Digital Hydraulic Design for Low-Speciﬁc-Speed Propeller Runners with Fixed Blades

: The operating point of a propeller hydropower station will deviate from the effective workspace while the discharge reduces excessively during dry seasons. It usually leads to a decrease in efﬁciency and even to being unable to work. To solve the above problem, a scheme named decreasing capacity to increase efﬁciency was presented in this article. A low-speciﬁc-speed propeller runner with ﬁxed blades that has the same installing dimensions as the original one was redesigned and equipped in dry seasons. A positive circulation at the outlet of the blades bigger than in conventional runners is allowed. Some key technologies about hydraulic design for runner blades were researched, which include distribution of velocity circulation at the inlet and outlet of the runner, thickening of the epiphyseal line of an aerofoil, unfolding aerofoil being converted to a cylindrical section, etc. In the section on digital modeling for runner blades, aerofoils on the cylindrical sections at the rim and at the hub were constructed employing the trend extrapolation method. Moreover, a blade digital model was built at one time according to the aerofoils on all cylindrical sections by means of a successful redevelopment to UniGraphics, and it has perfect symmetry. A case presented indicates that the method of decreasing capacity to increase efﬁciency is feasible. Using the method, the turbine efﬁciency increased from less than 28.6% to 83.4% while the discharge decreased from 3.20 m 3 s − 1 to 1.00 m 3 s − 1 , and then the hydropower unit was able to work properly.


Introduction
A propeller turbine is a kind of axial flow hydro-turbine in which the fixed blades have the advantages of large overflowing, high efficiency and good cavitation performance. Moreover, the processing cost of the unit is low. It is also convenient to be installed, operate and maintain. The propeller turbine is appropriate for a medium-mini size hydropower unit [1]. Currently, the standardization and serialization of medium-mini size hydropower units have been formulated, which have already formed one more complete system. In addition, several new components are currently being developed constantly. For example, Uchiyama et al. developed a micro-hydraulic turbine with excellent performance in the passage of foreign matter included in the water flow [2]. The runner has four blades, with a circular hollow around the rotating axis to pass foreign matter through the runner. Moreover, Ma et al. designed an inlet guide, vane and runner for developing a 2.5 kW hydraulic turbine. The efficiency and power of the turbine increased by 0.95% and 1.45%, respectively, compared to those of the reference model [3].
As for the development of propeller runners, at present, most scholars are absorbed in improving single capacity and increasing efficiency. For example, aiming at the problem of the insufficient output of the Huangtankou Hydropower Station, based on the existing conditions, a runner optimization design was implemented by CFD in reference [4], and non-circular curves in planes can be extended intelligently for a short distance. However, the method and models are not available for spatial non-circular curves or surfaces.
In view of the difficult problem of the propeller, the turbine runs properly while the discharge reduces excessively during dry seasons; a modification plan of decreasing capacity to increase efficiency has been presented, and several key technical problems in hydraulic design have been solved in this article. In addition, a digital design method for runner blades has been developed by applying trend extrapolation to the extension of spatial non-circular curves and surfaces. In the end, the research achievement is applied to an instantiation; the validity of the theoretical analysis is proved.

Reasons for Failure to Generate
The inlet of a propeller runner has no impact, yet the outlet is along a normal direction under design conditions [14]. However, the design condition corresponds only to a certain head and discharge. But in actual operation, due to the changes in reservoir level and external load, head and discharge can frequently change, which will often make the propeller turbine off its optimum operating condition. The farther the operating condition from the design condition, the more efficiency declines.
Assuming the optimal operating condition of an original runner under high discharge to be "A", while the operating condition unable to generate under low discharge to be "B".The velocity triangles at the inlet and outlet of the airfoil are shown in Figure 1, where the velocity circulation at the outlet of the cylindrical surface is zero. The blade inlet angle will decrease from α 1 to α 1 when the guide vane opening decreases under low-discharge conditions, due to the velocity circulation at the outlet on the cylindrical section being zero, v 2u = 0. The value of the absolute velocity at the inlet depends on H, hence is constant; the direction of it varies with the guide vane opening, namely v 1 = v 1 . The value of a circular velocity depends on the generated frequency; hence, it is constant. Therefore, u 1 = u 1 = u 2 = u 2 . According to the independence assumption among cylindrical layers [15], the value and direction of the meridional velocity at the outlet should be equal to that at the inlet, hence v 2m = v 1m , and v 2m = v 1m . The blade outlet angle of a propeller runner is unable to adjust, and then β 2 remains constant, namely β 2 = β 2 .
Symmetry 2022, 14, x FOR PEER REVIEW 3 of 15 According to the similarities between curve extension and information research, we studied several intelligent extension-models in Reference [13] by introducing trend extrapolation, with which non-circular curves in planes can be extended intelligently for a short distance. However, the method and models are not available for spatial non-circular curves or surfaces.
In view of the difficult problem of the propeller, the turbine runs properly while the discharge reduces excessively during dry seasons; a modification plan of decreasing capacity to increase efficiency has been presented, and several key technical problems in hydraulic design have been solved in this article. In addition, a digital design method for runner blades has been developed by applying trend extrapolation to the extension of spatial non-circular curves and surfaces. In the end, the research achievement is applied to an instantiation; the validity of the theoretical analysis is proved.

