International Journal of Rotating Machinery

Volume 2018 (2018), Article ID 2909546, 12 pages

https://doi.org/10.1155/2018/2909546

## Spatial Fluctuating Pressure Calculation of Underwater Counter Rotating Propellers under Noncavitating Condition

School of Naval Architecture and Ocean Engineering, Dalian Maritime University, No. 1 Linghai Street, Ganjingzi District, Dalian, Liaoning 116026, China

Correspondence should be addressed to L. X. Hou

Received 17 August 2017; Revised 25 December 2017; Accepted 4 January 2018; Published 19 February 2018

Academic Editor: Gerard Bois

Copyright © 2018 L. X. Hou and A. K. Hu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The spatial fluctuating pressure field (FPF) of counter rotating propeller (CRP) under noncavitating condition is investigated. The hydrodynamic performance and pressure distributions on the blade surfaces are obtained through low-order potential-based panel method, which is also used to analyze the hydrodynamic interaction between the front and rear propellers of CRP as well as the hydrodynamic interference between any solid surface and propeller. The interaction between the given solid spherical surface and propeller is used to simulate the spatial FPF of propeller, and the fluctuating pressure induced by a propeller over one revolution is analyzed in frequency domain through fast Fourier transform. The method proposed is validated through two given propellers by comparing the calculation results with test data. The FPFs of the front and rear propellers are calculated and compared with that of the corresponding single propeller. The result shows that the CRP produces weaker FPF compared with the single propeller.

#### 1. Introduction

Working in the nonuniform wake field behind ship hull, the propeller is subjected to unsteady surface loadings which lead to fluctuating pressure. It is well known that the fluctuating pressure can cause serious vibration problem on ship hull as well as the appendages behind the vessel. The fluctuating pressure induced by marine propeller can be classified into cavitation and noncavitation fluctuating pressure. The marine propeller cavitation is the most prevalent source of underwater fluctuating pressure in ocean. To the authors’ knowledge, a lot of work has been done on the fluctuating pressure analysis of marine propeller under cavitation condition. Breslin et al. [1] studied propeller-induced hull pressures arising from intermittent blade cavitation, loading, and thickness using theoretical method coupled with experiments. Numerical methods have been developed based on surface panel method to compute the fluctuating pressure of cavitating propellers and compared the computational results with experimental data [2]. Seol [3] and Berger et al. [4] addressed the pressure fluctuation induced by a propeller sheet cavitation. The developed time domain prediction methods provided reasonable results, and these results are in good agreement with the experimental results. Kanemaru and Ando [5] simulated unsteady sheet cavitation patterns, cavity volume evolution, and pressure fluctuations around marine propellers in nonuniform wake using the simple surface panel method SQCM with consideration of viscous effects to improve the calculation accuracy. The CFD technology has been widely used to predict the fluctuating pressure generated by cavitating propeller. Kawamura and Kiyokawa [6] simulated cavitating flows around a propeller rotating in a ship wake. Their results demonstrated that the magnitude of the pressure fluctuations increased greatly during cavitation, though the pressure fluctuations associated with the cavitation were still underestimated and the higher frequency components were not reproduced. Sato et al. [7] predicted the sheet cavitation behavior and pressure fluctuations using CFD software. Their simulations accurately predicted the 1st blade frequency component of the pressure fluctuations, while the high frequency components were severely underestimated due to the inability to simulate the tip vortex cavitation. Ji et al. [8] verified that the acceleration due to the cavity volume changes was the main source of the pressure fluctuations excited by the propeller cavitation by adopting the CFD technique. Lloyd et al. [9] calculated the pressure pulses inside the cavitation tunnel using the computational fluid dynamics code ReFRSCO. In order to predict the pressure pulse, Peralli et al. [10] investigated the wetted and cavitating flow around the INSEAN E779A propeller in a cavitation tunnel using the uRANS and BEM-BEM, respectively, and discussed the pro and cons of these two methods. Bensow and Gustafsson [11] investigated how hull forces and pressure are influenced by small propeller tip clearance by creating a setup where systematic variation of tip clearance could be achieved at similar propeller conditions. The simulations were performed using a scale resolved PANS approach combined with cavitation modelling considering the fluid as a mixture and incorporating mass transfer source terms.

