Method for evaluating diffraction hydrodynamic force of concentric cylinder wave containing annular plate

Abstract

The invention discloses an annular plate-containing concentric cylinder wave diffraction hydrodynamic force evaluation method, an annular plate-containing concentric cylinder comprises an inner cylinder, an outer cylinder and a perforated annular plate arranged between the inner cylinder and the outer cylinder, the method is characterized by comprising the following steps: establishing a cylindrical coordinate system at the intersection point of a still water surface and the center of the cylinder, and the z axis is vertically upward; the drainage basin is divided, the outer area of the outer cylinder is set as the outer area, the annular space area between the inner cylinder and the outer cylinder is set as the inner area, and the inner area comprises the area above the annular plate and the area below the annular plate; setting an opening boundary condition, defining a porous effect parameter of the opening annular plate as sum, and setting a related speed potential boundary condition; deriving an outer domain velocity potential and an inner domain velocity potential, and coupling velocity potential solutions through inner and outer domain boundary conditions; and solving the hydrodynamic load and the free surface elevation according to the velocity potential. The invention provides a wave diffraction hydrodynamic force evaluation method without a plurality of characteristic roots for a concentric cylinder structure containing a single-layer horizontal perforated ring plate. According to the method, the inner domain is further divided into two drainage basins, so that the trouble of a plurality of feature roots is avoided, a brand new feature function is established, and the solving efficiency can be greatly improved.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of ocean engineering, and particularly relates to a method for evaluating wave diffraction hydrodynamic force of a concentric cylindrical wave with a ring plate. BACKGROUND

[0002] Breakwater is an important coastal engineering structure, usually built near the coastline or estuary, aiming to protect the port, wharf and coastline from the erosion of sea waves, swells and currents. These structures are usually built of concrete, rock or other solid materials, and their main function is to absorb, reduce or resist the impact force of sea waves, thereby protecting the land, facilities and buildings behind from the erosion of marine power. Double-cylinder structure is a classic breakwater design, which adopts a double-cylinder breakwater unit composed of concentric outer cylinder and inner cylinder, and reduces the stress on the inner cylinder through the protection of the outer cylinder. Under the premise of ensuring the strength of the inner cylinder structure, how to reduce the self-weight, improve the applicability and reduce the manufacturing and installation difficulty becomes a problem to be solved. In view of the above problems, we propose a double-cylinder breakwater unit with energy dissipation plate and a diffraction hydrodynamic force evaluation method (ZL2023114660595). The introduction of this diffraction hydrodynamic force evaluation method provides a scientific and reliable basis for water dynamic evaluation for engineering design. Engineering designers can optimize the structure according to the evaluation results to ensure the stability and reliability of the breakwater under various water dynamic loads.

[0003] However, the previously proposed diffraction hydrodynamic force evaluation method can solve the basic problems that have been proposed, but the method needs to solve complex eigenvalues for the treatment of the inner domain. Research shows that it is still not easy to quickly and accurately obtain complex eigenvalues within a considerable parameter range, so it is necessary to further optimize the design of this method. SUMMARY

[0004] In view of the problems existing in the prior art, the present application provides a method for evaluating the wave diffraction hydrodynamic force of a concentric cylindrical wave with a ring plate, which aims to solve the problem that the current method for evaluating the wave diffraction hydrodynamic force of a concentric cylindrical wave with a ring plate needs to be optimized.

[0005] The present application is implemented as follows: a method for evaluating the wave diffraction hydrodynamic force of a concentric cylindrical wave with a ring plate, the concentric cylindrical wave with a ring plate comprising an inner cylinder, an outer cylinder and an open ring plate arranged between the inner cylinder and the outer cylinder, characterized by comprising the following steps: Establishing a cylindrical coordinate system at the intersection of the still water surface and the center of the cylinder, wherein the z-axis is vertically upward; dividing the flow domain, and setting the outer region of the outer cylinder as the outer domain Setting the annular region between the inner cylinder and the outer cylinder as the inner domain, the inner domain comprising the region above the ring plate And the region below the ring plate ; the open-hole boundary condition is set, and the porous effect parameters of the open-hole ring plate are defined as and , and the relevant velocity potential boundary conditions are set; the outer domain velocity potential and the inner domain velocity potential are derived, the velocity potential solutions are coupled through the inner and outer domain boundary conditions; the hydrodynamic load and the free surface elevation are solved according to the velocity potential.

[0006] Preferably, the outer domain is composed of the linear superposition of the wave incident velocity potential and the diffraction velocity potential .

