Numerical simulation of structural response during propeller-rudder interaction

Abstract

The propeller wake can cause vibrations on the rudder surface, which worsen the noise and reliability. The vibration monitoring of the rudder operating in the propeller wake with fluid-structure interaction (FSI) method is still challenging. In the present study, the structural response during propeller-rudder interaction is investigated using detached eddy simulation. Three-dimensional distributions of loads, stresses, and deformations are discussed. The leading and trailing edges exhibit the strongest deformations in opposite directions, which are S-shaped. The strongest lateral deformation occurs between the tip vortex and hub vortex regions. In the tip vortex region, the dominant lateral vibrations fluctuate at the blade passing frequency (BPF) and shaft frequency (SF). However, the 75 Hz-fluctuation becomes significant at the trailing edge of the rudder. In the hub vortex region, the lateral deformation fluctuates mainly at 75 Hz except the area near the leading edge. There are weak vibrations occurring at the natural frequencies of the rudder when the natural frequencies of the rudder are much higher than the SF and BPF. However, the plate in the propeller suffers intense vibrations at the frequencies near the natural frequencies, where the natural frequencies of the plate are close to SF and BPF.

KEYWORDS:

• Structural response
• vortex
• propeller-rudder interaction
• detached eddy simulation

Subject classification codes:

• 580.50

1. Introduction

The vibrations on the rudder operating in the propeller wake play an important role in the reliability and noise of the vessel. The wake of propellers represents a non-uniform flow field structure with intense periodic characteristics (Mascio et al., 2014; Wang et al., 2017). When the propeller wake vortex collides with a body or rudder, its fluid load fluctuations are transmitted via vortex-body interactions. This results in complex structural responses that directly relate to vibrations and noises. Accordingly, the fluid-structure interaction (FSI) cannot be overlooked during in-depth studies related to this phenomenon.

As the propeller has a finite number of blades, the flow field in its wake is generally non-uniform. This has been confirmed using the laser Doppler velocimetry method and numerical studies (Felli et al., 2008; Gong et al., 2018; Stella et al., 2000; Zhang & Jaiman, 2019). The properties of complex propeller wake fields are generally dependent on the number of blades or blade passing frequency (BPF). Felli et al. (2011) observed the double-helix breakdown of propeller vortices in the far field of a propeller wake field. Mascio et al. (2014) studied propeller wake vortexes. They demonstrated that the flows in the wake of a propeller exhibit intense, turbulent fluctuations, which were strongly correlated with the BPF of the propeller. Kumar and Mahesh (2017) analysed the axial evolution of the propeller wake and found that the coherent tip vortex breaks up immediately when the propeller wake begins to destabilize. However, the hub vortex remains coherent even in the far wake.

It has been long known that the propeller vortex developing aft of a propeller interacts with the rudder. Generally, the propeller wake can induce changes in hydrodynamic properties such as the lift and thrust. Abramowski et al. (2010) discovered that the thrust efficiency of a propeller can be improved by installing a streamline rudder in its wake. He and Kinnas (2017) found that the propeller vortex induces a loss of lift in the rudder. Sun et al. (2018) observed that rudders in the propeller wake experience a fluctuating force that oscillates at the BPF. Paik et al. (2012) showed that the wake flow field of a propeller affects the rudder cavitation. Using inviscid numerical simulations, Li (1996) found that the rudder splits the propeller wake vortex into two parts. Furthermore, the wake flow field was found to be misaligned on both sides of the rudder along its span. Felli et al. (2009) and Felli and Falchi (2011) used experimental methods to analyze such misalignments, which were found to be caused by the spanwise gradient of the circulation and asymmetric perturbations induced by the flow roll-up at the blade tip. Sun et al. (2018) demonstrated that the propeller wake does not exhibit any spanwise misalignments until it reaches the leading edge of the rudder.

Vortexes are important structures in propeller wake as their interactions with bodies can induced loads that cause structural vibrations (Rockwell, 1998). Such vortex-body interactions have been observed in a variety of settings including aerospace, water pumps, ships, and marine engineering applications. This issue dates back to the 1980s, when studies were focused on the noise produced by these interactions (Howe, 1988, 1989). Marshall (1994) and Marshall and Yalamanchili (1994) studied changes in the bending and cross-sectional plane of line vortexes when cut by a blade. They found that the bending of the line vortex is directly related to the ratio of the blade thickness to the ambient vortex core radius. In particular, the line vortex bends only slightly at small ratios. When a line vortex is cut by a blade, the vortex core radii changes. As a result, the vortex force on the blade depends on the radius of the vortex core in the interaction region. Marshall and Grant (1996) found that the vorticity field agglomerates into a thin vortex ribbon near the blade leading edge. They also found that the vortex-induced suction force acting on the blade is directly proportional to the square of the vortex circulation. The movements of the vortex also affect the boundary layer of solid structures. Studies have demonstrated that intense changes occur in the boundary layer during vortex-solid wall interactions (Hirsa et al., 2000; Shi et al., 1991; Wu et al., 2006). Affes and Conlisk (1993) studied the unsteady boundary layer during the collision between a vortex filament and circular cylinder. They found that the vortex motion leads to boundary layer separation. Furthermore, they observed that the secondary flow patterns of the boundary layer resemble a horseshoe vortex. Jiang et al. (2015) conducted viscous numerical simulations of the flow over a rod-foil configuration. They found that the collision between the upstream and downstream vortices is one of the main causes of turbulent fluctuations.

Analyses of the structural deformations and vibrations are highly relevant for hydrodynamic and structural design and several FSI models have been raised (Jiang et al., 2019; Salih et al., 2019). Ghalandari et al. (2019) investigated the flutter speed of a wing with presented differential quadrature method. The research was performed under subsonic regime which modeled by Wagner function. Many studies had proved that there are pressure fluctuations on the rudder surface induced by propeller wake. Muscari et al. (2017) employed detached eddy simulations (DES) to analyze the coupling between the rudder and wake of a simplified single-blade rotor. They installed probes on the rudder surfaces to monitor the pressure acting upon them. They found that pressure fluctuations on the rudder are predominantly determined by the BPF. A study by Yilmaz et al. (2019a) confirmed that intense pressure fluctuations occur on the rudder surfaces. Hu et al. (2019a) used large eddy simulations (LES) to study the wake of a contra-rotating propeller. They found that the cutting of the forward propeller tip vortex by the aft propeller reduces its axial vorticity. Wang et al. (2019) noted that fluctuations of turbulent kinetic energy are highly significant near the rudder surfaces and strongly correlate with the propeller BPF. Based on LES, Hu et al. (2019b) similarly found that pressure fluctuations over the rudder surface are strongly correlated with the BPF. They also found that the pressure fluctuations caused by the tip vortex of single-bladed rotors are more intense than those caused by four-bladed rotors. Propeller vortex-rudder collisions also involves intense fluid-structure interaction processes. For example, Turnock and Wright (2000) compared the results of coupled and uncoupled numerical simulations of such interactions. They found that a coupled numerical simulation is much more accurate in simulating the pressure distribution on the rudder. However, this research focused on the steady-state variations in the rudder’s stress and strain, rather than the structural vibration.

