# Turbulence Model Assessment in Compressible Flows around Complex Geometries with Unstructured Grids

## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

## 3. Solution Procedure

#### 3.1. Hybrid Unstructured Mesh Generation

#### 3.2. Spatial Discretization

#### 3.3. Relaxation Scheme

## 4. Solver Parallelization

## 5. Turbulence Modeling in RANS

#### 5.1. Turbulent Transport Equations

#### 5.2. Discretization of the Turbulent Transport Equations

## 6. Initial and Boundary Conditions

#### 6.1. Solid Wall Boundary

#### 6.2. Inflow and Outflow Boundaries

#### 6.3. Symmetry Boundary

#### 6.4. Engine Boundary

## 7. Results and Discussion

#### 7.1. Supersonic Flat Plate

#### 7.2. Transonic RAE 2822 Airfoil

#### 7.3. ONERA M6 Wing

#### 7.4. Generic F15 Aircraft Configuration

## 8. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Pope, S.B. Turbulent Flows; Cambridge University Press: Cambridge, UK, August 2000. [Google Scholar]
- Moin, P.; Mahesh, K. Direct Numerical Simulation: A Tool in Turbulence Research. Annu. Rev. Fluid Mech.
**1998**, 30, 539–578. [Google Scholar] [CrossRef] - Meneveau, C.; Katz, J. Scale-invariance and turbulence models for large-eddy simulation. Annu. Rev. Fluid Mech.
**2000**, 32, 1–32. [Google Scholar] [CrossRef] - Frohlich, J.; von Terzi, D. Hybrid LES/RANS methods for the simulation of turbulent flows. Prog. Aerosp. Sci.
**2008**, 44, 349–377. [Google Scholar] [CrossRef] - Catalano, P.; Amato, M. An evaluation of RANS turbulence modelling for aerodynamic applications. Aerosp. Sci. Technol.
**2003**, 7, 493–509. [Google Scholar] [CrossRef] - Knight, D.; Yan, H.; Panaras, A.G.; Zheltovodov, A. Advances in CFD prediction of shockwave turbulent boundary layer interactions. Prog. Aerosp. Sci.
**2003**, 39, 121–184. [Google Scholar] [CrossRef] - Manzari, M.T.; Hassan, O.; Morgan, K.; Weatherill, N.P. Turbulent flow computations on 3D unstructured grids. Finite Elem. Anal. Des.
**1998**, 30, 353–363. [Google Scholar] [CrossRef] - Sørensen, K.A.; Hassan, O.; Morgan, K.; Weatherill, N.P. A multigrid accelerated hybrid unstructured mesh method for 3D compressible turbulent flow. Comput. Mech.
**2003**, 31, 101–114. [Google Scholar] [CrossRef] - Peiró, J.; Peraire, J.; Morgan, K. FELISA System Reference Manual. Part 1—Basic Theory; Swansea Report C/R/821/94; University of Wales: Wales, UK, 1994. [Google Scholar]
- Peraire, J.; Morgan, K.; Peiró, J. Unstructured finite element mesh generation and adaptive procedures for CFD. In Applications of Mesh Generation to Complex 3D Configurations; AGARD: Paris, France, 1990. [Google Scholar]
- Hassan, O.; Morgan, K.; Probert, E.J.; Peraire, J. Unstructured tetrahedral mesh generation for three-dimensional viscous flows. Int. J. Numer. Methods Eng.
**1996**, 39, 549–567. [Google Scholar] [CrossRef] - Weatherill, N.P.; Hassan, O. Efficient three–dimensional Delaunay triangulation with automatic boundary point creation and imposed boundary constraints. Int. J. Numer. Methods Eng.
**1994**, 37, 2005–2039. [Google Scholar] [CrossRef] - Jameson, A.; Schmidt, W.; Turkel, E. Numerical simulation of the Euler equations by finite volume methods using Runge–Kutta timestepping schemes. In Proceedings of the 14th Fluid and Plasma Dynamics Conference, Palo Alto, CA, USA, 13–25 June 1981; AIAA Paper. pp. 81–1259. [Google Scholar]
- Harten, A.; Lax, P.D.; van Leer, B. On upstreaming differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev.
**1983**, 25, 35. [Google Scholar] [CrossRef] - Morgan, K.; Peraire, J.; Peiro, J.; Hassan, O. The computation of 3-dimensional flows using unstructured grids. Comput. Methods Appl. Mech. Eng.
**1991**, 87, 335–352. [Google Scholar] [CrossRef] - Sørensen, K.A. A Multigrid Accelerated Procedure for the Solution of Compressible Fluid Flows on Unstructured Hybrid Meshes. Ph.D. Thesis, University of Wales, Swansea, Wales, 2002. [Google Scholar]
- Wilcox, D.C. Turbulence Modeling for CFD; DWC Industries, Inc.: La Canada, CA, USA, 2006. [Google Scholar]
