2.1. Nomenclature and Software
The following sections deal with different types of pressure and cavitation volume time signals in order to distinguish the indices that are used to address each entity individually. Signals and signal sequences from the original pressure measurement are indexed with
m, such as a continuous signal
or the i
th sequence
cut from the original measurement. The volume signal modelled with the polynomial function described in
Section 2.3 is denoted with
p and with an additional
for the randomly manipulated sequences, e.g.,
. The control points of the polynomial function as pairs of time and volume
are denoted with
c and numbered with
j. The mean volume signal shown in
Figure 1 is referred to as
.
All calculations and analyses are performed with MATLAB® version R2021b.
2.2. Data Basis
The analysed sound pressure time signals were recorded during an Atlantic passage of a 3600 TEU container vessel whose particulars are given in
Table 1 [
8]. The measuring conditions are listed in
Table 2 specifying the two operating points used for the present study. The sound pressure was measured by a pressure sensor (PGMCA-200KP, Kyowa Electronic Instruments Co., Novi, MI, USA) mounted in the hull plating at the propeller plane on the vessel’s centre line with a sampling frequency of 2560 Hz and a measurement duration of 120 s each.
For each of the two operating points, the mean signal of sound pressure and cavitation volume are deduced as described in [
13]. Equation (
1) relates the volume acceleration to the sound pressure with the seawater density
of 1025 kg/m
3 and the distance between the sound source and receiver
r of 3.8 m. The simplifications of assuming a constant source–receiver distance, neglecting the free water surface, and considering the scattering effect of the hull plating with a solid boundary factor of 1.4, are outlined in [
13], and are in agreement with [
20]. Integrating the volume acceleration twice, with respect to the time, yields the volume time signal
.
As proposed in [
13], the
max−L method for the sequencing and aligning of all derived blade passages is applied for the calculation of mean signals for one representative blade passage. The
max−L method separates the blade passages by identifying the volume maxima which are unambiguously discernible in contrast to the volume minima which often have two troughs close together. All separated blade passages are stretched to a common signal length and the mean value for each sample point is calculated. The scaling in length helps to average over similar features among all sequences independent from their actual lengths. Thus, the different regions (maxima, collapse, minima) are aligned in order not to distort the mean signal. The resultant volume curve
begins at one volume maximum and ends at the next. In order to create the typical volume evolution of a blade passage the mean signal is cut at its lowest point (understood as the junction between adjacent passages). The second part containing the volume increase is set in front of the former part containing the collapse phase to assemble the volume signal as shown in
Figure 1. The original pressure signal is treated likewise, resulting in the respective mean signal.
Figure 1 also illustrates the strong variability of the individual sequences (light grey).
For the evaluation of individual features of the shape of each volume curve, a different sequencing method
max−cen is applied, which leaves the length of each passage unchanged. As the duration of the different blade passages is an evident feature of the cavitation evolutions it needs to be preserved. Similar to the previously described method the
max−cen method separates the volume signal at the maxima. Thus, the collapse process, already identified as significant for the noise generation [
16] and located between these maxima, remains unaffected by the separation.
Figure 1 shows the highest pressure peaks in the range of the collapsing volume.
2.3. Analytical Model for Systematic Manipulation
The evolution process of a cavity growing and collapsing on a propeller blade has both a repetitive and stochastic nature. So, the first objective of this study is to create a parametric model which takes both aspects into account when simulating a continuous volume signal. The repetitive aspect is represented by the use of the mean signal which ensures that all sequences bear the similarities of a common source. The stochastic nature of cavitation behaviour is taken into account by applying the Monte Carlo method in form of randomised variations to the initial model. By creating a continuous volume signal composed of varied blade passages the effect of stochastic variations on the noise spectrum shall be investigated.
Several requirements are placed on a model that represents the mean volume signal shown in
Figure 1. As it is primarily a means to investigate the noise generation of cavitation it is essential that the pressure signal can be correctly deduced from it. This requires the second derivative of the function in question to represent the mean pressure signal with sufficient accuracy. Further, the parameters defined in the previous section need to be embedded in a manipulable way to establish their effect on the noise spectrum.