Reasons for Failure to Generate
The inlet of a propeller runner has no impact, yet the outlet is along a normal direction under design conditions [14]. However, the design condition corresponds only to a certain head and discharge. But in actual operation, due to the changes in reservoir level and external load, head and discharge can frequently change, which will often make the propeller turbine off its optimum operating condition. The farther the operating condition from the design condition, the more efficiency declines.
Assuming the optimal operating condition of an original runner under high discharge to be "A", while the operating condition unable to generate under low discharge to be "B".The velocity triangles at the inlet and outlet of the airfoil are shown in Figure 1, where the velocity circulation at the outlet of the cylindrical surface is zero. The blade inlet angle will decrease from to when the guide vane opening decreases under lowdischarge conditions, due to the velocity circulation at the outlet on the cylindrical section being zero, = 0. The value of the absolute velocity at the inlet depends on , hence is constant; the direction of it varies with the guide vane opening, namely = . The value of a circular velocity depends on the generated frequency; hence, it is constant. Therefore, = = = . According to the independence assumption among cylindrical layers [15], the value and direction of the meridional velocity at the outlet should be equal to that at the inlet, hence = , and = . The blade outlet angle of a propeller runner is unable to adjust, and then remains constant, namely = .
(a) (b) As shown in Figure 1b, the value of is too large, which means the is too large. According to the fundamental equation = ( − )/2 , being too little will make a sharp drop in efficiency in the case that both and are constant. The propeller turbine will be unable to work because of too little if the discharge descends to a certain value. Moreover, being too large does not favor to energy recovery for draft tubes [16], which can not only decrease efficiency but also reduce the running As shown in Figure 1b, the value of v 2u is too large, which means the Γ 2 is too large. According to the fundamental equation H η s g = ω Γ 1 − Γ 2 /2π, ∆Γ being too little will make a sharp drop in efficiency in the case that both H and ω are constant. The propeller turbine will be unable to work because of too little η s if the discharge descends to a certain value. Moreover, v 2u being too large does not favor to energy recovery for draft tubes [16], which can not only decrease efficiency but also reduce the running stability of units. As shown in Figure 1a, β 1 < β 1 , so there is a great negative flow impact on the outlet under the condition "B". Thus, the hydraulic loss is enormous [17] and the hydraulic efficiency of unit will reduce. For example, the optimal flow of GOE430 runner in a hydropower station is 3.20 m 3 s −1 . When the discharge declines to 1.5 m 3 s −1 , the velocity

Solutions for Failure to Generate
To solve the above problem, we present a scheme as follows. A low-specific-speed propeller runner with fixed blades, named new runner, that has the same installing dimensions with the original one is redesigned. The new runner and the original one are used interchangeably, corresponding to dry seasons and rainy seasons. As in the following, the unit can work properly under different seasons. On account of the original flow passage components remain unchanged, the reconstruction budget is the lowest.
As for the new runner, the parameters of the velocity triangles of the aerofoils on the same cylindrical section, against the original one, at the Inlet and Outlet under the condition "B" does not make any mark at the top-right corner. According to Hgη s = ω(Γ 1 − Γ 2 )/2π [18], H and ω being constant, if the velocity circulation difference of the new runner between inlet and outlet under the condition "B" reaches or approaches that of the original runner under the condition "A", the efficiency of the new runner under the condition "B" reaches or approaches that of the original runner under the condition "A". That is to say, From the above analysis and Figure 2, some parameter relationships between the new runner under condition "B" and the original runner under condition "A" on the same cylindrical section are as follows.
stability of units. As shown in Figure 1a, < , so there is a great negative flow impact on the outlet under the condition "B". Thus, the hydraulic loss is enormous [17] and the hydraulic efficiency of unit will reduce. For example, the optimal flow of GOE430 runner in a hydropower station is 3.20 m 3 s −1 . When the discharge declines to 1.5 m 3 s −1 , the velocity circulation difference = 3.02 m 3 s −1 . The falls into 28.6%, and the units are unable to work completely.

Solutions for Failure to Generate
To solve the above problem, we present a scheme as follows. A low-specific-speed propeller runner with fixed blades, named new runner, that has the same installing dimensions with the original one is redesigned. The new runner and the original one are used interchangeably, corresponding to dry seasons and rainy seasons. As in the following, the unit can work properly under different seasons. On account of the original flow passage components remain unchanged, the reconstruction budget is the lowest.
As for the new runner, the parameters of the velocity triangles of the aerofoils on the same cylindrical section, against the original one, at the Inlet and Outlet under the condition "B" does not make any mark at the top-right corner. According to = ( − )/2 [18], and being constant, if the velocity circulation difference of the new runner between inlet and outlet under the condition "B" reaches or approaches that of the original runner under the condition "A", the efficiency of the new runner under the condition "B" reaches or approaches that of the original runner under the condition "A". That is to say, From the above analysis and Figure 2, some parameter relationships between the new runner under condition "B" and the original runner under condition "A" on the same cylindrical section are as follows.
(a) (b) As shown in Figure 2b, 2u is significantly less than 2u . Therefore, if the inlet angle and outlet angle of the new runner are approximately equal to the inflow angle and outflow angle, there is no or little positive impact at the entrance. Moreover, the velocity circulation of the outflow decreases greatly. In conclusion, the new runner can overcome the problem of the original runner under condition "B".

Digital Hydraulic Design for Blades
As for the new runner, the average cascade solidity should be great for the unit discharge is rather small and the specific speed is very low [19]. Secondly, because of power As shown in Figure 2b, v 2u is significantly less than v 2u . Therefore, if the inlet angle and outlet angle of the new runner are approximately equal to the inflow angle and outflow angle, there is no or little positive impact at the entrance. Moreover, the velocity circulation of the outflow decreases greatly. In conclusion, the new runner can overcome the problem of the original runner under condition "B".

