A method for evaluating hydrodynamic response of flexible floating structure considering equivalent effect of upper layer stagnant water

Abstract

The application discloses a kind of flexible floating structure hydrodynamic response evaluation method considering upper layer water equivalent effect.First, three-dimensional hydroelastic-porous side wall coupling model is established, and flexible structure is equivalent to elastic disc below static water surface, and outer impermeable boundary is equivalent to porous side wall with minimum permeability coefficient.By dividing external water area, internal limited overlying water layer and lower water body and defining velocity potential, semi-analytical solution of velocity potential in each region is constructed using separation of variables method and eigenfunction expansion method.Combined with fluid continuity condition and porous side wall matching condition with pressure jump compensation, flow field matching is carried out to solve each modal coefficient, and then wave excitation force, deflection response and free surface response are calculated.The application evaluates the influence of solid floating ring water resistance and limited overlying water layer on flexible structure hydrodynamic response through layered modeling and near impermeable boundary equivalent treatment, and provides theoretical basis for structure optimization and wind and wave resistance design of floating photovoltaic system under upper layer stagnant water condition.

Description

Technical Field

[0001] This invention belongs to the field of marine engineering hydrodynamic assessment technology, and in particular relates to a method for assessing the hydrodynamic response of flexible floating structures that considers the equivalent effect of upper layer water retention. Background Technology

[0002] Floating photovoltaic (FPV) systems, as a novel energy engineering structure utilizing vast water resources, play a crucial role in improving power generation efficiency. Among numerous structural forms, the floating ring membrane structure photovoltaic platform, composed of an outer high-density polyethylene (HDPE) floating ring and a central large-area flexible film, is widely used due to its advantages such as low cost and convenient installation. This structure typically lays photovoltaic modules directly on the flexible film, presenting an overall physical form similar to a "closed basin." However, under sea conditions such as heavy rainfall, shallow stagnant water easily forms on the surface of the flexible film due to the water-blocking effect of the outer solid floating ring. The presence of this internal stagnant water, combined with external waves, forms a complex three-dimensional fluid-structure interaction field of "internal stagnant water - elastic film - external waves," significantly altering the dynamic characteristics of the system.

[0003] In existing technologies, the hydrodynamic assessment of flexible floating structures is often simplified to the response of an elastic plate within a single fluid domain, failing to fully consider the coupling dynamic differences between the upper stagnant water and the lower water body after they are physically isolated by the flexible medium. Specifically, existing assessment methods typically ignore the constraint effect of the impermeable boundary of the outer floating ring on the internal stagnant water, resulting in an inability to accurately simulate the complex flow at the edge boundary. Secondly, traditional models often treat the fluid domain as a whole, lacking refined layered modeling and velocity potential definition for the internal upper stagnant water region and lower water body region, making it difficult to capture the additional dynamic pressure exerted by the stagnant water layer on the flexible membrane. Furthermore, when dealing with the flow domain matching, existing technologies often use continuous simplification, failing to introduce porous sidewall matching conditions that can compensate for pressure jumps and the corresponding eigenfunction expansion method, leading to significant deviations in calculating the deflection response of the flexible disk and the stress conditions of the platform. Therefore, how to accurately incorporate the impermeable sidewall constraint and layered fluid coupling effect into the theoretical model has become a key technical problem in assessing the hydrodynamic safety of flexible photovoltaic platforms with an upper stagnant water layer. Summary of the Invention

[0004] To address the problem that existing technologies struggle to reasonably characterize the water-blocking constraint effect of the outer solid floating ring within an analytical framework, and to effectively assess the layered hydroelastic coupling effect between the restricted overlying water layer, the flexible structure, and the lower water body, resulting in insufficient accuracy in evaluating the hydrodynamic response of flexible floating structures under upper-layer stagnant water conditions, this invention provides a method for evaluating the hydrodynamic response of flexible floating structures that considers the equivalent effect of upper-layer stagnant water.

[0005] This invention is implemented as follows: a method for evaluating the hydrodynamic response of a flexible floating structure considering the equivalent effect of upper-layer water retention, comprising the following steps: S1. Establish a three-dimensional hydroelastic-porous sidewall coupled dynamic theoretical model: The flexible floating structure is equivalent to an elastic disk located below the still water surface, the water above the elastic disk is equivalent to a restricted overlying water layer, and the near-impermeable water-blocking boundary of the outer perimeter of the structure is equivalent to a sidewall boundary that satisfies the matching conditions of the porous sidewall. The equivalent permeability parameter of the sidewall boundary is set to a minimum value to simulate the water-blocking effect of the outer solid floating ring. S2. Divide the fluid region and define the velocity potential: Establish a cylindrical coordinate system with the center of the still water surface as the origin, divide the fluid domain into an outer region, an inner upper stagnant water region, and an inner lower water body region, and establish the velocity potential equations for each region based on the linear potential flow theory. S3. Determine the boundary conditions and eigenfunction expansion: Using the method of separation of variables, combined with the free liquid surface condition, the impermeable seabed condition, the kinematic continuity condition of the upper and lower surfaces of the elastic disk, and the dynamic equation of the elastic disk, the eigenfunction expansion of the velocity potential in each region is obtained respectively. S4. Perform flow domain matching and solution: At the interface of each region, velocity potential matching is performed using the fluid continuity condition and the porous sidewall matching condition containing pressure jump compensation term. Combined with the elastic disk edge constraint condition, a system of linear algebraic equations is constructed and solved to obtain the unknown coefficients of each mode. S5. Calculate hydrodynamic response parameters: Based on the solved unknown coefficients, calculate the wave excitation force of the main body of the flexible floating structure, the deflection response of the elastic disk, and / or the free surface response of the overlying water layer.

