Experimental and Simulation Study on Flow-Induced Vibration of Underwater Vehicle

Abstract

At high speeds, flow-induced vibration noise is the main component of underwater vehicle noise. The turbulent fluctuating pressure is the main excitation source of this noise. It can cause vibration of the underwater vehicle’s shell and eventually radiate noise outward. Therefore, by reducing the turbulent pressure fluctuation or controlling the vibration of the underwater vehicle’s shell, the radiation noise of the underwater vehicle can be effectively reduced. This study designs a cone–column–sphere composite structure. Firstly, the effect of fluid–structure coupling on pulsating pressure is studied. Next, a machine learning method is used to predict the turbulent pressure fluctuations and the fluid-induced vibration response of the structure at different speeds. The results were compared with experimental and numerical simulation results. The results show that the deformation of the structure will affect the flow field distribution and pulsating pressure of the cylindrical section. The machine learning method based on the BP (back propagation) neural network model can quickly predict the pulsating pressure and vibration response of the cone–cylinder–sphere composite structure under different Reynolds numbers. Compared with the experimental results, the error of the machine learning prediction results is less than 7%. The research method proposed in this paper provides a new solution for the rapid prediction and control of hydrodynamic vibration noise of underwater vehicles.

1. Introduction

As human demand for marine resources increases, more and more ships and marine structures are also used to explore marine resources [1]. This also leads to an increasing impact on marine life [2,3]. Noise is a special pollutant produced by ships, which will affect the living environment of marine organisms in a certain range [4]. In addition, in the military field, there are higher requirements for the noise level of ships [5]. Underwater vehicles are a special type of vessel. Their appearance and internal structure are relatively complex, so the noise components of this type of ship are also quite complex. In general, ship noise can be divided into three types: noise caused by mechanical equipment, noise caused by fluids, and noise caused by propellers. Among them, fluid-induced noise involves the interaction between the fluid and structure. This problem belongs to the interdisciplinary field of fluid mechanics, solid mechanics, and acoustics. Therefore, it has become a recent research hotspot. A large underwater vehicle can be approximately regarded as a cylindrical thin-walled structure. Local deformation of such structures occurs under the action of water pressure and free flow, and this deformation changes continuously during the underwater vehicle’s voyage, potentially reducing the vehicle’s service life if not effectively controlled. In order to reduce the effects of fluid-induced vibration on underwater vehicles, many scholars have used various methods to study the characteristics of fluid-induced vibration. Ying et al. [6], taking the porous square cylinder model as the research object, studied the flow-induced vibration characteristics, influencing factors, and mechanism of the porous square cylinder via the numerical model method. Corcos et al. [7] first measured the pulsating pressure on a cylindrical surface in a wind tunnel and analyzed the loading characteristics of the cylindrical surface in different frequency ranges. Based on the experimental results, Chase et al. [8,9,10] proposed a turbulent pulsating pressure frequency wavenumber spectral model which approximately describes the pulsating pressure distribution on a flat plate. Liu et al. [11] investigated the characteristics of pulsating pressure on the surface of subscale submarines using large vortex simulation (LES) and frequency wavenumber spectral methods and also explored the mechanism of flow-induced vibration noise. Wei et al. [12] and Tian et al. [13] studied the excitation load characteristics of a stern propeller under unsteady flow. The finite element and boundary element methods are also used to analyze the structural response of the submarine under the excitation load of the stern shaft. Li et al. [14] observed the vortex structure on the surface of a ship propeller at different speeds through model tests. They also used computational fluid dynamics (CFD) methods to predict the pulsating pressure on the propeller surface, while evaluating the prediction accuracy of different frequency bands. Qin et al. [15] studied the hydrodynamic characteristics of underwater vehicles based on LES and Ffows Williams and Hawkings (FW–H) methods and proposed a shape of underwater vehicles with low noise level.

With the continuous application of machine learning methods in engineering, some researchers have begun to apply neural network models to turbulence feature prediction. A flow-induced vibration power output prediction model was established by Li et al. [16] through the mechanical learning method of a deep neural network (DNN). Based on this model, a power database was established, which significantly improved the prediction efficiency of the flow-induced vibration power response. Fukami et al. [17,18] reconstructed turbulent flow field data using a convolutional neural network model and improved the spatial resolution of turbulent flow fields. Rabault [19] designed a reinforcement learning model for active flow control and reduced surface resistance of cylinders at moderate Reynolds numbers. Maulik [20] applied machine learning methods to the classification problem of spatiotemporal dynamic turbulence models and verified the reliability of the model in predicting complex flow phenomena. Li et al. [21] also used the neural network model to accurately detect the turbulent and non-turbulent interfaces in the three-dimensional space flow field. This method improves the precision of interface prediction. Raissi et al. [22] combined nonlinear differential equations describing the laws of physics with neural networks. In solving physical problems, the convergence speed of the neural network model is accelerated, and its prediction accuracy is improved.

