#### Омские Новости

June 16th, 2019

Compressor surge and rotating stall vibrations place fundamental limitations on aircraft engine performance and remain persistent problems in the development of axial compressor and fan stages. Compressor surge and rotating stall are purely

fluid mechanic instabilities, while blade flutter, stall flutter, and surge flutter and their variants are aeroelastic instabilities involving both blade vibrations and fluid motion. Although both rotating stall flutter and rotating stall tend to occur when the blades of a compressor or fan are operating at high incidence angles and/or speed, and unsteady viscous flow separation plays a key roll in both of these phenomena, the various fluttering phenomenon are precursors to compressor surge.

Surge is characterized by large amplitude fluctuations of the pressure in unsteady, circumferentially uniform, annulus-averaged mass flow. It is a onedimensional instability that spreads through the compression system as a whole and culminates in a limit cycle oscillation in the compressor map. In most situations, surge is initiated in a compressor when the compressor mass flow is obstructed and throttled. The frequency of surge oscillations is relatively in a low – frequency band (<25-30 Hz) which could couple with the aeroelastic modes of vibration. The performance of the compressor in surge is characterized by a loss in efficiency leading to high aeroelastic vibrations in the blade as well as influences the stress levels in the casing. In jet engines, surge can lead to the so-called flame – out of the combustor which could involve reverse flow and chaotic vibrations.

Based on the amplitude of mass flow and pressure fluctuations, surge was classified into four distinct categories: mild surge, classical surge, modified surge, and deep surge by de Jager (1995). This classification is now widely accepted and is used to differentiate between different forms of surge and rotating stall vibrations. During mild surge, the frequency of oscillations is around the Helmholtz frequency associated with the resonance within a cavity, i. e., the resonance frequency of the compressor duct and the plenum volume connected to the compressor. This frequency is typically over an order of magnitude smaller than the maximal rotating stall frequency which is normally of the same order as the rotor frequency. Classical surge is a non-linear phenomenon such as bifurcation and chaos with larger oscillations and at a lower frequency than mild surge, but the mass flow fluctuations remain positive. Modified surge is a mix of both classical surge and rotating stall. Deep surge, which is associated with reverse flow over part of the cycle, is associated with a frequency of oscillation well below the Helmholtz frequency and is induced by transient non-linear processes within the plenum. Mild surge may be considered as the first stage of a complex non-linear phenomenon which bifurcates into other types of surge by throttling the flow to compressor to lower mean mass flows. Mild surge is generally a relatively low- frequency phenomenon (&5-10 Hz) while rotating stall is a relatively higher- frequency phenomenon (& 25-30 Hz).

There two modes of stable control of a compressor, the first is based on surge avoidance which involves by operating the compressor in instability free domain (Epstein et al. 1989; Gu et al. 1996). Most control systems currently used in industry are based on this control strategy. In this simple strategy, a control point is defined in parameter space with a redefined stability margin from the conditions for instability defined in terms of stall point. This stability margin is defined by (1) typical uncertainties in the location of the stall point, (2) typical disturbances including load variations, inlet distortions, and combustion noise, and (3) a consideration of the available sensors and actuators and their limitations. Generally, a bleed valve or another form of bleeding or recycling of the flow is used to negate the effect of throttling the flow. The control is either the valve position or if one employs an on/off approach as in pulse width modulation, the relative full opening times of the bleed valve in a cycle. Such an approach achieves stability at the expense of performance and the approach is not particularly suitable when the flow is compressible. In short, the surge avoidance approach is not performance optimal. There are also problems associated with the detection of instability. The second mode of control involves continuous feedback control of the mass flow by introducing a control valve or an independently controlled fan. This method involves stability augmentation as the changes in the mass flow will effectively change the conditions for instability and thus increase the stability margin. Rather than operating away from the domain of instability, the domain is pushed further away from the operating point. Based on the experiments performed by a number of earlier researchers (see for example Greitzer 2009), a 20 % increase in mass flow is deemed achievable by this means of stability augmentation.

