Boundary layers

An inviscid flow, the kind Bernoulli’s equation describes, is allowed to slide along a solid surface at whatever speed it likes. Nothing in the mathematics of an ideal fluid forbids it. Nothing in nature permits it. The gap between those two statements is not a footnote to fluid mechanics; it is most of the reason the subject needs a viscosity in the first place, and this article is about the thin region next to every solid surface where that gap gets closed.

The no-slip condition

Fluid in direct contact with a solid boundary takes the velocity of that boundary. Not approximately, not on average, but exactly: at a stationary wall the fluid velocity right at the surface is zero, and at a moving one it matches the wall's own motion. This is the no-slip condition, and it is worth being clear that it is not derived from anything more basic. It is an experimental fact about how real fluids behave at a molecular level, and every viscous flow calculation you will do takes it as a starting boundary condition rather than a result.

Its consequence is unavoidable. Somewhere between the wall, where the speed is zero, and the free stream, where it is whatever the flow far away is doing, the velocity has to get from one value to the other. That transition cannot happen instantaneously, because an instantaneous jump would mean an infinite velocity gradient and an infinite shear stress with it, which nothing in a real fluid supports. So it happens over some finite distance, and that distance, the region across which the velocity climbs from zero at the wall to essentially the free-stream value, is what the whole rest of this article calls the boundary layer.

The word essentially is doing real work in that last sentence. The approach to the free-stream speed is asymptotic, not exact, so the edge of the boundary layer needs an operational definition rather than a natural one. The usual convention fixes it at the height \(\delta\) where the velocity has recovered to 99 per cent of the free-stream value:

That 99 per cent is arbitrary in the way a Richter scale threshold is arbitrary: everyone uses the same convention, so it does not matter that a different number could have been chosen. What is not arbitrary is the physical picture underneath it — a viscous region hugging the wall, and, beyond it, a flow that for most purposes behaves as if it had no viscosity at all. Almost everything a student first learns about inviscid flow is a statement about that outer region, and it is only honest because the boundary layer is usually so thin.

Growth of the boundary layer

Follow the flow along a flat plate from its leading edge and the layer is not a fixed thing; it grows. At the very front, \(\delta = 0\), since no fluid has yet had time to feel the wall. Further along, more fluid has been in viscous contact with the surface for longer, and the affected region thickens. For the smooth, orderly flow near the leading edge, known as laminar, the exact solution to the governing equations — Blasius's solution, which this course will not derive but is worth knowing by name — gives

so the layer grows with the square root of distance: doubling \(x\) only grows \(\delta\) by a factor of \(\sqrt{2}\). That slow growth does not last. Past a critical local Reynolds number, usually taken as \(Re_{x,\text{cr}} \approx 5\times10^5\) for a smooth flat plate though it shifts with surface roughness and free-stream disturbance, small waves in the layer amplify rather than decay, and the flow trips into turbulence. A turbulent boundary layer mixes far more vigorously across its own height than a laminar one, and it grows faster as a result:

which is a milder power of \(x\) in exponent but a larger coefficient, and in practice produces a layer several times thicker than the laminar correlation would have given at the same station. The transition itself is not a knife-edge; it happens over a patch of plate where intermittent turbulent bursts appear and spread, but treating it as a single point at \(Re_{x,\text{cr}}\) is accurate enough for design work and is what the rest of this course will do.

Inside the turbulent layer the velocity profile itself has a different character: flatter across most of its height, with nearly all of the change crammed into a very thin sub-layer right against the wall, where a viscous, nominally laminar sliver persists even though the bulk of the layer above it is not. That shape is not a curiosity; it is the reason the next two sections come out differently for laminar and turbulent flow.

Thickness and displacement thickness

\(\delta\) answers one question: how far from the wall do you have to go before the flow is, for practical purposes, undisturbed. It is useful and it is also somewhat arbitrary, fixed by that 99 per cent convention rather than by anything the outer flow can actually detect. There is a second thickness that answers a sharper question, and the outer flow can detect it directly.

