An intuitive parameterization method with inherently high-order differentiability for compressor blade sections based on ellipse hierarchical deformation

Chunrui SI, Jinxin CHENG, Zhengqing ZHU,*, Zhitong CHEN,c,d,Qinglong HAO

a School of Mechanical Engineering and Automation, Beihang University, Beijing 100191, China

b School of Mechanical Engineering, University of Science and Technology Beijing, Beijing 100083, China

c Zaozhuang Beihang Machine Tool Innovation Research Institute Co., Ltd, Zaozhuang 277500, China

d Ningbo Institute of Technology, Beihang University, Ningbo 315832, China

KEYWORDS Aerodynamics;Blade sections;Compressors;Deformation;Ellipse;Parameterization

Abstract Shape parameterization has a crucial influence on the optimal solution of aerodynamic optimization.This paper proposes a novel parameterization method for compressor blade sections based on the three-level deformation of the ellipse, which simultaneously satisfies the requirements of flexibility,smoothness,intuitiveness,and compactness.In proposed method,the first-level deformation directly controls nine key geometric parameters to construct the blade section profile, and then the second- and third-level deformations are performed respectively to coarsely and finely modify the profile while keeping the key geometric parameters unchanged.These three levels of deformation effectively decompose the design space without destroying the ellipse’s infinite differentiability,allowing designers to work only with intuitive shape-related parameters to design blade sections with inherently high-order continuity.To verify the effectiveness,six existing blade sections are first fitted and then one of them is selected for a three-level optimization.The results show that the geometry and aerodynamic performance of the fitted and the original blade sections are in good agreement, and the loss coefficient of the optimized blade section is reduced by a total of 36.41%,with 27.34%, 8.45%, and 0.62% reductions for the first to the third level, respectively.Therefore,the proposed parameterization method facilitates the design of lower-loss and higher-load compressor blade sections.

1.Introduction

Blade parameterization, namely expressing the geometry by several design variables, is an important part of the compressor automated design workflow because it defines the design space in which optimization algorithms search for the optimal solution.1–3The aerodynamic design of the three-dimensional blade is generally decomposed into the design of a series of two-dimensional blade sections stacked along the radial direction.This paper focuses on the parameterization of the twodimensional blade sections, which is fundamental and crucial for the parameterization of the three-dimensional blades.

A well-behaved parameterization is to use as few design variables as possible to express a large enough design space to cover the potentially optimal solutions.4Flexibility,smoothness, intuitiveness, and compactness are the four main characteristics of a successful parameterization,5–7in which the smoothness should reach the curvature continuity and curvature derivative continuity to avoid velocity spikes that may lead to flow separation.8–10Circular arcs and polynomials are the main forms of early blade section parameterization.They usually used multiple circular arcs and polynomials to connect several key points on the blade section together in the light of C1or C2continuous conditions.11–13The polynomial parameterization allows designers to specify some key geometric parameters.Nevertheless, the polynomial coefficients hardly convey the geometric insight about the blade section shape, the determination of which fully depends on the solving of a linear equation system, limiting the flexibility and intuitiveness of the method.In addition,high-degree polynomials tend to produce undesirable inflection points, which makes it difficult to achieve C3continuity at junction points.

Now the parameterization of blade sections adopts mainly Bezier, B-spline, and NURBS curves.Corral and Pastor10defined two chains of quartic Bezier curves to respectively express the suction and pressure sides and achieved G3continuity from the leading-edge stagnation point to the blend points of the trailing edge.Gra¨sel et al.14utilized two Bspline curves to respectively represent the suction and pressure sides which are connected with the leading and trailing edges under the tangency condition.Koini et al.15used a single cubic(or higher) NURBS curve to describe the blade section.The control points of the NURBS curve are determined by the distribution points around the camber line.Agromayor et al.5,16defined a cubic B-spline curve to express the camber line,and then used two quartic B-spline curves to respectively represent the suction and pressure sides, which are connected at the stagnation points to form a G2continuous blade section.It is easy to connect several free-form curves to form a G1continuous curve; however, when extended to form curves with a higher degree of continuity, especially for G3continuity, it is not straightforward.In principle, a single B-spline or NURBS curve of degree four or higher can directly achieve G3continuity everywhere.However,when trying to control all the details of the blade section,especially for the leading and trailing edge regions with relatively large curvature, the single-curve methods usually require so many control points that it is difficult to meet the requirements of the compactness and intuitiveness.

In addition to the parameterization that directly defines the blade geometry, Korakianitis et al.17developed a CIRCLE method that defined the central regions of the suction and pressure sides by mapping the desirable curvature distribution and achieved continuous slope of curvature everywhere along the entire blade section.Nemnem et al.18used a cubic B-spline to directly control the second derivative of the camber line,and then superimposed the thickness distribution represented by a quartic B-spine on it.This method also achieved continuous slope of curvature everywhere.The curvature distribution is considered to be more directly related to the aerodynamic performance.It directly affects the pressure distribution of the blade surface and the blades with the same curvature distribution tend to exhibit similar aerodynamic performance.However, since the effect of the curvature change on the blade section shape is not intuitive, the curvature mapping based approach relies heavily on the experience of the designer.

