Lagrangian measurements

Gattuso, J.P., Pinchon, M., Delesalle, B., Canon, C. and Frankignoulle, M. (1996) Carbon fluxes in coral reefs. I. Lagrangian measurement of community metabolism and resulting air-sea C02 disequilibrium. Marine Ecology Progress Series, 145, 109-121. 

The problem thus stated is akin to the familiar question in fluid mechanics of how Eulerian observations, that are more conveniently made, can be converted to Lagrangian information that is often more relevant and the desired quantity. This is because Eulerian observations are made at a fixed point in space whereas Lagrangian measurements require tracking a specific particle in motion. 

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 . 

Figure 7.2. Response of elastic-plastic solid to planar impact at X = 0 u = longitudinal particle velocity. Measurements are made as a function of time at fixed Lagrangian position X.

Test data are available for two experiments at different impact velocities in this configuration. In one of the tests the projectile impact velocity was 1.54 km/s, while in the second the impact velocity was 2.10 km/s. This test was simulated with the WONDY [60] one-dimensional Lagrangian wave code, and Fig. 9.21 compares calculated and measured particle velocity histories at the sample/window interface for the two tests [61]. Other test parameters are listed at the top of each plot in the figure. 

The second model evaluated was the Lagrangian Photochemical Model LPM (54), a trajectory model. Backward trajectories were first determined so that starting positions could be used which would allow trajectories to reach station locations at the times of measurement. Measured concentrations ranged from 0.20 to 0.26 ppm and estimated concentrations from 0.05 to 0.53 ppm. 

Frieden s theory is that any physical measurement induces a transformation of Fisher information J I connecting the phenomenon being measured to intrinsic data. What we call physics - i.e. our objective description of phenomenologically observed behavior - thus derives from the Extreme Physical Information (EPI) principle, which is a variational principle. EPI asserts that, if we define K = I — J as the net physical information, K is an extremum. If one accepts this EPI principle as the foundation, the status of a Lagrangian is immediately elevated from that of a largely ad-hoc construction that yields a desired differential equation to a measure of physical information density that has a definite prior significance. 

The discreteness in this new theory refers to the discrete nature of all measurements. Each measurement fixes a particle s position Xn and time Both and tn are allowed to take on any value in the spectrum of continuous eigenvalues of the operators (xn)op and (tn)op. Notice also that in this discrete theory there is no Hamiltonian and no Lagrangian only Action. 

The above description refers to a Lagrangian frame of reference in which the movement of the particle is followed along its trajectory. Instead of having a steady flow, it is possible to modulate the flow, for example sinusoidally as a function of time. At sufficiently high frequency, the molecular coil deformation will be dephased from the strain rate and the flow becomes transient even with a stagnant flow geometry. Oscillatory flow birefringence has been measured in simple shear and corresponds to some kind of frequency analysis of the flow... 

P. Goix, P. Paranthoen, and M. Trinite 1990, A tomographic study of measurements in a V-shaped H2-air flame and a Lagrangian interpretation of the turbulent flame brush evolution. Combust. Flame 81 229-241. 

When the Lagrangian technique is used to solve the preceding problem, the least squares estimate of the measurement errors is given by... 

In the Lagrangian frame, droplet trajectories in the spray may be calculated using Thomas 2-D equations of motion for a sphere 5791 or the simplified forms)154 1561 The gas velocity distribution in the spray can be determined by either numerical modeling or direct experimental measurements. Using the uncoupled solution approach, many CFD software packages or Navier-Stokes solvers can be used to calculate the gas velocity distribution for various process parameters and atomizer geometries/configurations. On the other hand, somesimple expressions for the gas velocity distribution can be derived from... 

According to measurements made in the atmosphere, the Lagrangian time scale is of the order of 100 sec (Csanady, 1973). Using a characteristic particle velocity of 5 m sec", the above conditions are 100 sec and L > 500 m. Since one primary concern is to examine diffusion from point sources such as industrial stacks, which are generally characterized by small T and L, it is apparent that either one (but particularly the second one) or both of the above constraints cannot be satisfied, at least locally, in the vicinity of the point-like source. Therefore, in these situations, it is important to assess the error incurred by the use of the atmospheric diffusion equation. 

The Lagrangian integral time scale Jl can be considered as a measure of the time during which a fluid particle persists, in the mean, in a motion in a given direction. In Section VIU we will discuss Ti, quantitatively. Where unambiguous, 71 will be denoted simply by T. 

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,... 

Wingenter, O. W M. K. Kubo, N. J. Blake, T. W. Smith, D. R. Blake, and F. S. Rowland, Hydrocarbon and Halocarbon Measurements as Photochemical and Dynamical Indicators of Atmospheric Hydroxyl, Atomic Chlorine, and Vertical Mixing Obtained during Lagrangian Flights, . /. Geophys. Res., 101, 4331-4340... 

Helioseismic waves are detected by measuring the Doppler shift of lines in the solar spectrum due to vertical motion of the Sun s surface along the line of sight. With appropriate data-reduction techniques, the frequencies for global oscillations can be determined to an accuracy of 5 ppm. This extreme accuracy requires long-term, continuous observations that are best done by spacecraft such as the joint ESA/NASA SOHO spacecraft, which observes the Sun from the Lagrangian point where the Earth gravity balances that of the Sun. 

Measurements of the air mobility spectrum seem to add considerable information toward an understanding of aerosol formation and growth at sizes below a few nanometers. Hence, to characterize nucleation mechanisms more precisely, such data should be included in experimental designs. Lagrangian aerosol sampling techniques would also be favored, since this approach can yield data on microphysical evolution without the complicating effects of a changing air mass. Further laboratory studies should be undertaken to quantify the thermodynamic data that define ion properties under tropospheric conditions, at ion sizes and compositions relevant to aerosol nucleation. The sparseness of such data imposes a limitation on our ability to quantify ion-based nucleation mechanisms [19,33],... 

The hypothesis that the constituents of the mixture have a Lagrangian microstructure (in the sense of Capriz [3]) means that each material element of a single body reveals a microscopic geometric order at a closer look then it is there assigned a measure Vi(x) of the peculiar microstructure, read on a manifold Mi of finite dimension rnp e.g., the space of symmetric tensor in the theory of solids with large pores or the interval [0, v) of real number, with v immiscible mixture (see [5, 9]). We do not fix the rank of the tensor order parameter u%. 

The variational formalism makes it possible to postulate a relativistic Lagrangian that is Lorentz invariant and reduces to Newtonian mechanics in the classical limit. Introducing a parameter m, the proper mass of a particle, or mass as measured in its own instantaneous rest frame, the Lagrangian for a free particle can be postulated to have the invariant form A = mulxiilx = — mc2. The canonical momentum is pf, = iiiuj, and the Lagrangian equation of motion is... 

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