US20050232778A1

US20050232778A1 - Blade shape creation program and method

Info

Publication number: US20050232778A1
Application number: US11/092,132
Authority: US
Inventors: Naoya Kakishita; Itsuhei Kori
Original assignee: Mitsubishi Fuso Truck and Bus Corp
Current assignee: Mitsubishi Fuso Truck and Bus Corp
Priority date: 2004-03-30
Filing date: 2005-03-29
Publication date: 2005-10-20
Also published as: JP2005282492A; DE102005014871A1

Abstract

In a blade shape creation program and method, a blade thickness function defining equation is constructed by a cubic function as a first function defining a leading edge blade thickness function on a leading edge side of a maximum blade thickness point of the blade thickness function, and a cubic function as a second function defining a trailing edge blade thickness function on a trailing edge side of the maximum blade thickness point; is defined, with a camber line length, a position of maximum blade thickness, a maximum blade thickness value, a leading edge blade thickness change rate, a trailing edge blade thickness change rate, a leading edge blade thickness value, and a trailing edge blade thickness value being taken as design factors, and has a boundary condition that the first function and the second function have tangents continuous with each other at the maximum blade thickness point.

Description

CROSS REFERENCE TO RELATED APPLICATION

The entire disclosure of Japanese Patent Application No. 2004-099031 filed on Mar. 30, 2004, including specification, claims, drawings and summary, is incorporated herein by reference in its entirety.

BACKGROUND OF THE INVENTION

1. Field of the Invention
This invention relates to a blade shape creation program and method for creating the blade shape of a cooling fan or the like.
2. Description of the Related Art
When the blade shape of a cooling fan installed in a vehicle is to be created (drawn) in designing the cooling fan, for example, the first step is to create (draw) the cross-sectional shapes of a blade at a plurality of locations in the hub diameter direction of the blade. Then, based on these cross-sectional shapes of the blade, the entire shape of the blade (visible outline and exterior surface) is created (drawn) by spline interpolation or the like. A method using “Joukowski airfoil” shown, for example, in the following document is named as one of ordinary methods for drawing the cross-sectional shape of the blade:

- T. Fujimoto, “2nd Revision of Fluid Dynamics”, 2nd Revision, 6th Edition, YOKENDO Co., Ltd., published Jan. 20, 1992, p. 141”

An outline of this method will be shown in FIGS. 11(a) and 11(b). The “Joukowski airfoil” is an airfoil (cross-sectional shape of blade) 3 as shown in FIG. 11(b), which is obtained by-the coordinate transformation (mapping) of a combination of two circles 1 and 2 with centers M and M′, as shown in FIG. 11(a), by the equation (1) offered below. To change the airfoil profile (cross-sectional shape of blade) in this case, the shapes of the two circles 1 and 2 before coordinate transformation are adjusted. The method using “Joukowski airfoil” is one of general methods for drawing an “average camber curve (camber line)”, which is a basic skeleton of the cross-sectional shape of the blade. According to this method, the central points of the airfoil (cross-sectional shape of blade) 3 shown in FIG. 11(b) are connected to draw a camber line 4. $\begin{matrix} z = ζ + \frac{a^{2}}{ζ}, a = \frac{c}{4} & (1) \end{matrix}$
To improve the performance of the blade (lift performance and drag performance), it is necessary to change (adjust) the shape of the section of the blade contour (airfoil) (i.e., blade profile), and study influence on the performance of the blade. For this purpose, it is effective to individually change (adjust) a plurality of design factors (details to be described later), which determine the blade profile, thereby directly investigating the degree of contribution of each design factor to the performance of the blade. Particularly, the ability to change each design factor, independently of one another, on the leading edge side of a maximum blade thickness point (see FIG. 3, details to be described later) of a blade thickness function, which represents a change in the blade thickness at a section of the blade, and on the trailing edge side of the maximum blade thickness point would be very effective for studying the performance of the blade.
However, conventional methods, such as the method using “Joukowski airfoil”, pose difficulty in changing each design factor independently. Needless to say, changing each design factor, independently on the leading edge side and the trailing edge side of the blade thickness function, is also difficult.
The present invention has been accomplished in light of the above-described circumstances. It is an object of the present invention to provide a blade shape creation program and method capable of changing a plurality of design factors, which determine the blade profile (airfoil), on the leading edge side and the trailing edge side of the blade thickness function, with the leading edge side and the trailing edge side being separated from each other, in changing (adjusting) the airfoil.
It is another object of the present invention to provide a blade shape creation program and method capable of changing a plurality of design factors, which determine the blade profile (airfoil), independently on the leading edge side and the trailing edge side of the blade thickness function, in changing (adjusting) the airfoil, and also capable of reliably checking the created airfoil based on numerical values, without relying on visual checks.

