Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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The thread: The exact theory is wrong — page 20

Page 20 of 22, continuing through the 193 essays this motif runs through.

193 essays carry this thread — page 20 of 22.

One material loop, at five stages of being drawn out. A circle of fluid particles carried by four point vortices, drawn at equal intervals over fourteen time units. Its length grows by a factor of six and its shape becomes unrecognisable; the circulation round it does not move at all. Ideal flow

The drift was the instrument

Kelvin's theorem was checked on this site by carrying a loop and watching its circulation move by eight parts in a hundred thousand. That drift is not the flow forgetting. The exact number is a count of what is inside the loop, it does not move by anything at all, and the drift belongs entirely to the two instruments used to measure it.

 A band of growing waves that opens at 5772 and narrows as the viscosity goes. The wavenumbers at which a two-dimensional wave on plane Poiseuille flow neither grows nor decays, against the Reynolds number on a logarithmic axis. Inside the tongue waves grow; outside they decay. The tongue's tip is the critical point. Both edges slope downward and towards each other in wavenumber as the Reynolds number rises, so the band of unstable waves shrinks towards long waves — the direction in which the inviscid problem, which has no growing wave at all, is reached. Transition and turbulence

The profile Rayleigh cleared and viscosity did not

Flow between two plates has no inflection point, so without viscosity no wave on it can grow. With viscosity one does, above a Reynolds number of 5772. Taking the viscosity away again slows that wave and narrows the band it grows in, because the stress that feeds it is made by viscosity in the first place.

 Eight numbers, one construction. The dimensionless groups this collection has produced, placed on a logarithmic axis at a representative value. Each is a memory time divided by a process time, each was named separately in a different field, and each decides the same question: whether the past is still present. Regimes and numbers

Every memory number is one time over another

These essays produced eight dimensionless groups in eight different fields, named after eight different people, spanning a factor of four hundred in value. Written out, all eight are a memory time divided by a process time, and all eight govern the same curve.

 Momentum theory answers every descent rate except those between hover and twice the hover inflow. The induced velocity at a rotor disc against its climb speed, both in units of the hover induced velocity √(T/2ρA), at fixed thrust. The climb branch (thick) solves v(V + v) = 1 and is a streamtube for every climb and for hover, where v = 1. Continued into descent (dashed) it still has a root, but the air it describes leaves the tube at both ends. The windmill-brake branch (thin) solves v(V + v) = −1 and is real only for descent faster than two hover inflows, where it meets v = 1 again. Between V = −2 and V = 0 (shaded) neither is a streamtube. The faint diagonal is v = −V, where the rotor would need no power: it crosses the band and touches neither valid branch. Fluids at work

Between hover and twice the hover inflow

A rotor's momentum balance has an answer for every climb and for every fast descent, and none for descending at anything between zero and twice its own hover inflow. There one root sends air out of both ends of its streamtube and the other root is not a real number, and at each edge of the band one end of the tube stops moving — which is where the vortex ring state lives, and where a wind turbine's thrust coefficient of one sits.

 A cooled wall at β = 1: the linear relation is 158 K out inside the layer. The static temperature across a laminar layer at Mach 5 with an edge temperature of 220 K, over a wall whose total enthalpy is 0.5 of the edge's, at a pressure gradient β = 1: exact (thick) and from the Crocco–Busemann linear relation (thin), at a Prandtl number of one with constant properties. The wall is at 660 K in both. The exact profile peaks at 684 K and the linear one at 759 K; the largest difference, −158.3 K, is at η = 0.81. Compressible flow

The gradient the heat never hears

On a flat plate at a Prandtl number of one, a boundary layer's total enthalpy is a straight-line function of its velocity, whatever the wall's temperature. Put the same layer in a pressure gradient and the straight line fails everywhere except on an insulated wall, because the gradient enters the velocity's equation and not the enthalpy's — and a favourable gradient can leave a band of gas colder than the free stream above a wall three times hotter than it.

 The column in height and time: a falling interface, a rising shock, a fan. A batch settling test from a uniform φ₀ = 0.1, height above the bottom against time, both scaled on the column height and the single-particle settling time. The interface with clear water (thick) falls in a straight line at 0.4538; the sediment shock rises from the bottom at 0.1484 until the two meet at t = 1.661, height 0.2464; the thin lines are characteristics of the fan, each carrying one concentration between 0.317 and packing, and the interface bends as it crosses them. Dots are the finite-volume solve on 400 cells: the interface and the sediment front. Flows and fields

The column the chord rule cannot settle

A suspension settling in a closed column is a kinematic wave, and its flux curve bends both ways. At the top the chord rule works: clear water meets the suspension at a single falling front. At the bottom it does not, and the bed grows behind a shock that stops short of packing and a graded layer beneath it — so the interface, instead of arriving, slows for ever.

 Below a critical downstream pressure the flow rate stops listening. The flow rate through the meter against the downstream pressure, for upstream pressures of 3 bar, 5 bar, 7 bar. As the downstream pressure falls the flow rises — until the throat reaches vapour pressure, after which it is flat: 0.470 L/s from 3 bar, reached at 2.554 bar downstream; 0.608 L/s from 5 bar, reached at 4.254 bar downstream; 0.720 L/s from 7 bar, reached at 5.954 bar downstream. Everything to the left of each knee delivers the same flow, so the device holds its flow rate against any disturbance downstream. What is taught wrongly

The venturi that stops listening downstream

In a venturi with real walls, the narrowing-speeds-it-up story is exact: continuity and Bernoulli run forward from the drawing. Followed far enough, the same story predicts its own limit. The throat's pressure cannot fall below the liquid's vapour pressure, and once it gets there the flow rate stops responding to anything downstream — the meter has become a limiter, and its throat is supersonic for the vapour-laden mixture passing through it.

 There and back again. A blob of a hundred and twenty tracer particles at the start, after four time units of stirring, and after the same four run backwards. The third set is drawn over the first and the worst particle is 1.4·10⁻⁹ from where it began. Ideal flow

Reversible, and unusable

Ideal flow has no arrow of time in it. Run a stirring backwards and the dye comes back — here to 1.4 parts in a thousand million. Nudge the state by a hundred-millionth first and the same reversal returns a blob almost five hundred times further from home than the nudge was large.

 A force that depends on the molecular scale only through its logarithm. The force per unit length needed to move a 30° contact line of water at 1 mm/s, out to 1 mm, against the slip length on a logarithmic axis, from a picometre to a tenth of a millimetre. Each tenfold change in the slip length moves the force by the same fixed amount, so eight decades of the most uncertain length in the problem change the answer by a factor of about twenty — and at zero slip length the line keeps rising without end. The dashed curve takes the exact wedge's angle factor with a sharp cutoff; the solid one the thin-film wedge with Navier slip. What is taught wrongly

The drop a no-slip wall would never let spread

Liquid touching a solid moves with it, and nearly everywhere that is as close to exact as anything in fluid mechanics. At the edge of a spreading drop it cannot be: the stress in the corner rises as one over the distance from the edge, and the force needed to move the edge is infinite. Something slips over a nanometre, and because the answer depends on that length only through its logarithm, a drop spreads at almost the same rate whatever the something is.

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