Research on propeller cavitation wake characteristics based on multivariate statistical modeling method

Abstract

The radiation noise generated by cavitation has been extensively studied for underwater target recognition, but there are few reports on the related mechanism of the cavitation noise of ship propellers that attract attention in the field of hydroacoustics. In this paper, the RANS equations of the underwater propeller wake field are constructed, and numerically solved by combining the cavitation model and the turbulence model. The power spectrum is used to analyze the signal of the numerical calculation results of the propeller wake pressure. The feature estimation and extraction are carried out to obtain the characteristic values of the specific characteristic parameters. These eigenvalues not only reflect the flow field characteristics but also the geometric parameters and working conditions of the propeller. Therefore, two models are established around the relationship between them. Firstly, these eigenvalues are used for regression analysis in multivariate statistics to obtain a statistical model reflecting the characteristics of propeller cavitation wake. Secondly, the relationship between the propeller skew angle and the low frequency linear spectrum amplitude is obtained by using the power spectrum diagram. In this paper, the processing results of the experimental data of the cavitation water tunnel with controllable parameters and the radiation noise data of the actual target are used to verify and supplement each other with the processing results of the feature model.

Keywords

Propeller cavitation numerical calculation feature extraction regression modeling skew angle analysis

Introduction

Rotating ship propeller, the blade tip speed is often tens of meters per second. This high-speed rotation causes obvious local pressure drop in the seawater around the blade, which makes the blade tip, and even the blade surface near the hub often have strong cavitation. A simple acoustic ship model was proposed to study the physical process of cavitation noise generation by Wittekind and Schuster.¹ The tip vortex cavitation noise, including the primary and fully developed vortex cavitation and surface cavitation, was studied by using the hydrofoil experiment with elliptical plane shape.² In addition, the propeller works in the wake field of the ship, and the hull lines and appendages are not centrally symmetric, so the wake field of the ship has obvious circumferential inhomogeneity. In this way, the relative velocity and pressure of the periodic rotating blade and its contact with the non-uniform wake have periodic changes, so that the intensity of cavitation also occurs corresponding periodic fluctuations. Finally, the rotation of the blade near the propeller area of seawater also caused the blade frequency hydrodynamic pressure pulsation field. Under the action of this fluctuating pressure, a large number of bubbles in this region make forced volume pulsation with the periodic change of flow field environmental pressure. Therefore, the collapse and rebound of a large number of transient cavitation in the propeller blade area cause strong radiation noise, which has the characteristics of high frequency continuous spectrum. With the rotation of the propeller, the periodic forced vibration of a large number of bubbles also causes radiation noise and has the characteristics of low frequency linear spectrum. The pressure fluctuation characteristics of wake field and cavitation noise are also modulated by the propeller rotation beat, and the low frequency linear spectrum characteristics generated by the two are similar.

Radiation noise from propeller cavitation has been studied extensively in recent years.^3–8 Sharma et al.³ was verified that the leading type of cavitation was tip vortex cavitation, and accompanied by leading edge suction side plate cavitation. The noise generated depends on the advance coefficient, cavitation number, propeller geometry and other parameters. Sakamoto and Kamiirisa⁷ studied the complementarity between viscous CFD and semi-empirical formula, proposed and verified a practical calculation method for predicting cavitation noise of near-field propeller.

Experimental measurement noise has been used to study propeller cavitation, but due to many limitations of the experiment itself and limited available data, these studies still cannot fully meet the needs of practical engineering design. Therefore, many researchers prefer to use numerical methods to study the cavitation flow around the propeller. The numerical method with solving Reynolds Average Navier–Stokes (RANS) equations were one of the common methods to predict propeller cavitating flow.^9–13 Zhu⁹ has proved that there was a correlation between the pressure numerical signal of the simulated wake field obtained by the RANS equation and the actual measured noise signal. And the basic characteristics of the actual propeller noise can refer to the basic characteristics of the pressure numerical signal of the wake field. Wu et al.¹² carried out numerical study on cavitation flow around a marine propeller to explore the internal relationship between sheet cavitation and radiation noise. Bai et al.¹³ used the Ffowcs Williams-Hawkings (FW-H) equation to numerically study the cavitation noise induced by the sheet cavity and the tip leakage vortex cavity (TLVC), and obtained the correlation between their radiated noise. Wang et al.¹⁴ and Cheng et al.¹⁵ proposed to solve the Rayleigh-Plesset equation combined with Euler-Lagrangian method for numerical calculation, and achieved good results.