SUMMARY OF THE INVENTION

A first aspect of the present invention, for attaining the above object, is a blade shape creation program for creating a blade shape on a space virtually defined by a computer, wherein a blade thickness function defining equation for defining a blade thickness function representing a change in a blade thickness to be defined on a-cross section of the blade shape is constructed by a first function which defines a leading edge blade thickness function on a leading edge side of a maximum blade thickness point of the blade thickness function, and a second function which defines a trailing edge blade thickness function on a trailing edge side of the maximum blade thickness point of the blade thickness function.
A second aspect of the present invention is the blade shape creation program according to the first aspect, wherein the blade thickness function defining equation has the first function and the second function each defined by a cubic function, is defined, with a camber line length of a section of the blade shape, a position of maximum blade thickness, a maximum blade thickness value, a leading edge blade thickness change rate, a trailing edge blade thickness change rate, a leading edge blade thickness value, and a trailing edge blade thickness value being taken as design factors, and has a boundary condition that the first function and the second function have tangents continuous with each other at the maximum blade thickness point.
A third aspect of the present invention is a blade shape creation method for creating a blade shape on a virtually defined space, wherein a blade thickness function defining equation for defining a blade thickness function representing a change in a blade thickness to be defined on a cross section of the blade shape is constructed by a first function which defines a leading edge blade thickness function on a leading edge side of a maximum blade thickness point of the blade thickness function, and a second function which defines a trailing edge blade thickness function on a trailing edge side of the maximum blade thickness point of the blade thickness function.
A fourth aspect of the present invention is the blade shape creation method according to the third aspect, wherein the blade thickness function defining equation has the first function and the second function each defined by a cubic function, is defined, with a camber line length of a section of the blade shape, a position of maximum blade thickness, a maximum blade thickness value, a leading edge blade thickness change rate, a trailing edge blade thickness change rate, a leading edge blade thickness value, and a trailing edge blade thickness value being taken as design factors, and has a boundary condition that the first function and the second function have tangents continuous with each other at the maximum blade thickness point.
A fifth aspect of the present invention is a blade shape creation program for creating a blade shape on a space virtually defined by a computer, wherein a blade thickness function defining equation for defining a blade thickness function representing a change in a blade thickness to be defined on a cross section of the blade shape is constructed by a first function which defines a leading edge blade thickness function on a leading edge side of a maximum blade thickness point of the blade thickness function, and a second function which defines a trailing edge blade thickness function on a trailing edge side of the maximum blade thickness point of the blade thickness function; and in the first function and the second function of the blade thickness function defining equation, a value of the blade thickness is calculated over an entire region of the blade thickness function, and the calculated blade thickness value is compared with a maximum blade thickness value set as a design factor to check whether the blade thickness function has a blade thickness value larger than the maximum blade thickness value.
A sixth aspect of the present invention is a blade shape creation program for creating a blade shape on a space virtually defined by a computer, wherein a blade thickness function defining equation for defining a blade thickness function representing a change in a blade thickness to be defined on a cross section of the blade shape is constructed by a first function which defines a leading edge blade thickness function on a leading edge side of a maximum blade thickness point of the blade thickness function, and a second function which defines a trailing edge blade thickness function on a trailing edge side of the maximum blade thickness point of the blade thickness function; and the first function and the second function of the blade thickness function defining equation are differentiated to check over an entire region of the blade thickness function whether the blade thickness function has a maximum or minimum point or an inflection point at a position other than a position of maximum blade thickness set as a design factor.
A seventh aspect of the present invention is the blade shape creation program according to the fifth or sixth aspect, wherein the blade thickness function defining equation has the first function and the second function each defined by a cubic function, is defined, with a camber line length of a section of the blade shape, a position of maximum blade thickness, a maximum blade thickness value, a leading edge blade thickness change rate, a trailing edge blade thickness change rate, a leading edge blade thickness value, and a trailing edge blade thickness value being taken as design factors, and has a boundary condition that the first function and the second function have tangents continuous with each other at the maximum blade thickness point.
An eighth aspect of the present invention is the blade shape creation program according to any one of the fifth to seventh aspects, wherein results of checking whether the blade thickness function has a blade thickness value larger than the maximum blade thickness value, or results of checking whether the blade thickness function has a maximum or minimum point or an inflection point at a position other than the position of maximum blade thickness are displayed on a checklist window.
A ninth aspect of the present invention is a blade shape creation method for creating a blade shape on a virtually defined space, wherein a blade thickness function defining equation for defining a blade thickness function representing a change in a blade thickness to be defined on a cross section of the blade shape is constructed by a first function which defines a leading edge blade thickness function on a leading edge side of a maximum blade thickness point of the blade thickness function, and a second function which defines a trailing edge blade thickness function on a trailing edge side of the maximum blade thickness point of the blade thickness function; and in the first function and the second function of the blade thickness function defining equation, a value of the blade thickness is calculated over an entire region of the blade thickness function, and the calculated blade thickness value is compared with a maximum blade thickness value set as a design factor to check whether the blade thickness function has a blade thickness value larger than the maximum blade thickness value.
A tenth aspect of the present invention is a blade shape creation method for creating a blade shape on a virtually defined space, wherein a blade thickness function defining equation for defining a blade thickness function representing a change in a blade thickness to be defined on a cross section of the blade shape is constructed by a first function which defines a leading edge blade thickness function on a leading edge side of a maximum blade thickness point of the blade thickness function, and a second function which defines a trailing edge blade thickness function on a trailing edge side of the maximum blade thickness point of the blade thickness function; and the first function and the second function of the blade thickness function defining equation are differentiated to check over an entire region of the blade thickness function whether the blade thickness function has a maximum or minimum point or an inflection point at a position other than a position of maximum blade thickness set as a design factor.
An eleventh aspect of the present invention is the blade shape creation method according to the ninth or tenth aspect, wherein the blade thickness function defining equation has the first function and the second function each defined by a cubic function, is defined, with a camber line length of a section of the blade shape, a position of maximum blade thickness, a maximum blade thickness value, a leading edge blade thickness change rate, a trailing edge blade thickness change rate, a leading edge blade thickness value, and a trailing edge blade thickness value being taken as design factors, and has a boundary condition that the first function and the second function have tangents continuous with each other at the maximum blade thickness point.
A twelfth aspect of the present invention is the blade shape creation method according to any one of the ninth to eleventh aspects, wherein results of checking whether the blade thickness function has a blade thickness value larger than the maximum blade thickness value, or results of checking whether the blade thickness function has a maximum or minimum point or an inflection point at a position other than the position of maximum blade thickness are displayed on a checklist.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention will become more fully understood from the detailed description given hereinbelow and the accompanying drawings which are given by way of illustration only, and thus are not limitative of the present invention, and wherein:
FIG. 1 is an external outline view of a personal computer for executing a blade shape creation program according to an embodiment of the present invention;
FIG. 2A is a front view of a cooling fan, and FIG. 2B is a side view of the cooling fan (a view taken in the direction of A in FIG. 2A);
FIG. 3 is an explanation drawing of design factors for determining a blade profile (airfoil), and a coordinate system (blade thickness function drawing method) used when drawing a blade thickness function by a cubic function;
FIG. 4 is a view showing an example of drawing the blade thickness function when only a leading edge blade thickness change rate is changed;
FIG. 5 is a view showing an example of drawing a blade thickness function in which the blade thickness value of a blade thickness point other than a set maximum blade thickness point is greater than the maximum blade thickness value of the maximum blade thickness point;
FIG. 6 is a view showing an example of drawing a blade thickness function which has inflection points at blade thickness points other than a set maximum blade thickness point;
FIG. 7 is a view showing an example in which a blade section extends beyond a hub;
FIG. 8 is a view showing an example of a checklist window;
FIG. 9 is a view showing an example of drawing a blade thickness function of a shape in which there is no problem in a maximum blade thickness value;
FIG. 10 is a view showing an example of drawing a blade thickness function of a delicate shape in which the blade thickness value of a blade thickness point other than a set maximum blade thickness point is slightly greater than the maximum blade thickness value of the maximum blade thickness point; and
FIG. 11 is an explanation drawing showing a method of drawing a blade profile with the use of “Joukowski airfoil”.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

