Numerical calculation method and system for ship-ship interaction during approach stage of double ship cross replenishment based on cfd-mmg coupling

Abstract

This invention provides a numerical calculation method and system for inter-ship effects during the lateral replenishment approach phase of two ships based on CFD-MMG coupling, belonging to the field of ship hydrodynamic numerical simulation technology. The method includes: constructing an MMG ship motion mathematical model; acquiring the dynamic response time history data of the ship from start-up to stable navigation; and determining the stable navigation time based on the response convergence criterion; constructing a segmented adjustment strategy based on state event triggering, using propeller speed as the control input; reconstructing unsteady velocity and displacement data segmentally under overall relative displacement constraints, and adjusting the trajectory according to the navigation phase to achieve spatial closed-loop control of the two ships from overtaking to parallel operation; constructing a continuous-time velocity function to drive ship motion based on the segmented reconstruction results, while maintaining physical consistency and continuity; embedding the continuous velocity function into CFD numerical calculation, and using an overlapping grid method to simulate the relative motion of the two ships from overtaking to parallel operation, thereby obtaining the inter-ship hydrodynamic and flow field evolution results.

Description

Technical Field

[0001] This invention relates to the field of numerical simulation technology of ship hydrodynamics, and in particular to a numerical calculation method and system for inter-ship effects during the lateral replenishment approach phase of two ships based on CFD–MMG coupling. Background Technology

[0002] As vital vessels for ocean shipping and maritime operations, ships increasingly require replenishment of fuel and supplies at sea, especially when far from ports. Lateral replenishment, due to its high efficiency and wide applicability, is widely used in practical engineering. During lateral replenishment, the two ships typically need to move from rear-end approach to close-range parallel navigation to establish a stable replenishment operation.

[0003] When ships are in close relative motion, the flow fields around the hulls will significantly interfere with each other, leading to uneven velocity distribution and pressure field distortion, which in turn induces inter-ship effects such as additional lateral forces, longitudinal forces, and yaw moments. This effect has obvious unsteady characteristics, which are particularly prominent during the approach-parallel transition phase, and will directly affect the ship's maneuvering stability and the safety of lateral replenishment operations.

[0004] For the unsteady motion process of two ships transitioning from overtaking to parallel movement, existing numerical methods still have the following shortcomings:

[0005] (1) It is difficult to balance computational efficiency and accuracy: Although the CFD method using propeller or virtual disk model can reflect the real propulsion mechanism, it significantly increases the grid size and computational cost; at the same time, it is difficult to control the relative displacement of the ship and cannot accurately regulate its speed and displacement changes, and cannot ensure that the two ships can accurately transition from the chasing state to the parallel state. (2) Insufficient physical consistency of motion input: Some methods use preset speed or uniform speed drive, and do not generate motion trajectory based on ship dynamic response. In essence, it is a form of motion assumed by humans, which is difficult to accurately characterize the unsteady characteristics of the two-ship transition process, thus affecting the accuracy of hydrodynamic calculation and physical law analysis.

[0006] (3) Insufficient convergence characteristics during the transition from the initial speed and corresponding speed to the target working condition. Due to the influence of the dynamic balance of ship propulsion and resistance, the speed response exhibits obvious asymptotic convergence characteristics. The rate of change in the final stage decreases, which leads to a longer convergence process, increases the computation time cost, and reduces the comparison efficiency between different working conditions.

[0007] Therefore, accurately understanding the unsteady hydrodynamic characteristics during the transition between two ships is of significant engineering importance. Summary of the Invention

[0008] To address the aforementioned technical problems, this invention proposes a numerical calculation method and system for inter-ship effects during the lateral replenishment approach phase of dual-ship operations based on CFD-MMG coupling. This invention addresses the slow terminal convergence problem of traditional methods by introducing a staged control and displacement compensation strategy to achieve continuous and controllable construction of the relative motion process. This improves upon the shortcomings of existing technologies in terms of computational efficiency, motion realism, and coupling stability, thereby enhancing the reliability and engineering applicability of dual-ship unsteady process simulation.

[0009] The technical means employed in this invention are as follows: A numerical calculation method for inter-ship effects during the lateral replenishment approach phase of two ships based on CFD-MMG coupling includes: constructing an MMG ship motion mathematical model and ensuring that the dynamic response is consistent with the actual ship dynamics; obtaining the dynamic response time history data of the ship from start-up to stable navigation through single-ship straight-line CFD numerical simulation, and determining the stable navigation time based on the response convergence criterion; constructing a segmented adjustment strategy based on state event triggering with propeller speed as the control input to achieve segmented adjustment control of propeller speed at different navigation stages; under the constraint of overall relative displacement, reconstructing the unsteady velocity and displacement data in segments, and adjusting the trajectory according to the navigation stage to achieve spatial closed-loop control of the two ships from overtaking to parallel operation; constructing a continuous-time velocity function to drive ship motion based on the segmented reconstruction results, while maintaining physical consistency and continuity; embedding the continuous velocity function into CFD numerical calculation, and realizing the relative motion simulation of the overtaking to parallel operation process of the two ships through the overlapping grid method, thereby obtaining the hydrodynamic and flow field evolution results between the ships.

[0010] Furthermore, the MMG ship motion mathematical model is used to calculate the dynamic response of the ship during navigation and obtain a response curve that conforms to physical laws. The MMG ship motion mathematical model corresponds to the same ship as the numerical simulation ship, ensuring consistent dynamic characteristics. A four-quadrant modeling method is used to uniformly represent the operating state and describe the hydrodynamic characteristics of the propeller under complex operating conditions. The complex operating conditions include: propulsion, braking, and reversing. Experimental data is used to verify the ship motion mathematical model, and the scale effect is eliminated by a dimensionless method to ensure the comparability of motion characteristics under different ship sizes.

[0011] Furthermore, determining the stable navigation time specifically includes: Numerical simulations of a single ship's still-water straight-line navigation were used to obtain time-history data of the ship's acceleration from a standstill to stable navigation, and the changes in hull motion and hydrodynamic response over time were analyzed. The average value at the end of the time history was used as the basis for the analysis. For steady-state reference; for each time step signal Determine if the following conditions are met:

[0012] in, In response to fluctuation threshold Ensure that the response fluctuation is less than 6% and remains stable; when the condition is met for a continuous period of time and lasts for at least At each sampling point, the ship is determined to have reached a stable state, and the earliest time that meets the conditions is defined as the stable navigation moment. .

