## Текст наукової роботи на тему «Акустична нестійкість в камерах з усередненим потоком і виділенням тепла»

?Electronic Journal «Technical Acoustics» http: // webcenter. ru / ~ eeaa / ejta /

2004, 14 Konstantin I. Matveev

California Institute of Technology, Pasadena, CA, 91125, USA e-mail: Ця електронна адреса захищена від спам-ботів. Вам потрібно увімкнути JavaScript, щоб побачити її.

Vortex-acoustic instability in chambers with mean flow and heat release

Received 30.09.2004, published 18.10.2004

Acoustic instability appearing in chambers with isothermal or reacting mean flow is an important engineering problem. The subject of this work is the instability that is coupled with vortex shedding and impingement, which can also be accompanied by heat release. A reduced-order theory is formulated that includes the chamber acoustics, vortex-structure interaction, and unsteady heat addition. Assuming that acoustic sources are localized in space and time, the kicked oscillator concept is applied. Model results are compared with experimental data. Possible applications for flow control are discussed.

1. INTRODUCTION

Intensive pressure and flow fluctuations in the combustion chambers of rocket motors and similar unstable phenomena in industrial applications involving ducts with isothermal mean flow are important problems in mechanical engineering. In the development stage of rocket motors, practically all of them experience some kind of these instabilities. This effect is very harmful, since it may lead to intensive vibrations, unacceptable for navigation systems and payload, and to enhanced heat transfer, which can result in overheating the structure. In extreme cases, rockets are damaged or destroyed by mechanical or thermal mechanisms related to acoustic-combustion instability. The problem of the acoustic instability in rocket flows attracted a lot of attention by researchers, and extensive theoretical, experimental, and numerical studies were undertaken [1-6].

There is a variety of causes for acoustic instability in the systems with mean flow and heat addition. This paper is concerned only with those that are coupled with vortex shedding occurring inside the chambers. This type of instability can be approximately subdivided into two groups. In the first one, the vorticity impinging on the structure acts as an acoustic disturbance. A scheme of such a process is shown in Fig. 1 (a). There are baffles in the chamber that are analogous to restrictors between propellant segments in actual solid-fuel rocket motors. Vortices are generated on upstream baffles and impinge on downstream restrictors. Acoustic disturbances produced at the moments of vortex collisions feed back to the process of vortex shedding. A closed-loop system is formed that can exhibit self-excited oscillations at the suitable geometry and flow

conditions. This sound generation process is also relevant to one of the noise production mechanisms in ventilation systems.

(A)

Uo ^

> i-

(B) baffles

vortex

burning

\ I / -O / 1 \

Figure 1. General configurations of the chambers prone to acoustic instability:

(A) isothermal flow in a duct with restrictors; (B) vortex-driven instability in dump combustor

The second group of acoustic instabilities, involving vortex shedding, corresponds to the flow with significant unsteady heat release. These phenomena are common to liquid-fuel motors with premixed combustors, as well as to various industrial burners and gas turbines. A schematic view of the process is shown in Fig. 1 (b). A vortex generated at the flameholder (in Fig. 1b, it is a rearward-facing step) consists of cold unburnt reactants from an incoming flow and hot products from a recirculation zone. Upon the impingement of such a vortex on a downstream structure (or a wall of a combustion chamber) or after a certain induction time defined by chemical and hydrodynamic properties, a rapid mixing of hot and cold components takes place, followed by fast heat release due to vortex burning. This sudden heat addition acts as an acoustic disturbance, and the resulting acoustic waves influence the vortex shedding process, creating a system with a feedback similar to the isothermal acoustic instability. The energy released in combustion is typically much higher than that in vortex-structure interaction; therefore, at the favorable phasing between combustion and acoustics, the direct influence on acoustics by vorticity impingement can be neglected.

In both cases, acoustic sources can be considered localized in space and time if the characteristic vortex size is much smaller than the acoustic wavelength and the system dimensions in a longitudinal direction, and if the characteristic time of the vortex collision or combustion is much smaller than the vortex convection time and the acoustic cycle period. The

cold reactants

vortex shedding

hot products

general interaction scheme between the most important processes is depicted in Fig. 2. If flow is isothermal, these processes are vorticity production, vortex impingement, and chamber acoustics. The primary directions of the influences between these phenomena are designated by solid arrows in Fig. 2. If combustion at the vortex impingement becomes important, then an unsteady heat release plays an important role, and additional links between the processes appear (dashed lines in Fig. 2).

