Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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The thread: A smooth picture proves nothing — page 7

Page 7 of 20, continuing through the 172 essays this motif runs through.

172 essays carry this thread — page 7 of 20.

The same number, by two integrals that share no arithmetic. Three flows whose dissipation is in closed form both ways. The volume route integrates the dissipation function over the fluid; the boundary route multiplies a force or a torque by the speed of whatever is applying it. Neither calculation contains the other, and the residual column is what is left when they are subtracted. Viscosity

The price of a gradient

Viscosity does not charge for motion. It charges for the rate at which a parcel is being deformed, and a fluid in solid-body rotation at any speed whatever destroys nothing at all. What is charged for is a sum of squares, which is why the bill can be computed twice.

 Two flows with the same rate of strain, doing different things to a blob. A circle of fluid carried by two flows chosen to have exactly the same rate-of-strain magnitude, drawn at four times. Pure strain pulls it into an ellipse whose axes stay put; simple shear pulls it into an ellipse whose axes rotate as fast as they stretch. Both have zero divergence, so both preserve the area. The difference between them is not the strength of the straining but what the rotation does to the direction being stretched. Flows and fields

Longer, with nothing pulling it

Two flows with exactly the same rate of strain. In one a line of fluid grows by a factor of 148 in five time units; in the other it grows by 10. Turn the straining axes faster than the strain rate and no line grows at all, however hard the fluid is being strained.

 One sheared stream, and the total pressure across it. A parallel shear flow is an exact steady solution of the Euler equations, and the momentum equation requires its static pressure to be uniform. So the total pressure is entirely the dynamic pressure, which varies with the speed — by three and a fifth dynamic heads across this layer, on a flow where the static pressure does not vary at all. What is taught wrongly

Four Bernoullis and one name

"Bernoulli's equation" names at least four statements with four different constants, three domains of validity and one shared reputation for being misapplied. A single sheared stream separates the first three: its total pressure is constant along every streamline, varies by three dynamic heads across them, and its static pressure never moves at all.

 A falling drop is a bun, not a tear. The shape of a drop pressed on by the air it is falling through, computed by matching the Legendre component of a sphere's own potential-flow pressure distribution against the change in curvature it produces. The result is oblate — flattened along the direction of travel — because the pressure is high at the poles and low round the equator. The teardrop of every illustration has the wrong symmetry entirely: a drop in free fall has no up, and one at terminal speed is being pressed on from in front. Regimes and numbers

The drop that is not a tear

A falling raindrop is flattened along the direction it is going, by the pressure of the air passing it rather than by its own weight, and the group that decides is the Weber number. The teardrop of every illustration has the wrong symmetry entirely — there is no up in the problem it is drawn for.

 How far downstream the heat is still being made. The dissipation accumulated from a station ahead of a cylinder to a station behind it, as a fraction of the whole of what is made inside the frame, at five Reynolds numbers. At Reynolds number 1 the fluid has finished paying by about a diameter behind the body. At 100 it has not finished at five, and the curve is still climbing at the edge of the picture — the drag is a force on the body, and the heat it stands for is somewhere else. Viscosity

Where the heat of a drag is made

The power it takes to tow a body through a fluid becomes heat, all of it, eventually. None of the interesting words in that sentence are the first four. It is the "eventually" that decides how a wake behaves, how far a disturbance reaches, and why no box drawn round a body contains its own bill.

 Q and the vorticity along a radius of one vortex. Two candidate measures of where the vortex is, along a radius of a Lamb–Oseen vortex. The vorticity is a Gaussian: positive at every radius, so a threshold on it puts the edge wherever the threshold is put. Q — the excess of rotation over strain — changes sign exactly once, at 1.121 core radii, and that radius is a property of the flow rather than of the person drawing it. Inside it, 71.5 per cent of the circulation. Flows and fields

Where a vortex stops

Four criteria decide where a vortex ends, and in two dimensions three of them are the same criterion. The fourth is a knob. And the one that is not a knob is not objective: a co-rotating pair of vortices occupies two per cent of a window to one observer and twenty-five to another.

 Four sections, and what the shape story says about each. A flat plate, a symmetric section twelve per cent thick, a cambered one, and a section with a wavy upper skin. The first two have upper and lower surfaces of exactly equal length; the last has an upper surface two and a half per cent longer than its lower one, which is twice the cambered section's excess. What is taught wrongly

The wing that is flat, and flies

Refuting the equal-transit story by computing the parcels leaves its premise standing, and the premise is the part most readers believe: that the shape is what makes the lift. It is a claim about shapes, so it is tested with shapes — a flat plate, a symmetric section, and a cambered one flown upside down.

 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. Ideal flow

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.

 The relaxation time a particle actually has. Two quantities against the particle-to-fluid density ratio. β = 3ρ_f/(2ρ_p + ρ_f) is three for a bubble, one for a neutrally buoyant particle and nearly zero for anything heavy; it is the factor by which the fluid's own acceleration is felt. The other curve is the true relaxation time over the usual formula's, which is one for a heavy droplet, exactly three halves for a neutrally buoyant tracer, and unbounded for a bubble — the usual formula gives a bubble a relaxation time of zero, and therefore no dynamics at all. Regimes and numbers

The tracer that is not one

Every Stokes number is built on a relaxation time that counts the particle's own inertia and nothing else. Adding the two terms it leaves out gives a bubble a relaxation time where the usual formula gives zero, makes a neutrally buoyant tracer half as slow again as advertised, and sends bubbles into vortex cores that droplets are flung out of.

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