[0013] Furthermore, the segmented adjustment strategy, based on the MMG ship motion mathematical model, uses propeller speed as the control input to construct an event-triggered segmented control strategy based on state variables, thereby realizing adaptive segmented control of the speed during acceleration, steady state, deceleration, and convergence phases. An event-triggered speed control mechanism based on state variables is constructed at the input layer, representing the propeller speed as a piecewise function:

[0014] Among them, stage variables Triggered by system status, it is used to automatically switch between different control phases; Indicates the acceleration phase. This indicates the speed recovery phase. Indicates the deceleration phase. Indicates the convergence control phase; For the maximum propeller speed, This is the reference rotational speed for the constant speed range. Minimum propeller speed, The initial propeller speed. This is the speed correction factor. For the initial steady-state speed, Let be the speed of the ship at time t; when At that time, the control process terminates. This is the speed error tolerance.

[0015] Furthermore, during the segmented reconstruction, initial conditions for ship motion and event triggering thresholds are set, including initial state variables, initial propeller speed, and speed error tolerance parameters; the fourth-order Runge-Kutta method is used to numerically integrate the system to obtain the ship motion history response. Based on the Froude similarity criterion, the calculated ship motion history response is subjected to a scale-consistency transformation to achieve a dynamic similarity mapping between the scale of the actual ship and the model:

[0016]

[0017] in, and These represent the numerical simulation of the ship model's speed and the time of speed change. and These represent the actual ship's speed and the time of change, respectively. The scale ratio between the length of the actual ship and the numerical simulation ship model; Relative velocity variables are constructed based on the initial steady-state speed. :

[0018] in, Let be the speed of the ship at time t. The steady-state speed corresponding to the initial propeller speed; For the processed relative velocity variable By performing numerical integration, the relative displacement is obtained:

[0019]

[0020] in, and These represent the relative displacements during the acceleration and deceleration phases, respectively. , These are the end times of the acceleration and deceleration phases, respectively. This is the end time of the uniform velocity phase; For two-ship navigation, the longitudinal distance between the two ships during the initial uniform speed phase is defined. Displacement of the overtaking vessel during the uniform velocity phase is obtained based on displacement compensation. and duration of the uniform velocity phase :

[0021]

[0022] in, This is the maximum speed during the acceleration phase; By splicing the velocity curves during the stable navigation, acceleration, constant speed, deceleration, and parallel phases, the velocity is reconstructed to obtain a continuous and differentiable composite trajectory. :

[0023] The composite trajectory is a relative velocity trajectory. Indicates the time of stable navigation. Indicates the trajectory during the acceleration phase. Represents the trajectory during the uniform velocity phase. This represents the trajectory during the deceleration phase; the endpoint being zero indicates that the ship's absolute speed has converged to its initial steady-state speed. This satisfies the requirements of displacement conservation and spatial closure control.

[0024] Furthermore, constructing the continuous-time velocity function specifically includes: Obtain the relative velocity time series obtained from the phased motion reconstruction and corresponding time node sequences :

[0025] in, This represents the concatenated unified timeline. This represents the relative velocity value at the corresponding node. This represents the total number of discrete time nodes.

[0026] The discrete velocity sequence is reconstructed into a continuous, monotonic, and non-oscillating velocity function by employing piecewise conformal cubic Hermite interpolation. :

[0027] In each time interval Internally, Hermite interpolation function Defined as:

[0028]

[0029]

[0030] in, Indicates the first interpolation nodes The relative velocity function value at that location, Indicates the first interpolation nodes The relative velocity function value at that location, Indicates the first Nodes The first derivative value at that point, Indicates the first Nodes The first derivative value at; Indicates the first Time interpolation nodes Indicates the first Time interpolation nodes Indicates the first Each time interval step; Indicates time interval Continuous-time variables within, Represents normalized local coordinates, used to divide time intervals Mapping to standard interval Hermite basis functions are:

[0031]

[0032] A STAR-CCM+ ternary expression is constructed to directly drive ship speed in numerical simulation, expanding each Hermite polynomial into a standard cubic polynomial form:

[0033] In each polynomial segment:

[0034] coefficient The following is obtained from Hermite polynomial expansion:

[0035]

[0036]

[0037]

[0038] By integrating the relative velocity function over time, we can verify that its cumulative displacement satisfies the displacement conservation constraint, thus ensuring that the two ships achieve spatial position closure under the target operating conditions.

[0039]

[0040] in, This represents the integral of the relative velocity function.

[0041] Furthermore, the simulation of the relative motion of the two ships overtaking each other in parallel using the overlapping mesh method specifically includes: Overlapping and inter-ship meshing regions were established within the dual-ship computational domain. A cut-volume mesh generator and a prism layer mesh generator were used for mesh generation. A separate verification strategy was employed to systematically validate the CFD numerical method and determine its feasibility. The computational domain and boundary conditions for dual-ship motion simulation were constructed, and a gas-liquid two-phase region was defined. A velocity inlet, pressure outlet, no-slip wall, and far-field wave suppression were established. A velocity inlet corresponding to the initial steady-state speed was applied through the velocity inlet. The incoming flow boundary conditions are determined; the generated high-order continuous ternary expression is embedded into the overtaking vessel motion control to achieve a smooth transition from the overtaking process to the parallel process; typical working conditions are selected to carry out grid-independent simulation, calculate key hydrodynamic indicators and determine the optimal grid resolution for the dual-vessel simulation; the control parameters and lateral spacing are changed, multiple sets of velocity trajectories are repeatedly generated and imported into multi-working-condition simulation to analyze the influence of the forces between the vessels.

