General relation for stationary probability density functions

Abstract

A linear relation between a normalized, time (t) dependent, statistically stationary quantity (z) and the normalized conditional expectation (r) of ∂2z/∂t2 allows r to generally satisfy two conditions subject to the stationarity requirement. Experimental data for both temperature and vorticity in several turbulent flows indicate that this relation appears universal. As a result, the exact expression derived by Pope and Ching [Phys. Fluids A 5, 1529 (1993)] for the probability density function (PDF) of any stationary quantity should generally reduce to the simpler form obtained by Ching [Phys. Rev. Lett. 70, 283 (1993)].