Digital Hydraulic Design for Blades
As for the new runner, the average cascade solidity should be great for the unit discharge is rather small and the specific speed is very low [19]. Secondly, because of power output decreases, the load on the blade's declines compare to the original runners. Finally, as shown in Figure 1b, compare to the original runner, the relative velocity distribution in the vane passage of the new runner decreased greatly, but turbine setting, runner diameter and speed are the same with the original runners. As a result, the cavitation performances of the new runner depend exclusively on the designed aerofoils. Only if the aerofoils are reasonably well designed, can the cavitation performances be superior to the original runner's [4]. In the process of design, the epiphyseal lines of the aerofoils are designed by singularity distribution method firstly, and then the epiphyseal lines are thickened on comparing the thickness variation of a certain excellent aerofoil that has the most suitable hydraulic parameters and similar specific speed against the new runner [20]. Several key technical problems in digital hydraulic design for runner blades are solved as follows.

Distribution of Velocity Circulation
There are three kinds of distribution of velocity circulation at the inlet and outlet of the runner [21]. One of them is that the velocity circulation at the outlet being zero lies in the cylindrical section (denoted by M-M) in the middle locations between the rim and hub, which is employed in this article. The distribution of the velocity circulation at the inlet and outlet of the new runner are shown in Figure 3 under condition "B". Considering the specialty of the design condition, a positive Γ 0 should be added to the initial distribution of velocity circulation at the outlet of the new runner. Consequently, the velocity circulation at the outlet on the M-M is Γ 0 . as shown in Figure 1b, compare to the original runner, the relative velocity distribution in the vane passage of the new runner decreased greatly, but turbine setting, runner diameter and speed are the same with the original runners. As a result, the cavitation performances of the new runner depend exclusively on the designed aerofoils. Only if the aerofoils are reasonably well designed, can the cavitation performances be superior to the original runner's [4]. In the process of design, the epiphyseal lines of the aerofoils are designed by singularity distribution method firstly, and then the epiphyseal lines are thickened on comparing the thickness variation of a certain excellent aerofoil that has the most suitable hydraulic parameters and similar specific speed against the new runner [20]. Several key technical problems in digital hydraulic design for runner blades are solved as follows.

Distribution of Velocity Circulation
There are three kinds of distribution of velocity circulation at the inlet and outlet of the runner [21]. One of them is that the velocity circulation at the outlet being zero lies in the cylindrical section (denoted by M-M) in the middle locations between the rim and hub, which is employed in this article. The distribution of the velocity circulation at the inlet and outlet of the new runner are shown in Figure 3 under condition "B". Considering the specialty of the design condition, a positive should be added to the initial distribution of velocity circulation at the outlet of the new runner. Consequently, the velocity circulation at the outlet on the M-M is .  For the new runner, n 110 = n D 1 √ Hη s [21], According to the fundamental equation of the turbine [21], On the M-M, From Figure 1a, From Equations (3)-(6), the velocity circulation of the new runner at the outlet on the M-M is as follows.
Symmetry 2022, 14, 2250 6 of 15 From the basic equation of the turbine, According to the literature [21], the δΓ shown in Figure 3 can be described as follows.
If the velocity circulation at the outlet is regularly distributed in a straight line along the radius direction, its distribution is as follows.
From Equations (8) and (10), the distribution of the velocity circulation at the inlet along the radius direction is as follows.

Thickening of Epiphyseal Line of Aerofoil
Several discrete points (11 points in this article) on every epiphyseal line of an aerofoil in stream surface can be calculated during the hydraulic calculation of the runner blade. A continuous epiphyseal line of aerofoil can be fitted according to the above discrete points, which can be thickened to an aerofoil with a certain thickness. GOE430 [20] shown in Figure 4 is an aerofoil to thicken accordingly in this article. The position of every discrete point calculated on the epiphyseal line may not coincide with that of the discrete point selected (17 points in literatures [20]). Consequently, the epiphyseal line of aerofoil must be interpolated into 17 points from the 11 points by calculating it according to the value before they are thickened. Those points are signed, respectively, from the outlet to the inlet. There are several interpolation methods, one of which is B-Spline interpolation. It enables the fit curves interpolated to pass all the date points. In addition, the first and second derivatives in nodes are continuous and smooth, and have a good convergence [22]. The b-Spline interpolation method is applied in this article. After the 17 points are interpolated, the thickened value y B (marked as y B i ) and y H (marked as y H i ) corresponding to the points can be calculated respectively.
From Equations (3)-(6), the velocity circulation of the new runner at the outlet on the M-M is as follows.
From the basic equation of the turbine, According to the literature [21], the shown in Figure 3 can be described as follows.
If the velocity circulation at the outlet is regularly distributed in a straight line along the radius direction, its distribution is as follows.
From Equations (8) and (10), the distribution of the velocity circulation at the inlet along the radius direction is as follows.

Thickening of Epiphyseal Line of Aerofoil
Several discrete points (11 points in this article) on every epiphyseal line of an aerofoil in stream surface can be calculated during the hydraulic calculation of the runner blade. A continuous epiphyseal line of aerofoil can be fitted according to the above discrete points, which can be thickened to an aerofoil with a certain thickness. GOE430 [20] shown in Figure 4 is an aerofoil to thicken accordingly in this article. The position of every discrete point calculated on the epiphyseal line may not coincide with that of the discrete point selected (17 points in literatures [20]). Consequently, the epiphyseal line of aerofoil must be interpolated into 17 points from the 11 points by calculating it according to the value before they are thickened. Those points are signed, respectively, from the outlet to the inlet. There are several interpolation methods, one of which is B-Spline interpolation. It enables the fit curves interpolated to pass all the date points. In addition, the first and second derivatives in nodes are continuous and smooth, and have a good convergence [22]. The b-Spline interpolation method is applied in this article. After the 17 points are interpolated, the thickened value (marked as ) and (marked as ) corresponding to the points can be calculated respectively.  As shown in Figure 5, G i and G i−1 (i = 1, 2, . . . , 20) are the two adjacent points obtained from interpolating. Obviously, the relative curvature of the blade aerofoil is minor, and the epiphyseal line From Equations (3)-(6), the velocity circulation of the new runner at the outlet on the M-M is as follows.
From the basic equation of the turbine, According to the literature [21], the shown in Figure 3 can be described as follows.
If the velocity circulation at the outlet is regularly distributed in a straight line along the radius direction, its distribution is as follows.
From Equations (8) and (10), the distribution of the velocity circulation at the inlet along the radius direction is as follows.

