Kinematic modeling and numerical calculation method of manta ray submersible gliding propulsion

Abstract

The application relates to a kind of manta ray submarine gliding propulsion kinematics modeling and numerical calculation method;Belong to underwater bionic robot technical field.This method first carries out feature point marking and three-dimensional motion reconstruction to real manta ray biology, extracts its kinematics parameter, and then establishes a kind of unified kinematics equation which strictly satisfies the constraint of body length invariable, can accurately couple the deformation of pectoral fin spanwise and chordwise;By adjusting specific parameters, the mathematical description of multiple real motion modes such as flapping, gliding, gliding and compound, and maneuvering turn can be realized.Secondly, the high-fidelity kinematics model dynamic boundary input condition is combined with the immersed boundary method and the computational fluid dynamics solver to efficiently and stably calculate the hydrodynamic performance and flow field structure of the manta ray submarine in the complex motion process.The application effectively solves the problem that the simulation result deviates from the actual situation and it is difficult to capture the key vortex dynamics structure caused by the distortion of the kinematics model of the prior art.

Description

Technical Field

[0001] This invention belongs to the field of underwater biomimetic robots, specifically relating to a kinematic modeling and numerical calculation method for the sliding propulsion kinematics of a manta ray-inspired submersible. Background Technology

[0002] With the continuous advancement of my country's deep-sea strategy, the demand for underwater environmental monitoring, scientific research, and resource exploration is increasing. Manta ray-inspired submersibles, with their efficient propulsion and good biocompatibility, demonstrate unique advantages in these marine missions. In the design and optimization of biomimetic underwater vehicles, numerical simulation can simulate the real morphology and movement patterns of biological prototypes, with fewer constraints from objective conditions such as site and equipment. Among these, the definition of the kinematic equations of the physical model is crucial to the reliability of the simulation results. Establishing control equations that accurately describe the real movement of the fish is essential for propulsion mechanism analysis and the optimized design of manta ray-inspired submersibles.

[0003] Currently, for underwater vehicles like manta rays that employ a central fin / paired fin (MPF) propulsion mode, kinematic modeling and numerical calculation still face the following key technical bottlenecks: 1. Distortion of Kinematic Models and Lack of Modalities: Most existing methods only model a single flapping mode, and their kinematic equations fail to encompass the various modes that real manta rays exhibit during swimming, such as gliding, combined flapping and gliding, maneuvering turns, and asymmetrical vertical movements. This simplified description of a single mode leads to significant deviations between simulated motion patterns and real biological motion patterns, failing to provide realistic kinematic boundary conditions for mechanistic studies.

[0004] 2. Insufficient Motion Coupling and Lack of Physical Constraints: More fundamentally, existing modeling methods often treat spanwise and chordal motions in steps or decouple them when dealing with large pectoral fin deformations. This approach is not only computationally inefficient but also disrupts the kinematic consistency of the pectoral fin deformation as an organic whole, and usually fails to strictly satisfy the fundamental physical constraint of "constant body length" during motion. The distortion and decoupling of motion description directly lead to the inability of numerical simulations to accurately capture and analyze key vortex dynamic structures during propulsion (such as the generation, evolution, and interaction of tip vortices and trailing vortices), thus failing to correctly reveal the hydrodynamic mechanism of its efficient propulsion.

[0005] In summary, developing a kinematic modeling and numerical calculation method for the ski-throw propulsion of a manta ray-inspired submersible that can strictly satisfy physical constraints, accurately couple spanwise and chordal motions, fully cover multimodal motions, and efficiently combine with high-precision, high-stability flow field solution methods has become a key technical challenge that urgently needs to be overcome in this field. Existing technologies have not yet provided a systematic and effective solution. Summary of the Invention

[0006] The technical problem to be solved: To overcome the shortcomings of existing technologies, this invention provides a kinematic modeling and numerical calculation method for the ski-throw propulsion of a manta ray-inspired submersible. This method obtains real motion morphology data by marking characteristic points of the actual manta ray's movement process, establishes high-precision kinematic control equations to accurately describe the complex coupled motion of the manta ray's pectoral fins and the multimodal motion state of its ski-throw propulsion. Combined with numerical calculation methods, it achieves efficient and accurate solutions for hydrodynamic parameters and flow field information throughout the entire motion process. This solves the problems in existing manta ray-inspired submersible ski-throw propulsion simulations, where kinematic model distortion leads to significant deviations between simulation results and actual motion patterns, and makes it difficult to accurately capture and analyze key vortex structures during propulsion.

