Pressure-coupled Response Function Research Articles

Numerical study of triggered thermoacoustic instability driven by linear and nonlinear combustion response in a solid rocket motor

Thermoacoustic instability (TAI) has consistently presented challenges to the development of solid rocket motors (SRMs), making the prediction of TAI critically important. Most existing TAI predictions rely on linear instability theory, which is inadequate for predicting certain nonlinear TAI, such as triggered TAI. To address this challenge, this study has constructed the nonlinear response model for the burning rate, known as the nonlinear pressure-coupled response function (PCR). The nonlinear PCR is capable of considering the effects of both frequency and amplitude of pressure oscillations. By integrating the PCR into the computational fluid dynamics framework, this study successfully replicated the nonlinear triggered TAI. When exclusively employing the linear PCR, the model demonstrates typical multi-order resonant modes, and the stability map exhibits either persistent stability or persistent instability, contingent upon the distribution of the linear PCR function. However, by incorporating the nonlinear PCR, this study effectively reproduces nonlinear pulse-triggered instability. This instability arises only when the pulse intensity surpasses the threshold value due to SRM damping. The nonlinear response framework allows for the identification of the instability boundary, facilitating a more comprehensive assessment of SRM performance. This study fills a critical gap in predicting triggered TAI in SRMs, providing insights into nonlinear TAI mechanisms.

ACOUSTIC ADMITTANCE OF AN ALUMINIZED COMPOSITE SOLID PROPELLANT BY LASER DOPPLER VELOCIMETRY AT LOW PRESSURE

The present study involves the use of laser Doppler velocimetry (LDV) to determine the acoustic admittance of an aluminized composite solid propellant at low mean chamber pressure. The aim of this study is to measure the fluctuating velocity component perpendicular to the burning surface of the particles in the gaseous phase above the burning surface of the aluminized composite solid propellant for the imposed acoustic pressure oscillations. To check the accuracy and precision of the mini-LDV used in this study the acoustic mode shape was generated for the imposed acoustic oscillations with amplitudes of 215 and 6 Pa. The results obtained are very satisfying; they show that the predicted acoustic velocity mode shape matches with the generated mode shape using the LDV. The response function of a solid propellant indicates the inclination of the propellant to cause acoustic combustion instability. Both velocity fluctuations and imposed pressure fluctuations are measured experimentally and used to determine the acoustic admittance of the solid propellant and hence deducing the pressure coupled response function from acoustic admittance. The probe volume of the laser is maintained at a constant distance of 800-900 μm from the burning surface by a servomechanism using a linear actuator. The incorporation of the linear actuator dramatically increases the number of particles passing through the laser probe volume. The linear actuator is controlled using a microcontroller and the servomechanism is operated under an open loop system. A high-pressure window bomb is designed and fabricated. Low-frequency ranges (140-280 Hz) of the longitudinal acoustic modes are investigated in this paper. Tests were conducted at a mean chamber pressure of 0.5 MPa. To generate the acoustic pressure oscillations a rotary valve is designed and fabricated. The rotary valve generates a standing wave of the characterizing frequency depending upon the length of the chamber. Tests are conducted for two aluminized composite solid propellants. The results obtained are in good agreement with the literature for the pressure coupled response function at these mean chamber pressures.

AP/HTPB combustion response simulation with a new equivalent linear system method

In the study of solid rocket motor (SRM) combustion instability, modeling and analyzing the combustion response of the solid propellant is a laborious task. We propose a new idea, an equivalent linear system method, for modeling solid propellant combustion response with three steps: first, simulating the transient combustion process under pulse pressure excitation for only one CFD case; then, treating the combustion process as a black-box and obtaining the transfer function of the combustion process from the CFD result; finally, predicting and analyzing the combustion response based on the transfer functions. For the AP/HTPB sandwich propellant withp¯=3MPa, the amplitude, duration, and direction of excitation pulses are studied. The new method is also verified, with a maximum relative error of 4.3% for burning rate response under p˜≤1MPa (as high as one-third of the pressure DC). Applicability of this new method is demonstrated for another two AP/HTPB sandwich propellants with different configuration under p¯=8MPa and p¯=10MPa, showing that the system-simulated combustion responses of burning surface temperature and burning rate are consisted well with the CFD results and the maximum relative error of burning rate response is 2.1%. Asymmetric combustion response for positive and negative pressure perturbations is a new finding based on the CFD and simulation results. Besides, a new approach using dual transfer functions is proposed to ensure high accuracy for the simulation of solid propellant combustion. The pressure-coupled response function is predicted by this method for the sandwich propellant under p¯=5MPa and p˜=20kPa within a wide continuous frequency range (0–3000 Hz). The comparison with CFD results indicates that the relative bias of Rp is less than 2%. With the proposed method, the combustion response to arbitrary pressure excitations can be simulated with high accuracy, high time-resolution, and high speed. The method shows great advantage and potential in simulating and predicting combustion instability.

