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

Non dimensionalisation — where it appears

The rewriting of a problem in terms of ratios, so that the number of parameters falls to the number that can actually change the answer. It is what makes a wind-tunnel test mean anything, and the groups it leaves are the regimes the subject is organised by.

Named by 6 essays across 4 fields — each of them below, with the objects they name alongside it.

Reynolds number: one number, four different flows. Reynolds number is inertia ÷ viscosity. It is not a property of the fluid or of the shape but of the combination, and crossing a threshold changes the physics rather than the magnitude.

The Reynolds number, and the length in it

The most useful number in fluid mechanics has an arbitrary quantity buried in it, and quoting one without saying which length was used makes it meaningless. That detail is where most misuse comes from.

regimes · Reynolds The speed for least drag is not the speed for least power. Drag and power against airspeed, in units of the speed at which drag is least. Power is drag times speed, so its minimum sits slower — at the fourth root of a third of the least-drag speed, which is 0.76 of it. Flying for range and flying for endurance are therefore different speeds, and the difference is not a rule of thumb.

The cheapest way to stay up

There is a speed at which an aircraft's drag is least, and a different, slower speed at which its power is least. The ratio between them is the fourth root of a third — a number that does not depend on the aircraft, the altitude, or anything else about the flight.

viscous · Drag budget The pressure coefficient, read off the field. The closed-form surface pressure of a cylinder, and the same quantity computed from the speeds of the solved field at the same points. The coefficient is exactly one where the flow stops and exactly minus three at the shoulder, and those two numbers are properties of the shape rather than of the tunnel.

The number that does not depend on the tunnel

A pressure measured in a wind tunnel is a fact about that tunnel on that day. Divide it by the dynamic pressure and it becomes a fact about the shape — the same at any speed, in any fluid, at any scale, and equal to exactly one where the flow comes to rest.

inviscid · Pressure The correction belongs to the wing, not to the air. Lift-curve slope against Mach number for three aspect ratios, each computed by solving the lifting line for the wing the transformation actually implies — one of aspect ratio βAR — and dividing by β. The dashed curve is the two-dimensional rule, which is what an aerofoil section gets. At Mach 0.7 the aspect ratio of 20 has gained 35 per cent of slope and the aspect ratio of 4 has gained 24, against the 40 per cent the section rule promises both. The β in the finite-wing term cancels the β in front of it, so the shorter the wing the less compressibility does to it.

The wing the equation is really solving

The Prandtl–Glauert rule is usually quoted as a factor on the answer. It is a change of shape — and in three dimensions the shape it changes is the aspect ratio, so a short wing is far less affected by compressibility than a long one.

regimes · Similarity Storage and loss moduli, for one relaxation time and for a spectrum. The two moduli of a Maxwell fluid and of a Rouse chain, against frequency in units of the longest relaxation time. One relaxation time makes the storage modulus overtake the loss modulus at λω = 1 and then leave it behind without limit. A spectrum of 1000 modes makes them rise together as the square root of frequency and stay a fixed ratio apart, so the material never becomes the solid the single time predicts.

A solid, if it is not given time

The Deborah number is the only such group with no fluid in it at all — two times and nothing else — and it says a material is a solid or a liquid depending on how long anybody watches. What it throws away is that no real material has one time, and the spectrum it replaces changes the answer in kind rather than in degree.

regimes · Deborah Every ball's best angle is on one curve. The launch angle that carries a ball furthest over level ground, against β, the square of its launch speed measured in terminal speeds. At small β the air does nothing and the answer is 45°. It falls slowly as β grows: 41° at β = 1, 32.6° at 10 and 20.3° at 1000. A shot put sits at β = 0.009 and 44.9°, a golf ball struck without spin at β = 2.39 and 38.1°, a shuttlecock hit at 30 m/s at β = 19.5 and 30.2°.

The terminal speed sets the best launch angle

In a vacuum every ball goes furthest at forty-five degrees. In air the best angle is lower, and it is lower by an amount that depends on one number only: how many times its own terminal speed the ball is launched at. A shot put, a baseball and a shuttlecock are three points on one curve, which falls from 45° to 41° at once and then needs a thousandfold more speed to reach 20°.

applied · Sport ball

Named alongside it

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

DragLifting lineSimilarityAspect ratioBernoulli's equationBlasiusBoundary conditionCompressibilityConstitutive lawCritical machDimensional analysisDimensionless

All concepts