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

Entropy — where it appears

The thermodynamic quantity that cannot decrease in an isolated system, and the only reason a shock may compress a flow and not expand one. In compressible flow it is the book-keeping that separates the jumps nature allows from the jumps the conservation laws also permit.

Named by 21 essays across 2 fields — each of them below, with the objects they name alongside it.

The energy is all there the whole way: enthalpy spent to buy speed. At each Mach number, the fraction of the stagnation enthalpy still held as heat and the fraction converted into directed motion. The two always sum to the total, which is the compressible replacement for Bernoulli's equation — an energy statement rather than a pressure one. At Mach 2 more than four fifths of the heat has become speed.

Energy instead of pressure

Bernoulli's equation is a statement that pressure and speed trade against each other at fixed density. Take the fixed density away and the trade is between heat and speed instead — and what survives is not a pressure at all.

compressible · Bernoulli's equation A normal shock at Mach 2.00, and what crosses it unchanged. The state in front of the shock and the state behind it. Every ratio was computed from the standard jump relations and then substituted back into mass, momentum and energy, which is an independent route — a mistyped exponent in the total-pressure expression cannot survive a momentum balance it never appeared in. The residuals are printed below because a check nobody can see is a check nobody can audit.

The jump the equations allow

A shock is a discontinuity in a fluid, which sounds like a breakdown of the description rather than a solution of it. It is a solution: mass, momentum and energy can all be satisfied across a jump, and every ratio across one follows from that alone.

compressible · Shock The second law is the only thing that forbids the other half of this curve. Entropy change across a normal shock, against the Mach number in front of it. The solid branch is the compression shock that exists. The dashed branch below Mach one is the expansion shock, and it satisfies mass, momentum and energy exactly — the residuals are zero to machine precision. It is refused by the second law alone, the one statement in the problem that no conservation residual can show.

The only law that forbids it

The jump conditions permit a discontinuity in either direction. An expansion shock conserves mass, momentum and energy exactly — the residuals are zero to machine precision — and it does not exist. Nothing that can be drawn rules it out.

compressible · Shock Everything a normal shock does, against the Mach number in front of it. Four quantities across a normal shock, each scaled to fit one axis. Pressure and density rise without limit and without bound as the Mach number grows; the Mach number behind falls towards a floor it never passes; and the total pressure — the flow's ability to be turned back into speed — collapses. That last curve is why a supersonic intake is designed around avoiding a single strong shock.

What a shock costs

All the heat survives a shock and none of it is lost. What is lost is the ability to turn that heat back into speed — 27.9 per cent of it at Mach 2 and 93.8 per cent at Mach 5 — and every supersonic intake ever built is a scheme for paying less.

compressible · Shock Past 23.0° at Mach 2.00 there is no attached shock. The same wedge at two half-angles. On the left the θ–β–M relation has a root and the shock sits on the nose. On the right it has none, and the solver throws rather than returning the nearest thing — which matters, because a solver that quietly clamped to the maximum would draw a neat attached shock on a body that cannot carry one. The bow shock on the right is indicative: its shape is not solved here.

When the wedge is too blunt

Every curve of shock angle against deflection has a maximum. Past it there is no attached shock at any angle — the solver has no root to return and must say so, rather than quietly handing back the nearest thing and drawing a picture that cannot exist.

compressible · Oblique shock The same 10° turn, taken both ways. A supersonic stream turned away from itself expands through a fan of Mach waves and keeps every bit of its total pressure. Turned into itself through the same angle it shocks, and pays. Nothing in the equations distinguishes the two cases except the sign of the angle: compression waves converge and steepen into a front, expansion waves diverge and spread.

Turning the other way is free

Compression through ten degrees at Mach 2 costs 1.54 per cent of the total pressure. Expansion through the same ten degrees costs exactly nothing — not approximately nothing, nothing — and the two are the same equations with the sign of one angle changed.

compressible · Expansion fan Lift and wave drag on a flat plate at Mach 2, exactly and to first order. The lift and wave-drag coefficients of a supersonic section against incidence, computed face by face from shocks and fans, with Ackeret's linear result dashed over them. The two agree to three decimal places at small angles and part company slowly — which is what a first-order theory is supposed to do, and evidence rather than tautology, since the two routes share no algebra.

Drag with nothing to rub

d'Alembert's paradox says a closed body in a steady, inviscid flow has no drag, and four essays on this site argue it and none of them is wrong. Above Mach one it is false — the flow is still inviscid, still steady, and the drag is real, finite and quadratic in incidence.

compressible · Wave drag Three waves out of one discontinuity. The x–t diagram of the burst diaphragm. A shock runs right at a speed of its own, a contact surface follows it more slowly, and an expansion fan spreads left as a family of rays that opens with time — the three wave families the Euler equations possess, produced at once by an initial condition with no waves in it at all. Every straight line here is a speed the solution computed, and the fan is drawn as the rays it actually consists of.

