A high-efficiency hyperbolic rudder for ships and a modeling design method thereof

Abstract

The application relates to the ship technology field and discloses a marine hyperbolic high-efficiency rudder, which comprises a boss and a hyperbolic rudder, the hyperbolic rudder is installed on the boss, the boss center is located on the vertical center line of the hyperbolic rudder, the hyperbolic rudder comprises a rudder blade upper end surface and a rudder blade lower end surface, the rudder blade upper end surface and the rudder blade lower end surface are respectively the top and bottom of the hyperbolic rudder, and the hyperbolic rudder edge between the two is an arc-shaped part of a hyperbola. The marine hyperbolic high-efficiency rudder designed in the application ensures that the rudder blade chord length is unchanged along the length direction, the rudder blade upper end surface and the lower end surface thickness are unchanged, the rudder blade thickness chord length ratio changes along the length, the rudder body thickness near the boss center height is reduced, the rudder weight is reduced, the steel material is saved, the propelling efficiency under the same speed is improved, the rudder appendage resistance is reduced, and the demand for the main engine power is reduced.

Description

Technical Field

[0001] This invention relates to the field of marine technology, specifically to a high-efficiency hyperbolic rudder for ships and its modeling and design method. Background Technology

[0002] A rudder is a marine control device, and its design typically prioritizes its impact on ship maneuverability, specifically increasing the lift-to-drag ratio to enhance rudder efficiency. In the field of marine engineering, with the expansion of the "high-performance rudder" concept, rudder design now focuses not only on ship maneuverability but also on its impact on ship speed (including drag and propulsion efficiency). As a ship appendage, the rudder operates within the propeller wake, generating appendage drag. The interaction between the rudder, hull, and propeller determines the ship's speed performance during straight-line navigation. The propeller wake field at the stern is spatially uneven, exhibiting typical triaxial flow characteristics. Conventional rudders are usually symmetrical rudders with a certain airfoil profile and a fixed thickness-to-chord ratio along the span. For single-propeller, single-rudder ships, this chord line lies within the ship's mid-longitudinal section. From the perspective of the propeller wake field at the stern, this conventional rudder is not the optimal design for speed. With advancements in design capabilities, many high-performance rudders have been developed to maximize the ship's energy efficiency potential and improve its speed performance. Classified by design concept, it includes two aspects: increasing the lift-to-drag ratio to improve drag performance and changing the local flow angle of attack to improve propulsion efficiency.

[0003] In existing technologies, the main starting point is to improve propulsion efficiency by changing the local flow angle of attack. For example, the guide-edge twist rudder adapts to the direction of the incoming flow by twisting the guide edge, rectifyes the wake, suppresses the rotation of the wake, increases the axial induced velocity of the propeller, and generates additional thrust. This is equivalent to recovering the rotational kinetic energy of the wake, thereby improving propulsion efficiency. It is gradually being put into practical use on medium and large ships, especially large container ships with high speed and high fuel consumption, which have generally adopted the guide-edge twist rudder as the main energy-saving measure. Existing patent technologies are based on this, such as the "Design Method and Rudder of a Ship" disclosed in application number "CN115959256A", which proposes to divide the rudder blade into multiple sections along the extension direction of the rudder blade, at least one of which is an asymmetrical section, and adjust the angle of attack of the rudder blade at the location of the section according to the angle of attack of the incoming flow at the location of each section, so that the rudder can generate thrust in the forward direction when the hull moves forward, thereby improving the propulsion efficiency of the ship; the "Rudder with different angles of attack that is twisted up and down" disclosed in application number "KR1020220111365A" proposes to twist the upper and lower parts of the rudder in different directions to form an angle of attack to improve propulsion efficiency.

[0004] In summary, most existing technologies improve ship propulsion efficiency by changing the local angle of attack of the rudder blade profile, but this requires significant modifications to the overall shape and structure of the rudder, making the rudder shape very complex and unconventional. Summary of the Invention

[0005] The technical problem to be solved by the present invention is that the existing technology improves the ship propulsion efficiency by changing the local angle of attack of the rudder blade section, which makes the overall shape and structure of the rudder very complicated and unconventional.

