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

The no-slip condition — where it appears

The condition that fluid in contact with a solid surface moves with it, tangentially as well as normally. It is the one condition an inviscid fluid may not be told, and everything a boundary layer does follows from it.

Named by 28 essays across 5 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 Stokes' theorem on a solved wake at Re 40. A rectangle drawn in a viscous flow that was solved on a grid. The circulation round its boundary is computed by walking the four sides and adding up the velocity along them; the vorticity inside it is computed by adding up the stored vorticity cell by cell. The two computations share no sample point and the theorem says they must agree.

Circulation is vorticity, added up

One of these two quantities is measured by walking round a loop and one by summing over the area inside it, and a theorem says they are the same number. That reconciles the site's most confusable pair — and explains how a flow with circulation can have no spin in it anywhere.

kinematics · Vorticity 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 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 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 jet divides 75% to 25%. A jet striking a plate at 60 degrees. Both sheets leave at the jet's own speed, because their surfaces are at ambient pressure and Bernoulli allows nothing else, and the plate can exert no force along itself because the fluid has no viscosity. Momentum along the plate then fixes the split at (1 + cos β)/2 = 0.7500, and the normal force at ṁV sin β = 0.8660. Nothing about the plate's material, size or roughness enters either.

What a jet cannot push sideways

A jet striking a plate divides in two, and how it divides is fixed by a single sentence — an inviscid fluid exerts no force along a surface. That one statement, plus mass, gives the split exactly — and the same sentence turns a flat plate into a bucket worth twice as much.

applied · Jet The same flow, from two different physics. Solid lines: the depth-averaged flow between two plates a small distance apart, with an obstacle standing between them. Dashed: the ideal-flow solution for the same obstacle. They are the same field to a part in ten billion, because averaging Stokes flow across a narrow gap gives a velocity that is the gradient of a harmonic potential. The cell has no inertia at all, which is the one hypothesis the ideal theory cannot do without.

The exact theory, drawn by viscosity

Two flat plates a millimetre apart, syrup between them, an obstacle in the gap. The Reynolds number is a hundredth, inertia is absent, and the streamline pattern is the potential flow past that obstacle — exactly, to a part in ten billion. The one hypothesis ideal flow cannot do without is the one this flow most conspicuously breaks.

inviscid · Hele-shaw Slip follows the stripes more closely than the shear does. Plan views of a striped surface, with the stripes running across each panel, for a shear at 0°, 30°, 54.7° and 90° to them. The faint arrow is the direction of the shear; the dark one is the slip velocity it produces, whose component along the stripes is the along-stripe slip length times the shear and whose component across them is half that. The slip is turned towards the stripes by 0.0°, 13.9°, 19.5° and 0.0°. It is largest, 19.47°, for a shear at 54.74°, where tan θ = √2. A surface with a tensor for a boundary condition can push a flow sideways, which a scalar slip length never can.

Twice as slippery along as across

A surface of alternating gas and solid stripes lets a liquid slip, and a flow far above it sees one number in place of the pattern — but the number depends on which way the flow goes. Along the stripes it is Philip's logarithm; across them it is exactly half, for a reason that takes one substitution to show. And the logarithm means that the slip is bought by the pattern's period rather than by how much of it is gas.

kinematics · Boundary conditions A spun cylinder carries its whole circulation at once, and hides it until the vorticity has left. The circulation round circles of radius r about a cylinder of radius a started spinning at once, as a share of the circulation of its own surface, 2πa²Ω, against r/a on a logarithmic axis, at νt/a² = 0.01, 0.1, 1, 10 and 100. At the surface it is the whole of it from the first instant, because no slip makes the fluid there turn with the cylinder. Just outside, the spin-up has laid down an equal and opposite ring of vorticity, so the circulation round a larger circle is only what has diffused past it: at two radii 0.000, 0.035, 0.611, 0.936 and 0.993 of the surface's at the same five times. The circulation a Magnus rotor needs is in the fluid the moment it spins; the far field learns of it only as fast as the counter-vorticity moves out.

