Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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The thread: Taught wrongly, everywhere — page 8

Page 8 of 14, continuing through the 124 essays this motif runs through.

124 essays carry this thread — page 8 of 14.

Where a profile stops being a parabola. Two measures of how far Womersley's solution has left the quasi-steady parabola, against the Womersley number, both logarithmic. The amplitude deficit reaches a hundredth at α = 0.28 and the phase lag reaches a hundredth of a right angle at α = 0.22 — both well below α = 1, which is where the unsteady and viscous terms are equal and is the value the number is named for. By α = 1 itself the flow is already a tenth short and eight degrees late. Regimes and numbers

Where the parabola goes

A pipe carrying a steady flow has a parabolic profile, and the Womersley number is supposed to say when an oscillating one still does. For the flow rate it is very nearly honest — one per cent at α = 0.91 — and for the phase it is out by a factor of three, because a lag is second order in the number and an amplitude deficit is fourth.

 A cylinder in a uniform shear, K = 0.4. A stream whose velocity increases with height, meeting a circular cylinder. The oncoming profile is drawn at the left. The flow carries uniform vorticity −K, so it is a solution of Euler's equations and not of Laplace's, the pattern is no longer symmetric top to bottom, and the body feels a lift towards the fast side with no circulation anywhere. Ideal flow

Inviscid does not mean irrotational

Dropping viscosity gives Euler's equations. Assuming nothing is spinning gives Laplace's — one scalar, linear, unique. The second step is a separate hypothesis about the flow's history, and a flow that fails it is still an inviscid flow with exact solutions of its own.

 The chord and the tangent, which are the two speeds. The flux of a conserved quantity against its own density, for a wide river and for traffic. At any point the slope of the chord from the origin is the speed the material moves at, and the slope of the tangent is the speed a disturbance moves at. They are the same number only if the curve is a straight line through the origin. For the river the tangent is five-thirds of the chord at every depth; for traffic the tangent turns negative above half the jam density while the chord never does. Flows and fields

A wave nothing in it travels with

A flood crest moves at five-thirds the speed of the water it is made of, at every depth, whatever the roughness and whatever the slope. A traffic wave moves backwards through cars that are all going forwards. Neither result contains a momentum equation.

 The coefficient of the equation's second derivative, along a chord. The bracket multiplying the streamwise second derivative in the transonic small-disturbance equation, along a chord at Mach 0.85. Where it is positive the equation is elliptic and the flow is subsonic; where it is negative the equation is hyperbolic and the flow is supersonic. Which it is at a given point depends on the perturbation velocity there, which is the thing being solved for. Forty-two per cent of this chord is hyperbolic, and no amount of inspecting the problem beforehand could have said so. Compressible flow

The equation that changes type inside its own answer

Near Mach one the coefficient of the streamwise second derivative depends on the perturbation velocity, which is what is being solved for. Two solutions of the linear equation no longer add — the leftover is three times the term the linear theory keeps — and the critical Mach number approaches one as the two-thirds power of thickness.

 What the phase reaches, and what it does not. The difference between the two records, as a fraction, for five quantities. The variance and the autocorrelation are the same to machine precision because they are the spectrum. A narrow-band linear oscillator answers its own frequency and almost nothing else, so it is nearly phase-blind too. Everything extremal — the crest, the peak drag load, the range of the running integral — is not. What is taught wrongly

The same statistics, and a different load

A wind or wave specification is written as a spectrum, and a spectrum discards the phases. Two records built from one spectrum agree in variance to thirteen figures and in peak drag load by twenty per cent — and with the phases lined up, the same spectrum is a single impulse thirty-one times worse.

 Five numbers, one flow, one name. A hyperbolic-tangent shear layer with a matching density profile, measured five ways, each of which appears in the literature as "the Richardson number". The minimum gradient value is what Miles' and Howard's theorem is about and is the only one the quarter belongs to. The bulk numbers depend on which thickness and which velocity difference were used; the depth-averaged one depends on how far from the layer the measurement extended, and grows without limit as it extends further. Transition and turbulence

Five numbers, one name

The Richardson number has a threshold at a quarter, and the quarter belongs to one of the five quantities that go by the name. On a single tanh shear layer they run from J to seventeen J, and the largest of them grows without limit as the measurement is extended further from the layer.

 Two roots walking towards each other, and one of them crosses. The roots of the characteristic quartic in the complex plane as the airspeed is raised from nothing to 105 metres per second — growth rate across, frequency up. At rest the two sit on the imaginary axis at the uncoupled frequencies. As the speed rises, the aerodynamic coupling drags them towards each other in frequency while pushing one left and the other right, and at 80.8 metres per second the right-hand one crosses the axis. Everything about the failure is in this picture: the coalescence, the crossing, and the fact that the flutter frequency is neither of the two the structure started with. Circulation and lift

The shake that is not resonance

A steady airstream contains no oscillation at any frequency, so nothing is driving anything. What happens instead is that the aerodynamic forces couple two structural motions that were independent, drag their frequencies together, and turn one damping negative — and the wing takes the energy out of the air itself.

 The number that settles the argument. How many turns the earth's contribution gets through in the time the vessel takes to drain: the draining time divided by the rotation period at the drain. A bathtub manages a tenth of one and there is nothing to see. The apparatus that settled it manages nearly nineteen hundred, and that is the whole of what its enormous area-to-drain ratio was for. What is taught wrongly

The bath that was only ever a wait

The Coriolis force is far too weak to steer a draining bath, and it does steer a large enough tank that has been left alone long enough. Both are true, and the number that separates them is not a force ratio — it is how many turns the earth's contribution gets through before the vessel is empty.

 The one calculation in which the atmosphere really does push. How high a partial vacuum inside the tube will raise the liquid, against the absolute pressure achieved inside it. The lift is (p_atm − p_inside)/ρg and it is capped at the barometric height of 10.11 m, because below the vapour pressure the liquid boils and pulling harder buys nothing. The horizontal lines are five crown heights: a crown below a line's intersection with the curve can be primed by suction and one above it cannot, at any pump. This is the process the running siphon's argument explicitly excluded, and it is the one where the atmospheric account is the mechanism rather than a limit. What is taught wrongly

The one place the atmosphere pushes

The height of a siphon's hump is not in its flow rate, and the atmosphere holds the column together rather than driving it. That account excludes one process by name — starting. That is the process the atmospheric account describes correctly, and it is the only one in the whole device.

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