In this chapter, the complexities of combustion chamber phenomena found in liquid bipropellant thrust chambers are treated. We describe combustion behavior and its analysis in general terms, together with several types of combustion instability with their resulting undesirable effects, and discuss semiempirical remedies on how to avoid these effects. An objective here is to operate at very high combustion efficiencies while preventing any occurrence of disruptive or destructive combustion instabilities. Thrust chambers need to operate with stable combustion over a wide range of operating conditions. Further treatment of these subjects can be found in Refs. 9–1 to 9–7.

Liquid propellant combustion is very efficient in well‐designed thrust chambers. Combustion efficiencies of between 95 to 99.5% are typical, in contrast to turbojets or ordinary furnaces which may range from 50 to 97%. Such high efficiencies stem from the very fast reaction rates at high combustion temperatures resulting from the thorough mixing of fuel and oxidizer (from proper injection flow distributions, uniform atomization, and turbulent gas diffusion). Losses arise from nonuniform mixture ratios resulting in incomplete burning or inadequate mixing. Combustion efficiencies can be well below 95% in very small bipropellant thrust chambers or small gas generators because their injectors lack sufficient injection orifices or elements for proper mixing.

9.1 COMBUSTION PROCESS

For understanding combustion processes, it is helpful to divide the combustion chamber into a series of discrete zones as shown in Fig. 9–1. The configuration depicted has a flat injector face with many small injection orifices that introduce both fuel and oxidizer liquids into the combustion chamber and this results in many discrete liquid propellant streams, or thin sprays or sheets. The relative thickness, behavior, and transition of each discrete zone in the axial direction is strongly influenced by specific propellant combination, operating conditions (pressure, mixture ratio, etc.), and by injector design and chamber geometry. The boundaries between the zones shown in Fig. 9–1 are really neither flat nor steady surfaces. They can be undulating or movable irregular boundaries with local changes in velocity, temporary bulges, locally intense radiation emissions, and/or variable temperatures. Table 9–1 shows all major interacting physical and chemical processes that may occur in the chamber. This table is a modification of tables and data in Refs. 9–1, 9–2, and 9–7.

Combustion behavior is highly propellant injection dependent. With a hypergolic propellant combination there is an initial chemical reaction in the liquid phase as droplets of fuel impinge on droplets of oxidizer. Experiments show that such contacts can create local mini‐explosions with enough energy release to suddenly vaporize thin layers of fuel and oxidizer locally at the droplet's contact face—there immediately follows a vapor chemical reaction and droplets blow‐apart and breakup. Small local explosions have been documented (Refs. 9–8 and 9–9). In contrast, when a cryogenic fuel like hydrogen comes from cooling the thrust chamber, the injected hydrogen is gaseous and relatively warm (60 to 240 K); here, there are no hydrogen droplets to evaporate.

Combustion processes are typically analyzed as steady flow, but in reality this is not truly so. When observing any one location within the chamber, one finds turbulent flows with local fluctuations in pressure, flow, temperature, density, mixture ratio, and/or radiant emission. Some of the processes listed in the table occur sequentially, while others may occur simultaneously. Not all listed processes happen with each propellant. Three‐dimensional turbulence consistently forms in all parts of the combustion chamber and in the nozzle. Most combustion also generates a great deal of noise.

9.2 ANALYSIS AND SIMULATION

In analyzing the combustion process and its instabilities it has been convenient to divide the associated acoustical characteristics into their linear and nonlinear parts. A number of computer simulations with linear analyses have been developed over the past 55 years to understand combustion processes in liquid propellant combustion devices and to predict combustion oscillation frequencies. Nonlinear behavior (i.e., why does a disturbance suddenly change an apparently stable combustion to unstable?) is not well understood and has not been properly simulated. Mathematical descriptions require a number of simplifying assumptions to permit solutions (see Refs. 9–2 to 9–5, , and 24). Good models exist for isolated phenomena such as propellant droplet vaporization (see Refs. 9–1 and 9–10), burning in gaseous environments, and/or for steady‐state gas flows under heat release; some programs also consider turbulence and film cooling effects. The thermochemical equilibrium principles mentioned in Chapter 5 also apply here.

