Integrated,passage,design,based,on,extended,free-form,deformation,and,adjoint,optimization

Xin LI, Tongtong MENG, Weiwei Li,*, Ling ZHOU, Lucheng JI

aInstitute for Aero Engine, Tsinghua University, Beijing 100084, China

bSchool of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China

KEYWORDSTurbomachinery;Aerodynamic;Transonic flow;Integrated passage design;Extended free-form deformation;Adjoint optimization

AbstractInspired by the three-dimensional design of flow passages in turbomachinery, this study proposes the concept of integrated passage design.The capability of adjoint method for efficient optimization and the flexibility of the parameterization method based on extended free-form deformation have been considered to develop a feasible approach to design an integrated passage.This concept was applied to redesign a typical transonic fan,Rotor 67,and the results were analyzed by CFX.It is shown that the passage was adequately adjusted in all three dimensions and reduced the strength of shock wave and wake-induced flow.In particular,the secondary flow was appropriately reorganized and the corner separation was well controlled in the end wall region,leading to significant improvements in adiabatic efficiency and diffusion.

Improving the aerodynamic performance of turbomachinery is becoming increasingly challenging.1Multiple design spaces must be considered in design,such as blade sections,2stacking lines,3end wall contours,4,5and corner regions.6Although significant advances have been made in the design of each space,a comprehensive structure in which all these design spaces are robustly combined is still difficult to obtain and expensive to develop.Given that the four design spaces mentioned above constitute a major part of the flow passage, their combined design can be regarded as that of the passage itself.This study focuses on improving the passage by the process of coupled design, which is also called as Integrated Passage Design(IPD), and demonstrates its implementation.

Although it can be regarded as a combination of the designs in different design spaces, IPD is not a simple addition of relevant design techniques but a reasonable choice that follows flow characteristics in turbomachinery.The internal flow in turbomachinery is different from external flow owing to the restricted space.Except for inlet and outlet, most physical boundaries are viscous solid walls in aerodynamic simulation of turbomachinery.Therefore, the shape of physical solid boundaries (i.e., passage) is crucial to its overall performance.The traditional design focuses on the blade, which has led to significant performance improvements.However, although the blade is the most important component of passage, other parts,such as corner region and end wall,still need to be better designed.7,8In recent years,relevant design techniques,such as the non-axisymmetric end wall9and Blended-Blade and End Wall(BBEW),1,6have shown the potential to improve aerodynamic performance for aforementioned regions.A control technique that conforms to the shape of entire passage requires further investigation.

The IPD concept is influenced by the trend of complete Three-Dimensional(3D) passage design.For a long time after the development of turbomachinery,the design of blade profile was the key for improving aerodynamic performances of aeroengine.Owing to a lack of understanding of internal flow in the early stage of research,several series of blade profiles based on airfoils,e.g.,NACA 65 and C4,are used.This was followed by independent design methods, such as multi-circle design10and controllable diffusion airfoil,11,12which became mainstream in blade design.This weakens the connection between the blade design and wing design.However, these methods mainly focus on the blade design by considering its 2D sections.In this study,to distinguish them from subsequent techniques, designs that conform to blade profiles are known as‘‘the first generation of 3D blade design”.

Moreover, the other dimension of 3D blade, i.e., spanwise stacking, was considered.Studies on sweep and dihedral started in the 1940s, and the development has been long and winding.3,13The understanding of corresponding mechanisms was gradually deep at the 21st century.14With the help of blade profile design and stacking line control, the entire blade can be constructed in a 3D approach,although it appears to a simple combination of these two different design spaces.Designs that conform to the stacking line are known as ‘‘the second generation of 3D blade design”.

While the aforementioned two generations of blade design might appear to have completed 3D characteristics of passage,flow in the mainstream can be adequately organized and～30% of the aerodynamic loss occurs owing to the end wall.8Due to its complicated structure and flow characteristics, no systematic theory for the design of whole end wall has been developed.However, several active and passive control techniques have been effective, such as boundary layer removal technique,15,16non-axisymmetric end wall, and BBEW.As shown in Fig.1, the corresponding passage profiles are different from that of the first and second generations.In this study,designs that conform to the end wall are known as ‘‘the third generation of 3D blade design”.

