Contra-rotating open type rotor numerical simulation method and device considering nacelle coupling

Abstract

The invention belongs to the technical field of aero-engine simulation, and particularly relates to a contra-rotating open type rotor numerical simulation method and device considering nacelle coupling. The method comprises the following steps: calculating an induced velocity vector generated by a nacelle at a lift line position of two rows of paddle fan blades; according to the induced velocity vector of each propeller fan, the self-induced velocity of each propeller fan and the induced velocity between the two propeller fans, the total induced velocity of each propeller fan is calculated, and then the lift coefficient and the resistance coefficient are calculated based on the attack angle; and then circular rector distribution is determined, iterative loop control is carried out according to the convergence of the circular rector distribution, and in the loop process, the self-induced velocity of each propeller fan and the induced velocity between the two propeller fans are updated based on the circular rector distribution. The method can achieve the calculation of the characteristics of the contra-rotating propeller fan in a full working condition range, is high in calculation precision, and is high in calculation speed.

Description

Technical Field

[0001] This application belongs to the field of aero-engine simulation technology, and particularly relates to a numerical simulation method and device for a contra-rotating open rotor considering nacelle coupling. Background Art

[0002] As a new generation of high-efficiency and energy-saving green power, the open rotor engine has always been a research hotspot in recent years. During the R & D process of the open rotor engine, the overall performance design model is the basis for carrying out the overall engine matching research and component design. Therefore, in the preliminary design stage, establishing an overall performance design model of the open rotor engine with high accuracy, small calculation amount, high integration, and stable numerical calculation helps to accelerate the design process and is an urgent need in the field of future aircraft power plant design.

[0003] Currently, the overall simulation method of the open rotor engine is mainly based on the component-level zero-dimensional model. Although the zero-dimensional method based on the characteristic map can achieve the rapid calculation of the overall performance of the open rotor engine, this method has certain limitations: ① It is extremely dependent on the propeller-fan characteristics, and there are few publicly published test characteristic maps, and the tested characteristic maps are also relatively sparse, making it difficult to ensure the accuracy of model calculation. ② The semi-empirical model between the two rows of propeller-fans cannot fully reflect the complex interference effects between the contra-rotating propeller-fans, thus bringing certain errors. ③ For different propeller-fan design parameters and different working conditions, it is usually necessary to scale and interpolate the characteristic map, thus introducing additional errors, especially at off-design point conditions, the errors are very large. ④ The zero-dimensional model cannot reflect the influence of design parameters such as blade geometry, number of blades, diameter, rotational speed, and pitch angle on the aerodynamic performance, and can only be used as a module for evaluating thrust and cannot be used for the design of contra-rotating propeller-fans. In summary, the calculation accuracy of the contra-rotating propeller-fan model based on the single-propeller characteristic map is limited.

[0004] In order to improve the model accuracy, some scholars have used CFD technology to achieve three-dimensional calculation of the contra-rotating propeller-fan. However, three-dimensional simulation requires a large amount of computing resources and computing time, and it takes more than one hour to calculate a single point. Therefore, three-dimensional simulation is mainly used for mechanism analysis and is not suitable for the rapid design requirements in the preliminary design stage of the open rotor engine. Summary of the Invention

[0005] To solve the above problems, this application provides a numerical simulation method and device for a contra-rotating open rotor considering nacelle coupling, taking into account both calculation efficiency and calculation accuracy.

[0006] The first aspect of this application provides a numerical simulation method for a contra-rotating open rotor considering nacelle coupling, mainly including: Step S1: Divide each blade of the contra-rotating propeller-fan into multiple blade elements along the span direction, construct a three-dimensional model of each blade according to the input airfoil parameters, and determine the position coordinates of the lift line of each propeller-fan; Step S2: Calculate the induced velocity vector generated by the nacelle on the blades of the two rows of contra-rotating fans at the lift line positions based on the air flow velocity at the inlet of the contra-rotating fans, the grid point coordinates of the nacelle geometry, and the position coordinates of all lift lines of the two rows of fans. Step S3: Calculate the total induced velocity of each fan based on the air flow velocity at the inlet of the contra-rotating fans, the rotational speeds of the two rows of fans respectively, the induced velocity vector of the nacelle on each fan, the self-induced velocity of each fan, and the induced velocity between the two fans, where the initial values of the self-induced velocity of each fan and the induced velocity between the two fans are set to 0. Step S4: Perform coordinate transformation on the total induced velocity of the two fans to determine the resultant velocity, chordwise component velocity, and normal component velocity of each blade element. Step S5: Determine the angle of attack distribution of each blade element of the two rows of contra-rotating fans based on the chordwise component velocity and normal component velocity of each blade element. Step S6: Determine the lift coefficient and drag coefficient corresponding to the angle of attack of each blade element in the two-dimensional airfoil lift-drag characteristic database. Step S7: Determine the circulation distribution of the blade elements according to the lift coefficient and the resultant velocity of the blade elements. Step S8: Judge whether the circulation of each blade element converges. After the circulation of each blade element in the iterative calculation converges, calculate the performance of the contra-rotating fans according to the circulation distribution of the blade elements; otherwise, recalculate the new circulation distribution of the blade elements according to the relaxation factor. Step S9: Determine the induced velocity generated by each vortex filament segment of each wake vortex of each blade at the control point of each blade element according to the new circulation of the blade elements, so as to calculate the induced velocity of each wake vortex of each fan in the axial direction of the fan. Step S10: Determine the geometric coordinates of each vortex filament segment according to the induced velocity of the wake vortex filament in the axial direction of the fan, the air flow velocity at the inlet of the contra-rotating fans, and the rotational speed of the rotating shaft. Step S11: Calculate the self-induced geometric coefficient and mutual-induced geometric coefficient of each fan of the contra-rotating fans based on the lift line position coordinates of the two fans and the geometric coordinates of each vortex filament segment of the two fans. Step S12: Determine the self-induced velocity of each fan and the induced velocity between the two fans based on the self-induced geometric coefficient, the mutual-induced geometric coefficient, and the new circulation distribution of the blade elements of the two rows of fans, and return to Step S3, and loop to execute Step S3 - Step S12 until the circulation of the blade elements in the iterative calculation converges.