In analytical models, the following phenomena are usually not treated or are greatly simplified: cross flows; nonsymmetrical gradients; flow unsteadiness (such as time variations in local temperatures, local velocities, or local gas composition); local thermochemical reactions at off‐design mixture ratios and at different kinetic rates; vaporization enhancement by acoustic fields (see Ref. 9–8); uncertainties in sprayed‐droplets spatial and size distribution; and/or drag forces on the droplets. While computer program results give valuable information regarding any particular design and are useful guides in interpreting actual test results, by themselves they are not sufficient to define such designs, select specific injector patterns, or predict combustion instabilities occurrences.

Most computer software known to the authors is suitable only for steady‐state flow descriptions, usually at a predetermined average mixture ratio and chamber pressure. However, during engine starts, thrust changes (such as throttling), and stopping transients, local and overall mixture ratios and pressures may change sufficiently to make routine analyses of these transient conditions relatively inadequate.

As previously stated, combustion is strongly influenced by injector design. The following are some injection parameters which influence combustion behavior: injector spray or jet patterns and their impingement locations, injector hole sizes and/or hole distribution, chamber/injector geometry, angle of jets or injection sheets, and liquid injection pressure drops. Also of influence are: pressurant gas saturation, droplet evaporation, mixture ratio distributions, pressure and/or temperature gradients near the injector, and initial propellant temperature. Presently, attempts to analyze all these effects together require advanced computational methods.

Computational fluid dynamics (CFD) techniques allow for the most comprehensive description of interrelated fluid dynamic and thermodynamic phenomena. They may model the time history of all parameters and may include many nonlinear effects. For complex geometries the information is tracked with a multitude of discrete locations to account for changes in gas composition; thermodynamic conditions; equilibrium reactions; phase changes; viscous or nonviscous flow; one‐, two‐, or three‐dimensional flows; and steady‐state or transient conditions. CFD has been applied to resonant cavities in injectors and/or chambers and to the flow of burning gases through turbines.

9.3 COMBUSTION INSTABILITY

When rocket combustion processes are not well controlled, combustion instabilities may grow and very quickly cause excessive pressure‐induced vibrational forces (that may break engine parts) or excessive heat transfer (that may melt thrust chamber parts). The engineering aim here must be to prevent all occurrences of such damaging instabilities by proper design and to maintain reliable operation (see Refs. , and 9–11 to 9–13). Much progress has been made in understanding and avoiding combustion instabilities; since about 1975, hardly any thrust chambers have experienced destructive instabilities.

References 9–2 to 9–5 and 14 describe analyses of combustion instabilities in thrust chambers as practiced in the United States. References 9–4 and 9–13 present the same information but as practiced in Russia; there are some differences in the assumptions, analysis, and testing, but their basic approach is essentially similar. Any new or modified thrust chamber and/or rocket engine is tested to demonstrate that combustion instabilities will not occur. Combustion stability for hybrid propulsion systems is discussed in Chapter 16 and for solid propellant motors in Chapter 14.

Table 9–2 lists the principal types of combustion generated vibrations encountered in liquid rocket thrust chambers (see Refs. 9–2 and 9–9). Admittedly, liquid propellant rocket thrust chamber combustion is never perfectly smooth because minor fluctuations of pressure, temperature, and velocity are always present. When these fluctuations interact with the natural frequencies of the propellant feed system (with or without vehicle structure) or with the chamber acoustics, large periodic superimposed oscillations develop. In normal rocket propulsion practice, smooth combustion is said to occur when pressure fluctuations during operation do not exceed ± 5% of the mean chamber pressure. Combustion that generates larger pressure fluctuations, as measured at chamber wall locations and occurring at completely random intervals is called rough combustion. Unstable combustion, or combustion instability, displays organized oscillations occurring at well‐defined intervals with pressure peaks that may be maintained, may increase, or may die out. These periodic peaks represent fairly large concentrations of vibratory energy and can be easily recognized against the random‐noise background using high‐frequency pressure measurements (see Fig. 9–2).

