Boundary layer — where it appears
Named by 75 essays across 9 fields — each of them below, with the objects they name alongside it.
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.
The number that is not a number
Transition Reynolds numbers are quoted to three figures and vary by two decades. That is not sloppiness in the measurement — it is the honest report of a quantity that depends on the laboratory as much as on the fluid, and knowing which part is which decides what may be designed on it.
A layer with a kink in it
Rayleigh proved in 1880 that an inviscid shear flow cannot be unstable unless its velocity profile bends the other way somewhere. It is one line of algebra, it is necessary and not sufficient, and it ties instability to separation through the sign of a single derivative at the wall.
The roughness a wall cannot feel
A rough pipe and a polished one carry the same flow for the same pressure over three decades of Reynolds number, and then suddenly they do not. What changed is not the pipe. It is the thickness of the film of fluid at the wall, which is the only part of the flow that can see the roughness at all.
A guess with a constant in it
Prandtl's mixing length is one line — an eddy near a wall can only be as big as its distance from the wall. Integrate it and the whole structure of a turbulent wall profile falls out, sublayer and log region and all. That is a fact about the assumption, and the essay is careful about which.
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.
The pocket on top of the wing
An airliner cruising at Mach 0.85 has subsonic flow almost everywhere and a patch of supersonic flow over its wing. Closing that patch takes a shock, the shock separates the layer beneath it, and the correction that predicts all of this also predicts, in the plainest terms, where it stops being true.
Where the straight line stops
Ideal flow will report a lift coefficient at forty degrees of incidence without complaint. What ends the lift curve is the boundary layer refusing to follow the surface, and the estimate of when that happens joins two solvers that have nothing else in common.
The drag that falls as it speeds up
There is a band of speeds in which a smooth ball experiences less drag the faster it goes. Not a smaller coefficient — a smaller force. Dimples move that band down to where a golf ball actually flies, and they do it by making the friction worse.
A ball that swings without spinning
A cricket ball curves in flight with no spin about any useful axis. The mechanism is not the Magnus effect; it is a seam tripping the boundary layer on one side so that side lets go later. Which way the ball then goes depends on one borrowed number, and this site's own inviscid solver supplies the value that gets it wrong.
The gradient that does both
One line of the boundary-layer equations at the wall says the profile's curvature there equals the pressure gradient. That single sign causes separation and causes instability, and it causes the instability a long way before it causes the separation.
The cost of going turbulent
A turbulent boundary layer costs several times the friction of a laminar one, and the multiple is not a constant — it rises with Reynolds number, because the two laws have different exponents. That is why laminar flow is worth more on a long fast surface than on a short slow one.
The wall that shakes
Slide a wall back and forth in its own plane and the fluid above it does not follow — a wave travels upwards into the fluid and dies within one wavelength. The depth it reaches is √(2ν/ω), it contains no length from the geometry at all, and the whole thing is one of the very few exact solutions the Navier–Stokes equations have.
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.
The layer that stops growing
Blasius' boundary layer thickens as the square root of distance and never stops. Suck fluid through the wall at a uniform rate and it stops immediately — the profile becomes a single exponential with no x anywhere in it, and the friction comes out exactly equal to the momentum of the fluid that was taken away.
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.
The other layer, and the one number that separates them
A wall in a stream carries two boundary conditions and grows two layers. Their thicknesses differ by a factor of twenty across ordinary fluids, and at exactly one Prandtl number the two profiles are not similar but identical.
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.
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.
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.
A slot is not a nozzle
The gap between a wing and its flap is supposed to blow fast air into a tired boundary layer. A solver with no boundary layer in it at all — no viscosity, no wake, no mixing — produces most of the lift increment anyway, and produces it on the element nobody moved.
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.
The layer with no length in it
The logarithm in a turbulent wall profile does not come from any model of turbulence. It comes from a region where neither of the flow's two lengths is allowed to appear, and where a velocity gradient therefore has nothing to depend on but the distance to the wall. The constant in it has never been derived from anything.
The wall that heats itself
A surface told nothing about its temperature does not settle at the air's. It settles most of the way to the stagnation temperature, and the heat flux is driven from that invented temperature rather than from the free stream's — so a wall hotter than the air can be being heated by it.
How far before a duct forgets what was fed into it
A pipe is always drawn with its answer already in place. Getting there takes a distance proportional to the Reynolds number, which means a more viscous fluid is done sooner — and the entrance costs a fixed number of dynamic pressures however long the pipe is.
