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

Momentum — where it appears

Mass times velocity, whose rate of change is a force and whose flux through a surface is one too. Nearly every force in this collection is computed by counting it across a control volume rather than by integrating a stress over a body.

Named by 11 essays across 4 fields — each of them below, with the objects they name alongside it.

Ideal flow past a cylinder. A uniform stream past a circular cylinder in a fluid with no viscosity. The solution is exact and closed-form: streamlines part at a stagnation point, run round the surface and close up perfectly behind, and the pressure recovers to exactly what it was in front.

Fast means low pressure

The trade between speed and pressure is the most useful relation in the subject and the most misused. Where it comes from, what it costs, and why the pressure over a wing is negative almost everywhere.

inviscid · Pressure The speed for least drag is not the speed for least power. Drag and power against airspeed, in units of the speed at which drag is least. Power is drag times speed, so its minimum sits slower — at the fourth root of a third of the least-drag speed, which is 0.76 of it. Flying for range and flying for endurance are therefore different speeds, and the difference is not a rule of thumb.

The cheapest way to stay up

There is a speed at which an aircraft's drag is least, and a different, slower speed at which its power is least. The ratio between them is the fourth root of a third — a number that does not depend on the aircraft, the altitude, or anything else about the flight.

viscous · Drag budget The fluid a moving cylinder carries with it. Kinetic energy density around a cylinder moving through fluid that is at rest far away. The fluid is not dragged along in a lump: it is pushed aside in front and closes in behind, and the energy in that motion is what has to be supplied to change the body's speed.

The force of getting going

The exact theory says a body moving steadily through an ideal fluid feels no force at all. It does not say the fluid is free. Accelerating the body has to accelerate the fluid too, and the bill for that is exactly the mass of fluid the body displaces.

inviscid · Dalembert A steady pressure field, from a flow with no steady part. The time-averaged pressure round a cylinder in a stream that oscillates as U₀cos ωt. The mean velocity is exactly zero at every point — the flow spends as long going one way as the other — and the mean pressure is not, because pressure depends on the square of the speed and a square has no sign. The mean coefficient reaches -2.00 at the shoulders and averages -1.00 over the surface, and its resultant is 6.6e-16: a real field with no force in it. The pale lines are the instantaneous streamlines, which reverse every half cycle.

The mean is not the flow

Average an unsteady flow and the result is a new object with its own properties, and it is not a solution of anything. An inviscid stream oscillating about zero has a mean velocity of exactly nothing everywhere, a mean pressure that reaches minus two dynamic pressures at the shoulders, and a missing term in its own momentum equation that can be written down in closed form.

kinematics · Averaging The air a wing carries along, and how little of it there is. The Blasius profile, in the wing's frame, with the free stream at one. No slip says the air at the surface is at rest relative to the surface, which in the ground's frame means it is moving with the wing — but only exactly at the wall. The deficit, integrated across the layer, is the displacement thickness: at a Reynolds number of 1e+6 and a metre of chord it is 1.72 millimetres of air moving at flight speed, which is the whole of what is 'carried'. The step drawn on the axis is that same deficit as a solid slab. A wing does not drag a blanket of air with it; it leaves a boundary layer behind it, and the layer is made of air that keeps being replaced.

The air a wing does not carry

No slip says the air touching a surface moves with it, and the usual reading is that a wing drags a blanket of air along. The blanket is 1.7 millimetres thick per metre of chord, it is different air every instant, and the drag it costs falls as it gets thicker.

misconceptions · The no-slip condition The current at the surface is 45° from the wind, and nothing sets that angle. The Ekman spiral drawn as a hodograph: each point is the velocity at one depth, and depth runs along the curve. At the surface the flow is at exactly 45 degrees to the wind that drives it — not approximately, exactly, and independently of the wind, the viscosity and the latitude. By one Ekman depth the flow has turned another radian and lost 1/e of its speed; by three it is a hundredth of the surface value and pointing back the way it came. The angle is a property of the equation having two terms in it, and nothing else.