[0007] The technical solution of this invention is: a method for kinematic modeling and numerical calculation of the sliding propulsion kinematics of a manta ray-inspired submersible, comprising the following steps: Kinematic modeling steps: Based on feature extraction and regression analysis of three-dimensional spatiotemporal trajectory data of real manta ray biological movement processes, a unified set of pectoral fin kinematic control equations that strictly satisfy the constraint of constant body length is established. This set of equations can accurately couple and describe the spanwise and chordal deformation of the pectoral fins. By adjusting and combining specific kinematic parameters in the control equations, mathematical descriptions of various kinematic modes of the manta ray-inspired submersible, such as flapping, gliding, gliding-flapping composite, left-right asymmetric maneuvers, and up-down asymmetric offset, are realized. Numerical calculation steps: The control equations obtained from the kinematic modeling steps are used as dynamic boundary input conditions. The hydrodynamic parameters and flow field information of the manta ray-inspired submersible under the various motion modes are iteratively solved by combining the submerged boundary method and the fluid dynamics solver.

[0008] A further technical solution of the present invention is: the kinematic modeling step specifically includes: Step 1: Select multiple feature points on the pectoral fin of the manta ray that cover its key geometric locations; Step 2: Obtain the three-dimensional spatiotemporal trajectory data of the feature points during the motion process using high-speed camera and 3D reconstruction technology; Step 3: Based on the three-dimensional spatiotemporal trajectory data, extract the kinematic parameters reflecting the undulation characteristics of the pectoral fin, and use regression analysis to establish a quantitative relationship model between the kinematic parameters and swimming speed; Step 4: The Tukey-Kramer multiple comparison test is used to evaluate the significance of the regression analysis results to verify the reliability of the extracted motion laws. A further technical solution of the present invention is: the unified pectoral fin kinematic control equation, for the left wing, has the following form:

[0009] The right wing is symmetrical in form;

[0010] In the formula, A l , A r The amplitude is measured on the left and right sides; t For time; ( x 0, y 0, z 0) represents the initial position coordinates of the manta ray model; x , y , z (Time step) t Position coordinates; W l , W r For the wave numbers on the left and right sides; f l , f r The frequency of left and right movements; , These are the control parameters for left and right chordal oscillations; , This represents the maximum value of the left and right bending angles; u l , u r To control the parameters of the upward offset of the left and right wings, d l , d r Parameters for controlling the downward offset of the left and right wings; , To adjust the phase of movement of the left and right pectoral fins; ML The body length is designed to resemble that of a manta ray submersible model. MW The model is a manta ray-inspired submersible with a half-length span.

[0011] A further technical solution of the present invention is: by using the wavenumber in the governing equation W l , W r Set to 0, left and right pectoral fin movement phase , Setting the value to 0 and fixing the time parameter, the control equation is simplified to an equation describing the gliding mode, where the parameter controlling the maximum value of the left and right pectoral fin bending angle is defined as the gliding bow angle. β .

[0012] A further technical solution of the present invention is: by dividing the simulation time into two segments, and at time nodes... T hTo ensure the continuity of the pectoral fin position and model the gliding-flapping composite motion: when 0 ≤ t ≤ T h When, the gliding mode equations are used; when T h ≤t≤ T t At that time, the basic governing equations of the flapping mode are adopted. A further technical solution of the present invention is: by independently adjusting the amplitude of the left and right pectoral fins in the control equation. A l and A r ,frequency f l and f r , wave number W l and W r or phase and This enables the modeling of asymmetrical turning motions.

[0013] A further technical solution of the present invention is: by independently adjusting the vertical offset parameters of the left and right pectoral fins in the control equation. u l , u r , d l , d r This enables asymmetric control of the amplitude of the pectoral fin's up-and-down flapping motion, thereby completing the modeling of the asymmetric up-and-down offset motion; where, when u l , u r =0, d l , d r When =1, the asymmetric offset motion between the upper and lower parts degenerates into a flapping motion with symmetrical upper and lower amplitudes.

[0014] A further technical solution of the present invention is: the numerical calculation step specifically includes: Step 101: Establish a simulation computational physical model of the manta ray-inspired submersible and generate a surface mesh; Step 102: Construct a CFD computational domain containing a uniformly encrypted region and perform flow field mesh generation; Step 103: Assign the kinematic control equations to the Lagrange nodes of the surface mesh and calculate their motion velocity; Step 104: Using an improved Eulerian point search method, establish the mapping relationship between Lagrange nodes and Eulerian nodes in the flow field; Step 105: Iteratively solve the fluid dynamics equations containing the moving boundary force source using the prediction-correction method, and feed the solution results back to the Lagrange node; Step 106: Calculate the hydrodynamic forces, torques, input power, and efficiency based on the forces acting on the Lagrange nodes; Step 107: Repeat steps S103 to S106 until the target time step is reached.

[0015] A further technical solution of the present invention is: the improved Euler point search method is as follows: in the initial time step, a global carpet search is used to determine and store the mapping relationship; in subsequent time steps, the search range is limited to a finite number of adjacent grid scales of the Euler point positions stored in the previous time step. A further technical solution of the present invention is as follows: the process of solving the prediction-correction method is as follows: Prediction step: Solve the Navier-Stokes equations without submerged boundary force source terms, thus obtaining... t+ Density of 1 time step and prediction speed The expression is as follows:

[0016] in, For time step t density, For time step t velocity field, For the Laplace operator.

