Numerical evaluation of acoustic characteristics and their damping of a thrust chamber using a constant-volume bomb model

Jianxiu QIN,Huiqiang ZHANG,Bing WANG

School of Aerospace Engineering,Tsinghua University,Beijing 100084,China

1.Introduction

Combustion instabilities have been experienced in many development programs of liquid rocket engines,which are manifested as transient large-amplitude pressure oscillations in the thrust chamber exceeding around 10%of the averaged chamber pressure.Based on the characteristic frequency of pressure oscillations,combustion instabilities can be classified as low-frequency(chugging),intermediate-frequency(buzzing),and high-frequency(acoustical)modes.1High-frequency combustion instability generally results from thermalacoustic coupling,which is the most destructive for its extensive damage to the chamber and injector face.1,2

Research has shown that high-frequency combustion instability can generally occur when oscillatory energy supplied by unsteady combustion heat release is sufficiently greater than the loss of oscillatory energy damped in the chamber.1Acoustic combustion instability can be retrained by increasing damping,as well as by decreasing or breaking down thermalacoustic coupling for a concerned acoustic mode.Thus,it is important to determine the acoustical modes and their damping characteristics for a thrust chamber.The popular method for predicting acoustic characteristics of a combustion chamber is to solve linear or nonlinear pressure wave equations,3–7in which acoustic modes can be obtained by the Helmholtz analysis.For a linear model,the pressure disturbance grows without a bound and is unphysical,while nonlinear models can obtain the limit cycle behavior in real situations.5–7However,both linear and nonlinear models require an accurate definition of the mean flow and response functions,which are limited to apply to cases with complex chamber geometries.The Finite Element Method(FEM)can be employed to analyze acoustic fields in complex geometries by solving the Helmholtz equation with a Neumann boundary condition.8–10However,it is difficult for the FEM to handle the flow and combustion process occurring in a chamber,which in fact has an important effect on acoustic characteristics.

The detailed processes of flow,combustion,and propagation of pressure wave can be solved based on the Computational Fluid Dynamics(CFD),which is somehow superior to the Helmholtz analysis of the FEM.Both linear and nonlinear disturbances can be handled in this method,which can also predict the effects of geometry variations and mean flow on acoustic characteristics.Using the CFD method,pressure oscillations have to be excited in a chamber first,and then acoustic characteristics can be obtained by analyzing the recorded time series of the pressure oscillations.Therefore,artificial disturbance models are usually employed in CFD to excite pressure oscillations,which have been developed by different researchers known as models of oscillating velocity disturbance,11–13mass flow rate disturbance,14energy disturbance15,and pressure disturbances16–22as well as pressure pulse.20–22They have then been successfully applied to analyze chamber acoustic properties and acoustic damping of various passive devices by researchers such as Abdelkader16and Taro et al.,17and to stimulate combustion instability by Kim,18Grenda,19Habiballah,20Zhuang,21and Urbano et al.22However,it must be indicated that artificial disturbance is not indispensable for self-sustained combustion instabilities in a combustion chamber predicted by CFD.23–25

In summary,there are two main approaches to obtain pressure oscillations in a combustion chamber using the CFD method:one is to excite decaying oscillations by an artificial disturbance model,and the other is to realize self-excited combustion instability without artificial perturbations.Based on such pressure oscillations,the acoustic characteristics and the phenomenon of combustion instability in a combustion chamber can be investigated numerically.However,because it is hard to achieve self-excited combustion instability,artificial disturbances are more feasible to evaluate the acoustic characteristics and their damping in a thrust chamber,and often used in industries.For the existing artificial disturbance models,such as the models of mass flow rate,energy,velocity,and pressure,a specific acoustic mode is needed,and a resonantmode oscillation is then trigged.The amplitudes of pressure pulse models are limited to a small value,which are capable to excite a specific acoustic mode,usually a single-frequency mode.In practice,an artificial disturbance method that can simultaneously excite the multi-acoustic modes of a chamber is a pressing need to determine which acoustic mode is the easiest to be excited and which one is the most difficult to be attenuated in the acoustic modes in a given chamber.Therefore,an artificial model must be applied with a large-amplitude pressure pulse and then drive the nonlinear acoustics process of pressure propagations.However,all the existing disturbance models do not possess all of the above traits,which will be further analyzed in Section 2 of the present paper.