Embodiments of the present invention will now be described in detail with reference to the accompanying drawings. The application of the blade shape creation program according to the present invention to the creation of the blade shape of a cooling fan will be taken as an example for explanation.
FIG. 1 is an external outline view of a personal computer for executing the blade shape creation program according to an embodiment of the present invention. FIG. 2A is a front view of a cooling fan, and FIG. 2B is a side view of the cooling fan (a view taken in the direction of A in FIG. 2A).
As shown in FIG. 1, a personal computer 11 has a computer body 12, and peripheral instruments connected to the computer body 12, such as a keyboard 13 as an input means, and a display device 14 as a display means, for example, CRT or a liquid crystal display.
The computer body 12 is equipped with a CPU, a hard disk (HD) drive, and a compact disk (CD) drive, and the CPU executes a blade shape creation program P (software) stored in storage media such as HD and CD. The blade shape creation program P is a program for creating a blade shape on a space virtually defined by the personal computer 11. This program can change a plurality of design factors, which determine a blade profile (sectional shape of a blade; airfoil), independently of each other, in changing the blade profile, although details of the program will be described later.
The keyboard 13 is used to enter data for execution of the blade shape creation program P into the computer body 12. The display device 14 is used for displaying on a display screen 15 the data entered from the keyboard 13 into the computer body 12, and the results of execution of the blade shape creation program P in the computer body 12. For example, the display device 14 displays a checklist window 16 (details to be described later).
FIGS. 2A and 2B show an example of a cooling fan loaded on a vehicle. A cooling fan 21 illustrated in FIGS. 2A and 2B comprises a plurality of (eight in the illustrate example) blades 23 provided on an outer peripheral surface 22 a of a cylindrical hub 22. The cooling fan 21 has a rotating shaft (not shown) connected, for example, to a rotating shaft of an engine of the vehicle, and rotationally driven thereby. In the side view of FIG. 2B, each blade 23 is provided on the outer peripheral surface 22 a of the hub such that its chord is inclined at a predetermined blade inclination angle with respect to a hub center axis B (see FIG. 7). The exterior shape of the blade 23 is not limited to the illustrated one, but is available in various types.
In creating (drawing) the blade shape of each blade 23 of the cooling fan 21 for designing the cooling fan 21, the present embodiment is arranged to execute the blade shape creation program P on the personal computer 11, thereby deriving a blade thickness function representing a change in the blade thickness in a blade section, and creating (drawing) a blade profile having a blade thickness calculated by the blade thickness function in connection with a separately designated camber line.
The blade thickness function creation capability (program), blade thickness function checking capability (program), and checklist window display capability (program) of the blade shape creation program P will be described in detail based on FIGS. 3 to 10.
FIG. 3 is an explanation drawing of design factors for determining a blade profile (airfoil), and a coordinate system (blade thickness function drawing method) used when drawing a blade thickness function by a cubic function. FIG. 4 is a view showing an example of drawing the blade thickness function when only a leading edge blade thickness change rate is changed. FIG. 5 is a view showing an example of drawing an airfoil and a blade thickness function in which the blade thickness value of a blade thickness point other than a set maximum blade thickness point is greater than the maximum blade thickness value of the maximum blade thickness point. FIG. 6 is a view showing an example of drawing and airfoil and a blade thickness function which have inflection points at blade thickness points other than a set maximum blade thickness point. FIG. 7 is a view showing an example in which a blade section extends beyond a hub. FIG. 8 is a view showing an example of a checklist window. FIG. 9 is a view showing an example of drawing an airfoil and a blade thickness function of a shape in which there is no problem in a maximum blade thickness value. FIG. 10 is a view showing an example of drawing an airfoil and a blade thickness function of a delicate shape in which the blade thickness value of a blade thickness point other than a set maximum blade thickness point is slightly greater than the maximum blade thickness value of the maximum blade thickness point.
The blade thickness function creation capability of the blade shape creation program P will be described first of all.
In providing the blade thickness function creation (drawing) capability, the following seven design factors (1) to (7) were selected as optimal (basic) design factors for determining the blade profile (airfoil) (see FIG. 3):

- (1) Camber line length Lc
- (2) Position of maximum blade thickness x_Tmax
- (3) Maximum blade thickness value y_Tmax
- (4) Leading edge blade thickness change rate α
- (5) Trailing edge blade thickness change rate β
- (6) Leading edge blade thickness value Tf
- (7) Trailing edge blade thickness value Tb

As shown in FIG. 3, a camber line 31 is a line formed by connecting the centers of blade thicknesses B of a blade section (airfoil) 32, and a camber line length Lc refers to the length of the camber line 31. The blade thickness B of the blade section 32 is a blade thickness in a direction perpendicular to a tangent to the camber line 31 at each camber point SP on the camber line 31. A blade thickness function 33 represents a change in the blade thickness B, namely, a change over the range from a leading edge 31 a of the camber line 31 (leading edge 32 a of the blade section 32) to a trailing edge 31 b of the camber line 31 (trailing edge 32 b of the blade section 32). The leading edge 31 a of the camber line 31 is a site where airflow enters, while the trailing edge 31 b of the camber line 31 is a site where airflow exits.
To express the blade thickness function 33 by an x-y coordinate system, a coordinate axis representing the position of the camber line 31 in the camber line length direction is designated as an x-axis, the leading edge 31 a of the camber line 31 is taken as the origin of the x-axis, and a coordinate axis representing the magnitude of the blade thickness B is designated as a y-axis. A maximum blade thickness value y_Tmaxis the maximum value of the blade thickness B. Each point on the blade thickness function 33 is called a blade thickness point BP and, of these blade thickness points BP's, the point at which the blade thickness B takes the maximum blade thickness value y_Tmaxis called a maximum blade thickness point BPM. In the x-y coordinate system, the blade thickness B is expressed by the y-coordinate, and refers to the length of a perpendicular dropped from each blade thickness point BP on the blade thickness function 33 to the x-axis. The position of maximum blade thickness x_Tmaxis the position in the camber line direction (x-axis direction) at which the blade thickness B takes the maximum blade thickness value y_Tmax.
A leading edge blade thickness change rate α is the change rate of the blade thickness B at the leading edge 33 a of the blade thickness function 33, and refers to an angle which a tangent 33 c at the leading edge 33 a of the blade thickness function 33 makes with a line 34 a parallel to the x-axis. A trailing edge blade thickness change rate β is the change rate of the blade thickness B at the trailing edge 33 b of the blade thickness function 33, and refers to an angle which a tangent 33 d at the trailing edge 33 b of the blade thickness function 33 makes with a line 34 b parallel to the x-axis. A leading edge blade thickness value Tf is the value of the blade thickness B at the leading edge 33 a of the blade thickness function 33. The leading edge blade thickness value Tf may be zero when a leading edge portion of the blade section 32 is arcuate. Alternatively, the leading edge blade thickness value Tf may be some value when the leading edge portion is flat, as in the illustrated example. A trailing edge blade thickness value Tb is the value of the blade thickness B at the trailing edge 33 b of the blade thickness function 33. The trailing edge blade thickness value Tb may be zero when a trailing edge portion of the blade section 32 is at an acute angle. Alternatively, the trailing edge blade thickness value Tb may be some value when the trailing edge portion is flat, as in the illustrated example.
An equation for defining the blade thickness function 33, which represents a change in the blade thickness B to be defined on the cross section 32 of the blade shape, is constructed by a first function which defines a leading edge blade thickness function on the leading edge side of the maximum blade thickness point BPM of the blade thickness function 33, and a second function which defines a trailing edge blade thickness function on the trailing edge side of the maximum blade thickness point BPM on the blade thickness function 33. That is, as shown in FIG. 3, the blade thickness function 33 is divided into a leading edge side and a trailing edge side, with the maximum blade thickness point BPM as a boundary. A cubic function of an equation (2) is selected as a first function which defines (represents) a leading edge blade thickness function 33A on the leading edge side of the maximum blade thickness point BPM, while a cubic function of an equation (3) is selected as a second function which defines (represents) a trailing edge blade thickness function 33B on the trailing edge side of the maximum blade thickness point BPM.
y _L =a _L x _L ³ +b _L x _L ² +c _L x _L +d _L (2)
y _T =a _T x _T ³ +b _T x _T ² +c _T x _T +d _T (3)
The reason for selecting the cubic functions as the first function and the second function is that the aforementioned seven design factors are selected as the optimal design factors determining the shape of the blade section 32 (the shape of the blade thickness function 33), whereby the eight constraints (1) to (8) to be indicated below can be set based on these design factors. That is, of the following eight constraints (1) to (8), the four constrains (1), (3), (5) and (7) can be set for the leading edge side of the blade thickness function 33, while the other four constrains (2), (4), (6) and (8) can be set for the trailing edge side of the blade thickness function 33. In accordance with these constraints, therefore, the respective coefficients (a_L, b_L, c_L, d_L, a_T, b_T, C_T, d_T) of the cubic functions of the equations (2) and (3) can all be uniquely determined. The constrains (1) to (4) are the constraints concerned with the transit points of the blade thickness function 33, while the constraints (5) to (8) are the constraints about the gradient of the tangents at the transit points of the blade thickness function 33.
If the number of the design factors (constraints) is small, quadratic functions may be used as the first and second functions. If the number of the design factors (constraints) is large, functions of fourth or higher order may be used. However, if the number of the design factors (constraints) is small, sufficient adjustment of an airfoil cannot be made. Too large a number of the design factors (constraints) would wastefully render the equations of the functions complicated. Thus, it would be best to select, as the first function and the second function, cubic functions which are suitable for the seven design factors (camber line length Lc, position of maximum blade thickness x_Tmax, maximum blade thickness value y_Tmax, leading edge blade thickness change rate α, trailing edge blade thickness change rate β, leading edge blade thickness value Tf, trailing edge blade thickness value Tb) optimal as design factors for determining the blade profile (airfoil).

