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

Shape factor — where it appears

The ratio of a boundary layer's displacement thickness to its momentum thickness, which describes the profile's fullness in one number. It rises as a layer approaches separation, which makes it the standard indicator of how much trouble a layer is in.

Named by 15 essays across 4 fields — each of them below, with the objects they name alongside it.

The laminar boundary-layer profile. Speed against height through a laminar boundary layer on a flat plate, in the similarity variable that collapses every station along the plate onto one curve. The straight line is the slope at the wall, which is what the skin friction is proportional to.

How thick is thin

The boundary layer has no edge. It approaches the free stream and never arrives, so any thickness quoted for it is a convention — and the three conventions in use measure three different things, one of which is not a height at all.

viscous · Boundary layer The wall slope against pressure gradient, and where it runs out. How steeply the flow leaves the wall, plotted against the pressure gradient the layer is running into. A favourable gradient presses the profile against the surface and steepens it; an adverse one hollows it out. The curve reaches zero at a definite value, and beyond that there is no attached solution at all.

How much uphill a layer can take

A boundary layer running into rising pressure is climbing a hill on the last of its momentum. There is a definite steepness at which it can no longer do it, and the number is not a rule of thumb — it is where a family of solutions stops existing.

viscous · Separation Every wavelength grows, and the shortest grows fastest. Growth rate against wavenumber for a vortex sheet of zero thickness and for a layer of thickness δ. The sheet's curve rises without limit, which is the model announcing that it has no shortest scale in it; the layer's turns over and dies at kδ = 1, the neutral wavenumber computed from Rayleigh's equation rather than fitted.

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.

viscous · Separation A boundary layer with no x in it. The velocity profile over a porous wall with uniform suction: U(1 − e^{−Vy/ν}), exactly, at every station along the wall. The displacement thickness is ν/V, the momentum thickness is half of it, and the shape factor is two — all of them constants, none of them a function of distance. It is the cleanest demonstration there is that a boundary layer's thickness is a balance rather than an accumulation.

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.

viscous · Exact layer 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 wall the outer flow is really solving for. A flat plate, the edge of its boundary layer, and the line the outer flow behaves as though the plate were on. The displacement thickness is the mass deficit divided by ρU — checked here against the profile's own integral rather than quoted — and moving the wall out by that much reproduces exactly the flow rate the viscous layer lets past. It is a third of the visible thickness of the layer and it is the only part of the layer the outer problem knows about.

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.

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

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.

viscous · Energy thickness Four guesses at a boundary-layer profile. A straight line, a parabola, Pohlhausen's cubic and a quarter sine, each rising from zero at the wall to the free stream at the edge. Two of them also satisfy the conditions the true profile satisfies — no curvature at the wall, no slope at the edge — and two do not, which is what sorts them.

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.

viscous · Boundary layer Two external flows that agree where it matters and nowhere else. The velocity just outside the boundary layer, for two pressure distributions, against distance along the surface. They cross at the half-way station with the same value and the same gradient, and they have nothing else in common: one accelerates steadily and the other does most of its accelerating at once.

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.

viscous · Boundary layer Above the logarithm the profile lifts away: the wake. The mean velocity in wall units against the distance from the wall, for a flat-plate layer at Reθ = 10⁴ (δ⁺ = 3484), from Spalding's inner law alone and with Coles's wake of strength Π = 0.3, 0.55 and 1. Below about a fifth of the layer the curves coincide on the logarithm; above it the wake lifts the profile by up to 2Π/κ, 2.68 wall units for a flat plate — a tenth of the edge velocity, and the part of the profile a log law cannot describe.

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.

turbulence · Wall law An adverse gradient hollows the layer from the outside in. The velocity profile of an equilibrium layer, as a fraction of the edge velocity against y/δ, at Clauser's β = 0, 2, 10 and 50, with the wake strength from Das's fit. The inner part keeps the wall law; the outer part's wake grows — Π = 0.476, 1.59, 4.67 and 15.2 — and the profile sags until most of the layer is moving slowly over a thin fast wall region.

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.

turbulence · Wall law A quarter of the way into the layer, the lag follows the transitions along the chord. The transit lag a quarter of the way into the layer, as a multiple of the lag outside it, against chordwise station, at 0° and 4° on the thin cambered section at a chord Reynolds number of a million, with each surface's transition marked. At 0° the upper layer turns at 51% and the lower at 35%; the lag is −0.122 times the outside value at mid-chord and 0.61 at 95%. At 4° the upper layer turns at 11% and the lower not at all before the trailing edge; the lag is 2.91 times the outside value at mid-chord and 3.51 at 95%.

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.

misconceptions · Equal transit Let its shear stress lag, and a layer reaches its new equilibrium shape in about seventeen thicknesses. The shape factor downstream of a step from zero gradient to an outer speed falling as the power −0.2 of distance, in layer thicknesses at the step: with the shear stress lagging, by Green's lag-entrainment method; with the entrainment fixed by the layer's shape, as in Head's method; and Clauser's equilibrium layer at the lagged layer's own local β, through Coles's profile with Das's wake strength. Twenty thicknesses on, the lagged layer's shape factor is 1.479 and the shape-only layer's 1.445; eighty on, 1.439 and 1.432.

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.

turbulence · Wall law The lagged layer arrives at the release with the worse shape and leaves it with the better one. The shape factor of the layer from its flat-plate run through an outer flow decelerating from x = 1 m and released at 1.6 m into a flat one: Green's lagged layer and the layer whose entrainment follows its own shape. At the release the lagged layer's shape factor is 1.66 against 1.57; 11.8 thicknesses later it is below the shape-only layer's, and at the end, 4.5 m, it is 1.282 against 1.301.

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.

turbulence · Wall law The second gradient separates the layer soonest after a short release, latest after a dozen thicknesses, and sooner again after a long one. The distance past the second gradient's start at which the layer separates, in its thickness there, against the release's length in the thickness at the first release: Green's lagged layer, the same layer restarted at the second gradient with no memory, and the shape-only layer. Lagged: 2.91 after no release, 5.07 after 12 thicknesses, 3.95 after 60; with no memory, 3.16, 4.81 and 3.91.

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.

turbulence · Wall law

Named alongside it

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

Boundary layerSeparationDisplacement thicknessModel limitMomentum thicknessSkin frictionAdverse pressure gradientBlasiusEntrainmentFalkner–SkanIntegral methodPressure gradient

All concepts