The direct control on entire passage profile is considered to be reasonable.Against this backdrop, the IPD tries to integrate design spaces of all three generations of blade design techniques in a unified and harmonious approach.To this end, we used the parameterization method of Extended Free-

Form Deformation (EFFD)17in the aerodynamic design of turbomachinery.To validate the effectiveness of IPD,a typical transonic fan, Rotor 67,18was redesigned, and improvements in its aerodynamic performance were analyzed.

2.1.Reasonableness of EFFD

The parameterization method determines the geometric deformation required to realize IPD.We used an EFFD method.

Parameterization can be divided into two categories: (A)those involving a new design using a curve or a surface function,19,20and(B)those involving changes to the original design using a perturbation function.21,22The shape of passage is complicated for the first case,and requires a manual combination of such construction functions as the Bezier curve19and the NURBS surface to build a 3D passage.In this case, IPD is ineffective and is a simple addition to other design techniques.For the second category, the traditional perturbation function for the curve or surface, such as the Hicks–Henne bump function,21requires a manual combination to determine how each curve/surface changes.This procedure is complicated, leads to the loss of some design space, and is not compatible with the IPD concept.A 3D perturbation function is the most suitable approach for IPD.

A 3D perturbation function refers to FFD.23FFD is flexible enough and has been widely used in the aerodynamic design of turbomachinery.22However,its control volume must be cuboid,which makes it challenging to satisfy the periodicity requirement of end wall.In the relevant study on FFD, only part of the blade can be involved in the region of deformation.

The strict constraint on the cuboid control volume has driven improvements to the FFD algorithm.An important improvement that allows for an arbitrary shape of control volume is EFFD.We used this to render the entire 3D passage in the control volume to build a passage-shaped control volume.Thus,adjustments to the control volume by EFFD are directly operated on the passage.Moreover, not only the periodicity requirement can be satisfied, the passage-shaped control volume can also be changed in three dimensions to render its arbitrary deformation.

2.2.Implementation of parameterization

Fig.1 Trend of 3D design.

The theory of EFFD is detailed in Ref.17 and is not explained here.Although EFFD is the main algorithm for the parameterization of IPD, direct EFFD is inconvenient to implement owing to its iteration process and the reshaping of control volume.17Several operations are therefore applied to enhance the feasibility of this parameterization method by avoiding the iterations of EFFD and guaranteeing the satisfaction of periodicity requirement.Together with the EFFD algorithm,these operations constitute the entire parameterization method for IPD.

The deformation of the flow passage in Rotor 67 has been considered to demonstrate the detailed implementation of the parameterization method for IPD.First, the deformed physical region needs to be confirmed, which is a single passage in Fig.2.

The process of implementation can be expressed as follows:

Step 1.Normalization of the flow domain

According to Ref.17, the relative coordinates of the deformed physical node should be calculated using the hypersurface and Newton’s iterations for the use of EFFD, which significantly increases the deformation cost.Moreover,although the control volume of EFFD can be arbitrary in theory, it must be reconstructed according to the shape of deformed physical region for specific applications.To avoid these time-consuming procedures, a linear transformation was performed using Eq.(1) in parameterization.The linearized results of the scenario in Fig.2 are shown in Fig.3.Thus, the control volume of EFFD can maintain a cylindrical shape and the relative coordinates of control point can be directly calculated using Eq.(2).

In Eqs.(1)and(2),the superscript‘‘′”refers to the normalized value, ‘‘node”refers to each deformed physical node;‘‘hub”and ‘‘tip”refer to the corresponding positions of the hub and tip; ‘‘periodic1”and ‘‘periodic2”denote the corresponding positions at two periodic boundaries; ‘‘min”and‘‘max”denote the minimum and maximum axial positions,respectively; r refers to the radial coordinate, θ is the circumferential coordinate; z is the axial coordinate.u, v, and w are the relative coordinates of control point.

Fig.2 Deformed physical region.

Fig.3 Normalized physical region.