[0007] Preferably, the inner domain velocity potential and are respectively decomposed into vertical and horizontal components in the regions and .

[0008] Preferably, the region outside the vertical space where the outer cylinder is located is set as ; the annular gap region between the inner and outer cylinders is set as the inner domain, and the inner domain is divided into two flow regions and , wherein the region above the open-hole ring plate is set as , and the region below the open-hole ring plate is set as ; and are respectively the radii of the inner cylinder and the outer cylinder, h is the submergence depth of the open-hole ring plate fixed between the inner and outer cylinders.

[0009] Preferably, the open-hole boundary condition is composed of the following equations: (1) (2) (3) (4) (5) (6) wherein, is the wave number of the incident wave; g is the acceleration of gravity; represents the incident velocity potential; the symbols and respectively represent the upper surface and the lower surface of the open-hole ring plate; symbol and These represent the outer and inner surfaces of the outer cylinder, respectively. and These are the porosity parameters of the porous ring plate and the outer cylinder, respectively. when or When the value is close to 0, it indicates that the surface is completely impermeable; When their values ​​approach infinity, it indicates that the surface is completely open; Derivation of the outer velocity potential and the inner velocity potential: Outland velocity potential Wave incident velocity potential and diffraction velocity potential The linear superposition, i.e. (7) In the above formula, the wave incident velocity potential Represented as: (8) In the above formula, It is the acceleration due to gravity. It is the amplitude of a regularly incident wave. The angular frequency of the wave. Angular coordinates, It is the first category The first-order Bessel function is used to describe the radial ripple characteristics, where The incident wave number is... Radial coordinates, It is the imaginary unit; Let represent the external domain vertical characteristic function, i.e.: (9) In the above formula, The coordinates are vertical, pointing upwards. Water depth, which is the vertical distance from the water surface to the seabed. It is a hyperbolic cosine function; Outland diffraction velocity potential It can be represented as: (10) Substituting equations (8) and (10) into equation (7), we can obtain the outer domain. Total velocity potential for: (11) in, Indicates the first class Hankel function of order, Indicates the first class The modified Bessel function of order 1, These are unknown coefficients, and the external domain vertical characteristic function. Defined as: (12) in, These are the positive real roots of the dispersion relation shown below: (13) The inner region includes the watershed and For the watershed and The velocity potential can be decomposed into the following form: (14a) (14b) In the formula, For the region The vertical velocity potential component, For the region Horizontal velocity potential component, For the region The vertical velocity potential component, For the region The horizontal velocity potential component; velocity potential The boundary conditions for the impermeable plate surface, the free water surface, and the impermeable inner cylinder are as follows: (15a) (15b) (15c) The conditions for satisfying both the free water surface and the impermeability of the inner and outer cylinders are as follows: (16a) (16b) (16c) To satisfy the seabed conditions, the impermeability of the annular plate and the inner cylinder can be expressed as: (17a) (17b) (17c) The conditions for seabed conditions and the impermeability of the inner and outer cylinders must be met, namely: (18a) (18b) (18c) Furthermore, based on the above boundary conditions, the transmission condition equations (4)-(5) can be adjusted as follows: (19a) (19b) as well as: (20) velocity potential after decomposition It can be written as: (twenty one) It is the amplitude of a regularly incident wave. It is an unknown coefficient. It is a radial characteristic function. It is a vertical characteristic function; Radial characteristic function and vertical characteristic function They are respectively: (twenty two) (twenty three) in, These are characteristic roots. velocity potential It can be represented as: (twenty four) In the above formula, These are unknown coefficients, radial characteristic functions. It can be represented as: (25) in, It is an m-order von Neumann function; It is the first derivative of the m-th order von Neumann function; Eigenvalues It is the nth positive real root of the following equation: (26) velocity potential It can be represented as: (27) In the above formula, These are unknown coefficients; (28) It is a positive real root of the dispersion relation in the above equation; The radial characteristic function in formula (27) It can be represented as: (29) in yes m The first-order modified Bessel function of the order; It is the first derivative of the function; yes m The second-order modified Bessel function; It is its first derivative; The radial characteristic function in formula (27) It can be represented as: (30) velocity potential It can be represented as: (31) in, , These are unknown coefficients; The definition can be found in formula (25); S5. Solve for the velocity potential of the flow field: Based on the first equality sign of formulas (19a) and (19b), and the vertical characteristic function... In the interval The orthogonality of the equations gives us: (32) in, (33) (34) (35) (36) (37) Based on the second equality sign of equation (19a), and the vertical characteristic function In the interval The orthogonality yields: (38) in, (39) (40) (41) (42) Based on the second equality sign of equation (19b), and the vertical characteristic function In the interval The orthogonality yields: (43) in, (44) (45) (46) Based on the first equality sign of equation (20), and the radial characteristic function In the interval The orthogonality yields: (47) in, (48) Based on the second equality sign of equation (20), and the radial characteristic function In the interval The orthogonality yields: (49) in, (50) (51) (52) (53) By solving the system of linear equations (32)-(49), the unknown coefficients can be obtained. Thus, the velocity potential function of each watershed is obtained; S6. Solve for hydrodynamic loads and free surface elevation: Vertical excitation force of the perforated ring plate The solution can be obtained using the following formula: (54) Horizontal excitation force of inner cylinder Through The pressure at the wetted surface is integrated to obtain the result, i.e.: (55) Horizontal excitation force of outer cylinder Through The pressure difference over the wetted surface is integrated to obtain the result, i.e.: (56) Free surface elevation of the outer domain Calculated using the linear Bernoulli equation: (57) Inner Domain Free Surface Elevation Solve using the following formula: (58) Based on the design of breakwaters with concentric cylindrical structures, this invention proposes a wave diffraction hydrodynamic assessment method without complex eigenvalues. The advantages and effects of this method are as follows: First, by introducing a concentric cylindrical structure containing a single-layer horizontally perforated annular plate, this invention avoids the computational difficulties caused by solving for complex eigenvalues ​​in traditional methods. Building upon this, the invention further divides the inner domain into two watersheds, significantly simplifying the computational complexity of the evaluation method. This division not only avoids the trouble of complex eigenvalues ​​but also establishes entirely new eigenfunctions, resulting in a significant improvement in the solution efficiency of the wave diffraction hydrodynamic evaluation method.

