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

Boundary layer — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · viscous
  2. 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.

    part 2 · viscous
  3. 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.

    part 3 · viscous
  4. 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.

    part 4 · viscous
  5. 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.

    part 5 · viscous

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