Gary A. Flandro

Solid propellant acoustic admittance corrections

The acoustic behavior of solid rocket propellants is conveniently expressed in terms ofthe specific acoustic admittance of the burning surface. Numerous measurements of the admittance have been made in T-burners, and it is generally assumed that the data can be used directly in estimates of rocket motor acoustic stability. However, as this paper shows, the propellant response to incident acoustic waves is influenced by the orientation of the propellant surface to the waves. If there is a tangential component of acoustic velocity, then viscous effects give rise to the generation of a traveling shear wave which modifies the apparent propellant response. This wave is out of phase with the velocity fluctuations and obviously represent an energy loss to the system. It is demonstrated that the shear wave is strongly influenced by the mean flow due to propellant combustion; for larger values of burning surface Mach number, the speed of propagation approaches that of the combustion gases. Thus admittance data from T-burners with the propellant samples arranged with normal wave incidence must be corrected for the shear effects if they are to be used in motor calculations wherein acoustic modes with grazing incidence are of interest. For the same reason it is essential to interpret carefully T-burner data secured in devices employing extended or cup-shaped grains, since part of the propellant produces shear waves and thus exhibits a different effective admittance. The corrections are shown to be of the order of the mean flow Mach number; that is, they are of the same order as the real part of the admittance and must not be ignored in motor calculations. It is demonstrated that inclusion of the corrections brings about a marked improvement in agreement between the classical linearized theory of acoustic combustion instability and experimental observations of the phenomenon.

Vortex generated sound in cavities

Cavities in which geometric disturbances are present may produce acoustic waves due to the shedding of vortices from the disturbance in the range 300<Re<!04. A simple model is examined which represents a feedback mechanism between the vortex shedding process and the acoustic modes produced by the vortices. It is shown that the growth of acoustic waves is greatest when the characteristic frequency of vortex shedding is commensurate with a natural acoustic frequency of the cavity. For the model assumed, one nonlinear aspect of the phenomenon is the excitation of harmonics of the primary mode. Comparison with experiments are favorable in predicting growth rates.

Energy balance analysis of nonlinear combustion instability

To be of practical value, analytical models for nonlinear rocket motor combustion instability must adequately represent: 1) steep fronted waves, 2) limit cycle operation, and 3) triggering phenomena. Inclusion of all these effects in an approximate analysis based on perturbation expansion procedures requires retention of terms to at least the third order in the perturbation parameter representing the system amplitude. In this paper, the acoustic energy balance method is extended into the nonlinear regime and combined with a simplified geometrical representation of the wave structure to greatly simplify this procedure. A practical approximate model for axial pressure fluctuations in a tubular rocket motor results. Effects of nonlinear combustion and nonisentropic energy losses in the steep wave fronts are represented. Since the relative amplitudes of the Fourier components that comprise the traveling shock waves may be assumed to remain effectively fixed near amplitude limiting conditions, great mathematical simplifications accrue. The behavior of the system is governed by a simple polynomial expression which gives the rate of change of the composite system amplitude. The coefficients are integral expressions taken over the chamber volume and its bounding surfaces. The familiar exponential growth rate model appears as the first-order (linear) limiting case. The model demonstrates all of the nonlinear characteristics observed in motor data and shows promise as an easily used diagnostic tool. It has been successfully employed to simulate actual experimental results from pulser tests in inert chambers and from sonic end-vent burner tests.

Nonlinear Rocket Motor Stability Prediction: Limit Amplitude, Triggering, and Mean Pressure Shift

High-amplitude pressure oscillations in solid propellant rocket motor combustion chambers display nonlinear effects including: 1) limit cycle behavior in which the fluctuations may dwell for a considerable period of time near their peak amplitude, 2) elevated mean chamber pressure (DC shift), and 3) a triggering amplitude above which pulsing will cause an apparently stable system to transition to violent oscillations. Along with the obvious undesirable vibrations, these features constitute the most damaging impact of combustion instability on system reliability and structural integrity. The physical mechanisms behind these phenomena and their relationship to motor geometry and physical parameters must, therefore, be fully understood if instability is to be avoided in the design process, or if effective corrective measures must be devised during system development. Predictive algorithms now in use have limited ability to characterize the actual time evolution of the oscillations, and they do not supply the motor designer with information regarding peak amplitudes or the associated critical triggering amplitudes. A pivotal missing element is the ability to predict the mean pressure shift; clearly, the designer requires information regarding the maximum chamber pressure that might be experienced during motor operation. In this paper, a comprehensive nonlinear combustion instability model is described that supplies vital information. The central role played by steep-fronted waves is emphasized. The resulting algorithm provides both detailed physical models of nonlinear instability phenomena and the critically needed predictive capability. In particular, the true origin of the DC shift is revealed.

