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Mathematics in astronautics: rotational detonation engine



Sending someone or something beyond our planet to this day is an extremely difficult and costly pleasure. While space travelers from various sci-fi works of mass culture use repeaters (“Mass Effect”), warp engines (“Star Trek”) or even stargate (“Stargate”), in reality everything is much more prosaic. At the moment, we are not aware of such unrealistic technologies, because we use rocket fuel. Naturally, to launch one shuttle or a booster rocket, extremely much is needed. A new type of engine - rotational detonation - can solve this problem. While the process of its development is far from complete, scientists from the University of Washington decided to create a mathematical model of this device in order to better understand the principle of its operation.This will allow engineers to conduct accurate prototype tests and better understand which improvements need to be implemented. So, what does the rocket engine look like through the eyes of a mathematician, and what did you learn through modeling? The answers to these questions await us in the report of the research group. Go.


It is obvious that a huge amount of energy is required to take the spacecraft out of the Earth’s atmosphere. The amount of this energy depends on the fuel used and on the engine of the apparatus. There are many options for the first, but they are far from sci-fi equivalents in their effectiveness. Because a lot of attention is paid to the development of a new type of engine.

A classic rocket engine operates through an exothermic chemical reaction of fuel and oxidizer. When these two components of the fuel react, a lot of thermal energy and a gaseous working fluid are generated, which expands. This leads to the fact that its internal energy is converted into kinetic energy of the jet stream. At its core, this chemical process is deflagration, i.e. subsonic combustion process.

Deflagration can be replaced by detonation when a shock wave propagates through a substance, initiating chemical combustion reactions. The type of engine that implements such a model is called a pulsed detonation engine, but it is also still in development.

In this study, we are talking about a rotational detonation engine (RDE, i.e. a rotating detonation engine ) - a device that creates thrust in which self-sustaining shock waves caused by combustion (detonation) propagate azimuthally in an annular combustion chamber.

Fuel and oxidizing agent are injected into the channel, usually through small openings or slots (annular gaps). Due to the narrow annular gap, the density and pressure gradients caused by heat release are self-amplifying, eventually forming shock waves strong enough for self-ignition of the fuel.

The stable operation of RDE, which is the object of research, combines a balance of several aspects: combustion, injection and mixing, the release and release of energy. If these variables are not balanced, then there is a destabilization of the engine, which manifests itself in the form of a transition to a different number of waves or in the form of modulation of the wave velocity.


Image # 1: RDE schema.

Computational hydrodynamic modeling of RDE allows a detailed study of the wave structure and flow field * engine.
The flow field * is the distribution of the density and velocity of a liquid in space and time.

Vector field * - transformations of space, where each of its points is displayed as a vector with a beginning at this point.
However, previously such a procedure was very costly and complicated, as the scientists themselves say. In addition, previously created models could not isolate the factors affecting the formation of bifurcation * .
Bifurcation * - a qualitative change in the behavior of a dynamic system with an infinitely small change in its parameters.
Despite the expected difficulties, it was decided to carry out modeling, but using new experimental data on the nonlinear dynamics of rotating detonation waves. This allowed us to create a model that takes into account the most insignificant changes, thereby fixing the bifurcations observed in practice during the experiments.

experimental part


To conduct a full-fledged study and appropriate modeling, certain experiments were carried out. For this, an RDE and a test chamber were specially prepared for studying the dynamics of a rotating detonation wave. The engine used for this study is unique in that its internal components are modular. Engine parts can be replaced to obtain various annular clearances and the length of the combustion chamber. You can also replace the injector, which allows you to explore different options for connecting and mixing fuel.


Image No. 2 The

test camera is optically accessible, which allows recording the complete kinematic history of all detonation waves with high spatial-temporal resolution ( 2a ).

Each experiment represents a 0.5 second combustion of methane gas and oxygen with a given proportion and feed rate. In a successful experiment, a spark ignites the mixture and produces an accelerating flame, which transforms into a series of traveling detonation waves.

The basis of this study is the assumption that the observed luminosity in the experiments correlates with the progress of combustion. Therefore, brighter regions exhibit higher heat generation than darker regions. If this assumption is true, then you can consider several examples of waveforms extracted from data from a high-speed camera.

Kinematics of the wave can be obtained from camera data using an algorithm for integrating pixel intensities and recorded in the form of a diagram ( 2b ).


Image No. 3

Camera records can also be converted into a wave report system, in this case the phase difference between the waves will be clearly visible.

Graph 3a shows data from 2b in the form of a wave report system, and 3c shows the corresponding speed of the wave being tracked.

