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

Pressure gradient — where it appears

The rate at which pressure changes along the flow, which accelerates or decelerates the fluid near a wall. Its sign decides whether a boundary layer thickens or thins, and its integral decides whether the layer survives to the trailing edge.

Named by 17 essays across 8 fields — each of them below, with the objects they name alongside it.

Every curved streamline has a pressure gradient across it. Twelve points in the flow past a cylinder, with the arrow at each showing the pressure gradient across the streamline. It points away from the centre of curvature everywhere, and its size is ρq²κ: the fluid is being pushed round a bend, and something has to do the pushing. Both sides are computed here and they share no arithmetic — one is Bernoulli's pressure differenced across the flow, the other is the turning rate of the velocity direction along it — and they agree to 6.7e-5. This is the whole content of the effect usually named after Coandă, and it is happening on every curved streamline of every flow.

The effect that explains nothing

A jet follows a curved wall and the wall feels a suction. Both are real, both are famous, and naming them after Coandă explains neither — what is happening is the normal component of Euler's equation, and it holds on every curved streamline in every flow.

misconceptions · Coanda The pressure, relaxed rather than quoted. The pressure field round a cylinder, obtained by relaxing ∇²p = −ρ∇·(u·∇u) on a body-fitted polar lattice with the exact pressure on the surface and on a far circle. The interior was told nothing except the velocity gradients. It agrees with Bernoulli's closed form everywhere to under two parts in ten thousand of the dynamic pressure, which is the strongest statement this site can make that the elliptic equation is the pressure's own.

Pressure has no speed

Take the divergence of the momentum equation for an incompressible flow and the time derivative disappears, the viscosity disappears, and what is left is Poisson's equation. Pressure is not carried anywhere: it is whatever satisfies an elliptic equation everywhere at once, and that is a statement about a fluid nobody has.

inviscid · Pressure The wall makes vorticity at a rate with no viscosity in it. At a stationary wall the momentum equation collapses to ν ∂²u/∂y² = (1/ρ) ∂p/∂x, and the left-hand side is the diffusive flux of vorticity out of the surface. So the pressure gradient along the wall is the vorticity source, and the viscosity that made the no-slip condition necessary has cancelled out of what the condition produces. The curve is that flux across the Falkner–Skan family, computed from profiles solved by shooting and differenced at the wall; the straight line is the pressure gradient each of those flows has. They agree to 2.3e-14. At zero pressure gradient the flux is exactly zero: a flat plate creates no vorticity at all after its leading edge, and everything in its layer arrived from there.

Where vorticity comes from

Every scrap of vorticity in a flow past a body entered through its surface, and the rate at which it enters contains no viscosity at all — it is the pressure gradient along the wall. A flat plate makes none, and a closed body makes exactly as much of each sign.

kinematics · Vorticity 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 Where the suction on a cylinder is actually made. The share of the surface's pressure deficit generated within a given distance of it. Twelve per cent comes from the first twentieth of a radius, half from within a third, and a tenth of it from beyond one and a half radii. There is no thin layer doing the work: the suction is made by the curvature of the whole outer field.

The effect that is real, and where it stops

The Coandă effect has a genuine mechanism, and it is the same equation that makes the suction on a wing. What separates the two is where the integral comes from: a wall jet generates all of its pressure deficit inside a layer two per cent of the radius thick, and a cylinder in a stream needs a third of a radius for half of it.

misconceptions · Coanda The error in the balance is the number itself. The fractional error in the geostrophic wind, against the Rossby number, on logarithmic axes. It is a straight line of slope one through the origin, and that is not an approximation: keeping the centripetal term gives V_g/V = 1 ± Ro exactly, so the error and the number are the same quantity. The geostrophic wind is one per cent right at Ro = 0.01 and a hundred per cent wrong at Ro = 1, which is the value the number is named for and is quoted as the boundary of the approximation.

The balance that is its own error

Geostrophic balance is licensed by the Rossby number being small, and the fractional error in the geostrophic wind is the Rossby number — exactly, not approximately. So the balance everybody uses at Ro of order one is a hundred per cent wrong, and the same quadratic has a hard limit at a quarter that no anticyclone can pass.

regimes · Geostrophy The wall the outer flow is really solving for. A flat plate, the edge of its boundary layer, and the line the outer flow behaves as though the plate were on. The displacement thickness is the mass deficit divided by ρU — checked here against the profile's own integral rather than quoted — and moving the wall out by that much reproduces exactly the flow rate the viscous layer lets past. It is a third of the visible thickness of the layer and it is the only part of the layer the outer problem knows about.