A wall puts in exactly its own speed

A wall sliding in its own plane makes vorticity at a rate equal to its acceleration, with no viscosity in the rate. So however a wall is started, the vorticity it has put into the fluid is its speed, to the last digit; a wall that stops takes all of it back and leaves the fluid moving; and a spinning cylinder carries its whole circulation from the first instant, hidden behind an equal and opposite ring until viscosity carries the ring away.

kinematics · Vorticity 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 Where the inverse cube stops being an inverse cube. The force resisting a sphere's approach to a plane, against the gap, for four slip lengths. With no slip the force goes as one over the gap and has no limit. Any slip at all bends the curve over: below the slip length the force grows only as a logarithm. The curves separate where the gap falls through the slip length, which is exactly where a measurement has to be made and is why the apparatus has to close to nanometres rather than microns.

A length you can only measure by squeezing

No slip is a boundary condition rather than a law, and nothing in the equations requires it. The evidence that it is right for an ordinary liquid to within a nanometre comes from the one experiment whose small length is chosen rather than given — a squeeze film closed to tens of nanometres.

misconceptions · The no-slip condition Three dies, one melt, three flow curves. The apparent flow curve a capillary rheometer reports, for three dies whose radii differ by a factor of four. A material property cannot depend on the instrument, so the fact that these three do not lie on one another is a measurement of something other than a viscosity. The single curve drawn beneath them is what the same melt gives with the wall condition set to no slip.

The wall a melt really does slip on

For a simple liquid the slip length is under two nanometres, which is why no-slip is safe everywhere. For an entangled polymer melt it is a millimetre, which is larger than the die it is being pushed through — so the melt moves as a plug and the viscosity a rheometer reports is a property of the instrument.

misconceptions · The no-slip condition The speed past the flank, in the cell and in the ideal flow it is meant to show. The gap-averaged speed along the top of a cylinder, against the distance from its surface in cylinder radii, for three cells — the cylinder's radius 10, 35 and 100 times the Brinkman length h/√12 — and for ideal flow. Ideal flow slides along the wall at twice the stream speed. The cell's flow must stop at the wall and does so in a layer a few Brinkman lengths thick, outside which it rejoins the ideal curve.

The cell draws a larger cylinder

A Hele-Shaw cell draws the streamlines of ideal flow past an obstacle and cannot obey the one rule ideal flow breaks: the fluid must stop at the obstacle's wall. It does stop there, in a layer a third of the gap thick, and far away the whole correction amounts to one thing — the cell is drawing ideal flow past a cylinder one layer-thickness too big, and the obstacle has exactly that cylinder's drag.

inviscid · Hele-shaw A slip law implies a slip length that changes with the rate. The slip length — slip velocity over the true wall shear rate — against the wall shear rate, for a melt whose slip follows a velocity law in stress with exponent 1.5, 2 or 3, all matched to slip at 2 mm/s at a stress of 100 kPa, and for a melt with a constant slip length, dashed. With the exponent of 2 the implied length falls from 0.22 mm to 0.08 mm across the range of rates; only a law whose exponent is exactly 1/n, 2.5 here, implies a constant length.

A melt's slip length has an exponent it cannot choose

A polymer melt slips at the wall, and there are two ways to say by how much: a slip length, fixed, so the slip velocity follows the wall's shear rate; or a slip law, the slip velocity as a power of the wall's stress. The literature moves between them as though they were the same, and for a melt they are not. A constant slip length forces the slip law's exponent to be exactly one over the melt's flow index — 2.5 for a typical melt — so a measured exponent of 2 is a slip length that changes with the rate. And temperature separates them outright: a tenfold change in viscosity moves the slip velocity tenfold under one description and three-hundredfold under the other.

misconceptions · The no-slip condition A layer read as slip gives a slip length that drifts. The slip length a drainage measurement would fit at each gap to the force of a 1-nanometre viscous layer, using the slip formula, for layers three and ten times as viscous and immobile. Far off it is the first-order value, −0.667 nm for κ = 3; closer, it shrinks towards zero — −0.597 nm at 10 nm and −0.525 at 5. A slip length that depends on the gap is the layer's signature, and the only one a drainage curve carries.

A viscous nanometre reads as slip until the gap is ten of it

A drainage experiment turns a molecular length at a wall into a measurable force, and reports it as a slip length. A layer of liquid a nanometre thick and several times as viscous as the bulk changes the same force by the same amount, to first order. Solved with the stratified viscosity, the two pictures part by a per cent only when the gap is eleven to seventeen layers wide, and the part of the difference a measurement can see is a slip length that drifts with the gap. A layer that does not move at all is exactly a wall in a different place.

misconceptions · The no-slip condition Elasticity adds a spring at first order and a slip only at second. The damping's deficit, one minus the damping ratio, and the in-phase force against Λ on logarithmic axes. The in-phase force rises in proportion to Λ — a fitted slope of 1.0000 — and the damping's deficit as its square, 2.0000. A slip length changes the damping at first order and adds no spring at all, so a soft wall and a slipping one are distinct in kind: the soft wall announces itself first in the phase of the force.

