integration of cfd and low-order models for combustion ... - IWR

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factor describing nonuniformity of the enthalpy transport in the axial direction. According to the definition, it is determined by both axial velocity and temperature. That means it also has an influence on the transport of the entropy wave. It is evident from Fig. 13 that the position of maximum amplitude of J 1 is in the recirculation zone. This can be explained by the fact that fluctuation in the inlet air not only alters the fuel vapor distribution in this region, it also alters the vorticity, and thus effects the radial distribution of enthalpy. The results of neglecting J 1 on temperature and velocity perturbation are shown in Figs. 14 and 15, respectively. It is seen that there is large difference in the unsteady temperature and velocity distributions predicted with and without consideration of J 1 , particularly in combustion region. From Fig. 15, we can see that there is large difference in the phase of downstream fluctuation in velocity. This difference can lead to a large error downstream boundary condition and lead to significantly different resonant frequencies. Therefore, it is important to include the fluctuation of J 1 in any stability analysis. In fact, from our previous work, it is clear that the mechanism of rumble involves the resonant interaction of the entropy and acoustic waves. The effects of changes in the radial distribution of axial transport of enthalpy and hence of entropy are described by the non-dimensional factor Jˆ 1 . We see that in the prediction of entropy wave, it is not sufficient to treat the shape function J 1 as steady. An adjustment is needed to cope with time variations in radial non-uniformity, particularly due ¢ to the changing strength of the recirculation vortex. Fortunately, J 1 x¦ t£ ¢ is ¢ influenced primarily by the inlet air flow rate and Jˆ 1 ω ¦ x£¨ ˆm a ω£ can be identified from a single forced CFD calculation as described in section 3. In Eq. (8), F represents the variation of momentum flux when the oscillations changes the nonuniformity of the axial velocity. The magnitude and phase of the relationship between F and inlet air mass flow rate are shown in Fig. 16. It is seen x¦ t£ that there are three peaks in the magnitude plot. Comparison with the flow field contours shows that these ¢ three peaks correspond to the jets induced by inlets 1 and 2, the combustion recirculation zone and the downstream contraction, respectively. It means that the oscillation alters the size or intensity of the jets or vortex, therefore the axial uniformness of momentum transport. A similar conclusion can be drawn about the non-dimensional factor J 2 , which represents the effect of oscillation on the uniformness of kinetic energy in the radial direction. The comparison of the results from a one-dimensional analysis with and without considering F and J 2 are shown in Figs. 17 and 18. It is seen that, although there are some differences, they are small and the predicted the magnitude and phase of the flow parameters are quite comparable. Figures 19 and 20 give time traces of shape factors J 1 and F at the typical location for forcing at a frequency 50 Hz. It is seen that, in each position, the ratios of magnitude of oscillation to mean value for J 1 and F are of the same order. However, as we discussed above, the influence of J 1 on the one-dimensional flow parameters is much greater than that of F. This reinforces our argument that the unsteady entropy transport is more important that those of momentum and kinetic energy. Hence in a simplified analysis, it is justified to neglect any unsteadiness in the shape factors F and J 2 , although it should be retained in J 1 . 5 CONCLUSIONS In this work, a new approach is developed to give predications for instability onset and the frequencies of oscillation. This is based on integrating CFD and low-order models. The CFD provides the transfer function between the heat release rate per unit length and air flow rate through the atomizer by time-dependent calculations of the combustion processes with specified forced inlet boundary conditions. At the same time the transfer functions between non-dimensional shape factors and inlet air flow rate can be calculated. By integrating the ensemble-averaged conservation equations of motion, quasi-onedimensional equations for a linear stability analysis are derived. The effects of any radial non-uniformity on transport of flow parameters are considered by shape factors, which can be obtained from the same CFD calculation. Flow quantities can be determined quickly and straightforwardly by solving the mean flow and the linear perturbation equations. The amplitude and phase relationships of temperature and velocity perturbations, which play an important role in the downstream boundary condition, have been compared with CFD results for the same inlet boundary conditions. The results show good agreement. To simplify the perturbation analysis, the sensitivity to the non-dimensional shape factors was also investigated. It is shown that the inclusion of factor J 1 , which represents the effect of radial nonuniformity on enthalpy transport, is important if the combustion oscillations are to be predicted correctly. The shape factors F and J 2 , which represent the momentum and kinetic energy transports, can be neglected in the linear analysis of combustion oscillation. The strategy outlined here can be expected to have useful practical applications if the flame transfer function and the shape factors, determined from a small number of detailed CFD calculations, can subsequently be applied to linear stability analyses under a wider range of conditions. In the future, we will use the method developed here to predict the instability onset and the frequencies of oscillation. We anticipate the results will be useful to understand the combustion oscillation and be helpful in the development of control strategies. ACKNOWLEDGMENT This work was funded by the Engineering and Physical Sciences Research Council, Rolls-Royce and European Union GROWTH Program, Research project “ICLEAC: Instability Control of Low Emission Aero Engine Combustors”
1 2 / 0 Contract(G4RD-CT2000-0215), whose support are gratefully acknowledged. 3 4 5 REFERENCES [1] M. Zhu, A.P. Dowling and K.N.C. Bray. Combustion oscillations in burners with fuel spray atomiser. ASME paper 98–GT–302, Indianapolis, Indiana, June 1999. ASME International Gas Turbine and Aeroengine Congress and Exhibition. [2] M. Zhu, A.P. Dowling and K.N.C. Bray. Flame transfer function calculations for combustion oscillations. ASME paper 2000–GT–374, New Orleans, Louisiana, June 2001. ASME International Gas Turbine and Aeroengine Congress and Exhibition. [3] M. Zhu, A.P. Dowling and K.N.C. Bray. Self-excited Oscillations in Combustors with Spray Atomisers. ASME paper 2000–GT–108, Munich, Germany, May 2000. ASME International Gas Turbine and Aeroengine Congress and Exhibition. [4] A. P. Dowling, 1995, “The calculation of thermoacoustic oscillations”, Journal of Sound and Vibration, 180(4), pp. 557– 581. [5] A. P. Dowling, 1997, “Nonlinear self-excited oscillations of a ducted flame”, J. Fluid Mech., 346, pp. 271–290. [6] L. Ljung. System Identification Toolbox User’s Guide. The MathWorks Inc., 1991. r(m) 0.27 0.26 0.25 0.24 0.23 0.22 0.21 0.2 0.19 Figure 1. Schematic diagram of the geometry. 6 7 /98 0.18 0 0.05 0.1 0.15 0.2 x(m) /:/ 2000 1800 1600 1400 1200 1000 800 600 400 FIGURES Figure 2. Contour plot of the mean temperature distribution chamber at idle conditions. The black line indicates the mean position of the stoichiometric curve and arrows denote the direction and magnitude of the mean velocity. 4.5 4 3.5 3 mean F 2.5 2 1.5 1 0 0.05 0.1 0.15 0.2 0.25 axial coordinate x(m) Figure 3. The mean of dimensionless shape factor ¯F ¢ x£ , which is defined by Eq. (8).
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