Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Concept

Newtonian impact — where it appears

Newton's model of a fluid as a stream of particles that lose their normal momentum on striking a surface and slide along it. It gives a pressure coefficient of twice the squared sine of the incidence: close at hypersonic speed on windward faces, and zero on shaded ones.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

The lift curve that made flight look impossible. Lift coefficient against incidence, by Newton's impact theory and by thin-aerofoil theory. One is quadratic in the angle and the other linear, so at small incidence — which is where aircraft fly — they differ by more than an order of magnitude. Newton's version says a wing large enough to carry a man would need an engine nobody could build, and for a century that arithmetic was taken as settling the question.

The theory that forbade flight

Newton treated air as a hail of particles that give up their normal momentum on impact, and got a lift coefficient of 2sin²α cos α. At five degrees that is a thirty-sixth of what a wing actually makes, and the quadratic is why powered flight looked arithmetically impossible for two centuries. The same formula is exact at Mach twenty.

misconceptions · Newtonian The shaded face's share of the force is set by K, and it is not small until K is. The fraction of a flat plate's normal force carried by its leeward face against K = M sin α, at Mach 3, 5, 10, 20, with the hypersonic small-disturbance value and the share the leeward face would have at vacuum (dashed). Newtonian theory puts it at zero. The curves collapse on K: the small-disturbance share is 35.7 per cent at K = 0.5, 24.2 at 1, 11.2 at 2 and 3.4 at 4, and the vacuum bound is 28.8, 11.4 and 3.4 per cent at 1, 2 and 4. The zero is a good approximation only where K is large — which is also the only place the Newtonian windward pressure is itself accurate.

The face Newton left in shadow

Newtonian theory gives a surface turned away from the stream a pressure coefficient of exactly zero, and at hypersonic speed the rest of the theory is nearly right. The shaded face is not. Computed exactly on a flat plate, its share of the force depends on the similarity parameter K = M sin α rather than on the Mach number, it is a quarter of the force at K = 1, and it moves a hypersonic plate's best lift-to-drag ratio from 5 to 7 at Mach 10.

misconceptions · Newtonian The worst jet amplifies the stagnation pressure by about the Mach number. The largest amplification of the stagnation pressure, over every incident turn, against the free-stream Mach number, on a logarithmic axis: the type IV jet, the best single turning shock followed by a normal shock, and the lossless ceiling. The jet's peak runs close to the line equal to the Mach number itself, from 3.5 at Mach 4 to 12.4 at Mach 12. The ceiling grows as the Mach number to the power of three and a half and is never approached. The estimate with one turning shock falls further behind the jet as the Mach number rises.

The spot a local theory cannot see

Newtonian theory gives every panel of a hypersonic vehicle a pressure set by its own angle to the stream, and no panel more than the stagnation pressure behind a normal shock. Let a shock from one part cross the bow shock of another and a supersonic jet forms that reaches the surface through weaker shocks. At Mach 8 it stagnates at 8.6 times the ceiling — and the worst amplification at every Mach number is close to the Mach number itself.

misconceptions · Newtonian Behind a strong shock the layer is an exact rotational flow, squeezed into a tenth of the shock's radius. Streamlines of the constant-density shock layer between a spherical shock (radius 1) and a sphere at density ratio ε = 0.167, from the solution ψ = sin²χ(ar⁴ + br² + c/r): the body sits at 0.8944 of the shock's radius, a standoff of 0.118 body radii. The gas enters along the shock, turns, and runs round the body in a layer whose vorticity over r sin χ is the same everywhere.

The shock layer is Hill's vortex, and it heats the nose a tenth harder than Newton says

Behind a strong bow shock the gas is squeezed six-fold or more and turned round the nose in a thin layer, and if its density is taken as constant there the layer has an exact solution: the same family of rotational flows as Hill's spherical vortex, fitted to the shock's three conditions. It stands off further than a mass balance says, and its gas accelerates away from the stagnation point a fifth faster than Newton's pressure law implies. Since a nose's heating goes as the square root of that acceleration, the layer heats the nose about ten per cent harder than the estimate every first calculation of re-entry uses.

compressible · Shock layer

Named alongside it

The objects these essays reach for when they reach for this one.

HypersonicMisconceptionOblique shockPressure coefficientAerodynamic heatingLift coefficientModel limitStagnation pointCrocco theoremDragExact solutionInterference

All concepts