Several attempts have been made to incorporate the influence of blade dynamics into model for stall prediction. Compressor surge by itself places a fundamental limitation on performance. Hence, active control methods that tend to suppress the various forms of stall will allow the system to be effectively employed over the parameter space prior to the occurrence of surge. Moreover, it is important to consider the various forms of stall in a holistic and integrated fashion as it would be quite impossible to design individual control systems to eliminate each of the individual instabilities. To this end, it is also important to develop a holistic and integrated dynamic model. The model developed by Moore and Greitzer (1986) based on the assumptions that the system is incompressible except in a plenum which is assumed to enclose the compressor and turbine stages, and that radial variations are unimportant, represents compressor surge as a Helmholtz – type hydrodynamic instability. In the original Moore-Greitzer model, an empirical, semi-actuator disk representation of the compressor was used, incorporating Hawthorne and Horlock’s (1962) original actuator disk model of an axial compressor and it served as the basic model incorporating rotating stall. By introducing a semi-empirical actuator disk theory into the model, Moore and Greitzer were able to predict rotating stall and surge. The advantage of the Moore-Greitzer model is the analyst ability to incorporate a host of hysteresis models into the compressor characteristics that permit the prediction of a variety of limit cycle response characteristics. Gravdahl and Egeland (1997) extended the Moore-Greitzer model by including the spool dynamics and the input torque into the same framework as the original model, thus permitting the inclusion of the control inputs into the dynamics. The models may be derived by the application of finite volume type analysis and may also be extended to the case of rotating stall instability and rotating stall induced flutter. In the Moore-Greitzer model, the downstream flow field is assumed to be a linearized flow with vorticity, so a solution of a form similar to the upstream solution can be found. The plenum chamber is assumed to be an isentropic compressible chamber in which the flow is negligibly small and perturbations are completely mixed and distributed. Thus, the plenum acts merely as a ‘‘fluid spring.’’ The throttle is modeled as a simple quasisteady device across which the drop in pressure is only a function of the mass flow rate. Flow variations across the compressor are subject to fluid-inertia lags in both the rotor and the stator, and these lags determine the rotation rate of rotating stall. Stability of rotating stall is determined by the slope of the compressor total-to – static pressure rise map. Greitzer (2009) discussed the possibility of the active control of both stall and rotating stall by controlling the relevant Helmholtz cavity resonance frequencies which could be achieved by structural feedback.

Apart from the numerous methods of synthesizing control laws that have been proposed by the application of linear control law synthesis methods, which are only suitable for the guaranteed stabilization of mild surge, a few non-linear control law synthesis methods have also been proposed. In order to design an active feedback controller that can control deep surge, an inherently non-linear surge-control model is essential. A number of non-linear models have been proposed (Chen et al. 1988; Krstic et al. 1995; Nayfeh and Abed 1999; Paduano et al. 1993; Young et al. 1998), but almost all of these are oriented toward rotating stall control synthesis and include the dynamics of the amplitude of the leading circumferential mode. Many of these models (Gu et al. 1997; Hos et al. 2003) have been employed to perform a bifurcation analysis to explore the behavior of the post-instability dynamics.

In this section, an unsteady non-linear and extended version of the Moore – Greitzer model is developed to facilitate the synthesis of a surge and stall controller. The motivation is the need for a comprehensive and yet low-order model to describe the various forms of stall as well as the need to independently represent the transient disturbance and control inputs in the compressor pressure rise dynamics. Furthermore the extended version of the Moore-Greitzer model is developed by reducing the number of independent model parameters to a minimum. Our preliminary studies indicate that model can effectively capture the dynamics of the phenomenon of compressor surge and that its post-stall instability behavior is well representative of the observed behaviors in real axial flow compressors. The controller is synthesized in two steps. In the first step, the desired equilibrium throttle position and the desired equilibrium value of the ratio of the non-dimensional pressure rise at minimum flow to a quarter of the peak to peak variation of the pressure fluctuation at the compressor exit are established. This defines the equilibrium point and ensures that the desired equilibrium point is stable. In the second step, the margin of stability at the equilibrium point is tuned or increased by an appropriate feedback of change in the mass flow rate about the steady mass flow rate at the compressor exit. The first step may be considered to be an equilibrium point controller while the second corresponds to stability augmentation. Such a two-step process then ensures that both the desired equilibrium solution is reachable and that any perturbations about the equilibrium point are sufficiently stable.

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