Under the influence of turbulent pulsating pressures, underwater structures are susceptible to flow-induced vibrations, which can significantly affect their service life. Chen et al. [23] designed a flow-induced vibration test device for a flexible hydrofoil. Through this device, they studied the vibration characteristics of composite hydrofoils under turbulent pulsating pressure excitation and compared the accuracy of prediction via linear potential flow theory. Banafsheh [24] and Ma [25] et al. studied the influence of the wake flow of an upstream cylinder on vortex-induced vibration characteristics of a downstream cylinder in a tandem cylinder system through experimental tests. Xu [26] and Ma [27] et al. studied the vibration characteristics of a flexible cylinder under turbulent excitation, while Korkischko [28] adjusted the boundary conditions of the cylinder to analyze the vortex-induced vibration characteristics of the cylinder under different boundary conditions. Jin et al. [29,30] developed an energy-based formula to describe the vibration characteristics of submarine hulls in heavy fluids. Based on this formula, they explored the coupling of stiffness with the vibration and acoustic radiation characteristics of underwater structures. Zhang et al. [31] studied the unsteady flow characteristics of a propeller under stern wake conditions and analyzed the vibration characteristics of a propeller under unbalanced exciting force. There are two main methods for controlling typical flow-induced vibration in underwater structures: active control and passive control [32]. A velocity feedback controller was developed by Baz [33] to control vortex-induced vibration of a flexible cylinder, and the results were compared with theoretical predictions to verify the effectiveness of the controller. A method based on modal development techniques was proposed by Zhou et al. [34], which can effectively predict the vibration–acoustic behavior of immersion reinforced composite plates under turbulent boundary layer excitation. A semi-analytical method is proposed by Chen [35], which can predict the vibration and noise of annular ribbed cylindrical shells considering the internal structural coupling. A new semi-analytical method was proposed by Wang et al. [36,37] to predict the vibration characteristics of functionally graded composites. They also used this method to study the vibration characteristics of plate–shell structures under different boundary conditions.

From the literature review, it can be found that many scholars have studied the characteristics of turbulent pulsating pressure and flow-induced vibration of underwater structures, but there are few studies on the rapid prediction of pulsating pressure and flow-induced vibration. Machine learning methods have been applied in the field of flow field identification by some scholars, but this method has not been applied in the field of vibration prediction. The research shows that the neural network model in the machine learning method has good performance in solving the data prediction problem. Compared with other models, the BP neural network model has stronger approximation ability when solving nonlinear mapping relationships.

In this paper, the cone–cylinder–sphere combination structure is taken as the research object. An experimental model was designed to measure the pulsating pressure and vibration response of the structure surface at different flow velocities. The influence of sensor quality and water pressure on the calculation results of the model is analyzed. A neural network model is also established to predict the pulsating pressure and vibration response of the shell structure surface. The validity of the machine learning method is verified by comparing the predicted results with the experimental results.

2. Experiment Measurement

2.1. Experimental Model

The structure of the actual underwater vehicle is complicated, which is not suitable for direct theoretical and experimental research. Zou et al. [38] simplified the underwater vehicle into a cone–cylinder–sphere combined structure when studying the flow-induced noise in underwater vehicles, which could better represent the main characteristics of underwater vehicles. The same simplification method was used in this study. The specific geometric parameters are shown in Figure 1a, and the shape parameters of the combined structure can be represented by Equation (1).

$$\left\{\begin{array}{c}{\left(x-R\right)}^2+y^2{=R}^2\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} (0\le x\le R)\hfill \\ \left|y\right|=R \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} (R\le x\le 9R)\hfill \\ y=-0.24x+0.32\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} (9R\le x\le 11.1R)\hfill \\ {\left(x-11.1R\right)}^2+y^2={\left(0.5R\right)}^2\hspace{1em}\hspace{1em}\hspace{1em}(11.1R\le x\le 11.5R)\hfill \end{array}\right.$$

(1)

where L = 1.15 m represents the total length of the underwater vehicle, and R = 0.1 m is the radius of the cylindrical section of the underwater vehicle.

According to the equation of forced vibration for objects, the vibration response of the structure under complex flow is affected not only by the fluctuating pressure of the fluid, but also by its inherent modes. In order to obtain the relationship between vibration response and excitation load in different regions of the underwater vehicle, two cone–cylinder–sphere combined structural models of the same size are designed and fabricated. In order to install the pressure sensor, several circular holes were cut into the surface of one of the models. The installation position of the pressure sensor is shown in Figure 1b, and the parameters of the pressure sensor are consistent with those described in the literature [38]. The surface of another model is smooth with marked locations for the installation of acceleration sensors on the internal wall. These sensors are used to measure the vibration acceleration at these locations. However, the acceleration sensor may change the mass distribution of the model itself and change the inherent vibration characteristics of the cone–pillar–ball composite structure. Considering sensitivity, mass, and size, the piezoelectric waterproof acceleration sensor was chosen. Each sensor has a mass of approximately 39 g. During the simulation, each sensor is loaded to its corresponding position on the surface of the cone–cylinder–sphere combined structure.

