# A Coupling Method for the Design of Shape-Adaptive Compressor Blades

^{1}

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## Abstract

**:**

## 1. Introduction and Motivation

## 2. The Coupling Methodology

#### 2.1. Selection of the NASA 67 Rotor as Test Case and Reference Design

- Starting from the design point in red, the design point mass flow is shifted beyond the choke limit of the rotor (blue). This possibility is particularly interesting for the development of alternative energy concepts, such as fuel cells, which are highly dependent on different mass flows for their power output [3].
- By adjusting the design point pressure ratio, while keeping the mass flow constant, an alternative for variations in rotational speed is assessed (yellow).
- Extending the surge margin by moving along the performance curve (green) improves part-load performance, which is especially critical when the airplane takes off or accelerates during flight and the danger of compressor stall has to be avoided [26].

#### 2.2. Aerodynamic Design Methodology

#### 2.2.1. Meridional Design

_{1}as well as flow deflection requirement Δβ [13]. The required geometry boundary data are derived from the reference design point geometry, while the new design point conditions (mass flow and pressure ratio) are defined though the selected adaption scenarios.

#### 2.2.2. Blade Design

_{c}and its chordwise position x

_{c}a generalized parabolic mean arc is created to model the blade camber, following Schlichting [24] (Figure 6).

_{c}, f

_{c}), offering two adaption possibilities for the aerostructural coupling. While the blade turning increases linearly with the blade maximum camber, the impact of a maximum camber position variation on the achievable blade turning variation increases with the deviation from the symmetrical configuration at x

_{c}= 0.5. Although this effect is similar for a rearward and frontward shift, a rearward shift is expected to have a positive impact on the performance of the target designs, due to the suction-side curvature reduction towards the compression shock in the tip region of the rotor sections [29]. With a rearward location of the maximum camber position, the impact of f

_{c}variations is additionally amplified, which is especially relevant for a possible reference design adjustment. The resulting leading-edge camber angle α

_{1}directly influences the required blade stagger angle λ and increases with the maximum camber or a relocation of the maximum camber position towards a centered position (Figure 7).

_{1}is defined, and the required spanwise stagger angle is derived according to Figure 5.

_{1}and the rotor inflow conditions β

_{1}predicted by the SLC calculation. While the camber design parameters x

_{c}and f

_{c}influence the blade shape and the spanwise morphing of the blade turning, the stagger angle describes the spanwise variation of blade twist and therefore the third morphing parameter for the shape adaption and the coupled aerostructural design methodology.

#### 2.3. Structural Analysis Method and Coupling Approach

#### 2.3.1. Geometric Design and Parameterization

#### 2.3.2. Structural Modelling and Material Parameters

#### 2.3.3. Structural Analysis

#### 2.4. Aero-Structural Coupling in the Meridional Plane

_{i}due to the shape morphing, and the structurally feasible turning variation ΔΔφ can be calculated according to Figure 5. The feasible variation in section-wise blade turning is then directly compared with the desired aerodynamic target design requirements.

_{1}has to match the altered inflow angle β

_{1}, in order to achieve the mass flow required by the adaption scenarios (Figure 19).

_{1}, according to the SLC solution is compared to the structural variation of the leading-edge metal angle, Δκ

_{1}, which, again, is a result of the leading-edge camber angle variation Δα

_{1}as well as the stagger angle variation Δλ (Figure 5).

_{1}is dominant for this adaption scenario, this result indicates that the initially specified mass flow variation of 2.75 kg/s is not feasible through shape adaption. However, the selected actuation concept (n

_{act}= 2, o

_{1}= 45°, o

_{2}= 135°, both actuators expanded) achieves a reduction of shape adaption incidence in the rotor tip region and is therefore selected for a further simulative investigation.

