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 3

Page 3 of 20, continuing through the 172 essays this motif runs through.

172 essays carry this thread — page 3 of 20.

The dynamic pressure is not ½ρU², and by Mach 0.85 it is out by a fifth. The pressure difference a pitot tube measures, divided by the incompressible dynamic pressure ½ρU², against Mach number. The dashed curve is the two-term series 1 + Ma²/4 + Ma⁴/40 that the rule of thumb comes from. An airspeed inferred from ½ρU² alone reads high, and the error is entirely predictable — which is why it is corrected rather than tolerated. What is taught wrongly

What the airspeed indicator believes

A pitot tube measures the difference between two pressures, correctly, at every speed. Everything wrong with an airspeed reading is in the arithmetic applied to that difference — and the error is 2.3 per cent at Mach 0.3 and 19.4 per cent at Mach 0.85.

 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. Ideal flow

Bodies made out of nothing

Put a source in a stream and a solid nose appears in front of it. Put a sink downstream of the source and the nose closes into a finite body. Nothing was placed in the fluid — the surface is a level set of a function, and it behaves like a wall because nothing crosses it.

 Every rung of the hierarchy is worse than the last. Writing a transport equation for the Reynolds stress does not close the system: that equation contains the triple correlation, whose equation contains the quadruple. The bars are the number of independent components at each order and the number of new unknowns its own equations introduce. The second is always larger, and the gap widens. Transition and turbulence

The ladder that never closes

Being six equations short is a problem with an obvious remedy — derive six more. The remedy works, produces an exact equation for the Reynolds stress, and leaves ten new unknowns behind. The gap does not narrow at any level, and the counting says why.

 The number on a streamline is a flow rate. Two streamlines and a crooked line drawn between them. The volume of fluid crossing that line every second, integrated from the velocity field, is the difference between the two streamfunction values at its ends — which is what makes a streamline a label rather than merely a curve. Flows and fields

The number on a streamline is a flow rate

Streamlines get drawn as decoration — curves the flow follows, spaced however the plotting looked best. Each one carries a number, the difference between two of those numbers is the volume of fluid passing between them every second, and it does not matter what route the measurement takes.

 Finding the aerodynamic centre by sweeping the chord. How fast the pitching moment changes with incidence, plotted against where along the chord the moment is taken. The curve crosses zero once, and that crossing is the aerodynamic centre — the single point about which pitching the wing does not change the moment. Circulation and lift

Where the lift acts

Lift gets drawn as an arrow, and an arrow has to start somewhere. The pressure is spread over the whole surface, so the choice of where to put the arrow is free — except that exactly one point on the chord has the property of not changing its answer when the wing is pitched.

 Stokes' theorem on a solved wake at Re 40. A rectangle drawn in a viscous flow that was solved on a grid. The circulation round its boundary is computed by walking the four sides and adding up the velocity along them; the vorticity inside it is computed by adding up the stored vorticity cell by cell. The two computations share no sample point and the theorem says they must agree. Flows and fields

Circulation is vorticity, added up

One of these two quantities is measured by walking round a loop and one by summing over the area inside it, and a theorem says they are the same number. That reconciles the site's most confusable pair — and explains how a flow with circulation can have no spin in it anywhere.

 A normal shock at Mach 2.00, and what crosses it unchanged. The state in front of the shock and the state behind it. Every ratio was computed from the standard jump relations and then substituted back into mass, momentum and energy, which is an independent route — a mistyped exponent in the total-pressure expression cannot survive a momentum balance it never appeared in. The residuals are printed below because a check nobody can see is a check nobody can audit. Compressible flow

The jump the equations allow

A shock is a discontinuity in a fluid, which sounds like a breakdown of the description rather than a solution of it. It is a solution: mass, momentum and energy can all be satisfied across a jump, and every ratio across one follows from that alone.

 The second law is the only thing that forbids the other half of this curve. Entropy change across a normal shock, against the Mach number in front of it. The solid branch is the compression shock that exists. The dashed branch below Mach one is the expansion shock, and it satisfies mass, momentum and energy exactly — the residuals are zero to machine precision. It is refused by the second law alone, the one statement in the problem that no conservation residual can show. Compressible flow

The only law that forbids it

The jump conditions permit a discontinuity in either direction. An expansion shock conserves mass, momentum and energy exactly — the residuals are zero to machine precision — and it does not exist. Nothing that can be drawn rules it out.

 A circle of fluid, 0.90 of a gradient time later. A material circle in a uniform velocity gradient, carried by the exact matrix exponential of that gradient. It becomes an ellipse — always an ellipse, for every gradient — and the axes it stretches along are the eigenvectors of the symmetric half. In an incompressible flow the area is unchanged however extreme the distortion, which is the statement that stretching in one direction is squashing in the other. Flows and fields

What a parcel does in the first instant

Drop a circle of dye into a flow and it becomes an ellipse. The velocity gradient that did it splits into a stretch and a spin in exactly one way, the split is not a convention, and one half of it is the reason a fluid has any stress in it at all.

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