A soft wall is read as slip only at second order

A drainage force smaller than Taylor's has been read as the liquid slipping at the wall. A wall that gives under the drainage pressure lowers the force too, and it can be mistaken for slip. Solved together, the flow and the elastic wall say how: the softness first adds a spring to the force, in phase with the motion, and only at second order takes anything from the damping a slip length is read from. When it does, the slip it imitates grows as the square of the frequency and falls as the square of the gap.

misconceptions · The no-slip condition The source points along the span; the flow does not. Round a swept circular cylinder, the share of the wall's vorticity source that points along the local direction of the flow outside the layer, against the angle from the attachment line, at sweeps of 10°, 20°, 35° and 50°. The source lies along the span everywhere. At the attachment line the flow does too and the share is one; at the suction peak, 90°, it is 0.0878, 0.179, 0.33 and 0.512. What is not along the flow is the ordinary across-the-flow vorticity a two-dimensional layer has.

A swept wall makes vorticity along its isobars, not across its flow

A still wall in a pressure gradient puts vorticity into the fluid at a rate set by the gradient, and the source is a vector: it lies in the wall, along the isobars. On an unswept body the isobars run across the flow and so does the vorticity, which is the ordinary boundary layer. On a swept one the isobars run along the span and the flow does not, so part of every new vortex line points along the flow — half of it, over the accelerating front of a cylinder swept 35°. That part is where a swept wing's crossflow comes from.

kinematics · Vorticity A slender nose at incidence makes most of its vorticity pointing along the flow. The share of the wall's vorticity source that points along the local surface flow, integrated over the region where the flow accelerates, against incidence, for prolate spheroids of fineness 1 (a sphere), 2, 4 and 8. A sphere makes none at any incidence. At 10° a body of fineness 2 makes 20% of its source streamwise, fineness 4 55% and fineness 8 81%, with no sweep anywhere.

A nose at incidence makes its vorticity along the flow

A wall makes vorticity along its isobars, and on a swept wing that sends part of it along the flow. A fuselage at a few degrees of incidence has no sweep anywhere, and still most of the vorticity its wall makes points along the flow — a fifth for a body twice as long as it is wide, four-fifths for one eight times — because at incidence its pressure varies round the body faster than along it. A sphere, at any incidence, makes none. The lee vortices begin at the wall.

kinematics · Vorticity A free jet keeps its momentum; a wall jet keeps the flux of its momentum flux. Each quantity against distance, relative to its value at 2 cm, on logarithmic axes. The free jet's momentum flux is constant and its volume flux grows as x^(1/3). The wall jet's momentum flux falls as x^(−1/4) — the wall's shear takes it — while its volume flux grows as x^(1/4), and the one thing it conserves is Glauert's ∫u(∫u²)dy, the flux, through each height, of the momentum flux carried above it.

A jet along a wall keeps the flux of its momentum flux

A jet in open fluid keeps its momentum and gathers fluid as it goes. Blow the same jet along a wall and the wall takes its momentum away as drag, so it keeps neither. What it keeps is stranger: the flux, through each height, of the momentum flux carried above that height. That one invariant fixes everything else — the jet thickens as the three-quarter power of distance, slows as the inverse square root, and gathers fluid as the fourth root, more slowly than the free jet's cube root.

viscous · Free shear Hold the nozzle further from the plate and the jet arrives slower but leaves a stronger wall jet. For a laminar jet from a 2 mm slot at 5 cm/s in water, against the nozzle's distance from the plate, each over its value at 10 cm: the arriving centreline speed, which falls as h^−⅓; the inherited invariant F = 3KQ/32, which grows as h^⅓ because the jet has gathered fluid; and the wall jet's peak speed a metre along the plate, which grows as h^⅙. At 2 cm the arriving speed is 1.71 of its value at 10 cm and the far wall jet 0.765; at 2 m, 0.368 and 1.65.

