Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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The thread: One number decides the regime — page 3

Page 3 of 34, continuing through the 305 essays this motif runs through.

305 essays carry this thread — page 3 of 34.

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. Compressible flow

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.

 The chart, with one exact line on it. The friction factor of a pipe against Reynolds number, for five relative roughnesses. Every curve here except one is Colebrook's correlation, solved by iteration rather than read off a chart. The exception is the short straight line at the left: f = 64/Re is the laminar solution and it is exact. The curves flatten to the right because once the roughness pokes out of the viscous layer the Reynolds number has nothing left to change. Fluids at work

The roughness a wall cannot feel

A rough pipe and a polished one carry the same flow for the same pressure over three decades of Reynolds number, and then suddenly they do not. What changed is not the pipe. It is the thickness of the film of fluid at the wall, which is the only part of the flow that can see the roughness at all.

 The acceleration field of a steady flow. How hard the fluid is being accelerated at each point of a steady flow past a cylinder. The flow does not change with time anywhere in this picture, and yet almost nowhere in it is a parcel travelling at constant velocity — the pattern stands still while the fluid running through it is thrown about. Flows and fields

Steady does not mean nothing is happening

Photograph the flow past a cylinder twice and the two pictures are identical. Every parcel of air in them is being thrown about — braked to a dead stop, hauled round the shoulder at nearly twice the free-stream speed, braked again. Both statements are exactly true.

 Flow net — a stream past a cylinder. Two families of curves drawn over the same flow: the streamlines, along which the streamfunction is constant, and the equipotentials, along which the velocity potential is constant. They cross at right angles at every point, because they are the two parts of a single analytic function of position. Ideal flow

One function instead of two

A velocity field carries two numbers at every point and must satisfy two constraints. Both constraints can be solved once and for all by writing the whole flow as a single scalar function — and the two families of curves that function generates cross at right angles everywhere, for reasons that have nothing to do with fluids.

 Reynolds number: one number, four different flows. Reynolds number is inertia ÷ viscosity. It is not a property of the fluid or of the shape but of the combination, and crossing a threshold changes the physics rather than the magnitude. Regimes and numbers

The model that cannot be matched

A scale model behaves like the real thing when its dimensionless numbers agree. With one number that is a matter of choosing the tunnel speed. With two it is usually impossible, and every wind-tunnel result ever published has been obtained in spite of that.

 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. Compressible flow

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.

 A wing's two drags, and the lift at which they are equal. Friction drag and induced drag plotted against lift coefficient, with their sum above them. Friction is flat, because a surface costs the same whatever the wing is doing; induced drag rises as the square of the lift. The total is least where the two are equal, and that is also the point of best glide. Viscosity

The two drags a wing pays

A wing pays for having a surface, and it pays for making lift with a finite span. One of those bills falls as it flies faster and the other rises, so there is a speed at which the total is least — and the condition for it turns out to be that the two are equal.

 The dynamic pressure is not ½ρU², and by Mach 0.85 it is out by a fifth. The pressure difference a pitot tube measures, divided by the incompressible dynamic pressure ½ρU², against Mach number. The dashed curve is the two-term series 1 + Ma²/4 + Ma⁴/40 that the rule of thumb comes from. An airspeed inferred from ½ρU² alone reads high, and the error is entirely predictable — which is why it is corrected rather than tolerated. What is taught wrongly

What the airspeed indicator believes

A pitot tube measures the difference between two pressures, correctly, at every speed. Everything wrong with an airspeed reading is in the arithmetic applied to that difference — and the error is 2.3 per cent at Mach 0.3 and 19.4 per cent at Mach 0.85.

 The wall slope against pressure gradient, and where it runs out. How steeply the flow leaves the wall, plotted against the pressure gradient the layer is running into. A favourable gradient presses the profile against the surface and steepens it; an adverse one hollows it out. The curve reaches zero at a definite value, and beyond that there is no attached solution at all. Viscosity

How much uphill a layer can take

A boundary layer running into rising pressure is climbing a hill on the last of its momentum. There is a definite steepness at which it can no longer do it, and the number is not a rule of thumb — it is where a family of solutions stops existing.

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