Despite these early observations (Marshall, 1994; Marshall & Grant, 1996; Marshall & Yalamanchili, 1994; Muscari et al., 2017; Turnock & Wright, 2000), the mechanism of the rudder vibrations induced by propeller wake has remained unclear. In the present study, a fluid-structure interaction model is employed to investigate the vibrations in the propeller-rudder interactions. Simulations are carried out using STAR-CCM+, which allows direct simulation of the FSI. The fluids are simulated using DES, and the structural deformations and vibrations of a both ends fixed-rudder are simulated using the finite volume solid stress method. The dominant frequencies of the vibrations are obtained with fast Fourier transform (FFT). The distributions of deformations and vibrations are analysed and compared, which will enhance the design of the propeller-rudder system.

2. Mathematical models

2.1. Fluid model

The k–ω SST DES model is employed to simulate the turbulence in the flow filed. The transport equations of the turbulent kinetic energy (k) and turbulence dissipation rate (ω) of the k–ω SST model are respectively expressed as (Shur et al. 2008) (1) $\begin{array}{rl}& \frac{\mathrm{\partial }}{\mathrm{\partial }t}(\rho k)+\mathrm{\nabla }\cdot (\rho k\overline{\mathit{v}})\\ & =\mathrm{\nabla }\cdot [(\mu +{\sigma }_k{\mu }_t)\mathrm{\nabla }k]+P_k\\ & -\rho \beta \ast f_{\beta \ast }(\omega k-{\omega }_0{k}_0)+S_k,\end{array}$(1) (2) $\begin{array}{rl}& \frac{\mathrm{\partial }}{\mathrm{\partial }t}(\rho \omega )+\mathrm{\nabla }\cdot (\rho \omega \overline{\mathit{v}})\\ & =\mathrm{\nabla }\cdot [(\mu +{\sigma }_{\omega }{\mu }_t)\mathrm{\nabla }\omega ]+P_{\omega }\\ & -\rho \beta f_{\beta }({\omega }^2-{{\omega }_0}^2)+S_{\omega },\end{array}$(2) where $\overline{\mathit{v}}$ is the mean velocity, k is turbulent kinetic energy, μ is the dynamic viscosity, ρ is the density, f_β* is the free-shear modification factor, f_β is the vortex-stretching modification factor, ω₀ and k₀ are the ambient turbulence values for counteracting turbulence decay, P_k and P_ω are production terms, S_k and S_ω are source terms, and σ_k, σ_ω, α*, β*, φ₁, and φ₂ are model coefficients. The calculation of these model coefficients can be found in the user guide of STAR-CCM+ (SIEMENS, 2019).

2.2. Fluid-structure interactions

To calculate the structural stresses, the finite volume solid stress model is used whose governing equations are expressed as (3) $\begin{array}{rl}M& ={\int }_{V_s}{\rho }_s(t)\text{ d}V={\int }_{V_s}{\rho }_s\text{ d}V=\text{const,}\end{array}$(3) (4) $\begin{array}{rl}{\rho }_s{\mathit{u}}_s& =\mathrm{\nabla }\cdot \sigma +{\mathit{b}}_s,\end{array}$(4) where M, ρ_s, and V_s are the mass, density, and volume of the solid structure, respectively, u_s is the solid displacement, b_s is the total volume force per unit volume, and σ is the Cauchy stress tensor.

3. Numerical setup and uncertainty analysis

3.1. Numerical setup

An E779A propeller and a rudder having the NACA0020 profile, whose geometric characteristics are listed in Table 1, are selected for this study. The configurations are the same as that used by Hu et al. (2019b, 2020). The simulations are performed within a cubic computational domain of dimensions (l × w × h) 12.5D × 5.0D × 5.0D, where D is the diameter of the propeller. To simulate the rotations of the propeller and deformations of the rudder, the computational domain is divided into fluid domain, rotation domain, deformation buffer, and rudder domain (Figure 1). The distance between the propeller and rudder is set at 0.5D. The most intense interactions between the rudder and propeller vortex only occur in a small section of the rudder surface within the wake flow field. Therefore, the rudder domain is only applied over a 600 mm span of the rudder. The upper and lower end plates of the rudder are fixed. The rudder material is Al, whose density, Young’s modulus, and Poisson’s ratio are 2.702 × 10³ kg/m³, 6.8 × 10⁴ MPa, and 0.33, respectively. To buffer the rudder deformations, a deformation buffer is applied around the rudder domain. The reference frame is constructed on the propeller, and the X-axis is aligned with the axis of the propeller.

Figure 1. Configuration of the propeller and rudder. (a) General view; (b) Top view.

Table 1. Geometric characteristics.

Propeller	Diameter	D = 227 (mm)
	Number of blades	Z = 4
	Hub ratio	0.200
	Pitch ratio	P/D = 1.1
	Expanded area ratio	0.689
	Rake	4°35'
	Rotation speed	25 rps
Rudder	Chord length	c = 180 (mm)
	Span length	l = 300 (mm)

According to Mascio et al. (2014) and Muscari et al. (2017), a time step corresponding to 1° of propeller rotation is sufficient to accurately resolve the vortex of an E779A propeller. Moreover, Yilmaz et al. (2019b) reduced the time step to 0.81° of propeller rotation. Based on these results, the time step is set to Δt = 5.00 × 10⁻⁵ in this study, which corresponds to 0.45° of propeller rotation.

We have deliberately chosen to overlook the deformations of the propeller as it is assumed to be a fully rigid, non-deforming structure. A solid mesh is therefore not constructed for its internal regions. The fluid and rotation domains are meshed using hexahedral cells. Prism layers are constructed on the interfaces among the fluid domain, rotation domain, and deformation buffer. A transitional layer is formed by inserting pyramid cells and tetrahedral cells between these prism layers and the hexahedral cells. For an accurate calculation of boundary layer, the cells in the boundary layer need special treatment. The prism layer mesher is employed, and 10 prism layers are created for the boundary mesh. The first boundary layers near the propeller and rudder are obtained using y+ = 10 with a growth rate of 1.2. The mesh of the refined region is shown in Figure 2.

