Supersonic axial flow turbine blade profile design method for turbine pump

Abstract

The invention discloses a supersonic axial flow turbine blade profile design method for a turbine pump, and belongs to the field of turbine blade design. According to the method, a Laval spray pipe is replaced by a supersonic static blade designed based on a characteristic line method, so that an accelerated supersonic flow surface can be matched with an annular channel of a movable blade inlet, and circumferential uniform airflow is provided for the turbine movable blade inlet; and therefore, the pneumatic loss generated by air inlet degree configuration of similar parts in the turbine moving blades is reduced. The blade profile of the ultrasonic quick-acting blade designed based on the characteristic line and the free vortex does not depend on design experience, and the stability of the performance of the designed movable blade can be guaranteed. The blade profiles of the supersonic static blades and the moving blades designed through the method remarkably improve the performance of a supersonic turbine in a turbine pump, the fuel carrying amount of a rocket is reduced, and then the thrust-weight ratio of the rocket is increased.

Description

Technical Field

[0001] The present invention belongs to the field of turbine blade design, and in particular relates to the field of supersonic axial flow turbine blade profile design for a turbine pump. Background Art

[0002] As a core component of a rocket engine, the turbopump's performance directly determines the power output of the propellant supply system and the overall efficiency of the rocket system. With the increasing power density requirements for high-thrust rocket engines, high-load supersonic turbines have become the standard solution for driving fuel pumps. Currently, supersonic axial-flow turbines in turbopumps typically use a Laval nozzle to accelerate the airflow to supersonic speeds to drive the impulse rotors, achieving high-density power output. However, the Laval nozzle's elliptical outlet flow surface cannot fully cover the annular area at the rotor inlet, resulting in a flow configuration similar to a partial intake. This configuration results in circumferentially uneven airflow at the rotor inlet, significantly increasing secondary flow losses in the hub and casing areas and exacerbating mixing losses between high- and low-kinetic energy flows, ultimately deteriorating the turbine flow field and aerodynamic performance. Furthermore, the current lack of effective methods for supersonic impulse rotor profile design makes it difficult to ensure stable performance of supersonic rotors designed solely based on experience. This combined effect directly reduces the overall aerodynamic performance of the turbine, forcing the rocket to carry more fuel, which in turn increases the weight of the propulsion system and reduces the rocket's thrust-to-weight ratio.

[0003] Therefore, designing high-performance supersonic turbines for turbopumps has become an urgent issue to be solved. Summary of the Invention

[0004] To address these shortcomings, the present invention provides a supersonic axial-flow turbine blade design method for a turbopump. This method replaces the Laval nozzle with supersonic stators, eliminating the geometric mismatch between the supersonic airflow outlet and the rotor blade inlet, providing circumferentially uniform airflow at the rotor blade inlet. The rotor blade passageway is designed based on the governing equations of supersonic flow, ensuring blade performance stability.

[0005] The present invention is realized by the following technical solution: A supersonic axial flow turbine blade design method for a turbo pump includes the design of supersonic stationary blades and supersonic moving blades. The supersonic stationary blade design requires given parameters: blade pitch , Mach number at the stator blade outlet e , stator blade outlet geometric angle Specific heat ratio of working fluid , leading edge radius R L , leading edge arc angle and trailing edge radius R T The supersonic rotor blade design requires given parameters: blade pitch , inlet geometry angle , relative inlet Mach number Ma i Specific heat ratio of working fluid 、Suction surface free vortex Mach number Ma s 、Pressure surface free vortex Mach Ma p , leading edge thickness D r and the leading edge ellipse aspect ratio .

[0006] In the aforementioned supersonic axial-flow turbine blade design method for a turbopump, the supersonic stator blade profile is first generated using the characteristic line method based on a given exit Mach number to generate the expansion section nozzle profile. The discretized characteristic line equation is used to determine the node position coordinates, and the compatibility equation determines the node flow parameters. The characteristic line equation and compatibility equation are:

[0007]

[0008]

[0009] in is the slope of the characteristic line, is the geometric angle of the profile, is the Mach angle, is the fluid velocity, M n Calculate the Mach number at the point for the nozzle.

