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The thread: What is conserved — page 25

Page 25 of 33, continuing through the 292 essays this motif runs through.

292 essays carry this thread — page 25 of 33.

The shaft speed is a window, and cavitation closes the top of it. Two groups against shaft speed for the same duty: the specific speed, which must be inside a band for a runner to exist, and the suction specific speed, which must be below about 3 for the impeller not to cavitate. Both rise with the shaft speed, so raising it to reach a band is also raising it towards the cavitation limit. The window here runs from 274.98 rpm to 962.31 rpm and cavitation sets its top. A duty whose window is empty needs something other than a different machine — a booster, a lower installation, or an inducer.

The group with no head in it

The number that picks a machine says nothing about whether the machine can exist. A second group formed from the same variables, with the delivered head replaced by the margin available at the inlet, decides that — and the delivered head has left the expression entirely, so how far a pump lifts is irrelevant to whether it tears the liquid apart at its own entrance.

Friction lends the crown 28.4 kPa, and the tank takes it back. The absolute pressure at the crown of a siphon with friction in its hose, through a whole drain, for the crown placed at three positions along the hose, against the frictionless constant of 23.01 kPa. With the crown 0.3 of the way it starts at 63.3 kPa, with the crown 0.5 along it starts at 51.5 kPa and with the crown 0.7 of the way it starts at 40.4 kPa. Every curve is above the constant, every curve falls towards it as the level falls, and every curve reaches it at the end — 23.06 kPa with a centimetre of level left. Friction never brings a siphon nearer to breaking; it lends a margin, and the draining tank returns it pascal by pascal, so the worst the crown ever sees is the frictionless value.

What is taught wrongly

The margin friction lends a siphon

Without friction a draining siphon's crown pressure does not depend on the source level at all, and every real hose has friction. It turns out always to raise the crown pressure, by an amount the draining tank hands back pascal by pascal — and how much it lends is decided by where along the hose the crown sits, not by how rough or how narrow the hose is.

4 nodes and 2 saddles, and the difference is the shape. The skin-friction pattern on a body, drawn as the streaks a film of oil would leave, looking at the nose. Every critical point of the field is marked: 4 nodes, where the streaks converge or diverge, and 2 saddles, where two streaks cross. Their indices sum to 2, which is the Euler characteristic of a sphere, and nothing about the flow enters that number — the stream is at 0 and the crossflow at 0, and neither may change it. What the flow decides is where the points are and what they look like; what the body decides is how many there are of each.

Flows and fields

The count computed on a body

The rule for a closed surface is usually quoted and seldom solved for. This solves for one, and what the computation adds is not confirmation — it is the discovery that the count survives two events in which the number of stagnation points falls, and that both of them have closed forms.

$Wagner's function and Küssner's, from one solver and two inputs. Lift as a fraction of its steady value, against distance travelled in semichords. The step in incidence and the sharp-edged gust are the same unsteady problem with two different right-hand sides, and Jones's exponential fits to both are drawn over the solve. From four semichords on they are nearly the same curve — which is why they get interchanged.$ Circulation and lift

Two answers to one question

Unsteady aerofoil theory collapses a wing's whole history onto one function of one variable, and every quasi-steady gust calculation convolves something with it. There are two such functions, not one: a wing that is pitched changes its boundary condition everywhere at once, and a wing flying into a gust has not met most of the gust yet.

The patches' centroids, against the point-vortex circle. Two circular patches of uniform vorticity, advected by nothing but the velocity their own boundaries induce, over one full co-rotation. Their centroids stay within one per cent of a separation of the exact point-vortex orbit — which they must, because the exterior field of a circular patch is the point vortex's and a harmonic function's area average over a disc is its value at the centre.

Ideal flow

What a point vortex is not

Two circular patches of vorticity move exactly as two point vortices do — the centroid velocity is the point model's with no correction of any order, and that is a theorem rather than an approximation. The trouble starts the moment they stop being circular, which is immediately.

One shape, at every station and for every jet. The axial velocity of Schlichting's round jet at four distances, each scaled on its own centreline speed and its own half-width. The four curves are one curve: the shape function (1 + eta²/4)⁻² has no parameter in it at all, and every station of every jet — laminar or modelled, weak or strong — collapses onto it exactly.

Viscosity

Exactly similar, and one number short

The round jet has an exact solution of the Navier–Stokes equations, and a turbulent jet is modelled by the same formula with the viscosity replaced by a fitted constant. The shape is identical. Nothing a measurement of the profile can do will tell the two apart.

Fourteen pure numbers, and where each came from. Every one of these is dimensionless, exact and quoted as a fact about fluids. None of them comes from dimensional analysis, which says which numbers an answer may depend on and never what any of them is. They come in four kinds — algebra, an integral, a root and an optimum — and the last two are not equally knowable.

Regimes and numbers

Where a pure number comes from

Sixty-four for a round pipe, sixteen twenty-sevenths for a wind turbine, 0.332 for Blasius. Counting dimensions produces none of them — it produces the list of arguments, and the function has to be solved. Which of the four ways it was solved decides how many digits are worth printing.

Four kinds of critical point, and the curve that separates them. The invariants of a trace-free velocity gradient, with the discriminant curve 27R²/4 + Q³ = 0 drawn through them. Inside the two upper lobes the cubic has one real root and a complex pair, which is a spiral being stretched along its own axis on the left and squeezed on the right; below the curve all three roots are real and the point is a node with two saddle directions. Of 820 random incompressible gradients, 509 land in the spiral region and 311 in the real one. A plane flow is the vertical line R = 0 and nothing else, which is why a plane has two kinds and space has four. The marked points are the cases the calculation checks that fall inside this window; the two vortex cases it also checks sit at Q = 3.25 and |R| = 4.25, off the top corners, because a window wide enough to hold them would flatten the curve the figure is about.

Flows and fields

Two kinds is a plane flow's privilege

A plane incompressible flow has a saddle or a centre and nothing else, and the proof is one line about a trace. The same line in three dimensions constrains three numbers instead of two, which is far less, and what it leaves is four kinds of point separated by a curve — with the one a plane cannot have being the structure the whole of turbulence is made from.

Where a rotor's pressure rise comes from, as the radius moves. The static pressure rise across a rotor, split into the two terms rothalpy gives it. The diffusion term is held at the de Haller limit throughout — the blade is being asked to slow the relative flow as hard as a boundary layer will allow — so it is a flat 11558.4 Pa at every radius ratio. Everything above that line is the centrifugal term, which costs no diffusion and has no limit of its own. At a radius ratio of 2 it supplies 49.92 per cent of the rise and at 3, 72.66 per cent. An axial machine, at a ratio of exactly one, gets none of it.

Fluids at work

What a turning frame keeps

Euler's equation prices the work and says nothing about where the pressure comes from. In the frame turning with the blades — which is accelerating, and carries two fictitious forces — a Bernoulli-like quantity survives both of them, and it splits the pressure rise into a term a boundary layer limits and a term that is free if the radius moves.

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