Figure 2. Scheme of interactions between acoustics, vorticity, and combustion

Significant progress has been achieved in understanding and simulating vortex-acoustics-combustion interactions [1-6]. However, there is lack of fast and inexpensive methods that allow to model the system dynamics and build effective controllers for real-world systems. Most models are either capable of only qualitatively describing the steady-state characteristics [6, 7], or require complicated computations for determining the system behavior [8, 9]. In order to develop more practical engineering tools for design and active flow control, the construction of reduced-order models, simulating the system dynamics, becomes an important task. An idea of instantaneous burning of localized vortices in combustors with vortex shedding was proposed for modeling purposes [10]. In a research letter [11] the author suggested using a related scheme for acoustic mode excitation in ducts with baffles. In this paper, these approaches are generalized in the model capable of simulating acoustic instabilities in both isothermal-flow systems and those with unsteady heat addition, relevant to rocket flows.

In the following section, a theoretical model is described that represents flow fluctuations via acoustic eigen modes. The causes for acoustic perturbations are the vortex-baffle collisions or the vortex burning. The quasi-steady model for vortex shedding in the oscillatory flow is utilized. The model results are compared with experimental data in Section 3. Possible applications of this theory for flow control are discussed in Section 4.

2. THEORETICAL MODEL

A duct with a mean flow is considered as a general configuration for analyzing acoustic instabilities inside real-world chambers. Examples of the parts of the duct, where all major processes happen, are shown in Fig. 1 for two cases of isothermal flow and flow with heat addition. Only longitudinal acoustics is the subject of this work, and one-dimensional acoustic theory is applied to model disturbed motions. Mach numbers of both mean and oscillating flow components are considered small.

Two locations inside the motor chamber, which are important for the phenomena studied, are the point of vortex formation and the point of vortex impingement on the structure (or its position when it burns abruptly). The first location is associated with an upstream baffle (Fig. 1a) or a rearward-facing step (Fig. 1b). The second location is either the downstream baffle (Fig. 1a) or the point of vortex burning (Fig. 1b), which depends on chemical and hydrodynamic properties of the system.

Disturbed motions in the chamber are represented through the acoustic mode expansion in the following form [12]:

p '(xt) = Xp'n (xt) = po Y.Vn (t) Wn (x), (1)

,(*.,) =?, N (x, t) = 1 ^^ | (2)

where p0 is the mean undisturbed pressure, tfn (t) is the time-varying amplitude of the nth mode, / n (x) is the pressure mode shape, y is the gas constant, and kn is the modal wave number.

The dynamics of the modal time variables will be determined using a one-dimensional wave equation with spatially averaged gas properties over a chamber volume and two driving sources pertinent to the phenomena studied [13]:

d2p a2 d2p = pa2 dF + 2 dQ (3)

~ DF "a ~~ pa aT + pa ^ 7 '(3)

where x is the horizontal coordinate (starting from an upstream end of the duct), a is the speed of sound, p is the gas density, F is the force per unit of mass, and Q is the heat addition rate per unit of volume.

The driving sources on the right-hand side of Eq. (2) model the acoustic disturbances happening at the moments of vortex impingement (or burning). The force F represents the dipole source, which is the most effective radiator caused by vorticity impingement. The unsteady heat release rate Q is the dominant source in reacting flows.

Equations for mode dynamics can be obtained by substituting Eqs. (1) and (2) into Eq. (3), multiplying it by the mode shape, and integrating over a chamber length. Neglecting by the mode non-orthogonality and introducing a modal damping |, dynamics equations for individual modes are derived:

(4)

where El = /; dx .

To model the force F and the heat addition Q, the assumption of a shortness of the

impingement and burning, as well as the localized character of these acoustic sources, will be used. Also, the magnitude of these disturbances should grow with increasing vortex circulation r, which correlates with a vortex size. Thus, the force and heat release can be approximated by the following expressions, involving delta functions in space and time:

can be determined using theoretical, numerical or experimental results for particular situations. In

where u 0 is the mean flow velocity at the plane of the vortex formation (at the upstream baffle or at the step in a combustor).

