Skip to main content
Log in

Review of Numerical Simulations on Aircraft Dynamic Stability Derivatives

  • Original Paper
  • Published:
Archives of Computational Methods in Engineering Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29
Fig. 30
Fig. 31
Fig. 32
Fig. 33
Fig. 34
Fig. 35
Fig. 36
Fig. 37
Fig. 38
Fig. 39
Fig. 40
Fig. 41
Fig. 42

Similar content being viewed by others

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

\(a_{\infty }\) :

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

References

  1. Gursul I (2015) Unsteady flow phenomena over delta wings at high angle of attack. AIAA J 32(2):225–231

    Google 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):734–746

    Google Scholar 

  3. Huang W, Liu J, Wang ZG (2012) Investigation on high angle of attack characteristics of hypersonic space vehicle. Sci China 55(5):1437–1442

    Google Scholar 

  4. Dowell EH, Williams MH, Bland SR (2015) Linear/nonlinear behavior in unsteady transonic aerodynamics. AIAA J 21(1):38–46

    MATH  Google 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):472–484

    Google Scholar 

  6. Kramer B (2013) Experimental evaluation of superposition techniques applied to dynamic aerodynamics. In: AIAA aerospace sciences meeting & exhibit

  7. Greenwell DI (2015) Frequency effects on dynamic stability derivatives obtained from small-amplitude oscillatory testing. J Aircr 35(5):776–783

    Google 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 conference

  9. Dudley R (1999) Unsteady aerodynamics. Science 284(5422):1937–1939

    Google Scholar 

  10. Yukovich R, Liu D, Chen P (2013) State-of-the-art of unsteady aerodynamics for high performance aircraft. In: Aerospace sciences meeting & exhibit

  11. Queijo MJ, Wells WR, Keskar DA (1978) Influence of unsteady aerodynamics on extracted aircraft parameters. J Aircr 16(10):708–713

    Google 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):668–676

    Google Scholar 

  13. Bryan GH (1911) Stability in aviation. Macmillan, London

    MATH  Google Scholar 

  14. Boyd TJM (2011) One hundred years of G. H. Bryan’s stability in aviation. J Aeronaut Hist 4:97–115

    Google Scholar 

  15. Tong BG, Chen Q (1983) Some remarks on unsteady aerodynamics. Adv Mech 4:377–394

    Google Scholar 

  16. Etkin B, Reid LD (1996) Dynamics of flight: stability and control. Wiley, New York, p 107

    Google Scholar 

  17. Etkin B (2012) Dynamics of atmospheric flight. Dover Publications, New York, p 125

    Google Scholar 

  18. Tobak M, Schiff LB (1981) Aerodynamic mathematical modeling-basic concepts. AGARD Lect Ser 77(114):1–32

    Google Scholar 

  19. Tobak M, Schiff LB (1976) On the formulation of the aerodynamic characteristics in aircraft dynamics: NASA TRR-456. NASA, Washington, DC

    Google 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, DC

    Google Scholar 

  21. Theodorsen T, Garrick I (1933) General potential theory of arbitrary wing sections. US Government Printing Office, New York, pp 77–80

    MATH  Google Scholar 

  22. Theodorsen T (1935) General theory of aerodynamic instability and the mechanism of flutter. Rept. 496, NACA

  23. Wukelich SR, Williams HE (1979) The USAF stability and control digital Datcom. AFFDL-TR-79-3030

  24. Mattsaits GR (1982) An update of the digital Datcom computer code for estimating dynamic stability derivatives. AEDC-TR-81-30

  25. Blake WB (1985) Prediction of fighter aircraft dynamic derivatives using Digital Datcom. In: AIAA 3rd applied aerodynamics conference

  26. Jaslow H (2015) Aerodynamic relationships inherent in Newtonian impact theory. AIAA Journal 6(4):608–612

    Google Scholar 

  27. Tobak M, Wehrend WR (1956) Stability derivatives of cones at supersonic speeds. NACA TN 3788, NACA, Washington, DC

  28. Busemann A (1933) Handbook of natural sciences, IV, liquid and garvewegung. 2nd Jena: Gustav Fisher, 12–55

  29. Hui W, Tobak M (1981) Unsteady Newton-Busemann flow theory. Part 2: bodies of revolution. AIAA J 19(10):1272–1273

    MATH  Google Scholar 

  30. Ericsson LE (1968) Unsteady aerodynamics of an ablating flared body of revolution including effect of entropy gradient. AIAA J 6(5):2395–2401

    Google 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):309–330

    Google Scholar 

  32. Ashley H, Zartarian G (1956) Piston theory, a new aerodynamic tool for the aeroelastican. J Aeronaut Sci 23(12):1109–1118

    Google 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):469–476

    MathSciNet  Google Scholar 

  34. Zhang WW, Ye ZY, Zhang CA et al (2009) Supersonic flutter analysis based on a local piston theory. AIAA J 47(10):2321–2328

    Google Scholar 

  35. Ye C, Ma DL (2012) An aircraft steady dynamic derivatives calculation method. In: Proceedings of 2012 international conference on modelling, identification and control

  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):1142–1153

    Google Scholar 

  37. Stalnaker JF (2004) Rapid computation of dynamic stability derivatives. In: 42nd AIAA aerospace sciences meeting & exhibit