Chugging, the first type of combustion instability listed in Table 9–2, stems mostly from the elastic nature of the feed systems and vehicle's structures and/or from the imposition of propulsive forces upon the vehicle. Chugging in an engine or thrust chamber assembly may occur at a test facility or during flight, especially with low‐chamber‐pressure engines (100 to 500 psia). It may originate from propellant pump cavitation, gas entrapment in propellant flows, tank pressurization control fluctuations, and/or vibration of engine supports and propellant lines. It may also be caused by resonances in the engine feed system (such as oscillating bellows inducing periodic flow fluctuations) or by the coupling of structural and feed system frequencies.

When both the vehicle structure and the propellant liquid in the feed system have similar natural frequencies, then a coupling of forces occurs that may not only maintain but also strongly amplify existing oscillations. Propellant flow rate disturbances, usually at 10 to 50 Hz, give rise to such low‐frequency longitudinal combustion instabilities, producing longitudinal vibration modes in the vehicle. Pogo instability refers to a vehicle's long‐feed pipe instability since it is similar to a pogo jumping stick motion. Pogo instabilities can occur in the large, long propellant feed lines of large vehicles such as space launch vehicles or in ballistic missiles. See Refs. 9–14 and 9–15.

Avoiding objectionable engine–vehicle coupled oscillations is best accomplished during initial vehicle design, in contrast to applying “fixes” later, as had been the case with rocket engines of older large flight vehicles. Analytical methods exist for understanding most vibration modes and damping tendencies of major vehicle components, including propellant tanks, tank pressurization systems, propellant flow lines, engines, and basic vehicle structures. Figure 9–3, a simplified spring–mass model of a typical two‐stage vehicle, illustrates the complexity of the analytical problem. Fortunately, most vibrational characteristics of the rocket assembly can be substantially controlled by introducing damping into major components or subassemblies. Techniques for damping pogo instabilities include the use of energy absorption devices in fluid flow lines, perforated tank liners, special tank supports, and properly designed engine, interstage, and payload support structures.

A partially gas‐filled, spherical, small pogo accumulator attached to the main oxidizer feed line has been used as an effective damping device. It was used in the oxidizer feed line of the Space Shuttle main engine (SSME) between the oxidizer booster pump and the main oxidizer pump (see Figs. 6–1 and 6–11).

Along with the bending or flexing of pipes, joints, bellows, or long tanks, the dynamic characteristics of propellant pumps can further influence pogo‐type vibrations, as examined in Ref. 9–16. Pogo frequencies will also change as propellant is consumed and as remaining mass of propellant in the vehicle changes.

Buzzing, an intermediate frequency type of instability, seldom derives from pressure perturbations greater than 5% of the mean in the combustion chamber and is usually not accompanied by large vibratory energies. It is more often an annoying noise than a damaging effect, although the occurrence of buzzing may sometimes initiate high‐frequency instabilities. It is characteristic of a coupling between the combustion process and the flow in a portion of the propellant feed system. Buzzing initiation is thought to originate from the combustion process itself. Acoustic resonances of the combustion chamber with some critical portion of the propellant flow system, sometimes originating in a pump, promote continuation of these buzzing effects. This type of instability seems to be more prevalent in medium‐size engines (2000 to 250,000 N thrust or about 500 to 60,000 lbf) than in larger engines.

The third type of instability, screeching or screaming, produces high frequencies (4 to 20 kHz) and is the most perplexing and common feature in new engine development. Many liquid rocket engines and solid propellant motors experience some high‐frequency instability during their developmental phase. Since energy content increases with frequency, this type can be the most damaging, capable of destroying an engine in less than 1 sec. Once encountered, it is often difficult to prove that any newly incorporated “fixes” or improvements will render the engine “stable” under all launch and flight conditions. Screeching may be treated as a phenomenon solely related to the combustion chamber and not generally influenced by feed systems or structures.

