Derivation of a Revised Tsiolkovsky Rocket Equation That Predicts Combustion Oscillations

Our study identifies a subtle deviation from Newton’s third law in the derivation of the ideal rocket equation, also known as the Tsiolkovsky Rocket Equation (TRE). TRE can be derived using a 1D elastic collision model of the momentum exchange between the differential propellant mass element (dm) and the rocket final mass (m1), in which dm initially travels forward to collide with m1 and rebounds to exit through the exhaust nozzle with a velocity that is known as the effective exhaust velocity ve. We observe that such a model does not explain how dm was able to acquire its initial forward velocity without the support of a reactive mass traveling in the opposite direction. We show instead that the initial kinetic energy of dm is generated from dm itself by a process of self-combustion and expansion. In our ideal rocket with a single particle dm confined inside a hollow tube with one closed end, we show that the process of self-combustion and expansion of dm will result in a pair of differential particles each with a mass dm/2, and each traveling away from one another along the tube axis, from the center of combustion. These two identical particles represent the active and reactive sub-components of dm, co-generated in compliance with Newton’s third law of equal action and reaction. Building on this model, we derive a linear momentum ODE of the system, the solution of which yields what we call the Revised Tsiolkovsky Rocket Equation (RTRE). We show that RTRE has a mathematical form that is similar to TRE, with the exception of the effective exhaust velocity (ve) term. The ve term in TRE is replaced in RTRE by the average of two distinct exhaust velocities that we refer to as fast-jet, vx1, and slow-jet, vx2. These two velocities correspond, respectively, to the velocities of the detonation pressure wave that is vectored directly towards the exhaust nozzle, and the retonation wave that is initially vectored in the direction of rocket propagation, but subsequently becomes reflected from the thrust surface of the combustion chamber to exit through the exhaust nozzle with a time lag behind the detonation wave. The detonation-retonation phenomenon is supported by experimental evidence in the published literature. Finally, we use a convolution model to simulate the composite exhaust pressure wave, highlighting the frequency spectrum of the pressure perturbations that are generated by the mutual interference between the fast-jet and slow-jet components. Our analysis offers insights into the origin of combustion oscillations in rocket engines, with possible extensions beyond rocket engineering into other fields of combustion engineering.

A Passive Method to Control Combustion Instabilities with Perforated Liner

The effectiveness of perforated liner with bias flow on the control of combustion instability is investigated. Combustion instabilities result from the coupling between acoustic waves and unsteady combustion heat release. Sometimes the phenomenon happens in afterburners of aeroengine and rocket engine, and it always causes damage to flame holders, liner seetions and other engine components. Passive methods, such as perforated liner, are often used to suppress such instabilities in application. In this article, first, a burner testbed is built in order to study the characteristic of this phenomenon. The unstable frequencies and unsta- ble area are investigated experimentally. Then an analytical model, based on "transfer element method", is developed and the numerical results are compared with those from experiments. At last the perforated liner is applied to the burner to suppress the instabilities. The results show that the sound pressure can be greatly reduced by the perforated liner.

Study on Instable Combustion of Solid Rocket Motor with Finocyl Grain

The instable combustion or oscillation combustion which occurs in three high capacity solid rocket motors using high energy composite propellant with finocyl grain is studied. The reasons of the acoustic combustion instability are also discussed. Three engineering methods that can eliminate combustion instability are proposed and discussed. The study shows that the combustion instability mainly depends on the propellant grain shape and nozzle structure. Some measures to reduce the acoustic energy and mass generation rate of combustion gas can be adopted. The test results indicate that the modified rocket motors can significantly eliminate the instable combustion and improve the motor internal ballistic performance.

Mitigation of Combustion Instability and NO_(x)Emissions by Microjets in Lean Premixed Flames with Different Swirl Numbers

Swirl combustion serves as a helpful flame stabilization method,which also affects the combustion and emission characteristics.This article experimentally investigated the effects of CO_(2)microjets on combustion instability and NO_(x)emissions in lean premixed flames with different swirl numbers.The microjets’control feasibility was examined from three variables of CO_(2)jet flow rate,thermal power,and swirl angles.Results indicate that microjets can mitigate the combustion instability and NO_(x)emissions in lean premixed burners with different swirl numbers and thermal power.Still,the damping effect of microjets in low swirl intensity is better than that in high swirl intensity.The damping ratio of pressure amplitude can reach the maximum of 98%,and NO_(x)emissions can realize the maximum reduction of 10.1×10^(−6)at the swirl angle of 30°.Besides,the flame macrostructure switches from an inverted cone shape to a petal shape,and the flame length reduction at low swirl intensity is higher than that of high swirl intensity.This research clarified the control differences of mitigation of combustion instability and NO_(x)emissions by microjets in lean premixed flames with different swirl numbers,contributing to the optimization of microjets control and the construction of high-performance burners.