[0006] In the above technical solution, preferably, in step S2, the dynamic equation of the elastic disk is described using Kirchhoff's thin-plate theory:

[0007] in, The bending stiffness of the elastic disk. For a bitone operator, Let be the deflection of the elastic disk. For dynamic response time, Let be the mass per unit area of ​​the elastic disk. This is the net pressure difference applied to the elastic disk.

[0008] In the above technical solution, preferably, the dynamic boundary conditions of the elastic disk are transformed into a form containing only velocity potential:

[0009] in, For dimensionless stiffness, For dimensionless mass, It is a dimensionless frequency; The vertical coordinates are in cylindrical coordinates. The vertical distance between the elastic disk and the still water surface; The bending stiffness of the elastic disk. For water density, It is the acceleration due to gravity. The radius of the elastic disk, and These are the material density and thickness of the elastic disk, respectively. The incident wave angular frequency, and These are the complex velocity potentials of the upper stagnant water region and the lower water region, respectively.

[0010] In the above technical solution, preferably, in step S2, the velocity potential expansion of the external region is:

[0011] in, These are radial coordinates in cylindrical coordinates. The circumferential angle is in cylindrical coordinates. The vertical coordinates are in cylindrical coordinates. For the unknown coefficients in the expansion of the velocity potential in the external region. Let be the circumferential modal order. The vertical modal order is... This is a modified Bessel function of the second kind. For the vertical eigenvalues ​​of the external region, For circumferential modal functions, The vertical characteristic function of the external region. Let be the velocity potential of the incident wave.

[0012] In the above technical solution, preferably, in step S2, the velocity potential expansion formulas for the upper internal stagnant water region and the lower internal water region are as follows:

[0013] in, These are radial coordinates in cylindrical coordinates. The circumferential angle is in cylindrical coordinates. The vertical coordinates are in cylindrical coordinates. The unknown coefficients are the velocity potential expansions for the upper and lower water layers within the interior. Let be the circumferential modal order. The vertical modal order is... It is a modified Bessel function of the first kind. The depth-segmented vertical eigenfunctions for the upper and lower water layers are defined. For circumferential modal functions, The eigenvalues ​​are determined by the dispersion relation of the submerged elastic disk.

[0014] In the above technical solution, preferably, the edge constraint condition of the elastic disk adopts a fixed edge condition, which satisfies:

[0015]

[0016] in, The radius of the elastic disk, The vertical distance between the elastic disk and the still water surface. The vertical position corresponding to the elastic disk; For the order of the circumferential mode summation, The order of the summation of vertical modes; The unknown coefficients are those in the velocity potential expansion formula of the lower internal water region. These are the eigenvalues ​​determined by the dispersion relation of the submerged elastic disk; This is a modified Bessel function of the first kind. This is the first derivative of the modified Bessel function of the first kind; The depth of the upper stagnant water region and the lower water body region is segmented into piecewise vertical eigenfunctions. Regarding depth The first derivative.

[0017] In the above technical solution, preferably, in step S4, the condition for the outer region and the inner upper stagnant water region to match through the porous sidewall is:

[0018] in, and These are the complex velocity potentials of the outer region and the upper internal perched water region, respectively. These are radial coordinates in cylindrical coordinates. This is the partial derivative with respect to the radial coordinate; The imaginary unit, Let be the angular frequency of the incident wave. The dimensionless equivalent permeability parameter of the porous sidewall boundary. This represents the density of the water.

[0019] In the above technical solution, preferably, in step S5, the hydrodynamic response parameters include the horizontal excitation force. Total vertical force :

[0020]

[0021] in, and These represent the circumferential components of the velocity potential in the outer region and the upper internal perched water region, respectively. and These are the vertical characteristic functions of the upper and lower water layers, respectively. The unknown coefficients are the velocity potential expansions of the circumferential zero-order corresponding upper and lower water layers. It is a zero-order modified Bessel function of the first kind; The imaginary unit, For water density, The incident wave angular frequency, The radius of the elastic disk, The vertical distance between the elastic disk and the still water surface. The eigenvalues ​​are determined by the dispersion relation of the submerged elastic disk. The vertical coordinates are in cylindrical coordinates. These are radial coordinates in cylindrical coordinates. This represents the number of segments cut off in the vertical direction.

[0022] In the above technical solution, preferably, in step S5, the formula for calculating the deflection amplitude at the center of the elastic disk is:

[0023] In the formula, Let be the deflection of the elastic disk. These are radial coordinates in cylindrical coordinates. The circumferential angle is in cylindrical coordinates. These are the vertical coordinates in the cylindrical coordinate system; The imaginary unit, The incident wave angular frequency; The unknown coefficients are the velocity potential expansions of the circumferential zero-order corresponding upper and lower water layers. For the vertical characteristic functions of the upper and lower water layers within the interior, with respect to the vertical coordinates, [the following is a list of partial derivatives of the functions]. The vertical position corresponding to the elastic disk. This represents the number of segments cut off in the vertical direction.

[0024] In the above technical solution, preferably, it further includes step S6: truncating the spatial intrinsic expansion series with a finite number of terms, setting the vertical truncation number as... The upper limit of the circumferential modal cutoff is The hydrodynamic response of the entire field is reconstructed through numerical calculation.

