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

Viscosity — where it appears

The property by which a fluid resists being sheared, carrying momentum across streamlines. It is small enough in air and water to be neglected almost everywhere, and the places where it cannot be neglected decide the drag of everything.

Named by 33 essays across 6 fields — each of them below, with the objects they name alongside it.

Flow past a cylinder at Re 40. 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.

Everything happens in a layer you cannot see

Air has so little viscosity that ignoring it works almost everywhere. Almost everywhere leaves out a film next to the surface, perhaps a millimetre thick, and that film decides drag, stall and whether an aircraft flies at all.

viscous · Boundary layer 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.

regimes · Reynolds One more condition, and the price of it. The model problem ε u″ + u′ = 0 with a condition at each end, at three values of ε. The outer solution is the flat line at one — that is the whole of the answer when ε is zero, and it is a first-order equation that can meet one condition, so it meets the one at the far end and misses the one at the wall by the whole range. Restoring ε restores the second condition and pays for it with a layer of thickness ε, inside which the gradient is of order 1/ε. The product of those two — which is what a stress is — does not depend on ε at all: it is 1.0000 at every value tried, to nine decimal places. Drag does not vanish as viscosity does. It converges.

How many things a flow must be told

The equations of motion do not have one answer. They have as many as the conditions on the edge allow, and the number of those is decided by the order of the equation — which is why viscosity does not make the same problem harder, it makes a different problem.

kinematics · Boundary conditions 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.

regimes · Womersley 20.45 MPa out of a film 25 µm thick. The pressure along a tapered pad, from the closed-form solution of Reynolds' equation, with the same equation's tridiagonal grid solve drawn over it as points. The peak is 20.45 MPa — enough to yield mild steel — and it sits at 69 per cent of the way along rather than in the middle, because the pressure gradient vanishes where the film equals the harmonic mean of its two ends and the harmonic mean is biased towards the thinner one. The pad's own shape is drawn along the top, to a vertical scale of its own. Nothing pumps this oil: the runner drags it into a narrowing gap and the gap does the rest.

Nothing but the shape of the gap

A machine that holds a steel shaft off its bearing with a film of oil twenty-five microns thick has no pump in it, and the pressure it generates would yield mild steel. The mechanism is not the oil and not the speed; it is that the gap narrows.

viscous · Lubrication The wall makes vorticity at a rate with no viscosity in it. At a stationary wall the momentum equation collapses to ν ∂²u/∂y² = (1/ρ) ∂p/∂x, and the left-hand side is the diffusive flux of vorticity out of the surface. So the pressure gradient along the wall is the vorticity source, and the viscosity that made the no-slip condition necessary has cancelled out of what the condition produces. The curve is that flux across the Falkner–Skan family, computed from profiles solved by shooting and differenced at the wall; the straight line is the pressure gradient each of those flows has. They agree to 2.3e-14. At zero pressure gradient the flux is exactly zero: a flat plate creates no vorticity at all after its leading edge, and everything in its layer arrived from there.

Where vorticity comes from

Every scrap of vorticity in a flow past a body entered through its surface, and the rate at which it enters contains no viscosity at all — it is the pressure gradient along the wall. A flat plate makes none, and a closed body makes exactly as much of each sign.

kinematics · Vorticity 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.

The eddies nobody stirs

A slow flow past the mouth of a sharp corner does not simply fail to get in. The corner fills with an infinite sequence of counter-rotating eddies, each about three times smaller and sixteen hundred times weaker than the last, and the whole structure is decided by one complex number.

viscous · Corner The air a wing carries along, and how little of it there is. The Blasius profile, in the wing's frame, with the free stream at one. No slip says the air at the surface is at rest relative to the surface, which in the ground's frame means it is moving with the wing — but only exactly at the wall. The deficit, integrated across the layer, is the displacement thickness: at a Reynolds number of 1e+6 and a metre of chord it is 1.72 millimetres of air moving at flight speed, which is the whole of what is 'carried'. The step drawn on the axis is that same deficit as a solid slab. A wing does not drag a blanket of air with it; it leaves a boundary layer behind it, and the layer is made of air that keeps being replaced.