[0010] In the above-mentioned supersonic axial flow turbine blade design method for a turbo pump, the nozzle profile of the expansion section is symmetrical along the axis to generate a complete nozzle profile. The nozzle inlet is the throat of the stator blade, and the initial width d t = 1. After the axisymmetric expansion section nozzle is scaled to the specified throat width, its symmetry axis is rotated. The upper half of the nozzle is the initial profile of the suction surface, and the lower half is the translation pitch. As the initial profile of the pressure surface. Based on the given stator pitch, trailing edge radius, outlet geometry angle and nozzle expansion ratio, the actual throat width can be calculated as the scaling ratio of the nozzle. The calculation formula is:

[0011]

[0012] Where K is the ratio of the nozzle outlet width to the inlet width of the expansion section.

[0013] In the aforementioned supersonic axial-flow turbine blade design method for a turbopump, on the supersonic stator pressure side, the diverging section nozzle inlet, i.e., the stator throat, is connected to the leading edge via a circular arc and a polynomial, while the diverging section nozzle outlet is directly connected to the trailing edge. On the supersonic stator suction side, the diverging section nozzle inlet, i.e., the stator throat, is directly connected to the leading edge, while the diverging section nozzle outlet is connected to the trailing edge via a straight line. Both the leading and trailing edges are designed as circular arcs, with the leading edge arc angle based on the design value and the trailing edge as a semicircle.

[0014] In the above-mentioned supersonic axial turbine blade design method for a turbopump, the arc at the entrance of the expansion section of the pressure surface of the supersonic stator blade has the same radius as the leading edge. This design takes into account the two-dimensional flow effect at the nozzle inlet, ensures that the curvature on both sides of the throat inlet is the same, and realizes uniform airflow entering the expansion section.

[0015] In the above-mentioned supersonic axial flow turbine blade design method for a turbopump, the axial chord length and circumferential chord length of the supersonic stator blade can be expressed by analytical expressions:

[0016]

[0017]

[0018] In the above-mentioned supersonic axial turbine blade design method for a turbopump, the supersonic moving blade blade is impulsively designed using the characteristic line and free vortex method, and the relative inlet Mach number and inlet geometric angle of the moving blade are the same as the relative outlet Mach number and outlet geometric angle. The supersonic moving blade channel consists of an inlet transition zone, a free vortex flow zone, and an outlet transition zone. The inlet transition zone converts the inlet uniform flow into a free vortex flow, wherein the suction surface increases the inlet Mach number to a specified free vortex Mach number Ma by the characteristic line method. s , on the pressure surface, the inlet Mach number is reduced to the specified free vortex Mach number Ma by the characteristic line method p The free vortex flow area is composed of concentric arcs. The airflow in the free vortex flow area maintains free vortex flow, and the airflow on the suction surface and pressure surface maintains the specified Mach number Ma. s and Ma p The outlet transition zone converts the free vortex flow into a uniform flow at the outlet, and the free vortex Mach number Ma is converted to the uniform flow at the outlet by the characteristic line method on the suction surface. s Reduce to the outlet Mach number, and use the characteristic line method on the pressure surface to convert the free vortex Mach number Ma p Increase to the exit Mach number.

[0019] In the aforementioned supersonic axial turbine blade design method for a turbopump, the dimensionless arc radii of the suction and pressure surfaces in the free vortex flow region are determined based on the specified free vortex Mach numbers and free vortex equations for the suction and pressure surfaces. By specifying the free vortex Mach numbers for the suction and pressure surfaces, the load distribution within the rotor blade passage can be designed. The free vortex equation is:

[0020]

[0021]

[0022] in is the dimensionless velocity, is the dimensionless radius, is the Mach number in the free vortex flow region.

[0023] In the above-mentioned supersonic axial flow turbine blade design method for a turbo pump, the supersonic rotor relative inlet Mach number Ma is known. i and inlet geometry , the dimensionless radius in the free vortex channel The pressure and suction profiles of the inlet transition section are obtained by the characteristic line method using the Mach number and the Mach number as initial conditions. Similarly, the outlet transition section is known with the outlet Mach number and the outlet geometry angle, and the dimensionless radius in the free vortex channel is used. Using the Mach number as initial conditions, the characteristic line method was used to obtain the profiles of the pressure and suction sides of the outlet transition section. Finally, a straight line was used to connect the suction and pressure sides to complete the design of the supersonic rotor blade profile without leading edge thickness.

