# Multi-Row Turbomachinery Aerodynamic Design Optimization by an Efficient and Accurate Discrete Adjoint Solver

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Schemes and Boundary Conditions

#### 2.1. Numerical Schemes

#### 2.2. Boundary Conditions

## 3. Adjoint Principle

## 4. Automatic Differentiation

#### 4.1. Forward and Backward Modes

#### 4.2. Problem Description and Analysis

#### 4.2.1. Conditional Branches

#### 4.2.2. Reals/Integrals

## 5. Hybrid Automatic and Manual Differentiation

#### 5.1. Strategy One: Directly Evaluate Conditional Branches

#### 5.2. Strategy Two: Directly Evaluate Intermediate Variables

#### 5.3. Strategy for Real-Life Adjoint Codes

## 6. Rotor-Stator/Stator-Rotor Interface Treatment

#### 6.1. Mixing Plane Method

- (1)
- Average flux in the circumferential direction. It can be expressed in the following symbolic form$$({F}_{0}^{j})={f}_{1}({U}^{j},{X}^{j})(j=1,2)$$In Equation (11), the superscript j represents the upstream or downstream of a rotor-stator/stator-rotor interface. When j equals 1, it represents the upstream of the interface, otherwise, it represents the downstream of the interface. The subscript 0 represents the value before flux interpolation. X is the grid coordinate vector, and ${f}_{1}$ is the function related to flux averaging.
- (2)
- Interpolate flux in the radial direction. This step is a must as the grid distribution in the radial direction across an interface is often different as shown in Figure 8b. The interpolation operation is represented symbolically as follows$$\begin{array}{c}Row1:\left({F}_{1}^{1}\right)={f}_{2}({X}^{1},{X}^{2},{F}_{0}^{2})\end{array}$$$$\begin{array}{c}Row2:\left({F}_{1}^{2}\right)={f}_{2}({X}^{2},{X}^{1},{F}_{0}^{1})\end{array}$$
- (3)
- Update solution based upon the one-dimensional non-reflective boundary condition. This operation is represented as follows$$\begin{array}{c}Row1:\left({U}^{1,\ast}\right)={f}_{3}({U}^{1},{F}_{0}^{1},{F}_{1}^{1})\end{array}$$$$\begin{array}{c}Row2:\left({U}^{2,\ast}\right)={f}_{3}({U}^{2},{F}_{0}^{2},{F}_{1}^{2})\end{array}$$In the above equation, the superscript ∗ represents the updated solution, and ${f}_{3}$ represents the function related to the solution update.

