# Numerical Simulations on Unsteady Nonlinear Transonic Airfoil Flow

## Abstract

**:**

## 1. Introduction

## 2. Flow Solver: DLR TAU-Code

## 3. Numerical Mesh for the RAE2882 Airfoil

## 4. Test Cases and Numerical Settings

## 5. Results

#### 5.1. Steady CFD Results

#### 5.2. Unsteady Nonlinear Aerodynamic Responses

#### 5.2.1. Definition of Variables

#### 5.2.2. Time-Domain Representation

#### 5.2.3. Frequency Content

#### 5.2.4. Maximum Lift Coefficient

#### 5.2.5. Unsteady Shock Motion

#### 5.2.6. Examples of Instantaneous Flow Fields

#### 5.2.7. Influence of the Turbulence Model

#### 5.3. Possible Indicators for the Assessment of Nonlinear Responses

#### 5.3.1. Definition of Variables

#### 5.3.2. Harmonic Distortion

#### 5.3.3. Maximum Shock Motion

## 6. Conclusions and Outlook

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CFD | Computational Fluid Dynamics |

DFT | Discrete Fourier Transform |

DLM | Doublet-Lattice Method |

FRF | Frequency Response Function |

LFD | Linear Frequency-Domain |

RANS | Reynolds-Averaged Navier-Stokes |

ROM | Reduced-Order Model |

SA | Spalart-Allmaras |

SST | Shear-Stress Transport |

TE | Trailing Edge |

URANS | Unsteady Reynolds-Averaged Navier-Stokes |

## Appendix A. Numerical Sensitivities

#### Appendix A.1. Grid Sensitivity Study

**Figure A1.**Comparison of three CFD meshes at Mach 0.70. (

**a**) First harmonic of the lift FRF, (

**b**) Maximum lift.

#### Appendix A.2. Time Step Study

**Figure A2.**Comparison of different time step sizes at Mach 0.70. (

**a**) ${L}_{G}=125.5$ m (k = 0.05), (

**b**) ${L}_{G}=2.5$ m (k = 2.5).

## Appendix B. Verification of Linearity for Low-Amplitude Results

**Figure A3.**Verification of linearity for low-amplitude results at Mach 0.70. (

**a**) First harmonic of the lift, (

**b**) Harmonic deviation and and harmonic distortion of the lift for the two lowest excitation amplitudes.

## References

- Brink-Spalink, J.; Bruns, J.M. Correction of Unsteady Aerodynamic Influence Coefficients using Experimental or CFD Data; IFASD-2001-034. In Proceedings of the International Forum on Aeroelasticity and Structural Dynamics, Madrid, Spain, 5–7 June 2001. [Google Scholar]
- Karpel, M.; Moulin, B.; Chen, P.C. Dynamic Response of Aeroservoelastic Systems to Gust Excitation. J. Aircr.
**2005**, 42, 1264–1272. [Google Scholar] [CrossRef] - Weigold, W.; Stickan, B.; Travieso-Alvarez, I.; Kaiser, C.; Teufel, P. Linearized Unsteady CFD for Gust Loads with TAU. In Proceedings of the International Forum on Aeroelasticity and Structural Dynamics, IFASD-2017-187, Como, Italy, 25–28 June 2017. [Google Scholar]
- Albano, E.; Rodden, W.P. A doublet-lattice method for calculating lift distributions on oscillating surfaces in subsonic flows. AIAA J.
**1969**, 7, 279–285. [Google Scholar] [CrossRef] - Giesing, J.P.; Kalman, T.P.; Rodden, W.P. Correction Factor Techniques for Improving Aerodynamic Prediction Methods; NACA-CR-144967; Technical Report; McDonnell Douglas Corporation: St. Louis, MA, USA; NASA: Washington, DC, USA, 1976.
- Palacios, R.; Climent, H.; Karlsson, A.; Winzell, B. Assessment of Strategies for Correcting Linear Unsteady Aerodynamics Using CFD or Experimental Results. In Progress in Computational Flow-Structure Interaction; Haase, W., Selmin, V., Eds.; Springer: Berlin/Heidelberg, Germany, 2003; pp. 209–224. [Google Scholar]
- Thormann, R.; Dimitrov, D. Correction of aerodynamic influence matrices for transonic flow. CEAS Aeronaut. J.
**2014**, 5, 435–446. [Google Scholar] [CrossRef] - Banavara, N.K.; Dimitrov, D. Prediction of Transonic Flutter Behavior of a Supercritical Airfoil Using Reduced Order Methods. In New Results in Numerical and Experimental Fluid Mechanics IX; Notes on Numerical Fluid Mechanics and Multidisciplinary Design; Springer International Publishing: Berlin/Heidelberg, Germany, 2014; Volume 124, pp. 365–373. [Google Scholar] [CrossRef] [Green Version]
- Quero-Martin, D. An Aeroelastic Reduced Order Model for Dynamic Response Prediction to Gust Encounters. Ph.D. Thesis, Institute of Aeroelasticity, Berlin, Germany, 2017. [Google Scholar] [CrossRef]