One diaphragm, every wave

Two states of the same gas at rest, separated by nothing, is the simplest initial condition compressible flow admits — and its answer contains all three waves the equations possess at once: a shock one way, an expansion fan the other, and between them a surface across which the density jumps and the pressure does not.

compressible · Shock tube Two different physics, one destination. The Fanno and Rayleigh lines, each with temperature up the page and entropy across, each referred to its own sonic state. Neither is a curve anybody drew: the entropy was swept over Mach number and its maximum located, and it sits at Mach 1 on both to within the resolution of the sweep. The subsonic branch runs up to the nose from the left and the supersonic branch runs up from below, so whatever is being done to the flow, it can only be moved towards that point.

Two ways to choke

Friction and heat are different physics acting on different conservation laws, and they drive a duct's flow to exactly the same place. Both have their entropy maximum at Mach 1, so neither can push a subsonic flow past it and both drag a supersonic one down to it — and where their two curves cross is a shock.

compressible · Fanno rayleigh The Hugoniot of a gas that can burn, and the gap in the middle of it. Pressure against specific volume, both scaled on the unburnt gas. The lower curve is the ordinary shock Hugoniot, which passes through the initial state because a jump of zero strength satisfies mass, momentum and energy. Adding a heat release lifts it away, and the initial state now sits in a region no wave can reach: between the two branches a Rayleigh line would need a positive slope, and its slope is minus the square of the mass flux. A burning gas has no weak waves available to it at all.

The other branch of the same curve

Put heat into the jump conditions and the Hugoniot lifts away from the initial state, leaving a gap no wave can occupy. A burning gas has no weak waves: it must run supersonically or subsonically, and the conservation laws pick the first speed exactly and say nothing at all about the second.

compressible · Detonation A curved shock, and the entropy each streamline picks up crossing it. A parabolic bow shock ahead of a blunt nose at Mach six, with the streamlines drawn arriving horizontally and a marker at each crossing whose size is the total pressure lost there. The streamline through the nose crosses a normal shock and keeps three per cent of its total pressure; one four nose radii out crosses at fourteen degrees and keeps ninety-four per cent. Every streamline gets a different entropy, and the stagnation enthalpy is the same on all of them.

The spin a shock leaves behind

A curved shock gives every streamline a different entropy rise and the same stagnation enthalpy. Crocco's theorem then forces vorticity into a flow with no viscosity anywhere — and it scales as the inverse of the shock's radius of curvature, exactly, so a straight shock makes none.

compressible · Crocco γ for air, against temperature. The ratio of specific heats for air as a mixture of nitrogen and oxygen, with the vibrational mode filling according to the Einstein function. It is 1.400 at room temperature, where only translation and rotation are available; 1.337 at a thousand kelvin; and 1.288 at six thousand. Every compressible result on this site has used 1.4, and that is the value for a gas that is not hot — which, behind any shock worth drawing, it is not.

When gamma stops being a number

Every compressible result on this site has used γ = 1.4, which counts the ways a nitrogen molecule can hold energy at room temperature. Behind a Mach 10 shock the gas is at 3,800 kelvin and the count is different — and the pressure barely moves while the temperature falls by fifteen per cent.

compressible · Real gas The velocity through a shock at Mach two, as a function of position. The Becker profile, integrated outwards from its own inflection point. It runs from the upstream velocity to the downstream one — the two states the jump conditions give, which appear here as the equilibria of a first-order differential equation — and its steepest gradient matches the closed form to one part in ten thousand. The horizontal axis is in units of the thickness, which for this shock is 139 nanometres.

The discontinuity that has a thickness

The jump conditions do not contain the viscosity, which is why they are exact. The thickness is entirely viscosity — 289 nanometres at Mach 1.5, 35 at Mach 5, against a mean free path of 64. At Mach five the continuum equations have produced a structure thinner than the distance between collisions.

compressible · Shock So splitting a turn into N ramps costs one over N squared. A twelve-degree compression at Mach 3, done in one ramp and in up to sixty-four. The entropy is N times a cube of one Nth, so it falls as exactly the inverse square of the number of ramps — the measured exponent is −2.00 — and sixty-four ramps cost a two-hundred-and-fifty-sixth of what one costs.

A compression that costs nothing in the end

Turning a supersonic stream away from itself is free and turning it into itself is not. But the price of a compression is the cube of its strength, so splitting one turn into N turns costs one over N squared — and in the limit the compression is free too.

compressible · Expansion fan The coefficient Stokes threw away is not a correction. What absorbs sound in dry air, split into the three transport effects that add to make the diffusivity of sound. Shear viscosity is the largest single contribution and it is not most of it; the bulk viscosity — which Stokes set to zero in 1845, saying in the same paper that he had no argument for doing so — is a quarter, and thermal conduction is a fifth. Setting ζ to zero does not make a small error in the absorption; it makes a 24 per cent one.