- (1) When x_L=0, y_L=Tf: Leading edge position (leading edge blade thickness value)
- (2) When x_T=Lc, y_T=Tb: Trailing edge position (camber line length, trailing edge blade thickness value)
- (3) When x_L=x_Tmax, y_L=y_Tmax: Position of maximum blade thickness, maximum blade thickness value
- (4) When x_T=x_Tmax, y_T=y_Tmax: Position of maximum blade thickness, maximum blade thickness value
- (5) When x_L=0, dy_L/dx_L=tan α: Leading edge blade thickness change rate
- (6) When x_T=Lc, dy_T/dx_T=tan(−β): Trailing edge blade thickness change rate
- (7) When x_L=x_Tmax, dy_L/dx_L=0: Position of maximum blade thickness (gradient of tangent)
- (8) When x_T=x_Tmax, dy_T/dx_T=0: Position of maximum blade thickness (gradient of tangent)

The constraint (1) is a constraint on the leading edge position of the blade thickness function 33 (leading edge blade thickness value Tf) for the equation (2). When x_L=0, namely, at the position of the leading edge 33 a of the blade thickness function 33, the blade thickness value y_L=Tf. The constraint (2) is a constraint on the trailing edge position of the blade thickness function 33 (camber line length Lc, trailing edge blade thickness value Tb) for the equation (3). When x_T=Lc (camber line length), namely, at the position of the trailing edge 33 b of the blade thickness function 33, the blade thickness value y_T=Tb. The constraint (3) is a constraint on the position of maximum blade thickness x_Tmaxand the maximum blade thickness value y_Tmaxof the blade thickness function 33 for the equation (2). The constraint (4) is a constraint on the position of maximum blade thickness x_Tmaxand the maximum blade thickness value y_Tmaxof the blade thickness function 33 for the equation (3). The constraint (5) is a constraint on the leading edge blade thickness change rate α of the blade thickness function 33 for the equation (2), namely, a constraint on the gradient of the tangent at the position of the leading edge 33 a of the blade thickness function 33. The constraint (6) is a constraint on the trailing edge blade thickness change rate β of the blade thickness function 33 for the equation (3), namely, a constraint on the gradient of the tangent at the position of the trailing edge 33 b of the blade thickness function 33.
The constraint (7) is a constraint on the gradient of the tangent at the position of maximum blade thickness x_Tmax, i.e., at the maximum blade thickness point BPM of the blade thickness function 33, for the equation (2). The constraint (8) is a constraint on the gradient of the tangent at the position of maximum blade thickness x_Tmax, i.e., at the maximum blade thickness point BPM of the blade thickness function 33, for the equation (3). Under the constrains (7) and (8), the gradient of the tangent at the position of maximum blade thickness x_Tmax(maximum blade thickness point BPM) is zero, i.e., dy_L/dx_L=0. This is because unless the gradient of the tangent at the position of maximum blade thickness x_Tmax(maximum blade thickness point BPM) is zero, the value of the blade thickness B (y_L, y_T) at the set maximum blade thickness point BPM is not maximal. The constrains (7) and (8) also mean that the maximum blade thickness value at the position of maximum blade thickness x_Tmax(maximum blade thickness point BPM) is similarly y_Tmax, and the gradient of the tangent (dy_L/dx_L, dy_T/dx_T) is similarly zero, showing that the equation (2) of the first function and the equation (3) of the second function have the boundary condition that their tangents are continuous with each other at the maximum blade thickness point BPM.
Based on the above constraints (1) to (8), the respective design factors (camber line length Lc, position of maximum blade thickness x_Tmax, maximum blade thickness value y_Tmax, leading edge blade thickness change rate α, trailing edge blade thickness change rate β, leading edge blade thickness value Tf, trailing edge blade thickness value Tb) are set (changed) independently of each other to find the respective coefficients (a_L, b_L, c_L, d_L, a_T, b_T, c_T, d_T) of the cubic functions of the equations (2) and (3). By so doing, the leading edge blade thickness function 33A can be defined (drawn) based on the cubic function of the equation (2), and the trailing edge blade thickness function 33B can be defined (drawn) based on the cubic function of the equation (3). By combining the cubic functions of the equations (2) and (3), the whole of the blade thickness function 33 can be defined (drawn).
The relationships between the respective coefficients (a_L, b_L, c_L, d_L, a_T, b_T, c_T, d_T) of the cubic functions of the equations (2) and (3) and the respective design factors (camber line length Lc, position of maximum blade thickness x_Tmax, maximum blade thickness value y_Tmax, leading edge blade thickness change rate α, trailing edge blade thickness change rate β, leading edge blade thickness value Tf, trailing edge blade thickness value Tb) are as indicated by the equations (4) to (11) offered below. To avoid the complexity of the indications of the equations, the equations (9), (10) and (11) for b_T, c_Tand d_Tinclude a_T. However, since a_Tis a function involving only the design factors as in the equation (8), b_T, c_Tand d_Tcan also be regarded as functions composed of the design factors alone.
As the following equations (4) to (7) show, the respective coefficients (a_L, b_L, c_L, d_L) of the equation (2) for the cubic function on the leading edge side can be uniquely determined by determining the position of maximum blade thickness x_Tmax, maximum blade thickness value y_Tmax, leading edge blade thickness change rate α, and leading edge blade thickness value Tf as the design factors. As the following equations (8) to (11) show, the respective coefficients (a_T, b_T, c_T, d_T) of the equation (3) for the cubic function on the trailing edge side can be uniquely determined by determining the camber line length Lc, position of maximum blade thickness x_Tmax, maximum blade thickness value y_Tmax, trailing edge blade thickness change rate β, and trailing edge blade thickness value Tb as the design factors. The procedure for deriving the following relational expressions (4) to (11) will be described later. $\begin{matrix} a_{L} = \frac{- 2 y_{T \max} + x_{T \max} \cdot \tan α + T_{f}}{x_{T \max}^{3}} & (4) \\ b_{L} = \frac{y_{T \max} - T_{f}}{x_{T \max}^{2}} - \frac{\tan α}{x_{T \max}} - x_{T \max} (\frac{\begin{matrix} - 2 y_{T \max} + x_{T \max} \cdot \\ \tan α + T_{f} \end{matrix}}{x_{T \max}^{3}}) & (5) \\ c_{L} = \tan α & (6) \\ d_{L} = T_{f} & (7) \\ a_{T} = - \frac{(L_{c} - x_{T \max}) \cdot \tan (- β) + 2 y_{T \max} - 2 T_{b}}{{(x_{T \max} - L_{c})}^{3}} & (8) \\ b_{T} = - \frac{3}{2} (L_{c} + x_{T \max}) \cdot a_{T} + \frac{\tan (- β)}{2 (L_{c} - x_{T \max})} & (9) \\ c_{T} = a_{T} \cdot (\frac{1}{2} L_{c}^{} + 2 L_{c} \cdot x_{T \max} + \frac{1}{2} x_{T \max}^{2}) - & (10) \\ \frac{1}{L_{c} - x_{T \max}} (\frac{(L_{c} + x_{T \max}) \cdot \tan (- β)}{2} + y_{T \max} - T_{b}) \\ d_{T} = \frac{1}{6} (x_{T \max}^{3} - 2 x_{T \max} \cdot L_{c}^{2} - 5 x_{T \max}^{2} \cdot L_{c}) \cdot a_{T} + & (11) \\ \frac{x_{T \max}}{x_{T \max} - L_{c}} (\begin{matrix} \frac{x_{T \max} \cdot \tan (- β)}{6} - \\ \frac{2}{3} (\frac{(L_{c} + x_{T \max}) \cdot \tan (- β)}{2} + y_{T \max} - T_{b}) + \\ \frac{x_{T \max} - L_{c}}{x_{T \max}} y_{T \max} \end{matrix}) \end{matrix}$
