Method for predicting water outlet trajectory of supercavitation aircraft based on BO-FNN

Abstract

The invention discloses a supercavitation vehicle effluent trajectory prediction method based on BO-FNN. The method comprises the following steps: step 1, constructing a CFD basic mathematical model; 2, based on the mathematical model, boundary condition setting and mesh generation are carried out on the external flow field of the aircraft, and a simulation model is obtained; 3, verifying the simulation model to obtain a numerical model of the water outlet process; 4, an orthogonal test is designed, and a water outlet trajectory data set is obtained according to the numerical model; step 5, designing a feedforward neural network, and fitting the water-out trajectory data set to obtain a water-out trajectory prediction model presenting an over-fitting characteristic; and step 6, optimizing the water outlet trajectory prediction model to obtain a supercavitation vehicle water outlet trajectory prediction model based on the BO-FNN. According to the supercavitation aircraft water outlet trajectory prediction method based on the BO-FNN, the problems that the supercavitation aircraft water outlet process is high in nonlinearity, and prediction cannot be carried out under the multivariable coupling condition are solved.

Description

Technical Field

[0001] This invention belongs to the technical field of vehicle trajectory prediction methods, specifically relating to a method for predicting the water exit trajectory of supercavitating vehicles based on BO-FNN. Background Technology

[0002] Supercavitating vehicles are underwater weapons that utilize supercavitation technology. Through natural cavitation or artificial ventilation, the vehicle is largely or completely enveloped in cavitation bubbles, reducing drag by over 90% and achieving speeds exceeding 200 knots, resulting in superior penetration capabilities compared to traditional underwater vehicles. However, supercavitating vehicles cannot be guided solely by traditional underwater acoustic homing methods. This is due to several factors: the small area of ​​the cavitation duct at the nose makes it difficult to install transducers; the cavitation bubble envelops the vehicle, increasing its acoustic complexity; and the high speed results in significant self-noise. One solution is to surface the supercavitating vehicle and use electromagnetic waves to detect targets in the air, thereby correcting its course.

[0003] According to publicly available literature, research on the motion characteristics of supercavitating vehicles during the water exit process mainly focuses on unsteady hydrodynamics and water exit cavitation, while research on water exit trajectory prediction under multivariable coupling conditions is rare. Summary of the Invention

[0004] The purpose of this invention is to provide a method for predicting the water exit trajectory of supercavitating vehicles based on BO-FNN, which solves the problem that the supercavitating vehicle's water exit process is highly nonlinear and cannot be predicted under multivariate coupling conditions.

[0005] The technical solution adopted in this invention is a method for predicting the water exit trajectory of a supercavitating vehicle based on BO-FNN, comprising the following steps: Step 1: Construct the basic mathematical model of CFD; Step 2: Based on the basic CFD mathematical model, set boundary conditions and mesh the external flow field of the vehicle to obtain the simulation model; Step 3: Verify the simulation model to obtain a numerical model suitable for the water exit process of a supercavitating vehicle; Step 4: Design orthogonal experiments and obtain the water-ejection trajectory dataset based on the numerical model; Step 5: Design a feedforward neural network to fit the water-escape trajectory dataset and obtain a water-escape trajectory prediction model that exhibits overfitting characteristics. Step 6: Optimize the water exit trajectory prediction model using the Bayesian optimization method to obtain the water exit trajectory prediction model for supercavitating vehicles based on BO-FNN.

[0006] The invention is further characterized by: The basic mathematical models of CFD in step 1 include multiphase flow model, turbulence model, rigid body motion model and overlapping mesh model.

[0007] The multiphase flow model selected is the VOF model based on homogeneous equilibrium flow, which is used to obtain the phase interface; the basic governing equations of the VOF model include the continuity equation and the momentum equation. The continuity equations are shown in equations (1) and (2); (1); (2); in, α q For the fluid micro-element q Phase volume fraction; r q For the first q Phase density; v The velocity vector of the fluid element; n For the total number of phases, n =3, including air, water, and water vapor; interphase mass transfer rate and These represent the evaporation rate and condensation rate of water, respectively, described using a cavitation model; The momentum equations are shown in equations (3), (4) and (5); (3); (4); (5); in, p For static pressure, f It is a volume force; r m The average density of the fluid element; m m The average dynamic viscosity of the fluid element; The Schnerr-Sauer model based on transport equations and local pressure judgment is selected for multiphase flow model to deal with the phase change problem between water and water vapor in multiphase flow. The description of interphase mass transfer is shown in equations (6) and (7). (6); (7); in, r w and r v These are the densities of the water and vapor phases, respectively. α It represents the volume fraction of the vapor phase. p v This is the saturated vapor pressure; nThe number of voids per unit volume; R B Where is the bubble radius.

[0008] The turbulence model selected is the Realizable k-ε model based on the two-equation Reynolds time-averaged model. The Realizable k-ε model is used for turbulent kinetic energy. k and turbulent kinetic energy dissipation rate e The transport equations are shown in equations (8), (9) and (10); (8); (9); (10); in, r For fluid density, m t The turbulent viscosity coefficient; C μ The mean strain rate is a function of curl, in the equilibrium boundary layer inertial sublayer. C μ =0.09; s k and s ε For Prandtl numbers, s k =1.0, s ε =1.2; G k and G b These are the turbulent kinetic energy caused by the velocity gradient and buoyancy, respectively. Y M To illustrate the effect of compressible turbulent pulsating expansion on dissipation rate, for incompressible flow... Y M =0; u Kinematic viscosity; C 1, C 2, C 1ε , C 3ε For coefficients; The Realizable k-ε model is modified using wall functions. The scaled wall function is used to process the near-wall region. The scaled wall functions are shown in equations (11) and (12). (11); (12); in, U * For dimensionless velocity, y+ The distance is dimensionless; k =0.42, E =9.81; U a for a Average fluid velocity at point k a for a Average turbulent kinetic energy at point, y a for a Distance from point to wall; t w This represents the wall shear stress.

[0009] The rigid body motion model is as follows: the aircraft is regarded as a rigid body, and the free motion of the aircraft is divided into translation of the center of mass and rotation around the center of mass. The translation and rotation equations are solved in the ground coordinate system and the body coordinate system, respectively, as shown in equations (13) and (14). (13); (14); in, m For rigid body mass, J Let the moment of inertia of the rigid body be denoted as . v G For the velocity of the center of mass, oh It is the rotational angular velocity; F The resultant force acting on the rigid body, M B To obtain the resultant torque in the body coordinate system, we need to obtain the resultant torque from the ground coordinate system. M G Conversion, i.e. M B = R C M G ,in R C This is the transformation matrix; The overlapping mesh model is used to solve the independent background mesh and the overlapping mesh separately. The obtained data is then subjected to boundary interpolation to ensure that the mesh around the vehicle remains precise. At the same time, the free motion model is coupled to define the relative motion of the overlapping domain using the vehicle motion parameters obtained through iteration.