  38. Park MA, Green L(2000) Steady-state computation of constant rotational rate dynamic stability derivatives. In: 18th applied aerodynamics conference

  39. Guglieri G, Quagliotti FB (1993) Dynamic stability derivatives evaluation in a low-speed wind tunnel. J Aircr 30(3):421–423

    Google Scholar 

  40. Hanff ES, Orlikruckemann KJ (2015) Wind-tunnel measurement of dynamic cross-coupling derivatives. J Aircr 15(1):40–46

    Google 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 sciences

  42. Liu X, Liu W, Zhao YF (2016) Navier–Stokes predictions of dynamic stability derivatives for air-breathing hypersonic vehicle. Acta Astronaut 118:262–285

    Google Scholar 

  43. Moelyadi MA, Sachs G (2007) CFD based determination of dynamic stability derivatives in yaw for a bird. J Bionic Eng 4:201–208

    Google Scholar 

  44. Hui WH (1969) Stability of oscillating wedges and caret wings in hypersonic and supersonic flows. AIAA J 7(8):1524–1530

    Google Scholar 

  45. Roy JFL, Morgand S (2010) SACCON CFD static and dynamic derivatives using elsA. In: 28th AIAA applied aerodynamics conference

  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 conference

  47. Alemdaroglu N, Iyigun I, Altun M et al (2002) Determination of dynamic stability derivatives using forced oscillation technique. In: 40th aerospace sciences meeting & exhibit

  48. Ronch AD, Vallespin D, Ghoreyshi M et al (2012) Evaluation of dynamic derivatives using computational fluid dynamics. AIAA J 50(2):470–484

    Google Scholar 

  49. Ronch AD (2012) On the calculation of dynamic derivatives using computational fluid dynamics. University of Liverpool

  50. Moore FG, Swanson RC (1972) Dynamic derivatives for missile configurations to Mach number three. J Spacecr 15(2):65–66

    Google 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 & exhibit

  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 exposition

  53. Oktay E, Akay H (2002) CFD predictions of dynamic derivatives for missiles. In: 40th AIAA aerospace sciences meeting & exhibit

  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 conference

  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 47:674–694

    Google 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 conference

  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 dynamics

  58. Green LL, Spence AM, Murphy PC (2004) Computational methods for dynamic stability and control derivatives. In: 42nd AIAA aerospace sciences meeting & exhibit

  59. Mi BG, Zhan H (2017) Calculating dynamic derivatives of flight vehicle with new engineering strategies. Int J Aeronaut Space Sci 18(2):175–185

    Google Scholar 

  60. Ghoreyshi M, Lofthouse AJ (2017) Indicial methods for the numerical calculation of dynamic derivatives. AIAA J 55(7):2279–2294

    Google 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 4217217:1–11

    Google Scholar 

  62. Zhang WW, Gong YM, Liu YL (2018) Abnormal changes of dynamic derivatives at low reduced frequencies. Chin J Aeronaut 31(7):1428–1436

    Google 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 conference

  64. Liu X, Liu W, Zhao YF (2015) Unsteady vibration aerodynamic modeling and evaluation of dynamic derivatives using computational fluid dynamics. Math Probl Eng 813462:1–16

    MathSciNet  MATH  Google Scholar 

  65. Tuling S (2006) Modelling of dynamic stability derivatives using CFD. In: 25th international congress of the aeronautical sciences

  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):788–797

    Google 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):6206–6225

    MATH  Google Scholar 

  68. Mcmullen M, Jameson A, Alonso J (2006) Demonstration of nonlinear frequency domain methods. AIAA J 44(7):1428–1435

    Google 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):638–646

    Google 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):52–67

    MathSciNet  Google Scholar 

  71. Murman SM (2007) Reduced-frequency approach for calculating dynamic derivatives. AIAA J 45(6):1161–1168

    Google 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):694–706

    Google 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 exposition

  74. Cherif MA, Emamirad H, Mnif M (2012) Derivatives for time-spectral computational fluid dynamics using an automatic differentiation adjoint. AIAA J 50(12):2809–2819

    Google 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 99:1526–1530

    Google Scholar 

  76. Byushgens GS (1999) Aerodynamics, stability and control of supersonic aircraft. Science, Moscow

    Google Scholar 

  77. Coulter SM, Marquart EJ (1982) Cross and cross-coupling derivative measurements on the standard dynamics model at AEDC. In: 12th aerodynamic testing conference

  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):203–224

    Google Scholar 

  79. Bragg M, Hutchison T, Merret J et al (2000) Effect of ice accretion on aircraft flight dynamics. In: 38th aerospace sciences meeting & exhibit

  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 conferences

  81. Da X, Yang T, Zhao Z (2012) Virtual flight Navier–Stokers solver and its application. Procedia Eng 31(1):75–79

    Google 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 2:1–16

    Google 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 95:74–79

    Google Scholar 

Download references

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).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Baigang Mi.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mi, B., Zhan, H. Review of Numerical Simulations on Aircraft Dynamic Stability Derivatives. Arch Computat Methods Eng 27, 1515–1544 (2020). https://doi.org/10.1007/s11831-019-09370-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11831-019-09370-8

Navigation