The blade thickness function 33, which has been created (drawn) by the cubic functions of the equations (2) and (3), is combined with the camber line 31 created (drawn) beforehand. That is, the values of the blade thickness B (y_L, y_T) at the respective blade thickness points BP of the blade thickness function 33 are added to the respective camber points SP of the camber line 31 in a direction perpendicular to the tangents at the respective camber points SP. As a result, the shape of the blade section 32 is created (drawn). Such a shape of blade section (blade profile) is created (drawn) at each of a plurality of locations in the hub diameter direction of the blade. Based on the resulting blade profiles, spline interpolation is performed to create (draw) a spline curve (visible outline of the blade) and a spline surface (exterior surface of the blade), thereby creating (drawing) the entire shape of the blade (external diameter line, external diameter surface). In this case, the camber line 31 may be created by the aforementioned method using the “Joukowski airfoil”, or may be created by any method.
According to the present embodiment, as described above, under the blade shape creation program P, which creates a blade shape on a space virtually defined by the personal computer 11, the equation for defining the blade thickness function 33, which represents a change in the blade thickness B to be defined on the section 32 of the blade shape is composed of the first function (cubic function) which defines the leading edge blade thickness function 33A on the leading edge side of the maximum blade thickness point BPM of the blade thickness function 33, and the second function (cubic function) which defines the trailing edge blade thickness function 33B on the trailing edge side of the maximum blade thickness point BPM of the blade thickness function 33. Thus, with the exception of the design factors concerning the maximum blade thickness point at the boundary between the first function and the second function (i.e., position of maximum blade thickness x_Tmax, maximum blade thickness value y_Tmax), the design factors on the leading edge side of the blade thickness function 33 and those on the trailing edge side of the blade thickness function 33 can be independently set (changed) by the first function and the second function. Thus, the influence of each design factor on the site of flow can be systematically studied. This facilitates tuning of the site of flow, and enables an airfoil of higher performance to be developed. In connection with the maximum blade thickness point BPM on the boundary between the first function and the second function, it goes without saying that the first function and the second function are equal to each other in terms of the position of maximum blade thickness x_Tmaxand the maximum blade thickness value y_Tmax, with their tangents at BPM continuing, and the gradients of the tangents being zero.
In the present embodiment, in particular, the seven design factors (camber line length Lc, position of maximum blade thickness x_Tmax, maximum blade thickness value y_Tmax, leading edge blade thickness change rate α, trailing edge blade thickness change rate β, leading edge blade thickness value Tf, trailing edge blade thickness value Tb) were selected as optimal design factors for determining the blade profile (airfoil) and the cubic functions of the equations (2) and (3) were selected as the first function and the second function suited for these design factors. Thus, the respective design factors can be changed independently of each other. This makes it possible to directly grasp the degree of influence which each design factor exerts on the performance of the blade (lift performance and drag performance) (i.e., the degree of contribution to blade performance).
For example, FIG. 4 shows an example of the blade thickness function 33 created (drawn), with only the leading edge blade thickness change rate α being changed in three different ways, and an example of the blade section 32 created (drawn) based on the blade thickness function 33 and the camber line 31. In FIG. 4, only the leading edge blade thickness change rate α is changed, and the other design factors (camber line length Lc, position of maximum blade thickness x_Tmax, maximum blade thickness value y_Tmax, trailing edge blade thickness change rate β, leading edge blade thickness value Tf, trailing edge blade thickness value Tb) are not changed. Thus, the influence of the leading edge blade thickness change rate α on the performance of the blade can be grasped directly. Since each design factor can be changed independently of one another in this manner, the influence of each design factor on the site of flow can be systematically studied. Hence, tuning of the site of flow becomes easy, and an airfoil with higher performance can be developed.
The procedure for deriving the relationships between the respective coefficients (a_L, b_L, c_L, d_L, a_T, b_T, c_T, d_T) in the cubic functions of the equations (2) and (3) and the design factors (camber line length Lc, position of maximum blade thickness x_Tmax, maximum blade thickness value y_Tmax, leading edge blade thickness change rate α, trailing edge blade thickness change rate β, leading edge blade thickness value Tf, trailing edge blade thickness value Tb) will be shown.
First, the relations between the respective coefficients (a_L, b_L, c_L, d_L) of the cubic function equation (2) on the leading edge side of the blade thickness function and the design factors are derived in accordance with the following procedure:
From the equation (2) and the constraint (1),
d_L=T_f (12)
From the equation (2),
dy _L /dx _L=3a _L x _L ²+2b _L x _L +c _L (13)
From the equation (13) and the constrain (5),
c_L=tan α (14)
From the equation (2) and the constraint (3), the equation (12) and the equation (14),
y _Tmax =a _L ·x _Tmax ³ +b _L ·x _Tmax ² +x _Tmax·tan α+T _f (15)
Both sides are multiplied by 2 to give
2y _Tmax=2a _L ·x _Tmax ³+2b _L ·x _Tmax ²+2x _Tmax·tan α+T _f (16)
From the equation (13) and the equation (14), as well as the constraint (7),
0=3a _L ·x _Tmax ²+2b _L ·x _Tmax+tan α (17)
Both sides are multiplied by x_Tmaxto obtain
0=3a _L ·x _Tmax ³+2b _L ·x _Tmax ² +x _Tmax·tan α (18)
Subtraction of the equation (18) from the equation (16) gives
2y _Tmax =−a _L ·x _Tmax ³ +x _Tmax·tan α+T _f $\begin{matrix} ∴ a_{L} = \frac{- 2 y_{T \max} + x_{T \max} \cdot \tan α + T_{f}}{x_{T \max}^{3}} & (19) \end{matrix}$
From the equation (15), $\begin{matrix} b_{L} = \frac{y_{T \max} - T_{f}}{x_{T \max}^{2}} - \frac{\tan α}{x_{T \max}} - a_{L} \cdot x_{T \max} = \frac{y_{T \max} - T_{f}}{x_{T \max}^{2}} - \frac{\tan α}{x_{T \max}} - x_{T \max} (\frac{- 2 y_{T \max} + x_{T \max} \cdot \tan α + T_{f}}{x_{T \max}^{3}}) & (20) \end{matrix}$
Next, the relations between the respective coefficients (a_T, b_T, c_T, d_T) of the cubic function equation (3) on the trailing edge side of the blade thickness function and the design factors are derived in accordance with the following procedure:
From the equation (3),
dy _T /dx _T=3a _T ·x _T ²+2b _T ·x _T +c _T (21)
From the equation (21) and the constraint (6)
