Aerodynamic shape optimization of a transonic fan by an adjoint-response surface method

Accepted Manuscript Aerodynamic shape optimization of a transonic fan by an adjoint-response surface method

Xiao Tang, Jiaqi Luo, Feng Liu

PII: DOI: Reference:

S1270-9638(16)31391-8 http://dx.doi.org/10.1016/j.ast.2017.05.005 AESCTE 4018

To appear in:

Aerospace Science and Technology

Received date: Revised date: Accepted date:

29 December 2016 2 April 2017 3 May 2017

Please cite this article in press as: X. Tang et al., Aerodynamic shape optimization of a transonic fan by an adjoint-response surface method, Aerosp. Sci. Technol. (2017), http://dx.doi.org/10.1016/j.ast.2017.05.005

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Aerodynamic Shape Optimization of a Transonic Fan by an Adjoint-Response Surface Method Xiao Tanga,1,∗, Jiaqi Luoa,2 , Feng Liub,3 a Dept.

of Astronautics & Aerodynamics, Peking University, Beijing, China b Dept. of Mechanical & Aerospace Engineering, University of California, Irvine, United States

Abstract An adjoint-response surface method is developed to provide efficient surrogate model in a parametrized design space for aerodynamic optimization of turbomachinery blades. Our goal is to improve the adiabatic efficiency or equivalently reduce the entropy generation through blade row with a mass flow rate constraint. Firstly, an aerodynamic sensitivity analysis is conducted with a viscous adjoint method to find the suitable number of control points on the suction surface of the transonic NASA rotor 67. Then quadratic polynomial (QP) based response surfaces of 4, 6 and 8 parameters are examined to validate the advantages of the gradient-enhanced model. In the following 24-parameter aerodynamic design optimization case, a steepest descent optimization (SDO) based on adjoint gradient is conducted, then QP based response surface model is constructed using both the values of cost function and its adjoint gradients with respect to geometry control parameters. We present the geometric features, overall aerodynamic improvements and flow details of optimal designs given by SDO and gradient-enhanced response surface model (GERSM). The effects of blade reshaping on shock system, tip clearance flow and flow separation at hub are examined. Also, off-design performances are analysed regarding both adiabatic efficiency and stall margin. Keywords: Aerodynamic Design Optimization, Adjoint Method, Response Surface, Compressor Rotor, Transonic

1. Introduction The development of CFD techniques in aerospace engineering has significantly changed the design optimization methodology of transonic fan/compressor rotors, especially after efficient automatic design optimization tools came into being. With limited computational resources, early designers can only manage to roughly explore some design concepts. Later on, parametric study for key design features can be conducted, like blade profile maximum thickness and its location[1], blade sweep and dihedral[2, 3, 4] and cascade area ratios[5]. These design features have great influence on passage shock strength, boundary layer development, tip clearance flow, etc. and related aerodynamic losses. When CFD code and computer power reached a new level, multi-parameter comprehensive design optimization was realized, based on observation and aerodynamic analysis[6]. This is followed by methods based on gradient information obtained by finite differences[7] or evolutionary algorithms such as the genetic algorithm[8]. In the last decade, automatic design optimization for internal flow based on the adjoint method was introduced. ∗ Corresponding

author D. Candidate 2 Research Associate 3 Professor, AIAA Fellow 1 Ph.

Preprint submitted to Elsevier

Yang, Wu and Liu[9] and Wu and Liu[10] developed an adjoint method for 2-D cascades and 3-D turbine stator design optimizations. Recently, Luo et al.[11, 12, 13] further developed the adjoint method for 3-D internal flow in rotating turbomachinery components including multi-point design optimization and endwall contouring. By the adjoint method, gradient of the cost function can be obtained with only one flow calculation and one adjoint calculation, regardless of the number of design parameters. Instead of focusing on key design features, adjoint tools usually optimize the blade or passage with general shape functions, or in other words, focus on aerodynamic sensitive regions. The authors feel that firstly, the potential of the adjoint gradient method is not fully used if only incorporated into steepest descent optimization (SDO) steps. On the one hand, gradient-based search methods like SDO easily get trapped in nearby local optimum. On the other hand, by such methods, the distributive relations between the optimization target and design parameters are often ignored. Secondly, through an advanced gradient calculator like the adjoint CFD method, design sensitive regions can be efficiently determined. Designers should be able to connect them with conventional key design features which would help understand the design results. The response surface method was firstly applied by Box and Wilson[14] in chemistry research. Ahn et al.[15] used a response surface method for airfoil design in tranMay 4, 2017

