# Estimation and Separation of Longitudinal Dynamic Stability Derivatives with Forced Oscillation Method Using Computational Fluid Dynamics

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Moment Calculation

#### 2.2. Forced Simple Harmonic Oscillation

#### 2.2.1. Pitching Mode

#### 2.2.2. Plunging Mode

#### 2.2.3. Flapping Mode

## 3. Numerical Analysis

#### 3.1. Computational Fluid Dynamics

#### 3.1.1. Geometry Configuration

#### 3.1.2. Mesh Configuration

#### 3.1.3. Mesh Convergence Study

#### 3.1.4. Dynamic Mesh

#### 3.2. Solver Setting

## 4. Validation and Results

#### 4.1. Computational Fluid Dynamics (CFD) Validation

#### 4.1.1. Steady Case Validation

#### 4.1.2. Unsteady Case Data Filtering

^{−5}. To filter the result, the first cycle is omitted, and the last two cycles are taken into consideration. Figure 12 shows the comparison between the full three cycles and the filtered result of the last two cycles. The filtering process is undertaken to remove the transient region data which are not very well converged yet during the first cycle. Figure 13 shows the comparison for three sets of $\left({C}_{m}-{C}_{{m}_{o}}\right)$ for various angles of attack which have been filtered. The four angles of attack are shown to represent the whole alpha sweep region. The smooth ellipse curve shows that the moment coefficient values converged well and are fit to be used in the unsteady dynamic derivatives analysis.

#### 4.1.3. Unsteady Case Validation

_{R}matrix form [21] of the Equation (7) with fewer points for the CFD analysis than the wind tunnel test points. Lastly, for ${C}_{{m}_{o}}$, results from pitching mode are overlapping between the unsteady and the steady analysis as shown in Figure 14c.

#### 4.1.4. Steady vs. Unsteady Comparison

#### 4.2. Coefficient Separation Method

#### Dynamic Derivative Coefficients

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

${C}_{m}$ | Pitching moment coefficient |

${C}_{{m}_{o}}$ | Pitching moment coefficient in steady reference condition |

${C}_{{m}_{q}}$ | Pitching moment coefficient due to pitch rate |

${C}_{{m}_{\alpha}}$ | Pitching moment coefficient due to change in angle of attack |

${C}_{{m}_{\dot{\alpha}}}$ | pitching moment coefficient due to change rate of the angle of attack |

${C}_{{m}_{q}}+{C}_{{m}_{\dot{\alpha}}}$ | Direct pitching moment coefficient damping derivatives |

l | Length reference (m) |

d | Model diameter (m) |

f | Dimensional frequency (Hz) |

T | Period cycle |

k | Reduced oscillation frequency |

$q$ | Pitch rate |

${M}_{\infty}$ | Freestream Mach number |

$R{e}_{d}$ | Reynolds number based on the model diameter |

${P}_{0}$ | Total Pressure, Pa |

${T}_{0}$ | Total Temperature, K |

${q}_{\infty}$ | Freestream dynamic pressure, Pa |

${V}_{\infty}$ | Freestream velocity, m/s |

$\overline{c}$ | Mean chord |

t | Time(s) |

Δt | Time-step size(s) |

N | Number of iteration |

V | Velocity of the body relative to the fluid |

$\alpha $ | Angle of attack (deg) |

$\dot{\alpha}$ | Angle of attack change rate |

$\beta $ | Angle of sideslip |

$\rho $ | Density of the fluid |

$\theta $ | Inclination angle measured normal to the horizontal plane to aircraft longitudinal axis |

${\theta}_{A}$ | Inclination angle oscillation amplitude |

$\omega $ | Angular rate |

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**Figure 1.**Simple harmonic oscillation methods: (

**a**) pitching mode; (

**b**) plunging mode; (

**c**) flapping mode. Reproduced with permission from Baigang Mi et al, Review of Numerical Simulations on Aircraft Dynamic Stability Derivatives; published by Springer Nature, 2019 [25].

**Figure 2.**(

**a**) Axis system on unstructured surface mesh of ANF missile; (

**b**) 45 deg missile rotation along x-axis for tests-conduct condition.

**Figure 5.**(

**a**) Yplus distribution contour; (

**b**) Yplus distribution plot along directional symmetry plane.

**Figure 6.**Details of: (

**a**) Prism boundary layer at nose-body conjunction; (

**b**) far-field with small dynamic mesh sphere zone in the middle.

**Figure 7.**Mesh convergency test for coarse, medium, and fine mesh model; (

**a**) angle of attack 0 to 10 deg; (

**b**) detailed view of angle of attack of 5 deg.

**Figure 10.**Static pressure (P) profile of (

**a**) circumferential pressure at c.g.; (

**b**) Axial pressure from nose to base along with the symmetry plane cut.

**Figure 11.**Moment coefficient value for steady analysis alpha sweep from 0 to 85 deg: (

**a**) ${C}_{m}$ vs. α; (

**b**) ${C}_{{m}_{\alpha}}$ vs. α.

**Figure 12.**Filtered data using the last two cycles of the computational fluid dynamics (CFD) analysis.

**Figure 13.**History plot for (

**a**) ${C}_{m}-{C}_{{m}_{o}}$ vs. θ; (

**b**) ${C}_{m}-{C}_{{m}_{o}}$ vs. pitch rate.

**Figure 14.**Dynamic stability derivatives of forced pitching simple harmonic oscillation: (

**a**) (${C}_{{m}_{q}}+{C}_{{m}_{\dot{\alpha}}}$) vs. α; (

**b**) ${C}_{{m}_{\alpha}}$ vs. α; (

**c**) ${C}_{{m}_{o}}$ vs. α.

**Figure 16.**Comparison of combined dynamic stability derivatives of forced plunging and forced pitching simple harmonic oscillation.