#### 6.2. Discrete Adjoint Mixing Plane Method

- (1)
- Differentiate the subroutines related to solution update.$$\begin{array}{c}Row1:({U}_{a}^{1},{F}_{0,a}^{1},{F}_{1,a}^{1})={f}_{3}\_a({U}^{1},{U}^{1,\ast},{U}_{a}^{1,\ast},{F}_{0}^{1},{F}_{1}^{1})\end{array}$$$$\begin{array}{c}Row2:({U}_{a}^{2},{F}_{0,a}^{2},{F}_{1,a}^{2})={f}_{3}\_a({U}^{2},{U}^{2,\ast},{U}_{a}^{2,\ast},{F}_{0}^{2},{F}_{1}^{2})\end{array}$$$$\begin{array}{c}\begin{array}{cc}\hfill {U}_{a}^{j}& ={(\frac{\partial {f}_{3}}{\partial {U}^{j}})}^{T}{U}_{a}^{j,\ast}(j=1,2)\hfill \\ \hfill {F}_{0,a}^{j}& ={(\frac{\partial {f}_{3}}{\partial {F}_{0}^{j}})}^{T}{U}_{a}^{j,\ast}(j=1,2)\hfill \\ \hfill {F}_{1,a}^{j}& ={(\frac{\partial {f}_{3}}{\partial {F}_{1}^{j}})}^{T}{U}_{a}^{j,\ast}(j=1,2)\hfill \end{array}\end{array}$$In the above equations, ${U}_{a}^{1,\ast}$ and ${U}_{a}^{2,\ast}$ are initialized by 0.
- (2)
- Differentiate the subroutines related to flux interpolation.$$\begin{array}{c}Row1:({F}_{0,a}^{2})={f}_{2}\_a({X}^{1},{X}^{2},{F}_{0}^{2},{F}_{1}^{1},{F}_{1,a}^{1})\end{array}$$$$\begin{array}{c}Row2:({F}_{0,a}^{1})={f}_{2}\_a({X}^{2},{X}^{1},{F}_{0}^{1},{F}_{1}^{2},{F}_{1,a}^{2})\end{array}$$$$\begin{array}{c}\begin{array}{cc}\hfill {F}_{0,a}^{2}& ={F}_{0,a}^{2}+{(\frac{\partial {f}_{2}}{\partial {F}_{1}^{1}})}^{T}{F}_{1,a}^{1}\hfill \\ \hfill {F}_{0,a}^{1}& ={F}_{0,a}^{1}+{(\frac{\partial {f}_{2}}{\partial {F}_{1}^{2}})}^{T}{F}_{1,a}^{2}\hfill \end{array}\end{array}$$
- (3)
- Differentiate the subroutines related to flux averaging.$$({U}_{a}^{j})={f}_{1}\_a({U}^{j},{X}^{j},{F}_{0}^{j},{F}_{0,a}^{j})(j=1,2)$$$${U}_{a}^{j}={U}_{a}^{j}+{(\frac{\partial {f}_{1}}{\partial {F}_{0}^{j}})}^{T}{F}_{0,a}^{j}(j=1,2)$$In the above equations, the subscript a represents the backward mode of the AD.

## 7. Results

#### 7.1. NASA Stage 35

#### 7.1.1. Grid Independence Study

#### 7.1.2. Adjoint Sensitivity Verification

#### 7.1.3. Computational Efficiency

#### 7.1.4. Results of Design Optimizations

#### 7.2. Aachen Turbine

#### 7.2.1. Adjoint Sensitivity Verification

#### 7.2.2. Computational Efficiency

#### 7.2.3. Results of Design Optimizations

## 8. Conclusions

- (1)
- The hybrid ADJ almost has the same sensitivity convergence histories as those from the linearized solver and higher sensitivity accuracy than the CEV. The maximum relative difference of sensitivities between the FDM and the hybrid ADJ is no more than $1.0\%$ for the cases studied in the paper.
- (2)
- The hybrid ADJ has higher computational efficiency than the discrete adjoint solver purely developed by the AD tool. About $43\%$ CPU time and $9.0\%$ memory consumption can be saved for the single-stage NASA Stage 35, and $44\%$ CPU time and $9.6\%$ memory consumption can be saved for the 1.5-stage Aachen turbine.
- (3)
- The multi-row turbomachinery aerodynamic design optimizations can be effectively performed by the hybrid ADJ. For the single-stage NASA Stage 35, the isentropic efficiency over the entire operating range of $100\%$ design speed is significantly improved, and the stall margin is increased for the optimized blades. For the 1.5-stage Aachen turbine, the entropy generation rate is decreased after optimization. Moreover, the variations in mass flow rate and total pressure ratio are also acceptable.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 10.**Isentropic efficiency against the mass flow rate at 100% design speed (

**a**); Fine-grid solution, with discretization errors computed using the corresponding equations in reference [32] (

**b**).

**Figure 11.**Total pressure ratio against the mass flow rate at 100% design speed (

**a**); fine-grid solution, with discretization errors computed using the corresponding equations in reference [32] (

**b**).

**Figure 12.**Flow and adjoint fields at 50% span of the NASA Stage 35. (

**a**) Turbulence quantity. (

**b**) Adjoint quantity corresponding to the turbulence model equation.

**Figure 13.**Comparison of sensitivity convergence histories among LIN, CEV, and hybrid ADJ for the NASA Stage 35.

**Figure 15.**RMS residual convergence histories of both the flow and adjoint solvers (

**a**); Comparison of CPU time among different solvers for the NASA Stage 35 (

**b**).

**Figure 16.**Evolution histories of the weighted objective functions and constraints for the NASA Stage 35.

**Figure 20.**Radial distributions of rotor aerodynamic parameters at the peak efficiency point for the NASA Stage 35.

**Figure 21.**Radial distributions of stator aerodynamic parameters calculated across the stator only at the peak efficiency point for the NASA Stage 35.