The air a wing does not carry

No slip says the air touching a surface moves with it, and the usual reading is that a wing drags a blanket of air along. The blanket is 1.7 millimetres thick per metre of chord, it is different air every instant, and the drag it costs falls as it gets thicker.

misconceptions · The no-slip condition The jet has to be fed from the sides. The velocity field of the plane jet with streamlines integrated through it. The seven central streamlines run down the jet and spread; the ten started at the top and bottom edges bend inwards and join it, which is entrainment and is a consequence of the solution rather than an addition to it. The dashed lines are the half-speed edges, widening as x^{2/3}. The transverse velocity far from the axis is 5.70e-3 m/s at this station, inward on both sides — a jet is a sink as seen from a distance, which is why two parallel jets pull together.

What a jet keeps, and what it collects

A jet leaving a nozzle into still fluid has no boundary anywhere and one conserved quantity. Its momentum flux is exactly the same at every station downstream; its mass flux is not conserved at all and grows without limit, because a jet is a machine for acquiring fluid it did not start with.

viscous · Free shear One number instead of a porous medium. The velocity through the bottom of a channel whose lower wall is a porous block of permeability 1e-4. Inside the block the flow decays over the pore scale √K = 1.0e-2 to Darcy's seepage velocity; above it the channel profile arrives at the interface with a slip velocity rather than at rest. A channel told nothing but u = √K du/dy at a flat wall reproduces that profile to 0.058 per cent of the flow rate, against 3.03 per cent for a wall told to hold the fluid still. The grid solve of the coupled problem agrees with the closed form to 0.0077 per cent.

A wall that is not quite there

A porous surface has structure on every scale below the pore, and no calculation resolves it. The whole of it can be replaced by one length — the square root of the permeability — and the replacement is exact to first order, with what it leaves out identifiable as the flow the wall itself carries.

kinematics · Boundary conditions The core spreads and the outside never notices. The swirl velocity at four times a factor of four apart, with the free vortex Γ/2πr drawn behind them. Every curve leaves the free-vortex line at its own core radius and turns over into solid-body rotation inside it; outside the core all four are the same curve, to the precision of the plot. Viscosity has rounded off the singularity and changed nothing else. The peak swirl falls from 52.4 to 6.6 m/s across the four, and the circulation is identical for all of them.

What viscosity cannot take away

Leave a vortex alone in a viscous fluid and every local measure of it falls — the peak spin, the peak velocity, the enstrophy. The circulation round a large loop does not move at all, ever, and the far field is identical to the line vortex it started as.

viscous · Diffusion 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.

regimes · Knudsen The current at the surface is 45° from the wind, and nothing sets that angle. The Ekman spiral drawn as a hodograph: each point is the velocity at one depth, and depth runs along the curve. At the surface the flow is at exactly 45 degrees to the wind that drives it — not approximately, exactly, and independently of the wind, the viscosity and the latitude. By one Ekman depth the flow has turned another radian and lost 1/e of its speed; by three it is a hundredth of the surface value and pointing back the way it came. The angle is a property of the equation having two terms in it, and nothing else.

The layer that stops at a depth

Every other boundary layer grows. This one does not — rotation supplies a frequency, the balance against diffusion supplies a length, and the transport that comes out contains the stress on the surface and not the viscosity underneath it.

viscous · Rotating 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.

An oscillation with somewhere to go

Shake a fluid back and forth over a body and it develops a steady circulation that never reverses. The driving flow has no mean at all; the mean of its own nonlinear term does, and integrating that twice across the oscillatory layer gives a slip velocity of exactly three-quarters of U dU/dx over the frequency.

viscous · Streaming The same number, by two integrals that share no arithmetic. Three flows whose dissipation is in closed form both ways. The volume route integrates the dissipation function over the fluid; the boundary route multiplies a force or a torque by the speed of whatever is applying it. Neither calculation contains the other, and the residual column is what is left when they are subtracted.