High‐frequency instabilities occur in two basic modes, longitudinal and transverse. The longitudinal mode (sometimes called organ pipe mode) propagates along axial planes of the combustion chamber, and its pressure waves are reflected at the injector face and at the converging nozzle entrance section. The transverse modes propagate along planes perpendicular to the chamber axis and can be broken down into tangential and radial modes. Transverse mode instabilities predominate in large liquid rockets, particularly in the vicinity of the injector. Figure 9–4 shows pressure distributions at various time intervals in a combustion chamber of cylindrical cross section that is encountering transverse mode instabilities. Two kinds of wave form have been observed in tangential vibrations; one can be considered a standing wave because it remains in a fixed position while its pressure amplitude fluctuates; the other is a spinning or traveling tangential wave with an associated rotation of the whole vibratory system—this waveform can be visualized as maintaining constant amplitude while the wave rotates. Combinations of transverse and longitudinal modes may also occur and their frequency can also be estimated. See Refs. 9–2 to 9–6.

Screeching is believed to be predominantly driven by acoustical‐energy‐stimulated variations in droplet vaporization and/or mixing, by local detonations, and by acoustic changes in combustion rates. Thus, with any assisting acoustic properties, once triggered, high‐frequency combustion instabilities can rapidly induce destructive modes. Invariably, any distinct boundary layer may disappear and heat transfer rates may increase by an order of magnitude causing metal melting and wall burn‐throughs, sometimes within less than 1 sec. Here tangential modes appear to be the most damaging, with heat transfer rates during an instability often increasing 2 to 10 times and instantaneous pressure peaks about twice as high as with stable operation.

One source that possibly triggers high‐frequency pressure‐wave instabilities is a rocket combustion phenomenon called popping. Popping is an undesirable random high‐amplitude pressure disturbance that arises during the steady‐state operation of rocket engines that use hypergolic propellants. These “pops” exhibit some of the characteristics of a detonation wave. The pressure rise times are a few microseconds and the pressure ratios across the wave can be as high as 7:1.

Some combustion instabilities are induced by pulsations in the liquid flow originating at the turbopumps. Unsteady liquid flows may result from irregular cavitation at the leading edge of inducer impellers or main pump impellers (Ref. 9–16). Also, when an impeller's trailing edge passes a rib or stationary vane in the volute, a small pressure perturbation always results in the liquid as it travels downstream to the injector. These two types of pressure fluctuation can be greatly amplified if they coincide with the natural frequencies of combustion induced vibrations in the chamber. Also, liquid propellant feed systems may generate oscillations under certain conditions. See Ref. 9–17. Ground stability studies with liquid‐oxygen/liquid‐methane thrust chambers are reported in Refs. 9–18 and 9–19.

An estimate for the natural combustion gas frequencies can be determined from the wavelength l and the distance traveled per cycle together with the acoustic velocity a (see Eq. 3–10). The frequency (number of cycles per second) is

9–1

where k is the specific heat ratio, R^′ the universal gas constant, the effective average molecular mass of the hot chamber gases, and T the local average gas absolute temperature. The wavelength l depends on the type of vibrational mode, as shown in Fig. 9–4. Smaller chambers give higher frequencies.

Table 9–3 lists estimated vibrational frequencies for one version of the French Vulcain HM 60 rocket thrust chamber; it operates with liquid hydrogen and liquid oxygen propellants and has a vacuum thrust of 1008 kN, a nominal chamber pressure of 10 MPa, and a nominal mixture ratio of 5.6 (see Ref. 9–20). Values in the table are based on acoustic measurements at ambient conditions (with corrections for an appropriate sonic velocity correlation); since the chamber has a shallow conical shape with no discrete converging nozzle section, any purely longitudinal vibration modes would be weak; in fact, no pure longitudinal modes have been detected.

Figure 9–5 shows time‐sequenced diagrams of frequency–pressure–amplitude measurements taken with the Vulcain HM 60 rocket engine during the first 8 sec of a static thrust chamber test while operating at off‐nominal design conditions. Chugging can be seen at low frequencies (up to 500 Hz) during the first few seconds; the action around 1500 Hz is attributed to the natural resonance frequency of the oxygen injector dome structure where a high‐frequency pressure transducer was mounted. The continued oscillations observed at about 500 and 600 Hz are likely resonances associated with the feed system. The frequency–pressure–amplitude diagram shown in Fig. 9–5 for a liquid propellant engine is very similar to that shown in Fig. 14–6 for a solid propellant rocket motor.