[0010] Secondly, the new characteristic function can more accurately describe the wave diffraction hydrodynamic characteristics of concentric cylindrical structures, thereby improving the accuracy and reliability of the evaluation results. This improvement allows engineers to obtain more scientific and reliable hydrodynamic assessment data, helping to ensure the stability and reliability of breakwaters under various hydrodynamic loads.

[0011] Furthermore, the simplification and increased efficiency of this evaluation method provide greater flexibility in engineering design, facilitating the optimized design of breakwater structures. By eliminating the problems associated with complex eigenvalues, the evaluation method becomes easier to understand and apply in practical engineering designs. This will help reduce the self-weight of breakwater structures, improve their applicability, and reduce manufacturing and installation difficulties, thus providing a more sustainable and economical solution for coastal engineering. Attached Figure Description

[0012] Figure 1 This is a front view of the breakwater unit in this application; Figure 2 This is a top view of the breakwater unit in this application; Figure 3 This is a top view of the analytical model for diffraction hydrodynamic analysis in this application; Figure 4 This is a side view of the analytical model of the diffraction hydrodynamic analysis theory in this application. Detailed Implementation

[0013] 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 and not intended to limit the invention.

[0014] To address the shortcomings of current methods for evaluating the hydrodynamic effects of wave diffraction in concentric cylinders containing ring plates, this invention provides a new method for evaluating the hydrodynamic effects of wave diffraction in concentric cylinders containing ring plates. This method not only simplifies computational complexity and improves evaluation efficiency and accuracy, but also provides greater flexibility in engineering design, promoting the optimized design and sustainable development of breakwater structures. To further illustrate the structure of this invention, a detailed description is provided below in conjunction with the accompanying drawings: Please see Figure 1 and Figure 2 A double-cylindrical breakwater unit containing an energy-dissipating plate includes an outer cylinder 1 and an inner cylinder 2 concentrically located within the outer cylinder. An annular gap is formed between the outer and inner cylinders. An annular energy-dissipating plate is disposed within the annular gap. The annular energy-dissipating plate 3 has energy-dissipating permeable holes 3-1, which connect to the annular gaps on the upper and lower sides of the annular energy-dissipating plate. The energy-dissipating permeable holes are circular. The outer cylinder has circular external permeable holes 1-1.

[0015] In this embodiment, the diffraction hydrodynamic assessment method for a double-cylindrical breakwater unit containing an energy dissipation plate adopts the following steps: S1. Establish a coordinate system.