Figure 2. Meshing details of the refined domains, propeller, and near-rudder region. (a) Distribution of meshes in the refined domains, (b) Boundary layer mesh near the propeller, (c) Meshes of the rudder and transitional region.

3.2. Uncertainty analysis

The standard uncertainty analysis proposed by Roache (1997) and ITTC (2002) is employed to verify the reliability of present meshing strategy. Thus, results of fine, medium, and coarse meshes are evaluated with a grid refinement ratio of $r=\sqrt{2}$. All three meshes are constructed with the same boundary layer meshing and time step. Owing to a lack of experimental data, the comparison shown in Table 2 is conducted using data reported by Hu et al. (2020). This table provides the discrepancy (ϵ₂₁ and ϵ₃₂), convergence ratio (R_G), order of accuracy (P_G), error estimation (δ), correction factor (C_G), and uncertainty (U_G) among the results of adjacent meshes. In the present work, the theoretical order of accuracy P_Gest is 2. These variables can be calculated using Equations (5–11): (5) $\begin{array}{rl}{\epsilon }_{21}& =S_{i-\text{Medium}}-S_{i-\text{Fine}},\end{array}$(5) (6) $\begin{array}{rl}{\epsilon }_{32}& =S_{i-\text{Coarse}}-S_{i-\text{Medium}},\end{array}$(6) (7) $\begin{array}{rl}{R}_G& =\frac{{\epsilon }_{21}}{{\epsilon }_{32}}\end{array}$(7) (8) $\begin{array}{rl}{P}_G& =\frac{\text{ln}({\epsilon }_{32}/{\epsilon }_{21})}{\text{ln}(r_G)}\end{array}$(8) (9) $\begin{array}{rl}\delta & =\frac{{\epsilon }_{21}}{{r_G}^{P_G}-1}\end{array}$(9) (10) $\begin{array}{rl}{C}_G& =\frac{{r_G}^{P_G}-1}{{r_G}^{P_{G\text{est}}}-1}\end{array}$(10) (11) $\begin{array}{rl}{U}_G& =|C_G\delta |+|(1-C_G)\delta |\end{array}$(11) where S_i represents K_T, 10K_Q and C_F, ϵ₂₁ represents the discrepancy between the results of fine grid and medium grid, and ϵ₃₂ represents that between the results of medium grid and coarse grid. As R_G < 1, the discrepancies decrease with cell size.

Table 2. Grid resolution uncertainty analysis.

	Number of cells	Number of cells in the refinement area	KT	10KQ	CF (×10^−3)
Hu et al. (2020)	–	–	0.1704	0.3190	3.93 × 10^−3
Fine	9.33 M	6.59 M	0.1729	0.3292	3.89 × 10^−3
Medium	6.58 M	4.63 M	0.1735	0.3326	3.08 × 10^−3
Coarse	5.52 M	3.29 M	0.1745	0.3563	1.57 × 10^−3
ϵ21	–	–	0.0006	0.0034	8.11 × 10^−4
ϵ32	–	–	0.0010	0.0237	1.51 × 10^−3
RG	–	–	0.6000	0.1435	0.5388
PG	–	–	1.4740	5.6026	1.7846
δ	–	–	9.00 × 10^−4	5.69 × 10^−4	9.47 × 10^−4
CG	–	–	0.6667	5.9707	0.8561
UG	–	–	9.00 × 10^−4	6.23 × 10^−3	9.47 × 10^−4

As the structural deformations and structural vibrations occurring in the rudder due to the propeller vortex are the focus of this study, comparing only the hydrodynamic coefficients is insufficient. Therefore, we extend the grid convergence analysis to the wake of the propeller. Figure 3 illustrates the distribution of the vertical vorticity of the rudder in each grid, and Figure 4 shows the vortex structures of the propeller, obtained using a Q-criterion of 1000/S². The comparison is carried out between the results of present simulation and those by Hu et al. (2020), in which the Q-criterion used is 1000/S². The same value of Q-criterion is employed in present simulation. Vertical vorticity ‘Vorticity-z’ and Q-criterion can be obtained using Equations (12–13). In these figures, the vortex core and vortex tube are better defined with an increasingly refined grid. The relationship between the misalignment of the propeller tip vortex and the direction of the propeller’s rotation is consistent with that observed by Li (1996). (12) $\begin{array}{rl}& \text{Vorticity -}z=\frac{\mathrm{\partial }v}{\mathrm{\partial }x}-\frac{\mathrm{\partial }u}{\mathrm{\partial }y},\end{array}$(12) (13) $\begin{array}{rl}& Q=\frac{1}{2}({||\mathrm{\Omega }||}^2-{||S_{\omega }||}^2)=-\frac{1}{2}\left(\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}-\frac{\mathrm{\partial }{u}_j}{\mathrm{\partial }{x}_i}\right),\end{array}$(13)

Figure 3. Distribution of vertical vorticity near the rudder. (a) Hu et al. (2020), (b) Fine, (c) Medium, (d) Coarse.

Figure 4. Vortex structures of the propeller in grids with varying levels of refinement. (a) Hu et al. (2020), (b) Fine, (c) Medium, (d) Coarse.

From Table 2 and Figures 3 and 4, the results using the fine grid are similar to those of Hu et al. (2020) in terms of hydrodynamic properties and the near-rudder flow field. Hence, the fine mesh is reliable for this study. This flow domain configuration and fine cell size are used in following sections.

4. Analysis of flow field

The wake of a propeller is composed of spatial non-uniform flows causing non-uniform stresses and deformations on the rudder. Therefore, the research focus on the flow field near the rudder should be applied at first. It is necessary to study the temporal and frequency characteristics of the pressure fluctuations generated by the propeller wake.

4.1. Distribution of pressure and velocity

 Figure 5 shows the distributions of pressure and velocity near the rudder at the instant of 6 T (where T is the period of one propeller rotation). The flow-field cross sections correspond to Z = 0.50D, 0.25D, 0, −0.25D, and −0.50D. A reference pressure of 101 325 Pa is used. The difference in pressures between the two sides of the rudder reaches its maximum at Z = ±0.25D. The pressure of the flow field is significantly lowered on both sides of the rudder at Z = 0. The velocity in the X direction (V_x) reaches its maximum at Z = ±0.25D. The axial velocities V_x in the flow field are symmetric about the X-Z plane at Z = 0, while the distributions of V_x are not completely symmetrical at Z = ±0.5D and Z = ±0.25D. At Z = 0.25D, V_x is larger on the Y < 0 side than on the Y > 0 side, and vice versa occurs at Z = −0.25D. A similar asymmetric distribution of V_x is also observed at Z = ±0.50D. A high-V_x wake flow only appears on one side of the rudder at these positions.