The viscosity nobody uses

A fluid has two viscosities. One resists a change of shape and appears in every figure on this site; the other resists a change of volume, was set to zero by Stokes in 1845 in a paper that says he had no argument for doing so, and for carbon dioxide is fifteen hundred times larger than the one everybody quotes.

viscous · Bulk viscosity Seven shock structures, and the two states they all connect. The velocity through the shock for seven dissipation models — Prandtl numbers from a quarter to two, viscosities from constant to linear in temperature. Each curve is shifted so its midpoint sits at the origin. They start at the same speed, end at the same speed, and are nothing alike in between.

The jump does not ask what made it

Seven different dissipation mechanisms are made to smear the same shock. Their interiors are a factor of two and a third apart in thickness, their entropies overshoot the final value by between a quarter and a doubling, and the state they all arrive at agrees to seven parts in ten billion — because the end states are conservation and the interior is transport.

compressible · Shock The four waves, and the one that never goes away. The shock tube in space and time: a shock running right, an expansion fan running left, and the contact surface between them. The shock and the fan are travelling disturbances that leave; the contact is made of fluid, so it is carried along and is there for ever.

A surface that remembers the diaphragm

Between the shock and the expansion in a shock tube there is a surface across which the pressure and the velocity are identical and the temperature differs by a factor of two. It is made of fluid, so it never goes away, and nothing in the pressure field says it is there.

compressible · Shock tube The Fanno line, which has only one direction on it. The entropy relative to the sonic state, against Mach number, for both branches. Friction raises the entropy, so a duct flow moves to the right along this curve whichever branch it is on — up in Mach number from below and down from above — and it stops at the sonic point.

A duct that cannot be run backwards

Friction drives a compressible duct flow towards the speed of sound from either side, and the entropy rises the whole way. So the state of the gas at a station is an odometer: it records how much duct is behind it, and no amount of further duct can take it back.

compressible · Fanno rayleigh Two totals across a shock. The ratio of the total temperature and the ratio of the total pressure across a normal shock, against the shock's Mach number. One of them is one at every Mach number, to the last bit of double precision; the other falls to under a hundredth by Mach eight.

Two totals, one of which a shock cannot touch

Across a normal shock the total temperature ratio is 1.000000000000000 at every Mach number, and the total pressure ratio falls to 0.0085 by Mach 8. One of the two records the energy that has been added to the gas and nothing else; the other records every irreversibility on the way.

compressible · Bernoulli's equation With friction the flow goes sonic after the throat, and leaves slower. The Mach number along a convergent–divergent nozzle of exit area ratio 2.5, for friction lengths 4fL/Dₜ of 0, 0.2 and 1, each solution passing smoothly through Mach 1 at the point where the sonic condition holds. Without friction that point is the throat, at x = 0.42. With friction it moves downstream — to 0.4357 and 0.4996 — and the throat itself is subsonic, at Mach 0.965 and 0.852. The exit Mach number falls from 2.443 to 2.247 and 1.747. The throat is where the area is least; the sonic point is where the widening has caught up with the friction, and the two coincide only when there is none.

Friction moves the sonic point past the throat

A choked nozzle is sonic at its throat — in a nozzle with frictionless walls. With friction the flow reaches Mach one where the section's widening rate has caught up with the friction, which is downstream of the throat, and the throat itself is subsonic. The solution through that point is a saddle that can only be found from the inside, and the mass flow it passes is less than the throat's area allows.

compressible · Area mach A 20° cone at Mach 2: the shock at 37.80°, and a flow still compressing behind it. The conical flow round a cone of 20° half-angle at Mach 2, from the Taylor–Maccoll equation. The shock sits at 37.80° from the axis and turns the flow crossing it through only 8.57°, leaving it at Mach 1.693. Between the shock and the surface every ray from the apex carries its own state: the Mach number falls from 1.693 just behind the shock to 1.568 at the surface and the pressure rises from 1.586 to 1.912 times the free stream's — the rest of the turn, made without a shock. Nothing depends on the distance from the apex, so the rays are lines of constant state.

A cone finishes its turn after the shock

A wedge turns a supersonic stream all at once, at its shock. A cone of the same angle does not: its shock turns the flow only part of the way and leaves the rest to a smooth compression between the shock and the surface. Solved from Taylor and Maccoll's equation, the cone's shock is weaker, keeps more of the total pressure, carries less than half the wedge's surface pressure, and stays attached to 40.7° at Mach 2 where the wedge gives up at 23°.

compressible · Oblique shock

Named alongside it

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

Total pressureShock waveIrreversibilityStagnation temperatureNormal shockThe second law of thermodynamicsDiscontinuityMeasurementModel limitModel validityRankine–Hugoniot conditionsConservation

All concepts