[0024] In the above-mentioned supersonic axial flow turbine blade design method for a turbo pump, the dimensionless coordinates of the supersonic moving blades are obtained by the characteristic line method and the free vortex method. The dimensionless pitch of the moving blade is then calculated. According to the specified blade pitch S r and dimensionless pitch The blade profile is scaled to obtain the designed supersonic rotor blade coordinates.

[0025] In the above-mentioned supersonic axial flow turbine blade design method for a turbo pump, in order to design a supersonic rotor blade with leading edge and trailing edge thickness, the suction surface and the pressure surface are moved in opposite directions by a certain distance, which is equal to the leading edge thickness D r Half of the ellipse. Then, according to the aspect ratio Taking the leading edge thickness as the minor axis, a complete supersonic blade profile is obtained by using a half elliptical arc to close the blade profile.

[0026] In the above-mentioned supersonic axial flow turbine blade design method for a turbopump, after the supersonic stationary and moving blade blade designs are completed, the leading edges of blades with different cross-sections are aligned radially, and the overall three-dimensional configuration of the blades is determined by interpolation.

[0027] The beneficial effects of this invention are as follows: in the supersonic axial flow turbine used in the turbopump, the supersonic stator blades designed based on the characteristic line method replace the Laval nozzle, so that the accelerated supersonic flow surface can match the annular channel at the rotor blade inlet, providing a circumferentially uniform airflow for the turbine rotor blade inlet. This reduces the aerodynamic losses caused by similar partial air intake configurations in the turbine rotor blades. The supersonic rotor blade profile designed based on the characteristic line and free vortex does not rely on design experience and can ensure the stability of the designed rotor blade performance. The supersonic stator and rotor blade profiles designed by this method significantly improve the performance of the supersonic turbine in the turbopump, reduce the amount of rocket fuel carried, and thus improve the thrust-to-weight ratio of the rocket. BRIEF DESCRIPTION OF THE DRAWINGS

[0028] Figure 1 This is a comparison diagram of the supersonic turbine of the present invention and a typical supersonic turbine in a turbopump;

[0029] Figure 2 This is a schematic diagram of a nozzle based on the characteristic line method of the present invention;

[0030] Figure 3 This is a schematic diagram of the supersonic stator blade profile design of the present invention;

[0031] Figure 4 Schematic diagram of a supersonic rotor blade passage based on characteristic lines and free vortex design according to the present invention;

[0032] Figure 5 This is a schematic diagram of the supersonic rotor blade profile design of the present invention;

[0033] Figure 6 This is a schematic diagram of a specific embodiment of a supersonic stator blade according to the present invention;

[0034] Figure 7 It is a schematic diagram of a specific embodiment of the supersonic rotor blade of the present invention. DETAILED DESCRIPTION

[0035] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the examples described are only part of the embodiments of the invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention. The embodiments of the present invention are described in detail below. Based on the technical solutions of the present invention, detailed implementation methods and specific operating processes are given, but the scope of protection of the present invention is not limited to the following embodiments.

[0036] Figure 1 This is a comparison diagram of the supersonic turbine of the present invention and a typical supersonic turbine in a turbopump. In the present invention, the Laval nozzle is replaced with a supersonic stator blade to eliminate the geometric mismatch between the supersonic airflow outlet and the moving blade inlet, providing circumferentially uniform airflow for the moving blade inlet.

[0037] The supersonic stator blade profile design requires given parameters: blade pitch , Mach number at the stator blade outlet e , stator blade outlet geometric angle Specific heat ratio of working fluid , leading edge radius R L , leading edge arc angle and trailing edge radius R T .

[0038] Figure 2 For a nozzle schematic based on the characteristic line method, supersonic vane profile design first uses the characteristic line method to generate the nozzle profile for the expansion section based on a given exit Mach number. The discretized characteristic line equation is used to determine the node position coordinates, and the compatibility equation determines the node flow parameters. The characteristic line equation and compatibility equation are:

[0039]

[0040]

[0041] in is the slope of the characteristic line, is the geometric angle of the profile, is the Mach angle, is the fluid velocity, Calculate the Mach number at the point for the nozzle.