[0007] Preferably, in Step S3, the total induced velocity of each fan is calculated by the following formula: ; where is the total induced velocity of the i-th blade element of the front row of fans, is the total induced velocity of the i-th blade element of the rear row of fans, is the air flow velocity at the inlet of the contra-rotating propeller fan, is the rotational speed of the front row propeller fan's rotating shaft, is the rotational speed of the rear row propeller fan's rotating shaft, is the distance from the i-th blade element of the front row propeller fan to the rotating shaft of the front row propeller fan, is the distance from the i-th blade element of the rear row propeller fan to the rotating shaft of the rear row propeller fan, is the induced velocity vector of the nacelle on the i-th blade element of the front row propeller fan, is the induced velocity vector of the nacelle on the i-th blade element of the rear row propeller fan, is the self-induced velocity of the front row propeller fan at its i-th blade element, is the induced velocity of the rear row propeller fan at the i-th blade element of the front row propeller fan, is the self-induced velocity of the rear row propeller fan at its i-th blade element, is the induced velocity of the front row propeller fan at the i-th blade element of the rear row propeller fan.

[0008] Preferably, in step S8, when the norm of the difference between the circulation of each blade element calculated in the current iteration and the circulation of each blade element calculated in the previous iteration is less than or equal to 10 -6 , it is determined that the circulation of the blade element calculated by the iterative calculation converges; otherwise, a new circulation of the blade element is calculated through the following formula : ; wherein, is the circulation of the blade element calculated in the previous iteration, is the circulation of the blade element calculated in the current iteration, is the relaxation factor.

[0009] Preferably, in step S8, the performance of the contra-rotating propeller fan is calculated through the following steps: Step S81: Calculate the lift and drag of each blade element according to the circulation of the blade element, the resultant velocity, the lift coefficient and the drag coefficient; Step S82: Calculate the component forces of each blade element in each direction according to the lift and drag of each blade element; Step S83: Obtain the thrust and torque of the front row propeller fan and the rear row propeller fan through integration.

[0010] Preferably, in step S10, the geometric coordinates of each vortex filament segment in the cylindrical coordinate system are determined through the following formula , , : ; wherein, is the radius position of the j-th segment of the k-th wake vortex filament, is the radius position of the i-th blade element, is the azimuth angle of the wake vortex filament, is the number of turns allowing the wake vortex filament to develop, is the number of segments into which the wake vortex filament is divided, is the number of each segment into which the wake vortex filament is divided, is the rotational speed of the shaft of the propeller fan, is the induced velocity of the wake vortex filament in the axial direction of the propeller fan, is the parameter for controlling the grid level.

[0011] The second aspect of the present application provides a contra-rotating open rotor numerical simulation device considering nacelle coupling, mainly including: A lift line determination module, configured to divide each blade of the contra-rotating propeller fan into multiple blade elements along the span direction, construct a three-dimensional model of each blade according to the input airfoil parameters, and determine the position coordinates of the lift line of each propeller fan; A nacelle induced velocity vector calculation module, configured to calculate the induced velocity vector generated by the nacelle on the blades of the two rows of propeller fans at the lift line position according to the air flow velocity at the inlet of the contra-rotating propeller fan, the grid point coordinates of the nacelle geometry, and the position coordinates of all lift lines of the two rows of propeller fans; A total induced velocity calculation module, configured to calculate the total induced velocity of each propeller fan according to the air flow velocity at the inlet of the contra-rotating propeller fan, the rotational speeds of the two rows of propeller fans respectively, the induced velocity vector of the nacelle on each propeller fan, the self-induced velocity of each propeller fan, and the induced velocity between the two propeller fans, wherein the initial values of the self-induced velocity of each propeller fan and the induced velocity between the two propeller fans are set to 0; A blade element velocity determination module, configured to perform coordinate transformation on the total induced velocity of the two propeller fans to determine the resultant velocity, chordwise component velocity, and normal component velocity of each blade element; A blade element angle of attack determination module, configured to determine the angle of attack distribution of each blade element of the two rows of propeller fans based on the chordwise component velocity and normal component velocity of each blade element; A lift coefficient and drag coefficient determination module, configured to determine the lift coefficient and drag coefficient corresponding to the angle of attack of each blade element in the two-dimensional airfoil lift-drag characteristic database; A blade element circulation determination module, configured to determine the blade element circulation distribution according to the lift coefficient and the resultant velocity of the blade element; A circulation control module, configured to determine whether the circulation of each blade element converges. After the circulation of each blade element converges in the iterative calculation, calculate the performance of the contra-rotating propeller fan according to the blade element circulation distribution, otherwise recalculate the new blade element circulation distribution according to the relaxation factor; A propeller fan axial direction induced velocity determination module, configured to determine the induced velocity generated by each vortex filament segment of each wake vortex filament of each blade at the control point of each blade element according to the new blade element circulation, so as to calculate the induced velocity of each wake vortex filament of each propeller fan in the axial direction of the propeller fan; The propeller fan wake vortex filament geometric coordinate determination module is used to determine the geometric coordinates of each vortex filament segment according to the induced velocity of the wake vortex filament in the axial direction of the propeller fan, the air flow velocity at the inlet of the contra-rotating propeller fan, and the rotational speed of the rotating shaft; The geometric coefficient determination module is used to calculate the self-induced geometric coefficient and the mutual-induced geometric coefficient of each propeller fan of the contra-rotating propeller fan based on the lift line position coordinates of the two propeller fans and the geometric coordinates of each vortex filament segment of the two propeller fans; The induced velocity update module is used to determine the self-induced velocity of each propeller fan and the induced velocity between the two propeller fans based on the self-induced geometric coefficient, the mutual-induced geometric coefficient, and the new blade element circulation distribution on the two rows of propeller fans.

[0012] Preferably, in the total induced velocity calculation module, the total induced velocity of each propeller fan is calculated by the following formula: ; Where, is the total induced velocity of the i-th blade element of the front row propeller fan, is the total induced velocity of the i-th blade element of the rear row propeller fan, is the air flow velocity at the inlet of the contra-rotating propeller fan, is the rotational speed of the rotating shaft of the front row propeller fan, is the rotational speed of the rotating shaft of the rear row propeller fan, is the distance from the i-th blade element of the front row propeller fan to the rotating shaft of the front row propeller fan, is the distance from the i-th blade element of the rear row propeller fan to the rotating shaft of the rear row propeller fan, is the induced velocity vector of the nacelle on the i-th blade element of the front row propeller fan, is the induced velocity vector of the nacelle on the i-th blade element of the rear row propeller fan, is the self-induced velocity of the front row propeller fan at its i-th blade element, is the induced velocity of the rear row propeller fan at the i-th blade element of the front row propeller fan, is the self-induced velocity of the rear row propeller fan at its i-th blade element, is the induced velocity of the front row propeller fan at the i-th blade element of the rear row propeller fan.

