Acoustic Characterization of Virtual Thrust Chamber Demonstrators

Transcription

1 Sonderforschungsbereich/Transregio 4 Annual Report Acoustic Characterization of Virtual Thrust Chamber Demonstrators By A. Chemnitz AND T. Sattelmayer Lehrstuhl für Thermodynamik, Technische Universität München Boltzmannstr. 5, Garching b. München The acoustics of three thrust chamber configurations are studied numerically with a hybrid approach. Eigenfrequencies and mode shapes of the basic configurations are calculated with the Linearized Euler Equations. The underlying quasi-one-dimensional mean flow is based on single flame simulations. Thereto an approach to account for the effect of the nozzle on the heat release in the procedure for the mean flow calculation is proposed. The test cases comprise two chamber designs, one fed with Hydrogen/Oygen and one with Methane/Oygen or Hydrogen/Oygen alternatively. The single flame simulations of the different configurations are analyzed in terms of the basic flame properties. For the mean flow the sound speed distributions and cut-on frequencies of the configurations are compared with each other, yielding influences of the propellant combination as well as the chamber design. An analysis of the chambers eigenmodes of first transverse type shows a trend to lower damping rates for the CH 4 configuration compared to the H 2 case. For all configurations the first transverse and the combined first transverse/first radial/first longitudinal modes are found to be weakly damped. The amplitude distributions of these modes are considered to assess the potential for flame interaction with the chamber acoustics.. Introduction Combustion instabilities arise from the coupling of pressure fluctuations in the chamber with the heat release. Their occurrence depends on the overall engine design, including combustion process and eternal components. The accurate prediction of a rocket engine s stability behavior in early stages of the design process is thus of high importance. Concepts for net generation engines include design choices that can negatively affect the combustor s thermoacoustic stability. In this contet, virtual thrust chamber demonstrators (TCD) have been proposed [ 3], which include key elements of the current development. Two configurations are of special relevance regarding combustion stability: TCD2 features a low pressure drop across the injector head, which reduces the required power of the turbomachinery but promotes the acoustic coupling of injection system and combustion chamber. TCD3 is designed to run on the propellant combinations H 2 /O 2 as well as CH 4 /O 2. Especially the latter combination is susceptible to combustion instabilities, while being of increased interest due to its performance and storability. All cases are designed for chamber pressures around bar. The main dimensions of the two demonstrator geometries are given in Tab.. Three aspects are of relevance for the assessment of an engine s stability: first, the chamber acoustics, which are determined by the mean-flow. They are described by the

2 7 A. Chemnitz & T. Sattelmayer TCD2 TCD3 radius (r cyl ), mm 95 2 length (l c), mm TABLE. Demonstrator main dimensions spatial distribution of oscillation amplitudes as well as the eigenfrequencies. The mean flow has been found to significantly contribute to the mode damping rates [4]. Second, the flame feedback is decisive. It represents the destabilizing mechanisms of combustion while it has only a minor influence on the mode shape and frequency. Finally, the coupling to the propellant dome is of relevance, particular for TCD2. In the present analysis we will consider the first aspect, the eigenmodes, for the three cases TCD2, TCD3 with H 2 and TCD3 with CH 4. This provides the basis for future studies of the full configurations. This report is structured as follows: First, the basic numerical approach is outlined. Thereby an etension of the procedure for the generation of the mean-flow from single flame simulations is introduced to account for the heat release in the nozzle section. Then, the single flame results and the mean flow are discussed before the acoustic analysis for the different configurations is presented. 2. Numerical Setup The numerical approach is based upon an eigenvalue analysis of the Linearized Euler Equations (LEE). These are transformed into frequency space and discretized using stabilized finite elements. To keep the computations efficient, the mean flow for the LEE is set up as quasi-one-dimensional. The basis are single flame simulations, whose aial profiles of certain radial averaged acoustic quantities are reproduced by the onedimensional flow. In this section, the modeling of the acoustics is described first. Thereafter, the setup for the single flame simulations is outlined followed by the procedure for the mean flow generation. 2.. Acoustic Modeling The LEE describe the perturbations φ of the flow fields from a specified mean state φ. Their finite element discretization is stabilized following the Galerkin Least Squares approach [5]. The harmonic behavior of the oscillating flow fields is described as with the comple amplitude ˆφ. The comple eigenfrequency φ = ˆφ ep (iωt) (2.) Ω = ω + iα (2.2) is composed of the eigenfrequency ω and the damping rate α and is obtained directly from the eigenvalue analysis. For further reduction of the computational cost, two-dimensional computations are carried out with the dependence of the solution on

