Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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Series — page 5

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
  2. 2 A thousandth of a monolayer holds a bubble still
  3. 3 The warm side pulls a bubble with no force on it
  4. 4 A cap that slows a rising bubble stops a migrating one
4 essays · viscous The lift curve that made flight look impossible. Lift coefficient against incidence, by Newton's impact theory and by thin-aerofoil theory. One is quadratic in the angle and the other linear, so at small incidence — which is where aircraft fly — they differ by more than an order of magnitude. Newton's version says a wing large enough to carry a man would need an engine nobody could build, and for a century that arithmetic was taken as settling the question.

Newtonian

  1. 1 The theory that forbade flight
  2. 2 The only theory simple enough to optimise
  3. 3 The face Newton left in shadow
  4. 4 The spot a local theory cannot see
4 essays · misconceptions Seven fluids in one pipe at one pressure gradient. Velocity profiles of power-law fluids in a round pipe, all at the same pressure gradient and the same consistency, scaled to the fastest. A shear-thinning fluid (n below one) is flatter in the middle and steeper at the wall; a shear-thickening one is the reverse. The flattening is often called plug-like, which invites the reading that the fluid is moving more freely — what has actually happened is that all the shear has been pushed into a thin annulus at the wall, which is the expensive place to put it.

Non-newtonian

  1. 1 A viscosity that depends on the question
  2. 2 The core that does not move
  3. 3 The stress a pipe knows
  4. 4 The fluid that has not finished its last deformation
4 essays · viscous Ideal flow past a cylinder. A uniform stream past a circular cylinder in a fluid with no viscosity. The solution is exact and closed-form: streamlines part at a stagnation point, run round the surface and close up perfectly behind, and the pressure recovers to exactly what it was in front.

Pressure

  1. 1 Fast means low pressure
  2. 2 The number that does not depend on the tunnel
  3. 3 What the airspeed indicator believes
  4. 4 Pressure has no speed
4 essays · inviscid 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.

Reynolds

  1. 1 One number decides which physics applies
  2. 2 The Reynolds number, and the length in it
  3. 3 The world with no inertia
  4. 4 How small is small enough
4 essays · regimes 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.

Separation

  1. 1 When the flow lets go
  2. 2 How much uphill a layer can take
  3. 3 Where the straight line stops
  4. 4 The gradient that does both
4 essays · viscous The one exact law, and where viscosity takes it back. The two terms of the Kármán–Howarth relation against separation, measured in Kolmogorov lengths. The four-fifths term rises linearly with r and is the whole of the law at large separation; the viscous term, 6ν dS₂/dr, falls as r^(−1/3) and takes over below 5.64η. Nothing in the calculation was told what η is: the crossing is at the same multiple of it at every viscosity tried, which is what makes the dissipation scale the lower end of the inertial range rather than a separate assumption.

Structure function

  1. 1 The one exact result
  2. 2 The fraction that is really four thirds
  3. 3 A relation with no turbulence in it
  4. 4 At higher orders the bottleneck hides intermittency
4 essays · turbulence What a barometer on the wing would read at 60 m/s. The absolute pressure along an aerofoil's surface, in kilopascals, with the atmosphere put back. Nothing on the wing is near zero: the deepest point of the so-called suction peak is a few per cent below a hundred kilopascals, and the air there is still pushing on the wing hard enough to hold up a column of itself eight kilometres tall. The lift is the difference between two large pushes, not a pull.

Suction

  1. 1 Nothing sucks
  2. 2 A force forgets the datum, a stress cannot
  3. 3 Where a liquid does pull
  4. 4 A breaking strength that is the size of a flaw
4 essays · misconceptions Elliptic loading rolls up to πb/4, and it is π that puts it there. Three loadings on the same span carrying the same lift, with the rolled-up core positions marked below each. The core sits at the centroid of the shed vorticity, which is a quadrature over the loading and needs nothing about the roll-up itself. Elliptic loading gives πb/8 from the centreline, so the pair ends up 0.7854 of the span apart — the number every wake-separation rule is written against, and one of the few places in this subject where π turns up in an answer an engineer uses directly. The bell rolls up to 0.586 and a nearly rectangular loading to 0.978, because it sheds at the tips.

Tip vortex

  1. 1 Where the wake ends up
  2. 2 A wake that says what made it
  3. 3 A wake ends by bending, not by fading
  4. 4 Four vortices bend faster, and bend the wrong one
4 essays · circulation The laminar line does not end; the flow leaves it. Friction factor against Reynolds number in a pipe. The laminar law f = 64/Re is exact and is drawn continuing past the transitional Reynolds number, faintly, because it remains a solution there — the flow simply stops taking it. The turbulent branch is Blasius' correlation and begins where experiments find transition, not where any calculation puts it.