Combustion instability of pilot flame in a pilot bluff body stabilized combustor

Combustion instability of pilot flame has been investigated in a model pilot bluff body stabilized combustor by running the pilot flame only. The primary objectives are to investigate the pilot flame dynamics and to provide bases for the study of the interaction mechanisms between the pilot flame and the main flame. Dynamic pressures are measured by dynamic pressure transduc- ers. A high speed camera with CH＊ bandpass filter is used to capture the pilot flame dynamics. The proper orthogonal decomposition （POD） is used to further analyze the high speed images. With the increase of the pilot fuel mass flow rate, the pilot flame changes from stable to unstable state grad- ually. The combustion instability frequency is 136 Hz when the pilot flame is unstable. Numerical simulation results show that the equivalence ratios in both the shear layer and the recirculation zone increase as the pilot fuel mass flow rate increases. The mechanism of the instability of the pilot flame can be attributed to the coupling between the second order acoustic mode and the unsteady heat release due to symmetric vortex shedding. These results illustrate that the pilot fuel mass flow rate has significant influences on the dynamic stability of the pilot flame.

Analysis of combustion instability via constant volume combustion in a LOX/RP-1 bipropellant liquid rocket engine

Turbulent two-phase reacting flow in the chamber of LOX/RP-1 bipropellant liquid rocket engine is numerically investigated in this paper. The predicted pressure and mean axial velocity are qualitatively consistent with the experimental measurements. The self-excited pressure oscillations are obtained without any disturbance introduced through the initial and boundary conditions. It is found that amount of abrupt pressure peaks appear frequently and stochastically in the head regions of the chamber, which are the important sources to drive and strengthen combustion instability. Such abrupt pressures are induced by local constant volume combustion, because local combustible gas mixtures with high temperature are formed and burnt out suddenly due to some fuel droplets reaching their critical state in a rich oxygen surrounding. A third Damkhler number is defined as the ratio of the characteristic time of a chemical reaction to the characteristic time of a pressure wave expansion to measure the relative intensity of acoustic propagation and combustion process in thrusters. The analysis of the third Damkhler number distributions in the whole thrust chamber shows that local constant volume combustion happens in the head regions, while constant pressure combustion presents in the downstream regions. It is found that the combustion instability occurs in the head regions within about 30 mm from the thruster head.

3D convolutional selective autoencoder for instability detection in combustion systems

While analytical solutions of critical(phase)transitions in dynamical systems are abundant for simple nonlinear systems,such analysis remains intractable for real-life dynamical systems.A key example is thermoacoustic insta-bility in combustion,where prediction or early detection of the onset of instability is a hard technical challenge,which needs to be addressed to build safer and more energy-efficient gas turbine engines powering aerospace and energy industries.The instabilities arising in combustion chambers of engines are mathematically too complex to model.To address this issue in a data-driven manner instead,we propose a novel deep learning architecture called 3D convolutional selective autoencoder(3D-CSAE)to detect the evolution of self-excited oscillations using spatiotemporal data,i.e.,hi-speed videos taken from a swirl-stabilized combustor(laboratory surrogate of gas turbine engine combustor).3D-CSAE consists of filters to learn,in a hierarchical fashion,the complex visual and dynamic features related to combustion instability from the training videos(i.e.,two spatial dimensions for the image frames and the third dimension for time).We train the 3D-CSAE on frames of videos obtained from a limited set of operating conditions.We select the 3D-CSAE hyper-parameters that are effective for characterizing hierarchical and multiscale instability structure evolution by utilizing the dynamic information available in the video.The proposed model clearly shows performance improvement in detecting the precursors and the onset of instability.The machine learning-driven results are verified with physics-based off-line measures.Advanced active control mechanisms can directly leverage the proposed online detection capability of 3D-CSAE to mitigate the adverse effects of combustion instabilities on the engine operating under various stringent requirements and conditions.

N_(2)and Ar dilution on the premixed biogas jet flame under external acoustic enforcement

In this study,combustion instabilities and flue gas emission changes under different dilutions of N_(2)(nitrogen)and Ar(argon)of a promising biogas mixture(70%CH4-30%CO_(2))in the fight against greenhouse gas emissions were investigated.In the experiments,additions were made from 0%to 50%at intervals of 10%for both gases.In order to detect the instability of the flame,external acoustic enforcements at different frequencies was applied through the speakers placed in the combustion chamber arms.The dynamic pressure fluctuation values were recorded.The results showed that low dilution ratios were effective in reducing flame instability for both inert gases.However,as the dilution ratio increased,the fuel/air mixture became leaner and blowoff occurred.In the case of comparing two different gases,it has been observed that the effect of argon gas on reducing dynamic pressure fluctuation is higher.Burner outlet temperature and brightness values of the flame decreased in both Ar and N_(2)dilution.CO and NOx emissions increased with increasing diluent volume for all dilution conditions.When the emissions of the two diluent gases are compared,the CO emission,which was 3134 ppm in the undiluted condition,increased up to 4949 ppm in 50%Ar dilution,while it increased to 4521 ppm in 50%N_(2)dilution.