Preliminary analyses [
21,
22] show that Gauß and Cauchy distribution functions which resemble the overall shape of the volume evolution well-fulfill the first requirement regarding the differentiability, but lack the possibility of manipulating the collapse region sufficiently. Therefore, the presented model is based on a 14-degree polynomial function instead. The high degree is necessary firstly to represent the volume curve in detail and secondly to ensure an accurate fit of the second derivative presenting the pressure signal; 15 control points are spread within the time span of the mean signals as well as outside of it. The former serves to identify significant features, such as the start, maximum, and end of the curve, and flatten the flanks of growth and collapse. The latter resembles the adjacent volume curves with identical shapes to adjust the curvature at the signal ends. Two control points
and
are chosen in accordance with the evaluation parameters listed in the next section.
Figure 2 shows the original mean signals and the modelled volume curve with its derived pressure signal. The points
and
are marked by the black rectangle.
Since the model’s objective is to be manipulated and lined up in a new artificial continuous signal, special attention has to be paid to the junction of adjacent sequences in order to avoid discontinuities in the resultant pressure signal. The first measure is the definition of control points before and after the sequence time itself (t < 0 s and t > 0.123 s in
Figure 2). Further smoothing of the junctions is described in
Section 2.5 as it is implemented in the setup of the Monte Carlo simulations.
2.4. Evaluation of Shape Parameters
Building on the observation of variations among the large number of individual blade passages [
13] the second objective of this study is to identify influencing parameters that determine the sound pressure produced by the growing and collapsing cavity.
Figure 1 illustrates the correlation between the curvature (second derivative) of the volume curve and the sound pressure signal defined by Equation (
1). Experimental and numerical studies [
9] already attributed higher pressure peaks to more rapid collapse events.
In order to grasp this correlation with a geometrically describable entity the position of the pressure maximum
and the volume minima within each blade passage are analysed.
Figure 3 shows their respective position within each sequence. The diagonal line from the origin to the upper right corner marks where pressure maximum and volume minimum occur at the same position within the sequence. Markers below and right from this line represent sequences where the pressure minimum occurs before the respective volume minimum. Each marker represents one blade passage. The passage with a single volume minimum is marked with
+. Large amounts of blade passages bear two volume minima close together, as discernible in the exemplary sequences of
Section 2.5. In case of these double-volume minima, the positions of both minima are considered (
∘ and
·), those passages produce two markers accordingly but are still counted only once for the histograms. The proximity of the double minima is visible in the horizontal clustering of the respective markers (
∘ and
·). Note that the blade passages with only one volume minimum often bear some kind of plateau within the collapse region just before the single volume minimum as shown in the black rectangle of
Figure 2. This causes the position of the single minimum to coincide with the position of the second of two minima from other passages. Independent from the number of volume minima the blade passages cluster (vertically in
Figure 3) into three main groups depending on the position of the pressure maximum:
- (A)
at t > 0.05 s: pressure peak occurs at the beginning of the subsequent bubble growth
- (B)
at 0.02 s < t < 0.05 s: pressure peak occurs near the end of the collapse phase
- (C)
at t < 0.02 s: pressure peak occurs near the volume maximum (beginning of sequence)
For 96.5 rpm, group B is the largest by far, containing most of the individual passages and thus dominating the mean signals from
Figure 1 showing the pressure maximum near the end of the large bubble’s collapse. For 87.5 rpm, the distribution into the three groups in not as distinct, see
Figure 4. Group A contains roughly half of the passages. While the occurrence of the pressure maximum in course of the bubble collapse (group B) agrees with reported observations, a physical explanation for the pressure maximum at the beginning of the bubble growth (group A) is lacking. Therefore and with a focus on the finding for 96.5 rpm, passages from group B are chosen for the stochastic evaluation.
The distinct separation between passages with a single or a double volume minimum in
Figure 3 and
Figure 4 provides an additional criterion for the identification of shape parameters. The upper histograms show that the two types occur equally often.
Figure 5 shows that passages with a double minimum tend to produce higher pressure peaks than those with a single minimum. A similar distribution is found for 87.5 rpm. This emphasises the influence of the collapse region that is outlined in
Section 3.1. Additional to the influence of the number of volume minima, it allows also the conclusion that an increasing volume amplitude itself also tends to increase the corresponding pressure peak [
13].