Transition

  1. 1 The solutions stop being chosen
  2. 2 The number that is not a number
  3. 3 Every mode decays and it grows anyway
  4. 4 A puff that does not know how old it is
4 essays · turbulence 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
  2. 2 One curve for every thickness
  3. 3 A drag that needs a discontinuity
  4. 4 A shock that will not stand still
4 essays · compressible Two of opposite sign go somewhere. Two vortices of equal and opposite strength. Each is carried by the other's field, both are carried the same way, and the pair travels in a straight line at Γ/2πd forever, keeping its separation exactly. The speed is a consequence of one vortex's field evaluated at the other, and nothing else.

Vortex dynamics

  1. 1 Vortices move each other
  2. 2 Three is the most that can be predicted
  3. 3 What a point vortex is not
  4. 4 Reversible, and unusable
4 essays · inviscid Kirchhoff's rotation rate, which a point vortex does not have. A patch of uniform vorticity bounded by an ellipse turns rigidly at omega a b/(a+b)², a rate that depends on the shape alone. It is largest for a circle, where it is unobservable, and falls away as the patch is drawn out. A point vortex has no shape and therefore no entry on this axis at all.

Vortex patch

  1. 1 The shape a vortex keeps
  2. 2 Past three, an ellipse is a shear layer
  3. 3 A strained vortex holds until it has no shape to hold
  4. 4 A strained vortex loses its ring at the limit of what it holds
4 essays · inviscid 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 a decade in Reynolds number is checked to move 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.

Atomisation

  1. 1 Where a jet stops being a jet
  2. 2 Viscosity lets a jet break, later and into bigger drops
  3. 3 The air shortens a jet's fastest ripple, and the drops follow
3 essays · regimes A steady pressure field, from a flow with no steady part. The time-averaged pressure round a cylinder in a stream that oscillates as U₀cos ωt. The mean velocity is exactly zero at every point — the flow spends as long going one way as the other — and the mean pressure is not, because pressure depends on the square of the speed and a square has no sign. The mean coefficient reaches -2.00 at the shoulders and averages -1.00 over the surface, and its resultant is 6.6e-16: a real field with no force in it. The pale lines are the instantaneous streamlines, which reverse every half cycle.

Averaging

  1. 1 The mean is not the flow
  2. 2 Two averages of one flow
  3. 3 What a mean profile cannot tell anybody
3 essays · kinematics The coefficient Stokes threw away is not a correction. What absorbs sound in dry air, split into the three transport effects that add to make the diffusivity of sound. Shear viscosity is the largest single contribution and it is not most of it; the bulk viscosity — which Stokes set to zero in 1845, saying in the same paper that he had no argument for doing so — is a quarter, and thermal conduction is a fifth. Setting ζ to zero does not make a small error in the absorption; it makes a 24 per cent one.

Bulk viscosity

  1. 1 The viscosity nobody uses
  2. 2 The largest bulk viscosity is the first to expire
  3. 3 A bulk viscosity holds a shock together until it splits
3 essays · viscous The wall is what cancels the waves it made. The characteristic net of a minimum-length nozzle designed for Mach 2.4, with 18 waves. The pale lines run from the sharp throat down to the axis, reflect there by symmetry, and run back up to the wall; the wall turns through exactly the angle needed to cancel each one as it arrives, so nothing reflects back into the flow and the exit is uniform at Mach 2.400 and parallel to 0.0 degrees. The area ratio decides the Mach number and this net decides the shape, and the two agree on the exit height to 0.51 per cent at this resolution.

Characteristics

  1. 1 The wall that cancels its own waves
  2. 2 Every compression becomes a shock in the end
  3. 3 Two numbers that do not change
3 essays · compressible Q and the vorticity along a radius of one vortex. Two candidate measures of where the vortex is, along a radius of a Lamb–Oseen vortex. The vorticity is a Gaussian: positive at every radius, so a threshold on it puts the edge wherever the threshold is put. Q — the excess of rotation over strain — changes sign exactly once, at 1.121 core radii, and that radius is a property of the flow rather than of the person drawing it. Inside it, 71.5 per cent of the circulation.

Coherent structures

  1. 1 Where a vortex stops
  2. 2 A boundary that only exists over a window
  3. 3 An edge no observer can move still depends on what it is averaged against
3 essays · kinematics 45° of corner, and an infinite number of eddies in it. The creeping flow in a corner of 45 degrees, drawn from Moffatt's similarity solution. Each eddy turns the opposite way to its neighbours and is 3.17 times smaller and 1.6e+3 times weaker than the one outside it. The contour levels are rescaled inside each eddy, because they differ in strength by three orders of magnitude per step and a single set of levels would show the first and nothing else — which is itself the reason nobody has seen the third. The dividing lines between eddies are drawn where the stream function changes sign, and the dots are the centres, both found from the solved field.

Corner

  1. 1 The eddies nobody stirs
  2. 2 A scraper held flat is pushed off the wall it scrapes
  3. 3 A pressed blade floats on its land and leaves half the gap behind
3 essays · viscous 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.

Crossover

  1. 1 What "of order one" is worth
  2. 2 One formula for both ends
  3. 3 The three that never converge
3 essays · regimes

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