[0042] This invention also provides a numerical calculation system for inter-ship effects during the approach phase of lateral replenishment between two ships based on CFD–MMG coupling, comprising: a ship dynamics modeling module, a continuous velocity function generation module, a steady-state determination and simulation input module, a multi-condition import and response analysis module, and a displacement conservation-constrained unsteady ship velocity reconstruction module, wherein: The ship dynamics modeling module is used to construct the MMG ship dynamics mathematical model, obtain the dynamic response time history data during the navigation process of the two ships, and perform dimensionless processing of different ship sizes using the Froude similarity criterion to achieve physical consistency feature extraction. The continuous velocity function generation module is used to generate a composite velocity curve based on the initial longitudinal spacing, acceleration, constant speed and deceleration stages. It generates a continuous, monotonic and non-oscillating high-order velocity function through piecewise conformal interpolation and outputs a STAR-CCM+ ternary expression to directly drive the changes in ship motion in numerical simulation. The steady-state determination and simulation input module is used to analyze the ship's motion and hydrodynamic response through single-ship straight-course simulation analysis to determine the stable navigation time. Determine the initial conditions for unsteady simulation and realize overtaking and parallel simulation starting from physical steady state; The multi-condition import and response analysis module is used to import the high-order continuous ternary expression generated by different speed curves into the simulation, analyze the changes in inter-ship force and flow field during the process of two ships overtaking each other to parallel under different navigation conditions, and complete the multi-condition sensitivity analysis and inter-ship hydrodynamic response characteristic analysis. The displacement conservation constraint ship unsteady velocity reconstruction module integrates the output velocity of the dynamic model and inversely calculates the intermediate stage time to ensure that different operating conditions meet the unified displacement condition.

[0043] Compared with the prior art, the present invention has the following advantages: This invention transforms the unsteady motion of ships from being driven by a propeller propulsion model to being driven by a continuous velocity function. It eliminates the need to explicitly build a propeller or virtual propulsion model in CFD calculations, reducing the pressure of local mesh refinement and multiphysics coupling solution. While ensuring the authenticity of motion characteristics, it reduces the computational scale and improves computational efficiency and engineering applicability.

[0044] This invention generates unsteady response data based on the MMG ship dynamics model and obtains a continuous velocity function through staged motion reconstruction, enabling the motion input to reflect the dynamic evolution under propulsion-drag balance. Compared with methods that directly preset velocity curves, this method reduces the idealization and simplification of motion input, thereby improving the rationality and reliability of inter-ship hydrodynamic calculations during the unsteady phase.

[0045] This invention uses relative displacement conservation as a global constraint to uniformly reconstruct the acceleration, constant speed, and deceleration stages, enabling the discrete dynamic response to achieve spatial closed-loop control under the overall displacement constraint. This ensures that the two ships meet the target longitudinal spacing requirements during the process of overtaking and paralleling, avoiding problems such as trajectory deviation accumulation and uncontrollable final state in traditional methods, thereby improving the physical consistency and controllability of the motion process.

[0046] This invention achieves a unified mapping between dynamic response and ship motion control input by constructing a velocity function that satisfies stage continuity constraints and converting discrete dynamic response into a continuous velocity expression suitable for CFD dynamic motion solving.

[0047] This invention determines the start time of unsteady simulation by using a steady-state determination method based on the convergence characteristics of dynamic response, allowing the simulation to start from the physical steady-state stage, reducing the impact of initial transient disturbances on the calculation results. At the same time, it can shorten the computational domain length and reduce the mesh size, thereby further reducing computational costs and improving the reliability of numerical results. Attached Figure Description

[0048] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0049] Figure 1 This is a flowchart of the numerical calculation method for the inter-ship effect during the approach phase of lateral replenishment of two ships based on CFD–MMG coupling in this invention.

[0050] Figure 2 This is a schematic diagram illustrating the definition of the four-quadrant operating conditions for the propeller in this invention.

[0051] Figure 3 This is a comparison chart of the results of the left 35° rotation test and the right 35° rotation test in an embodiment of the present invention.

[0052] Figure 4 This is a schematic diagram illustrating the verification results of the zigzag maneuvering of the ship motion mathematical model in this embodiment of the invention.

[0053] Figure 5 This is a schematic diagram showing the hydrodynamic and kinematic response results of a ship under different forward speeds in an embodiment of the present invention.

[0054] Figure 6 This is a flowchart of the ship unsteady navigation speed reconstruction method based on event-triggered speed control in an embodiment of the present invention.

[0055] Figure 7 This is a schematic diagram of the segmented velocity time-history curves of the ship overtaking process based on displacement conservation constraints in an embodiment of the present invention.

[0056] Figure 8 This is a schematic diagram illustrating the comparison and verification of the relative velocity curve fitting method in the embodiments of the present invention.

[0057] Figure 9 This is a comparison chart of wave height capture results and experimental data during parallel navigation in this embodiment of the invention.

[0058] Figure 10 This is a comparison chart of the calculated results of the additional resistance coefficient under different ship length ratios in the embodiments of the present invention and the experimental data.

[0059] Figure 11 This is a schematic diagram illustrating the verification results of the hydrodynamic response characteristics of the overtaken vessel during the overtaking process in an embodiment of the present invention.

[0060] Figure 12 This is a schematic diagram of the computational domain layout and motion stages of the transition process from overtaking to parallel navigation in an embodiment of the present invention. Detailed Implementation

[0061] It should be noted that, unless otherwise specified, the embodiments and features described in the present invention can be combined with each other. The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0062] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. The following description of at least one exemplary embodiment is merely illustrative and is in no way intended to limit the present invention or its application or use. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0063] It should be noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit the scope of exemplary embodiments according to the invention. As used herein, the singular form is intended to include the plural form as well, unless the context clearly indicates otherwise. Furthermore, it should be understood that when the terms "comprising" and / or "including" are used in this specification, they indicate the presence of features, steps, operations, devices, components, and / or combinations thereof.

[0064] Unless otherwise specifically stated, the relative arrangement, numerical expressions, and values ​​of the components and steps described in these embodiments do not limit the scope of the invention. It should also be understood that, for ease of description, the dimensions of the various parts shown in the drawings are not drawn to actual scale. Techniques, methods, and devices known to those skilled in the art may not be discussed in detail, but where appropriate, such techniques, methods, and devices should be considered part of the specification. In all examples shown and discussed herein, any specific values ​​should be interpreted as merely exemplary and not as limitations. Therefore, other examples of exemplary embodiments may have different values. It should be noted that similar reference numerals and letters in the following figures denote similar items; therefore, once an item is defined in one figure, it need not be further discussed in subsequent figures.