[0025] Compared with the prior art, the present invention has the following beneficial effects: This invention establishes a three-dimensional hydroelastic-porous sidewall coupled dynamic theoretical model, equating the flexible floating structure to an elastic disk located below the still water surface, and equating the near-impermeable water-blocking boundary of the platform to a sidewall boundary satisfying the matching conditions of the porous sidewall. By setting the equivalent permeability parameter of the sidewall boundary to a minimum value, this invention can approximately characterize the lateral constraint effect of the outer solid floating ring on the overlying water layer in the analytical model, overcoming the problem that traditional analytical models cannot handle the influence of the solid floating ring water-blocking boundary. This provides a calculable theoretical model for evaluating the hydrodynamic response of flexible floating structures with the equivalent effect of upper layer water retention.

[0026] This invention employs a layered watershed partitioning method, dividing the fluid domain into an external region, an internal upper perched water region, and an internal lower water region, and establishing velocity potential equations for each region based on linear potential flow theory. This layered modeling approach can describe the hydroelastic coupling between the confined overlying water layer, the elastic disk, and external waves, helping to assess the influence of the overlying water layer on the wave load and elastic deformation response of flexible structures, thus overcoming the shortcomings of traditional single-fluid-domain models in reflecting the separation effect between upper and lower water layers.

[0027] This invention introduces a porous sidewall matching condition with a pressure jump compensation term into the watershed matching process. Combining the eigenfunction expansion method and Kirchhoff's thin-plate theory, a system of linear algebraic equations is constructed to solve for the unknown modal coefficients. Based on the obtained modal coefficients, this invention can quantitatively calculate the horizontal excitation force, total vertical excitation force, elastic disk deflection response, and free surface response of the overlying water layer of a flexible structure. This provides a theoretical basis for the structural design, parameter optimization, and hydrodynamic safety assessment of flexible floating structures under perched water conditions.

[0028] The method described in this invention employs a solution approach combining eigenfunction expansion and finite term truncation, enabling the reconstruction of the entire field hydrodynamic response while ensuring computational feasibility. This method can be used not only for the hydrodynamic response assessment of circular flexible floating structures but also as a reference for floating flexible engineering structures with similar characteristics of external water-blocking boundaries and overlying confined water layers. It has application value for the wind and wave resistance design and engineering safety assessment of floating photovoltaic platforms and similar structures. Attached Figure Description

[0029] Figure 1 This is a top view schematic diagram of the model of the flexible floating structure simulated by the present invention; Figure 2 This is a side view schematic diagram of the model of the flexible floating structure simulated by the present invention; Figure 3 This is a comparison diagram of the distribution of dimensionless amplitude contour lines above a submerged rigid disk when the model described in this invention degenerates into a submerged rigid disk; Figure 4 This is a comparative verification diagram showing the change of dimensionless wave amplitude with relative wave number at three measuring points when the model described in this invention degenerates into a submerged rigid disk. Figure 5 This is a comparative verification diagram showing the vertical excitation force of the model described in this invention degenerating into a submerged rigid disk and the relevant research results; Figure 6 This is a comparative verification diagram showing the horizontal and vertical excitation forces of the model described in this invention degenerating into a truncated cylinder, and related research results. Detailed Implementation

[0030] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention. To further illustrate the structure of the invention, a detailed description is provided below with reference to the accompanying drawings: This embodiment provides a method for evaluating the hydrodynamic response of a flexible floating structure that considers the equivalent effect of upper-layer water retention, specifically including the following steps: S1. Establish a three-dimensional hydroelastic-porous sidewall coupled dynamic theoretical model: The main body of the flexible floating structure is equivalent to an elastic disk located below the still water surface, the water above the elastic disk is equivalent to a restricted overlying water layer, and the near-impermeable water-blocking boundary of the platform is equivalent to a sidewall boundary that satisfies the matching conditions of the porous sidewall. The equivalent permeability parameter of the sidewall boundary is set to a minimum value to approximately simulate the water-blocking effect of the outer solid floating ring.

[0031] S2. Divide the fluid region and define the velocity potential: Establish a cylindrical coordinate system with the center of the still water surface as the origin. , These are radial coordinates in cylindrical coordinates. The circumferential angle is in cylindrical coordinates. These are the vertical coordinates in the cylindrical coordinate system.

[0032] The fluid domain is divided into an outer region (Region I) and an inner region (Region II). The inner region (Region II) is further divided into an upper perched water region (Region IIa) and an inner lower water region (Region IIb). Velocity potential equations are established for each region based on linear potential flow theory. The dynamic equations of the elastic disk are described using Kirchhoff's thin-plate theory.

[0033] in, The bending stiffness of the elastic disk. For a bitone operator, Let be the deflection of the elastic disk. For dynamic response time, Mass per unit area Let be the net pressure difference applied to the elastic disk. The dynamic boundary conditions of the elastic disk are transformed into a form containing only velocity potential:

[0034] in, For dimensionless stiffness, For dimensionless mass, It is a dimensionless frequency.

[0035] For the bending stiffness of the elastic disk, For water density, It is the acceleration due to gravity. The radius of the elastic disk, and These are the density and thickness of the elastic disc material, respectively. The incident wave angular frequency, and These are the complex velocity potentials of the upper stagnant water region and the lower water region, respectively.

[0036] S3. Determine Boundary Conditions and Eigenfunction Expansion: Using the method of separation of variables, combined with the free surface condition, the impermeability of the seabed condition, the kinematic continuity condition of the upper and lower surfaces of the elastic disk, and the dynamic equations of the elastic disk, the eigenfunction expansions of the velocity potential in each region are obtained. The velocity potential expansion of the external region is:

[0037] in, For the unknown coefficients in the expansion of the velocity potential in the external region. Let be the circumferential modal order. The vertical modal order is... This is a modified Bessel function of the second kind. For the vertical eigenvalues ​​of the external region, For circumferential modal functions, The vertical characteristic function of the external region. Let be the velocity potential of the incident wave.

