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

Riemann problem — where it appears

The initial-value problem in which two uniform states are placed side by side and released. Its solution is a fan of waves — a shock, an expansion and a contact — and it is both the shock tube's answer and the building block of most compressible flow solvers.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

Three waves out of one discontinuity. The x–t diagram of the burst diaphragm. A shock runs right at a speed of its own, a contact surface follows it more slowly, and an expansion fan spreads left as a family of rays that opens with time — the three wave families the Euler equations possess, produced at once by an initial condition with no waves in it at all. Every straight line here is a speed the solution computed, and the fan is drawn as the rays it actually consists of.

One diaphragm, every wave

Two states of the same gas at rest, separated by nothing, is the simplest initial condition compressible flow admits — and its answer contains all three waves the equations possess at once: a shock one way, an expansion fan the other, and between them a surface across which the density jumps and the pressure does not.

compressible · Shock tube The four waves, and the one that never goes away. The shock tube in space and time: a shock running right, an expansion fan running left, and the contact surface between them. The shock and the fan are travelling disturbances that leave; the contact is made of fluid, so it is carried along and is there for ever.

A surface that remembers the diaphragm

Between the shock and the expansion in a shock tube there is a surface across which the pressure and the velocity are identical and the temperature differs by a factor of two. It is made of fluid, so it never goes away, and nothing in the pressure field says it is there.

compressible · Shock tube The column in height and time: a falling interface, a rising shock, a fan. A batch settling test from a uniform φ₀ = 0.1, height above the bottom against time, both scaled on the column height and the single-particle settling time. The interface with clear water (thick) falls in a straight line at 0.4538; the sediment shock rises from the bottom at 0.1484 until the two meet at t = 1.661, height 0.2464; the thin lines are characteristics of the fan, each carrying one concentration between 0.317 and packing, and the interface bends as it crosses them. Dots are the finite-volume solve on 400 cells: the interface and the sediment front.

The column the chord rule cannot settle

A suspension settling in a closed column is a kinematic wave, and its flux curve bends both ways. At the top the chord rule works: clear water meets the suspension at a single falling front. At the bottom it does not, and the bed grows behind a shock that stops short of packing and a graded layer beneath it — so the interface, instead of arriving, slows for ever.

kinematics · Kinematic waves

Named alongside it

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

CharacteristicsContact surfaceEntropyExpansion fanShock tubeConservationConserved quantityConstitutive lawDiscontinuityKinematic-waveMass conservationMaterial line

All concepts