sonic flow. Madsen et al.[16] studied a diffuser shape optimization problem with response surface techniques. Recently, based on a polynomial response surface approximation technique, Pehlivanoglu et al.[17] conducted different cases of aerodynamic optimization in genetic algorithm architecture. For some response surface models (RSM), the basis functions are continuous and the gradient with respect to control parameters can be determined analytically. In such cases, both the values and gradients of cost function can be regarded as system responses. With the same total number of pieces of information for constructing RSM, the number of support points can be significantly reduced. The adjoint method is used herein due to its excellent performance on gradient calculation. In the present study, an aerodynamic sensitivity analysis is firstly conducted to determine the suitable number of control points on the suction surface of the transonic NASA Rotor 67. Then a series of RSMs/GERSMs are examined to demonstrate the advantages of a gradientenhanced model. In the design optimization part, a 24parameter case is studied with both SDO and GERSM. The design space is parametrized with the Class/Shapefunction Transform (CST) method[18]. Quadratic polynomial (QP) based models are constructed using both cost function values and adjoint gradients. For multi-parameter applications, Latin hypercube sampling (LHS)[19] is adopted, the QP based GERSM gives satisfactory accuracy and cuts down the computational cost in model construction compared with traditional RSM. Optimized blade geometries are compared, with emphasis on profile maximum thickness and its location. Overall aerodynamic improvements and flow details are presented. How blade reshaping influences the shock system, tip leakage flow and flow separation at hub are investigated. Off-design performances at design speed are also discussed, considering both adiabatic efficiency and stall margin. 2. Methods and Algorithms

Introduce the adjoint variable Ψ as Lagrangian multiplier and rearrange  T T       ∂I ∂I ∂R T ∂R − ΨT δΓ δw+ −Ψ δI = ∂w ∂w ∂Γ ∂Γ (3) The following step is to cancel the variation of flow variables in Eqn. (3) by solving adjoint equations     ∂I ∂R − Ψ=0 (4) ∂w ∂w such that the variation of I only depends on the variation of Γ.     ∂I T ∂R −Ψ δΓ (5) δI = ∂Γ ∂Γ Thus the gradient with respect to the geometric parameters is     ∂I ∂R δI = − ΨT (6) G= δΓ ∂Γ ∂Γ 2.2. Gradient-Enhanced Response Surface Method Detailed examination of RSM/GERSM used in the authors’ works can be found in [21], including model construction, cost, accuracy and robustness. Here in this paper, a brief description on RSM/GERSM is given since the emphasis is aerodynamic design. Considering order of complexity and time cost, we use quadratic polynomial (QP) based response surface model in the current research work. Mathematical formula of QP based RSM is f (x) =

nd  i=1

αi x2i +

n d −1

nd 

i=1 j=i+1

βi,j xi xj +

nd 

γi xi + θ (7)

i=1

where x is the vector of nd control variables. There are totally

nd (nd − 1) (nd + 1) (nd + 2) nd + + nd + 1 = (8) 2 2

terms in Eqn. (7), each with an unknown coefficient. In constructing the RSM, we mean to solve for the coefficient 2.1. Adjoint Method for Internal Flow set {αi , βi,j , γi , θ} with at least 12 (nd + 1) (nd + 2) equaThe adjoint method was firstly used in airfoil design tions from a set of x and f (x) pairs called samples, using optimization by Jameson[20]. In recent decades, adjoint least squares. method was successfully applied to turbomachinery components[9, In traditional response surface method, each sample 10, 11, 12, 13]. gives just one response, i.e., one equation for {αi , βi,j , γi , θ}.  In design optimization, the cost function I depends on So the computation cost is of O nd 2 numbers of flow soboth the flow field variables w and the geometric boundary lutions. We introduce gradients obtained by an adjoint Γ, i.e., I = I (w, Γ). Variation of the cost function is method into the construction of response surface, i.e., the response surface now fits both the cost function value and  T   ∂I ∂I the gradient value at the sampling points such that evδI = δΓ (1) δw + ∂w ∂Γ ery flow and adjont calculation provides (nd + 1) responses (Eqn. (7) and (9)). In steady state, the flow variables and geometry are con ∂f nected through R (w, Γ) = 0, which by variation gives = 2xl αl + xi βi,l + γl , (l = 1, 2, ..., nd ) (9) ∂xl T    i=l ∂R ∂R δΓ = 0 (2) δw + δR = Thus the computational cost, i.e., the number of sampling ∂w ∂Γ points is reduced to O (nd ). 2

3. Descriptions of Design Optimization

has been proved to be robust and versatile in many applications. In the present work, the class/shape functions are used as the perturbation bumps added to the blade surface in sensitivity calculation and geometry reshaping. The blade surface perturbation bump is a function y = y (x, z) of axial coordinate x and radial coordinate z, or its counterpart ξ = ξ (ψ, η) in computational domain. The class/shape functions can be expressed as

NASA Rotor 67 is an axial-flow transonic fan rotor with a design tip relative Mach number of 1.38. The rotor has 22 blades and an aspect ratio of 1.56. Its design pressure ratio is 1.63 at a mass flow of 33.25 kg/sec with a design rotational speed of 16043 rpm and the running tip clearance is about 1.0 mm. Z X

Y

F (ψ, η) =

Nx  Nz  i=0 j=0

N1 fi,j (ψ, η) = CN (ψ) 2

Nx  Nz 

[Bi,j Szj (η) Sxi (ψ)]

i=0 j=0

(10) where F (ψ, η) is the summary of all the bumps fi,j (ψ, η) and N1 (ψ) = ψ 1/2 (1 − ψ) CN 2 Nz ! N −j η j (1 − η) z Szj (η) = j! (Nz − j)! Nx ! N −i ψ i (1 − ψ) x Sxi (ψ) = i! (Nx − i)! 1

Figure 1: Computation grid of rotor 67

Notice that each perturbation bump consists of a pubN1 lic class function CN (ψ) times the unique shape func2 tion Szj (η) Sxi (ψ), with coefficient Bi,j , and F (0, η) = F (1, η) = 0, which means that the coordinates of the LE/TE will not change.