[0010] Step 2 is as follows: Step 2.1: Divide the external flow field of the vehicle into a cylindrical background domain and an overlapping domain; the axial length of the background domain is 8 times the length of the vehicle, and the radial length of the background domain is 3.5 times the length of the vehicle; the axial length of the overlapping domain is 1.5 times the length of the vehicle, and the radial length of the overlapping domain is 5 times the diameter of the vehicle; the background domain is stationary, the angle between the axis of the overlapping domain and the axis of the background domain is defined as the exit angle, the velocity of the overlapping domain is defined as the initial velocity of the vehicle, the angle between the velocity direction of the overlapping domain and the axis of the overlapping domain is defined as the initial angle of attack, the angular velocity of the overlapping domain is defined as the initial angular velocity, and the deflection angle of the cavitation disk at the front of the vehicle is defined as the bow deflection angle δ, the sign of which is determined by the right-hand rule; Step 2.2: Set the boundary conditions for the background domain and the overlapping domain; set the background domain inlet and sidewall to 0 velocity inlet, the background domain outlet to standard atmospheric pressure outlet, the overlapping domain boundary to overlapping interface, and the vehicle wall to stationary wall. Step 2.3: Perform structured mesh generation on the background domain and overlapping domain respectively to obtain the simulation model; the mesh scale of the motion passage area is no higher than 50 mm, the near free liquid surface starts from 20 mm and increases exponentially, the number of mesh layers in the cavitation region is no less than 30, and the boundary layer meets the requirement of the scaled wall function y+=30~300. The mesh scale of the first layer of the solid boundary is 0.2 mm.

[0011] Step 3 includes verification of cavitation morphology and verification of effluent operating conditions. The cavitation morphology verification specifically involves: calculating the steady-state cavitation flow field of the horizontal motion of the vehicle, comparing the cavitation profile of the flow field with the cavitation profile obtained by the Logvinovich empirical formula, ensuring that the cavitation diameter error of the vehicle body section does not exceed 10%, and comparing the cavitation drag coefficient obtained from the cavitation flow field with the cavitation drag coefficient of the Logvinovich empirical formula. If the error does not exceed 5%, it indicates that the cavitation model meets the simulation requirements. The verification of the water discharge condition specifically involves: performing flow field simulation on the free water discharge process of the hollow sphere in the experiment, comparing the flow field phase diagrams at different times with the experimental photographs, and ensuring that the maximum height error of the water mound does not exceed 10%. Furthermore, comparing the changes in the water discharge velocity between the simulation and the experiment, and ensuring that the maximum velocity error does not exceed 5%, indicates that the simulation model can effectively simulate the water discharge process.

[0012] The specific process of step 4 is as follows: Step 4.1: Design an orthogonal array based on different factors and the number of levels; different factors include initial velocity. v 0. Water outlet angle i 0. Initial angle of attack α 0. Initial angular velocity ω0 and bow deflection angle d At least two groups should be selected from different factors; the number of levels can be any number of groups, and the values ​​of each level should be as even as possible. Step 4.2: Input the orthogonal array data into a numerical model suitable for the water emergence process of a supercavitating vehicle for calculation, and organize the orthogonal test results as the water emergence trajectory dataset; the orthogonal test results include terminal velocity. v 1. Final pitch angle i 1. Final angle of attack α 1. Final pitch angular velocity oh 1 and the final state mass center y coordinate y1.

[0013] The specific process of step 5 is as follows: Step 5.1: Process the water jet trajectory dataset; The water jet trajectory dataset was normalized using a normalization method. The dataset order was shuffled, and the first 80% was used as the training set, while the remaining 20% ​​was used as the test set. The normalization method used was either the Z-score method or the min-max method. Step 5.2: Set up the feedforward neural network structure; Set the input layer: The input variables are several processed datasets containing velocity, angle of attack, angular velocity, pitch angle, and rudder angle. Set the number of neurons in the input layer to the number of factors. Set the output layer: Set the number of neurons in the output layer (Nou) to 1, and build a single-output prediction model for each final state parameter. Set the hidden layer: Set the number of hidden layer nodes, learning rate, L2 regularization coefficient, training method, and activation function to arbitrary values. Set the objective function: Define the objective function as the mean squared error, calculate the error of the objective function in network training, and evaluate it. Step 5.3: Learn from the training set data, calculate the mean square error of the test set data, and obtain the water exit trajectory prediction model that exhibits overfitting characteristics.

[0014] The specific process of step 6 is as follows: Step 6.1: Divide the 20 training sets into 15 sub-training sets and 5 sub-validation sets, and define the validation set MSE as the BO objective function; Step 6.2: Construct a Gaussian process surrogate model, use the expected-improvement-plus function for data collection, set the exploration weight to 0.5, the number of iterations to 50, and the convergence condition to be validation set MSE < 0.001, and calculate the BO objective function; Step 6.3: Determine whether the number of iterations and convergence conditions meet the requirements. If not, repeat the acquisition process. Once the requirements are met, obtain the supercavitating vehicle water exit trajectory prediction model based on BO-FNN.

[0015] The beneficial effects of this invention are: This invention provides a method for predicting the emergence trajectory of supercavitating vehicles based on BO-FNN. A high-precision flow-motion coupling model suitable for the emergence process of supercavitating vehicles is established, and the flow field characteristics, hydrodynamic characteristics, and motion characteristics under typical emergence conditions are studied. During the emergence phase, the cavitation on the upstream side begins to collapse first, while the collapse speed on the downstream side is faster. When 50% of the vehicle's length emerges from the water, the hydrodynamics of the vehicle increases, and the directions of lift and pitch moment are opposite to those before. Multiple sets of orthogonal test conditions were designed and calculated. Through variance analysis, it was found that the initial velocity has a significant impact on all four test indicators; the emergence angle has a significant impact on the terminal velocity and terminal pitch angle; the initial angle of attack has a significant impact on the terminal velocity and terminal angular velocity; and the bow deflection angle only has a significant impact on the terminal pitch angle. A BO-FNN model was established to fit the initial and final ballistic parameters of the aircraft. The model was trained and the R² of the prediction model on all test sets was above 0.9, indicating strong generalization. Except for the pitch angle model, the MSE of the model on the test set was not more than 0.07, and except for the angle of attack model, the MAPE was not more than 11%. The prediction of velocity and angular velocity was more accurate than that of pitch angle and angle of attack. The comparison with the FNN model shows that the prediction results of the BO-FNN model are better. Attached Figure Description