Propeller cavitation has become a research hotspot in the field of fluid mechanics. There are many related works, but the radiation noise caused by cavitation was rarely involved. Although some scholars used statistical theory to study the cavitation noise of propeller, they did not combine the specific characteristics of propeller. And because the cavitation formation mechanism has not been fully revealed, the calculation of propeller cavitation noise based on spherical cavitation theory is difficult and challenging. Based on the work of Zhu et al.,^9,10,17 the further research in these paper were carried out to study the relationship between propeller cavitation noise and propeller operating parameters.

Theories and models

The characteristics of propeller wake include the characteristics of flow field structure and important physical parameters such as pressure pulsation in the flow field. The characteristics of wake field structure mainly include the distribution characteristics of axial velocity and vortex in multiple axial cross sections and longitudinal axial planes. Setting unsteady numerical calculation parameters of propeller cavitation wake in whole basin, and obtaining numerical calculation signal x(i) of propeller cavitation wake pressure fluctuation. The numerical calculation of propeller wake pressure pulsation using modern computational fluid dynamics professional numerical calculation software to build RANS equations of the underwater propeller wake field, and combined with the turbulence model and cavitation model to solve the RANS equations, so as to obtain the relevant information of the volume fraction of the vapor phase around the blade surface of the underwater propeller and the pressure pulsation in the wake field.

Constructing RANS equations

In this paper, the cavitation phenomenon of propeller was studied by using the vapor-liquid mixing uniform two-phase flow model method. The flow field was introduced into the cavitation model by the phase transition rate in the phase transition process. The flow field density is composed of the mixed two-phase flow of the vapor and liquid, which is a function of the volume fraction of the vapor phase. Assuming that the mass fraction of the vapor phase is $f_{v}$ , the mass fraction of the liquid phase is $f_{l}$ . The density $ρ_{m}$ of the mixed flow field can be expressed as:

\frac{1}{ρ_{m}} = \frac{f_{v}}{ρ_{v}} + \frac{f_{l}}{ρ_{l}}

(1)

\frac{\partial}{\partial t} (ρ_{m} f_{v}) + \nabla \cdot (ρ_{m} {\bar{ν}}_{M} f_{v}) = \nabla \cdot (\frac{μ_{t}}{σ_{v}} \nabla f_{v}) + R_{e} - R_{c}

(2)

Where ${\bar{ν}}_{M}$ is mixture velocity, $σ_{v}$ is Prandt1 number of the vapor turbulence, $ρ_{l}$ and $ρ_{v}$ are the density of liquid and vapor respectively. $R_{e}$ and $R_{c}$ are the rates of the vapor generation and condensation, which is expressed by the Full Cavitation Model.¹⁶

The vapor-liquid mixture flow is taken as a flow, and then the governing equations can be written as the mass and momentum conservation of the mixture flow, such that

\frac{\partial}{\partial t} (ρ_{m}) + \nabla \cdot (ρ_{m} v_{m}) = 0

(3)

\begin{matrix} \frac{\partial}{\partial t} (ρ_{m} v_{m}) + \nabla \cdot (ρ_{m} v_{m} v_{m}) \\ = - \nabla \cdot p + \nabla \cdot μ_{m} [(\nabla \cdot v_{m} + v_{m}^{T}) + \frac{2}{3} \nabla \cdot v_{m} I] \end{matrix}

(4)

Where $I$ is unit tensor, $v_{m} = (a_{l} ρ_{l} v_{l} + a_{v} ρ_{v} v_{v}) / ρ_{m}$ is average speed for mass, $ρ_{m} = a_{l} ρ_{l} + a_{v} ρ_{v}$ is the mixture density, $μ_{m} = a_{l} μ_{l} + a_{v} μ_{v}$ is the mixture viscosity.¹⁷

Cavitation model

$R_{e}$ and $R_{c}$ are expressed by full cavitation model, vaporization coefficient and condensation coefficient are obtained by empirical method.¹⁸ When pressure $p < p_{v}$ (vapor pressure), liquid phase become into vapor phase, and bubbles appear. We obtain net vapor condensation rate $R_{e}$ as follows:

R_{e} = C_{e} \frac{k}{γ} ρ_{l} ρ_{v} \sqrt{\frac{2}{3} \cdot \frac{p_{v} - p}{ρ_{l}}} (1 - f_{v})

(5)

When pressure $p > p_{v}$ , vapor phase become into liquid phase. In the same way, we obtain net vapor condensation rate $R_{c}$ as follows:

R_{c} = C_{c} \frac{k}{γ} ρ_{l} ρ_{v} \sqrt{\frac{2}{3} \cdot \frac{p - p_{v}}{ρ_{l}}} f_{v}

(6)

Where $p_{v}$ is saturated vapor pressure, $γ$ is surface tension coefficient, $k$ is turbulence kinetic energy.