- Spalart, P.R.; Allmaras, S.R. A one equation turbulence model for aerodynamics flows. In Proceedings of the 30th Aerospace Science Meeting and Exhibit, Reno, NV, USA, 6–9 January 1992. AIAA Paper 92-0439. [Google Scholar]
- Menter, F.R. Review of the shear-stress transport turbulence model experience from an industrial perspective. Int. J. Comput. Fluid Dyn.
**2009**, 23, 305–316. [Google Scholar] [CrossRef] - Geuzaine, P. An Implicit Upwind Finite Volume Method for Compressible Turbulent Flows on Unstructured Meshes. Ph.D. Thesis, Université de Liège, Liège, Belgium, 1999. [Google Scholar]
- Spalart, P.R.; Rumsey, C.L. Effective Inflow Conditions for Turbulence Models in Aerodynamic Calculations. AIAA J.
**2007**, 45, 2544–2553. [Google Scholar] [CrossRef][Green Version] - White, F.M. Viscous Fluid Flow; McGraw Hill: New York, NY, USA, 1974. [Google Scholar]
- Schlichting, H. Boundary Layer Theory; McGraw Hill: New York, NY, USA, 1968. [Google Scholar]
- Araya, G.; Jansen, K. Compressibility effect on spatially-developing turbulent boundary layers via DNS. In Proceedings of the 4th Thermal and Fluids Engineering Conference (TFEC2019), Las Vegas, NV, USA, 14–17 April 2019. [Google Scholar]
- Cook, P.; McDonald, M.; Firmin, M. Aerofoil RAE2822 Pressure Distributions and Boundary Layer and Wake Measurements; Report AR-138; AGARD: Paris, France, 1979. [Google Scholar]
- Hirschel, E.H. Finite Approximations in Fluid Mechanics II: DFG Priority Research Program, Results 1986–1988; Notes on Numerical Fluid Mechanics; Vieweg: Braunschweig, Germany; Wiesbaden, Germany, 1989; Volume 25. [Google Scholar]
- Swanson, R.C.; Rossow, C.C. An efficient solver for the RANS equations and a one-equation turbulence model. Comput. Fluids
**2011**, 42, 13–25. [Google Scholar] [CrossRef] - Schmitt, V.; Charpin, F. Pressure Distributions of the ONERA M6 Wing at Transonic Mach Numbers; Report AR-138; AGARD: Paris, France, 1979. [Google Scholar]
- LeMoigne, A.; Qin, N. Variable-fidelity aerodynamic optimization for turbulent flows using a discrete adjoint formulation. AIAA J.
**2004**, 42, 1281–1192. [Google Scholar] - Nielsen, E.J.; Anderson, W.K. Recent improvements in aerodynamic sesign optimization on unstructured meshes. AIAA J.
**2002**, 40, 1155–1163. [Google Scholar] [CrossRef] - Huang, J.C.; Lin, H.; Yang, J.Y. Implicit preconditioned WENO scheme for steady viscous flow computation. J. Comput. Phys.
**2009**, 228, 420–438. [Google Scholar] [CrossRef] - Jakirlić, S.; Eisfeld, B.; Jester-Zurker, R.; Kroll, N. Near-wall, Reynolds-stress model calculations of transonic flow configurations relevant to aircraft aerodynamics. Int. J. Heat Fluid Flow
**2007**, 28, 602–615. [Google Scholar] [CrossRef] - Webb, L.; Varda, D.; Whitmore, S. Flight and Wind-Tunnel Comparisons of the Inlet/Airframe Interaction of the F-15 Airplane; Technical Paper 2374; NASA: Washington, DC, USA, 1984.

**Figure 1.**Mesh schematic (

**a**); close-up of the near wall region (

**b**); and iso-contours of turbulent kinetic energy from Menter SST (

**c**) in the supersonic flat plate.

**Figure 3.**Mean streamwise velocity (

**a**); and temperature distribution (

**b**) in the supersonic flat plate.

**Figure 6.**Distance distribution of the first off-wall point in wall units along the RAE airfoil for the Menter SST model.

**Figure 10.**Iso-surfaces of streamwise velocity over the upper surface in the ONERA M6 wing at $\alpha ={3.06}^{\circ}$ (flow from left to right).