#### 2.5. Representative Simulative Evaluation

_{c}, x

_{c}or λ is specified, defining the others through the required blade turning and inflow angle. With the highest deformations concentrated in the blade tip region, leaving the lower blade sections basically undeformed, the transonic behavior of the blade profiles plays a major role, emphasizing the maximum camber position x

_{c}as the active design parameter. In order to control the flow acceleration towards a compression shock and to reduce suction-side curvature beyond the leading edge, a rearward shift of x

_{c}is implemented for the tip sections of the rotor. Between 70% rotor height and blade hub, the maximum camber position is kept constant. Combined with an increase or decrease in the maximum camber, depending on the specified flow deflection requirement, the camber design is derived for all sections (Equation (5)). With x

_{c}and f

_{c}defined through the turning requirement, the blade camber leading-edge angle is fixed, and the required spanwise stagger angle can be calculated for each section (Equation (9)). With the projection of the profile sections on the meridional reference streamlines, aerodynamic target rotor designs are derived for the three selected adaption scenarios. In Figure 20, Figure 21 and Figure 22, the redesigned target shapes are represented by Section 2 at 95% of the rotor height and compared to the reference design as well as to the best suited structural deformation, selected in the first coupling step.

_{act}= 2, o

_{1}= 45°, o

_{2}= 135°), the actuation mechanism can be switched from expansion to compression, reversing the deformation of the leading-edge metal angle. This increases the total achievable variation of the unique incidence angle to 0.52°, with a compressed actuation corresponding to a MF adaption scenario with reduced design point mass flow (Figure 21).

## 3. Conclusions and Discussion

## 4. Outlook

## Author Contributions

## Funding

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Variable | Declaration |
---|---|

a, b | CSM design parameters |

f_{c} | maximum camber |

fc(x) | dimensionless profile camber function |

ft(x) | dimensionless profile thickness function |

l | length |

$\dot{m}$ | mass flow |

n | vortex law design parameter |

nact | number of actuators |

n1, n2 | class factors |

o_{1}, o_{2} | fiber orientation |

x_{c} | maximum camber position |

y(x) | profile function value |

y+ | dimensionless wall distance |

C(x) | class function |

F | force |

KR | CSM thickness parameter |

S(x) | shape function |

V_{θ} | circumferential velocity |

X | x-coordinate (machine axis) |

α_{i} | camber angle |

ꞵ_{i} | relative flow angle |

Δ | variation |

Δβ | flow turning |

Δφ | profile turning |

iSA | shape adaption incidence |

κi | metal angle |

ꞷ | pressure loss coefficient |

λ | stagger angle |

Π | pressure ratio |

Index | Declaration |
---|---|

θ | circumferential |

c | camber |

t | tip |

1 | leading edge |

2 | trailing edge |

Abbreviation | Extended Meaning |
---|---|

comp. | compression |

exp. | expansion |

ref | reference/original |

ACP | Ansys Composite PrepPost |

AVDR | Axial Velocity Density Ratio |

CSM | Class Shape/Class Form Function |

CFD | Computational Fluid Dynamics |

CFRP | Carbon-Fiber-Reinforced Plastics |

CMC | Ceramic Matrix Composite |

DP | Design Point |

FEA | Finite Element Analysis |

IGV | Inlet Guide Vane |

LE | Leading Edge |

MF | Mass Flow |

MFC | Macro-Fiber Composite |

OGV | Outlet Guide Vane |

PM | Performance Map |

PR | Pressure Ratio |

PS | Pressure Side |

Q3D | Quasi-Three-Dimensional |

SA | Shape Adaption |

SLC | Streamline Curvature |

SMA | Shape Memory Alloy |

SS | Suction Side |

TE | Trailing Edge |

3D | Three-Dimensional |

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**Figure 2.**Exemplary actuation configurations for the NASA 67 compressor rotor blade (grey) with the pressure sides on the right and the suction sides on the left.

**Top**: one large surface actuator per side (blue and green);

**middle**: one large surface actuator on the suction side (orange) and two smaller actuators on the pressure side (green and blue);

**bottom**: four smaller actuators on each side (colored surfaces).

**Figure 3.**Simulated speed line for the NASA 67 front stage for design speed, with the literature design point (DP, red), a performance map adaption well beyond the simulated surge margin (PM, green), a mass flow adaption (MF, blue) and a variation of the pressure ratio (PR, yellow).

