This video details many of the spatio-temporal dynamics exhibited in rotating detonation rocket engines (RDEs), including combustion front interactions that behave like solitons. A complete bifurcation analysis of the dynamics is performed showing that our multi scale, dynamical model captures all the salient features of the combustion dynamics.

## The video lecture is about rotating detonation engines (RDEs).

The speaker, James Cook, discusses the following topics:

- Why RDEs are interesting: They are a potential alternative to traditional rocket engines and can be more efficient.
- How RDEs work: They use detonation waves traveling around a circular channel to create thrust.
- Challenges of RDEs: One challenge is that the detonation waves can be unstable.
- Experimental methods to study RDEs: The speaker describes the University of Washington’s 3-inch RDE.
- Computational modeling of RDEs: The speaker introduces a mathematical model to simulate the behavior of detonation waves in an RDE.
- Key findings from the model: The model can capture some of the same wave dynamics that are observed in experiments, such as wave nucleation and mode-locking.

Overall, the lecture provides an introduction to RDEs, their potential benefits, and some of the scientific challenges associated with developing this technology.

https://gemini.google.com *Retell the text. Here is the text*

## Расшифровка видео

Intro

0:00

hello everyone my name is James cook

0:03

with the University of Washington in

0:04

Seattle today I want to talk to you

0:06

about the rotating detonation engine or

0:09

more specifically the nonlinear dynamics

0:12

of the ways we see inside of these

0:14

engines the work I’m going to be showing

0:18

here it was featured in two articles

0:20

first is physical review II by cooker

0:24

Osaka Nolen and cuts and in the second

0:27

we posted to the archive it’s going to

0:29

be submitted sometime soon so you might

What is the rotating detonation engine?

