# Review of Numerical Simulations on Aircraft Dynamic Stability Derivatives

- 27 Downloads

## Abstract

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.

## List of Symbols

## Abbreviations

- CFD
Computational Fluid Dynamics

- DATCOM
Data Compendium

- DLR
Deutsches Zentrum für Luft- und Raumfahrt

- DNW
German–Dutch wind tunnel

- HBS
Hyper Ballistic Shape

- NACA
National Advisory Committee for Aeronautics

- NASA
National Aeronautics and Space Administration

- NS
Navier Stokes

- RANS
Reynolds-averaged Navier–Stokes

- SACCON
Stability and Control Configuration

- SDM
Standard Dynamic Model

- TCR
Transcruiser configuration

## Notation

- \(a_{\infty }\)
Velocity of sound

*b*Body

- \(\alpha\)
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, \(i = L,D,m,l,n\)

- \(C_{1} \cdots C_{15} \cdots\)
Coefficients

- \(C_{l}\)
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

- \(p\)
Pressure

- \(\rho\)
Density

*q*Pitch rate

- \(\bar{q}\)
Non dimensional pitch rate

*Q*Conservative variable

*r*Yaw rate

*R*Flux vector

- \(\gamma\)
Specific heat ratio

- \(S\)
Area

- \(V\)
Velocity

*v*Additional velocity

*w*Wind

- \(\omega\)
Angular frequency

*y*Displacement

- \(\varepsilon\)
Angle between the body and wind axis

- \(\Delta\)
Difference

- \(\hat{\Delta }\)
High-order terms

- \(\Delta \alpha\)
Additional angle of attack

- \(\Delta V\)
Additional velocity

## Notes

### Acknowledgements

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).

### Compliance with Ethical Standards

### Conflict of interest

The authors declare that they have no conflict of interest.

