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

Similarity solution — where it appears

A solution that has the same shape at every station once the coordinates are stretched by the right power. The stretching turns a partial differential equation into an ordinary one, which is why so many exact results in viscous flow are of this kind.

Named by 21 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 The wall slope against pressure gradient, and where it runs out. How steeply the flow leaves the wall, plotted against the pressure gradient the layer is running into. A favourable gradient presses the profile against the surface and steepens it; an adverse one hollows it out. The curve reaches zero at a definite value, and beyond that there is no attached solution at all.

How much uphill a layer can take

A boundary layer running into rising pressure is climbing a hill on the last of its momentum. There is a definite steepness at which it can no longer do it, and the number is not a rule of thumb — it is where a family of solutions stops existing.

viscous · Separation 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 One layer, and the four others inside and outside it. The velocity profile over a flat plate, and the temperature profile in the same layer at four Prandtl numbers: a liquid metal at 0.01, air at 0.71, water at 7 and a heavy oil at 100. The equations differ by one number and the profiles differ by a factor of twenty in thickness. At Pr = 1 the two are the same function — not similar, identical, to eight decimal places — because the equations and the conditions are then the same, which is what every statement called a Reynolds analogy rests on.

The other layer, and the one number that separates them

A wall in a stream carries two boundary conditions and grows two layers. Their thicknesses differ by a factor of twenty across ordinary fluids, and at exactly one Prandtl number the two profiles are not similar but identical.

regimes · Peclet 45° of corner, and an infinite number of eddies in it. The creeping flow in a corner of 45 degrees, drawn from Moffatt's similarity solution. Each eddy turns the opposite way to its neighbours and is 3.17 times smaller and 1.6e+3 times weaker than the one outside it. The contour levels are rescaled inside each eddy, because they differ in strength by three orders of magnitude per step and a single set of levels would show the first and nothing else — which is itself the reason nobody has seen the third. The dividing lines between eddies are drawn where the stream function changes sign, and the dots are the centres, both found from the solved field.

The eddies nobody stirs

A slow flow past the mouth of a sharp corner does not simply fail to get in. The corner fills with an infinite sequence of counter-rotating eddies, each about three times smaller and sixteen hundred times weaker than the last, and the whole structure is decided by one complex number.

viscous · Corner The jet has to be fed from the sides. The velocity field of the plane jet with streamlines integrated through it. The seven central streamlines run down the jet and spread; the ten started at the top and bottom edges bend inwards and join it, which is entrainment and is a consequence of the solution rather than an addition to it. The dashed lines are the half-speed edges, widening as x^{2/3}. The transverse velocity far from the axis is 5.70e-3 m/s at this station, inward on both sides — a jet is a sink as seen from a distance, which is why two parallel jets pull together.

What a jet keeps, and what it collects

A jet leaving a nozzle into still fluid has no boundary anywhere and one conserved quantity. Its momentum flux is exactly the same at every station downstream; its mass flux is not conserved at all and grows without limit, because a jet is a machine for acquiring fluid it did not start with.

viscous · Free shear The core spreads and the outside never notices. The swirl velocity at four times a factor of four apart, with the free vortex Γ/2πr drawn behind them. Every curve leaves the free-vortex line at its own core radius and turns over into solid-body rotation inside it; outside the core all four are the same curve, to the precision of the plot. Viscosity has rounded off the singularity and changed nothing else. The peak swirl falls from 52.4 to 6.6 m/s across the four, and the circulation is identical for all of them.

What viscosity cannot take away

Leave a vortex alone in a viscous fluid and every local measure of it falls — the peak spin, the peak velocity, the enstrophy. The circulation round a large loop does not move at all, ever, and the far field is identical to the line vortex it started as.

viscous · Diffusion The current at the surface is 45° from the wind, and nothing sets that angle. The Ekman spiral drawn as a hodograph: each point is the velocity at one depth, and depth runs along the curve. At the surface the flow is at exactly 45 degrees to the wind that drives it — not approximately, exactly, and independently of the wind, the viscosity and the latitude. By one Ekman depth the flow has turned another radian and lost 1/e of its speed; by three it is a hundredth of the surface value and pointing back the way it came. The angle is a property of the equation having two terms in it, and nothing else.