0:35

ask what is the RTE exactly now to

0:39

properly motivate this I want to show

0:41

first the historical context so this

0:44

here is a picture of the f1 rocket

0:47

injector plate and you might notice that

0:50

there are a lot of very interesting

0:52

features all around the engine so we can

0:55

see here baffles line the engines

0:59

circumferentially we also have baffles

1:01

that prevent maybe some radial motion

1:05

here and that’s exactly the motivation

1:06

for these structures so these are

1:10

baffles to prevent the formation of

1:12

thermal acoustic instabilities

1:15

associated with the heat release process

1:17

for this periodic geometry so we have

1:21

for example here baffles perpendicular

1:25

to the first circumferential mode of

1:27

oscillation and also against the radial

1:30

mode of acoustic propagation now around

1:35

the same time as engineers we’re

1:37

developing this engine there was a

1:40

another effort to perhaps simplify the

1:45

the methodology used to mitigate

1:48

thermoacoustic instabilities so

1:51

engineers proposed an alternative and

1:53

this alternative was perversely not to

1:56

mitigate these thermoacoustic

1:58

instabilities but actually to saturate

2:01

them such that they quickly transition

2:04

to perhaps stable structures or

2:09

predictable behavior so the easiest way

2:12

one can do that

2:13

is actually to remove the baffles and

2:16

actually isolate one of the acoustic

2:20

modes such that we promote its as

2:23

amplification so this here is the same

2:27

injector face but I’ve overlaid just the

2:29

circle to show you what amplifying the

2:32

first tangential mode might look like

2:34

right so in the limit what we can expect

2:38

is that these circumferential

2:39

instabilities might transition to a

A Dichotomy of Time Scales

2:42

number of traveling detonation waves all

2:45

right so this is pretty special I’ve

2:47

shown here is sketch of the rotating

2:50

detonation engine flow field starting

2:53

from over here we have some discretes

2:56

fuel and oxidizer injection ports that

2:59

go into the annular combustion chamber

3:01

and then we also have the detonation

3:04

wave itself right here right but what’s

3:08

interesting is that it’s actually this

3:11

dichotomy of timescales that gives rise

3:13

to this stable flow field structure we

3:17

have the detonation time scale which is

3:21

incredibly short both in space and time

3:24

that really really fast time scale of

3:27

combustion is what drives the wave

3:29

motion forward as per detonation theory

3:33

classical detonation theory we have

3:35

another time scale prescribed by the

3:38

time it takes for a detonation wave to

3:39

circumnavigate our entire annulus and

3:43

likewise within that one period we have

3:47

to have sufficient fuel re-introduction

3:50

and mixing for the detonation wave to

3:53

Staveley propagate that little period of

3:56

mixing and reintroduction of fuel is

3:58

shown here in this sliver of this

4:01

olive-colored

4:02

region here so that represents the

4:05

regeneration of propellant for the

4:07

detonation wave to propagate through now

Experimente

4:12

one of the nice things is that because

4:14

this is an annular combustion chamber

4:16

with no moving parts it’s remarkably

4:19

easy to make and test in a laboratory

4:21

setting so what I’ve shown here are two

4:24

images this is the University of

4:26

Washington

4:27

three-inch rde this is the end of the

4:31

designed and and tested for my doctoral

4:33

work here I’ve shown in the upper right

4:36

hand corner this is an isometric view of

4:38

the CAD and the bottom image is if I

4:41

were to take a section cut of that top

4:43

image down the axis to show you the

4:46

internals so a couple features heater

4:49

note my propellant enters here and mixes

4:52

after introduction through these

4:54

orifices in this combustion chamber

4:56

right so after mixing a detonation wave

4:59

can come on by and just that propellant

5:02

and it’s expand the hot gases downstream

5:06

which is going to produce thrust in this

5:09

case one of the other nice things is

High-Speed Imaging

5:12

that we can directly image our

5:15

combustion chamber during experiments

5:17

which give us remarkable images like

5:20

this so this is an ignition phenomena

5:23

that I filmed in one of our experiments

5:26

so after this nice deflagration plume we

5:28

see the rapid transition to detonation

5:31

waves that wrap around the annulus we

5:33

have this nice flame out consuming the

5:36

leftover propellant downstream of the

5:39

combustor and now we see the stable

5:41

formation or the rather the formation of

5:44

stable pulses through time after this

5:48

flame out so right now I see three waves

5:52

who through time are mode locking

5:55

they’re approaching the same stable

5:57

speed and same phase difference between

6:00

the different waves now instead of

Space-Time History

6:03

watching videos which is entertaining

6:05

its own right what we can do is perhaps

6:08

recast the videos in terms of just a

6:11

series or snapshots of the system so

6:14

what I can do is take every video frame

6:17

and I can find the location of the

6:19

annulus for each video frame and I can

6:21

integrate the pixel intensity around

6:24

that annulus so I’m going to get this

6:25

nice column vector representing the

6:28

state of the domain for that point in

6:31

time now what I can do is I can stack

6:33

those one D vectors up into a 2d array

6:35

and display that as a simple pseudo

6:38

color plot or as an image