## References

- 1.Gursul I (2015) Unsteady flow phenomena over delta wings at high angle of attack. AIAA J 32(2):225–231CrossRefGoogle Scholar
- 2.Adams RJ, Buffington JM, Banda SS (2012) Design of nonlinear control laws for high-angle-of-attack flght. J Guid Control Dyn 17(4):734–746Google Scholar
- 3.Huang W, Liu J, Wang ZG (2012) Investigation on high angle of attack characteristics of hypersonic space vehicle. Sci China 55(5):1437–1442CrossRefGoogle Scholar
- 4.Dowell EH, Williams MH, Bland SR (2015) Linear/nonlinear behavior in unsteady transonic aerodynamics. AIAA J 21(1):38–46CrossRefzbMATHGoogle Scholar
- 5.Davis MC, White JT (2008) X-43A flight-test-determined aerodynamic force and moment characteristics at mach 7.0. J Spacecr Rockets 45(3):472–484CrossRefGoogle Scholar
- 6.Kramer B (2013) Experimental evaluation of superposition techniques applied to dynamic aerodynamics. In: AIAA aerospace sciences meeting & exhibitGoogle Scholar
- 7.Greenwell DI (2015) Frequency effects on dynamic stability derivatives obtained from small-amplitude oscillatory testing. J Aircr 35(5):776–783CrossRefGoogle Scholar
- 8.Dan DV, Huber KC, Loeser TD, et al (2014) Low-speed dynamic wind tunnel test analysis of a generic 53°swept UCAV configuration. In: AIAA applied aerodynamics conferenceGoogle Scholar
- 9.Dudley R (1999) Unsteady aerodynamics. Science 284(5422):1937–1939CrossRefGoogle Scholar
- 10.Yukovich R, Liu D, Chen P (2013) State-of-the-art of unsteady aerodynamics for high performance aircraft. In: Aerospace sciences meeting & exhibitGoogle Scholar
- 11.Queijo MJ, Wells WR, Keskar DA (1978) Influence of unsteady aerodynamics on extracted aircraft parameters. J Aircr 16(10):708–713CrossRefGoogle Scholar
- 12.Junkins JL, Bang H (1993) Maneuver and vibration control of hybrid coordinate systems using Lyapunov stability theory. J Guid Control Dyn 16(4):668–676CrossRefGoogle Scholar
- 13.Bryan GH (1911) Stability in aviation. Macmillan, LondonzbMATHGoogle Scholar
- 14.Boyd TJM (2011) One hundred years of G. H. Bryan’s stability in aviation. J Aeronaut Hist 4:97–115Google Scholar
- 15.Tong BG, Chen Q (1983) Some remarks on unsteady aerodynamics. Adv Mech 4:377–394Google Scholar
- 16.Etkin B, Reid LD (1996) Dynamics of flight: stability and control. Wiley, New York, p 107Google Scholar
- 17.Etkin B (2012) Dynamics of atmospheric flight. Dover Publications, New York, p 125Google Scholar
- 18.Tobak M, Schiff LB (1981) Aerodynamic mathematical modeling-basic concepts. AGARD Lect Ser 77(114):1–32Google Scholar
- 19.Tobak M, Schiff LB (1976) On the formulation of the aerodynamic characteristics in aircraft dynamics: NASA TRR-456. NASA, Washington, DCGoogle Scholar
- 20.Tobak M, Schiff LB (1978) The role of time-history effects in the formulation of the aerodynamics of aircraft dynamics: NASA TM 78471. NASA, Washington, DCGoogle Scholar
- 21.Theodorsen T, Garrick I (1933) General potential theory of arbitrary wing sections. US Government Printing Office, New York, pp 77–80zbMATHGoogle Scholar
- 22.Theodorsen T (1935) General theory of aerodynamic instability and the mechanism of flutter. Rept. 496, NACAGoogle Scholar
- 23.Wukelich SR, Williams HE (1979) The USAF stability and control digital Datcom. AFFDL-TR-79-3030Google Scholar
- 24.Mattsaits GR (1982) An update of the digital Datcom computer code for estimating dynamic stability derivatives. AEDC-TR-81-30Google Scholar
- 25.Blake WB (1985) Prediction of fighter aircraft dynamic derivatives using Digital Datcom. In: AIAA 3rd applied aerodynamics conferenceGoogle Scholar
- 26.Jaslow H (2015) Aerodynamic relationships inherent in Newtonian impact theory. AIAA Journal 6(4):608–612CrossRefGoogle Scholar
- 27.Tobak M, Wehrend WR (1956) Stability derivatives of cones at supersonic speeds. NACA TN 3788, NACA, Washington, DCGoogle Scholar
- 28.Busemann A (1933) Handbook of natural sciences, IV, liquid and garvewegung. 2nd Jena: Gustav Fisher, 12–55Google Scholar
- 29.Hui W, Tobak M (1981) Unsteady Newton-Busemann flow theory. Part 2: bodies of revolution. AIAA J 19(10):1272–1273CrossRefzbMATHGoogle Scholar
- 30.Ericsson LE (1968) Unsteady aerodynamics of an ablating flared body of revolution including effect of entropy gradient. AIAA J 6(5):2395–2401CrossRefGoogle Scholar
- 31.Ericsson LE (1973) Unsteady embedded Newtonian flow (as basis for nose bluntness effect on aerodynamics of hypersonic slender bodies). Astronaut Acta 18(3):309–330Google Scholar