**Figure 11.**Iso-contours of streamwise velocity in Menter SST at $y/(b/2)=0.9$ (

**a**) and zoom over the leading edge (

**b**) in the ONERA M6 wing at $\alpha ={3.06}^{\circ}$ (flow from right to left).

**Figure 13.**Iso-contours of the pressure coefficient in the F15 aircraft configuration (C

_{p}range of color contour as in Figure 14).

**Figure 15.**Variation of the pressure coefficient in the top surface along $x/{L}_{x}$ of F15 aircraft for present numerical data (open symbols) experimental data (closed symbols).

**Figure 16.**Skin friction coefficient in the upper surface of F15 at $y/{L}_{y}=0.15$ (

**a**); and zoom of the separated flow zone (

**b**).

**Figure 18.**Pressure coefficients of F15 in the upper (

**a**) and lower (

**b**) surfaces at $y/{L}_{y}=0.35$.

**Figure 19.**Pressure coefficients of F15 in the upper (

**a**) and lower (

**b**) surfaces at $y/{L}_{y}=0.59$.

SA | k-$\mathit{\omega}$ | SST | Exp. [25] | Num. [26] | Num. [27] | |
---|---|---|---|---|---|---|

${C}_{L}$ | $0.801$ | $0.811$ | $0.792$ | $0.803$ | $0.794-0.824$ | $0.824$ |

${C}_{D\phantom{\rule{3.33333pt}{0ex}}total}$ | $0.0175$ | $0.0178$ | $0.0182$ | $0.0168$ | $0.0164-0.0165$ | $0.0165$ |

${C}_{D\phantom{\rule{3.33333pt}{0ex}}friction}$ | $0.0054$ | $0.0054$ | $0.0064$ | $N/A$ | $0.0054-0.0055$ | $0.0055$ |

${C}_{m\phantom{\rule{3.33333pt}{0ex}}25\%}$ | $-0.1121$ | $-0.1144$ | $-0.1099$ | $-0.099$ | $-0.0874$ | $N/A$ |

Location | Parameter | SA | k-$\mathit{\omega}$ | SST | Exp. [25] |
---|---|---|---|---|---|

$x/c=0.65$ | ${\delta}^{*}$ | $0.004782$ | $0.005029$ | $0.005304$ | $0.004956$ |

${\theta}^{*}$ | $0.001816$ | $0.001794$ | $0.002010$ | $0.002043$ | |

H | $2.633$ | $2.803$ | $2.638$ | $2.426$ | |

$x/c=0.75$ | ${\delta}^{*}$ | $0.006571$ | $0.006630$ | $0.007276$ | $0.006884$ |

${\theta}^{*}$ | $0.003032$ | $0.003013$ | $0.003204$ | $0.002999$ | |

H | $2.167$ | $2.201$ | $2.271$ | $2.296$ |

Aerodynamic Coeff. | SA | k-$\mathit{\omega}$ | SST | Num. [29] | Num. [30] |
---|---|---|---|---|---|

${C}_{L}$ | $0.260$ | $0.262$ | $0.253$ | $0.270$ | $0.253$ |

${C}_{D\phantom{\rule{3.33333pt}{0ex}}total}$ | $0.0175$ | $0.0179$ | $0.0189$ | $0.0174$ | $0.0168$ |

${C}_{D\phantom{\rule{3.33333pt}{0ex}}friction}$ | $0.0048$ | $0.0051$ | $0.0057$ | $0.0050$ | N/A |

Aerodynamic Coeff. | Spalart–Allmaras | Wilcox k-$\mathit{\omega}$ | Menter SST |
---|---|---|---|

${C}_{L}$ | $0.181$ | $0.174$ | $0.175$ |

${C}_{D\phantom{\rule{3.33333pt}{0ex}}total}$ | $0.0397$ | $0.0374$ | $0.0398$ |

© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Araya, G.
Turbulence Model Assessment in Compressible Flows around Complex Geometries with Unstructured Grids. *Fluids* **2019**, *4*, 81.
https://doi.org/10.3390/fluids4020081

**AMA Style**

Araya G.
Turbulence Model Assessment in Compressible Flows around Complex Geometries with Unstructured Grids. *Fluids*. 2019; 4(2):81.
https://doi.org/10.3390/fluids4020081

**Chicago/Turabian Style**

Araya, Guillermo.
2019. "Turbulence Model Assessment in Compressible Flows around Complex Geometries with Unstructured Grids" *Fluids* 4, no. 2: 81.
https://doi.org/10.3390/fluids4020081