6:39

which is shown here in the bottom right

6:40

so this is actual data of an experiment

6:43

that exhibited one wave that was stable

6:46

propagating in space and in time so what

6:50

I like to do now is show you all some

6:53

examples of wave transients I’ll start

6:57

before playing this video and showing

6:58

you what the XT or X theta sorry theta

7:02

as he diagrams look like and the top

7:04

figure here I have the raw pixel

7:08

intensity through time so again if I

7:11

take a vertical cut that vertical cut is

7:13

going to show through the entire 2 pi

7:16

the annulus the integrated pixel

7:19

intensity the bottom is instead of in

7:23

the laboratory reference frame so

7:24

instead of what the camera is saying I

7:26

am now going to attach myself to one of

7:29

the detonation ways I’m going to look

7:31

forward through the annulus until I

7:33

reach my own tail right so in this way I

7:35

am NOT in the laboratory reference frame

7:37

I am in the reference frame of the

7:39

detonation wave and this reference frame

7:42

my phase difference between the

7:44

different waves now is an explicit

7:46

output so let’s play this video right

7:50

now I have a single detonation wave and

7:52

appears to be traveling around the of

7:56

the annulus I don’t know if it’s stable

7:57

or unstable yet I don’t know the speed

7:59

but I want you to pay attention to

8:02

what’s happening in other parts of the

8:04

annulus we might start to see some

8:06

background luminosity changes reflecting

8:09

different regions of combustion in

8:11

different portions so you might notice

8:14

that as I as I move along at some point

8:17

in criticality I’m going to form a

8:19

second detonation wave or rather a

8:21

second luminous blob because of the

8:24

nature of the annulus I have some really

8:26

tight channels and geometric confinement

8:28

I’m going to promote the self steepening

8:30

of pressure and density gradients it’s

8:32

eventually going to lead to shock

8:34

formation and as soon as that shock

8:36

forms I’m going to have coupling of of

8:38

the shock front and heat release that’s

8:40

going to transition to another

8:42

detonation wave that’s exactly what’s

8:44

happening in the video and around these

8:46

points in time and through time these

8:50

different waves approach

8:52

again a nice stable phase difference

8:55

between them that’s the phase difference

8:57

of Pi radians or 180 degrees now I can

9:02

show you another similar video going

9:06

from a set of 1 to 2 waves how about

9:08

from 2 to 1 wave as well it’s the same

9:11

style plot I have raw pixel intensity

9:14

that I have in the wave reference frame

9:16

through time what’s really nice about

9:18

this is you can see a really really

9:21

clear and explicit exchange of wave

9:23

strength amplitude speed and their phase

9:28

difference or the difference in in phase

9:31

between the waves is also sila tori and

9:33

seemingly growing exponentially in time

9:35

until eventually another point of

9:37

criticality occurs where there’s a

9:40

destructive bifurcation the stronger of

9:42

the two waves during one of the large

9:45

amplitude modulation x’ actually

9:47

overruns the weaker wave this is the

9:49

transition from two to one so we can let

9:52

this video play I see that they are

9:55

dancing around each other you know the

9:58

phase differences go goes from less than

10:01

PI to greater than PI really an

10:03

interesting process now now I can see

10:08

that the waves are beginning to have

10:11

even further a large amplitude

10:12

modulation so one of these we will see

10:15

the larger wave overrun the weaker yeah

10:22

there it is now just a fantastic process

10:27

now lastly I’ve shown wave destruction

10:31

I’ve shown wave nucleation but there’s

10:34

also an opportunity for what appears to

10:37

be almost like simple harmonic motion

10:39

right if you’re to think of just a

10:40

simple oscillator now these are two

10:44

waves playing this cat-and-mouse game

10:46

and it’s not our chamber where one

10:48

accelerates the other one catches up the

10:51

preceding wave catches up to the tail

10:52

the other way they just keep going in

10:54

this interesting cycle now I’ve shown

10:57

here same two style of plots laboratory

11:00

reference frame and in the reference

11:02

frame in one of the waves and you can

11:03

see this really large amplitude

11:05

modulation

11:06

of phase difference but it does appear

11:09

to be stable in time right so what I

A Qualitative Model

11:13

would like to do now is I have a

11:15

collection of of nonlinear dynamics I

11:18

have a collection of bifurcations of the

11:19

system

11:20

I would like to formulate a model to

11:22

capture these phenomena so what I’m

11:25

going to do is just a really simple

11:27

control volume approach and I really

11:29

hope that I can I can recover things

11:31

like a detonation like structure so

11:34

those of you who are more familiar with

11:35

detonation community or the detonation

11:36

literature this would be like the Zelda

11:38

Fontenoy meandering model for for

11:41

detonation

11:42

I also want the interaction of time

11:44

scales I mentioned before that there’s a

11:46

dichotomy of scales between injection

11:48

the round-trip time of the wave the

11:50

timescale of combustion

11:52

I do want a model to appropriately watch

11:56

or observe the interplay of these

11:58

different time scales so to get started

Control Volume

12:02

I’m going to show a really simple 2d

12:04

control volume a couple things to note

