# Application of a Novel Weighted Essentially Non-Oscillatory Scheme for Reynolds-Averaged Stress Model and Reynolds-Averaged Stress Model/Large Eddy Simulation (RANS/LES) Coupled Simulations in Turbomachinery Flows

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## Abstract

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## Featured Application

**This study demonstrates the application of high-order WENO-ZQ schemes in aerodynamic simulations of aircraft engine components. By integrating WENO-ZQ into NUAA-Turbo2.0 and employing RANS and hybrid RANS/LES turbulence models, we conducted detailed computations and simulations of aircraft engine compressors, high-pressure turbines, and turbine film cooling. Using WENO-ZQ enhances simulation accuracy and efficiency, effectively capturing critical flow details such as turbine wakes and film cooling distribution. This work highlights the importance of advanced numerical schemes in guiding the design and analysis of modern turbomachinery.**

## Abstract

## 1. Introduction

- The computational cost is very high, and the calculation process is complex.
- The optimal (linear) weights depend on the geometry of the mesh and may become negative in some cases, meaning they lack robustness.
- The drawbacks become more pronounced with an increase in spatial dimension.

## 2. WENO-ZQ Schemes

- Choose the big central spatial stencil ${T}_{1}=\left\{{I}_{i-2},{I}_{i-1},{I}_{i},{I}_{i+1},{I}_{i+2}\right\}$ and the other two smaller stencils ${T}_{2}=\left\{{I}_{i-1},{I}_{i}\right\}$ and ${T}_{3}=\left\{{I}_{i},{I}_{i+1}\right\}$ to reconstruct the polynomials ${p}_{1}(x)$, ${p}_{2}(x)$, and ${p}_{3}(x)$. Additionally, the generalized expression for the reconstructed polynomial on nonuniform meshes provided by Shu [25] is adopted in this paper.
- Compute the smoothness indicators ${\beta}_{l},l=1,2,3$, which are obtained through a multiple of the local grid spacing and the difference in polynomial values at adjacent points:$${\beta}_{l}={\displaystyle \sum _{k=1}^{r}{\displaystyle {\int}_{{I}_{i}}{h}^{2k-1}}}{\left(\frac{{d}^{k}{p}_{l}(x)}{d{x}^{k}}\right)}^{2}dx,l=1,2,3$$
- Calculate the nonlinear weights based on the linear weights and the smoothness indicators. An adaptive formula for $\tau $ is written based on the difference between ${\beta}_{1}$, ${\beta}_{2}$, and ${\beta}_{3}$ as follows:$$\tau ={\left(\frac{\left|{\beta}_{1}-{\beta}_{2}\right|+\left|{\beta}_{1}-{\beta}_{3}\right|}{2}\right)}^{2}$$$${\omega}_{l}=\left(\frac{{\overline{\omega}}_{l}}{{\overline{\omega}}_{1}+{\overline{\omega}}_{2}+{\overline{\omega}}_{3}}\right),\text{}{\overline{\omega}}_{l}={\gamma}_{l}\left(1+\frac{\tau}{{\beta}_{l}+\epsilon}\right),\text{}l=1,2,3$$
- The final reconstruction formulation of conservative values $u(x,t)$ at the point ${x}_{i+1/2}$ of the target cell ${I}_{i}$ is given by ${\beta}_{l},l=1,2,3$.$${u}_{i+\frac{1}{2}}^{L}={\omega}_{1}\left(\frac{1}{{\gamma}_{1}}{p}_{1}(x)-\frac{{\gamma}_{2}}{{\gamma}_{1}}{p}_{2}(x)-\frac{{\gamma}_{3}}{{\gamma}_{1}}{p}_{3}(x)\right)+{\omega}_{2}{p}_{2}(x)+{\omega}_{3}{p}_{3}(x)$$