[0017] Correction step: Combining the constructed linear equations including the contribution of the submerged boundary force source, the predicted velocity field is corrected to obtain the corrected velocity field at time step t+1; the expression of the linear equations is:

[0018] In the formula, For the density of the fluid, u For speed, p For pressure, The coefficient of dynamic viscosity, f The additional force term representing the boundary effect, I Unit tensor; The correction speed is obtained by calculating the correction step using the following formula. :

[0019] Final velocity field: The predicted velocity field is combined with the corrected velocity field to obtain the final velocity field at time step t+1. : .

[0020] Beneficial effects The beneficial effects of this invention are as follows: This invention provides a kinematic modeling and numerical calculation method for the sliding propulsion of a manta ray-inspired submersible. This kinematic modeling method captures the true morphological data of the movement process by marking feature points on real manta ray organisms, and establishes control equations that strictly satisfy physical constraints and precisely couple spanwise and chordal motions. This method can efficiently and accurately reproduce the complex kinematic characteristics of manta ray organisms, thus providing kinematic conditions for numerical simulation that closely resemble the swimming morphology of real organisms. Combining this kinematic model with the submerged boundary method allows for efficient and accurate calculation of the hydrodynamic parameters and flow field information of the sliding propulsion. Specific effects are analyzed as follows: 1. The unified control equations established in this invention can flexibly and accurately describe various realistic motion modes such as flapping, gliding, combined flapping and gliding, maneuvering turns, and vertical offsets through parameter adjustment. This provides kinematic boundary conditions that are highly close to the biological prototype for simulation, enabling the simulation results to truly reflect the complex motion laws of the manta ray-inspired submersible.

[0021] 2. This invention combines the Immersed Boundary Method (IBM) with a high-efficiency fluid solver, avoiding problems such as mesh distortion and computational crashes encountered by traditional dynamic mesh methods when simulating large deformations of pectoral fins, thus ensuring computational stability and robustness. Through improved Eulerian point search strategies (such as global initial search combined with local follow-up search) and other algorithmic optimizations, the efficiency of data interaction and flow field solution is significantly improved, making the simulation of long-duration, multi-period motions more feasible.

[0022] 3. The key flow field data, such as hydrodynamic parameters, pressure field and vortex structure evolution, obtained through this kinematic modeling and numerical calculation method, provide a reference for correctly and profoundly revealing the fluid dynamic mechanism of efficient biological propulsion. It can also be used to guide the optimized design of manta ray-inspired submersibles and improve research and development efficiency. Attached Figure Description

[0023] Figure 1 This is a flowchart illustrating the overall calculation process of the sliding propulsion kinematics modeling and numerical calculation method of the present invention.

[0024] Figure 2 This is a schematic diagram of the flow field mesh for the numerical calculation method of this invention.

[0025] Figure 3 This is a schematic diagram illustrating the feature point marking and analysis performed in this invention.

[0026] Figure 4 This is a schematic diagram of the spanwise deformation of the motion equation of the manta ray-inspired submersible using the present invention.

[0027] Figure 5This is a schematic diagram of the flapping deformation of the manta ray-inspired submersible using the motion equations of this invention.

[0028] Figure 6 This is a schematic diagram of the gliding parameters of the manta ray-inspired submersible using the present invention.

[0029] Figure 7 This is a schematic diagram of the gliding bow-shaped angle β deformation of the manta ray-inspired submersible using the present invention.

[0030] Figure 8 is a schematic diagram of the flapping and gliding hydrodynamic coefficients and vortex structure of the manta ray-inspired submersible obtained using the numerical calculation method of the present invention. Detailed Implementation

[0031] The embodiments described below with reference to the accompanying drawings are exemplary and intended to explain the invention, and should not be construed as limiting the invention.

[0032] Currently, kinematic modeling methods for fish propulsion vary depending on their propulsion mode. For fish using the BCF (body / tail flare) propulsion mode, the motion equations are typically based on the wave motion projection principle, simplifying the fish's body sway as a traveling wave propagating along the axis. This model effectively captures core motion features under the assumption of small amplitude and has high computational efficiency. However, for manta rays and their biomimetic submersibles that propel themselves using the MPF (central fin / pair fin flare) mode, their pectoral fin flapping exhibits large-amplitude flexible deformation and involves coupled spanwise and chordal motions, including gliding, flapping, and combined gliding-flapping modes. The aforementioned models are no longer applicable. Therefore, developing a novel kinematic equation that strictly satisfies body length constraints and accurately describes span-chordal coupled motion has become an urgent technical requirement.

[0033] The existing technology, "A Mathematical Description Method for the Pectoral Fin Deformation Motion of a Manta Ray-Inspired Underwater Vehicle," Chinese Patent CN118194445 A, uses chordal motion equations, Bessel spanwise motion equations, and torsional equations to complete the motion. However, the chordal and spanwise motions are obtained step by step, resulting in low computational efficiency; moreover, it only models the flapping mode and does not cover the various motion modes of real manta rays, such as gliding and flapping; furthermore, the mathematical model is not logically consistent and it is difficult to directly solve for spatial coordinates. Currently, the kinematic description of manta ray-inspired underwater vehicles is still incomplete. Developing a computationally efficient, versatile, and mathematically rigorous method for modeling the kinematics of flapping propulsion has become a key technical problem that urgently needs to be solved in this field.