[0042] Figure 3 This is a schematic diagram of the supersonic stator blade design. The nozzle profile of the expansion section is symmetrical along the axis to generate a complete nozzle profile. The nozzle inlet is the stator blade throat, and the initial width d t = 1. After the axisymmetric expansion section nozzle is scaled to the specified throat width, its symmetry axis is rotated. The upper half of the nozzle is the initial profile of the suction surface, and the lower half is the translation pitch. As the initial profile of the pressure surface. Based on the given stator pitch, trailing edge radius, outlet geometry angle and nozzle expansion ratio, the actual throat width can be calculated as the scaling ratio of the nozzle. The calculation formula is:

[0043]

[0044] Where K is the ratio of the nozzle outlet width to the inlet width of the expansion section.

[0045] On the supersonic vane pressure surface, the diverging section nozzle inlet (dc), also known as the vane throat, connects to the leading edge (ae) via a circular arc (cb) and a polynomial curve (ba). The diverging section nozzle outlet (cd) is directly connected to the trailing edge (dg). On the supersonic vane suction surface, the diverging section nozzle inlet (fe), also known as the vane throat, connects directly to the leading edge (ea), while the diverging section nozzle outlet (ef) connects to the trailing edge (gd) via a straight line (fg). Both the leading edge (ea) and the trailing edge (dg) are designed as circular arcs, with the leading edge arc angle based on the design value and the trailing edge being a semicircle.

[0046] The arc (bc) at the entrance of the expansion section of the pressure surface of the supersonic stator blade has the same radius as the leading edge (ae). This design takes into account the two-dimensional flow effect at the nozzle inlet, ensures that the curvature on both sides of the throat inlet (ec) is the same, and realizes uniform airflow entering the expansion section.

[0047] Axial chord length of supersonic stator blade ( ) and circumferential chord length ( ) can be expressed analytically as:

[0048]

[0049]

[0050] The supersonic rotor blade design requires given parameters: blade pitch S r , inlet geometry angle , relative inlet Mach number Ma i Specific heat ratio of working fluid 、Suction surface free vortex Mach number Ma s 、Pressure surface free vortex Mach Ma p , leading edge thickness D r and the leading edge ellipse aspect ratio .

[0051] Figure 4 The figure is a schematic diagram of a supersonic blade channel designed based on characteristic lines and free vortexes. The relative inlet Mach number and inlet geometry angle of the blade are the same as the relative outlet Mach number and outlet geometry angle. The supersonic blade channel consists of an inlet transition zone (DCW and ABW), a free vortex flow zone (CWAFVH) and an outlet transition zone (HVI and FVG). The inlet transition zone converts the inlet uniform flow into a free vortex flow, and increases the inlet Mach number to the specified suction surface free vortex Mach number Ma on the suction surface (BA) through the characteristic line method. s , on the pressure surface (DC) the inlet Mach number is reduced to the specified pressure surface free vortex Mach number Ma by the characteristic line method pThe airflow in the free vortex flow area (CWAFVH) maintains free vortex flow, and the airflow on the suction surface and pressure surface maintains the specified Mach number Ma s and Ma p The outlet transition zone converts the free vortex flow into a uniform flow at the outlet, and the free vortex Mach number Ma on the suction surface (FG) is converted to the uniform flow at the outlet by the characteristic line method. s Reduce to the outlet Mach number, and use the characteristic line method on the pressure surface (HI) to convert the pressure surface free vortex Mach number Ma p Increase to the exit Mach number.

[0052] In the free vortex flow region, the dimensionless arc radius of the suction and pressure surfaces is determined according to the arc Mach number and free vortex equation specified by the suction and pressure surfaces ( and By specifying the Mach number of the free vortex flow area on the suction side and the pressure side, the load distribution in the moving blade channel can be designed. The free vortex equation is:

[0053]

[0054]

[0055] in is the dimensionless velocity, is the dimensionless radius, M f is the Mach number in the free vortex flow region.

[0056] The supersonic blade inlet Mach number Ma is known i and inlet geometry , the dimensionless radius in the free vortex channel (CWAFVH) The profiles of the pressure surface (DC) and suction surface (BA) of the inlet transition section are obtained by the characteristic line method using the Mach number and the Mach number as the initial conditions. Similarly, the outlet transition section is known with the outlet Mach number and the outlet geometry angle, and the dimensionless radius in the free vortex channel (CWAFVH) is used. The profiles of the pressure surface (HI) and suction surface (FG) of the outlet transition section are obtained respectively by the characteristic line method with the Mach number and the initial conditions.