Because the fluid inside the boundary layer moves slower than the free stream, less mass crosses any given station near the wall than would cross it if the fluid there were moving at \(U_\infty\). That deficit, integrated over the height of the layer, is a missing volume flow rate, and displacement thickness \(\delta^*\) is defined as the height of a strip of the free stream that would carry exactly that missing amount:

For a typical laminar profile \(\delta^*\) works out to roughly a third of \(\delta\); for a turbulent one, with its flatter core, the fraction is smaller, since less of the layer's height is actually short-changing the flow. Either way \(\delta^*\) is not a bookkeeping curiosity. The flow outside the layer cannot tell that it is passing over a slow-moving region rather than a slightly thicker solid body; as far as it is concerned, the wall might as well have been built up by \(\delta^*\) and the fluid past that new surface been perfectly inviscid.

That equivalence is what makes the quantity practical rather than merely tidy. A wind-tunnel test section is built slightly wider than the nominal test geometry requires, specifically to leave room for the tunnel walls' own displacement thickness without choking the flow. A duct or a compressor inlet, sized only on its geometric, wall-to-wall area, is quietly narrower than the drawing says once the boundary layer has grown along it:

with \(P\) the wetted perimeter. Ignore that correction on a long enough duct and the flow arriving at the far end is faster, and at a lower pressure, than the geometry alone would predict — not because the calculation was wrong, but because it was answering a question about the wrong cross-section.

Wall shear stress and skin friction

Everything about the shape of the velocity profile away from the wall is, for one particular purpose, irrelevant. What drags on the surface is set entirely by the gradient of velocity right at \(y=0\), through Newton's law of viscosity:

It is worth pausing on how counter-intuitive that is the first time you see it. The wall shear stress is largest exactly where the fluid is moving slowest, at the wall itself, because that is where the velocity is changing fastest over the shortest distance — a thin laminar layer packs its entire speed change into a small \(\delta\), and a thin layer with a large speed change has a steep gradient. A thick layer, other things equal, spreads the same change over more height and drags less.

Nondimensionalise \(\tau_w\) against the dynamic pressure of the free stream and the result is the skin-friction coefficient, and the two boundary-layer correlations from earlier translate directly into it:

Both fall as the layer grows and the velocity gradient at the wall relaxes, but the turbulent correlation sits well above the laminar one at any given Reynolds number it is compared against, because a turbulent layer's flatter core still funnels the whole velocity change through a very thin sub-layer at the wall, keeping the local gradient there sharp even though the layer as a whole has grown thick. Integrate \(\tau_w\) over the whole wetted surface of a body and the result is its skin-friction drag, one of the two contributions — the other is pressure — that make up the total drag force taken up properly in the next topic.

Separation

On the upstream face of a curved body — the front of a cylinder, the nose of an aerofoil, the leading half of any rounded shape — the external, nominally inviscid flow is accelerating, because the body's curvature is opening up room for it to speed into. By Bernoulli, an accelerating flow is one whose pressure is falling along the direction of travel:

A pressure gradient like that is called favourable precisely because it helps the boundary layer along: it is pushing the fluid forward in the same direction the flow is already moving, and a layer under a favourable gradient stays thin and firmly attached to the surface. Past the widest point of the body, the geometry reverses. The external flow now has to decelerate, room to move into is running out, and by the same Bernoulli argument the pressure climbs back up along the direction of travel:

That rising pressure now pushes backward against every particle of fluid it meets, including the ones deep inside the boundary layer that have already been robbed of momentum by friction along the whole front half of the body. Fluid in the free stream, with momentum to spare, can fight through a moderate adverse gradient without much trouble. Fluid right against the wall, moving slowly to begin with, cannot. Given a strong enough adverse gradient and enough distance to act over, the near-wall velocity is driven to zero and then reversed, flowing briefly backward against the main stream. The point where the wall gradient first reaches zero,

is the point of separation. Beyond it the boundary layer is no longer a thin film clinging to the surface; it has lifted off entirely, and the region behind the body fills instead with a wide, low-pressure, highly unsteady wake. Whether a given body separates early or late, and how large the resulting wake is, turns out to be one of the most consequential questions in the whole of practical fluid mechanics — it is most of what decides how much a car, a golf ball or an aircraft wing actually costs to push through the air, which is exactly where the next topic picks the story up.