Lalit Patidar
Lesson 9 of 14

Inside the Annulus

Ride along with the wave and the RDE's flowfield stands still: a wedge of fresh mixture, a detonation front, an oblique shock, a slip line, a strip of parasitic burning, and products turning toward the exit. What each feature does, and the rules of thumb for how big an engine must be.

6 min2 live simulations
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The question

Stand on the chamber wall of a running RDE, and every few hundred microseconds a detonation hits you. Ride with the wave instead, and the picture freezes: the same shapes, in the same places, lap after lap. What does a parcel of gas see as the wave sweeps past?

A steady picture in a moving frame

In the laboratory frame, an RDE's flow is violently unsteady: every point on the wall sees a pressure spike thousands of times a second. But in a frame that rotates with the wave, a well-behaved single-wave engine is almost steady. That is why its flowfield has a classic, recognisable anatomy, first sketched from Bykovskii's photographs and later computed in detail by groups like Schwer and Kailasanath's at the US Naval Research Laboratory.

Live solverRiding with the wave
Fill height0% of the chamber
Wave speed—
Injectors blocked0% of the lapred strip under the chamber
Burned away from the front0% of the heatat the contact, or as it is injected
Challenges0/3
  • Find the oblique shock in the schlieren view. (The slip line is too smeared on this coarse grid to see.)
  • Use the heat-release view to see where the mixture burns — and where it burns without the wave.
  • With tracers on, follow parcels from the injector plate, through the front, to the exit, for two laps of the wave.
The same live simulation as Lesson 8, but now the camera rides with the detonation: the wave stays put, and the gas streams through the picture from right to left at the wave speed. Dots are tracer parcels injected at the bottom — mint while fresh, orange once burnt. Switch to schlieren (density gradients) to see the shocks, or to heat release to see where the mixture actually burns.

Starting at the injector plate and moving with the gas:

  • The fill layer. Fresh mixture enters through the plate and piles up in a wedge. Just behind the wave, the injectors are blocked by the high pressure; then they reopen. At first the new mixture enters products that are still well over 1000 K and burns as it arrives: in the course's solver, a fresh layer only survives from about a quarter of the lap onward, and from there it grows roughly linearly until the next wave arrives. Its height at the front — the Fill height hThe 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. hh — is the most important number in the engine.
  • The detonation front. Only as tall as the fill layer. It turns the fresh mixture into hot, high-pressure products and throws them sideways, in the wave's direction, and upward.
  • The oblique shock. Above the fill layer there is no fuel, only the previous lap's products. The high-pressure gas behind the front pushes into them, and an Oblique shockThe angled shock that trails from the top of the detonation front into the products of the previous cycle. trails from the top of the front back into the old products.
  • The slip line. Between the freshly detonated gas and the older products lies a Slip lineA boundary between gases moving at different speeds but the same pressure — here between freshly detonated products and older ones.: the same pressure on both sides, but different velocities, temperatures and densities.
  • Contact burning. Along the top of the fill wedge, fresh mixture touches hot products from the last lap. Some of it burns there as an ordinary flame — Contact burningParasitic deflagration where fresh mixture touches hot products from the previous cycle. Burned this way, the mixture contributes no pressure gain. — which is pure loss: that fuel releases its heat at low pressure and leaves less for the wave.
  • Expansion to the exit. The products expand and turn toward the exit, still carrying a swirl in the direction of the wave.

How tall must the fill be?

A detonation front that is too short loses too much energy through its top, where there is no fuel to support it — just as a detonation fails in a tube that is too narrow. Bykovskii, Zhdan and Vedernikov's experiments gave the rule:

h∗≈(12±5) λ.h^* \approx (12 \pm 5)\,\lambda.

From the same experiments came the shape of the engine around it: neighbouring waves sit about 7h∗7h^* apart along the circumference, the chamber must be at least about 2h∗2h^* long, and the channel at least about 0.2h∗0.2h^* wide. All of them scale with the cell width — which is why the cell size of Lesson 5 is the RDE designer's ruler.

Design rulesWill this engine run?
unrolled annulus, 471 mm around × 150 mm longcritical fill height h* ≈ 12 λ = 66 mm
1.00 · stoichiometric
2.0 atm
150 mm
150 mm
15.0 mm
Cell width λ5.5 mm
Most waves that fit1each needs a full-height fill layer
Room for at least one wave✓1.02 waves fit · circumference ≥ 7 h*
Chamber long enough✓150 mm · ≥ 2 h* = 132 mm
Channel wide enough✓15 mm · ≥ 0.2 h* = 13 mm
This engine can sustain up to 1 wave.
Challenges0/3
  • Design a hydrogen–air engine under 10 cm in diameter that passes every rule.
  • Make a methane–air engine of under 30 cm diameter pass — what does it take?
  • Fit four waves in a hydrogen–air engine.
The empirical rules of Bykovskii and colleagues, from decades of experiments: the fresh layer ahead of a wave must be about h* ≈ (12 ± 5) λ tall; neighbouring waves sit about 7 h* apart around the circumference; the chamber must be at least about 2 h* long and the channel at least about 0.2 h* wide. The ruler for all of them is the cell width λ. The drawing is to scale.
Estimate first
How tall must the fill layer be for a stoichiometric hydrogen–air RDE at atmospheric pressure, in centimetres?
0.101101001,000
Drag along the line to place your guess — each tick is ten times the last.
Going deeper
Going deeper: why the rules are only rules

The fill-height criterion is empirical, with a scatter of ±40%, and the cell width it uses is itself a measured average with a factor-of-two uncertainty. Real engines also inject fuel and oxidizer separately, so the "fill layer" is never perfectly mixed: its effective height for the detonation is lower than its geometric height. Many engines have run with fill heights apparently below h∗h^*, and many have failed above it. Treat the rules as a way to get the right order of magnitude — the starting point of a design, not its proof.

Check your intuition
Which feature of the RDE flowfield separates gas burned by this lap's detonation from gas burned in the previous lap?
Recap
What to carry forward
  • In the wave's frame, the RDE flowfield is nearly steady: fill wedge, detonation front, oblique shock, slip line, contact burning, and products expanding to the exit.
  • Behind the wave the injectors are briefly blocked; they reopen as the pressure falls.
  • The fill height must exceed about h∗≈(12±5)λh^* \approx (12 \pm 5)\lambda; waves sit about 7h∗7h^* apart; the chamber needs L≳2h∗L \gtrsim 2h^* and a channel δ≳0.2h∗\delta \gtrsim 0.2h^*.
  • Contact burning wastes fuel at low pressure and starves the wave.

Next: the battle at the bottom of the chamber — how injectors refill a chamber that has just exploded.

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Up next · Lesson 10Injection & MixingHow do you refill a chamber a hundred microseconds after it exploded?Keep going →