Incorporation of Nonlinear Capabilities in the Standard Stability Prediction Program

Predictive algorithms now in general use cannot characterize high-amplitude pressure oscillations that are frequently observed in solid propellant rocket motor combustion chambers. In fact, programs such as the Standard Stability Prediction (SSP) code are based on a linear theory, which has serious shortcomings. Therefore, it is necessary to address both correction of the flawed linear theory and incorporation of models to allow prediction of important nonlinear effects. These include: 1) limit cycle behavior in which the pressure fluctuations may dwell for a considerable period of time near their peak amplitude, 2) elevated mean chamber pressure (DC shift), and 3) a triggering amplitude above which pulsing may cause an apparently stable system to transition to violent oscillations. Culick’s wellestablished nonlinear model provides useful guidance in dealing with the system limit cycle transition. It is demonstrated in this paper that his calculations represent the classical steepening mechanism by which the wave system evolves from an initial set of standing acoustic modes into a shock-like, traveling, steep-fronted wave. However, a very important missing element is the ability to predict the accompanying mean pressure shift; clearly, the program user requires information regarding the maximum chamber pressure that might be experienced during operation of the motor, as well as the peak amplitudes reached by the pressure oscillations. Recent theoretical work has resulted in a firm foundation upon which to build the required predictive capabilities. These are described in detail, and it is demonstrated that the new theory yields results that are in excellent agreement with experimental data.

UCDS Nozzle Acoustic Dynamics and Stability Modeling

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.

Irregular Burning: On the Origin of the DC Shift

*The irregular burning (or DC shift) effect, characterized by unexpected and sometimes very large changes in the mean operating pressure, has been an oft-observed and muchfeared feature of nonlinear combustion instability in solid propellant rockets. It poses an obvious threat to the structural integrity and performance of the motor. Elevated pressures modify the burning rate, local surface mean flow Mach number, and the apparent response function of the propellant. It is necessary that the physical mechanism of the DC shift be fully understood and accurately modeled if useful predictive/interpretive computational tools for combustion instability are to be constructed. Previous attempts to understand the DC shift mechanism were based on ad hoc velocity coupling mechanisms involving flowreversal, rectification of burning surface unsteady cross-flow velocity, and acoustic erosivity effects. For the most part these mechanisms are not only experimentally unsupportable, but also are unnecessary in accounting for irregular burning. A new model based on a new nonlinear stability analysis is demonstrated in this paper. The correct mechanism is revealed in a nonlinear system mass balance calculation. Only known and measurable properties of the propellant and motor geometry are required in evaluating the results. The acoustic admittance of the propellant is the single system parameter requiring experimental closure. Experimental data from several programs involving a wide range of motor configurations and propellants serve to guide the analysis and to establish its validity. It is demonstrated that the effect of the pressure shift mechanism is greatly strengthened when nonlinear interactions are accounted for. Multiple harmonics appear as the wave system approaches finite amplitude. Their presence explains cases in which a depression rather than the usual increase of the mean pressure is observed. The validity of the irregular burning predictions described in this paper is established by direct comparison to experimental data.

Thrust Oscillations in Large Solid Rocket Boosters

*† Large solid rocket boosters frequently exhibit undesirable oscillations in thrust and associated fluctuations in chamber pressure. These are related to the classical combustion instability phenomenon in which there exists a positive net unsteady energy input from a complex balance of gain and loss mechanisms. Recent improvements in the analytical framework for combustion instability show the origins of the driving effects that tip the energy balance. In particular effects of unsteady vortex shedding are found to be a major feature of these problems. A new nonlinear combustion instability model is used to highlight the features of the key driving mechanisms. The new algorithm allows estimation of the limit amplitude as well as the linear stability features of the system. Of most importance is the demonstration that thrust oscillation effects can be minimized by application of simple changes in the combustion chamber configuration. A combined analytical/computational approach is described that allows the formulation of effective corrective procedures.

Hydrodynamic Stability Analysis of Solid Rocket Motors with Arbitrary Grain Design

Solid rocket motor combustion stability analysis has historically focused on the linear acoustic stability problem. Acoustic instability presumes that the perturbed flowfield is irrotational and compressible. Recent efforts by Abu-Irshaid, Majdalani, and Casalis (AbuIrshaid, E. M., Majdalani, J., and Casalis, G., “Hydrodynamic Stability of Rockets with Headwall Injection,” Physics of Fluids, Vol. 19, No. 2, 2007, pp. 024101-11) and Chedevergne, Casalis and Majdalani (Chedevergne, F., Casalis, G., and Majdalani, J., “Biglobal Linear Stability Analysis and DNS Investigation of the Flow Induced by Wall Injection,” AIAA Paper 2007-5796, July 2007) have demonstrated that hydrodynamic instability, based on a complementary rotational incompressible perturbed flowfield, can be incorporated into the linear stability analysis and significantly impact the predicted stability of large L/D solid rocket motors (SRMs). The intrinsic perturbed fluid motions connected with hydrodynamic instability can play a major role in inducing large acoustic and subsequent thrust oscillations in long segmented SRMs. As in acoustic stability analysis, hydrodynamic stability analysis requires the computation of mode shapes corresponding to the geometric shape of the motor cavity of interest. Software and Engineering Associates, Inc. (SEA) has presented several articles detailing the derivation and computation of acoustic mode shapes, and herein presents a similar approach and results for the computation of hydrodynamic mode shapes using the ARPACK eigensolver.

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

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