The H-type grid is shown in Fig. 1. Sharp LE/TE and one-point approximation for blade tip are adopted. Compared with experimental data[22], also verified by Storer and Cumpsty[23], the one-point approximation for rotor tip has no significant influence on flow field and overall performance. A 3-D Navier-Stokes solver by Liu and Jameson[24, 25] is used with classical JST scheme[26] and S-A turbulence model[27]. The choke mass flow rate reached by the solver is 34.99kg/s while the measured is 34.96kg/s. Figure 2 shows the computed working characteristics together with experimental data.

3.2. Cost Function and Adjoint Boundary Conditions In internal flow aerodynamic optimizations, the major concern is to increase efficiency or equivalently reduce losses under a certain working condition. Here we take the entropy generation, which measures aerodynamic losses, as the optimization target. Cost function I is chosen to be a combination of entropy generation sgen and half the square of mass flow rate deviation.

0.94 0.92

1 2 ˙ −m ˙ 0) I = sgen + ξ (m 2

0.90

0.84 1.70

0.82

1.65 1.60 1.55 1.50 1.45 1.40 1.35

π CFD π Exp η CFD η Exp 0.92

0.94

0.96 Relative Massflow

0.98

(12)

This penalty function approach transforms a constraint problem into a non-constraint one such that no matter how many constraints there are, only one adjoint calculation is required. ξ = 104 in this paper. In an adjoint method, different cost functions lead to different (δI)Γ and (δI)w , which lead to different boundary conditions[11, 13]. With the mass flow constraint in Eqn. (12), the outlet boundary condition is

Adiabatic Efficiency

Total Pressure Ratio

0.88 0.86

(11)

T

T c = c1 , c2 , c3 , c4 , c5 = ξ (m ˙ −m ˙ 0 ) 0, ni Si1 , ni Si2 , ni Si3 , 0 (13)

1.00

3.3. Adjoint Sensitivity Analysis In the current study, the authors redesign the suction surface (SS) of Rotor 67 and leave the pressure surface (PS) unchanged. This is because the SS is much more sensitive than the PS for this transonic blade. Also, there is a balance between the number of total control points

Figure 2: Working characteristics of Rotor 67.

3.1. CST Parameterization for Geometry Perturbation The CST parametrization method[18] offers analytically smooth expressions for aerodynamic geometries. It 3

0.02

0.01

0.03 0.01 Gradient

dynamic sensitivity distribution can still be recognized. In the 17×9 CST bump configuration, the axial index is from 0 to 17 while the spanwise index from 0 to 9. For convenience, we denote the negative partial derivative region on the SS near the LE from one quarter to three quarters span as R0-1,2-7 . Subscript 0-1 corresponds to CST bump index 0 to 1 in the axial direction and 2-7 means index 2 to 7 in the span direction. The same terminology is adopted for regions such as R3-9,4-7 , corresponding to the hump near the middle of the SS, and R3-12,9 , which demonstrates the sensitive tip region.

0

−0.01 −0.01 −0.03 −0.02 −0.05 30 20 CS T( rad 10 ial)

0

0

10

20

50

40

30

60

−0.03

l)

axia

( CST

4. RSM vs GERSM

(a) 65 × 33 CST bumps

To demonstrate the advantage of the gradient-enhanced response surface over traditional response surface, we compare a series of RSMs and GERSMs. Three cases R3-4,6-7 , R3-4,5-7 and R3-4,5-8 are adopted, with 4, 6 and 8 design parameters respectively. For the nd -parameter case, the QP based response surface contains 12 (nd + 1) (nd + 2) unknowns, see Eqn. (8). The required number of sampling points in RSM and GERSM are listed in Tab. 1. Here Dim (equals to nd ) is the dimension of design space. In GERSM, one sample point can provide (nd + 1) responses (1 cost function value and nd gradient components), so only 12 (nd + 2) sample points are needed.

Gradient

0.03 0.04

0.02

0.02

0.01

0.00

0

−0.02

−0.01

−0.04

−0.02

−0.06

−0.03

−0.08

−0.04

8 6 CS T( rad 4 ial)

2 0

0

2

4

12 14 10 8 6 l) (axia CST

16

−0.05 −0.06

(b) 17 × 9 CST bumps

Table 1: Number of RSM/GERSM Sample Points

Figure 3: Adjoint sensitivities on suction surface of rotor 67.

