A Functional Treatise on Statistical Hydromechanics with Random Force Action

Abstract

Statistical hydromechanics originated by Hopf is generalized to take into account the influence of an external random force field with an arbitrary (not necessarily Gaussian) distribution. Starting from the space-time functional formalism for turbulence recently established, a general expression for the characteristic functional describing the statistical behavior of fluid at each instant is derived in the form of a functional integral. For the case that the random force field is temporally correlated in the delta-function manner, a functional-differential equation is obtained which governs the time-evolution of the characteristic functional. This equation, being a counterpart of the Hopf equation which was given for the case of no random excitation, involves Novikov's equation, as a special case.