Step 2.Determination of control points, sub-control volumes, and virtual control points

Once the total control volume is confirmed in Step 1,EFFD can be simply implemented.The geometric deformation of FFD relies on the adjustment of control points, which are averaged distributed by Eq.(3).In Eq.(3), the variables ‘‘i,j,k”are the serial numbers of control points in each direction,and the subscript‘‘all”is the total number of control points in corresponding direction.The main difference between FFD and EFFD lies in the concept of partial deformation, which implies that the physical node is deformed by the sub-control volume but not the entire control volume.In IPD parameterization, the sub-control volumes were selected as the domain inside two sets of circumferential control points.For partial deformation,the virtual control points are needed to guarantee the continuity of curve, where adjustments should vary with changes in the control points.The results of this step are shown in Fig.4.

Fig.4 Control points/volumes in normalized domain.

Step 3.Deformation by perturbing control points

EFFD builds a relationship between control points and the physical domain.Once the control points are established(Fig.4), each actual control point can be manually adjusted,and each virtual control point should be accordingly adjusted.In IPD parameterization,the control points along the periodic boundary should undergo the same changes and those at the minimum/maximum axial position should remain unchanged.The other control points can be changed in all three directions.Under the aforementioned settings, the passage can be subjected to arbitrary 3D deformation and the requirements for periodicity and continuity can be satisfied.In this study, the control points at the shroud were left unchanged to avoid the influence of complex flow in tip region.Fig.5 shows a set of random perturbations and its deformation result of the normalized passage is shown in Fig.6.

Step 4.Transformation back to a normal passage

This process is the inverse of Step 1.Using Eq.(1), the deformed passage can be transformed into its original shape.For example, the passage shown in Fig.5 is deformed by random perturbations(Fig.7),where the requirements of flexibility, continuity and periodicity can be easily checked.

Although the aforementioned parameterization method can satisfy the requirements for IPD, it has several disadvantages.For example, a sufficient number of control points is required to better control the 3D passage, which incurs a large calculation cost.Moreover,it is difficult to implement direct geometric constraints during the design process.However, the advantages of this method are evident.Moreover, the grid in the passage can be deformed using this parameterization tool.In addition,the capability of flexible deformation is sufficiently attractive to address or overcome the aforementioned disadvantages.Specifically, the calculation cost can be reduced using the adjoint method, whereas we temporarily ignore the geometric constraint.

Fig.5 Randomly perturb control points.

Fig.6 Deformed normalized domain.

Fig.7 Deformation of passage domain.

As shown in Fig.5, a large number of design variables are involved in the control volume of EFFD, leading to a prohibitive calculation cost for general design methods.Traditional manual designs, global optimization methods, and general gradient-based optimization methods are therefore limited.The adjoint method is mainly considered here because its calculation cost has almost no relationship with the number of design variables.

Proposed by Lions in 1971,24the adjoint method was first used for external aerodynamic optimization by Jameson in 1988.25And it was not until the 2000s that several continuous and discrete adjoint optimization systems were developed to optimize aerodynamic shape of turbomachinery by Dreyer,26Liu,27,28and Duta29et al.This has led to considerable advances in multi-point optimization,30,31multi-disciplinary optimization,32,33high-precision adjoint analysis,34,35highly flexible parameterization method,22robust design,36and uncertainty analysis.37–39

Given the remarkable capability of the adjoint method, we have developed a continuous adjoint optimization tool known as OptiTurbo21and a discrete adjoint optimization tool known as TurboSim_un.In this study, the latter discrete one was used.TurboSim_un is a complete adjoint optimization system that contains a flow solver based on the RANS equations and a corresponding turbulent adjoint solver.The flow governing equations are spatially discretized in the framework of a cellcentered finite volume method with inviscid flux calculated using the Roe scheme.GMRES (Generalized Minimal Residual method) is used to solve the discretized equations.The adjoint solver was manually developed based on the SA turbulence model.The tetrahedral and hexahedral grids are supported by TurboSim_un.The grids were structurally generated but treated in an unstructured manner in this study.Simulation and sensitivity analysis capabilities are discussed in the next section.

4.1.Numerical validation for TurboSim_un and CFX on Rotor 67

According to the location of experimental measurements in Ref.40, the computational domain was selected as the range between Station 1 and Station 2 in Fig.8.The grid was generated by AutoGrid 5, and the total number of nodes was 1.18 million with a 0.001 mm minimum wall distance, as shown in Fig.9.The simulation condition was the peak-efficiency condition which had detailed experimental data and the SST(Shear Stress Transport) turbulence model was adopted.The boundary conditions in the simulation were carefully adjusted so that the averaged boundary parameters matched well with experimental data,40as shown in Fig.10.The patmin Fig.10 refers to the static pressure for atmosphere, which is also 101325 Pa.p0refers to the total pressure.T0refers to the total temperature and p refers to the pressure.