[0038] The velocity potential expansion formulas for the upper stagnant water region and the lower stagnant water region are as follows:

[0039] in, The unknown coefficients are the velocity potential expansions for the upper and lower water layers within the interior. Let be the circumferential modal order. The vertical modal order is... It is a modified Bessel function of the first kind. For the depth segmentation of the internal region, use vertical eigenfunctions. For circumferential modal functions, The eigenvalues ​​are determined by the dispersion relation of the submerged elastic disk.

[0040] The edge constraint condition of the elastic disk adopts the fixed edge condition, which satisfies:

[0041] and

[0042] in, The radius of the elastic disk, This represents the vertical distance between the elastic disk and the still water surface, and corresponds to the thickness of the restricted overlying water layer in the equivalent model. The vertical position corresponding to the elastic disk; For the order of the circumferential mode summation, The order of the summation of vertical modes; The unknown coefficients are those in the velocity potential expansion formula of the lower internal water region. These are the eigenvalues ​​determined by the dispersion relation of the submerged elastic disk; This is a modified Bessel function of the first kind. This is the first derivative of the modified Bessel function of the first kind; For the Region II vertical characteristic function with respect to depth The first derivative.

[0043] S4. Perform flow domain matching and solution: At the interfaces of each region, velocity potential matching is performed using the fluid continuity condition and the porous sidewall matching condition including pressure jump compensation. Combined with the elastic disk edge constraint condition, a system of linear algebraic equations is constructed and solved to obtain the unknown coefficients of each mode. The matching condition between the outer region and the inner upper perched water region through the porous sidewall is as follows:

[0044] in, and These are the complex velocity potentials of the outer region and the upper internal perched water region, respectively. These are radial coordinates in cylindrical coordinates. This is the partial derivative with respect to the radial coordinate; The imaginary unit, Let be the angular frequency of the incident wave. The dimensionless equivalent permeability parameter of the porous sidewall boundary. This represents the density of the water.

[0045] S5. Calculate hydrodynamic response parameters: Based on the solved unknown coefficients, calculate the wave excitation force, elastic disk deflection response, and / or overlying water layer free surface response of the flexible floating structure. The hydrodynamic response parameters include the horizontal excitation force. Total vertical force :

[0046]

[0047] in, and These represent the circumferential components of the velocity potential in the outer region and the upper internal perched water region, respectively. and These are the vertical characteristic functions of the upper and lower water layers, respectively. The unknown coefficients are the circumferential zero-order velocity potential expansion of the internal region. It is a zero-order modified Bessel function of the first kind; The imaginary unit, For water density, The incident wave angular frequency, The radius of the elastic disk, The vertical distance between the elastic disk and the still water surface. The eigenvalues ​​are determined by the dispersion relation of the submerged elastic disk.

[0048] The formula for calculating the deflection amplitude of the elastic disk is:

[0049] In the formula, Let be the deflection of the elastic disk. These are radial coordinates in cylindrical coordinates. The circumferential angle is in cylindrical coordinates. These are the vertical coordinates in the cylindrical coordinate system; The imaginary unit, The incident wave angular frequency; The unknown coefficients are the circumferential zero-order velocity potential expansion of the internal region. Let be the partial derivative of the vertical characteristic function of the interior region with respect to the vertical coordinate. The vertical position corresponding to the elastic disk.

[0050] S6. The spatial intrinsic expansion series is truncated with a finite number of terms. The vertical truncation number is set to N, and the upper limit of the circumferential modal truncation is set to S. The full-field hydrodynamic response is reconstructed through numerical calculation.

[0051] In this embodiment, for details, please refer to [link / reference]. Figure 1 and Figure 2 Taking a floating photovoltaic platform at sea as an engineering background, a three-dimensional hydroelastic-porous sidewall coupled dynamic theoretical model was established. This model equates the flexible floating structure to an elastic disk located below the still water surface, and the water above the elastic disk to a confined overlying water layer. The water-blocking effect of the outer floating ring is approximated by the sidewall boundary, which takes the minimum value of the equivalent permeability parameter. Thus, the phenomenon of water stagnation in the upper layer caused by waves or extreme rainfall is transformed into a fluid-structure interaction problem between the fluid domain inside the impermeable boundary and the elastic structure. The computational domain uses a cylindrical coordinate system with the origin located at the center of the still water surface. . The axis is positively oriented upwards. The seabed is located... Among them The water depth is constant. The radius of the elastic disk is... The submersion depth of the disk is The dimensionless porous effect parameter is .

[0052] The entire fluid domain is divided into two sub-regions: ;

[0053] Based on linear potential flow theory, the fluid is assumed to be irrotational, inviscid, and incompressible. Under this assumption, the fluid motion can be described by velocity potential: (1) The spatial velocity potential satisfies Laplace's equation: (2) about The deflection of the elastic disk is: (3) in, To represent taking the real part, the function and Let represent the time-independent portions of the complex velocity potential and the time-independent deflection of the elastic disk, respectively. Time harmonic factor: is the base of the natural logarithm. The imaginary unit, Angular frequency, is the time constant.

