Abstract
It is shown that the so-called statistical theory of turbulence, which can be exactly incorporated by the Hopf functional equation, is imperfect in that it fails to ensure the irreversible approach to a unique ultimate steady state of turbulence (for a steady boundary condition) expected from observation; and that this imperfection is removed if a stochastic random-force term is added into the Navier-Stokes equation. The ensemble mechanics for the random-forced Navier-Stokes flow is formulated by taking into account the natural random force, which has usually been neglected in the Navier-Stokes equation.
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