Navier-Stokes Equations and Turbulence

Abstract

This book aims to bridge the gap between practising mathematicians and the practitioners of turbulence theory. It presents the mathematical theory of turbulence to engineers and physicists, and the physical theory of turbulence to mathematicians. The book is the result of many years of research by the authors to analyse turbulence using Sobolev spaces and functional analysis. In this way the authors have recovered parts of the conventional theory of turbulence, deriving rigorously from the Navier–Stokes equations what had been arrived at earlier by phenomenological arguments. The mathematical technicalities are kept to a minimum within the book, enabling the language to be at a level understood by a broad audience. Each chapter is accompanied by appendices giving full details of the mathematical proofs and subtleties. This unique presentation should ensure a volume of interest to mathematicians, engineers and physicists.

Inertial manifolds and finite-dimensional reduction for dissipative PDEs

“…In particular, even the strange word inertial in the title of inertial manifold (which replaces/duplicates more precise and more natural notion of a center manifold) comes from the Navier-Stokes equations, namely, it is related with the so-called inertial term in the equations as well as the associated inertial scale in the empiric theory of turbulence, see [20,19] and reference therein. However, despite many attempts there are still no progress in finding the inertial manifolds for that problem or proving their non-existence, so the problem remains completely open.…”

When does eddy viscosity damp subfilter scales sufficiently?

“…2-3. From that we conclude that the eddy viscosity is to be taken such that (12) and (17) are satisfied. The essential difference between these two equalities is the sign of the transport term: if we would simply disregard the difference between δv with ω and name them both φ, then (12) and (17) become…”

“…2, we have determined the eddy viscosity such that the energy that is transferred from the large eddies (scales of size ≥ ) to the subfilter scales is dissipated so fast that the production of subfilter scales by the nonlinear mechanism in the left-hand side of (3) becomes dynamically irrelevant. Condition (12) ensures that the transfer of energy from the large eddies to the subfiler scales is balanced properly by the eddy dissipation. This condition is necessary, but not sufficient, to limit the dynamics governed by (3) to scales of size ≥ .…”

“…Notice that a number of terms vanish here, because the convective operator is skew symmetric, i.e., (v · ∇) * = −(v · ∇) [12]. The term δv · S(v)δv dx represents the energy transfer from v to δv.…”

“…where the plus sign corresponds to (12) and the minus sign to (17). From a physical point of view the plus sign represents the requirement that the forward cascade of energy stops at the scale set by the filter (that is, the production of smaller scales stops at the filter scale); the minus sign expresses that there is no backward cascade too (that is, scales of size < cannot become dynamically relevant, since any subfilter perturbations decay exponentially fast).…”