[0054] Referring to the hydroelastic model of wave-elastic structure coupling interaction by Meylan et al. (1994), Kirchhoff's thin plate theory is used to describe the motion of the elastic plate, and its dynamic equations are as follows: (4) in, Represents bending stiffness. Let the biharmonic operator represent the fourth-order spatial partial derivative. Let be the deflection of the elastic disk. Mass per unit area This represents the net pressure difference applied to the plate. Furthermore, the interface between the elastic plate and the fluid must satisfy linearized kinematic boundary conditions: (5) After separating the time variable, we can obtain: (6) The dynamic boundary conditions of the elastic disk, reduced to a form containing only velocity potential, can be written as: (7) In the formula, Represents dimensionless stiffness, which is the relative ratio between the elastic bending restoring force of a structure and the buoyancy restoring force of a fluid. D The bending stiffness of the elastic disk. For water density, The radius of the elastic disk, It is the acceleration due to gravity; This represents dimensionless mass, indicating the relative ratio between the surface mass of the structure and the mass of the fluid. For disk density, The thickness of the disk; It represents the dimensionless frequency, which is the relative ratio between the dynamic inertial force generated by the high-frequency vibration of the system and the gravity.

[0055] The velocity potential of different watersheds satisfies the following boundary conditions.

[0056] Free surface conditions: (8) Seabed impermeability conditions: (9) Kinematic continuity conditions for the upper and lower surfaces of an elastic disk: (10) Regarding the velocity potential expansion of Region I, the fluid velocity potential of this basin must strictly satisfy the governing equations mentioned above, satisfying the free surface condition equation (8) and the seabed impermeability condition equation (9). Based on the above boundary conditions, the eigenfunction expansion of the velocity potential of this basin can be obtained using the method of separation of variables: (11) in, (12) Represents the incident potential; In the formula For unknown coefficients, The modified Bessel function of the second kind describes the physical laws governing the radiation of waves towards infinity and their gradual decay. Let I be the vertical characteristic function of Region I. The phase factor describes the change in the circumferential direction. The vertical modal order is... Let be the order of the circular modes; For the incident wave amplitude, This is a modified Bessel function of the first kind, describing the radial undulation of the wave. The eigenvalues ​​of Region I are determined by the dispersion relation of Region I: (13) In the formula, The positive real roots satisfying the dispersion relation equation (13) correspond to the locally attenuated wave mode; while These are negative pure imaginary roots, corresponding to the traveling wave modes propagating towards the far field. The vertical eigenfunctions are: (14) in, Orthogonal normalization coefficients: (15) The vertical eigenfunction In water depth range This forms a complete set of functions that satisfy the following orthogonality relation. (16) Regarding the velocity potential expansion for Region II, the fluid velocity potential satisfies the free surface condition equation (8), the seabed condition equation (9), and the kinematic continuity condition equation (10) for the upper and lower surfaces of the elastic disk. Its general solution expansion is: (17) In the formula, For unknown coefficients, It is a modified Bessel function of the first kind. The phase factor varies in the circumferential direction. The vertical characteristic function of Region II. The eigenvalues ​​for Region II are determined by the dispersion relation of the submerged elastic disk: (18) Based on this eigenvalue, a piecewise vertical eigenfunction satisfying the boundary conditions of each water depth can be constructed: (19) In the formula, , representing the distance from the structure to the seabed. For typical plate parameters, the solution of the dispersion relation equation (18) can be divided into three groups in the complex plane: two negative pure imaginary roots corresponding to the propagating waves (denoted as ). and ); an infinite number of positive real roots corresponding to local decay modes (denoted as ); ); and two complex roots corresponding to the local bending deformation of the elastic disk (denoted as ); and Therefore, in the general solution equation (17), the characteristic mode summation subscript in the depth direction is... from Start the calculation.

[0057] The model in this paper adopts a fixed-edge condition; therefore, the elastic disk at the edge must satisfy the constraints of zero vertical displacement and zero radial rotation. Based on the kinematic relationships, the following conditions must be met: (20) (twenty one) In the formula, Let the radius of the elastic disk be . Let be the distance between the elastic disk and the free liquid surface. These are the eigenvalues ​​of Region II. For the first kind of modified Bessel function, and at the boundary The value at this point represents the radial amplitude of each wave mode at the edge. It is the first derivative of the modified Bessel function of the first kind. For the Region II vertical characteristic function with respect to depth The first derivative.

[0058] Based on the continuity of fluid, the velocity potentials of Region I and Region IIb are... Continuous existence at: (twenty two) Region I and Region IIa are continuous via a porous wall. According to the classical theory of wave interaction with thin porous walls proposed by Chwang and Allen T (1983), when fluid flows from Region IIa to Region I through the porous sidewall, the flow velocity is proportional to the pressure difference between the inside and outside of the pore, i.e.: (twenty three) After rearranging and simplifying, the velocity potential matching conditions for Region I and Region IIa, including the pressure jump compensation term, can be obtained: (twenty four) Integrating formulas (22) and (24), the velocity potential matching at full depth can be expressed as: (25) Substituting the velocity potential equation (11) of Region I and the velocity potential equation (17) of Region II into equation (25), we get: (26) Multiply both sides of equation (26) Integrating over the interval [-h,0], we can obtain the following using equation (16): (27) in: (28) (29) (30) (31) In addition, Region I and Region II are in The radial velocities are equal at the point where they exist. (32) Substituting the partial derivatives of equations (11) and (17) with respect to r into equation (32), we get: (33) Eliminate the coefficients of the unknown terms according to equations (27) and (33). get: (34) To numerically compute the analytical solution equations for the infinite-dimensional fluid-structure interaction described above, the spatial eigenvalue expansion series must be truncated with a finite number of terms. Due to the axisymmetric nature of the circular structure geometry, the angular modes along the circumference are decoupled from each other in the governing equations and boundary conditions. Therefore, the full-field problem can be transformed into solving the problem for each circumferential mode. Solve a series of linear algebraic equations independently.