Numerical analysis on combustion instabilities in end-burning-grain solid rocket motors utilizing pressure-coupled response functions

The combustion instability has been one of severe problems suffered by solid rocket motors (SRMs) for a long term. This paper proposes a burning rate model utilizing the pressure-coupled response function for description of the spatiotemporal burning of AP-HTPB composite propellant grains in SRMs. The model is successfully incorporated to the axisymmetric internal ballistic simulation by the source terms of gas-phase governing equations. The prediction of propellant combustion connected with acoustic pressures can be realized by means of an in-house high-order numerical solver, in which the numerical fluxes are reconstructed by the fifth-order WENO scheme and the viscous terms are discretized by a sixth-order compact scheme. The thermoacoustic combustion instabilities in the longitudinal modes are triggered by a burning rate pulse imposed to the steady flow. The analysis indicates that the pressure oscillations grow as a primary symptom of combustion instabilities. The influences of several factors on production of the instability symptom are discussed. It is shown that the pressure-coupled response function, the pressure index, and the reaction heat of the propellant, and the specific heat ratio of the burnt gas promote the pressure oscillation growth process evidently, but the magnitude and the imposed region of burning rate pulses have less effects.

Modeling and analysis of triggering pulse to thermoacoustic instability in an end-burning-grain model solid rocket motor

Occurrences of pulse-triggered instabilities in combustion chambers always trouble solid rocket motors (SRMs) in application. This paper performs numerical simulation of the pulse-triggered nonlinear instabilities in the end-burning-grain SRM, in use of the pressure-coupled response function to describe the spatiotemporal burning rate of AP-HTPB composite propellant grains. A high-order numerical solver realizes the cooperation of the burning rate model into the axisymmetric internal ballistic simulation of SRMs via the source terms in gas-phase governing equations. The analysis of high-amplitude pressure oscillations, which are the primary symptom of nonlinear thermoacoustic instabilities, provides the insight into the nature of the longitudinal instability growth process after the pulse triggering. The change rate of pressure fluctuation over the pressure fluctuation in the chamber is independent of the pulse intensity and triggering position. The growth characteristics of pressure oscillations for the present combustor system is neither affected by the pressure pulse intensity nor by the imposed regions in the flow field, while the higher pressure-coupled response function of the propellant promotes the growth process evidently. The present study is expected for the guidance of grain design in SRMs.

Computational Fluid Dynamics Simulation of Combustion Instability in Solid Rocket Motor : Implementation of Pressure Coupled Response Function

<p class="NomenclatureClauseTitle">Combustion instability in solid propellant rocket motor is numerically simulated by implementing propellant response function with quasi steady homogeneous one dimensional formulation. The convolution integral of propellant response with pressure history is implemented through a user defined function in commercial computational fluid dynamics software. The methodology is validated against literature reported motor test and other simulation results. Computed amplitude of pressure fluctuations compare closely with the literarture data. The growth rate of pressure oscillations of a cylindrical grain solid rocket motor is determined for different response functions at the fundamental longitudinal frequency. It is observed that for response function more than a critical value, the motor exhibits exponential growth rate of pressure oscillations.</p>

Improved method of measuring pressure coupled response for composite solid propellants

Pressure coupled response is one of the main causes of combustion instability in the solid rocket motor. It is also a characteristic parameter for predicting the stability. The pressure coupled response function is usually measured by different methods to evaluate the performance of new propellant. Based on T-burner and “burning surface doubled and secondary attenuation”, an improved method for measuring the pressure coupled response of composite propellant is introduced in this article. A computational fluid dynamics (CFD) study has also been conducted to validate the method and to understand the pressure oscillation phenomenon in T-burner. Three rounds of tests were carried out on the same batch of aluminized AP/HTPB composite solid propellant. The experimental results show that the sample propellant had a high response function under the conditions of high pressure (~11.5 MPa) and low frequency (~140 Hz). The numerically predicted oscillation frequency and amplitude are consistent with the experimental results. One practical solid rocket motor using this sample propellant was found to experience pressure oscillation at the end of burning. This confirms that the sample propellant is prone to combustion instability. Finally, acoustic pressure distribution and phase difference in T-burner were analyzed. Both the experimental and numerical results are found to be associated with similar acoustic pressure distribution. And the phase difference analysis showed that the pressure oscillations at the head end of the T-burner are 180° out of phase from those in the aft end of the T-burner.