The wind a swept wing feels
Sweeping a wing back is usually justified by saying it meets a slower wind. It does not meet a slower wind. The equations split exactly in two, and the flow along the span is a passenger that exerts no force and changes nothing — until a pressure gradient breaks the split, and then it becomes the reason a swept wing is a different problem rather than a harder one.
What a code says to a wall
A calculation that cannot afford to resolve the viscous sublayer has to tell the wall something else instead, and what it tells it is the law of the wall — an asymptotic result, applied at one grid point, on the assumption that the point lies in a region the calculation has not checked exists. Where it does, the answer is exact. Where it does not, the friction is out by tens of per cent, and refining the grid makes it worse.
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.
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.
The spin a shock leaves behind
A curved shock gives every streamline a different entropy rise and the same stagnation enthalpy. Crocco's theorem then forces vorticity into a flow with no viscosity anywhere — and it scales as the inverse of the shock's radius of curvature, exactly, so a straight shock makes none.
Turbulent some of the time
At the edge of a jet or a wake a probe is inside turbulent fluid for part of the time and in perfectly smooth flow for the rest, and an ordinary time average mixes the two. Most of the fluctuation it records there is not turbulence at all — it is the switching between two states, and it peaks where the switching is most even rather than where the turbulence is strongest.
The body the outer flow actually sees
A boundary layer lets less fluid past than an inviscid one would. The outer flow can be given exactly the same reduced flow rate by leaving the fluid inviscid and moving the wall out — so the potential flow that matters is not the flow past the body, but the flow past the body plus a thickness the boundary layer computes.
The third thickness
A boundary layer has no edge, so every thickness quoted for it is an integral of the profile against some weight. Two of them are famous. The third answers a question the other two cannot — how much of the power spent towing a plate has actually become heat by the time the fluid leaves it — and the answer is 78.6 per cent.
The half that carries nothing
Thin-aerofoil theory splits a section into a camber line that carries all the lift and a thickness distribution that carries none — at any incidence, exactly none. That is very nearly true, and what it discards decides the peak suction, the critical Mach number and where the boundary layer gives up.
The number that is an answer
Almost every dimensionless group is a hypothesis: somebody sets the speed, the size and the fluid, and the number licenses a model. The Nusselt number is not. It is what the experiment produces, it sits on the left of the equals sign, and a regime diagram drawn on it is a category error.
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.
A speed nobody imposed
Every regime number in this collection contains a velocity somebody chose. Natural convection has none: a warm plate makes its own flow, and the Grashof number is what is left when the speed is taken out. The Reynolds number of the result — six thousand, on an ordinary radiator — is an output of the solution rather than a setting on an apparatus.
The singularity a layer makes for itself
March Prandtl's equations into an adverse pressure gradient and the wall shear reaches zero with an infinite slope at a finite station, and the solution cannot be continued past it. The singularity is real, it is not a numerical difficulty, and it belongs to the boundary condition rather than to the equations.
One formula for both ends
Two limits, each with its own description, neither valid everywhere. The composite is the sum less the part they agree about, and it is uniformly good — but the overlap region that justifies the construction does not exist at ε = 0.01, and the composite is still accurate to two per cent there.
How far downwind a surface is remembered
Walk from a field into a wood and the wind ten metres above your head is still the field's wind. It takes about a kilometre of trees before a ten-metre measurement is measuring the trees — a hundred times the height it is made at, and a great deal more than most masts are given.
What a turning frame keeps
Euler's equation prices the work and says nothing about where the pressure comes from. In the frame turning with the blades — which is accelerating, and carries two fictitious forces — a Bernoulli-like quantity survives both of them, and it splits the pressure rise into a term a boundary layer limits and a term that is free if the radius moves.
Four profiles, one drag
The momentum integral is exact and asks nothing about the shape of the velocity profile. Four guesses at that shape span twenty-three per cent in the drag they give — and the two that satisfy the conditions the true profile satisfies are within three, which says the freedom belongs to the family rather than to the constraint.
The length the limit invents
Prandtl's equations are parabolic, so nothing at one station can depend on anything downstream of it. Every experiment shows the pressure rising ahead of a shock or a step. The resolution is a region three eighths of a power of the Reynolds number long, which the limit that produced the equations was supposed to have removed.
