# A New Method for Rapid Optimization Design of a Subsonic Tandem Blade

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## Abstract

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## Featured Application

**The investigation in this article puts forward an innovative and rapid way to design a subsonic tandem blade and shows a good application prospect for a subsonic/transonic axial compressor with tandem blades.**

## Abstract

## 1. Introduction

## 2. CFD Setup and Procedure

_{M}is the sum of body forces, u

_{eff}is the effective viscosity accounting for turbulence, and p’ is the modified pressure, as defined in Equation (3) [19,20]. The convergence of the simulation for the 2D tandem blade model is shown in Figure 3. The mass flow and momentum residuals decline by three orders of magnitude, and the inlet mass flow rate becomes stable in outlet value. Mesh independence and validation has been done in previous research to improve simulation accuracy [21].

#### 2.1. Setup for 2D Tandem Blade Model

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_{11}, total pressure—Pt

_{11}, three velocity directions (V

_{z}≡ 0, V

_{y}/V

_{x}= tanβ

_{11}).

#### 2.2. Setup for Double-Stage Transonic Compressor with Tandem Blades

_{11}= 101325Pa, Tt

_{11}= 288.15K, and axial flow direction) were imposed as the inlet reference plane. Additionally, outlet average static pressure was increased gradually to achieve the compressor characteristics in 100%, 90%, and 80% design speeds.

## 3. Analysis of the Aerodynamic Interaction between the Forward and Aft Blades

#### 3.1. The FB Flow Characteristic Influenced by AB

#### 3.2. The AB Flow Characteristic Influenced by FB

## 4. Optimization for Tandem Blades

#### 4.1. Optimization for a 2D Tandem Blade Model

#### 4.1.1. The Optimization Procedure

- Modification for the forward blade

- Modification for the aft blade

#### 4.1.2. Numerical Simulation Results and Analyses

#### 4.2. Optimization for a Double-Stage Transonic Compressor with Tandem Blades

#### 4.2.1. The Optimization Procedure

#### 4.2.2. Numerical Simulation Results and Analyses

- Discussions for Flow Details At 80% Design Speed, CO Point

- Discussions for flow details at design speed, DE point

## 5. Conclusions

- (1)
- Under subsonic (shock-free) conditions, the traces of aerodynamic interaction between FB and AB can be reflected in blade surface isentropic Mach number distributions. Compared to the single-blade configuration, the FB trailing edge local aerodynamic load and the AB incidence increased under the FB and AB relative position investigated in this paper (AO ~ −0.1, PP ~ 0.7). Furthermore, this phenomenon will lead to additional total pressure loss in the tandem configuration.
- (2)
- To fully utilize the aerodynamic interaction in tandem blades, the optimization principles and the “camber line modification method” are presented in this paper: in a tandem configuration, the FB and AB surface isentropic Mach number distributions (i.e., the local aerodynamic load) can be revised to rapidly and conveniently approach the CD airfoil by using this method.
- (3)
- To validate the effectiveness and feasibility of this method, two tandem blade optimization examples were performed and simulated:
- Optimization for a 2D tandem blade model: with an inflow Mach number of 0.8, the minimum loss decreases by 6%, and the incidence range extends approximately 2° compared to the baseline;
- Optimization for a double-stage transonic compressor with tandem blades: at 80% design speed, isentropic efficiency increases by 2 points, with an increase of mass flow by 1.6% at the choke point simultaneously. In addition, compressor characteristics at design speed remain almost unchanged.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

FB | Forward blade |

AB | Aft blade |

LE | Leading edge |

TE | Trailing edge |

DE point | Design point |

NS point | Near stall point |

CO point | Cooperating point |

S | Pitch |

C | Chord |

t | Pitch space between FB TE and AB LE |

Ma | Mach number |

Subscripts | |

11 | Station for the forward blade inlet |

12 | Station for the forward blade outlet |

21 | Station for the aft blade inlet |

22 | Station for the aft blade outlet |

Symbols | |

x | Axial direction |

y | Circumferential direction |

z | Radial direction |

V | Air flow velocity |

χ | Blade metal angle (measured from axial direction, °) |

τ | Blade normalized thickness, τ_{Local}/τ_{Max} |

κ_{1} | Forward blade normalized camber, (χ11 − χ)/(χ11 − χ12) |

κ_{2} | Aft blade normalized camber, (χ_{21} − χ)/(χ21 − χ22) |

m | Mass flow rate (kg/s) |

Pt | Total pressure (Pa) |

Ps | Static pressure (Pa) |

Tt | Total temperature (K) |

β | Flow angle to the axial direction (°) |

A | Area (m^{2}) |

ω_{Total} | Tandem blade total pressure loss, $\left(P{t}_{11}-P{t}_{22}\right)/\left(P{t}_{11}-P{s}_{11}\right)$ |

ω_{FB} | Forward blade total pressure loss, $\left(P{t}_{11}-P{t}_{12}\right)/\left(P{t}_{11}-P{s}_{11}\right)$ |

i | Incidence, β – χ_{11} (°) |

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**Figure 3.**Convergence of the simulation for the 2D tandem blade model; incidence = −1°, inlet Mach = 0.8. (