Fig.8 Location of experiment measurement.40

Fig.9 Grid used in Rotor 67 validation.

Under the aforementioned conditions, the axial distributions of the Mach number in a particular circumferential position from the experiment40and TurboSim_un are shown in Fig.11.The horizontal axis ‘‘chord-wise”in Fig.11 refers to the nondimensionalized axial position where the position at leading edge is 0 and the position at trailing edge is 1.The results have matched,although there are still some differences.These differences are more evident in the Mach number contours in Fig.12.

Objectively, the isolines of the Mach number indicate that the computational domain is significantly different from the experiment.Especially at the 30% span, there is an obvious separation area around the trailing edge on the suction side,whereas it is almost nonexistent in the experiment.To identify the source of these differences, a similar numerical simulation was performed in a commercial simulation software, CFX,whose boundary status is also shown in Fig.10,and the corresponding Mach number contour is shown in Fig.13.

As we can observe, though still different from the experiment, the two simulation results are almost the same, especially for the separation area in Fig.13(a).The differences are mainly distributed at the 90% span, which, as per our assumption, come from the different inlet boundaries.

According to Ref.40,the boundary conditions measured in Fig.8 were not from the real inlet and outlet.In another word,the measured inlet and outlet data must be circumferentially different and this difference cannot be specifically applied in simulation.In our opinion, different boundary conditions are major reasons for different flow fields.To solve this,a possible approach is to extend the computational domain and keep modifying the boundary condition until the averaged parameters at Station 1 and Station 2 are the same as those in experiment.However, we believe there is high uncertainty about this approach and the validation may be less rigorous.As the simulation results of TurboSim_un are similar to those of CFX,which can prove the simulation capability of the flow solver, no further attempts are made in this section.

4.2.Numerical validation for both TurboSim_un and CFX on Stage 35

However, the mismatch with the experiment in Rotor 67 validation is defective.To further guarantee the simulation capability of TurboSim_un, another numerical validation on Stage 35 is provided.

Fig.10 Boundary conditions in Rotor 67 validation.

Fig.11 Axial distributions of Mach number.

Overall,the numerical validation is similar to that provided in Ref.41.Fig.14 shows that 5.94 million grids were used for a domain containing two rotors and three stator blades,and the minimum distance to the wall was 0.001 mm.Under the boundary conditions for the inlet listed in Table 1, Fig.15 illustrates the validation results of the SST turbulence model,where Prrefers to the ratio of total pressure.

Under the near peak-efficiency condition,the spanwise distributions of the exit angle and total pressure ratio by Turbo-Sim_un well match the CFX results.However, the differences in the total temperature ratio and efficiency were relatively large.As per our analysis, all the results by TurboSim_un are closer to the experimental distributions than the other two tools,41SKE (Standard k-ε turbulence model) and CTE(CMOTT modified k-ε turbulence model), which implies the simulation accuracy of TurboSim_un is acceptable.

4.3.Gradient validation for adjoint solver in TurboSim_un on a tube

The correctness of the adjoint solver in TurboSim_un was validated based on a gradient comparison with the Finite Difference Method (FDM).To avoid the interference of complex flow phenomena(separation phenomena will cause much trouble for the theoretical accuracy of the adjoint method42), the validation test case is chosen as a simple duct, whose basic shape and grid are shown in Fig.16 and Fig.17.The boundary conditions for the subsonic flow with no separation are listed in Table 2.

The outer wall of the duct was parameterized by the Hicks-Henne bump function43and only three control variables (α1-α3) were applied during validation.The recovery coefficient of the total pressure is chosen as the objective function.Fig.18 shows the distribution of the gradient by FDM and TurboSim_un,where the dotted line is the result of the adjoint method and discrete points are the results of FDM.The horizontal axis δα is the perturbation size and the vertical axis‘‘Gradient”is the sensitivity result.