[0006] To solve the above-mentioned technical problems, the technical solution of the present invention is to provide a marine hyperbolic high-efficiency rudder, including a propeller hub and a hyperbolic rudder. The hyperbolic rudder is mounted on the propeller hub, and the center of the propeller hub is located on the vertical center line of the hyperbolic rudder. The hyperbolic rudder includes an upper end face and a lower end face of the rudder blade. The upper end face and the lower end face of the rudder blade are respectively the top and bottom of the hyperbolic rudder. The edge of the hyperbolic rudder between the two is an arc shape that is part of a hyperbola. Above the point of minimum thickness on both sides of the hyperbolic rudder are the upper half branch of the port side and the upper half branch of the starboard side of the hyperbolic rudder. Below the point of minimum thickness are the lower half branch of the port side and the lower half branch of the starboard side of the hyperbolic rudder.

[0007] Optionally, the point where the thickness of the two sides of the hyperbolic rudder is minimum is located above the center of the propeller hub.

[0008] Optionally, a dummy rudder is connected to the upper end face of the rudder blade.

[0009] Optionally, both the upper and lower surfaces of the rudder blade are symmetrical about the vertical centerline of the hyperbolic rudder.

[0010] A modeling and design method for a high-efficiency hyperbolic rudder for marine applications includes the following steps:

[0011] S1. Define the height of the upper and lower end faces of the rudder blade and the thickness of the rudder blade.

[0012] S2. Define the minimum value of the maximum thickness of the hyperbolic rudder along the span direction and the location where the minimum value occurs;

[0013] S3. Calculate the thickness distribution of the rudder blades along the span direction for the port and starboard portions of the hyperbolic rudder.

[0014] S4. Calculate the thickness distribution along the span direction of a conventional rudder with a constant thickness chord ratio.

[0015] S5. Based on the thickness ratio of the hyperbolic rudder and the conventional rudder at the same height, perform unidirectional scaling modeling of the conventional rudder in the lateral direction.

[0016] S6. Obtain the geometric model of the hyperbolic rudder.

[0017] Optionally, in step S1, endpoints A and B, and endpoints D and C are defined on the upper and lower surfaces of the hyperbolic rudder blade, respectively, and the thickness of the upper surface of the rudder blade, i.e., the length of line segment AB, is set to the rudder blade thickness t. U; The height from the upper end face of the rudder blade to the center of the propeller hub is H U ; The thickness of the lower end face of the rudder blade, that is, the length of the line segment CD, is the thickness t of the rudder blade L ; The height from the lower end face of the rudder blade to the center of the propeller hub is H L .

[0018] Optionally, the one formed by the line segments AB, BC, CD, and DA is a conventional rudder

[0019] Optionally, in step S2, the endpoints E and F are defined as the vertices of the hyperbolas on both sides of the hyperbolic rudder, and the length of the line segment EF is the minimum value of the maximum thickness of the hyperbolic rudder along the span direction

[0020] Optionally, in step S3, the calculation method for the thickness distribution of the rudder blade of the left and right sides of the hyperbolic rudder along the span direction is as follows

[0021] S31. Set the vertical height from the vertices E and F of the hyperbola to the center of the propeller hub as H0 (-H L / 3 < H0 < H U / 3), and the thickness of the rudder blade of the conventional rudder at the height of the vertices E and F of the hyperbola is t M ;

[0022] S32. Set the thickness reduction ratio of the left side of the hyperbolic rudder at the vertex E compared with the conventional rudder as R1 (0 < R1 ≤ 2 / 3); the thickness reduction ratio of the right side of the hyperbolic rudder at the vertex F compared with the conventional rudder as R2 (0 < R2 ≤ 2 / 3);

[0023] S33. Calibrate the coordinates of each endpoint and vertex. Endpoint A: (-t U / 2, H U ); Endpoint B: (t U / 2, H U ); Endpoint C: (t L / 2, -H L ); Endpoint D: (-t L / 2, -H L ); Vertex E: (-(1 - R1)t M / 2, H0); Vertex F: ((1 - R2)t M / 2, H0);

[0024] S34. Obtain the function equations of the upper half branch of the left side of the hyperbolic rudder, the upper half branch of the right side of the hyperbolic rudder, the lower half branch of the left side of the hyperbolic rudder, and the lower half branch of the right side of the hyperbolic rudder according to the coordinates of each endpoint and vertex

[0025] Upper half branch of the left side of the hyperbolic rudder

[0026]