[0016] Figure 1 This is a schematic diagram of the supercavitating vehicle in Embodiment 6 of the present invention; Figure 2 This is a schematic diagram of the computational domain and boundary conditions in Embodiment 6 of the present invention; Figure 3 This is a schematic diagram of the overlapping domain mesh division in Embodiment 6 of the present invention; Figure 4 This is a schematic diagram of the background domain grid division in Embodiment 6 of the present invention; Figure 5 This is a comparison diagram of the cavitation profiles of numerical calculations and empirical formulas in Embodiment 6 of the present invention; Figure 6 This is a schematic diagram of the numerical calculation results in Embodiment 6 of the present invention; Figure 7 This is a schematic diagram of the rate decay in numerical calculations and experiments in Embodiment 6 of the present invention; Figure 8 This is a schematic diagram of the drag coefficients of different mesh sizes in Embodiment 6 of the present invention; Figure 9 This is the phase cloud diagram in Embodiment 6 of the present invention; Figure 10 This is the pressure cloud map in Embodiment 6 of the present invention; Figure 11 This is a schematic diagram illustrating the change of hydrodynamic coefficients with the displacement of the vehicle in Embodiment 6 of the present invention; Figure 12 This is a schematic diagram of the velocity and pitch angular velocity in Embodiment 6 of the present invention; Figure 13 This is a schematic diagram of the pitch angle, trajectory tilt angle, and angle of attack in Embodiment 6 of the present invention; Figure 14 This is a schematic diagram of a single-layer FNN structure in Embodiment 6 of the present invention; Figure 15 This is a flowchart of the BO implementation in Embodiment 6 of the present invention; Figure 16 This is a schematic diagram of the pitch angle θ1 of the BO-FNN model in Embodiment 6 of the present invention; Figure 17 This is a schematic diagram of the angle of attack α1 of the BO-FNN model in Embodiment 6 of the present invention; Figure 18 This is a schematic diagram of the angular velocity ω1 of the BO-FNN model in Embodiment 6 of the present invention; Figure 19 This is a schematic diagram of the pitch angle θ1 of the FNN model in Embodiment 6 of the present invention; Figure 20 This is a schematic diagram of the angle of attack α1 of the FNN model in Embodiment 6 of the present invention; Figure 21 This is a schematic diagram of the angular velocity ω1 of the FNN model in Embodiment 6 of the present invention. Detailed Implementation

[0017] The present invention will now be described in detail with reference to specific embodiments.

[0018] Example 1 The method for predicting the water-emergence trajectory of a supercavitating vehicle based on BO-FNN proposed in this embodiment includes the following steps: Step 1: Construct the basic mathematical model of CFD; Step 2: Based on the basic CFD mathematical model, set boundary conditions and mesh the external flow field of the vehicle to obtain the simulation model; Step 3: Verify the simulation model to obtain a numerical model suitable for the water exit process of a supercavitating vehicle; Step 4: Design orthogonal experiments and obtain the water-ejection trajectory dataset based on the numerical model; Step 5: Design a feedforward neural network to fit the water-escape trajectory dataset and obtain a water-escape trajectory prediction model that exhibits overfitting characteristics. Step 6: Optimize the water exit trajectory prediction model using the Bayesian optimization method to obtain the water exit trajectory prediction model for supercavitating vehicles based on BO-FNN.

[0019] Example 2 The method for predicting the water-emergence trajectory of a supercavitating vehicle based on BO-FNN proposed in this embodiment includes the following steps: Step 1: Construct the basic mathematical model of CFD; The basic mathematical models of CFD in step 1 include multiphase flow model, turbulence model, rigid body motion model and overlapping mesh model; The multiphase flow model selected is the VOF model based on homogeneous equilibrium flow, which is used to obtain the phase interface; the basic governing equations of the VOF model include the continuity equation and the momentum equation. The continuity equations are shown in equations (1) and (2); (1); (2); in, α q For the fluid micro-element q Phase volume fraction; r q For the first q Phase density; v The velocity vector of the fluid element; n For the total number of phases, n =3, including air, water, and water vapor; interphase mass transfer rate and These represent the evaporation rate and condensation rate of water, respectively, described using a cavitation model; The momentum equations are shown in equations (3), (4) and (5); (3); (4); (5); in, p For static pressure, f It is a volume force; r m The average density of the fluid element; m m The average dynamic viscosity of the fluid element; The Schnerr-Sauer model based on transport equations and local pressure judgment is selected for multiphase flow model to deal with the phase change problem between water and water vapor in multiphase flow. The description of interphase mass transfer is shown in equations (6) and (7). (6); (7); in, r w and r v These are the densities of the water and vapor phases, respectively. α It represents the volume fraction of the vapor phase. p vThis is the saturated vapor pressure; n The number of voids per unit volume; R B Where is the bubble radius; The turbulence model selected is the Realizable k-ε model based on the two-equation Reynolds time-averaged model. The Realizable k-ε model is used for turbulent kinetic energy. k and turbulent kinetic energy dissipation rate e The transport equations are shown in equations (8), (9) and (10); (8); (9); (10); in, r For fluid density, m t The turbulent viscosity coefficient; C μ The mean strain rate is a function of curl, in the equilibrium boundary layer inertial sublayer. C μ =0.09; s k and s ε For Prandtl numbers, s k =1.0, s ε =1.2; G k and G b These are the turbulent kinetic energy caused by the velocity gradient and buoyancy, respectively. Y M To illustrate the effect of compressible turbulent pulsating expansion on dissipation rate, for incompressible flow... Y M =0; u Kinematic viscosity; C 1, C 2, C 1ε , C 3ε For coefficients; The Realizable k-ε model is modified using wall functions. The scaled wall function is used to process the near-wall region. The scaled wall functions are shown in equations (11) and (12). (11); (12); in, U * For dimensionless velocity,y + The distance is dimensionless; k =0.42, E =9.81; U a for a Average fluid velocity at point k a for a Average turbulent kinetic energy at point, y a for a Distance from point to wall; t w This refers to the wall shear stress. The rigid body motion model is as follows: the aircraft is regarded as a rigid body, and the free motion of the aircraft is divided into translation of the center of mass and rotation around the center of mass. The translation and rotation equations are solved in the ground coordinate system and the body coordinate system, respectively, as shown in equations (13) and (14). (13); (14); in, m For rigid body mass, J Let the moment of inertia of the rigid body be denoted as . v G For the velocity of the center of mass, oh It is the rotational angular velocity; F The resultant force acting on the rigid body, M B To obtain the resultant torque in the body coordinate system, we need to obtain the resultant torque from the ground coordinate system. M G Conversion, i.e. M B = R C M G ,in R C This is the transformation matrix; The overlapping mesh model is used to solve the independent background mesh and the overlapping mesh separately. The obtained data is then subjected to boundary interpolation to ensure that the mesh around the vehicle remains precise. At the same time, the free motion model is coupled to define the relative motion of the overlapping domain using the vehicle motion parameters obtained through iteration. Step 2: Based on the basic CFD mathematical model, set boundary conditions and mesh the external flow field of the vehicle to obtain the simulation model; Step 3: Verify the simulation model to obtain a numerical model suitable for the water exit process of a supercavitating vehicle; Step 4: Design orthogonal experiments and obtain the water-ejection trajectory dataset based on the numerical model; Step 5: Design a feedforward neural network to fit the water-escape trajectory dataset and obtain a water-escape trajectory prediction model that exhibits overfitting characteristics. Step 6: Optimize the water exit trajectory prediction model using the Bayesian optimization method to obtain the water exit trajectory prediction model for supercavitating vehicles based on BO-FNN.

