocean upwind of the modeled area were used instead. The axes are given in terms of the percentage of VOC or NOx emissions compared to the baseline level on August 28, 1987, rather than in terms of absolute concentrations as for... 

Equation 2.4-4 states the mass conservation principle as measured by a stationary observer. The derivative (d/dt) is evaluated at a. fixed position in space (this is referred to as the Eulerian point of view) whereas, Eq 2.4-5 states the conservation principle, as... 

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-... 

Fig. 20.2 Surface-level concentrations of sulphate in Varrio during the peak event, modeled with SILAM (a) Lagrangian kernel, (b) Eulerian kernel with ECMWF and HIRLAM meteorological input, compared to the measured concentrations of aerosol particles with diameter below 1 m...

In general, the Eulerian scheme (version 4) produced much wider horizontal spread than the Lagrangian (Fig. 20.4), despite the higher concentration peaks. Runs with both meteorological drivers showed the sulphate polluted plume over Varrio, but with about 5-h delay compared to both measurements and Lagrangian (HIRLAM data) run. 

In this case study the Lagrangian run with HIRLAM data appears to match the observations better than the others. However, its plume was too narrow and its dispersion velocity seems to be somewhat over-predicted (Fig. 20.2a). Also, it is obvious that the Lagrangian run showed higher variability of the concentrations than the Eulerian one. The Eulerian plume appears much wider, traveling slower due to larger mass fraction near the surface and, thus, matching the measurement site more certainly. It also resulted in a rather smoother shape of concentration time series (Fig. 20.2b). 

Recent efforts to distinguish between the terms burning velocity and flame speed on the basis of Eulerian and Lagrangian coordinate systems appear to introduce confusion. Therefore, the terms are used interchangeably here, as synonyms for such terms as deflagration velocity, wave speed, and propagation velocity. They all refer to velocities measured with respect to the gas ahead of the wave. 

Unfortunately, not all texts describing the tnrbnlence correlation fimctions are using the same notation. Hinze ([66], p. 45) find it more logical to rename these functions using the opposite notation. That is, considering the correlation between the values of a velocity component in a given direction at a fixed point in the flow field at two different instants t and t + t, the resnlting correlation was called Eulerian as the measurements were performed at one fixed point in space. 

According to Taylor [159] [160] [161], the properties of the Lagrangian scales are similar to the Eulerian correlation scales. Although the statistical turbulence theory is derived in terms of the Eulerian correlation functions, accurate measurements of the Lagrangian scales and correlations are easier and direct. In contrast, the measurements of Eulerian correlations requires two probes simultaneously working at two different locations. 

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. 

Balzer G, Simonin O (1993) Extensions of Eulerian gas-solid flow modelling to dense fluidized bed prediction. In Proc 5th Int Symp on refined flow modelhng and turbulence measurements, ed Viollet PL, Paris, pp 417-424... 

Several length scales of turbulence can also be determined from the measured correlation functions. The Eulerian length scales obtained from f(t, x) and g t,x) are the longitudinal and transverse integral scales (1.312) and (1.313), respectively. These length scales are characteristics of the larger turbulence vortices in the flow. 

Some attempts have been made to determine these time scales, in one set of experiments with a homogeneous grid-generated turbulence, the ratio of the Lagrangian to the Eulerian scales was reported to be three (Snyder and Lumley. 1971). Thus the ratio of the eddy diffusivity of a panicle that is too heavy to follow the gas motion to the eddy dilTusivity of the gas would he 1/3. What is meant by a bemy particle in this context One measure is the ratio of the characteristic particle time to the smallest time scale ofthe fluid... 

Environmental problem solving creates major conflict areas with the politicians in developing countries. For example. Environmental Impact Assessment Studies is a good Eulerian approach to predict the effects of transport phenomena and a very convenient planning tool for future activities. However, in its application local politicians may block the scientific opinions from reaching the public. Thus new technologies such as pollution prevention measures are sometimes refused. Thus they cause local people to lose an economic benefit, although it ensures the future of politicians near election time. 

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