**Figure 4.**Calculation domain and grid for the SLC calculation with rotor, rotor reference design sections and stator geometry in the meridional plane.

**Figure 5.**Schematic overview of flow and blade angle definitions, following [28].

**Figure 6.**Exemplary decomposition of an aerodynamic profile section (

**b**) into the camber modelling, defined by the maximum camber f

_{c}, its position x

_{c}(

**a**) and the thickness modelling with the free parameter KR (

**c**).

**Figure 7.**Profile turning Δφ variation (

**a**) and leading-edge camber angle α

_{1}variations (

**b**) for a relative adaption of f

_{c}and its chordwise position x

_{c}in relation to the reference parameters f

_{c,ref}and x

_{c,ref}.

**Figure 8.**Exemplary wireframe structures and derived CAD models for the structural design based on the aerodynamic design streamline sections with different actuator configurations. (

**a**) Matlab blade contour basis model following SLC aerodynamic curves, (

**b**) and (

**c**) Matlab blade contour models with two possible actuator configurations – the blue lines representing the actuator limits for the pressure side and the red lines those for the suction side, (

**d**) CAD basis geometry with streamlines and actuator limits, (

**e**) CAD solid blade body model with actuators limits for an exemplary configuration.

**Figure 9.**Blade architecture example,

**left**: actuators set with suction side on top and pressure side at bottom (without blade);

**right**: blade’s body with markings for actuators’ mounting positions.

**Figure 10.**Schematic representation of an MFC actuator structure—Adapted with permission from ref. [20], Smart Material Corp.

**Figure 11.**SMA actuators (

**left**) and working principle (

**right**). l and b indicate the size of the actuator, s the deformation path, F the load carrying capacity and R the electric resistance applied to the actuator as a source of heat—Adapted with permission from ref. [21], CompActive.

**Figure 12.**ACP model for MFC actuators. The green surfaces represent the actuating surfaces, and the green arrows represent the fiber direction, which corresponds with the actuation direction.

**Figure 13.**Bonded contact illustration for one actuator. (

**a**) Blade body with highlighted actuating surfaces for one actuator, (

**b**) Contact actuator surfaces for exemplary actuator, (

**c**) Target body surfaces on blade for exemplary actuator.

**Figure 14.**Boundary conditions for the model of the morphing blade. A, in red, shows the embedded actuator surface and the area to which the thermal analogy boundary constraint is applied. B, in blue, shows the fixed support constraint and represents the behavior of the blade at its root.

**Figure 15.**Ansys Workbench static structural analysis results for deformation. The black cable structure represents the undeformed blade.

**Figure 16.**Ansys Workbench static structural analysis results for deformation at the original aerodynamic sections. These sections enable the coupling with aerodynamics. The black cable structure represents the undeformed blade.

**Figure 17.**Qualitative representation of possible morphing shapes. The black wireframe represents the undeformed blade. On the

**top left**, a larger camber modification can be observed; the

**bottom left**shows a change in the twist angle, and the

**right**image shows a displacement in the position of the maximum camber.

**Figure 18.**Comparison of the flow deflection angle variations in the blade profile turning angle morphing for different vortex designs and actuation concepts, separated according to the selected adaption scenario (

**left**: PM,

**middle**: PR,

**right**: MF).

**Figure 19.**Comparison of the inflow angle variations in the leading-edge metal angle morphing for different vortex designs and actuation concepts, separated according to the selected adaption scenario (

**left**: PM,

**middle**: PR,

**right**: MF).

**Figure 20.**Q3D calculation domain for the representative evaluation of Section 2 in relation to the streamlines of the design point geometry and the pre-defined reference sections (

**left**) and the comparison of the SLC AVDR with the hyperbolic remodeling approach (

**right**).

**Figure 21.**Representative Q3D evaluation (Section 2) of the deformed profile sections (

**left**), comparison of PM and PR target design performances with selected actuation configuration (

**middle**) and comparison of MF target design with selected actuation concept (

**right**).

**Figure 22.**Comparison of target design and morphed shape to the reference design (Section 2, PM scenario, profile staggering relative to reference design).