Figure 2 shows the influence of additional mass caused by the acceleration sensors on the natural vibration characteristics of the structure. As can be seen from the figure, the additional mass has almost no effect on the first three-order natural frequencies, but its effect gradually emerges as the frequency increases. Figure 2c,d shows the structural modes at different frequencies. It can be seen that with the increase in frequency, the structural mode becomes more complex, and the local mode begins to dominate. This shows that the additional mass has a significant effect on the local mode. It also leads to a large error in the modal characteristics of the structure in the middle- and high-frequency range.

2.2. Experiment System

The experiment was conducted in a gravity-type water tunnel at the Key Laboratory of Underwater Acoustic Technology, Harbin Engineering University. The experimental system, detailed in the literature (Zou), met the testing requirements for this study. According to the installation requirements of the experimental system, the experimental model was processed and the installation interface was left in the upper part of the model. The basic principle of the experimental test is shown in Figure 3. The test model was fixed to the test section through the top cover plate, and the acceleration sensor was installed to the inner wall of the model through magnetic suction. The sensor was connected to the signal acquisition system via cable. Two sets of acceleration sensors were also installed on the top cover plate of the test model to measure the background vibration of the experimental system. In order to study the fluid-induced vibration characteristics of the cone–column–sphere combined structures, the vibration responses of the structures at different flow velocities were measured.

The fluctuating pressure load and flow-induced vibration response of the cone–cylinder–sphere combined structure were separately measured. The layout of the pulsating pressure sensor requires mounting holes in the model surface. The vibration acceleration sensor is adsorbed on the inner surface of the structure via a magnetic base. On the same model, there is no way to install a vibration acceleration sensor and a pulsating pressure sensor in the same location. Two models with consistent structural parameters are designed to maintain the consistency of pulsating pressure and vibration response measurements. Model 1 was only used to measure the fluctuating pressure on the surface of the structure, while model 2 was only used to measure the vibration response on the inner wall of the structure. During the experiment, the model was lifted with a crane and waterproofed at the connection between the model and the test system. At least 3 measurements were taken for each flow velocity to reduce testing errors. After completing all the tests for all operating conditions with model 1, it was replaced with model 2, and the above process was repeated for measurement.

The uncertainty of the test results under the same working condition was compared during the test. For the measurement of pulsating pressure, it is considered that the measurement results with a dispersion of less than 3 dB are valid. In all the repeated tests of each working condition, three groups of test data that meet the requirements are selected as the basis for analysis.

The measurement points of the model are numbered 1# to 15# from the nose to the tail. Figure 4 shows the measured results of the acceleration of the model and the top cover plate at some flow velocities. The experimental results show that there is a significant difference in vibration acceleration between the model surface and the top cover plate in the frequency range of 10 Hz–1000 Hz. This finding provides evidence that the vibration of the top cover plate has no noticeable effect on the model surface. The results show that it is reasonable to truncate the top cover plate in the subsequent analysis.

The mechanism of cavitation noise is that when the fluid moves at high speed, the local pressure decreases, and cavitation bubbles are formed. When the bubble bursts, it produces a strong pulsing noise. The premise of cavitation noise is that the flow is cavitation.

The cavitation number ($\sigma$) can be used to measure whether cavitation occurs in the fluid surrounding a structure and the degree of development of cavitation. The cavitation number can be expressed as follows:

$$\sigma ={\displaystyle \frac{p-p_v}{\rho v_0^2/2}}$$

(2)

In the formula, $p$ represents the absolute pressure of the reference point, $v_0$ represents the free flow velocity, $\rho$ represents the liquid density, and $p_v$ represents the saturated vapor pressure of the liquid at the corresponding temperature.

In this study, the maximum flow rate is 9.81 m/s, and the flow temperature is less than 30 °C. According to the above formula, the cavitation number is about 2.2, and it is difficult to have a stable cavitation phenomenon.

3. Theoretical Methods

3.1. Turbulence Model

The large eddy simulation (LES) method is a technique that lies between direct numerical simulation (DNS) and Reynolds-averaged Navier–Stokes (RANS) methods. It provides high accuracy and computational efficiency in simulating unsteady flow. The filtered continuity equation and the incompressible Navier–Stokes equation can be expressed as follows:

$${\displaystyle \frac{\partial {\overline{u}}_i}{\partial x_i}}=0$$

(3)

$${\displaystyle \frac{\partial {\overline{u}}_i}{\partial t}}+{\displaystyle \frac{\partial }{\partial x_j}}({\overline{u}}_i{\overline{u}}_j)=-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial \overline{p}}{\partial x_i}}+\nu {\displaystyle \frac{{\partial }^2{\overline{u}}_i}{\partial x_j^2}}-{\displaystyle \frac{\partial }{\partial x_j}}{\tau }_{\mathit{ij}}^S$$