The parameterization methods mentioned above can be regarded as constructive methods as they are used to create a completely new design.In fact, parameterization methods can also be used to modify an already existing design, and these methods are referred to as deformation methods.These methods mainly include the displacements of mesh points19or control points,20superposition disturbing functions such as the Hicks-Henne function21and weighted Bernstein polynomials,22and space morphing method based on free-form deformation23or radial basis function interpolation.24The most prominent advantage of deformation methods is their high flexibility so that these methods can explore a rich design space.Nevertheless,the design variables of deformation methods usually have no intuitive meaning, making it difficult to deal with geometric constraints to obtain practical shapes.

From the above analysis, it can be concluded that most of the existing parameterization methods of blade sections are difficult to simultaneously satisfy the requirements of flexibility, smoothness, intuitiveness, and compactness.To solve this problem,a novel parameterization method based on the ellipse hierarchical deformation is proposed in this paper.This method is inspired by the Class/Shape function Transformation(CST)technique6that is used to represent the airfoil shape in the external flow field.In the CST method,the class function is used to describe the profile of the geometry, and then the shape function is used to modify the profile to obtain a specific shape.The ellipse itself integrates the large and small curvature geometries together and thus has a natural similarity to the curvature distribution of the blade section.Similar to the CST method, the proposed method selected the ellipse as the basic shape, and then performed three levels of deformation on it to obtain the desirable blade section.The three levels of deformation effectively decompose the design space,balancing the requirements of flexibility and compactness.At the same time, the infinite differentiability of the ellipse is preserved during the deformation and thus the high-order continuity will be provided as a byproduct of the proposed method.Finally, this method allows the designer to directly specify nine key geometric parameters of the blade section,such as the maximum thickness and leading- and trailingedge radii, and the other design variables are also shaperelated parameters.Therefore, this method enables the designer to work only with intuitive shape-related parameters to design blade sections with inherently high-order continuity.

The rest of the paper is organized as follows.Section 2 presents the preparations for the ellipse deformation, mainly including basic definitions and functions.Section 3 is the focus of this paper, elaborating the method of the three-level deformation of the ellipse.Section 4 verifies the effectiveness of the proposed method from both inversion and optimization aspects.Conclusions are made in Section 5.

2.Preparations for ellipse deformation

2.1.Data preprocessing

Fig.1 Key geometric parameters of cascade and blade section.

The key geometric parameters of the cascade and blade section are shown in Fig.1(a) and Fig.1(b), respectively.These parameters are commonly used in the general aerodynamic theory and therefore play an important role in the shape definition of the blade section.In Fig.1(a), ε is the stagger angle,βinand βoutare the inlet and exit metal angles respectively, and αleand αteare the leading- and trailing-edge angles respectively.ε, βin, βoutand αleare positive in sign and αteis negative.Consequently, the metal angles βinand βoutare defined as βin=ε+αleand βout=ε+αterespectively.In this study, the stagger angle and leading- and trailing-edge angles are directly controlled and so do the metal angles.To facilitate the introduction of the proposed method, the blade section posture in Fig.1(a) is converted to that shown in Fig.1(b)in which both the leading- and trailing-edge points L and T are located on the x-axis and symmetrical about the y-axis.In the following sections,the proposed method will be demonstrated in Fig.1(b),and the desired posture of the blade section in the cascade will be determined by the chord length d, the stagger angle ε, and the position of the leading-edge point L in the cascade.

The key geometric parameters of the blade section shown in Fig.1(b) can be divided into two categories: those related to the thickness distribution and those belonging to the camber line.The chord length d, the maximum thickness tmaxand its chordwise location dt, and the leading- and trailing-edge radii rleand rteare all geometric parameters related to the thickness distribution.Meanwhile, the maximum deflection cmaxand its chordwise location dc, and leading- and trailing-edge angles αleand αteare all geometric parameters belonging to the camber line.It is noted that the chordwise location dtrepresents the distance measured from the leading-edge point to the maximum thickness location and thus is always positive in sign.It is similar for the chordwise location dc.The proposed method in this paper provides the ability to directly control these geometric parameters, the details of which will be presented in Section 3.

2.2.Definitions of non-uniform transformations of sine and cosine functions

The standard ellipse parametric equation is:

where a and b represent the major and minor semi-axes of the ellipse, respectively.In this study, the ellipse deformation is performed by applying several designed deformation functions to Eq.(1).For the convenience of introducing the deformation functions later,the definitions of non-uniform transformations of sine and cosine functions are given first.

Taking y=sin x as an example,y=M（x)sin［x+P（x)]+D（x) can be obtained by transforming its amplitude, phase and vertical position.If M（x),P（x) and D（x) are functions of the variable x rather than constants, the amplitude, phase or vertical position of y=sin x are said to be non-uniformly transformed.The definition for cosine functions is the same.Under the non-uniform transformations, the graphs of sine and cosine functions will be deformed because the amount of stretching, shrinking or shifting of each point on the graphs is not constant but varies with variable x.The deformation of the ellipse is precisely based on the non-uniform transformations of the x and y components in Eq.(1).The focus of this study is to construct several appropriate functions M（x),P（x)and D（x)to deform the ellipse into the blade section.The functions M（x),P（x) and D（x) are collectively referred to as the deformation functions and the construction method will be presented in detail in Section 3.

