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

Transition — where it appears

The change from laminar to turbulent flow, which is not a property of the fluid but of the disturbances reaching it. A quiet enough pipe stays laminar to a hundred thousand, so the Reynolds number usually quoted for it describes the environment.

Named by 17 essays across 7 fields — each of them below, with the objects they name alongside it.

The laminar line does not end; the flow leaves it. Friction factor against Reynolds number in a pipe. The laminar law f = 64/Re is exact and is drawn continuing past the transitional Reynolds number, faintly, because it remains a solution there — the flow simply stops taking it. The turbulent branch is Blasius' correlation and begins where experiments find transition, not where any calculation puts it.

The solutions stop being chosen

Hagen and Poiseuille's pipe profile is an exact solution of the Navier–Stokes equations at every Reynolds number, and it is linearly stable at every Reynolds number. Something else happens at 2300 anyway, and it is not that the solution stopped being one.

turbulence · Transition Reynolds number: one number, four different flows. Reynolds number is inertia ÷ viscosity. It is not a property of the fluid or of the shape but of the combination, and crossing a threshold changes the physics rather than the magnitude.

The Reynolds number, and the length in it

The most useful number in fluid mechanics has an arbitrary quantity buried in it, and quoting one without saying which length was used makes it meaningless. That detail is where most misuse comes from.

regimes · Reynolds Two drag laws, one derived and one fitted. Flat-plate drag coefficient against Reynolds number, laminar and turbulent, on log axes. The laminar curve is Blasius' similarity solution, solved by shooting; the turbulent one is the 1/7-power correlation, which was fitted to experiment and is drawn dashed-in-kind to keep the difference visible. Their slopes differ — −1/2 against −1/5 — so the gap between them widens rather than staying put.

The number that is not a number

Transition Reynolds numbers are quoted to three figures and vary by two decades. That is not sloppiness in the measurement — it is the honest report of a quantity that depends on the laboratory as much as on the fluid, and knowing which part is which decides what may be designed on it.

turbulence · Transition 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 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 The cliff a rough ball reaches sooner. The drag coefficient of a sphere against Reynolds number, on log axes. The smooth curve is Morrison's correlation, which is a fit to measurements and is drawn in the colour this site reserves for a borrowed claim. The other is the same curve shifted along the Reynolds axis by a factor of 6 — a stated model of what roughness does, which is to trip the boundary layer early, and not a measurement of any real ball.

The drag that falls as it speeds up

There is a band of speeds in which a smooth ball experiences less drag the faster it goes. Not a smaller coefficient — a smaller force. Dimples move that band down to where a golf ball actually flies, and they do it by making the friction worse.

applied · Sport ball Two sides of one ball, at different pressures. The surface pressure coefficient round a ball, measured from the front stagnation point, on the side the seam trips and on the side it does not. Up to separation both follow the exact potential-flow distribution 1 − (9/4)sin²θ. After it both take the same wake pressure, which is what a manometer measures rather than what the ideal theory predicts. The asymmetry is the shaded area between them, and integrating it gives a side force of 0.2949 towards the later-separating side.

A ball that swings without spinning

A cricket ball curves in flight with no spin about any useful axis. The mechanism is not the Magnus effect; it is a seam tripping the boundary layer on one side so that side lets go later. Which way the ball then goes depends on one borrowed number, and this site's own inviscid solver supplies the value that gets it wrong.

applied · Sport ball 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 Two drag laws, one derived and one fitted. Flat-plate drag coefficient against Reynolds number, laminar and turbulent, on log axes. The laminar curve is Blasius' similarity solution, solved by shooting; the turbulent one is the 1/7-power correlation, which was fitted to experiment and is drawn dashed-in-kind to keep the difference visible. Their slopes differ — −1/2 against −1/5 — so the gap between them widens rather than staying put.

The cost of going turbulent

A turbulent boundary layer costs several times the friction of a laminar one, and the multiple is not a constant — it rises with Reynolds number, because the two laws have different exponents. That is why laminar flow is worth more on a long fast surface than on a short slow one.

viscous · Drag budget 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 Stable in every mode, and 100 times larger first. The energy of the worst-case disturbance against time, on a logarithmic scale, at four Reynolds numbers and at the one the slider selects. Time is in units of Re, which is what makes the four curves the same shape; what changes with Reynolds number is the height, and it changes as the square. Every eigenvalue of this operator is negative throughout, so nothing that grows here is an instability in the sense a stability analysis reports.