As shown in Fig.18, the sign and relative size of gradient results of TurboSim_un are similar to those of FDM.When the perturbation size in FDM is properly chosen,the differences between the two results were sufficiently small in our opinion.Therefore, it can be concluded that the numerical accuracy of the gradient calculation in TurboSim_un can satisfy the need for aerodynamic optimization.

Fig.12 Contours of Mach number in validation results of Rotor 67.

4.4.Gradient validation for adjoint solver in TurboSim_un on Rotor 67

Although the gradient validation in the simple flow above agrees well with our expectations,the capability of the adjoint solver in general compressor blades needs to be tested.Before validation, it can be predicted in advance that, for a complex flow with an obvious separation region, the flow field cannot converge fully, as shown in Fig.19.According to Ref.42,the corresponding gradient of the adjoint method deteriorates regardless of the turbulence model used in the adjoint solver.What is more, for FDM, the gradient will be quite different at different virtual iterative states, which implies that the numerical accuracy of FDM will deteriorate too.

Adiabatic efficiency was selected as the objective function for this validation.Other settings,including the grid,boundary conditions, and parameterization method, are the same as the optimization in Section 5.However, in contrast to the validation in the tube,the perturbation size is chosen as 1.0×10–4to reduce the calculation cost.Fig.20 shows the gradient distribution for several design variables.

Fig.13 Contours of Mach number in CFX results.

Fig.14 Grids for Stage 35.

Table 1 Boundary conditions for numerical validation.

Fig.15 Numerical validation of Stage 35 using CFX.

Fig.16 Basic shape of validation duct.

Fig.17 Grid for validation.

Table 2 Boundary conditions for duct.

Fig.18 Gradient distribution in tube validation.

Fig.19 Typical iterative history for flow with separation region.

The two series of gradient distribution coincide well in some locations, whereas differences in other regions are evident.However, the overall distribution is analogous in which the sign and relative sizes of the two gradient distributions are similar.Given that the FDM and adjoint method are influenced by the separation flow, the results in Fig.20 can prove that the gradient by adjoint solver in TurboSim_un can satisfy the requirement for aerodynamic optimization in turbomachinery.Although the capability to locate the local optimum design decreases, the adjoint optimization can still make improvement, which can be proved by the following design in Section 5.

Fig.20 Gradient distribution in Rotor 67 validation.

5.1.Simulation and optimization settings

The typical transonic fan, Rotor 67, is redesigned using IPD.

The range of axial deformation is shown in Fig.21.There are 15 axial control points (excluding the virtual control points), 9 radial control points, and 8 circumferential control points.The distribution range of each is within its original local control volume.The total number of design variables is 2184.

The grid used for optimization and verification includes 1.2 million nodes, and the minimum wall distance is 0.001 mm,which is selected based on the validation of grid independence in Fig.22.

The boundary conditions for aerodynamic adjoint optimization are listed in Table 3 and are also considered as the peak efficiency conditions at 100% rotational speed.The SA turbulence model is used in TurboSim_un for both the flow solver and adjoint analysis.The SST turbulence model is used in CFX for the final verification.

The objective function of adjoint optimization is as follows:

where η is the adiabatic efficiency,m is the mass flow,Pris the total pressure ratio, ω is the weighted coefficient, and the subscript "0" represents the parameters of the prototype.ω1and ω2are set to 100 and 200, respectively.

Fig.21 Computation and design domain of Rotor 67.

Fig.22 Grid independence of Rotor 67 in CFX.

Table 3 Boundary conditions for IPD-based redesigning of Rotor 67.

During optimization,only the grids on solid boundaries are perturbed by EFFD,and the inner grids are updated by linear elastic deformation to maintain the grid quality.42The perturbation on grids can be flexibly adjusted.

5.2.Optimization results

Fig.23 Flow field and adjoint field at 50% span in prototype.

Using the aforementioned settings,the Rotor 67 design can be automatically optimized by the adjoint method.Fig.23 presents the contours of axial Mach number (Maz) and corresponding adjoint variables at the 50% span.The isolines of wake shown in Fig.23(a) exhibit a similar distribution in Fig.23(b), which is located around the leading edge.The opposite spatial distribution is consistent with the experience of the adjoint field.

The overall performance of Rotor 67 during optimization is shown in Fig.24.