A jet striking a plate leaves a stronger wall jet the further it travelled

A plane jet that strikes a wall splits into two wall jets, and each keeps not the jet's momentum but Glauert's flux of momentum flux. Turned without loss at the stagnation point, each half arrives laid along the wall with its axis on the wall, and its invariant is the momentum it brought times the fluid it had gathered on the way: F = 3KQ/32. Hold the nozzle further away and the jet arrives slower but leaves a stronger wall jet, by the cube root of the distance — and the wall jet takes a distance of about the Reynolds number in jet widths to forget it was ever free.

viscous · Free shear The paint pushes a scraper off the wall, and roughly as one over the square of the angle it is held at. The force the fluid exerts on a still blade at angle α to a wall sliding into the corner, per unit span over μU ln(R/ε): its component pushing the blade away from the wall and the size of its component dragging it along with the wall, towards the apex. At 10° they are 196 and 22.9; at 45°, 8.56 and 4.88; at 90°, 1.36 and 2.14; at 150°, 0.0757 and 0.924.

A scraper held flat is pushed off the wall it scrapes

A corner stirred from far away fills with an endless sequence of eddies. Drive the corner at the corner instead — let one of its walls slide under a still blade, as paint slides under a scraper — and the eddies vanish: every streamline turns once, and the stress rises into the corner without limit. G. I. Taylor solved it in 1962. The paint pushes the blade away from the wall with a force that grows as one over the square of the blade's angle, so a blade held flat rides up on what it was meant to remove, and the force is finite only because something too small to appear in the problem ends the corner.

viscous · Corner

A mica sheet hides its glue only at the smallest gaps, and changes how the false slip falls

A drainage measurement against a soft wall reads a slip that is not there, falling as the square of the gap. The wall in a surface-force apparatus is not a soft half-space but a stiff mica sheet on a soft glue, and the sheet spreads the drainage pressure over its own bending length before the glue feels it. Where the pressed region is wider than that length the sheet changes nothing; where it is narrower the sheet sinks as one piece, a spring in series with the liquid. That cuts the false slip only by the square of the bending length over the pressed width — threefold at three nanometres for three micrometres of mica — and it turns the slip's fall with the gap from the inverse square towards the inverse, which is the test an experimenter would have used to see it.

misconceptions · The no-slip condition

A stagnation point can leave the flow through the wall, and the count is kept by halves

A no-slip wall is a sheet of zeros, so the index count that holds for a flow in open space has nothing to count there. Divide the velocity by the height and the wall's zeros shrink to the skin friction's nodes, foci and saddles; reflect the result through the wall and each of them is counted once while every point in the fluid is counted twice. That is why the points on a wall are halves, and an exact swirling flow onto a plate shows the halves doing their job: a stagnation point in the fluid can disappear without a partner by sinking into the wall, where the skin friction's focus changes from attachment to separation to pay for it. A section through the flow sees none of this, because in the section every one of the points is a saddle.

kinematics · Topology

A stagnation point off the axis can land only on a saddle or a node, and turns it inside out

In a swirling flow onto a plate every stagnation point lived on the axis, so one sinking to the wall always met the wall's single critical point. Without the symmetry it might seem free to land anywhere — on an ordinary stretch of skin-friction line, leaving a pair of new points behind. It cannot. Divided by the height, the velocity is continuous down to the wall, where it becomes the skin friction, so a point can only reach the wall where the skin friction already vanishes. It lands on a saddle or a node, and the count does the rest: the point it lands on turns from separation to attachment. An oil-flow picture of the wall, which shows the same pattern before and after, cannot tell which it is looking at.

kinematics · Topology

A pressed blade floats on its land and leaves half the gap behind

A scraper or a doctor blade is not held at a gap; it is pressed on with a load, and it rises until the liquid it lets through carries that load. Without a flat land under its edge the liquid can carry it only through the corner's logarithm, so the gap falls exponentially with the load, and a blade steeper than about forty-four degrees cannot float at a working load at all — its edge is in contact. Give it a land a tenth of a millimetre long and the film becomes half the gap, falls as one over the load, and is a quarter of a micrometre at sixty degrees. The land is also why a blade's film grows as its edge wears.

viscous · Corner

Named alongside it

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

Model limitBoundary conditionViscosityBoundary layerSlipMeasurementCreeping flowLubricationVorticityPotential flowReynolds equationSeparation

All concepts