(4)

where $u_i$ represents the velocity component associated with $x_i$, $\overline{p}$ represents the filtered pressure of the fluid, ${\overline{u}}_i$ and ${\overline{u}}_j$ denote the averaged velocity components after filtering, $\rho$ represents the density of the fluid, and $\nu$ represents the dynamic viscosity coefficient of the fluid. ${\tau }_{\mathit{ij}}^S={\overline{u}}_i{\overline{u}}_j-\overline{u_i{u}_j}$ represents sub-grid scale Reynolds stress. This physical quantity reflects the interaction between large-scale eddies and small-scale eddies, such as energy and momentum exchange, feedback of small-scale eddies on large-scale eddies, and others. The Smagorinsky sub-grid scale model for turbulence was originally proposed in LES and remains one of the most commonly used sub-grid scale models. In this study, the adaptive wall local vortex viscosity (WALE) sub-grid scale model is selected. According to Boussinesq’s assumption, the sub-grid scale stress tensor can be expressed as follows:

$${\tau }_{\mathit{ij}}^S-{\displaystyle \frac{1}{3}}{\delta }_{\mathit{ij}}{\tau }_{\mathit{kk}}=-2{\nu }_T\overline{S_{\mathit{ij}}}$$

(5)

$$\overline{S_{\mathit{ij}}}={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial {\overline{u}}_i}{\partial x_j}}+{\displaystyle \frac{\partial {\overline{u}}_j}{\partial x_i}}\right)$$

(6)

$${\nu }_T={\left(C_W\Delta \right)}^2{S}_w$$

(7)

$$S_w={\displaystyle \frac{{\left(S_{\mathit{ij}}^d{S}_{\mathit{ij}}^d\right)}^{3/2}}{{\left(S_{\mathit{ij}}^d{S}_{\mathit{ij}}^d\right)}^{5/4}+{\left(\overline{S_{\mathit{ij}}^d}\overline{S_{\mathit{ij}}^d}\right)}^{5/2}}}$$

(8)

$$S_{\mathit{ij}}^d={\displaystyle \frac{1}{2}}\left[\nabla \nu ·\nabla \nu +{\left(\nabla \nu ·\nabla \nu \right)}^T\right]-{\displaystyle \frac{1}{3}}\mathrm{tr}\left(\nabla \nu ·\nabla \nu \right)I$$

(9)

where ${\delta }_{\mathit{ij}}$ represents the Kronecker delta function, and $\overline{S_{\mathit{ij}}}$ represents the sub-grid scale Reynolds stress tensor. ${\nu }_T$ represents the sub-grid scale eddy viscosity coefficient, and $\Delta$ represents the filter width. $C_W$ represents the model coefficient. $S_w$ represents deformation coefficient. $S_{\mathit{ij}}^d$ represents the traceless, symmetric part of the squared filtered velocity gradient tensor. $I$ represents the identity tensor.

The WALE sub-grid scale model uses a new form of velocity gradient tensor that does not require any form of near-wall damping. It can provide accurate proportional scaling automatically at the wall. Posa and Balaras [39] used the WALE model coupled with the standard equilibrium wall-layer model for the simulation of the DARPA Suboff at very high Reynolds number, obtaining a good agreement with the experimental data.

3.2. Modal Superposition Method

The vibration equation of a multi-degree-of-freedom elastic body under external forces can be expressed as follows:

$$\left[m\right]\left|\ddot{x}\left(t\right)\right|+\left[c\right]\left|\dot{x}\left(t\right)\right|+\left[k\right]\left|x\left(t\right)\right|=\left|F\left(t\right)\right|$$

(10)

where $\left[m\right]$ represents the mass matrix of the system, $\left[c\right]$ represents the damping matrix of the system, $\left[k\right]$ represents the stiffness matrix of the system, $\left|x\left(t\right)\right|$ represents the generalized displacement vector of the system, and $\left|F\left(t\right)\right|$ represents the excitation load of the system.

When the system is under proportional damping, the damping matrix can be expressed as a linear combination of $\left[m\right]$ and $\left[k\right]$, and $\left[c\right]$ can be expressed as follows:

$$\left[c\right]=\alpha \left[m\right]+\beta \left[k\right]$$

(11)

In the formula, $\alpha$ and $\beta$ represent rayleigh damping coefficients.

Define the modal matrix of a system as $\left[u\right]=\left(\left|u^{\left(1\right)}\right|,\left|u^{\left(2\right)}\right|,\cdots ,\left|u^{\left(n\right)}\right|\right)$. Therefore, Equation (9) can be transformed as follows:

$$\left({\left[u\right]}^T\left[m\right]\left[u\right]\right)\left|\ddot{x}\left(t\right)\right|+\left({\left[u\right]}^T\left[c\right]\left[u\right]\right)\left|\dot{x}\left(t\right)\right|+\left({\left[u\right]}^T\left[k\right]\left[u\right]\right)\left|x\left(t\right)\right|={\left[u\right]}^T\left|F\left(t\right)\right|$$

(12)

$${\left[u\right]}^T\left[m\right]\left[u\right]=\left[I\right]$$

(13)

$${\left[u\right]}^T\left[k\right]\left[u\right]=\left[\begin{array}{ccc}\bcancel{}& & \\ & {\omega }_r^2& \\ & & \bcancel{}\end{array}\right]$$

(14)

Therefore, Equation (10) can be further expressed as follows:

$$\ddot{x}\left(t\right)+2{\xi }_r{\omega }_r\dot{x}\left(t\right)+{\omega }_r^{2}x\left(t\right)=\left|u^{\left(r\right)}\right|\left|F\left(t\right)\right|\left(r=1, 2,\cdots ,n\right)$$

(15)

$${\xi }_r={\displaystyle \frac{\alpha +\beta {\omega }_r^2}{2{\omega }_r}}$$

(16)

where ${\omega }_{\mathrm{r}}$ represents the r-th order modal frequency of the system. When each component of the excitation load $\left|F\left(t\right)\right|$ is a determined function, each equation in Formula (15) can be decomposed into independent single-degree-of-freedom systems.