Thus,a numerical constant-volume bomb model for a highamplitude pressure pulse is proposed in this paper.The nonreactive turbulent flows in a chamber are numerically simulated by CFD.Multi-frequency pressure oscillations are excited by the present numerical constant-volume bomb model imposed on a limited region in the chamber.Acoustic modes and their corresponding damping characteristics are analyzed by both the Fast Fourier Transformation(FFT)analysis and the half-power bandwidth method.Furthermore,the effects of the forced geometrical regions on the excited pressure oscillations are further discussed.

2.Numerical method

Acoustic properties in a typical configuration of a thrust chamber in a liquid rocket engine are numerically investigated in this paper.The chamber is initially filled with quiescent air with a temperature of 298 K and a pressure of 0.1 MPa.A pressure pulse is then imposed on the mean pressure in a small region,and then fluid flows and pressure oscillations in the chamber are induced.Such turbulent flows with pressure oscillations are numerically simulated by using Unsteady Reynolds-Averaged Navier-Stokes(URANS)equations,and the acoustic properties of the chamber can be obtained by analyzing the excited pressure oscillations.Based on the k-ε twoequation turbulent model,the governing equations in a general form for the turbulent flows in the chamber are written as follows23:

where φ,Γφ,Sφand ujrepresent the conservation variables,the convective flux coefficient,the source terms,and the velocity in the jth direction,respectively;ρ,t,xjdenote density,time and coordinate axis in the jth direction in the Cartesian coordinate system,respectively.These variables are shown in details in Table 1.

In Table 1,p,T,k,ε,and Yirepresent pressure,temperature,turbulent kinetic energy,turbulent dissipation rate and mass fraction of the ith species,respectively;λ denotes heat conductivity coefficient,while cpis specific heat at constant pressure;μedenotes the effective viscous coefficient,which contains the laminar viscous coefficient μ and the turbulentviscous coefficient μt.The detailed expressions are shown as follows:

Table 1 Variables,coefficients,and source terms in Eq.(1).

where A1and A2are constants,taken as 1.457×10-5and 110,respectively.

The expression of Gkis shown as follows:

The other parameters are turbulence model coefficients,shown in Table 2.

The governing equations are discretized by the finite volume method,in which the diffusion and convection terms are discretized by a second-order central difference scheme and a second-order upwind scheme,respectively.The numerical dispersion as well as numerical dissipations are minimized by these schemes.The Semi-Implicit Method for Pressure-Linked Equations(SIMPLE)is employed to solve the discretized equations.26Time marching is approximated by an explicit first-order Eulerian scheme.To guarantee a computation stability,the time step is specified as 5×10-7s,but it is set as 1×10-8s when a constant-volume bomb is imposed on the steady flow in the chamber.The grid resolution is specified as 1 mm in the simulation.Further refinement of the mesh does not improve the prediction results of eigenfrequencies of the acoustic mode and their damping characteristics significantly.

Adiabatic and non-slip conditions are adopted to the injector face and sidewall of the thrust chamber.An open outletcondition is applied for the nozzle exit.For that,the outlet pressure is set as the ambient pressure,while the velocity components are set equal to those of the logical inside neighbor vertexes.A choked condition and supersonic flow are not achieved in the nozzle for the cases investigated in this paper,and an open outlet condition means that the pressure at the nozzle exit is set as the ambient pressure.The turbulent flow and pressure oscillations described by the above models are induced by the initial pressure pulse,so how to realize such a suitable pressure pulse is the key problem,which is described in the following section.

Table 2 Turbulence model coefficients.

