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

Series — page 9

A field says what an essay is about. A series follows one idea essay by essay — from the question that introduces it to the one that assumes all the others.
Two-thirds to one, and nothing outside it. The drag on a spherical drop as a fraction of Stokes' drag on a rigid sphere of the same size, against the ratio of the viscosities. It runs from exactly two-thirds for a clean gas bubble to exactly one for a rigid particle and takes no value outside that range, however light or heavy the drop is — the formula has no density in it anywhere. A drop of water in air and a drop of mercury in water sit at opposite ends of it.

Mobile interface

  1. 1 The surface that moves with the flow
1 essay · viscous Nu/√Re against the Prandtl number, over eight decades. The whole of the flat plate's heat transfer, as one curve. It is not a power law: it goes as Pr^½ at the bottom, where the thermal layer is far thicker than the viscous one, and as Pr^⅓ at the top, where it is buried inside it. The Pr^⅓ everybody quotes is the upper half. The two asymptotes are drawn beside it, and the low one is √(Pr/π) in closed form.

Nusselt

  1. 1 The number that is an answer
1 essay · regimes Every one of these is a solution, and they lift different amounts. Lift coefficient against the circulation the aerofoil was told to carry, with the Kutta condition's own answer marked. Each point is a complete solve: the sources were found for that circulation and the surface is a wall to within 1.3e-4 at every collocation point. Every member satisfies the equations of motion and the boundary condition, and the lift runs through them at exactly 2Γ/Uc — Kutta–Joukowski appearing as a property of the family rather than as a result about any member of it. Ideal flow round a closed body does not have a unique answer, and the Kutta condition is the extra sentence that picks one.

Panel

  1. 1 Nothing but the edge
1 essay · circulation A sphere does not go at the speed of the flow it is in. How far a force-free sphere on the axis of a round tube lags the fluid at its own centre, against its size. The lag is two-thirds of the square of the size ratio and comes from the reciprocal theorem in one line: the sphere moves at the average of the ambient flow over its own surface, and a parabolic profile is slower everywhere on that surface than at the middle. At a fifth of the tube's radius the lag is 2.7 per cent, which is the difference between a tracer and a thing being measured.

Reciprocity

  1. 1 A force without the flow that makes it
1 essay · viscous The density ratio and the nose pressure coefficient, against Mach number. Two quantities across a normal shock, on a logarithmic Mach axis. Both approach limits that depend on γ and on nothing else: six for the density ratio and 1.8394 for the pressure coefficient at the stagnation point. By Mach five the second is within three per cent of its limit and by Mach twenty within a tenth of a per cent. Above that the flow round a blunt body has stopped depending on how fast it is going and started depending on what the gas is.

Shock layer

  1. 1 A shock that lies on the body
1 essay · compressible 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.

Slender body

  1. 1 A body with no lift, and a moment anyway
1 essay · inviscid Three exponents for one dimensionless group. The local slope of each error, measured over one decade at a time. The duct's is exactly 1, the long wave's is exactly 2, and the slender body's drifts from 1.900 to 1.733 across the range and never reaches either. The same geometric ratio, in three problems that look alike, and the third one has no exponent at all.

Slenderness

  1. 1 One group, three exponents
1 essay · regimes A slotted flap at 30°, in a flow with no viscosity anywhere. Streamlines through a main element and a flap, computed by a two-body panel solve. Each element carries its own circulation and its own Kutta condition, and the two interfere through their velocity fields and through nothing else — there is no boundary layer here, no wake, no mixing region and no high-energy air. The system's lift coefficient is 2.757 against 0.698 for the main element alone at the same incidence, and the main element itself is carrying 3.98 times the circulation it carries by itself.

Slot

  1. 1 A slot is not a nozzle
1 essay · circulation A film with nothing useful to show for itself. The pressure under two discs squeezed together, as a fraction of the pressure at the centre. It is a parabola, and it holds an enormous load — but nothing is sliding, so there is no output at all and every joule put in becomes heat in the oil. The two routes to that heat share no arithmetic: one integrates the dissipation function over the film, the other multiplies the force by the approach speed.

Squeeze film

  1. 1 The last of the oil
1 essay · viscous The only candidate the far field allows, and the wall it slips past. The general Stokes solution has four constants; the condition at infinity kills two of them and fixes a third, leaving one to satisfy two conditions at the wall. Setting the stream function to zero there uses it up, and the tangential velocity that remains is exactly twice the free stream — for every radius, every speed, and every fluid.

Stokes' paradox

  1. 1 The flow with no solution
1 essay · viscous Ideal flow past a sphere, drawn in a meridional plane. The Stokes stream function of ideal flow past a sphere, contoured at equal intervals. Its relations to the velocity carry factors of r sin θ that the plane stream function does not have, and differencing it reproduces the closed-form velocity to 10⁻¹¹. The surface speed at the equator is exactly one and a half times the free stream, against twice for a circular cylinder: a three-dimensional body lets the flow past in two directions rather than one.