[0027] in,

[0028]

[0029] c 21 =H0

[0030] Hyperbolic rudder starboard upper branch:

[0031]

[0032] in,

[0033]

[0034]

[0035] Hyperbolic rudder lower port branch:

[0036]

[0037] in,

[0038]

[0039] c 23 =H0

[0040] Hyperbolic rudder starboard lower branch:

[0041]

[0042] in,

[0043]

[0044] c 24 =H0;

[0045] S35. Based on the above functional equations, calculate the thickness distribution of the rudder blades on the port and starboard sides along the span direction:

[0046] Arc BC:

[0047]

[0048] Arc DA:

[0049]

[0050] Optionally, in step S5, based on the thickness of the hyperbolic rudder and the conventional rudder at the same height along the span direction, the ratio of the thickness of the hyperbolic rudder to the thickness of the conventional rudder at the same height is calculated. Then, keeping the rudder blade chord length at each height of the conventional rudder unchanged, the conventional rudder is scaled unidirectionally in the y-direction according to the thickness ratio, thereby completing the modeling of the entire hyperbolic rudder.

[0051] In summary, the high-efficiency hyperbolic rudder designed in this invention has the same side shape as a conventional rudder, which is obtained by varying the thickness at different heights along the span of a conventional rudder. While ensuring that the rudder blade chord length remains constant along the span and the thickness of the upper and lower end faces of the rudder blade remains constant, the thickness-to-chord ratio of the rudder blade varies along the span, and the thickness-to-chord ratio and the profile height satisfy a hyperbolic function relationship. This reduces the thickness of the rudder body near the center height of the propeller hub, reduces the weight of the rudder, saves steel, improves the propulsion efficiency at the same speed, reduces the appendage drag of the rudder, and reduces the power requirement of the main engine. Attached Figure Description

[0052] Figure 1 This is a schematic diagram of the overall layout of the hyperbolic rudder of the present invention;

[0053] Figure 2 This is a schematic diagram of the hyperbolic rudder modeling process of the present invention;

[0054] Figure 3 A schematic diagram of the geometric model of a ship with a conventional rudder;

[0055] Figure 4 A schematic diagram of the geometric model of the hyperbolic rudder of the present invention for a ship;

[0056] In the diagram: 11. Dummy rudder; 14. Upper end face of rudder blade; 15. Lower end face of rudder blade; 21. Upper half branch of hyperbolic rudder on port side; 22. Upper half branch of hyperbolic rudder on starboard side; 23. Lower half branch of hyperbolic rudder on port side; 24. Lower half branch of hyperbolic rudder on starboard side; 3. Propeller hub. Detailed Implementation

[0057] The following combination Figure 1-4 The present invention will be described in further detail below.

[0058] This invention discloses a high-efficiency hyperbolic rudder for marine applications. By altering the rudder's thickness-to-chord ratio without changing the angle of attack of each rudder blade section, the thickness distribution of the rudder is changed along its span. This minimizes modifications to the rudder's shape while improving propulsion efficiency, maintaining a constant rudder area, reducing the impact on maneuverability, and decreasing the overall weight of the rudder. (Refer to...) Figure 1The system includes a rotor hub 3 and a hyperbolic rudder. The hyperbolic rudder is mounted on the rotor hub 3, with the center of the rotor hub 3 located on the vertical centerline of the hyperbolic rudder. The hyperbolic rudder includes an upper end face 14 and a lower end face 15 of the rudder blade. The upper end face 14 and the lower end face 15 of the rudder blade are the top and bottom of the hyperbolic rudder, respectively. The edge of the hyperbolic rudder between the two is an arc shape that is part of a hyperbola. Above the point of minimum thickness on both sides of the hyperbolic rudder are the upper half branch 21 on the port side and the upper half branch 22 on the starboard side of the hyperbolic rudder. Below the point of minimum thickness are the lower half branch 23 on the port side and the lower half branch 24 on the starboard side of the hyperbolic rudder. The point of minimum thickness on both sides of the hyperbolic rudder is located above the center of the rotor hub 3. A dummy rudder 11 is connected to the upper end face 14 of the rudder blade. Both the upper end face 14 and the lower end face 15 of the rudder blade are symmetrical about the vertical centerline of the hyperbolic rudder.