The price of a gradient

Viscosity does not charge for motion. It charges for the rate at which a parcel is being deformed, and a fluid in solid-body rotation at any speed whatever destroys nothing at all. What is charged for is a sum of squares, which is why the bill can be computed twice.

viscous · Dissipation Two things moving by a million, and their product standing still. Viscosity, the squared velocity gradient at the dissipation scale, and their product, across six decades of Reynolds number at a fixed large-scale flow. The viscosity falls by a factor of 1e+6; the squared gradient rises by exactly the same factor, because η falls as Re^(−3/4) and u_η as Re^(−1/4); and the dissipation ν(u_η/η)² does not move at all. That is the dissipation anomaly stated as arithmetic: the limit of the dissipation as viscosity vanishes is not the value it takes when viscosity is zero.

The limit that is not the value

Dissipation is viscosity times the square of a velocity gradient, so it ought to vanish as the viscosity does. It does not. The gradient rises by exactly the factor the viscosity falls by, the product stands still, and a fluid with no viscosity at all would dissipate nothing — which is why the limit and the value are different numbers.

turbulence · Dissipation The parabola is the cheapest shape the walls allow. The dissipation of the profile u = A(1 − |y/h|ⁿ) carrying a fixed flux between fixed walls, against the exponent n, divided by the parabola's. Every member of the family satisfies the same boundary conditions and carries the same fluid; they differ only in shape. The minimum is at n = 2 exactly, which is not a coincidence — it is Helmholtz's theorem, and the parabola is a solution of the equations because it is the least dissipative shape rather than the other way round.

The cheapest shape the walls allow

The parabola in a pipe is usually presented as what the equations give. It is better understood the other way round — of every profile that could carry that flow between those walls, it is the one that destroys the least energy, and the equations give it for that reason.

viscous · Minimum-dissipation 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.

The last of the oil

A wet plate on a table is held down by nothing but the difficulty of getting air in underneath it. The force required to separate two flat surfaces with a film between them goes as the inverse cube of the gap, which means there is no finite energy that removes the last of the film and no finite time in which it drains away.

viscous · Squeeze film How hot seven films get. The peak temperature rise inside the film itself, on a logarithmic axis, for seven lubricated contacts with ordinary engineering numbers. A journal bearing runs tens of kelvin above its own housing, which is why lubrication systems are designed around heat rather than around load; a water-lubricated bush runs at a hundredth of a kelvin, because water conducts six hundred times better per unit of viscosity. The air bearing is marked: its gap is smaller than air's mean free path, so its film is not a continuum at all.

The film that heats itself

The oil in a bearing is not at the temperature of the metal around it. It is tens of kelvin hotter, the rise contains no length whatever, and whether the heating runs away or settles down depends on something that is not a property of the oil at all — whether the machine driving it holds the speed or holds the force.

viscous · Viscous heating Twice as fast is not twice as thick. The film a plate carries out of water, against the speed it is withdrawn at, both logarithmic. The slope is exactly two-thirds, so doubling the speed thickens the film by 58.7 per cent and never by more. The open marks are outside the range the derivation holds in — above a capillary number of about a hundredth the film is no longer thin against the capillary length, and the measured thickness leaves this line.

What a plate takes with it

Pull a plate out of a bath and it comes out wet. How wet is not set by the plate, the bath or how much liquid there is, but by a competition in a region a fraction of a millimetre long that nobody looking at the plate can see — and the film goes as the two-thirds power of the speed, never as the first.

viscous · Coating 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 group on this site with no fluid in it — 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 The velocity through a shock at Mach two, as a function of position. The Becker profile, integrated outwards from its own inflection point. It runs from the upstream velocity to the downstream one — the two states the jump conditions give, which appear here as the equilibria of a first-order differential equation — and its steepest gradient matches the closed form to one part in ten thousand. The horizontal axis is in units of the thickness, which for this shock is 139 nanometres.