In Fig.24,the horizontal axis represents the number of iterations of the adjoint optimization.The vertical axis to the right represents changes in the adiabatic efficiency δη, and that to the left represents the rate of change of mass δm and total pressure ratio δPr.All results are analyzed by TurbuSim_un.The definition of those quantities are as follows:

As shown in Fig.24, adjoint optimization achieves significant performance improvement.The adiabatic efficiency is increased by more than 0.55%, whereas the change in mass flow rate and total pressure ratio is lower than 0.65%.

To accurately simulate the complex flow in IPD by a more popular numerical tool, the final design is going to be further validated in CFX with the SST turbulence model.

5.3.Verification of results

Although the increase in adiabatic efficiency shown in Fig.24 is evident, the results are still uncertain because the numerical accuracy of the flow solver in TurboSim_un is not widely accepted.The final results are verified using CFX to ensure the effectiveness of the optimization.The overall performance of the prototype and redesigned Rotor 67 after verification are illustrated in Fig.25.

The choked mass flow rate is slightly increased by 0.18%,whereas the peak efficiency is increased by 0.68%.The efficiency under near-stall conditions is significantly increased by 0.57%.A mismatch with Fig.24 is occurred owing to different simulation tools used and because the flow solver in TurboSim_un is relatively less accurate than CFX.The corresponding change in the total pressure ratio in Fig.25(b)is similar to the change in efficiency.Owing to the strong constraints in the objective function, minor changes in the overall performance occur under the near-choked and near-peak-efficiency conditions.

Fig.25 verifies the effectiveness of redesigned Rotor 67 in CFX.In this section, the detailed geometric and aerodynamic changes are examined to reveal the control mechanisms of IPD.

Fig.24 Process of redesigning Rotor 67.

Fig.25 Efficiency and total pressure ratio over mass flow under different working conditions.

6.1.Geometric characteristics of IPD

In EFFD parameterization, all structures in the blade passage are indistinguishably placed in the control volume.All control points, except those on the volume boundaries, are indistinguishably adjusted using the automatic optimization algorithm.Therefore, the entire passage can be simultaneously deformed.Fig.26 shows the blade deformation of the redesigned Rotor 67, where the contours represent geometric changes (ΔX) in its 2D profiles.

The major changes in the blade profile are intensively distributed around the hub and the middle part around the 70% span, particular along the leading edge around the hub.Given that the maximum change is<1.0 mm and the changes around the hub are still influenced by the radial deformation of the end wall, the overall changes in blade profile are minor.Fig.27 shows several profiles of the prototype and redesigned blade.

Fig.26 Changes in 2D blade profile in Section S1.

By combining Figs.26 and 27, we see that for the region below the 50% span, the leading edge is deflected toward the pressure side and the deflection becomes more severe with the decrease in the spanwise location.For the profile at the 75% span, the suction and pressure surfaces are deflected toward the pressure side, although this change is relatively minor.In practice, constraints on the structural strength and direction of outflow should be applied in automatic optimization to guarantee the quality of the results.However, no constraint is used in this study, and the distribution of the thickness and region around the trailing edge has been maintained.Fig.28 shows the thickness distribution of several 2D blade profiles and the spanwise distribution of maximum thickness, where the thickness has been simplified as the distance between the corresponding nodes on the suction and pressure sides.The‘‘L*”in Fig.28 refers to the nondimensionalized position of midline.

Although the thickness is slightly decreased in Fig.28(a)before the maximum thickness location over 25% span, the reduction is minor, and the maximum thickness is not decreased at all.We believe that this is adequate for guaranteeing the structural strength of the blade.

Although the region around the clearance of the tip can be redesigned by IPD,it was excluded from the domain of parameterization to avoid the influence of complex flow in this area.Thus, only the contour of the hub was redesigned.Fig.29 shows the radial deformation (ΔR) of the hub.

It can be concluded from the contour layer in Fig.29 that the geometric deformation of hub is larger than that of the blade profiles, and the maximum change is 1.2 mm.Specifically,the forward part of suction surface and the region behind trailing edge are the major areas where the hub is uplifted.The region around the pressure surface and the latter part of the suction side are the main downward areas.After redesigning,the revolving structure of hub is eliminated, as shown in Fig.30, where the vertical axis represents the relevant height and the horizontal axis (θ*) is the circumferential coordinate normalized by the circumferential range of a single passage.