3.Artificial constant-volume bomb model

Firstly,the existing oscillating velocity disturbance,11–13mass flow rate disturbance,14energy disturbance,15and pressure disturbance16–18as well as the pressure pulse20–22are compared,which are shown in Table 3.The variables in Table 3 are explained in detail in the corresponding references.

These models can be classified as two types.One is organized disturbance models,in which resonant acoustic frequencies are requested in prior such as velocity disturbance,11–13mass flow rate disturbance,14energy disturbance,15and organized pressure disturbance.16,18Only the resonant acoustic mode can be excited in most of the cases.Thus,the characteristics on which mode is the easiest to be excited and which mode is the most difficult to be damped cannot be determined.Moreover,it is hard to get the acoustics frequencies of the chamber in advance for a complex geometry.The other is pulse disturbance models,such as those proposed by Habiballah,20Zhuang,21and Urbano et al.22The overpressure coefficient(αpin the Habiballah’s pressure pulse model)is defined as the ratio of the pulse amplitude to the mean chamber pressure,usually taken as a small value in these models.

Taking the pressure pulse model developed by Habiballah and Lourme20as an example,it is applied to a combustion chamber with a pressure of 1.15 MPa and a temperature of 3000 K.A pressure pulse is imposed on the region at the chamber axis and near the head of the combustor with a duration of 0.1 μs,and its overpressure coefficient is taken in a range from 0.1 to 1.5.As shown in Fig.1,a pressure peak appears while the density keeps constant,and a significant temperature peak is therefore induced.The larger the overpressure coefficient of the pressure pulse is,the higher the temperature peak is.It can be seen that the temperature peak is greater than 5000 K when the amplitude of the pressure pulse takes a value of 1.5 times of the mean chamber pressure.It is unreasonable for such a high local temperature,and dramatically leads to a divergent computation with an increasing overpressure coefficient.

Such a pressure pulse is imposed on the flow field in a small region.It propagates spherically at the initial stage,and then transforms to propagate in specific directions through re flections on the chamber wall,and finally multi-mode acoustic pressure oscillations are formed.Two factors are very important for this process.One is that the pressure waves are still strong enough at the end of the initial propagation stage,and the other is that after the transformation to specific modes,the pressure oscillations can maintain long enough to observe their decay.It is very important to increase the initial peak value of the pressure pulse.However,it is difficult to excite multi-mode acoustic pressure oscillations using the existingpressure disturbance model due to the limited low value of the overpressure coefficient,especially when it is applied to a real case of the combustion chamber.Inconsistencies of pressure and temperature may happen in the imposed regions for the existing model,which can be explained as that such a pressure pulse is realized based on a constant-volume or constantdensity process.Energy in a form of pressure is introduced to the flow field without any carrier mass,so that only a small overpressure coefficient can be acceptable.

Table 3 Artificial disturbance models in references.

Fig.1 Time histories of pressure,temperature,and density in imposed regions with Habiballah’s pressure pulse model.

Then,a model of the numerical constant-volume bomb is therefore proposed to achieve a large-amplitude disturbance.Differed from the existing models,the corresponding mass as a carrier of the pressure pulse is also introduced in the imposed regions.The model is applied as the following based on the CFD scheme:

(1)An imposed region Ω(x,y,z)is determined at a time instant t0,usually according to the research purpose,and the pressure,temperature,and density are denoted as p0(x,y,z),T0(x,y,z),and ρ0(x,y,z),respectively;

(2)A gas that has the same components as in the original gas in the region Ω(x,y,z)with pressure γp0(x,y,z),temperature αT0(x,y,z),and density βρ0(x,y,z)is determined in the same region Ω(x,y,z);

(3)The imposed and original gases are immediately mixed in the constant-volume region Ω(x,y,z).The tuned temperatureT(x,y,z),pressurep(x,y,z),and density ρ(x,y,z)in the same region are obtained as below:

(4)The temperature,pressure,and density of the imposed region Ω(x,y,z)are set as Eq.(6)to achieve a largeamplitude disturbance from the initial time t0to time t0+Δt in numerical simulations.