3 Thrust Chamber Demonstrator Acoustics 7 the angular coordinate θ being described via the analytical solution of cylindrical ductflow acoustics [6]: ˆφ = φ ep (iθ). (2.3) Since no swirl is present in the mean flow, the resulting conservation equations for mass momentum and energy in cylindrical coordinates (r, θ, ) are: ( ρ iω ρ + ũ r r + ) ( r ρ ρ + ũ + ũ i θ r ρ + ρ ur r + r u r + u ) + ρ r ũr + ρ (2.4) + u ũ ρ + u r r ρ =, u r iωũ r + ũ r r + ũ u r ρ p ρ 2 r + u ũ r r r + u ũ r ũ φ iωũ θ + u r r + ũ θ u iωũ + ũ r r + ũ u ρ p ρ 2 + u r u r r + u ũ r + u + p ρ r = ũ θ + i ρr p = ũ + p ρ =, ( κp iω p + ũ r r + p r p ) ( κ iκp p + ũ θ κ r r + ũ p ) κ κ ( ( ur + p κ r + u r r + u ) ( )) κ u r κ r + u κ ( ũr + κp r + ũ ) p + u r r + u p =. (2.5) (2.6) Thereby u denotes velocity, ρ density, p pressure and κ isentropic compressibility. The boundary conditions in the simulation are set as energy neutral at the inlet, nonreflecting at the supersonic outlet and hard-walled at the chamber contour (cf. section 2.3) Single Flame Simulation The single flame simulation provides the aial profiles that are used in the calculation of the one-dimensional mean flow. It comprises a single injector element attached to a cylindrical domain (cf. Fig. ). The radius of the domain is chosen to preserve the area ratio between injection elements and combustion chamber of the full configuration. The aial dimension of the domain covers the length of the chamber up to the nozzle throat. The flow is described via the steady Reynolds-Averaged Navier-Stokes equations (RANS). Turbulence closure is obtained by the k ɛ model, following the findings of Chemnitz et al [7]. Thereby turbulent species and energy transport are modeled via turbulent Schmidt numbers and a unity Lewis-number. The turbulent Schmidt number is selected according to an eperimentally validated simulation of an H 2 and a CH 4 fueled chamber [8] to Sc t =. and Sc t =.7 respectively. Combustion is accounted for with a semi-diabatic flamelet model: The species mass fractions are obtained at adiabatic conditions, while the temperature calculation accounts for the local enthalpy in the flow. The flamelets are isobaric at a pressure of bar. In the simulation, the low injection

4 72 A. Chemnitz & T. Sattelmayer symmetry ṁ o ṁ f r p out TCD2 TCD3 FIGURE. Computational domain for single flame simulation FIGURE 2. Computational domain for D mean flow temperatures (T O2 = 95 K, T H2 = K, T CH4 = 23 K) are accounted for via the Soave-Redlich-Kwong real-gas equation of state. The setup is two-dimensional ais-symmetric with the boundary conditions shown in Fig.. The value of the outlet pressure p out is determined iteratively with the mean-flow simulations described below, based on the pressure at the end of the cylindrical chamber section One-Dimensional Flow The mean flow is the reference state for the solution of the LEE. It covers the complete chamber (cf. Fig. 2). Since the resolution of the single injectors and resulting flame structures is computationally epensive, a quasi-one-dimensional flow is calculated. It reproduces the radially averaged distribution of sound speed (c) and isentropic compressibility from the single flame. Heat release due to combustion is represented by a one-dimensional heat source profile. Some adaptions have to be made in order to apply the procedure to the present case. In previous studied configurations [9], the subsonic nozzle part was considerably shorter than the cylindrical chamber part. The properties in the nozzle were treated as frozen and the heat release could be neglected. However, for the technology demonstrators the subsonic nozzle part takes up more than 5 % of the combustion chamber length. Accordingly, the processes in the convergent nozzle section are of higher importance for the generation of the mean flow. To account for this aspect, two corrections are applied in this study: one concerns the heat release in the subsonic nozzle flow, the second the properties at the nozzle throat. In the following, the term nozzle is used to refer to the subsonic part of the full convergent-divergent nozzle Heat Release The flowfield in the nozzle is subjected to aial acceleration, deflection due to the chamber contour and a decreasing pressure and temperature. These factors potentially affect the combustion process. To keep the approach computationally efficient, the nozzle heat release is based on the single flame simulations, combined with corrections for relevant influencing factors. An estimation using the CEA code [] shows that the influence of pressure on the equilibrium composition in the convergent nozzle section is small. Furthermore, the heat release in the aial region of the single flame domain that corresponds to the nozzle is found to be still dominated by miing rather than by the reduction of non-equilibrium. Correspondingly, the influence of the decreasing pressure is neglected. Likewise, the influence of the temperature change is not included in the model. Based on the typically low isentropic eponents of the combustion products, the temperature change due to

