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

Every essay — page 15

Page 15 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 section the pressure asked for. The designed section over the one it started from, both drawn to their own chords. Asking for 28 per cent more speed over the forward 62 per cent of the upper surface produces a section 15.0 per cent thick against the original's 10, with the extra thickness forward and the camber changed — none of which was asked for, and all of which is what that pressure distribution is. The pale outline is the baseline. The one thing the method cannot be told is where along the chord any of it happens: the speed is prescribed against the circle's parameter, and where a given station ends up is an output of the same solve that produces the shape.

Ask for the pressure, and see what shape that is

A designer knows what the pressure distribution has to do long before knowing what shape does it. Running the problem that way round is possible, it is exact, and it refuses more asks than it grants.

8 figures Three contours, one force, three different accounts of it. The same vortex, and the same total force on it, computed by a momentum balance over three contours of the same area. All three give ρUΓ to eight figures. What differs is the bookkeeping: the tall box gets 16 per cent of it from pressure and the rest from momentum flux, the wide box gets 84 per cent from pressure, and the circle gets exactly half. Neither part converges on its own as the contour is enlarged — each falls off like 1/r while the contour grows like r — so the split is a property of the shape of the limit rather than of the flow.

Where the reaction to a wing's lift is

The force on a body can be computed on any contour drawn round it, and the answer is the same every time. How much of that answer is pressure and how much is momentum flux is not — it runs from three per cent to ninety-seven, and the difference is the shape of the contour.

7 figures 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.

8 figures The two bodies the far field cannot tell apart. A circular cylinder and the Rankine oval that has the same doublet strength: 1.17 radii long against the circle's one, and 0.94 tall against its one, with a source and a sink 1.2 apart inside it. On the pale ring, one and a half radii out, the two flows differ by 19 per cent of the disturbance; at six radii by one per cent; at infinity not at all. What a far field records of a body is three numbers — its circulation, its net outflow and its doublet — and nothing else survives the journey.

What the far field remembers

Three numbers survive the journey to infinity — a circulation, a net outflow and a dipole — and nothing else about a body does. Two shapes with nothing in common can therefore make the same flow a few radii away, and the difference between them dies two orders faster than the disturbance either one makes.

8 figures Two divergence-free fields with the same boundary conditions. On the left, the exact potential flow round a cylinder moving through fluid at rest. On the right, the same flow with a divergence-free eddy added — one that has no normal velocity on the body or on the outer circle, so it changes nothing about what crosses a boundary. Both fields conserve mass, both satisfy the wall condition, and only one is the flow. Nothing in the drawing says which.

The flow with the least energy in it

Draw a flow that conserves mass and does not go through the walls, and it will look exactly like a solution. There are infinitely many of them and one is the flow. What separates it from the others is not visible anywhere in the picture — it is a number, and the number is an energy.

8 figures Flow net — a stream past a cylinder. Two families of curves drawn over the same flow: the streamlines, along which the streamfunction is constant, and the equipotentials, along which the velocity potential is constant. They cross at right angles at every point, because they are the two parts of a single analytic function of position.

The mirror that is a circle

A flat wall is made by reflecting everything in it. A round one is made the same way, except that the mirror is an inversion — the image of a point at distance d sits at a²/d, and a vortex acquires a second image at the centre that nothing about the wall requires.

8 figures The Kirchhoff flow past a flat plate. A uniform stream meeting a flat plate held across it, with two streamlines leaving the edges and never returning. Between them is a wake of fluid at rest at a constant pressure. The equations solved are the same equations that give d'Alembert's paradox for a closed body, and this flow has a drag coefficient of 0.8798.

Drag in the theory that forbids it

d'Alembert's paradox is a theorem about flows that close behind the body. Stop requiring that, let two streamlines leave the edges and never come back, and the same equations — no viscosity, no vorticity — produce a drag coefficient of 0.8798.

9 figures The hodograph plane, where the unknown boundary is the known one. The same flow drawn in the plane of its own velocity, ζ = (u − iv)/U. The plate, whose shape is known in the physical plane, becomes a segment of the imaginary axis; the axis of symmetry becomes a segment of the real one; and the free streamline — whose shape nobody knows — becomes an arc of the unit circle, because the speed on it is exactly the free stream. The unknown and the known have changed places, which is why the problem can be solved at all.

Where the unknown boundary is the known one

A free surface is the hardest kind of boundary — its shape is part of the answer, so the region the problem is posed in is not known until the problem is solved. Draw the same flow in the plane of its own velocity and the shape becomes an arc of a circle, known in advance and exactly.

7 figures The momentum of the fluid, against the shape of the region it is added up over. Momentum of the fluid around a cylinder moving through it, divided by the body's hydrodynamic impulse, against the aspect ratio of the rectangle the integral was taken over. Every rectangle has the same area and contains the same body. A tall region gives minus the impulse, a long one gives plus it, a square gives exactly zero, and the limit of a large region is whichever of those the region was shaped like. The momentum of an unbounded ideal flow is not a number.

The momentum with no value

A cylinder is pushed from rest to a steady speed. Work was done, energy went into the fluid, something was pushed. How much momentum does the fluid carry? The integral converges, the answer is finite, and it is a different finite number for every shape of region it is summed over.

7 figures A cylinder in a uniform shear, K = 0.4. A stream whose velocity increases with height, meeting a circular cylinder. The oncoming profile is drawn at the left. The flow carries uniform vorticity −K, so it is a solution of Euler's equations and not of Laplace's, the pattern is no longer symmetric top to bottom, and the body feels a lift towards the fast side with no circulation anywhere.

Inviscid does not mean irrotational

Dropping viscosity gives Euler's equations. Assuming nothing is spinning gives Laplace's — one scalar, linear, unique. The second step is a separate hypothesis about the flow's history, and a flow that fails it is still an inviscid flow with exact solutions of its own.

8 figures Hill's spherical vortex. A sphere of rotating fluid travelling steadily through fluid at rest, drawn in the frame that moves with it. Outside the sphere the flow is the ordinary potential flow past a sphere; inside, the vorticity is proportional to the distance from the axis and the fluid recirculates. The two solutions match in value and in slope across the surface, and there is no body anywhere — the boundary is a streamline and nothing else.

The one rotational solution anybody can write down

A sphere of spinning fluid travelling steadily through fluid at rest, with no body anywhere in it — the boundary is a streamline and nothing else. It is exact, it is two lines long, and the reason it is the famous one turns out to be the reason it is the only one a real fluid can settle into.

8 figures Two exact solutions of the same problem. Two steady Euler flows in the same square cell with the same boundary condition, differing only in the function relating vorticity to streamfunction. On the left the vorticity is proportional to the streamfunction, which is the textbook cellular flow; on the right it is uniform. Both satisfy the equations exactly. Nothing in the ideal theory prefers either, and at the same peak streamfunction their kinetic energies differ by thirty-one per cent.

The vorticity nothing decides

A streamline that comes from upstream carries its vorticity with it. A closed one comes from nowhere, so nothing determines what it carries — the ambiguity is not one number per body but a whole function. What closes it is a limit, and setting the viscosity to zero gives a different answer from letting it go to zero.

7 figures