tan(−β)=3a _T ·L _c ²+2b _T ·L _c +c _T (22)
From the equation (21) and the constraint (8),
0=3a _T ·x _Tmax ²+2b _T ·x _Tmax +c _T (23)
Subtraction of the equation (23) from the equation (22) gives $\begin{matrix} \tan (- β) = 3 a_{T} \cdot (L_{c}^{2} - x_{T \max}^{2}) + 2 b_{T} \cdot (L_{c} - x_{T \max}) ∴ b_{T} = - \frac{3}{2} (L_{c} + x_{T \max}) \cdot a_{T} + \frac{\tan (- β)}{2 (L_{c} - x_{T \max})} & (24) \end{matrix}$
From the equation (3) and the constraint (2),
T _b =a _T ·L _c ³ +b _T ·L _c ² +c _T ·L _c +d _T (25)
From the equation (3) and the constraint (4),
y _Tmax =a _T ·x _Tmax ³ +b _T ·x _Tmax ² +c _T ·x _Tmax +d _T (26)
Subtraction of the equation (26) from the equation (25) gives
T _b −y _Tmax =a _T·(L _c ³ −x _Tmax ³)+b _T·(L _c ² −x _Tmax ²)+c _T·(L _c −x _Tmax) (27)
Substitution of the equation (24) into the equation (27), followed by arrangement, yields $\begin{matrix} ∴ c_{T} = a_{T} \cdot (\frac{1}{2} L_{c}^{} + 2 L_{c} \cdot x_{T \max} + \frac{1}{2} x_{T \max}^{2}) - \frac{1}{L_{c} - x_{T \max}} (\frac{(L_{c} + x_{T \max}) \cdot \tan (- β)}{2} + y_{T \max} - T_{b}) & (28) \end{matrix}$
Subtraction of (the equation (26)×3) from (the equation (23)×x_Tmax) gives $\begin{matrix} d_{T} = - \frac{x_{T \max}^{2}}{3} b_{T} - \frac{2 x_{T \max}}{3} c_{T} + y_{T \max} & (29) \end{matrix}$
Substitution of b_Tand c_Tinto the equation (29) followed by arrangement, yields $\begin{matrix} ∴ d_{T} = \frac{1}{6} (x_{T \max}^{3} - 2 x_{T \max} \cdot L_{c}^{2} - 5 x_{T \max}^{2} \cdot L_{c}) \cdot a_{T} + \frac{x_{T \max}}{x_{T \max} - L_{c}} (\frac{x_{T \max} \cdot \tan (- β)}{6} - \frac{2}{3} (\frac{(L_{c} + x_{T \max}) \cdot \tan (- β)}{2} + y_{T \max} - T_{b}) + \frac{x_{T \max} - L_{c}}{x_{T \max}} y_{T \max}) & (30) \end{matrix}$
Substitution of b_T, c_Tand d_Tinto the equation (23), followed by arrangement, yields $\begin{matrix} ∴ a_{T} = - \frac{(L_{c} - x_{T \max}) \cdot \tan (- β) + 2 y_{T \max} - 2 T_{b}}{{(x_{T \max} - L_{c})}^{3}} & (31) \end{matrix}$
Next, the blade thickness function checking capability and the checklist window display capability in the blade shape creation program P will be described.
In creating (drawing) the blade thickness function 33 by the blade shape creation program P (cubic functions of the equations (2) and (3)), the following cases may be encountered, depending on a combination of the seven design factors (camber line length Lc, position of maximum blade thickness x_Tmax, maximum blade thickness value y_Tmax, leading edge blade thickness change rate α, trailing edge blade thickness change rate β, leading edge blade thickness value Tf, trailing edge blade thickness value Tb) determining the blade profile (airfoil), even if the eight constraints (1) to (8) to be satisfied are fulfilled: There may be a blade thickness function, like the blade thickness function 33 illustrated in FIG. 5, which, at a blade thickness point BP other than a set maximum blade thickness point BPM, has a value of blade thickness B (y_L, x_L) greater than a maximum blade thickness value y_Tmaxat the set maximum blade thickness point BPM. There may be another blade thickness function, like the blade thickness function 33 illustrated in FIG. 6, which, at blade thickness points BP other than the set maximum blade thickness point BPM, has inflection points (may have a maximum or minimum point).
Under the blade shape creation program P, therefore, a numerical check is made for such cases (i.e., whether a blade thickness value greater than the set maximum blade thickness value is present, and whether a maximum or minimum point or an inflection point is present at a blade thickness point other than the set maximum blade thickness point) at the time of creating the blade thickness function 33. A further check is performed of whether the blade section 32 does not extend beyond the hub 22. The results of these checks are displayed on the checklist window. A concrete procedure is as follows:
<Method of Checking Whether a Blade Thickness Value Greater than a Set Maximum Blade Thickness Value is Present>
In the first function (cubic function) and the second function (cubic function) of the blade thickness function defining equation, whose coefficients were determined by setting the design factors (constraints), the value of the blade thickness B (y_L, x_L) is calculated over the entire region of the blade thickness function 33 in the camber line direction (x-axis direction of FIG. 3). That is, in connection with the cubic function of the equation (2), each coefficient is determined based on the design factors (constrains), and then a blade thickness value y_Lat each position (each blade thickness point BP) over the range from x_L=0 to x_L=x_Tmaxis calculated. In connection with the cubic function of the equation (3) as well, each coefficient is determined based on the design factors (constrains), and then a blade thickness value y_Tat each position (each blade thickness point BP) over the range from x_T=x_Tmaxto x_T=Lc is calculated.
These calculated blade thickness values y_Land y_Tare compared with the maximum blade thickness value y_Tmaxset as a design factor to check whether the blade thickness function 33 has blade thickness values y_Land y_Tgreater than the maximum blade thickness value y_Tmax.
<Method of Checking Whether a Maximum, Minimum or Inflection Point other than a Set Maximum Blade Thickness Point is Present>
The first function (cubic function) and the second function (cubic function) of the blade thickness function defining equation, whose coefficients were determined by setting the design factors (constraints), are subjected to differentiation (differentiation of first order, or differentiation of second or higher order). By so doing, whether the blade thickness function 33 has a maximum or minimum point or an inflection point at a position other than the position of maximum blade thickness x_Tmax(blade thickness point BP other than the maximum blade thickness point BPM) set as a design factor is checked over the entire region of the blade thickness function 33.
For example, in the first function (cubic function) and the second function (cubic function) of the blade thickness function defining equation, whose coefficients were determined by setting the design factors (constraints), the gradient of the tangent to the blade thickness function 33 (dy_L/dx_L, dy_T/dx_T) is calculated over the entire region of the blade thickness function 33 in the camber line direction (x-axis direction of FIG. 3). That is, in connection with the cubic function of the equation (2), each coefficient is determined based on the design factors (constrains), and then the gradient of the tangent (dy_L/dx_L) at each position (each blade thickness point BP) over the range from x_L=0 to x_L=x_Tmaxis calculated. In connection with the cubic function of the equation (3) as well, each coefficient is determined based on the design factors (constrains), and then the gradient of the tangent (dy_T/dx_T) at each position (each blade thickness point BP) over the range from x_T=x_Tmaxto x_T=Lc is calculated. Then, a check is made of whether the positivity or negativity of the sign of the calculated gradient of the tangent (dy_L/dx_L, dy_T/dx_T) is reversed before and after a position other than the set position of maximum blade thickness (blade thickness point BP other than the maximum blade thickness point BPM) (namely, whether there is a maximum or minimum point).
