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

Womersley number — where it appears

A pipe's radius measured in the depth viscous information reaches in one cycle of an oscillating flow. Below about a third the profile is the quasi-steady parabola; above about ten the core moves as a plug a quarter of a cycle behind the pressure.

Named by 10 essays across one field — each of them below, with the objects they name alongside it.

α = 13.0: a plug in the middle and everything happening at the wall. The velocity profile at eight phases of one cycle, at a Womersley number of 13.0 — the human aorta's at rest. The core moves almost as a solid plug, because viscosity cannot reach it within a cycle; all the shear is in a layer of thickness √(ν/ω) = 0.77 mm against a radius of 10.0 mm. At some phases the fluid near the wall is moving backwards while the core still moves forward, which is the reversal a Poiseuille profile can never show and which is routinely measured in arteries.

Too fast for a profile

A pipe carrying a steady flow has a parabolic profile. Make the pressure oscillate and one number decides whether it still does — and above about ten the core moves as a plug, a quarter of a cycle behind the pressure, with the fastest fluid in a ring near the wall rather than on the axis.

regimes · Womersley Where a profile stops being a parabola. Two measures of how far Womersley's solution has left the quasi-steady parabola, against the Womersley number, both logarithmic. The amplitude deficit reaches a hundredth at α = 0.28 and the phase lag reaches a hundredth of a right angle at α = 0.22 — both well below α = 1, which is where the unsteady and viscous terms are equal and is the value the number is named for. By α = 1 itself the flow is already a tenth short and eight degrees late.

Where the parabola goes

A pipe carrying a steady flow has a parabolic profile, and the Womersley number is supposed to say when an oscillating one still does. For the flow rate it is very nearly honest — one per cent at α = 0.91 — and for the phase it is out by a factor of three, because a lag is second order in the number and an amplitude deficit is fourth.

regimes · Womersley 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 What an oscillation carries, against how fast it is asked. The extra axial diffusivity of a zero-mean oscillation, at fixed velocity amplitude, against the Womersley number. At low frequency it is exactly Taylor's dispersion evaluated at the mean square velocity — the profile has time to be Poiseuille's at every instant. At high frequency the shear is confined to a Stokes layer, the tracer in the core is never sheared, and the transport falls as the cube of the Womersley number.

Transport with nothing transported

Taylor's mechanism needs shear and diffusion and nothing else — and in particular it does not need a mean flow. Oscillate a tube about a fixed position and the tracer still spreads along it, by hundreds of times the molecular rate, which is how a patient is ventilated with a tidal volume smaller than the airway it goes down.

regimes · Dispersion The exponent a pulsing tree wants is between two and three. The tree's reflection at its root, averaged over the pulse's ten harmonics with the pulse's own weights, against the branching exponent k in r_parent^k = 2 r_daughter^k — two is area-preserving, three is Murray's law. With a wave speed that does not change with radius the best exponent is 2.15, just above the inviscid match of two; with one that rises as smaller arteries stiffen, 2.58, just above 2.5. Viscosity in the smallest branches pushes the best exponent a little towards Murray's.

Murray's law is not the rule for a pulse

Murray's law sizes a branching vessel for the cheapest steady flow, and at every junction built to it a pressure pulse is partly reflected. The rule that makes a junction transparent to a pulse is a different exponent, set by how the wave speed changes with radius. A tree is not the sum of its junctions, either: a Murray tree six generations deep reflects a third of the pulse at the heart rate, three times what one of its junctions does, and viscosity in the smallest branches means no area rule can make it transparent at every frequency. The best a pulsing tree can do lies between area-preserving and Murray.

regimes · Womersley Area-preserving in the aorta, Murray's cube in the small arteries. The branching exponent k, in r_parent^k = r₁^k + r₂^k, that makes one junction reflect least of the heart's pulse, against the parent artery's radius, for a symmetric split and for side branches a half and 0.15 of the continuing trunk. In the aorta, where the pulse is carried by inertia, the transparent rule is close to 2 — area preserved — whatever the asymmetry. In arteries under a millimetre, where viscosity carries it, it is 3: Murray's law, the rule for the cheapest steady flow, is exactly the rule that passes the pulse. Asymmetry moves the answer only in between.