[0059] In the numerical solution, the upper limit of modal cutoff in the circumferential direction is set as follows: For a given circular mode Furthermore, a unified truncation is applied to the vertical eigenfunctions of the water depth: the truncation number for the attenuated wave mode is set as... Based on this truncation, RegionI contains one traveling wave mode and... There are 10 attenuated wave modes, totaling 1000. In Region II, because the dispersion equation of the elastic disk contains two additional complex roots (physically corresponding to the local bending deformation of the elastic disk), the total number of its vertical modes is... item.

[0060] As mentioned earlier, at the flow field interface ( At point ), the front of the external watershed is taken sequentially. One vertical eigenfunction (i.e., taking the modal order) Multiplying this by the continuity equation and integrating it eliminates the coefficients of the external watershed. For each given... Both can derive an unknown coefficient for only the Region II watershed. Independent linear equations.

[0061] By order By performing the above matching, the fluid continuity conditions can provide a total of Several linearly independent equations. However, the Region II watershed system still contains... unknown coefficients To ensure system closure, the two additional governing equations are provided by the boundary conditions at the edge of the elastic disk. These are provided by the matching conditions. The three equations, together with the two equations provided by the disk edge conditions, constitute a equation concerning the Region II watershed. A system of linear algebraic equations with unknown coefficients. In practical calculations, for each truncated circular mode... It only requires independent assembly and solving for the size. * Solving the complex coefficient matrix Then the coefficients of the unknown terms in the external field can be calculated. Finally, by linearly superimposing the results from each mode, the hydrodynamic response of the entire field can be reconstructed. After calculating the unknown coefficients, relevant parameters, such as the horizontal excitation force, can be calculated. The calculation is based on the pressure difference between Region I and Region IIa, and its expression is as follows: (35) Total vertical force acting on the structure It is determined by the fluid pressure difference between the upper and lower surfaces of the disk, and its expression is as follows: (36) The deflection response of an elastic disk is determined by the kinematic boundary conditions of the fluid-structure interaction interface. At the physical interface where the structure and fluid are in contact ( The vertical velocity of the fluid particles must be continuous with the vertical velocity of the local structure (eq(6)). Based on this continuity condition, the complex amplitude distribution of the deflection of the elastic disk can be deduced from the local fluid velocity potential gradient. The formula for calculating the deflection amplitude of the disk is as follows: (37) The free surface elevation is a key physical quantity characterizing wave motion and the degree of wave rise. It is based on the dynamic boundary conditions of a linearized free surface, assuming a still water surface (…). The hydrodynamic pressure of the fluid at a given location is equal to atmospheric pressure, which allows us to establish the complex amplitude of the free surface elevation. The direct algebraic relationship with the fluid velocity potential at that location yields the expression for the free surface elevation amplitude: (38) Model validation: Before conducting a systematic analysis of parameter influences, the accuracy and convergence of the analytical model and numerical solution program established in this paper must be verified. This section will demonstrate the reliability of the model in this paper through series truncation convergence tests and degeneracy comparisons with classical analytical solutions.

[0062] When performing numerical solutions and convergence tests on this analytical model, it is necessary to set the vertical truncation number respectively. Number of cut-offs in the circumferential direction Due to the orthogonality of the circumferential characteristic functions and the asymptotic properties of the first-kind modified Bessel function at the origin, the series solutions of the macroscopic load and the central response of the system will naturally undergo modal truncation.

[0063] Specifically, the total vertical excitation force and the deflection at the center are only affected by the axisymmetric mode ( The total horizontal excitation force is controlled by only the first-order antisymmetric mode. Therefore, in evaluating the vertical cutoff number... When considering convergence, the calculation formulas for the corresponding physical quantities (Formulas 36, 36, 37) can be directly simplified to the following cutoff form: (39) (40) (41) The above equation shows that the contribution of higher-order angular modes is strictly zero when evaluating the overall load and the response at the center. Therefore, subsequent vertical modes... The convergence test can be selected independently. or As an evaluation indicator, the number of circumferential cutoffs was effectively excluded. Cross-interference. The basic geometric parameters fixed in the calculation are: , The dimensionless test parameters are defined as follows: (42) Regarding the convergence of the number of eigenmodes N in the vertical direction.

[0064] When calculating the vertical excitation force and the deflection at the center point, set... When calculating the horizontal excitation force, set The convergence of the vertical mode under different structural parameters was examined. The specific results are shown in Tables 1 and 2.

[0065] Table 1: In , , , Convergence of vertical excitation force, horizontal excitation force, and deflection at the center point of the structure under different cutoff numbers.

[0066] Table 2: In , , , Convergence of vertical excitation force, horizontal excitation force, and deflection at the center point of the structure under different cutoff numbers.

[0067] According to Tables 1 and 2, when the number of vertical modes reaches... At that time, the numerical precision of each dynamic response index reached four decimal places. In all subsequent numerical calculations, the vertical characteristic mode was uniformly selected. .

[0068] Regarding the circumferential orientation angle mode number Convergence.

[0069] Unlike the global stress, the local wave height at the wave-facing edge of the structure ( It exhibits strong non-uniformity in spatial distribution and requires the superposition of higher-order angular modes. Only by determining the vertical truncation number can a precise description be achieved. Therefore, once the vertical truncation number is determined... Based on this, this paper selects the dimensionless wave height at this location as the observation index to test the circumferential direction angle mode cutoff number. The convergence of the results is shown in Table 3.

