On the theory of turbulence for incompressible fluids

The theory of turbulence, based upon the Reynolds equations of mean motion and the dynamical equations of the velocity correlations of successive orders derived from the equations of the turbulent velocity fluctuation by using the condition of pseudo-similarity and the hypotheses on the viscous dissipation terms in the correlation equations, is developed by a method of successive approximation. As examples in the first order approximation, we have solved the turbulent flows through a channel, in a plane wake and in jets by using the equations of mean motion and of double correlation, while the terms in the triple correlation have been neglected. The agreements between the calculated values and the experiments are satisfactory. In the present paper, the equations of the triple and quadruple correlations in addition to those used in the first order approximation are solved for the plane turbulent wake in the second order approximation by the method of substitution, starting from the solution of the first order approximation. Agreements between theory and existing experiments for the triple velocity correlation are also satisfactory. The theory has also yielded the components of the quadruple correlation which can be tested by experiment. Presented at the Third Asian Congress of Fluid Mechanics on September 1, 1986, in Tokyo, Japan.