[0020] Example 3 The method for predicting the water-emergence trajectory of a supercavitating vehicle based on BO-FNN proposed in this embodiment includes the following steps: Based on Example 2; Step 2 is as follows: Step 2.1: Divide the external flow field of the vehicle into a cylindrical background domain and an overlapping domain; the axial length of the background domain is 8 times the length of the vehicle, and the radial length of the background domain is 3.5 times the length of the vehicle; the axial length of the overlapping domain is 1.5 times the length of the vehicle, and the radial length of the overlapping domain is 5 times the diameter of the vehicle; the background domain is stationary, the angle between the axis of the overlapping domain and the axis of the background domain is defined as the exit angle, the velocity of the overlapping domain is defined as the initial velocity of the vehicle, the angle between the velocity direction of the overlapping domain and the axis of the overlapping domain is defined as the initial angle of attack, the angular velocity of the overlapping domain is defined as the initial angular velocity, and the deflection angle of the cavitation disk at the front of the vehicle is defined as the bow deflection angle δ, the sign of which is determined by the right-hand rule; Step 2.2: Set the boundary conditions for the background domain and the overlapping domain; set the background domain inlet and sidewall to 0 velocity inlet, the background domain outlet to standard atmospheric pressure outlet, the overlapping domain boundary to overlapping interface, and the vehicle wall to stationary wall. Step 2.3: Perform structured mesh generation on the background domain and overlapping domain respectively to obtain the simulation model; the mesh scale of the motion passage area is no higher than 50 mm, the near free liquid surface starts from 20 mm and increases exponentially, the number of mesh layers in the cavitation region is no less than 30, and the boundary layer meets the requirement of the scaled wall function y+=30~300. The mesh scale of the first layer of the solid boundary is 0.2 mm.

[0021] Example 4 The method for predicting the water-emergence trajectory of a supercavitating vehicle based on BO-FNN proposed in this embodiment includes the following steps: Based on Example 3; Step 3 includes verification of cavitation morphology and verification of effluent operating conditions. The cavitation morphology verification specifically involves: calculating the steady-state cavitation flow field of the horizontal motion of the vehicle, comparing the cavitation profile of the flow field with the cavitation profile obtained by the Logvinovich empirical formula, ensuring that the cavitation diameter error of the vehicle body section does not exceed 10%, and comparing the cavitation drag coefficient obtained from the cavitation flow field with the cavitation drag coefficient of the Logvinovich empirical formula. If the error does not exceed 5%, it indicates that the cavitation model meets the simulation requirements. The verification of the water discharge condition is as follows: the flow field simulation is carried out on the free water discharge process of the hollow sphere in the experiment, the flow field phase diagram at different times is compared with the experimental photos, the maximum height error of the water mound does not exceed 10%, and the water discharge velocity change of the simulation and the experiment is compared. When the maximum velocity error does not exceed 5%, it shows that the simulation model can effectively simulate the water discharge process. The specific process of step 4 is as follows: Step 4.1: Design an orthogonal array based on different factors and the number of levels; different factors include initial velocity. v 0. Water outlet angle i 0. Initial angle of attack α 0. Initial angular velocity ω0 and bow deflection angle d At least two groups should be selected from different factors; the number of levels can be any number of groups, and the values ​​of each level should be as even as possible. Step 4.2: Input the orthogonal array data into a numerical model suitable for the water emergence process of a supercavitating vehicle for calculation, and organize the orthogonal test results as the water emergence trajectory dataset; the orthogonal test results include terminal velocity. v 1. Final pitch angle i 1. Final angle of attack α 1. Final pitch angular velocity oh 1 and the final state mass center y coordinate y1.

[0022] Example 5 The method for predicting the water-emergence trajectory of a supercavitating vehicle based on BO-FNN proposed in this embodiment includes the following steps: Based on Example 4, The specific process of step 5 is as follows: Step 5.1: Process the water jet trajectory dataset; The water jet trajectory dataset was normalized using a normalization method. The dataset order was shuffled, and the first 80% was used as the training set, while the remaining 20% ​​was used as the test set. The normalization method used was either the Z-score method or the min-max method. Step 5.2: Set up the feedforward neural network structure; Set the input layer: The input variables are several processed datasets containing velocity, angle of attack, angular velocity, pitch angle, and rudder angle. Set the number of neurons in the input layer to the number of factors. Set the output layer: Set the number of neurons in the output layer (Nou) to 1, and build a single-output prediction model for each final state parameter. Set the hidden layer: Set the number of hidden layer nodes, learning rate, L2 regularization coefficient, training method, and activation function to arbitrary values. Set the objective function: Define the objective function as the mean squared error, calculate the error of the objective function in network training, and evaluate it. Step 5.3: Learn from the training set data, calculate the mean square error of the test set data, and obtain the water exit trajectory prediction model that exhibits overfitting characteristics. The specific process of step 6 is as follows: Step 6.1: Divide the 20 training sets into 15 sub-training sets and 5 sub-validation sets, and define the validation set MSE as the BO objective function; Step 6.2: Construct a Gaussian process surrogate model, use the expected-improvement-plus function for data collection, set the exploration weight to 0.5, the number of iterations to 50, and the convergence condition to be validation set MSE < 0.001, and calculate the BO objective function; Step 6.3: Determine whether the number of iterations and convergence conditions meet the requirements. If not, repeat the acquisition process. Once the requirements are met, obtain the supercavitating vehicle water exit trajectory prediction model based on BO-FNN.