An investigation of combustion stability in liquid‐oxygen/liquid‐methane thrust chamber (5,000 lbf thrust) is reported in Ref. 9–17 and a transverse combustion instability analysis is found in Ref. 9–18. Reference 9–19 uses a particular analytical model to simulate combustion instabilities in liquid rocket engines.

Control of Instabilities

Control (or really complete elimination) of instabilities is an important task during the design and development phases of a rocket engine. Designers rely on both prior experience with similar engines and on tests of new experimental engines; analytical tools are also utilized. The design selection has to be proven to be free of instabilities in actual experiments over a wide range of transient and steady‐state operating conditions. Some of these experiments may be accomplished with subscale rocket thrust chambers that have similar injectors, but most tests have to be done on the full‐scale engine or thrust chamber.

Design features needed to control instabilities are different for each of the three types of vibrations described in Table 9–2. Chugging is usually avoided by eliminating resonances in the propellant feed system and/or its couplings with elastic vehicle structures or any one of their components. Increased injection pressure drops and the addition of artificial damping devices in the propellant feed lines have been successfully used. Chugging and other acoustical instabilities can sometimes be traced to the natural frequency of a particular feed system component that is free to oscillate, such as a loop of piping that vibrates, an injector dome, or a bellows whose oscillations cause a pumping effect.

With the choice of the propellant combination usually fixed early in the planning of a new engine, the designer may alter combustion feedbacks (depressing the driving mechanism) by altering injector details (such as changing the injector hole pattern, hole sizes, or increasing the injection pressure drop), or alternatively by increasing acoustical damping within the combustion chamber. Of the two methods, the second has been favored in recent years because of its greater effectiveness—it is also better understood. This leads to the application of injector face baffles, discrete acoustic energy absorption cavities, and combustion‐chamber liners, and/or to other changes in injector design from trial‐and‐error approaches.

Injector face baffles (see Fig. 9–6) have been a widely accepted anti‐vibration remedy since the 1960s for overcoming or preventing high‐frequency instabilities mostly in medium and larger size rocket engines. Baffle design is predicated on the assumption that, along with its driving source, the most severe instability oscillations are found in or near the atomization zone at the injector‐end of the combustion chamber. Baffles minimize any influential coupling and amplification of gas dynamic forces within the chamber particularly with transverse oscillations. Obviously, these baffles must be strong, have excellent resistance to the high combustion temperatures (they are usually cooled internally by the propellants), and must protrude enough into the chamber to be effective, yet not so far as to act as a separate combustion chamber with its own acoustical characteristics. The identical injector of Fig. 9–6 has been used in several Russian engines. It has nine circular sets of injector elements operating at chamber pressures above 3500 psi and with relatively high area ratios for a booster engine (producing an I_s above 330 sec. in a vacuum). In addition to the baffle pattern, Fig. 9–6 shows other hardware remedies for combustion instabilities such as a large oxygen‐rich gas cavity which reduces feed system instability and a fuel manifold pattern which results in different mixture ratios at different burn locations thus minimizing any high frequency instabilities. Chapter 8.4 of Ref. 9–16 shows additional information on these injector elements (in Fig. 8–4–20) and the injector face (in Fig. 8–4–19).

As stated, all combustion processes generate abundant noise and large thrust chambers and large plumes produce acoustically painful levels. Strong noise may also induce combustion instabilities the exact mechanisms of which are still controversial (see Ref, 2).

Various energy absorption or vibration damping mechanisms already exist in thrust chambers (although damping by wall friction in the combustion chambers is considered to be relatively insignificant). The nozzle acts as a main damper for longitudinal mode oscillations; wave reflections at a convergent nozzle entrance depart significantly from those predicted for an ideal closed end. The principal damping mechanism affecting wave propagation in a transverse plane is the combustion itself. The great volumetric change in going from liquid to burned gases and the momentum imparted by the fast moving gas to the particles (solid and/or liquid) both constitute damping mechanisms in that they absorb energy fromthe combustion. Unfortunately, the combustion process itself may generate a great deal more pressure‐driven oscillatory energy than is absorbed by its inherent damping mechanisms.

