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

Adverse pressure gradient — where it appears

A pressure that rises in the direction the flow is going, decelerating the fluid nearest the wall the most. It is what makes a boundary layer separate, and how much of it a layer can survive is the whole of the design problem for a wing.

Named by 14 essays across 4 fields — each of them below, with the objects they name alongside it.

Flow past a cylinder at Re 100. A real fluid past a circular cylinder. At low Reynolds number the flow closes up behind the body much as the ideal theory says; as it rises the flow separates and a region of reversed flow appears behind, which is where drag comes from.

When the flow lets go

Every body asks the air behind it to slow down and climb back up to the pressure it started at. Sometimes the air cannot, and the moment it refuses is separation — the source of most drag, the cause of stall, and the reason a golf ball has dimples.

viscous · Separation A shear layer, and the point of inflection in it. The velocity profile U = tanh y across a layer of finite thickness, with the inflection point located by searching for a sign change in the second derivative rather than by reading it off the algebra. Rayleigh's theorem says an inviscid parallel flow can only be unstable if such a point exists — a necessary condition, not a sufficient one.

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.

turbulence · Instability 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 Pressure recovery along the upper surface at 6°. Surface speed and the local Falkner–Skan pressure-gradient parameter, plotted along the upper surface from the nose. The speed peaks near the leading edge and then falls, which is the layer climbing back up to the pressure it started at, and the parameter crosses the separation value where that climb becomes too steep.

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.

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 The section the pressure asked for. The designed section over the one it started from, both drawn to their own chords. Asking for 28 per cent more speed over the forward 62 per cent of the upper surface produces a section 15.0 per cent thick against the original's 10, with the extra thickness forward and the camber changed — none of which was asked for, and all of which is what that pressure distribution is. The pale outline is the baseline. The one thing the method cannot be told is where along the chord any of it happens: the speed is prescribed against the circle's parameter, and where a given station ends up is an output of the same solve that produces the shape.

Ask for the pressure, and see what shape that is

A designer knows what the pressure distribution has to do long before knowing what shape does it. Running the problem that way round is possible, it is exact, and it refuses more asks than it grants.

inviscid · Inverse design Where the first grid point may go. The error in the friction a wall treatment infers, against the height of the first grid point, at Re_τ = 20,000. The velocity fed to each treatment is the closure's own, so what is plotted is the modelling of the boundary condition with nothing else in it. The log-law function is exact between y⁺ 30 and 10261 and is 64 per cent wrong at y⁺ = 1; the sublayer treatment is exact below y⁺ = 5 and hopeless above it; and the blend that most codes ship is within a few per cent everywhere and exact nowhere.

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.

turbulence · Wall law The wall shear, marched to the station where it stops. Howarth's linearly retarded outer flow, marched with an implicit finite-difference scheme from a Blasius profile. The wall shear falls, its slope steepens, and at x = 0.11983 it reaches zero — against Howarth's 0.1198, which is a quarter of a per cent. There is nothing downstream of it: the solution does not continue.

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.

viscous · Boundary layer 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 A cooled wall at β = 1: the linear relation is 158 K out inside the layer. The static temperature across a laminar layer at Mach 5 with an edge temperature of 220 K, over a wall whose total enthalpy is 0.5 of the edge's, at a pressure gradient β = 1: exact (thick) and from the Crocco–Busemann linear relation (thin), at a Prandtl number of one with constant properties. The wall is at 660 K in both. The exact profile peaks at 684 K and the linear one at 759 K; the largest difference, −158.3 K, is at η = 0.81.

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.

compressible · Recovery The profile a diverging channel flattens into, and then cannot hold. Five purely outward profiles in a wedge of 0.2 radians, at rising flux, each normalised to its own centreline value. As the flux rises the profile flattens in the middle and steepens at the walls — and then it stops. The last one has zero slope at the wall, which is separation, and beyond it no purely outward profile of this form exists at all. Nothing was added to the equation to make that happen: the wall shear is the square root of a cubic and the cubic runs out.

One channel, one flux, two flows

Flow between two plane walls meeting at a line has an exact solution. Past a threshold that turns out to be a ratio of gamma functions, it has two — the same wedge carrying the same flux, once outward everywhere and once with the fluid running backwards along both walls, and nothing in the equations chooses.

viscous · Exact layer The wall lets go long before the textbook's shock arrives. Where the flow leaves the nozzle's wall, as a fraction of the divergent section's length, against the ratio of chamber to ambient pressure. The inviscid picture runs full to the exit until the pressure ratio falls to 24.5, where a normal shock stands at the exit, and then moves the shock inside. The wall flow separates instead, when its pressure falls to Summerfield's 0.4 of ambient — at a pressure ratio of 192 — or to Schmucker's Mach-dependent value, at 138: six to eight times higher. Across that whole band the textbook's nozzle is full and the real one is not.

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.

compressible · Area mach A cold wall holds its layer on longer; a hot one lets it go sooner. The pressure-gradient parameter β at which the wall shear vanishes, against the wall's total enthalpy over the edge's. With the velocity deaf to the temperature it is −0.1988 for every wall. Coupled, it is -0.3264 for a wall at absolute zero, -0.2623 for a wall at half the edge's total enthalpy, -0.1988 at one, -0.1573 at one and a half and -0.1295 at two. Cooling delays separation by more than the same heating hastens it.

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.

compressible · Recovery 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

Named alongside it

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

SeparationBoundary layerFalkner–SkanSimilarity solutionModel limitShape factorSkin frictionBoundary conditionTransitionAerodynamic heatingAnalogyClosure

All concepts