Turbulence model

When calculating the Reynolds-averaged Navier-Stokes equation, the turbulent model should be used to close the Reynolds stress term. As another important factor affecting the numerical results of cavitation flow, each turbulence model has different advantages for the simulation of propeller cavitation flow. Common turbulence models include the Standard $k - ω$ , SST $k - ω$ ,¹⁹ and RNG $k - ε$ ²⁰ turbulence models .The SST $k - ω$ turbulence model was improved on the basis of $k - ω$ and $k - ε$ , and has good adaptability to near wall and far field.

In the SST $k - ω$ turbulence model, the transformation of calculation from the inner boundary layer by the Standard $k - ω$ turbulence model to the out boundary layer by the $k - ω$ turbulence model was carried out through a mixture function. The two equations are

\begin{matrix} \frac{\partial}{\partial t} (ρ_{m} k) + \frac{\partial}{\partial x_{i}} (ρ_{m} k u_{i}) = \frac{\partial}{\partial x_{j}} [(μ + \frac{μ_{t}}{σ_{k}}) \frac{\partial_{k}}{\partial x_{j}}] \\ + G_{k} - ρ_{m} β^{*} k ω \end{matrix}

(7)

\begin{matrix} \frac{\partial}{\partial t} (ρ_{m} ω) + \frac{\partial}{\partial x_{i}} (ρ_{m} ω u_{i}) = \frac{\partial}{\partial x_{j}} [(μ + \frac{μ_{t}}{σ_{ω}}) \frac{\partial_{ω}}{\partial x_{j}}] \\ + \frac{ρ_{m}}{v_{t}} G_{k} - ρ_{m} β ω^{2} \\ + 2 (1 - F_{1}) ρ_{m} σ_{ω, 2} \frac{1}{ω} \frac{\partial_{k}}{\partial x_{j}} \frac{\partial_{ω}}{\partial x_{j}} \end{matrix}

(8)

Where $G_{k} = - ρ_{m} \bar{u_{i}^{'} u_{j}^{'}} \frac{\partial u_{i}}{\partial x_{j}}$ is turbulence kinetic energy, $F_{1}$ is mixture functions, $v_{t} = \frac{a_{1} k}{max (a_{1} ω, Ω F_{2})}$ is eddy viscosity coefficient, $β^{*} = 0.09$ , $a_{1} = 0.31$ . The first group of parameters in the model near the wall is $σ_{k, 1} = 1.176$ , $σ_{ω, 1} = 2.0$ , $β_{1} = 0.075$ . The second group of parameters faraway from the wall is $σ_{k, 2} = 1.0$ , $σ_{ω, 2} = 1.168$ , $β_{2} = 0.0828$ . The numerical method is about computational domain, grid generation, grid independence studies, boundary condition, and numerical arithmetic. The details can be found in Zhu.⁹

Numerical calculation

Firstly, the flow grid around the propeller was established, and the number of grid cells is 5 million. Then the digital model of the propeller was imported into the GAMBIT meshing software for meshing, and the meshing method adopts the unstructured meshing method. Secondly, the mesh model was imported into CFD fluid calculation software, and then the initial conditions were input into CFD software, including inflow velocity $U_{\infty}$ , fluid reference pressure $p_{\infty}$ , saturated vapor pressure $p_{v}$ , liquid fluid density $ρ_{1}$ , revolution speed of propeller $n$ , airscrew diameter $D$ . According to these conditions, we can determine the value of the working condition, wherein the advance coefficient is $J = \frac{U_{\infty}}{(nD)}$ and the cavitation number is $σ_{n} = 2 (p_{\infty} - p_{v}) / (ρ_{1} n^{2} D^{2})$ . Thirdly, find a point A with rich propeller information in the wake field of CFD software. After many experiments, the results are shown in Figure 1, point A is located at the radial $r = 0.5 R$ and axial $x = 2 R$ of the propeller wake, the pressure fluctuation test data $p (i)$ at this point is used as the result of numerical calculation test. Finally, the numerical calculation signal $x (i) = \frac{p (i) - p_{\infty}}{ρ m n^{2} D^{2}}$ of the pressure fluctuation of the propeller cavitation wake is obtained.

Figure 1.

Pressure pulsation detection point.

