ON THE SOLUTIONS OF NAVIER-STOKES EQUATIONS AND THE THEORY OF HOMOGENEOUS ISOTROPIC TURBULENCE

Abstract

The present paper is a further development of our previous work in solving the wholeproblem of the homogeneous isotropic turbulence from the nitial period to the final period ofdecay. An expansion method is developed to obtain the axinlly symmetrical solution of theNavier-Stokes equations of motion in the form of an infinite set of nonlinear partial differen-tial equations of the second order. For the present we solve the zeroth order approximation.By using the method of Fourier transform, we get a nonlinear nitegro-differential equationfor the amplitude function in the wave number space.It is also the dynamical equation forthe energy spectrum. By choosing a suitable initial condition, we solve this equation numerically. The energyspectrum function and the energy transfer spectrum function thus calculated satisfy the spec-trum form of the karman-Howarth equation exactly. We Lave computed the energy spectrumfunction, the energy transfer function the decay of turbulent energy, the integral scale, Taylormicroscale, the double and triple velocity correlations on the whole range from the initialperiod to the final period of decay. As a whole all these calculated statistical physicalquantities agree with experiments very wall except a few cases with small discrepancies at largeseparations.