5 Thrust Chamber Demonstrator Acoustics 73 the epansion in the nozzle can be estimated to stay below %, while the argument of the miing dominated reaction process is valid here as well. The stratification of the flow at the nozzle inlet is small, the flames are compact with respect to the radial chamber dimensions and the contour of the nozzle wall changes the flow direction smoothly. Thus, it is assumed that the radial velocity components inside the nozzle do not significantly affect the miing process. So the volumetric heat release used for the mean flow generation is calculated as q V,D = q V,nz η geo (2.7) with q V,nz denoting the heat release of the single flame simulation after correction for the accelerated flow in the nozzle, while η geo accounts for the shape and dimensions of the nozzle geometry. To access the influence of the flow acceleration on the heat release, a Lagrangian point of view is taken: A mass element containing fuel and oidizer travels in aial direction through a one-dimensional nozzle and releases heat at a constant mass specific rate q L. This converts to a volumetric heat release in an Eulerian coordinate system as q V V ṁ = q L t = q L L u. (2.8) With the fied nozzle volume V that corresponds to the length L which the reacting flow passes in a time t. Since the mass flow is constant in aial direction while the velocity changes, the heat release is proportional to the inverse velocity: q V,nz = q V,ref u,sf u,nz u,nz u,sf. (2.9) e The inde sf denotes the single flame simulation, u,nz refers to the radially averaged distribution of aial velocity in the nozzle and q V,ref is the heat release interpolated from the single flame simulation (see below). The normalization of the velocity profile with the values at the nozzle entrance e is done to ensure a steady transition of the heat release from cylinder to nozzle. As stated above, the combustion process is dominated by miing. In the case of the nozzle flow, the distance passed by the fluid during the miing process is larger than in the single flame simulation due to the higher flow velocity. The energy that is not released at a certain position due to the reduction according to Eq. 2.9 does not just leave the system but will be released further downstream. Thus the interpolation of the heat release from the single flame simulation q V,sf to the full chamber q V,ref can not be done via aial position but needs to be based on the amount of heat released upstream. Between nozzle entrance and throat, e th, the reference heat release is taken from the single flame simulation at position sf : q V,ref = q V,sf sf, e q V,nz dˇ = sf e q V,sf dˇ (2.) This equation is solved iteratively together with Eq. 2.9 in a pre-processing step. To obtain the velocity distribution u,nz of the nozzle flow, the full chamber is simulated without heat release in the nozzle. The resulting overall procedure is shown in Fig. 3. Finally, since a volumetric heat release is used, a correction for the chamber geometry

6 74 A. Chemnitz & T. Sattelmayer aial velocity heat release throat properties aial velocity Single flame D mean flow (no nozzle heat release) D mean flow (w. nozzle heat release) speed of sound isentropic compressibility st simulation 2 nd simulation 3 rd simulation FIGURE 3. Schematic of mean flow generation needs to be applied to obtain the heat release prescribed in the simulation: η geo = ( rcyl r nz,wall ) 2 (2.) with r cyl the radius of the cylindrical part of the combustion chamber and r nz,wall the local radius of the nozzle wall Outlet Properties The chamber pressure is determined by the total pressure that is necessary to pass the injected mass flow through the choked nozzle throat. The throat mass flow can be calculated from basic ideal gas relations: κ p ṁ th = A th ρc = A th. (2.2) R T R denotes the specific gas constant, A th is the cross-sectional area of the nozzle throat and T the temperature. Thus at a given total heat release the values of the isentropic eponent and the specific gas constant at the outlet are decisive for the pressure level in the combustor. The procedure applied in the cylindrical part of the chamber accounts for the profiles of sound speed and isentropic compressibility from the single flame simulation in the calculation of R and κ. In the nozzle, this is blended smoothly to the value obtained at the outlet of the single flame simulation, as are the coefficients used for the calculation of the specific heat capacity c p.