[0034] Existing numerical computation methods face challenges related to large deformations: at the numerical implementation level, methods based on traditional dynamic meshing and mesh reconstruction techniques (such as those in CN111241662B and CN118194444A) are prone to computational crashes or accuracy degradation due to mesh distortion or negative volume when dealing with the large amplitude and flexible coupled deformations unique to the pectoral fins of manta rays. Computational stability and efficiency face severe challenges. Although immersion boundary methods (such as those used in the paper "Formation effects…") offer a new approach to handling large deformation boundaries, their computational accuracy and efficiency are highly dependent on the coupled flow field solver and the ability to accurately describe complex kinematic boundary conditions.

[0035] Therefore, based on the above problems, this invention proposes a kinematic modeling and numerical calculation method for the sliding propulsion kinematics of a manta ray-inspired submersible, comprising the following steps: Kinematic modeling steps: Based on feature extraction and regression analysis of three-dimensional spatiotemporal trajectory data of real manta ray biological movement processes, a unified set of pectoral fin kinematic control equations that strictly satisfy the constraint of constant body length is established. This set of equations can accurately couple and describe the spanwise and chordal deformation of the pectoral fins. By adjusting and combining specific kinematic parameters in the control equations, mathematical descriptions of various kinematic modes of the manta ray-inspired submersible, such as flapping, gliding, gliding-flapping composite, left-right asymmetric maneuvers, and up-down asymmetric offset, are realized. Numerical calculation steps: The control equations obtained from the kinematic modeling steps are used as the motion boundary conditions. The hydrodynamic parameters and flow field information of the manta ray-inspired submersible under the various motion modes are iteratively solved by combining the submerged boundary method and the fluid dynamics solver.

[0036] The above technical solution will be further analyzed below with reference to the accompanying drawings: In one embodiment, refer to Figure 1 As shown, a method for kinematic modeling and numerical calculation of the sliding propulsion kinematics of a manta ray-inspired submersible includes the following steps: Step 1: Establish a simulation computational physical model and divide the surface into triangular meshes. This embodiment performs numerical simulation on a manta ray-inspired submersible model. Reverse engineering techniques are used to establish the physical model. By measuring the dimensions of the actual object, three-dimensional point data is obtained. Then, three-dimensional curves are constructed from the point data, further constructing three-dimensional surfaces, and ultimately reconstructing the CAD model of the physical object. In this embodiment, the body length of the manta ray-inspired submersible model is used as... ML It indicates that the half-length extension is used MW It means that among them ML =1.85m, MW =1.45m. The model was imported into ICEM software to generate a triangular surface mesh, and the Lagrange point data was output as initial values ​​for this numerical calculation method.

[0037] Step 2: Initialize simulation settings.

[0038] Configure the overall environment for the numerical simulation. Set the inlet velocity. Determine the individual computation time step based on the Courant number (CFL) condition (CFL=0.5 in this embodiment) and mesh size. Set the total number of iteration computation time steps, ensuring coverage of more than 5 complete thrashing cycles to guarantee that the hydrodynamic performance reaches a periodic steady state. Set the boundary conditions of the computational domain, such as velocity inlet, pressure outlet, and no-slip walls.

[0039] Step 3: Refer to Figure 2 As shown, the scope of the flow field computational domain is defined. The flow field mesh generation method includes the following steps: Step 3.1: Settings x , y , z The computational domain lengths in the three directions are set. x , y , z The coordinates of the center point of the uniformly encrypted region in three directions and the length of the uniformly encrypted region, as well as the length of the computational domain and the length of the uniformly encrypted region, can be adjusted according to the CFD calculation requirements. Step 3.2: Settings x , y , z The number of grid cells in the uniformly encrypted region in three directions is set. x , y , z The grid numbers located at the center point of the uniformly dense region in three directions; the number of grids in the uniformly dense region can be adjusted according to the object's feature size. Step 3.3: Obtain the uniformly encrypted region for each Euler node. x , y , z Coordinates in three directions; to ensure the continuity of the mesh transition between the uniformly encrypted and unencrypted regions, the coordinates of each Euler node in the unencrypted region are set by a function. x , y , z Coordinates in three directions; Step 3.4: Add two layers of ghost mesh outside the outer mesh, with the length of each mesh determined by the size of its adjacent internal mesh; Step 3.5: Calculate the value of each grid cell. x , y , z The coordinates of the center point in three directions, the interpolation relationship with the two grids before and after it, and the generated grid output.

[0040] In this embodiment, the fluid domain mesh is uniformly generated using a mesh generation method, and the surrounding flow field range is determined based on the model size. The total flow field size is 16.2. ML ×16.2 ML ×16.2 ML The uniformly encrypted area size is 1.62. ML ×1.95 ML ×1.62 ML The mesh size of the uniformly encrypted zone is 0.01. ML .