The coefficient a accounts for a reduced velocity of the vortex convection due its motion along the boundary of the recirculation zone. For example, this coefficient is commonly in the range 0.5-0.6 for the solid-fuel rocket motors with cavities between propellant segments [6].

Substituting Eqs. (5-6) into Eq. (4) and integrating over a chamber length, we find the dynamics equation for the nth mode amplitude in time interval (t, tj + 1):

where CQn = CqY / El and CFn = CpJ / E ^.

Differential equations with forces containing delta functions in time describe dynamics of the systems that belong to the family of kicked oscillators [14]. The system behaves as an ordinary linear oscillator almost all the time except time moments t ,, where jumps in variables occur. The

p = Cp Xr - «(L _ x,) S (1 _ tj),

(5)

Q = Cq Xr-5 (L _ x,) S (t _ t,),

(6)

where summation is carried on the number of vortices shed, L is the location of the vortex impingement (burning) point in the chamber, x, is the position of the, th vortex, and t, is the

moment of impingement (burning) of the jth vortex. Numerical values for coefficients cp and cQ

this study, we will determine them empirically by fitting model results to experimental data. The vortex instantaneous velocity is approximated by the formula

dx ,

- = au 0 (x,) + u '(x ,, t), dt 0 , ,

(7)

(8)

influence of delta function is analogous to the instantaneous increment in the variable velocity [15], while a time derivative of the delta function modifies the variable itself [10]. If the modal damping is small enough, then the following jump conditions are satisfied at time tj:

V "(t j +) -n" (t, -) = c0nyn (x2) r

0n n 2 j

dVn

dt

(T ")

dnn (tj _) = cjL_ (x, ^..

dt

dx

(9)

(10)

To model the process of the vortex generation and separation in the flow with an oscillating component, a quasi-steady hypothesis for the vortex shedding is applied [10]. The growth rate of the vortex circulation at the edge is approximated by integrating shedding vorticity over a boundary layer thickness (Fig. 3) [16]:

dr

dt

f dun \ u ^ dy |

dy

1

2 '

(11)

where us is the velocity at the outer edge of a boundary layer, which is taken as the sum the mean flow velocity and acoustic component, us (t) = u0 + u '(t). The influence of the shed vorticity on the edge velocity is neglected.

Figure 3. Transformation of the boundary layer vorticity into the vortex forming at the edge

By analogy with a steady case, the vortex separation is assumed to occur at the moment t when vortex circulation reaches the critical level proportional to the momentary flow velocity:

r

steady

-u0 dt | 20

u0 D ~ 2St

r,

unsteady V sep

(Tsep):

us (tsep) D

2St

(12)

(13)

where St is the steady-flow Strouhal number and D is the characteristic dimension (usually the diameter of the orifice).

2

s

The quasi-steady vortex shedding hypothesis was verified against experimental data [10, 17]. However, this hypothesis is not universally held in all possible cases of the vortex shedding in unsteady flow, and a caution is needed to apply it for particular cases. To simplify the modeling, another assumption is imposed on the vortex dynamics: its circulation remains constant between the moments of the vortex detachment and impingement.

The formulation of the vortex dynamics sub-model completes the mathematical theory for acoustic instability in chambers involving vortex shedding. The system behavior can now be studied by integrating numerically the dynamics equations for the modes and vortices.

3. EXPERIMENTAL AND MODEL RESULTS

The theory outlined in the previous section contains some non-obvious assumptions, such as spatially and timely localized acoustic sources caused by vortex impingement and burning. To prove that the model has meaning and can be used in practice, a verification of the model results against test data is necessary. For a valid prediction, the accurate identification of the system geometry, hydrodynamics, acoustics, and combustion is required. In this section, three well-defined experimental situations, two dealing with isothermal flow and one with reacting flow, are modeled.