Study on Pulse Triggering Combustion Instability in a Combustion Chamber

Based on a T-burner, pulse triggering, and ‘burning surface doubled and secondary attenuation’ algorithm, different kinds of solid propellants’ oscillation combustion features in the acoustic field were studied experimentally and numerically. In the experimental part, the effectiveness of the pulse excitation modes was explored, and the combustion instability characteristic of different solid propellants was researched. In the numerical part, the difference of pressure pulse and mass release rate pulse was studied, and performance parameters of propellant, such as burning rate and flame temperature, how to effect on the pressure coupled response function was discussed. The results shown that pulse source burning rate must be intense enough to excite pressure oscillation. The thermo-acoustic oscillation combustion mechanism in T-burner is analyzed. The uneven oxidant particle and flue mixing of composite propellant caused the mass release rate periodic variation, that respond the pressure are the main reasons of thermo-acoustic coupling effect.

Ultrasonic Measurement of the Pressure-Coupled Response Function for Composite Solid Propellants

Transient phenomena associated with solid-propellant combustion are examined by means of an oscillatory burner designed to create pressure oscillations through the use of a rotary nozzle. The oscillatory burner is designed to operate at pressures up to 20.7 MPa (3000 psi), with the target operating pressure achieved through prepressurization with an inert gas (helium) and manipulation of constant and variable-area nozzle diameters. A spherically focused 1-MHz ultrasound transducer is pulsed at 2.5 kHz through the bottom of the propellant to obtain the thickness-time profile of the propellant, which is then converted to a burning rate by taking the time derivative. Two piezoelectric pressure transducers simultaneously record the pressure-time history of each experiment. The burning rate and pressure are then used to calculate the pressure-coupled response of each test. The experiments presented here were motivated by three primary goals. The first was to investigate the effect of propellant composition on the response function. The second was to evaluate effect of mean pressure on the response of propellants, and the third was to investigate the response of a propellant within a plateau region. These experiments targeted mean pressures of 2.0, 5.0, and 12.5 MPa at frequencies ranging from 20-200 Hz.

Evaluation of Ultrasound Technique for Solid-Propellant Burning-Rate Response Measurements

The development of an ultrasound technique for precisely measuring the instantaneous regression rate of a solid-rocket propellant under transient conditions is reviewed. The technique is used to measure the burning-rate response of several solid propellants to an oscillatory chamber pressure with a frequency of up to 300 Hz. This measurement is known as the propellant's pressure-coupled response function and is used as an input into rocket stability prediction models. The ultrasound waveforms are analyzed using cross-correlation and other digital signal processing techniques to determine burning rate. Digital methods are less prone to bias and offer greater flexibility than other techniques previously used. The resulting data are corrected for compression effects. The effects of a changing thermal profile on the measurement are discussed. Other phenomena that may corrupt the measurement, such as surface roughness, are also covered. Results of the experiments are compared to data from two other measurement techniques: a T-burner and a magnetic flowmeter.

Solid-Propellant Combustion Response Function from Direct Measurement Methods: ONERA Experience

Usually, the pressure-coupled response function of a solid propellant is determined using the traditional indirect methods such as T-burner or modulated exhaust jet technique. A few direct methods, based on wave propagation through the solid-propellant grain, exist: microwave and ultrasound techniques. There is also the magnetohydrodynamic technique derived from magnetic velocimetry through the gas phase. The theoretical approaches of each method are explained and are coupled with an acoustic analysis of the pressure oscillations. The response function is determined using these direct and nonintrusive methods for the Ariane V solid propellant (HTPB‐ AP‐ Al) and a nonmetalized propellant. Results obtained for composite propellants, such as that of the Ariane V, show the accuracy of each method and underline the advantages or disadvantages of each method as far as ONERA’ s experience can carry it out. Direct methods are nonintrusive but are limited because of their principle itself. There is a frequency limit above which the measurement cannot be performed. Finally, the objectives of future ONERA work are proposed.

Leading-Edge Flame Detachment: Effect on Pressure-Coupled Combustion Response

An experiment was conducted to establish the critical role of a detailed aspect of the e amelet structure in composite propellants in determining the overall combustion response to pressure oscillations during combustor instability. Nomenclature ab = velocity of sound of gases in T-burner gc = units conversion factor L = length of interior cavity of T-burner Rp = pressure-coupled response function r = burning rate Sb = cross-sectional area of cavity of T-burner Sc = burning surface area of propellant sample c = combustion amplie cation d = burner damping during burning 1 = exponential growth rate during burning, decay rate in pulse test 2 = decay rate after burnout 2(t1) = decay rate corrected to frequency during burning s = density of solid propellant

Examination of chemical approaches to stabilizing composite-propellant combustion

Acoustic-mode combustion instability has long plagued the solid-propellant industry, and increased requirements for reduced-smoke propellants, with the elimination of metal oxide participate products that damp acoustic oscillations, is expected to exacerbate this problem. One strategy for alleviating the problem involves identifying approaches to decrease a major source of acoustic energy, the transient burning-rate response of the propellant to pressure oscillations. In this study, a model was developed and used to define potential effects of varying either oxidizer or fuel ablation activation energy on the pressurecoupled response function. Based on calculations with unimodal-oxidizer-size formulations, it appears that increasing oxidizer ablation activation energy at a fixed value of binder ablation activation energy or increasing binder ablation activation energy at fixed oxidizer ablation activation energy can result in a significant reduction of this pressure-coupled burning rate response.