Acoustical absorbers are applied usually as discrete cavities along or inside the combustion chamber walls near the injector end. See Refs. 9–1 and 9–23. These cavities act as Helmholtz resonator arrays (i.e., as enclosed cavities with small passage entries) that remove vibratory energy which would otherwise act to maintain the pressure oscillations. Figure 9–7 shows the application of discrete cavities (interrupted slots) at the “corner” or periphery of an injector face. A corner location usually minimizes fabrication problems, and it is the only location in a combustion chamber where pressure antinodes exist for all resonant modes of vibration (including longitudinal, tangential, radial, and combinations of these). Velocity oscillations are also minimal at this location, which favors absorber effectiveness; hence transverse modes of instability are best damped by locating absorbers at corners. Figure 9–7 also shows a Helmholtz resonator cavity and its working principles in simplified form. For each resonator element, the mass of gas in the orifice together with the volume of gas behind it form an oscillatory system analogous to the spring–mass system shown in the figure (see Refs. 9–18 and 9–23).

Absorption cavities designed as Helmholtz resonators placed in or near the injector face offer relatively high absorption bandwidths and energies per cycle. Such resonators dissipate energy twice each cycle because jets are formed upon inflow and outflow. Modern design practices favor acoustic absorbers over baffles. The rocket engine shown in Fig. 8–2 has acoustic absorption cavities in the chamber wall at a location next to the injector.

The resonance frequency f of a simple Helmholtz cavity can be estimated as

9–2

Here, a is the local acoustic velocity, A is the restrictor area [], and other symbols are as shown in Fig. 9–7. ΔL is an empirical factor between 0.05 and 0.9L to allow for any additional oscillating gas mass. It varies with the L/d ratio and the edge condition of the restricted orifice (sharp edge, rounded, chamfered). Resonators in thrust chambers are tuned or designed to perform their maximum damping at predicted frequencies.

Small changes in injector geometry may cause unstable combustion to become stable and vice versa. New injectors, therefore, use geometry designs of proven, stable prior units with the same propellants. For example, the individual pattern of concentric tube injector elements used with gaseous hydrogen and liquid oxygen (shown in Fig. 8–4d) are likely to be more stable when the hydrogen gas is relatively warm and its injection velocity is at least 10 times larger than that of the liquid oxygen.

There are other combustion‐driven vibration remedies besides baffles and resonant cavities. They include higher injector pressure drops or higher injection velocities, avoidance of resonances from certain key engine component structures, changes of critical injector dimensions, and changing or modifying the propellants. References 9–2 and 9–6 describe several other methods, such as cooled flame holders, critical dimensions of injector elements, varying the number of injector elements and their distribution over the chamber cross section, making the mixture ratio nonuniform near the face of the injector, and/or adding temporary consumable baffles, which are effective only during starting. Small solid particulates in the reaction gas may cause some damping in nozzle oscillating flows.

In summary, the designer needs to (1) be familiar with and use data from prior successful engines and simulation programs to establish key design features in order to estimate all likely resonances of the new engine and its major components, (2) design the feed system and structure to avoid these identified resonances, (3) use an injector design robust enough to provide appropriate mixing and propellant dispersion and be resistant to disturbances, and if needed, (4) include tuned damping devices (such as cavities) to overcome acoustic oscillations. To validate that a particular thrust chamber is stable, it is necessary to test it over the entire range of likely operating conditions without encountering instabilities. Analyses are needed (e.g., Ref. 9–24) to determine the maximum and minimum likely propellant temperatures, the maximum and minimum probable chamber pressures, and the highest and lowest mixture ratios, using a propellant budget such as shown in Section 11.1. Determining these ranges then establishes the variations of test conditions for the experiments. Presently, because of an improved understanding, the amount of testing needed to validate stability has been greatly reduced compared to that required 30 to 50 years ago.