Figure 5. Pressure and velocity distributions. (a) Pressure, (b) V_x, (c) V_y, (d) V_z.

The difference of V_x between the two sides of the rudder is caused by the misalignment of the propeller tip vortex with the rudder, as shown in Figure 6. The propeller tip vortex undergoes spanwise and chordwise displacements as it collides with the rudder. This is consistent with the findings reported by Johnston and Sullivan (1992). The X-position function P_x in Figure 6 is colored for easily determining the direction of the tip vortex misalignment, which is towards the -Z and + Z directions on the Y < 0 and Y > 0 sides, respectively. The rudder surfaces can be divided into pressure and suction sides based on the direction of the propeller tip vortex and the pressure distribution shown in Figure 5(a). In particular, on the suction sides, the tip vortex tube bends towards the smaller circle. On the pressure sides, the tip vortex tube develops along the outer circle. Comparing Figures 5(c) and 6, the lateral velocity, V_y, is higher around the leading edge of the rudder. At the same spanwise position, the suction and pressure sides exhibit V_y in the opposite direction. At the same chordwise position, the V_y on the same side of the rudder always point in the same direction. As shown in Figure 5(d), the direction of V_z corresponds to the misalignment of the propeller tip vortex shown in Figure 6, where the maximum V_z occurs in the hub vortex flow.

Figure 6. Displacement of the propeller vortex.

The distributions of V_x, V_y, and V_z at X = 0.6D and X = 0.9D are shown in Figures 7 and 8. At X = 0.6D, V_x shows a spanwise misalignment (in the Z direction) along the rudder, and the maximum V_x occurs between the tip vortex and hub vortex. Moreover, the axial velocities (V_x) on the suction sides are greater than those on the pressure sides. At these sections, V_y and V_z on different sides of the rudder spread in opposite directions. At X = 0.6D, the lateral velocities (V_y) are higher on the suction side, whereas they are higher on the pressure side at X= 0.9D. In the near-rudder flow field, V_x is observed to be the dominant component of the wake flow. The maximum V_x occurs between the tip and hub vortices. Although the V_z values at X = 0.6D and X = 0.9D are similar, V_x and V_y both decrease along the X axis.

Figure 7. Distribution of velocities near the rudder (X = 0.6D). (a) V_x, (b) V_y, (c) V_z.

Figure 8. Distribution of velocities near the rudder (X = 0.9D). (a) V_x, (b) V_y, (c) V_z.

4.2. Pressure fluctuations on the rudder

Mascio et al. (2014) and Wang et al. (2017) both noted that the propeller wake vortex is temporally non-uniform. To study the periodicity of the pressure fluctuations and structural vibrations, virtual probes are placed on the rudder. These virtual probes will not impact the flow filed and the structural characteristics of the rudder. A two-dimensional m-n coordinate system is constructed so that the position of each probe can be distinguished using its (m, n) coordinates (Figure 9). In this coordinate system, m (m = 0–6) and n (n = −7–7) indicate the chordwise and spanwise positions along the rudder, respectively. Values for n = ±6, ±5, ±4, ±2, and ±1 correspond to ±0.50D, ±0.41D, ±0.35D, ±0.26D, and ±0.09D spanwise locations (where D is the diameter of the propeller). The probes at n = ±6 are located directly behind the blade tip of propeller (Z = ±0.5D). The probes at n = 0 are placed directly behind the propeller hub. The probes at m = 0 and m = 6 are located at the leading and trailing edges of the rudder, respectively. A value of m = 1 represents the 0.08C chordwise position, and m = 2, 3, 4, and 5 correspond to the 0.18C, 0.28C, 0.53C, and 0.78C chordwise positions (where C is the chord length of the rudder). To facilitate subsequent comparisons and analysis, we define the n = 0 and n = ±6 positions as the hub vortex and tip vortex regions, respectively.

Figure 9. Layout of the probes.

The pressure fluctuations shown in Figure 10 are monitored by probes (0–6, 6) located in the tip vortex region. The results obtained by modeling the rudder as a flexible and as a fully rigid structure are compared. The pressure fluctuates in an intense, periodic manner with different amplitudes. The pressure fluctuations of adjacent probes at coordinates (1, 6) and (2, 6) are observed to exhibit a phase difference. This will exacerbate the deformation of the rudder. Comparing the coupled and uncoupled results, the pressure fluctuations on the rudder surface are clearly smoother when the rudder is assumed to be a fully rigid. Figure 11 illustrates the FFT results on the pressure fluctuations recorded by probe (0, 6). Both FFT results are dominated by pressure fluctuations whose frequencies correspond to the BPF of the propeller (100 Hz) and its harmonics. However, the flexible rudder also exhibits intense pressure fluctuations at non-integer multiples of the BPF. Additionally, the pressure fluctuations corresponding to the 5th and 6th order harmonics of the BPF (500 and 600 Hz, respectively) are significantly weaker in the flexible case than that in the rigid case. Peaks and valleys are prominent in the FFT results for the rigid rudder, whereas the results are generally smoother for the flexible rudder.

Figure 10. Pressure fluctuations monitored by probes (0–6, 6). (a) Flexible rudder, (b) Rigid rudder.

Figure 11. Comparison between the frequency domain features of pressure fluctuations for a rigid and a flexible rudder.

The FFT is used to extract the power spectral density (PSD) of the pressure fluctuations given in Figure 10(a), as shown in Figure 12. The maximum amplitude of fluctuation always occurs at a BPF (100 Hz) for all of the probes. The amplitude at this frequency decreases along the chord of the rudder. Although large amplitudes of fluctuation also occur at the harmonics of the BPF, they decrease rapidly with increasing frequencies.

Figure 12. PSD of the pressure fluctuations at probes (0–6, 6).

Figure 13 shows the PSD of the pressure fluctuations at n = 0 and n = −6. Comparing Figure 13(a,b), and 12, most of the pressure fluctuations caused by the tip vortex aggregate around the BPF and its harmonics. Although pressure fluctuations also occur at the shaft frequency (SF, 25 Hz), their amplitudes are much smaller than those at the BPF. The pressure fluctuations caused by the hub vortex are much more complex. Figure 13(a) shows that the pressure fluctuations at the SF are much more intense than those occurring at the BPF. Furthermore, intense hub vortex-induced pressure fluctuations occur at the harmonics of the SF (50, 75, and 125 Hz). Although the hub and tip vortices generate pressure fluctuations at different frequencies, these fluctuations generally weaken along the chord of the rudder.