[0041] The flow field meshing method described in this invention automatically generates a structured background Eulerian mesh containing a refinement zone, a transition zone, and an outer zone. Two virtual meshes (Ghost Mesh) are added to the outermost layer to handle the boundary. The coordinates of all Eulerian mesh nodes and the topological relationships between the meshes are stored.

[0042] Step 4: Refer to Figure 3 As shown, real manta ray biokinematic information is obtained based on feature point labeling.

[0043] First, five feature points were selected at key geometric locations on the pectoral fins of manta rays. Then, multi-view photography was conducted using a high-speed camera to extract the two-dimensional pixel coordinates of the feature points, and three-dimensional spatiotemporal trajectory data was obtained through three-dimensional reconstruction technology.

[0044] Based on this, we further extracted the kinematic parameters related to pectoral fin undulation and used regression analysis to establish a relationship model.

[0045] To verify the reliability of the model, the Tukey-Kramer test was used to evaluate the significance of the regression results, thereby ensuring that the established motion laws are statistically significant and can be applied to the description of different motion conditions.

[0046] Step 5: Refer to Figure 4 , Figure 5 As shown, the manta ray flapping equation is established. The manta ray flapping can be viewed as a coupling of spanwise and chordwise deformation. Based on the actual biological movement patterns with a constant body length, the spanwise deformation is expressed as the rotation of points on the pectoral fin around the body's longitudinal axis. During the rotation, the maximum rotation angle differs at different spanwise positions, decreasing linearly from the fin tip to the fin root. Considering the contour of the pectoral fin at the point of maximum amplitude as part of a circular arc, the motion equation can be obtained as follows: The equation of motion for the left wing is:

[0047] The equation of motion for the right wing is:

[0048] In the formula, A l, A r The amplitude is measured on the left and right sides; t For time; ( x 0, y 0, z 0) represents the initial position coordinates of the manta ray model; x , y , z (Time step) t Position coordinates; W l , W r For the wave numbers on the left and right sides; f l , f r The frequency of left and right movements; , These are the control parameters for left and right chordal oscillations; , This represents the maximum value of the left and right bending angles; u l , u r To adjust the parameters, the upward offset from the pectoral fin is used. d l , d r To adjust the parameters for downward offset of the left and right pectoral fins; , To adjust the phase of movement of the left and right pectoral fins.

[0049] Based on this equation of motion, the gliding bow angle can be quickly obtained through simplification and adjustment. β Control equations and gliding-flapping multimodal motion equations, left-right asymmetric motion equations, and up-down amplitude asymmetric motion equations.

[0050] 1. To perform bow gliding, refer to... Figure 6 , Figure 7 As shown, the gliding bow angle is obtained based on the aforementioned equations of motion. β The governing equation process. Define the maximum left and right bending angles. , The arched angle β of the pectoral fin is taken as... W l , W r All are 0. t Hengwei t 1, , When the value is 0, the above equation can be quickly adjusted to a gliding equation.

[0051] 2. To perform gliding-flapping motion, the flapping equation and gliding equation need to be segmented along the simulation time step, ensuring that the pectoral fins are in the same position at the moment before and after the segment to maintain motion continuity. 0≤t≤ T h At that time, in gliding mode, it is only necessary to... Adjustments can be made:

[0052]

[0053] T h ≤t≤ T t At that time, it is in the flapping mode, and the governing equations are the basic motion equations. Among them, T h Indicates the time of the gliding portion. T t Indicates the total time.

[0054] 3. To perform a maneuvering turn, the asymmetrical movement of the left and right pectoral fins generates a continuous turning torque during the maneuver, thus achieving the turn. The process of obtaining the asymmetrical motion equations from the aforementioned equations is as follows: by adjusting... A l , A r Asymmetric amplitudes can be obtained from left and right sides, and adjustments can be made. W l , W r Asymmetric wavenumbers can be obtained from the left and right sides, and adjustments can be made. f l , f r Asymmetric frequencies can be obtained from the left and right sides, and adjustments can be made. , Asymmetric phases can be obtained from the left and right sides; these are collectively referred to as asymmetric motion equations.

[0055] 4. To perform offset motion, the process of obtaining the upper and lower asymmetric motion equations from the aforementioned motion equations is as follows: By adjusting... u l , u r , d l , d r It can obtain the upper and lower amplitude offset, when u l , u r =0, d l ,d r When the amplitude is 1, the asymmetric motion with vertical amplitude degenerates into symmetric motion with vertical amplitude.

[0056] The angle of attack is changed through coordinate transformation. The angle of attack is defined as the angle between the length axis of the body and the horizontal plane. By establishing a three-dimensional coordinate system in the manta ray's head and performing a rotational coordinate transformation:

[0057] During the calculation, a carpet search strategy is adopted in the initial search of Euler points, and the obtained Euler points are stored. In subsequent time step searches, the search range is changed to extend by two grid scales in each of the six directions of the Euler point position stored in the previous time step.

