Book Description: This book presents a non-traditional approach to the theory of turbulence. Its objective is to prove that Newtonian mechanics is fully equipped for the description of turbulent motions without the 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?
This book is about formulation rather than computation of turbulence. It is inspired by the conceptual differences between mathematicians and physicists in their views on the origin of instability in mechanics.
The main message of this book is that stability of solutions of the Navier-Stokes equations is not a physical invariant: it depends upon the class of functions in which the solution is sought, upon the frame of reference in which the equations are formulated, etc.
The fundamental departure from the established views on turbulence in this book is the removal of instability of solutions of the Navier-Stokes equations via introduction of a multivalued velocity field. The central role in the implementation of this removal is played by the Stabilization Principle that represents a bridge between stable and unstable models. Another fundamental departure from classical views is relaxing the Lipchitz condition that reveals the origin of randomness in "fully deterministic" Navier-Stokes equations.
Finally, the idea of a forced stabilization implemented by a feedback from the Liouville equation provides a computational strategy for numerical solution of the Navier-Stokes equations, as well as for solution of chaos with application to Lagrangian turbulence and n-body problem.
The most practical achievement of this book is formulation of the closure in turbulence without an experimentally based hypothesis, that has been the last unsolved problem in classical physics. (Imprint: Novinka) |