Every mode decays and it grows anyway

A stability analysis asks whether any mode of a flow grows, and for pipe flow the answer is no, at every Reynolds number, which the pipe disagrees with. The missing ingredient is that the modes are not perpendicular — a disturbance made of two nearly parallel decaying pieces can grow by a factor of Re²/16 before it dies.

turbulence · Transition Two profiles, and they are the same profile. The chordwise and spanwise velocity profiles in the boundary layer of a yawed flat plate, each as a fraction of its own edge velocity. They are computed by different code — the chordwise one by shooting a third-order nonlinear equation, the spanwise one by a single pass through a linear second-order one — and they agree to 2.1e-8 over the whole layer. They are the same function of η, because the two equations reduce to the same equation. A swept flat plate has no crossflow at any sweep angle, and that is the independence principle in the only form that has no wriggle room in it.

The wind a swept wing feels

Sweeping a wing back is usually justified by saying it meets a slower wind. It does not meet a slower wind. The equations split exactly in two, and the flow along the span is a passenger that exerts no force and changes nothing — until a pressure gradient breaks the split, and then it becomes the reason a swept wing is a different problem rather than a harder one.

circulation · Sweep Two lifetimes, crossing. The mean time for a turbulent puff to decay and the mean time for it to split into two, against Reynolds number, on a logarithmic axis spanning thirty orders of magnitude. Below the crossing puffs die faster than they multiply; above it they multiply faster than they die, and the flow stays turbulent.

A puff that does not know how old it is

A patch of turbulence in a pipe below the critical Reynolds number dies at random, and its chance of dying in the next second does not depend on how long it has already lasted. The flow that contains it has a memory anyway, because the patches multiply — and where multiplying overtakes dying is a Reynolds number.

turbulence · Transition Past transition the cube law gives way to two and a half. The exponent k in Q ∝ r^k obeyed by the cheapest vessel, against that vessel's Reynolds number, for a smooth wall and for a wall with a millimetre of roughness. While the flow is laminar k is 3, Murray's law. Past transition it drops to 2.45 at Re = 2·10⁴ and 2.4 at 1.01·10⁷, near Blasius's 27/11 = 2.455. A fixed absolute roughness barely moves it, because the cheapest pipes for large flows are large and a millimetre is a small fraction of them.

The cheapest turbulent pipe follows two and a half, not three

Murray's cube law comes from Poiseuille's resistance, and it fails the moment the flow in a vessel turns turbulent. The same cost minimised with turbulent friction gives a flow that grows as the radius to the power 27/11 in a smooth pipe and 7/3 in a rough one. Murray's invariant, a wall shear the same in every vessel, goes with it, and so does the local rule that made the law credible as biology: a vessel that holds its shear at a set point is twice too wide.

applied · Branching A quarter of the way into the layer, the lag follows the transitions along the chord. The transit lag a quarter of the way into the layer, as a multiple of the lag outside it, against chordwise station, at 0° and 4° on the thin cambered section at a chord Reynolds number of a million, with each surface's transition marked. At 0° the upper layer turns at 51% and the lower at 35%; the lag is −0.122 times the outside value at mid-chord and 0.61 at 95%. At 4° the upper layer turns at 11% and the lower not at all before the trailing edge; the lag is 2.91 times the outside value at mid-chord and 3.51 at 95%.

The side that turns turbulent first decides who arrives first near the wall

Inside a laminar layer the upper parcel's lead over the lower grows with depth. Real wings are not laminar to the trailing edge: one surface's layer turns turbulent first, and a turbulent layer is full almost to the wall. At the incidences a wing flies at, the upper side goes first and the exaggerated lead survives to the trailing edge, smaller than the laminar one and shrinking with the Reynolds number. Below about one degree the lower side goes first, and a tenth of the way into the layer the lower parcel arrives first.

misconceptions · Equal transit 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 Followed along their streamlines, deep parcels at 4° arrive nearly together, and at 0° the upper leads by more than outside. The arrival lag at 95% chord, lower parcel minus upper, as a multiple of the lag outside both layers, against the streamline both parcels follow, as a share of the upper layer's mass flux at the trailing edge. At 4° it is 0.46 for the deepest streamline drawn, −0.16 at a twentieth, 0.47 at a fifth and 1 at the layer's edge; at 0°, 1.47, 1.68, 1.33 and 1.

Followed rather than held, the parcels near a wing nearly tie

Parcels held at a fixed fraction of a boundary layer's thickness made the side that turns turbulent first decide which arrives first near the wall, and at a wing's ordinary incidence they gave the upper parcel a lead ten times the one outside the layer. Real smoke is not held at a depth: it follows a streamline. Near the attachment point the layer is thinner than any filament's height, so every filament starts outside it and is swallowed later, and on the upper surface, whose turbulent layer grows faster, it is swallowed deeper. Followed this way, the deep parcels at 4° arrive nearly together, and at a Reynolds number of a million the lower one can come first.

misconceptions · Equal transit

Named alongside it

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

Boundary layerReynolds numberModel limitSeparationCorrelationFalkner–SkanLinear stabilityAdverse pressure gradientRoughnessShape factorSkin frictionBlasius

All concepts