7 Thrust Chamber Demonstrator Acoustics 75 TCD2, H 2 r/rinj T, K 3,5 TCD3, H 2 r/rinj,8 TCD3, CH 4 r/rinj..2, m f stoich FIGURE 4. Temperature distribution and flame contour of the single flame simulations 3. Results With the setup described in the previous section, the three demonstrator configurations are analyzed. First, the reacting flow field of the different cases is studied based on the single flame simulation. Then, the mean flow is considered and used for a first assessment of the chamber s acoustic behavior. Finally, the eigenvalue analysis of the LEE is included to give an overview of the eigenmodes in the chamber. 3.. Single Flame The single flame simulations provide an insight into the flow structure inside the combustion chamber. Fig. 4 shows the temperature field plotted along with the line of stoichiometric miture fraction f stoich, which is taken as indicator for the flame contour. All three cases show the typical flow field structure: A region of hot combustion gases forms along the diffusion flame located between oidizer core and fuel. As miing proceeds with increasing distance from the injector, the temperature distribution becomes more uniform and the contour of stoichiometric miture fraction closes at the ais due to the fuel rich operating conditions. However, the configurations show notable differences regarding the development of the flame. TCD2 and TCD3 are both fueled with H 2 but differ in the injector and chamber design as well as operating conditions. This results in a shorter flame length for TCD3 compared to TCD2. At the same time, the cylindrical part of TCD2 is shorter than that of TCD3; as a consequence the flame closes in the nozzle part of the chamber, while for TCD3 it is well confined to the cylindrical section. The influence of the propellant combination on the flame shape can be seen from the two TCD3 configurations. Despite the lower turbulent Schmidt number the flame length is considerably higher for the CH 4 than for the H 2 case. At the same time, the region of steep temperature rise is located close to that of the TCD3 H 2 case and even upstream of that of TCD2, showing the impact of the chamber design on the combustion process.

8 76 A. Chemnitz & T. Sattelmayer 5 Config. TCD2 TCD3 Fuel H 2 H 2 CH 4 p c,sim, bar p c,nom, bar 7 TABLE 2. Chamber pressure c, m/s 5 TCD2 H 2 TCD3 H 2 TCD3 CH , m FIGURE 5. One-dimensional distribution of speed of sound 3.2. One-Dimensional Flow The pressure level of the one-dimensional flow allows for an assessment of the overall adequacy of the approach for the mean flow calculation. The chamber pressures reached for the different configurations p c,sim are given in Tab. 2. The values are calculated as the mean of the total pressure in the cylindrical chamber section. They are in very good agreement with the nominal operating pressures p c,nom, indicating the reasonability of the modeling approach. Building upon this basis, the acoustic mean flow properties can be discussed. The sound speed distributions in the chambers are given in Fig. 5. The two H 2 cases possess equal maimum values. For TCD3 this maimum is part of a plateau of the speed of sound, which is less distinct for TCD2. In the front part of the chamber, the sound speed of the TCD3 H 2 configuration eceeds that of TCD2. However, the shape of the profiles in the inlet region is quite similar with a steep increase in the beginning, followed by a gradual decrease before the transition to the main plateau. The most significant effect of a fuel change to CH 4 is the lower level of the sound speed. In contrast, the shape in the rear section of the chamber is similar for both TCD3 cases. In the front part, the gradual decrease observed for the H 2 case is replaced by a short sharp decrease followed by a continuous increase. The initial region of lowest speed of sound directly at the faceplate is even shorter than for the H 2 cases. The sound speed directly influences the cut-on frequencies. Its development along the chamber is shown in Fig. 6. In the front part the cut on frequency follows the sound speed, while in the nozzle region the wall contour and Mach number become relevant. Close to the nozzle throat the effect of the Mach number approaching unity eceeds the decrease of the radius and leads to a rapid decline of the cut-on frequency Acoustic Characterization The acoustics of the combustor are governed by its eigenmodes. Especially transverse modes are prone to become unstable. The current analysis provides an assessment of the damping of the modes of first transverse order (T X). The computed damping rates are influenced by the transport of acoustic energy through the nozzle throat as well as

