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

Vortex sheet — where it appears

A surface across which the tangential velocity jumps, being the zero-thickness limit of a shear layer at fixed circulation per unit length. It is unstable at every wavelength, and its nonlinear evolution develops a curvature singularity at a finite time from a smooth start — so a computation using one needs a length put back before it can be integrated.

Named by 9 essays across 4 fields — each of them below, with the objects they name alongside it.

Every wavelength grows, and the shortest grows fastest. Growth rate against wavenumber for a vortex sheet of zero thickness and for a layer of thickness δ. The sheet's curve rises without limit, which is the model announcing that it has no shortest scale in it; the layer's turns over and dies at kδ = 1, the neutral wavenumber computed from Rayleigh's equation rather than fitted.

Every wavelength at once

A vortex sheet of zero thickness is unstable at every wavelength, and the shorter the wavelength the faster it grows. The answer has no smallest scale in it, which is not a fact about fluids — it is the model reporting that it left something out.

turbulence · Instability Four camber lines, and the angle at which each stops lifting. Four mean lines on the same chord: symmetric, a circular arc, a four-digit line with its crest at forty per cent, and a reflexed line whose tail turns up. The zero-lift angle beside each is computed from that line's own slope by quadrature and is a property of the shape alone — no incidence, no speed, no thickness enters it. The symmetric line's is exactly zero, the arc's is −2m to ten decimal places, and the reflexed line's is positive: it needs to be pointed up before it stops lifting.

Where lift starts

A wing at zero incidence is not a wing making no lift. The angle at which a section stops lifting is a property of its camber line and of nothing else — not of its thickness, not of its speed, not of the air — and it is an integral anybody can take.

circulation · Thin-aerofoil Every one of these is a solution, and they lift different amounts. Lift coefficient against the circulation the aerofoil was told to carry, with the Kutta condition's own answer marked. Each point is a complete solve: the sources were found for that circulation and the surface is a wall to within 1.3e-4 at every collocation point. Every member satisfies the equations of motion and the boundary condition, and the lift runs through them at exactly 2Γ/Uc — Kutta–Joukowski appearing as a property of the family rather than as a result about any member of it. Ideal flow round a closed body does not have a unique answer, and the Kutta condition is the extra sentence that picks one.

Nothing but the edge

Cut an aerofoil into panels, put a singularity on each, and require the surface to be a wall. The system that comes out has one more unknown than it has equations, and the row that is missing is not a bookkeeping slip — it is the fact that ideal flow round a closed body has no unique answer at all.

circulation · Panel The two ways a shock can meet a wall. Left: the incident shock from the wedge reaches the wall and a second shock turns the flow back parallel to it, meeting at a point. Right: at a larger wedge angle no reflected shock can turn the flow that far, and the intersection lifts off the wall into a triple point with a nearly normal Mach stem standing on the surface and a slip line trailing downstream. The two configurations are drawn at the angles the solver returns, with every shock angle computed rather than sketched.

When a shock cannot bounce

A shock reflects off a wall until the reflected shock runs out of turning, which happens at a wedge angle well below the free stream's own limit. Between the two boundaries both configurations exist, both are stable, and which one appears depends on which direction the experiment came from.

compressible · Mach reflection Why a flat plate has no drag, drawn as a triangle. A flat plate at incidence with the three forces that must balance. Pressure can only act along the plate's normal, so the pressure force is the arrow perpendicular to the plate. Kutta–Joukowski says the resultant is perpendicular to the free stream. The difference between the two directions is the suction force at the leading edge, which acts forwards along the plate and is exactly L sin α. Without it the plate would have a drag of L sin α, and an inviscid fluid does not permit one.

A finite force from an infinite speed

Pressure on a flat plate can only act along the plate's normal. Kutta–Joukowski says the force is perpendicular to the free stream. Those two directions differ by the angle of attack, and the discrepancy is made up at a single point where the velocity is infinite and the area is zero.

circulation · Leading edge suction 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 Two theories, one composite, and the aspect ratio between them. The lift-curve slope against aspect ratio. Prandtl's lifting line is exact as the aspect ratio goes to infinity and Jones's slender-wing theory is exact as it goes to zero, and each is generous outside its own limit. Helmbold's formula reduces to both with no free constant, which is what a composite expansion is, and runs under them where they disagree.

Where the line stops being a line

Prandtl's lifting line replaces a wing with a single bound vortex and its trailing sheet, and the formula that comes out is the most quoted in low-speed aerodynamics. Solved numerically at aspect ratio one it returns its own closed form to sixteen decimals — and the answer is forty-one per cent too high.

circulation · Finite wing 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.

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.

circulation · Unsteady lift A vortex sheet rolling up, at five stages. One period of an initially flat vortex sheet with a small perturbation on it, drawn at equal intervals. The perturbation grows, the sheet steepens, and the ends wind into a spiral. Nothing is added to the sheet after the first instant.

A spiral is a legible record

When a vortex sheet rolls up, the fluid in it can never change places: two points on a sheet cannot pass one another. So the arms of the spiral are a map of the initial sheet, in order, and the picture is a record of the roll-up rather than a snapshot of it.

inviscid · Vortex sheet

Named alongside it

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

Model limitCirculationShear layerSuperpositionConformal mapConvergenceDiscontinuityDiscretisationIll-posedInduced dragKelvin helmholtzKutta condition

All concepts