Ordinary differential equations of the first order linear differential equations complex numbers and linear algebra simultaneous linear differential equations numerical methods the descriptive theory of nonlinear differential equations mechanical systems and electric circuits Fourier series Fourier integrals and Fourier transforms the Laplace transformation partial differential equations Bessel functions and Legendre polynomials applications and further properties of matrices vector analysis the calculus of variations analytic functions of a complex variable infinite series in the complex plane the theory of residues conformal mapping.

Advanced Engineering Mathematics. By C. R. Wylie. Pp. xiv, 813. 1966. (McGraw-Hill.)

Rotating detonation combustors and their similarities to rocket instabilities

Nonlinear, transverse-mode, liquid-propellant-rocket-motor combustion instability is examined with a two-dimensional model. The three-dimensional equations are integrated over the axial direction, for a multi-orifice short nozzle. Nonlinear transverse-wave oscillations in the circular combustion chamber are examined with the primary flow in the axial direction. Turbulent mixing of methane and gaseous oxygen with coaxial injection is analyzed. The combustion has two characteristic times, one for mixing and the other for chemical kinetics, producing a time lag in the energy-release rate relative to pressure. Then, the coupled combustion process and wave dynamics are calculated for a 10-injector chamber with methane and gaseous-oxygen propellants. The linear first tangential mode is imposed initially. Nonlinear triggering occurs; above a critical initial amplitude, the amplitude grows; otherwise, it decays with time. The second tangential mode also develops, and the nonlinear resonance creates a subharmonic ...

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Two-Dimensional Model for Liquid-Rocket Transverse Combustion Instability

The combustion stability of a liquid-propellant rocket engine experiencing a random, finite perturbation from steady-state conditions is examined. The probability is estimated for a nonlinear resonant limit-cycle oscillation to be triggered by a random disturbance. Transverse pressure waves are considered by using a previously published two-dimensional nonlinear pressure wave equation coupled with Euler equations governing the velocity components. The cylindrical combustion chamber is a complex system containing multiple co-axial methane‐oxygen injectors; each co-axial jet is analysed for mixing and burning on its own local grid scheme, with the energy release rate coupled to the wave oscillation on the more global grid. Two types of stochastic forcing for the random disturbance are explored: a travelling Gaussian pressure pulse and an oscillating pressure dipole source. The random variables describing the pulse are magnitude, location, duration and orientation of the disturbances. The polynomial chaos expansion (PCE) method is used to determine the long-time behaviour and infer the asymptote of the solution to the governing partial differential equations. Depending on the random disturbance, the asymptote could be the steady-state solution or a limit-cycle oscillation, e.g. a first tangential travelling wave mode. The asymptotic outcome is cast as a stochastic variable which is determined as a function of input random variables. The accuracy of the PCE application is compared with a Monte Carlo calculation and is shown to be significantly less costly for similar accuracy.

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Stochastic modelling of transverse wave instability in a liquid-propellant rocket engine

Combustion instability (CI) has been persistent in all forms of propulsion since their inception. CI is characterized by pressure oscillations within the propulsion system. If even a small fraction of the dense energy within the system is converted to acoustic oscillations the system vibrations can be devastating. The coupling of combustion and fluid dynamic phenomena in a nonlinear system poses CI as a significant engineering challenge. Drawing from previous analysis, second order acoustic energy models are taken to third order. Second order analysis predicts exponential growth. The addition of the third order terms capture the nonlinear acoustic phenomena (such as wave steepening) observed in experiments. The analytical framework is derived such that the energy sources and sinks are properly accounted for. The resulting third order solution is compared against a newly performed simplified acoustic closed tube experiment. This experiment provides the interesting result that in a forced system, as the 2nd harmonic is driven, no energy is transferred back into the 1st mode. The subsequent steepened waveform is a summation of 2nd mode harmonics (2, 4, 6, 8...) where all odd modes are nonexistent. The current third order acoustic model recreates the physics as seen in the experiment. Numerical experiments show the sensitivity of the pressure wave limit cycle amplitude to the second order growth rate, highlighting the importance of correctly calculating the growth rates. The sensitivity of the solution to the third order parameter is shown as well. Exponential growth is found if the third order parameter is removed, and increased nonlinear behavior is found if it retained and as it is increased. The solutions sensitivity to this term highlights its importance and shows the need for continued analysis via increasing the models

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A Study of Nonlinear Combustion Instability

