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

Every essay — page 18

Page 18 of 61, 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.

A cone sends 1.7 per cent of its jet's momentum sideways at 15°. A conical divergent section of 15° half-angle and exit area ratio 25, drawn to scale from its throat to its lip, with its virtual apex to the left. The gas leaves as a source flow: straight streamlines from the apex, each at its own angle, and a Mach number uniform on spheres centred there. On the spherical cap through the lip the flow is normal to the surface and uniform, at Mach 3.925 for a gas with γ = 1.2; its axial momentum is ρV² times the cap's projection onto the exit disc, while the mass crossing it is ρV times the cap itself. The ratio of the two areas is (1 + cos α)/2 = 0.9830, and it is the only thing the cone's shape does to the momentum. On the flat exit plane the flow is not uniform: its edge is further from the apex than its centre, and the Mach number there is higher.

The jet a cone sprays sideways

The one-dimensional nozzle sends all its gas straight out along the axis. A real divergent section is a cone, and the gas leaves it as a spray of straight lines from the cone's apex. Only the axial part of that momentum pushes, and the share that does is (1 + cos α)/2 — 98.3 per cent at fifteen degrees, 93.3 at thirty — whatever the gas, the Mach number or the area ratio, and it touches the momentum and never the pressure.

6 figures 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.

6 figures 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°.

7 figures Every output of the air-data reduction, and how hard it leans on each reading. The logarithmic sensitivity of each derived quantity to each of the three readings — the total pressure, the static pressure and the indicated total temperature — at Mach 0.3 and Mach 0.85, at 11 km with a probe recovering 0.98: a one per cent error in a reading times the number is the per cent error it puts into the output. The Mach number leans on the two pressures by 8.08 and −8.08 at Mach 0.3 and by 1.13 and −1.13 at Mach 0.85, and not at all on the temperature. The static temperature leans on its probe by exactly one and on the pressures by −0.280 at Mach 0.3 and −0.281 at Mach 0.85 — almost the same. The true airspeed leans on the probe by exactly one half and the density by exactly minus one, at every speed.

Three readings, and the one each answer leans on

An aircraft works out the air it flies through from three readings — a static pressure, a total pressure and a probe's temperature — and every derived number inherits their errors through one small table of sensitivities. Written out, the table says the airspeed is afraid of the pressure sensors almost to Mach one at cruise altitude and only to Mach 0.52 at sea level, and that the density belongs to the thermometer at every speed.

7 figures 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.

7 figures Below Mach 1.153 the boom turns back before it reaches the ground. The ray leaving the Mach cone straight down from an aeroplane at 11 km, at Mach 1.1, 1.15, 1.2, 1.5, 2, traced through a standard atmosphere whose sound speed rises from 295.1 m/s at the aeroplane to 340.3 m/s at the ground. A ray bends back upward where the local sound speed equals the aeroplane's speed, so Mach 1.1: turns at 4.00 km; Mach 1.15: turns at 0.25 km; Mach 1.2: lands 24.0 km on; Mach 1.5: lands 11.4 km on; Mach 2: lands 7.1 km on. The dividing speed, Mach 1.1533, is the ratio of the two sound speeds.

The boom that turns back before the ground

Sound is faster in the warm air near the ground, so a sonic boom's rays bend back upward on the way down. Whether any of them arrive is one comparison — the aeroplane's speed against the fastest sound beneath it — and the ray that just grazes the ground sets the edge of the carpet, which the uniform air of the ageing calculation cannot give it.

7 figures The arrival map, and the place where it goes backwards. Where each boom ray lands on the ground, against the Mach number the aeroplane was doing when it launched it, for four accelerations from 15 km. In level flight this would be a straight line rising at the aeroplane's own speed. Here it falls before it rises: a ray launched at a higher Mach number is shorter and steeper, and near the cut-off it shortens faster than the aeroplane advances. Every minimum in these curves is a fold — two emission times delivering to one place — and on the fold itself neighbouring rays converge onto a single line.

