Rotating Detonation EnginesA First-Principles Course

How this course works

Each lesson opens with one question and builds the intuition to answer it before formalising anything. Along the way you will find:

• Live explorables. Around thirty interactive simulations — racing a flame against a detonation, dragging a Rayleigh line until it kisses the Hugoniot, carving detonation cells into a soot foil, firing a pulse detonation tube, and, at the heart of the course, a real reactive-flow solver that computes a rotating detonation wave live in your browser. Predict first, then drag.
• Pause and ponder. Questions worth a minute of thought before you reveal the answer.
• Check your intuition. Short quizzes whose explanations are often the best part.
• Watch the idea. Many simulations have a narrated tour that moves the sliders for you. Press it, then take the controls.
• Challenges. Small goals under the simulations ("can you keep two waves alive at once?") that tick themselves off when you succeed.
• Estimate first. Before a big number is revealed, you guess it on a logarithmic scale.
• Going deeper. The equations engineers actually use, colour-coded to match the pictures: $\cfresh{\text{fresh mixture}}$ is always mint, $\cshock{p}$ and shocks always purple, $\cdet{D}$ and the detonation front always hot orange.
• Dotted words are defined on hover — the full glossary is at the bottom of this page.

Throughout the course, colours mean the same thing everywhere:

• The Design Studio. The course ends with a capstone: pick an application — a rocket thrust chamber, a gas-turbine combustor, a ramjet — and design an RDE that starts, stays alive and earns its keep.

What you will be able to do

• Explain why detonating a fuel can, in principle, extract more work from it than burning it, and where that gain comes from
• Use the Rankine–Hugoniot relations and the Chapman–Jouguet condition to predict a detonation's speed, pressure and temperature
• Describe the structure of a real detonation front — the ZND profile and the cellular pattern — and use the cell size λ as a design ruler
• Reason about initiation, deflagration-to-detonation transition, and the critical conditions for a detonation to survive
• Explain how a rotating detonation engine works: the fill layer, the flowfield, the injector's battle with the wave, and why engines choose their own number of waves
• Judge the performance claims for RDEs — pressure gain, specific impulse — and the engineering problems still in the way

Prerequisites

Comfort with undergraduate thermodynamics helps. Shock waves and compressible flow are built up from scratch in Lesson 2. No prior knowledge of detonation is assumed.

A note on the approach

Unlike the other courses on this site, this one does not follow a single textbook: rotating detonation engines are young enough that the field lives mostly in research papers. The course stitches together the classical theory of detonation — Chapman and Jouguet, Zel'dovich, von Neumann and Döring, the work collected in Lee's The Detonation Phenomenon and Fickett and Davis's Detonation — with the experimental and computational literature on rotating detonation engines from Voitsekhovskii and Bykovskii onward.

Three threads run through it: the wave sets its own speed, the cell size is the ruler, and the injector must refill what the wave destroys.

The simulations are reduced-order models. The detonation calculator is calibrated to land within a few percent of equilibrium Chapman–Jouguet calculations for eight fuel–oxidizer pairs. The reactive-flow solver is a genuine second-order finite-volume code, but with a single-step model chemistry on a coarse grid. Both are built for intuition, not for design.

Glossary

DeflagrationLesson 1

Combustion that spreads by heat and radicals diffusing into fresh gas: a subsonic flame, metres per second, at almost constant pressure.

DetonationLesson 1

A shock wave and a flame locked together: the shock heats the gas enough to ignite it, and the energy released keeps the shock going. Supersonic — about 2 km/s — with a pressure rise of 15–30×.

Pressure-gain combustionLesson 1

Any combustion process whose products leave at higher total pressure than the reactants entered with. Constant-pressure burners always lose a little; detonations can gain.

Humphrey cycleLesson 1

The ideal cycle with heat added at constant volume. More efficient than the constant-pressure Brayton cycle at the same compression.

Fickett–Jacobs cycleLesson 1

The ideal thermodynamic cycle of a detonation engine: compression, a detonation, isentropic expansion back to ambient pressure, and heat rejection.

Entropy riseLesson 1

The measure of lost work. For the same heat added, a detonation produces the smallest entropy rise of the three ideal cycles — which is the thermodynamic case for it.

Shock waveLesson 2

A jump in pressure, density and temperature only a few molecular mean free paths thick, travelling faster than sound into the gas ahead of it.

Rankine–Hugoniot relationsLesson 2

Conservation of mass, momentum and energy across a shock or detonation. They fix every jump once the wave speed is known.

Mach numberLesson 2

Speed divided by the local speed of sound. A shock moves into its gas at M > 1 and leaves it moving at M < 1 relative to the wave.

Rayleigh lineLesson 3

A straight line in the pressure–volume plane, set by mass and momentum conservation, of slope −(ρ₁D)². Every state behind a steady wave of speed D lies on it.

