# Uncertainty Quantification of Compressor Map Using the Monte Carlo Approach Accelerated by an Adjoint-Based Nonlinear Method

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Adjoint-Based Nonlinear/Linear Method (Adj-Nonlinear/Linear)

**U**represents the flow solution, a 6×N vector for an unstructured Reynolds-averaged Navier—Stokes (RANS) flow solver, whose turbulence equation is the Spalart—Allmaras turbulence equation. The number ’6’ presents the primal variables and ’N’ is the nodes of the mesh. $\mathit{\alpha}$ is the geometric variable vector, which is used to described the geometry of the blade. It can be the vector of coordinate points at the blade surface or the vector of design parameters like chord, thickness and so on.

**U**and $\mathit{\alpha}$ always satisfy the following symbolic flow governing equation:

**R**represents the nonlinear residual. The adjoint method can be used to calculate the sensitivity of the performance functional to the geometric variable vector as follows:

- 1.
- Obtain the flow solution ${\mathit{U}}_{0}$ and the adjoint solution ${v}_{0}^{T}$ for a specific performance functional for the baseline geometry;
- 2.
- Calculate sensitivity for all geometric variables. The sensitivity calculation involves mesh perturbation and the objective function J & residual $\mathit{R}$ evaluation, based on the perturbed mesh ${\mathit{\alpha}}_{0}+{\u03f5}_{i}$ and the baseline flow solution ${\mathit{U}}_{0}$. The finite difference method is also involved here to determine the sensitivities, see the green box in Figure 1. Therefore, the perturbation size of each design variable is quite important, as too big or too small a value introduces big truncation or rounding errors. The operation has to be performed for each geometric variable (for 1:M, M represents the amount of the geometric variables). In this investigation, there were, in total, 398 geometric variables. This meant that the number of mesh perturbations and residual evaluations was 398;
- 3.
- Calculate the performance metrics of all samples. The geometry perturbations of all samples($\Delta {\mathit{\alpha}}_{i}$) (for i = 1:N, N represents the amount of the samples) are reflected in changes to the baseline geometry variable vector ${\mathit{\alpha}}_{0}$. Then, Equation (9) is used to calculate the performance metric J.

## 3. Uncertainty Quantification

#### 3.1. Test Case

#### 3.2. Adjoint Solution Verification

#### 3.3. Verification of the MC−Adj−Nonlinear Method at Design and Near-Stall Conditions

#### 3.4. Full Map UQ of Aerodynamic Performance at Four Speeds

## 4. Conclusions

- (1)
- At 100% speed, compared with the MC−adj−linear method, the UQ results predicted by the MC−adj−nonlinear method were more accurate, especially for the near-stall condition, where the nonlinear dependence of performance functionals on geometric variables was stronger. At 50% speed, the differences in the UQ results predicted by the two adjoint-based methods were much smaller, due to the weaker nonlinearity of the flow. The MC−adj−nonlinear method required nearly 30 times less time than the MC−CFD method. Hence, the MC−adj−nonlinear approach provides a satisfactory balance between precision and time cost for UQ.
- (2)
- Aerodynamic performance is more sensitive to geometric deviations at high speeds than at low speeds. For this particular case, the geometric deviations produced an increased mean of mass flow rate and pressure at low speeds, while incurring a reduced mean at high speeds. The geometric deviations were generally detrimental to the mean efficiency over the four speeds. The reduction of the mean of mass flow rate, pressure ratio, and efficiency became more with increase in shaft speed.
- (3)
- The standard deviation of performance generally increased with increase in shaft speed. Along a speedline, the standard deviation also increased with increase in pressure ratio. The difference in standard deviation between a near choke point and a near stall point along a speedline was much larger at high speeds than at low speeds.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 6.**The nominal blade, average blade, and the average blades with twice the positive and negative standard deviation.

**Figure 7.**The eddy viscosity adjoint field with the mass flow rate as the performance functional at the design point.

**Figure 9.**Performance characteristic with PDFs of mass flow rate and pressure ratio using the three methods.

**Figure 10.**Performance characteristic with PDFs of mass flow rate and efficiency using the three methods.

**Figure 11.**The mean of normalized performance variations at the design (

**left**) and near-stall conditions (

**right**) computed using the three methods.

**Figure 12.**The normalized standard deviation of performance functionals variation at the design (

**left**) and near-stall conditions (

**right**) computing using the three methods.

**Figure 13.**The mean of normalized performance deviations at the design (

**left**) and near-stall conditions (

**right**) of 100% speeds and that of low pressure ratio (

**left**) and high pressure ratio (

**right**) operating points at 50% speed by the MC−adj−linear and MC−adj−nonlinear methods.

**Figure 14.**The normalized standard deviation of performance functionals at the design (

**left**) and near-stall conditions (

**right**) at 100% speeds and that of low pressure ratio (

**left**) and high pressure ratio (

**right**) operating points at 50% speed by the MC−adj−linear and MC−adj−nonlinear methods.

**Figure 15.**Time cost versus accuracy in calculating the mean mass flow rate at the design condition using the three methods.

**Figure 18.**Mach number distribution of the three blades at the near-stall condition at 50% speed: sample_B (

**left**), baseline (

**middle**) and sample_W (

**right**).

**Figure 19.**Mach number distribution of the three blades at the near-stall condition at 80% speed: sample_B (

**left**), baseline (

**middle**) and sample_W (

**right**).

**Figure 20.**Mach number distribution of the three blades at the near-stall condition at 100% speed: sample_B (

**left**), baseline (

**middle**) and sample_W (

**right**).

**Figure 21.**Mach number distribution of the three blades at the near-stall condition at 120% speed: sample_B (

**left**), baseline (

**middle**) and sample_W (

**right**).

**Table 1.**The mean aerodynamic performance computed using the three methods at the design and near-stall points.

Operating Point | Numerical Method | Mass Flow Rate (kg/s) | Pressure Ratio | Efficiency |
---|---|---|---|---|

the design condition | MC−CFD | 0.023428 | 1.420063 | 0.910661 |

MC−adj−linear | 0.023457 | 1.420703 | 0.912055 | |

MC−adj−nonlinear | 0.023431 | 1.420036 | 0.911103 | |

the near-stall condition | MC−CFD | 0.021610 | 1.476226 | 0.882060 |

MC−adj−linear | 0.021681 | 1.477472 | 0.884464 | |

MC−adj−nonlinear | 0.021614 | 1.476129 | 0.882412 |

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**MDPI and ACS Style**

Xu, S.; Zhang, Q.; Wang, D.; Huang, X.
Uncertainty Quantification of Compressor Map Using the Monte Carlo Approach Accelerated by an Adjoint-Based Nonlinear Method. *Aerospace* **2023**, *10*, 280.
https://doi.org/10.3390/aerospace10030280

**AMA Style**

Xu S, Zhang Q, Wang D, Huang X.
Uncertainty Quantification of Compressor Map Using the Monte Carlo Approach Accelerated by an Adjoint-Based Nonlinear Method. *Aerospace*. 2023; 10(3):280.
https://doi.org/10.3390/aerospace10030280

**Chicago/Turabian Style**

Xu, Shenren, Qian Zhang, Dingxi Wang, and Xiuquan Huang.
2023. "Uncertainty Quantification of Compressor Map Using the Monte Carlo Approach Accelerated by an Adjoint-Based Nonlinear Method" *Aerospace* 10, no. 3: 280.
https://doi.org/10.3390/aerospace10030280