[0013] Preferably, in the loop control module, when the norm of the difference between the blade element circulation calculated in the current iteration and the blade element circulation calculated in the previous iteration is less than or equal to 10 -6 , it is determined that the blade element circulation calculated by the iterative calculation converges, otherwise the new blade element circulation is calculated by the following formula : ; Where, is the blade element circulation calculated in the previous iteration, is the blade element circulation calculated in the current iteration, is the relaxation factor.

[0014] Preferably, the loop control module includes: A lift and drag calculation unit for calculating the lift and drag of each blade element according to the blade element circulation, the resultant velocity, the lift coefficient, and the drag coefficient; A component force calculation unit for calculating the component forces of each blade element in each direction according to the lift and drag of each blade element; A thrust and torque calculation unit for obtaining the thrust and torque of the front row of paddle fans and the rear row of paddle fans through integration.

[0015] Preferably, in the paddle fan wake vortex filament geometric coordinate determination module, the geometric coordinates of each vortex filament segment in the cylindrical coordinate system are determined by the following formula , , : ; where is the radius position of the j-th segment of the k-th wake vortex filament, is the radius position of the i-th blade element, is the azimuth angle of the wake vortex filament, is the number of turns allowed for the development of the wake vortex filament, is the number of segments into which the wake vortex filament is divided, is the number of each segment into which the wake vortex filament is divided, is the rotational speed of the axis of the paddle fan, is the induced velocity of the wake vortex filament in the axial direction of the paddle fan, is the parameter for controlling the grid level.

[0016] This application improves the simulation accuracy of the contra-rotating paddle fan, increases the calculation speed, and can quickly obtain the characteristics of the contra-rotating paddle fan in the full operating condition range. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] Figure 1 is a flowchart of a preferred embodiment of the numerical simulation method for a contra-rotating open rotor considering nacelle coupling in this application.

[0018] Figure 2 This is for this application Figure 1 a schematic diagram of the blade lift line positions of the front and rear rows of paddle fans in the illustrated embodiment.

[0019] Figure 3 This is for this application Figure 1 a schematic diagram of the wake vortex filament structure and coordinate positions of the contra-rotating paddle fan in the illustrated embodiment.

[0020] Figure 4 This is for this application Figure 1Schematic diagram of the velocity triangle of the contra-rotating propeller fan of the illustrated embodiment. Detailed implementation mode

[0021] To make the purpose, technical solutions and advantages of the implementation of this application clearer, the technical solutions in the implementation mode of this application will be described in more detail below in conjunction with the accompanying drawings in the implementation mode of this application. In the accompanying drawings, the same or similar reference numerals represent the same or similar elements or elements with the same or similar functions from beginning to end. The described implementation mode is a part of the implementation modes of this application, rather than all of the implementation modes. The implementation mode described below by referring to the accompanying drawings is exemplary and is intended to explain this application and should not be construed as a limitation to this application. Based on the implementation modes in this application, all other implementation modes obtained by those of ordinary skill in the art without creative efforts belong to the scope of protection of this application. The implementation mode of this application will be described in detail below in conjunction with the accompanying drawings.

[0022] The first aspect of this application provides a numerical simulation method for a contra-rotating open rotor considering nacelle coupling. Refer to Figure 1 , mainly including: Step S1: Divide each blade of the contra-rotating propeller fan into multiple blade elements along the span direction, construct a three-dimensional model of each blade according to the input airfoil parameters, and determine the position coordinates of the lift line of each propeller fan.

[0023] This application first constructs a three-dimensional model of the contra-rotating propeller fan in step S1, using the curve distribution of the airfoil parameters of the contra-rotating propeller fan along the radial direction as input. These airfoil parameters include: the designed lift coefficient , the maximum relative thickness ratio TOC, the chord length c, the twist angle θ, and the sweep angle Λ along the radial direction. Divide each propeller fan blade into N blade elements, determine the airfoil of each blade element according to the above airfoil parameters, and obtain the airfoil coordinates of the calculated blade element. First stack the airfoil coordinates of each blade element by the centroid, then translate the centroid position according to the sweep angle Λ, then rotate around its centroid according to the twist angle θ, and finally install according to the given installation angle β to obtain the final propeller fan blade coordinates. Take the 1 / 4 chord length position of each blade element as the lift line position of the blade. As Figure 2 shown, Figure 2 The lift lines of the front and rear rows of propeller fans shown are displayed in the global xyz coordinate system. This application realizes the construction of the global xyz coordinate system, the local scn coordinate system and the cylindrical coordinate system, and at the same time clarifies the conversion relationship between the coordinate systems. Among them, the global xyz coordinate system is for the entire contra-rotating propeller fan, xy is the rotation plane of the propeller fan, z is the rotation axis, the local scn coordinate system is for each blade element, s is the span direction of the blade / blade element, c is the chord line direction of the blade element, and n is the normal direction.

[0024] Given the axial spacing between two rows of propeller fans , translate the blade coordinates of the front row of propeller fans along the z-axis to obtain the blade coordinates in all coordinate systems of the rear row of propeller fans when the phase angle φ = 0°. If the phase angle φ of the front and rear rows of propeller fans is not equal to 0°, the blade coordinates of the rear row of propeller fans translated along the z-axis can be further rotated according to the phase angle φ and then coordinate transformation can be performed, and the blade coordinates in all coordinate systems of the rear row of propeller fans can also be obtained quickly.

[0025] Step S2: Calculate the induced velocity vector generated by the nacelle on the blades of the two rows of propeller fans at the lift line positions according to the air velocity at the inlet of the contra-rotating propeller fans, the grid point coordinates of the nacelle geometry, and the coordinates of all lift line positions of the two rows of propeller fans.

[0026] The air velocity in this step can be quickly calculated according to the atmospheric conditions (altitude, flight Mach number, temperature, etc.) at the given design point. For the nacelle, the induced effect of the two rows of propeller fans on the nacelle is not considered. Input the air velocity, the grid point coordinates of the nacelle geometry, and the coordinates of all lift line positions of the two rows of propeller fans into the Panair code developed by Boeing based on the panel method, and the induced velocity vectors generated by the nacelle on the blades of the two rows of propeller fans at the lift line positions can be obtained respectively. and .

