Estimating a monotone trend

Currently, waveguide probes are widely used in several turbomachinery applications ranging from the analysis of flow instabilities to the investigation of thermoacoustic phenomena. There are many advantages to using a waveguide probe. For example, the same sensor can be adopted for different measurement points, thus reducing the total number of sensors or a cheaper sensor with a lower operating temperature capability can be used instead of a more expensive one in case of high temperature applications. Typically, a waveguide probe is made up of a transmitting duct which connects the measurement point with a sensor housing and a damping duct which attenuates the pressure fluctuations reflected by the duct end. If properly designed (i.e., with a very long damping duct), the theoretical response of a waveguide has a monotone trend with an attenuation factor that increases with the frequency and the length of the transmitting duct. Unfortunately, the real geometry of the waveguide components and the type of connection between them have a strong influence on the behavior of the system. Even the smallest discontinuity in the duct connections can lead to a very complex frequency response and a reduced operating range. The geometry of the sensor housing itself is another element which contributes to increasing the differences between the expected and real frequency responses of a waveguide, since its impedance is generally unknown. Previous studies developed by the authors have demonstrated that the replacement of the damping duct with a properly designed termination could be a good solution to increase the waveguide operating range and center it on the frequencies of interest. In detail, the termination could be used to balance the detrimental effects of discontinuities and sensor presence. In this paper, an innovative waveguide system leading to a further increase of the operating range is proposed and tested. The system is based on the measurement of the pressure oscillations propagating in the transmitting duct by means of three sensors placed at different distances from the pressure tap. The pressures measured by the three sensors are then combined and processed to calculate the pressure at the transmitting duct inlet. The arrangement of the sensing elements and the geometry of the termination are designed to minimize the error of this estimation. The frequency response achieved with the proposed arrangement turns out to be very flat over a wide range of frequencies. Thanks to the minor errors in the estimation of pressure modulus and phase, the probe is also suitable for the signal reconstruction both in frequency and time domain.

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