Figure 13. PSD of the pressure on the rudder surface. (a) hub vortex region, (b) tip vortex region.

Hu et al. (2019b) noted that the pressure fluctuations are relatively weak in the area between the hub and tip vortex regions. However, they still can affect the structural deformations of the rudder. Therefore, the pressure fluctuations in the area between the hub and tip vortex regions are investigated. The amplitudes of the pressure fluctuations, A_p, on the rudder surface are shown in Figure 14. Pressure fluctuations are intense at the leading edge. The most intense pressure fluctuations occur at probe (1, 0), which is directly behind the propeller hub. Relatively intense pressure fluctuations are also detected by the probes at (0, 5–6) and (0, −5 to −6), which are behind the propeller tip vortex. Significant pressure fluctuations are also detected by probes (6, 2–3) and (−6, 2–3) on the trailing edge of the rudder. The frequencies corresponding to the A_p in Figure 14 are shown in Figure 15. The results show that the pressure fluctuations mainly occur at 100 Hz in the tip vortex region and at 25 Hz in the hub vortex region. The region with intense pressure fluctuations close to probe (1, 0) extends upwards and rearwards. The region dominated by 25 Hz fluctuations also extends in the upwards–rearwards direction.

Figure 14. Distributions of pressure fluctuations.

Figure 15. Dominant frequencies of pressure fluctuations.

4.3. Summary

The propeller vortex induces intense spatiotemporal inhomogeneity in the flow field near the rudder. High pressures can be observed between the tip vortex and hub vortex. The V_z on opposite sides of a rudder act in opposite directions. V_y is higher on the suction side of the leading edge, whereas it is higher on the pressure side of the trailing edge. The pressures on the sides of the rudder fluctuate significantly, especially in the hub and tip vortex regions. The most intense pressure fluctuations occur in the hub vortex region. The dominant pressure fluctuations in the hub and tip vortex regions fluctuate at 25 and 100 Hz.

5. Analysis of structural response

Section 4 analyses the non-uniform pressure distributions on the rudder surface as well as the flow fields near the rudder. These non-uniformities induce deformations and vibrations on the rudder. In this section, the stress and deformation of the rudder are studied. As a basement, the modal analysis is carried out in Figure 16. In the modal analysis, the rudder is fixed at its two ends. The constraint conditions are shown in Figure 16(a). The first mode is primarily bending, while the second mode contains twisting. The vibrations on the trailing edge are more severe. The natural frequencies of these two fixed ends of rudder is much larger than the BPF of the propeller.

Figure 16. Natural frequencies of the rudder. (a) Mode 1 (1596.3 Hz), (b) Mode 2 (2391.6 Hz), (c) Mode 3 (3756.2 Hz), (d) Mode 4 (5055.4 Hz).

5.1. Evolution of the propeller vortex

The propeller tip vortex passes the leading edge of the rudder with a period of 0.25T. We therefore analyze the propeller tip vortex and stresses of the rudder at various instances within a 0.25T period, as shown in Figure 17. As the propeller tip vortex approaches the leading edge of the rudder, the vortex tube begins to bend and deform. These observations are consistent with the findings of Marshall (1994). When the propeller tip vortex flows past the leading edge of the rudder, the tip vortex branch in suction side separates from the rudder surface. In contrast, the tip vortex branch in the pressure side remains in contact with the rudder surface and joins the boundary layer to form a ‘continuously adhering’ structure. The stresses of the rudder thus undergo a set of changes that are temporally related to the evolution of the tip vortex. However, the largest stresses in the rudder always occur at the sections Z = ±0.25D and Z = 0. These results are consistent with the pressure and velocity distributions shown in Figure 5. Figure 17 shows that the characteristics of the tip vortex are different on the pressure and suction sides, and these variations lead to different structural stresses. Considering the differences between the suction and pressure sides is important when analyzing the structural deformations of a rudder.

Figure 17. Vortex evolutions (left) and distributions of structural stresses (right). (a) 5.75T, (b) 5.8125T, (c) 5.875T, (d) 5.9375T, (e) 6T

5.2. Stress fluctuations on the rudder

In Section 4.2, the rudder surface is observed to experience intense pressure fluctuations oscillating at 25, 50, 75, and 100 Hz, which are a major cause of the structural vibrations. With the time-averaged stresses in Figure 18, the average stress, s_ave, is found to be highest at a chordwise position of 0.08C. The s_ave at 0.26D is larger than that at −0.26D. The s_ave at the area directly behind the propeller hub are close to 0. A high stress also appears at 0.59D, which is near the fixed end. The direction of s_ave occurring in that area is opposite that at 0.26D.

Figure 18. Distribution of time-averaged stresses on the rudder surface.

The PSDs of the stresses corresponding to n = 0 and n = ±4 acting on the rudder surface are compared in Figure 19. Note that most of the stress fluctuations oscillate at frequencies between 0 and 200 Hz. Extremely intense stress fluctuations occurring at 25 and 50 Hz are observed in the hub vortex region. However, probes ((0, 4) and (0, −4)) in the tip vortex region report intense stress fluctuations only at 100 Hz. In the tip vortex region, the stress fluctuation at 75 Hz begins to intensify after 0.08C and its amplitude eventually exceeds that of the BPF stress fluctuation at 0.28C (m = 3). The 125 Hz stress fluctuation also begins to intensify after 0.18C. The low-frequency stress fluctuations (e.g. the 25 Hz stress fluctuation) begin to grow and strengthen after 0.53C (m = 4). They culminate at the trailing edge of the rudder. At this position, the 25, 50, and 75 Hz stress fluctuations are more intense than the 100 Hz stress fluctuation, and their amplitudes decrease with increasing frequency. In the hub vortex region, the stress fluctuations are especially intense near the leading edge of the rudder (m = 0–1). Additionally, the 75 Hz stress fluctuation becomes intense only around 0.28C and 0.78C. The stress fluctuations in the hub vortex region mainly occur within the 25–75 Hz frequencies. However, intense stress fluctuations at higher frequencies (>200 Hz) also occur at the trailing edge of the rudder (m = 5–6).

Figure 19. PSD of the structural stress, s. (a) m = 0, (b) m = 1, (c) m = 2, (d) m = 3, (e) m = 4, (f) m = 5, (g) m = 6

A comparison of Figures 19 and 13 clearly indicates that the spectral features of the pressure and stress on the rudder surface are not entirely identical. Therefore, the maximum stress fluctuation amplitudes of the rudder surface, A_s, are shown in Figure 20. The most intense stress fluctuations occur around a chordwise position of 0.08C in the hub vortex region. Additionally, intense stress fluctuations also occur around a spanwise position of 0.26D.