- Katzenmeier, L.; Vidy, C.; Breitsamter, C. Correction Technique for Quality Improvement of Doublet Lattice Unsteady Loads by Introducing CFD Small Disturbance Aerodynamics. J. Aeroelasticity Struct. Dyn.
**2017**, 5, 17–40. [Google Scholar] [CrossRef] - Bekemeyer, P.; Thormann, R.; Timme, S. Frequency-Domain Gust Response Simulation Using Computational Fluid Dynamics. AIAA J.
**2017**, 55, 2174–2185. [Google Scholar] [CrossRef] - Bekemeyer, P.; Ripepi, M.; Heinrich, R.; Görtz, S. Nonlinear Unsteady Reduced-Order Modeling for Gust-Load Predictions. AIAA J.
**2019**, 57, 1839–1850. [Google Scholar] [CrossRef] - Halder, R.; Damodaran, M.; Khoo, B.C. Deep Learning Based Reduced Order Model for Airfoil-Gust and Aeroelastic Interaction. AIAA J.
**2020**, 58, 4304–4321. [Google Scholar] [CrossRef] - Lucia, D.J.; Beran, P.S.; Silva, W.A. Reduced-order modeling: New approaches for computational physics. Prog. Aerosp. Sci.
**2004**, 40, 51–117. [Google Scholar] [CrossRef] [Green Version] - Hall, K.C.; Thomas, J.P.; Clark, W.S. Computation of Unsteady Nonlinear Flows in Cascades Using a Harmonic Balance Technique. AIAA J.
**2002**, 40, 879–886. [Google Scholar] [CrossRef] [Green Version] - Leishman, J.G.; Beddoes, T.S. A Generalised Model for Airfoil Unsteady Aerodynamic Behaviour and Dynamic Stall Using the Indicial Method. In Proceedings of the American Helicopter Society (AHS) Annual Forum, New York, NY, USA, 2–5 June 1986. [Google Scholar]
- Reddy, T.S.R.; Kaza, K.R.V. Comparative Study of Some Dynamic Stall Models; Technical Report; NASA: Washington, DC, USA, 1987.
- Goman, M.; Khrabrov, A. State-space representation of aerodynamic characteristics of an aircraft at high angles of attack. J. Aircr.
**1994**, 31, 1109–1115. [Google Scholar] [CrossRef] - Dowell, E.H.; Williams, M.H.; Bland, S.R. Linear/Nonlinear Behavior in Unsteady Transonic Aerodynamics. AIAA J.
**1983**, 21, 38–46. [Google Scholar] [CrossRef] - Mallik, W.; Raveh, D.E. Kriging-Based Aeroelastic Gust Response Analysis at High Angles of Attack. AIAA J.
**2020**, 58, 3777–3787. [Google Scholar] [CrossRef] - Kaiser, C.; Quero, D.; Nitzsche, J. Quantification of Nonlinear Effects in Gust Load Prediction. In Proceedings of the International Forum on Aeroelasticity and Structural Dynamics, Savannah, GA, USA, 10–13 June 2019. [Google Scholar]
- Mallik, W.; Raveh, D.E. Gust Response at High Angles of Attack. AIAA J.
**2019**, 57, 3250–3260. [Google Scholar] [CrossRef] - Friedewald, D.; Thormann, R.; Kaiser, C.; Nitzsche, J. Quasi-steady doublet-lattice correction for aerodynamic gust response prediction in attached and separated transonic flow. CEAS Aeronaut. J.
**2017**, 9, 53–66. [Google Scholar] [CrossRef] - European Aviation Safety Agency (EASA). Certification Specifications and Acceptable Means of Compliance for Large Aeroplanes CS-25, Amendment 12; Technical Report; European Aviation Safety Agency (EASA): Cologne, Germany, 2012.
- Gerhold, T.; Galle, M.; Friedrich, O.; Evans, J. Calculation of complex three-dimensional configurations employing the DLR-TAU-code. In Proceedings of the 35th Aerospace Sciences Meeting and Exhibit, San Diego, CA, USA, 5–9 January 1997. [Google Scholar] [CrossRef]
- Schwamborn, D.; Gerhold, T.; Heinrich, R. The DLR TAU-Code: Recent Applications in Research and Industry. In Proceedings of the European Conference on Computational Fluid Dynamics (ECCOMAS), Egmond aan Zee, The Netherlands, 5–8 September 2006. [Google Scholar]
- Jameson, A.; Schmidt, W.; Turkel, E. Numerical Solutions of the Euler Equations by the Finite Volume Methods Using Runge Kutta Time Stepping Schemes. In Proceedings of the 14th Fluid and Plasma Dynamics Conference, AIAA 81–1259, Palo Alto, CA, USA, 23–25 June 1981. [Google Scholar] [CrossRef] [Green Version]
- Jameson, A. Time Dependent Calculations using Multigrid, with Applications to Unsteady Flows past Airfoils and Wings. In Proceedings of the 10th Computational Fluid Dynamics Conference, AlAA 91-1596, Honolulu, HI, USA, 24–26 June 1991. [Google Scholar] [CrossRef]