[0059] This invention also discloses a modeling and design method for a marine hyperbolic high-efficiency rudder, referring to... Figure 2 This includes the following steps:

[0060] S1. Define the height of the upper and lower end faces of the rudder blade and the thickness of the rudder blade.

[0061] Specifically, endpoints A and B, and endpoints D and C are defined on the upper end face 14 and lower end face 15 of the hyperbolic rudder, respectively, and the thickness of the upper end face 14, i.e., the length of line segment AB, is set to the rudder thickness t. U The height from the upper end face 14 of the rudder blade to the center of the propeller hub 3 is H. U The thickness of the lower end face of the rudder blade is 15 mm, which is the length of line segment CD, and the thickness of the rudder blade is t. L The height from the lower end face 15 of the rudder blade to the center of the propeller hub 3 is H. L The rudder is enclosed by line segments AB, BC, CD, and DA.

[0062] S2. Define the minimum value of the maximum thickness of the hyperbolic rudder along the span direction and the location where the minimum value occurs;

[0063] Specifically, the endpoints E and F are defined as the vertices of the hyperbolic rudder on both sides. The length of line segment EF is the minimum value of the maximum thickness of the hyperbolic rudder along the span direction.

[0064] S3. Calculate the thickness distribution of the rudder blades along the span direction for the port and starboard portions of the hyperbolic rudder.

[0065] S31. Set the vertical height from the vertices E and F of the hyperbola to the center of the rotor hub 3 as (H0-H). L / 3 <H0<H U / 3), the thickness of the conventional rudder blade at the heights of the hyperbola vertices E and F is t. M ;

[0066] S32. Set the thickness reduction ratio of the left - hand side part of the hyperbolic rudder at the vertex E compared to the conventional rudder as R1 (0 < R1 ≤ 2 / 3); set the thickness reduction ratio of the right - hand side part of the hyperbolic rudder at the vertex F compared to the conventional rudder as R2 (0 < R2 ≤ 2 / 3).

[0067] S33. Refer to Figure 1 , take the center of the propeller hub 3 as the origin, establish a coordinate system, with the upward direction as the positive y - axis direction and the right - hand direction as the positive x - axis direction, and mark the coordinates of each endpoint and vertex. Endpoint A: (-t U / 2, H U ); Endpoint B: (t U / 2, H U ); Endpoint C: (t L / 2, -H L ); Endpoint D: (-t L / 2, -H L ); Vertex E: (-(1 - R1)t M / 2, H0); Vertex F: ((1 - R2)t M / 2, H0), where point A and point B are symmetric about the longitudinal mid - plane of the hull (i.e., symmetric about the y - axis), and point C and point D are symmetric about the longitudinal mid - plane of the hull (i.e., symmetric about the y - axis).

[0068] S34. Obtain the function equations of the upper - half branch 21 (arc AE) of the left - hand side of the hyperbolic rudder, the upper - half branch 22 (arc BF) of the right - hand side of the hyperbolic rudder, the lower - half branch 23 (arc DE) of the left - hand side of the hyperbolic rudder, and the lower - half branch 24 (arc CF) of the right - hand side of the hyperbolic rudder according to the coordinates of each endpoint and vertex:

[0069] The upper - half branch 21 of the left - hand side of the hyperbolic rudder:

[0070]

[0071] Among them,

[0072]

[0073] c 21 = H0

[0074] The upper - half branch 22 of the right - hand side of the hyperbolic rudder:

[0075]

[0076] Among them,

[0077]

[0078] c 22 = H0

[0079] The lower - half branch 23 of the left - hand side of the hyperbolic rudder:

[0080]

[0081] in,

[0082]

[0083] c 23 =H0

[0084] Hyperbolic rudder starboard lower half branch 24:

[0085]

[0086] in,

[0087]

[0088] c 24 =H0;

[0089] S35. Based on the above functional equations, calculate the thickness distribution of the rudder blades on the port and starboard sides along the span direction:

[0090] Arc BC:

[0091]

[0092] Arc DA:

[0093]

[0094] S4. Calculate the thickness distribution along the span direction of a conventional rudder with a constant thickness chord ratio:

[0095] Given that the chord length distribution of a conventional rudder blade along its span is C = f(z), where z represents the span direction (from top to bottom), and the thickness-to-chord length ratio of a conventional rudder is a constant K = t / C = constant, multiplying the two formulas gives the thickness distribution of a conventional rudder with a constant thickness-to-chord length ratio along its span: t = f(z) × k.