- Choose the big spatial stencil ${T}_{1}=\left\{{I}_{i-2},{I}_{i-1},{I}_{i},{I}_{i+1},{I}_{i+2}\right\}$ and the other three equidistant stencils ${T}_{1}^{\mathrm{js}}=\left\{{I}_{i-2},{I}_{i-1},{I}_{i}\right\}$,${T}_{2}^{\mathrm{js}}=\left\{{I}_{i-1},{I}_{i},{I}_{i+1}\right\}$, and ${T}_{3}^{\mathrm{js}}=\left\{{I}_{i},{I}_{i+1},{I}_{i+1}\right\}$ to reconstruct the polynomials ${p}_{1},{p}_{1}^{js},{p}_{2}^{js}\mathrm{and}{p}_{3}^{js}$.
- Compute the linear weights based on the polynomials ${p}_{1},{p}_{1}^{js},{p}_{2}^{js}\mathrm{and}{p}_{3}^{js}$ as follows:$${p}_{1}={\gamma}_{1}^{js}{p}_{1}^{js}+{\gamma}_{2}^{js}{p}_{2}^{js}+{\gamma}_{3}^{js}{p}_{3}^{js},$$
- Compute the smoothness indicators ${\beta}_{l},l=1,2,3$ based on Formula (4):
- Calculate the nonlinear weights based on the linear weights and the smoothness indicators.$${\alpha}_{l}=\frac{{\gamma}_{l}^{js}}{{\left(\epsilon +{\beta}_{l}\right)}^{2}},{\overline{\omega}}_{l}^{js}=\frac{{\alpha}_{k}}{{\displaystyle \sum _{l=0}^{r-1}{\alpha}_{l}}},l=1,2,3$$
- The final reconstruction formulation of conservative values $u(x,t)$ at the point ${x}_{i+1/2}$ of the target cell ${I}_{i}$ is given by$${u}_{i+\frac{1}{2}}^{L}={\overline{\omega}}_{1}^{js}{p}_{1}^{js}+{\overline{\omega}}_{2}^{js}{p}_{2}^{js}+{\overline{\omega}}_{3}^{js}{p}_{3}^{js}.$$

## 3. Classic Numerical Scheme Test Cases

#### 3.1. Solving the One-Dimensional Euler Equations under Riemann Initial Conditions

#### 3.2. Numerical Simulation of the Double Mach Reflection Problem

#### 3.3. Numerical Simulation of Rayleigh–Taylor Instability (RT Problem)

- Left and right boundaries: anti-symmetric boundary conditions.
- Bottom boundary: $(\rho ,u,v,p)=(2,0,0,1)$.
- Top boundary: $(\rho ,u,v,p)=(1,0,0,2.5)$.

## 4. Application Examples for Turbomachinery

#### 4.1. RANS

#### 4.1.1. NASA Stage 35

#### 4.1.2. Pratt and Whitney Energy-Efficient Engine High-Pressure Turbine

#### 4.2. Hybrid RANS/LES Model Menter SST-SAS

#### 4.2.1. Numerical Calculation of LS89

#### 4.2.2. Simulation of Film Cooling Based on C3X Cascade

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The results of solving the Lax problem using different numerical schemes. (