Case R3-4,6-7 R3-4,5-7 R3-4,5-8

and model accuracy in the construction of the response surface. Figure 3 shows the adjoint sensitivities on the SS. The presented partial derivatives are scaled by the maximal values of the perturbation bumps. In the first sub-figure, the number of CST bumps equals to the number of grid nodes on the SS, providing high resolution sensitivities. Several interesting properties are observed. Firstly, from about one quarter to three quarters span, the gradient components near the LE are negatively large, indicating that here greater curvature of SS is preferred. Secondly, sensitivities in regions before the passage shock are positively large. Thus to reduce sgen , the curvature of this part should be smaller. Explanation is that for the transonic Rotor 67, local flow here is supersonic, reduced curvature enlarges the flow passage such that the compression waves become weaker, resulting in weaker passage shock. Thirdly, the gradient components at the blade tip around mid chord are relatively large which indicates promising aerodynamic improvement if the tip blade profile is properly redesigned. In Fig. 3(b), a 17 × 9 CST bump layout is adopted (and used in the following model verification and design optimization) such that the number of control parameters becomes much smaller, yet the features of aero-

Dim 4 6 8

ns (RSM) 15 28 45

ns (GERSM) 3 4 5

Both RSM and GERSM are constructed for all three cases. Because of the mathematical property of quadratic polynomials, the local details of the actual response surface are smoothed to some extent while the global distribution is modelled. In order to avoid the rank deficient problem of the coefficient matrix when applying least squares, we need the number of samples to be larger than listed in Tab. 1. For every case, 60 to 100 samples by Latin Hypercube Sampling (LHS) are computed, among which ns samples are used for model construction and the rest for model performance calculation. Comparisons between RSMs and GERSMs are given in Fig. 4 in terms of the coefficient of determination R2 = 1.0−RSS/TSS. Here RSS is the residual sum of squares, TSS is the total sum of squares. In QP based models, the RSM and GERSM both give accurate predictions for function values in 4, 6, 8-parameter cases. However, RSMs are obviously not capable to provide the correct gradient responses while GERSMs succeed in gradient prediction. 4

1.50

multi-parameter optimization. It also acts as a comparison in demonstrating how GERSM is different from a step by step gradient-based optimization. In sensitivity analysis, in order to know the relative magnitude of the gradient components, we scale them with local bump sizes. But during optimization, different extents of reshaping is permitted since different regions represent different physical phenomena. For example, the region at 50% chord length permits much larger reshaping than in the LE area. A better choice is the unscaled gradients because scaling enlarges relative gradient magnitude near the LE and at blade tip. With 17×9 CST bumps, the magnitude of gradient component is ∼ 10−3 . Let Cmid be the chord length at the middle span. The CST bump size (maximum height) is about 10% of Cmid with Bi,j =1.0. The grid cell width is 5 × 10−4 Cmid at wall. In generation of a new mesh for each optimization step, Δl represents the dimensionless optimization step length, the CST bump added to the SS has a size of Δl × gradient × unit bump size, which equals to 10−4 Δl Cmid . So one CST bump changes the geometry by (10−4 Δl Cmid )/(5 × 10−4 Cmid ) = Δl/5 of wall cell width in the maximum places. The overlaying of CST bumps gives a factor of 2.0 (at most) in geometry change, thus every optimization approximately changes the SS by Δl/2.5 of wall cell width at the maximum places. The optimization step length Δl is set to be 2.5 such that the geometric change in every step is bounded by 1 wall cell width. The aerodynamic improvement after 60 steps of SDO is shown in Tab. 2. Figure 5 gives the total perturbations on all the 24 control parameters, which are obtained through integration on the gradients and the optimization step length. In this optimal design, the 3rd , 5th , 9th , 11th , 12th and 13th control parameters have almost zero values. However, they are all considered very sensitive at the start, see Fig. 3(b). This happens because the design space is highly non-linear and the optimization path is probably extremely twisted. This design is just a local minimum in a 24 dimensional design space.

R2 of Function Value 2

R of Gradient 1.00

R

2

4−p

6−p

8−p

0.50

0.00

−0.50 15

20

25

30

35 ns

40

45

50

55

(a) R2 of RSMs 1.10

R2 of Function Value 4−p

2

R of Gradient

R2

1.00

0.90 6−p

8−p

0.80

0.70 4

6

8

10 ns

12

14

16

(b) R2 of GERSMs Figure 4: R2 of 4,6,8-Parameter Models with Increasing ns

5. Application of Design Optimization In this section, we study a 24-parameter optimization case using both SDO and GERSM. Notice that traditional RSM would require at least 325 sample points (see Eqn. 8) to build a model, which is impractical for industrial applications. The optimization is conducted at a point near peak efficiency, with back pressure equals to 1.04 of the ambient static pressure. According to the previous sensitivity analysis, region R1,2-5 ∪ R4-8,4-7 , which contains the most sensitive and robust 24 control parameters, is chosen for optimization. R0,3-6 is not included because it is very close to the thin LE such that too much reshaping would easily cause structure failure. It is the same for R3-12,9 at blade tip.