[0023] Example 6 The method for predicting the water-emergence trajectory of a supercavitating vehicle based on BO-FNN proposed in this embodiment includes the following steps: Step 1: Construct the basic mathematical model of CFD; Multiphase flow model and cavitation model; The hypercavitating flow field involves multiphase flow problems of water-air-water vapor. The multiphase flow model selected is the VOF model based on homogeneous equilibrium flow. This model can obtain clear phase interfaces and has a small computational memory footprint. Since water has low compressibility, the temperature distribution of the flow field is almost unaffected by the flow, so the energy equation can be ignored. The basic governing equations include the continuity equation and the momentum equation: (1); in, α q For the fluid micro-element q The volume fraction of the phase satisfies the following relationship: (2); in, r q For the first q Phase density; vThe velocity vector of the fluid element; n For the total number of phases, n =3, including air, water, and water vapor; interphase mass transfer rate and These represent the evaporation rate and condensation rate of water, respectively, described using a cavitation model; Momentum equation: (3); in, p For static pressure, f It is a volume force; r m The average density of the fluid element. m m Let be the average dynamic viscosity of the fluid element, satisfying the following relationship. (4); (5); In multiphase flow problems involving phase transitions between water and water vapor, the Schnerr-Sauer model, based on transport equations and local pressure assessments, is selected. This model is computationally stable and converges rapidly. Its description of interphase mass transfer is as follows: (6); in, r w and r v These are the densities of the water and vapor phases, respectively. α It represents the volume fraction of the vapor phase. p v This is the saturated vapor pressure; n The number of voids per unit volume; R B Let the bubble radius be , which satisfies , (7); Turbulence models and wall functions; High-speed vehicles have high Reynolds numbers, and the accuracy of simulations of strong turbulence directly affects the generation and development of cavitation, thus influencing the hydrodynamic forces acting on the vehicle. The turbulence model selected is the Realizable k-ε model based on the two-equation Reynolds-averaged (RANS) model, which exhibits good robustness. This model is suitable for turbulent kinetic energy... k and turbulent kinetic energy dissipation rate e The transport equations are as follows: (8); (9); in, r For fluid density, m tLet be the turbulent viscosity coefficient, satisfying the relationship, (10); in, C μ The mean strain rate is a function of curl, in the equilibrium boundary layer inertial sublayer. C μ =0.09; s k and s ε For Prandtl numbers, s k =1.0, s ε =1.2; G k and G b These are the turbulent kinetic energy caused by the velocity gradient and buoyancy, respectively. Y M To illustrate the effect of compressible turbulent pulsating expansion on dissipation rate, for incompressible flow... Y M =0; u Kinematic viscosity; C 1, C 2, C 1ε , C 3ε For coefficients; While the Realizable k-ε model is effective for fully developed high Reynolds number turbulence, the near-wall region has low velocities and large velocity gradients, necessitating modification of the turbulence model using wall functions. A scaled wall function is chosen to handle the near-wall region; this model, based on an improvement of the standard wall function, is better suited for simulating complex flows. According to the mean velocity wall law, (11); (12); in, U * For dimensionless velocity, y + The distance is dimensionless; k =0.42, E =9.81; U a for a Average fluid velocity at point k a for a Average turbulent kinetic energy at point, y a for a Distance from point to wall; t wThis refers to the wall shear stress. Rigid body motion model and overlapping mesh technique; Treating the vehicle as a rigid body, its free motion can be divided into translational motion around its center of mass and rotational motion about its center of mass. The translational and rotational motions are solved separately in the ground coordinate system and the body coordinate system. (13); (14); in, m For rigid body mass, J Let the moment of inertia of the rigid body be denoted as . v G For the velocity of the center of mass, oh It is the rotational angular velocity; F The resultant force acting on the rigid body, M B To obtain the resultant torque in the body coordinate system, we need to obtain the resultant torque from the ground coordinate system. M G Conversion, i.e. M B = R C M G ,in R C This is the transformation matrix; To address the issue of water emergence, refining the mesh across the entire area of ​​the vehicle's motion is not only computationally intensive but also leads to mesh deformation and reduced mesh quality. Therefore, an overlapping mesh technique is employed to solve the independent background mesh and the overlapping mesh separately. The resulting data undergoes boundary interpolation to maintain mesh precision in the area surrounding the vehicle. Simultaneously, a free motion model is coupled, defining the relative motion of the overlapping domain using the iteratively obtained vehicle motion parameters. Step 2: Based on the basic CFD mathematical model, set boundary conditions and mesh the external flow field of the vehicle to obtain the simulation model; Boundary conditions and mesh generation; The research uses a simplified design of a real supercavitating vehicle, with an appearance as follows: Figure 1 As shown. The aircraft body includes a cavitation section, a cone section, and a cylindrical section. The diameter of the disc cavitation section is Dn, the diameter of the cylindrical section is Dc = 5Dn, and the slenderness ratio is L / Dc = 14. The computational domain includes a background domain and an overlap domain, both of which are cylindrical computational domains with dimensions as shown. Figure 2 As shown. The background domain is stationary; the position of the overlapping domain is changed to define the water angle; the motion state of the overlapping domain is changed to define the initial motion parameters of the vehicle. Since the vehicle speed is defined by the overlap domain, the background domain inlet and sidewall are set to 0 speed inlet, the background domain outlet is set to standard atmospheric pressure outlet, the overlap domain boundary is set to overlap interface (overset), and the vehicle body wall is set to stationary wall. Structured meshes were applied to the background domain and the overlapping domain respectively, and the meshing results are shown below. Figure 3 , 4 As shown. The mesh size in the motion zone is no higher than 50 mm, and it increases exponentially from 20 mm near the free liquid surface. The number of mesh layers in the cavitation zone is no less than 30. The boundary layer satisfies the requirement of the scaled wall function y+=30~300. The mesh size of the first layer of the solid boundary is found to be 0.2 mm. Step 3: Verify the simulation model to obtain a numerical model suitable for the water exit process of a supercavitating vehicle; Cavitation morphology verification; Calculate the cavitation number of the aircraft s =0.023 Steady-state cavitation flow field of horizontal motion, and compared with Logvinovich empirical formula for cavitation, as follows: Figure 5 As shown in the figure, the cavitation profile obtained by the numerical model is basically consistent with the cavitation profile obtained by the empirical formula, and the error of the cavitation diameter of the airfoil section does not exceed 5%. In addition, the calculated cavitation drag coefficient is 0.858, which has an error of 1.7% compared with the drag coefficient of 0.844 obtained by the empirical formula, thus meeting the simulation requirements.

[0024] Verification of water output conditions; Flow field simulation was performed on the aeration and water discharge process of the ball-head projectile in the water discharge experiment. The projectile was 160 mm long and 26 mm in diameter, with an initial water depth of 0.45 m, an initial velocity of 12 m / s, and an aeration pressure of 0.5 MPa. The flow field phase diagram was compared with experimental photographs, as follows: Figure 6 As shown. Figure 6 The results show that numerical calculations can effectively simulate the cavitation morphology and free liquid surface changes during the water effluent process.

[0025] The simulation and experiment showed a difference in the attenuation of the outflow velocity, such as... Figure 7 As shown. Figure 7 The results show that the trends in water velocity observed in the numerical simulation and the actual experiment are consistent. This verifies the rationality of using numerical calculations to simulate the water effluent operating conditions. Mesh independence verification; Keeping the mesh growth method unchanged, the number of meshes in the background and overlapping regions was varied to obtain models with a total mesh count of 1.25 million, 2.5 million, and 5 million respectively. Under the same settings, a vertical water outflow simulation with an initial velocity of 100 m / s was performed to obtain the axial forces as follows: Figure 8 As shown. Figure 8This indicates that the drag coefficients obtained from 2.5 million and 5 million grids are basically equal, while the results from 1.25 million grids show a relatively larger error. A total grid size of 2.5 million satisfies the grid independence requirement. Step 4: Design orthogonal experiments and obtain the water-ejection trajectory dataset based on the numerical model; Orthogonal experimental design; The establishment of a ballistic prediction model using a neural network requires experimental data that is as large and representative as possible. In order to obtain a wide dataset using limited computing power, a four-factor, five-level orthogonal experiment was designed. The orthogonal factors and levels are shown in Table 1.

[0026] Table 1 Summary of Orthogonal Factors and Levels

[0027] The orthogonal array includes 25 sets of effluent operating conditions, namely L 25 (5 4 Orthogonal array, orthogonal experimental factors include initial velocity v 0. Water outlet angle i 0. Initial angle of attack α 0. Bow deflection d Taking into account launch requirements, stability control, and water safety, the upper and lower limits of each factor level were obtained. The levels of each factor were made as uniform as possible to ensure the completeness of the experiment.