[0065] like Figure 1 As shown, this invention provides a numerical calculation method for inter-ship effects during the lateral replenishment approach phase of two ships based on CFD-MMG coupling. The method includes: constructing an MMG ship motion mathematical model and ensuring that the dynamic response is consistent with the actual ship's dynamic characteristics. Specifically, in a preferred embodiment, the MMG ship motion mathematical model is used to calculate the dynamic response during ship navigation, obtaining a response curve that conforms to physical laws. The MMG ship motion mathematical model and the numerically simulated ship correspond to the same ship, ensuring consistent dynamic characteristics. In implementation, the MMG ship motion mathematical model includes dynamic modules such as hull inertial hydrodynamics, viscous hydrodynamics, propeller propulsion, and rudder force. The corresponding ship's main dimensions and geometric configuration are consistent with the ship's geometric model used in the subsequent CFD numerical simulation, ensuring that the dynamic response characteristics match the flow field calculation. This model can obtain time-history response data such as the ship's speed, heading angle, and angular velocity under different control inputs, providing basic data support for subsequent unsteady ship motion reconstruction.

[0066] like Figure 2 As shown, a four-quadrant modeling method is used to uniformly represent the operating states and describe the hydrodynamic characteristics of the propeller under complex operating conditions. These complex operating conditions include propulsion, braking, and reversing. The four-quadrant operating conditions are based on the propeller speed. With speed The system is divided into symbol combinations, including typical operating states such as forward propulsion, reverse braking, and reverse propulsion. Experimental data is used to validate the mathematical model of ship motion, and a dimensionless method is used to eliminate scale effects, ensuring the comparability of motion characteristics for ships of different sizes. By constructing a nonlinear mapping relationship between propeller speed and thrust, a continuous description of thrust changes under the above operating conditions is achieved, ensuring the continuity and physical consistency of thrust output during operating condition switching. This enables unified modeling of the ship's dynamic response during acceleration, deceleration, and reverse motion phases.

[0067] Figure 3 In the diagram, (a) and (b) show a comparison of the trajectories of the left 35° turn test and the right 35° turn test, respectively; (c) and (d) show a comparison of the longitudinal and lateral velocity time histories during the left 35° and right 35° turns, respectively; (e) and (f) show a comparison of the rudder angle changes during the left 35° and right 35° turns, respectively; and (g) and (h) show a comparison of the bow roll rate time histories during the left 35° and right 35° turns, respectively. Figure 3 It can be seen that under the ±35° turning condition, the simulation results can accurately reproduce the turning motion characteristics of the ship, and the turning trajectory and velocity response are in high agreement with the experimental data.

[0068] like Figure 4 As shown, (a) and (b) are comparisons of the bow angle and rudder angle time histories during ±20° and ±10° Z-shaped maneuvers, respectively; (c) and (d) are comparisons of the longitudinal velocity and lateral velocity time histories during ±20° and ±10° Z-shaped maneuvers, respectively; (e) and (f) are comparisons of the motion trajectories during ±20° and ±10° Z-shaped maneuvers, respectively; and (g) and (h) are comparisons of the bow roll rate time histories during 20° and 10° Z-shaped maneuvers, respectively. In the Z-shaped maneuver test, the simulation and experimental results showed good consistency under small rudder angle (±10°), while under large rudder angle (±20°), there was a certain deviation in the response amplitude and phase. This deviation mainly originated from propeller load changes and nonlinear scaling effects.

[0069] Overall, the established ship motion model can reasonably reflect the speed and attitude changes during ship maneuvering, and the obtained speed time history data has good physical consistency, which can provide reliable input conditions for subsequent CFD numerical simulation, ensuring the scientific nature and engineering practicality of the entire technical solution.

[0070] Under calm water and straight-line conditions, since the hull geometry is symmetrical about the longitudinal center plane and no rudder angle or lateral external forces are applied, the sway and yaw degrees of freedom theoretically do not produce significant responses. Therefore, this embodiment mainly selects longitudinal motion-related variables for steady-state analysis, including sailing speed, resistance, and heave and pitch responses, to characterize the longitudinal dynamic convergence process of the ship. Through the time evolution of the above variables, the dynamic response characteristics of the hull from the unsteady acceleration stage to the steady-state straight-line stage are analyzed.

[0071] The dynamic response time history data of a ship from start-up to stable navigation is obtained through single-ship straight-line CFD numerical simulation, and the stable navigation time is determined based on the response convergence criterion. In a preferred embodiment of the invention, the time history data of the ship from acceleration from rest to stable navigation is obtained through single-ship still-water straight-line numerical simulation, and the variation law of hull motion and hydrodynamic response over time is analyzed; the average value at the end of the time history is used as the basis for the analysis. For steady-state reference; for each time step signal Determine if the following conditions are met:

[0072] in, In response to fluctuation threshold Ensure that the response fluctuation is less than 6% and remains stable; when the condition is met for a continuous period of time and lasts for at least At each sampling point, the ship is determined to have reached a stable state, and the earliest time that meets the conditions is defined as the stable navigation moment. .

[0073] During implementation, a single-ship still-water straight-line test was conducted to obtain the time-history curves of the ship's heave, pitch, and resistance. The results of determining the stable navigation time at different speeds are as follows: Figure 5 As shown. Figure 5 (a) Comparison of heave duration at different speeds; (b) Comparison of pitch duration at different speeds; (c) Comparison of drag duration at different speeds. As the initial conditions for acceleration, uniform speed, and deceleration phases, this ensures that unsteady simulations begin from a physically steady state, reducing the impact of transient disturbances on the flow field and ship motion response, while also reducing the computational grid size and numerical simulation costs.

