Fluid Dynamics

Before the friction term existed, Jean le Rond d'Alembert had already derived something impossible. Solve the ideal-flow equations around a body moving at constant speed, and the pressure distribution comes out symmetric: whatever pushes back on the front pushes forward on the rear by exactly as much. The net force is zero. An ideal fluid offers no resistance.

Every rower, swimmer and sailor knew otherwise. D'Alembert said plainly that he could not explain it and left it "to geometers". The gap between theory and observation was not small — the theory predicted zero — and it split the subject in two for a century and a half. Mathematicians developed an elegant theory of ideal flow that engineers could not use; engineers compiled empirical tables of resistance with no theory behind them. Water, in the British physicist Horace Lamb's later account of the situation, was studied by one group and hydraulics by another, and the two did not meet.

Ludwig Prandtl closed it in 1904, in eight pages, with an argument about where an approximation fails. Viscosity is small for air and water, in the sense that the Reynolds number is large, so throwing the viscous term away looks safe. But the no-slip condition forces the velocity to zero at the surface while the outer flow is moving at full speed, which means the velocity gradient near the wall is enormous — and the viscous stress is proportional to that gradient. In a thin layer, viscosity is never negligible, however large the Reynolds number is. The layer gets thinner as $\mathrm{Re}$ grows, and the stress in it does not vanish correspondingly.

Three consequences follow, and they are the whole of applied aerodynamics. The layer carries the skin-friction drag. The layer can separate from the surface when pressure rises downstream, at which point a large wake forms and the drag rises by an order of magnitude — the difference between a streamlined and a blunt body. And the way the layer leaves a wing's trailing edge fixes the circulation around the wing, which by the Kutta–Joukowski theorem fixes the lift.

Non-dimensionalise the Navier–Stokes equations for a flow of speed $U$ over an object of size $L$ in a fluid of kinematic viscosity $\nu$, and exactly one parameter survives:

$$\mathrm{Re} = \frac{UL}{\nu} = \frac{\text{inertial forces}}{\text{viscous forces}}.$$

Two flows with the same $\mathrm{Re}$ are the same flow, scaled. That is why a model tells you anything about an aircraft, and it is why a single number organises a subject that spans bacteria and hurricanes.

A car on a motorway. $U = 30$ m/s, $L = 4$ m, and for air $\nu = 1.5\times10^{-5}$ m²/s:

$$\mathrm{Re} = \frac{30 \times 4}{1.5\times10^{-5}} = 8\times10^{6}.$$

Prandtl's estimate puts the boundary layer thickness at roughly $5L/\sqrt{\mathrm{Re}}$:

$$\delta \approx \frac{5 \times 4}{\sqrt{8\times10^{6}}} = \frac{20}{2830} = 7 \text{ mm}.$$

Seven millimetres out of four metres. All of the friction drag, and the decision whether the flow separates behind the rear window, happens in that 0.2% of the length scale. This is why the inviscid theory can be simultaneously correct about the pressure field and useless about the force.

A swimming bacterium. $U = 30$ µm/s, $L = 1$ µm, and for water $\nu = 10^{-6}$ m²/s:

$$\mathrm{Re} = \frac{(3\times10^{-5})(10^{-6})}{10^{-6}} = 3\times10^{-5}.$$

Eleven orders of magnitude below the car. Inertia does not merely matter less; it does not exist. Ask how far such an organism coasts if it stops swimming. Its mass is about

$$m = \tfrac{4}{3}\pi(0.5\times10^{-6})^{3} \times 1000 = 5.2\times10^{-16} \text{ kg},$$

and Stokes drag gives a friction coefficient $6\pi\mu a = 6\pi(10^{-3})(0.5\times10^{-6}) = 9.4\times10^{-9}$ kg/s. The coasting distance is $mU$ divided by that coefficient:

$$\frac{(5.2\times10^{-16})(3\times10^{-5})}{9.4\times10^{-9}} \approx 2\times10^{-12} \text{ m},$$

two picometres — a fiftieth of the width of an atom. A bacterium that stops beating its flagellum stops instantly and absolutely. It cannot glide, cannot throw anything, and cannot swim by any stroke that is merely reversed on the return, because at $\mathrm{Re} \to 0$ the equations are time-reversible and a reciprocal motion returns the organism exactly where it began. The corkscrew flagellum and the breaststroke-with-a-twist of a flagellate are solutions to a constraint that has no analogue at human scale.

Between the two lie everything else: $\mathrm{Re} \approx 10^{2}$ for a swimming tadpole, $10^{4}$ for a sparrow, $10^{8}$ for a whale, $10^{9}$ for a weather system. The transition from the viscous regime to the inertial one is not a smooth loss of accuracy. Around $\mathrm{Re} \approx 2000$ in a pipe, steady flow stops being stable at all, and the subject changes into turbulence.

The foundations of this field have been secure for 180 years, which makes its unsolved problems unusual in character. Nobody doubts the Navier–Stokes equations; the question is whether they have solutions. For smooth initial data in three dimensions it has never been proved that the velocity stays finite — the nonlinear term pumps energy into ever smaller scales, viscosity removes it there, and whether dissipation always wins is open and carries a million-dollar prize. Two-dimensional flow is proved well behaved, which is of no comfort, since the world is not two-dimensional and the mechanism in question exists only in three.

This is an odd situation for an engineering science. Aircraft are certified using numerical solutions of equations that may, for all anyone can prove, develop infinities. In practice the computations agree with wind tunnels, and the agreement is the justification. How those computations are done, and why the cost grows so steeply with Reynolds number, belongs to numerical methods for partial differential equations and to the next field in this thread.