Kolmogorov structure function

The / coefficient vaiue in the dropiet rms veiocity reiation (9.31) obtained by Kuboi et al [53, 54] is accidentally about the same as the C r ( )C fc value in the Kolmogorov structure function relation (9.14), that is / C 2.0. However, it was commented by Kuboi et al [53, 54] that the comparison of these relations cannot be very decisive, in view of the fact that there is a large difference between the processes and kind of fluid used to obtain these relations. Kuboi et al [53, 54] also investigated the particle-fluid density difference effect producing non-neutrally buoyant particle flows and concluded that the parameter value discussed above is very sensitive to the density ratio. Therefore, the application of the above relation as an approximation for the bubble velocity is highly questionable. 

Luo and Svendsen [80] did not distinguish between the experimentally determined relation (9.32) and the Kolmogorov structure function (9.15). In their work the theoretical parameter value C was... 

Abstract This is a tutorial about the main optical properties of the Earth atmosphere as it affects incoming radiation from astrophysical sources. Turbulence is a random process, of which statitical moments are described relying on the Kolmogorov model. The phase structure function and the Fried parameter ro are introduced. Analytical expressions of the degradation of the optical transfer function due to the turbulence, and the resulting Strehl ratio and anisoplanatism are derived. 

For Kolmogorov statistics, it turns out that the power spectrum is infinite at the origin, which means that the variance is infinite. The structure function can be used instead of the co-variance to overcome this problem. It is defined as... 

The phase structure function for a separation r is defined as the mean value of the square difference of phase for all points with that separation. For wave-fronts affected by Kolmogorov turbulence it is given by... 

Kolmogorov [83] [84] introduce the concept of structure functions describing processes with stationary - or homogeneous increments. 

By definition, the second order velocity structure function is the covariance of the difference in velocity between two points in space. A consequence of isotropy is that the structure function can be expressed in terms of a single scalar function. According to the similarity hypotheses of Kolmogorov, the scalar function can be expressed as Bdd x) = Sv Sv is a derived velocity scale sometimes... 

The results of this dimensional analysis are supported by an exact result obtained by Kolmogorov for the third-order longitudinal structure function. The structure functions are moments of velocity differences measured at a given distance. More specifically the structure function of order n, Sn(l), is defined as... 

Using the result of Kolmogorov for the longitudinal velocity structure function... 

The space-fractional equation (3.91) can be derived from the Kolmogorov-Feller equation (3.74) by using the assumption that the random jump Z has a Levy-stable PDF Wg,(z), symmetric with respect to zero, with power-law tails as z -> oo. There is no general explicit form for u (z), but the characteristic function of iv z), the structure function, is... 

Kolmogorov second order velocity structure function (m /s ) Capacitance matrix... 

The Local Structure Operator By the Kolmogorov consistency theorem, we can use the Bayesian extension of Pn to define a measure on F. This measure -called the finite-block measure, /i f, where N denotes the order of the block probability function from which it is derived by Bayesian extension - is defined by assigning t.o each cylinder c Bj) = 5 G F cti = 6i, 0 2 = 62, , ( j — bj a value equal to the probability of its associated block ... 

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