[0027] Step S3: Calculate the total induced velocity of each propeller fan according to the air velocity at the inlet of the contra-rotating propeller fans, the rotational speeds of the two rows of propeller fans respectively, the induced velocity vectors of the nacelle on each propeller fan, the self-induced velocity of each propeller fan, and the induced velocity between the two propeller fans, where the initial values of the self-induced velocity of each propeller fan and the induced velocity between the two propeller fans are set to 0.

[0028] It can be understood that in step S3, in order to calculate the total induced velocity, the present application not only considers the induced velocity of the nacelle on the two rows of propeller fans, but also considers the self-induced velocity of each propeller fan and the induced velocity between the two propeller fans, thereby improving the calculation accuracy.

[0029] In some alternative embodiments, in step S3, referring to Figure 4 the inlet velocity triangles of the two rows of propeller fans shown, calculate the total induced velocity of each propeller fan through the following formula: ; where is the total induced velocity of the i-th element of the front row of propeller fans, is the total induced velocity of the i-th element of the rear row of propeller fans, is the air velocity at the inlet of the contra-rotating propeller fans, is the rotational speed of the shaft of the front row of propeller fans, is the rotational speed of the shaft of the rear row of propeller fans, is the distance from the i-th element of the front row of propeller fans to the shaft of the front row of propeller fans, is the distance from the i-th blade element of the rear row propeller fan to the axis of the rear row propeller fan, is the induced velocity vector of the nacelle on the i-th blade element of the front row propeller fan, is the induced velocity vector of the nacelle on the i-th blade element of the rear row propeller fan, is the self-induced velocity of the front row propeller fan at its i-th blade element, is the induced velocity of the rear row propeller fan on the i-th blade element of the front row propeller fan, is the self-induced velocity of the rear row propeller fan at its i-th blade element, is the induced velocity of the front row propeller fan on the i-th blade element of the rear row propeller fan.

[0030] Step S4: Perform coordinate transformation on the total induced velocity of the two propeller fans to determine the resultant velocity, chordwise component velocity, and normal component velocity of each blade element.

[0031] Since the velocity at each blade element in the global xyz coordinate system is calculated in Step S3, according to the coordinate systems constructed in Step S1 and the transformation relationships between the coordinate systems, it is easy to obtain the resultant velocity of the cn airfoil plane of each blade element in the local scn coordinate system in Step S4 , chordwise component velocity and normal component velocity . Among them, the subscript i has the same meaning as in Step S3 and is the blade element number.

[0032] Step S5: Determine the angle of attack distribution of each blade element of the two rows of propeller fans based on the chordwise component velocity and normal component velocity of each blade element.

[0033] In this step, according to the formula , calculate the angle of attack distribution of the airflow of the two rows of propeller fans respectively, that is, calculate the angle of attack of the i-th blade element of the front row propeller fan and the angle of attack of the i-th blade element of the front row propeller fan.

[0034] Step S6: Determine the lift coefficient and drag coefficient corresponding to the angle of attack of each blade element in the two-dimensional airfoil lift-drag characteristic database.

[0035] This step makes use of the two-dimensional airfoil lift-drag characteristic database, referring to Figure 1 , from the design lift coefficient , the maximum relative thickness ratio TOC, the angle of attack calculated in Step S5, the Reynolds number Re, and the Mach number Ma. Through the two-dimensional airfoil lift-drag characteristic database using kriging surrogate, obtain the lift coefficient and drag coefficient on the i-th blade element. Here, the lift coefficients and drag coefficients of the NACA16 series and NACA65 series airfoil databases are specifically used, and the surrogate model is: 。

[0036] Among them, shape is the airfoil type.

[0037] Step S7: Determine the circulation distribution of the blade element according to the lift coefficient and the resultant velocity of the blade element.

[0038] This step calculates the circulation of each blade element according to the circulation equation .

[0039] Among them, is the circulation of the i-th blade element, is the chord length of the i-th blade element, is the resultant velocity of the i-th blade element in the cn airfoil plane, is the lift coefficient of the i-th blade element.

[0040] Step S8: Determine whether the circulation of each blade element converges (it is defaulted that it does not converge in the first iteration). When the circulation of each blade element calculated by iterative calculation converges, calculate the performance of the contra-rotating propeller fan according to the circulation distribution of the blade element; otherwise, recalculate the new circulation distribution of the blade element according to the relaxation factor.

[0041] This step determines whether to perform simulation iteration through the convergence coefficient. For example, in some alternative embodiments, in step S8, when the norm of the difference between the circulation of each blade element calculated in the current iteration number and the circulation of each blade element calculated in the previous iteration number is less than or equal to 10 -6 , it is determined that the circulation of the blade element calculated by iterative calculation converges; otherwise, calculate the new circulation of the blade element through the following formula : ; Among them, is the circulation of the blade element calculated in the previous iteration number, is the circulation of the blade element calculated in the current iteration number, is the relaxation factor. In the first iteration, the induced velocity is set to 0 in step S3, and the circulation is obtained through steps S3 to S7, and it is defaulted that the circulation does not converge, and it is used as the for the second iteration to calculate. In the second iteration, the induced velocity in step S3 needs to be calculated according to and the wake vortex filaments.

[0042] Step S9: Determine the induced velocity generated by each vortex filament segment of each wake vortex filament of each blade at the control point of each blade element according to the new circulation of the blade element, so as to calculate the induced velocity of each wake vortex filament of each propeller fan in the axial direction of the propeller fan.

[0043] In this step, first, according to the Biot-Savart law, the induced velocity generated by the j-th segment of the k-th wake vortex filament of the m-th blade at the control point of the i-th blade element is given by: ; where the regularization factor is calculated according to the Lamb-Oseen vortex core model . is the vector from the end point A of the vortex filament segment to the control point P, is the vector from the end point B of the vortex filament segment to the control point P, is the vector from the end point A to the end point B of the vortex filament segment, is the distance from the vortex filament segment AB to the control point P, is the Lamb-Oseen constant, with a value of 1.25463, is the vortex core radius.

[0044] After that, based on the induced velocity generated by the j-th segment of the k-th wake vortex filament of the m-th blade at the control point of the i-th blade element, calculate the induced velocity of each wake vortex filament of each propeller fan in the axial direction of the propeller fan .