[0070] Table 3: In , , , In this case Wave height measured against waves

[0071] As shown in Table 3, the wave number The larger the value, the more drastic the change in the flow field along the circumference, and the more angular modes are required to achieve convergence. When the cutoff number... At that time, the wave height values ​​under various high-frequency operating conditions had all tended to stabilize. Therefore, in subsequent calculations, the maximum angular mode number in the circumferential direction was determined to be... .

[0072] Model degradation verification.

[0073] Rigid Disk Degeneration Verification: To verify the accuracy of this analytical model, it is first degenerated into a classic submerged, impermeable rigid disk. During calculation, the equivalent permeability parameters of the sidewall boundaries are used. The dimensionless bending stiffness of the disk is set to a maximum value, making it equivalent to an approximately open boundary; at the same time, the dimensionless bending stiffness of the disk is set to a maximum value, making it equivalent to an indeformable rigid disk; and the mass effect of the disk is ignored (i.e., the mass parameter is taken). Under these limiting parameters, the model established in this study can be precisely degenerated into the submerged rigid disk physical model studied by Yu and Chwang (1993).

[0074] Please see Figure 3 (Left: Results of this model; Right: Yu and Chwang (1993)), showing the model degradation results. , , , , , A contour map of the dimensionless wave amplitude distribution in the watershed above the disk. To further verify this quantitatively, this paper selects three specific spatial locations on the edge of the disk: , as well as The dimensionless wavefront elevation amplitudes at these points were calculated and extracted. Please refer to [link / reference]. Figure 4 The above measuring points were plotted while maintaining a dimensionless constant water depth. Under the condition that the dimensionless wave height varies with the relative wavenumber parameter The variation curves are shown. Comparison reveals that the results of this scheme are in perfect agreement with the classical analytical data of Yu and Chwang (1993), verifying the accuracy of the model in amplitude calculation.

[0075] Furthermore, to verify the reliability of the force calculation, this paper further calculates the force at dimensionless water depth. Dimensionless potential , , , The dimensionless vertical excitation force on the degraded rigid submerged disk under the working condition was calculated and compared with the results of Zhao et al. (2017), as follows: Figure 5 As shown in the figure. The results prove the correctness of the calculation of the vertical excitation force.

[0076] Degeneracy verification of truncated cylinder: Equivalent permeability parameters of the sidewall boundary of the model in this study. The stiffness of a disk tends towards infinity. When approaching infinity, the structure resembles a truncated cylinder containing water. When calculating the horizontal and vertical excitation forces, neglecting the velocity potential of Region IIa, i.e., when using formulas (35) and (36), the formulas are modified as follows: (43) (44) The excitation force results of truncated cylinders with similar geometric dimensions can then be obtained.

[0077] The geometric dimensions are calculated here as follows: , , , , The vertical and horizontal excitation forces of the model under the following conditions, such as Figure 6 As shown (where (a) is a comparison and verification diagram of the horizontal excitation force of the model degenerated into a truncated cylinder and related research results; (b) is a comparison and verification diagram of the vertical excitation force of the model degenerated into a truncated cylinder and related research results), when the bending stiffness of the disk in the model of this paper approaches infinity and the equivalent permeation parameter of the sidewall boundary is set to a minimum value, the model of this paper successfully degenerates into a classic truncated rigid cylinder diffraction model. The calculated excitation force curve is in complete agreement with the semi-analytical solution results of Garrett (1970), verifying the accuracy of the model of this paper in calculating the horizontal and vertical wave excitation forces.

[0078] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for evaluating the hydrodynamic response of a flexible floating structure considering the equivalent effect of upper-layer water retention, characterized in that, Includes the following steps: S1. Establish a three-dimensional hydroelastic-porous sidewall coupled dynamic theoretical model: The flexible floating structure is equivalent to an elastic disk located below the still water surface, the water above the elastic disk is equivalent to a restricted overlying water layer, and the near-impermeable water-blocking boundary of the outer perimeter of the structure is equivalent to a sidewall boundary that satisfies the matching conditions of the porous sidewall. The equivalent permeability parameter of the sidewall boundary is set to a minimum value to simulate the water-blocking effect of the outer solid floating ring. S2. Divide the fluid region and define the velocity potential: Establish a cylindrical coordinate system with the center of the still water surface as the origin, divide the fluid domain into an outer region, an inner upper stagnant water region, and an inner lower water body region, and establish the velocity potential equations for each region based on the linear potential flow theory. S3. Determine the boundary conditions and eigenfunction expansion: Using the method of separation of variables, combined with the free liquid surface condition, the impermeable seabed condition, the kinematic continuity condition of the upper and lower surfaces of the elastic disk, and the dynamic equation of the elastic disk, the eigenfunction expansion of the velocity potential in each region is obtained respectively. S4. Perform flow domain matching and solution: At the interface of each region, velocity potential matching is performed using the fluid continuity condition and the porous sidewall matching condition containing pressure jump compensation term. Combined with the elastic disk edge constraint condition, a system of linear algebraic equations is constructed and solved to obtain the unknown coefficients of each mode. S5. Calculate hydrodynamic response parameters: Based on the solved unknown coefficients, calculate the wave excitation force of the main body of the flexible floating structure, the deflection response of the elastic disk, and / or the free surface response of the overlying water layer.

2. The method for evaluating the hydrodynamic response of a flexible floating structure considering the equivalent effect of upper-layer water retention, as described in claim 1, is characterized in that... In step S3, the dynamic equations of the elastic disk are described using Kirchhoff's thin-plate theory: in, The bending stiffness of the elastic disk. For a bitone operator, Let be the deflection of the elastic disk. For dynamic response time, Let be the mass per unit area of ​​the elastic disk. This is the net pressure difference applied to the elastic disk.