Hugoniot curveLesson 3

All the end states that conserve energy across a wave for a given heat release. Where it meets the Rayleigh line is the state behind the wave.

Chapman–Jouguet (CJ) pointLesson 3

The single point where the Rayleigh line just touches the reactive Hugoniot. There, the products move away from the front at exactly the local speed of sound, so no disturbance from behind can catch the wave: it selects its own speed D_CJ.

CJ speed D_CJLesson 3

The speed of a self-sustained, steady, planar detonation. About 2 km/s for hydrogen–air and 1.8 km/s for hydrocarbon–air mixtures at 1 atm.

ZND modelLesson 4

Zel’dovich, von Neumann and Döring’s picture of a detonation: an inert shock, then an induction zone where little happens, then a fast reaction zone that releases the heat.

von Neumann spikeLesson 4

The pressure peak just behind the leading shock, before any heat is released — roughly twice the CJ pressure.

Induction zoneLesson 4

The region behind the shock where radicals build up but almost no heat is released yet. Its length Δ is extraordinarily sensitive to the post-shock temperature.

Activation energyLesson 4

How strongly a reaction rate depends on temperature (k ∝ e^(−Ea/RT)). High activation energy makes detonations unstable and their cells irregular.

Detonation cellLesson 5

The diamond-shaped pattern traced by triple points as transverse waves sweep across a detonation front. The cell width λ is the detonation’s fundamental length scale.

Triple pointLesson 5

Where an incident shock, a Mach stem and a transverse wave meet. Triple points are the hottest spots on the front and leave tracks on a sooted foil.

Soot foilLesson 5

A metal sheet coated with soot and placed on the wall of a detonation tube. The triple points scour fish-scale tracks into it, recording the cell structure.

Cell width λLesson 5

The width of a detonation cell, roughly 10–50 induction lengths. About 10 mm for hydrogen–air and 300 mm for methane–air at 1 atm; it shrinks roughly as 1/p.

DDTLesson 6

Deflagration-to-detonation transition: a flame accelerates by turbulence and confinement until it can form a shock strong enough to ignite the gas on its own.

Shchelkin spiralLesson 6

A coil of wire or a row of obstacles inside a tube that makes a flame turbulent and speeds up the transition to detonation.

Critical initiation energyLesson 6

The minimum energy that starts a detonation directly — without a run-up — by a blast. It scales roughly with λ³ for spherical waves: grams of explosive for hydrogen–air, kilograms for methane–air.

Critical tube diameterLesson 6

The smallest tube from which a detonation can emerge into open space and survive: about 13 cell widths.

Pulse detonation engineLesson 7

A tube that is filled, detonated, emptied and refilled many times a second. The precursor of the rotating detonation engine.

BlowdownLesson 7

The part of a pulse detonation cycle in which the high-pressure products rush out of the open end while an expansion wave runs back up the tube.

Rotating detonation engineLesson 8

An engine in which one or more detonation waves run continuously around an annular chamber, consuming fresh mixture injected through the bottom.

AnnulusLesson 8

The ring-shaped gap between two coaxial cylinders that forms an RDE’s combustion chamber, typically 5–20 mm wide.

Fill height hLesson 9

The height of the layer of fresh mixture ahead of a rotating detonation wave. It must exceed roughly 12 ± 5 cell widths for the wave to survive.

Oblique shockLesson 9

The angled shock that trails from the top of the detonation front into the products of the previous cycle.

Slip lineLesson 9

A boundary between gases moving at different speeds but the same pressure — here between freshly detonated products and older ones.

Contact burningLesson 9

Parasitic deflagration where fresh mixture touches hot products from the previous cycle. Burned this way, the mixture contributes no pressure gain.

Injector stiffnessLesson 10

The ratio of plenum to chamber pressure. Higher stiffness isolates the plenum from the wave but costs total pressure — the main penalty an RDE must beat.

Injector recoveryLesson 10

The time after the wave passes before the injectors can flow again, while the local chamber pressure is still above the plenum pressure.

BackflowLesson 10

Hot, high-pressure products pushed back into the injector passages after the wave passes. It delays refilling and can damage the injector.

Wave modeLesson 11

The number and direction of the detonation waves running in the annulus. Engines often add waves as mass flow rises, and may run in co- or counter-rotating modes.

Counter-rotating wavesLesson 11

A mode in which waves travel in both directions and pass through each other, often weaker than co-rotating detonations and closer to a strongly coupled acoustic mode.

Equivalent available pressureLesson 12

The pressure of a steady, uniform flow that would produce the same ideal thrust as the RDE’s unsteady exhaust. Comparing it with the plenum pressure measures true pressure gain.

Velocity deficitLesson 12

Measured RDE waves usually run at 70–95% of the CJ speed, because of imperfect mixing, contact burning and losses to the walls.