In this paper, the types 4381, 4382, 4383, and 4384 propellers of the DTMB series were used, Figure 2 shows the geometric appearance of the four types of propellers and the cavitation of the propeller under the same cavitation conditions. And Table 1 shows some physical parameters of four types of propellers. Keep the propeller speed $n$ at 25 rps, the inlet velocity was determined by the inlet advance coefficient, the fluid reference pressure is an atmospheric pressure, the saturated vapor pressure is 2368 $Pa$ , the fluid density is 998 $kg / m^{3}$ , the propeller diameter $D$ is 0.992 m. These initial data were input into the CFD fluid calculation software to obtain the mixed fluid density $ρ$ , and the pressure pulsation detection data $p (i)$ at point A was extracted. The data were dimensionless by using Formula $x (i) = \frac{p (i) - p_{\infty}}{ρ n^{2} D^{2}}$ . When the determined numerical calculation was adopted, the unsteady numerical calculation of the wake field of the propeller was carried out under the required working conditions. In the unsteady calculation, the time step is set as $T = 0.005 T_{p}$ and $T_{p}$ are the rotation period of the propeller. The time length involved in the data is 30 $T_{p}$ , and $T_{p}$ is 0.04 s, and the time step is $T = 0.005 T_{p} = 0.2 ms$ . Therefore, the time length involved in the numerical calculation of pressure pulsation is 1.2 s, and a total of 6000 data lengths were collected for the numerical calculation of pressure pulsation.

Figure 2.

DTMB series propeller models.

Table 1.

DTMB series propeller geometric shape parameters.

Paddle model number	4381	4382	4383	4384
Number of blade	5	5	5	5
Disk area ratio	0.725	0.725	0.725	0.725
Hub diameter ratio	0.20	0.20	0.20	0.20
Side angle (°)	0	36	72	108
Diameter (m)	0.992	0.992	0.992	0.992

The numerical calculation results (part of the data) of DTMB series propellers under two working conditions are shown in the pressure pulsation waveform at point A in Figure 3. The five wave crest curves regularly show the characteristics of five-blade propeller. The flow field in this region is also affected by hub vortex, tip vortex, and blade wake, and the characteristic information of wake field is significant and comprehensive. The pressure fluctuation signal also contains abundant spectrum information, which is related to propeller characteristic information.

Figure 3.

Pressure fluctuation waveform of point A.

Feature extraction

The dimensionless pressure pulsation value $x (i)$ collected in the numerical calculation was used for power spectrum analysis and demodulation spectrum calculation, in which the direct method was used for power spectrum analysis. The direct method, also known as the periodic graph method, treats the $N$ observed data of the random sequence $x (n)$ as a sequence with limited energy, directly calculates the discrete Fourier transform of $x (n)$ to obtain $x (k)$ , and then divides the square of its amplitude by $N$ to obtain the real power spectrum estimation of the sequence $x (n)$ . The flowchart of the periodic diagram method for power spectrum analysis is shown in Figure 4.

Figure 4.

Flow chart of periodic diagram method.

The power spectrum of periodic graphs is estimated as follows:

P (k) = \frac{1}{6000} | X {(k)}^{2} |, X (k) = \sum_{n = 0}^{n = 5999} x (i) e^{- ji \frac{2 π k}{6000}}

(9)

The demodulation spectrum adopts the absolute value method. The absolute value of the collected signal $x (i)$ was taken, and then the low-pass filter was carried out. Finally, the spectrum analysis was carried out by DFT. The power spectral density transformation program was written in MATLAB software to realize the signal transformation from time domain to frequency domain. And according to the power spectrum, the amplitude of propeller shaft frequency, double shaft frequency, triple shaft frequency, quadruple shaft frequency, blade frequency, and double blade frequency were extracted, and the logarithm of the extracted amplitude was taken as the characteristic parameter reflecting the characteristics of propeller cavitation flow field.

The obtained pressure pulsation signal $x (i)$ was input into MATLAB for power spectrum analysis, and the signal was transformed from time domain to frequency domain. The amplitude parameters of low frequency linear spectrum were extracted, and then the logarithm of the parameters was taken to obtain the power spectrum of $x (i)$ , as shown in Figure 5. The amplitude parameters of linear spectrum in the spectrum were extracted, including the amplitude of shaft frequency, double shaft frequency, triple shaft frequency, quadruple shaft frequency, blade frequency, and double blade frequency. After taking the logarithm, the fine feature extraction analysis was carried out, and the specific characteristic parameters were obtained. Then a new pressure pulsation signal was obtained by changing the advance coefficient J, and the required low frequency linear spectrum amplitude parameters were extracted by power spectrum analysis. Repeat this process, the amplitude analysis of low frequency linear spectrum under five operating conditions was obtained.

