Stationary functions and their applications to turbulence II. Turbulent solutions of the Navier-Stokes equations

Abstract The first part of this article is devoted to the theory of stationary functions (functions which have a temporal mean value and a correlation function), and to the construction of stationary functions. The class of pseudorandom functions is closely related to the uniformly distributed sequences modulo 1, which constitute a sort of simulation of random variables uniformly distributed over (0, 1) [J. Bass, Stationary Functions and Their Applications to the Theory of Turbulence. I. Stationary Functions, J. Math. Anal. Appl. 47 (1973)] . The second part consists of the application of the above theory to the turbulent solutions of the Navier-Stokes equations. In a preliminary discussion, the various attempts at a theory of turbulence are compared. The nature of probabilistic concepts is discussed, and the role of temporal mean values is illustrated. The importance of the local structure of the turbulent oscillations is emphasized, as much for its technical necessity in the resolution of differential equations as for its physical significance. The construction of a good function-space, containing the “turbulent functions,” is explained. For a flow “permanent in the mean” this function-space corresponds to the asymptotic properties (when t → ∞) of the turbulence, and it does not depend on the boundary conditions. The method of resolution is then applied to the model of the Burgers equation, which can be completely solved. In the case of the general Navier-Stokes equations for an incompressible fluid, the same method gives a class of turbulent solutions, which are deduced from analytic solutions of the equations of permanent motion by a transformation due to R. Berker. The turbulent velocity is given by an expansion in powers of a pseudorandom function. But the solutions obtained are not purely turbulent. They contain a turbulent component and a periodic component, which results from the nonlinear character of the given equation; this seems to be unavoidable. In the case of the linearized Navier-Stokes equations, it is easy to write purely turbulent solutions. They are the sum of a potential term, and of a term containing vorticity, a solution of the homogenous equations (without pressure).