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Lift on a 2D Series of Flat Plates

The following example describes the RANS CFD results for a series of 2D flat plates. The plates were created by splitting a 2D cylinder in half and inserting sections of various lengths between the cylinder halfs. The plate thickness, and thus cylinder diameter, is based on the approximate trailing edge thickness of a NACA 0012 airfoil. The freestream is at Mach 0.1 and the angle of attack is 1 degree. The Reynolds number is based on a value of 3e6 for the original NACA 0012 chord length. The overall case objective is to try to understand the Kutta condition, one of the assumptions required for the generation of lift for an airfoil, a little more.

The plate thickness, non-dimensionalized by the NACA 0012 chord, is equal to 0.00252. Therefore, the Reynolds number based on the diameter is 7560. Ten inserts were added. The lengths, non-dimensionalized by the NACA 0012 chord, were 0.0025, 0.005, 0.01, 0.02, 0.04, 0.08, 0.16 0.32, 0.64, and 0.99748 The first four geometries appear as follows. For the CFD runs, the Spalart-Allmaras (SA) turbulence model was used. The number of points around the leading edge semi circle was 301 and the trailing edge semi-circle had the same amount. The number of points along the length of each insert were respectively: 101, 201, 201, 301, 401, 601, 801, 1001, 1001, and 1001. The number of points towards the outer boundary was 401 for all cases. The outer boundary was 150 NACA 0012 chords away from the surface. The first off body grid spacing was 2.4e-6 and represents a y+ of a little less than one based on the plate thickness. The code, Aero Troll CFD, is a 2nd order spatial discretized central difference method with artificial dissipation. The following table shows the results. The reference length for cl is the chord of each geometry. For example, the reference chord for the first insert, 0.0025, was 0.00502. The cl was calculated with the pressure only, skin friction is not included. All results are fully converged.Insert Length | cl | Max Mach |

0.00250 | 0.091964 | 0.155 |

0.00500 | 0.102742 | 0.154 |

0.01000 | 0.108749 | 0.154 |

0.02000 | 0.111401 | 0.154 |

0.04000 | 0.111867 | 0.155 |

0.08000 | 0.111353 | 0.159 |

0.16000 | 0.110546 | 0.162 |

0.32000 | 0.109828 | 0.165 |

0.64000 | 0.109356 | 0.164 |

0.99748 | 0.109173 | 0.162 |

0.005 Case | Slip | κ(4) | cl | cd | Max Mach |

RANS | N | 0.04 | 0.102742 | 0.118787 | 0.154 |

RANS | Y | 0.04 | 0.051508 | 0.000916 | 0.169 |

Laminar | Y | 0.04 | 0.052769 | 0.000774 | 0.169 |

Euler | Y | 0.04 | 0.064271 | 0.000375 | 0.171 |

Euler | Y | 0.02 | 0.063937 | 0.000275 | 0.171 |

Euler | Y | 0.01 | 0.063218 | 0.000212 | 0.171 |

References

1) Pulliam, T.H., "A Computational Challenge : Euler Solution for Ellipses," AIAA J., Vol. 28,No. 10, Oct. 1990.