Resumo

MERCADO, José Roberto et al. The Discharge Coefficient and the Beta Density. Tecnol. cienc. agua [online]. 2014, vol.5, n.2, pp.161-175. ISSN 2007-2422.

Discharge coefficient and turbulence intensity distribution are studied. With Torricelli's theorem and the approach of probability theory, flow discharge and discharge coefficient equation are derivate, following an unimodal Beta density function, renormalized, with two shape parameters. A multifractals model for the kinetic energy cascade in the turbulence was build, starting from the methods of Pearson and Kolmogorov. For turbulence intensity, with the first method, a Beta distribution was created; with the second, a power law. The multifractals model is completed, recognizing the structure function as a Kummer function. The compatibility between the two models are searched and so the identification of its parameters. It is found that the two shape parameters determine the cascade model resolution. Local dimension and dimension spectra are determine for the two states that produces Torricelli theorem. Redefining the structure function, resolution is defined by the water depth for the regime change. Analogously, different prototypes could be define, which we have call: the four experimentals, three channels, Kolmogorov, Kármán, Taylor, Verhulst (logistic), Cauchy-Manning, and Euclides (golden proportion). We conclude that the discharge coefficient is a renormalized Beta; turbulence intensities distribution is a Beta; Torricelli prototype results representative for the four experimentals and the Euclides, far away from the Gaussian distribution that is contained in von Karman model, meanwhile the Taylor's model yield the Dirac function.

Palavras-chave : Discharge equations; self-similarity; turbulence models; density function; Kummer function; multifractals; kinetic energy.

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