12:07

here this is this is my model so I get

12:10

to choose the fluid right so you might

12:12

notice that I have flux functions

12:15

but these flux functions are up to me to

12:18

determine their functional form right so

12:21

let’s start by looking at what I’m

12:23

actually tracking so I’m going to do the

12:25

the evolution of an intensive fluid

12:29

property perhaps analogous to internal

12:32

energy which I’ll denote by lowercase u

12:35

now I’d also know that my reaction waves

12:38

are supported by energy input from

12:41

combustion right so I’m also going to

12:43

have a source term somewhere in my model

12:46

that’s going to mimic chemical heat

12:48

release Q via the progression of a

12:52

combustion progress variable which I’m

12:54

going to denote as lambda right now in

12:57

my model I’m free to choose my flux

13:00

function I’m gonna choose burgers flux

13:02

so burgers flux is one-half u squared

13:05

what’s nice about burgers flux is that

13:07

it’s mathematically pretty tractable

13:10

right but also burgers flux guarantees

13:14

that I will have shock formation with

13:17

any concavity change in my domain

13:19

eight so if i have burgers flux and I

13:22

correctly incorporated a source term I

13:25

can I can begin to have a model that

13:28

might that might have the interplay of

13:33

chemistry and flow which is what I want

13:36

I want that detonation structure you

13:38

might also notice that I have here what

13:40

have 1/2 new square is equal to P so P

13:43

is indeed pressure just by analogy here

13:47

and that’s because I want to enforce

13:50

that the gradient of my flux function is

13:52

what drives the flow right so it’s a

13:54

pressure gradient that drives the flow

13:56

and I need to be consistent with my my

13:58

fluid analogy for this model so if I do

Spatial Derivatives

14:04

the standard control volume approach and

14:06

I take the limits as my as my control

14:09

volume to mention is go to zero right in

14:11

both x and y space spatial dimensions I

14:13

guess something looks like this right so

14:15

I have that some poor LEvolution of my

14:18

internal energy property with my spatial

14:22

derivative terms is equal to my source

14:24

term or the heat release my x-direction

14:27

is going to be my periodic dimension

14:30

this is gonna be my 1d domain about the

14:33

circumferential dimension of my annulus

14:36

my axial gradient so that’s my partial U

14:41

squared over 2 partial Y I’m going to

14:44

model this so this is my like an axial

14:46

pressure gradient that’s going to be

14:48

modeled and you might notice that I have

14:50

here this minus epsilon u squared so the

14:55

reason this is a simplifies to a

14:57

polynomial is that in our lab we

14:59

actually observed that if you pump

15:01

enough energy into this small annular

15:05

chamber it’s actually pretty easy to

15:07

thermally choke the device and it turns

15:09

out that thermally choked devices or

15:11

back pressurized devices have

15:13

self-similar combustor profiles in terms

15:16

of static pressure so I’m enforcing the

15:18

same behavior via this really simple

15:21

quadratic loss term now lastly we need

15:26

to talk about the dynamics of my

15:28

combustion progress variable now in my

15:31

previous source term I just had

15:34

this partial lambda partial T modifying

15:37

my my heat release little keel but

15:40

that’s not sufficient I need to

15:41

introduce a competition between

15:44

injection and combustion right they go

15:48

head-to-head and I also need to

15:49

introduce the competition between energy

15:52

input and energy output so what this

15:54

ends up looking like is I have gain

15:57

depletion by combustion that’s what this

15:59

term represents this is similar to our

16:02

Irenaeus kinetics and I have gained

16:05

recovery where gain recovery is an

16:08

injection model right so this is how I’m

16:09

introducing the chemical potential into

16:12

my domain likewise for the evolution of

16:16

little yield I have input output energy

16:20

balance right so all my energy is input

16:22

through chemical reactions and my energy

16:25

is dissipated through in this case

16:29

exhaust but what’s nice about this is

16:32

it’s an input output energy balance but

16:35

with a nonlinear medium right this

16:37

burgers flux that’s going to give us

16:38

shock formation now the last piece to

Zero-Order Injection Model

16:43

put this all together is an injection

16:45

model so I call this like a zero order

16:48

injection model and our experiments we

16:51

use gaseous propellants with choked

16:54

orifices what I mean by that is we have

16:57

a really really high pressure ratio

16:59

between upstream of the orifice or the

17:02

gas injector and downstream which would

17:04

be the combustion chamber but as these

17:06

detonation waves pass over our office

17:08

locations the detonation wave is a

17:11

really really high pressure so it turns

17:13

out that the detonation wave imposes a

17:15

blockage or even worse a backflow of

17:18

propellant back into our propellant feed

17:21

system so I’ve defined here this beta

17:24

this is an injection excuse me an

17:27

injection model that’s based on an

17:30

activation function so you can imagine I

17:32

have either state of my domain little u

17:34

that’s acting as an on/off switch

17:38

imposed on the injection scheme so if

17:40

the state of my combustors is high

17:43

energy a high you i can’t flow any more

17:45

propellant into

17:47

domaine so that’s what this activation

17:48

