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

Acoustic analogy — where it appears

Lighthill's rearrangement of the exact equations of motion into a wave equation for a still medium, with everything the flow does moved to the source side. It turns the sound of a flow into the sound of known sources, and its sources are quadrupoles unless a surface adds dipoles.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

Each cancellation costs two powers of the Mach number. Radiated power against compactness for three source clusters: a single monopole, two of opposite sign, and four on a square with alternating signs. The fitted slopes are 0.00, 2.00, 4.00 — zero, two and four in (kd), measured by integrating the far field over a sphere rather than assumed. A turbulent eddy turns over in about the time sound crosses it, so kd is of order the Mach number, and those exponents become the fourth, sixth and eighth powers of speed. A flow with no moving surfaces has no monopole and no dipole available to it, which is Lighthill's whole argument, and the eighth power is what is left.

The sound that only leaves

A flow is a catastrophically bad radiator, and the reason is that it has no monopole and no dipole available to it. What is left is the eighth power of speed — and the equation is equally happy with sound converging on a jet, which is ruled out by a condition imposed at infinity.

compressible · Aeroacoustics The body's dipole outshouts the flow's quadrupoles by one over the Mach number squared. The power of a wire's lift tone over the generous bound on its wake's quadrupole noise, against the wind's Mach number, for rms lift coefficients 0.2, 0.45 and 0.6 — the spread of the measurements. At Mach 0.03, a 10 m/s wind, the ratio is 9.4·10⁴; it falls as one over the Mach number squared and is still 104 at Mach 0.9, where a real cylinder's shedding has long since changed.

A wire sings with the force on it, not the flow round it

A free flow can only radiate as a quadrupole, four sources nearly cancelling, which is why a jet's noise grows as the eighth power of its speed and so little of its power becomes sound. Put a body in the flow and the cancellation is broken: the fluctuating force the body exerts is a dipole, its sound grows as the sixth power, and at the speed of any wind it outshouts the turbulence that made it by a factor of tens of thousands. The whistle of a wire is the force of its own wake, radiated.

compressible · Aeroacoustics A steady force carried round a circle sings, and each harmonic rises steeply with the tip's Mach number. The loudest level, 30 m away, of the first four blade-passing harmonics of a two-bladed 1.88 m propeller giving 2 kN of thrust from 127 kW, against its tip Mach number. At a tip Mach number of 0.4 the fundamental is 89.8 dB and the fourth harmonic 51; at 0.78, the climb, 98.2 and 88.6; at 0.95, 101 and 98.9.

More blades hush a propeller only while its tips are slow

A propeller's thrust and torque are steady in the blade's own frame, and a steady force held still makes no sound. Carried round a circle it sings at the blade-passing frequency, and Gutin's formula says how loudly: through a Bessel function whose order is the harmonic's and whose argument is that order times the tip's speed towards the listener over the speed of sound. Such a function is exponentially small until its argument nears its order. So spreading the same thrust over more blades, which raises the order of the fundamental, silences it by twenty-five decibels at a tip Mach number of 0.4 and by one at 0.95.

compressible · Aeroacoustics

Named alongside it

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

AeroacousticsMultipoleCorrelationDopplerFar fieldNoiseScalingSound powerSpeed of soundBoundary conditionCancellationCompactness

All concepts