Next-order structure-function equations

Kolmogorov's equation [Dokl. Akad. Nauk SSSR 32, 16 (1941)] relates the two-point second- and third-order velocity structure functions and the energy dissipation rate. The analogous next higher-order two-point equation relates the third- and fourth-order velocity structure functions and the structure function of the product of pressure-gradient difference and two factors of velocity difference, denoted T-ijk. The equation is simplified on the basis of local isotropy. Laboratory and numerical simulation data are used to evaluate and compare terms in the equation, examine the balance of the equation, and evaluate components of T-ijk. Atmospheric surface-layer data are used to evaluate T-ijk in the inertial range. Combined with the random sweeping hypothesis, the equation relates components of the fourth-order velocity structure function. Data show the resultant error of this application of random sweeping. The next-order equation constrains the relationships that have been suggested among components of the fourth-order velocity structure function. The pressure structure function, pressure-gradient correlation, and mean-squared pressure gradient are related to T-ijk. Inertial range formulas are discussed. (C) 2001 American Institute of Physics.