[0028] Analysis of typical working conditions; The aforementioned flow-motion coupled simulation method was used to conduct numerical simulation analysis of the water discharge process for 25 sets of orthogonal experimental conditions. Among them... v 0 = 106m / s , i 0 = 10° α 0 = 0.5° d Using 0 as an example, we can demonstrate its cavitation and motion characteristics.

[0029] Cavitation characteristics during high-speed water exit: Water volume fraction cloud diagrams of the longitudinal section of the vehicle at different times under typical operating conditions are shown below. Figure 9 As shown in the figure, the motion of the vehicle can be divided into three stages: the first is the underwater navigation stage, during which the vehicle generates a stable underwater cavitation bubble, and the displacement at this time is - L ~0; Starting from a displacement of 0 (cavitation disk is on a horizontal surface), the water surface is significantly lifted by the vehicle due to viscosity, resulting in a clear asymmetry in the vehicle's cavitation and forces. The free liquid surface has a significant effect on the vehicle, and the water exit phase begins at this point; when the displacement is 1.5... L ~2 L The spacecraft completely leaves the free surface, the cavitation bubble completely collapses, and the spacecraft is only slightly wetted; this is the flight phase. The pressure cloud diagrams corresponding to each moment in the phase diagram are as follows: Figure 10As shown in the figure. During the underwater navigation stage, the vehicle moves at high speed. There is a pressure stagnation point of 5.5 MPa on the cavitation disk. Cavitation occurs in the high-speed flow field behind the cavitation disk where p < pv. Due to the positive angle of attack and the influence of the free surface, the cavitation on the upstream side of the vehicle is thicker, and the vehicle is not wetted except for the cavitation disk. As the depth decreases, the cavitation on the upstream side gradually squeezes the middle water layer together with the air. During the water exit stage, the pressure stagnation point of the vehicle disappears rapidly, and the cavitation collapses from front to back. At the same time, the cavitation on the upstream side is connected to the air, keeping the front end of the upstream side non-wetted. The cavitation collapse speed is greater than the vehicle speed. When the cavitation collapses, high-pressure regions are generated on both sides of the vehicle, and the hydrodynamic force acts greatly at this time. The cavitation on the upstream side starts to collapse first, and the cavitation on the downstream side collapses faster. The shoulder cavitation of the vehicle completely collapses when the displacement is L, and the tail cavitation also collapses when the displacement is 1.5L. The free surface on the upstream side is disturbed greatly, while there is almost no disturbance on the downstream side. During the airborne flight stage, the vehicle is affected by a small hydrodynamic force, and the attitude and speed change little; Hydrodynamic characteristics during the high-speed water exit process: The directions are two translations and one rotation, that is, it is subject to resistance, lift, and pitching moment. The hydrodynamic coefficients of the vehicle change with displacement as Figure 11 shown. Analysis Figure 11 shows that when the vehicle moves underwater, the drag coefficient stabilizes at 0.034, which is equivalent to that of horizontal straight navigation. The lift and pitching moment of the vehicle are close to 0; when approaching the free surface, due to the disappearance of the pressure stagnation point at the head, the drag of the vehicle decreases, and the high-pressure region generated by the collapse of the shoulder cavitation results in larger lift and pitching moment; when the displacement reaches 0.5 L , the wetted area of the vehicle increases, causing the drag to rise. At the same time, the cavitation collapse changes from being mainly on the upstream side to the downstream side, and the larger high-pressure region on the downstream side makes the directions of the lift and pitching moment opposite to those before; after the shoulder cavitation collapses, as the vehicle exits the water, the wetted area decreases, the drag decreases, the high-pressure regions on both sides of the vehicle disappear, and the lift and pitching moment tend to 0; Motion characteristics during the high-speed water exit process: Obtain the vehicle speed v , pitching angular velocity oh , pitching angle i , ballistic inclination angle Θ, angle of attack α changing with displacement as Figure 12 , 13 shown. From Figure 12 , 13 it can be seen that the vehicle speed decreases by 3 m / s, and the change is relatively gentle due to the influence of the drag; the angular velocity first increases and then decreases due to the pitching moment that is first positive and then negative, and stabilizes at 0.12 rad / s at the end of the water exit; the ballistic inclination angle first decreases and then increases due to the influence of the lift, and finally stabilizes at 9.3°; the change of the pitching angle is consistent with the angular velocity, first decreasing and then increasing, and the final state θ1 = 10.7°; the change of the angle of attack is similar to that of the pitching angle, which is because the ballistic inclination angle is relatively stable, and the final state α1 = 1.4°; Calculation results of orthogonal test; For L 25 (54 Simulation calculations were performed using orthogonal test conditions from the orthogonal array, and the results are shown in Table 2. In Table 2, A~D represent the four initial motion parameters from Table 1. v 0、 i 0、 α 0、 d The calculation conditions are represented by levels 1 through 5. The level sequence contains 25 combinations of factors and levels, such as A5B3C2D1. v 0 takes the 5th level. i 0 takes the 3rd level. α 0 takes the second level. d Take level 1, i.e., the typical working condition in Section 2.2, and so on. Table 2... v 1 represents the final velocity. i 1 represents the final pitch angle. α 1 represents the final angle of attack. oh 1 represents the final pitch angular velocity.

[0030] Table 2. Calculation Results of Orthogonal Experiment

[0031] An analysis of variance was performed on the results in Table 2 to obtain the ratio of the mean square of each factor to the mean square of the error, i.e., the F-value. The corresponding p-values ​​can be obtained from the F-distribution function, as shown in Table 3. The p-values ​​can explain the significance of each factor's influence on each indicator; * indicates p < 0.05, meaning the factor has a relatively significant effect on the indicator; ** indicates p < 0.01, meaning the effect is quite significant.

[0032] Table 3 Summary of p-values ​​for orthogonal experimental variance analysis

[0033] As shown in Table 3, the initial velocity v 0 has a significant impact on all four test indicators, including the outlet angle. i 0 relative to final velocity v 1 and final pitch angle i 1. Has a significant impact on the initial angle of attack. α 0 relative to final velocity v 1 and final angular velocity oh 1. Significant impact on rudder deflection. d Only for the final pitch angle i 1. Has a significant impact; Step 5: Design a feedforward neural network to fit the water-escape trajectory dataset and obtain a water-escape trajectory prediction model that exhibits overfitting characteristics. Feedforward neural network; Feedforward neural networks (FNNs) have been widely used because they implement arbitrary nonlinear mappings from input to output with a simple structure. They typically have one or more hidden layers from the input layer to the output layer. Theoretically, a single-hidden-layer FNN with a sufficient number of neurons can approximate any continuous function with arbitrary precision. For example... Figure 14 As shown, this is the structure of a single hidden layer FNN. The input / output data in the figure needs to be obtained from orthogonal experimental data after normalization.

