Navier-Stokes equations

The traditional model of fluids used in physics is based on a set of partial differential equations known as the Navier-Stokes equations. These equations were originally derived in the 1840s on the basis of conservation laws and first-order approximations. But if one assumes sufficient randomness in microscopic molecular processes they can also be derived from molecular dynamics, as done in the early 1900s, as well as from cellular automata of the kind shown on page 378, as I did in 1985 (see below). For very low Reynolds numbers and simple geometries, it is often possible to find explicit formulas for solutions to the Navier-Stokes equations. But even in the regime of flow where regular arrays of eddies are produced, analytical methods have never yielded complete explicit solutions. In this regime, however, numerical approximations are fairly easy to find. Since about the 1960s computers have been powerful enough to allow computations at least nominally to be extended to considerably higher Reynolds numbers. And indeed it has become increasingly common to see numerical results given far into the turbulent regime—leading sometimes to the assumption that turbulence has somehow been derived from the Navier-Stokes equations. But just what such numerical results actually have to do with detailed solutions to the Navier-Stokes equations is not clear. For in particular it ends up being almost impossible to distinguish whatever genuine instability and apparent randomness may be implied by the Navier-Stokes equations from artifacts that get introduced through the discretization procedure used in solving the equations on a computer. One of the key advantages of my cellular automaton approach to fluids is precisely that it does not require any such approximations.

At a mathematical level analysis of the Navier-Stokes has never established the formal uniqueness and existence of solutions. Indeed, there is even some evidence that singularities might almost inevitably form, which would imply a breakdown of the equations, and perhaps a need to account for underlying molecular processes.

In turbulent flow at higher Reynolds numbers there begin to be eddies with a wide range of sizes. And to capture all these eddies in a computation eventually involves prohibitively large amounts of information. In practice, therefore, semi-empirical models of turbulence tend to be used—often "eddy viscosities"—with no direct relation to the Navier-Stokes equations. In airflow past an airplane there is however typically only a one-inch layer on each surface where such issues are important; the large-scale features of the remainder of the flow, which nevertheless accounts for only about half the drag on the airplane, can usually be studied without reference to turbulence.

The Navier-Stokes equations assume that all speeds are small compared to the speed of sound—and thus that the Mach number giving the ratio of these speeds is much less than one. In essentially all practical situations, Mach numbers close to one occur only at extremely high Reynolds numbers—where turbulence in any case would make it impossible to work out the detailed consequences of the Navier-Stokes equations. Nevertheless, in the case of cellular automaton fluids, I was able in 1985 to work out the rather complicated next order corrections to the Navier-Stokes equations.

Above the speed of sound, fluids form shocks where density or velocity change over very small distances (see below). And by Mach 4 or so, shocks are typically so sharp that changes occur in less than the distance between molecular collisions—making it essential to go beyond the continuum fluid approximation, and account for molecular effects.