# High-Order Accurate Numerical Simulation of Supersonic Flow Using RANS and LES Guided by Turbulence Anisotropy

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Nozzle Geometry and Boundary Conditions

#### 2.2. PIV Experimental Data from Literature

#### 2.3. Governing Equations

#### 2.4. Baseline RANS

#### 2.5. Mesh Sensitivity

#### 2.6. Effect of CFD Domain Size in RANS Simulations

## 3. Results—Improved Accuracy RANS

#### 3.1. Inlet Modeling

#### 3.1.1. Turbulence Intensity

#### 3.1.2. Effect of Upstream Air Supply Duct

#### 3.2. Wall Modeling

#### 3.2.1. Prism Layer Sensitivity

#### 3.2.2. Adiabatic vs. Isothermal Walls

#### 3.3. Wall Surface Roughness

#### 3.4. Laminar to Turbulent Transition

## 4. Results—LEVM, NLEVM, RSM and LES

#### 4.1. Turbulence Capture in RANS

#### 4.1.1. Streamwise Vorticity in Linear vs. Non-Linear Eddy Viscosity RANS

#### 4.1.2. Turbulent Viscosity in RANS

#### 4.2. Comparisons with LES

#### 4.2.1. Jet Self-Similarity Assessment

#### 4.2.2. Boundary Layer Growth in RANS and LES

_{ref}is the reference temperature, P is the absolute pressure in Pa and P

_{ref}is the reference pressure. Star-CCM+ assumes the standard state temperature for an ideal gas as 298.15 K and standard state entropy as 0. Therefore, the entropy calculations are relative to the standard conditions and negative values are possible. Entropy production through boundary layer can be clearly seen from Figure 21a,b. In LES (Figure 21b), the boundary separates due to the throat shock wave just downstream of the throat. We anticipate that this is due to the near-wall grid as well as accounting of the turbulence in RANS vs. LES. In this case, LES resolves the boundary layer since the height of first prism layer falls within the viscous sublayer, leading to wall y+ ~1. Figure 21c shows a line plot at x/De = −1 comparing the boundary layer profiles from RANS and LES which clearly indicates separated boundary layer in LES downstream of the throat.

## 5. Discussion on Anisotropy

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CFD | Computational Fluid Dynamics |

DNS | Direct numerical simulation |

HPC | High performance computing |

LES | Large Eddy Simulation |

LEVM | Linear eddy viscosity model |

NLEVM | Non-linear eddy viscosity model |

PIV | Particle image velocimetry |

QCR | Quadratic constitutive relation |

TKE | Turbulent Kinetic Energy |

RANS | Reynolds Averaged Navier Stokes |

RSM | Reynolds stress model |

SST | Shear Stress Transport |

WALE | Wall adapting local eddy viscosity |

WSS | Wall shear stress |

u | Axial component of velocity |

uj | Jet centerline velocity at nozzle exit |

uc | Jet centerline velocity along jet axis |

De | Nozzle equivalent diameter |

h | Height at nozzle exit from minor axis plane view |

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**Figure 1.**Rectangular nozzle geometry taken from [60], dimensions in inches.

**Figure 2.**Centerline velocity comparison for baseline RANS with PIV data [60].

**Figure 4.**Mesh sensitivity study for jet centerline velocity compared with PIV data from [60].

**Figure 5.**Domain dependency study compared with PIV data from [60].

**Figure 6.**Effect of inflow conditions compared with PIV data from [60]. (

**a**) Effect of turbulence intensity (TI). (

**b**) Effect of upstream air supply duct.

**Figure 7.**Contours of normalized streamwise vorticity along crossflow planes inside the nozzle. (

**a**) Without the upstream supply duct. (

**b**) With upstream supply duct.

**Figure 9.**Prism layer sensitivity. (

**a**) Jet centerline velocity compared with PIV data from [60]. (

**b**) Boundary layer profiles at nozzle exit.

**Figure 10.**Effect of adiabatic and isothermal wall modeling on exit temperature profile and jet centerline velocity compared with PIV data from [60]. (

**a**) Nozzle exit temperature profile. (

**b**) Jet centerline velocity.

**Figure 12.**Effect of nozzle wall surface roughness on jet centerline velocity compared with PIV data from [60].

**Figure 13.**Effect of transition model on jet centerline velocity compared with PIV data from [60].

**Figure 14.**Streamwise vorticity contours along cross-flow planes at x/De = 1 (

**top**) and x/De = 8.9 (

**bottom**).

**Figure 17.**Jet centerline velocity comparison with PIV data from [60]—Boussinesq SST RANS, QCR, RSM and LES.

**Figure 18.**Jet centerline velocity comparison with PIV data from [60]—LES with and without upstream duct.

**Figure 21.**Boundary layer growth downstream of the nozzle throat, indicating (

**a**) attached boundary layer in RANS, (

**b**) separated boundary layer in LES, (

**c**) Line plot at x/De = −1, comparing RANS and LES.

**Figure 22.**TKE comparisons on minor- and major-axis shear layers with PIV data from [60].

Category | Parameters |
---|---|

Mesh sensitivity | Mesh refinements in nozzle and jet |

Domain dependency | Baseline domain, full experimental facility size domain |

Inlet modeling | Turbulence intensity Effect of upstream supply duct |

Wall modeling | Prism layer sensitivity Isothermal vs. adiabatic walls |

Surface roughness | Smooth vs. rough walls |

Transition modeling | Gamma transition model |

Turbulence capture | Boussinesq (linear) k-omega SST RANS, Quadratic Constitutive relation (non-linear) k-omega SST RANS Linear pressure–strain Reynolds stress model, WALE LES |

Case Abbreviation | Number of Cells (Million) | Refinement Size (m) | Total Pressure at Nozzle Exit (Normalized by P_{in}) |
---|---|---|---|

Coarse | 6 | D_{e}/25 | 0.965 |

Medium | 8 | D_{e}/30 | 0.964 |

Fine | 10 | D_{e}/35 | 0.964 |

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**MDPI and ACS Style**

Bhide, K.; Abdallah, S. High-Order Accurate Numerical Simulation of Supersonic Flow Using RANS and LES Guided by Turbulence Anisotropy. *Fluids* **2022**, *7*, 385.
https://doi.org/10.3390/fluids7120385

**AMA Style**

Bhide K, Abdallah S. High-Order Accurate Numerical Simulation of Supersonic Flow Using RANS and LES Guided by Turbulence Anisotropy. *Fluids*. 2022; 7(12):385.
https://doi.org/10.3390/fluids7120385

**Chicago/Turabian Style**

Bhide, Kalyani, and Shaaban Abdallah. 2022. "High-Order Accurate Numerical Simulation of Supersonic Flow Using RANS and LES Guided by Turbulence Anisotropy" *Fluids* 7, no. 12: 385.
https://doi.org/10.3390/fluids7120385