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

Inviscid — where it appears

Having no viscosity, and therefore no mechanism for a fluid to exert a shear stress. The assumption is excellent away from surfaces and catastrophic at them, since it removes the one condition that produces drag, vorticity and separation.

Named by 11 essays across 4 fields — each of them below, with the objects they name alongside it.

Ideal flow past a cylinder. A uniform stream past a circular cylinder in a fluid with no viscosity. The solution is exact and closed-form: streamlines part at a stagnation point, run round the surface and close up perfectly behind, and the pressure recovers to exactly what it was in front.

The theory that solves everything

Throw away viscosity and assume nothing is spinning, and fluid mechanics collapses into a linear problem with closed-form answers. The price is one term, and the term turns out to matter more than everything kept.

inviscid · Ideal flow d'Alembert's paradox, measured. Surface pressure round a cylinder in ideal flow, plotted against angle. The distribution is symmetric front to back, so every push on the front is matched by an equal push on the back, and the total force along the stream is exactly zero.

The exact theory says nothing has any drag

Solve the flow past a body in a fluid with no viscosity and the answer is beautiful, closed-form, and predicts that a cyclist needs no legs and an airliner no engines. This is not a small error, and it is the most useful failure in the subject.

inviscid · Dalembert Creeping flow, and the same body with inertia. The exact creeping-flow solution beside a solved field at a Reynolds number where inertia matters. The creeping flow is a mirror image of itself front to back — a photograph of it run backwards is a photograph of it — and the field with inertia has a wake, which is what a direction of time looks like.

The world with no inertia

Drop the viscosity and the equations become exactly solvable and wrong about drag. Drop the inertia instead and they become exactly solvable again — and for a cylinder in an unbounded fluid there is no solution at all, which took fifty years to notice and longer to fix.

regimes · Reynolds The fluid a moving cylinder carries with it. Kinetic energy density around a cylinder moving through fluid that is at rest far away. The fluid is not dragged along in a lump: it is pushed aside in front and closes in behind, and the energy in that motion is what has to be supplied to change the body's speed.

The force of getting going

The exact theory says a body moving steadily through an ideal fluid feels no force at all. It does not say the fluid is free. Accelerating the body has to accelerate the fluid too, and the bill for that is exactly the mass of fluid the body displaces.

inviscid · Dalembert One more condition, and the price of it. The model problem ε u″ + u′ = 0 with a condition at each end, at three values of ε. The outer solution is the flat line at one — that is the whole of the answer when ε is zero, and it is a first-order equation that can meet one condition, so it meets the one at the far end and misses the one at the wall by the whole range. Restoring ε restores the second condition and pays for it with a layer of thickness ε, inside which the gradient is of order 1/ε. The product of those two — which is what a stress is — does not depend on ε at all: it is 1.0000 at every value tried, to nine decimal places. Drag does not vanish as viscosity does. It converges.

How many things a flow must be told

The equations of motion do not have one answer. They have as many as the conditions on the edge allow, and the number of those is decided by the order of the equation — which is why viscosity does not make the same problem harder, it makes a different problem.

kinematics · Boundary conditions Lift and wave drag on a flat plate at Mach 2, exactly and to first order. The lift and wave-drag coefficients of a supersonic section against incidence, computed face by face from shocks and fans, with Ackeret's linear result dashed over them. The two agree to three decimal places at small angles and part company slowly — which is what a first-order theory is supposed to do, and evidence rather than tautology, since the two routes share no algebra.

Drag with nothing to rub

d'Alembert's paradox says a closed body in a steady, inviscid flow has no drag, and four separate arguments establish it, none of them wrong. Above Mach one it is false — the flow is still inviscid, still steady, and the drag is real, finite and quadratic in incidence.

compressible · Wave drag Wound up, and worth exactly what it started with. A material loop in a steady cellular flow — an exact solution of Euler's equations — drawn at four times. Each streamline in the cell has its own period, so the loop is stretched steadily into a spiral: by the last frame its perimeter is 5.7 times what it started as. The circulation round it is 0.903741 at the start and 0.903666 at the end. Nothing about the curve survives except the number.

What survives being wound up

Draw a loop of marked fluid particles and let the flow carry it. It will be stretched, folded and wound into a spiral until nothing about its shape is recognisable, and the circulation round it will not have moved at all — provided three conditions hold, each of which can be broken on purpose.

inviscid · Kelvin A loading that integrates to nothing and turns the aeroplane over. The loading a slender body carries at 6 degrees, drawn along it. It is positive over the front half and negative over the back, in equal measure, so the lift integrates to -3.6e-18 — nothing, which is d'Alembert's paradox for a body of revolution. The moment does not: the two halves act at different stations, and the couple that survives is nose-up, grows with the square of the speed, and is what a tail is sized against.

A body with no lift, and a moment anyway

A fuselage in ideal flow carries no lift at any incidence and still tries to turn the aeroplane over. The couple is computable in one line, it is why tails are the size they are, and the line comes from applying the wall condition to a place where there is no wall.

inviscid · Slender body A curved shock, and the entropy each streamline picks up crossing it. A parabolic bow shock ahead of a blunt nose at Mach six, with the streamlines drawn arriving horizontally and a marker at each crossing whose size is the total pressure lost there. The streamline through the nose crosses a normal shock and keeps three per cent of its total pressure; one four nose radii out crosses at fourteen degrees and keeps ninety-four per cent. Every streamline gets a different entropy, and the stagnation enthalpy is the same on all of them.

The spin a shock leaves behind

A curved shock gives every streamline a different entropy rise and the same stagnation enthalpy. Crocco's theorem then forces vorticity into a flow with no viscosity anywhere — and it scales as the inverse of the shock's radius of curvature, exactly, so a straight shock makes none.

compressible · Crocco Three aircraft, and the only thing the drag can see. The cross-sectional area against station, for a clean body, the same body with a wing added, and the body cut away where the wing is. The wave drag depends on this curve and on nothing else about the three shapes — so the middle one pays for its lump and the third does not, although the third has the same wing.

The drag that can only see one curve

A slender body's far field is a line of sources of strength A′(x), and nothing else about its shape reaches it. So its supersonic wave drag depends on the cross-sectional area distribution alone — and a fuselage cut away where the wing joins it can carry the wing for nothing.

inviscid · Slender body Each collapsed form of the plate's heat transfer reaches its limit decades away from air. The exact local Nusselt number of a vertical isothermal plate in its two collapsed forms against the Prandtl number: over (Gr·Pr²)^¼, which tends to Le Fevre's 0.600 as Pr → 0, and over (Gr·Pr)^¼, which tends to 0.503 as Pr → ∞. At Pr = 10⁻⁵ the first reads 0.5994; at 10⁵ the second reads 0.5023. The two limit laws cross at Pr = 0.494, where the exact value is 0.736 of either; air, at 0.72, reads 0.421 and 0.387.

Over a plate in mercury the warm layer flows as if inviscid

The exact solution for a warm vertical plate has been tabulated since 1953 and is usually shot from the wall, which fails in a liquid metal unless the shot starts within a thousandth of the answer. Posed instead as one algebraic system, it solves from a Prandtl number of a hundred-thousandth to a hundred thousand — and shows that in mercury the layer's speed is set without viscosity, that the integral method runs low there rather than high, and that air sits where neither limiting law is within a fifth.

regimes · Grashof

Named alongside it

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

d'Alembert's paradoxPotential flowModel limitBoundary layerAdded massBoundary conditionDoubletDragEntropyEulerian and LagrangianIrrotationalLaplace's equation

All concepts