**1. Introduction**

A reversible pump can either supply energy to the fluid or it can obtain energy from the fluid, depending on the direction of rotation of the impeller and the direction of flow. When the pump draws energy from the fluid, it is known as pump as turbine (PAT). The interest in this type of turbomachine is because power generation is less expensive in equipment than a conventional turbine for low power ranges [1–8]. Even though the use of PATs may be the best option for harnessing small hydro resources [9], especially in rural and remote areas with power supply problems [7, 8], the study of this type of turbomachines has not been extensive. This could be verified by a quick search on the Web of Science database, revised on November 1, 2022. There, 259,547 references are reported on pumps, 132,230 references on turbines, and only 281 references on PAT's. Of these 281 references, a general classification was made according to some keywords of interest in the field of turbomachines, which can be seen in **Table 1**.


#### **Table 1.**

*References on the web of science database, revised November 1, 2022.*

**Table 1** shows that the most studied issue related to PAT performance is efficiency, which is obvious given the importance of this aspect in power generation, and the least studied are cavitation and flow instabilities. Cavitation is an important source of instabilities, so it was included in the same query as instabilities. In this group, the following topics are addressed: geometry, efficiency, cavitation [4, 7–21], flow structures [8, 22–25], pressure fluctuations [16, 21, 26, 27], vortex rope [27], energy losses when switching from pump to turbine mode [28], and application of entropy production theory for energy losses [29]. None of them addresses the measurement of instabilities.

The study of instabilities is important because they cause several problems in the performance of turbomachines, such as efficiency losses, noise, and vibrations [12, 14, 16, 30–33] and can even threaten their structural integrity [16, 33]. Given this scenario, it is deemed important to identify the hydrodynamic phenomena that can affect the performance of a PAT and to estimate its level of instability. Brennen's classification [33] is used to establish a framework for flow instabilities in turbomachines. According to this author, hydrodynamic instabilities causing vibrations can be classified into three classes: global flow oscillations, local flow oscillations, and radial and rotodynamic forces. These three classes bring together at least 12 flow instabilities. The total instability at an operating point of a turbomachine may be the combination of several types of instabilities.

Instability analysis can be done using theoretical models; however, when the flows are very complex, it is necessary to use techniques that include experimental information [34]. One of those techniques is the frequency domain analysis of signals from sensors installed in the turbomachine. This is also known as Fourier analysis. An advantage of Fourier analysis is that the signals of the variables of interest obtained in the time domain can be converted into individual frequency components and vice versa. In addition, variables of interest (such as velocity, acceleration, and pressure, ), can be expressed as a sum of their mean and a complex component incorporating the amplitude and phase of the fluctuation [34]. In this paper, the words fluctuation, perturbation, and instability are synonymous.

Although the total instability at an operating point can be regarded as a sum of instabilities of different origins under the linearity assumption [34], not all these sources of instability have periodic characteristics. This means that, using Fourier analysis, only periodic or quasi-periodic instabilities can be clearly identified. Therefore, the main objective of this research is to estimate the instability due to periodic phenomena in a low-specific-speed pump working as a turbine. Concerning periodic instabilities, it is relevant to point out that they can be divided into two groups. The first group corresponds to those that depend on the rotation frequency of the

turbomachine, and the second to those that do not. In this research, only those of the first group were considered.