Here α,β,and γ= αβ are model coefficients,which determine the peak value of the pressure pulse.t0is the initial imposed time,while Δt is the imposed time period.This model can realize a pressure pulse with a large overpressure coefficient,which is verified to excite pressure oscillations corresponding to multi-frequency acoustics modes.We call the present model as a numerical constant-volume bomb model,because the imposed and original gases in the specified region are supposed to be immediately mixed in the unchanged constant-volume region,and such a pressure pulse with a large amplitude is achieved.

The pressure,temperature,and density in the imposed regions for α =1, γ=0.5–9,and β = γ are shown in Fig.2.Different from the case shown in Fig.1 where γ is taken as 1.5,the over-temperature phenomenon does not appear in the present model even for the overpressure coefficient γ=9.A sharp rise is observed on the density,which is characterized by the constant-volume mixing process,so the present model is much more reasonable with an advantage of achieving a pressure pulse with a very large amplitude.

The model coefficients γ and β (or α),especially the overpressure coefficient γ,are now in posterior for this study.The values of the model coefficients are applicable if multimode acoustic pressure oscillations can be excited.The larger the overpressure coefficient is,the more easily the evolution of multi-mode acoustic pressure oscillations is triggered.The selected value of γ yields to the below limitation:no divergence of the flow field in numerical simulations and competition over a possible damping of the flow field.Therefore,the parameter can be estimated during the implementation of numerical simulations,which is several times or dozens of the mean pressure of the engine chamber.Therefore,γ can be determined inde-pendently based on how much the amplitude of the pressure pulse is required to excite multi-mode pressure oscillations,while β is used to adjust the temperature in the imposed region.

Fig.2 Time histories of pressure,temperature,and density in imposed regions with constant-volume bomb model.

4.Quantification of damping capacity of chamber

Pressure oscillations can be excited by the above numerical constant-volume bomb,which can be expressed as27

where An,maxis the initial maximum amplitude of each mode,αnis the corresponding damping rate,fnis the acoustic resonant frequencies,and φnis the initial phase of each mode.

Based on the results of FFT analysis,the damping factor for a peak frequency fn,peakis defined by the half-power bandwidth method as follows:2,10

where fn,peakis the resonance frequency of each mode,fn,1and fn,2are the frequencies(here,fn,2＞fn,1),of which the corresponding amplitudes areis the amplitude of the frequency fn,peak).Once ηnis determined,the damping rate αncan be obtained by the following relation:28,29

where Δfn=fn，2-fn，1.

A broadened bandwidth indicates a higher damping factor,followed by a larger damping rate.2,10Thus,the damping factor and half-power bandwidth have the same physical interpretation of acoustic damping as the damping rate.2,30The damping factor ηnand half-power bandwidth Δfnof each mode correspond to the damping rate of that mode.

Through FFT analyzing the recorded excited pressure oscillations,multi-acoustic modes can be obtained by identifying and matching the peak frequencies with the corresponding theoretical acoustic eigenfrequencies.The acoustic mode that is the easiest to be excited can be determined based on the amplitudes of the peak frequencies;the acoustic mode that is the most difficult to be damped can be determined through the damping factor ηnand half-power bandwidth Δfnfor that mode.Therefore,the acoustics characteristics and their damping of the chamber can be compared and evaluated after the numerical constant-volume bomb is applied.

5.Results and discussion

5.1.Validation and verification of numerical methods

Firstly,the numerical dissipation is investigated for a cylindrical chamber with a diameter of 32 mm and a length of 100 mm.The chamber has two closed ends,and the wall condition is slip and adiabatic.The grid-resolution is taken as 1 mm here.The chamber is filled with non-viscous quiescent air with a pressure of 0.1 MPa and a temperature of 298 K.A forcing is provided with a frequency of the first longitudinal mode 1735 Hz and an amplitude of 400 Pa.After the forcing is switched off,the pressure oscillation in the chamber is recorded.The damping is theoretically zero for such a case.31The amplitude decays from 399.1 Pa to 305.8 Pa during 19 periods.Thus,the damping rate is estimated about 0.104 s-1.The corresponding half-power bandwidth is 0.033 Hz,which is sufficiently smaller than that in a thrust chamber in next section.It proves that the numerical dissipation is well controlled in the present numerical simulations through a second-order discrete scheme and fine grids.