However, submarines and torpedoes are usually operated deep enough under the sea to avoid cavitation [12], and some low speed vessels do not have noticeable cavitation phenomenon. It is of great significance to have investigations about the fluctuating pressure generated by marine propeller under noncavitating condition. Early researches about fluctuating pressure generated by propeller under noncavitating condition mainly obtained some empirical formulas according to a large quantity of accumulated data. Garguet and Lepeix [13], Tsakonas et al. [14], and Hu et al. [15] considered the influences of the hull and the free surface by introducing solid wall correction factor. Chen and Zhou [16] calculated the noncavitating fluctuating pressure of propeller in given wake field through CFD software. Güngör and Bedii Özdemir [17] investigated the performance of an inclined propeller in both noncavitating and cavitating conditions using a finite volume based solver and compared the results with the experimental data. The sliding mesh technique was used to implement the rotations in URANS solver with the renormalization group (RNG) -*ε* turbulent model. In the noncavitating case, the amplitudes of the pressure fluctuations were in agreement with the experimental data, but those of pressure pulses in the cavitating condition were underpredicted.

As the world's energy shortage problem gets increasingly serious and the energy efficiency design index (EEDI) for new ship came into effect on January 1, 2013, reducing fuel consumption and building green ship not only relate directly to the operating costs but also help to deal with other risk factors [18]. As the power source of ships, efficient propulsion machinery will reduce fuel consumption and operation costs. The application of the counter rotating propeller (CRP) has been developed significantly in last decades. Compared with single propeller, both of the front and rear propellers of CRP have lower loadings while supplying the same thrust as the single propeller. Therefore, the CRP has better cavitation performance under the same operating condition. Thus, it is significant to investigate the noncavitation fluctuating pressure of CRP theoretically. Nowadays, works concerning the fluctuating pressure field of CRP are hard to find.

In the present paper, the spatial FPF of CRP under noncavitating condition is investigated in detail. The low-order potential based panel method is adopted throughout this study, and the computation formulas for fluctuating pressure prediction are proposed. The calculation program based surface panel method is validated by comparing the calculation results and test data of DTRC P4118. Two given single propellers’ fluctuating pressures are calculated and the results are compared with the test data to validate the fluctuating pressure prediction method. Then the FPFs of a set of CRP and the corresponding single propeller are calculated. The hydrodynamic interaction between the front and rear propellers of CRP is considered. Through analysis, the FPF characteristics of CRP can be obtained, which will provide a basis for proper stern vibration control strategies under noncavitating condition.

#### 2. Methods

The panel method has been proved to be effective for hydrodynamic performance prediction of propeller [19]. Studies by Hsin [20] and Kinnas and Fine [21] have given specific description about the details and fundamentals of panel method. Thus, this paper only gives a brief description. This method derived from Green’s theorem and the velocity potential at a fixed point located anywhere in the flow field can be expressed as follows:where , represent the propeller surface and the wake surface respectively, is the potential jump across the wake sheet, represents the velocity potential at any point on , and denotes Green’s function. In the case of unbounded three-dimensional fluid domain, is given as with being the distance between points and . in (1) has values as follows:(i), if* p* lies in the flow field, but not on .(ii), if lies on .(iii), if lies outside .

The propeller surface and wake sheet are discretized with hyperboloidal panels. A constant source and a constant dipole are then distributed on each panel. During unsteady calculation, the integral equation (1) is solved at each time step, and time-dependent terms are updated at the next time step. The specific unsteady treatment is given in literature [20]. The time domain is discretized into equal time intervals , and (1) at each time step can be discretized as follows:where is the number of blades. For each blade, is the number of chordwise panels, is the number of spanwise panels, is the total number of panels, and is the number of chordwise panels in the wake. The influence coefficients and are defined as the potentials induced at panel by unit (constant) strength dipole and source distributions, respectively, located at panel on blade . The wake influence coefficients are defined similarly. The definitions of the influence coefficients are given in the dissertation of Hsin [20].

