Nozzle Admittance Research Articles

Modified computation of the nozzle damping coefficient in solid rocket motors

In solid rocket motors, the bulk advection of acoustic energy out of the nozzle constitutes a significant source of damping and can thus influence the thermoacoustic stability of the system. In this paper, we propose and test a modified version of a historically accepted method of calculating the nozzle damping coefficient. Building on previous work, we separate the nozzle from the combustor, but compute the acoustic admittance at the nozzle entry using the linearized Euler equations (LEEs) rather than with short nozzle theory. We compute the combustor's acoustic modes also with the LEEs, taking the nozzle admittance as the boundary condition at the combustor exit while accounting for the mean flow field in the combustor using an analytical solution to Taylor–Culick flow. We then compute the nozzle damping coefficient via a balance of the unsteady energy flux through the nozzle. Compared with established methods, the proposed method offers competitive accuracy at reduced computational costs, helping to improve predictions of thermoacoustic instability in solid rocket motors.

Nozzle Admittance and Damping Analysis Using the LEE Method

AbstractThe nozzle admittance is very important in the theoretical analysis of nozzle damping in combustion instability. The linearized Euler equations (LEE) are used to determine the nozzle admittance with consideration of the mean flow properties. The acoustic energy flux through the nozzle is calculated to evaluate the nozzle damping upon longitudinal oscillation modes. Then the parametric study, involving the nozzle convergent geometry, convergent half angle and nozzle size, is carried out. It is shown that the imaginary part of the nozzle admittance plays a non-negligible role in the determination of the nozzle damping. Under the conditions considered in this work (

Validation of Transverse Instability Damping Computations for Rocket Engines

The reliable numerical evaluation of thermoacoustic stability is one of the most important unresolved problems in the development of liquid rocket engines. To provide data for code validation, a transonic cold-flow test facility was built, which comprises a rocket engine model for acoustic measurements. It consists of a cylindrical chamber with inflow through a perforated plate representing the faceplate and a choked nozzle at the exit. The flow is excited with a siren and complex eigenfrequencies, and the nozzle admittances for the most critical first-transverse mode are extracted from dynamic pressure measurements. These data are used to benchmark three stability assessment tools of widely varying numerical complexity. Results obtained with one- and three-dimensional methods based on the solution of the linearized Euler equations and results from nonlinear Navier–Stokes computations (unsteady Reynolds-averaged Navier–Stokes) are compared with the experimental data. All three codes reproduce the measured first-transverse-mode frequency, but only the two linearized Euler equation methods provide satisfactory values for first-transverse-mode damping and nozzle admittance. The nozzle admittances calculated from the unsteady Reynolds-averaged Navier–Stokes results exhibit large scatter, but the average of the data from 11 monitor points is close to the measured values and mode decomposition during data analysis is expected to reduce scatter further. However, the employed unsteady Reynolds-averaged Navier–Stokes method overpredicts nozzle damping.

A finite element approach for predicting nozzle admittances

A finite element method is used to predict the admittances of axisymmetric nozzles. It is assumed that the flow in the nozzle is isentropic and irrotational, and the disturbances are small so that linear analyses apply. An approximate, two dimensional compressible model is used to describe the steady flow in the nozzle. The propagation of acoustic disturbances is governed by the complete linear wave equation. The differential form of the acoustic equation is transformed to an integral equation by using Galerkin's method, and Green's theorem is applied so that the acoustic boundary conditions can be introduced through the boundary residuals. The boundary conditions are described for both straight and curved sonic lines. A two dimensional FEM with linear elements is used to solve the acoustic equation. A one dimensional FEM is also used to solve the reduced equation of Crocco, and the solution verifies the sufficiency of the boundary residual formulation. Comparison between computed admittances and experimental data is shown to be quite good.

Scaling of Rocket Nozzle Admittances

To apply the admittance data measured experimentally for small-scale nozzles, the necessary scaling criteria must be established. The results of an investigation undertaken to determine these criteria are presented and the data indicate that under cold-flow conditions the damping of axial instabilities provided by full-scale nozzles can be determined from the attenuation data obtained experimentally for geometrically similar small-scale nozzles. The scaling of both short and long nozzles has been investigated. The experimental data also support the scaling criteria proposed by Crocco in his analytical investigation of nozzle damping.

Damping of Axial Instabilities by Small-Scale Nozzles Under Cold-Flow Conditions

The damping of axial instabilities by a variety of small-scale solid rocket nozzles has been determined experimentally under cold-flow conditions using the modified impedance tube technique. The dependence of the damping upon the cavity depth and the secondary flow rate issuing from the cavity of a submerged nozzle, the geometry of the convergent section of a single-ported nozzle, and the number of nozzles present in a multipleported nozzle cluster has been determined. Measured data indicate that in the case of the submerged nozzle the cavity depth surrounding the nozzle has a significant effect upon the nozzle admittance while the secondary flow rate issuing from the cavity has negligible effect on the admittance. Tests conducted with conical, equal-radii-ofcurvature and linear-velocity-profile nozzles showed that conical nozzles provide the most damping. Tests with multiple-ported nozzles indicated that quadruple-ported nozzles provide less damping for axial instabilities than single and dual-ported nozzles whose damping capabilities are approximately the same.