Thickening of Epiphyseal Line of Aerofoil
Several discrete points (11 points in this article) on every epiphyseal line of an aerofoil in stream surface can be calculated during the hydraulic calculation of the runner blade. A continuous epiphyseal line of aerofoil can be fitted according to the above discrete points, which can be thickened to an aerofoil with a certain thickness. GOE430 [20] shown in Figure 4 is an aerofoil to thicken accordingly in this article. The position of every discrete point calculated on the epiphyseal line may not coincide with that of the discrete point selected (17 points in literatures [20]). Consequently, the epiphyseal line of aerofoil must be interpolated into 17 points from the 11 points by calculating it according to the value before they are thickened. Those points are signed, respectively, from the outlet to the inlet. There are several interpolation methods, one of which is B-Spline interpolation. It enables the fit curves interpolated to pass all the date points. In addition, the first and second derivatives in nodes are continuous and smooth, and have a good convergence [22]. The b-Spline interpolation method is applied in this article. After the 17 points are interpolated, the thickened value (marked as ) and (marked as ) corresponding to the points can be calculated respectively. As shown in Figure 5, and ( = 1, 2, . . . ,20) are the two adjacent points obtained from interpolating. Obviously, the relative curvature of the blade aerofoil is minor, and the epiphyseal line is very short. Therefore, can be replaced by a line segment . A point on the epiphyseal line is interpolated along its normal direction. G i G i−1 is very short. Therefore, From Equations (3)-(6), the velocity circulation of the new runner at the outlet on the M-M is as follows.
From the basic equation of the turbine, According to the literature [21], the shown in Figure 3 can be described as follows.
If the velocity circulation at the outlet is regularly distributed in a straight line along the radius direction, its distribution is as follows.
From Equations (8) and (10), the distribution of the velocity circulation at the inlet along the radius direction is as follows.

Thickening of Epiphyseal Line of Aerofoil
Several discrete points (11 points in this article) on every epiphyseal line of an aerofoil in stream surface can be calculated during the hydraulic calculation of the runner blade. A continuous epiphyseal line of aerofoil can be fitted according to the above discrete points, which can be thickened to an aerofoil with a certain thickness. GOE430 [20] shown in Figure 4 is an aerofoil to thicken accordingly in this article. The position of every discrete point calculated on the epiphyseal line may not coincide with that of the discrete point selected (17 points in literatures [20]). Consequently, the epiphyseal line of aerofoil must be interpolated into 17 points from the 11 points by calculating it according to the value before they are thickened. Those points are signed, respectively, from the outlet to the inlet. There are several interpolation methods, one of which is B-Spline interpolation. It enables the fit curves interpolated to pass all the date points. In addition, the first and second derivatives in nodes are continuous and smooth, and have a good convergence [22]. The b-Spline interpolation method is applied in this article. After the 17 points are interpolated, the thickened value (marked as ) and (marked as ) corresponding to the points can be calculated respectively. That is to say, that is a line of the two points and on the front and the back of the aerofoil thickened is perpendicular to . Due to | | = and | | = , the following two equations can be obtained in RtΔ and RtΔ according to Pythagoras theorem with the above analysis.
The first point on the epiphyseal line of the aerofoil is not necessarily thickened because it is on the aerofoil surface [21]. The coordinate values ( , ) and ( )  (12) and (13). Therefore, the coordinate values of 33 points on the aerofoil surface have been determined. A smooth curve can be fitted from these points by using B-Spline. Thus, an unfolded aerofoil with a certain thickness can be obtained.

Unfolding Aerofoil Converted to Cylindrical Section
The unfolding aerofoil should be converted to the corresponding cylindrical section in 3D solid modeling. According to the rotation-axis position determinacy [21], a line connecting all rotation axes of the aerofoil is perpendicular to the runner axis. Likewise, a line connecting all outlets of the aerofoil is also perpendicular to the runner axis. As shown in Figure 6, as for the airfoil i ( = 1，2, . . . ,5), the rotation axis ( ) of it is perpendicular to the runner axis, and intersect in the point with it. The outlet ( ) is also perpendicular to the runner axis, and intersect in the point with it. It is obvious that all the rotation axes lie on and outlets on . The radius of the aerofoil on cylindrical section is . As shown in Figure 6, is a coordinate system for hydraulic calculation; moreover, its origin lies in the outlet. On the other hand, each coordinate axis of is parallel to that of . -axis is coincident with the runner axis. In addition, its origin lies on the intersection of -axis and . The coordinate values of a discrete point on the aerofoil can be transformed into those in using affine transformation [23]. See Equation (14), in which ( ， ， ) are the coordinate values of in . As shown in Figure 5, the coordinates value of i respectively. From the right triangle AG i H i (marked as Rt∆AG i H i ) in Figure 5, we can see: Due to |H i G i | = y Hi and |B i G i | = y Bi , the following two equations can be obtained in Rt∆AG i H i and Rt∆CB i G i according to Pythagoras theorem with the above analysis.
The first point on the epiphyseal line of the aerofoil is not necessarily thickened because it is on the aerofoil surface [21].  (12) and (13). Therefore, the coordinate values of 33 points on the aerofoil surface have been determined. A smooth curve can be fitted from these points by using B-Spline. Thus, an unfolded aerofoil with a certain thickness can be obtained.

