The collection

Every essay — page 11

Page 11 of 39, continuing through the fields in the same order.

Flows and fields Ideal flow Circulation and lift Viscosity Regimes and numbers Compressible flow Transition and turbulence Fluids at work What is taught wrongly Series Concepts Regimes Refutations Search

Compressible flow

What changes when density stops being a constant. Shocks, expansion fans, nozzles that work backwards, and the one place on this site where an inviscid flow has drag.

Each cancellation costs two powers of the Mach number. Radiated power against compactness for three source clusters: a single monopole, two of opposite sign, and four on a square with alternating signs. The fitted slopes are 0.00, 2.00, 4.00 — zero, two and four in (kd), measured by integrating the far field over a sphere rather than assumed. A turbulent eddy turns over in about the time sound crosses it, so kd is of order the Mach number, and those exponents become the fourth, sixth and eighth powers of speed. A flow with no moving surfaces has no monopole and no dipole available to it, which is Lighthill's whole argument, and the eighth power is what is left.

The sound that only leaves

A flow is a catastrophically bad radiator, and the reason is that it has no monopole and no dipole available to it. What is left is the eighth power of speed — and the equation is equally happy with sound converging on a jet, which is ruled out by a condition imposed at infinity.

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The sound is what fails to cancel. A compact quadrupole of kd = 0.01, followed outwards. The upper curve is what one of its four sources produces on its own at each radius — every one of them is as loud as a monopole — and the lower curve is what all four produce together. Inside the source they barely cancel at all; by one radian of wavelength the sum is 9.3e-5 of what a single source is doing, and it is falling as 1/R from there on because that is what radiation does. Between the two the field falls as the cube of the distance, which is a near field rather than a sound.

The sound is what does not cancel

A quadrupole is not a weak source. Every one of the four monopoles in it is as loud as a monopole of the same strength, and what makes the assembly quiet is that they very nearly cancel — one part in ten thousand survives at a hundredth of a wavelength. The eighth-power law is a statement about how nearly, not about how little.

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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.

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The two ways a shock can meet a wall. Left: the incident shock from the wedge reaches the wall and a second shock turns the flow back parallel to it, meeting at a point. Right: at a larger wedge angle no reflected shock can turn the flow that far, and the intersection lifts off the wall into a triple point with a nearly normal Mach stem standing on the surface and a slip line trailing downstream. The two configurations are drawn at the angles the solver returns, with every shock angle computed rather than sketched.

When a shock cannot bounce

A shock reflects off a wall until the reflected shock runs out of turning, which happens at a wedge angle well below the free stream's own limit. Between the two boundaries both configurations exist, both are stable, and which one appears depends on which direction the experiment came from.

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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.

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The density ratio and the nose pressure coefficient, against Mach number. Two quantities across a normal shock, on a logarithmic Mach axis. Both approach limits that depend on γ and on nothing else: six for the density ratio and 1.8394 for the pressure coefficient at the stagnation point. By Mach five the second is within three per cent of its limit and by Mach twenty within a tenth of a per cent. Above that the flow round a blunt body has stopped depending on how fast it is going and started depending on what the gas is.

A shock that lies on the body

Above about Mach eight the flow round a blunt body stops depending on how fast it is going. The density ratio, the nose pressure coefficient and the shock standoff all reach limits set by γ alone — and going from a perfect gas to a dissociating one halves the standoff.

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The coefficient of the equation's second derivative, along a chord. The bracket multiplying the streamwise second derivative in the transonic small-disturbance equation, along a chord at Mach 0.85. Where it is positive the equation is elliptic and the flow is subsonic; where it is negative the equation is hyperbolic and the flow is supersonic. Which it is at a given point depends on the perturbation velocity there, which is the thing being solved for. Forty-two per cent of this chord is hyperbolic, and no amount of inspecting the problem beforehand could have said so.

The equation that changes type inside its own answer

Near Mach one the coefficient of the streamwise second derivative depends on the perturbation velocity, which is what is being solved for. Two solutions of the linear equation no longer add — the leftover is three times the term the linear theory keeps — and the critical Mach number approaches one as the two-thirds power of thickness.

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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.

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One signature, aged four times. The pressure signature of a slender body at four distances, computed by the exact Lax formula for the nonlinear propagation. Each point of the waveform moves forward in proportion to its own overpressure, so the compression at the front catches the undisturbed air and a shock forms there, while the expansion at the rear falls behind and forms a second one. What is left is an N-wave: two discontinuities and a straight line between them, spreading and weakening.

The signature that forgets the shape

The pressure field near a supersonic aeroplane depends on every part of it. What reaches the ground has two parameters. Two bodies whose near-field signatures differ by fifty-five per cent in peak and by their whole shape age into the same N-wave, to two and a half per cent.

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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.

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Four bodies of identical length and volume, and their wave drags. Each body has the same length and the same volume; only the distribution of area along it differs. The Sears–Haack body — the spindle whose area goes as the three-halves power of x(L−x) — has the least wave drag of the four, and every other shape pays between thirty-seven per cent and a hundred and seventy-five per cent more for carrying the same volume the same distance. Nothing about the cross-sections' shape enters: only the area distribution does.

The least drag a volume can have

A body's supersonic wave drag depends on nothing about it except how its cross-sectional area is distributed along its length. Minimising that for a given volume gives one shape — and the answer goes as the volume squared over the fourth power of the length.

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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.

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