On the condition of pseudo-similarity and the theory of turbulence

Reynolds (1895) has developed the theory of turbulence on the basis of the Navier-Stokes equations of motion for incompressible viscous fluids. For more than 40 years after the publication of Reynolds' paper, research regarding turbulence was mainly concerned with investigations related to Reynolds' equations of mean motion. A new stage in turbulence research began with the application of the equations of turbulent fluctuations by Rotta (1951). Because of the nonlinearity of the Navier-Stokes equations, the dynamical equations of velocity correlations are not closed. Zhou (1940) has treated the closure problem by making assumptions regarding the relation between the quadruple and double velocity correlations, the correlation of velocity and pressure fluctuations, and the terms in viscous decay. Zhou (1945) considered also views regarding the rigorous solution of the turbulence problem. In the present paper the condition of pseudosimilarity is generalized for homogeneous isotropic turbulence to general turbulent shear flows.