**Figure 17.**Comparison of ${C}_{{m}_{\alpha}}$ of forced plunging and forced pitching simple harmonic oscillation.

Mode | Pitching | Plunging | Flapping |
---|---|---|---|

y-axis rotation (t) | ${\alpha}_{A}\mathrm{sin}\left(\omega t\right)$ | - | ${\alpha}_{A}\mathrm{cos}\left(\omega t\right)$ * |

z-axis translation (t) | - | ${z}_{A}\mathrm{sin}\left(\omega t\right)$ | ${z}_{A}\mathrm{sin}\left(\omega t\right)$ |

$\alpha \left(t\right)$ | ${\alpha}_{0}+{\alpha}_{A}\mathrm{sin}\left(\omega t\right)$ | ${\alpha}_{0}+{\alpha}_{A}\mathrm{sin}\left(\omega t\right)$ | ${\alpha}_{0}$ |

$\theta \left(t\right)$ | ${\theta}_{A}\mathrm{sin}\left(\omega t\right)$ | 0 | ${\theta}_{A}\mathrm{sin}\left(\omega t\right)$ |

$\mathit{R}{\mathit{e}}_{\mathit{d}}$ | ${\mathit{M}}_{\mathbf{\infty}}$ | ${\mathit{P}}_{\mathbf{0}}$ | ${\mathit{T}}_{\mathbf{0}}$ | ${\mathit{q}}_{\mathbf{\infty}}$ | ${\mathit{V}}_{\mathbf{\infty}}$ |
---|---|---|---|---|---|

(Pa) | (K) | (Pa) | (m/s) | ||

0.086 × 10^{6} | 1.96 | 22,753 | 306.11 | 8342.7 | 516.97 |

**Table 3.**Comparison of dynamic stability derivatives of forced pitching (A), forced plunging (B), and forced flapping (C).

α | ${\mathit{C}}_{{\mathit{m}}_{\dot{\mathit{\alpha}}}}$ (B) | ${\mathit{C}}_{{\mathit{m}}_{\mathit{q}}}$ (C) | Total (B + C) | ${\mathit{C}}_{{\mathit{m}}_{\dot{\mathit{\alpha}}}}\mathbf{+}{\mathit{C}}_{{\mathit{m}}_{\mathit{q}}}$ (A) | Error |
---|---|---|---|---|---|

0 | 47.231 | −316.149 | −268.918 | −268.700 | −0.08% |

10 | 27.708 | −369.019 | −341.311 | −331.720 | −2.89% |

45 | 14.321 | −600.932 | −586.611 | −570.105 | −2.90% |

75 | −49.649 | −564.955 | −614.604 | −588.397 | −4.45% |

**Table 4.**Comparison of ${C}_{{m}_{\alpha}}$ between forced pitching (

**A**), forced plunging (

**B**), and forced flapping (

**C**).

α | ${\mathit{C}}_{{\mathit{m}}_{\mathit{\alpha}}}$ (B) | ${\mathit{C}}_{{\mathit{m}}_{\mathit{\alpha}}}$ (C) | Total (B + C) | ${\mathit{C}}_{{\mathit{m}}_{\mathit{\alpha}}}$ (A) | Error |
---|---|---|---|---|---|

0 | −17.099 | 0.655 | −16.444 | −16.465 | 0.13% |

10 | −16.115 | −1.356 | −17.470 | −17.375 | −0.55% |

45 | −18.965 | 23.789 | 4.824 | 3.234 | −49.18% |

75 | −13.259 | 28.019 | 14.760 | 13.550 | −8.93% |

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**MDPI and ACS Style**

Juliawan, N.; Chung, H.-S.; Lee, J.-W.; Kim, S.
Estimation and Separation of Longitudinal Dynamic Stability Derivatives with Forced Oscillation Method Using Computational Fluid Dynamics. *Aerospace* **2021**, *8*, 354.
https://doi.org/10.3390/aerospace8110354

**AMA Style**

Juliawan N, Chung H-S, Lee J-W, Kim S.
Estimation and Separation of Longitudinal Dynamic Stability Derivatives with Forced Oscillation Method Using Computational Fluid Dynamics. *Aerospace*. 2021; 8(11):354.
https://doi.org/10.3390/aerospace8110354

**Chicago/Turabian Style**

Juliawan, Nadhie, Hyoung-Seog Chung, Jae-Woo Lee, and Sangho Kim.
2021. "Estimation and Separation of Longitudinal Dynamic Stability Derivatives with Forced Oscillation Method Using Computational Fluid Dynamics" *Aerospace* 8, no. 11: 354.
https://doi.org/10.3390/aerospace8110354