Nozzle Damping in Solid Rocket Instabilities

Present understanding regarding the damping provided by solid-propellant rocket motor nozzles, during axial instabilities, is reviewed. Expressions describing the various modes of wave energy losses to the nozzle and the nozzle decay coefficient are derived and their use in practice is discussed. Available theories for the prediction of the nozzle admittance and available methods for the experimental determination of nozzle admittances are evaluated. Experimental nozzle admittance data obtained by use of the impedance tube method, for two different solid-propellant rocket nozzles, is presented and discussed. An analysis of the experimental nozzle admittance data shows that 1) the admittances of short nozzles are independent of the frequency when the wavelength of the oscillation is much longer than the length of the nozzle convergent section, 2) the admittances of short nozzles are practically independent of the geometrical details of their convergent sections, and 3) the measured nozzle admittances are larger than those predicted by the short nozzle theory. The reported experimental data are used to derive expressions for the nozzle decay coefficients for cylindrical combustors experiencing axial instabilities. These expressions are compared with a corresponding expression derived in related experimental studies and good agreement is shown.

Experimental and theoretical determination of the admittances of a family of nozzles subjected to axial instabilities

The interaction between the oscillations in the combustor and the wave system in the nozzle (required in combustion instability analysis of rocket engines) can be determined once the nozzle admittance is known. Experimental and theoretical methods of determining the admittance of nozzles subjected to axial oscillations are discussed. One-dimensional nozzle admittances were measured by an impedance tube technique modified to account for the presence of a mean flow. Crocco's nozzle admittance theory is applied to the prediction of admittances for several test nozzles.

Experimental determination of three-dimensional liquid rocket nozzleadmittances.

The three-dimensional nozzle admittance, an important parameter in combustion instability studies, was experimentally measured for several nozzle configurations. The admittance values were obtained using a modification of the classical impedance tube technique. The modified impedance tube method measures the admittance of a duct termination in the presence of one-dimensional mean flow and three-dimensional oscillations. Values of the nozzle admittance were obtained from pressure amplitude measurements taken at discrete points along the length of the tube. To determine the effects of nozzle geometry, nozzles were tested with half-angles of 15, 30, and 45 deg and entrance Mach numbers of 0.08, 0.16, and 0.20. The admittance results are presented as functions of nondimensional frequency for mixed first tangential-longitudinal modes. These results are compared with available theoretical predictions, and good agreement between theory and experiment is shown.

Stability of Combustors with Partial Length Acoustic Liners

An analytical technique for the evaluation of combustion stability in rocket motors with partial length acoustic absorbers is presented. The combustors considered have concentrated combustion zones at the injector, finite mean flows, cylindrical cross sections, and acoustic liners of arbitrary length and impedance. Linear three dimensional oscillations in such combustion chambers are analyzed using an integral equation-iteration technique. Stability limits in terms of a combustion response factor are calculated for several values of Mach number, length to radius ratio, liner impedance, liner length, liner location, and nozzle admittance. Results indicate that increasing liner length increases combustor stability substantially at low Mach numbers but has a substantially smaller effect for larger Mach numbers. Increasing Mach numbers or length to radius ratio have destabilizing effects while liner location has only a minor effect on stability.

A theoretical study of three-dimensional combustion instability in liquid-propellant rocket engines

A theoretical study of three-dimensional linear combustion instability in liquid-propellant rocket motors is presented. A concentrated combustion model is used, and the mathematical analysis is presented as a boundary-value problem in which the solutions describing the flow oscillations in the combustor are required to satisfy a combustion-zone boundary condition at one end of the chamber and a Nozzle Admittance Relation at the other end. Crocco's time-lag hypothesis is used to describe the combustion process, and an analysis of the nonsteady nozzle flow is used to derive the Nozzle Admittance Relation. The model used in this study can predict stability behavior of combustors with low and high Mach number flows over a wide frequency range. An investigation of stability behavior over a wide frequency range predicts the possibility of spontaneous combustion instability with respect to the pure transverse mode and the mixed acoustic modes (which are associated with a given pure transverse mode). In addition, this study predicts that each of the following effects: (a) increasing the combustor's length, (b) decreasing the Mach number of the mean flow, and (c) “smoothing” the convergence of the nozzle are destabilizing with respect to the pure transverse modes. No general conclusions regarding the stability of the combustor with respect to the various mixed acoustic modes can be drawn. It appears that the stability behavior of each of these modes should be considered separately. Experimental evidence supporting some of these theoretical predictions is presented.

Verification of Nozzle Admittance Theory by Direct Measurement of the Admittance Parameter

Verification of Nozzle Admittance Theory by Direct Measurement of the Admittance Parameter