- 32.Ashley H, Zartarian G (1956) Piston theory, a new aerodynamic tool for the aeroelastican. J Aeronaut Sci 23(12):1109–1118CrossRefGoogle Scholar
- 33.Chen JS (1991) Pitching derivatives of wing in supersonic and hypersonic stream—method for local flow piston theory. Acta Aerodyn Sin 9(4):469–476Google Scholar
- 34.Zhang WW, Ye ZY, Zhang CA et al (2009) Supersonic flutter analysis based on a local piston theory. AIAA J 47(10):2321–2328CrossRefGoogle Scholar
- 35.Ye C, Ma DL (2012) An aircraft steady dynamic derivatives calculation method. In: Proceedings of 2012 international conference on modelling, identification and controlGoogle Scholar
- 36.Despirito J, Silton SI, Weinacht P (2015) Navier–Stokes predictions of dynamic stability derivatives: evaluation of steady-state methods. J Spacecr Rockets 46(6):1142–1153CrossRefGoogle Scholar
- 37.Stalnaker JF (2004) Rapid computation of dynamic stability derivatives. In: 42nd AIAA aerospace sciences meeting & exhibitGoogle Scholar
- 38.Park MA, Green L(2000) Steady-state computation of constant rotational rate dynamic stability derivatives. In: 18th applied aerodynamics conferenceGoogle Scholar
- 39.Guglieri G, Quagliotti FB (1993) Dynamic stability derivatives evaluation in a low-speed wind tunnel. J Aircr 30(3):421–423CrossRefGoogle Scholar
- 40.Hanff ES, Orlikruckemann KJ (2015) Wind-tunnel measurement of dynamic cross-coupling derivatives. J Aircr 15(1):40–46CrossRefGoogle Scholar
- 41.Mi BG, Zhan H, Wang B (2014) Computational investigation of simulation on the dynamic derivatives of flight vehicle. In: 29th international conference on aerospace sciencesGoogle Scholar
- 42.Liu X, Liu W, Zhao YF (2016) Navier–Stokes predictions of dynamic stability derivatives for air-breathing hypersonic vehicle. Acta Astronaut 118:262–285CrossRefGoogle Scholar
- 43.Moelyadi MA, Sachs G (2007) CFD based determination of dynamic stability derivatives in yaw for a bird. J Bionic Eng 4:201–208CrossRefGoogle Scholar
- 44.Hui WH (1969) Stability of oscillating wedges and caret wings in hypersonic and supersonic flows. AIAA J 7(8):1524–1530CrossRefGoogle Scholar
- 45.Roy JFL, Morgand S (2010) SACCON CFD static and dynamic derivatives using elsA. In: 28th AIAA applied aerodynamics conferenceGoogle Scholar
- 46.Roy JFL, Morgand S, Farcy D (2014) Static and dynamic derivatives on generic UCAV without and with leading edge control. In: 32nd AIAA applied aerodynamics conferenceGoogle Scholar
- 47.Alemdaroglu N, Iyigun I, Altun M et al (2002) Determination of dynamic stability derivatives using forced oscillation technique. In: 40th aerospace sciences meeting & exhibitGoogle Scholar
- 48.Ronch AD, Vallespin D, Ghoreyshi M et al (2012) Evaluation of dynamic derivatives using computational fluid dynamics. AIAA J 50(2):470–484CrossRefGoogle Scholar
- 49.Ronch AD (2012) On the calculation of dynamic derivatives using computational fluid dynamics. University of LiverpoolGoogle Scholar
- 50.Moore FG, Swanson RC (1972) Dynamic derivatives for missile configurations to Mach number three. J Spacecr 15(2):65–66CrossRefGoogle Scholar
- 51.Sahu J (2007) Numerical computations of dynamic derivatives of a finned projectile using a time-accurate CFD method. In: AIAA atmospheric flight mechanics conference & exhibitGoogle Scholar
- 52.Bhagwandin VA, Sahu J (2012) Numerical prediction of roll damping and magnus dynamic derivatives for finned projectiles at angle of attack. In: 50th AIAA aerospace sciences meeting including the new horizons forum and aerospace expositionGoogle Scholar
- 53.Oktay E, Akay H (2002) CFD predictions of dynamic derivatives for missiles. In: 40th AIAA aerospace sciences meeting & exhibitGoogle Scholar
- 54.Mialon B, Khrabrov A, Ronch AD et al (2010) Benchmarking the prediction of dynamic derivatives: wind tunnel tests, validation, acceleration methods. In: AIAA atmospheric flight mechanics conferenceGoogle Scholar
- 55.Mialon B, Khrabrov A, Khelil SB et al (2011) Validation of numerical prediction of dynamic derivatives: the DLR-F12 and the Transcruiser test cases. Prog Aerosp Sci 47:674–694CrossRefGoogle Scholar
- 56.Silton SI (2011) Navier–Stokes predictions of aerodynamic coefficients and dynamic derivatives of a 0.50-cal projectile. In: 29th AIAA applied aerodynamics conferenceGoogle Scholar
- 57.Forsythe JR, Fremaux CM, Hall RM (2004) Calculation of static and dynamic stability derivatives of the F/A-18E in abrupt wing stall using RANS and DES. In: 3rd international conference on computational fluid dynamicsGoogle Scholar