Unfolding Aerofoil Converted to Cylindrical Section
The unfolding aerofoil should be converted to the corresponding cylindrical section in 3D solid modeling. According to the rotation-axis position determinacy [21], a line connecting all rotation axes of the aerofoil is perpendicular to the runner axis. Likewise, a line connecting all outlets of the aerofoil is also perpendicular to the runner axis. As shown in Figure 6, as for the airfoil i(i = 1, 2, . . . , 5), the rotation axis (o 1 o i ) of it is perpendicular to the runner axis, and intersect in the point o with it. The outlet (A 1 A i ) is also perpendicular to the runner axis, and intersect in the point o A with it. It is obvious that all the rotation axes lie on o 1 o i and outlets on A 1 A i . The radius of the aerofoil i on cylindrical section is r i . As shown in Figure 6, x i y i z i is a coordinate system for hydraulic calculation; moreover, its origin lies in the outlet. On the other hand, each coordinate axis of x y z is parallel to that of x i y i z i . z -axis is coincident with the runner axis. In addition, its origin lies on the intersection of z -axis and o 1 o i . The coordinate values of a discrete point P i on the aerofoil can be transformed into those in x y z using affine transformation [23]. See Equation (14)   = - The is a cylindrical coordinate system, its -axis is coincident with the of the , and its polar axis is just the rotation axis of the runner. As for the | | = . is a point that ( ) is projected onto the -axis. According to t tions between real 3D aerofoil and unfolding one, | | equals to the arc length o As a result, the coordinate value of ( ) that is converted to the cylindrical se is shown in Equation (15). According to the conversion relationship between lindrical coordinate system and spatial Cartesian coordinate system, the coordinat of ( ) in the cylindrical section in is shown in Equation (16).

Digital Hydraulic Design for Runner Blades
Aerofoils on cylindrical section obtained by a hydraulic design for the runne are finite, usually are five in the middle of the blade, as shown in Figure 7. The a on cylindrical sections are saved to an EXCEL file in the form of point coordinat EXCEL file is converted into a ".csv" format ready for import into Imageware [24 map of the blade can be obtained, and then saved it as a ". igs" format, put it into UniGraphics. Finally, the points of aerofoil on cylindrical section are grouped in order and form a close curve by the "Spline Fitting" [25]. The zθρ is a cylindrical coordinate system, its z -axis is coincident with the z -axis of the x y z , and its polar axis is just the rotation axis of the runner. As for the airfoil i, oo i = r i . P iv is a point that P i P i is projected onto the y -axis. According to the relations between real 3D aerofoil and unfolding one, o i P iv equals to the arc length of along the radius direction is as follows. = 2 + 1u − − 60 2 + 1.6 ( − ) ( − ) + 2

Thickening of Epiphyseal Line of Aerofoil
Several discrete points (11 points in this article) on every epiphyseal line o in stream surface can be calculated during the hydraulic calculation of the ru A continuous epiphyseal line of aerofoil can be fitted according to the abo points, which can be thickened to an aerofoil with a certain thickness. GOE430 in Figure 4 is an aerofoil to thicken accordingly in this article. The position o crete point calculated on the epiphyseal line may not coincide with that of point selected (17 points in literatures [20]). Consequently, the epiphyseal lin must be interpolated into 17 points from the 11 points by calculating it acco value before they are thickened. Those points are signed, respectively, from t the inlet. There are several interpolation methods, one of which is B-Spline in It enables the fit curves interpolated to pass all the date points. In addition, t second derivatives in nodes are continuous and smooth, and have a good c [22]. The b-Spline interpolation method is applied in this article. After the 1 interpolated, the thickened value (marked as ) and (marked as sponding to the points can be calculated respectively. As shown in Figure 5, and ( = 1, 2, . . . ,20) are the two adjacent tained from interpolating. Obviously, the relative curvature of the blade aerof and the epiphyseal line is very short. Therefore, can be replac segment . A point on the epiphyseal line is interpolated along its norm o i P iv . As a result, the coordinate value of P i P i that is converted to the cylindrical section in zθρ is shown in Equation (15). According to the conversion relationship between the cylindrical coordinate system and spatial Cartesian coordinate system, the coordinate value of P i P i in the cylindrical section in x y z is shown in Equation (16).

Digital Hydraulic Design for Runner Blades
Aerofoils on cylindrical section obtained by a hydraulic design for the runner blade are finite, usually are five in the middle of the blade, as shown in Figure 7. The aerofoils on cylindrical sections are saved to an EXCEL file in the form of point coordinates. The EXCEL file is converted into a ".csv" format ready for import into Imageware [24]. A bitmap of the blade can be obtained, and then saved it as a ". igs" format, put it into UniGraphics. Finally, the points of aerofoil on cylindrical section are grouped in proper order and form a close curve by the "Spline Fitting" [25]. Because the actual flow of the turbine in the rim and hub cannot meet the independence assumption among cylindrical layer. Therefore, the aerofoil of the cylindrical section is difficult to be obtained through hydraulic design and calculation. It brings great difficulties for the 3D model of runner blades. With respect to this problem, two possible solutions are provided, and then their performance characteristics are analyzed and compared.