**a**) Momentum and mass; (

**b**) mass flow rate on INLET.

**Figure 8.**Comparison of the FB isentropic Mach number in single-blade and tandem configurations. (

**a**) Incidence = −5°; (

**b**) incidence = −1°; (

**c**) incidence = +5°.

**Figure 9.**Comparison of static pressure contours in single-blade and tandem configurations; incidence −1°, inflow Mach number 0.8. (

**a**) FB single-blade configuration; (

**b**) tandem configuration.

**Figure 10.**Comparison of the AB isentropic Mach number in single-blade and tandem configurations. (

**a**) Incidence = −5°; (

**b**) incidence = −1°; (

**c**) incidence = +5°.

**Figure 12.**Comparison of Mach contours in single-blade and tandem configurations. FB incidence −1°; FB inflow Mach number 0.8. (

**a**) AB single-blade configuration; (

**b**) tandem configuration.

**Figure 13.**Comparison of Mach contours in single-blade and tandem configurations. FB incidence +5°; FB inflow Mach number 0.8. (

**a**) AB single-blade configuration; (

**b**) tandem configuration.

**Figure 17.**Comparison of baseline and optimized blade performances. (

**a**) Incidence—loss coefficient; (

**b**) incidence—diffusion factor.

**Figure 18.**Baseline blade Mach contours in typical operating conditions. (

**a**) i = −5°; (

**b**) i = −1°; (

**c**) i = +5°.

**Figure 19.**Optimized blade Mach contours in typical operating conditions. (

**a**) i = −5°; (

**b**) i = −1°; (

**c**) i = +5°.

**Figure 20.**Comparison of the baseline and optimized blade surface isentropic Mach numbers. (

**a**) i = −5°; (

**b**) i = −1°; (

**c**) i = +5°.

**Figure 21.**Operating conditions of the tandem stator at design speed and off design speed. (

**a**) Incidence range; (

**b**) inlet Mach number.

**Figure 24.**FB surface isentropic Mach number at 80% design speed at the CO point. (

**a**) 10% blade height; (

**b**) 50% blade height; (

**c**) 90% blade height.

**Figure 25.**Mach contours near the blade surface at 80% design speed at the CO point. (

**a**) Original; (

**b**) modified.

**Figure 26.**FB surface isentropic Mach number at 100% design speed at the DE point. (

**a**) 10% blade height; (

**b**) 50% blade height; (

**c**) 90% blade height.

**Figure 27.**Mach contours near the blade surface at 100% design speed at the DE point. (

**a**) Original; (

**b**) modified.

FB | AB | Tandem | |
---|---|---|---|

S (mm) | 17.1 | 17.1 | 18.5 |

C (mm) | 20.5 | 19.7 | 39.3 |

Inlet Metal Angle (°) | 56.1 | 51.3 | - |

Outlet Metal Angle (°) | 43.4 | 9.7 | - |

Solidity | 1.20 | 1.18 | 2.36 |

AO | - | - | −0.10 |

PP | - | - | 0.70 |

Forward Blade | Aft Blade | |||||
---|---|---|---|---|---|---|

Hub | Mid | Tip | Hub | Mid | Tip | |

Camber (°) | 21.4 | 12.7 | 23.6 | 38.2 | 41.6 | 45.3 |

Stagger (°) | 46.9 | 42.8 | 52.1 | 17.1 | 14.9 | 18.6 |

Solidity | 1.53 | 1.32 | 1.36 | 1.20 | 1.09 | 1.03 |

DF | 0.44 | 0.42 | 0.45 | 0.49 | 0.45 | 0.48 |

i (°) | Ma_{21} | β_{21} (°) |
---|---|---|

−5 | 0.705 | 43.863 |

−4 | 0.659 | 43.852 |

−3 | 0.630 | 43.911 |

−2 | 0.607 | 43.970 |

−1 | 0.585 | 44.081 |

0 | 0.566 | 44.234 |

+1 | 0.549 | 44.373 |

+2 | 0.533 | 44.605 |

+3 | 0.519 | 44.850 |

+4 | 0.509 | 45.157 |

+5 | 0.503 | 45.577 |

+6 | 0.501 | 45.936 |

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**MDPI and ACS Style**

Tao, Y.; Yu, X.; Liu, B.
A New Method for Rapid Optimization Design of a Subsonic Tandem Blade. *Appl. Sci.* **2020**, *10*, 8802.
https://doi.org/10.3390/app10248802

**AMA Style**

Tao Y, Yu X, Liu B.
A New Method for Rapid Optimization Design of a Subsonic Tandem Blade. *Applied Sciences*. 2020; 10(24):8802.
https://doi.org/10.3390/app10248802

**Chicago/Turabian Style**

Tao, Yuan, Xianjun Yu, and Baojie Liu.
2020. "A New Method for Rapid Optimization Design of a Subsonic Tandem Blade" *Applied Sciences* 10, no. 24: 8802.
https://doi.org/10.3390/app10248802