# Influence of Fluid Viscosity and Compressibility on Nonlinearities in Generalized Aerodynamic Forces for T-Tail Flutter

## Abstract

**:**

## 1. Introduction

## 2. Approach and Simulation Models

#### 2.1. Extended Modal Approach

#### 2.2. Obtaining Quadratic Mode Shape Components

#### 2.3. Time Domain Harmonic Forced Motion

#### 2.4. Aerodynamic Stiffness and Damping from Time Domain Results

#### 2.5. CFD Models

#### 2.5.1. Inviscid Flow

#### 2.5.2. Viscous Flow

^{6}to 30 × 10

^{6}and the boundary layer thickness $\delta $ at the trailing edge amounts to roughly $0.027$ $\mathrm{m}$ and $0.024$ $\mathrm{m}$, respectively.

#### 2.6. Temporal Discretization

## 3. Results

#### 3.1. Inviscid Flow

#### 3.1.1. GAF Hystereses

#### 3.1.2. Aerodynamic Stiffness and Damping

- Aerodynamic stiffness:
- –
- Nonlinear for low and high reduced frequencies with linear displacements;
- –
- Only marginally nonlinear for low and high reduced frequencies with quadratic displacements.

- Aerodynamic damping:
- –
- Marginally nonlinear for low reduced frequencies and both displacement descriptions;
- –
- Nonlinear for high reduced frequencies and both displacement descriptions;
- –
- Increased nonlinearity with quadratic displacements.

#### 3.2. Impact of Fluid Viscosity

#### 3.2.1. GAF Hystereses

#### 3.2.2. Aerodynamic Stiffness and Damping

- Aerodynamic stiffness:
- –
- Significant and amplitude-independent offset with respect to inviscid flow results;
- –
- No remarkable impact on nonlinear character.

- Aerodynamic damping:
- –
- Marginally larger damping at small displacement amplitudes and low reduced frequencies;
- –
- Marginally lower damping at small displacement amplitudes and high reduced frequencies;
- –
- Increased nonlinearity for linear modal displacement at low reduced frequencies;
- –
- Decreased nonlinearity for extended modal displacement at low reduced frequencies;
- –
- Decreased nonlinearity for linear and extended modal displacement at high reduced frequencies.

#### 3.3. Impact of Fluid Compressibility

#### 3.3.1. GAF Hystereses

#### 3.3.2. Aerodynamic Stiffness and Damping

- Aerodynamic stiffness:
- –
- Increased nonlinearity for low and high reduced frequencies;
- –
- Reduced impact of geometric nonlinearity, especially at high reduced frequencies.

- Aerodynamic damping:
- –
- Nonlinearity is almost entirely canceled;
- –
- No impact of geometric nonlinearity.

## 4. Discussion

#### 4.1. Physical Sources for Aerodynamic Nonlinearities

#### 4.2. Impact of Fluid Compressibility

#### 4.3. Summary of Aerodynamic Coupling Term Nonlinearity and Implications for T-Tail Flutter

## 5. Conclusions and Outlook

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Linear rigid body displacements (blue) and quadratic displacement components (orange) against undisplaced geometry (gray). (

**a**) Roll motion; (

**b**) yaw motion.

**Figure 7.**Deviations in magnitudes and phase angles of first harmonic GAF contents with respect to fine mesh (inviscid flow, Mach 0.4, reduced frequency 0.231, $5.0$${}^{\circ}$ roll angle amplitude).

**Figure 9.**GAF hystereses (inviscid flow, Mach 0.4). (

**a**) ${Q}_{hh}(1,1)$; (

**b**) ${Q}_{hh}(1,2)$; (

**c**) ${Q}_{hh}(2,1)$; (

**d**) ${Q}_{hh}(2,2)$.

**Figure 10.**Time history of ${Q}_{hh}(2,1)$ for one period of oscillation (inviscid flow, Mach 0.4, reduced frequency 0.231): (

**a**) $0.01\xb0$ roll motion amplitude; (

**b**) $5.0\xb0$ roll motion amplitude.

**Figure 11.**Aerodynamic stiffness and damping over amplitude for ${Q}_{hh}(2,1)$ (inviscid flow, Mach 0.4).

**Figure 12.**Impact of fluid viscosity on GAF hystereses (Mach 0.4). (

**a**) Inviscid flow—${Q}_{hh}(1,1)$; (

**b**) viscous flow—${Q}_{hh}(1,1)$; (

**c**) inviscid flow—${Q}_{hh}(2,1)$; (

**d**) viscous flow—${Q}_{hh}(2,1)$.

**Figure 13.**Impact of fluid viscosity on aerodynamic stiffness and damping of ${Q}_{hh}(2,1)$ (Mach 0.4).

**Figure 14.**Impact of fluid compressibility on GAF hystereses (viscous flow). (

**a**) Mach 0.4—${Q}_{hh}(1,1)$; (

**b**) Mach 0.8—${Q}_{hh}(1,1)$; (

**c**) Mach 0.4—${Q}_{hh}(2,1)$; (

**d**) Mach 0.8—${Q}_{hh}(2,1)$.

**Figure 15.**Impact of fluid compressibility on aerodynamic stiffness and damping of ${Q}_{hh}(2,1)$ (viscous flow).

**Figure 16.**Spatial locations of nonlinear components of ${Q}_{hh}(2,1)$ (viscous flow, Mach 0.4. Positive semi-span: linear modal, negative semi-span: extended modal). (

