Wave Modes

Every wave needs its share

The injectors deliver a fixed flow of fresh mixture, spread around the whole circumference. With $n$ waves running, each wave only gets the mixture injected in the $1/n$ of a lap since the previous wave passed, so the fill layer in front of it is thinner. The simple estimate says $n$ times thinner:

$h_n = \frac{\dot m''\,\pi d}{n\, \cdet{D}\, \rho_{fill}} = \frac{h_1}{n},$

where $\dot m''$ is the mass flux through the annulus, $d$ its mean diameter and $\rho_{fill}$ the fill density: the mixture injected while the wave travels $\pi d/n$, spread into a wedge. Each of those layers must still be taller than the critical height $h^* \approx 12\lambda$ of Lesson 9. That gives the engine an upper limit on its wave count.

The estimate is an upper bound on the thinning. In the course's solver, two waves each meet a layer about 68% as tall as a lone wave's, not 50%: with the plate at high pressure for less of each wave's period, the injectors take in a little more mixture. Start the same engine with three or four waves, though, and every one of them dies within a lap.

There is also a reason to add waves. If the fill layer grows much taller than a wave needs, a lot of fresh mixture sits for a long time against hot products. Contact burning grows, hot spots form, and sometimes one of them runs away into a new detonation: the engine nucleates an extra wave. The waves then reorganise until they share the mixture evenly.

More flow, more waves

Here is the surprising part. In an engine whose exit is choked, raising the mass flow raises the chamber pressure in proportion. The fill density rises with it, so the single-wave fill height $h_1$ barely changes. But the cell size shrinks as $1/p$, and the critical fill height with it. So as the flow goes up, the same fill layer becomes "taller" relative to what a wave needs — and room appears for more waves.

The model above makes both mechanisms explicit: a wave starves below $h^*$, and a new wave is born when the layer exceeds about $2.4\,h^*$. Because those two limits overlap for neighbouring wave counts, the engine shows hysteresis: the number of waves at a given operating point depends on the path taken to reach it. The course's solver shows it too: raise its supply pressure slowly and a second wave appears at about +29%; lower it again and the engine keeps both waves down to about +21%. Real engines show exactly this, and their wave counts are reported on "operating maps" of mass flow against equivalence ratio, with fuzzy, history-dependent boundaries.

The zoo

Real engines do not only change their number of co-rotating waves. They also fall into stranger modes:

• Counter-rotating waves. Two families of waves run in opposite directions and pass through each other twice a lap. They are typically weaker than co-rotating detonations, with lower pressure ratios — partway between a detonation and a strong acoustic mode of the annulus — and they often appear when the mixing is poor or the injectors are strongly coupled to the plenums.
• Longitudinal pulsing. At low flow rates, or with poor injection, the whole annulus can burn and refill in unison, axially, a few kilohertz at a time — an accidental pulse detonation (or pulse deflagration) engine.
• Unsteady switching. Near a mode boundary, the engine may hop between modes every few milliseconds, with waves appearing, merging and dying.

Next: the question every RDE ultimately has to answer — does it actually gain pressure?