An acoustic test under an ambient condition for a smallthrust Liquid Rocket Engine(LRE)chamber is conducted as shown in Fig.3.The diameter of the cylindrical chamber is 2rc=32 mm,while that of the throat is 2r*=14.3 mm.The length of the cylindrical section Lchis 38.4 mm,while that of the contraction section Lcvis 27.9 mm.The injector face is a solid wall,while the chamber exit is open.A mimicked bomb is placed near the sidewall,5 mm away from the injector face,while the acoustic sensor is near the chamber wall on the opposite side to the bomb.The time trace of dynamical pressure is recorded by a sensor after the bomb is detonated.The bomb in the test is realized by an explosion of gunpowder.

Fig.3 Schematic diagram of experimental chamber.

Table 4 Comparison of theoretical,experimental and numerical results.

Numerical simulation is also conducted for an experimental test case to verify the above numerical methods.The computational region is chosen from the injector face to the throat plane.The chamber is initially filled with quiescent air with a pressure of 0.1 MPa and a temperature of 298 K.The eigenfrequencies of acoustic modes for this chamber are approximately calculated theoretically,in which the sound speed is taken as 347 m/s.The theoretical,experimental and numerical results are compared and shown in Table 4,in which 1L1T represents the combinations of the First Tangential(1T)and First Longitudinal(1L)mode.

An artificial constant-volume bomb with γ=19 and β = γ is imposed into the chamber for 0.1 μs in a cylindrical region with a diameter of 4 mm and a length of 3 mm.The imposed position is taken as the same as that in the experimental case.After the constant-volume bomb is switched off,pressure oscillations are recorded in the opposite direction of the imposed region and near the chamber sidewall,which is the same position as that of an acoustic sensor fixed in the experiments.The corresponding FFT analysis results of pressure oscillations are shown in Fig.4.Several peak frequencies are observed as 2950 Hz,6545 Hz,and 7376 Hz in the experiments as well as 3300 Hz,7000 Hz,and 8100 Hz in numerical simulations.Compared to the theoretical frequencies shown in Table 4,2950 Hz in the experimental case and 3300 Hz in the numerical case are identified as 1L mode,while 6545 Hz in the experimental case and 7000 Hz in the numerical case are identified as 1T mode.7376 Hz and 8100 Hz are 1L1T mode.The frequencies of acoustic modes obtained by the simulations and experiments are a little bit higher than those of the theoretical acoustic eigenfrequencies due to the effects of the converging section of the thruster on acoustic characteristics,which has also been verified by Farshichi et al.8The relative errors of eigenfrequencies between the experimental and simulation cases are less than 11%,which may be caused by the effects of holes for equipping with the acoustic sensors and the excitation device,the initial pressure peak,and the outlet boundary conditions.In general,it can be concluded that multi-mode acoustic pressure oscillations are stimulated by the present numerical constant-volume bomb model,and the resonant acoustic frequencies obtained by the present simulations agree well with the experimental and theoretical results.

