Separability into coherent and chaotic time dependencies of turbulent fluctuations

Theoretical background is given for the separability of turbulent unsteadiness into the coherent and the chaotic. A method of nonlinear separation of variables applied to the turbulent correlation equation yields a fluctuation equation having two temporal operators representing coherent and chaotic unsteadiness. This equation is compared with the equation of fluid dynamic stability with regard to similarity in form and difference in physical content. It is stressed that the equation is essentially complex, and that the nonlinear term representing triple correlation substantiates continuous energy cascade towards lower frequencies as well as higher ones. A Fourier transform of this equation yields a Navier–Stokes equation for variables with physical meaning distinct from the ordinary ones.