The curve that measures a gradient
Three of the four curves drawn through a flow answer the same question — where did the fluid go. The fourth answers a different one. A line of particles released together is displaced by the local velocity and by nothing else, so its shape is the velocity profile, and dividing the elapsed time back out returns that profile exactly rather than approximately.
The vorticity a clean surface cannot refuse
A clean bubble's surface cannot hold a shear stress, and it is easy to conclude that it makes no vorticity. On a curved surface it must carry exactly 2κu — three times the speed over the radius at a sphere's equator, whatever the Reynolds number. That is so much weaker than a rigid wall's that the flow stays irrotational to leading order, and the bubble's drag is the dissipation of that irrotational flow: 48/Re, three to ten times below a rigid sphere's.
The truncation that cannot carry three times the heat
Lorenz's three modes give a convecting layer's heat flux in one line — one plus twice (r − 1) over r — and it can never reach three times what conduction carries. The same rolls computed with forty-two modes agree with that line exactly at onset, carry 7.1 per cent more heat at twice the critical Rayleigh number, and twice as much at thirty times it. The ceiling is not in the convection; it is in having one sine to draw the temperature with.
A layer that is an integral of everything upstream
Two surfaces are given external velocity distributions that agree exactly at one station — the same speed and the same gradient. The boundary layers there differ by 38 per cent in momentum thickness, and the two surfaces separate five per cent of their length apart.
How long a fluid takes to forget it was not rotating
Spin a container of water and the fluid inside reaches solid-body rotation in a hundred seconds rather than the three hours diffusion would need. The shortcut is the thin layers on the end walls, and the advantage they give is exactly the reciprocal of the square root of the Ekman number.
The gradient the heat never hears
On a flat plate at a Prandtl number of one, a boundary layer's total enthalpy is a straight-line function of its velocity, whatever the wall's temperature. Put the same layer in a pressure gradient and the straight line fails everywhere except on an insulated wall, because the gradient enters the velocity's equation and not the enthalpy's — and a favourable gradient can leave a band of gas colder than the free stream above a wall three times hotter than it.
The solution that keeps its nonlinear term
Every exact solution before this one has been exact because the nonlinear term vanished. A rotating disc's does not vanish — at the wall it is the whole of the balance — and the reduction is exact anyway, because the radius divides out of all three momentum equations at once.
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.
An overexpanded nozzle lets go before its shock arrives
The textbook's overexpanded nozzle runs full until its back pressure is high enough to hold a normal shock at the exit, and then draws the shock walking inside. A real nozzle never shows that sequence. Its wall's boundary layer cannot climb the pressure rise a normal shock imposes; it separates at a wall pressure of about four-tenths of ambient, which for a rocket nozzle happens at six to eight times the pressure ratio the textbook's shock needs. The separation is not a failure: it is what keeps the nozzle's thrust, and it is why a sea-level engine can be built twice the size it expands to.
The stern's wake puts the bulb at the bow
Michell's thin-ship integral cannot tell a ship's bow from its stern: reverse any hull and its wave resistance is unchanged, so the least-resistance hull is symmetric and a bulb is worth as much at the stern as at the bow. Real ships are fuller aft and put their bulbs forward. Let the stern's waves be made by water the hull's own boundary layer has slowed, and both follow: the optimum hull leans aft, and a bulb at the bow cuts the waves by far more than the same bulb at the stern.
The sheath outlasts the body
A blunt hypersonic body wraps itself in a sheath of hot, thin gas from its nose's shock, and its boundary layer grows by eating that gas from the inside. The usual picture has the boundary layer through the sheath within a few nose diameters. It is not: a laminar boundary layer takes about a third of a Reynolds number of diameters to swallow the sheath at Mach 15, which is thousands to millions of diameters, and a turbulent one hundreds. On any body of ordinary length the boundary layer never sees the cooler gas outside, and it is heated by the sheath's gas all the way down.
The gas a reflected shock refuses is inside the layer
A reflected shock in a shock tube bifurcates when the gas in the wall's boundary layer cannot be pushed into the reservoir behind it: its stagnation pressure, in the shock's frame, is below the reservoir's. Mark's criterion asks that of the gas at the wall, and in air it says the bifurcation stops above an incident Mach number of 6.5. But the gas that fails at high Mach numbers is not at the wall. It is inside the layer, heated by friction until it meets the shock too slowly for its sound speed, and on that measure a reflected shock in air bifurcates at every Mach number above 1.3.