Figure 20. Distributions of stress fluctuations.

5.3. Vibrations on the rudder

The distribution of deformations is detected by the probes on the rudder surface, as shown in Figure 9. First, the distribution of time-averaged deformations in the Y direction, d_yave, is analysed in Figure 21. Similar to the time-averaged stresses, the maximum d_yave occurs in the first 0.28C of the rudder. The spanwise position corresponds to the region between the tip vortex and hub vortex regions. Significant d_yave are also can be observed in these spanwise positions at the trailing edge of the rudder. The deformations d_yave in the pressure side and suction side are opposite each other. The deformations d_yave at same spanwise position on the leading and trailing edges of the rudder also show this trend. In general, the deformations acting on the rudder seem to twist around its center.

Figure 21. Time-averaged deformations in the Y direction.

 Figure 22 illustrates the frequency domain characteristics of d_y recorded by the probes. It can be seen that the fluctuations on the pressure and suction sides are very similar. In the tip vortex region near the leading edge (m = 0–1), intense lateral vibrations can be observed at 25 Hz (SF) and 100 Hz (BPF). The 75 Hz-fluctuation of d_y becomes more apparent after 0.18C, attaining an amplitude similar to the 100 Hz fluctuation at 0.28C (m = 3). At 0.53C and 0.78C, the amplitudes of the 75 Hz fluctuation exceed that of the 100 Hz fluctuation. At the trailing edge of the rudder, the amplitudes of the 75 and 100 Hz (BPF) fluctuations are very similar. Intense 25, 50, 75, and 125 Hz vibrations occur in the hub vortex area. The 25 Hz-vibration is the dominant vibration in the first 0.18C of the rudder (m = 0–2). For the remainder of the rudder, the 75 Hz vibration is dominant, and its amplitude increases along the rudder chord. At probe (6, 6), the amplitude of the 75 Hz vibration is 0.025 μm. Although no large lateral deformation can be observed at hub vortex region, the lateral vibrations at hub vortex region are more intense than that at tip vortex region.

Figure 22. PSD of lateral vibrations. (a) m = 0, (b) m = 1, (c) m = 2, (d) m = 3, (e) m = 4, (f) m = 5, (g) m = 6

The maximum d_y amplitude at each probe, A_dy, is plotted in Figure 23 for comparing the intensities of their fluctuations across the rudder surface. The largest A_dy values occur behind the propeller hub, especially near the trailing edge, while those behind the tip vortex are much lower. Note that the A_dy values are symmetrical about the X-axis (m-axis). The stresses on the upper half (suction side) of the rudder are larger than those in the lower half. Besides, A_dy is larger on the pressure side in the area near the trailing edge.

Figure 23. Distributions of lateral vibrations.

The time-averaged deformations of the rudder surface d_xave are shown in Figure 24. The largest deformation in the X direction (d_xave) appears in the first 0.28C of the rudder. Furthermore, d_xave is larger on the suction side than that on the pressure side. Figure 18 shows that there is no time-averaged stress in the hub vortex region of the rudder surface. Similarly, d_xave is also 0 in the hub vortex region. As a whole, d_xave appears to be lower than d_yave. The frequency-domain characteristics shown in Figure 25 indicate that the dominant d_x fluctuation at the leading edge oscillates at 50 Hz in both the tip and hub vortex regions. Behind the leading edge, the most significant d_x fluctuations on the rudder surface oscillate at 25, 50, 75, 100, and 125 Hz. In the front part of the rudder (0.08C–0.28C), the dominant d_x fluctuations on the rudder surface behind the hub and tip oscillate at 25 and 50 Hz, respectively. However, the tip vortex region also exhibits fluctuations at 25 and 75 Hz with amplitudes similar to the 50 Hz-fluctuation. Moving further down the chord of the rudder (0.53C–0.78C), the dominant d_x fluctuations behind the hub oscillate at 75 and 50 Hz. At the trailing edge of the rudder, the rudder surface behind the hub and tip exhibits oscillations in the X direction only at 50 and 100 Hz.

Figure 24. Distribution of time-averaged deformations in the X direction.

Figure 25. PSD of axial vibrations. (a) m = 0, (b) m = 1, (c) m = 2, (d) m = 3, (e) m = 4, (f) m = 5, (g) m = 6

The maximum amplitudes of the axial vibrations on the rudder surface are shown in Figure 26. The most intense axial vibrations occur at 0.28C, directly behind the propeller hub. Deformation fluctuations with amplitudes greater than 0.00125 μm occur throughout the range of −0.35D–0.35D. Although significant pressure fluctuations are detected by the probes behind the tip vortex ((0, 5–6) and (0, −5 to −6)), d_x at these locations are small. The stress fluctuations at most of the positions along the trailing edge are relatively weak. However, intense axial vibrations are detected at probes (6, ±3) and (5, ±2) (Figure 26). These intense axial vibrations correlate well with the observed stress fluctuations.

Figure 26. Distributions of axial vibrations.

Large deformations in the Z direction are obtained in the first 0.28C of the rudder (Figure 27). The maximum positive d_zave occurs directly behind the blade tip (on the suction side at n = 6 and on pressure side at n = −6). The maximum negative d_zave value is found in the hub vortex region. The region exhibiting large Z direction deformations ranges from n = 6 to probe (5, 4), and from n = −6 to probe (5, −7). However, the Z direction deformations measured by probes (5, −4) and (5, 7) are relatively weak. This misalignment in the distribution of d_z is generally consistent with the misalignment of the propeller wake vortex. Comparing the d_x and d_z values shown in Figures 24 and 27, it can be observed that the values of d_x and d_z are close to each other, and both smaller than d_y. Furthermore, the positive d_x values are larger in magnitude than the negative d_y values. Additionally, the negative d_z values are larger in magnitude than the positive d_z values.

Figure 27. Distribution of time-averaged deformations in the Z direction.

