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

Oscillation — where it appears

A motion that repeats, whether imposed on a flow or generated by it. Its frequency compared with whatever time the flow itself carries decides whether the flow follows it, ignores it, or locks to it.

Named by 11 essays across 3 fields — each of them below, with the objects they name alongside it.

Stokes' drag stops being the answer almost at once. The damping on a millimetre sphere oscillating in air, divided by Stokes' steady drag on the same sphere, against frequency. The extra term is the sphere's radius over the layer thickness, so it takes over as soon as the layer is thinner than the body — which for a millimetre sphere in air is below one hertz. At a kilohertz the damping is fifteen times the steady value, and the exponent is a half rather than zero.

What a fluid takes out of a swing

The damping a body feels from the air around it is not Stokes' drag, and stops being it far earlier than anybody expects — a millimetre sphere in air is already forty-six per cent above the steady answer at one hertz. Past that the damping rises as the square root of the frequency, and the fluid it is fighting is a shell a fraction of its own size.

viscous · Damping A 204 m hammer traded for a 8.57 m swing over 299 seconds. The water level in a 10 m surge tank at the end of a 2 km tunnel 3 m across, carrying 2 m/s, after the turbine is shut off at once, with the level measured from the reservoir's. Without friction it rises to V₀√(L Aₜ/g Aₛ) = 8.57 m and swings with a period 2π√(L Aₛ/g Aₜ) = 299.1 s, the integration agreeing with both closed forms. With the tunnel's 5 m of friction the level starts 5 m below the reservoir, peaks at 5.61 m after 98 s, falls to −3.70 m, and decays. The same tunnel shut at its end with no tank would take the Joukowsky rise of 204 m. The tank does not remove the column's momentum; it gives it a free surface to push against, slowly.

A tank that turns a hammer into a swing

Shut a turbine at the end of a two-kilometre tunnel in two seconds and the valve takes a rise of 256 metres of head. Put a shaft open to the air beside it and the rise is 51, the tunnel never carries the closure as a wave at all, and its water slows instead against a level that climbs for a minute and a half — to a height that is a closed form with the tank's area under a square root.

applied · Water hammer Two forces on one cylinder, a quarter of a cycle apart. The inertia and drag terms of Morison's equation over one wave period, at a Keulegan-Carpenter number of ten. The inertia term follows the acceleration and peaks where the velocity is zero; the drag term follows the velocity and peaks where the acceleration is. They are a quarter of a cycle apart and they are different kinds of quantity.

Two forces, and only one of them remembers

Morison's equation adds an inertia term to a drag term and is usually presented as an empirical patch. It is not: the two terms are two different kinds of memory, one a function of the present acceleration and one a function of the wake left by the previous half cycle.

regimes · Keulegan–Carpenter number Below Thoma's 6.07 m² the governed tank's swing grows; above it, it dies. The tank level after the turbine's power demand drops by two per cent, with a governor holding the power constant, for tanks of 0.7 and 1.3 times Thoma's area of 6.07 m² — a tank 2.78 m across. The smaller tank's swing grows by a factor of 1.47 every 74 s cycle and has reached −12.20 m by 427 s; the larger one's keeps 0.75 of itself every 100 s and is barely visible. Carried on, the smaller tank's run is refused at 794 s, where the head at the turbine has fallen below a quarter of its design value and the governor would be asking for a flow no turbine passes. The instability has nothing to do with the tank's height: it is the governor drawing more water as the level falls, which feeds the swing, against the tunnel's friction, which is the only thing damping it.

The better tunnel needs the bigger tank

A turbine governed to hold its power opens further when the head at it falls, and draws the tank down harder. That makes it a negative resistance, the tunnel's friction is the only thing damping the swing against it, and so the smallest stable tank grows as the friction shrinks — 2.78 metres across for five metres of friction, 6.09 for one.

applied · Water hammer A wave that travels and a wave that spreads. A harmonic pressure wave's amplitude and its instantaneous value along a tube, over two wavelengths of the inviscid wave, at four Womersley numbers. At α = 15 the wave marches on, a little weaker each wavelength. At α = 5 it is visibly damped. At α = 2 it is nearly gone within a wavelength. At α = 0.5 there is no wave to speak of: the disturbance falls away within a small fraction of the inviscid wavelength, as heat does into a wall.

The pulse that has to travel

In a rigid tube the Womersley number decides the shape of an oscillating flow. Make the wall elastic and the pressure pulse has to travel, a second number appears — the tube's length in wavelengths — and the first number turns out to decide something more basic than the profile: whether the tube carries a wave at all, or only a disturbance that spreads like heat.

regimes · Womersley One number decides which pulse grows. The pressure pulse and the flow pulse at the far end of the tube, each as a multiple of its value at the entrance, against the load's reflection coefficient. A load that reflects pressure with the same sign — a stiffer or narrower continuation — amplifies the pressure pulse and damps the flow pulse. One that reflects it inverted — a wider continuation, or many branches — does the opposite. With no reflection both fall slightly, by the wave's own attenuation. The two curves cross near Γ = 0 and pull apart on either side.