Figure 5.

Power spectrum diagram.

In this paper, the power spectrum of propeller cavitation under five advance coefficients (0.71, 0.789, 0.887, 1.014, 1.183) was analyzed. Tables 2 and 3 are the spectrum amplitude of pressure pulsation calculation signal line when the advance coefficients are 0.71 and 0.887. Through the analysis of the data, it is found that the amplitude of the linear spectrum of the double blade frequency changes significantly with the change of the propeller advance coefficient, while other shape parameters such as the angle and size of the propeller cannot see obvious changes, and the correlation is small. Then, the relationship model between the amplitude of the double blade frequency and the advance coefficient is analyzed. Table 4 shows the change of the linear spectrum amplitude of the 4382 propeller under five kinds of advance coefficients.

Table 2.

J = 0.71 Comparison of amplitude of low-frequency linear spectrum.

J = 0.71	Shaft frequency (lg)	Double shaft frequency (lg)	Triple shaft frequency (lg)	Quadruple shaft frequency (lg)	Blade frequency (lg)	Double blade frequency (lg)
n = 25 rps	Shaft frequency (lg)	Double shaft frequency (lg)	Triple shaft frequency (lg)	Quadruple shaft frequency (lg)	Blade frequency (lg)	Double blade frequency (lg)
4381 propeller	2.42	2.80	1.59	0.43	4.22	0.22
4382 propeller	1.86	2.18	1.29	1.35	4.35	0.23
4383 propeller	1.89	2.19	1.63	1.10	4.32	0.33
4384 propeller	2.74	2.08	1.67	1.24	4.15	−1.71

Table 3.

J = 0.887 Comparison of amplitude of low-frequency linear spectrum.

J = 0.887	Shaft frequency (lg)	Double shaft frequency (lg)	Triple shaft frequency (lg)	Quadruple shaft frequency (lg)	Blade frequency (lg)	Double blade frequency (lg)
n = 20 rps	Shaft frequency (lg)	Double shaft frequency (lg)	Triple shaft frequency (lg)	Quadruple shaft frequency (lg)	Blade frequency (lg)	Double blade frequency (lg)
4381 propeller	1.87	2.45	1.52	1.89	3.90	1.2
4382 propeller	1.73	1.94	1.48	1.44	4.10	0.87
4383 propeller	1.24	1.53	0.89	0.36	3.92	0.86
4384 propeller	1.91	1.69	0.89	1.21	3.97	0.10

Table 4.

Double blade frequency amplitude of 4382 propeller.

4382 propeller	J = 0.71	J = 0.789	J = 0.887	J = 1.014	J = 1.183
Double blade frequency amplitude (lg)	0.23	0.68	0.87	1.17	1.42

Modeling analysis

Regression analysis

Various characteristic parameters reflecting the characteristics of the cavitation flow field of the propeller were extracted from the power spectrum. Then the correlation between them is analyzed by comparing with the corresponding operating parameters of the propeller and the geometric shape parameters of the propeller, so as to establish the characteristic relationship model from the characteristic parameters of the propeller target to the characteristic parameters of the cavitation flow field. The characteristic relationship model includes the relationship model between the key geometric shape parameters of propeller such as blade number, diameter, skew angle, and disk-to-plane ratio and the fine characteristic parameters of cavitation flow field, including a characteristic relationship model between propeller operating parameters such as advance coefficient and cavitation number (including ship speed, propeller speed, and reference pressure) and fine characteristic parameters of cavitation flow field is included. This paper uses two models to analyze the regression curves between them.

Model 1

The linear spectrum amplitude extracted from the power spectrum was used as the dependent variable $y$ , and the propeller working condition parameters or propeller geometric parameters were used as the independent variable $x$ for the univariate quadratic polynomial regression analysis to obtain the regression model.

y = a_{2} x^{2} + a_{1} x + a_{0}

(10)

The confidence interval with significance of 95% is made, and the regression model curve was fitted. The above model is used to predict the characteristics of propeller linear spectrum amplitude, and then the noise modulation related characteristics are referred and guided.

The confidence interval calculation method is as follows:

P_{r} (c 1 < μ < c 2) = 1 - α

(11)

In the formula, $α$ is significant level, $P_{r}$ is probability, $(1 - α)$ is confidence level.