The vibration response of a structure subjected to turbulence-induced pressure fluctuations can be solved using the modal superposition method. This method assumes that the surface displacement of the structure can be represented by the linear combination of modal shapes and uses modal frequencies and structural shapes to decouple the fluid–structure coupling vibration equation. A modal coordinate system is established to calculate the response in the modal coordinate system, and the structural response in the original physical coordinate system is obtained via combination. This method can be expressed by Equation (17).

$$y\left(x,t\right)={\displaystyle \sum _{i=1}^n{\varphi }_i{q}_i\left(t\right)}$$

(17)

where $y\left(x,t\right)$ represents the displacement vector corresponding to each node degree of freedom, ${\varphi }_i$ represents the i-th order modal shape vector, and $q_i\left(t\right)$ represents the i-th order modal coordinate. The relationship between the cross power spectral density function ($S_{q_j{q}_k}\left(\omega \right)$) in generalized coordinates and the cross power spectral density function ($S_{f_j{f}_k}\left(\omega \right)$) of the excitation load can be expressed as follows:

$$S_{q_j{q}_k}\left(\omega \right)=H_j\left(\omega \right)H_k^{\ast }\left(\omega \right)S_{f_j{f}_k}\left(\omega \right)$$

(18)

where $H_j\left(\omega \right)$ and $H_k^{\ast }\left(\omega \right)$ represent the system transformation matrix.

By substituting Equation (18) into Equation (17), the structural vibration response can be further expressed as follows:

$$\overline{y^2\left(x,t\right)}={\displaystyle \sum _j{\displaystyle \sum _k{\varphi }_j\left(x\right)}}{\varphi }_k\left(x\right){\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}{H}_j\left(\omega \right)H_k^{\ast }\left(\omega \right)S_{f_j{f}_k}\left(\omega \right)d\omega }$$

(19)

$$S_x\left(\omega \right)={\displaystyle {\int }_{-\infty }^{+\infty }{R}_x\left(\tau \right)}{e}^{-i\omega \pi }d\tau$$

(20)

In the equation, $S_x$ represents power spectral density, and $R_x\left(\tau \right)=E\left[x\left(t\right)x\left(t+\tau \right)\right]$ is defined as the autocorrelation function of power spectral density. By using this definition, the power spectral density function of the structural response can be calculated.

3.3. Machine Learning Framework

The internal relationships in complex dynamic problems can be effectively described with neural networks. In this study, a neural network framework is proposed. As shown in Figure 5, the framework consists of a feature input, activation function layer, batch normalization, hidden layer, and feature output.

The feature input layer is the data collector of the neural network and performs normalization of the global data. In this study, when the surface pulsation pressure of the cone–cylinder–sphere combined structure is predicted, the input data $X=[f,u,l,r,R,p_i,p_j]$, where f represents the frequency, u represents the distance between the measuring point and the head of the model, l represents the total length of the model, r represents the radius of the cross section where the measuring point is located, R represents the maximum radius of the model cross-section, $p_i$ represents the pressure at the head of the cylinder, and $p_j$ represents the pressure at the end of the cylinder. The output variable is the pulsating pressure at the measuring point. When the surface fluid-induced vibration response of the cone–cylindric–spherical composite structure is predicted, the input data $X=\left[{f,u,l,r,R,m}_d,p\right]$, where $m_d$ represents the participation factor of the structural mode, and $p$ represents the pulsating pressure at the measuring point. The output variable is the fluid-induced vibration response at the measuring point.

The activation function layer applies the selected activation function to the feature input layer to obtain a high-dimensional representation. The activation function plays a very important role in the neural network framework, as it greatly affects the gradient features during the training process. In this study, the tanh-sigmoid function and linear unit function were chosen as the activation functions, which can be expressed as follows:

$$\begin{gathered} {\mathit{f}}_1\left(x\right)={\displaystyle \frac{e^x-e^{-x}}{e^x+e^{-x}}} \\ {f}_2(x)=x \end{gathered}$$

(21)

To enhance the robustness of the neural network, batch normalization was incorporated into its framework to mitigate the problem of gradient explosion. The input feature data set was divided into a training set and a testing set, which were, respectively, utilized for neural network model training and effectiveness verification. The hidden layer, which represents the core computation module of the neural network architecture, processes the input feature parameters with Equation (22) to obtain the input of the h-th hidden layer neuron. Its output is then processed by Equation (23) to generate the input of the j-th output neuron.

$${\alpha }_h={\displaystyle \sum _{i=1}^d{v}_{ih}{x}_i}$$

(22)

$${\beta }_j={\displaystyle \sum _{h=1}^d{w}_{hj}{b}_h}$$

(23)

In the formula, ${\alpha }_h$ represents the input received by the h-th neuron in the hidden layer, and ${\beta }_j$ represents the input received by the j-th neuron in the output layer.