[0045] Specifically, by rearranging the calculation formula of the induced velocity , it can be written as , is the geometric coefficient calculated according to the coordinates of the j-th segment of the k-th wake vortex filament of the m-th blade and the coordinates of the control point of the i-th blade element. Integrating all blades and all vortex filaments, the circulation on the k-th wake vortex filament is , then the induced velocity of all wake vortex filaments on the i-th blade element is , where is the circulation on the blade element when i = k, is the geometric coefficient of the k-th wake vortex filament on the i-th blade element. Finally, perform coordinate transformation, and according to obtain the induced velocity of each wake vortex filament of each propeller fan in the axial direction of the propeller fan.

[0046] In this step, the axial direction of the propeller fan is also the z direction in the global xyz coordinate system.

[0047] Step S10: Determine the geometric coordinates of each vortex filament segment according to the induced velocity of the wake vortex filament in the axial direction of the propeller fan, the air flow velocity at the inlet of the contra-rotating propeller fan, and the rotational speed of the rotating shaft.

[0048] In some alternative embodiments, in step S10, the geometric coordinates of each vortex filament segment in the cylindrical coordinate system are determined by the following formula 、 , : ; Among them, is the radius position of the j-th segment of the k-th wake vortex filament, is the radius position of the i-th blade element. Since the wake vortex filaments are generated from the boundaries of the blade elements and the radius of each filament is constant, the radii of all filament segments of the k-th wake vortex filament are the same, and the radius position of the k-th wake vortex filament is determined by the average value of the radius positions of its adjacent two blade elements. is the azimuth angle of the wake vortex filament, is the number of turns allowed for the development of the wake vortex filament, is the number of segments into which the wake vortex filament is divided, is the numbering of each segment into which the wake vortex filament is divided, is the rotational speed of the shaft of the propeller fan, is the induced velocity of the wake vortex filament in the axial direction of the propeller fan, is the parameter for controlling the grid level.

[0049] In this step, according to the specified wake model, the coordinates of all wake vortex filament segments generated by all the propeller fan blades after discretization of the front and rear rows of propeller fans are calculated respectively, as shown in Figure 3 .

[0050] Step S11: Calculate the self-induced geometric coefficient and the mutual-induced geometric coefficient of each propeller fan of the contra-rotating propeller fan based on the lift line position coordinates of the two propeller fans and the geometric coordinates of each filament segment of the two propeller fans.

[0051] In this step, the isolated wake model described above is used for wake superposition for both rows of propeller fans, and the wake vortex filament structure changes with the relative phase angle φ of the two rows of propeller fans to consider the periodic influence. The mutual influence of the wake models between the two rows of propeller fans is considered by adding the axial components of the self-induced velocity and the mutual-induced velocity of the two rows of propeller fans (i.e., the induced velocity ).

[0052] Specifically, the self-induced geometric coefficient generated by the k-th filament of all the blades of the front propeller on the i-th blade element of the front propeller is calculated from the lift line coordinates of the front propeller and the coordinates of the wake vortex filaments of the front propeller, and the self-induced geometric coefficient generated by the k-th filament of all the blades of the rear propeller on the i-th blade element of the rear propeller is calculated from the lift line coordinates of the rear propeller and the coordinates of the wake vortex filaments of the rear propeller. The induced geometric coefficient of the rear propeller on the front propeller , calculating the induced geometric coefficient of the front row of blades on the rear row of blades generated by the k-th vortex filament of all blades of the front row on the i-th element of the rear row according to the coordinates of the rear row of blade lift lines and the coordinates of the front row of blade wakes .

[0053] Step S12: Determine the self-induced velocity of each row of blade fans and the induced velocity between the two rows of blade fans based on the self-induced geometric coefficient, the mutual-induced geometric coefficient, and the new element circulation distribution on the two rows of blade fans, return to step S3, and loop through steps S3 - S12 until the element circulation calculated by iteration converges.

[0054] This step calculates each induced velocity through the following formula: ; where, as mentioned before, here is the circulation on the k-th element of the front row of blade fans, is the circulation on the k-th element of the rear row of blade fans. As mentioned before, is the circulation of the i-th element. Since the circulation on the k-th wake vortex filament is equal to , that is, the k-th wake vortex filament is determined by the difference in circulation between the k-1-th element and the k-th element, and the circulation on different wake vortex filaments is different. The induced velocity on the i-th element is . In special cases, when k = 1 or k = N + 1, . Therefore, after expanding and organizing this formula, it becomes , where is the circulation on the element when i = k, and is used to calculate the induced velocity generated by the k-th wake vortex filament.

[0055] After obtaining the above-mentioned induced velocities, return to step S3, and iteratively update the wake geometry coordinates, induced velocity distribution, total induced velocity distribution, angle of attack distribution, lift / drag coefficient, etc. in sequence, and determine whether to terminate the iteration by whether the norm of the difference between the new element circulation and the circulation of the previous step is less than 10 -6 .

[0056] In some alternative embodiments, in step S8, calculate the performance of the contra-rotating blade fans through the following steps: Step S81: Calculate the lift and drag of each element according to the element circulation, resultant velocity, lift coefficient, and drag coefficient; Step S82: Calculate the component forces of each element in each direction according to the lift and drag of each element; Step S83: Obtain the thrust and torque of the front row of blade fans and the rear row of blade fans through integration.

[0057] In this embodiment, according to the circulation distribution, velocity distribution, and lift coefficient distribution, calculate the lift distribution and drag distribution of the blade fan blades, and then calculate the component forces in the local coordinate system: ; ; In the above formula, is the angle of attack of the i-th blade element, is the lift of the i-th blade element, is the drag of the i-th blade element, is the resultant velocity of the i-th blade element on the cn airfoil plane, is the chord length of the i-th blade element, is the length of the lift line segment of the i-th blade element, is the air density. are the component forces in the local scn coordinate system respectively.

[0058] For different propfans, different angles of attack are adopted. As mentioned above, the angle of attack of the i-th blade element of the front row propfan and the angle of attack of the i-th blade element of the front row propfan ; The lift

[0059] Convert the component forces in the local coordinate system to the component forces in the cylindrical coordinate system , and , and finally perform integration. Calculate the thrust and torque of the front and rear row propfans respectively according to the following formula, and finally obtain the total power of the contra-rotating propfan , total thrust and efficiency .

[0060] .

[0061] In the formula, T is the thrust, Q is the torque, is the number of blades, is the distance from the position of the i-th blade element to the propfan rotation axis.