The half-power bandwidth method is used to evaluate the damping characteristics.For the 1T acoustic mode,the halfpower bandwidths obtained in the numerical simulations and experiments are 146 Hz and 159 Hz,respectively,while the damping factors are 2.23%and 2.27%,respectively.For the 1L acoustic mode,the half-power bandwidths obtained in the numerical simulations and experiments are 130 Hz and 50 Hz,respectively,while the damping factors are 1.69%and 3.94%,respectively.The outlet boundary condition really has an effect on the eigenfrequencies and their damping rates.The chamber exit is open in the experiments,while the pressure is set as the ambient pressure and velocity components are set equal to those of the logical inside neighbor vertexes at the nozzle exit in the simulations.There are differences on the outlet condition between the experimental and numerical cases,and such differences have effects on the longitudinal acoustic modes.That is the reason why there are obvious differences for the damping rate of the 1L acoustic mode between the experimental and numerical cases.In order to avoid such differences on the outlet boundary between the experimental and numerical cases,a large region downstream from the nozzle exit should be included in the simulations,and the pressure boundary condition can be set on the boundary of this additional region.However,it will induce much expensive computational consumption.The main purpose of this paper is to introduce the numerical bomb,which can excite multi-mode acoustic pressure oscillations.Then the acoustic characteristics of the combustion chamber is investigated.Therefore,we do not try to avoid such differences on the outlet boundary.Fortunately,such differences have a little effect on the 1T acoustic mode.Therefore,we ignore the difference on the acoustic characteristics of the 1L acoustic mode,and validate the present works only using the experimental data of the 1T acoustic mode.It can be found that the predicted half-power bandwidth and damping factor of the 1T acoustic mode agree well with those obtained by experiments,and the errors between them are less than 10%.Thus,the numerical method in this paper can capture acoustic and damping characteristics of the thrust chamber correctly.

Fig.4 FFT analysis of pressure time series.

5.2.Effects of imposed position on excited pressure oscillations

In order to evaluate the acoustics characteristics and their damping of a considered thrust chamber,effects of the imposed position on excited pressure oscillations are discussed in this section.Here,the constant-volume bomb model is applied in a different geometry of an LRE combustor.The diameter and length of the cylindrical part are 35.25 mm and 7.1 mm,respectively,while the length of the contraction part is 25.63 mm.The diameter of the throat is 10.22 mm,and the total length of the thrust chamber is 89.73 mm.The theoretical acoustic eigenmodes are shown in Table 5,in which 2T and 1R represent the Second Tangential mode and the First Radial mode,respectively.

For a numerical evaluation,the chamber is initially filled with quiescent air with a pressure of 0.1 MPa and a temperature of 298 K.Then,a constant-volume bomb with γ=19 and β = γ is imposed on a cylinder-shaped region with a radius of 3.6 mm and a length of 3 mm.The positions of the center of the imposed region for different cases are shown in Table 6.The imposed cylinder region is located on the axis of the chamber for Cases 1,2 and 3,but off-axis for Cases 4 and 5,respectively.The former and latter are called as the central and offcentered constant-volume bombs,respectively.

Table 5 Acoustic eigenmodes by theoretical calculations.

Table 6 Positions of center of imposed region.

The pressure oscillations p′at an observation point in the imposed region are shown in Fig.5(a)for Cases 1,2,and 3,respectively,and their corresponding FFT analyses are shown in Fig.5(b).The amplitudes are normalized by the mean chamber pressure pmean.The peak frequencies and their amplitudes are shown in Table 7.Compared to theoretical eigenfrequencies,these peak frequencies are identified and the corresponding modes are marked in Fig.5(b).The pressure oscillations characterized by 1L and 1R acoustic modes are excited for all the three cases.The half-power bandwidths of the 1R acoustic mode are larger than those of the 1L acoustic mode,which means that the 1R pressure oscillations decay faster than the 1L pressure oscillations.The half-power bandwidths of the 1L and 1R acoustic modes for Case 3 are both larger than those for Cases 1 and 2.Because the constant-volume bomb for Case 3 is imposed on the nozzle convergent section which can attenuate pressure oscillations.The amplitudes of 1L and 1R are weaker with the imposed region away from the injector face.