If the neural network’s output for the training set $(x_k,y_k)$ is ${\widehat{y}}_k=({\widehat{y}}_1^k,{\widehat{y}}_2^k,\cdots ,{\widehat{y}}_l^k)$, then the mean squared error of the neural network on the training set can be expressed as follows:

$$E_k={\displaystyle \frac{1}{2}}{\displaystyle \sum _{j=1}^l{({\widehat{y}}_j^k-y_j^k)}^2}$$

(24)

In order to make the training result closer to the real result, the error is backpropagated, and the weight and threshold are adjusted according to the error of the hidden layer neurons. After repeated iterations, the training stops when the error meets the set requirements. The cumulative error of the entire training set can be expressed as Formula (25). To avoid overfitting of the neural network, which may result in training low set error but test set high error, a component describing the complexity of the network was added to the error function. Then Formula (25) can be represented as Formula (26).

$$E={\displaystyle \frac{1}{m}}{\displaystyle \sum _{k=1}^m{E}_k}$$

(25)

$$E=\lambda {\displaystyle \frac{1}{m}}{\displaystyle \sum _{k=1}^m{E}_k}+(1-\lambda ){\displaystyle \sum _i{w}_i^2}$$

(26)

4. Numerical Simulation Model

4.1. Flow Field Simulation

Based on the parameters of the experimental model, a corresponding numerical model was established in STAR CCM. As shown in Figure 6a, the right boundary is set as the flow inlet and the left boundary as the outlet. The model surface is set to a smooth wall. Zou et al. [38] studied the grid resolution of the numerical model, and based on this research, a structured mesh with a length of 2 mm was adopted for the surface mesh of the cone–cylinder–sphere. The height of the first layer grid nodes had a significant impact on the numerical simulation results. Ren et al. [40] investigated the influence of the height of the first layer grid on wall pressure fluctuations in their literature; for the model in this study, the same y+ value as that in the literature can be used. In this study, the analysis frequency range of fluctuating pressure and vibration response was set to be 10 Hz to 1000 Hz. The maximum time step in unsteady flow calculations can be determined with Formula (27).

$$f_s={\displaystyle \frac{1}{2\Delta t}}$$

(27)

In the formula, $f_s$ represents the upper limit of the calculation frequency, and $\Delta t$ represents the computational time step. $\Delta t=5\times {10}^{-4}s$.

As shown in Figure 6b, the flow field mesh around the model was refined. A second-order upwind discretization scheme was used for time discretization, and the hybrid Gaussian-LSQ method was selected for gradient calculation. Due to the special installation form, the upper part of the test model is close to the top of the working section. In order to analyze whether the close distance of the wall will affect the flow simulation, the influence of the range of the flow field on the simulation results is studied. Pressure monitoring points were set on the surface of the numerical model, and their positions were consistent with those of the pressure monitoring points in the experimental model. This research method can better explore the relationship between flow simulation results and the flow field domain range.

Figure 6d,e show the distribution of the surface pressure coefficient of the cone–cylinder–sphere combined structure under different flow field ranges. The variation of flow field range has no significant effect on the pressure distribution in most areas of the cone–cylinder–sphere combined structure. For the enlarged flow field area, a larger high-pressure region will be formed behind the flange region. This region is marked with a red dashed circle. Figure 6c shows the generation, development, and separation characteristics of the vortex structure around the cone–cylinder–sphere combined structure. The vortex structure is captured by the Q criterion and displayed as Q = 500. The surface color represents the velocity amplitude of the vortex structure. The results show that large eddy simulation method can capture the flow characteristics of the fluid on the surface of the cone–cylinder–sphere combined structure well.

A significant stagnation point appears at the head of the structure, where the velocity is close to zero and the pressure coefficient is close to its maximum. The boundary layer in this region is layered, and it gradually transitions from laminar flow to turbulent flow in a length range of about 0.09 L and finally achieves complete separation at 0.09 L. In the cylinder section, the vortex distribution remains almost the same in the absence of protrusions. A connecting vortex is generated in front of the protrusion, and two separate horseshoe vortices are generated on either side of the protrusion. Then, a backward spike vortex forms above the protrusion, eventually evolving into a larger vortex that separates downstream of the protrusion. The vortex appears at the tail of the structure, where it interacts in the form of a hairpin vortex with the shed wake behind the protrusions.

From the perspectives of surface pressure coefficient distribution on the structure and development of vortex surrounding the structure, different approaches to dividing the flow field region can influence the numerical simulation results in the area behind the protrusion. To ensure that the numerical simulation results are more consistent with reality, a flow field region of the same size as the experimental area was selected for computation and analysis in subsequent numerical calculations.