2.3.Weight functions design

To preserve the ellipse’s infinite differentiability, all the designed deformation functions are based on cosine and sine functions.However,the symmetry and periodicity of the above two types of functions lead to their poor local controllability.To tackle this, two weight functions W1（θ) and W2（θ) are designed in advance, which will be used as weight coefficients of the deformation functions to improve the local controllability.The functions W1（θ) and W2（θ) are designed as follows:

where w1and w2represent two weight parameters.The effect of the parameter w1on the function W1（θ) and that of the parameter w2on the function W2（θ) are shown in Fig.2(a)and Fig.2(b), respectively.It can be seen that both weight functions are symmetrical about θ=π and W1（θ),W2（θ)?（0,1].For the function W1（θ) in θ ?［0,π], its value at θ=π is always equal to one and the values at other positions tend to be zero with the increase of the weight parameter w1.Moreover,the range of θ at which the value tends to be zero extends continuously from θ=0 to θ=π.Therefore, when the function W1（θ)acts as the weight coefficient of a deformation function,the main influence scope of the deformation function will be increasingly concentrated around θ=π as the parameter w1increases.The parameter w2has a similar effect on the function W2（θ), and the main influence scope of the deformation function will be relatively concentrated around θ=0 when the function W2（θ) is used as the weight coefficient.In the next section, the above two weight functions will be reused three times with different weight parameters to satisfy the requirements of different deformation functions on the influence scope.The details will be presented later.

Fig.2 Weight functions W1(θ) and W2(θ).

3.Three-level deformation of ellipse

The ellipse deformation is divided into three levels.In the first level, nine key geometric parameters as shown in Fig.1(b) are directly controlled to obtain the profile of the blade section.The second- and third-level deformations are respectively coarse and fine adjustments on the profile while keeping the key geometric parameters unchanged.The specific deformation method is elaborated below.

3.1.The first-level deformation—direct control of key geometric parameters

As mentioned in Section 2.1,the key geometric parameters can be divided into those related to the thickness distribution and those related to the camber line.The control methods of these two parts will be presented separately below.

3.1.1.Control key geometric parameters of thickness distribution

For Eq.(1),let a=d/2,b=tmax/2,and Eq.(1)can be rewritten as

In this way, the major and minor axes of the ellipse naturally establish a one-to-one correspondence with the chord length d and maximum thickness tmax, respectively.Since the thickness distribution discussed in this paper is symmetrical,all the following discussions about the thickness distribution are carried out within θ ?［0,π].For the part of θ ?［π,2π],the x- and y-components of the thickness distribution should always be symmetrical about θ=π.

In Eq.(3), the maximum thickness is always located at xt（θ)=0 which corresponds to θ=π/2.To adjust the location of the maximum thickness,a simple and intuitive method is to change the value of xt（θ) at θ=π/2.According to the nonuniform phase transformation of cosine functions mentioned in Section 2.2, the control of the maximum thickness location is shown as follows:

where tlocrepresents the adjustment parameter of the maximum thickness location and its effect on the maximum thickness location is shown in Fig.3.The blue, orange and purple lines above and below in Fig.3 (a) represent the graphs of tlocsin θ and xt（θ) when tloc>0, tloc=0 and tloc<0, respectively.As mentioned above, we assume that the distance between the maximum thickness location and the leading edge point L is dt,which means that the value of xt（θ)at θ=π/2 is equal to dt-d/2.Accordingly, the value of tloccan be solved as

From Eq.(5),it can be observed that when tloc>0,namely dt

Fig.3 Control of the maximum thickness location.

Eq.(4) has controlled the chord length d, maximum thickness tmaxand its location dt.The remaining geometric parameters of the thickness distribution that need to be controlled are the leading- and trailing-edge radii rleand rte.Considering that the leading- and trailing-edge regions are mainly around θ=0 and θ=π, the radii rleand rtecan be controlled by adjusting the values of yt（θ)in these two regions.For this reason,the non-uniform amplitude transformation of yt（θ)in Eq.(4) is implemented.The amplitude deformation function R（θ)is designed as follows:

where Rle（θ) and Rte（θ) are the two subfunctions,trleand trterepresent two adjustment parameters for the leading- and trailing-edge radii,Wrle（θ) and Wrte（θ) represent the weight functions W1（θ) and W2（θ) respectively and their corresponding weight parameters are denoted as wrleand wrte.To mainly adjust the values around θ=0 and θ=π, cos θ is selected as the basic function.The working principle of Eq.(6) will be explained later on.

