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

Objectivity — where it appears

The property of a quantity that does not change when the observer moves or rotates. The rate of strain is objective and the vorticity is not, so a criterion built on the vorticity gives a different answer to an observer who is turning.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

One flow, two observers, two pictures. The same ideal flow past a circular cylinder, drawn in the frame of the tunnel and in the frame of the undisturbed air. The two are related by subtracting one constant velocity. On the left the flow arrives from infinity, divides at a stagnation point on the nose and closes at another on the tail. On the right the air is at rest far away, the body pushes through it, the streamlines are closed loops, and there is no stagnation point anywhere in the field. Every force, every pressure and every measurement either observer can make is identical.

The picture belongs to whoever is watching

Photograph the flow past a cylinder from the tunnel and it has two stagnation points. Photograph the same flow from a frame moving with the air and it has none at all, and its surface speed is exactly the free stream at every angle. Both pictures are correct and no measurement distinguishes them.

kinematics · Frames Two flows with the same rate of strain, doing different things to a blob. A circle of fluid carried by two flows chosen to have exactly the same rate-of-strain magnitude, drawn at four times. Pure strain pulls it into an ellipse whose axes stay put; simple shear pulls it into an ellipse whose axes rotate as fast as they stretch. Both have zero divergence, so both preserve the area. The difference between them is not the strength of the straining but what the rotation does to the direction being stretched.

Longer, with nothing pulling it

Two flows with exactly the same rate of strain. In one a line of fluid grows by a factor of 148 in five time units; in the other it grows by 10. Turn the straining axes faster than the strain rate and no line grows at all, however hard the fluid is being strained.

kinematics · Material lines Q and the vorticity along a radius of one vortex. Two candidate measures of where the vortex is, along a radius of a Lamb–Oseen vortex. The vorticity is a Gaussian: positive at every radius, so a threshold on it puts the edge wherever the threshold is put. Q — the excess of rotation over strain — changes sign exactly once, at 1.121 core radii, and that radius is a property of the flow rather than of the person drawing it. Inside it, 71.5 per cent of the circulation.

Where a vortex stops

Four criteria decide where a vortex ends, and in two dimensions three of them are the same criterion. The fourth is a knob. And the one that is not a knob is not objective: a co-rotating pair of vortices occupies four per cent of a window to one observer and more than half of it to another.

kinematics · Coherent structures Every particle leaves a flow that is a vortex at every instant. Eight fluid particles starting on a small circle in Haller's rotating-saddle flow, followed for 2.4 time units. At every instant the velocity gradient has complex eigenvalues and a positive Q — the flow is a vortex by every criterion that reads a snapshot — and every particle spirals outward, its distance growing as e to the time. The fluid is not held; it is flung.

Every snapshot says vortex, and every particle leaves

There is a flow in which the velocity gradient has complex eigenvalues at every point and every instant — a vortex by every criterion that reads a snapshot, in the frame the flow is measured in — and in which every fluid particle is flung away exponentially. The snapshot is not wrong about the gradient. It is wrong about the fluid, and it is wrong exactly when the strain's axes turn.

kinematics · Frames In the pair's own frame, two eddies with no vorticity. The flow round two equal co-rotating vortices (orange dots), seen from the frame that turns with them, where it is steady; shading is the stream function. Round each vortex a lobe of fluid circulates; a band round both is bounded by a figure-eight through the saddle at the centre; and above and below, centred on the two points that make equilateral triangles with the vortices (blue dots), are two large eddies of fluid that circulate in this frame and carry no vorticity at all, bounded by the streamline through the two outer saddles at √5 half-separations.

A vortex pair carries eddies no snapshot can see

Two equal vortices circling each other are, by every snapshot of the velocity gradient, two small vortices in a straining flow: outside their cores the flow is irrotational, and the gradient there is pure strain. Seen from the frame that turns with them, the same flow contains two large eddies, one on each side, centred where a third point would complete an equilateral triangle with the pair. Their fluid goes round with the pair for ever, and it carries no vorticity at all. They hold seventy times the area of the cores, and the only way to see them is to follow the fluid or to turn with it.

kinematics · Frames The two observers disagree twelvefold about where Q says the vortices are, and not at all about the LAVD edge. The Q = 0 contours and the outermost convex LAVD level over one orbit, in the laboratory and in the frame turning with the pair. Q > 0 covers 2.00 of the window in the laboratory and 24.94 in the turning frame; the LAVD edges enclose 6.29 in the laboratory and 6.29 in the turning frame, and the two LAVD fields differ nowhere by more than 8.0e-8 of their peak.

An edge no observer can move still depends on what it is averaged against

The Q-criterion puts a co-rotating vortex pair in four per cent of a window for one observer and in more than half of it for another. The cure is to average, along each fluid path, how far the vorticity departs from its mean — a number no observer's rotation can change — and take the outermost convex level of it round each core. Computed, it works: both observers find two discs of radius 1.00, twice the core radius, and no fluid inside them strays from its core in an orbit. But the edge has a setting of its own. The mean must be taken over some region, and as that region grows from a radius of 4 to 20 the edge moves from 0.90 to 1.13.

kinematics · Coherent structures

Named alongside it

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

Model limitQ-criterionRotating frameVorticityStrain rateVelocity gradientCoherent structuresEigenvalueMeasurementPathlineVortexVortex core

All concepts