The pulse that grows as it leaves the heart

The pressure pulse measured at the wrist is larger than the pulse in the aorta that drives it, and the flow pulse is smaller. Nothing downstream is pumping. A wave reflected from the end of an elastic tube arrives back in step with the outgoing wave near the end and out of step near the start, and a single number — the reflection coefficient — decides whether it is the pressure or the flow that grows.

regimes · Womersley The first three ways a drop can ring. A drop's first three modes of oscillation, each drawn at the two ends of its swing: the shaded outline and the red one are half a period apart. The fundamental, l = 2, alternates between a lemon and a lentil. The next two ring at 1.94 and 3.00 times its pitch, with three and four lobes. The swings are drawn at a fifth of the radius so they can be seen; the frequencies belong to swings much smaller than that.

A drop rings like a bell

Disturb a drop and it swings between a lemon and a lentil at a pitch set by its surface tension and its size, and it keeps swinging for dozens of cycles because water is nearly frictionless at that scale. The usual story has a falling raindrop's wake ringing it. The wake strikes several times too fast for the note the drop plays.

regimes · Drop shape Surge is a loop round the characteristic's peak. The two runs in the plane of flow coefficient and plenum pressure rise, over the characteristic continued to reversed flow. With B = 0.5 the state slides off the peak and settles on the stalled characteristic. With B = 2 it traces a large loop: the flow collapses at nearly constant pressure, reverses while the plenum empties, recovers to the right-hand branch at low pressure, and climbs back up it while the plenum fills, round and round.

The plenum decides whether a compressor surges

Throttle a compressor past the peak of its characteristic and it does one of two things. It settles into rotating stall, a steady state with a cell of dead flow running round the annulus, or it surges, the whole flow through the machine collapsing, reversing and recovering over and over. Which one is not decided by the blades. It is decided by the volume the compressor discharges into, against the inertia of the air in its duct — one number, Greitzer's B, which grows with the blade speed, so the same machine stalls at part speed and surges at full speed.

applied · Turbomachine Every rotor lands on the same hot state, later and more smoothly. The centre temperature against time, on a logarithmic axis, for a soft drive five per cent past its fold switched on from rest, with rotors of four inertias. Each climbs, lingers and jumps, and each ends at the hot operating point without overshooting it. The time to pass three e-folds is 7, 16, 101, 948 diffusion times for M = 0, 0.1, 1, 10.

A rotor's inertia slows the jump and cannot make it ring

Give the motor driving a self-heating oil film a rotor that has to spin up, and there are two clocks: the film's diffusion time and the rotor's. Two clocks are what an oscillator is usually made of, and this pair cannot make one. Every eigenvalue stays real at every inertia, because the film and the rotor only ever push each other the same way. What inertia does instead is add its own delay to the film's, and in a real machine, where the rotor is hundreds of times slower, the delay past the fold is almost entirely the rotor's.

viscous · Viscous heating An S-shaped set of states and a slow variable that crosses it. The film's steady mean temperature against the housing's, on the cool and hot branches of operating points, with the line on which the housing is in equilibrium with the film, at a derating of 0.22 per unit. Between housing temperatures 1.24 and 1.59 both branches exist. At the derating the essay uses, 0.22, neither meets the equilibrium line there: on the hot branch the housing is always warming, on the cool branch always cooling, so the film runs round the loop drawn over them, jumping at each fold.

A slow sensor makes the bearing cycle

Derate the motor driving a self-heating oil film on the temperature of its housing, which warms over minutes, and a protection meant to hold the bearing cool instead makes it cycle for ever: jump hot, warm the housing until the hot state is lost, drop cool, let the housing cool until the cool state is lost, jump again. The period belongs to the housing, the amplitude belongs to the film, and the derating strength decides only whether the cycle happens at all.

viscous · Viscous heating With a small plenum, stall feedback closes the hysteresis loop. The stall amplitude against the throttle as it is closed slowly past the peak and opened again, at B = 0.2, with no stall feedback and with feedback at twice the critical gain; closing solid, opening dashed. Without feedback the cell appears suddenly near γ₀ = 0.6094 and vanishes only at 0.6428, and the loop between the two legs has an area of 0.078. With feedback the cell grows gradually as the throttle closes and shrinks along the same curve as it opens: the area is 0.0014.

A throttle can hold the peak only in a small plenum

A compressor makes its most pressure at the peak of its characteristic, and at the peak it either surges or stalls. A throttle moved by feedback can fight both. Driven by the plenum's pressure it damps surge, with a gain that grows as the square of Greitzer's B until it meets a ceiling B does not move; driven by the flow, it cannot. Driven by the stall cell's amplitude it turns the sudden jump into deep stall into a gradual one and closes the hysteresis loop — but only while the plenum is small. Above B of about a third the same feedback turns the jump into a cycle.

applied · Turbomachine

Named alongside it

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

Model limitBifurcationStabilityDampingDimensionless numberHysteresisUnsteady flowWave speedAdded massDynamical systemFriction factorImpedance

All concepts