[0074] Based on the established MMG ship dynamics model, with propeller speed n(t) as the only external input and rudder angle always being 0, a state-space form ship motion control system is constructed:

[0075] in:

[0076] Using propeller speed as the control input, a segmented adjustment strategy based on state event triggering is constructed to achieve segmented adjustment and control of propeller speed during different navigation stages of the ship. Specifically, in a preferred embodiment of this invention, the segmented adjustment strategy, based on the MMG ship motion mathematical model, uses propeller speed as the control input to construct an event-triggered segmented control strategy based on state variables, achieving adaptive segmented control of speed during acceleration, steady state, deceleration, and convergence phases. An event-triggered speed control mechanism based on state variables is constructed at the input layer, representing propeller speed as a piecewise function:

[0077] Among them, stage variables Triggered by system status, it is used to automatically switch between different control phases; Indicates the acceleration phase. This indicates the speed recovery phase. Indicates the deceleration phase. Indicates the convergence control phase; For the maximum propeller speed, This is the reference rotational speed for the constant speed range. Minimum propeller speed, The initial propeller speed. This is the speed correction factor. For the initial steady-state speed, Let be the speed of the ship at time t; when At that time, the control process terminates. This is the speed error tolerance.

[0078] Apply higher speed input during acceleration. This allows the system to transition from its initial state to the target speed range; during the intermediate stage, the rotational speed is adjusted to the reference rotational speed corresponding to the constant speed stage. This allows the speed to maintain a relatively stable evolution trend near the target; the rotational speed is reduced during the deceleration phase. To achieve velocity decay; a feedback correction based on velocity error is introduced in the terminal phase. This allows the system to gradually converge to the initial reference state, ensuring that the terminal speed meets the convergence requirements.

[0079] Stage variables Triggered by the following state events: :

[0080] :

[0081] :

[0082] Termination conditions:

[0083] in, This is the maximum speed throughout the entire process, and also serves as the speed during the constant speed phase. This is the threshold for switching the deceleration phase, used to change the speed from... The rotational speed smoothly transitions to the convergence control phase. It can be represented as: , where 'a' is an adjustable constant, preferably ranging from 1.01 to 1.1; an excessively large value will prolong the system convergence time, while an excessively small value may result in a negative relative ship speed in the low-speed range. This mechanism ensures that control switching is entirely driven by the system state, rather than by a preset time division.

[0084] Under the constraint of overall relative displacement, unsteady velocity and displacement data are reconstructed in segments, and the trajectory is adjusted according to the navigation stage to achieve spatial closed-loop control of the two ships from overtaking to parallel operation; the flowchart of the ship unsteady navigation speed reconstruction method based on event-triggered speed control is as follows. Figure 6 As shown. In a specific implementation, as a preferred embodiment of the present invention, when performing segmented reconstruction, the initial conditions for ship motion and the event triggering judgment threshold are set, including initial state variables, initial propeller speed and speed error tolerance parameters; the fourth-order Runge-Kutta method is used to numerically integrate the system to obtain the ship motion time history response; Based on the Froude similarity criterion, the calculated ship motion history response is subjected to a scale-consistency transformation to achieve a dynamic similarity mapping between the scale of the actual ship and the model:

[0085]

[0086] in, and These represent the numerical simulation of the ship model's speed and the time of speed change. and These represent the actual ship's speed and the time of change, respectively. The scale ratio between the length of the actual ship and the numerical simulation ship model; Relative velocity variables are constructed based on the initial steady-state speed. :

[0087] in, Let be the speed of the ship at time t. The steady-state speed corresponding to the initial propeller speed; For the processed relative velocity variable By performing numerical integration, the relative displacement is obtained:

[0088]

[0089] in, and These represent the relative displacements during the acceleration and deceleration phases, respectively. , These are the end times of the acceleration and deceleration phases, respectively. The end time of the uniform velocity phase is given by the segmented velocity time-history curve of the ship overtaking process based on displacement conservation constraints, as shown in the figure. Figure 7 As shown.

[0090] For two-ship navigation, the longitudinal distance between the two ships during the initial uniform speed phase is defined. Displacement of the overtaking vessel during the uniform velocity phase is obtained based on displacement compensation. and duration of the uniform velocity phase :

[0091]

[0092] in, This is the maximum speed during the acceleration phase; By splicing the velocity curves during the stable navigation, acceleration, constant speed, deceleration, and parallel phases, the velocity is reconstructed to obtain a continuous and differentiable composite trajectory. :

[0093] The composite trajectory is a relative velocity trajectory. Indicates the time of stable navigation. Indicates the trajectory during the acceleration phase. Represents the trajectory during the uniform velocity phase. This represents the trajectory during the deceleration phase; the endpoint being zero indicates that the ship's absolute speed has converged to its initial steady-state speed. This satisfies the requirements of displacement conservation and spatial closure control.

[0094] An event-triggered, displacement-compensated mechanism was constructed to establish a motion-time reconstruction method based on displacement conservation constraints and state-triggered control coupling. By integrally decomposing the displacement contributions during acceleration, constant speed, and deceleration phases, the time scale adaptive allocation under the constraint of total motion displacement consistency was achieved. At the same time, an event-triggered propeller speed control strategy based on state variables was introduced, and combined with the segmented MMG dynamic response and continuous trajectory reconstruction method, a differentiable continuous expression of unsteady navigation trajectory was realized, thereby improving the consistency and numerical stability of the motion process under different operating conditions.

[0095] Based on the segmented reconstruction results, a continuous-time velocity function is constructed to drive the ship's motion while maintaining physical consistency and continuity. Specifically, as a preferred embodiment of this invention, the relative velocity time series obtained from the segmented motion reconstruction is acquired. and corresponding time node sequences :

[0096] in, This represents the concatenated unified timeline. This represents the relative velocity value at the corresponding node. This represents the total number of discrete time nodes.

[0097] To ensure physical monotonicity and avoid non-physically reversed velocities during acceleration or deceleration, a piecewise conformal cubic Hermite interpolation method is used to reconstruct the discrete velocity sequence as a continuous function, thus constructing a continuous, monotonic, and oscillatory velocity function. :

[0098] In each time interval Internally, Hermite interpolation function Defined as:

[0099]

[0100]

[0101] in, Indicates the first interpolation nodes The relative velocity function value at that location, Indicates the first interpolation nodes The relative velocity function value at that location, Indicates the first Nodes The first derivative value at that point, Indicates the first Nodes The first derivative value at; Indicates the first Time interpolation nodes Indicates the first Time interpolation nodes Indicates the first Each time interval step; Indicates time interval Continuous-time variables within, Represents normalized local coordinates, used to divide time intervals Mapping to standard interval Hermite basis functions are:

[0102]

[0103] The velocity function constructed using the above method satisfies continuity, monotonicity, and smoothness constraints across the entire time domain. It can be used for a unified description of acceleration, constant velocity, and deceleration phases, and meets the requirements of CFD numerical computation for continuous boundary inputs. For example... Figure 8 As shown, (a) compares the original speed data with the PCHIP fitting curves under different test conditions; (b) compares the speed curve fitting methods. This method shows good consistency in all 7 typical test conditions (Test1–Test7). The original data and PCHIP fitting results are highly consistent in peak speed, acceleration / deceleration process and constant speed stage, indicating that the method has strong cross-condition applicability and stability.