The S3 view shows that the redesigned hub has a clear nonaxisymmetric characteristic, which is reflected in the phenomenon of uplifted hub in the middle of the circumferential passage.The uplifted structure helps restrain the spread of transverse secondary flow.Another geometric change in the corner region shown in Fig.30 is evident, where the original right angle between the blade and hub becomes an acute or obtuse angle.To quantify the changes in the corner region,Fig.31 shows the stream-wise distribution of dihedral angle,which is defined as the angle between the blade surface and hub contour.All the dihedral angles are larger than before,except for the forward part of suction surface.This confirms the control principles of the BBEW technique.1

Fig.27 Geometric changes in several profiles of blade.

Fig.28 Changes in thickness distribution of the blade.

Fig.29 Radial deformation of contours of hub.

The blade profiles,end wall contour,and corner region are adjusted in a unified parameterization mode by IPD in the redesign of Rotor 67.The continuous and smooth deformations of the hub and corner region in the redesigned result are similar to the initial IPD concept in Fig.1.Thus,the combination of EFFD parameterization and adjoint optimization is useful to attain an IPD.

6.2.Aerodynamic improvement

The total pressure (pt) distribution at the outlet is shown in Fig.32 to identify the locations at which performance is improved.As the clearance of the tip has not been redesigned,the total pressure distribution has remained the same in this region.The total pressure of the mainstream is improved,whereas the basic flow topology is maintained.The area around the hub exhibits the largest aerodynamic change.Although the range of the lowest total pressure is slightly increased and its circumferential position is shifted, the average total pressure is increased as shown in Fig.33.

Fig.33 shows the spanwise distribution of the massaveraged total pressure ratio and adiabatic efficiency.It confirms that the adiabatic efficiency is almost improved over the entire span.Moreover,the total pressure ratio is improved except in the range above the 60% span and around the 25%span.

The pressure coefficient (Cp) along the blade surface at the locations where the aerodynamic performance improved, is shown in Fig.34.This implies that the difference in static pressure between the suction and pressure sides is increased,indicating that the diffusion capability is increased after the IPD-based redesign.

Fig.30 S3 view of solid wall around corner at several axial locations.

Fig.31 Distribution of dihedral angle.

Fig.32 Total pressure at outlet.

Fig.33 Spanwise aerodynamic improvements.

The Mach contours at the 5%and 50%spans are shown in Fig.35, which are obtained by extracting the flow field at positions where the performance is improved as shown in Fig.33(b).The main flow in the passage of Rotor 67 includes the wake,corner separation,and shock wave.At the 5%span,the overall velocity of flow is increased and the range of separation is significantly decreased.Thus, the wake-induced flow is considerably improved.At the 50% span, the Mach number before the shock wave is decreased, judging by the range of the Ma = 1.2 contour line.Together with the phenomenon that the static pressure at suction side at 50% span in Fig.34 is decreased, the shock strength is weakened.

Although the aerodynamic performance is improved as shown in Fig.33(b), the improvement is much better below the 25% span.Fig.36 shows the limiting surface streamlines on the suction surface and hub, where the contour on solid wall represents the static pressure (p) and the contour in Section S3 represents the axial velocity (Vz).A large corner separation exists in the rear part of suction surface.After redesigning, the range of separation is significantly decreased,and the original two vertices in the hub become one.The improvement in corner separation is evident in the contours of velocity, where the original region of negative velocity does not exist anymore in the redesign.The circumferential position of the region of lower velocity is shifted, which is consistent with the distribution of total pressure in Fig.32.

Fig.34 Distribution of static pressure along blade surface.

Fig.35 Mach contours at several spanwise locations.

Corner separation is influenced by the intersection of the boundary layer between the blade and hub, secondary flow,and non-uniform inflow.1The first two factors are improved by the IPD.The improvement in the intersection of the boundary layer relies on the change in the dihedral angle in Fig.31.To prove this,Fig.37 shows the total pressure contours at several sections, where each section is normal to the suction surface around the hub, and the contour at hub represents the static pressure.For the front two sections, the total pressure around the corner is higher after redesigning, which is expected.Judging by the range of the 90 kPa contour line,the total pressure is relatively lower in the third and fourth sections of the redesign.However, the fluid with lower energy is restricted to the lower spanwise location,which helps to reduce the size of the separation.