[0058] Step 6: Detailed iterative calculation process. The effect of the boundary on the flow field is expressed through a force source. f The form is reflected in the Navier-Stokes equations:

[0059] In the formula, For the density of the fluid, u For speed, p For pressure, The coefficient of dynamic viscosity, f The additional force term representing the boundary effect, I It is a unit tensor.

[0060] Based on the conventional Navier-Stokes equations:

[0061]

[0062] The prediction-correction method is used to solve the problem. The prediction step solves the conventional Navier-Stokes equations to obtain the solution. t+ Density of 1 time step and prediction speed :

[0063] The correction speed is obtained by calculating the correction step using the following formula. ,

[0064] Based on the predicted speed and correction speed get t+ The velocity field at time step 1 is :

[0065] The data interaction between Lagrange nodes and Euler nodes is as follows:

[0066]

[0067] Step 7: The force applied to the object can be... x , y , z The thrust is obtained by decomposing the material in three directions. T Lift L Dimensionless thrust coefficient C T and lift coefficient C L as follows:

[0068]

[0069] in, For density, U This refers to the inlet velocity.

[0070] In one embodiment, the sliding propulsion kinematics modeling, feature point labeling, and motion feature extraction include the following parts: 1. First, feature points were selected. Specifically, five feature points were selected: the root of the anterior edge of the pectoral fin, the halfway point of the anterior edge of the pectoral fin, the tip of the pectoral fin, the halfway point of the posterior edge of the pectoral fin, and the root of the posterior edge of the pectoral fin. These points cover the key geometric locations of pectoral fin movement and can effectively characterize the morphological changes of its wavy movement. 2. Using a high-speed camera to capture real manta ray movements, obtain continuous time-series images, extract the two-dimensional pixel coordinates of the aforementioned feature points, and further convert them into three-dimensional spatial coordinates through three-dimensional reconstruction technology combined with calibration parameters, thereby obtaining the spatiotemporal trajectory data of each feature point during the movement process; 3. Based on the reconstructed 3D trajectory data, extract kinematic parameters. These include, but are not limited to, statistics reflecting the undulation characteristics of the pectoral fin, such as the sum of vertical amplitudes and the ratio of vertical amplitudes. Regression analysis is employed to obtain regression equations by fitting kinematic parameters and swimming speed, and the explanatory power of the regression model is evaluated using the coefficient of determination of the equations. 4. To verify the reliability of the relationships obtained from the regression analysis and to rigorously evaluate the significance of parameter differences among different motion modes, the Tukey-Kramer multiple comparison test was used to detect the p-values ​​of the regression analysis and comparisons between groups. This test controlled the overall error rate in the multiple comparisons. It ensured that the extracted motion parameters and the established kinematic model were not caused by random errors, thus confirming the reliability of the model in describing the motion under different amplitudes and frequencies.

[0071] In one embodiment, the kinematic modeling of the sliding propulsion is based on a unified kinematic equation and includes the following parts: 1. Kinematic modeling of the flapping process. During the active propulsion of the manta ray, the pectoral fin exhibits significant flexible deformation in both the spanwise and chordwise directions. This deformation can be decomposed into spanwise flexible deformation and chordwise flexible deformation. The spanwise flexible deformation is manifested as a deformation along the longitudinal axis... x The up-and-down movement of the axis as the center can be controlled by amplitude. A The description is as follows: chordal flexible deformation manifests as sinusoidal oscillations along the chord direction. This modeling method utilizes chordal wave parameters... Describe it, and use wavenumber W n Measure the spatial position of the chord wave. During the movement, the trajectory of the fin tip is approximately a sine function. When swimming rapidly with large amplitude, the amplitudes of the fin tip's upward and downward movements are basically equal, and its sine function is a standard sine function with the body's longitudinal axis as the axis of symmetry. 2. Kinematic modeling of the gliding process. During gliding, the manta ray's pectoral fins are in an upward-curved state, with the degree of curvature varying depending on the gliding speed and angle. The wingtips can point outwards, vertically upwards, or inwards, corresponding to the maximum left and right bending angles controlled in the equations. , During gliding, the manta ray's posture will change, and its head may rise or fall. This model uses the angle of attack... Describe it; 3. Kinematic modeling of the gliding-flapping process. The manta ray's gliding-flapping process can be described as follows: The pectoral fins glide at the apex of the upward curve for a period of time (0 ≤ t ≤ t). T h Then, from this position, perform normal flapping movements for a period of time. T h ≤t≤ T t This process modeling can be completed based on the modeling of the two processes mentioned above.

[0072] 4. Modeling the asymmetrical left-right motion process. This process mainly occurs during the manta ray's turning maneuver. Through the asymmetrical movement of the left and right pectoral fins, a certain lateral force is continuously generated during the movement, as well as torque caused by the imbalance of forces on both sides, enabling flexible turning maneuvers. This is mainly divided into four modeling methods: asymmetrical left-right amplitude... A l , A r Left and right asymmetric wavenumbers W l , W r Left and right asymmetric frequencies f l , f r and left and right asymmetric phases , .