[0104] Compared with the Smoothing Spline method and the maximum jerk planning method based on S-curve, the PCHIP method is more stable in maintaining velocity monotonicity and suppressing local oscillations, while better preserving the local shape features of the original data.

[0105] A STAR-CCM+ ternary expression is constructed to directly drive ship speed in numerical simulation, expanding each Hermite polynomial into a standard cubic polynomial form:

[0106] In each polynomial segment:

[0107] coefficient The following is obtained from Hermite polynomial expansion:

[0108]

[0109]

[0110]

[0111] By constructing ternary expressions through segmented conditional judgments, they can be directly imported into STAR-CCM+ to achieve speed-driven operation at arbitrary time resolutions.

[0112] By integrating the relative velocity function over time, we can verify that its cumulative displacement satisfies the displacement conservation constraint, thus ensuring that the two ships achieve spatial position closure under the target operating conditions.

[0113]

[0114] in, This represents the integral of the relative velocity function.

[0115] Output higher-order continuous velocity function It can be used for CFD boundary condition input and time-varying drive of MMG-CFD coupled systems.

[0116] By embedding continuous velocity functions into CFD numerical calculations and employing an overlapping mesh method, the relative motion of the overtaking and parallel processes of two ships is simulated, thereby obtaining the hydrodynamic and flow field evolution results between the ships. In a preferred embodiment of this invention, the IGS files of the overtaking and overtaken ships are imported into STAR-CCM+, ensuring the model's geometric integrity and consistency with the MMG ship dynamics model. Overlapping regions and inter-ship refinement regions are set within the dual-ship computational domain. Mesh generation is performed using a cut-body mesh generator and a prism layer mesh generator. The distance between the first layer walls of the prism layer mesh is set according to the initial speed, with a growth factor of 1.2. The SST k-ω turbulence model is used in conjunction with the full The wall treatment method adopts an adaptive mode that takes into account both the solution of the viscous sublayer and the approximation of the wall function, ensuring the stability of the flow field solution and the accuracy that meets engineering requirements throughout the entire motion process.

[0117] A multi-mechanism verification strategy was adopted to systematically verify the CFD numerical method and determine its feasibility. Since there is a lack of unified experimental data for the entire process of ship overtaking and parallel operations, this embodiment selects publicly available experimental databases and standard benchmark examples to verify three key physical processes: parallel interference between two ships, wave-making drag increase, and unsteady overtaking response. This allows for independent evaluation of different modules of the numerical method.

[0118] Based on parallel dual-ship experimental data provided by Flanders Hydraulics Research (FHR), the hydrodynamic response under different water depths and UKC conditions was validated. Comparison indicators included longitudinal force X, lateral force Y, roll moment K, and bow moment N, and cross-validation was performed with publicly available numerical results from Jin, Chalmers, and IOWA. Table 1 shows that the calculated results (Scur) of this method are generally consistent with the experimental data (D) and results from multiple literature studies; Figure 9 It can be seen that the wave height distribution of the free liquid surface between the ships is basically consistent with the experiment, which verifies the applicability of the numerical method to the flow field and free liquid surface changes between the ships.

[0119] Table 1

[0120] Wave drag enhancement verification: Using regular wave drag test data from the KVLCC2 ship model, combined with the published research results of Guo et al. (2012) and Ozdemir et al. (2017), the effects of still water drag and wave-making drag enhancement by regular waves were verified. Figure 10 It can be seen that the trend of wave drag variation with wavelength is consistent with the experiment, and the drag evolution law under wave interference can be stably reproduced.

[0121] Verification through overtaking: Using benchmark examples published at the MASHCON international conference, the applicability of the method under strongly unsteady ship-to-ship interactions is verified. Comparison metrics include longitudinal force coefficients, transverse force coefficients, bow moment coefficients, and roll moment coefficients. Figure 11 In the figure, (a) is the longitudinal force coefficient; (b) is the sway force coefficient; (c) is the yaw moment coefficient; and (d) is the roll moment coefficient. Figure 11 It can be seen that this method can effectively capture the phased changes in hydrodynamics between ships during the overtaking process.

[0122] In summary, the numerical method presented in this example has been verified to be effective through multiple typical cases, demonstrating its reliability and engineering application foundation for simulating hydrodynamic processes involving two ships overtaking each other in parallel.

[0123] In this embodiment, a numerical computation domain and boundary conditions are set to construct a numerical simulation environment for ship motion. The computation domain is divided into a gas phase region above and a liquid phase region below. The far-field boundaries are set as the velocity inlet and pressure outlet boundaries, respectively, where the inlet velocity is based on the initial steady-state speed. Given that all hull surfaces are assumed to be non-slip walls. To ensure stability in unsteady calculations, wave damping is applied in the far-field region to attenuate boundary reflections of waves and pressure disturbances.

[0124] Based on this, the ship's navigation is divided into acceleration, constant speed, and deceleration stages according to the relative motion process, and the staged motion control of the overtaking ship is realized through a continuous-time velocity function. The above-mentioned computational domain structure, boundary condition settings, and motion stage division are as follows: Figure 12 As shown.

[0125] The generated high-order continuous ternary expression is embedded into the straight-line motion of the overtaking ship, ensuring that the two ships begin to accelerate and decelerate to overtake each other in parallel after the initial stable navigation, thus achieving a natural connection between the simulation and the physical steady state.

[0126] For different grid numbers and sizes, multiple sets of numerical simulations were performed on typical two-ship overtaking conditions to calculate key hydrodynamic parameters and determine the optimal grid resolution for two-ship overtaking and parallel simulation.