Frequency domain analysis provides an intuitive way to understand rudder vibrations. The d_z fluctuations are accordingly subjected to a frequency domain analysis, with the results shown in Figure 28. Probe (0, 0) detected d_z fluctuations oscillating at the SF, but small-amplitude fluctuations of 0.0001 μm occur widely at a frequency range of 0–700 Hz. Probe (6, 0) also detected intense high-frequency fluctuations more intense than those at (0, 0). However, the strongest d_z fluctuation at (6, 0) oscillates at a frequency of 75 Hz. In Figure 22, the 100 Hz oscillation is not the dominant d_x fluctuation in the hub vortex region. However, in Figure 28, the most intense d_z fluctuation oscillates at the BPF (100 Hz) in the hub vortex region. In the area behind the blade tip, the d_z fluctuations are weak at the leading and trailing edges, but intense in the middle of the rudder. Probes (0, ±6) and (6, ±6) exhibit weak d_z fluctuations that have maximum amplitudes <0.00025 μm. At probes (1, ±6), the 50, 75, and 125 Hz fluctuations begin to increase in strength. The amplitudes corresponding to these frequencies are all >0.0004 μm. Note that the BPF (100 Hz) fluctuation is still dominant at this position. At 0.28C, the amplitude of the 75 Hz fluctuation exceeds that of the BPF fluctuation. Furthermore, the amplitudes of the 50 and 125 Hz fluctuations also begin to approach that of the BPF fluctuation. This occurs at the position corresponding to the thickest part of the rudder. At 0.53C–0.78C, the d_z fluctuations of the rudder surface behind the tip vortex mainly oscillate at 75 Hz.

Figure 28. PSD of structural displacements in the Z direction. (a) m = 0, (b) m = 1, (c) m = 2, (d) m = 3, (e) m = 4, (f) m = 5, (g) m = 6.

 Figure 29 shows the distribution of maximum d_z amplitudes, A_dz, on the rudder surface. The largest A_dz values occur behind the blade tip instead of the propeller hub. Note that A_dz is especially high on the pressure side. The area behind the propeller hub does show intense d_z fluctuations, but only at (1–2, 1) and (5, 0). The largest A_dz values on the trailing edge occur at probes (6, ±3) and (6, 0), while they are almost 0 in the area between these probes.

Figure 29. Distributions of vertical vibrations.

 Figure 30 illustrates the overall deformations of the rudder scaled by a factor of 100,000 for easy visibility. The figure indicates that the deformations of the first half of the rudder occur in the same direction as the propeller rotation. The propeller wake causes the leading and trailing edges of the rudder to bend in opposite directions in an S-shape along the pressure and suction sides. Lateral deformation is dominantly observed in the rudder. The maximum d_y occurs at the ±0.26D spanwise positions. Other than at the rudder’s fixed ends, the lateral deformations are the smallest directly behind the propeller hub. Additionally, S-shaped deformations are induced in the leading and trailing edges in X direction. The trailing edge in the upper part of the rudder deforms in the -X direction, while the leading edge in the lower part of the rudder deforms in the X direction. Furthermore, the deformations of the leading edge of the rudder are in the same directions as those of the propeller vortex rotation. Although v_y is generally weaker than both v_x and v_z, the deformations of the rudder surface are greatest in the Y direction.

Figure 30. Deformation of the rudder (green: initial state; red: deformation). (a) +X view, (b) −X view, (c) +Y view.

The deformation of the rudder is similar to the fourth mode shape in Figure 16. The fourth natural frequencies of the two ends fixed rudder is 5055.4 Hz. Hence, the PSD of lateral vibrations obtained by (0–6, 2) are discussed in Figure 31. Lateral vibrations occurring at the frequencies>2000Hz are weak.

Figure 31. PSD of lateral vibrations (probes (0–6, 2)).

5.4. Structural response of the plate

The vibrations of the rudder have been obtained in the propeller-rudder interaction. However, the natural frequencies are much larger than the excitation frequencies. Hence, a plate fixed at two ends has been used to replace the rudder. The structural response of a plate operating in the propeller wake has been study with same propeller, same rotation speed, and advance velocity. The configuration of the propeller-plate system layout of the probes are shown in Figure 32. The plate is fixed at two ends, and natural frequencies of the plate are much smaller than that of the two ends-fixed rudder (Figure 33).

Figure 32. Configuration of the propeller and plate.

Figure 33. Mode shapes and natural frequencies of the plate. (a) Mode 1 (244.36 Hz), (b) Mode 2 (354.14 Hz), (c) Mode 3 (673.18 Hz), (d) Mode 4 (838.07 Hz),

The propeller tip vortex also undergoes spanwise and chordwise displacements when the tip vortex encounters the plate (Figure 34). The relationship between the rotation and displacement directions is same as that in the propeller-rudder interaction.

Figure 34. Vortex evolution in the propeller-plate interaction.

The pressure fluctuations on the probes (0–6, 6) are shown in Figure 35. For the probe placed at the front of the leading edge of the plate (probe (0, 6)), the pressure fluctuation is similar to that in the Figure 10. However, for probes placed at the side of the plate, the pressure fluctuations are different from those obtained by probes in the propeller-rudder interaction.

Figure 35. Pressure fluctuations monitored by probes (0–6, 6).

For a further comparison, frequency domain features are performed by FFT (Figure 36). More intense fluctuations at SF and BPF can be observed at the leading edge of the plate. The two ends-fixed plate has smaller natural frequencies than the two ends fixed rudder, while the BPF are fixed at 100 Hz. In the propeller-plate interaction, the SF and BPF are closer to natural frequencies of the plate. At the first three natural frequencies, intense pressure fluctuations can be observed, especially for probes (1–4, 6). Pressure fluctuations on the leading edge of the plate are dominant at SF and BPF. At the side of the plate, the pressure fluctuations at the SF and BPF are much weaker than the pressure fluctuations at the first three natural frequencies. The most intense fluctuation appears at the third natural frequency.

Figure 36. PSD of the pressure on the probes (0–6, 6).

In the propeller-plate interaction, the pressure fluctuations on the plate surface can be impacted by the natural frequencies of the plate. Then the stress fluctuations and vibrations with this BPF are be investigated (Figure 37). For the probe located at the leading edge of the plate (probe (0, 6)), more intense fluctuations can be observed in the SF and the frequencies near the natural frequencies. The fluctuations at the BPF are weak, while the most intense fluctuation appears at 687.50 Hz, which is close to the third natural frequency (673.18 Hz). For the stress fluctuations on the rudder side, intense fluctuations also occur at the 687.50 Hz.

Figure 37. PSD of the stress fluctuations on the probes (0–6, 6).

For the plate surface in the tip vortex region, intense stress fluctuations can be observed at 925 Hz, which is near the fourth natural frequency. However, at this frequency, no apparent d_y fluctuation or lateral vibration can be obtained (Figure 38). The d_y fluctuations are more intense at SF and frequencies near the first/second/third natural frequencies. The most intense lateral vibration appears at the frequencies near the third natural frequencies. In the hub vortex region, the dominant lateral vibration occurs near the second natural frequency (Figure 39). In contrary, the lateral vibrations occurring at 600–1000 Hz are weak. Consistent with Figure 38, the d_x and d_z fluctuations in the tip vortex region are also dominant at the frequencies near the third natural frequencies (Figure 40).