[0096] S5. Based on the thickness ratio of the hyperbolic rudder and the conventional rudder at the same height, perform unidirectional scaling modeling of the conventional rudder in the lateral direction.

[0097] Specifically, based on the thickness distribution of the rudder blades on the port and starboard sides along the span direction obtained in steps S3 and S4, and the thickness distribution of the conventional rudder with the same thickness chord ratio along the span direction, the thickness of the hyperbolic rudder and the conventional rudder at the same height along the span direction is calculated. The thickness ratio of the hyperbolic rudder to the conventional rudder at the same height along the span direction is calculated. Then, keeping the rudder blade chord length at each height of the conventional rudder unchanged, the conventional rudder is scaled unidirectionally in the y-direction according to the thickness ratio, thereby completing the modeling of the entire hyperbolic rudder.

[0098] S6. Obtain the geometric model of the hyperbolic rudder.

[0099] Example

[0100] Reference Figure 3 and Figure 4 This embodiment specifically illustrates the invention. The hyperbolic high-efficiency rudder and conventional rudder of the invention are installed on a certain type of gas ship. The dummy rudder 11 of the two rudders are completely identical. The upper end face 14 and the lower end face 15 of the rudder blade are completely overlapped. The thickness of the hyperbolic rudder at the height of the propeller axis is reduced.

[0101] The rapid performance of the ship at its design draft and design speed was compared and calculated using the CFD viscous flow numerical calculation method. The calculation adopted the industry-recognized model-scale volume force self-propulsion calculation method and converted the actual ship's rapid performance according to a reasonable conversion method. When comparing the calculations of the two rudders, the calculation grid, parameters, and settings were kept consistent. When predicting the actual ship performance of the two rudders, the conversion process, method, and coefficients were kept consistent.

[0102] Table 1 compares the predicted high-speed performance of actual ships with two different rudder configurations:

[0103] Table 1. Energy-saving effect analysis of implementation cases

[0104] drinking water speed Improve efficiency effective power Received power conventional rudder 9.5m 16.5kn 0.747 8195.6kW 10976.8kW Hyperbolic rudder 9.5m 16.5kn 0.760 8173.4kW 10759.7kW In comparison / / +1.7％ -0.3％ -2.0％

[0105] As can be seen from the table, the hyperbolic rudder improves propulsion efficiency by 1.7% and reduces power consumption by 2% compared to the conventional rudder, resulting in significant energy savings.

[0106] It should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the technical solutions of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A marine hyperbolic high-efficiency rudder, characterized in that, It includes a hub (3) and a hyperbolic rudder. The hyperbolic rudder is installed on the hub (3), and the center of the hub (3) is located on the vertical center line of the hyperbolic rudder. The hyperbolic rudder includes an upper end face (14) of the rudder blade and a lower end face (15) of the rudder blade. The upper end face (14) and the lower end face (15) of the rudder blade are respectively the top and bottom of the hyperbolic rudder. The edge of the hyperbolic rudder between the two is an arc that is a part of a hyperbola. Above the minimum thickness at both sides of the hyperbolic rudder are the upper half branches (21) of the left side of the hyperbolic rudder and the upper half branches (22) of the right side of the hyperbolic rudder, and below the minimum thickness are the lower half branches (23) of the left side of the hyperbolic rudder and the lower half branches (24) of the right side of the hyperbolic rudder.

2. The marine hyperbolic high-efficiency rudder according to claim 1, characterized in that, The location where the minimum thickness at both sides of the hyperbolic rudder is located above the center of the hub (3).

3. The marine hyperbolic high-efficiency rudder according to claim 1, characterized in that, A false rudder (11) is connected to the upper end face (14) of the rudder blade.

4. The marine hyperbolic high-efficiency rudder according to claim 1, characterized in that, Both the upper end face (14) and the lower end face (15) of the rudder blade are symmetric about the vertical center line of the hyperbolic rudder.