[0034] Figure 14 The input layer has 4 nodes, representing the number of experimental factors; the output layer has a single node, indicating that a mapping model is established for each of the four experimental indicators separately; the number of hidden layer nodes is variable, satisfying the following relationship. (15); (16); in x i For input layer data, y To output data, u j This is intermediate data for the hidden layer; v ij and w j These are the input layer and hidden layer weights, respectively. and These are the biases for the input layer and the hidden layer, respectively. f h and f o These are the activation functions for the hidden layer and the output layer, respectively. A single-hidden-layer FNN can meet the training requirements of the small dataset in this paper. The hyperparameters that need to be determined include the number of hidden layer nodes, learning rate, regularization coefficient, training method, and activation function. The number of hidden layer nodes can be initially estimated using the empirical formula (17). (17); in, N i , N h , N oThe number of nodes in the input layer, hidden layer, and output layer are respectively determined, resulting in 3 to 13 hidden layer nodes. A learning rate that is too low may lead to overfitting, while a rate that is too high may cause the model to diverge; generally, it should not exceed 0.1. L2 regularization adds a sum of squared weights penalty term to the loss function to alleviate overfitting; the coefficient λ controls the magnitude of this term's influence, generally less than 0.01. The training method is selected from the LM algorithm, Bayesian regularization (BR) algorithm, and momentum adaptive gradient descent (GDX) algorithm. Hidden layer activation function options include sigmoid, tanh, and ReLU; the output layer activation function uses the purelin function. Step 6: Optimize the water exit trajectory prediction model using the Bayesian optimization method to obtain the water exit trajectory prediction model for the supercavitating vehicle based on BO-FNN. Bayesian optimization neural networks; Bayesian optimization (BO) is a strategy for global optimization of functions. Its core idea is to construct a surrogate model (such as a Gaussian process) and progressively select the optimal parameters to effectively find the global optimum, thus solving high-dimensional, non-convex, and computationally expensive optimization problems. The optimization process of the BO method is as follows: Figure 15 As shown.

[0035] The ability of Boolean optimization to find the global optimum, combined with the tendency of FNN (Fundamental Neural Network) to get trapped in local minima (based on gradient descent), can help FNN escape local minima and obtain a training model with better generalization.

[0036] Since the FNN validation set is not used in training, the dataset is first divided into an 80% training set (20 sets) and a 20% test set (5 sets). The BO objective function cannot leak test set data and thus "cheat," so the 20 training sets are further divided into 15 sub-training sets and 5 sub-validation sets, with the validation set MSE defined as the BO objective function. Figure 16 , 17 Figure 18 shows the prediction results of the BO-FNN sub-model. Note that the BR training algorithm uses Bayesian regularization, while L2 regularization is ineffective. As shown in the figure, the data points in both the training and test sets are located near the ideal fitting line. Except for one data point at the angle of attack, the prediction error is within 20% for all other data points, indicating that the BO-FNN predicts the trajectory of the projectile relatively accurately.

[0037] Table 4 shows the hyperparameter optimization results and prediction metrics for the four sub-models. As shown in Table 4, the FNN hyperparameters obtained through BO have a wide selection range, and all three candidate training algorithms and activation functions were applied. The number of hidden layer nodes ranges from 3 to 11, indicating that BO can find optimal variables within a large range. The MSE, MAPE, and R values ​​on the training and test sets are also shown. 2 Both are good; R scores for all test and training sets are good. 2All values ​​are above 0.9, indicating that the BO-FNN model has high prediction accuracy and strong generalization ability. Comparing several sub-models, except for the pitch angle model, the MSE on the test set does not exceed 0.07, and except for the angle of attack model, the MAPE does not exceed 11%, indicating that the prediction model is more accurate in predicting velocity and angular velocity than in predicting pitch and angle of attack.

[0038] Table 4. Hyperparameters and prediction metrics of the BO-FNN sub-model

[0039] BO validity verification; The prediction results of the unoptimized FNN model and the BO-FNN model are compared. The established FNN model has 4 hidden layer nodes, a learning rate of 0.001, is trained using the LM method without regularization, and uses the sigmoid function as the hidden layer activation function. The results are as follows. Figure 19 , 20 As shown in Figure 21, compared to the FNN model optimized by the BO method, it can be seen that the entire training set of the FNN model lies on the ideal fitting line, that is, the predicted data and the actual data are exactly the same. However, the error of multiple points on the test set exceeds 20%, which indicates that the FNN model without BO exhibits overfitting characteristics and cannot effectively predict the trajectory of water-borne projectiles.

[0040] Further analysis of the prediction metrics of the FNN model, as shown in Table 5, reveals that the MSE and R of the FNN model training set... 2 It is "perfect," exhibiting overfitting characteristics, while the test set angle of attack and angular velocity R... 2 A value less than 0.7 indicates a worse fit compared to velocity and pitch angle. This is because the angle of attack and angular velocity are more significantly and nonlinearly affected during the vehicle's emergence from the water. BO-FNN no longer exhibits overfitting characteristics due to the addition of a suitable regularization term, which helps it escape local minima. BO-FNN's predictions on the test set are more accurate, indicating that BO can optimize the FNN model, resulting in a ballistic prediction model with better generalization.

[0041] Table 5. Prediction Indicators of FNN Model

Claims

1. A method for predicting the ejection trajectory of a supercavitating vehicle based on a BO-FNN, characterized in that, The method comprises the following steps: Step 1, constructing a CFD basic mathematical model; Step 2, based on the CFD basic mathematical model, setting boundary conditions and dividing grids for the outer flow field of the vehicle to obtain a simulation model; Step 3, verifying the simulation model to obtain a numerical model suitable for the water exit process of the supercavitating vehicle; Step 4, designing an orthogonal test, and obtaining a water exit trajectory data set according to the numerical model; Step 5, designing a feedforward neural network, fitting the water exit trajectory data set, and obtaining a water exit trajectory prediction model with overfitting characteristics; Step 6, optimizing the water exit trajectory prediction model through a Bayesian optimization method to obtain a supercavitating vehicle water exit trajectory prediction model based on BO-FNN.

2. The BO-FNN-based supercavitating vehicle out-of-water trajectory prediction method of claim 1, wherein, The CFD basic mathematical model in step 1 comprises a multiphase flow model, a turbulence model, a rigid body motion model and an overlapping grid model.

3. The BO-FNN-based supercavitating vehicle out-of-water trajectory prediction method of claim 2, wherein, The multiphase flow model selects a VOF model based on homogeneous equilibrium flow to obtain a phase interface; the basic control equation of the VOF model comprises a continuity equation and a momentum equation; The continuity equation is shown in equations (1) and (2); （1）； （2）； where, α q is the volume fraction of the fluid element, q The momentum equation is shown in equations (3), (4) and (5); q is the density of the phase, q v is the velocity vector of the fluid element, n is the total number of phases, n = 3, including air, water, and water vapor; the mass transfer rate between phases and are the evaporation and condensation rates of water, respectively, described by a cavitation model;​​ The multiphase flow model selects a Schnerr-Sauer model based on a transport equation and local pressure judgment to handle the phase change problem between water and water vapor in the multiphase flow, and the description of the interphase mass transfer is shown in equations (6) and (7); （3）； （4）； （5）； wherein, p is the static pressure, f is the body force; The Schnerr-Sauer model is shown in equations (8) and (9); m is the average density of the fluid element; The Schnerr-Sauer model is shown in equations (8) and (9); m is the average dynamic viscosity of the fluid element; The Schnerr-Sauer model is shown in equations (8) and (9); （6）； （7）； where, The Schnerr-Sauer model is shown in equations (8) and (9); w and The Schnerr-Sauer model is shown in equations (8) and (9); v are the water and vapor phase densities, respectively; α is the vapor volume fraction; p v is the saturation vapor pressure; n is the number of voids per unit volume; R B is the bubble radius.