The Universal Combustion Device Stability™ (UCDS™) Process is the culmination of more than 40 years of research and provides the means to understand the complex dynamics and processes inside any chemical propulsion system, including rockets, turbojets and scramjets. With UCDS, it is now possible to accurately predict the time history of the pressure oscillation amplitudes during operation. This paper will provide an overview of the UCDS process as it is applied to liquid rocket engines. Using the UCDS nonlinear models a series of simulated bomb tests are used to explore the range of dynamic responses for a generic liquid rocket engine. These results reveal a wide range of complex dynamic responses that explain the widely varying instability effects seen in actual liquid rocket tests. When applied to the Corporal liquid rocket, this case study shows that the oscillatory behavior seen in the extensive testing of the Corporal rocket performed by JPL can be matched and clearly explained.

Nonlinear Liquid Rocket Combustion Instability Behavior using UCDS™ Process

In the analysis of rocket combustion stability, there are various sources and sinks of unsteady energy, ranging from propellant response to the interaction of the flow with vortices. As stability models are improved, additional fidelity is required in previously adequate models to reduce the error bandwidth. In this paper, we present a new technique for obtaining the linear nozzle damping exponent and compare it with a historically accepted method. Nozzle damping allows rocket motors and engines to literally exhaust unsteady energy from the interior, and an improved understanding of this process both improves our ability to predict overall stability and design rockets for improved stability. Previous techniques focused on computing the admittance or response of a nozzle to an incident wave, and encapsulating the average wave reflection as a simple function of frequency. In contrast, in the approach presented herein, the combined chamber-nozzle acoustic modes are computed, taking into account the chamber’s steady flow field influence on the acoustics throughout the motor or engine. The flux of unsteady energy through the nozzle plane is computed to derive the nozzle damping exponent. Three sample motors/engines are evaluated to investigate the characteristics of varying length to diameter ratio and to examine the influence of nozzle shape using regular and submerged nozzles.

UCDS Nozzle Acoustic Dynamics and Stability Modeling

Finite Amplitude Nonlinear Waves in Liquid Rocket Combustion Chambers

With the increase in computational resources, time-accurate reacting flow, computational fluid dynamics have improved to the point where periodic, unsteady behavior associated with hydrodynamic instability and acoustic resonance can be accurately modeled. However, computational fluid dynamics solutions, like experiments, only reveal the manifestation of unstable behavior such as pressure fluctuations or vortex generation. They do not readily divulge the sources of energy transfer leading to and sustaining instabilities. Because of this, it can be arduous to correlate design decisions with overall system stability using computational fluid dynamics alone. Combining the well-established unsteady energy transport theory and a choice of viable data analysis techniques, a rigorous approach for analyzing time-accurate computational fluid dynamics solutions is established. Through this approach, the sources of driving and damping can be visualized and quantified. Thereby, the aforementioned issue is rectified an...

Determining Sources of Unsteady Energy Transfer in Time-Accurate Computational Fluid Dynamics

High amplitude pressure waves in rocket engine combustion chambers often exhibit nonlinear effects such as mean pressure shift and shock-like waveforms. These effects both act to cause large mechanical vibrations, thrust variations, internal damage and catastrophic failure of the engine. Understanding these phenomena analytically and experimentally is critical if rocket engines are to be designed reliably. The use of the experimental data in this paper will serve to elucidate which analytical approach is the most appropriate. A closed tube with a length of ten feet was oscillated via a piston, driven by a DC motor, at frequencies near to the fundamental harmonics. The pressure waveform and mean pressure were recorded at the end of the tube and the mean pressure was measured at the middle of the tube. When the tube was excited near to its 1 st and 2 nd modes non-linear features such as shock-like waveforms, cascading of energy to higher modes and mean pressure shift were observed. These results are only possible to predict by using a nonlinear analytical approach. Work by Chester 5 in 1964 successfully predicts the shock-like waveform phenomenon; however, it fails to predict the mean pressure shifts. Recent work by Flandro 6,16-17

Eric J. Jacob

Papers

Nonlinear Liquid Rocket Combustion Instability Behavior using UCDS™ Process

UCDS Nozzle Acoustic Dynamics and Stability Modeling

Finite Amplitude Nonlinear Waves in Liquid Rocket Combustion Chambers

Determining Sources of Unsteady Energy Transfer in Time-Accurate Computational Fluid Dynamics

Forced Resonant Shock Waves and Mean Pressure Shift in a Closed Tube