4.2. Vibration Calculation Model

The experimental model used in this study has a thickness of 2 mm. The fluid–structure coupling method is used to transfer the flow field pressure and structural deformation. The solution of this process was completed through STAR CCM and Abaqus co-simulation. Figure 7a shows the calculation procedure involved in the method, while Figure 7b shows the boundary conditions and loads imposed on the structural model during the calculation. In the interface region between the flow field domain and the structural domain, perfect matching of the mesh partitioning technique was adopted for ensuring a smooth exchange of both flow and structural information. This method was used to minimize transmission errors and ensure higher accuracy in the numerical calculations.

In actual experimental conditions, the model was subjected to an initial deep-water pressure due to the high free water surface of the gravity-type water tank. In addition, the interior of the model is filled with air, which makes the model susceptible to additional deformation due to differences between internal and external pressures. If this deformation is ignored in the calculation, the results of the calculation can be significantly inaccurate. Figure 8 shows the displacement of the structure’s surface and the distribution of flow velocity at a velocity of 5.71 m/s. The results reveal that under certain pressure differences, the test model exhibits remarkable deformation in the cylindrical region (the black dashed line represents the pre-deformation shape, while the red dashed line represents the post-deformation shape). In addition, the fluid–structure coupling effect causes significant changes in the flow field around the test model. When the shell surface of the cylindrical section of the model is subjected to pressure, the curvature of this part changes. Under the effect of an adverse pressure gradient, the fluid velocity decreases, and a large number of fluid micro-groups gather in this area, affecting the original vortex formation and shedding. By comparing the wakes behind the model in both cases, it can be concluded that the vortex size in Figure 8a is significantly larger than that in Figure 8b for the same area. This may cause the fluctuating pressures on the surface to migrate towards low frequencies and increase the vibration response induced by flow forces on the surface of the structure. Figure 9 shows the distribution of pressure coefficients on the surface of the structure under two conditions. The results further demonstrate the influence of the deformation of the cylinder region on the pressure distribution of the structure.

5. Results and Discussion

5.1. Analysis of Pulsating Pressure

This study compares three methods for obtaining the fluctuating pressure of a cone–cylinder–sphere combined structure: numerical calculation, machine learning prediction, and experimental measurement. To improve the reliability of the machine learning algorithm, all input data were divided into a training set, validation set, and test set. Among them, the training set accounts for 70% of the total data, and the validation set and the test set account for about 15% each. However, in the frequency range of 10 Hz–1600 Hz, the agreement between the initial prediction and the actual results of the pulsation pressure is poor, especially in the low frequency range of 10 Hz–100 Hz. By comparing the measured pulsation pressure spectrum, it can be found that the pulsation pressure changes significantly in the range of 10 Hz–100 Hz and is approximately exponential in the range of 100 Hz–1600 Hz. Training the entire frequency range uniformly would have impacted the low-frequency prediction results considerably. Therefore, further training was conducted on low frequency fluctuating pressure, emphasizing its characteristics through enhanced training. Take pressure test points 5#, 8#, 11#, and 14# for example. Figure 10 shows the results obtained via different methods when the flow velocity is 5.71 m/s.

The simulation results are in good agreement with the experimental results, which indicates that both CFD and machine learning are effective methods to predict the surface pulsation pressure of the experimental model. The CFD method has higher accuracy in predicting the waveform characteristics of the fluctuating pressure, particularly in the low-frequency range. The machine learning method exhibits greater accuracy in predicting the fluctuating pressure behavior above 100 Hz, where an exponential decay trend is visible. However, for a given boundary condition, the time required to perform a CFD method prediction is much longer than the time required for a machine learning-based prediction.

By calculating the sum of energy in the specified frequency band, the total pressure level in the frequency range of 10 Hz–1600 Hz can be obtained. Figure 11 shows the error of the machine learning prediction method at different flow velocities. At lower flow velocities, the error of the total pressure level is less than 3%, and the predicted value is slightly less than the experimental value. As the flow velocity increases and the turbulence on the surface of the vehicle becomes more complex, the error in the prediction results usually increases slightly. The maximum error of the pulsating pressure of the vehicle is about 3.2%, which still meets the practical requirements of engineering. This shows the feasibility of the pulsating pressure prediction method based on machine learning. In the case of a large number of training samples, the machine learning method can predict the pulsating pressure on the surface of the underwater vehicle faster and more accurately.

5.2. Analysis of Fluid-Induced Vibration Response

The pulsating pressure predicted in this study is taken as the input load applied to the surface of the structural model. Then, the modal superposition method is used to solve the vibration response of the structure surface. Using the second-, third-, and fourth-order natural modes as examples, Figure 12 shows the vibration response of the structure surface at different speeds. It can be observed that when the frequency is low, the vibration displacement response of the aircraft surface is mainly concentrated in the parallel, middle, and tail regions. The response near the connection area was smaller than that in the belly area of the underwater vehicle. With the increase in the Reynolds number, the vibration displacement of the underwater vehicle surface also increases noticeably, which is due to the higher fluctuating pressure under high flow velocity compared to that under low flow velocity. At different flow velocities, the vibration response of the second-order mode is much larger than that of the third-order and fourth-order modes. However, in some cases, the vibration response of the fourth mode is greater than that of the third mode. This suggests that the fourth-order mode contributes more to the overall vibration than the third-order mode at these flow velocities.