[0062] It should be noted that in the spatial period of the contra-rotating propfan of the present application, P relative positions of the propfans are divided, then P wake geometric structures of the contra-rotating propfans are obtained, corresponding to P relative phase angles. By sequentially changing the relative phase angle φ, rotating the blades and wake geometric coordinates of the rear row propfan, and repeating the steps of the present application P times, the thrust and torque of the contra-rotating propfan at different phases can be obtained. Finally, the thrust and torque obtained at all phases are integrated and averaged to obtain the thrust and torque of the contra-rotating propfan considering the periodic influence.

[0063] The present application can calculate the induction effect of the nacelle on the two rows of propfans and obtain the nacelle installation performance of the contra-rotating propfan.

[0064] The contra-rotating propeller fan lift-line model developed in this application can calculate the characteristics of the contra-rotating propeller fan over the entire operating range, can reflect the influence of design parameters such as blade geometry, number of blades, diameter, rotational speed, and pitch angle on aerodynamic performance, can directly obtain the true characteristics, thus avoiding the errors caused by characteristic scaling, has high calculation accuracy, fast calculation speed, and good convergence and stability. Compared with the wind tunnel test data of the F7-A7 contra-rotating propeller fan, the error is only about 0.5%, and it only takes about 20 s to calculate a working point with an ordinary computer.

[0065] The contra-rotating propeller fan lift-line model developed in this application can be used as a separate characteristic diagram calculation program. By specifying the propeller fan blade profile geometry, characteristic diagrams over the entire operating range under different combined parameters of the pitch angles of the two rows can be obtained, solving the problem of the scarcity of contra-rotating propeller fan characteristic diagrams, and can also be used for contra-rotating propeller fan design, parameter sensitivity analysis, research on the matching characteristics of the two rows of propeller fans, etc.

[0066] In the second aspect of this application, a numerical simulation device for a contra-rotating open rotor considering nacelle coupling corresponding to the above method is provided, mainly including: A lift-line determination module, configured to divide each blade of the contra-rotating propeller fan into multiple blade elements along the span direction, construct a three-dimensional model of each blade according to the input blade profile parameters, and determine the position coordinates of the lift-line of each propeller fan; A nacelle induced velocity vector calculation module, configured to calculate the induced velocity vector generated by the nacelle on the blades of the two rows of propeller fans at the lift-line positions according to the air flow velocity at the inlet of the contra-rotating propeller fan, the grid point coordinates of the nacelle geometry, and the position coordinates of all the lift-lines of the two rows of propeller fans; A total induced velocity calculation module, configured to calculate the total induced velocity of each propeller fan according to the air flow velocity at the inlet of the contra-rotating propeller fan, the rotational speeds of the two rows of propeller fans respectively, the induced velocity vector of the nacelle on each propeller fan, the self-induced velocity of each propeller fan, and the induced velocity between the two propeller fans, wherein the initial values of the self-induced velocity of each propeller fan and the induced velocity between the two propeller fans are set to 0; A blade element velocity determination module, configured to perform coordinate transformation on the total induced velocity of the two propeller fans to determine the resultant velocity, chordwise component velocity, and normal component velocity of each blade element; A blade element angle of attack determination module, configured to determine the angle of attack distribution of each blade element of the two rows of propeller fans based on the chordwise component velocity and normal component velocity of each blade element; A lift coefficient and drag coefficient determination module, configured to determine the lift coefficient and drag coefficient corresponding to the angle of attack of each blade element in the two-dimensional airfoil lift-drag characteristic database; A blade element circulation determination module, configured to determine the blade element circulation distribution according to the lift coefficient and the resultant velocity of the blade element; A circulation control module, which is used to determine whether the circulation of each blade element converges. After the circulation of each blade element converges in the iterative calculation, the performance of the contra-rotating propeller fan is calculated according to the circulation distribution of the blade element; otherwise, a new circulation distribution of the blade element is recalculated according to the relaxation factor. An induced velocity determination module in the axial direction of the propeller fan, which is used to determine the induced velocity generated by each vortex filament segment of each wake vortex of each blade at the control point of each blade element according to the new circulation of the blade element, so as to calculate the induced velocity of each wake vortex of the propeller fan in the axial direction of the propeller fan. A geometric coordinate determination module for the wake vortex filaments of the propeller fan, which is used to determine the geometric coordinates of each vortex filament segment according to the induced velocity of the wake vortex filaments in the axial direction of the propeller fan, the air flow velocity at the inlet of the contra-rotating propeller fan, and the rotational speed of the rotating shaft. A geometric coefficient determination module, which is used to calculate the self-induced geometric coefficient and the mutual-induced geometric coefficient of each propeller fan of the contra-rotating propeller fan based on the lift line position coordinates of the two propeller fans and the geometric coordinates of each vortex filament segment of the two propeller fans. An induced velocity update module, which is used to determine the self-induced velocity of each propeller fan and the induced velocity between the two propeller fans based on the self-induced geometric coefficient, the mutual-induced geometric coefficient, and the new circulation distribution of the blade elements on the two rows of propeller fans.

[0067] In some alternative embodiments, in the total induced velocity calculation module, the total induced velocity of each propeller fan is calculated by the following formula: ; Wherein, is the total induced velocity of the i-th blade element of the front row propeller fan, is the total induced velocity of the i-th blade element of the rear row propeller fan, is the air flow velocity at the inlet of the contra-rotating propeller fan, is the rotational speed of the rotating shaft of the front row propeller fan, is the rotational speed of the rotating shaft of the rear row propeller fan, is the distance from the i-th blade element of the front row propeller fan to the rotating shaft of the front row propeller fan, is the distance from the i-th blade element of the rear row propeller fan to the rotating shaft of the rear row propeller fan, is the induced velocity vector of the nacelle on the i-th blade element of the front row propeller fan, is the induced velocity vector of the nacelle on the i-th blade element of the rear row propeller fan, is the self-induced velocity of the front row propeller fan at its i-th blade element, is the induced velocity of the rear row propeller fan at the i-th blade element of the front row propeller fan, is the self-induced velocity of the rear row propeller fan at its i-th blade element, is the induced velocity of the front row propeller fan at the i-th blade element of the rear row propeller fan.

[0068] In some alternative embodiments, in the loop control module, when the norm of the difference between the circulation of each blade element calculated for the current iteration number and the circulation of each blade element calculated for the previous iteration number is less than or equal to 10 -6 it is determined that the circulation of the blade element calculated by the iterative calculation converges; otherwise, the new circulation of the blade element is calculated by the following formula : ; wherein, is the circulation of the blade element calculated for the previous iteration number, is the circulation of the blade element calculated for the current iteration number, is the relaxation factor.