By substituting Eq.(6) into Eq.(4), the control of the leading- and trailing-edge radii is given as follows:

where C1and C2are the two parameters determined as follows:

From Eq.(8) and Eq.(9), it can be concluded that on the premise that the chord length d, maximum thickness tmaxand its location parameter tlochave been determined, for the given radii rleand rte, the weight parameters wrleand wrteneed to be specified first to calculate the corresponding adjustment parameters trleand trte.Typically, both the leading- and trailing-edge regions are small so that both the weight parameters wrleand wrteare set to 50 in the first-level deformation.Taking trle,trte>0 as an example, the deformation function R（θ)and its corresponding thickness distribution are displayed as the orange lines in Fig.4 (a) and Fig.4 (b), respectively.Fig.4(a) shows that the main influence scope of the function R（θ)is around θ=0 and θ=π,which means that the function R（θ)only affects the leading-and trailing-edge regions.Meanwhile, the values of the function R（θ) at θ=0 and θ=π are mutual independent and thus it can control the leading- and trailing-edge radii independently.These conclusions can also be obtained by comparing the thickness distributions represented by the orange and blue lines in Fig.4(b), in which the blue line represents the thickness distribution obtained from Eq.(4).

Eq.(7) is the preliminary model of the thickness distribution.It directly controls the chord length d, maximum thickness tmax, maximum thickness location dt, and leading- and trailing-edge radii rleand rte,and allows each of the parameters to adjust independently.In the second-level deformation, the thickness distribution on the suction and pressure sides will be adjusted, and the weight parameters wrleand wrte, which are both set to 50 in the first level, will be regarded as two degrees of freedom to accommodate the desired thickness distribution in the leading- and trailing-edge regions on the condition that the given radii are not changed.The details will be presented in the second level.

3.1.2.Control of key geometric parameters of camber line

Eq.(7)can be regarded as a blade section with zero deflection.From this point of view,the original model of the camber line can be expressed as

Fig.4 Control of leading- and trailing-edge radii.

where yc（θ) represents the vertical coordinate of the camber line.In fact, when θ ?［0,π], Eq.(10) can fully describe the shape of the camber line.However, in order to facilitate the subsequent superposition of the thickness distribution on the camber line, the x-component of the camber line is intentionally aligned with that of the thickness distribution.In the following discussion, the camber line is mainly discussed in θ ?［0,π].For the part of θ ?［π,2π],the discussion will be carried out under the condition that the y-component of the camber line should always be symmetrical about θ=π.

By referring to the ellipse standard parametric equation,the camber line with deflection is constructed as follows:

Eq.(11)can be treated as an ellipse folded along the x-axis.Consequently, the control methods of the geometric parameters of the camber line can refer to those of the geometric parameters of the thickness distribution.

In Eq.(11), the location of the maximum deflection is always the same as that of the maximum thickness.To adjust the maximum deflection location while not affecting the maximum thickness location, the non-uniform phase transformation of yc（θ) in Eq.(11) is carried out.The control of the maximum deflection location is shown as follows:

where clocis the adjustment parameter of the maximum deflection location and its effect on the location is shown in Fig.5.The blue,orange and purple lines above and below in Fig.5(a)represent the graphs of clocsin θ and yc（θ) when cloc>0,cloc=0 and cloc<0, respectively.As mentioned above, we suppose that the distance between the maximum deflection location and the leading-edge point L is dc.Then the θ value corresponding to the maximum deflection location, denoted as θc, should satisfy

Fig.5 Control of maximum deflection location.

From Eq.(14),it is easy to know that when cloc=0,namely θc=π/2, dcwill be equal to dtand the location of the maximum deflection will align with that of the maximum thickness,as shown by the orange line in Fig.5 (b).Meanwhile, when cloc>0, namely θc<π/2, according to the monotonic decrease of xt（θ) with respect to θ, it can be obtained that dc>dtand the maximum deflection location will be to the right of the maximum thickness location,as shown by the blue line in Fig.5(b).Similarly,when cloc<0,namely θc=π/2,the location of the maximum deflection will be to the left of the maximum thickness location, as shown by the purple line in Fig.5 (b).In this way, the parameter clocdirectly controls the maximum deflection location.

Eq.(12)has controlled the maximum deflection cmaxand its location dc.The rest of key geometric parameters of the camber line that need to be controlled are the leading-and trailingedge angles αleand αte.The control method of these two parameters is similar to that of the leading- and trailing-edge radii and the non-uniform amplitude transformation of yc（θ)in Eq.(12) is performed.The deformation function A（θ) is designed as follows:

where Ale（θ) and Ate（θ) are the two subfunctions,cαleand cαterepresent the adjustment parameters of the leading- and trailing-edge angles,Wαle（φ) and Wαte（φ) represent the weight functions W1（θ) and W2（θ) respectively and their corresponding weight parameters are denoted as wαleand wαte,φ=θ+clocsin θ.The reason for using ‘‘cos φ”instead of‘‘cos θ”is to have the zeros of both subfunctions at θ=θcand thus the value of the function A（θ) at θ=θcis equal to one.Consequently,the function A（θ)does not affect the maximum deflection and its location.

Substitute Eq.(15) into Eq.(12), and the control of the leading- and trailing-edge angels is shown as follows:

From Eq.(17) and Eq.(18), it can be observed that on the condition that the chord length d, maximum deflection cmaxand it location parameter clocas well as the maximum thickness location parameter tlochave been determined, for the given leading-and trailing-edge angles αleand αte,it is also necessary to first specify the values of the weight parameters wαleand wαteto calculate the corresponding adjustment parameters cαleand cαte.In the first-level deformation, both wαleand wαteare set equal to 3 due to the relatively large influence scope of the function A（θ).Taking cαle>0 and cαte<0 as an example,the deformation function A（θ) and its corresponding camber line are shown by the orange lines in Fig.6(a) and Fig.6(b),respectively.The blue line in Fig.6 (b) represents the camber line obtained from Eq.(12).The effect of the deformation function A（θ) on the angles αleand αteis similar to that of the deformation function R（θ) on the radii rleandrte, which will not be repeated here.