The effects of the radial position of the imposed region on the pressure oscillations are investigated through Cases 2,4 and 5.The pressure oscillations for these cases are shown in Fig.6(a)and their FFT analyses are shown in Fig.6(b).The shapes of pressure oscillations for Cases 4 and 5 are different from that for Case 2.Compared to Case 2,more peak frequen-cies are found for Cases 4 and 5,which are identified based on the theoretical eigenfrequencies,and the corresponding modes are marked in Fig.6(b).For the central constant-volume bomb,the major acoustic modes are 1L and 1R,while 1T for the off-centered constant-volume bomb.

Table 7 Acoustic and damping characteristics of Cases 1–3.

Fig.5 Pressure oscillations and their FFT analyses in Cases 1–3.

Fig.6 Pressure oscillations and their FFT analyses in Cases 2,4 and 5.

In order to reveal the propagating process of pressure waves,the temporal behaviors of the velocity vector in a transverse section of the chamber with a distance of 4 mm away from the injector face are given in Figs.7–9.The red color represents high pressure,while the blue and green colors represent low and middle pressures,respectively.The arrow displays the propagating direction of pressure waves.

As shown in Fig.7(a)for Case 2,a high-pressure region is formed at the center of the transverse section,because the constant-volume bomb is imposed on the axis of the chamber.Pressure waves propagate to the chamber wall along the radial direction as shown in Fig.7(a)–(d).When the pressure waves reach the wall,they are re flected and propagate to the center as shown in Fig.7(e)and(f).Thus,the central high-pressure region is formed again.A typical radial acoustic mode is therefore established,which is marked as 1R in Fig.5(b).

For Case 4,the center of the imposed region is located at the half of the chamber radius as shown in Fig.8(a).Fig.8(b)–(h)show the propagating process of the pressure waves due to a constant-volume bomb.They propagate in both the tangential and radial directions because of low pressure in the other regions of the chamber.The pressure waves propagate to the right side first and reach one side of the chamber wall.Then they propagate along the wall due to re flection.Tangential acoustic modes are triggered,shown as Fig.8(e)and(f).Thus,the 1T,2T,and 1R acoustic modes are observed for Case 4 as shown in Fig.6(b).

For Case 5,a constant-volume bomb is imposed very close on the chamber wall as shown in Fig.9(a).Comparing Fig.9(b)to Fig.8(b),the pressure wave propagating along the wall in Case 5 is much stronger than that in Case 4.The tangential acoustic modes are therefore enhanced,and the radial acoustic mode is suppressed,which can also be observed in Fig.6(b).Fig.9(e)and(f)show the 1T and 2T modes occurring in this case.

Fig.7 Temporal behavior of velocity vector for Case 2.

Fig.8 Temporal behavior of velocity vector for Case 4.

Fig.9 Temporal behavior of velocity vector for Case 5.

A high-pressure region is formed where a numerical bomb is imposed.The pressure waves travel from the high-pressure region to the low-pressure region spherically.For the central constant-volume bomb,the high-pressure region is always axially symmetrical for the chamber.The pressure varies in the radial direction and keeps constant in the azimuthal direction.Thus,the pressure waves propagate only in the radial direction in the transverse section.Pure radial modes occur in the transverse section,and tangential modes do not appear.For an offcentered constant-volume bomb,the high-pressure region is not axially symmetrical in the transverse section.Thus,the pressure waves propagate in both the radial and tangential directions.The tangential modes are excited.The pressure waves also propagate in the longitudinal direction for thesetwo types of constant-volume bomb.Thus,the 1L mode occurs in all the cases.When the imposed constant-volume bomb moves from the axial center of the chamber to the sidewall,the tangential acoustic mode becomes stronger,while the radial and longitudinal acoustic modes become weaker.

Table 8 Acoustic and damping characteristics of Cases 2,4 and 5.