With regard to the analytic model described in the previous section the following parameters defining the shape of the volume curve are analysed further. The third and fourth parameters are defined to enable the simulation of single or double minima (see the black rectangle in
Figure 2).
Volume amplitude , taken at the start of each sequence;
Length of each sequence , reciprocal to the blade passing frequency;
Volume at the pressure peak at the collapse region , i.e., the 10th control point of the polynomial function;
Volume in the middle between the pressure peak and the only or second volume minimum , i.e., the 11th control point.
These four features are regarded as stochastic random variables
where
and
are calculated from
n blade passages by
while
and
X represent the above four parameters for clarity whose calculated values are listed in
Table 3. A
–test is performed to ensure the correct assumption of normal distributions. The values of
agree with the corresponding features of the mean signal
, whose polynomial model is used as a basis for the generation of randomly varied sequences (
Section 2.5). Only the height of the 11
th control point is overestimated by the stochastic analysis in comparison to the mean signal. The normalised standard deviation is found to be of equal size in each measuring conditions. The entire process from the raw signal to the manipulable model is displayed schematically in
Figure 6.
2.5. Monte Carlo Simulations
In order to understand the mechanisms that lead to the characteristic noise spectrum of propeller cavitations, artificially varied signals are composed by lining up many single sequences created by manipulating the described polynomial function. Each sequence is altered by factors controlling the feature parameters described in
Section 2.4. For each parameter, as many random numbers are generated based on a normal distribution described by the standard deviation
, such blade passages are to be strung together for the artificial signals. For the purpose of scaling, the modelled signal the stochastic random factors
a and
l have to be generated around a mean of unity and a standard deviation normalised by the mean given in
Table 3. The process of sequence manipulation and the line-up is shown in
Figure 7. The four parameters take effect according to the following equations:
Volume amplitude–scaling with a constant factor
Sequence length–scaling and resampling
Volume at control point
Volume at control point
The scaling of the sequence length
described by Equation (
7) is numerically implemented by changing the number of samples contained by the individual sequence. Depending on the factor
, more or less samples are considered to describe the sequence. Having setup an auxiliary time vector of appropriate length, the original sequence is resampled by interpolation. Since the sampling frequency is assumed to be constant, the reduction of the sample number is effectively the shortening of the sequence and vice versa.
As mentioned in
Section 2.3, the junction between the sequences is a source for discontinuities, especially in the pressure signal which is proportional to the second derivative of the cavitation volume. In order to provide a continuous pressure signal for the spectral analysis, differentiability class C
2 is needed for the volume signal. Since the mean signal starts and ends with zero values, which are unchangeable by the scaling according to Equations (
5) to (
11) only C
0 continuity is provided by the lining-up itself. It leads to artificial distortions of the derived pressure signal as visible in the blue line of
Figure 8. Ideally, C
∞ would ensure that no additional broadband component is created. However, this would require a significant number of samples of both adjacent sequences to be manipulated to obtain a smooth transition changing the actually desired features of each sequence. Therefore, C
2 continuity is chosen as a balance. Adjustments of the curvature (by polynomial control points before and after the actual sequence time span) do not guarantee a smooth transition between the sequences. Therefore, after each amendment of the artificially composed signal by the next manipulated sequence the last four sample points on either side of the junction are deleted and recalculated by spline interpolation for the respective time samples. The spline creates a C
2 connection between the volume sequences. The effect of this smoothing is shown in the orange lines in
Figure 8.
Despite these efforts, a signal assembled of unvaried sequences still bears remaining junction problems that produce a visible though low broadband component which can be observed in the blue curves shown in all Figures of the next section. By applying a Hanning window [
23] to the assembled signal, the effect of a rectangular time window can be reduced to create a spectrum more similar to the ideally clean line spectrum. In the frequency range of interest, the remaining noise floor is approximately 30 dB lower than the broadband noise in the originally measured spectrum (
Section 3.2) and is, therefore, assumed not to affect the results presented in the next sections.