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.
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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