Eq.(16)is the preliminary model of the camber line.It controls the maximum deflection cmax, maximum deflection location dc, and leading- and trailing-edge angles αleand αtedirectly, and allows each of the parameters to adjust independently.In the second-level deformation,the weight parameters wαleand wαtewill also be regarded as two degrees of freedom to adapt the shape of the camber line.The details will be introduced later.

3.1.3.Profile of blade section

Based on Eq.(7) and Eq.(16), the preliminary model of the blade section with thickness distribution along the normal direction of the camber line can be obtained as follows:

where γ（θ) = arctan（yc（θ)/xt（θ)),γ ?（-π/2,π/2) is the angle between the tangent of the camber line and the x-axis.As shown in Fig.7,after specifying the nine key geometric parameters, the ellipse with the first-level deformation starts taking shape.However,the detailed thickness distribution on the suction and pressure sides as well as on the leading- and trailingedge regions need to be further controlled.This is the same for the shape of the camber line.Therefore, in the second-level deformation, the preliminary thickness distribution and camber line, respectively expressed by Eq.(7) and Eq.(16), will be further adjusted without affecting the key geometric parameters.

Fig.6 Control of leading- and trailing-edge angles.

Fig.7 Profile of blade section.

3.2.The second-level deformation—coarse adjustments to blade section profile

In this section,the coarse adjustments to the preliminary thickness distribution and camber line will be carried out separately to provide more control over the blade section shape.The details are presented below.

3.2.1.Coarse adjustment of thickness distribution

The non-uniform amplitude transformation of yt（θ)in Eq.(7)is adopted to adjust the thickness distribution of the suction and pressure sides.It is known from Section 3.1 that the value of θ corresponding to the maximum thickness location is equal to π/2.Therefore, in order to adjust the thickness distribution from the leading-edge point L to the maximum thickness location and from the maximum thickness location to the trailingedge point T,the deformation function T（θ)should be able to independently adjust the values of yt（θ) in θ ?（0,π/2) and θ ?（π/2,π).Based on this, the deformation function T（θ) is designed as follows:

where Tf（θ) and Tb（θ) are the two subfunctions;tfand tbrepresent the adjustment parameters of the thickness distribution;Wf（θ) and Wb（θ) represent weight functions W1（θ) and W2（θ) respectively and their corresponding weight parameters are denoted as wfand wb;sin2（2θ) is the basic function.Since the expected influence scope of function T（θ) is θ ?（0,π),the weight parameters wfand wbare generally not more than 5.Meanwhile, a large number of specific examples show that the thickness distribution represented by Eq.(7) is generally greater than the actually required one.Consequently, the adjustment parameters tfand tbare generally less than zero.The graphs of subfunctions Tf（θ) and Tb（θ) with the different values of weight parameters wfand wbare shown in Fig.8(a)and Fig.8(b) respectively, and their corresponding functions T（θ) are shown in Fig.8 (c).As can be seen from Fig.8(a)and Fig.8(b), both functions Tf（θ) and Tb（θ) can adjust the asymmetry of the basic function sin2（2θ),and with the increase of the weight parameters,the asymmetry becomes more noticeable.Therefore,the values of the deformation function T（θ)in θ ?（0,π/2) and θ ?（π/2,π) can be relatively independent, as shown in Fig.8 (c).

By substituting Eq.(20)into Eq.(7),the coarse adjustment of the thickness distribution on the suction and pressure sides is obtained as follows:

The orange and blue lines in Fig.9 represent the thickness distributions expressed by Eq.(21)and Eq.(7),respectively.It can be seen that the deformation function T（θ) can relatively independently adjust the thickness distribution from the leading-edge point L to the maximum thickness location and from the maximum thickness location to the trailing-edge point T, and the key geometric parameters determined in the first-level deformation remain constant.At the same time,Fig.9 also shows that the function T（θ) is not sensitive to the leading-and trailing-edge regions.Hence,the thickness distribution in these two regions requires additional attention,which will be discussed below.

Fig.8 Adjustment function for thickness distribution on suction and pressure sides.

Fig.9 Adjustment of thickness distribution of suction and pressure sides.

Fig.10 Adjustment of thickness distribution in leading- and trailing-edge regions.

