Differential Equation for Turbulence Power Losses and Energy Spectra Based on Consolidated Empirical Results

Abstract

A second order differential equation for the energy dissipation rate of turbulence is presented. The derivation procedure is explained. The obtained governing equation is a Euler equation, which integration naturally conduces to power laws for the energy dissipation rate as a function of the wavenumber, a result that is extended to the energy spectrum of turbulence. Power laws are obtained for the cases of two equal and two different real roots. For the case of two conjugate complex roots, the solution is a sum of sine and cosine functions of the normal logarithm of the wavenumber. The differential equation accrues from a more basic equation obtained through thermodynamic-type steps that joint part of already consolidated empirical and semi-empirical information on turbulence existing in the literature, and is formally analogue to the Thermodynamics equation of thermal radiation. It is also shown that parameters of turbulence like length and velocity scales may be related to this formulation.

Keywords

• turbulence spectra
• thermodynamic analogy
• euler differential equation
• empirical laws
• constitutive equations