[0069] In some alternative embodiments, the loop control module includes: a lift and drag calculation unit configured to calculate the lift and drag of each blade element according to the circulation of the blade element, the resultant velocity, the lift coefficient and the drag coefficient; a component force calculation unit configured to calculate the component forces of each blade element in each direction according to the lift and drag of each blade element; a thrust and torque calculation unit configured to obtain the thrust and torque of the front row of paddle fans and the rear row of paddle fans through integration.

[0070] In some alternative embodiments, in the blade fan wake vortex filament geometric coordinate determination module, the geometric coordinates of each vortex filament segment in the cylindrical coordinate system are determined by the following formula , , : ; wherein, is the radius position of the j-th segment of the k-th wake vortex filament, is the radius position of the i-th blade element, is the azimuth angle of the wake vortex filament, is the number of turns allowing the wake vortex filament to develop, is the number of segments into which the wake vortex filament is divided, is the number of each segment into which the wake vortex filament is divided, is the rotational speed of the axis of the paddle fan, is the induced velocity of the wake vortex filament in the axial direction of the paddle fan, is the parameter for controlling the grid level.

[0071] The above are only the specific embodiments of the present application, but the protection scope of the present application is not limited thereto. Any changes or substitutions that can be easily thought of by those skilled in the art within the technical scope disclosed in the present application should be covered by the protection scope of the present application. Therefore, the protection scope of the present application should be subject to the protection scope of the claims.

Claims

1. A numerical simulation method for counter-rotating open rotors considering nacelle coupling, characterized in that: include: Step S1, dividing each blade of the counter-rotating propeller fan into a plurality of blade elements along the span direction, constructing a three-dimensional model of each blade according to the input blade profile parameters, and determining the lift line position coordinates of each propeller fan; Step S2, calculating the induced velocity vector generated by the nacelle on the two rows of propfan blades at the lift line position according to the airflow velocity at the inlet of the counter-rotating propfan, the grid point coordinates of the nacelle geometry, and the coordinates of all lift line positions of the two rows of propfans; Step S3, calculating the total induced speed of each propeller fan according to the airflow speed at the inlet of the counter-rotating propeller fan, the rotation speed of each row of propeller fans, the induced speed vector of the nacelle to each propeller fan, the self-induced speed of each propeller fan and the induced speed between the two propeller fans, wherein the initial values ​​of the self-induced speed of each propeller fan and the induced speed between the two propeller fans are set to 0; Step S4, performing coordinate conversion on the total induced velocity of the two propeller fans to determine the combined velocity, chord-wise component velocity and normal component velocity of each blade element; Step S5, determining the attack angle distribution of each blade element of the two rows of propellers based on the chordwise component velocity and the normal component velocity of each blade element; Step S6, determining the lift coefficient and drag coefficient corresponding to the angle of attack of each blade element in the two-dimensional airfoil lift-drag characteristic database; Step S7, determining the blade element circulation distribution according to the lift coefficient and the combined velocity of the blade element; Step S8, judging whether the circulation of each blade element converges, and when the iteratively calculated circulation of each blade element converges, calculating the performance of the counter-rotating propeller according to the blade element circulation distribution, otherwise recalculating a new blade element circulation distribution according to the relaxation factor; Step S9, determining the induced velocity of each vortex filament segment of each trail vortex filament of each blade at the control point of each blade element according to the new blade element annulus, thereby calculating the induced velocity of each trail vortex filament of each propeller fan in the axial direction of the propeller fan; Step S10, determining the geometric coordinates of each vortex filament segment according to the induced velocity of the trail vortex filament in the axial direction of the propeller, the airflow velocity at the inlet of the counter-rotating propeller, and the rotation speed of the shaft; Step S11, calculating the self-induced geometric coefficient and the mutual induced geometric coefficient of each propeller fan of the counter-rotating propeller fans based on the lift line position coordinates of the two propeller fans and the geometric coordinates of each vortex filament segment of the two propeller fans; Step S12, determine the self-induced speed of each propeller fan and the induced speed between the two propeller fans based on the self-induced geometric coefficient, the mutual induced geometric coefficient and the new blade annulus distribution on the two rows of propeller fans, return to step S3, and loop through steps S3-S12 until the iteratively calculated blade annulus converges.

2. The method for numerical simulation of counter-rotating open rotors considering nacelle coupling according to claim 1, characterized in that: In step S3, the total induced speed of each propeller fan is calculated by the following formula: ； in, is the total induced velocity of the i-th blade element of the front row propeller fan, is the total induced velocity of the i-th blade element of the rear propeller fan, is the air velocity at the inlet of the counter-rotating propeller fan, is the shaft speed of the front propeller fan, is the shaft speed of the rear propeller fan, is the distance between the i-th blade element of the front row propeller and the rotation axis of the front row propeller, is the distance between the i-th blade element of the rear propeller fan and the rear propeller fan shaft, is the induced velocity vector of the nacelle on the i-th blade element of the front propeller fan, is the induced velocity vector of the nacelle on the i-th blade element of the rear propeller fan, is the self-induced speed of the front propeller at its ith blade element, is the induced velocity of the rear propeller fan at the i-th blade element of the front propeller fan, is the self-induced speed of the rear propeller fan at its ith blade element, is the induced velocity of the front propeller fan at the i-th blade element of the rear propeller fan.

3. The method for numerical simulation of counter-rotating open rotors considering nacelle coupling according to claim 1, characterized in that: In step S8, the difference norm of each blade element ring quantity calculated by the current iteration number and each blade element ring quantity calculated by the previous iteration number is less than or equal to 10 -6 When the blade element annulus calculated iteratively converges, otherwise the new blade element annulus is calculated by the following formula : ； in, is the blade ring quantity calculated for the last iteration, is the blade ring quantity calculated for the current iteration number, is the relaxation factor.

4. The method for numerical simulation of counter-rotating open rotors considering nacelle coupling according to claim 1, characterized in that: In step S8, the performance of the counter-rotating propeller fan is calculated by the following steps: Step S81, calculating the lift and drag of each blade element according to the blade element annulus, the resultant velocity, the lift coefficient and the drag coefficient; Step S82, calculating the component force of each blade element in each direction according to the lift and drag of each blade element; Step S83, obtain the thrust and torque of the front propeller fans and the rear propeller fans by integration.