PROBLEMS

1. For a certain liquid propellant thrust chamber the following data are given:

   Chamber pressure	68 MPa
   Chamber shape	Cylindrical
   Internal chamber diameter	0.270 m
   Length of cylindrical section	0.500 m
   Nozzle convergent section angle	45°
   Throat diameter and radius of wall curvature	0.050 m
   Injector face	Flat
   Average chamber gas temperature	2800 K
   Average chamber gas molecular weight	20 kg/kg‐mol
   Specific heat ratio	1.20

   Assume the gas composition and temperature to be uniform in the cylindrical chamber section. Stating any other assumptions that may be needed, determine the approximate resonance frequencies in the first longitudinal, radial, and tangential modes.
2. Discuss how the three frequencies from Problem 1 will change with changes of combustion temperature, chamber pressure, chamber length, chamber diameter, and throat diameter.
3. Discuss why heat transfer increases during combustion instabilities?
4. Prepare a list of steps for undertaking a series of tests to validate the stability of a new pressure‐fed medium sized liquid bipropellant rocket engine. State your assumptions.
5. Estimate the resonant frequency for a set of each of nine cavities similar to Fig. 9–7. Here, the chamber diameter , the slot width is 1.0 mm, and the width and height of the cavity are each 20.0 mm. The walls separating the individual cavities are 10.0 mm thick. Assume , , and .

