In isotropic turbulence the Kármán–Howarth equation (after Theodore von Kármán and Leslie Howarth 1938), which is derived from the Navier–Stokes equations, is used to describe the evolution of non-dimensional longitudinal autocorrelation.
[1][2][3][4][5] Consider a two-point velocity correlation tensor for homogeneous turbulence For isotropic turbulence, this correlation tensor can be expressed in terms of two scalar functions, using the invariant theory of full rotation group, first derived by Howard P. Robertson in 1940,[6] where
is the root mean square turbulent velocity and
uniquely determines the two-point correlation function.
Theodore von Kármán and Leslie Howarth derived the evolution equation for
uniquely determines the triple correlation tensor L.G.
Loitsianskii derived an integral invariant for the decay of the turbulence by taking the fourth moment of the Kármán–Howarth equation in 1939,[7][8] i.e., If
Lev Landau and Evgeny Lifshitz showed that this invariant is equivalent to conservation of angular momentum.
Reid showed that this invariant does not hold always since
is not in general zero, at least, in the initial period of the decay.
[10][11] In 1967, Philip Saffman showed that this integral depends on the initial conditions and the integral can diverge under certain conditions.
[12] For the viscosity dominated flows, during the decay of turbulence, the Kármán–Howarth equation reduces to a heat equation once the triple correlation tensor is neglected, i.e., With suitable boundary conditions, the solution to above equation is given by[13] so that,