Similarity phenomena of lean swirling flames at different bulk velocities with acoustic disturbances

In this study,flame responses to acoustic disturbances with different frequencies and amplitudes were experimentally investigated in a lean premixed swirl-stabilized combustor operating at different bulk velocities.The total heat release rate fluctuations and spatial CH*chemiluminescence distributions were captured using a photomultiplier tube and high-speed camera,respectively.The results indicate that the heat release rate exhibits a relatively drastic oscillation and high-order harmonics for low-frequency disturbances.When the bulk velocity and forcing frequency were doubled simultaneously,similar flame structures were observed in the CH*chemiluminescence distributions.As the bulk velocity increases,the gain of the Flame Describing Function(FDF)extends toward the higher frequencies,and the delay time of the flame response decreases.The similarity among FDFs at different bulk velocities was effectively captured by introducing a non-dimensional parameter,defined as the ratio of the flame response delay to the forcing time scale,to replace the dimensional forcing frequency.Furthermore,the availability of the newly defined non-dimensional parameter was verified for flames with different swirl numbers,as this played an important role in determining the flame structures and associated unsteady heat release rate.

Effect of ramp-cavity on hydrogen fueled scramjet combustor

Sustained combustion and optimization of combustor are the two challenges being faced by combustion scientists working in the area of supersonic combustion.Thorough mixing,lower stagnation pressure losses,positive thrust and sustained combustion are the key issues in the field of supersonic combustion.Special fluid mechanism is required to achieve good mixing.To induce such mechanisms in supersonic inflows,the fuel injectors should be critically shaped incurring less flow losses.Present investigations are focused on the effect of fuel injection scheme on a model scramjet combustor performance.Ramps at supersonic flow generate axial vortices that help in macro-mixing of fuel with air.Interaction of shocks generated by ramps with the fuel stream generates boro-clinic torque at the air&liquid fuel interface,enhancing micro-mixing.Recirculation zones present in cavities increase the residence time of the combustible mixture.Making use of the advantageous features of both,a ramp-cavity combustor is designed.The combustor has two sections.First,constant height section consists of a backward facing step followed by ramps and cavities on both the top and bottom walls.The ramps are located alternately on top and bottom walls.The complete combustor width is utilized for the cavities.The second section of the combustor is diverging area section.This is provided to avoid thermal choking.In the present work gaseous hydrogen is considered as fuel.This study was mainly focused on the mixing different fuel injection locations.It was found that injecting fuel upstream of the ramp was beneficial from fuel spread point of view.

Influence of thermal inhibitor position and temperature on vortex-shedding-driven pressure oscillations

Vortex-acoustic coupling is one of the most important potential sources of combustion instability in solid rocket motors （SRMs）. Based on the Von Karman Institute for Fluid Dynamics （VKI） experimental motor, the influence of the thermal inhibitor position and temperature on vortex-shedding-driven pressure oscillations is numerically studied via the large eddy simulation （LES） method. The simulation results demonstrate that vortex shedding is a periodic process and its accurate frequency can be numerically obtained. Acoustic modes could be easily excited by vortex shedding. The vortex shedding frequency and second acoustic frequency dominate the pressure oscillation characteristics in the chamber. Thermal inhibitor position and gas temperature have little effect on vortex shedding frequency, but have great impact on pressure oscillation amplitude. Pressure amplitude is much higher when the thermal inhibitor locates at the acoustic velocity anti-nodes. The farther the thermal inhibitor is to the nozzle head, the more vortex energy would be dissipated by the turbulence. Therefore, the vortex shedding amplitude at the second acoustic velocity antinode near 3/4L （L is chamber length） is larger than those of others. Besides, the natural acoustic frequencies increase with the gas temperature. As the vortex shedding frequency departs from the natural acoustic frequency, the vortex-acoustic feedback loop is decoupled. Consequently, both the vortex shedding and acoustic amplitudes decrease rapidly.