- Heinrich, R. Simulation of Interaction of Aircraft and Gust Using the TAU-Code. In New Results in Numerical and Experimental Fluid Mechanics IX; Notes on Numerical Fluid Mechanics and Multidisciplinary Design; Springer International Publishing: Berlin/Heidelberg, Germany, 2014; Volume 124, pp. 503–511. [Google Scholar] [CrossRef]
- Spalart, P.; Allmaras, S. A One-Equation Turbulence Model for Aerodynamic Flows. In Proceedings of the 30th Aerospace Sciences Meeting and Exhibit, AIAA-92-0439, Reno, NV, USA, 6–9 January 1992. [Google Scholar] [CrossRef]
- Deutsches Zentrum für Luft- und Raumfahrt (DLR). TAU-Code User Guide 2016.1.0; Technical Report; Deutsches Zentrum für Luft- und Raumfahrt (DLR): Braunschweig, Germany, 2016. [Google Scholar]
- Giannelis, N.F.; Vio, G.A.; Levinski, O. A review of recent developments in the understanding of transonic shock buffet. Prog. Aerosp. Sci.
**2017**, 92, 39–84. [Google Scholar] [CrossRef] - Menter, F.R. Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications. AIAA J.
**1994**, 8, 1598–1605. [Google Scholar] [CrossRef] [Green Version] - Cook, P.H.; McDonald, M.A.; Firmin, M.C.P. Aerofoil RAE 2822: Pressure Distributions, and Boundary Layer and Wake Measurements; Technical Report AGARD-AR-138, AGARD; North Atlantic Treaty Organization: Brussels, Belgium, 1979. [Google Scholar]
- Bardina, J.E.; Huang, P.G.; Coakley, T.J. Turbulence Modeling Validation, Testing and Development; NASA Technical Memorandum 110446; Ames Research Center: Moffett Field, CA, USA, 1997. [Google Scholar]
- Hellström, T.; Davidson, L.; Rizzi, A. Reynolds Stress Transport Modelling of Transonic Flow around the RAE2822 Airfoil. In 32nd Aerospace Sciences Meeting and Exhibit; American Institute of Aeronautics and Astronautics: Reno, NV, USA, 1994. [Google Scholar] [CrossRef]
- Knopp, T. Validation of the Turbulence Models in the DLR TAU Code for Transonic Flows—A Best Practice Guide; Forschungsbericht 2006-01; Institut für Aerodynamik und Strömungsmechanik: Göttingen, Germany, 2006. [Google Scholar]
- McCroskey, W.J.; McAlister, K.W.; Carr, L.W.; Pucci, S.L.; Lambert, O.; Indergrand, R.F. Dynamic Stall on Advanced Airfoil Sections. J. Am. Helicopter Soc.
**1981**, 26, 40–50. [Google Scholar] [CrossRef] - Iovnovich, M.; Raveh, D. Transonic Unsteady Aerodynamics in the Vicinity of Shock-Buffet Instability. J. Fluids Struct.
**2012**, 29, 131–142. [Google Scholar] [CrossRef] - Thormann, R.; Timme, S. Application of Harmonic Balance Method for Non-linear Gust Responses. In AIAA SciTech Forum; American Institute of Aeronautics and Astronautics: Reston, VA, USA, 2018. [Google Scholar] [CrossRef]
- Ericsson, L.; Reding, J. Dynamic stall overshoot of static airfoil characteristics. In 12th Atmospheric Flight Mechanics Conference; American Institute of Aeronautics and Astronautics: Snowmass, CO, USA, 1985. [Google Scholar] [CrossRef]
- Harper, P.W.; Flanigan, R.E. The Effect of Rate of Change of Angle of Attack in the Maximum Lift of a Small Model; Technical Report NACA Technical Note 2061; NACA TN 2061; NACA: Alabama Birmingham, AL, USA, 1950. [Google Scholar]
- Dimitrov, D. Unsteady aerodynamics of wings with an oscillating flap in transonic flow. In Proceedings of the 8th PEGASUS-AIAA Student Conference, Poitiers, France, 11–13 April 2012. [Google Scholar]
- Shmilovitz, D. On the definition of total harmonic distortion and its effect on measurement interpretation. IEEE Trans. Power Deliv.
**2005**, 20, 526–528. [Google Scholar] [CrossRef] - Davis, S.; Malcolm, G. Unsteady Aerodynamics of Conventional and Supercritical Airfoils. In Structures, Structural Dynamics, and Materials and Co-Located Conferences; American Institute of Aeronautics and Astronautics: Reston, VA, USA, 1980. [Google Scholar] [CrossRef]
- Thormann, R.; Widhalm, M. Linear-Frequency-Domain Predictions of Dynamic-Response Data for Viscous Transonic Flows. AIAA J.
**2013**, 51, 2540–2557. [Google Scholar] [CrossRef] - Kaiser, C.; Thormann, R.; Dimitrov, D.; Nitzsche, J. Time-Linearized Analysis of Motion-Induced and Gust-Induced Airloads with the DLR TAU Code; Deutscher Luft-und Raumfahrtkongress: Rostock, Germany, 2015. [Google Scholar]