The layer that stops at a depth

Every other boundary layer grows. This one does not — rotation supplies a frequency, the balance against diffusion supplies a length, and the transport that comes out contains the stress on the surface and not the viscosity underneath it.

viscous · Rotating The wall's condition, on its way to the middle. Five profiles across the half-channel, from just inside the entrance to fully developed, each drawn at the station where it occurs. The march starts from a slab of uniform flow and never assumes a shape: what arrives at the far end is a parabola, with a centre-line speed of 1.4979 times the mean against the exact 3/2 and a momentum flux of 1.1995 against 6/5. Notice what the middle does while the edges are being slowed: it speeds up, because the flow rate is held, and that acceleration is what the entrance's extra pressure drop pays for.

How far before a duct forgets what was fed into it

A pipe is always drawn with its answer already in place. Getting there takes a distance proportional to the Reynolds number, which means a more viscous fluid is done sooner — and the entrance costs a fixed number of dynamic pressures however long the pipe is.

viscous · Entrance 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 The shock radius against time, from an equation that was given no exponent. The thin-shell energy balance integrated forward from a small initial radius, on logarithmic axes. The two-fifths power is not put in: the ordinary differential equation is Ṙ = √(E/AρR³), and the straight line is what it does. The fitted slope is 0.39983 and the fitted prefactor is 0.90721 against a closed form of 0.90702 — the small residuals being the integration's memory of where it started, which the similarity solution has no equivalent of.

A radius that gives the energy away

Four quantities, three dimensions, one group. The radius of a strong blast must be a constant times (Et²/ρ)¹⁄⁵, and nothing about the device, the chemistry or the initial size can appear. The exponent is free and the constant is not.

compressible · Blast wave 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 A vortex has no energy. The kinetic energy of a Lamb–Oseen vortex inside a circle, per metre of its length, against the logarithm of that circle's radius. It is a straight line and it does not stop: outside the core the swirl is Γ/2πr, the energy density falls as 1/r², and the area grows as r², so every decade of radius adds the same amount. There is no such thing as the energy of a line vortex without a stated cutoff, and no cutoff is physical.

The energy a vortex cannot have

A line vortex has infinite kinetic energy. Not a large amount — infinite, growing without limit as the logarithm of however far out the counting stops. And it is losing that energy at a rate that is finite, exactly known, and contains no cutoff at all.

viscous · Diffusion Nu/√Re against the Prandtl number, over eight decades. The whole of the flat plate's heat transfer, as one curve. It is not a power law: it goes as Pr^½ at the bottom, where the thermal layer is far thicker than the viscous one, and as Pr^⅓ at the top, where it is buried inside it. The Pr^⅓ everybody quotes is the upper half. The two asymptotes are drawn beside it, and the low one is √(Pr/π) in closed form.

The number that is an answer

Almost every dimensionless group is a hypothesis: somebody sets the speed, the size and the fluid, and the number licenses a model. The Nusselt number is not. It is what the experiment produces, it sits on the left of the equals sign, and a regime diagram drawn on it is a category error.

regimes · Nusselt 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 27 per cent high 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 The convergence exponent depends on the gas, which a dimensional exponent cannot. R ∝ (−t)^α for a converging shock, against the ratio of specific heats, for cylindrical and spherical symmetry. Guderley's exact values are marked and the agreement is to four figures. The Sedov blast's two-fifths is drawn beside them: it is the same for every gas, because it comes from dimensions and a conserved energy, and γ is dimensionless.

An exponent dimensions cannot give

A blast wave's radius goes as the two-fifths power of time, and the two-fifths is arithmetic: count the dimensions and it falls out. A shock converging on a point goes as the 0.717 power, and no amount of counting will produce that number — because it depends on the gas, and γ is dimensionless.

regimes · Similarity One shape, at every station and for every jet. The axial velocity of Schlichting's round jet at four distances, each scaled on its own centreline speed and its own half-width. The four curves are one curve: the shape function (1 + eta²/4)⁻² has no parameter in it at all, and every station of every jet — laminar or modelled, weak or strong — collapses onto it exactly.