The layer that stops at a depth

Every other boundary layer grows. This one does not — rotation supplies a frequency, the balance against diffusion supplies a length, and the transport that comes out contains the stress on the surface and not the viscosity underneath it.

viscous · Rotating The momentum of the fluid, against the shape of the region it is added up over. Momentum of the fluid around a cylinder moving through it, divided by the body's hydrodynamic impulse, against the aspect ratio of the rectangle the integral was taken over. Every rectangle has the same area and contains the same body. A tall region gives minus the impulse, a long one gives plus it, a square gives exactly zero, and the limit of a large region is whichever of those the region was shaped like. The momentum of an unbounded ideal flow is not a number.

The momentum with no value

A cylinder is pushed from rest to a steady speed. Work was done, energy went into the fluid, something was pushed. How much momentum does the fluid carry? The integral converges, the answer is finite, and it is a different finite number for every shape of region it is summed over.

inviscid · Impulse A tube opened at the foot of a reservoir. Speed in a two-metre tube fed by a one-metre head, from the instant it is opened. The quasi-steady answer — Torricelli's √(2gh), which is what dropping the time derivative from Bernoulli's equation gives — is reached at the instant of opening, before any fluid has moved. The true answer is a hyperbolic tangent with a time constant of 2L/√(2gh), and it takes 2.4 seconds to come within one per cent. The whole of that transient is one term.

The pressure that depends on the past

Bernoulli's equation for an unsteady flow has a term nobody writes down and a right-hand side that is a function of time rather than a constant. The term is exactly zero once a flow has started and is the whole of the flow while it is starting, which is why it never appears in an answer and is never negligible in getting to one.

inviscid · Unsteady bernoulli Two integrands, and the wrong one claims 9.58 per cent more drag. The two things that get integrated across a wake, each scaled to its own peak so the shapes can be compared. The momentum integrand u(U − u)/U² is the drag; the mass integrand (U − u)/U is the displacement thickness, and it is not a drag at all. They differ by a factor of u/U inside them, so the mass one is fatter wherever the deficit is deep — and its integral here is 1.1 times the momentum one's. That ratio is decided by how deep the wake is rather than by how wide: at a twentieth of this momentum thickness it falls to 1, and at twice it rises to 1.25. The error is smallest exactly where a survey is properly done, far downstream where the wake has spread and shallowed.

Weighing what is missing

A control volume drawn round a wing gets the lift out of it, on a flow that was solved exactly. The wake survey asks the same box for the drag on a flow nobody has solved, and it is how a real aerofoil's drag is known — with three assumptions, all of which are checkable, and one integral standing next to it that is wrong.

misconceptions · Momentum lift A body of no volume with a finite added mass. Thin an ellipse towards a plate and the fluid it displaces goes to nothing while its broadside added mass does not move at all — it stays at πρa² to the last digit. Whatever added mass measures, it is not how much fluid a body carries with it: the ratio of the two diverges as the reciprocal of the thickness, reaching a million at a thickness ratio of a millionth.

The mass a body has to borrow

Accelerate a sphere through water and it resists as though it were half again as heavy. The half is exact, it is a rational number rather than a measurement, and almost everything a reader infers from it about carried fluid is false.

inviscid · Impulse The trailing sheet rolls up into two vortices, and nothing it carries is lost. The trailing vortex sheet behind an elliptically loaded wing, seen in a plane across the wake, at times 0, 0.05, 0.2, 0.6 in units of b²/Γ₀, represented by 160 point vortices with a smoothing length of 0.03 of the span. The tips curl up first and the sheet winds into two concentrated vortices while the whole system sinks under its own induced velocity; by t = 0.6 the pair's centroid has descended 0.122 of the span. Through all of it the crossflow energy — the induced drag — and the separation of the two halves' centroids, 0.7854 of the span, stay exactly what they were.

The drag a wake keeps however it rolls up

A plane drawn across the wake of a finite wing contains its induced drag as the kinetic energy of the swirling crossflow. The trailing sheet then rolls up into two vortices, and the energy does not change at all — roll-up moves the drag around the plane without spending any of it. What does spend it is viscosity, which turns crossflow energy into a total-pressure defect, so a plane farther back reads less induced drag, more profile drag, and the same total.

misconceptions · Momentum lift

Named alongside it

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

Added massPotential flowMeasurementBernoulli's equationControl volumeDoubletDragKinetic energyModel limitBlasiusBoundary conditionBoundary layer

All concepts