[0073] 5. Modeling the asymmetrical vertical motion process. This process often occurs when manta rays swim forward to increase thrust or enhance maneuverability. This modeling method achieves the offset of the symmetry axis of the sinusoidal function of the fin tip trajectory by adjusting the offset parameters, and uses the vertical offset parameters of the left and right pectoral fin movements to achieve the same effect. u l , u r , d l , d r To achieve modeling of asymmetric vertical motion processes.

[0074] The hydrodynamic parameters of the manta ray flapping process obtained using the kinematic modeling and calculation method given in this invention are shown in Figure 8(a), and the vortex structure of the manta ray flapping process is shown in Figure 8(b). It can be seen from the figures that as the amplitude increases, the vortex intensity of the manta ray tail flow field is greater. Due to the viscosity effect of the fluid, the vortex will slowly dissipate as it propagates in the flow direction. For strong vortex structures, the dissipation time is relatively long. Therefore, it can be seen from the figures that the T1 vortex profile becomes clearer as the amplitude increases. Increased amplitude leads to increased vortex intensity, which inevitably leads to increased thrust. However, high thrust does not necessarily mean high propulsion efficiency. A u =0.4 ML At this amplitude, the vortex breaking phenomenon in the flow field is obvious, and the fin tip vortex and trailing edge vortex interfere with each other greatly, which inevitably leads to energy waste. Therefore, the propulsion efficiency decreases at this amplitude.

[0075] The hydrodynamic parameters of the manta ray gliding process obtained using the gliding equation, angle-of-attack transformation equation, and calculation method given in this invention are shown in Figure 8(c), and the vortex structure of the manta ray gliding process is shown in Figure 8(d). The upward deformation of the pectoral fins can effectively reduce gliding drag, and the greater the deformation, the greater the drag reduction effect, because upward deformation can increase the span vortex energy and improve the lateral change of the streamline behind the pectoral fins. Small upward deformation of the pectoral fins slightly increases lift, but large upward deformation leads to a lower lift value because of the increased pressure above the pectoral fins. Within a large angle-of-attack range, small upward deformation leads to a slight increase in the lift-to-drag ratio of the manta ray, while large upward deformation leads to a larger increase in the lift-to-drag ratio at larger angles of attack.

[0076] In summary, this invention presents a kinematic modeling and numerical calculation method for the flapping propulsion of a manta ray-inspired submersible. Compared to existing technologies, in describing the motion process, it establishes a set of kinematic equations based on biological observations marked with feature points. Based on this set of equations and its variations, it completes the description of the entire flapping propulsion process of the manta ray-inspired submersible, including flapping, gliding, and gliding-flapping, capturing the core kinematic features more realistically and accurately. In the numerical calculation method, this kinematic modeling, combined with the submerged boundary method, calculates the hydrodynamic performance and flow field information of the manta ray-inspired submersible's flapping propulsion. This provides a reference for clarifying the kinematic characteristics of manta rays and revealing the hydrodynamic mechanism of manta ray propulsion, and can also guide the optimized design of manta ray-inspired submersibles, improving research and development efficiency.

[0077] Although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those skilled in the art can make changes, modifications, substitutions and variations to the above embodiments within the scope of the present invention without departing from the principles and spirit of the present invention.

Claims

1. A method for kinematic modeling and numerical calculation of the sliding propulsion kinematics of a manta ray-inspired submersible, characterized in that, Includes the following steps: Kinematic modeling steps: Based on feature extraction and regression analysis of three-dimensional spatiotemporal trajectory data of real manta ray biological movement processes, a unified set of pectoral fin kinematic control equations that strictly satisfy the constraint of constant body length is established. This set of equations can accurately couple and describe the spanwise and chordal deformation of the pectoral fins. By adjusting and combining specific kinematic parameters in the control equations, mathematical descriptions of various kinematic modes of the manta ray-inspired submersible, such as flapping, gliding, gliding-flapping composite, left-right asymmetric maneuvers, and up-down asymmetric offset, are realized. Numerical calculation steps: The control equations obtained from the kinematic modeling steps are used as dynamic boundary input conditions. The hydrodynamic parameters and flow field information of the manta ray-inspired submersible under the various motion modes are iteratively solved by combining the submerged boundary method and the fluid dynamics solver.

2. The method for kinematic modeling and numerical calculation of the manta ray-inspired submersible's sliding propulsion according to claim 1, characterized in that: The kinematic modeling steps specifically include: Step 1: Select multiple feature points on the pectoral fin of the manta ray that cover its key geometric locations; Step 2: Obtain the three-dimensional spatiotemporal trajectory data of the feature points during the motion process using high-speed camera and 3D reconstruction technology; Step 3: Based on the three-dimensional spatiotemporal trajectory data, extract the kinematic parameters reflecting the undulation characteristics of the pectoral fin, and use regression analysis to establish a quantitative relationship model between the kinematic parameters and swimming speed; Step 4: The Tukey-Kramer multiple comparison test is used to evaluate the significance of the regression analysis results to verify the reliability of the extracted motion laws.

