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

Beltrami flow — where it appears

A flow whose vorticity is everywhere parallel to its velocity, so that the nonlinear term is a gradient and can be absorbed into the pressure. Such flows are steady solutions of Euler's equations and carry the largest helicity their energy and enstrophy permit.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

One streamline, sectioned, in two steady flows. Every time a single streamline crosses the plane z ≡ 0 going upwards, a point is plotted. On the left the flow is integrable and the points lie on a curve, however long the trajectory is run. On the right one coefficient of the same exact solution has been changed and the same single streamline scatters over a sixth of the plane. Both flows are steady, both are incompressible to machine precision, and both are exact solutions of the Euler equations.

Steady, three-dimensional, and mixing anyway

A steady flow that solves the Euler equations exactly, with its vorticity equal to its velocity to six parts in ten thousand million — and one of its streamlines wanders through a sixth of the box while another, started nearby, lies on a curve for ever.

kinematics · Advection A double integral that comes out an integer. The Gauss linking integral evaluated on six pairs of closed curves. It is not constrained to be a whole number by anything in its own definition — it is a double integral of a smooth kernel — and it returns one to within two parts in ten thousand on two hundred points per curve, because what it is computing is a topological count.

The knot a flow cannot untie

Ideal flow conserves energy, circulation and impulse, and all three are what they look like. It conserves a fourth quantity that is not: a volume integral of the velocity dotted into the vorticity, whose value counts how many times the vortex lines are linked through one another.

kinematics · Helicity

Named alongside it

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

Model limitVorticityThe Biot–Savart lawCirculationConserved quantityDynamical systemExact solutionHelicityIrrotationalLinking numberMixingPathline

All concepts