The Myth of the Continuum

Predicted and observed tracks of Hurricane Matthew (John Droz).

A turbulent region of a fluid is a region which is less ordered than streamline flow, i.e. where the Boltzmann entropy density is higher than for streamline flow.  The stochastic character of turbulence cannot be represented by the Navier–Stokes equations and their finite difference approximations. Numerical models based on them are unstable and eddy viscosity must be set high everywhere in order to suppress turbulence. In the real world eddy viscosity is only non-zero where turbulence already exists. Like Maxwell’s equations, the Navier–Stokes equations fail at high energy densities because they are Newtonian – they are partial differential equations and, as such, describe a mythical continuum which is deterministic, continuous and differentiable. They do not allow for the granularity and stochastic behaviour of the systems they are supposed to describe. They do not accommodate the atomic theory nor the granularity of action space. Like Planck’s formula for  black-body radiation, Kolmogorov’s turbulence spectrum is based on probabilistic assumptions, not on Navier-Stokes.

Models fail because no real fluid is a continuum.