The body the outer flow actually sees

A boundary layer lets less fluid past than an inviscid one would. The outer flow can be given exactly the same reduced flow rate by leaving the fluid inviscid and moving the wall out — so the potential flow that matters is not the flow past the body, but the flow past the body plus a thickness the boundary layer computes.

inviscid · Interaction 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.

viscous · Boundary layer The Earth turns a straight river into a very gentle bend. The transverse circulation through the depth of a straight river running at 1.2 m/s at latitude 50°N, in thousandths of the main flow, against height above the bed — and the circulation of the same river without rotation but bent to the left with a radius of 2V/f, 21 kilometres. The two nearly coincide: the Earth's rotation pushes the fast surface water right and the slow bed water left, exactly as a bend does, and for a river it is a bend of about twenty kilometres' radius everywhere at once.

The Earth bends every river a little

In 1860 Karl Ernst von Baer claimed that the Earth's rotation makes rivers in the northern hemisphere cut into their right banks. The Coriolis force on a river does drive a helix that throws surface water towards the right bank, the same helix a bend drives. For a mountain stream it is a ten-thousandth of what the stream's own bends do. For the great slow rivers of the far north it is a fifth to two-fifths, and there Baer's claim is not absurd.

misconceptions · Coriolis scale Time per litre rises in a straight line with the litres already filtered. The elapsed time divided by the filtrate collected, against the filtrate, for a constant-pressure filtration of a silica slurry through a cloth: integrated step by step, and Ruth's closed form. The line's slope is the cake's resistance, which grows with every litre, and its intercept is the cloth's, which does not. After a quarter of a cubic metre the cake is already the larger.

A filter is slowed by what it has caught

Push a slurry through a cloth and within minutes the cloth no longer matters: the particles it has caught form a cake, the cake is a packed bed, and every litre filtered makes the bed thicker for the next. The filtrate grows only as the square root of time. And if the cake is soft, it packs tighter against the cloth the harder it is pushed, until more pressure buys almost nothing.

applied · Porous At a coast the swell's water goes back, or goes along. The Lagrangian mean current — the Stokes drift plus the Eulerian current — under an 8-second swell arriving straight at a coast, in seas 8, 25 and 60 metres deep, each as a fraction of its own surface drift against depth as a fraction of the water's. Left, the part across the shore, positive onshore; right, the part along it. The cross-shore parts carry no net water. At 8 metres, inside the Ekman layer, the whole exchange is across the shore; at 25 the return is shared with a current along the coast; at 60 the open shelf's own rotation has already taken the transport back, and the coast has little left to do.

A coast sends the drift back, or sends it along

A shelf sea under a swell keeps part of the Stokes drift's transport, turned to the right by the rotating Earth, and near a coast some of it points at the land. The sea answers with a slope in its surface. In water shallower than an Ekman depth the slope drives the water straight back as an undertow, as it would without rotation; a few Ekman depths down the coast turns the transport into a current along the shore instead, and there the sea can slope down towards the land it faces.

kinematics · Stokes drift The current along the shore runs fastest three Ekman depths down. The depth-averaged Eulerian current along the shore, positive to the left looking shoreward, against the local depth in Ekman depths, for a swell arriving 20° off the shore-normal at the shelf edge and for one arriving head-on. Both make a jet to the right, fastest at 2.9 Ekman depths for the oblique swell (-1.42 mm/s at 37.6 m) and 2.6 for the head-on one (-0.948 mm/s), and a weak current the other way in water shallower than one Ekman depth. Rotation turns the transport the shelf returns, so even a swell with no alongshore component drives one.

A sloping shelf puts the swell's current three Ekman depths down

A swell's Stokes transport arrives at a coast and the sea must send it back or turn it aside. On a flat shelf one depth decided which. On a real shelf, deepening from the beach to its edge, every depth is present at once, and the sea does both: it sets up a few centimetres against the beach, dips a tenth of a millimetre where the depth passes two Ekman depths, and runs a current along the shore that is fastest about three Ekman depths down — wherever the friction puts it.

kinematics · Stokes drift The source points along the span; the flow does not. Round a swept circular cylinder, the share of the wall's vorticity source that points along the local direction of the flow outside the layer, against the angle from the attachment line, at sweeps of 10°, 20°, 35° and 50°. The source lies along the span everywhere. At the attachment line the flow does too and the share is one; at the suction peak, 90°, it is 0.0878, 0.179, 0.33 and 0.512. What is not along the flow is the ordinary across-the-flow vorticity a two-dimensional layer has.