3. The method for evaluating the hydrodynamic response of a flexible floating structure considering the equivalent effect of upper-layer water retention, as described in claim 2, is characterized in that... The dynamic boundary conditions of the elastic disk are transformed into a form containing only velocity potential: in, For dimensionless stiffness, For dimensionless mass, For dimensionless frequency, The vertical coordinates are in cylindrical coordinates. The vertical distance between the elastic disk and the still water surface; The bending stiffness of the elastic disk. For water density, It is the acceleration due to gravity. The radius of the elastic disk, and These are the material density and thickness of the elastic disk, respectively. The incident wave angular frequency, and These are the complex velocity potentials of the upper stagnant water region and the lower water region, respectively.

4. The method for evaluating the hydrodynamic response of a flexible floating structure considering the equivalent effect of upper-layer water retention, as described in claim 1, is characterized in that... In step S2, the velocity potential expansion of the external region is: in, These are radial coordinates in cylindrical coordinates. The circumferential angle is in cylindrical coordinates. The vertical coordinates are in cylindrical coordinates. For the unknown coefficients in the expansion of the velocity potential in the external region. Let be the circumferential modal order. The vertical modal order is... This is a modified Bessel function of the second kind. For the vertical eigenvalues ​​of the external region, For circumferential modal functions, The vertical characteristic function of the external region. Let be the velocity potential of the incident wave.

5. The method for evaluating the hydrodynamic response of a flexible floating structure considering the equivalent effect of upper-layer water retention, as described in claim 1, is characterized in that... In step S2, the velocity potential expansions of the upper internal stagnant water region and the lower internal water region are as follows: in, These are radial coordinates in cylindrical coordinates. The circumferential angle is in cylindrical coordinates. The vertical coordinates in cylindrical coordinate system The unknown coefficients are the velocity potential expansions for the upper and lower water layers within the interior. Let be the circumferential modal order. The vertical modal order is... It is a modified Bessel function of the first kind. The depth-segmented vertical eigenfunctions for the upper and lower water layers are defined. For circumferential modal functions, The eigenvalues ​​are determined by the dispersion relation of the submerged elastic disk.

6. The method for evaluating the hydrodynamic response of a flexible floating structure considering the equivalent effect of upper-layer water retention, as described in claim 5, is characterized in that... The edge constraint condition of the elastic disk adopts the fixed edge condition, which satisfies: in, The radius of the elastic disk, The vertical distance between the elastic disk and the still water surface. The vertical position corresponding to the elastic disk; For the order of the circumferential mode summation, The order of the summation of vertical modes; The unknown coefficients are the velocity potential expansions for the upper and lower water layers within the interior. These are the eigenvalues ​​determined by the dispersion relation of the submerged elastic disk; This is a modified Bessel function of the first kind. This is the first derivative of the modified Bessel function of the first kind; The depth of the upper stagnant water region and the lower water body region is segmented into piecewise vertical eigenfunctions. Regarding depth The first derivative.

7. The method for evaluating the hydrodynamic response of a flexible floating structure considering the equivalent effect of upper-layer water retention, as described in claim 1, is characterized in that... In step S4, the condition for matching between the outer region and the inner upper stagnant water region through the porous sidewall is: in, and These are the complex velocity potentials of the outer region and the upper internal perched water region, respectively. These are radial coordinates in cylindrical coordinates. This is the partial derivative with respect to the radial coordinate; The imaginary unit, Let be the angular frequency of the incident wave. The dimensionless equivalent permeability parameter of the porous sidewall boundary. This represents the density of the water.

8. The method for evaluating the hydrodynamic response of a flexible floating structure considering the equivalent effect of upper-layer water retention, as described in claim 1, is characterized in that... In step S5, the hydrodynamic response parameters include the horizontal excitation force. Total vertical force : in, and These represent the circumferential components of the velocity potential in the outer region and the upper internal perched water region, respectively. and These are the vertical characteristic functions of the upper and lower water layers, respectively. The unknown coefficients are the velocity potential expansions of the circumferential zero-order corresponding upper and lower water layers. It is a zero-order modified Bessel function of the first kind; The imaginary unit, For water density, The incident wave angular frequency, The radius of the elastic disk, The vertical distance between the elastic disk and the still water surface. The eigenvalues ​​are determined by the dispersion relation of the submerged elastic disk. The vertical coordinates are in cylindrical coordinates. These are radial coordinates in cylindrical coordinates. This represents the number of segments cut off in the vertical direction.

9. The method for evaluating the hydrodynamic response of a flexible floating structure considering the equivalent effect of upper-layer water retention, as described in claim 1, is characterized in that... In step S5, the formula for calculating the deflection amplitude at the center of the elastic disk is: In the formula, Let be the deflection of the elastic disk. These are radial coordinates in cylindrical coordinates. The circumferential angle is in cylindrical coordinates. These are the vertical coordinates in the cylindrical coordinate system; The imaginary unit, The incident wave angular frequency; The unknown coefficients are the velocity potential expansions of the circumferential zero-order corresponding upper and lower water layers. For the vertical characteristic functions of the upper and lower water layers within the interior, with respect to the vertical coordinates, [the following is a list of partial derivatives of the functions]. The vertical position corresponding to the elastic disk. This represents the number of segments cut off in the vertical direction.

10. The method for evaluating the hydrodynamic response of a flexible floating structure considering the equivalent effect of upper-layer water retention, as described in claim 1, is characterized in that... It also includes step S6: truncating the spatial eigenvalue expansion series with a finite number of terms, setting the vertical truncation number as... The upper limit of the circumferential modal cutoff is The hydrodynamic response of the entire field is reconstructed through numerical calculation.

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