Figure 38. PSD of the d_y on the probes (0–6, 6).

Figure 39. PSD of the d_y on the probes (0–6, 0).

Figure 40. PSD of the d_x and d_z on the probes (0–6, 6). (a) d_x, (b) d_z.

According to Figure 33, the second mode shape is twisting in Z axis, and the third mode shape is bending. In the third mode, the most intense vibration occur at the tip vortex region, while the vibrations in the hub vortex region are close to zero (Figure 33(c)). In the second mode shape, the vibrations concentrate at the leading edge and trailing edge. For the plate surface in the tip vortex region, the lateral vibrations occurring near the third natural frequencies are much more intense at the leading edge (probe (0, 6)) and trailing edge (probe (6, 6)), which consistent with third mode shape. The dominant lateral vibrations on the probes (0, 0) and (6, 0) are more intense than that on the probes (1–5, 0). Hence, in the present propeller-plate interaction, the natural frequency plays a more important role in the propeller-induced vibrations of the plate.

5.5. Summary

The rudder exhibits high stresses and intense stress fluctuations in the area between the hub and tip vortex regions. However, despite these significant fluctuations, the time-averaged stress of the hub vortex region is zero. In the tip vortex region, the stress fluctuations are dominated by BPF oscillations at the leading edge, 75 Hz oscillations at positions between 0.28C and 0.53C, and SF oscillations at positions beyond that. Unlike the tip vortex region, the axial-frequency stress fluctuations are entirely dominant in the hub vortex region.

Lateral deformation is the main component of rudder deformation. The largest lateral and axial deformations both occur in the first 0.28C of the rudder. These deformations are also larger on the suction side than that on the pressure side. At 0.28C, the axial deformation fluctuations are significant, but the lateral vibrations are weak. The lateral deformations of the trailing edge are opposite those of the leading edge. Although the lateral deformations at the trailing edge are small, their fluctuations are quite intense. The lateral vibrations in the tip vortex region are dominated by vibrations at the BPF and SF near the leading edge. The 75 Hz vibration gradually becomes dominant after the 0.28C position. The axial deformations in the tip vortex region exhibit significant vibrations at the SF, the second/third-order harmonic of the SF, and the BPF. The lateral and axial deformations of the hub vortex region both average to almost zero, though the axial deformations exhibit intense fluctuations. The lateral and axial vibrations of the hub vortex region mainly occur at SF at the area near the leading edge of the rudder. However, the lateral vibrations gradually become dominated by the 75 Hz vibrations after 0.28C.

In the first 0.28C of the rudder, the vertical deformation is apparent in the hub vortex region, and that in the tip vortex region is significant. Additionally, the vertical deformations of the rudder surface are offset in the same direction as the misalignment of the tip vortex. In the tip vortex region, the vertical vibrations in the first 0.18C are dominating at the SF and BPF. After 0.28C, the 75 Hz vertical vibration becomes dominant. In the hub vortex region, the vertical vibrations oscillate at the SF, 75 Hz, and the BPF. The interaction between the rudder and propeller tip vortex leads to S-shaped deformations in the rudder. These deformations are most significant in the trailing and leading edges, where they act in opposite directions.

Spanwise and chordwise displacements of the tip vortex of the propeller are also can be observed in the propeller-plate interaction. Due to the natural frequencies of the plate are closer to the excitation frequencies, the natural frequency plays a more important role in the propeller induced vibration of the plate than the propeller.

6. Conclusion

DES is used to analyze the structural responses of the rudder in the propeller-rudder interaction. Next, frequency domain analyses are used to conduct three-dimensional comparisons of the pressure, stresses, and vibrations acting on the rudder. The relationship between the natural frequencies and structural response have also been studied with a plate.

The results indicate that the flow field pressure near the rudder is high in the area between the tip vortex and hub vortex regions. Additionally, the two sides of the rudder exhibit vertical velocities in opposite directions. Pressure fluctuations are much more intense in the hub vortex region than tip vortex region. Moreover, they mainly oscillate at the SF and BPF in the hub vortex and tip vortex regions.

The maximum stress on the rudder is located in the area between the hub vortex and tip vortex regions. The stress fluctuations are especially intense in this area and the propeller hub region. The stress fluctuations in the tip vortex region mainly oscillate at the BPF and SF near the leading and trailing edges. Intense axial-frequency fluctuations occur in the middle of the rudder. The area between the tip vortex and hub vortex regions also exhibits the highest deformation velocities at the leading and trailing edges. The lateral deformation velocities are the largest in the first 0.53C of the rudder, and the vertical deformation velocities are the largest at the trailing edge of the rudder.

In the tip vortex region, the lateral vibrations are dominated by the SF and BPF oscillations at the leading edge. However, the 75 Hz fluctuation becomes quite significant at positions beyond 0.53C. In the hub vortex region, the lateral vibrations near the leading edge mostly occur at the SF. These vibrations mainly oscillate at 75 Hz at all other positions. The deformation fluctuations of the tip vortex and hub vortex regions are intense at the trailing edge but weak at the leading edge. The amplitudes of the axial and vertical vibrations are generally weaker than those of lateral vibrations. The maximum amplitude of the lateral vibration occurs in the area between the tip vortex and hub vortex regions. However, that of the vertical vibration occurs in both the tip vortex and hub vortex regions. In particular, vertical deformations in the tip vortex region also exhibit offsets in the same direction as the misalignment of the tip vortex. The most significant deformations of the rudder occur at its leading and trailing edges, both of which exhibit S-shaped deformations in opposite directions. When the natural frequencies are close to the excitation frequencies, the former play a more important role in causing vibrations and deformations.

Limited by the length of the paper, only the two ends-fixed rudder has been considered. However, since the support conditions also have strong influence on the vibration of a rudder operating in propeller wake, their influence can be studied in future.

Acknowledgements

This research was supported in part with computational resources at the Harbin Engineering University. We would like to thank Xu Wang and Yingzhu Wang, and so on, for their help and guidance. We would also like to thank the anonymous reviewers and the editors for their comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is financially supported by the National Natural Science Foundation of China [grant numbers 52071105, 51679045 and 51579052], the Fundamental Research Funds for the Central Universities [grant number 3072020CFT0103], Pre-Research Fund of Marine Equipment [grant number 6141B42862, 61402100201, 61402070304] and Pre-Research Fund of Key Laboratory [grant number 6142204180408, 6142407180108].