The case of the sound induced by mean flow in the duct with baffles, similar to the arrangement shown in Fig. 1 (a), is modeled based on information found in the paper by Huang and Weaver [18]. They studied flow oscillations inside a duct with baffles, and possibilities for active control of acoustic instabilities. Mean flow velocity was a variable parameter. At certain values of the mean velocity, acoustic eigen modes of the duct were excited, coupled with vortex shedding at the upstream baffle and vortex impingement at downstream baffle. Experimentally obtained, the dominant frequency in the spectrum of the acoustic signal is shown for some values of mean velocity in Fig. 4 by the cross symbols. The natural frequencies of the acoustic eigen modes are represented by dotted lines. There is another characteristic frequency in the system, corresponding to hydrodynamic instability at a baffle. This is the Strouhal frequency, defined as the vortex shedding frequency that occurs in a steady flow:

f0 = StD-, (14)

where St is the Strouhal number, a constant in a wide range of the Reynolds number, and D is the diameter of the orifice at a baffle. For the system considered, the dependence of the Strouhal frequency on a mean flow velocity is shown by a solid line in Fig. 4.

Mean flow velocity [m / s]

Figure 4. Dominant frequency of the sound produced in isothermal flow through the duct with baffles: o - model results, x - test data [18]. Dotted lines - natural frequencies of acoustic modes; solid line - mean-flow Strouhal frequency

At the resonances observed, there was the identifiable integer number of vortices convected between the baffles. In the regions close to intersections of the Strouhal frequency line with acoustic eigen mode frequencies, there was only one vortex. At the mean velocities of about 10 and 12 m / s, there were two and three vortices between the baffles. The attraction of the dominant frequency to the natural frequencies of the system and weak variation of the sound frequency with a mean velocity in these regions is known as the lock-in phenomenon, which is often observed in self-oscillating systems.

The model for simulating this experiment has the parameters identical to those of the actual test system. The first natural frequency is 57 Hz. Higher frequencies are multiple integers of the first one. The six lowest modes are accounted for in the mathematical modeling. The duct is open-ended. The Strouhal number is 0.87. The coefficient a (Eq. 7) for the reduced convection velocity is 0.6. Damping coefficients are calculated from the standard expressions for sound attenuation due to a boundary layer and sound radiation from the open ends [13]. The forcing coefficient cFn is empirically chosen for all modes to be equal 0.1 m-1. More accurate identification of this coefficient would require computational fluid dynamics studies of the vortex impingement process. Since the flow is non-reacting, the other forcing coefficient, cQn, is zero.

Numerical results are obtained by integrating dynamics equations given in Section 2. At the initial time moment, there are no flow disturbances. Due to hydrodynamic instability at the

upstream baffle, vortices are generated, and their impingement on the downstream baffle leads to excitation of the acoustic eigen modes of the duct. Integration is carried out until the system reaches a steady state with well-defined periodic behavior of the system variables.

Numerical results for the dominant frequency in the acoustic signal at resonances are shown in Fig. 4 by the open circle symbols. The model results generally agree with the test data, predicting correctly the lock-in regions. Even the excitation of higher modes at low Strouhal frequency (in the mean velocity range of 10-13 m / s) is captured well by the modeling. One of the possible reasons for the excitation of higher modes, in particularly the third mode in this case, is the remoteness of the mean-flow Strouhal frequency from the closest natural frequencies, while the Strouhal frequency becomes a subharmonic of a higher mode. A fair agreement between model results and test data demonstrates that the theory developed can be used for practical design and control of the sound-vortex interaction in isothermal systems.

Let us now consider a system with heat release, similar to that shown in Fig. 1 (b). Experiments on a dump combustor built at California Institute of Technology were described by Smith [19]. For certain values of the mean flow velocity and the fuel-to-oxidizer ratio, acoustic modes of the system were excited and accompanied by regular vortex shedding and cyclic heat release rate with periods of acoustic oscillations. Heat release had articulate maximums correlated with vortex impingement on a lower wall of the combustor chamber. An example of the spectrum of the acoustic signal measured is given in Fig 5 (a) for a mean flow velocity at the dump plane of 22 m / s. The two sharp peaks, dominating a broadband noise, have frequencies 188 and 457 Hz, corresponding to the first and fourth acoustic eigen modes of the system.