5. The method for numerical simulation of counter-rotating open rotors considering nacelle coupling according to claim 1, characterized in that: In step S10, the geometric coordinates of each vortex segment on the cylindrical coordinate system are determined by the following formula: , , : ； in, is the radius position of the jth vortex of the kth wake vortex, is the radius position of the ith blade element, is the azimuth of the wake vortex, The number of turns to allow the development of the wake vortex filaments is: is the number of segments into which the wake vortex filament is divided, Number the segments into which the wake vortex filaments are divided, is the shaft speed of the propeller fan, is the induced velocity of the wake vortex filament in the axial direction of the propeller fan, Parameters that control the grid level.

6. A numerical simulation device for counter-rotating open rotors considering nacelle coupling, characterized in that: include: A lift line determination module is used to divide each blade of the counter-rotating propeller fan into multiple blade elements along the span direction, construct a three-dimensional model of each blade according to the input blade profile parameters, and determine the lift line position coordinates of each propeller fan; A nacelle induced velocity vector calculation module is used to calculate the induced velocity vector generated by the nacelle on the two rows of propeller fan blades at the lift line position according to the airflow velocity at the inlet of the counter-rotating propeller fan, the grid point coordinates of the nacelle geometry, and the coordinates of all lift line positions of the two rows of propeller fans; A total induced speed calculation module is used to calculate the total induced speed of each propeller fan according to the airflow speed at the inlet of the counter-rotating propeller fan, the rotation speed of each row of propeller fans, the induced speed vector of the nacelle to each propeller fan, the self-induced speed of each propeller fan and the induced speed between the two propeller fans, wherein the initial values ​​of the self-induced speed of each propeller fan and the induced speed between the two propeller fans are set to 0; The blade element speed determination module is used to perform coordinate conversion on the total induced speed of the two propeller fans and determine the combined speed, chord-wise component speed and normal component speed of each blade element; A blade element angle of attack determination module is used to determine the angle of attack distribution of each blade element of two rows of propellers based on the chord-wise component velocity and the normal component velocity of each blade element; A lift coefficient and drag coefficient determination module is used to determine the lift coefficient and drag coefficient corresponding to the angle of attack of each blade element in a two-dimensional airfoil lift and drag characteristic database; A blade element annular quantity determination module, used to determine the blade element annular quantity distribution according to the lift coefficient and the combined velocity of the blade element; A loop control module is used to determine whether the circulation of each blade element converges. When the iteratively calculated circulation of each blade element converges, the performance of the counter-rotating propeller is calculated according to the blade element circulation distribution. Otherwise, a new blade element circulation distribution is recalculated according to the relaxation factor. A propeller fan axial direction induced velocity determination module is used to determine the induced velocity generated by each vortex filament segment of each trail vortex filament of each blade at the control point of each blade element according to the new blade element annulus, thereby calculating the induced velocity of each trail vortex filament of each propeller fan in the propeller fan axial direction; A propeller fan wake vortex filament geometric coordinate determination module is used to determine the geometric coordinates of each vortex filament segment according to the induced velocity of the wake vortex filament in the axial direction of the propeller fan, the airflow velocity at the inlet of the counter-rotating propeller fan, and the rotation speed of the shaft; A geometric coefficient determination module, used for calculating the self-induced geometric coefficient and the mutual induced geometric coefficient of each propeller of the counter-rotating propeller based on the lift line position coordinates of the two propellers and the geometric coordinates of each vortex filament segment of the two propellers; The induced speed update module is used to determine the self-induced speed of each propeller fan and the induced speed between the two propeller fans based on the self-induced geometric coefficient, the mutual induced geometric coefficient and the new blade ring distribution on the two rows of propeller fans.

7. The numerical simulation device for counter-rotating open rotors considering nacelle coupling according to claim 6, characterized in that: In the total induced speed calculation module, the total induced speed of each propeller fan is calculated by the following formula: ； in, is the total induced velocity of the i-th blade element of the front row propeller fan, is the total induced velocity of the i-th blade element of the rear propeller fan, is the air velocity at the inlet of the counter-rotating propeller fan, is the shaft speed of the front propeller fan, is the shaft speed of the rear propeller fan, is the distance between the i-th blade element of the front row propeller and the rotation axis of the front row propeller, is the distance between the i-th blade element of the rear propeller fan and the rear propeller fan shaft, is the induced velocity vector of the nacelle on the i-th blade element of the front propeller fan, is the induced velocity vector of the nacelle on the i-th blade element of the rear propeller fan, is the self-induced speed of the front propeller at its ith blade element, is the induced velocity of the rear propeller fan at the i-th blade element of the front propeller fan, is the self-induced speed of the rear propeller fan at its ith blade element, is the induced velocity of the front propeller fan at the i-th blade element of the rear propeller fan.

8. The numerical simulation device for counter-rotating open rotors considering nacelle coupling according to claim 6, characterized in that: In the loop control module, the difference norm of each blade element ring quantity calculated by the current iteration number and each blade element ring quantity calculated by the previous iteration number is less than or equal to 10 -6 When the blade element annulus calculated iteratively converges, otherwise the new blade element annulus is calculated by the following formula : ； in, is the blade ring quantity calculated for the last iteration, is the blade ring quantity calculated for the current iteration number, is the relaxation factor.

9. The numerical simulation device for counter-rotating open rotors considering nacelle coupling according to claim 6, characterized in that: The circulation control module comprises: A lift and drag calculation unit is used to calculate the lift and drag of each blade element according to the blade element annulus, resultant velocity, lift coefficient and drag coefficient; A component force calculation unit, used to calculate the component forces of each blade element in each direction according to the lift and drag of each blade element; The thrust and torque calculation unit is used to obtain the thrust and torque of the front and rear propeller fans through integration.

10. The numerical simulation device for counter-rotating open rotors considering nacelle coupling according to claim 6, characterized in that: In the propeller fan trail vortex geometric coordinate determination module, the geometric coordinates of each vortex segment on the cylindrical coordinate system are determined by the following formula: , , : ； in, is the radius position of the jth vortex of the kth wake vortex, is the radius position of the ith blade element, is the azimuth of the wake vortex, The number of turns to allow the development of the wake vortex filaments is: is the number of segments into which the wake vortex filament is divided, Number the segments into which the wake vortex filaments are divided, is the shaft speed of the propeller fan, is the induced velocity of the wake vortex filament in the axial direction of the propeller fan, Parameters that control the grid level.

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