Stream function

  1. 1 The one number that runs out at three dimensions
1 essay · kinematics An oscillation with no mean, and the steady flow it drives. The steady second-order velocity through a Stokes layer, in units of U U′/ω. The first-order flow averages to zero at every height; the average of its own nonlinear term does not, and the pale curve is that forcing. Integrating it twice across the layer, with no slip at the wall and no stress at the top, gives a steady velocity that rises through the layer and settles at -0.749998 — Rayleigh's −3/4, which was not put in anywhere. Beyond about five layer thicknesses nothing more happens, which is why the number is a boundary condition for the flow outside.

Streaming

  1. 1 An oscillation with somewhere to go
1 essay · viscous Where Einstein's line stops being the measurement. The viscosity of a suspension of rigid spheres relative to the liquid's, against the volume fraction. Einstein's 1 + 5φ/2 is exact for one sphere and holds while the spheres cannot feel one another, which is up to about five per cent by volume. Batchelor and Green's two-sphere term takes it a little further; beyond about a fifth nothing derived works and the curve drawn is a fit, which diverges at a maximum packing that is itself a measurement.

Suspension

  1. 1 A viscosity made of particles
1 essay · viscous A swimmer that dissipates the same everywhere. Taylor's waving sheet, with the wave drawn along the bottom and the dissipation drawn against height above it. The dissipation function works out at 4μb²c²k⁶y²e^{−2ky} — with no x in it at all, so the sheet is destroying energy at the same rate under every part of the wave and at every instant of the cycle. It peaks one radian of wavelength above the sheet and is gone within about three.

Swimming

  1. 1 A swimmer that cannot go backwards
1 essay · viscous The two long-wave speeds, and the swirl at which one of them stops. For uniform axial velocity the wave speeds follow from the criticality condition by a Galilean boost: c = W(1 ± 2S/j), with j the first zero of J1. The upstream-running root crosses zero exactly at S = j/2 = 1.9159, and above that swirl no disturbance can travel upstream — which is what subcritical and supercritical mean here and in an open channel.

Swirl

  1. 1 The swirl that holds a wave still
1 essay · inviscid Three answers for the lift-curve slope, with three different signs. Thin-aerofoil theory has no thickness term at all and returns 2π for every section. The exact potential solution rises: 2π(1 + 0.766 t/c), measured off the Joukowski map. Real sections do the opposite, because the boundary layer thickens towards the trailing edge and decambers the section. Two of these curves are computed here; the third is what measurement says.

Thickness

  1. 1 The half that carries nothing
1 essay · circulation The coefficient of the equation's second derivative, along a chord. The bracket multiplying the streamwise second derivative in the transonic small-disturbance equation, along a chord at Mach 0.85. Where it is positive the equation is elliptic and the flow is subsonic; where it is negative the equation is hyperbolic and the flow is supersonic. Which it is at a given point depends on the perturbation velocity there, which is the thing being solved for. Forty-two per cent of this chord is hyperbolic, and no amount of inspecting the problem beforehand could have said so.

Transonic

  1. 1 The equation that changes type inside its own answer
1 essay · compressible How the weight is shared between the two surfaces, against the centre of gravity. The wing's load and the tail's, for an aeroplane in level flight, against the static margin. The two must sum to the weight and their moments must cancel, and those two equations decide the split. At a forward centre of gravity the tail carries down and the wing carries more than the weight; the crossing is where the tail carries nothing.

Trim

  1. 1 More lift than weight
1 essay · circulation The wall is a boundary condition, solved for rather than reflected. A model's trailing vortices in a closed working section, with the sheet of sources that makes the wall a wall drawn as a displacement of the wall itself. The interference is an upwash — the model looks better than it is, and the correction factor comes to 0.1242 against the exact value 0.1250 that the image at the inverse point gives for a circle. The arrows are the interference velocity alone, with the model's own downwash removed, drawn to a common scale set by the longest of them.

Tunnel interference

  1. 1 The walls are in the answer
1 essay · circulation A tube opened at the foot of a reservoir. Speed in a two-metre tube fed by a one-metre head, from the instant it is opened. The quasi-steady answer — Torricelli's √(2gh), which is what dropping the time derivative from Bernoulli's equation gives — is reached at the instant of opening, before any fluid has moved. The true answer is a hyperbolic tangent with a time constant of 2L/√(2gh), and it takes 2.4 seconds to come within one per cent. The whole of that transient is one term.

Unsteady bernoulli

  1. 1 The pressure that depends on the past
1 essay · inviscid

All essays