A theory of turbulence

Abstract

In this paper an attempt is made to develop a deductive physical theory of turbulence. The basic idea in this theory is the introduction of correlations in the velocity components (u i ) at two different points (r' and r" say) and at two different times ( t' and t" say). It is assumed that in stationary, homogeneous and isotropic turbulence these correlations depend, apart from the vector joining the points r' and r" only on | t' — t" |. With one additional statistical hypothesis that the fourth-order moments, Q ij;kl = u i (r', t' ) u j (r', t' ) u k (r", t" ) u l (r", t" ), are related to the second-order moments, Q ij = u i (r', t' ) u j (r", t" ) as in a joint normal distribution, a nonlinear partial differential equation is derived for the defining scalar, Q(r, t) ( r = | r' — r" | and t = | t' — t" | ) of Q ij . This differential equation gives precision to the idea that in some sense, the inertial and the viscous terms in the equations of motion act alike with effects which are additive. In the limiting case of infinite Reynolds number the equation for Q is discussed in some detail. It is shown how the theory enables one to follow explicitly the initial evolution of Q , if at t = 0, it has the form one supposes on Kolmogoroff’s similarity principles. For t →∞, it appears that the solution is separable in the variables r and t ; the nature of these solutions (which form a one-parameter family) are discussed and illustrated.