1

0.8 0.6 0.4 0.2 0

Figure 5. Normalized spectra of the sound produced in a rearward-facing step combustor. Dump-plane mean flow velocity 22 m / s; (A) experimental data [19], (b) model results [10]

Modeling of this experiment was accomplished by retaining the heat release forcing term in Eq. (3) [10]. The dipole source due to vorticity impingement is negligible in this situation in comparison with unsteady heat addition. The system parameters selected for modeling are based

on information available for the combustor studied [19, 20]. The model results in a form of the acoustic spectrum at the given mean velocity are shown in Fig. 5 (b). The two dominant components have frequencies 177 and 513 Hz. There is also another noticeable peak, although with a smaller magnitude. The approximate agreement between experimental data and model results, as seen in Fig. 5, manifests the relevance of the model to real combustors. This allows using the theory developed here for preliminary design and analysis of combustion systems prone to acoustic instability coupled with vortex shedding. The discrepancy between model and test data is due to an attempt to describe a very complicated real process with a fairly simple model. Further model developing (and complicating) may improve its accuracy with regard to unstable combustion devices.

The third example of the experiment selected for modeling by our theory has an arrangement different from the previous situations. This case is supposed demonstrate a general applicability of the model to various configurations. Nelson et al. [21] examined a flow excited resonance of the system schematically shown in Fig. 6. The steady grazing flow was provided across a slot backed by a rectangular cavity to form a Helmoltz resonator. Vortices were shed at the upstream lip and impinged on the downstream lip, generating acoustic motions dependent on the cavity acoustic properties.

mean flow

vortex

shedding

Helmholtz

resonator

Figure 6. System arrangement for studying a flow excited Helmholtz resonator [21]

Different from situations previously considered, the primary time variable in this case is the gas particle displacement y in the resonator neck. The dynamics equation for this variable has a form of the oscillator motion with nonlinear damping:

d2 y dt2

dy k dy

dt

+

d0 dt

dy

dt

+ ®0y '|

MS

(15)

where y coordinate is directed downwards, | and k are linear and nonlinear damping coefficients, d0 is the slot width, co0 is the natural frequency of the Helmholtz resonator, p is the driving pressure outside the resonator, M is the inertance of the mass of air in the resonator neck, and S is the area of the neck.

Nonlinear damping term is important at the resonance condition, when particle velocity becomes large. From the system description available [21], it follows that the natural frequency is 605 Hz, the slot width is 0.01 m, the linear damping coefficient is 0.049, the inertance of air in the neck is 22 kg / m4, the neck area is 10-3 m2, and the vortex convection coefficient a (Eq. 7) is 0.27.

The vortex shedding frequency, corresponding to the mean flow velocity, was empirically found as the linear function of the velocity far from resonance:

u

f = Std + A f, (16)

d

where Strouhal number St is 0.12 and the frequency shift A f is 335 Hz.

It was also found that acoustic particle velocity at the neck was much smaller than the mean flow velocity, so the circulation growth (Eq. 11) and a moment of the vortex detachment (Eq. 13) were influenced mostly by mean flow.

The driving pressure term on the right-hand side of Eq. 15 was chosen in the form analogous to Eqs. (5, 6):

P '= cp Xr; 5 (d - xj) 5 (t - tj),

(17)

where cp is the appropriate constant, the x coordinate is directed from the upstream to downstream lip of the resonator neck, the upstream lip being the origin of the x axis; and the other parameters are the same as in Eqs. (5, 6).

Numerical values for two parameters, cp and k, were selected empirically to be equal to 1.3 kg / m and 1.5, respectively. Experimental data and computed results, corresponding to excited steady states, are shown in Fig. 7 for the sound pressure level inside the cavity and for the dominant frequency. The results are given versus a variable parameter - the mean flow velocity outside the resonator. The agreement between test data and model results is good, except for the magnitude of pressure fluctuations at mean flow velocities much smaller that that corresponding to a resonance. This comparison demonstrates the relevance of the model even to the systems different from those upon which the theory was derived (Fig. 1). Other researchers have modeled the same experiment [21] using other approaches: the describing-fun cti on theory [22] and the computational fluid dynamics method based of the kinetic (Boltzmann) equation [23]. Our simple model produces results no worse than those obtained by more complicated methods.

