#### 3.3. Flow Instabilities and Unsteady Fluid Forces

The systematic evaluation of the flow instabilities during the cavitating experiments highlighted the presence of rotating and axial phenomena for different flowrates, as briefly summarized in

Table 7. In the table, at the same flowrate, oscillating phenomena showing analogous characteristics (rotating with 1 lobe, axial, etc.) are grouped together regardless of the operating regimes (

$\sigma $, frequency).

The presence of cavitation instabilities leads to unwanted forces on the shaft even at $\Phi ={\Phi}_{D}$. However, at design condition the intensity of the detected oscillating forces is minor than at lower flowrates. For this reason, the present study only focuses on $\Phi $ = 0.108 $\left(\Phi /{\Phi}_{D}=0.9\right)$ in order to show some of the typical pressure-force spectra relationships found at all the tested regimes.

Figure 8 reports the cavitating performance at a nominal

${\Phi}_{N}=0.108$ as well as the flow coefficient behavior during the inlet pressure decay. At the end of the experiment, the massive presence of cavitation leads to the breakdown which, in turn, leads to the flow coefficient drop.

The flow instabilities illustrated in this paper are basically connected with the presence of cavitation. According to [

31], cavitation inception usually starts in the tip vortex generated at the blade inlet. Therefore, it is of no surprise that the major part of the found phenomena is clearly visible by the upstream pressure transducers, becoming less visible while moving downstream. On the other hand, axial phenomena effectively propagate from the upstream to the downstream and vice versa, becoming clearly visible in both stations. In the figures, Ω is intended as the rotating frequency.

Figure 9 reports the energy frequency content of three pressure transducers as representative of the three different stream locations: upstream, midstream, and downstream (according to

Figure 2 and

Figure 3) versus the cavitation number

$\sigma $. The frequency energy content

$\left({E}_{f}\right)$ is directly related to the amplitude

$\left({A}_{f}\right)$ of the acting oscillating phenomenon as follows:

where

${N}_{s}$ is the number of samples considered for the fast Fourier transform (FFT) while

${f}_{s}$ is the sampling frequency exploited during the experiment.

While the blade passage frequency is clearly visible at

$6\Omega $ per each station, the other relevant phenomena are generally of major interest at a single station. In order to understand the physical nature of such oscillating phenomena,

Figure 10,

Figure 11 and

Figure 12 report the phase of the cross-spectrum of the upstream, midstream, and downstream pressure transducers, respectively. The figures show only the phenomena of major interest characterized by an amplitude

${A}_{f}\ge 100$ Pa and a coherence

${\gamma}_{xy}\ge 0.95$, focusing on the range of

$f/\Omega =\left[0,4\right]$ where interesting phenomena have been found. The phase values allow for understanding the nature of the phenomenon [

16,

17]. The main outcomes of this analysis are summarized in

Table 8.

Figure 13 shows the frequency energy content of the force components measured by the dynamometer with a force amplitude oscillation greater than 0.5 N. In particular, the figure reports the force components independent of the chosen reference frame (i.e., rotating or fixed), as described previously.

RC2 and RC3 do not generate relevant effects on the forces sensed by the dynamometer, therefore they won’t be further discussed. On the contrary, some of the instabilities reported in

Table 8 clearly lead to unwanted oscillations of the force acting on the shaft. In particular, the two axial instabilities A1 and A2 generate oscillations of

${F}_{z}$ for corresponding values of

$\sigma -f$, while they are not visible at all on a plane perpendicular to the rotational axis

$\left(\mathrm{XY},xy\right)$.

Like A1 and A2, the pressure distribution connected with the rotating cavitation-induced instability RC1 generates a fluctuating component on the rotational axis.

However, RC1 also leads to a fluctuating component on the plane $\mathrm{XY}\text{}\left(xy\right)$ at the same operating regimes $\left(\sigma \right)$ and frequencies, and with an intensity directly connected to the corresponding energy value.

In order to understand the RC1 effects on the plane

$\mathrm{XY}\text{}\left(xy\right)$, it is useful to analyze the frequency content of the force component

${F}_{X}$ and

${F}_{x}$ defined in the fixed frame and in the rotating one, respectively (

Figure 14). Let’s consider the schematic proposed in

Figure 5 and in the following. The frequency energy content of

${F}_{xy}$ is influenced only by the amplitude and the acting frequency of

$\tilde{{F}_{1}}$. The pressure distribution due to the presence of cavitation in the form of RC1 generates a rotating force imbalance with a rotational velocity corresponding to the rotational velocity of the phenomenon itself, which is given by the pressure transducer analysis. Moreover, when this pressure distribution interacts with the (static) volute tongue at the impeller exit, it may lead to an oscillating unbalanced force at the same frequency as the phenomenon itself. The above considerations may be summarized as

${\omega}_{2}={\omega}_{1}=2\pi {f}_{RC1}$, therefore

which is coherent with the frequency content in

Figure 14. Moreover, the corresponding force on the rotating frame is given by

where there are only three acting frequencies (as shown in

Figure 14, center):

$\left({\omega}_{1}-\Omega \right)/2\pi $, whose intensity depends on the combination of ${\tilde{F}}_{1},{F}_{2},{\theta}_{0}$;

$\left({\omega}_{1}+\Omega \right)/2\pi $;

$\Omega /2\mathsf{\pi}$.