Exactly similar, and one number short

The round jet has an exact solution of the Navier–Stokes equations, and a turbulent jet is modelled by the same formula with the viscosity replaced by a fitted constant. The shape is identical. Nothing a measurement of the profile can do will tell the two apart.

viscous · Free shear A cooled wall at β = 1: the linear relation is 158 K out inside the layer. The static temperature across a laminar layer at Mach 5 with an edge temperature of 220 K, over a wall whose total enthalpy is 0.5 of the edge's, at a pressure gradient β = 1: exact (thick) and from the Crocco–Busemann linear relation (thin), at a Prandtl number of one with constant properties. The wall is at 660 K in both. The exact profile peaks at 684 K and the linear one at 759 K; the largest difference, −158.3 K, is at η = 0.81.

The gradient the heat never hears

On a flat plate at a Prandtl number of one, a boundary layer's total enthalpy is a straight-line function of its velocity, whatever the wall's temperature. Put the same layer in a pressure gradient and the straight line fails everywhere except on an insulated wall, because the gradient enters the velocity's equation and not the enthalpy's — and a favourable gradient can leave a band of gas colder than the free stream above a wall three times hotter than it.

compressible · Recovery Three profiles that do not depend on the radius. The radial, azimuthal and axial velocities of the flow above a rotating disc, as functions of one similarity variable. The radial one is a jet: fluid thrown outward by the swirl it has picked up, peaking at 0.181 of the local disc speed a fifth of the way through the layer. The azimuthal one falls from the disc's own speed to nothing. And the axial one is the surprise — it does not vanish far from the disc but tends to a constant, so the disc draws fluid down onto itself at 0.8845 times the square root of the viscosity times the rotation rate, at every radius and for ever.

The solution that keeps its nonlinear term

Every exact solution before this one has been exact because the nonlinear term vanished. A rotating disc's does not vanish — at the wall it is the whole of the balance — and the reduction is exact anyway, because the radius divides out of all three momentum equations at once.

viscous · Exact layer The profile a diverging channel flattens into, and then cannot hold. Five purely outward profiles in a wedge of 0.2 radians, at rising flux, each normalised to its own centreline value. As the flux rises the profile flattens in the middle and steepens at the walls — and then it stops. The last one has zero slope at the wall, which is separation, and beyond it no purely outward profile of this form exists at all. Nothing was added to the equation to make that happen: the wall shear is the square root of a cubic and the cubic runs out.

One channel, one flux, two flows

Flow between two plane walls meeting at a line has an exact solution. Past a threshold that turns out to be a ratio of gamma functions, it has two — the same wedge carrying the same flux, once outward everywhere and once with the fluid running backwards along both walls, and nothing in the equations chooses.

viscous · Exact layer Six that are symmetries and five that look like them. Each transformation applied to an exact solution, with the Navier-Stokes residual recomputed from the transformed field by finite differences — nothing differentiated by hand. The six symmetries leave the residual at the differencing floor, a few parts in 10^8. The five near-misses leave between 0.048 and 4.3, which is six to nine orders of magnitude larger. The gap is what makes this a test rather than an illustration: a transformation that is nearly a symmetry does not exist here, and every one of the five is something a reader might reasonably believe.

Why the list is this long

Every textbook list of exact solutions of the Navier–Stokes equations is about a dozen long, and the usual explanation is that the equations are hard. It is not the reason. A similarity reduction is a solution invariant under a subgroup of the equations' own symmetries, so the catalogue of possible reductions is the catalogue of subgroups — and that is a finite, countable object.

viscous · Exact layer

Named alongside it

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

Boundary layerModel limitMeasurementSelf-similarViscosityBoundary conditionConservationExact solutionFalkner–SkanScalingAdverse pressure gradientBlasius

All concepts