<Method for Checking Whether the Blade Section Does Not Extend Beyond the Hub>
A check is made of whether the blade section 32, created (drawn) based on the blade thickness function 33 by the blade thickness function defining equation (cubic function), does not extend beyond the hub 22 in a side view (plan view), when its inclination angle with respect to the hub center axis B is also taken into consideration. FIG. 7 shows an example in which a leading edge portion C of the blade section 32 extends beyond the hub 22 in a side view (plan view) of the cooling fan.
<Method for Display of Checklist Window>
The results of the checks made by the above checking methods are displayed on a checklist window 16 on a display screen 15 as shown in FIG. 8. Curve-1 to Curve-3 in a column of the checklist window 16 represent blade thickness functions created (drawn) for the blade section at each position of the blade in the hub diameter direction. The number of the created blade thickness functions is not limited to 3 in the illustrated example, but may be 2 or 4 or more in accordance with the shape of the blade to be created.
Error-1 to Error-4 in a row of the checklist window 16 represent items checked by the above-described checking methods. Error-1 shows the results of the check of whether the blade thickness function 33 as a whole has a blade thickness value greater than the set maximum blade thickness value. When values y_Land y_Tgreater than the maximum blade thickness value y_Tmaxare not present, a judgment “no problem” is made, and a circle “◯” meaning no problem is displayed. If blade thickness values y_Land y_Tgreater than the maximum blade thickness value y_Tmaxare present, this means that the conditions for setting (preconditions) the maximum blade thickness value and the position of maximum blade thickness are not fulfilled. Since a judgment “problematical” is made, “warning” is displayed.
Error-2 shows the results of the check of whether the leading edge blade thickness function 33A has a maximum or minimum point or an inflection point. When there is no maximum or minimum point or no inflection point, a judgment “no problem” is made, and a circle “◯” meaning no problem is displayed. If there is a maximum or minimum point or an inflection point, the presence of a maximum or minimum point or an inflection point on the leading edge side (leading edge blade thickness function 33A) is considered to affect, often adversely, the performance of the blade. Thus, a judgment “problematical” is made, and “warning” is displayed. Error-3 shows the results of the check of whether the trailing edge blade thickness function 33B has a maximum or minimum point or an inflection point. When there is no maximum or minimum point or no inflection point, a judgment “no problem” is made, and a circle “◯” meaning no problem is displayed. If there is a maximum or minimum point or an inflection point, “caution” is displayed. The reason why “caution”, rather than “warning,” is displayed here is that the presence of a maximum or minimum point or an inflection point on the trailing edge side (trailing edge blade thickness function 33B) does not necessarily exert an adverse influence on the performance of the blade, but is rather considered to exert a favorable influence on the performance of the blade. Anyway, a display of “caution” enables the developer to recognize reliably that a maximum or minimum point or an inflection point is present.
Error-4 shows the results of the check of whether the blade section 32 does not extend beyond the hub 22. When the blade section 32 does not extend beyond the hub 22, a judgment “no problem” is made, and a circle “◯” meaning no problem is displayed. If the blade section 32 extends beyond the hub 22, this is not necessarily a problem, and it suffices to have the developer recognize that the blade section 32 extends beyond the hub 22. Thus, “caution” is displayed.
A “Close” button 42 displayed on the display screen 16 of FIG. 8 is a button to be pushed (for example, to be clicked by a mouse) for closing (erasing) the checklist window 16.
According to the present embodiment described above, in the first function (cubic function) and the second function (cubic function) of the blade thickness function defining equation, the value of the blade thickness B (y_L, y_T) is calculated over the entire region of the blade thickness function 33. This calculated blade thickness value (y_L, y_T) is compared with the maximum blade thickness value y_Tmaxset as a design factor to check whether the blade thickness function 33 has a blade thickness value (y_L, y_T) greater than the maximum blade thickness value y_Tmax. Hence, the presence or absence of a delicate blade thickness value (y_L, y_T), which is difficult to confirm visually, can be numerically checked with reliability when creating the blade thickness function 33. Thus, the efficiency of blade development increases. For example, the blade thickness function 33 of FIG. 9 poses no problem about blade thickness values. In regard to the blade thickness function 33 of FIG. 10, on the other hand, a value of the blade thickness B (y_L) at a blade thickness point BP nearer to the leading edge is slightly larger than a maximum blade thickness value y_Tmaxat the set maximum blade thickness point BPM. The problem of such a delicate blade thickness value (y_L) can be checked reliably.
According to the present embodiment, moreover, the first function (cubic function) and the second function (cubic function) of the blade thickness function defining equation are differentiated. By so doing, whether the blade thickness function 33 has a maximum or minimum point or an inflection point at a position other than the position of maximum blade thickness set as a design factor is checked over the entire region of the blade thickness function 33. Hence, the presence or absence of a maximum or minimum point or an inflection point, which is difficult to confirm visually, can be numerically checked with reliability when creating the blade thickness function 33. Thus, the efficiency of blade development increases.
According to the present embodiment, moreover, the results of the checks of whether the blade thickness function has a greater blade thickness value than the maximum blade thickness value, whether the blade thickness function has a maximum or minimum point or an inflection point at a blade thickness point other than the maximum blade thickness point, and whether the blade section does not extend beyond the hub are displayed on the checklist window 16. Accordingly, these checking results are clear at a glance, and the efficiency of blade development increases.
While the present invention has been described by the above embodiment, it is to be understood that the invention is not limited thereby, but may be varied or modified in many other ways. Such variations or modifications are not to be regarded as a departure from the spirit and scope of the invention, and all such variations and modifications as would be obvious to one skilled in the art are intended to be included within the scope of the appended claims.