**Figure 23.**Comparison of target design and morphed shape to the reference design (Section 2, PR scenario, profile staggering relative to reference design).

**Figure 24.**Comparison of target design and morphed shape to the reference design (Section 2, MF scenario, profile staggering relative to reference design).

**Table 1.**Material properties for the exemplary compressor blade—Titanium Alloy [31].

Property | Value | Unit |
---|---|---|

Density | 4620 | kg/m^{3} |

Young’s Modulus | 9.6 × 10^{10} | Pa |

Poisson’s Ratio | 0.36 | - |

Shear Modulus | 3.5294 × 10^{10} | Pa |

Tensile Yield Strength | 9.3 × 10^{8} | Pa |

**Table 2.**Material properties for exemplary MFC piezoelectric actuators [20].

Property | Value | Unit |
---|---|---|

Density | 4700 | kg/m^{3} |

Young’s Modulus X | 3.00 × 10^{10} | Pa |

Young’s Modulus Y | 1.55 × 10^{10} | Pa |

Young’s Modulus Z | 1.55 × 10^{10} | Pa |

Poisson’s Ratio XY | 0.35 | - |

Poisson’s Ratio YZ | 0.4 | - |

Poisson’s Ratio XZ | 0.35 | - |

Shear Modulus XY | 1.07 × 10^{10} | Pa |

Shear Modulus YZ | 5.70 × 10^{9} | Pa |

Shear Modulus XZ | 1.07 × 10^{10} | Pa |

Coef. of Thermal Expansion X | 8.36 × 10^{−7} | 1/°C |

Coef. of Thermal Expansion Y | −3.96 × 10^{−7} | 1/°C |

Coef. of Thermal Expansion Z | −3.96 × 10^{−7} | 1/°C |

Tensile Yield Strength | 9.3 × 10^{8} | Pa |

**Table 3.**Overview of selected actuation concepts and aerodynamic design parameters for exemplary morphing scenarios.

Scenario | Aerodynamic Design | Actuation Configuration |
---|---|---|

PM | Π = 1.69, $\dot{m}$ = 32 kg/s, n = 0.79 | n_{act} = 2, o_{1} = 0°, o_{2} = 0°, SS = exp, DS = comp |

PR | Π = 1.69, $\dot{m}$ = 33.25 kg/s, n = 0.79 | n_{act} = 2, o_{1} = 0°, o_{2} = 0°, SS = exp, DS = comp |

MF | Π = 1.63, $\dot{m}$ = 36 kg/s, n = 1.1 | n_{act} = 2, o_{1} = 45°, o_{2} = 135°, SS = exp, DS = exp |

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## Share and Cite

**MDPI and ACS Style**

Montano, Z.; Seidler, M.; Riemenschneider, J.; Friedrichs, J. A Coupling Method for the Design of Shape-Adaptive Compressor Blades. *Appl. Mech.* **2022**, *3*, 182-209.
https://doi.org/10.3390/applmech3010014

**AMA Style**

Montano Z, Seidler M, Riemenschneider J, Friedrichs J. A Coupling Method for the Design of Shape-Adaptive Compressor Blades. *Applied Mechanics*. 2022; 3(1):182-209.
https://doi.org/10.3390/applmech3010014

**Chicago/Turabian Style**

Montano, Zhuzhell, Marcel Seidler, Johannes Riemenschneider, and Jens Friedrichs. 2022. "A Coupling Method for the Design of Shape-Adaptive Compressor Blades" *Applied Mechanics* 3, no. 1: 182-209.
https://doi.org/10.3390/applmech3010014