5.2. Optimization by GERSM 5.2.1. Model Construction and Performances Since only 6 of the CST control parameters of the SDO optimized solution exceeds ±0.01, as shown in Fig. 5, the design space in which we build the response surfaces is 24 defined to be D [-0.01, 0.01] . Here 24 is the dimension of design space as well as the number of control parameters. For the nd -parameter case, the QP based response surface model contains 12 (nd + 1) (nd + 2) unknowns, see Eqn. (8). With the incorporation of the adjoint gradient, only 12 (nd + 2) sample points are needed. In this research work, we use Latin hypercube sampling (LHS) method to distribute sample points in the design space D. Totally 200 samples are computed, among which ns points are used to construct the GERSM model (solve system of Eqn. (7) and (9) with

5.1. Optimization by SDO A steepest descent optimization (SDO) is performed with the adjoint method. The optimum design by SDO works as a baseline in proving the ability of GERSM in 5

5.3. Design Optimization Improvements at Optimization Point 5.3.1. General Performances Table 2 presents the overall performances of three blades at the optimization point: the original Rotor 67 as reference, the SDO optimum and the GERSM optimum. Cost function I and entropy generation sgen are non-dimensional values. The SDO optimum reduces the I by 4.14% while the GERSM gives 4.83% reduction. Both the entropy generation and mass flow rate deviation of GERSM are superior than SDO. The adiabatic efficiency of GERSM is slightly higher than SDO.

least square) and the rest for model performance calculation. In this 24-parameter optimization case, the SDO takes 60 steps, i.e., 60 times of flow solution and adjoint solution. The GERSM takes 30-60 samples (flow and adjoint solutions) to construction the model and 10-30 successive calibration steps. So here for this 24-parameter case, SDO and GERSM are comparable in computational cost. 5.2.2. Design Optimization Besides its ability as a quick surrogate model for timeconsuming CFD calculation, the GERSM provides the distribution trend of cost function in design space and gives a model-indicated optimal point, which either equals to the extreme point of the quadratic polynomials or lies on the border of the design space. This is valuable in leading designer to find the optimum in a global (or non-local) sense. The authors adopt a genetic algorithm to search for the global optimum in the design space. The number of individuals is 104 and 100 generations of evolution is used. Every individual has 24 genes corresponding to the 24 control parameters. Each gene is transformed into 10 bits of binary codes. The mutate and cross probability are both 0.01. To calibrate and further improve the design, 30 steps of successive steepest descent is conducted based on the optimum found by the genetic algorithm. 0.02

Table 2: Overall Aerodynamic Performances

Blade Reference SDO GERSM

SDO GERSM

Total Perturbations

0.01

0.00

−0.01

−0.02 0

5

10 15 Control Parameters

20

25

Figure 5: Optimum Designs by SDO and GERSM

Diamonds in Fig. 5 show the optimum found by GERSM, in terms of the total changes of control parameters. The designs of SDO and GERSM demonstrate some similarity in the distribution trend of control parameters, but are quite different in their values. Especially the 3rd , 4th , 5th , 8th , 11th , 12th and 13th control parameters, which lie on the 5% to 40% chord-wise and 40% to 60% spanwise region.

6

 I 10-4 1.862 1.785 1.772

 sgen 10-4 1.862 1.688 1.681

m ˙ (kg/s) 34.46 34.80 34.79

η (%) 91.71 92.58 92.61

π 1.641 1.650 1.650

The spanwise distributions of the profile maximum thickness and its chordwise location are shown in Fig. 6. The maximum thickness value is non-dimensionalized by the length of local camber-line. The hub maximum thickness of the reference blade is about 0.08 and decreases to around 0.03 at 95% span. This curve drops to zero value at r = 1.0 because the blade tip is modelled by onepoint approximation[22, 23]. Both the SDO and GERSM designs reduce the maximum thickness above 40% span. The reduction in maximum thickness is about 0.005 from 60% to 95% span. Below 30% span, the GERSM design increases maximum thickness while SDO design gives almost the same thickness as the reference blade. Fig. 6(b) demonstrates the radial distribution of the location of blade profile maximum thickness for the three blades. Both designs move the maximum thickness location towards the TE in the outer span and towards the LE in the inner span. The GERSM design presents larger geometric modifications in above 80% span and below 50% span. As indicated by experimental and numerical results of Wadia[1], in transonic fan/compressor rotor, reduced profile maximum thickness and aft maximum thickness location at tip region reduce the LE wedge angle such that lower shock loss can be achieved. The thinner blade tip of SDO/GERSM obviously increases the mouth area[5] of the flow passage. But the throat area almost does not change because of the aft maximum thickness location. The reduced shock strength causes lesser blockage in tip region and is responsible for the approximately one percent increase in mass flow rate near peak efficiency. Throat area below 40% span is effectively reduced by the forward-moved maximum thickness location (as in Fig. 6(b)), especially for the GERSM design because its maximum thickness below 40% span is increased, compared to the

