Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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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.
A lift curve that keeps climbing to 49 degrees. The lift of a slender delta of aspect ratio 1, split into the potential term that an attached flow would give and the vortex term the separation adds, with a conventional wing's curve behind them. The delta's is nonlinear from the start and reaches 1.68 at 49 degrees, where a conventional wing stalled at fifteen. That is the whole design case for the shape: not that it is efficient — it is not — but that it still has lift at incidences where an ordinary wing has none, which is what a delta-winged aircraft needs on approach and in a turn.

Vortex lift

  1. 1 Lift out of a failure
  2. 2 Two lifts at one incidence
2 essays · circulation The growth rate a discretised sheet has, at every wavelength it can carry. Kelvin–Helmholtz gives a growth rate proportional to the wavenumber and without bound. A sheet represented by N point vortices has pi m (1 − m/N) instead — the same rate at long waves and half of it at the shortest wave the grid carries, with the fastest-growing mode at the grid scale itself. Smoothing the kernel over a length delta moves that mode back to a wavelength the physics chose.

Vortex sheet

  1. 1 A sheet that cannot stay a sheet
  2. 2 A spiral is a legible record
2 essays · inviscid The street, as two rows of point vortices. The exact velocity field of a staggered double row of point vortices at the stable spacing ratio, with the row spacing and the circulation of one core printed from a line integral of the field rather than from the number that built it. This is a model of a wake. It contains no body, no viscosity and no mechanism that would shed anything, and the viscous stepper used here does not produce a street at any Reynolds number.

Wake

  1. 1 The street a steady calculation cannot draw
  2. 2 A wake that keeps the drag and forgets the body
2 essays · turbulence Four aeroplanes, one number. Four biplanes with the same gap and the same loadings, staggered by nothing, by four tenths, by nine tenths and by one and six tenths of a chord. Their induced drags agree to the last bit of the arithmetic — the calculation cannot even express the stagger, because the Trefftz plane is a cross-section and everything drawn here projects onto the same one. That is Munk's stagger theorem, and stating it as the drag is unchanged understates it: there is no place in the computation where the stagger could be entered.

Biplane

  1. 1 Two wings and it does not matter where
1 essay · circulation Drop deformation in simple shear, against the capillary number. The shape model's steady deformation for six viscosity ratios. Every curve is linear in the capillary number at small Ca — which is Taylor's result — and every one of them saturates, at 5/2(2λ+3), because the shear's own rotation turns the drop out of the stretching direction. A drop in simple shear cannot be deformed beyond that however hard it is sheared.

Capillary

  1. 1 The number that cannot break a drop
1 essay · regimes The trailing-edge speed against circulation, for a sharp edge and a round one. Sampled one grid point off the trailing edge. The sharp edge is singular at every circulation but one: the speed there is about U at the Kutta value and 11.3U half a Kutta circulation away, and it grows without bound as the sample approaches the edge. The round edge has no such point. That is the whole of the Kutta condition's justification, and it needs the corner.

Circulation control

  1. 1 The condition that can be bought
1 essay · circulation Flow past a cylinder at Re 100. A real fluid past a circular cylinder. At low Reynolds number the flow closes up behind the body much as the ideal theory says; as it rises the flow separates and a region of reversed flow appears behind, which is where drag comes from.

Comparison

  1. 1 The two theories, side by side
1 essay · viscous A curved shock, and the entropy each streamline picks up crossing it. A parabolic bow shock ahead of a blunt nose at Mach six, with the streamlines drawn arriving horizontally and a marker at each crossing whose size is the total pressure lost there. The streamline through the nose crosses a normal shock and keeps three per cent of its total pressure; one four nose radii out crosses at fourteen degrees and keeps ninety-four per cent. Every streamline gets a different entropy, and the stagnation enthalpy is the same on all of them.

Crocco

  1. 1 The spin a shock leaves behind
1 essay · compressible Stokes' drag stops being the answer almost at once. The damping on a millimetre sphere oscillating in air, divided by Stokes' steady drag on the same sphere, against frequency. The extra term is the sphere's radius over the layer thickness, so it takes over as soon as the layer is thinner than the body — which for a millimetre sphere in air is below one hertz. At a kilohertz the damping is fifteen times the steady value, and the exponent is a half rather than zero.

Damping

  1. 1 What a fluid takes out of a swing
1 essay · viscous A fifth of the drag power is not heat yet. The energy account of towing a flat plate a metre long through air at thirty metres a second. The power it takes is the drag times the speed. The heat made inside the boundary layer is half the free-stream energy times the energy thickness, and it is less — 78.6 per cent of what was paid. The rest has not been destroyed: it is kinetic energy still in the wake, which will become heat somewhere downstream, in fluid that is no longer touching the plate.

Energy thickness

  1. 1 The third thickness
1 essay · viscous A double integral that comes out an integer. The Gauss linking integral evaluated on six pairs of closed curves. It is not constrained to be a whole number by anything in its own definition — it is a double integral of a smooth kernel — and it returns one to within two parts in ten thousand on two hundred points per curve, because what it is computing is a topological count.

Helicity

  1. 1 The knot a flow cannot untie
1 essay · kinematics One field, and the two parts the theorem splits it into. A velocity field made of a smooth source, a smooth vortex and a uniform stream, and the two fields the Helmholtz decomposition returns for it. The first carries the whole divergence and has no curl anywhere; the second carries the whole curl and has no divergence. They are computed by solving two Poisson problems on a grid, with the divergence and the vorticity differenced from the field rather than taken from the expressions that built it. Adding the two back together does not recover the field.

Helmholtz

  1. 1 Every flow is two flows
1 essay · kinematics The section the pressure asked for. The designed section over the one it started from, both drawn to their own chords. Asking for 28 per cent more speed over the forward 62 per cent of the upper surface produces a section 15.0 per cent thick against the original's 10, with the extra thickness forward and the camber changed — none of which was asked for, and all of which is what that pressure distribution is. The pale outline is the baseline. The one thing the method cannot be told is where along the chord any of it happens: the speed is prescribed against the circle's parameter, and where a given station ends up is an output of the same solve that produces the shape.

Inverse design

  1. 1 Ask for the pressure, and see what shape that is
1 essay · inviscid The two ways a shock can meet a wall. Left: the incident shock from the wedge reaches the wall and a second shock turns the flow back parallel to it, meeting at a point. Right: at a larger wedge angle no reflected shock can turn the flow that far, and the intersection lifts off the wall into a triple point with a nearly normal Mach stem standing on the surface and a slip line trailing downstream. The two configurations are drawn at the angles the solver returns, with every shock angle computed rather than sketched.

Mach reflection

  1. 1 When a shock cannot bounce
1 essay · compressible 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 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 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

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