For this data, time was defined as τ = t (D wave / L), where L is the length of the periodic region and D wave is the wave velocity in the mode-locked state.

On 3athe transition from one wave to two is visible during the launch process. With such a mode transition, a second detonation wave is formed after the critical point, which begins to propagate around the ring. However, the distance between the two waves in the annular space is asymmetric, which causes an imbalance in the amount of fuel consumed by each of the waves. A wave with coordinate θ 1 , following the previous wave θ 2 , exists with a phase difference Ψ = θ 2 - θ 1<π (). At this point, if we assume that the frequency of fuel renewal is approximately constant, less than half of the available fuel in the chamber for its consumption remains in the tail wave. Since the heat of rocket fuel directly affects the speed of detonation, the delay wave begins to slow down. However, the previous wave can process the rest of the available fuel and accelerate due to this excess. Thus, these two waves behave dispersively when they tend to a stable state with a maximum and symmetric phase difference.

For single wavelength 3aa quasistationary wave has a speed of 20-30% lower than the Chapman-Jouguet speed for rocket fuel. This metric is the direct observable energy necessary to maintain the detonation wave, which is subject to dissipation and restoration of gain in the combustion chamber. When a transition to two waves occurs and the dynamics is set to a stable state, the wave velocity decreases to approximately 90% of the velocity of a single wave.


Image No. 4

If you slow down the fuel supply at the end of the experiment, then the opposite situation is observed. On 4a shows a gradual transition from the wave 2 to wave 1 for about 10 ms. Two waves compete for an increasingly scarce fuel, unlike the case of excess fuel shown in 3a .

Due to the initial perturbation of the phase difference, the waves begin to regularly exchange force (speed and amplitude), causing an exponential increase in instability. As the oscillations of the phase difference increase, a catastrophic interaction between the waves occurs, when the lagging wave overtakes the ahead one during one of the oscillations with a large amplitude. After bifurcation, the velocity of the remaining wave is approximately 10% higher than that of the wave before instability.


Image No. 5

Also quite often wave instabilities were observed that did not lead to a change in the number of waves. Image No. 5 shows the periodic wave velocity and amplitude observed in an experiment with three rotating waves. This is a clear modulation instability, since the spectral sidebands accompany the carrier frequency corresponding to the average speed of the traveling wave in the combustion chamber. This mode of operation is stable in the sense that it does not lead to a bifurcation of the number of waves if the flow state is not substantially disturbed.

If the injector area was increased with respect to the area of ​​the annular chamber, a pulsed mode of operation was observed, which is characterized by a “on / off” mode of operation of the injectors.


Image No. 6

The image above shows vibrational plane waves in a pulsed mode.

Mathematical model


To understand exactly which physical aspects dominate in the process of wave formation, a mathematical model was created in the mode synchronization and mode bifurcations that reflects the nuances of combustion, fuel injection and energy dissipation, the structure of which is determined by the following formulas:
∂η / ∂t + η (∂η / ∂x) = (1 - λ) ω (η) q 0 + ϵξ (η)
∂λ / ∂t = (1 - λ) ω (η) - β (η, η p , s) λ
η (x, t) - properties of the working fluid;
λ - burning variable (λ = 0 - there was no burning, λ = 1 - complete combustion);
ω (η) is the heat release function;
q 0 - heat release and proportionality constant;
ϵξ (η) is the energy loss function;
ϵ is the loss constant;
β (η, η p , s) - injection model;
η p and s are injection parameters.

Experiment Results


Having prepared the mathematical model, the scientists conducted a series of numerical simulations (i.e. simulations) with the following parameters:



At the first stage of modeling, it was decided to consider the existence of planar solutions for the model system, including the behavior of the limit cycle * .
The limit cycle * is one of the possible variants of the stationary state of the system. The limit cycle of a vector field is a closed (periodic) trajectory of a vector field in the vicinity of which there are no other periodic trajectories.
The Cauchy problem * was solved using the initial conditions η (x, 0) = 1 and λ (x, 0) = 0.75.
The Cauchy problem * is the search for a solution to a differential equation that will satisfy the initial conditions (initial data).
A plane wave oscillates near a point in the phase space, where the burning gain depletion and injection gain recovery coincide [βλ = (1 - λ) ω (η)], provided the input energy is balanced, the energy deviates and the energy dissipates [ξ = (1 - λ )) ω (η) q 0 ].

Low-energy vibrations damp to a flat deflagration front without oscillations.

Pulsating fronts, similar to those observed in earlier experiments, are characterized by periodic “activation” and “deactivation” of the injectors, which first resonate with the release of heat and then are saturated with loss mechanisms. An example of a pulsating plane wave front is presented at 6d .

