**------------------ WORK IN PROGRESS ------------------ **

- Purpose
- Derivative calculation
- Test case
- Stability derivatives
- Control derivatives
- Conclusion and recommendations

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The goal of this study is to

- Benchmark flow5's prediction of stability and control derivatives with the predictions made by AVL.
- Perform parametric analyses to determine the sensivity of the derivatives to variables such as the angle of attack, the inclusion of the fuselage in the model and the type of panel method.
- Establish guidelines for the calculation of the derivatives.

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The methods to calculate the derivatives differ in AVL and flow5. To the author's understanding, the methods implemented in AVL are the ones described in "Flight Vehicle Aerodynamics", Mark Drela 2014.

The method takes advantage of the linear dependency of the panel forces with the 6 degrees of freedom in motion. This allows to express the stability derivatives explicitly, and to and solve them by direct back-substituion and linear combination which makes their calculation fast and accurate.

The derivatives are evaluated using finite differences.

The method used by AVL could be implemented for flow5's VLM. However it is not transposable to the thick surface panel methods in which the pressure forces are derived from the surface gradient of the doublet densities and do not depend linearly on the freestream's angle of attack and sideslip.

Solutions to this problem are being explored, but are not implemented as of flow5 v7.13.

The rotations of the control surfaces change both the linear system's matrix and the right hand side (RHS) vector. Strictly speaking, the problem should therefore be built and solved entirely for each derivative. This however would be computationally very expensive. A standard procedure is to assume that the change to the influence matrix is small, so that the LU-factorized matrix can be reused as-is, and that only the change to the RHS needs to be implemented. This simplification reduces the computations to the construction of the RHS and to its back-substitution.

This is how AVL, flow5 and other panel codes proceed. A numerical experiment was carried out with flow5 to evaluate the error implicit in this approximation and is presented herefafter.

The boundary condition are linearized since only the first order development is required, and the derivatives are calculated explicitely and solved by direct back-substitution and linear combination of deflections.

The problem cannot be linearized for the same reason as in the case of the stability derivatives. The derivatives are calculated by finite differences, with an RHS vector being constructed and back-substituted for each derivative.

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The test case is the simple 1.5 m span fictitious plane illustrated in the image on the right.

To keep the comparisons with AVL relevant:

- The fuselage was not included in the model.
- All flow5 analyses are inviscid.
- The first set of calculations in flow5 was done with the VLM2.

The corresponding flow5 and AVL project files can be downloaded here.

The purpose of these first runs is to check that the analyses are set up identically in flow5 and AVL.

Both AVL and flow5 predict that the piching moment is zero at α = 1.07°.

The intent of this first run is to compare the stability derivatives using the VLM common to both programs.

The stability derivatives are compared at the zero pitching moment angle, i.e. α = 1.08° for both programs.

AVL and flow5 do not use the same sign conventions, so the deviation is calculated with respect to the normalized values.

AVL 3.27 | flow5 v7.13 | deviation | |
---|---|---|---|

CLa | 5.469955 | -5.4931 | 0.4% |

Cma | -1.386379 | -1.4447 | -4.0% |

AVL 3.27 | flow5 v7.13 | deviation | |
---|---|---|---|

CYb | -0.111854 | -0.12915 | -13.4% |

Clb | -0.063762 | -0.056344 | 13.2% |

Cnb | 0.035501 | 0.042813 | -17.1% |

AVL 3.27 | flow5 v7.13 | deviation | |
---|---|---|---|

CYp | -0.072684 | -0.093622 | -22.4% |

Clp | -0.549236 | -0.55689 | -1.4% |

Cnp | -0.016272 | -0.0024056 | 576.4% |

AVL 3.27 | flow5 v7.13 | deviation | |
---|---|---|---|

CLq | 9.338631 | -9.4825 | -1.5% |

Cmq | -15.26148 | -15.848 | -3.7% |

AVL 3.27 | flow5 v7.13 | deviation | |
---|---|---|---|

CYr | 0.091196 | 0.096858 | -5.8% |

Clr | 0.055554 | 0.0074663 | 644.1% |

Cnr | -0.030879 | -0.035698 | -13.5% |

The derivatives are in reasonable agreement except for `Cnp` and `Clr`. To understand the discrepancy, the derivatives have been plotted as a function of the angle of attack.

