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

Every essay — page 22

Page 22 of 61, 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 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.

7 figures Two of opposite sign go somewhere. Two vortices of equal and opposite strength. Each is carried by the other's field, both are carried the same way, and the pair travels in a straight line at Γ/2πd forever, keeping its separation exactly. The speed is a consequence of one vortex's field evaluated at the other, and nothing else.

Vortices move each other

A vortex alone in an infinite fluid sits exactly still, forever — its own field is antisymmetric about it and there is nothing at its centre to be carried by. Everything a vortex does, another vortex did, and two of them already exhaust what can be written down.

7 figures The pressure, relaxed rather than quoted. The pressure field round a cylinder, obtained by relaxing ∇²p = −ρ∇·(u·∇u) on a body-fitted polar lattice with the exact pressure on the surface and on a far circle. The interior was told nothing except the velocity gradients. It agrees with Bernoulli's closed form everywhere to under two parts in ten thousand of the dynamic pressure, which is the strongest statement this site can make that the elliptic equation is the pressure's own.

Pressure has no speed

Take the divergence of the momentum equation for an incompressible flow and the time derivative disappears, the viscosity disappears, and what is left is Poisson's equation. Pressure is not carried anywhere: it is whatever satisfies an elliptic equation everywhere at once, and that is a statement about a fluid nobody has.

9 figures 1.5U at the equator, and no drag at all. The exact ideal flow past a sphere, in the meridional plane, with speed contoured behind the streamlines. The fastest fluid is at the equator at 1.5U — a cylinder's is at 2U — and the field is fore-and-aft symmetric, so the pressure integral over the surface gives a drag of -1.2e-16 against a dynamic scale of order one. The streamline spacing here does not measure speed the way it does in a plane flow: the flux between two meridional streamlines depends on the distance from the axis as well, which is why the speed is contoured rather than left to be inferred.

Three dimensions are kinder

The ideal flows usually drawn first are plane, and plane flow is the harsh case. Put the third dimension back and the fastest surface speed drops from twice the free stream to one and a half times, the disturbance dies as the cube of distance instead of the square, and the body cannot carry circulation at all.

8 figures 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.

8 figures 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