The detailed acoustic and damping characteristics are shown in Table 8.When the off-centered constant-volume bomb is imposed on the chamber,such as in Cases 4 and 5,the 1T and 2T modes can be stimulated.The 1T mode is the strongest acoustic mode compared to the other modes,while the 2T mode is very weak.Therefore,the most inspirable mode is the 1T mode with the off-centered constant-volume bomb.When the central constant-volume bomb is imposed,the dominant modes are 1L and 1R,and the amplitudes are higher than those in Cases 4 and 5.The half-power bandwidths of the transverse modes are larger than that of the 1L mode,so the 1L pressure oscillations keep for the longest time,and they are the most difficult to be attenuated.In the transverse modes,the half-power bandwidth of the 1R mode is larger than that of the 1T mode.

The half-power bandwidth of the higher acoustic mode(2T)is larger than that of the low acoustic mode(1T).The damping capacities of tangential acoustic modes are almost the same in Cases 4 and 5.When the imposed region is closer to the chamber sidewall,pressure waves in the longitudinal direction are easier to be weakened by the nozzle,and thus the damping factor should be larger as shown in Table 8.

From Case 1 to Case 5,three acoustic modes have been encountered in the LRE model:the longitudinal,tangential,and radial acoustic modes.The frequencies of the acoustic modes and the corresponding damping capacities numerically obtained by the present CFD method agree well with those by the experiments,but they are reasonably larger than those obtained by the theoretical method,because the effects of the converging section on the acoustic characteristics of the chamber are considered in the simulations.The 1L mode appears in all the cases,whose half-power bandwidth is the smallest.For these simulation cases,the 1L acoustic mode is the most difficult to be damped for the present chamber configuration,which is affected by its lowest eigenfrequency and the outlet boundary condition.The amplitudes of the 1R mode are larger for Cases 1–3,while they are smaller in the other cases.The tangential acoustic modes appear in Cases 4 and 5.The 1T mode has been proven to be the most violent and harmful mode,whose amplitude is found to be the largest in Cases 4 and 5.The half-power bandwidth of the 1R mode is wider than that of the 1T mode.In practice,pressure disturbances due to injection or violence combustion in the head region of the chamber usually are not located at the center of the chamber.This work explains why the 1T highfrequency combustion instability occurs frequently and is hard to be eliminated without passive devices or active control.

6.Conclusions

This paper proposes a new artificial constant-volume bomb model and numerically mathematizes it,considering the existing disturbance models with the shortcoming of a small overpressure coefficient,which cannot excite multi-frequency oscillations inside the chamber.Based on this model,the acoustic characteristics of a small-thrust LRE are investigated by the CFD method.A constant-volume bomb is imposed on the steady flow to activate pressure oscillations in the chamber.Peak frequencies are obtained through the FFT analyses of the temporally recorded pressure oscillations.Their corresponding acoustic modes are identified by comparing to the theoretical acoustic modes.The damping capacities of the identified acoustic modes are evaluated by the half-power bandwidth.The effects of the position of the constant-volume bomb on excited pressure oscillations are further investigated.

It is found that the 1L mode can be triggered easily.The half-power bandwidth of the 1L mode is the smallest.The amplitude of the 1L mode is smaller but its half-power bandwidth is larger as the center of the imposed region is closer to the chamber sidewall.The 1R mode can be excited by a central constant-volume bomb,because the pressure disturbances keep constant in the azimuthal direction,such as in Cases 1,2,and 3.In the other cases,the 1R mode amplitudes are too small to be identified.The tangential modes can only be excited by an off-centered constant-volume bomb,in which the pressure disturbances are non-axisymmetric about the axis in the chamber.Moreover,the 1T mode is verified as the fiercest mode and therefore the most inspirable one.The damping factor of the 1T mode is independent of the radial position of the bombed region.The half-power bandwidth of the 2T mode is larger than that of the 1T mode,indicating a faster decay.

This work establishes a CFD method coupled with a numerical constant-volume bomb model to evaluate acoustic characteristics,which can be applied to,but not limited to,small-thrust LREs.It somehow verifies and explains why the 1T mode in high-frequency combustion instabilities has been encountered frequently in engineering.

Acknowledgements

Financial support from the National Natural Science Foundation of China(Nos.51676111 and 11628206)is acknowledged.

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