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

Singularity — where it appears

A point or curve where a solution of the flow equations becomes infinite or loses smoothness, such as the centre of a point vortex or the cusp of a front. It usually marks where the model has left out a physical effect that would round it off in a real fluid.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

The growth rate a discretised sheet has, at every wavelength it can carry. Kelvin–Helmholtz gives a growth rate proportional to the wavenumber and without bound. A sheet represented by N point vortices has pi m (1 − m/N) instead — the same rate at long waves and half of it at the shortest wave the grid carries, with the fastest-growing mode at the grid scale itself. Smoothing the kernel over a length delta moves that mode back to a wavelength the physics chose.

A sheet that cannot stay a sheet

Let a shear layer's thickness go to zero and it becomes a surface across which the velocity jumps. The model is used everywhere in this subject, it is unstable at every wavelength, and the thing it does next is worse: it develops a singularity in its own shape, at a finite time, from a smooth start.

inviscid · Vortex sheet A wrinkled flame settles into arcs meeting at a cusp. The steady front of a flame in a periodic domain 5, 10 and 20 neutral wavelengths wide, from the exact pole solution of the Michelson–Sivashinsky equation, with the burnt gas below and the flame advancing upwards; each is drawn across one period, scaled to the same width. Every one is a single smooth arc bulging into the fresh gas, meeting its neighbour in a sharp cusp pointing back into the burnt gas, and in these units the three arcs nearly coincide: only the cusp sharpens as the domain widens. In physical units the arc's depth grows in proportion to the domain's width, so the three flames are the same shape at three sizes.

A wrinkled flame has one cusp and a speed limit

The linear theory of a flame says every long wrinkle grows and none is favoured. The weakly nonlinear theory — the Michelson–Sivashinsky equation — says where the growth goes: small wrinkles merge, the front settles into smooth arcs bulging into the fresh gas and meeting in sharp cusps, and in a domain of any width it ends with a single arc and a single cusp. That front is an exact solution made of poles in the complex plane, and its speed is a closed form that rises in steps as the domain admits more poles and then stops: beyond about five neutral wavelengths a wider flame is no faster, because it is the same shape at a larger size.

kinematics · Dilatation Posts slip as one over the root of their fraction. The slip length of a surface of posts, over the period of their square array, against the fraction of the surface that is solid, beside Philip's stripes along and across and the dilute limit in which each post is a lone disc dragged edgewise, (3/16)√(π/φ). At a hundredth solid the posts slip 2.87 periods against the stripes' 1.32; at a tenth 0.597 against 0.591, nearly equal; at a half 0.0663 against 0.11 along the stripes and 0.0552 across them.

A post holds liquid back by its radius, a stripe by its length

A surface of no-slip posts standing in a gas-filled texture lets liquid slip over it, and the slip length grows as one over the square root of the solid fraction, where a surface of stripes grows only as its logarithm. The reason is the oldest contrast in slow flow: a disc dragged through a viscous liquid has a drag proportional to its radius, which is Stokes' law, and a line has no finite drag at all, which is Stokes' paradox. But sparse posts beat stripes only while they are sparse.

kinematics · Boundary conditions The drag follows the surfactant load, and a surface pressure of μU pays nearly all of it. How far the drag has climbed from the clean bubble's to the rigid sphere's, against the mean surfactant load over the whole bubble as a surface pressure in units of μU. A load of 0.2 μU makes a 60° cap and a third of the climb; 0.56 a 90° cap and 70 per cent of it; 1 a 120° cap and 94 per cent. The dashed line is the cap's share of the surface for the same caps: the drag runs ahead of the area covered.

A thousandth of a monolayer holds a bubble still

A clean bubble rising slowly through water feels two-thirds of a rigid sphere's drag, and real bubbles almost never do, because surfactant swept to the rear holds the surface still over a cap there. Solving the flow with the cap in it shows how little that takes. The drag runs ahead of the area covered — half-way to rigid with a third of the surface held — and the surfactant needed is set by the viscous stress, not by the surface tension. For a bubble a tenth of a millimetre across, a thousandth of a monolayer, spread as a cap, makes it rise within a few per cent of a solid ball.

viscous · Mobile interface The water climbs the body, and the wetted width outruns the drawing. The wetted half-width against penetration for a wedge of 10° deadrise and a circular cylinder of unit radius: where the body crosses the undisturbed level, von Kármán's width, and where the risen water meets it, Wagner's. For the wedge Wagner's width is π/2 = 1.571 times the geometric one at every depth; for the circle it is √2 = 1.414 times, at small penetration.

Water climbs a falling wedge, and the load comes from the climbing

A hull that strikes the sea sets the water under it moving, and the force is the rate at which it does so. The obvious estimate measures the wetted width where the hull crosses the undisturbed surface. But the water does not wait: pushed aside, it rises up the hull and wets it sooner, a factor π/2 wider for a wedge and √2 for a round bottom. The force carries that factor twice, and the peak pressure, where a thin jet leaves the hull, carries it squared on a cotangent that grows without bound as the bottom flattens.

inviscid · Unsteady bernoulli The same cap costs a rising bubble a quarter of its speed and a migrating one four-fifths. Speed as a fraction of the clean bubble's against the half-angle of a stagnant cap over the rear: for a bubble migrating in a temperature gradient, and for the same bubble rising under its weight. A 90° cap leaves the rising bubble 74 per cent of its speed and the migrating one 21 per cent; a 120° cap, 68 and 4. The rising bubble can only fall to a rigid sphere's two-thirds; the migrating one falls to nothing.

A cap that slows a rising bubble stops a migrating one

Surfactant swept to the back of a bubble holds the surface still there, and for a bubble rising under its weight the worst this can do is turn it into a solid sphere and cost it a third of its speed. A bubble migrating in a temperature gradient has no such floor. Its own surface is the engine, and a cap removes the engine as well as raising the drag. A cap over the cold hemisphere leaves a rising bubble three-quarters of its speed and a migrating one a fifth; a cap of 120° leaves the migrating bubble four per cent. The surfactant it takes is set by the tension difference that drives the migration, and that is tiny.

viscous · Mobile interface

Named alongside it

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

Model limitBoundary conditionStokes flowBubbleDrag coefficientHadamard rybczynskiMarangoniNonlinearitySurfactantAdded massAnalyticityCirculation

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