# Influence of Fluid–Thermal–Structural Interaction on Boundary Layer Flow in Rectangular Supersonic Nozzles

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Governing Equations & Computational Methodology

_{i}, i = 1, 2, 3 are the components of the position vector x, t is the time, ${\mathsf{\sigma}}_{ij}^{s}$ is the Cauchy stress tensor, f

_{i}are the body forces, ${\mathsf{\rho}}_{\mathrm{s}}$ is the mass density, u

_{i}is the displacement vector, ${\mathsf{\u03f5}}_{\mathrm{ij}}$ is the strain, ${\mathsf{\u03f5}}^{th}$ is the thermal part of the strain tensor and

**C**is the elasticity tensor.

_{ijkl}^{−4}order of residual of the continuity equation, depending on the CFD mesh size of approximately 4 million cells.

#### 2.2. Grid Resolution Study

_{e}≈ 0.82 and not at the nozzle exit i.e., x/D

_{e}= 0. The error in the experimental measurements is ±1%. Figure 2 shows that the error between experimental data and simulated data is within 5% until 10 D

_{e}, which is calculated using the following equation:

#### 2.3. Turbulence Model Study

_{e}to 20 D

_{e}. k-omega SST is known for its better performance under adverse pressure gradients in boundary layers [11], hence it is used for all simulations.

#### 2.4. Improved Nozzle Design

#### 2.5. Fluid Thermal Structural Interaction

#### 2.5.1. Challenges in FTSI

_{in}–T

_{ex}) in fluid flow through the existing nozzle design is approximately 92 K. Hence, heat transfer due to conduction is pronounced and not as trivial. Therefore, FTSI is accurate from a physics perspective since it accounts for the conductive heat transfer in the nozzle. Convective heat transfer is not considered in the thermal analysis, as conduction is the main contributor to thermal loads. Moreover, imported pressure loads represent the net force due to shear and normal pressure [18], hence using appropriate y+ is essential.

#### 2.5.2. FTSI Workflow

_{e}downstream of the nozzle exit. 15 prism layers with tetrahedral unstructured mesh are used to ensure y+ < 5. The mass-flow rate at the inlet and exit planes is monitored to ensure convergence. The nozzle inlet is modeled as a stagnation inlet, the nozzle walls are no-slip adiabatic walls and the domain is modeled as a free stream with atmospheric pressure and temperature, which corresponds to 101,325 Pa and 300 K.

## 3. Results

#### 3.1. Structural Deformation and Boundary Layer Thickness

#### 3.2. Existing and Improved Design

#### 3.3. Multiphysics Modeling as a Prelude to Prototype

## 4. Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

CAD | Computer-Aided Design |

CAE | Computer-Aided Engineering |

CFD | Computational Fluid Dynamics |

FTSI | Fluid Thermal Structural Interaction |

LES | Large Eddy Simulation |

NPR | Nozzle Pressure Ratio |

RANS | Reynolds Averaged Navier Stokes |

SBLI | Shock wave Boundary Layer Interaction |

SST | Shear Stress Transport |

TR | Temperature Ratio |

y+ | Wall y plus |

## Nomenclature

δ | Boundary layer thickness |

Δ | Structural deformation |

e | Error |

D_{e} | Nozzle equivalent diameter |

$\dot{\mathrm{m}}$ | Mass flow rate |

Pa | Pascal |

T_{in} | Static temperature at nozzle inlet |

T_{ex} | Static temperature at nozzle exit |

V_{exp} | Experimentally measured velocity |

V_{sim} | Numerically calculated velocity |

x_{in} | Nozzle wall X coordinate at inlet |

x_{ex} | Nozzle wall X coordinate at exit |

x_{th} | Nozzle wall X coordinate at throat |

y_{in} | Nozzle wall Y coordinate at inlet |

y_{ex} | Nozzle wall Y coordinate at exit |

y_{th} | Nozzle wall Y coordinate at throat |

θ_{in} | Nozzle wall inlet angle with X axis |

θ_{ex} | Nozzle wall exit angle with X axis |

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**Figure 1.**AR 2 rectangular nozzle geometry (all dimensions are in mm) taken from Viswanath et al. [2].

**Figure 11.**Total deformation (exaggerated 3× for visualization) and thermal strain in improved design.

Units | Coarse | Medium | Fine | |
---|---|---|---|---|

Cell count | Million | 4.56 | 9.56 | 12 |

Base size | mm | 50 | 50 | 50 |

Refinement cell size | mm | 0.5 | 0.35 | 0.3 |

Throat Pt | (Pa) | 370,884 | 370,887 | 370,805 |

Mach | 0.8834 | 0.8832 | 0.8859 | |

$\dot{\text{}\mathrm{m}}$ | (kg/s) | 0.2398 | 0.2397 | 0.2394 |

Exit Pt | (Pa) | 363,594 | 363,707 | 363,437 |

Mach | 1.4857 | 1.4865 | 1.4894 | |

$\dot{\text{}\mathrm{m}}$ | (kg/s) | 0.241 | 0.241 | 0.2404 |

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**MDPI and ACS Style**

Bhide, K.; Siddappaji, K.; Abdallah, S. Influence of Fluid–Thermal–Structural Interaction on Boundary Layer Flow in Rectangular Supersonic Nozzles. *Aerospace* **2018**, *5*, 33.
https://doi.org/10.3390/aerospace5020033

**AMA Style**

Bhide K, Siddappaji K, Abdallah S. Influence of Fluid–Thermal–Structural Interaction on Boundary Layer Flow in Rectangular Supersonic Nozzles. *Aerospace*. 2018; 5(2):33.
https://doi.org/10.3390/aerospace5020033

**Chicago/Turabian Style**

Bhide, Kalyani, Kiran Siddappaji, and Shaaban Abdallah. 2018. "Influence of Fluid–Thermal–Structural Interaction on Boundary Layer Flow in Rectangular Supersonic Nozzles" *Aerospace* 5, no. 2: 33.
https://doi.org/10.3390/aerospace5020033