**Figure 1.**Hybrid CFD mesh for the RAE2822 airfoil with about 240.000 nodes. (

**a**) Far-field mesh, (

**b**) Quads in the near-field and the wake, (

**c**) Close-up of the near-field mesh.

**Figure 4.**Local coefficients for different Mach numbers at 3 deg. (

**a**) Pressure coefficients, (

**b**) Skin friction coefficients.

**Figure 6.**Global coefficients for two gust lengths and various amplitudes at Mach 0.70. (

**a**) ${L}_{G}=125.5$ m (k = 0.05), (

**b**) ${L}_{G}=31.5$ m (k = 0.20), (

**c**) ${L}_{G}=125.5$ m (k = 0.05), (

**d**) ${L}_{G}=31.5$ m (k = 0.20).

**Figure 7.**Normalized frequency content of the lift coefficient. (

**a**) ${L}_{G}=125.5$ m (k = 0.05), (

**b**) ${L}_{G}=31.5$ m (k = 0.20).

**Figure 8.**First harmonic of lift and moment frequency response functions (FRFs) for all simulated excitations at Mach 0.70. (

**a**) Magnitude of lift first harmonic, (

**b**) Magnitude of moment first harmonic, (

**c**) Phase of lift first harmonic, (

**d**) Phase of moment first harmonic.

**Figure 10.**Maximum lift during one period of excitation at Mach 0.70, compared to $\Delta {C}_{L}^{Max,Lin}$.

**Figure 11.**Maximum lift increment during one period of excitation. (

**a**) Mach 0.66, (

**b**) Mach 0.68, (

**c**) Mach 0.72.

**Figure 12.**Shock motion for two gust lengths and various amplitudes at Mach 0.70. (

**a**) ${L}_{G}=125.5$ m (k = 0.05), (

**b**) ${L}_{G}=31.5$ m (k = 0.20).

**Figure 13.**Minimum and maximum x-coordinates for the shock position during one period of excitation at Mach 0.70.

**Figure 14.**Maximum shock motion during one period of excitation. (

**a**) Mach 0.66, (

**b**) Mach 0.68, (

**c**) Mach 0.72.

**Figure 15.**(

**a**–

**g**): Snapshots of Mach number contours including velocity vectors. Additionally, the yellow line in the field plots shows the contour for x-velocity = 0. (

**h**): Lift, shock motion and normalized gust velocity ${\underset{W}{~}}_{G}=0.5({W}_{G}/{\underset{W}{^}}_{G}+1)$ during one period of excitation.