The discontinuity that has a thickness

The jump conditions do not contain the viscosity, which is why they are exact. The thickness is entirely viscosity — 289 nanometres at Mach 1.5, 35 at Mach 5, against a mean free path of 64. At Mach five the continuum equations have produced a structure thinner than the distance between collisions.

compressible · Shock 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.

The number that cannot break a drop

The capillary number sets the stress that stretches a drop against the stress that holds it round, and it predicts the deformation beautifully. It cannot predict the breakup, because above a viscosity ratio of about four a drop in simple shear cannot be broken at any shear rate — and the theory that says the ratio hardly matters is the same theory that gets the deformation right.

regimes · Capillary A vortex has no energy. The kinetic energy of a Lamb–Oseen vortex inside a circle, per metre of its length, against the logarithm of that circle's radius. It is a straight line and it does not stop: outside the core the swirl is Γ/2πr, the energy density falls as 1/r², and the area grows as r², so every decade of radius adds the same amount. There is no such thing as the energy of a line vortex without a stated cutoff, and no cutoff is physical.

The energy a vortex cannot have

A line vortex has infinite kinetic energy. Not a large amount — infinite, growing without limit as the logarithm of however far out the counting stops. And it is losing that energy at a rate that is finite, exactly known, and contains no cutoff at all.

viscous · Diffusion Seven shock structures, and the two states they all connect. The velocity through the shock for seven dissipation models — Prandtl numbers from a quarter to two, viscosities from constant to linear in temperature. Each curve is shifted so its midpoint sits at the origin. They start at the same speed, end at the same speed, and are nothing alike in between.

The jump does not ask what made it

Seven different dissipation mechanisms are made to smear the same shock. Their interiors are a factor of two and a third apart in thickness, their entropies overshoot the final value by between a quarter and a doubling, and the state they all arrive at agrees to seven parts in ten billion — because the end states are conservation and the interior is transport.

compressible · Shock 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.

What a fluid takes out of a swing

The damping a body feels from the air around it is not Stokes' drag, and stops being it far earlier than anybody expects — a millimetre sphere in air is already forty-six per cent above the steady answer at one hertz. Past that the damping rises as the square root of the frequency, and the fluid it is fighting is a shell a fraction of its own size.

viscous · Damping The core that does not move, and where its edge is. Four velocity profiles at the same pressure gradient and four yield stresses. The shear stress in a pipe is G r/2 whatever the fluid is — that is a force balance and not a constitutive law — so the fluid is unyielded exactly inside r = 2 tau_y/G, and the plug radius is known before anything is solved.

The core that does not move

Toothpaste in a tube has a region in the middle that is not being sheared at all, and its edge is at a radius you can write down before solving anything. Four things about that core are exact, and one of them is that the fluid does not flow below a threshold — not slowly, at all.

viscous · Non-newtonian The cushion changes its physics 0.36 mm from the ground. The two forces on a plate 10 cm across closing on a plane at 1 m/s in air at 20 °C, per metre of span, against the gap on logarithmic axes. The viscous squeeze film, Reynolds' lubrication result μVc³/h³, rises as the cube of the closeness; the inertial one, ρV²c³/24h² from the potential flow's added mass, as the square. They are equal where the gap Reynolds number ρVh/μ is exactly 24, at 0.361 mm, where each is 3.84e+2 N/m. Above that gap the cushion is the fluid's inertia and below it the fluid's viscosity — and at the crossover neither formula is accurate, since it is where one limit hands over to the other rather than a solution of the flow between them.

A cushion that changes its physics

A plate closing on a plane is resisted by the fluid it has to squeeze out, and the resistance is two different forces with two different laws — one from the fluid's inertia and one from its viscosity. They hand over at a gap of twenty-four kinematic viscosities per unit of closing speed, which for a wing in air is a third of a millimetre, and the two films disagree about whether the plate ever lands at all.

misconceptions · Ground cushion Four fluids under the same stress, and the four profiles that result. Each fluid carries the same linear stress and answers it differently: a Newtonian fluid with a parabola, a shear-thinning one with a blunter profile, a shear-thickening one with a sharper one, and a Bingham plastic with a plug in the middle where the stress is below its yield point.