5. A modeling and design method for a marine hyperbolic high-efficiency rudder, characterized in that, It includes the following steps: S1. Define the height of the upper end face and the lower end face of the rudder blade and the thickness of the rudder blade. S2. Define the minimum value of the maximum thickness of the hyperbolic rudder along the span direction and the position where the minimum value appears. S3. Calculate the thickness distribution of the rudder blade of the left and right sides of the hyperbolic rudder along the span direction. S4. Calculate the thickness distribution of a conventional rudder with an equal thickness chord ratio along the span direction. S5. According to the thickness ratio at the same height of the hyperbolic rudder and the conventional rudder, perform one-way scaling modeling on the conventional rudder in the lateral direction. S6. Obtain the geometric model of the hyperbolic rudder.

6. The modeling and design method for a marine hyperbolic high-efficiency rudder according to claim 5, characterized in that, In step S1, endpoints A and B, and endpoints D and C are defined on the upper end face (14) and lower end face (15) of the hyperbolic rudder, respectively, and the thickness of the upper end face (14), i.e., the length of line segment AB, is set to the rudder thickness t. U The height from the upper end face (14) of the rudder blade to the center of the propeller hub (3) is H. U The thickness of the lower end face (15) of the rudder blade, i.e., the length of line segment CD, is the rudder blade thickness t. L The height from the lower end face (15) of the rudder blade to the center of the propeller hub (3) is H. L .

7. The modeling and design method for a marine hyperbolic high-efficiency rudder according to claim 6, characterized in that, What is enclosed by line segment AB, line segment BC, line segment CD, and line segment DA is a conventional rudder.

8. The modeling and design method for a marine hyperbolic high-efficiency rudder according to claim 7, characterized in that, In step S2, the endpoints E and F are defined as the vertices of the hyperbola on both sides of the hyperbolic rudder, and the length of line segment EF is the minimum value of the maximum thickness of the hyperbolic rudder along the span direction.

9. The modeling and design method for a marine hyperbolic high-efficiency rudder according to claim 8, characterized in that, In step S3, the calculation method for the thickness distribution of the rudder blade of the left and right sides of the hyperbolic rudder along the span direction is as follows: S31. Set the vertical height from the vertices E and F of the hyperbola to the center of the hub (3) as H0 (-H L / 3 <H0<H U / 3), the thickness of the conventional rudder blade at the heights of the hyperbola vertices E and F is t. M ; S32. Set the thickness reduction ratio of the left side of the hyperbolic rudder at vertex E compared to the conventional rudder as R1 (0 < R1 ≤ 2 / 3); the thickness reduction ratio of the right side of the hyperbolic rudder at vertex F compared to the conventional rudder as R2 (0 < R2 ≤ 2 / 3). S33. Define the coordinates of each endpoint and vertex. Endpoint A: (-t) U / 2,H U Endpoint B: (t) U / 2,H U Endpoint C: (t) L / 2,-H L Endpoint D: (-t) L / 2,-H L Vertex E: (-(1-R1)t M / 2,H0); Vertex F: ((1-R2)t M / 2,H0); S34. According to the coordinates of each endpoint and vertex, obtain the function equations of the upper half branches (21) of the left side of the hyperbolic rudder, the upper half branches (22) of the right side of the hyperbolic rudder, the lower half branches (23) of the left side of the hyperbolic rudder, and the lower half branches (24) of the right side of the hyperbolic rudder: [[ID=]16]The upper half branches (21) of the left side of the hyperbolic rudder: Where, c 21 =H0 The upper half branches (22) of the right side of the hyperbolic rudder: Where, c 22 =H0 The lower half branches (23) of the left side of the hyperbolic rudder: Where, c 23 =H0 The lower half branches (24) of the right side of the hyperbolic rudder: Where, c 24 =H0; S35. According to the above function equations, calculate the thickness distribution of the rudder blade of the left and right sides along the span direction: Arc BC: Arc DA:

10. The modeling and design method for a marine hyperbolic high-efficiency rudder according to claim 9, characterized in that, In step S5, based on the thickness of the hyperbolic rudder and the conventional rudder at the same height along the span direction, the ratio of the thickness of the hyperbolic rudder to that of the conventional rudder at the same height along the span direction is calculated. Then, keeping the rudder blade chord length at each height of the conventional rudder unchanged, the conventional rudder is scaled unidirectionally in the y-direction according to the thickness ratio, thereby completing the modeling of the entire hyperbolic rudder.

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