REFERENCES

1. 9–1. K. K. Kuo, Principles of Combustion, 2nd ed., John Wiley & Sons, Hoboken, NJ, 2005.
2. 9–2. V. Yang and W. Anderson (Eds.), Liquid Rocket Engine Combustion Instability, Vol. 169 of Progress in Astronautics and Aeronautics, AIAA, 1995, in particular Chapter 1, F.E.C. Culick and V. Yang, “Overview of Combustion Instabilities in Liquid Propellant Rocket Engines.”
3. 9–3. M. S. Natanzon and F. E. C. Culick, Combustion Instability, Progress in Astronautics and Aeronautics, Vol. 222, AIAA, Reston, VA, 2008.
4. 9–4. W. Sirigano, A. Merzhanov, and L. De Luca, Advances in Combustion Science: In Honor of Ya. B. Zel'dovich, Progress in Astronautics and Aeronautics Series, V‐173, American Institute of Aeronautics and Astronautics, 1997.
5. 9–5. D. T. Harrje (Ed.), “Liquid Propellant Rocket Combustion Instability,” NASA SP‐194, U.S. Government Printing Office, No. 3300–0450, 1972; NASA NTRS Doc. ID 19720026079; http://hdl.handle.net/2060/19720026079
6. 9–6. G. P. Sutton, History of Liquid Propellant Rocket Engines, AIAA, 2006, Chapter 4.10, “Combustion and Vibrations,” and Chapter 8.4, “NPO Energomash.”
7. 9–7. V. Yang, M. Habiballah, J. Hulka and M. Popp (Eds.), Liquid Rocket Thrust Chambers: Aspects of Modeling, Analysis and Design, Progress in Astronautics and Aeronautics (server), Vol. 200, American Institute of Aeronautics and Astronautics, Reston, VA, 2004; K. Kobayashi et al., “Studies on Injection‐Coupled Instability for Liquid Propellant Rocket Engines,” AIAA Paper 2015‐3843, Orlando, FL, 2015.
8. 9–8. B. R. Lawver, “Photographic Observations of Reactive Stream Impingement,” Journal of Spacecraft and Rockets, Vol. 17, No. 2, Mar.—Apr. 1980, pp. 134–139.
9. 9–9. M. Tanaka W. Daimon and I. Kimura, “Explosion Phenomena from Contact of Hypergolic Liquids,” Journal of Propulsion and Power, Vol. 1, No. 4, 1985, pp. 314–316; doi: 10.2514/3.22801; B. R. Lawver, “Photographic Observations of Reactive Stream Impingement,” Journal of Spacecraft and Rockets, Vol. 17, No. 2, Mar.–Apr. 1980, pp. 134–139; doi: 10.2514/3.57719.
10. 9–10. R. I. Sujith, G. A. Waldherr, J. I. Jagoda, and B. T. Zinn, “Experimental Investigation of the Evaporation of Droplets in Axial Acoustic Fields,” Journal of Propulsion and Power, Vol. 16, No. 2, Mar.–Apr. 2000, pp. 278–285; doi: 10.2514/2.5566.
11. 9–11. P. Y. Liang, R. J. Jensen, and Y. M. Chang, “Numerical Analysis of the SSME Preburner Injector Atomization and Combustion Process,” Journal of Propulsion and Power, Vol. 3, No. 6, Nov.–Dec. 1987, pp. 508–513; doi: 10.2514/3.23018.
12. 9–12. M. Habiballah, D. Lourme, and F. Pit, “PHEDRE—Numerical Model for Combustion Stability Studies Applied to the Ariane Viking Engine,” Journal of Propulsion and Power, Vol. 7, No. 3, May–Jun. 1991, pp. 322–329; doi: 10.2514/3.23330.
13. 9–13. M. L. Dranovsky (author), V. Yang, F. E. C. Culick, and D. G. Talley (Eds.), Combustion Instability in Liquid Rocket Engines, Testing and Development; Practices in Russia, Progress in Astronautics and Aeronautics, Vol. 221, AIAA, Reston, VA, 2007.
14. 9–14. B. W. Oppenheim and S. Rubin, “Advanced Pogo Analysis for Liquid Rockets,” Journal of Spacecraft and Rockets, Vol. 30, No. 3, May–Jun. 1993, pp. 360–373; doi: 10.2514/3.25524.
15. 9–15. K. W. Dotson, S. Rubin, and B. M. Sako, “Mission Specific Pogo Stability Analysis with Correlated Pump Parameters,” Journal of Propulsion and Power, Vol. 21, No. 4. Jul.–Aug. 2005, pp. 619–626; doi: 10.2514/1.9440.
16. 9–16. T. Shimura and K. Kamijo, “Dynamic Response of the LE‐5 Rocket Engine Liquid Oxygen Pump,” Journal of Spacecraft and Rockets, Vol. 22, No. 7, Mar.–Apr. 1985, pp. 195–200; doi: 10.2514/3.25731.
17. 9–17. J. C. Melcher and R. L. Morehead, “Combustion Stability Characteristics of the Project Morpheus Liquid‐Oxygen/Liquid‐Methane Main Engine,” AIAA Paper 2014‐3681, July 2014.
18. 9–18. K. Shipley and W. Anderson, “A Computational Study of Transverse Combustion Instability Mechanism,” AIAA Paper 2014‐3679, Cleveland, OH, Jul. 2014.
19. 9–19. J. Pieringer, T. Settelmayer, and J. Fassel, “Simulation of Combustion Instabilities in Liquid Rocket Engines with Acoustic Perturbation Equations,” Journal of Propulsion and Power, Vol. 25, No. 5, 2009, pp. 1020–1031.
20. 9–20. E. Kirner, W. Oechslein, D. Thelemann, and D. Wolf, “Development Status of the Vulcain (HM 60) Thrust Chamber,” AIAA Paper 90‐2255, Jul. 1990.
21. 9–21. Guidelines for Combustion Stability Specifications and Verification Procedures for Liquid Propellant Rocket Engines,” CPIA Publication 655, Chemical Propulsion Information Agency, John Hopkins University, Jan. 1997.
22. 9–22. V. Yang, D. D. Ku, M. L. R. Walker, and L. T. Williams, Daniel Guggenheim School of Aerospace Engineering, Georgia Institute of Technology, “Liquid Oxygen (LOX)/Kerosene Rocket Engine with Oxidizer –Rich Preburner, Staged Combustion Cycles”, for NASA Marshall Apace Flight Center/Jacobs Technology, December 10, 2013.
23. 9–23. T. L. Acker and C. E. Mitchell, “Combustion Zone–Acoustic Cavity Interactions in Rocket Combustors,” Journal of Propulsion and Power, Vol. 10, No 2, Mar.–Apr. 1994, pp. 235–243; doi: 10.2514/3.23734.
24. 9–24. J. Garford, B. Barrewitz, S. Rari, and R. Frederic, “An Analytical Investigation Characterizing the Application of Single Frequency/Acoustic Modulation for High Frequency Instability Suppression,” AIAA Paper 2014–3679, Cleveland, OH, July 2014.