Using the data in Table 4, a polynomial regression analysis is made for the advance coefficient and the amplitude of the double blade frequency linear spectrum. The first model takes the advance coefficient as the independent variable $x$ and the amplitude of the double blade frequency amplitude as the dependent variable $y$ . According to the formula of Model 1, a regression model about the advance coefficient and the amplitude of the double blade frequency linear spectrum is obtained.

y = - 3.756 x^{2} + 9.492 x - 4.565

(12)

As shown in Figure 6, the real curve is the curve fitting of the regression model, and the range between the two imaginary lines is 95% confidence interval $y \pm 0.4219$ .

Figure 6.

Model 1 regression curve.

The regression model can predict that the amplitude of the double blade frequency corresponding to the advance coefficient of 0.946 is 1.0531. The amplitude of double blade frequency increases gently with the increase of the advance coefficient. The Model 1 can be used to predict the amplitude of double blade frequency of five-blade propeller.

Model 2

Taking the propeller working condition parameters or propeller geometry parameters as dependent variable $y$ , and the linear spectrum amplitude extracted from the power spectrum diagram as independent variable $x$ , a quadratic polynomial regression analysis is performed to obtain the regression model .

y = b_{2} x^{2} + b_{1} x + b_{0}

(13)

The confidence interval with significance of 95% was made, and the regression model curve was fitted. The above model can be used to estimate and judge the characteristics of propeller working condition parameters or propeller geometric shape parameters, and then the linear spectrum amplitude can be used to refer and guide the characteristics of propeller working condition or propeller geometric shape.

The second model takes the double blade frequency amplitude as the independent variable $x$ and the advance coefficient as the dependent variable $y$ . According to the second formula, a regression model about the double blade frequency amplitude and the advance coefficient is obtained:

y = 0.2603 x^{2} - 0.03207 x + 0.7021

(14)

As shown in Figure 7, the real line is the curve fitting of the regression model, and the range between the two imaginary lines is a 95% confidence interval of significance $y \pm 0.0782$ . The regression model can predict that the advance coefficient corresponding to the double blade frequency amplitude of 1.0531 is 0.957. The advance coefficient increases with the increase of the amplitude of the double blade frequency. Model 2 can be used to estimate and judge the characteristics of the advance coefficient of the five-blade propeller according to the amplitude of the double blade frequency.

Figure 7.

Model 2 regression curve.

Skew angle analysis

The propellers of ships are divided into non-skew propeller, low skew propeller and high skew propeller. And the wake field characteristic information of propellers with different skew angles is also different. Long et al.²¹ used large eddy simulation to simulate the cavitation flow around the traditional propeller and the highly skewed propeller. And the results show that the vortex structure and cavitation phenomenon of the traditional propeller and the high skew propeller were very different. Therefore, this paper wants to establish a model that can predict the skew angle of the propeller through the relevant information of the cavitation wake field. And correctly predicting the propeller skew angle is conducive to ship target recognition.

In this paper, the amplitude values of shaft frequency, double shaft frequency, triple shaft frequency and quadruple shaft frequency obtained by power spectrum analysis and the known propeller skew angle types were used for judgment and analysis, as shown in Figure 4. It is found that when the propeller is a high skew propeller (the skew angle is higher than 100°), the amplitude of the low frequency linear spectrum has a linear decrease with the increase of frequency. When the propeller is low skew propeller or non-skew propeller (the skew angle is less than 100°), the double shaft frequency linear spectrum is obvious, and the amplitude of the low frequency linear spectrum does not linearly decrease with the increase of frequency.

For the power spectrum of five-blade propeller, the low frequency linear spectrum amplitude approximately linearly decreases with the increase of frequency, that is the linear spectrum amplitude satisfies the following formula.

\begin{matrix} (\lg f_{1} - \lg f_{2}) / (\lg 50 - \lg 25) = (\lg f_{2} - \lg f_{3}) / (\lg 75 - \lg 50) \\ = (\lg f_{3} - \lg f_{4}) / (\lg 100 - \lg 75) \end{matrix}

(15)

Then the propeller is judged to be high skew propeller. On the contrary, when the linear spectrum of the double shaft frequency is obvious, it is a low skew propeller or a non-skew propeller. In which the $f_{1}$ is shaft frequency amplitude, $f_{2}$ is the double shaft frequency amplitude, $f_{3}$ is the triple shaft frequency amplitude, and $f_{4}$ is the quadruple shaft frequency amplitude.

For the power spectrum of four-blade propeller, the low frequency linear spectrum amplitude approximately linearly decreases with the increase of frequency, that is the linear spectrum amplitude satisfies the following formula.