[0057] Figure 5 This is a schematic diagram of the supersonic rotor blade design. Figure 3 The suction and pressure surface lines are obtained from the calculation, and the supersonic rotor blade profile is connected by straight lines (EB and GJ) to complete the supersonic rotor blade profile design without leading edge thickness. The dimensionless pitch of the rotor blade is then calculated. . According to the specified blade pitch and dimensionless pitch The blade profile is scaled to obtain the designed supersonic rotor blade coordinates.

[0058] To design a supersonic rotor blade with leading and trailing edge thickness, the suction side and the pressure side are moved in opposite directions by a distance equal to the leading edge thickness D. r Half of the ellipse. Then, according to the aspect ratio Taking the leading edge thickness as the minor axis, a complete supersonic blade profile is obtained by using half elliptical arcs (ED and IJ) to close the blade profile.

[0059] Example 1-3, given supersonic stator blade pitch , Mach number at the stator blade outlet e , stator blade outlet geometric angle Specific heat ratio of working fluid , leading edge radius R L , leading edge arc angle and trailing edge radius R T Table 1 shows the specific data corresponding to three groups of different design parameters of three embodiments. Figure 6 The profiles of supersonic stator blades with three different design parameters are given.

[0060] Table 1 Design parameters of supersonic stator blades in Examples 1-3

[0061]

[0062] Example 4-6, given the supersonic rotor blade pitch S r , inlet geometry angle 、Inlet Mach number Ma i Specific heat ratio of working fluid 、Suction surface free vortex Mach number Ma s 、Pressure surface free vortex Mach Ma p and leading edge thickness D r Table 2 shows the specific data corresponding to three different sets of design parameters of three embodiments. Figure 7 The profiles of supersonic rotor blades under three different design parameters are given.

[0063] Table 2 Design parameters of supersonic rotor blades for Examples 4-6

[0064]

[0065] The above embodiments are only used to illustrate the present invention. Any equivalent transformations and improvements based on the technical solution of the present invention should not be excluded from the protection scope of the present invention.