Effects of Tabular Stratified CO_(2)/O_(2)Jets on Dynamic and NO_(x)Emission Characteristics of a Model Gas Turbine Combustor

The effects of tabular stratified CO_(2)/O_(2)jet in cross flow on thermoacoustic instability and NO_(x)emission were experimentally studied.To explore the dependence of injection positions on flame stability,two factors were taken:the injection height and the injection direction of CO_(2)/O_(2)gas.Results show that the injection positions seriously affect the control effectiveness.The optimum acoustic amplitude-damped ratio of thermoacoustic instability can reach 76.61%with the first layer of horizontal direction.The sound pressure amplitude declined from 56 Pa to 13.1 Pa.The concentration-damped ratio of NO_(x)emission can achieve 66.67%with the first layer of vertical direction.The concentration of NO_(x)emission declined from 50.4 mg/m^(3)to 16.8mg/m^(3)as the jet in cross flow rate increased.Higher oxygen ratio of stratified CO_(2)/O_(2)jets can produce lower NO_(x)emission but higher combustion instability.The descending gradient of NO_(x)emissions is different among different injection positions.Frequency shifting of the sound pressure and flame CH*chemiluminescence emerged.The oscillation frequency declined as the flow rate of CO_(2)/O_(2)jets increased.The unsteady long and compact flame was dispersed after CO_(2)/O_(2)injection.The macrostructure of flame was characterized as flatter and short under jet in cross flow.The variation curves of the flame length and top view area are similar to the shape of half saddle lines.This research proved the optimal control of thermoacoustic instability and NO_(x)emissions with a passive method,which could be conducive to the realization of clean and secure combustion in industrial lean premixed combustors.

Effect of baffle injectors on the first-order tangential acoustic mode in a cylindrical combustor

Combustion instability is a very important issue in the development of the propulsion systems used in aerospace. It is very important to associate the high frequency combustion instabilities with the acoustic characteristics of the combustion chamber. In this paper, the effects of various baffle injectors which were installed on the injector faceplate on the first-order tangential acoustic mode were investigated theoretically and experimentally. The effects of the gap between adjacent injectors on the first-order tangential acoustic mode in a cylindrical chamber were considered. The acoustic admittance of the injectors was derived. The results showed that the amplitude and frequency of the first-order tangential acoustic mode increase with the increase in the gap between adjacent injectors, but decrease with the increase in the number and height of the baffles.The baffle injectors have a greater influence on the amplitude and frequency of the first-order tangential acoustic mode than the baffle blades.

Theoretical analysis of the conical premixed flame response to upstream velocity disturbances considering flame speed development effects

The effect of upstream velocity pertuibations on the response of a premixed flame was investigated in terms of the flame transfer function dependency on excitation frequency.In this study,the assumption of constant flame speed was extended and the effect of flame speed development was considered;i.e.,the flame speed would grow with the time after ignition or with the distance from a flame-holder.In the present study,the kinematics of a conical flame was investigated by linearization of the front tracking equation of flame to uniform and convected fluctuations of the flow velocity and the response was compared with that of a V-shaped flame and the experimental data in the previous studies.The results show that the effect of flame speed development could influence a decreasing gain and increase the phase of the flame response to the uniform velocity oscillations in low and moderate frequencies.Comparing the variations in the gain of flame response upon normalized frequency,show that a conical flame has lower values than the V-flame.In other woods,these flames might be less susceptible to combustion instabilities than the V-flames.Furthennore,the variations in phase of the V-flames responses,which show a quasi-linear behavior with normalized frequency,have higher values than the saturated behavior in phase of the conical flame responses.Also,considering that the flame speed development induces an increase in the gain and phase of the conical flame response to the convected velocity oscillations in certain frequencies;because the developed flame front has longer length in comparison to the flame front in constant flame speed model.Therefore,the flame length may be longer than convective wavelength and the heat release would be generated in different points of the flame;consequently the flow oscillations might exert a stronger impact on the unsteady heat release fluctuations.

Dense core response to forced acoustic fields in oxygen-hydrogen rocket flames

Oscillatory combustion representative of thermo-acoustic instability in liquid rockets is simulated by experiment and LES calculation to investigate the flame behavior in detail.In particular,we focus on how the velocity and pressure fluctuations affect the behavior of the dense oxygen jet,or‘LOx core’.The test case investigated is a high pressure,multiinjector,oxygen-hydrogen combustor with a siren for acoustic excitation.First,the LES calculation is validated by the resonant frequencies and average flame topology.A precise frequency correction is conducted to compare experiment with LES.Then an unforced case,a pressure fluctuation case,and a velocity fluctuation case are investigated.LES can quantitatively reproduce the LOx core shortening and flattening that occurs under transverse velocity excitation as it is observed in the experiments.On the other hand,the core behavior under pressure excitation is almost equal to the unforced case,and little shortening of the core occurs.The LOx core flattening is explained by the pressure drop around an elliptical cylinder using the unsteady Bemoulli equation.Finally,it is shown that the shortening of the LOx core occurs because theflattening enhances combustion by mixing and increase of the flame surface area.