A swept wall makes vorticity along its isobars, not across its flow

A still wall in a pressure gradient puts vorticity into the fluid at a rate set by the gradient, and the source is a vector: it lies in the wall, along the isobars. On an unswept body the isobars run across the flow and so does the vorticity, which is the ordinary boundary layer. On a swept one the isobars run along the span and the flow does not, so part of every new vortex line points along the flow — half of it, over the accelerating front of a cylinder swept 35°. That part is where a swept wing's crossflow comes from.

kinematics · Vorticity A slender nose at incidence makes most of its vorticity pointing along the flow. The share of the wall's vorticity source that points along the local surface flow, integrated over the region where the flow accelerates, against incidence, for prolate spheroids of fineness 1 (a sphere), 2, 4 and 8. A sphere makes none at any incidence. At 10° a body of fineness 2 makes 20% of its source streamwise, fineness 4 55% and fineness 8 81%, with no sweep anywhere.

A nose at incidence makes its vorticity along the flow

A wall makes vorticity along its isobars, and on a swept wing that sends part of it along the flow. A fuselage at a few degrees of incidence has no sweep anywhere, and still most of the vorticity its wall makes points along the flow — a fifth for a body twice as long as it is wide, four-fifths for one eight times — because at incidence its pressure varies round the body faster than along it. A sphere, at any incidence, makes none. The lee vortices begin at the wall.

kinematics · Vorticity A flat plate's layer is pushed along by the pressure its own warmth leaves, strongest at the wall. Across the layer of a horizontal plate warm side up, in air: the velocity scaled to its peak, the temperature excess, and the along-plate push −∂p′/∂x scaled to its wall value, against the similarity distance η = (y/x)·Gr^⅕. The push is the hydrostatic deficit under the warm fluid, (2/5)(ηθ − P), 0.694 at the wall in units of gβΔT·Gr^−⅕, falling to nothing at the layer's edge; the velocity peaks at η = 1.58 and the warm layer ends near 5.93.

Laid flat, a warm plate is pushed by the pressure of its own layer

A warm vertical plate drives its layer with buoyancy acting along it. Lay the plate flat and buoyancy acts across the layer instead, so the only push along it is the pressure deficit its own warm fluid leaves under the cold — deeper where the layer is thicker. That push gives a heat transfer growing as the fifth root of the Grashof number, not the fourth, works only on the plate's upper face, and survives a tilt of a degree for a third of a metre.

regimes · Grashof Let its shear stress lag, and a layer reaches its new equilibrium shape in about seventeen thicknesses. The shape factor downstream of a step from zero gradient to an outer speed falling as the power −0.2 of distance, in layer thicknesses at the step: with the shear stress lagging, by Green's lag-entrainment method; with the entrainment fixed by the layer's shape, as in Head's method; and Clauser's equilibrium layer at the lagged layer's own local β, through Coles's profile with Das's wake strength. Twenty thicknesses on, the lagged layer's shape factor is 1.479 and the shape-only layer's 1.445; eighty on, 1.439 and 1.432.

A layer whose stress lags the gradient separates sooner

A turbulent boundary layer that meets an adverse pressure gradient suddenly cannot answer it at once: the large eddies that carry its shear stress were made upstream, on a flat plate, and they draw in too little fluid for the new gradient for the first twenty or so layer thicknesses. The deficit does not protect the layer. It lets the gradient pile momentum deficit into it, so the layer reaches its new equilibrium shape quickly, its friction falls through the equilibrium value on the way, and in a strong gradient it separates in little more than half the distance a calculation without the lag predicts.

turbulence · Wall law The lagged layer arrives at the release with the worse shape and leaves it with the better one. The shape factor of the layer from its flat-plate run through an outer flow decelerating from x = 1 m and released at 1.6 m into a flat one: Green's lagged layer and the layer whose entrainment follows its own shape. At the release the lagged layer's shape factor is 1.66 against 1.57; 11.8 thicknesses later it is below the shape-only layer's, and at the end, 4.5 m, it is 1.282 against 1.301.

A layer released from a gradient overshoots the shape it left

A turbulent layer that meets an adverse gradient suddenly entrains only half what the gradient asks, because its large eddies were made upstream, and it separates sooner than a layer that answers at once. Release it — let the outer flow stop decelerating, as at a diffuser's exit or a wing's trailing edge — and the same lag runs the other way. The eddies made in the gradient keep entraining after it has gone. The layer arrives at the release with the worse shape and, twelve thicknesses later, has the better one, its shape factor falling past the flat plate's and its friction rising past the shape-only layer's.

turbulence · Wall law

Named alongside it

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

Model limitBoundary layerCoriolisEddy viscosityEntrainmentMeasurementSeparationShape factorBernoulli's equationIntegral methodMisconceptionThe no-slip condition

All concepts