- 58.Green LL, Spence AM, Murphy PC (2004) Computational methods for dynamic stability and control derivatives. In: 42nd AIAA aerospace sciences meeting & exhibitGoogle Scholar
- 59.Mi BG, Zhan H (2017) Calculating dynamic derivatives of flight vehicle with new engineering strategies. Int J Aeronaut Space Sci 18(2):175–185CrossRefGoogle Scholar
- 60.Ghoreyshi M, Lofthouse AJ (2017) Indicial methods for the numerical calculation of dynamic derivatives. AIAA J 55(7):2279–2294CrossRefGoogle Scholar
- 61.Mi BG, Zhan H, Chen BB (2017) New systematic methods to calculate static and single dynamic stability derivatives of aircraft. Math Probl Eng 4217217:1–11CrossRefGoogle Scholar
- 62.Zhang WW, Gong YM, Liu YL (2018) Abnormal changes of dynamic derivatives at low reduced frequencies. Chin J Aeronaut 31(7):1428–1436CrossRefGoogle Scholar
- 63.Stetson KF, Sawyer FM (1977) A comparison of hypersonic wind tunnel data obtained by static and free oscillation techniques. In: AIAA 10th fluid & plasmadynamics conferenceGoogle Scholar
- 64.Liu X, Liu W, Zhao YF (2015) Unsteady vibration aerodynamic modeling and evaluation of dynamic derivatives using computational fluid dynamics. Math Probl Eng 813462:1–16MathSciNetzbMATHGoogle Scholar
- 65.Tuling S (2006) Modelling of dynamic stability derivatives using CFD. In: 25th international congress of the aeronautical sciencesGoogle Scholar
- 66.Dufour G, Sicot F, Puigt G et al (2010) Contrasting the harmonic balance and linearized methods for oscillating-flap simulations. AIAA J 48(4):788–797CrossRefGoogle Scholar
- 67.Ekici K, Hall KC, Dowell EH (2008) Computationally fast harmonic balance methods for unsteady aerodynamic predictions of helicopter rotors. J Comput Phys 227(12):6206–6225CrossRefzbMATHGoogle Scholar
- 68.Mcmullen M, Jameson A, Alonso J (2006) Demonstration of nonlinear frequency domain methods. AIAA J 44(7):1428–1435CrossRefGoogle Scholar
- 69.Thomas JP, Dowell E, Hall K et al (2002) Nonlinear inviscid aerodynamic effects on transonic divergence, flutter, and limit-cycle oscillations. AIAA J 40(4):638–646CrossRefGoogle Scholar
- 70.Hall K, Ekici K, Thomas J et al (2013) Harmonic balance methods applied to computational fluid dynamics problems. Int J Comput Fluid Dyn 27(2):52–67CrossRefMathSciNetGoogle Scholar
- 71.Murman SM (2007) Reduced-frequency approach for calculating dynamic derivatives. AIAA J 45(6):1161–1168CrossRefGoogle Scholar
- 72.Ronch AD, McCracken AJ, Badcock KJ et al (2013) Linear frequency domain and harmonic balance predictions of dynamic derivatives. J Aircr 50(3):694–706CrossRefGoogle Scholar
- 73.Hassan D, Sicot F (2011) A time-domain harmonic balance method for dynamic derivatives predictions. In: 49th AIAA aerospace sciences meeting including the new horizons forum and aerospace expositionGoogle Scholar
- 74.Cherif MA, Emamirad H, Mnif M (2012) Derivatives for time-spectral computational fluid dynamics using an automatic differentiation adjoint. AIAA J 50(12):2809–2819CrossRefGoogle Scholar
- 75.Xie L, Yang Y, Zhou L et al (2015) High-performance computing of periodic unsteady flow based on time spectral method. Procedia Eng 99:1526–1530CrossRefGoogle Scholar
- 76.Byushgens GS (1999) Aerodynamics, stability and control of supersonic aircraft. Science, MoscowGoogle Scholar
- 77.Coulter SM, Marquart EJ (1982) Cross and cross-coupling derivative measurements on the standard dynamics model at AEDC. In: 12th aerodynamic testing conferenceGoogle Scholar
- 78.Mi BG, Zhan H, Chen BB (2018) Numerical simulation of static and dynamic aerodynamics for formation flight with UCAVs. J Eng Res 6(3):203–224Google Scholar
- 79.Bragg M, Hutchison T, Merret J et al (2000) Effect of ice accretion on aircraft flight dynamics. In: 38th aerospace sciences meeting & exhibitGoogle Scholar
- 80.Ratcliff CJ, Bodkin DJ, Clifton J et al (2016) Virtual flight testing of high performance flighter aircraft using high-resolution CFD. In: AIAA atmospheric flight mechanics conferencesGoogle Scholar
- 81.Da X, Yang T, Zhao Z (2012) Virtual flight Navier–Stokers solver and its application. Procedia Eng 31(1):75–79CrossRefGoogle Scholar
- 82.Mazuroski W, Berger J, Oloveira RCLF et al (2018) An artificial intelligence-based method to efficiently bring CFD to building simulation. J Build Perform Simul 2:1–16Google Scholar
- 83.Sotgiu C, Weigand B, Semmler K (2018) A turbulent heat flux prediction framework based on tensor representation theory and machine learning. Int Commun Heat Mass Trans 95:74–79CrossRefGoogle Scholar