**Figure 23.**Flow and adjoint fields at 50% span of the Aachen turbine. (

**a**) Turbulence quantity. (

**b**) Adjoint quantity corresponding to the turbulence model equation.

**Figure 26.**RMS residual convergence histories of both the flow and adjoint solvers (

**a**); comparison of CPU time among different solvers for the Aachen turbine (

**b**).

**Figure 27.**Evolution histories of the weighted objective function and constraints for the Aachen turbine.

**Figure 28.**Comparison of original and optimized blade profiles at three different spans for the Aachen turbine.

**Figure 29.**Radial distributions of total pressure ratio, inlet, and outlet flow angles in each row for the Aachen turbine.

Axial | Circumferential | Radial | Total | ||
---|---|---|---|---|---|

Row 1 | Grid1 | 133 | 37 | 57 | ∼280 k |

Grid2 | 149 | 57 | 73 | ∼620 k | |

Grid3 | 161 | 85 | 89 | ∼1218 k | |

Row 2 | Grid1 | 145 | 33 | 57 | ∼273 k |

Grid2 | 161 | 49 | 61 | ∼481 k | |

Grid3 | 169 | 61 | 81 | ∼835 k |

Flow | Original ADJ | Hybrid ADJ | |
---|---|---|---|

time | 1 | 6.16 | 3.49 (−43%) |

memory | 1 | 2.52 | 2.29 (−9%) |

**Table 3.**Comparison of aerodynamic performances between the original and optimized blade profiles for the NASA Stage 35.

m(kg/s) | $\mathit{\pi}$ | $\mathit{\eta}(\%)$ | ||
---|---|---|---|---|

choke | original | 21.02 | 1.745 | 83.35 |

optimized | 21.05 (+0.14%) | 1.745 (0%) | 84.00 (+0.65) | |

peak | original | 20.97 | 1.835 | 83.91 |

optimized | 21.00 (+0.14%) | 1.835 (0%) | 84.73 (+0.82) | |

stall | original | 20.60 | 1.924 | 82.19 |

optimized | 20.55 (−0.24%) | 1.922 (−0.10%) | 82.67 (+0.48) |

Flow | Original ADJ | Hybrid ADJ | |
---|---|---|---|

time | 1 | 6.16 | 3.43 (−44%) |

memory | 1 | 1.94 | 1.77 (−9.6%) |

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## Share and Cite

**MDPI and ACS Style**

Wu, H.; Da, X.; Wang, D.; Huang, X.
Multi-Row Turbomachinery Aerodynamic Design Optimization by an Efficient and Accurate Discrete Adjoint Solver. *Aerospace* **2023**, *10*, 106.
https://doi.org/10.3390/aerospace10020106

**AMA Style**

Wu H, Da X, Wang D, Huang X.
Multi-Row Turbomachinery Aerodynamic Design Optimization by an Efficient and Accurate Discrete Adjoint Solver. *Aerospace*. 2023; 10(2):106.
https://doi.org/10.3390/aerospace10020106

**Chicago/Turabian Style**

Wu, Hangkong, Xuanlong Da, Dingxi Wang, and Xiuquan Huang.
2023. "Multi-Row Turbomachinery Aerodynamic Design Optimization by an Efficient and Accurate Discrete Adjoint Solver" *Aerospace* 10, no. 2: 106.
https://doi.org/10.3390/aerospace10020106