The Kármán-Howarth equation evolves two-point correlations in homogeneous isotropic turbulence. In trace form it is . Integrating with radial volume weight reduces conservation of the Saffman integral to vanishing nonlinear and viscous boundary fluxes.
For unforced homogeneous turbulence with isotropic turbulence statistics, put and . The longitudinal Kármán-Howarth equation is . Since , regularity at zero givesNeglecting the last term in a locally equilibrated small-scale range gives the displayed equation. The Kolmogorov four-fifths law follows where the viscous term is negligible as well.
The signed third-order longitudinal structure function has this leading inertial range form under homogeneity, isotropy and the usual local-equilibrium scale separation. Its coefficient follows from the Kármán-Howarth equation and the mean kinetic energy budget, without assuming the Kolmogorov two-thirds law. Its negative sign encodes a forward energy cascade; it does not establish every higher-order similarity exponent.
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The Kármán–Howarth equation is a fundamental relation in fluid dynamics, particularly in the study of turbulence. It describes the evolution of the second-order velocity correlation function in an incompressible flow. The equation provides insight into the relationships between different scales of motion in turbulent flows. In turbulent fluid mechanics, the velocity field can be characterized using correlation functions, which measure the statistical relationships between the velocities at different points in space.