The stress a pipe knows

A capillary viscometer measures a pressure drop and a flow rate and reports a viscosity. The first half of that inference is a force balance and is exact for any fluid there is; the second needs the slope of a whole flow curve, which is the experiment the instrument was bought to avoid.

viscous · Non-newtonian What the fluid at one height is listening to. The weight the fluid two millimetres above a moving wall gives to the wall's velocity a given delay earlier, in water. It peaks at two thirds of a second and has a tail that falls as the delay to the power minus three halves — so the fluid is responding to a broad stretch of the wall's past rather than to a moment of it.

The wall the fluid is listening to

Water two millimetres above a moving wall is responding to what the wall did two thirds of a second ago — most likely. Half of its response is older than four and a half seconds, a tenth is older than two minutes, and the average age of what it is responding to does not exist at all.

viscous · Exact layer A band of growing waves that opens at 5772 and narrows as the viscosity goes. The wavenumbers at which a two-dimensional wave on plane Poiseuille flow neither grows nor decays, against the Reynolds number on a logarithmic axis. Inside the tongue waves grow; outside they decay. The tongue's tip is the critical point. Both edges slope downward and towards each other in wavenumber as the Reynolds number rises, so the band of unstable waves shrinks towards long waves — the direction in which the inviscid problem, which has no growing wave at all, is reached.

The profile Rayleigh cleared and viscosity did not

Flow between two plates has no inflection point, so without viscosity no wave on it can grow. With viscosity one does, above a Reynolds number of 5772. Taking the viscosity away again slows that wave and narrows the band it grows in, because the stress that feeds it is made by viscosity in the first place.

turbulence · Instability A force that depends on the molecular scale only through its logarithm. The force per unit length needed to move a 30° contact line of water at 1 mm/s, out to 1 mm, against the slip length on a logarithmic axis, from a picometre to a tenth of a millimetre. Each tenfold change in the slip length moves the force by the same fixed amount, so eight decades of the most uncertain length in the problem change the answer by a factor of about twenty — and at zero slip length the line keeps rising without end. The dashed curve takes the exact wedge's angle factor with a sharp cutoff; the solid one the thin-film wedge with Navier slip.

The drop a no-slip wall would never let spread

Liquid touching a solid moves with it, and nearly everywhere that is as close to exact as anything in fluid mechanics. At the edge of a spreading drop it cannot be: the stress in the corner rises as one over the distance from the edge, and the force needed to move the edge is infinite. Something slips over a nanometre, and because the answer depends on that length only through its logarithm, a drop spreads at almost the same rate whatever the something is.

misconceptions · The no-slip condition Six that are symmetries and five that look like them. Each transformation applied to an exact solution, with the Navier-Stokes residual recomputed from the transformed field by finite differences — nothing differentiated by hand. The six symmetries leave the residual at the differencing floor, a few parts in 10^8. The five near-misses leave between 0.048 and 4.3, which is six to nine orders of magnitude larger. The gap is what makes this a test rather than an illustration: a transformation that is nearly a symmetry does not exist here, and every one of the five is something a reader might reasonably believe.

Why the list is this long

Every textbook list of exact solutions of the Navier–Stokes equations is about a dozen long, and the usual explanation is that the equations are hard. It is not the reason. A similarity reduction is a solution invariant under a subgroup of the equations' own symmetries, so the catalogue of possible reductions is the catalogue of subgroups — and that is a finite, countable object.

viscous · Exact layer

Named alongside it

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

DissipationBoundary conditionBoundary layerModel limitThe no-slip conditionConservationMeasurementReynolds numberSimilarity solutionVorticityLubrication filmRegime

All concepts