[0016] Establish such as Figure 3 and Figure 4 The coordinate system shown is used to establish a cylindrical coordinate system at the intersection of the still water surface and the center of the cylinder. ,in z The axis is vertically upward. H This indicates the water depth. The radii of the inner and outer cylinders are respectively... and The submerged depth of the perforated annular plate fixed between the inner and outer cylinders is... h .

[0017] S2. Divide the watershed. Define the region outside the outer cylinder as the outer domain. The annular gap region between the inner and outer cylinders is defined as the inner region, which includes the area above the annular plate. and the area below the ring plate .

[0018] The area outside the vertical space where the outer cylinder is located Set as an outer domain, called The annular gap region between the inner and outer cylinders is designated as the inner domain, which is then divided into two flow domains. and The area above the perforated ring plate Set as The area below the perforated ring plate Set as .

[0019] S3. Set the opening boundary conditions.

[0020] Define the porosity parameter of the perforated ring plate as follows: and The relevant velocity potential boundary conditions are set; the outer domain velocity potential and the inner domain velocity potential are derived, and the velocity potential solution is coupled through the boundary conditions of the inner and outer domains; the hydrodynamic load and free surface elevation are solved based on the velocity potential. The outer domain... The total velocity potential is derived from the wave incident velocity potential. With diffraction velocity potential It is composed of linear superposition. Inner domain velocity potential. and In the respective regions and It is decomposed into vertical and horizontal components.

[0021] The boundary conditions for the opening consist of the following equations: (1) (2) (3) (4) (5) (6) in, It is the wave number of the incident wave; g is the acceleration due to gravity; Represents the incident velocity potential; symbol and These represent the upper and lower surfaces of the perforated ring plate, respectively. symbol and These represent the outer and inner surfaces of the outer cylinder, respectively. and These are the porosity parameters of the porous ring plate and the outer cylinder, respectively. when or When the value is close to 0, it indicates that the surface is completely impermeable; When their values ​​approach infinity, it indicates that the surface is completely open; Derivation of the outer velocity potential and the inner velocity potential: Outland velocity potential Wave incident velocity potential and diffraction velocity potential The linear superposition, i.e. (7) In the above formula, the wave incident velocity potential Represented as: (8) In the above formula, It is the acceleration due to gravity. It is the amplitude of a regularly incident wave. The angular frequency of the wave. Angular coordinates, It is the first category The first-order Bessel function is used to describe the radial ripple characteristics, where The incident wave number is... Radial coordinates, It is the imaginary unit; Let represent the external domain vertical characteristic function, i.e.: (9) In the above formula, The coordinates are vertical, pointing upwards. Water depth, which is the vertical distance from the water surface to the seabed. It is a hyperbolic cosine function; Outland diffraction velocity potential It can be represented as: (10) Substituting equations (8) and (10) into equation (7), we can obtain the outer domain. Total velocity potential for: (11) in, Indicates the first type Hankel function of order, Indicates the first type The modified Bessel function of order 1, These are unknown coefficients, and the external domain vertical characteristic function. Defined as: (12) in, These are the positive real roots of the dispersion relation shown below: (13) The inner region includes the watershed and For the watershed and The velocity potential can be decomposed into the following form: (14a) (14b) In the formula, For the region The vertical velocity potential component, For the region The horizontal velocity potential component, For the region The vertical velocity potential component, For the region The horizontal velocity potential component; velocity potential The boundary conditions for the impermeable plate surface, the free water surface, and the impermeable inner cylinder are as follows: (15a) (15b) (15c) The conditions for satisfying both the free water surface and the impermeability of the inner and outer cylinders are as follows: (16a) (16b) (16c) To satisfy the seabed conditions, the