Claims

1. A method for designing a supersonic axial flow turbine blade profile for a turbopump, characterized in that: Including supersonic stator and supersonic rotor blade design; S1. Supersonic stator blade profile design specifically includes the following sub-steps: Given parameters for supersonic stator blade design: blade pitch , Mach number at the stator blade outlet e , geometric angle of stator blade outlet Specific heat ratio of working fluid , leading edge radius R L , leading edge arc angle and trailing edge radius R T ; S1.1 Based on a given stator blade outlet Mach number Ma e , the characteristic line method is used to generate the nozzle profile of the expansion section; the discretized characteristic line equation is used to determine the node position coordinates, and the compatibility equation determines the node flow parameters; S1.2 The nozzle profile of the expansion section is symmetrical along the axis to generate a complete nozzle profile; the nozzle inlet is the throat of the stator blade, and the initial width d t =1; after the axisymmetric expansion section nozzle is scaled to the specified throat width, its symmetry axis is rotated After the degree is reached, the direction is kept consistent with the geometric angle of the stator blade outlet; The upper half of the nozzle is the initial profile of the suction surface, and the lower half is the translation blade pitch. As the initial profile of the pressure surface; according to the given blade pitch, trailing edge radius, stationary blade outlet geometric angle and nozzle expansion ratio, the actual throat width d is calculated as the scaling ratio of the nozzle; S1.3 On the supersonic vane pressure surface, the nozzle inlet of the diverging section, i.e., the vane throat, is connected to the leading edge via a circular arc and a polynomial curve, and the nozzle outlet of the diverging section is directly connected to the trailing edge; On the suction side of the supersonic stator blade, the inlet of the diverging section nozzle, i.e. the throat of the stator blade, is directly connected to the leading edge, while the outlet of the diverging section nozzle is connected to the trailing edge through a straight section; S1.4 The arc at the inlet of the expansion section of the pressure surface of the supersonic stator blade has the same radius as the leading edge, ensuring that the curvature on both sides of the throat inlet is the same; The axial chord length and circumferential chord length of the supersonic stator blade profile are expressed by analytical expressions: ； ； in, ; d is the throat width, K is the ratio of the nozzle outlet width to the inlet width of the expansion section; S2. Supersonic rotor blade profile design specifically includes the following sub-steps: Given parameters for supersonic rotor blade design: blade pitch , inlet geometry angle , relative inlet Mach number Ma i Specific heat ratio of working fluid 、Suction surface free vortex Mach number Ma s 、Pressure surface free vortex Mach Ma p , leading edge thickness D r and the leading edge ellipse aspect ratio ; S2.1 The supersonic rotor passage consists of an inlet transition zone, a free vortex flow zone, and an outlet transition zone. The rotor's relative inlet Mach number and inlet geometry are the same as their relative outlet Mach number and outlet geometry. The inlet transition zone converts the inlet uniform flow into free vortex flow, where the suction surface increases the inlet Mach number to the suction surface free vortex Mach number Ma by the characteristic line method. s , on the pressure surface, the inlet Mach number is reduced to the pressure surface free vortex Mach number Ma by the characteristic line method p ; The airflow in the free vortex flow area maintains free vortex flow, and the airflow on the suction surface and pressure surface respectively maintains the specified suction surface free vortex Mach number Ma s and pressure surface free vortex Ma p Complete the turning of airflow; The outlet transition zone converts the free vortex flow into a uniform flow at the outlet, and the free vortex Mach number Ma on the suction surface is calculated by the characteristic line method. s Reduce to the outlet Mach number, and use the characteristic line method on the pressure surface to convert the pressure surface free vortex Mach number Ma p Increase to the exit Mach number; S2.2 In the free vortex flow region, determine the dimensionless arc radius of the suction and pressure surfaces based on the specified free vortex Mach numbers and free vortex equations on the suction and pressure surfaces; design the load distribution in the rotor blade passage by specifying the free vortex Mach numbers on the suction and pressure surfaces; S2.3 Given the supersonic rotor relative inlet Mach number and inlet geometry, the dimensionless radius in the free vortex channel and Mach number as initial conditions and the profiles of the pressure surface and suction surface of the inlet transition section are obtained by the characteristic line method. Given the supersonic blade outlet Mach number and outlet geometry angle, the dimensionless radius in the free vortex channel and Mach number as initial conditions and the profiles of the pressure surface and suction surface of the outlet transition section are obtained by the characteristic line method; Finally, the suction side and the pressure side are connected by a straight line to complete the supersonic rotor blade design with no leading edge thickness. S2.4 Obtain the dimensionless coordinates of the supersonic rotor blades using the characteristic line method and the free vortex method; calculate the dimensionless pitch of the rotor blades ; According to the specified blade pitch S r and dimensionless pitch Compare and scale the blade profile to obtain the designed supersonic rotor blade coordinates; S2.5 designs supersonic rotor blades with leading and trailing edge thicknesses, moving the suction and pressure surfaces in opposite directions by the leading edge thickness D r Half the distance; based on the leading edge ellipse aspect ratio , taking the leading edge thickness as the minor axis, a complete supersonic rotor blade profile is obtained by using a half-elliptical arc to close the blade profile; S3. After the supersonic stationary and supersonic rotor blade profiles are designed, the leading edges of the blades with different cross-sections are aligned radially, and the overall three-dimensional configuration of the blades is determined by interpolation.

2. The design method according to claim 1, wherein: In step S1.1, the discretized characteristic line equation and compatibility equation are: ； ； in is the slope of the characteristic line, is the geometric angle of the profile, is the Mach angle, is the fluid velocity, M n Calculate the Mach number at the point for the nozzle.

3. The design method according to claim 1, wherein: In step S1.2, the scaling ratio of the nozzle is calculated as: ； Where K is the ratio of the nozzle outlet width to the inlet width of the expansion section.

4. The design method according to claim 1, wherein: In step S1.3, both the leading edge and the trailing edge are designed with arcs, where the leading edge arc angle , the leading edge radius is R L ; The trailing edge is a semicircle with a radius of R T .

5. The design method according to claim 1, wherein: In step S2.2, the free vortex equation is: ； ； in is the dimensionless velocity, is the dimensionless radius, M f is the Mach number in the free vortex flow region.

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