4. The BO-FNN-based supercavitating vehicle out-of-water trajectory prediction method of claim 2, wherein, The turbulent flow model selects a Realizable k-ε model based on a two-equation Reynolds time average model, and the transport equations of turbulent kinetic energy k and turbulent kinetic energy dissipation rate The Schnerr-Sauer model is shown in equations (8) and (9); are shown in equations (8), (9) and (10). （8）； （9）； （10）； where The Schnerr-Sauer model is shown in equations (8) and (9); is the fluid density, The Schnerr-Sauer model is shown in equations (8) and (9); t is the turbulent viscosity coefficient; C μ is a function of the average strain rate and the curl of the velocity, in the equilibrium boundary layer inertial sublayer C μ = 0.09; The Schnerr-Sauer model is shown in equations (8) and (9); k and The Schnerr-Sauer model is shown in equations (8) and (9); ε is the Prandtl number, The Schnerr-Sauer model is shown in equations (8) and (9); k = 1.0, The Schnerr-Sauer model is shown in equations (8) and (9); ε = 1.2; G k and G b are the velocity gradient and the buoyancy-induced turbulent kinetic energy, respectively; Y M is the effect of compressibility turbulent fluctuation expansion on the dissipation rate, for incompressible flow Y M = 0; The Schnerr-Sauer model is shown in equations (8) and (9); is the kinematic viscosity; C 1, C 2, C 1ε , C 3ε is a coefficient; The Schnerr-Sauer model is shown in equations (8) and (9); （11）； （12）； where, U * is the dimensionless velocity, y + is the dimensionless distance; The Schnerr-Sauer model is shown in equations (8) and (9); = 0.42, E = 9.81; U a is the a average fluid velocity at the point, k a is the a average turbulent kinetic energy at the point, y a is the a distance from the point to the wall; The Schnerr-Sauer model is shown in equations (8) and (9); w is the wall shear stress.

5. The BO-FNN-based supercavitating vehicle out-of-water trajectory prediction method of claim 2, wherein, The Schnerr-Sauer model is shown in equations (8) and (9); （13）； （14）； where m is the mass of the rigid body, J is the moment of inertia of the rigid body; v G is the velocity of the center of mass, The Schnerr-Sauer model is shown in equations (8) and (9); is the angular velocity of the rotation; F is the resultant force acting on the rigid body, M B is the resultant moment on the body coordinate system, which needs to be converted from the resultant moment on the ground coordinate system M G is converted, i.e. M B = R C M G where R C is the conversion matrix; The Schnerr-Sauer model is shown in equations (8) and (9); 6. 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Step 2.3, respectively structure the grid of the background domain and the overlapping domain to obtain the simulation model; the grid size of the motion area is not higher than 50 mm, the grid size of the near free surface starts from 20 mm and increases exponentially, the grid layer number of the cavitation area is not less than 30, the boundary layer satisfies the size scaling wall function y+=30~300 requirement, and the first layer grid size of the solid boundary is 0.2 mm.

7. The BO-FNN-based supercavitating vehicle out-of-water trajectory prediction method of claim 2, wherein, The verification in the step 3 includes cavitation shape verification and water ejection working condition verification; The cavitation shape verification specifically includes: calculating the steady-state cavitation flow field of the horizontal motion of the vehicle, comparing the cavitation profile of the flow field with the cavitation profile obtained by the Logvinovich empirical formula, the error of the cavitation diameter of the vehicle section is not more than 10%, and comparing the cavitation resistance coefficient obtained by the cavitation flow field with the cavitation resistance coefficient of the Logvinovich empirical formula, the error is not more than 5%, which indicates that the cavitation model meets the simulation requirements; The water ejection working condition verification specifically includes: simulating the flow field of the motion process of the hollow sphere free water ejection in the experiment, comparing the flow field phase diagram at different times with the experimental photos, the maximum height error of the water mound is not more than 10%, and comparing the water ejection speed change of the simulation and the experiment, the maximum speed error is not more than 5%, which indicates that the simulation model can effectively simulate the water ejection process.

8. The BO-FNN-based supercavitating vehicle out-of-water trajectory prediction method of claim 2, wherein, The specific process of the step 4 is: Step 4.1, designing orthogonal table according to different factors and level number; the different factors include initial velocity v 0, water outlet angle θ 0, initial angle of attack α 0, initial angular velocity ω0 and rudder deflection angle δ , at least 2 groups are selected from the different factors; the level number is any group, and the values of each level are as uniform as possible, Step 4.2, inputting the orthogonal table data into a numerical model suitable for the supercavitating vehicle water-exit process for calculation, and arranging the orthogonal test results as the water-exit trajectory data set; the orthogonal test results include the terminal velocity v 1, the terminal pitch angle θ 1, the terminal attack angle α 1, the terminal pitch angle velocity ω 1, and the terminal mass center y coordinate y1.

9. The BO-FNN-based supercavitating vehicle out-of-water trajectory prediction method of claim 2, wherein, The specific process of the step 5 is: Step 5.1, processing the water ejection trajectory data set; The data set of the water ejection trajectory is standardized by using the normalization method, the order of the data set is disturbed, and the first 80% is used as the training set and the remaining 20% is used as the test set; the normalization method selects the Z-score method or the min-max method; Step 5.2, setting the structure of the feedforward neural network; Setting the input layer: the input variables are a plurality of processed data sets including speed, attack angle, angular velocity, pitch angle and rudder angle, and the number of input layer neurons is set to the number of factors; setting the output layer: setting the number of output layer neurons Nou to 1, establishing a single output prediction model for each final state parameter; setting the hidden layer: setting the number of hidden layer nodes, learning rate, L2 regularization coefficient, training method and activation function to any value; setting the objective function: defining the objective function as the mean square error, calculating the error in network training and evaluating the objective function; Step 5.3, learning the training set data, calculating the mean square error of the test set data, and obtaining the water ejection trajectory prediction model with overfitting characteristics.

10. The BO-FNN-based supercavitating vehicle out-of-water trajectory prediction method of claim 9, wherein, The specific process of the step 6 is: Step 6.1, dividing 20 groups of training sets into 15 sub-training sets and 5 sub-validation sets, and defining the validation set MSE as the BO objective function; Step 6.2, build a Gaussian process surrogate model, use the expected-improvement-plus function for acquisition, set the exploration weight to 0.5, the number of iterations to 50, and the convergence condition to the verification set MSE<0.001, and calculate the BO target function; Step 6.3, judge whether the number of iterations and the convergence condition meet the requirements, if not, repeat the acquisition process; if the requirements are met, the supercavitating vehicle ejection trajectory prediction model based on BO-FNN is obtained.

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