1.0

1.6

Isentropic Mach

At 95.1% Span

0.8

r

0.6

1.4

1.2

1.0

0.4

0.8 0.0

0.0 0.00

0.02

0.2

0.4

0.6

0.8

1.0

x/c

Reference SDO GERSM

(a) 95.1% Span 1.4

0.04 0.06 0.08 Fraction of Local Camber−Line

0.10

At 75.0% Span

Isentropic Mach

0.2

(a) Maximum Thickness 1.0

0.8

0.6

1.2

1.0

0.8

0.6 0.0

Reference SDO GERSM

0.2

0.4

0.6

0.8

1.0

r

x/c (b) 75.0% Span

0.4

Figure 7: Isentropic Mach Number on Blade Surface 0.2

0.0 40

Reference SDO GERSM 50

60 Fraction of Local Chord

70

relative Mach number in front of the passage shock could be reduced, resulting in less aerodynamic loss. Similar phenomena can be observed at the 75% span. Generally in the blade tip region, the SDO and GERSM designs shift part of the aerodynamic loading from the rear part of the blade profile to the front part, relieving the passage shock loss. The loading shift and pre-compression also influence the tip clearance flow. The tip leakage flow is considered harmful for the mixing and blockage when the tip leakage vortex (TLV) interacts with the shock. Moreover, when the flow is throttled, oscillating TLV and its unsteady interaction with the passage shock cause additional instability to the machine, bad for stall margin[29, 30]. Pandya and Lakshminarayana[31] and Dawes[32] pointed out that the pressure difference between the SS and PS plays an important role in driving the fluid through the tip gap. The leakage flow then roll up to form the TLV. To demonstrate the behaviour of tip clearance flow as well as its causes, we plot the axial velocity and static pressure contours in a blade to blade plane between tip and casing in Fig. 8. The three contour plots in Fig. 8(a) show that the TLV initiates at the LE. The tip clearance flow is obviously stronger in the SDO and GERSM designs from the LE to one quarter chord length. This is because in the tip region, the SDO and GERSM designs shift part of the loading around the passage shock forward to the front part

80

(b) Maximum Thickness Location Figure 6: Spanwise Distribution of Blade Profile Maximum Thickness and Location

Reference and SDO designs. Since the flow is subsonic here, the reduced throat area mainly controls flow diffusion. 5.3.2. Flow Details In transonic fan/compressor rotors, the tip region shock system is necessary to achieve the required total pressure ratio. However, the strength of passage shock need to be well controlled such that the losses are not high. Figure 7 shows the isentropic Mach number on blade surfaces at about 95% and 75% span. Fig. 7(a) shows that the SDO and GERSM designs significantly reduce the shock strength on the rear part of the SS. The shock strength on the front part of the PS is also reduced, indicating a weakening of the shock system. The isentropic Mach number close to the LE on the SS becomes higher after the redesign. This is caused by decreased local profile thickness as well as aft moved maximum thickness location, called ‘de-camber’ effect, which works as precompression (or external compression)[5, 28] such that the 7

SS pressure remains approximately the same) results in a large difference in static pressure across the blade at this location. Such a sudden increase in static pressure after passage shock can be clearly observed in Fig. 8(b). As marked by the blue dashed circle, the reference blade suffers from a much larger pressure rise on the PS after passage shock compared with the SDO/GERSM designs, giving rise to a stronger second TLV. Suder et al.[36, 37] showed that the TLV-shock interaction accounts for a notable part in losses. Since the shock wave presents a strong inverse pressure gradient across the flow passage, severe diffusion happens when the TLV runs into a shock. The generated low momentum fluid mixes with the main flow, further increases the blockage and causes additional loss. In order to know how much the TLV-shock interaction diminishes, we examine the relative Mach number at 40%, 80% and 110% axial chord length positions. Fig. 9

-0.50 -0.40 -0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40 0.50

Reference

SDO

GERSM

(a) Axial Velocity

0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35

SS

Reference

SDO (b) Static Pressure

GERSM

5

0.90

1.3

15 1.

5

0.90

15 1.

PS 5

PS 0.95

Reference

05 1.

PS

0.9

5

0.60

05 1.

0.9

5

0.65

0.70

0.70

0.9

5

SDO

5

70 0.

0.5

0.7

05 1. 0 .95

5

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Figure 9: Relative Mach Number at Three Axial Positions Figure 8: Contours Between Tip and Casing

compares the relative Mach number contours in the outer span of the Reference, SDO and GERSM blades. At 40% axial chord position, the relative Mach number on the SS of the reference blade is higher than that of the SDO and GERSM optimums, indicating stronger shocks. However, the lower relative Mach number area near the casing close to the PS says that the TLV there is strengthened in the SDO and GERSM cases. Then the TLV interacts with the shock system and suffers from significant diffusion which greatly reduces the fluid momentum. The second row in Fig. 9 shows the contours of relative Mach number after the TLV-shock interaction. Because of the reduced shock strength and suppressed tip clearance flow in the SDO and GERSM optimums, the relative Mach number behind the shock is higher than that of the reference design. The last row in this figure further confirms that the two optimum designs cut down the TLV-shock interaction loss as well as losses in the wake. In Fig. 10, the spanwise distributions of total pressure ratio, total temperature ratio, adiabatic efficiency and flow turning are examined to help manifest the radial loading

of blade profile, increasing the pressure difference there between the PS and the SS. Chima[33] proved that though the TLV mainly initiates near the tip LE, it is continuously strengthened by downstream tip clearance flow. The final strength of the TLV depends on the ‘chordwise integration’ of the pressure difference between the PS and the SS. In our case, the loading shift increases the pressure difference near the LE but reduces it in the rear part of blade profile (see Fig. 7). This cuts down the tip clearance flow after the LE area, as proved in Fig. 8(a). The reduced shock strength favours the weakening of the TLV. In the first contour plot of Fig. 8(a), the positions of the LE bow shock and passage shock are marked by blue solid lines. The blue dashed line marks where the passage shock ends. As investigated by Biollo and Benini[34] and Hah[35], a second TLV initiates where the passage shock meets the SS. This is because after the passage shock, the sudden increase in static pressure on the PS (while the 8

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mization. This total pressure deficit happens because the flow separation pattern at the hub corner near the SS trailing edge is changed. Though 3D separations are complex in turbomachinery flow, Gbadebo et al.[38] proved that they can be well predicted with RANS codes, particularly the patterns of the limiting streamlines.

