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

Every essay — page 9

Page 9 of 39, continuing through the fields in the same order.

Flows and fields Ideal flow Circulation and lift Viscosity Regimes and numbers Compressible flow Transition and turbulence Fluids at work What is taught wrongly Series Concepts Regimes Refutations Search

Circulation and lift

Where lift actually comes from. The Kutta condition, the Joukowski aerofoil, and lift derived rather than asserted.

One incidence, two lifts. Lift coefficient against incidence for a wing pitched sinusoidally through the stall, with the static curve for comparison. The loop is traversed anticlockwise: at twelve degrees the wing carries 0.22 more lift going up than coming down.

Two lifts at one incidence

A wing pitched up and down through the stall does not retrace its own lift curve. At twelve degrees it carries 0.22 more lift going up than coming down, and the loop that opens between the two is the work the airstream does on it — which is where the energy for a stall flutter comes from.

8 figures What is left behind, and what it is made of. The two rolled-up vortices behind a large aircraft, drawn to scale against its span. They sit at 78.5 per cent of the span apart, each carries 508 square metres a second of circulation, and the pair descends at 1.72 metres a second under its own induction.

A wake that says what made it

Two vortices sitting behind an aeroplane carry its weight and its span in a form that can be read back out: circulation times separation times density times speed is the lift, exactly. The reading is exact for about a minute and worth nothing after four.

8 figures A blade passing the vortex it shed a passage ago. The lift the blade section feels as it passes a tip vortex at five per cent of the radius. The pulse is a doublet rather than a bump — upwash on one side and downwash on the other — so the blade is pushed one way and then the other in the width of a few chords.

A blade that flies through what it shed

A rotor blade meets the tip vortex the blade in front of it left, a fifth of a second earlier, at a few per cent of the radius. What it feels is a doublet — five and a half degrees of upwash and then five and a half of downwash within a few chords — and the peak goes as one over the miss distance.

8 figures What a downstream blade sees going past. The axial velocity a blade in the second row meets, over one revolution, as it passes through the wakes of thirty upstream blades. Each dip is one wake, and the blade meets all thirty of them every time it goes round.

A row that meets the row before it

A compressor blade is loaded and unloaded thirty times a revolution by geometry it does not have. The forcing sits at the blade count of the row in front of it and at multiples of that, and how far up the harmonics it reaches is decided by how far apart the two rows are.

8 figures The induced velocity, after a step in thrust. A rotor's induced velocity following a thirty per cent increase in thrust applied at fifty milliseconds. It does not jump: the air the disc has to accelerate has an apparent mass, and the response is a first-order climb to the new momentum-theory value.

The inflow that takes time to arrive

Momentum theory gives a rotor's induced velocity from its thrust, instantly. It does not arrive instantly: the air the disc has to accelerate has a mass, and the response is a first-order climb with a time constant of 33 milliseconds — a twentieth of the time the wake itself takes to convect a radius.

8 figures What every wing pays to bend its root less. Least induced drag against the root bending moment of the lift, both as fractions of the elliptic monoplane of the same span and lift, for the monoplane and for box wings with gaps of a tenth, a fifth and two-fifths of the span. Each curve is a parabola with its minimum at that wing's unconstrained optimum. The box wings' minima sit to the right of the monoplane's — they load their lift further out — and their parabolas are shallower: the gap-of-a-fifth box can bend its root 21 per cent less than the elliptic monoplane and still match its drag.

A lighter spar turns a box wing into a biplane

Prandtl's best wing system has two-thirds of a monoplane's induced drag at a gap of a fifth of the span, and part of that saving is carried by circulation turning the corner into its fins. Ask the box to bend its root less and it pays about half what a monoplane pays — but a biplane with no fins pays nearly as little, and by the time the spar is a fifth lighter the fins carry almost nothing and the box has become the biplane it was built from.

7 figures At a fixed impulse every orbit is a curve of constant energy. Axial separation of two coaxial rings against the radius of the ring that started smaller, for six pairs with the same total impulse, all starting in one plane. Pairs starting nearer equal (inner curves) trace closed loops: the separation swings from one sign to the other as the rings take turns, and the radius swings with it. Pairs starting further apart run off to one side and never return. The pale curves are the separatrix, the energy of two equal free rings — it passes through the plane of the start at a ratio of 0.340 and its arms reach out to infinite separation.

Two rings leapfrog only if they start alike

Two coaxial smoke rings passing through each other in turn is the most famous thing vortices do, and it is not what two rings do in general. Set them off from one plane with different radii and below a definite ratio the smaller one draws ahead and never comes back. Which of the two happens is decided before either ring moves, by whether two separate rings could hold the pair's energy.

7 figures

Compressible flow

What changes when density stops being a constant. Shocks, expansion fans, nozzles that work backwards, and the one place on this site where an inviscid flow has drag.

A source at Mach 0.55, and the sound it has already made. Each circle is one pressure pulse, expanding at the speed of sound from the point the source was at when it left. Below the speed of sound every circle still contains the source, so the pressure disturbance reaches every point of the fluid before the source does. Above it the circles have an envelope, and outside that envelope nothing has been heard at all.

What a signal travels at

Air finds out that an aeroplane is coming, and it finds out at a definite speed. That speed does not depend on how hard the air is squeezed or on how much of it there is — only on how hot it is — and every result in compressible flow is downstream of that one fact.

6 figures A source at Mach 2.00, and the sound it has already made. Each circle is one pressure pulse, expanding at the speed of sound from the point the source was at when it left. Below the speed of sound every circle still contains the source, so the pressure disturbance reaches every point of the fluid before the source does. Above it the circles have an envelope, and outside that envelope nothing has been heard at all.

When the warning cannot arrive

Supersonic flow is not fast flow. It is flow in which the fluid ahead has been told nothing, because the body is outrunning its own pressure signals — and that single change turns an equation of one type into an equation of another.

6 figures The energy is all there the whole way: enthalpy spent to buy speed. At each Mach number, the fraction of the stagnation enthalpy still held as heat and the fraction converted into directed motion. The two always sum to the total, which is the compressible replacement for Bernoulli's equation — an energy statement rather than a pressure one. At Mach 2 more than four fifths of the heat has become speed.

Energy instead of pressure

Bernoulli's equation is a statement that pressure and speed trade against each other at fixed density. Take the fixed density away and the trade is between heat and speed instead — and what survives is not a pressure at all.

6 figures A duct does the opposite thing above Mach one. The four cases of dA/A = (Ma² − 1) dV/V. Below the speed of sound a narrowing duct accelerates the flow, which is what continuity leads anyone to expect. Above it the sign of the bracket flips, and a narrowing duct decelerates: density is falling faster than the speed is rising, so the stream tube needs more room rather than less.

The duct that works backwards

Squeeze a pipe and the flow speeds up. Everyone knows this, it follows from continuity, and above the speed of sound it is false — a narrowing duct decelerates a supersonic stream, because the density is falling faster than the speed is rising.

6 figures The throat stops listening: mass flow against back pressure. Mass flow through a convergent nozzle, normalised on its choked value, as the back pressure is lowered. It rises until the throat reaches Mach one and then stops, exactly. Below that pressure the throat is sonic and nothing downstream can send a signal upstream to ask for more, so the flow does not respond however far the back pressure falls.

The throat that stops listening

Lower the pressure downstream of a nozzle and more gas flows through it. Keep lowering it and, at a definite point, the flow stops responding — not gradually, but completely, because the news that the pressure has fallen can no longer travel upstream.

6 figures