Projection
The curve of the nearest aerofoil 1 on the cylindrical section to the rim is projected onto the rim cylinder, which is taken as an approximate section curve of the aerofoil on the rim cylinder. Similarly, the curve of the nearest aerofoil 5 on the cylindrical section to the hub is projected onto the hub cylinder, which is taken as an approximate section curve of the aerofoil on the hub cylinder. A 3D model of the runner blade can be built according to the seven section curves by means of "Through Curves" in UniGraphics. The single blade model is shown in Figure 8a. The integrated runner digital model is shown in Figure  8b. In projection method, the aerofoil on cylindrical section at the rim or hub is formed by projecting the nearest aerofoil to the rim or hub cylinder. That is to say, the projected aerofoil is zoomed in or out based on the runner axes. As shown in Figure 8a, the positions A and B on the inlet of the runner are not in accordance with the trend of its middle part obtained by hydraulic calculation. Therefore, it is difficult to conform to the hydrodynamic rule, which will lead to a decrease in hydraulic efficiency.

Trend Extrapolation
Three kinds of intelligent extension-models [13] such as multinomial extension model, exponential extension-model and Gompertz extension-model had been built by Because the actual flow of the turbine in the rim and hub cannot meet the independence assumption among cylindrical layer. Therefore, the aerofoil of the cylindrical section is difficult to be obtained through hydraulic design and calculation. It brings great difficulties for the 3D model of runner blades. With respect to this problem, two possible solutions are provided, and then their performance characteristics are analyzed and compared.

Projection
The curve of the nearest aerofoil 1 on the cylindrical section to the rim is projected onto the rim cylinder, which is taken as an approximate section curve of the aerofoil on the rim cylinder. Similarly, the curve of the nearest aerofoil 5 on the cylindrical section to the hub is projected onto the hub cylinder, which is taken as an approximate section curve of the aerofoil on the hub cylinder. A 3D model of the runner blade can be built according to the seven section curves by means of "Through Curves" in UniGraphics. The single blade model is shown in Figure 8a. The integrated runner digital model is shown in Figure 8b. In projection method, the aerofoil on cylindrical section at the rim or hub is formed by projecting the nearest aerofoil to the rim or hub cylinder. That is to say, the projected aerofoil is zoomed in or out based on the runner axes. As shown in Figure 8a, the positions A and B on the inlet of the runner are not in accordance with the trend of its middle part obtained by hydraulic calculation. Therefore, it is difficult to conform to the hydrodynamic rule, which will lead to a decrease in hydraulic efficiency. Because the actual flow of the turbine in the rim and hub cannot meet the independence assumption among cylindrical layer. Therefore, the aerofoil of the cylindrical section is difficult to be obtained through hydraulic design and calculation. It brings great difficulties for the 3D model of runner blades. With respect to this problem, two possible solutions are provided, and then their performance characteristics are analyzed and compared.

Projection
The curve of the nearest aerofoil 1 on the cylindrical section to the rim is projected onto the rim cylinder, which is taken as an approximate section curve of the aerofoil on the rim cylinder. Similarly, the curve of the nearest aerofoil 5 on the cylindrical section to the hub is projected onto the hub cylinder, which is taken as an approximate section curve of the aerofoil on the hub cylinder. A 3D model of the runner blade can be built according to the seven section curves by means of "Through Curves" in UniGraphics. The single blade model is shown in Figure 8a. The integrated runner digital model is shown in Figure  8b. In projection method, the aerofoil on cylindrical section at the rim or hub is formed by projecting the nearest aerofoil to the rim or hub cylinder. That is to say, the projected aerofoil is zoomed in or out based on the runner axes. As shown in Figure 8a, the positions A and B on the inlet of the runner are not in accordance with the trend of its middle part obtained by hydraulic calculation. Therefore, it is difficult to conform to the hydrodynamic rule, which will lead to a decrease in hydraulic efficiency.

Trend Extrapolation
Three kinds of intelligent extension-models [13] such as multinomial extensionmodel, exponential extension-model and Gompertz extension-model had been built by introducing trend extrapolation, which can be selected and used reasonably according to

Trend Extrapolation
Three kinds of intelligent extension-models [13] such as multinomial extension-model, exponential extension-model and Gompertz extension-model had been built by introducing trend extrapolation, which can be selected and used reasonably according to the feature points of curves. Almost all non-circular curves in plane can be extended intelligently. The study also found that a curve extended near the starting point is proving well by comparison with the theoretical curve. The closer the distance from the starting point, the higher the accuracy is. 33 homonymous points on the five aerofoil curves gained by hydraulic calculation need to be fitted using a curve. It is obvious that the fitting curve is a spatial curve. Therefore, the extension technology for planar curves based on trend extrapolation in reference [13] is not immediately available.

Extending Spatial Curve
33 homonymous points on the five aerofoil curves are marked as P k i , where k equals to 1, 2, . . . , 5; i equals to 1, 2, . . . , 33. As shown in Figure 9, S is a curve fitted by the homonymous points i on the five aerofoil curves. Its expressions are as follows.
Symmetry 2022, 14, x FOR PEER REVIEW 10 of 15 the feature points of curves. Almost all non-circular curves in plane can be extended intelligently. The study also found that a curve extended near the starting point is proving well by comparison with the theoretical curve. The closer the distance from the starting point, the higher the accuracy is. 33 homonymous points on the five aerofoil curves gained by hydraulic calculation need to be fitted using a curve. It is obvious that the fitting curve is a spatial curve. Therefore, the extension technology for planar curves based on trend extrapolation in reference [13] is not immediately available.