Claims

1. A blade shape creation program for creating a blade shape on a space virtually defined by a computer, wherein

a blade thickness function defining equation for defining a blade thickness function representing a change in a blade thickness to be defined on a cross section of the blade shape is constructed by

a first function which defines a leading edge blade thickness function on a leading edge side of a maximum blade thickness point of the blade thickness function, and

a second function which defines a trailing edge blade thickness function on a trailing edge side of the maximum blade thickness point of the blade thickness function.

2. The blade shape creation program according to claim 1, wherein the blade thickness function defining equation

has the first function and the second function each defined by a cubic function,

is defined, with a camber line length of a section of the blade shape, a position of maximum blade thickness, a maximum blade thickness value, a leading edge blade thickness change rate, a trailing edge blade thickness change rate, a leading edge blade thickness value, and a trailing edge blade thickness value being taken as design factors, and

has a boundary condition that the first function and the second function have tangents continuous with each other at the maximum blade thickness point.

3. A blade shape creation method for creating a blade shape on a virtually defined space, wherein

4. The blade shape creation method according to claim 3, wherein the blade thickness function defining equation

5. A blade shape creation program for creating a blade shape on a space virtually defined by a computer, wherein

a blade thickness function defining equation for defining a blade thickness function representing a change in a blade thickness to be defined on a cross section of the blade shape is constructed by a first function which defines a leading edge blade thickness function on a leading edge side of a maximum blade thickness point of the blade thickness function, and a second function which defines a trailing edge blade thickness function on a trailing edge side of the maximum blade thickness point of the blade thickness function, and

in the first function and the second function of the blade thickness function defining equation, a value of the blade thickness is calculated over an entire region of the blade thickness function, and the calculated blade thickness value is compared with a maximum blade thickness value set as a design factor to check whether the blade thickness function has a blade thickness value larger than the maximum blade thickness value.

6. A blade shape creation program for creating a blade shape on a space virtually defined by a computer, wherein

the first function and the second function of the blade thickness function defining equation are differentiated to check over an entire region of the blade thickness function whether the blade thickness function has a maximum or minimum point or an inflection point at a position other than a position of maximum blade thickness set as a design factor.

7. The blade shape creation program according to claim 5, wherein the blade thickness function defining equation

8. The blade shape creation program according to claim 6, wherein the blade thickness function defining equation

9. The blade shape creation program according to claim 5, wherein

results of checking whether the blade thickness function has a blade thickness value larger than the maximum blade thickness value, or results of checking whether the blade thickness function has a maximum or minimum point or an inflection point at a position other than the position of maximum blade thickness are displayed on a checklist window.

10. The blade shape creation program according to claim 6, wherein

11. A blade shape creation method for creating a blade shape on a virtually defined space, wherein

12. A blade shape creation method for creating a blade shape on a virtually defined space, wherein

13. The blade shape creation method according to claim 11, wherein the blade thickness function defining equation

14. The blade shape creation method according to claim 12, wherein the blade thickness function defining equation

15. The blade shape creation method according to claim 11, wherein

results of checking whether the blade thickness function has a blade thickness value larger than the maximum blade thickness value, or results of checking whether the blade thickness function has a maximum or minimum point or an inflection point at a position other than the position of maximum blade thickness are displayed on a checklist.

16. The blade shape creation method according to claim 12, wherein