**a**) Full range; (

**b**) interrupted position.

**Figure 2.**The flow fields of double shock wave reflections computed using different improved finite-volume schemes: (

**a**) 3rd-order; (

**b**) 4th-order; (

**c**) 5th-order; (

**d**) 6th-order.

**Figure 3.**Results of different numerical formats for solving RT problems. (

**a**) WENO4; (

**b**) WENO6; (

**c**) WENO8; (

**d**) WENO-ZQ.

**Figure 12.**Comparison of numerical Schlieren calculation results under different WENO schemes. (

**a**) WENO_ZQ; (

**b**) WENO_JS.

**Figure 15.**Contour of average vorticity Ω and average strain rate S. (

**a**) Mean vorticity Ω; (

**b**) mean strain rate S.

Parameters | Values |
---|---|

Rotor rpm at 100% speed | 17,188.7 rpm |

Rotor aspect ratio | 1.19 |

Stator aspect ratio | 1.26 |

Number of rotor blades | 36 |

Number of stator blades | 46 |

Mass flow rate | 20.2 kg/s |

Total pressure ratio | 1.8 |

Parameters | Values |
---|---|

Total inlet temperature | 431.17 K |

Total inlet pressure | 375.56 Kpa |

Speed | 9789 rpm |

Total pressure ratio | 4.12 |

Efficiency | 90.8% |

Correct rotor clearance | 0.3683 mm |

Parameters | Values |
---|---|

Chord length (mm) | 67.647 |

Axial chord length (mm) | 36.985 |

Pitch-to-chord ratio (-) | 0.850 |

Throat-to-chord ratio (-) | 0.2207 |

Flow inlet angle (degree) | 0 |

Stagger angle (degree) | 55.0 |

Trailing-edge diameter (mm) | 1.42 |

Flow Conditions | ${\mathit{T}}_{\mathbf{inl}}^{*}(\mathit{K})$ | ${\mathit{P}}_{\mathbf{inl}}^{*}(\mathit{P}\mathit{a})$ | ${\mathit{\alpha}}_{\mathbf{inl}}(\mathit{degree})$ | ${\mathit{P}}_{\mathbf{out}}(\mathit{P}\mathit{a})$ | $\mathit{M}{\mathit{a}}_{\mathbf{ex}}$ |
---|---|---|---|---|---|

MUR129 | 409.2 | 184,900 | 0 | 116,500 | 0.84 |

MUR235 | 413.3 | 182,800 | 0 | 104,900 | 0.927 |

Parameters | Values |
---|---|

Chord length (mm) | 144.93 |

Axial chord length (mm) | 78.16 |

Pitch-to-chord ratio (-) | 0.812 |

Throat-to-chord ratio (-) | 0.227 |

Flow inlet angle (degree) | 0 |

Stagger angle (degree) | 55.47 |

Exit flow angle (degree) | 72.4 |

$\mathit{M}{\mathit{a}}_{\mathbf{in}}$ | ${\mathit{T}}_{\mathbf{inl}}^{\mathbf{*}}(\mathit{K})$ | ${\mathit{P}}_{\mathbf{inl}}^{\mathbf{*}}(\mathit{P}\mathit{a})$ | $\mathit{M}{\mathit{a}}_{\mathbf{ex}}$ | ${\mathit{P}}_{\mathbf{out}}(\mathit{P}\mathit{a})$ | ${\mathit{T}}_{\mathbf{cool}}(\mathit{K})$ | VR | DR |
---|---|---|---|---|---|---|---|

0.0793 | 297.34 | 97262 | 0.2702 | 92451 | 317.39 | 0.49 | 0.94 |

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**MDPI and ACS Style**

Wang, H.; Zhong, D.; Zhang, S.; Wu, X.; Ge, N.
Application of a Novel Weighted Essentially Non-Oscillatory Scheme for Reynolds-Averaged Stress Model and Reynolds-Averaged Stress Model/Large Eddy Simulation (RANS/LES) Coupled Simulations in Turbomachinery Flows. *Appl. Sci.* **2024**, *14*, 5085.
https://doi.org/10.3390/app14125085

**AMA Style**

Wang H, Zhong D, Zhang S, Wu X, Ge N.
Application of a Novel Weighted Essentially Non-Oscillatory Scheme for Reynolds-Averaged Stress Model and Reynolds-Averaged Stress Model/Large Eddy Simulation (RANS/LES) Coupled Simulations in Turbomachinery Flows. *Applied Sciences*. 2024; 14(12):5085.
https://doi.org/10.3390/app14125085

**Chicago/Turabian Style**

Wang, Hao, Dongdong Zhong, Shuo Zhang, Xingshuang Wu, and Ning Ge.
2024. "Application of a Novel Weighted Essentially Non-Oscillatory Scheme for Reynolds-Averaged Stress Model and Reynolds-Averaged Stress Model/Large Eddy Simulation (RANS/LES) Coupled Simulations in Turbomachinery Flows" *Applied Sciences* 14, no. 12: 5085.
https://doi.org/10.3390/app14125085