As mentioned in Section 3.1.1, the weight parameters wrleand wrtewill be regarded as two degrees of freedom to accommodate the desired thickness distribution in the leading- and trailing-edge regions.The effect of the weight parameters wrleand wrteon the deformation function R（θ)and on the thickness distribution are shown in Fig.10 (a) and Fig.10 (b),respectively.From Fig.10 (a), it is known that the different values of the weight parameters mainly affect the actual influence scope of the function R（θ) and the larger the weight parameters are,the smaller the actual influence scope is.Moreover,by comparing the blue and orange lines in Fig.10(a)and Fig.10 (b), which represent wrle=50,wrte=10 and wrle=50,wrte=30, respectively, it can be concluded that the thickness distribution of the leading-and trailing-edge regions can be adjusted almost independently of each other.Finally,the values of the function R（θ)at θ=0 and θ=π remain constant by updating the corresponding adjustment parameters trleand trteexpressed by Eq.(8).Therefore,the deformation function R（θ) with different values of the weight parameters can effectively adjust the thickness distribution in the leadingand trailing-edge regions while not changing the leading- and trailing-edge radii.In addition, since the aerodynamic performance of the blade section is sensitive to the shape of the leading-edge region, in some cases, in order to improve the accuracy of the leading-and trailing-edge regions,the designer can use the deformation function R（θ) more than once.

3.2.2.Coarse adjustment of camber line

The curvature variation of the camber line is relatively small compared with that of the thickness distribution.Accordingly,the shape of the camber line from the leading-edge point L to the maximum deflection location and from the maximum deflection location to the trailing-edge point T is directly controlled by the weight parameters wαleand wαte, as shown in Fig.11.This is similar to the adjustment of the thickness distribution in leading- and trailing-edge regions.The difference is that the values of the weight parameters wαleand wαteare usually smaller than those of the parameters wrleand wrtedue to the fact that the expected influence scope of the deformation function A（θ) is larger than that of the function R（θ).

The ellipse after the two levels of deformation is shown by the orange line in Fig.12,in which the blue line represents the profile obtained from the first-level deformation.It shows that the ellipse after the two levels of deformation can basically represent a real blade section shape, and the coarse adjustments effectively modify the profile while not affecting the key geometric parameters.In the third-level deformation, fine-tuning degrees of freedom will be performed on the basis of the coarse adjustments to enhance the detail control ability of ellipse deformation.

3.3.The third-level deformation—fine adjustments

Fig.11 Adjustment of shape of camber line by modifying weight parameters wαle and wαte.

Fig.12 Blade section derived from the second-level deformation of ellipse.

The fine-tuning degrees of freedom are mainly provided for the deformation functions A（θ) and T（θ), because they both have a large influence scope but relatively few adjustment parameters.The fine-tuned degrees of freedom are obtained by applying non-uniform phase transformation to the basic functions of these two deformation functions.Thus, the function A（θ)with fine-tuned degrees of freedom is given as follows:

where clefand ctefare the fine-tuning parameters of the camber line.Taking the function A（θ) in Fig.6(a) as an example, the effect of the parameters clefand ctefon the function A（θ) is shown Fig.13(a).It can be seen that the parameters clefand ctefonly fine-tune the values of the function A（θ) in θ ?（0,θc) and θ ?（θc,π), and do not influence the values at θ=0, θ=θcand θ=π.Therefore, the parameters clefand ctefonly fine-tune the shape of the camber line without affecting its key geometric parameters.

Similarly, the deformation function T（θ) with fine-tuned degrees of freedom is shown as follows:

where tffand tbfare the fine-tuning parameters of the thickness distribution.Taking the function T（θ)represented by the blue line in Fig.8 (c)as an example,the effect of the parameters tffand tbfon the function T（θ) is shown in Fig.13(b).It shows that the parameters tffand tbfonly fine-tune the values of the function T（θ) in θ ?（0,π/2) and θ ?（π/2,π), and do not affect the values at θ=0, θ=π/2 and θ=π.Therefore, the parameters tffand tbfonly fine-tune the thickness distribution without affecting its key geometric parameters.

It is noted that Fig.13 only reveals the cases where both the fine-tuning parameters are less or greater than zero at the same time.For other value combinations of the parameters, the effects are similar, which are not depicted in the figure.

4.Method validation

To verify the effectiveness of the proposed method,six existing blade sections with different geometric features are first fitted,and then one of the fitted blade sections is selected as an example to perform a three-level optimization with the multi-island genetic algorithm.The details are given below.

4.1.Inversion

High-accuracy inversion of the existing blade sections is one of the requirements for a successful parameterization method,and the accuracy should be evaluated from geometric and aerodynamic perspectives.Three rotor blade sections, two fan blade sections, and one stator blade section are chosen as representative blade sections to illustrate the inversion performance of the proposed method for different geometric features.The three rotor blade sections include one flat top section and two middle sections with elliptical and circular leading and trailing edges, respectively.They are denoted as RTS,RMSE,and RMSC,respectively.The two fan blade sections include an S-shaped top section and a heavily angled bottom section, denoted as FTS and FBS, respectively.The one stator section is a bottom section represented by SBS.It is noted that the leading and trailing edges of the RTS, FTS,FBS and SBS are also circular arcs.

Since the key geometric parameters shown in Fig.1(b) are generally not available in the given data of the existing blade sections,the‘‘equidistant line”method25is adopted first to calculate the key geometric parameters of the six instances, as listed in Table 1.According to Table 1, all modelingparameters in the first-level deformation can be directly determined.The modeling parameters included in the second- and third-level deformations are obtained by the nonlinear least squares fitting method.The solving of the least squares fitting is carried out with the numerical nonlinear global optimization function‘‘NMinimize”of Wolfram Mathematica 12.0,and the chosen optimization method is the Differential Evolution(DE).It is worth noting that the fitting result is sensitive to the parameters such as the cross probability, scaling factor,and random seed.The final fitting result with the smallest deviation is selected from the results obtained from the different value combinations of the parameters.