650

? 600

ct 550

0

500

140

135

CQ

T3

130

CL

125

120

16

20 22 24

Velocity [m / s]

_ (A) X X O X 5 ° 6 -

O

16 18 20 22 24 26 28

Velocity [m / s]

(B)

X 5 ® X ff

X X o

O O

26

28

Figure 7. Dominant frequency (a) and sound pressure level (b) of the pressure fluctuations inside the resonator: o - model results, x - test data [21]

4. POTENTIAL APPLICATION FOR FLOW CONTROL

Acoustic instability in real-world devices is usually a harmful phenomenon that needs to be suppressed. Both passive and active control methodologies are used in practice. Since the interactions among vorticity dynamics, acoustics, and combustion are complex, the control of acoustic instabilities poses a challenging task. An introduction of the reduced-order model considered in this paper improves the understanding of the relevant process, and makes it possible to optimize the design and to apply methods of the control theory for a well-defined system.

Passive control for suppressing acoustic instabilities is usually based on the system geometry modification. The objectives are to make the processes generating sound fade and to enhance the acoustic damping. In the general configurations considered here, the first objective is achieved by selecting the proper horizontal locations for the upstream baffle (or a step in dump combustors) and the downstream baffle (or a point of vortex burning), as well as the shapes of these system components. Due to lack of reduced-order models for the sound-producing mechanisms, passive control methods are usually implemented on a costly trial-and-error approach. The model proposed in this work allows incorporating explicitly the influence of the system geometry on the processes of vortex formation and impingement and their interaction with acoustic modes. The other objective of the passive control, an increase of sound attenuation, is achieved by placing

additional baffles, resonators, and acoustic liners in the chamber. However, low effectiveness of passive damping devices in a wide range of operating conditions limits a usefulness of these methods, and shows an importance of the active flow control for suppressing acoustic instabilities.

Active control is based on perturbing the relevant processes in real time. The two most popular active techniques for suppressing acoustic instability are to impose additional sound (usually by means of a loudspeaker) and to apply secondary injections of a fluid in certain points (the chemical composition of the injected fluid is important in reacting flows). Examples of the common implementation of these types of flow control are shown in Fig 8. The variant with a loudspeaker is found to be suitable for suppressing low-amplitude instabilities and convenient for laboratory testing. A properly implemented controller affects the acoustic velocity at the point of the vortex formation, attenuating the shear-layer natural instability [18]. Also, at the vicinity of the downstream baffle, acoustic energy being generated is influenced by the local momentary velocity. The existing control systems tend to ignore determining the details of the mechanisms of acoustic instabilities and concentrate on trial and error ad hoc efforts to control the dynamics. The model developed in this paper allows to include all important mechanisms, such as an externally imposed acoustic field, directly into consideration, providing the knowledge of how the added sound affects the vortex formation, convection and impingement (or burning).

Figure 8. Laboratory arrangements for active control of vortex-induced sound: (a) isothermal flow, (b) reacting flow

In the case of acoustic-vortex-combustion instabilities, more effective and practical way to suppress instability is to vary the fuel-oxidizer ratio of the fluid injected into combustion chambers and to utilize secondary injection ports. Variation of the equivalence ratio can be accounted in the model by changing the magnitude of the heat addition term. Fluid injected into the flow (Fig. 8b) affects both the process of the vortex formation and the amount of heat to be released at the vortex burning.

Combustion systems are often characterized by the hysteretic boundary between stable and unstable operating regimes, i.e. the system state is determined by the history of parameter variation. Pulses of the fuel at the flame-holding step can initiate a fast transition from the unstable to stable condition, with other system parameters kept fixed [24]. This is an example of nonlinear control that may cost less than other control methods, since only a few pulses are needed for the transition, instead of a continuing action of common controllers. The properties of this control approach, dealing with complex underlying physical processes, can be studied in a simple manner using the model developed in this paper.

5. CONCLUSIONS

Acoustic instabilities in ducts, especially those coupled with heat release in combustion chambers, pose a serious technical problem. Understanding and description of these phenomena for the systems with vortex shedding is provided in this work on a base of the reduced-order approach. Model results satisfactorily agree with experimental data in both isothermal and reacting flows. For successful application of the theory, the system acoustics and hydrodynamics, as well as the chemistry when combustion is involved, must be known with high accuracy. The necessary condition for using this model is the compact and time-localized character of acoustic sources, caused by vortex impingement on the structure or vortex burning.

To make the model developed here be usable in routine engineering practice, the well-defined quantified criteria for limitations of the model applicability still have to be formulated. To widen a range of the systems modeled by the theory, the assumptions on time and space compactness of acoustic sources can be relaxed. However, the challenge will be to stay within a reduced-order approach. The explicit description of all relevant processes suggests a straightforward application of the model for flow control. Both passive and active control means can be directly incorporated into mathematical formulation.

