Review of Numerical Simulations on Aircraft Dynamic Stability Derivatives


The dynamic stability derivatives of flight vehicle are directly related to unsteady aerodynamics during maneuverability, and are considered as key parameters to both the aerodynamics, control system design and flight qualities evaluation. As the rapid development of new conceptual aircraft configurations, higher precision and efficiency calculations of the dynamic stability derivatives are urgently required. Limited by high costs and risks, the traditional experimental ways of flight test and wind tunnel test cannot be widely used, thus the numerical calculation on dynamic stability derivatives have become the primarily approaches. This paper reviews the numerically methods applied in the estimation of aircraft dynamic stability derivatives, includes the early analytical method, the empirical and semi-empirical methods and the widely used modern time domain and frequency domain methods, deeply highlighting the advantages and drawbacks of these methods of the actual applications on various aircrafts. The early analytical method by using potential theory can be used to calculate dynamic stability derivatives and will never work for current geometries. The empirical and semi-empirical methods are available with simple mathematical model and existed data. They have the advantage to be simple and rapid to compute and can be well adapted for initial evaluation of aircraft conceptual design. The time domain analysis based on solving Euler or NS equations with CFD technique are the most widely used methods to obtain the aircraft dynamic stability derivatives. With a variety of different strategies, they can calculate the combined and single dynamic derivatives to satisfy the demand of aircraft design in each stage. Even they are accurate and better in adaptability, the huge time cost on the periodic unsteady flow limits the application. To overcome this problem, the frequency domain methods based on harmonic oscillation are developed. They only use the results at several sample points during the unsteady cycle to reconstruct the periodic unsteady flows to further efficiently obtain the dynamic stability derivatives. These frequency domain methods are currently available only in harmonic oscillation cases. This paper also discusses and analyses the existing problems and possible development directions of the numerical methods to calculate aircraft dynamic stability derivatives from four aspects: theory, calculating elaboration, efficiency and accuracy, and application.

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Computational Fluid Dynamics


Data Compendium


Deutsches Zentrum für Luft- und Raumfahrt


German–Dutch wind tunnel


Hyper Ballistic Shape


National Advisory Committee for Aeronautics


National Aeronautics and Space Administration


Navier Stokes


Reynolds-averaged Navier–Stokes


Stability and Control Configuration


Standard Dynamic Model


Transcruiser configuration

\(a_{\infty }\) :

Velocity of sound

b :


\(\alpha\) :

Angle of attack

\(\dot{\alpha }\) :

Angle of attack rate

\(\bar{\dot{\alpha }}\) :

Non dimensional angle of attack rate

\(\beta\) :

Angle of sideslip

\(\dot{\beta }\) :

Angle of sideslip rate

\(c\) :

Reference length

\(C_{i}\) :

Aerodynamic force or moment coefficients, \(i = L,D,m,l,n\)

\(C_{1} \cdots C_{15} \cdots\) :


\(C_{l}\) :

Rolling moment coefficient

\(C_{{l\dot{\alpha }}}\) :

Rolling moment coefficient derivative due to angle of attack rate

\(C_{{l\dot{\beta }}}\) :

Rolling moment coefficient derivative due to angle of sideslip rate

\(C_{lp}\) :

Rolling moment coefficient derivative due to roll rate

\(C_{lq}\) :

Rolling moment coefficient derivative due to pitch rate

\(C_{lr}\) :

Rolling moment coefficient derivative due to yaw rate

\(C_{m}\) :

Pitching moment coefficient

\(C_{m\alpha }\) :

Pitching moment coefficient derivative due to angle of attack

\(C_{{m\dot{\alpha }}}\) :

Pitching moment coefficient derivative due to angle of attack rate

\(C_{{m\dot{\beta }}}\) :

Pitching moment coefficient derivative due to angle of sideslip rate

\(C_{mp}\) :

Pitching moment coefficient derivative due to roll rate

\(C_{mq}\) :

Pitching moment coefficient derivative due to pitch rate

\(C_{mr}\) :

Pitching moment coefficient derivative due to yaw rate

\(C_{n}\) :

Yawing moment coefficient

\(C_{{n\dot{\alpha }}}\) :

Yawing moment coefficient derivative due to angle of attack rate

\(C_{{n\dot{\beta }}}\) :

Yawing moment coefficient derivative due to angle of sideslip rate

\(C_{np}\) :

Yawing moment coefficient derivative due to roll rate

\(C_{nq}\) :

Yawing moment coefficient derivative due to pitch rate

\(C_{nr}\) :

Yawing moment coefficient derivative due to yaw rate

\(C_{p}\) :

Pressure coefficient

D :

Coefficient matrix

\(\theta\) :

Pitch angle

\(\dot{\theta }\) :

Pitch angle rate, equal to q

\(I_{z}\) :

Moment of inertia to z axis

\(Ma\) :

Mach number

\(n_{z}\) :

Damping coefficient

\(N_{H}\) :

Number of harmonics

\(N_{T}\) :

Number of sample points

p :

Roll rate

\(p\) :


\(\rho\) :


q :

Pitch rate

\(\bar{q}\) :

Non dimensional pitch rate

Q :

Conservative variable

r :

Yaw rate

R :

Flux vector

\(\gamma\) :

Specific heat ratio

\(S\) :


\(V\) :


v :

Additional velocity

w :


\(\omega\) :

Angular frequency

y :


\(\varepsilon\) :

Angle between the body and wind axis

\(\Delta\) :


\(\hat{\Delta }\) :

High-order terms

\(\Delta \alpha\) :

Additional angle of attack

\(\Delta V\) :

Additional velocity


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The authors would like to acknowledge the support of National Natural Science Foundation of China (Grant No. 11672236) and Project funded by China Postdoctoral Science Foundation (Grant No. 2018M641381).

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Correspondence to Baigang Mi.

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Mi, B., Zhan, H. Review of Numerical Simulations on Aircraft Dynamic Stability Derivatives. Arch Computat Methods Eng 27, 1515–1544 (2020).

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