Table 1 Key geometric parameters of blade sections.

From Table 2,it is known that the minimum and maximum relative mean errors of the proposed method with the 21 design variables are 0.0038% and 0.0343%, respectively, as shown in the sixth column of Table 2.When the number of the design variables increases to 23, the maximum relative mean error deceases from 0.0343% to 0.0254%, as shown in the last column of row 7 of Table 2.Ref.5used two quartic B-spline curves and 22 design variables to directly define a blade section with continuous curvature and continuous slope of curvature.Its minimum and maximum relative mean errors are 0.020%and 0.087%, respectively.Therefore, the proposed method achieves higher geometric fitting accuracy with basically the same number of design variables.The reason for this is exactly what was mentioned in the introduction section.Although a single B-spline or NURBS curve of degree four or higher can achieve G3continuity directly, when trying to control all the details of the blade section, especially for the leadingand trailing-edge regions whose curvature is typically orders of magnitude greater than that of the suction and pressure sides, the single-curve methods usually require more control points to achieve the required fitting accuracy.

Table 2 reveals that the rotor and stator blade sections,i.e.,the RTS, RMSE, RMSC, and SBS, are fitted with high accuracy, while the fitting accuracy of the two fan blade sections,i.e., the FTS and FBS, is relatively poor.The fitting accuracy depends to a large extent on the shape of the camber line.From Table 1 and Fig.14 that compares the fitted blade sections and the original data, it is known that the FTS camber line is an S-shaped curve, the FBS camber line is a curve with a larger maximum deflection,and the camber lines of the other sections are relatively flat C-shaped curves.Therefore, the camber line model proposed in this paper is more suitable for representing the relatively flat C-shaped camber line.

Table 2 Matching results of precision.

As mentioned earlier, for the RTS, SBS, FTS, and FBS,they used the subfunction Rle（θ) or Rte（θ) or both twice to improve the fitting accuracy in the leading- and trailing-edge regions.The common feature of these four blade sections is that their leading and trailing edges are circular arcs.Taking the leading-edge region of the RTS as an example, as shown by the purple line in the lower right corner of Fig.14 (a), the fitted blade section with 21 design variables matches the original circular leading edge well.However, bulges, albeit small,tend to form at the connection between the leading edge and the suction and pressure sides.It is well known that the aerodynamic performance of a blade section is sensitive to the shape of the leading-edge region.26,27To eliminate these small bulges,two subfunctions Rle（θ)with different values of weight parameters wrleand wrteare used to fit the circular leading edgemore reasonably and precisely,as shown by the blue line in the enlarged image of Fig.14(a).The other three blade sections,SBS, FTS and FBS, are similar to the RTS and will not be repeated here.In fact, since the proposed parameterization method takes the ellipse as the basic geometry, it is more suitable for expressing the elliptical leading and trailing edges.As shown by the cyan line in Fig.14 (a), the fitted blade section with 21 design variables can easily construct an elliptical leading edge.The fitted blade section RMSE with 21 design variables, as shown in Fig.14 (b), also shows the similar result.

Table 3 Boundary condition setting.

Overall, Fig.14 shows that each fitted blade section is in good geometric agreement with the original section, especially in the leading-and trailing-edge regions,and the maximum fitting errors are all located on the suction and pressure sides.The curvature distributions of the RTS and RMSE are shown in Fig.15, in which Fig.15(a) and Fig.15(c) show the overviews of the curvature distribution, respectively, and the enlarged views in the leading edge region are shown in Fig.15(b) and Fig.15(d), respectively.They indicate that the curvature distribution of each fitted blade section is inherently continuous and smooth, and there are no high-frequency undulations.The curvature distributions of the remaining four blade sections with circular leading edge are similar to those of the RTS, only the values are different, and therefore they are not displayed in the figure.

To compare the aerodynamic performance of the fitted and the original blade sections, the boundary conditions are set as shown in Table 3, where Mainrepresents the inlet Mach number, γindenotes the inlet angle, Psoutis the outlet static pressure, and Ptinrepresents total inlet pressure.From the comparisons of the pressure ratio and Mach number shown in Fig.16 and Fig.17,respectively,it can be seen that the aerodynamic performance of the fitted blade sections is consistent well with that of the original one.

Fig.16 Pressure ratio comparison of original and fitted blade sections.

4.2.Optimization

Fig.17 Relative Mach number comparison of original and fitted blade sections.

Fig.18 Adjustment of key geometric parameters of blade section.

Table 4 Performance comparison of original and optimized blade section by each level.