function is mimicking I have a time

17:51

constant defined here as little as times

17:54

use of P where you see P is this

17:56

threshold for injection or no injection

18:00

so I’m directly modifying an injection

18:02

time constant and then I’m normalizing

18:04

this by an activation like term where

18:08

eventually if my domain is high enough I

18:11

can no longer inject propellant so what

18:15

you guys say that we start to get into

18:17

some numerical experiments I’m going to

18:19

show you guys one of the first runs I

18:22

did so this is a simulation output that

18:27

I can just display motion the same way

18:28

that we saw the videos so after an

18:31

initial pulse I see some really

18:35

interesting transients at the beginning

18:38

of the simulation though eventually

18:40

through time I have two waves that

18:42

approach the same stable speed and phase

18:45

differences they become mode locks so

18:47

it’s very very exciting and promising

18:48

results I can show much like I did in

18:51

the experiments I can show the same

18:53

style of theta through time diagrams

18:56

showing the space time history and I’ve

18:58

shown here a couple of different

18:59

snapshots at different points in time so

19:01

this is my initial condition just a

19:04

really nice set pulse to start off the

19:07

reactions and then I have an immediate

19:10

transition excuse me immediate

19:11

transition to a detonation wave a

19:15

traveling jjigae huge a detonation wave

19:17

you can see a really really sharp peaked

19:21

spike in the state of my domain and are

19:24

really nice to cang tail afterwards now

19:27

as soon as this detonation wave reaches

19:29

its tail there has not been enough

19:31

propellant reintroduction to sustain

19:34

that high speed and high strength so

19:37

what you see is that the detonation wave

19:39

immediately starts to decay but then

19:42

also because now the detonation wave has

19:44

decay in a slowed down the deflagration

19:47

or the slow scale combustion that’s not

19:50

associated with the wave deflagration

19:52

actually starts to play an important

19:54

role so you see the Declaration of the

19:56

domains start to self steepen and form a

19:59

second detonation

20:00

wave because the other way even that in

20:02

the chamber was not strong enough to

20:06

prevent that from happening so a really

20:08

interesting interplay of time skills

20:10

that were already starting to see in

20:11

this model

20:13

what’s really cool is that I can recover

Numerical Experiments

20:17

a lot of the same dynamics that we saw

20:19

in experiments

20:20

here’s wave nucleation is the same video

20:23

that I showed before and I can show you

20:26

the same style of wave nucleation just

20:30

from the prior slide

20:31

it’s that same data just showing you in

20:34

terms of the wave reference frame

20:36

instead of in the laboratory reference

20:37

frame so immediately after nucleation of

20:40

a second wave we see the phase

20:42

difference oscillations approach pi

20:44

right pi radians and we have stabili

20:48

mode-locked pulses in the bottom line I

20:52

have here a this fraction D over D c.j

20:56

this corresponds to the fraction of wave

21:00

speed related to the challenge of gauge

21:03

speed so this is this is essentially

21:05

saying that prior to wave nucleation I

21:09

have a certain speed call around 75% of

21:12

the Chapmans your gauge speed but after

21:14

wave nucleation my speeds decrease in

21:17

both experiments in the model by about

21:19

10% I can show the same thing for wave

21:25

destruction again this is my experiment

21:28

I can show the same thing for a similar

21:33

or representative case from the

21:34

simulation and again I have my wave

21:37

speeds through time really encouraging

21:40

stuff so lastly you might ask well what

21:44

kind of do is such a model the

21:46

possibilities are really endless and

21:48

this is a fantastic first step to

21:50

looking into the future of stability

21:52

analysis control different actuation

21:55

schemes might be able to Cape be

21:57

conceptualized and introduced in such a

22:00

model so as an example this is this is a

22:05

sub study I did of a bifurcation

22:08

analysis showing what could happen with

22:10

steadily propagating

22:13

detonation waves this is an experiment

22:17

shown in the mean velocity reference

22:19

frame through time non-dimensional time

22:22

showing this characteristic exchange of

22:25

wave strength amplitude phase difference

22:29

and speed right so a fantastic

22:30

repeatable structure here now just by

22:33

doing a bifurcation style analysis I can

22:36

extract a qualitatively identical

22:39

repeatable kinematic trace showing

22:41

qualitatively the same features right

22:44

but what’s even better is that through

22:46

such a model and through such analysis I

22:48

know the route it takes to get to this

22:51

specific condition which is a huge first

22:54

step so in terms of wave stability or

22:57

wave dynamics through a set of

23:00

engineering parameters we now have what

23:04

I would call the first step to producing

23:07

or reproducing these dynamics so last

23:12

that I do want to go back and recognize

23:13

that these works and these figures are

23:17

featured in these two papers they can be

23:19

found in the video description and if

23:23

there’s if there’s anything else there

23:27

any other questions you might have

23:28

you’re free to email me my email address

23:31

is found attached to these two papers as

23:34

well thank you so much