As it appears, the value of the derivatives `Cnp` and `Clr` estimated by AVL depend strongly on the angle of attack, which makes them very sensitive to the point of evaluation. No obvious explanation could be found for this difference of sensitivity exhibited by both programs.

The other lateral derivatives are sensitive to the angle of attack to various degrees.

On the other hand, the longitudinal derivatives are not sensitive to the angle of attack.

Quads thick | Quads thin | Tri-uni thick | Tri-uni thin | VLM2 | |
---|---|---|---|---|---|

CXu | -0.038923 | -0.014213 | -0.0034009 | -0.0010699 | -0.0018653 |

CLu | -0.00090185 | -0.00013729 | -9.2337E-05 | -8.8636E-06 | 2.6189E-05 |

Cmu | -1.7561E-06 | -6.6029E-08 | -0.0047468 | -6.4618E-09 | 1.0353E-07 |

CXa | -0.64907 | -0.33282 | 0.16031 | 0.085434 | 0.12196 |

CLa | -5.5324 | -5.4843 | -5.7636 | -5.6188 | -5.4932 |

Cma | -1.4773 | -1.132 | -1.6562 | -1.2149 | -1.4485 |

CXq | -1.2981 | -0.31383 | -0.38311 | -0.1022 | -0.25743 |

CLq | -9.2422 | -9.1713 | -9.7862 | -9.5477 | -9.4901 |

Cmq | -17.133 | -10.815 | -16.677 | -10.693 | -15.856 |

CYb | -0.13164 | -0.12969 | -0.12031 | -0.11863 | -0.12915 |

Clb | -0.075016 | -0.054906 | -0.058189 | -0.03846 | -0.056348 |

Cnb | 0.048835 | 0.028517 | 0.04353 | 0.026953 | 0.042822 |

CYp | -0.097317 | -0.096193 | -0.098104 | -0.094298 | -0.093629 |

Clp | -0.60218 | -0.48104 | -0.57314 | -0.38189 | -0.55689 |

Cnp | -0.011015 | -0.0040144 | -0.011683 | -0.0029192 | -0.002397 |

CYr | 0.0975 | 0.096827 | 0.088485 | 0.088861 | 0.096879 |

Clr | 0.015167 | 0.0081044 | -0.023376 | -0.010809 | 0.0074863 |

Cnr | -0.037425 | -0.024704 | -0.035301 | -0.02339 | -0.035711 |

The scattering of the predictions can be quite large depending on the derivative.

A slender fuselage was added to the flow5 model to evaluate its influence on the derivatives. The inertia properties were kept identical for the models with and without the fuselage.

The comparisons were performed using the tri-uniform method.

The calculated zero-pitching moment angles are close: 1.55° for the model without the fuselage, and 1.40° with the fuselage.

no fuselage | with fuselage | deviation | |
---|---|---|---|

CXu | -0.0034175 | -0.004320 | 26.4% |

CLu | -0.000093 | -0.000106 | 13.2% |

Cmu | -0.004868 | -0.004418 | -9.2% |

CXa | 0.160790 | 0.099582 | -38.1% |

CLa | -5.763000 | -5.669000 | -1.6% |

Cma | -1.652300 | -1.752500 | 6.1% |

Cxq | 0.214170 | 0.154110 | -28.0% |

CLq | -9.840400 | -9.692200 | -1.5% |

Cmq | -16.849 | -16.865 | 0.1% |

CYb | -0.12026 | -0.14292 | 18.8% |

Clb | -0.058095 | -0.059846 | 3.0% |

Cnb | 0.043516 | 0.047060 | 8.1% |

CYp | -0.097563 | -0.090962 | -6.8% |

Clp | -0.57233 | -0.57528 | 0.5% |

Cnp | -0.011734 | -0.01299 | 10.7% |

CYr | 0.1012 | 0.11863 | 17.2% |

Clr | 0.106980 | 0.101010 | -5.6% |

Cnr | -0.03494 | -0.037437 | 7.1% |

The inclusion of the fuselage changes the derivatives to various degrees. The variations are not easy to interpret although some can be expected such as:

- the increase of
`Cxq`due to the perturbation of the flow when the fuselage is not aligned with the x axis - the increase of
`CYb`due to the fuselage's lateral surface

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The derivatives have been evaluated in flow5 using the VLM2 method to be consistent with AVL.

The control derivatives are only evaluated in flow5 as part of a T7 analysis and are therefore calculated at the zero-pitching moment angle.

**Aileron**

AVL | flow5 | deviation | |
---|---|---|---|

CXd | |||

CYd | -0.000725 | 0.00076154 | 5.0% |

CZd | |||

Cld | -0.005344 | 0.0054668 | 2.3% |

Cmd | |||

Cnd | -0.000032 | -0.00000049 | -101.5% |

**Elevator**

AVL | flow5 | deviation | |
---|---|---|---|

CXd | |||

CYd | |||

CZd | 0.005236 | -0.0057114 | 9.1% |

Cld | |||

Cmd | -0.02009 | -0.021882 | 8.9% |

Cnd |

**Rudder**

AVL | flow5 | deviation | |
---|---|---|---|

CXd | |||

CYd | 0.001242 | 0.0014878 | 19.8% |

CZd | |||

Cld | 0.000067 | 0.000075661 | 12.9% |

Cmd | |||

Cnd | -0.000489 | -0.00058663 | 20.0% |

The derivative which stands out is again `Cnd`, for the same reason as in the case of the stability derivatives. This is illustrated by a plot of `Cnd` vs. the angle of attack.

Not only is the absolute value of the derivative small, but it also exhibits in AVL an important variation with the angle of attack. This makes its estimation very sensitive to the setup of the model and of the analysis.

A numerical experiment was made in flow5 to evaluate the error implicit in the assumption that the linear system's matrix is not modified by the deflection of the control surfaces. The influence matrix was rebuilt for each of the aileron, elevator and rudder controls, and the problem was entirely solved for the derivatives in each case. The results were compared to those presented above in the VLM case.

**Aileron**

Ref. | Mod. matrix | deviation | |
---|---|---|---|

CXd | |||

CYd | 0.00076154 | 0.0007525 | 1.2% |

CZd | |||

Cld | 0.0054668 | 0.0054548 | 0.2% |

Cmd | |||

Cnd | -0.00000049 | -0.00000046 | 6.6% |

**Elevator**

Ref. | Mod. matrix | deviation | |
---|---|---|---|

CXd | |||

CYd | |||

CZd | -0.0057114 | -0.0057132 | 0% |

Cld | |||

Cmd | -0.021882 | -0.021889 | 0% |

Cnd |

**Rudder**

Ref. | Mod. matrix | deviation | |
---|---|---|---|

CXd | |||

CYd | 0.0014878 | 0.0014955 | -0.5% |

CZd | |||

Cld | 0.000075661 | 0.000076071 | -0.5% |

Cmd | |||

Cnd | -0.00058663 | -0.00058967 | -0.5% |

It appears that the differences are minor and no larger than the variations seen beteween the VLM2 and the tri-uniform method, which justifies the approximation.

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- the derivatives exhibit various degrees of variability to the type of analysis
- the evaluation of the derivatives using panel methods are less precise than are the estimation of the lift, drag and moment coefficients.

- use preferably the estimations in a relative way to plot tendencies rather than in an absolute way
- use the same type of models and panel methods when comparing derivatives.

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