**a**) Longitudinal GAF component; (

**b**) lateral GAF component.

**Figure 17.**Nonlinear components of ${Q}_{hh}(2,1)$ along spanwise and chordwise slices (viscous flow, Mach 0.4). (

**a**) Longitudinal component, spanwise distribution at $x/c=0.725$; (

**b**) longitudinal component, chordwise distribution at $y/s=0.7$; (

**c**) lateral component, spanwise distribution at $x/c=0.725$; (

**d**) lateral component, chordwise distribution at $y/s=0.7$.

**Figure 18.**Spatial locations of nonlinear components of ${Q}_{hh}(2,1)$ (viscous flow, Mach 0.8. Positive semi-span: linear modal, negative semi-span: extended modal). (

**a**) Longitudinal GAF component; (

**b**) lateral GAF component.

**Figure 19.**Nonlinear components of ${Q}_{hh}(2,1)$ along spanwise and chordwise slices (viscous flow, Mach 0.8). (

**a**) Longitudinal component, spanwise distribution at $x/c=0.725$; (

**b**) longitudinal component, chordwise distribution at $y/s=0.7$; (

**c**) lateral component, spanwise distribution at $x/c=0.725$; (

**d**) lateral component, chordwise distribution at $y/s=0.7$.

Parameter | Symbol | Value |
---|---|---|

Mach numbers/- | $Ma$ | 0.4|0.8 |

Reduced frequencies/- | k | 0.056, 0.231 |

Frequencies/$\mathrm{Hz}$ | f | 1.213, 5.004 ${}^{\u2020}$|2.426, 10.008 ${}^{\u2021}$ |

Rotational amplitudes/${}^{\circ}$ | $\phi \mid \psi $ | 0.01, 0.917, 1.834, 3.669, 5.000 |

Temperature/$\mathrm{K}$ | T | 288.15 |

Density/$\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3}$ | $\rho $ | 1.225 |

Dynamic viscosity/$\mathrm{N}\mathrm{s}/{\mathrm{m}}^{-2}$ | $\nu $ | 17.89 × 10^{−6} |

Ratio of specific heats/- | $\kappa $ | 1.4 |

Ideal gas constant/$\mathrm{J}/\mathrm{k}\mathrm{g}/\mathrm{K}$ | R | 287 |

Reduced frequency reference length/$\mathrm{m}$ | $\overline{c}$ | 1.0 |

Reynolds number reference length/$\mathrm{m}$ | L | 2.0 |

Reynolds numbers/- | $Re$ | 15.216 × 10^{6}${}^{\u2020}$|30.432 × 10^{6} ${}^{\u2021}$ |

Relative Cauchy error for ⋯ | ||

⋯lift coefficient/- | ${\u03f5}_{{C}_{L}}$ | 1 × 10^{−6} |

⋯drag coefficient/- | ${\u03f5}_{{C}_{d}}$ | 1 × 10^{−6} |

⋯lateral force/- | ${\u03f5}_{Fy}$ | 1 × 10^{−4} |

⋯longitudinal moment coefficient/- | ${\u03f5}_{{C}_{{m}_{x}}}$ | 1 × 10^{−3} |

⋯lateral moment coefficient/- | ${\u03f5}_{{C}_{{m}_{y}}}$ | 5 × 10^{−6} |

^{†}Mach 0.4;

^{‡}Mach 0.8.

Parameter | Coarse | Medium | Fine |
---|---|---|---|

Number of grid points | 0.432 million | 0.865 million | 1.412 million |

Number of surface triangles | 85.732 k | 178.066 k | 412.094 k |

Number of volume tetrahedrons | 2.407 million | 4.816 million | 7.689 million |

Parameter | Value |
---|---|

First layer thickness | $1.4\times {10}^{-6}$ $\mathrm{m}$ |

Number of layers | 38 |

Stretching ratio | 1.25 |

Total prism layer height | $0.027$ $\mathrm{m}$ |

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**MDPI and ACS Style**

Schäfer, D.
Influence of Fluid Viscosity and Compressibility on Nonlinearities in Generalized Aerodynamic Forces for T-Tail Flutter. *Aerospace* **2022**, *9*, 256.
https://doi.org/10.3390/aerospace9050256

**AMA Style**

Schäfer D.
Influence of Fluid Viscosity and Compressibility on Nonlinearities in Generalized Aerodynamic Forces for T-Tail Flutter. *Aerospace*. 2022; 9(5):256.
https://doi.org/10.3390/aerospace9050256

**Chicago/Turabian Style**

Schäfer, Dominik.
2022. "Influence of Fluid Viscosity and Compressibility on Nonlinearities in Generalized Aerodynamic Forces for T-Tail Flutter" *Aerospace* 9, no. 5: 256.
https://doi.org/10.3390/aerospace9050256