Murray's law passes the pulse where the pulse is viscous

The rule that makes an arterial junction transparent to the heart's pulse was an exponent, and real arteries do not split evenly: the aorta sheds side branches a fraction of its size and carries on. For a lopsided junction the transparent rule is still an exponent — the same one, exactly, while viscosity is negligible. With viscosity it is a single function of the parent's Womersley number: close to area-preserving in the aorta, and exactly Murray's cube in arteries under a millimetre, where the pulse moves as the steady flow does. The asymmetry matters only in between, and a lopsided tree reflects far less than an even one.

regimes · Womersley Local transparency is the best tree only when the ends absorb. The root's reflection at the heart rate against the reflection at the tree's leaves, for the map tree, for the single exponent that is best at that leaf reflection, and for Murray's. With leaves that reflect nothing the map tree is the best of all, 0.035 against 0.0419: making every junction transparent is then the whole job. As the leaves start to reflect, the best single exponent pulls ahead, because it leaves its junctions slightly mismatched in the way that cancels the echo coming back from the ends; at a leaf reflection of 0.5 it reflects 0.0735 to the map tree's 0.13.

A tree transparent at every junction is not the quietest

The branching rule that lets a heart's pulse through one junction without reflection is set by that junction's Womersley number: area-preserving in the aorta, Murray's in the small arteries. Build a whole arterial tree that way, every junction at its own transparent rule, and it reflects more of the pulse than the best tree built to a single exponent. The single exponent wins by leaving its junctions slightly mismatched, in the way that cancels the echo from the tree's own ends — the trick an antireflection coating plays on light.

regimes · Womersley Between fixed ends the transparent-junction tree is the quiet one. The magnitude of the pulse reflected at the aortic root at the heart rate, 1.2 Hz, and its first four harmonics, for trees grown from the same aorta to the same 0.5 mm vessels by four branching rules. The Womersley-map tree reflects 0.028 at the heart rate and 0.0046 or less at the harmonics; the best single exponent 0.035; Murray's 0.083; and the area-preserving tree 0.21 at the heart rate though little at the harmonics.

Between fixed ends the transparent tree is the quiet one

Grown from one aorta for six generations, an arterial tree whose every junction passes the pulse reflected more than one built to a single compromise exponent. Grown instead from the same aorta down to the same small vessels, however many generations each rule needs, the order turns over: the junction-by-junction tree is the quietest, and Murray's tree, which looked cheapest for the heart, becomes the dearest. The earlier ranking was a comparison between trees that fed different beds.

regimes · Womersley Shaken at the right frequency, a liquid with one relaxation time moves sixteen hundred times the flow its viscosity allows. The amplitude of the oscillating flow rate over the steady Poiseuille flow the same pressure-gradient amplitude drives through a Newtonian liquid of the same zero-shear viscosity, against the frequency, in a tube 1 cm in radius. A Maxwell liquid with λ = 1 s and η₀ = 30 Pa·s peaks at 1660 times at 6.63 Hz and again at 317 times near 15.2 Hz; a Rouse spectrum with the same longest time and viscosity reaches 20.2 at 19.8 Hz; the Newtonian liquid only falls, to 0.99 at 30 Hz.

A liquid with one relaxation time rings in a tube, and a spectrum of times silences it

Push a viscoelastic liquid back and forth along a rigid tube and, if its relaxation time is long against the cycle, the tube becomes a cavity for shear waves in the liquid itself. At the cavity's resonance a wormlike-micelle solution in a tube one centimetre across moves sixteen hundred times the flow a Newtonian liquid of the same viscosity would. The Deborah number says when the liquid is elastic enough to ring; it cannot say whether it will, because a polymer with a spectrum of relaxation times and the same longest one damps the same resonance to a factor of twenty.

regimes · Deborah

Named alongside it

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

Model limitImpedanceBlood flowPulsatile flowReflectionWave speedDimensionlessMurray's lawOptimisationPhase lagScalingUnsteady

All concepts