Kolmogorov hypotheses

The description of turbulent mixing of passive scalars is based on an extension of the turbulence theory developed for the momentum transfer processes, described in sect 1.2.7. In particular, the energy cascade idea introduced by Richardson [76] and the Kolmogorov hypotheses [47] are adopted. 

The turbulence community refers to this work as Kolmogorov 41. A second article (Kolmogorov 1962) referred to as Kolmogorov 62 contains the refined Kolmogorov hypothesis. ... 

Note that the Kolmogorov hypothesis, and deductions drawn from them, apparently have no direct connection to the Navier-Stokes equations. 

Approximately stated Kolmogorov s first similarity hypothesis yields ([84] see also [122], p. 185) In every turbulent flow at sufficiently high Reynolds number, the statistics of the small scale motions have a universal form that is uniquely determined by o and s. The phrases similarity hypothesis and universal form refer to a mathematical consequence of the Kolmogorov hypothesis denoting that on the small scales aU high-Reynolds-number turbulent velocity fields are statistically similar. That is, they are statistically identical when they are scaled by the Kolmogorov velocity scale... 

The algorithm for estimating the LDC and LDM for teehniques of test analysis with visual indieation is suggested. It ineludes the steps to eheek the suffieieney of experimental material [1]. The hypothesis ehoiee about the type of frequeney distribution in unreliable reaetion (UR) region is based on the ealeulation of eriteria eomplex Kolmogorov-Smirnov eriterion,... 

Sleiched278 has indicated that this expression is not valid for pipe flows. In pipe flows, droplet breakup is governed by surface tension forces, velocity fluctuations, pressure fluctuations, and steep velocity gradients. Sevik and Park 279 modified the hypothesis of Kolmogorov, 280 and Hinze, 270 and suggested that resonance may cause droplet breakup in turbulent flows if the characteristic turbulence frequency equals to the lowest or natural frequency mode of an... 

Besides the hypothesis of spatially homogeneous processes in this stochastic formulation, the particle model introduces a structural heterogeneity in the media through the scarcity of particles when their number is low. In fact, the number of differential equations in the stochastic formulation for the state probability keeps track of all of the particles in the system, and therefore it accounts for the particle scarcity. The presence of several differential equations in the stochastic formulation is at the origin of the uncertainty, or stochastic error, in the process. The deterministic version of the model is unable to deal with the stochastic error, but as stated in Section 9.3.4, that is reduced to zero when the number of particles is very large. Only in this last case can the set of Kolmogorov differential equations be adequately approximated by the deterministic formulation, involving a set of differential equations of fixed size for the states of the process. 

In terms of the statistical methods of the partial life cycle whole-effluent tests, survival, growth, and reproduction data from the 7 day cladoceran or fish exposure are often analyzed using hypothesis testing to determine acceptable concentrations. In order to determine the appropriateness of using parametric statistical methods, the data are first tested for normality of distribution and homogeneity of variance, for which the US EPA recommends the use of Shapiro-Wilk s test and Bartlett s test, respectively. Kolmogorov test for normality and Levine s test for homogeneity can be also used for these purposes. Dunnett s anova test is typically used for a... 

Approximately stated Kolmogorov s hypothesis of local isotropy yields ([83] see also [121], p. 184) At sufficiently high Reynolds number, the small scale turbulent motions are statistically isotropic. 

The corresponding dissipation rate term is approximated using (1.405), the Prandtl-Kolmogorov relation (1.403), the eddy viscosity hypothesis (1.380), and the non-equilibrium boundary layer shear stress approximation (1.441) ... 

Stolovitzky G, Sreenivasan KR. (1994) Kolmogorov s refined similarity hypothesis for turbulence and general stochastic processes. Rev. Mod. Phys., 66 229-236. 

Use the Kolmogorov method to test the hypothesis that serum immunoglobulin G levels are normally distributed. 

Table 3 shows the results of Kolmogorov-Smirnov distances for different equities and states, and the difference in statistical proprieties (Skewness and Kurtosis) of simulated data from the estimated model with respect to those of the real dataset. Note that the Kolmogorov-Smirnov test always leads to the null hypothesis HO which means that the distribution of the simulated data is... 

One-sample Kolmogorov-Smimov test was performed to confirm the assumption of normality in the 2 groups for 4 outcome variables. Results are expressed as the mean value SD. Differences were considered to be statistically significant if the null hypothesis could be rejected with 95% confidence. Correlations were assessed with the Pearson test. A p 0.05 was... 

Approximately stated Kolmogorov s second similarity hypothesis yields ([84] see also [122], p. 186) In every turbulent flow at sufficiently high Reynolds numbers, the statistics of the motions of scale I in the range L A Z 3> have a universal... 

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