The Energy Operator and New Scaling Relations for the Incompressible Navier--Stokes Equations

Abstract

The incompressible Navier--Stokes equations have been widely studied due to their vast ability to model hydrodynamic phenomena. The nonlinear character of these equations produces a hierarchical system of dynamical scales that offer challenging barriers to computation simulation and scale elimination modeling. Here, the incompressible Navier--Stokes equations are decomposed and studied under the context of global eigensystems that have a direct relation to the dynamical character of the scales-of-motion. In particular, the self-adjoint linear Reynolds--Orr operator is shown to correspond to the energy operator of this nonlinear system of equations and has the distinguishing property of having an eigensystem that orders the time derivative of the integral kinetic energy for the nonlinear Navier--Stokes operator. The eigensystem of the Reynolds--Orr operator is shown to yield a dynamical range of scales that can be characterized by an energy-containing range, an inertial range, and a dissipation range. It is shown that a canonical decomposition of the incompressible Navier--Stokes equations has only a finite number of scales that are responsible for the unsteady character of the nonlinear system. Furthermore, a time-dependent growth of the kinetic energy is characterized without the need for a classical linearized instability. Discussions are presented and estimates are made on the various underlying scaling characteristics, and a contrast is made to empirical eigenfunctions found by ordering the integral kinetic energy.