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

Momentum thickness — where it appears

The thickness of free stream whose momentum equals the momentum a boundary layer has lost. It is the quantity the momentum integral equation transports, and the drag on a plate is exactly the density times the square of the speed times its value at the trailing edge.

Named by 8 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 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 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 Nu/Gr^¼ against the Prandtl number, with the exact solution's points on it. The closed form 0.508 Pr^½(20/21 + Pr)^−¼, over eight decades, with Ostrach's exact similarity values marked. The integral method is two to eight per cent high from Pr = 0.7 upwards and 10 per cent low at Pr = 0.01 — which is where the thermal layer is ten times the momentum layer and giving them one thickness stops being an approximation to anything.

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.

regimes · Grashof 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 Bluntness delays a cone's transition by a factor that peaks, and the correlation's Mach exponent sets how high. The distance at which a blunted 8° cone's layer transitions, over the sharp cone's 0.3 m, against the nose Reynolds number, at Mach 6 and 2.6·10⁷ per metre, for three correlations of the form Reθ/Mₑⁿ = C. Reθ alone: a largest delay of 4.96 at a nose of 43.9 mm, nose Reynolds number 1.14·10⁶; Reθ / √Mₑ: a largest delay of 2.57 at a nose of 20.1 mm, nose Reynolds number 5.23·10⁵; Reθ / Mₑ: a largest delay of 1.41 at a nose of 7.29 mm, nose Reynolds number 1.9·10⁵. Past a nose Reynolds number of 8.81·10⁵ the Mach-ratio criterion transitions within ten nose radii of the tip, where the model's edge is least trustworthy.

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.

compressible · Blast wave

Named alongside it

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

Boundary layerShape factorDisplacement thicknessModel limitSimilarity solutionSkin frictionBlasiusDragFalkner–SkanMeasurementRegimeReynolds number

All concepts