The fully- detailed derivation of the universal physical equation ofturbulence has been given. Instead of the conventional Reynolds average

a new method

named off-center ensemble average

is used to obtain the equation. Through some mathematical approximations and without introducing any empirical coefficients

the equation may provide a secondary order of accuracy

which satisfies the needs of engineering study of turbulence. In one-dimensional case

the momentum equation of turbulence is simplified as the mixed Burgers-Korteveg de Vries equation-the canonical equation of turbulence. The analysis of the Burgers-Kerteweg-de Vries equation gives the criterion of transition. The physical meaning of the equations is briefly discussed. It shows that dissipation and dispersion coexist as two basic physical principles of turbulence. The development of dissipation is monotonic

describing the conversion of the kinetic turbulence energy into heat due to viscosity. Dispersion can be either positive or negative. It represents the energy transformation between the mean flow and the large and micro-eddies before the energy is finally dissipated into molecular heat. Positive dispersion corresponds to cascading down process and negative dispersion means collection of energy. The mechanison of intermittency and anisotropy

and the effects of noise and developing history on turbulence are also discussed.