A cold wall holds its layer on longer
A compressible boundary layer's pressure gradient pushes on its density, and its density is its temperature. Solve the velocity and the enthalpy together and the wall's temperature reaches the velocity: a cold wall holds its layer on to an adverse gradient two-thirds steeper, a hot one lets it go a third sooner, the stagnation point's heat-to-friction ratio more than doubles between them, and the cold band an uncoupled layer put above a cooled nose disappears.
A wake held at the stern keeps the bulb and loses the lean
A wake that slows the water steadily from bow to stern makes Michell's least-resistance hull fuller aft and puts its bulb at the bow. A real ship's wake is nearly nothing along most of the hull and strong only in its last few metres, deepest at the keel. Held there, with the same wake at the propeller, it moves the hull's volume aft by a twentieth to a half as much, and it keeps a third to two-thirds of the bulb's preference for the bow. The lean was a property of the water along the run, and the bulb a property of the water at the stern.
The wake is a tenth of the velocity and a third of the displacement
The logarithmic law describes a band in the middle of a turbulent boundary layer, and above it the profile lifts away by an amount Coles called the wake. It is a tenth of the edge velocity, so it looks like a correction. It is not: it carries a third of the layer's displacement, it is what turns the log law into a friction law for a boundary layer, and with it the friction comes out within a few per cent of a measured correlation that contains no logarithm at all.
A point force makes a jet only when it is strong
Push on a fluid at one point and there is an exact solution of the full Navier–Stokes equations for what follows, at any strength. Weak, it is Stokes's point force: fluid pushed forward over a whole hemisphere and drawn in behind. Strong, it is Schlichting's slender jet, with fluid drawn in from every direction outside a narrowing cone. One constant joins them, and it says how strong a push must be before the boundary-layer jet everyone uses is right — at the axis by a jet Reynolds number of a hundred, at the edges only by a thousand.
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.
An adverse gradient grows the wake, not the log law
Push a turbulent boundary layer up a rising pressure and its profile changes from the outside in. The wall region keeps its law; the wake grows, the layer hollows, the shape factor climbs and the friction falls. Clauser found the layers that stay similar while this happens, and their arithmetic has a surprise in it: there is a steepest pressure rise any such layer can climb, close to a free stream falling as the inverse fourth root of the distance, and the friction approaches zero only as the layer becomes all wake.
Inside the layer the upper parcel leads by more
Two parcels parting at a wing's attachment point arrive at different times, and outside the boundary layer the difference is the circulation over the speed squared. Inside the layer one might expect the slow, sticky fluid to wash the difference out. It does the opposite: deep in the layer the upper parcel's lead is several times the inviscid one, because the suction side's layer is fuller than the pressure side's. Only the layer's outer quarter reads the inviscid lead, and on the suction side the laminar layer gives out before the trailing edge.
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.
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.
Over a plate in mercury the warm layer flows as if inviscid
The exact solution for a warm vertical plate has been tabulated since 1953 and is usually shot from the wall, which fails in a liquid metal unless the shot starts within a thousandth of the answer. Posed instead as one algebraic system, it solves from a Prandtl number of a hundred-thousandth to a hundred thousand — and shows that in mercury the layer's speed is set without viscosity, that the integral method runs low there rather than high, and that air sits where neither limiting law is within a fifth.
Laid flat, a warm plate is pushed by the pressure of its own layer
A warm vertical plate drives its layer with buoyancy acting along it. Lay the plate flat and buoyancy acts across the layer instead, so the only push along it is the pressure deficit its own warm fluid leaves under the cold — deeper where the layer is thicker. That push gives a heat transfer growing as the fifth root of the Grashof number, not the fourth, works only on the plate's upper face, and survives a tilt of a degree for a third of a metre.
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.
A blunted cone eats its nose's sheath in a few hundred radii
A sphere-cylinder's boundary layer takes a third of a Reynolds number of diameters to get through the hot sheath its nose leaves, which on any real body is never. Put the same nose on a slender cone and the answer collapses: the cone's widening perimeter feeds the boundary layer so fast that it swallows the sheath in tens to hundreds of nose radii, growing only as the cube root of the Reynolds number. On a two-metre cone at Mach 15 that makes a nose of about two centimetres the line between a base still bathed in the nose's gas and one that has forgotten it.