impermeability of the annular plate and the inner cylinder can be expressed as: (17a) (17b) (17c) The conditions for seabed conditions and the impermeability of the inner and outer cylinders must be met, namely: (18a) (18b) (18c) Furthermore, based on the above boundary conditions, the transmission condition equations (4)-(5) can be adjusted as follows: (19a) (19b) as well as: (20) velocity potential after decomposition It can be written as: (twenty one) It is the amplitude of a regularly incident wave. It is an unknown coefficient. It is a radial characteristic function. It is a vertical characteristic function; Radial characteristic function and vertical characteristic function They are respectively: (twenty two) (twenty three) in, These are characteristic roots. velocity potential It can be represented as: (twenty four) In the above formula, These are unknown coefficients, radial characteristic functions. It can be represented as: (25) in, It is an m-order von Neumann function; It is the first derivative of the m-th order von Neumann function; Eigenvalues It is the nth positive real root of the following equation: (26) velocity potential It can be represented as: (27) In the above formula, These are unknown coefficients; (28) It is a positive real root of the dispersion relation in the above equation; The radial characteristic function in formula (27) It can be represented as: (29) in yes m The first-order modified Bessel function of order; It is the first derivative of the function; yes mThe second-order modified Bessel function; It is its first derivative; The radial characteristic function in formula (27) It can be represented as: (30) velocity potential It can be represented as: (31) in, , These are unknown coefficients; The definition can be found in formula (25); S5. Solve for the velocity potential of the flow field: Based on the first equality sign of formulas (19a) and (19b), and the vertical characteristic function... In the interval The orthogonality of the equations gives us: (32) in, (33) (34) (35) (36) (37) Based on the second equality sign of equation (19a), and the vertical characteristic function In the interval The orthogonality yields: (38) in, (39) (40) (41) (42) Based on the second equality sign of equation (19b), and the vertical characteristic function In the interval The orthogonality yields: (43) in, (44) (45) (46) Based on the first equality sign of equation (20), and the radial characteristic function In the interval The orthogonality yields: (47) in, (48) Based on the second equality sign of equation (20), and the radial characteristic function In the interval The orthogonality yields: (49) in, (50) (51) (52) (53) By solving the system of linear equations (32)-(49), the unknown coefficients can be obtained. Thus, the velocity potential function of each watershed is obtained; S6. Solve for hydrodynamic loads and free surface elevation: Vertical excitation force of the perforated ring plate The solution can be obtained using the following formula: (54) Horizontal excitation force of inner cylinder Through The pressure at the wetted surface is integrated to obtain the result, i.e.: (55) Horizontal excitation force of outer cylinder Through The pressure difference over the wetted surface is integrated to obtain the result, i.e.: (56) Free surface elevation of the outer domain Calculated using the linear Bernoulli equation: (57) Inner Domain Free Surface Elevation Solve using the following formula: (58) 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 effects of wave diffraction in a concentric cylinder containing an inner ring plate, wherein the concentric cylinder includes an inner cylinder, an outer cylinder, and an open ring plate disposed between the inner and outer cylinders, characterized in that... Includes the following steps: Establish a cylindrical coordinate system at the intersection of the still water surface and the center of the cylinder, with the z-axis pointing vertically upwards; divide the watershed, defining the outer region of the outer cylinder as the outer domain. The annular gap region between the inner and outer cylinders is defined as the inner region, which includes the area above the annular plate. and the area below the ring plate ; Set the opening boundary conditions and define the porosity effect parameters of the opening ring plate as follows: and The relevant velocity potential boundary conditions are set; the velocity potential in the outer domain and the velocity potential in the inner domain are derived, and the velocity potential solution is coupled through the boundary conditions in the inner and outer domains; the hydrodynamic load and the elevation of the free surface are solved based on the velocity potential.

