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The thread: The exact theory is wrong — page 7

Page 7 of 22, continuing through the 193 essays this motif runs through.

193 essays carry this thread — page 7 of 22.

The street, as two rows of point vortices. The exact velocity field of a staggered double row of point vortices at the stable spacing ratio, with the row spacing and the circulation of one core printed from a line integral of the field rather than from the number that built it. This is a model of a wake. It contains no body, no viscosity and no mechanism that would shed anything, and the viscous stepper used here does not produce a street at any Reynolds number.

The street this site cannot draw

The alternating wake behind a cylinder is the most photographed structure in fluid mechanics, and this site's solver does not produce one. What can honestly be drawn instead is a model of it — and the model settles one thing exactly, which is the spacing.

Ideal flow

Three dimensions are kinder

Every ideal flow solved on this site so far is plane, and plane flow is the harsh case. Put the third dimension back and the fastest surface speed drops from twice the free stream to one and a half times, the disturbance dies as the cube of distance instead of the square, and the body cannot carry circulation at all.

Stable in every mode, and 100 times larger first. The energy of the worst-case disturbance against time, on a logarithmic scale, at four Reynolds numbers and at the one the slider selects. Time is in units of Re, which is what makes the four curves the same shape; what changes with Reynolds number is the height, and it changes as the square. Every eigenvalue of this operator is negative throughout, so nothing that grows here is an instability in the sense a stability analysis reports.

Transition and turbulence

Every mode decays and it grows anyway

A stability analysis asks whether any mode of a flow grows, and for pipe flow the answer is no, at every Reynolds number, which the pipe disagrees with. The missing ingredient is that the modes are not perpendicular — a disturbance made of two nearly parallel decaying pieces can grow by a factor of Re²/16 before it dies.

A lift curve that keeps climbing to 49 degrees. The lift of a slender delta of aspect ratio 1, split into the potential term that an attached flow would give and the vortex term the separation adds, with a conventional wing's curve behind them. The delta's is nonlinear from the start and reaches 1.68 at 49 degrees, where a conventional wing stalled at fifteen. That is the whole design case for the shape: not that it is efficient — it is not — but that it still has lift at incidences where an ordinary wing has none, which is what a delta-winged aircraft needs on approach and in a turn.

Circulation and lift

Lift out of a failure

Separation is what ends a wing's lift curve everywhere else on this site. A slender delta with sharp leading edges separates on purpose, rolls the shed sheet into a pair of vortices above its upper surface, and takes most of its lift from the suction they induce — with a curve that climbs to forty-nine degrees.

Three waves out of one discontinuity. The x–t diagram of the burst diaphragm. A shock runs right at a speed of its own, a contact surface follows it more slowly, and an expansion fan spreads left as a family of rays that opens with time — the three wave families the Euler equations possess, produced at once by an initial condition with no waves in it at all. Every straight line here is a speed the solution computed, and the fan is drawn as the rays it actually consists of.

Compressible flow

One diaphragm, every wave

Two states of the same gas at rest, separated by nothing, is the simplest initial condition compressible flow admits — and its answer contains all three waves the equations possess at once: a shock one way, an expansion fan the other, and between them a surface across which the density jumps and the pressure does not.

$Ninety microseconds, and most of it spent barely moving. The radius of a collapsing cavity against time, both as fractions of their own totals. The bubble spends most of the collapse near its original size and the last few per cent of the radius in the last fraction of a per cent of the time. The total is 91.47 microseconds for a millimetre cavity at one bar, computed by quadrature and agreeing with the closed form in gamma functions to a part in 10⁹.$ Fluids at work

The bubble that hammers

A vapour cavity swept into higher pressure does not deflate. It collapses, in ninety microseconds for a millimetre bubble, and the model that describes the collapse predicts a wall speed that reaches the speed of sound in water at three per cent of the original radius — which is to say it predicts its own failure, and locates it.

$What the ground actually gives a wing. Induced drag near the ground as a fraction of the same wing's in free air, against height in spans, at constant lift. The ground is a plane of symmetry, so the wake is joined by a mirrored wake of opposite circulation below it, and the upwash from that image is what takes the drag away. At a tenth of a span the wing keeps 0.516 of its induced drag — a saving of 48 per cent — and by a span and a half the effect is 1.3 per cent and going. The lift is held fixed all the way along this curve: what the ground gives here is not more lift, it is the same lift for less drag, and the two are different claims about a landing aeroplane.$ What is taught wrongly

The cushion that is not there

Ground effect is usually explained as air trapped under the wing and compressed into a cushion. At the heights an aeroplane actually flies at, the air under the wing is moving at four fifths of flight speed and most of what the ground gives is a drag saving the story never mentions.

45° of corner, and an infinite number of eddies in it. The creeping flow in a corner of 45 degrees, drawn from Moffatt's similarity solution. Each eddy turns the opposite way to its neighbours and is 3.17 times smaller and 1.6e+3 times weaker than the one outside it. The contour levels are rescaled inside each eddy, because they differ in strength by three orders of magnitude per step and a single set of levels would show the first and nothing else — which is itself the reason nobody has seen the third. The dividing lines between eddies are drawn where the stream function changes sign, and the dots are the centres, both found from the solved field.

Viscosity

The eddies nobody stirs

A slow flow past the mouth of a sharp corner does not simply fail to get in. The corner fills with an infinite sequence of counter-rotating eddies, each about three times smaller and sixteen hundred times weaker than the last, and the whole structure is decided by one complex number.

The correction belongs to the wing, not to the air. Lift-curve slope against Mach number for three aspect ratios, each computed by solving the lifting line for the wing the transformation actually implies — one of aspect ratio βAR — and dividing by β. The dashed curve is the two-dimensional rule, which is what an aerofoil section gets. At Mach 0.7 the aspect ratio of 20 has gained 35 per cent of slope and the aspect ratio of 4 has gained 24, against the 40 per cent the section rule promises both. The β in the finite-wing term cancels the β in front of it, so the shorter the wing the less compressibility does to it.

Regimes and numbers

The wing the equation is really solving

The Prandtl–Glauert rule is usually quoted as a factor on the answer. It is a change of shape — and in three dimensions the shape it changes is the aspect ratio, so a short wing is far less affected by compressibility than a long one.

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