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(b) Adiabatic Efficiency and Flow Turning Figure 10: Spanwise Distributions at Outlet (b) Streamlines on The 7th Grid Layer

redistribution in the design optimizations. The increase of the total pressure ratio near hub, as also observed by Hah[2] and Biollo[34] in transonic rotors, is caused by the secondary flows. The SDO and GERSM designs both increase the total pressure ratio from 10% to 95% span. The GERSM design gives higher π from 10% to 50% span but lower π from 70% to 95% span compared with the SDO. This is because GERSM shifts more loading from the outer span to inner span. Total temperature plot in Fig. 10(a) confirms the loading shift from above 80% span to below it. The spanwise distribution of η shows that the major improvement of adiabatic efficiency happens in the outer span. The second plot in Fig. 10(b) shows that the flow turning before and after design optimization are almost the same except in the tip region. The flow turning decreases above 80% span due to the reduced flow blockage. In the π plot of Fig. 10(a), the total pressure ratio near hub (below 10% span) becomes smaller after design opti-

Figure 11: Flow Separation in Hub Corner of SS

Fig. 11(a) presents the limiting streamlines on the SS in accompany with static pressure contours. As also examined by Arnone[39], the NASA Rotor 67 blade presents a relatively small flow separation region near hub on the SS close to the TE. The first sub-plot shows that the original blade has a separation line starting from the hub and ending at 35% blade height. In the SDO and GERSM optimums, the separation area is reduced. Meanwhile, the topology of the separation line has been changed. After optimization, the flow separation on the SS falls into two parts. One is still on the blade profile but suppressed. The other one turns out to be more related to a hub corner boundary layer separation, as shown in Fig. 11(b), where streamlines on the 7th grid layer (about 0.3 mm from wall) are drawn. The flow separation pattern change is attributed to the forward moved blade profile maximum 9

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0.76 0.78 0.80 Relative Massflow

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Figure 13: Working Characteristics at 70% Speed

shows the right part of the working characteristics of the three blades at 70% speed. The same scales are used for the x-axis in Fig. 12 and Fig. 13. The choke mass flow rate rises by 1.2 percent in 100% speed while only 0.4 percent in 70% speed case. This is because entropy-generationaimed optimization mainly focus on the reduction of shock strength, which is already much weaker, with much less flow blockage at 70% speed than at the 100% speed. The analysis above indicates that when redesigning a certain blade row (or one stage) of a compression system, forcing the mass flow rate to be the same with the reference design while reducing the shock-caused entropy generation may not be a good idea. Because such strict mass flow rate constraint probably changes the area ratios in or outside of the tip region so as to compensate the mass flow rate increment caused by shock strength reduction at design speed. However, this inevitably leads to mass flow rate deficit in low speed where the shock is weak. This deficit deteriorates the mismatching between blade rows at part speed and is very harmful to part speed performance. It must be pointed out also that a blade optimized at the design condition may change its shape at part-speed due to reduced centrifugal force. This effect is not considered in the present study but could complete the aeromechanical design.

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5.4. Off-Design Performances For transonic fan/compressor design optimization, good performance at key off-design operating conditions is very important. For the compression system of an aero-engine, maximum flow occurs once any blade row becomes choked[28]. Surge occurs when some stage reaches its maximum value of useful work or reaches its stalling point. The back pressure at the optimization point (near peak efficiency at design speed) is 1.04 the ambient static pressure. The working characteristics of the three blades are computed by both increasing and decreasing the back pressure with an interval of 0.01, as shown in Fig. 12. The choke point is determined when the mass flow rate increment is less than 0.1% of the last calculated point. All mass flow rate values are normalized by the choke mass flow rate of the original blade row. When approaching the stall condition, the performances of all three blades are calculated until the mass flow rate drops to around 92%. As suggested by Strazisar et al.[22], the Rotor 67 blade row stalls at about 92.1% mass flow rate at the design speed.

moved choke point favours the total pressure ratio of the compression system as well as the net thrust of the whole engine, especially when this component turns out to be the bottle-neck. At low speed, the shock strength becomes weaker. Fig. 13

Total Pressure Ratio

thickness near hub (see Fig. 6), which increases the flow diffusion after it.

