Cascade Representations for the Navier–Stokes Equations

Abstract

The basic equations governing the motion of a fluid are well understood. For simplicity, we shall refer to the case of an incompressible, constant density, viscous Newtonian fluid; the velocity vector field $u\left (t,x\right )$ and pressure scalar field $p\left (t,x\right )$ satisfy the classical Navier–Stokes equations (in dimension 3) with viscosity ν > 0

$$\displaystyle{ \begin{array}{rl} \partial _{t}u + (u \cdot \nabla )u + \nabla p& =\nu \varDelta u, \\ \mathop{\mathrm{div}}\nolimits u& = 0,\end{array} }$$

(15.1)

with appropriate initial and boundary conditions depending on the problem. For relatively simple fluid motions, these equations give us a very good tool for simulations and physical understanding. But there are complex fluid motions, those usually called turbulent, where special features are experimentally or numerically observed which do not have a clear explanation yet from the Navier–Stokes equations. In a sense, there is something at the foundation of fluid dynamics that is still unclear. For later reference, let us mention that this happens when a certain parameter R, called Reynolds number, is very large.