As the propeller hub is nonlifting, the dipoles on the panels of hub are set to zero, so just a constant source is distributed on each panel of hub. Once the velocity potential is determined, the velocities on the propeller surfaces can be obtained by differentiating the resulting velocity potential. Then the pressure distribution is calculated through Bernoulli’s equation.

In this paper, the solid surface is used for spatial fluctuating pressure field analysis of propeller. The local intensity of the distribution is denoted by , where the source point now denotes a general point of the surface , and then the normal velocity boundary condition must be satisfied at any point on the surface where is the distance between and , is the inflow velocity of point , is the velocity induced by the propeller at point , and can be obtained through panel method.

As the solid surface just has source distribution, the velocity induced by the solid surface at any point on the propeller surface can be expressed as follows:where is the distance between and .

The solid surface is divided into a number of quadrilateral source panels; the dimensions of which are small in comparison with the surface. The solution is constructed in terms of the source strengths on the surface. The integral equation for the source strengths is approximated by a matrix equation on the assumption of uniform strength on each panel. Equations (4)~(5) can be discretized as follows:Here, is the panel number of the solid surfaces, is the inflow velocity of the th panel, and is the velocity induced by the propeller on the th panel of the solid surface. For (6), the second term is zero when the th and th panels are on the same plane; namely, . is the distance between the point and the th panel of the solid surface.

The velocity induced by the solid surface on the propeller surface can be obtained through (8). By adding the velocities induced by the solid surface to the inflow velocities the unsteady hydrodynamic performance of the propeller as well as the velocities induced by the propeller on the solid surface panels are recalculated. Solving (6) can get new source strength . Through this iterative calculation, the stable source strengths and velocities on the panels of the solid surface can be obtained at each time step. The pressure distribution on the solid surface is obtained through Bernoulli’s equation. The hydrodynamic performance coefficients can be defined as follows:where is the advance coefficient, is the ambient fluid density, and , respectively, denote the rotational speed and the diameter of the propeller, is the ship speed, and are the thrust and torque of the propeller, and and are the thrust and torque coefficients of propeller, respectively.

As the propeller works in the nonuniform flow field, the pressure on the solid surface changes periodically. The Fourier transform is adopted to transform the pressure signals from the time domain to frequency domain, and the fluctuating pressure coefficient can be obtained:where is the th blade frequency harmonic component of the fluctuating pressure (zero to peak) and represents the fluctuating pressure coefficient of the th blade frequency harmonic component.

#### 3. Method Validation

The surface panel method proposed in this paper is applied to the propeller DTRC 4118 whose specific parameters are given by Hsin [20]. The panel arrangements of propeller model and slipstream of the key blade are illustrated in Figure 1. The cosine spacing is employed in the chordwise direction and spanwise direction of the propeller blade. The calculation accuracy will be much satisfying when the propeller blades are discretized using more than 14 chordwise panels and 14 spanwise panels [19]. The panel number of the propellers studied in this paper is much more than 14 × 14, and the calculation accuracy can be ensured. What is more, the slipstream model is linear. The linear slipstream model can satisfy the calculation accuracy required well [22] and is also used for other calculations of this paper. The incoming wake is assumed to be axial, with a once per revolution circumferential variation:with being the amplitude of the wake variation, taken equal to 0.2. The advance coefficient is 0.833, and the propeller is operated at 120 rpm with a forward velocity of 1.6 m/s. The density of the undisturbed medium of standard water is 1026 kg/m^{3}. In order to validate the panel method given in this paper, the calculation results are compared with the results of Hsin. The comparison results are shown in Figure 2, which gives the circulation distributions versus blade angle at two propeller radiuses; namely, and . It can be known that the calculation results show excellent agreement with the results of Hsin.