1.02

Figure 12: Working Characteristics at Design Speed

5.4.1. From Near Peak Efficiency to Choke As shown in Fig. 12, at the optimization point, the mass flow rate is increased by about 1.2%. Because the blade passage throat area and discharge area[5] are not reduced, the increase in mass flow rate is evidently a byproduct of shock strength reduction since by reducing the shock strength, the flow blockage due to shock-boundary layer and shock-TLV interaction is relieved. After design optimization, the aerodynamic improvement from 0.98 relative mass flow to choke point is large. For example, at 1.00 relative mass flow rate, the original blade gives 1.55 total pressure ratio and 89.5% adiabatic efficiency. However, at this mass flow rate, the optimized rotors achieve 1.65 π and 92.5% η. Meanwhile, the right

5.4.2. Near Stall Though rotating stall inception is essentially unsteady, the related time-averaged physical phenomena or convergence criteria in steady-state numerical simulation can be used to compare near-stall performance of different designs or approximately determine the ‘numerical stall point’ in steady CFD-based design optimizations[40, 41, 42]. 10

In the current study, the major concern is that after design optimization, the near stall performance at design speed should not be worse than the original design. Otherwise, the stall margin would probably be harmed. In Fig. 12, the near stall operating points of three blades are marked by the blue Ellipses A and B. The measured stall point of Rotor 67 locates at about 92.1% mass flow rate, corresponding to the triangle in A. The other two points in A belong to the SDO and GERSM designs. The three points in ellipse B are calculated with the back pressure reduced by one interval (0.01 ambient pressure) compared with A. Since rotating stall inception in axial compressor is featured with tip region back-flow and some criteria have been proposed by researchers based on time-averaged behaviour of tip clearance flow[29, 30], we investigate the tip region axial velocity fields in regions A and B of Fig. 12, aiming at comparing the performances of the three blades near stall condition and if possible, finding out whether the stall margins are harmed or not in the SDO and GERSM designs. Figure 14 shows the blade to blade axial velocity field near the stall condition just below the blade tip. Plot 14(a) compares the contours of the three points in Ellipse A. In the SDO and GERSM designs, the effect of back-flow in the middle pitch becomes stronger than the reference design, forcing the TLV moving closer to the LE of the next blade. As marked out by the blue dashed circles, there seems to be signal of forward spillage of tip clearance flow in the SDO and GERSM designs. Moreover, the axial velocity of the incoming flow has been notably reduced at this operating condition. These phenomena suggest that the SDO and GERSM designs are already stalled in region A. Figure 14(b) compares the axial velocity contours of the reference design in region A and the SDO and GERSM designs in region B. Now the back-flow regions in the middle pitch are slightly suppressed. The TLV paths are almost the same in the three designs. No signal of tip clearance flow spillage is observed. Meanwhile, the incoming flow has a higher axial velocity, good for operating stability. Based on these observations, it can be stated that the SDO and GERSM designs are under stable operation in region B. According to the above analysis, the stall points of the SDO and GERSM designs locate somewhere between region A and B. Using point B as the stall point will then yield a more conservative estimate of the stall margin, as defined in Eqn. (14), for the SDO and GERSM designs.

mpeak πstall SM = × − 1 × 100% (14) mstall πpeak

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are SMSDO,B = 9.4% and SMGERSM,B = 9.4%, slightly larger than that of the reference blade, SMReference,A = 9.1%. Since the true stable operating ranges of the SDO and GERSM designs extend further than points in Ellipse B, the corresponding stall margins can be expected to be larger after the optimization. 6. Conclusions A gradient-enhanced response surface model (GERSM) is built to act as a quick surrogate model which avoids time-consuming CFD calculation. It also provides a global representation of the cost function in a design space such that non-local optimum can be reached. Compared with traditional response surface, GERSM proves to be far more robust and accurate in the prediction of both function value and gradient. And the computational cost is cut down significantly. The transonic NASA Rotor 67 blade row is optimized at the design speed near peak efficiency using both the SDO and GERSM. The major advantage of GERSM is its

After optimization, the peak efficiency points at the design speed (see Fig. 12) are moved rightward by about 1 percent mass flow rate. The computed stall margins by assuming points in Ellipse B are stall points for the SDO and GERSM designs 11

ability to provide the non-local optimum while SDO could not. In the 24-parameter case adopted in this paper, SDO and GERSM are comparable in computational efficiency. The resulting two designs are different in geometry while both achieves notable aerodynamic improvements. Corresponding flow details are investigated and summarized below. In the tip region, the aft moved maximum thickness location of the blade profile together with the reduced maximum thickness de-cambers the front part of the blade, resulting in pre-compression of the supersonic flow. This reduces the passage shock strength. The loading shift in the tip region is accompanied by redistribution of the static pressure on the blade surface, thus affecting the strength of tip clearance flow along the blade tip. In particular, the tip clearance flow is strengthened near the LE but suppressed in the rear part of the blade tip. Also, the weakened passage shock favours the reduction of tip clearance flow at middle chord. Off-design performances at the design speed are analysed too. Both the adiabatic efficiency and total pressure ratio from the peak efficiency point to near-choke are increased. The choke mass flow rates are enlarged due to the reduced flow blockage in the tip region. Based on analysis of the axial velocity field, the tip clearance flow and TLV, it is proved that the current design optimizations at least maintains the stall margin of the original Rotor 67 blade.

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