[0127] For the overtaking and parallel motion processes, under the condition of uniform longitudinal displacement constraint, with the initial sailing speed Maximum speed during acceleration phase Target acceleration speed Minimum speed during deceleration phase and speed switching threshold Key control parameters were used as research variables, combined with different lateral spacings. A single-variable scanning experiment was conducted under different control parameter combinations and operating conditions. The trajectory generation process was executed multiple times to obtain the corresponding high-order continuous expressions of velocity / displacement. Then, the influence of different conditions on the inter-ship forces during the process of two ships overtaking and running in parallel was analyzed.

[0128] This invention also provides a numerical calculation system for ship-to-ship effects during the lateral replenishment approach phase based on CFD–MMG coupling, comprising: a ship dynamics modeling module, a continuous velocity function generation module, a steady-state determination and simulation input module, a multi-condition import and response analysis module, and a displacement conservation constraint ship unsteady velocity reconstruction module.

[0129] The ship dynamics modeling module is used to construct the MMG ship dynamics mathematical model, obtain the dynamic response time history data during the navigation process of the two ships, and perform dimensionless processing of different ship sizes using the Froude similarity criterion to achieve physical consistency feature extraction.

[0130] The continuous velocity function generation module is used to generate composite velocity curves based on the initial longitudinal spacing, acceleration, constant speed and deceleration stages. It generates continuous, monotonic and non-oscillating high-order velocity functions through piecewise conformal interpolation and outputs STAR-CCM+ ternary expressions to directly drive the changes in ship motion in numerical simulation.

[0131] The steady-state determination and simulation entry module is used to analyze the ship's motion and hydrodynamic response through single-ship straight-line simulation analysis to determine the stable navigation time. The initial conditions for unsteady simulation are determined, enabling overtaking and parallel simulation starting from physical steady state.

[0132] The multi-condition import and response analysis module is used to import high-order continuous ternary expressions generated from different speed curves into the simulation, analyze the changes in inter-ship forces and flow fields during the overtaking and paralleling process of two ships under different navigation conditions, and complete multi-condition sensitivity analysis and inter-ship hydrodynamic response characteristic analysis.

[0133] The displacement conservation constraint ship unsteady velocity reconstruction module integrates the output velocity of the dynamic model and inversely calculates the intermediate stage time to ensure that different operating conditions meet the unified displacement condition.

[0134] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A numerical calculation method for inter-ship effects during the approach phase of lateral replenishment between two ships based on CFD–MMG coupling, characterized in that, include: Construct a mathematical model of MMG ship motion and ensure that the dynamic response is consistent with the actual ship dynamics characteristics; The dynamic response time history data of a ship from startup to stable navigation was obtained by single-ship straight-line CFD numerical simulation, and the stable navigation time was determined based on the response convergence criterion. Using propeller speed as the control input, a segmented adjustment strategy based on state event triggering is constructed to realize the segmented adjustment and control of propeller speed at different navigation stages of the ship. Under the constraint of overall relative displacement, the unsteady velocity and displacement data are reconstructed in segments, and the trajectory is adjusted according to the navigation stage to achieve spatial closed control of the two ships from overtaking to parallel operation. Based on the segmented reconstruction results, a continuous-time velocity function is constructed to drive the ship's motion while maintaining physical consistency and continuity. The continuous velocity function is embedded into CFD numerical calculation, and the relative motion simulation of the overtaking and parallel process of the two ships is realized by using the overlapping grid method, thereby obtaining the hydrodynamic and flow field evolution results between the ships.

2. The numerical calculation method for inter-ship effects during the lateral replenishment approach phase of two ships based on CFD-MMG coupling, as described in claim 1, is characterized in that... The MMG ship motion mathematical model is used to calculate the dynamic response of the ship during navigation and obtain a response curve that conforms to physical laws; the MMG ship motion mathematical model corresponds to the same ship as the numerical simulation ship to ensure consistent dynamic characteristics; A four-quadrant modeling method is used to uniformly represent the operating state and describe the hydrodynamic characteristics of the propeller under complex operating conditions, including propulsion, braking and reversing. Experimental data were used to validate the mathematical model of ship motion, and the scale effect was eliminated by the dimensionless method to ensure the comparability of motion characteristics under different ship sizes.

3. The numerical calculation method for inter-ship effects during the approach phase of lateral replenishment for two ships based on CFD-MMG coupling, as described in claim 1, is characterized in that... The determination of stable navigation times specifically includes: Numerical simulations of a single ship's still-water straight-line navigation were used to obtain time-history data of the ship's acceleration from a standstill to stable navigation, and the changes in hull motion and hydrodynamic response over time were analyzed. The average value at the end of the time history was used as the basis for the analysis. For steady-state reference; for each time step signal Determine if the following conditions are met: in, In response to fluctuation threshold Ensure that the response fluctuation is less than 6% and remains stable; when the condition is met for a continuous period of time and lasts for at least At each sampling point, the ship is determined to have reached a stable state, and the earliest time that meets the conditions is defined as the stable navigation moment. .

4. The numerical calculation method for inter-ship effects during the approach phase of lateral replenishment for two ships based on CFD-MMG coupling, as described in claim 1, is characterized in that... The segmented adjustment strategy, based on the MMG ship motion mathematical model, uses propeller speed as the control input to construct an event-triggered segmented control strategy based on state variables, thereby realizing adaptive segmented control of speed during acceleration, steady state, deceleration, and convergence phases. An event-triggered speed control mechanism based on state variables is constructed at the input layer, representing the propeller speed as a piecewise function: Among them, stage variables Triggered by system status, it is used to automatically switch between different control phases; Indicates the acceleration phase. This indicates the speed recovery phase. Indicates the deceleration phase. Indicates the convergence control phase; For the maximum propeller speed, This is the reference rotational speed for the constant speed range. Minimum propeller speed, The initial propeller speed. This is the speed correction factor. For the initial steady-state speed, Let be the speed of the ship at time t; when At that time, the control process terminates. This is the speed error tolerance.