The pulsating plane-wave solutions of the complete model are stable for planar initial conditions, but are unstable to perturbations, since they develop into traveling detonation waves.

The initial conditions of the Cauchy problem for a traveling wave were: η (x, 0) = (3/2) sech 2 (x - x 0 ) and λ (x, 0) = 0 and λ (x, 0) = 0.

The Chapman – Jouguet velocity (CJ) was determined for this system (an inviscid stable wave in which all the energy was transferred into the wave in an infinitely thin reaction zone). This constant wave velocity is defined as the minimum velocity that satisfies the Rankin-Hugoniot * conditions for a given heat release. In the absence of losses, this minimum velocity is equal to DCJ = (η 1 + q 0) + √ q 0 (q 0 + 2η 1 ). In the case η 1 = 0, the wave velocity becomes equal to 2q 0 .
The Rankin-Hugoniot adiabat * is a mathematical relation that relates thermodynamic quantities before and after a shock wave.
This speed is the metric by which traveling waves are measured in the model under consideration.


Image No. 8

The image above shows the evolution of standard experimental modeling. Since the initial sech pulse is much higher than ηc, the medium emits heat locally and quickly. The wave becomes "sharper" and forms a detonation. This initial impulse propagates at a speed CJ until it reaches its tail, and at this moment the wave begins to quickly dissipate and slow down: a limited amount of combustion cannot continue to support the wave at DCJ = 2q 0 . In addition, the rapid heat release (compared with the energy scattering time scale) of the initial wave CJ leads to an increase in the average value of η in the region significantly exceeding the value of η 0environmental and η c ignition value .

Thus, the effective activation energy of the active medium decreases, and parasitic deflagration or slow heat generation unrelated to traveling waves increases in the entire region. Since the propagation time of the initial traveling wave was increased due to scattering, parasitic deflagration has enough time to complete the process of deflagration-detonation (DDT, i.e. deflagration-to-detonation ) and the formation of many detonation waves with a smaller amplitude.

In order to induce a mode transition process when there is a stable mode state, a step change in s was used , which caused bifurcation. An example of such a transition is shown in 4b., where the two initially rotating detonation waves with mode locking become unstable and destructively split.

Low-amplitude phase differences increase exponentially, which was also observed during the experiments ( 4a ). During the oscillation period, two waves exchange force (amplitude) and speed. For a given injection function β and losses, the instability growth rate and the oscillation period are parameterized by the degree of the applied step in changing the parameters s and η p .

When a new wave is generated or the existing one is destroyed, the set of waves in the test chamber acts dispersively, eventually forming a state with mode synchronization.


Image No. 9

Above are bifurcation diagrams showing the dependence of the number of waves, wave velocity and wave amplitude on s and the loss value. With increasing s from zero, stable flat deflagration fronts are formed for small values. As soon as the value of s can contribute to the formation of a traveling wave, the waves begin to show stairs, where their speed gradually increases until another bifurcation occurs. These waves result from parasitic deflagration during the DDT process. With each bifurcation, with increasing number of waves, the wave velocity decreases. When the value of s becomes sufficiently large, the number of waves increases until the wave fronts become small in amplitude and merge into a planar deflagration front.

For a more detailed acquaintance with the nuances of the study, I recommend that you look into the report of scientists .

Epilogue


Spacecraft are incredibly complex mechanisms that combine the knowledge of many scientific fields, physics, chemistry, mathematics, mechanics, etc. Currently used rocket engines use a whole string of control mechanisms and control of the combustion reaction, so that it can successfully provide the movement of a multi-ton colossus trying to take off the ground. In the case of a rotary engine, most of the responsibilities on this issue are assumed by the shock wave. This greatly reduces the amount of fuel consumed (given that an approximate estimate of the detonation efficiency is ~ 25% higher than that of classical deflagration), however, there are a number of problems. The main one is the instability of such waves. As the scientists themselves say, any detonation is an uncontrolled process that proceeds as he pleases.

In order to understand this chaotic process, at first glance, the scientists created a mathematical model. The model was based on practical experiments with the engine, the duration of which was only half a second, but this was enough to obtain the data necessary for the formation of the model.

Researchers say their model is the first of its kind. It makes it possible to understand whether this type of engine will work stably or not, as well as evaluate the operation of a particular engine used during the practical part of the experiments.

In other words, the model reveals maps of what physical processes occur during system operation. In the future, scientists intend to improve their creation so that it can be used already to determine certain aspects that require special attention to implement a working and stable rotary engine.

Thank you for your attention, remain curious and have a good working week, guys. :)

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