The carpet an accelerating aeroplane folds

Level flight launches every boom ray with the same invariant, so they run parallel and each place hears one boom. Accelerate, and each successive ray is shorter than the last — shorter, near the cut-off, than the aeroplane's own advance — so later rays overtake earlier ones and the arrival map folds onto a line.

8 figures What the rays say the edge is: nearly as loud, and then nothing. The overpressure across the carpet relative to the value under the track, from ray-tube spreading alone, at three Mach numbers from 15 km. It falls gently and then stops: at Mach 1.8 it is 0.92 ten kilometres out, 0.77 at twenty-three and 0.65 on the last ray, 36 km out, with silence beyond. Geometrical acoustics says the carpet ends at a cliff, and the boom at the edge of a real carpet fades as a rumble. The cliff is what the model says; it is also where the model stops.

The edge is a rumble, not a quieter bang

The rays that reach the outer half of a sonic-boom carpet arrive nearly horizontally, having travelled almost three times as far as the one under the track. Ray theory says they still carry two-thirds of the overpressure, right up to a line beyond which there is nothing. Neither half of that is what is heard — which is the useful result, because it says the edge's loudness is not a ray quantity at all.

7 figures A boom gathers most of its age in the thin air near the aeroplane. The share of the total age gathered above each height, for the ray under the track and the last ray computed near the carpet's edge, with the share of the path length travelled above each height for comparison. Under the track 66 per cent of the age is gathered above the tropopause in 26 per cent of the path. The edge ray spends most of its path in the lowest few kilometres and gathers only 11 per cent of its age below 3 km, because the same pressure distorts dense air far more slowly than thin air.

A boom is aged in the thin air it starts in

The rays that reach the edge of a sonic-boom carpet travel two and a half times as far as the one under the track, and it is natural to expect their signatures to have aged accordingly. They have not. A pressure wave distorts thin air far faster than dense air, so two-thirds of a boom's ageing is done in the stratosphere near the aeroplane, and the extra kilometres near the ground add little — which decides how far out a boom shaped to be quiet stays quiet.

7 figures Three different flows, drawn on top of one another. The surface pressure of three parabolic arcs of different thickness, each at the Mach number that gives them the same transonic similarity parameter, divided by the scale the similarity rule prescribes. They lie on one curve. The three solutions were computed separately from an equation with different coefficients in each case, and nothing in the solver knows what the parameter is — the scaled peaks differ by 0.01 per cent.

One curve for every thickness

Near Mach one there is no linear theory, so there is no superposition, no scaling of a solution by the thing that caused it, and no shortcut of any kind. There is exactly one thing left — a change of variables in which the thickness and the Mach number vanish and only their combination survives — and it is worth more than the theory that was lost.

7 figures A drag that is nothing until there is something to be discontinuous about. The scaled wave drag of a parabolic arc against the similarity parameter. Above about 1.4 it is not small but zero to five decimal places — d'Alembert's paradox, which the small-disturbance equation inherits. Below it the drag rises by four orders of magnitude over a range of the parameter that corresponds to a few hundredths in Mach number, and the only thing that changed is that the flow now contains a shock.

A drag that needs a discontinuity

A closed body in a steady inviscid flow has no drag, and the transonic equation obeys that as faithfully as any other — until a shock appears inside its own answer. Then the same pressure integral that was zero to five decimal places becomes four decades larger, with no viscosity added and nothing about the body changed.

8 figures A frequency set almost entirely by how nearly sonic the flow behind the shock is. The Strouhal number of the loop against the local Mach number behind the shock. The band drawn across it is what is measured on aerofoils in buffet, 0.06 to 0.08, and the arithmetic meets it at a post-shock Mach number of 0.971. Two per cent either side of that number changes the frequency by a third, which is why buffet onset is predicted by correlation rather than by calculation.

A shock that will not stand still

A converged transonic solution puts the shock in a definite place. Over a band of conditions the real one sweeps back and forth across a fifth of the chord at a fixed frequency, and the period is set almost entirely by how nearly sonic the flow just behind it is — a quantity that is small, hard to compute, and multiplies everything.

8 figures