(\lg f_{1} - \lg f_{2}) / (\lg 50 - \lg 25) = (\lg f_{2} - \lg f_{3}) / (\lg 75 - \lg 50)

(16)

Then the propeller is judged to be high skew propeller. On the contrary, when the linear spectrum of the double shaft frequency is obvious, it is a low skew propeller or non-skew propeller. In which the $f_{1}$ is the shaft frequency amplitude, $f_{2}$ is the double shaft frequency amplitude, and $f_{3}$ is the triple shaft frequency amplitude.

The DTMB series propellers used in this paper are all five-blade propellers. Therefore, combined with the data in the table, the skew angle formula (15) is substituted. The results show that the 4384 propeller with high skew angle satisfies the skew angle formula, that is, the amplitude of low-frequency linear spectrum of power spectrum (shaft frequency, double shaft frequency, triple shaft frequency, and quadruple shaft frequency) decreases linearly with the increase of frequency. The 4381 propeller without skew angle and the 4382 and 4383 propellers with low skew angle do not meet the skew angle formula, and their double shaft frequency spectrum is obvious.

Measurement noise analysis

In order to verify the rationality of the skew angle formula, the propeller noise signal in the flume experiment is analyzed in this paper. The propeller used in this test is E779A, it is a four-blade low skew propeller. The parameter setting of the collected propeller noise signal as follows: the sampling frequency is about 200 kHz, the inflow velocity is 3.25 m/s, and the fluid pressure is 113 kPa. The noise signal of propeller speed from 11 to 29 rps is collected in the experiment, where 11–25 rps is non-cavitation noise signal and 25–29 rps is cavitation noise signal. Therefore, we collected noise signals at propeller speeds of 15 and 28 rps for analysis.

The power spectrum analysis of the noised signal is carried out, and the low frequency information is extracted. As shown in Figures 8 and 9, the amplitude of each shaft frequency and the amplitude of multiple shaft frequencies are very prominent, of which the amplitude of the double shaft frequency is the largest, and the amplitude of the one shaft frequency and the triple shaft frequency and the double blade frequency are similar. Table 5 lists the amplitude of each shaft frequency, then combined with the skew angle formula for calculation. Formula (17) is the skew angle formula of propeller speed $n = 15 r / s$ , and formula (18) is the skew angle formula of propeller speed $n = 28 r / s$ .

(\lg f_{1} - \lg f_{2}) / (\lg 30 - \lg 15) = (\lg f_{2} - \lg f_{3}) / (\lg 45 - \lg 30)

(17)

(\lg f_{1} - \lg f_{2}) / (\lg 56 - \lg 28) = (\lg f_{2} - \lg f_{3}) / (\lg 84 - \lg 56)

(18)

Table 5.

Low-frequency linear spectrum amplitude of four-blade propeller.

Low frequency amplitude	Shaft frequency	Double shaft frequency	Triple shaft frequency	Blade frequency	Double blade frequency
n = 15 rps	5.649	6.038	5.748	4.902	4.267
n = 28 rps	5.384	6.302	5.669	5.504	5.465

Figure 8.

The measured noise power spectrum of N = 15 rps.

Figure 9.

The measured noise power spectrum of N = 28 rps.

The calculation formula is not equal at both ends, so it is judged that this propeller is a low skew propeller or a non-skew propeller. This result is consistent with the physical parameters of the experimental propeller.

Conclusion

In this paper, the multivariate statistical modeling method of fine characteristics of propeller cavitation wake is studied. The double model is successfully applied to establish the relationship curve between the amplitude of double blade frequency and the advance coefficient, as well as the analysis of propeller skew angle. The following conclusions are obtained:

The pressure fluctuation signal characteristic model of multi-domain cavitation flow field is constructed, and the mapping relationship between characteristic parameter space of cavitation flow field and propeller and flow field parameter space is clarified. Compared with the traditional statistical analysis and extraction method based on propeller noise samples, the regression model is obtained by numerical calculation of cavitation flow field, power spectrum feature extraction and multivariate statistical modeling method. Through Model 1, the advance coefficient can be used to predict the characteristic of the double blade frequency amplitude of the propeller, and then the noise modulation related characteristics can be referred. Through the correlation analysis of the double blade frequency amplitude obtained by the power spectrum analysis and the propeller advance coefficient, it is concluded that the advance coefficient increases with the increase of the double blade frequency amplitude, and then the relevant information of the propeller is obtained.