The side that turns turbulent first decides who arrives first near the wall
Inside a laminar layer the upper parcel's lead over the lower grows with depth. Real wings are not laminar to the trailing edge: one surface's layer turns turbulent first, and a turbulent layer is full almost to the wall. At the incidences a wing flies at, the upper side goes first and the exaggerated lead survives to the trailing edge, smaller than the laminar one and shrinking with the Reynolds number. Below about one degree the lower side goes first, and a tenth of the way into the layer the lower parcel arrives first.
A layer whose stress lags the gradient separates sooner
A turbulent boundary layer that meets an adverse pressure gradient suddenly cannot answer it at once: the large eddies that carry its shear stress were made upstream, on a flat plate, and they draw in too little fluid for the new gradient for the first twenty or so layer thicknesses. The deficit does not protect the layer. It lets the gradient pile momentum deficit into it, so the layer reaches its new equilibrium shape quickly, its friction falls through the equilibrium value on the way, and in a strong gradient it separates in little more than half the distance a calculation without the lag predicts.
A blunted nose delays its cone's transition only so far, and the correlation decides how far
A blunted tip leaves a sheath of hot, slow gas along a hypersonic cone, and the boundary layer growing inside it sees a lower Reynolds number and a lower Mach number than a sharp cone's. That is why bluntness is said to push transition back. Read the layer's edge through the sheath and the push has a limit: the delay grows with the nose, peaks, and falls again once the transition point lies inside the sheath. How large the peak is depends on whether the transition correlation divides by the edge Mach number — five times the sharp distance at Mach 6 for the momentum-thickness Reynolds number alone, 1.4 for its ratio to the Mach number — and every peak sits forty to sixty-five nose radii from the tip.
A layer released from a gradient overshoots the shape it left
A turbulent layer that meets an adverse gradient suddenly entrains only half what the gradient asks, because its large eddies were made upstream, and it separates sooner than a layer that answers at once. Release it — let the outer flow stop decelerating, as at a diffuser's exit or a wing's trailing edge — and the same lag runs the other way. The eddies made in the gradient keep entraining after it has gone. The layer arrives at the release with the worse shape and, twelve thicknesses later, has the better one, its shape factor falling past the flat plate's and its friction rising past the shape-only layer's.
Followed rather than held, the parcels near a wing nearly tie
Parcels held at a fixed fraction of a boundary layer's thickness made the side that turns turbulent first decide which arrives first near the wall, and at a wing's ordinary incidence they gave the upper parcel a lead ten times the one outside the layer. Real smoke is not held at a depth: it follows a streamline. Near the attachment point the layer is thinner than any filament's height, so every filament starts outside it and is swallowed later, and on the upper surface, whose turbulent layer grows faster, it is swallowed deeper. Followed this way, the deep parcels at 4° arrive nearly together, and at a Reynolds number of a million the lower one can come first.
The layer every airway restarts bends Murray's cube towards the turbulent pipe's
A laminar fork does not lose energy the way a turbulent one does, in a burst of mixing at the split. It loses it all the way down each daughter, because the boundary layer starts again at every flow divider and an airway a few diameters long ends before the layer has grown across it. Pedley measured that extra dissipation in 1970, and in the form he wrote it the radius cancels: it depends on the flow and the length alone. Price it into the cheapest tree and Murray's cube law gives way to a flow growing as the radius to the power twelve-fifths or thirteen-fifths, the band the turbulent pipe's exponent sits in, across the first eleven generations of a resting lung.
A layer's memory of one gradient helps it through the next, unless the next comes too soon
A turbulent layer released from an adverse gradient keeps the gradient's eddies for a dozen thicknesses, entraining more than its shape asks. Decelerate it again before they fade — behind the next compressor stage, or past a flap — and the question is whether that memory helps or hurts. Both, depending on when. A second gradient that arrives within about three thicknesses of the release meets a layer whose entrainment is still catching up, and it separates eight per cent sooner than a layer of the same shape with no history. One that arrives later meets eddies still working from the first gradient, and the layer holds on up to five per cent longer. The crossing sits at three thicknesses for every pair of gradients tried.
Named alongside it
The objects these essays reach for when they reach for this one.
Model limitSeparationSimilarity solutionReynolds numberSkin frictionShape factorTransitionAdverse pressure gradientEntrainmentThe no-slip conditionViscosityWall shear