5. The numerical calculation method for inter-ship effects during the approach phase of lateral replenishment for two ships based on CFD-MMG coupling, as described in claim 1, is characterized in that... During the segmented reconstruction, initial conditions for ship motion and event triggering thresholds are set, including initial state variables, initial propeller speed, and speed error tolerance parameters; the fourth-order Runge-Kutta method is used to numerically integrate the system to obtain the ship motion time history response. Based on the Froude similarity criterion, the calculated ship motion history response is subjected to a scale-consistency transformation to achieve a dynamic similarity mapping between the scale of the actual ship and the model: in, and These represent the numerical simulation of the ship model's speed and the time of speed change. and These represent the actual ship's speed and the time of change, respectively. The scale ratio between the length of the actual ship and the numerical simulation ship model; Relative velocity variables are constructed based on the initial steady-state speed. : in, Let be the speed of the ship at time t. The steady-state speed corresponding to the initial propeller speed; For the processed relative velocity variable By performing numerical integration, the relative displacement is obtained: in, and These represent the relative displacements during the acceleration and deceleration phases, respectively. , These are the end times of the acceleration and deceleration phases, respectively. This is the end time of the uniform velocity phase; For two-ship navigation, the longitudinal distance between the two ships during the initial uniform speed phase is defined. Displacement of the overtaking vessel during the uniform velocity phase is obtained based on displacement compensation. and duration of the uniform velocity phase : in, This is the maximum speed during the acceleration phase; By splicing the velocity curves during the stable navigation, acceleration, constant speed, deceleration, and parallel phases, the velocity is reconstructed to obtain a continuous and differentiable composite trajectory. : The composite trajectory is a relative velocity trajectory. Indicates the time of stable navigation. Indicates the trajectory during the acceleration phase. Represents the trajectory during the uniform velocity phase. This represents the trajectory during the deceleration phase; the endpoint being zero indicates that the ship's absolute speed has converged to its initial steady-state speed. This satisfies the requirements of displacement conservation and spatial closure control.

6. The numerical calculation method for inter-ship effects during the approach phase of lateral replenishment for two ships based on CFD-MMG coupling, as described in claim 5, is characterized in that... Constructing the continuous-time velocity function specifically includes: Obtain the relative velocity time series obtained from the phased motion reconstruction and corresponding time node sequences : in, This represents the concatenated unified timeline. This represents the relative velocity value at the corresponding node. This represents the total number of discrete time points; The discrete velocity sequence is reconstructed into a continuous, monotonic, and non-oscillating velocity function by employing piecewise conformal cubic Hermite interpolation. : In each time interval Internally, Hermite interpolation function Defined as: in, Indicates the first interpolation nodes The relative velocity function value at that location, Indicates the first interpolation nodes The relative velocity function value at that location, Indicates the first Nodes The first derivative value at that point, Indicates the first Nodes The first derivative value at; Indicates the first Time interpolation nodes Indicates the first Time interpolation nodes Indicates the first Each time interval step; Indicates time interval Continuous-time variables within, Represents normalized local coordinates, used to divide time intervals Mapping to standard interval Hermite basis functions are: A STAR-CCM+ ternary expression is constructed to directly drive ship speed in numerical simulation, expanding each Hermite polynomial into a standard cubic polynomial form: In each polynomial segment: coefficient The following is obtained from Hermite polynomial expansion: By integrating the relative velocity function over time, we can verify that its cumulative displacement satisfies the displacement conservation constraint, thus ensuring that the two ships achieve spatial position closure under the target operating conditions. in, This represents the integral of the relative velocity function.

7. The numerical calculation method for inter-ship effects during the lateral replenishment approach phase of two ships based on CFD-MMG coupling, as described in claim 1, is characterized in that... The simulation of relative motion during the parallel overtaking process of two ships using the overlapping mesh method specifically includes: Overlapping and inter-ship meshing regions were established within the dual-ship computational domain. A cut-volume mesh generator and a prism layer mesh generator were used for mesh generation. A separate verification strategy was employed to systematically validate the CFD numerical method and determine its feasibility. The computational domain and boundary conditions for dual-ship motion simulation were constructed, and a gas-liquid two-phase region was defined. A velocity inlet, pressure outlet, no-slip wall, and far-field wave suppression were established. A velocity inlet corresponding to the initial steady-state speed was applied through the velocity inlet. The incoming flow boundary conditions are determined; the generated high-order continuous ternary expression is embedded into the overtaking vessel motion control to achieve a smooth transition from the overtaking process to the parallel process; typical working conditions are selected to carry out grid-independent simulation, calculate key hydrodynamic indicators and determine the optimal grid resolution for the dual-vessel simulation; the control parameters and lateral spacing are changed, multiple sets of velocity trajectories are repeatedly generated and imported into multi-working-condition simulation to analyze the influence of the forces between the vessels.

8. A numerical calculation system for inter-ship effects during the approach phase of dual-ship lateral replenishment based on CFD-MMG coupling, implemented using the numerical calculation method for inter-ship effects during the approach phase of dual-ship lateral replenishment based on any one of claims 1-7, characterized in that, include: The system includes modules for ship dynamics modeling, continuous velocity function generation, steady-state determination and simulation initiation, multi-condition import and response analysis, and displacement-constrained unsteady ship velocity reconstruction. The ship dynamics modeling module is used to construct the MMG ship dynamics mathematical model, obtain the dynamic response time history data during the navigation process of the two ships, and perform dimensionless processing of different ship sizes using the Froude similarity criterion to achieve physical consistency feature extraction. The continuous velocity function generation module is used to generate a composite velocity curve based on the initial longitudinal spacing, acceleration, constant speed and deceleration stages. It generates a continuous, monotonic and non-oscillating high-order velocity function through piecewise conformal interpolation and outputs a STAR-CCM+ ternary expression to directly drive the changes in ship motion in numerical simulation. The steady-state determination and simulation input module is used to analyze the ship's motion and hydrodynamic response through single-ship straight-course simulation analysis to determine the stable navigation time. Determine the initial conditions for unsteady simulation and realize overtaking and parallel simulation starting from physical steady state; The multi-condition import and response analysis module is used to import the high-order continuous ternary expression generated by different speed curves into the simulation, analyze the changes in inter-ship force and flow field during the process of two ships overtaking each other to parallel under different navigation conditions, and complete the multi-condition sensitivity analysis and inter-ship hydrodynamic response characteristic analysis. The displacement conservation constraint ship unsteady velocity reconstruction module integrates the output velocity of the dynamic model and inversely calculates the intermediate stage time to ensure that different operating conditions meet the unified displacement condition.

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