Extending Spatial Curve
33 homonymous points on the five aerofoil curves are marked as , where equals to 1, 2, …, 5; equals to 1, 2, …, 33. As shown in Figure 9, is a curve fitted by the homonymous points on the five aerofoil curves. Its expressions are as follows.
As shown in Figure 9, the radiuses of the cylinders from Section 1 to Section 5 are decreasing at an arithmetic progression rate. That is to say, − are all constant. On the other hand, the difference of that between the rim and Section 1 is ( − ) 2 ⁄ , the same as between Section 5 and the hub. As for the plane curves [ ] and [ ] , due to the parts of them to be extended are approximately 10% of the length of the original curves, they only need to be extended for a short distance. According to the literature [13], the extension technology for plane curves based on trend extrapolation can be adopted. As shown in Figure 9, the curves [ ] and [ ] can be extended respectively in rim direction using the extension technology for plane curves based on trend extrapolation in literature [13]. Thus, the plane curves [ ] and [ ] can be obtained, and their expressions are shown in Equations (20) and (21). The P k i in the curve S are projected onto the plane xoy, the projected points of which are marked as P k [xoy]i ; similarly, the P k i are projected onto the plane xoz, the projected points of which are marked as P k [xoz]i . As shown in Figure 9, the projected points P k [xoy]i can be fitted into a plane curve S [xoy] , see in Equation (18); likewise,P k [xoz]i can be fitted into S [xoz] , see in Equation (19).
As shown in Figure 9, the radiuses of the cylinders from Section 1 to Section 5 are decreasing at an arithmetic progression rate. That is to say, r i+1 − r i are all constant. On the other hand, the difference of that between the rim and Section 1 is (r i+1 − r i )/2, the same as between Section 5 and the hub. As for the plane curves S [xoy] and S [xoz] , due to the parts of them to be extended are approximately 10% of the length of the original curves, they only need to be extended for a short distance. According to the literature [13], the extension technology for plane curves based on trend extrapolation can be adopted.
As shown in Figure 9, the curves S [xoy] and S [xoz] can be extended respectively in rim direction using the extension technology for plane curves based on trend extrapolation in literature [13]. Thus, the plane curves S [xoy] and S [xoz] can be obtained, and their expressions are shown in Equations (20) and (21).
Taking S [xoy] and S [xoz] as two generatrixes, two cylindrical surfaces can be formed by them being stretched along the z-axis and the y-axis respectively. Eventually, they are crossing on the curve S . Their expressions are shown in Equation (22).
In the same way, the curves S [xoy] and S [xoz] can be extended respectively in hub direction, and then the plane curves S [xoy] and S [xoz] can be obtained. Taking S [xoy] and S [xoz] as two generatrix, two cylindrical surfaces can also be formed by them being stretched along the z-axis and the y-axis respectively. They are crossing on the curve S . The curves S and S are the intelligent extension parts, respectively, of the spatial curve S along the rum and the hub direction. The intersections of the curve S with the rim cylinder are marked as P 6 i ; the intersections of the curve S with the hub cylinder are marked as P 0 i .

Digital Modeling for Runner
A hydraulic design program for runner blades has been developed by using MATLAB, in which the 33 discrete points on each of the five aerofoil curves gained by hydraulic calculation have been fitted into a closed curve by using a B-Spline [22]. By means of the above method of extending spatial curve using trend extrapolation, the homonymous points P 6 i and P 0 i on the aerofoils on the rim and hub separately can be obtained. Similarly, those points can be fitted into hub aerofoil and rim one separately by using B-Spline. The top view of the extending curves of aerofoils on the cylindrical section is shown in Figure 10a; the spatial diagram of the extending curves of aerofoils on the cylindrical section is shown in Figure 10b. The same as projection method, a 3D model of the runner blade can also be built according to the seven section curves by means of "Through Curves" in UniGraphics. The single-blade model is shown in Figure 11a. The integrated runner digital model is shown in Figure 11b. Unlike Figure 8, the aerofoils on the inlet of the runner are accord with the trend of its middle part. Therefore, there is a high hydraulic efficiency for the runner.

Digital Modeling for Runner
By means of the manual modeling method "Through Curves" in UNIGRAPHICS, two aerofoil curves are selected each time to generate part of the aerofoil surface, which is partial and fractional. Thus, the surface is discontinuous in junction. As shown in Figure  8 and Figure 11, the surface obtained is not smooth, which will necessarily lead to a decrease in hydraulic efficiency. In order to make the blade surface became smooth and continuous, an optimization program was developed and integrated into UNIGRAPHICS using Visual C++ in this article. The seven aerofoil curves can be imported into the program running in the NX Open Grip environment at one time, and then those curves are picked up in turn using sphere.grx to fit into a B-surface at once [22]. The single blade model optimized is shown in Figure 12a. The integrated runner digital model optimized is shown in Figure 12b. Compared with the runner model shown in Figure 11, the blade surface of that shown in Figure 12 is continuous and smooth, and has better flow-field stability and higher hydraulic efficiency.  By means of the manual modeling method "Through Curves" in UNIGRAPHICS, two aerofoil curves are selected each time to generate part of the aerofoil surface, which is partial and fractional. Thus, the surface is discontinuous in junction. As shown in Figures 8 and 11, the surface obtained is not smooth, which will necessarily lead to a decrease in hydraulic efficiency. In order to make the blade surface became smooth and continuous, an optimization program was developed and integrated into UNIGRAPHICS using Visual C++ in this article. The seven aerofoil curves can be imported into the program running in the NX Open Grip environment at one time, and then those curves are picked up in turn using sphere.grx to fit into a B-surface at once [22]. The single blade model optimized is shown in Figure 12a. The integrated runner digital model optimized is shown in Figure 12b. Compared with the runner model shown in Figure 11, the blade surface of that shown in Figure 12 is continuous and smooth, and has better flow-field stability and higher hydraulic efficiency.

