Foundation of Turbulence Revisited.

This paper presents a non-traditional approach to theory of turbulence. Its objective is to prove that Newtonian mechanics is fully equipped for description of turbulent motions without help of experimentally obtained closures. Turbulence is one of the most fundamental problems in theoretical physics that is still unsolved. The term “unsolved “ here means that turbulence cannot be properly formulated, i.e. reduced to standard mathematical procedure such as solving differential equations. In other words, it is not just a computational problem: prior to computations, a consistent mathematical model must be found. Although applicability of the Navier-Stokes equations as a model for fluid mechanics is not in question, the instability of their solutions for flows with supercritical Reynolds numbers raises a more general question: is Newtonian mechanics complete? The problem of turbulence (stressed later by the discovery of chaos) demonstrated that the Newton’s world is far more complex than those represented by classical models. It appears that the Lagrangian or Hamiltonian formulations do not suggest any tools for treating postinstability motions, and this is a major flaw of the classical approach to Newtonian mechanics. The explanation of that limitation is proposed in this paper: the classical formalism based upon the Newton’s laws exploits additional mathematical restrictions (such as space–time differentiability, and the Lipchitz conditions) that are not required by the Newton’s laws. The only purpose for these restrictions is to apply a powerful technique of classical mathematical analysis. However, in many cases such restrictions are incompatible with physical reality, and the most obvious case of such incompatibility is the Euler’s model of inviscid fluid in which absence of shear stresses are not compensated by a release of additional degrees of freedom as required by the principles of mechanics. It has been recently demonstrated, [3], that according to the principle of release of constraints, absence of shear stresses in the Euler equations must be compensated by additional degrees of freedom, and that led to a Reynolds-type enlarged Euler equations (EE equations) with a doublevalued velocity field that do not require any closures. In the first part of the paper, the theory is applied to turbulent mixing and illustrated by propagation of mixing zone triggered by a tangential jump of velocity. A comparison of the proposed solution with the Prandtl’s solution is performed and discussed. In the second part of the paper, a semi-viscous version of the Navier-Stokes equations is introduced. The model does not require any closures since the number of equations is equal to the number of unknowns.