Linear burning rate dynamics of solids subjected to pressure or external radiant heat flux oscillations

Linearized analysis of burning solids subjected to pressure or external radiant heat flux oscillations results in relatively simple expressions for the burning rate response, particularly when combined with quasisteady gas and surface zone assumptions. In this article, a linear expression for the radiant heat flux response function Rq as a function of primary experimental parameters is obtained. The implications of this expression and its relationship to the pressure coupled response function Rp are examined. In particular, the linearized effects of the mean radiant heat flux level and in-depth absorption on Rq are investigated. The linear response to a series of radiant pulses is also presented to suggest an alternate method of experimentally measuring Rq. The effects of nonlinearitie s are investigated using numerical calculations. It is shown that, in general, the relationship between Rp and Rq is more complicated than a constant scaling factor. The results demonstrate that in-depth absorption of the thermal radiant energy in the solid significantly affects Rq for many practical conditions. Furthermore, even when the equivalence principle holds in the steady case, an equivalent change in the initial temperature does not have the same effect on Rq, in general, as an equivalent mean radiant flux. Also, the mean radiant flux is seen to have a significant effect on Rq. An attempt is made throughout this article to further clarify the relationship and differences between the flame modeling (FM) and the Zeldovich-Novozhilov (ZN) phenomenological approaches in the prediction of Rq and Rp.

Direct measurement of high frequency, solid propellant, pressure-coupled admittances

This paper presents an experimental method that is capable of directly measuring solid propellant pressurecoupled responses at the high frequencies associated with tangential mode instabilities inside solid propellant rocket motors. The method utilizes a magnetic flowmeter to measure the velocity oscillation above a burning propellant surface simultaneously with a pressure oscillation measurement within an externally excited combustion chamber. A magnetic flowmeter burner was designed and constructed to evaluate this method of pressurecoupled response measurement. Response measurements were obtained for two formulations of AP/HTPB composite propellant at pressure oscillation frequencies of 4000 and 8000 Hz. The measurement data displayed repeatable trends in both the real and imaginary parts of the pressure-coupled response function.

Application of the impedance tube technique in the measurement of the driving provided by solid propellants during combustion instabilities

This paper describes the adaptation of the impedance tube technique for the measurement of solid propellant response functions. The experimental setup consists of a tube with a disc-shaped solid propellant sample placed at one end and an additional propellant sample at the side wall of the tube at some distance downstream of the end propellant sample. When the pressure coupled response function is measured only the disc-like sample of the propellant is needed while determination of the velocity coupled response function requires the use of both propellant samples. During an experiment, an acoustic driver located at the opposite end of the tube is utilized to establish a standing wave of a desired frequency within the impedance tube and an array of pressure transducers mounted at preselected locations along the impedance tube walls is utilized to measure the continuously varying wave structure within the tube throughout the course of the experiment. Upon the completion of the experiment, the measured data together with solutions of the impedance tube wave equations are utilized to determine the propellant response function. The paper discusses the different data reduction procedures that have been developed for the determination of both the pressure and velocity coupled response functions. Furthermore, the paper describes the results of pressure coupled response measurements for both aluminized and nonaluminized propellants and the results of a theoretical study that show that the impedance tube method could also be used in the measurement of the velocity coupled response function.

MHD Measurement of Acoustic Velocities in Rocket Motor Chambers

A linear analysis of longitudinal acoustic waves (either forced or self-induced) at or near the fundamental frequency in solid rocket chambers revealed that the acoustic velocity phase angle relative to the head-end pressure at the chamber midpoint is strongly dependent on the real part of the overall pressure-coupled response function and the imaginary part of the overall velocity-coupled response function. A flowmeter based on Faraday's Law was devised to demonstrate the feasibility of such velocity measurements under rocket motor conditions. The interaction of an externally-excited magnetic field with the unsteady velocity of hightemperature combustion gases results in a corresponding unsteady electrical potential whose magnitude is proportional to the flow velocity. The system is readily calibrated since electrical output does not depend on knowledge of ionization level and electrical conductivity (within certain limits) and measurements are independent of propellant type. Motor firing data has been used to deduce the real part of the pressure-coupled response from the phase angles between head-end pressures and midpoint velocities and the imaginary part of the velocity-coupled response from the phase angles between headand nozzle-end pressures. The overall response functions measured in the device are compatible to the form used in stability prediction models.

Microwave Measurement of the Solid-Propellant Pressure-Coupled Response Function

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