Combined with the skew angle formula and the linear spectrum amplitude, it can be judged whether the propeller is high skew propeller or low skew propeller or non-skew propeller. The DTMB series propellers studied in this paper are five-blade propeller, of which 4381 propeller is non-skew propeller, 4382 and 4383 propellers are low-skew propeller, and 4384 propeller is high-skew propeller. Finally, the experimental verification proves that only 4384 propeller conforming to the skew angle formula are high skew propeller, while the other three types do not conform to the skew angle formula, and their double shaft frequency spectrum is obvious, so they are low skew propeller or non-skew propeller.

In this paper, the relevant research results in the fields of modern fluid mechanics, cavitation dynamics, statistics, and signal processing are introduced, which reflects the interdisciplinary nature of multidisciplinary and multidisciplinary, and has important application value and application prospect.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Key Research and Development Projects of Anhui Province [No. 2022107020012].

ORCID iD

Kun Zhou

References

Wittekind

Schuster

. Propeller cavitation noise and background noise in the sea. Ocean Eng 2016; 120: 116–121.

Higuchi

Arndt

REA

Rogers

. Characteristics of tip vortex cavitation noise. J Fluids Eng 1989; 111(4): 495–501.

Sharma

Mani

Arakeri

. Cavitation noise studies on marine propellers. J Sound Vib 1990; 138(2): 255–283.

Bark

. Prediction of propeller cavitation noise from model tests and its comparison with full scale data. J Fluids Eng 1985; 107(1): 112–119.

Bertetta

Brizzolara

Gaggero

, et al. CPP propeller cavitation and noise optimization at different pitches with panel code and validation by cavitation tunnel measurements. Ocean Eng 2012; 53: 177–195.

Lafeber

Bosschers

van Wijngaarden

. Computational and experimental prediction of propeller cavitation noise. In: OCEANS 2015 - Genova, Genova, Italy, 18–21 May 2015, pp.1–9. New York, NY: IEEE.

Sakamoto

Kamiirisa

. Prediction of near field propeller cavitation noise by viscous CFD with semi-empirical approach and its validation in model and full scale. Ocean Eng 2018; 168: 41–59.

Aktas

Atlar

Turkmen

, et al. Propeller cavitation noise investigations of a research vessel using medium size cavitation tunnel tests and full-scale trials. Ocean Eng 2016; 120: 122–135.

Zhu

. Characteristic correlation between propellers cavitating wake and cavitation noise. Appl Acoust 2014; 81: 31–39.

10.

Zhu

Fang

. Numerical investigation of cavitation performance of ship propellers. Journal of Hydrodynamics 2012; 24(3): 347–353.

11.

Yang

Wang

Zhang

. Scale effects on propeller cavitating hydrodynamic and hydroacoustic performances with non-uniform inflow. Chin J Mech Eng 2013; 26: 414–426.

12.

Huang

Wang

, et al. Numerical modelling of unsteady cavitation and induced noise around a marine propeller. Ocean Eng 2018; 160: 143–155.

13.

Bai

Cheng

. LES investigation of the noise characteristics of sheet and tip leakage vortex cavitating flow. Int J Multiphase Flow 2022; 146: 103880.

14.

Wang

Cheng

. Euler–Lagrange study of cavitating turbulent flow around a hydrofoil. Phys Fluids 2021; 33(11): 112108.

15.

Cheng

Long

, et al. A new Euler-Lagrangian cavitation model for tip-vortex cavitation with the effect of non-condensable gas. Int J Multiphase Flow 2021; 134: 103441.

16.

Menter

. Two-equation eddy-viscosity turbulence models for engineering applications. AIAA J 1994; 32(8): 1598–1605.

17.

Zhu

Fang

Wang

. Characteristic analysis of unsteady viscous flow around a cavitating propeller. Sci China Technol Sci 2010; 53: 1983–1992.

18.

Singhal

Athavale

, et al. Mathematical basis and validation of the full cavitation model. J Fluid Eng 2002; 124(3): 617–624.

19.

Orszag

Yakhot

. Renormalization group modeling and turbulence simulations, near wall turbulent flows. Amsterdam: Elsevier, 1993.

20.

Zhu

W R

Gao

Z X

Tang

Y J

, et al. Adaptability of turbulence models to predict the performance and blade surface pressure prediction of a Francis turbine. Eng Comput 2016; 33(1): 238–251.

21.

Long

Han

, et al. Verification and validation of large eddy simulations of turbulent cavitating flow around two marine propellers with emphasis on the skew angle effects. Appl Ocean Res 2020; 101: 102167.