Fig.36 Limiting surface streamlines on suction side.

In addition to the change of dihedral angle in the corner region, a clear geometric change in the end wall occurs in the non-axisymmetric hub, which is generally considered to be effective in controlling the secondary flow.To verify this, the limiting surface streamlines at the hub are illustrated in Fig.38.The contour represents the relative height along the circumferential direction in Eq.(6)and Ravgis the circumferentially averaged radial position.The front five streamlines in Fig.38(a)have almost the same starting locations and the sixth one has a similar ending location to those in Fig.38(b).Owing to the force caused by the uplifted hub around the leading edge, the original first and second streamlines that quickly reach the suction side become relatively gently after the passage was redesigned.In addition, the ending locations of the third and fourth streamlines (E3, E4) is moved downward.The change in the front four streamlines indicates an improvement in the secondary flow.However, the secondary flow is not showed any significant improvement after the fifth streamline,where corner separation becomes the major phenomenon.

Fig.37 Section of dihedral angle and its total pressure contours.

Fig.38 Limiting surface streamlines at hub.

The variation in the limiting surface streamlines is closely related to the geometric changes in hub.The direction of secondary flow has been reorganized along the bulge owing to the uplifted region around the suction side.The nearby flow struggles to escape from the downward region along the pressure side, and the corner separation is further improved.In particular, the sixth streamline, which is finally developed as a part of the separation as shown in Fig.38(a),is not directly related to the separation illustrated in Fig.38(b).The effect of the two major variations around the trailing edge is more prominent,where the uplifted region has separated the flow from the suction and pressure surfaces and the downward region has organized the direction of the seventh streamline, which represents the boundary of corner separation.The range of corner separation is squeezed, and the aerodynamic performance of the end wall region is improved.

Although the redesign of Rotor 67 in this manuscript is a simple attempt without considering the tip region, the results still well verify the reliability of IPD.The blade profiles,corner region, and hub contour are jointly optimized in three dimensions.Owing to the geometric changes,the adiabatic efficiency is improved over the entire span,and the diffusion is increased as well.The adjustments made to blade profiles have reduced the strength of shock wave and wake-induced flow.This is the main reason for the improvement in the mainstream.The changes in the distribution of dihedral angle and hub contour have improved the boundary layer intersection in the end wall region and reasonably reorganized the direction of secondary flow.This has significantly improved the corner separation.

This study proposed the IPD concept and provided a feasible approach for its realization.The IPD considers the blade passage in turbomachinery as the main object of design and multiple design spaces in a unified and harmonious approach.The ability of the IPD to enable flexible geometric deformation of the entire 3D passage to improve aerodynamic performance was verified by redesigning Rotor 67.The main conclusions are as follows:

(1) The IPD is a magnification of the aerodynamic design space and follows the trend of the development of other aerodynamic design techniques.Multiple design spaces can be jointly designed in all three dimensions by using it.IPD is thus an important design concept to improve the aerodynamic performance of turbomachinery.

(2) The parameterization method based on EFFD can be used to make geometric adjustments to the entire passage in three dimensions while guaranteeing the constraints on continuity and periodicity.Adjoint optimization can be used to reduce the high calculation cost of EFFD parameterization method.

(3) The profiles of the blade, corner region, and contour of the hub of Rotor 67 are appropriately adjusted using the IPD to significantly improve performance.The adiabatic efficiency is increased by 0.68%,and this improvement is reflected over the entire span.Moreover, the diffusion capability considerably improved,and the whole performance was better at different rotational speeds.

(4) The strengths of the shock wave and wake-induced flow is decreased owing to changes in the curve and curvature of the blade profiles.The boundary layer intersection is improved because the dihedral angle is increased.The direction of the nearby secondary flow is adequately reorganized by changing the hub around the leading edge.The restricted space squeezes the separation range of the hub around the trailing edge.These changes significantly weaken the corner separation and improved the performance in end wall region.

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 work reported was supported by the National Science and Technology Major Project of China (Nos.2017-II-0006-0020 and J2019-II-0003-0023).

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