3. The method for kinematic modeling and numerical calculation of the manta ray-inspired submersible's sliding propulsion according to claim 2, characterized in that: The unified kinematic control equations for the pectoral fin, for the left wing, are in the following form: The right wing is symmetrical in form; In the formula, A l , A r The amplitude is measured on the left and right sides; t For time; ( x 0, y 0, z 0) represents the initial position coordinates of the manta ray model; x , y , z (Time step) t Position coordinates; W l , W r For the wave numbers on the left and right sides; f l , f r The frequency of left and right movements; , These are the control parameters for left and right chordal oscillations; , This represents the maximum value of the left and right bending angles; u l , u r To control the parameters of the upward offset of the left and right wings, d l , d r Parameters for controlling the downward offset of the left and right wings; , To adjust the phase of movement of the left and right pectoral fins; ML The body length is designed to resemble that of a manta ray submersible model. MW The model is a manta ray-inspired submersible with a half-length span.

4. The method for kinematic modeling and numerical calculation of the manta ray-inspired submersible's sliding propulsion according to claim 3, characterized in that: By using the wavenumber in the governing equation W l , W r Set to 0, left and right pectoral fin movement phase , Setting the value to 0 and fixing the time parameter, the control equation is simplified to an equation describing the gliding mode, where the parameter controlling the maximum value of the left and right pectoral fin bending angle is defined as the gliding bow angle. β .

5. The method for kinematic modeling and numerical calculation of the manta ray-inspired submersible's sliding propulsion according to claim 4, characterized in that: By dividing the simulation time into two segments and at time nodes T h To ensure the continuity of the pectoral fin position and model the gliding-flapping composite motion: when 0 ≤ t ≤ T h When, the gliding mode equations are used; when T h ≤t≤ T t At that time, the basic governing equations of the flapping mode are adopted.

6. The method for kinematic modeling and numerical calculation of the manta ray-inspired submersible's sliding propulsion according to claim 5, characterized in that: By independently adjusting the amplitude of the left and right pectoral fins in the control equation. A l and A r ,frequency f l and f r , wave number W l and W r or phase and This enables the modeling of asymmetrical turning motions.

7. The method for kinematic modeling and numerical calculation of the manta ray-inspired submersible's sliding propulsion according to claim 6, characterized in that: By independently adjusting the vertical offset parameters of the left and right pectoral fins in the control equation. u l , u r , d l , d r This enables asymmetric control of the amplitude of the pectoral fin's up-and-down flapping motion, thereby completing the modeling of the asymmetric up-and-down offset motion; where, when u l , u r =0, d l , d r When =1, the asymmetric offset motion between the upper and lower parts degenerates into a flapping motion with symmetrical upper and lower amplitudes.

8. The method for kinematic modeling and numerical calculation of the manta ray-inspired submersible's sliding propulsion according to claim 1, characterized in that: The numerical calculation steps specifically include: Step 101: Establish a simulation computational physical model of the manta ray-inspired submersible and generate a surface mesh; Step 102: Construct a CFD computational domain containing a uniformly encrypted region and perform flow field mesh generation; Step 103: Assign the kinematic control equations to the Lagrange nodes of the surface mesh and calculate their motion velocity; Step 104: Using an improved Eulerian point search method, establish the mapping relationship between Lagrange nodes and Eulerian nodes in the flow field; Step 105: Iteratively solve the fluid dynamics equations containing the moving boundary force source using the prediction-correction method, and feed the solution results back to the Lagrange node; Step 106: Calculate the hydrodynamic forces, torques, input power, and efficiency based on the forces acting on the Lagrange nodes; Step 107: Repeat steps S103 to S106 until the target time step is reached.

9. The method for kinematic modeling and numerical calculation of the manta ray-inspired submersible's sliding propulsion according to claim 8, characterized in that: The improved Euler point search method is as follows: in the initial time step, a global carpet search is used to determine and store the mapping relationship; in subsequent time steps, the search range is limited to a finite number of adjacent grid scales of the Euler point positions stored in the previous time step.

10. The method for kinematic modeling and numerical calculation of the manta ray-inspired submersible's sliding propulsion according to claim 9, characterized in that: The process of solving the prediction-correction method is as follows: Prediction step: Solve the Navier-Stokes equations without submerged boundary force source terms, thus obtaining... t+ Density of 1 time step and prediction speed The expression is as follows: in, For time step t density, For time step t velocity field, For the Laplace operator; Correction step: Combining the constructed linear equations including the contribution of the submerged boundary force source, the predicted velocity field is corrected to obtain the corrected velocity field at time step t+1; the expression of the linear equations is: In the formula, For the density of the fluid, u For speed, p For pressure, The coefficient of dynamic viscosity, f The additional force term representing the boundary effect, I Unit tensor; The correction speed is obtained by calculating the correction step using the following formula. : Final velocity field: The predicted velocity field is combined with the corrected velocity field to obtain the final velocity field at time step t+1. : 。

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