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

Every essay — page 16

Page 16 of 39, continuing through the fields in the same order.

Flows and fields Ideal flow Circulation and lift Viscosity Regimes and numbers Compressible flow Transition and turbulence Fluids at work What is taught wrongly Series Concepts Regimes Refutations Search

Ideal flow

The exact theory of a fluid with no viscosity — closed-form, elegant, and predicting no drag at all. Its failure is the most useful thing in the subject.

The same flow, from two different physics. Solid lines: the depth-averaged flow between two plates a small distance apart, with an obstacle standing between them. Dashed: the ideal-flow solution for the same obstacle. They are the same field to a part in ten billion, because averaging Stokes flow across a narrow gap gives a velocity that is the gradient of a harmonic potential. The cell has no inertia at all, which is the one hypothesis the ideal theory cannot do without.

The exact theory, drawn by viscosity

Two flat plates a millimetre apart, syrup between them, an obstacle in the gap. The Reynolds number is a hundredth, inertia is absent, and the streamline pattern is the potential flow past that obstacle — exactly, to a part in ten billion. The one hypothesis ideal flow cannot do without is the one this flow most conspicuously breaks.

8 figures The wall the outer flow is really solving for. A flat plate, the edge of its boundary layer, and the line the outer flow behaves as though the plate were on. The displacement thickness is the mass deficit divided by ρU — checked here against the profile's own integral rather than quoted — and moving the wall out by that much reproduces exactly the flow rate the viscous layer lets past. It is a third of the visible thickness of the layer and it is the only part of the layer the outer problem knows about.

The body the outer flow actually sees

A boundary layer lets less fluid past than an inviscid one would. The outer flow can be given exactly the same reduced flow rate by leaving the fluid inviscid and moving the wall out — so the potential flow that matters is not the flow past the body, but the flow past the body plus a thickness the boundary layer computes.

8 figures Four vortices, and the end of prediction. The same three vortices as before with a fourth, weaker one added near the middle. Three point vortices have three independent invariants for three degrees of freedom and cannot be chaotic; four have the same three invariants and one more degree of freedom, and generically are. The energy and the impulses are conserved here to fourteen digits over the whole run, which is what makes the tangle a property of the system rather than of the arithmetic.

Three is the most that can be predicted

Point vortices are the simplest dynamical system fluid mechanics has — no cores, no viscosity, no approximations, four exactly conserved quantities. Three of them are integrable and cannot be chaotic. Add a fourth and the same equations, conserving the same quantities to fourteen digits, stop being predictable at all.

8 figures Four corners, and what the flow does in each. The local flow in corners of four different interior angles, drawn from the exact local solution ψ = r^(π/α) sin(πθ/α). The exponent of the speed is π/α − 1 and depends on nothing else: at a right angle the corner is stagnant, at a flat wall nothing happens, and at any angle greater than a straight line the speed has no bound at the corner. The last panel, at 360 degrees, is the flow round the edge of a plate.

Nothing turns a sharp corner

Near a corner the flow is fixed by the angle and by nothing else — not by the size of the corner, not by the flow far away, not by the fluid. The exponent is π/α − 1, and every sharp edge in aerodynamics is the one case where it comes out at minus one half.

7 figures A tube opened at the foot of a reservoir. Speed in a two-metre tube fed by a one-metre head, from the instant it is opened. The quasi-steady answer — Torricelli's √(2gh), which is what dropping the time derivative from Bernoulli's equation gives — is reached at the instant of opening, before any fluid has moved. The true answer is a hyperbolic tangent with a time constant of 2L/√(2gh), and it takes 2.4 seconds to come within one per cent. The whole of that transient is one term.

The pressure that depends on the past

Bernoulli's equation for an unsteady flow has a term nobody writes down and a right-hand side that is a function of time rather than a constant. The term is exactly zero once a flow has started and is the whole of the flow while it is starting, which is why it never appears in an answer and is never negligible in getting to one.

8 figures The pressure of a sum against the sum of the pressures. Ten points around a cylinder with circulation, with the pressure coefficient of the combined flow plotted against what adding the two flows' separate coefficients would give. Nothing lies on the diagonal. The gap is exactly −1 − 2u_A·u_B/U², an identity checked to the last digit at every point, and it is not small: at one of these points the two answers differ by 1.92, which is more than the whole range of a suction peak.

The one thing that does not add up

Laplace's equation is linear, so flows can be laid on top of one another and almost every classical result is built that way. The two things anybody actually wants out of a flow — the pressure and the force — are quadratic in the velocity, and neither of them adds at all.

8 figures 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.

8 figures Kirchhoff's rotation rate, which a point vortex does not have. A patch of uniform vorticity bounded by an ellipse turns rigidly at omega a b/(a+b)², a rate that depends on the shape alone. It is largest for a circle, where it is unobservable, and falls away as the patch is drawn out. A point vortex has no shape and therefore no entry on this axis at all.

The shape a vortex keeps

Outside a circular patch of uniform vorticity the flow is exactly the point vortex's — not nearly, exactly — so replacing one by the other looks free. It is not. The patch has a shape, the shape has a rotation rate of its own, and there is a strain above which no shape exists at all.

9 figures The singularity a thin aerofoil drives onto its own nose. The Joukowski map has a critical point that maps to a place inside the body, a distance 4 mu²/(1 + 2 mu) from the leading edge. Against thickness that distance is a clean square: a twelve per cent section is analytic only within eight thousandths of a chord of its own nose, and the thin-aerofoil limit is the limit in which the singularity arrives on the surface.

The part of the flow inside the body

A potential flow outside a body is an analytic function, and an analytic function does not stop at the boundary it was defined on. It continues inward until it meets a singularity — and every body in this collection has at least one inside it, in a place that decides how the flow behaves outside.

8 figures The two long-wave speeds, and the swirl at which one of them stops. For uniform axial velocity the wave speeds follow from the criticality condition by a Galilean boost: c = W(1 ± 2S/j), with j the first zero of J1. The upstream-running root crosses zero exactly at S = j/2 = 1.9159, and above that swirl no disturbance can travel upstream — which is what subcritical and supercritical mean here and in an open channel.

The swirl that holds a wave still

A swirling flow down a pipe carries waves, and above a certain swirl one of them stops moving. Below it, a disturbance downstream can send information upstream; above it, the flow has outrun its own waves. The words are open-channel flow's words, and they are the same words for the same reason.

8 figures Five flows past one cylinder, every one of them a solution. Surface pressure round a cylinder in a stream, at five circulations. Each satisfies Laplace's equation, the tangency condition on the body and the condition at infinity, and each has a different lift. Nothing in the problem chooses between them: the domain has a hole in it, so the potential is many-valued and the circulation is a free constant.

The constant a hole leaves behind

In a region without holes, Laplace's equation and the boundary values have exactly one solution. Cut a hole and they have a one-parameter family. Nothing in the mathematics chooses between its members, which is why the Kutta condition has to exist and why it cannot be derived.

9 figures The free surface a submerged body leaves behind it, and the flat water in front. The linearised free-surface problem solved as a Fourier integral with a radiation condition. Behind the body a wave train of the wavelength that stands still relative to it, 2 pi U²/g; ahead of it, an amplitude a hundred and twenty times smaller. The asymmetry is the drag: an ideal fluid with a free surface can carry energy away.

The drag that is made of waves

D'Alembert's paradox says a body in a steady, irrotational, incompressible, inviscid flow feels no drag. Put a free surface above it and every one of those words still holds — and the drag is not zero. It is the energy walking away in the wave train behind.

9 figures