Digital Modeling for Runner
By means of the manual modeling method "Through Curves" in UNIGRAPHICS, two aerofoil curves are selected each time to generate part of the aerofoil surface, which is partial and fractional. Thus, the surface is discontinuous in junction. As shown in Figure  8 and Figure 11, the surface obtained is not smooth, which will necessarily lead to a decrease in hydraulic efficiency. In order to make the blade surface became smooth and continuous, an optimization program was developed and integrated into UNIGRAPHICS using Visual C++ in this article. The seven aerofoil curves can be imported into the program running in the NX Open Grip environment at one time, and then those curves are picked up in turn using sphere.grx to fit into a B-surface at once [22]. The single blade model optimized is shown in Figure 12a. The integrated runner digital model optimized is shown in Figure 12b. Compared with the runner model shown in Figure 11, the blade surface of that shown in Figure 12 is continuous and smooth, and has better flow-field stability and higher hydraulic efficiency.

Case
The basic parameters of runner GOE430 in a hydropower station are shown in Table 1. Rated discharge during rainy seasons is 3.20 m 3 s −1 . While the discharge decreased to 1.00 m 3 s −1 during dry seasons, the hydropower unit is unable to work. A new runner used in dry seasons has been designed according to the plan of decreasing capacity to increase efficiency, in which part of the intermediate and final results are as follows: n 110 = 172.1 r/min; Q 110 = 0.4483 m 3 s −1 ; n s = 322.6 m·kW; incremental velocity circulation Γ 0 = 1.46 m 2 s −1 ; (l/t) pj = 1.5; Z = 6. Based on GOE430, a new runner that has been designed and manufactured is shown in Figure 13, which has been put into operation. By running in experiment test and practical work, η = 83.4%, it can fulfill power properly.  Figure 13. New runner designed and manufactured.

Conclusions
The reason for the unit being unable to generate is that the velocity circulation at th outlet of the original runner is too large under low-discharge conditions during dry sea sons, and then its hydraulic efficiency would reduce sharply. To solve the problem, a low specific-speed propeller runner with fixed blades has been redesigned and equipped in dry seasons according to the scheme of decreasing capacity to increase efficiency. The new runner has the same installing dimensions, the same runner type, and some parameter with the original one. However, a positive circulation at the outlet of the blades bigge than in a conventional runner is allowed. Moreover, the new runner has bigger cascad solidity, a greater number of blades, and then has better cavitation characteristics. Com puter-aided design technologies are employed in hydraulic design. Such methods a "thickening of epiphyseal line of aerofoil" and "unfolding aerofoil being converted to cy lindrical section" have been researched and worked out. Hence it becomes possible t reduce R & D cycle, and improve design efficiency greatly. The runner digital model pro vides a foundation for CFD analysis and NC manufacturing. A method to establish th aerofoils on the cylindrical sections at the rim and hub has been provided. Employing trend extrapolation in this article, a spatial curve can be extended successfully. Thus, th aerofoils on the cylindrical sections can also be extended along the rum and hub direction according to their tendency. Comparing to projection method, this method can gain a run ner blade with its surface being more in line with the trend of the middle part, and bein more accord with fluid dynamics. Thus, the hydraulic efficiency is higher. The blad curves have been established through secondary development of UNIGRAPHICS with Visual C++ according to the aerofoils on the cylindrical sections, and picked up in turn t fit into a B-surface at once. Comparing to "Through Curves", the blade surface obtained is continuous and smooth, and has better flow-field stability and higher hydraulic effi ciency. A new runner in the case has been successfully applied in a hydropower station The result shows that the technical route is reasonable and scheme is rational and feasible Using the method, the turbine efficiency increased from less than 28.6% to 83.4% whil the discharge decreased from 3.20 m 3 s −1 to 1.00 m 3 s −1 . The redevelopment came to a com plete success.

Conclusions
The reason for the unit being unable to generate is that the velocity circulation at the outlet of the original runner is too large under low-discharge conditions during dry seasons, and then its hydraulic efficiency would reduce sharply. To solve the problem, a low-specific-speed propeller runner with fixed blades has been redesigned and equipped in dry seasons according to the scheme of decreasing capacity to increase efficiency. The new runner has the same installing dimensions, the same runner type, and some parameters with the original one. However, a positive circulation at the outlet of the blades bigger than in a conventional runner is allowed. Moreover, the new runner has bigger cascade solidity, a greater number of blades, and then has better cavitation characteristics. Computer-aided design technologies are employed in hydraulic design. Such methods as "thickening of epiphyseal line of aerofoil" and "unfolding aerofoil being converted to cylindrical section" have been researched and worked out. Hence it becomes possible to reduce R & D cycle, and improve design efficiency greatly. The runner digital model provides a foundation for CFD analysis and NC manufacturing. A method to establish the aerofoils on the cylindrical sections at the rim and hub has been provided. Employing trend extrapolation in this article, a spatial curve can be extended successfully. Thus, the aerofoils on the cylindrical sections can also be extended along the rum and hub direction according to their tendency. Comparing to projection method, this method can gain a runner blade with its surface being more in line with the trend of the middle part, and being more accord with fluid dynamics. Thus, the hydraulic efficiency is higher. The blade curves have been established through secondary development of UNIGRAPHICS with Visual C++ according to the aerofoils on the cylindrical sections, and picked up in turn to fit into a B-surface at once. Comparing to "Through Curves", the blade surface obtained is continuous and smooth, and has better flow-field stability and higher hydraulic efficiency. A new runner in the case has been successfully applied in a hydropower station. The result shows that the technical route is reasonable and scheme is rational and feasible. Using the method, the turbine efficiency increased from less than 28.6% to 83.4% while the discharge decreased from 3.20 m 3 s −1 to 1.00 m 3 s −1 . The redevelopment came to a complete success.