9 Thrust Chamber Demonstrator Acoustics fco, Hz 2 contour , m fco, Hz 2 H 2 CH 4 contour , m (a) TCD2 (b) TCD3 FIGURE 6. Cut-on frequency of the T mode the interaction of the mean flow and its gradients with the oscillating fields, which has been referred to as field damping by Schulze [4]. The eigenfrequencies and damping rates for pure and mied modes are shown in Fig. 7 up to the first transverse/first radial/first longitudinal (T R L ) mode. Thereby only modes with a damping rate less than 2 rad/s are included. As discussed before, the cut-on frequency changes along the chamber, which leads to different transverse mode frequencies in different chamber sections; the cut-on frequency distribution for the T and T R modes are included in Fig. 7. As a consequence, the transverse modes do not posses constant amplitude distributions along the chamber. This is not only the case if a mode is cut-off locally but also if a transverse mode occurs in regions of higher sound speed. Then regions of lower sound speed are cut-on as well and multiple modes that show an amplitude distribution similar to that of a combined T L mode can occur. In the considered range of comple eigenfrequencies, this is e.g. the case for the T L mode of the CH 4 fueled TCD3 case. For all configurations the T mode possesses the weakest damping. The cut-on frequency distribution shows that this mode is cut on along the first 3 to 4% of the chamber, depending on the case. This considerably eceeds the initial region of lowest sound speed directly at the faceplate. The net modes are of combined T L type. For the H 2 cases these are damped significantly stronger than the pure T mode. For TCD2 the first T L mode even eceeds the considered range of damping rates. For the TCD3 CH 4 case the first T L mode shows a notable increase in damping, while the damping rate of the second one is again very close to that of the T mode. The subsequent modes are again damped more strongly, however the overall level of damping rates is lower than for the H 2 cases. For all demonstrators, after the increasing damping of the combined T L modes with frequency, the T R mode shows lower damping, which even decreases for the T R L mode. The pressure amplitude distributions of the most unstable modes of each configuration are shown in Fig. 8. All cases show the typical shape of the partial cut-on T mode: In the region close to the faceplate, where the mode is cut-on, high amplitudes are present which then decay in aial direction. However, a clear influence of the propellant combination on the mode shape is visible, as the amplitudes for the H 2 fueled configurations decay considerably slower than those of the CH 4 case. This can be of

10 78 A. Chemnitz & T. Sattelmayer 2 3 T T R α, rad/s.5.5 /lc f co T T L T L 2 T L 3 T R T R L f, Hz (a) TCD α, rad/s.2.25 /lc f, Hz f, Hz (b) TCD3, H 2 (c) TCD3, CH 4 FIGURE 7. Eigenfrequencies and damping rates for modes of first transverse order relevance for the strength of flame feedback, since a considerable amount of heat is released about halfway of the cylindrical chamber section. The same applies to the T R L mode that has a region of high amplitudes in the middle part of the chamber for all cases. In general, for the T L and T R L modes no such significant influence of propellant combination or chamber geometry as for the T mode is observed. 4. Conclusions The acoustics of three demonstrator configurations have been studied numerically by conducting an eigenvalue analysis of the Linearized Euler Equations with a quasi-onedimensional mean flow. The approach for the mean flow calculation has been adapted

11 79 Thrust Chamber Demonstrator Acoustics T T L T R L (a) TCD2 T T L T R L (b) TCD3, H2 T T L T R L (c) TCD3, CH4 p n F IGURE 8. Pressure amplitude distributions to account for the heat release in the nozzle and the influence of the flow properties at the throat on the chamber pressure. Thereto an efficient procedure to correct the heat release from single flame simulations for flow acceleration has been proposed. The obtained pressure levels of the mean flow are in good agreement with the nominal chamber pressures, indicating the applicability of the modified procedure for the mean flow calculation. The one-dimensional distributions of speed of sound and the associated cut-on frequencies show influences of both, demonstrator design and propellant combination. While the level of sound speed as well as its qualitative development in the front section of the chamber are governed by the choice of fuel, the shape of the sound speed profile in the rear section depends on the demonstrator configuration. Among the modes of first transverse order, the T, the T R as well as the T R L mode possess the lowest damping rates. In addition, the TCD3 CH4 configuration showed an additional weakly damped T L mode. Altogether, the level of damping is lower for the CH4 than the H2 cases. However, the T mode shapes of the H2 cases showed significant amplitudes across a wider portion of the chamber, potentially increasing the interaction with the heat release. This applies to the T R L modes of all cases as well. Future work will cover an enhanced analysis of the stability behavior under the consideration of flame feedback, configuration specific aspects like the low pressure loss of TCD2 and, where required, additional modes.

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