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

Viscous sublayer — where it appears

The region closest to a wall in which the turbulent stress is negligible and the velocity rises linearly. It is a few tens of microns thick on an aircraft, and whether a surface's roughness is buried in it decides whether the flow can feel the roughness at all.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

The chart, with one exact line on it. The friction factor of a pipe against Reynolds number, for five relative roughnesses. Every curve here except one is Colebrook's correlation, solved by iteration rather than read off a chart. The exception is the short straight line at the left: f = 64/Re is the laminar solution and it is exact. The curves flatten to the right because once the roughness pokes out of the viscous layer the Reynolds number has nothing left to change.

The roughness a wall cannot feel

A rough pipe and a polished one carry the same flow for the same pressure over three decades of Reynolds number, and then suddenly they do not. What changed is not the pipe. It is the thickness of the film of fluid at the wall, which is the only part of the flow that can see the roughness at all.

applied · Internal flow u⁺ = (1/κ) ln y⁺ + B, integrated rather than asserted. The velocity profile in wall units, produced by integrating the mixing-length closure outward from the wall. The straight portion is the log law and the constants beside it were least-squares fitted to the integrated curve over 50 < y⁺ < 500 — so the 1/κ is a measurement on the drawing rather than the number that was fed in. The viscous sublayer u⁺ = y⁺ comes out rather than being pasted on.

A guess with a constant in it

Prandtl's mixing length is one line — an eddy near a wall can only be as big as its distance from the wall. Integrate it and the whole structure of a turbulent wall profile falls out, sublayer and log region and all. That is a fact about the assumption, and the essay is careful about which.

turbulence · Closure The roughness function, and the asymptote in which the viscosity has gone. The whole effect of a rough wall on a turbulent boundary layer is one number: the downward shift of the logarithmic profile. It vanishes on a smooth wall, rises through a transitional band, and becomes (1/kappa)ln(k+) + B − 8.5 — at which point substituting it back leaves u+ = (1/kappa)ln(y/k) + 8.5, with the fluid's own length gone from the answer entirely.

A second length at the wall

The logarithm in a turbulent wall profile exists because a region of the flow is not allowed to know about any length except the distance to the wall. Roughen the surface and there is one it does know about, which belongs neither to the fluid nor to the flow — and the slope does not change at all.

turbulence · Roughness Four wall models, through the buffer layer. Van Driest's damped mixing length, Reichardt's fit, Spalding's implicit law, and a control with no buffer layer at all — the viscous sublayer joined straight to the logarithm where they cross, at y+ = 11.6. The three fitted models agree with each other to a per cent and a half; the control is nineteen per cent above them at y+ = 10.

Three buffer layers, one friction

Four wall models are put through a pipe. The one with no buffer layer at all is nineteen per cent wrong where the turbulence production peaks and two and a half per cent wrong in the friction; the three respectable ones are within two per cent of each other in the buffer layer and spread by seven in the friction. The answer is not where it was expected.

turbulence · Wall law Slip follows the stripes more closely than the shear does. Plan views of a striped surface, with the stripes running across each panel, for a shear at 0°, 30°, 54.7° and 90° to them. The faint arrow is the direction of the shear; the dark one is the slip velocity it produces, whose component along the stripes is the along-stripe slip length times the shear and whose component across them is half that. The slip is turned towards the stripes by 0.0°, 13.9°, 19.5° and 0.0°. It is largest, 19.47°, for a shear at 54.74°, where tan θ = √2. A surface with a tensor for a boundary condition can push a flow sideways, which a scalar slip length never can.

Twice as slippery along as across

A surface of alternating gas and solid stripes lets a liquid slip, and a flow far above it sees one number in place of the pattern — but the number depends on which way the flow goes. Along the stripes it is Philip's logarithm; across them it is exactly half, for a reason that takes one substitution to show. And the logarithm means that the slip is bought by the pattern's period rather than by how much of it is gas.

kinematics · Boundary conditions

Named alongside it

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

The law of the wallSkin frictionThe von Kármán constantBoundary layerEddy viscosityFriction factorFriction velocityMixing lengthModel limitReynolds numberRoughnessWall units

All concepts