2. The method for evaluating wave diffraction hydrodynamics of a concentric cylindrical structure containing an inner ring plate according to claim 1, characterized in that: The outer domain The total velocity potential is derived from the wave incident velocity potential. With diffraction velocity potential It is composed of linear superposition.

3. The method for evaluating wave diffraction hydrodynamics of a concentric cylindrical structure containing an inner ring plate according to claim 1, characterized in that: Inner Domain Velocity Potential and In the respective regions and It is decomposed into vertical and horizontal components.

4. The method for evaluating wave diffraction hydrodynamics of a concentric cylinder containing an inner ring plate according to claim 1, characterized in that: Set the area outside the vertical space where the outer cylinder is located as The annular gap region between the inner and outer cylinders is designated as the inner domain, which is then divided into two flow domains. and The area above the perforated ring plate is set as The area below the perforated ring plate is set as ; and These are the radii of the inner and outer cylinders, respectively. h The submersion depth of the perforated annular plate fixed between the inner and outer cylinders.

5. The method for evaluating wave diffraction hydrodynamics of a concentric cylindrical structure containing an inner ring plate according to claim 1, characterized in that: The boundary conditions for the opening consist of the following equations: (1) (2) (3) (4) (5) (6) in, It is the wave number of the incident wave; g is the acceleration due to gravity; Represents the incident velocity potential; symbol and These represent the upper and lower surfaces of the perforated ring plate, respectively. symbol and These represent the outer and inner surfaces of the outer cylinder, respectively. and These are the porosity parameters of the porous ring plate and the outer cylinder, respectively. when or When the value is close to 0, it indicates that the surface is completely impermeable; When their values ​​approach infinity, it indicates that the surface is completely open; Derivation of the outer velocity potential and the inner velocity potential: Outland velocity potential Wave incident velocity potential and diffraction velocity potential The linear superposition, i.e. (7) In the above formula, the wave incident velocity potential Represented as: (8) In the above formula, It is the acceleration due to gravity. It is the amplitude of a regularly incident wave. The angular frequency of the wave. Angular coordinates, It is the first category The first-order Bessel function is used to describe the radial ripple characteristics, where The incident wave number is... Radial coordinates, It is the imaginary unit; Let represent the external domain vertical characteristic function, i.e.: （9） In the above formula, The coordinates are vertical, pointing upwards. Water depth, which is the vertical distance from the water surface to the seabed. It is a hyperbolic cosine function; Outland diffraction velocity potential It can be represented as: (10) Substituting equations (8) and (10) into equation (7), we can obtain the outer domain. Total velocity potential for: (11) in, Indicates the first type Hankel function of order, Indicates the first type The modified Bessel function of order 1, These are unknown coefficients, and the external domain vertical characteristic function. Defined as: (12) in, These are the positive real roots of the dispersion relation shown below: (13) The inner region includes the watershed and For the watershed and The velocity potential can be decomposed into the following form: (14a) (14b) In the formula, For the region The vertical velocity potential component, For the region Horizontal velocity potential component, For the region The vertical velocity potential component, For the region The horizontal velocity potential component; velocity potential The boundary conditions for the impermeable plate surface, the free water surface, and the impermeable inner cylinder are as follows: (15a) (15b) (15c) The conditions for satisfying both the free water surface and the impermeability of the inner and outer cylinders are as follows: (16a) (16b) (16c) To satisfy the seabed conditions, the impermeability of the annular plate and the inner cylinder can be expressed as: (17a) (17b) (17c) The conditions for seabed conditions and the impermeability of the inner and outer cylinders must be met, namely: (18a) (18b) (18c) Furthermore, based on the above boundary conditions, the transmission condition equations (4)-(5) can be adjusted as follows: (19a) (19b) as well as: (20) velocity potential after decomposition It can be written as: (21) It is the amplitude of a regularly incident wave. It is an unknown coefficient. It is a radial characteristic function. It is a vertical characteristic function; radial characteristic function and vertical characteristic function They are respectively: (22) (23) in, These are characteristic roots. velocity potential It can be represented as: (24) In the above formula, These are unknown coefficients, radial characteristic functions. It can be represented as: (25) in, It is an m-order von Neumann function; It is the first derivative of the m-th order von Neumann function; Eigenvalues It is the nth positive real root of the following equation: (26) velocity potential It can be represented as: (27) In the above formula, These are unknown coefficients; （28） It is a positive real root of the dispersion relation in the above equation; The radial characteristic function in formula (27) It can be represented as: （29） in yes m The first-order modified Bessel function of order; It is the first derivative of the function; yes m The second-order modified Bessel function; It is its first derivative; The radial characteristic function in formula (27) It can be represented as: （30） velocity potential It can be represented as: (31) in, , These are unknown coefficients; The definition can be found in formula (25); S5. Solve for the velocity potential of the flow field: Based on the first equality sign of formulas (19a) and (19b), and the vertical characteristic function... In the interval The orthogonality of the equations gives us: (32) in, （33） （34） （35） （36） （37） Based on the second equality sign of equation (19a), and the vertical characteristic function In the interval The orthogonality yields: （38） in, （39） （40） （41） （42） Based on the second equality sign of equation (19b), and the vertical characteristic function In the interval The orthogonality yields: （43） in, （44） （45） （46） Based on the first equality sign of equation (20), and the radial characteristic function In the interval The orthogonality yields: （47） in, （48） Based on the second equality sign of equation (20), and the radial characteristic function In the interval The orthogonality yields: （49） in, （50） （51） （52） （53） By solving the system of linear equations (32)-(49), the unknown coefficients can be obtained. Thus, the velocity potential function of each watershed is obtained; S6. Solve for hydrodynamic loads and free surface elevation: Vertical excitation force of the perforated ring plate The solution can be obtained using the following formula: (54) Horizontal excitation force of inner cylinder Through The pressure at the wetted surface is integrated to obtain the result, i.e.: (55) Horizontal excitation force of outer cylinder Through The pressure difference over the wetted surface is integrated to obtain the result, i.e.: (56) Free surface elevation of the outer domain Calculated using the linear Bernoulli equation: (57) Inner Domain Free Surface Elevation Solve using the following formula: (58)。

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