Before introducing the application of the proposed method in aerodynamic shape optimization, the mutual independence of the key geometric parameters, which means that modifying one of the parameters does not affect the others,is first demonstrated in Fig.18.Fig.18 takes the fitted blade section with the elliptical leading edge as an example, and Fig.18(a)-Fig.18(f)show the independent modification of the maximum thickness tmax,maximum thickness location dt,leading-and trailing-edge radii rleand rte,maximum deflection cmax,maximum deflection location dc, and leading- and trailing-edge angles αleand αte,respectively.It is worth emphasizing that it is only necessary to update the relevant expressions in the first-level deformation to maintain the mutual independence.For example, it only needs to update Eq.(8) and Eq.(9) to vary the maximum thickness without affecting other key geometric parameters.This property significantly improves the efficiency and stability of blade section shape modification compared to the methods that require iterative updating and solving of high-dimensional equations.10,11–13,17In addition,the effect of modifying the key geometric parameters on the blade section shape is intuitive and predictable, and the resulting blade sections in all cases are continuous and smooth.Consequently, the proposed parameterization method provides an excellent ability to directly and independently control the key geometric parameters, which is beneficial for understanding the relationship between the key geometric parameters and aerodynamic performance.

A three-level optimization corresponding to the three-level deformation of the ellipse is adopted to verify the effectiveness of the proposed method in aerodynamic shape optimization.In each level of optimization, the multi-island genetic algorithm is used to optimize the design variables in order to minimize the loss coefficient under the constraints of not reducing the mass flow, pressure ratio, and maximum thickness.The main reason for restricting the maximum thickness is to ensure the strength of the blade section.The number of iterations for the first to third level of optimization is set to 1200, 900, and 200, respectively, proportional to the number of design variables in each level, equal to 10, 8, and 4, respectively.It is noted that the number of variables in the first-level optimization is 10 instead of 9 because the stagger angle is also regarded as a variable.The total time spent on the optimization is about 6.4 h.

Table 4 lists the performance comparisons between the original blade section,denoted as Ori,and the optimized blade sections, denoted as Opt1, Opt2, and Opt3, respectively,obtained by each level of optimization.It can be seen that the loss coefficient is reduced by a total of 36.41 %.The 27.34 % reduction in the first-level optimization is the most significant and the reductions in the second- and third-level optimizations decrease sequentially, reducing by 8.45 % and 0.62%,respectively.The loss coefficient reduction is positively correlated with the ability of design variables to change the shape of the blade section.Specifically, the design variables in the first level are the key geometric parameters and stagger angle,the variations of which will produce the most noticeable changes to the blade section shape.However, changes in the design variables of the second and third levels will produce coarse and fine adjustments, respectively, on the profile obtained from the first level of deformation.The ability of the three levels of design variables to change the blade section shape decrease sequentially, and that is true for the loss coefficient.This conclusion will also be confirmed by the following geometry comparison between the original and the optimized blade sections.

Fig.19(a) compares the geometry of the original and the optimized blade sections.It shows that the first-level optimization produces the most significant changes to the blade section shape, which are mainly reflected in the longer chord length,the smaller leading- and trailing-edge radii and angles, the rearward shift of the maximum deflection location,and the larger stagger angle.The geometry obtained from the secondlevel optimization becomes a little thinner in the leading- and trailing-edge regions and the third-level optimization changes the shape so little that it is even difficult to distinguish.The magnitude of the geometry change reflects the degree of reduction of the loss coefficient.

Fig.19 Geometry and pressure ratio comparison of original and optimized blade sections.

Fig.20 Relative Mach number comparison of original and optimized blade sections.

The comparisons of pressure ratio and Mach number are displayed in Fig.19 (b) and Fig.20, respectively.They show that the thinning of the leading-edge geometry reduces the detached shock wave intensity.Meanwhile, due to the smaller positive incidence angle of the incoming flow, the acceleration of the airflow at the suction side is slowed down and the Mach number of the wavefront is reduced,resulting in the weakening of the shock wave intensity and converting the normal shock wave in the channel to an oblique one.All these significantly reduce the loss caused by the shock wave and the interaction between the shock wave and the boundary layer.In addition,it can be seen that there is a weak shock wave at the pressure side of the optimized blade sections, which will slightly increase the local shock wave loss.However,overall,the fitted blade section parameterized by the proposed method can be optimized by three levels to obtain an optimized shape with a smaller loss coefficient.

5.Conclusions

(1)A novel parameterization method for compressor blade sections based on the three-level deformation of the ellipse is proposed.This method provides direct control over nine key geometric parameters and enables the designer to work only with intuitive shape-related parameters to design blade sections with inherently high-order continuity.

(2) Six existing blade sections with different geometric features are fitted by the proposed method.The results show that the geometry and aerodynamic performance of the fitted blade sections are in good agreement with those of the original ones.Furthermore, compared with Ref.5, the proposed method achieves higher geometric fitting accuracy with basically the same number of variables.

(3)The mutual independence of the key geometric parameters is illustrated, and then a three-level optimization is performed.The optimization results show that the loss coefficient of the optimized blade section is reduced by a total of 36.41%,with reductions of 27.34%,8.45%and 0.62%for the first to the third level, respectively.

(4)The proposed method in this paper focuses on the compressor blade sections.In future work, it will be extended to the parameterization of three-dimensional compressor blades.The application of the proposed method to turbine blades will also be carried out.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This study was partially supported by the Fundamental Research (No.JCKY2021206B057) which is undertaken by Zaozhuang Beihang Machine Tool Innovation Research Institute Co., Ltd, China.