REFERENCES

1. Raushenbakh, B. V. Vibratory Combustion. Fizmatgiz, Moscow, 1961.

2. Harrje, D. T., Reardon, F. H. Liquid propellant rocket combustion instability. NASA SP-194,

1972.

3. Natanzon, M. S. Combustion Instability. Mashinostroenie, Moscow, 1986.

4. Flandro, G. A. Vortex driving mechanism in oscillatory rocket flows. J. Propulsion and Power, 1986, 2, 206-214.

5. Culick, F. E. C. Combustion instabilities in liquid-fuelled propulsion systems - an overview. AGARD-CP-450, 1988.

6. Dotson, K. W., Koshigoe, S., Pace, K. K. Vortex shedding in a large solid rocket motor without inhibitors at the segment interfaces. J. Propulsion and Power, 1997, 13, 197-206.

7. Rossiter, J. E. Wind tunnel experiments on the flow over rectangular cavities at subsonic and transonic speeds. Aeronautical Research Council, Report and Memorandum, No. 3438, 1964.

8. Bruggeman, J. C., Hirschberg, A., van Dongen, M. E. H., Wijnands, A. P. J., Gorter, J. Flow induced pulsations in gas transport systems: analysis of the influence of closed side branches. J. Fluids Eng., 1989, 111, 484-491.

9. Hourigan, K., Welsh, M. C., Thompson, M. C., and Stokes, A. N. Aerodynamic sources of acoustic resonance in a duct with baffles. J. Fluids and Structures, 1990, 4, 345-370.

10. Matveev, K. I., and Culick, F. E. C. A model for combustion instability involving vortex shedding. Combust. Sci. and Tech., 2003 175, 1059-1083.

11. Matveev, K. I. Reduced-order modeling of vortex-driven excitation of acoustic modes. Acoust. Res. Let. Online. In press.

12. Culick, F. E. C. Nonlinear behavior of acoustic waves in combustion chambers. Acta Astronautica, 1976, 3, 714-757.

13. Howe, M. S. Acoustics of Fluid-Structure Interactions. Cambridge University Press, Cambridge, 1998..

14. Andronov, A. A., Vitt, A. A., and Khaikin, S. E. Theory of Oscillators. Dover Publications, New York, 1987.

15. Landau, L. D., Lifshitz, E. M. Mechanics. Pergamon Press, Oxford, 1996..

16. Clements, R. R. An inviscid model of two-dimensional vortex shedding. J. Fluid Mech.,

1973 57, 321-336.

17. Castro, J. P. Vortex shedding from a ring in oscillatory flow. J. Wind Eng. Ind. Aerodyn., 1997, 71, 387-398.

18. Huang, X. Y., Weaver, D. S. On the active control of shear layer oscillations across a cavity in the presence of pipeline acoustic resonance. J. Fluids Struct., 1991, 5, 207-219.

19. Smith, D. A. An Experimental Study of Acoustically Excited, Vortex Driven, Combustion Instability within a Rearward Facing Step Combustor. Ph. D. dissertation, Caltech, Pasadena, CA, 1985.

20. Sterling, J. D., Zukoski, E. E. Nonlinear dynamics of laboratory combustor pressure oscillations. Combust. Sci. and Tech., 1991, 77, 225-238.

21. Nelson, P. A., Halliwell, N. A., and Doak, P. E. Fluid dynamics of a flow excited resonance, Part I: experiment. J. Sound Vibr., 1981, 78, 15-38.

22. Mast, T. D., Pierce, A. D. Describing-function theory for flow excitation of resonators. J. Acoust. Soc. Am., 1995, 97 (1), 163-172.

23.Mallick, S., Shock, R., Yakhot, V. Numerical simulation of the excitation of a Helmholtz resonator by a grazing flow. J. Acoust. Soc. Am., 2003 114 (4), 1833-1840.

24. Knoop, P., Culick, F. E. C., Zukoski, E. E. Extension of the stability of motions in a combustion chamber by nonlinear active control based on hysteresis. Combust. Sci. and Tech., 1997, 123, 363-376.

Завантажити оригінал статті: