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

Every essay — page 27

Page 27 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

Regimes and numbers

Reynolds, Mach, Froude, Strouhal. One number decides whether a flow creeps, separates or shocks, and the same shape behaves differently at each.

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 model that cannot be matched

A scale model behaves like the real thing when its dimensionless numbers agree. With one number that is a matter of choosing the tunnel speed. With two it is usually impossible, and every wind-tunnel result ever published has been obtained in spite of that.

7 figures Creeping flow, and the same body with inertia. The exact creeping-flow solution beside a solved field at a Reynolds number where inertia matters. The creeping flow is a mirror image of itself front to back — a photograph of it run backwards is a photograph of it — and the field with inertia has a wake, which is what a direction of time looks like.

The world with no inertia

Drop the viscosity and the equations become exactly solvable and wrong about drag. Drop the inertia instead and they become exactly solvable again — and for a cylinder in an unbounded fluid there is no solution at all, which took fifty years to notice and longer to fix.

7 figures Where the flow over the section first reaches Mach one. Two curves running towards each other. The falling one is the section's peak suction, solved exactly in incompressible flow and then deepened by the Prandtl–Glauert factor as the free-stream Mach number rises. The rising one is the pressure coefficient at which the local flow would be exactly sonic. Where they cross is the critical Mach number, and above it there is a pocket of supersonic flow on a subsonic aeroplane.

The pocket on top of the wing

An airliner cruising at Mach 0.85 has subsonic flow almost everywhere and a patch of supersonic flow over its wing. Closing that patch takes a shock, the shock separates the layer beneath it, and the correction that predicts all of this also predicts, in the plainest terms, where it stops being true.

6 figures 5 quantities, 3 rows, 2 left over. The dimension matrix for the drag on a sphere: one column per quantity, one row per base dimension, and every entry an exponent. Buckingham's theorem is a statement about this matrix and nothing else — the number of independent dimensionless groups is the number of columns minus the rank, computed here by elimination. Nothing about fluids enters until somebody decides which columns to write down.

Counting what matters

Five quantities decide the drag on a sphere, and the experiment that measures it has one curve in it rather than a five-dimensional table. The reason is a rank: the matrix of dimensions has three independent rows, and what is left over is the number of dimensionless groups the answer can possibly depend on.

7 figures Particles at St = 1, against the flow that carries them. Particle paths and the streamlines they were released on, in this site's exact cylinder solution. At small Stokes number the two are indistinguishable and the body catches nothing; as the particles get heavier their paths straighten, cross the streamlines, and begin to strike. The paths are integrated with Stokes drag and nothing else — no gravity, no lift, no effect of the particles on the flow.

Whether the droplet turns

The air goes round the wing. Whether what is carried in it goes round too is decided by one number — and below a critical value of that number the body collects nothing at all, however many droplets are thrown at it, because the flow turns every one of them in time.

8 figures The Ohnesorge diagram, with the boundaries where they belong. The classical map of jet break-up: the Ohnesorge number against the jet Reynolds number, with the five nozzles placed on it. The three sloping lines are Reitz's transitions in the gas Weber number, and their geometry is computed rather than sketched — a fixed We_g means Oh·Re is fixed, which is a straight line of slope exactly −1 in these coordinates, and the assertion checks that a decade in Reynolds number moves each line by exactly one decade. Where they sit is borrowed; that they are straight and parallel is not. A nozzle below and to the right of the last line atomises.

Where a jet stops being a jet

A tap makes drops a few centimetres down, a garden hose makes a stream that carries, a sprayer makes a mist and a diesel injector makes fog. Same liquid, same mechanism, four regimes — and the number that separates them is not the jet's inertia but the surrounding air's.

8 figures A million times faster, and the constant is 48.0. The effective diffusivity along a pipe, divided by the molecular one, against the Péclet number — both logarithmic. Below Pe ≈ 7 the tracer simply diffuses and the curve is flat at one. Above it the dispersion is all Taylor's, rising as the square of the Péclet number, so a thousandfold Péclet number is a millionfold enhancement. The constant in D(1 + Pe²/48) is not quoted here: it is recovered from a numerical solution of the cell problem across the section, giving 48.0000 for a tube and 52.5 for a plane channel, which is Aris' 2/105.

Two slow things make a fast one

Shear stretches a slug of dye and mixes nothing, because it is reversible. Molecular diffusion is hopeless at any scale bigger than a hair. Put the two together in a pipe and the dye spreads along it with an effective diffusivity two million times the molecular one — which gets larger as the molecular one gets smaller.

8 figures α = 13.0: a plug in the middle and everything happening at the wall. The velocity profile at eight phases of one cycle, at a Womersley number of 13.0 — the human aorta's at rest. The core moves almost as a solid plug, because viscosity cannot reach it within a cycle; all the shear is in a layer of thickness √(ν/ω) = 0.77 mm against a radius of 10.0 mm. At some phases the fluid near the wall is moving backwards while the core still moves forward, which is the reversal a Poiseuille profile can never show and which is routinely measured in arteries.

Too fast for a profile

A pipe carrying a steady flow has a parabolic profile. Make the pressure oscillate and one number decides whether it still does — and above about ten the core moves as a plug, a quarter of a cycle behind the pressure, with the fastest fluid in a ring near the wall rather than on the axis.

8 figures One layer, and the four others inside and outside it. The velocity profile over a flat plate, and the temperature profile in the same layer at four Prandtl numbers: a liquid metal at 0.01, air at 0.71, water at 7 and a heavy oil at 100. The equations differ by one number and the profiles differ by a factor of twenty in thickness. At Pr = 1 the two are the same function — not similar, identical, to eight decimal places — because the equations and the conditions are then the same, which is what every statement called a Reynolds analogy rests on.

The other layer, and the one number that separates them

A wall in a stream carries two boundary conditions and grows two layers. Their thicknesses differ by a factor of twenty across ordinary fluids, and at exactly one Prandtl number the two profiles are not similar but identical.

8 figures 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.

8 figures Where the number says, and where it happens. Fourteen dimensionless groups on one logarithmic axis. The open circle on each row is the value at which the two terms the group compares are equal, which is one by the way the group is formed; the filled mark is the value at which the thing a reader cares about first changes by 1%. The bar between them is the distance the folklore phrase "of order one" hides, and it runs from nothing at all to a factor of 594.

What "of order one" is worth

A dimensionless group is built by comparing two terms, so it is one when the terms are equal — and that is the only thing it says. Where the behaviour actually changes is a separate question with a separate answer, and across fourteen groups on this site the two numbers differ by factors from one to five hundred and ninety-four.

10 figures How wrong the ordinary answer already is. The error in a no-slip continuum calculation of the flow through a channel, against the Knudsen number, both logarithmic. The rule at Kn = 1 is where a molecule crosses the whole channel between collisions — the value the number is named for. The error is one per cent at Kn = 1/594, five per cent at 1/114 and ten at 1/54, so the continuum regime of the usual classification, which runs to Kn = 0.01, is a region in which the continuum answer is already six per cent out at its far end.

Where a fluid stops being one

The Knudsen number is the mean free path over the size of the thing, and at one a molecule crosses the whole channel between collisions. The continuum equations with a no-slip wall are already one per cent wrong at one part in five hundred and ninety-four, which is a factor nothing about the definition would suggest.

8 figures