In analyzing the flow-induced vibration response of underwater vehicles, it is essential to consider the influence of the structural natural vibration modes. To integrate this into machine learning predictions, the wave pressure and the structure’s natural vibration characteristics are included as part of the feature input set.

$${\rho }_{\mathrm{x},y}={\displaystyle \frac{\mathrm{cov}(x,y)}{{\sigma }_x{\sigma }_y}}={\displaystyle \frac{E\left(\left(x-{\mu }_x\right)\left(y-{\mu }_y\right)\right)}{{\sigma }_x{\sigma }_y}}={\displaystyle \frac{E(xy)-\mathrm{E}(x)E\left(y\right)}{\sqrt{E(x^2)-E^2(x)}\sqrt{E(y^2)-E^2(y)}}}$$

(28)

In the equation, $\mathrm{cov}(x,y)$ represents the covariance between variables x and y, while ${\sigma }_x$ and ${\sigma }_y$ represent the standard deviation of variables x and y, respectively. The symbol E denotes mathematical expectation. It should be noted that both variables x and y have been normalized prior to calculating the Pearson correlation coefficient. When using a BP neural network, a higher correlation coefficient implies a stronger correlation between the input and output variables. In this study, the input and output variables are plugged into the above formula and calculated; ${\rho }_{x,y}$ = 0.918. Figure 13 displays the regression results of the training and testing datasets during the neural network simulation process. The figure demonstrates that the neural network model designed in this study possesses good prediction and generalization capabilities, as the training set, validation set, and test set exhibit excellent regression performance.

After completing the training process, the neural network model was used to predict the surface vibration response of the structure at different flow velocities. The predicted results at some measurement points when the flow velocity was 7.79 m/s are presented in Figure 14. In the experimental results, distinct characteristic peaks were observed at 40 Hz, 80 Hz, and 125 Hz for all measurement points. These peak frequencies are highly consistent with the first three natural frequencies of the model. This indicates that the inherent vibration characteristics of the structure are one of the key factors influencing the flow-induced vibration response. From the perspective of trend, the neural network model can reflect the change in vibration response well. The predicted and experimental values show good agreement across all frequency bands. However, fluctuations were observed in the experimental values in the 100 Hz–1000 Hz frequency range, while the predicted values remained relatively stable. This indicates that the designed neural network model still has accuracy issues when handling rapid fluctuations, which will be further optimized in future research.

In engineering applications, the total vibration level is defined as the sum of all vibration energies in a specified frequency band. With the increase in velocity, the surface vibration response of underwater vehicles also increases. The 8th measurement point located in the middle region of the parallel body exhibits the highest total vibration level, while the 14th measurement point located in the conical section exhibits the lowest. The accuracy of prediction is deemed to meet engineering application needs when the error between the predicted and experimental values of the total vibration level falls below 3 dB. Figure 15 displays the predicted and experimental values of the total vibration levels of some measurement points at different flow velocities. The results show that when the flow velocity is 7.79 m /s, the prediction error of the maximum total vibration level is the largest. The maximum error is 8.9 dB, which occurs at the fifth measurement point. In contrast, the prediction error is minimal when the flow velocity is 3.57 m/s. The minimum error is 0.3 dB, which occurs at the 14th measurement point. When the prediction error of the total vibration level is less than 3 dB, the prediction result is considered to be accurate. According to the selected experimental data, the prediction accuracy of the neural network model reaches 93.75%.

6. Conclusions

In this paper, the unsteady flow characteristics of the flow field around the structure and the structural vibration response of the shell under turbulent excitation are studied by using a simplified model of the underwater vehicle. Firstly, an experimental model was designed, and the surface pulsation pressure and vibration response data of cone–cylinder–sphere combined structure were obtained through experimental tests. Subsequently, numerical calculation methods are used to simulate the unsteady flow characteristics of the surrounding flow field. Finally, in order to solve the problem of rapid prediction of pulsating pressure and fluid-induced vibration response of underwater vehicles. A neural network model is used to rapidly predict the pulsating pressure and vibration responses of cone–pillar–ball structures. The main research findings are as follows:

(1)

The additional mass caused by the acceleration sensor can reduce the natural frequency of the higher-order mode of the structure, but has little effect on the natural frequency of the lower-order mode;

(2)

The presence of protrusion changes the original vortex structure around the cone–cylinder–sphere combined structure, resulting in a large number of small and medium scale vortices in the middle region of the cylindrical section. Under the action of deep-water pressure, the shell structure undergoes slight deformation. This deformation is mainly concentrated in the cylindrical section. After deformation of the shell, the vortex structure around the wall also changes;

(3)

The neural network model proposed in this study can quickly predict the surface pulsation pressure and vibration response of cone–cylinder–sphere combined structures. The predicted results of the pulsating pressure and vibration response are basically consistent with the test results. With the error of 3 dB as the threshold, the accuracy of the forecast method proposed in this study is greater than 93%. Moreover, the required computational time is far less than that of numerical simulation methods. Therefore, it can be used as a new method for rapidly predicting flow-induced vibration noise of underwater vehicles.