**Figure 16.**Global variables in comparison between Spalart-Allmaras (solid lines) and Menter SST (symbols) at Mach 0.70. (

**a**) First harmonic of the lift FRF, (

**b**) Maximum lift.

**Figure 17.**Shock motion in comparison between Spalart-Allmaras (solid lines) and Menter SST (symbols) at Mach 0.70, (

**a**) Minimum and maximum x-coordinates for the shock position, (

**b**) Maximum shock motion during one period of excitation.

**Figure 18.**Harmonic distortion for Mach 0.70, (

**a**) ${\mathrm{HD}}_{CL}$ (Note: lines for n = 50 and n = 20 coincide with n = 10 and are not displayed), (

**b**) ${\mathrm{HD}}_{CMy}$ (Note: the line for n = 50 coincides with n = 20 and is not displayed).

**Figure 19.**Harmonic deviation $\delta {C}_{L}^{1}$ as a function of the harmonic distortion of the lift ${\mathrm{HD}}_{CL}$, which is computed using 10 harmonics. (

**a**) Mach 0.70, (

**b**) ${\widehat{W}}_{G}=15$ m/s.

**Figure 20.**Relative maximum lift $\delta {C}_{L}^{Max}$ as a function of the harmonic distortion of the lift ${\mathrm{HD}}_{CL}$, which is computed using 10 harmonics. (

**a**) Mach 0.70, (

**b**) ${\widehat{W}}_{G}=15$ m/s.

**Figure 21.**Harmonic deviation of lift and moment in comparison to the maximum shock motion at Mach 0.70. (grey dots denote the computed test cases). (

**a**) $\delta {C}_{L}^{1}$, (

**b**) $\delta {C}_{My}^{1}$.

**Figure 22.**Harmonic deviation of the moment in comparison to the maximum shock motion. (

**a**) Mach 0.66, (

**b**) Mach 0.68, (

**c**) Mach 0.72.

Mach | ${\mathit{U}}_{\mathit{\infty}}$ [m/s] | dt [s] | Inner Iterations |
---|---|---|---|

0.66 | 218.65 | $1.91\times {10}^{-4}$ | 400 |

0.68 | 225.28 | $1.85\times {10}^{-4}$ | 400 |

0.70 | 231.90 | $1.80\times {10}^{-4}$ | 400 |

0.72 | 238.52 | $1.75\times {10}^{-4}$ | 400 |

**Table 2.**Numerical settings for different gust lengths, independent of the Mach number (SPP: steps per period, ndt: total number of time steps).

${\mathit{L}}_{\mathit{G}}$ [ m] | k | SPP | nr. of Periods | ndt |
---|---|---|---|---|

2.5 | 2.51 | 60 | 20 | 1200 |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

18.0 | 0.35 | 432 | 20 | 8640 |

21.0 | 0.30 | 504 | 10 | 5040 |

25.0 | 0.25 | 600 | 10 | 6000 |

31.5 | 0.20 | 756 | 10 | 7560 |

42.0 | 0.15 | 1008 | 10 | 10,080 |

62.5 | 0.10 | 1500 | 5 | 7500 |

125.5 | 0.05 | 3012 | 5 | 15,060 |

**Table 3.**Gust velocities and gust-induced angles of attack at Mach 0.70, using ${\widehat{\alpha}}_{G}=arctan({\widehat{W}}_{G}/{U}_{\infty})$.

${\widehat{\mathit{W}}}_{\mathit{G}}$ [ m/s] | ${\widehat{\mathit{\alpha}}}_{\mathit{G}}$ [${}^{\circ}$] |
---|---|

$1\times {10}^{-6}$ | $2.5\times {10}^{-7}$ |

0.1 | 0.025 |

1.0 | 0.25 |

3.0 | 0.74 |

5.0 | 1.24 |

7.5 | 1.85 |

10.0 | 2.47 |

12.5 | 3.09 |

15.0 | 3.70 |

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**MDPI and ACS Style**

Friedewald, D.
Numerical Simulations on Unsteady Nonlinear Transonic Airfoil Flow. *Aerospace* **2021**, *8*, 7.
https://doi.org/10.3390/aerospace8010007

**AMA Style**

Friedewald D.
Numerical Simulations on Unsteady Nonlinear Transonic Airfoil Flow. *Aerospace*. 2021; 8(1):7.
https://doi.org/10.3390/aerospace8010007

**Chicago/Turabian Style**

Friedewald, Diliana.
2021. "Numerical Simulations on Unsteady Nonlinear Transonic Airfoil Flow" *Aerospace* 8, no. 1: 7.
https://doi.org/10.3390/aerospace8010007