RANS study of Strouhal number effects on the stability derivatives of an autonomous underwater vehicle

  • M. H. Shojaeefard
  • A. Khorampanahi
  • M. Mirzaei
Technical Paper


This paper presents a numerical procedure to calculate the stability derivatives of an Autonomous Underwater Vehicle (AUV). In addition, the effects of Strouhal number on the stability derivatives are investigated. The stability derivatives are obtained by finding the body hydrodynamic responses to some specified time variant motions. Here, three distinct oscillating maneuvers: surge, pure-heave, and pure-pitch are proposed based on the linearized equations of motion. A Computational Fluid Dynamics (CFD) method based on Reynolds Averaged Navier–Stokes (RANS) equations with dynamic mesh technique is used to simulate the specified maneuvers. To verify the numerical scheme, computational results for simulation of the flow field are then validated by comparison with experimental data. In addition, a comparison between quasi-steady (Theodorsen’s theory) and RANS approaches is taken into account, which shows that the quasi-steady methods do not guarantee the solution accuracy for the purpose of stability derivative calculation. Finally, the effects of Strouhal number variations on these coefficients are investigated which shows that the dimensionless damping coefficients (except longitudinal damping coefficient) are evidently dependent on the Strouhal number, specially at high frequencies. However, the Strouhal number variation effects on the added-mass coefficients are small.


Stability derivatives Theodorsen’s theory Strouhal number Dynamic mesh 

List of symbols


Wake width, approximated as 2z0


Mean thrust force coefficient


Moment of inertia in pitch


Thrust force


Hankel functions of the first kind and the second kind


Reynolds number


Strouhal number


Period of oscillations (s)


Surge and heave forces and pitch moment

X′, Z′, M

Dimensionless surge, heave forces, and pitch moment


Hydrofoil chord


Frequency of oscillation (1/s)


Amplitude of surge, heave, and pitch motions


Reference length


Mass of the body


Time variable


Surge, heave, and pitch velocities


Steady velocities in surge, heave, and pitch directions

u′, w′, q

Fluctuating velocities in surge, heave, and pitch directions


Dimensionless velocities in surge, heave, and pitch directions


Surge and heave positions


Position of center of gravity in xz plane


Pitch angle




Frequency of oscillation (rad/s)


  1. 1.
    Azarsina F (2009) Experimental hydrodynamics and simulation of maneuvering of an axisymmetric underwater vehicle [D]. Memorial University of Newfoundland, Faculty of Engineering and Applied Science, NewfoundlandGoogle Scholar
  2. 2.
    Mirzaei M, Alishahi MM (2015) EGHTESAD M. High-speed underwater projectiles modeling: a new empirical approach [J]. J Braz Soc Mech Sci Eng 37(2):613–626CrossRefGoogle Scholar
  3. 3.
    Greenwell DI (1998) Frequency effects on dynamic stability derivatives obtained from small-amplitude oscillatory testing [J]. J Aircr 35(5):776–783CrossRefGoogle Scholar
  4. 4.
    Pamadi BN (1998) Performance, stability, dynamics, and control of airplanes [M], AIAA Education SeriesGoogle Scholar
  5. 5.
    Ellison DE, Malthan LV (1965) USAF stability and control DATCOM [M], Douglas Aircraft Company, Inc.Google Scholar
  6. 6.
    Ronch AD (2012) On the calculation of dynamic derivatives using computational fluid dynamics [D]. University of Liverpool, LiverpoolGoogle Scholar
  7. 7.
    Saeidinezhad A, Dehghan AA, MANSHADI MD (2015) Experimental investigation of hydrodynamic characteristics of a submersible vehicle model with a non-axisymmetric nose in pitch maneuver. Ocean Eng 100:26–34CrossRefGoogle Scholar
  8. 8.
    Randeni PAT, Leong ZQ, Ranmuthugala D, Forrest AL, Duffy J (2015) Numerical investigation of the hydrodynamic interaction between two underwater bodies in relative motions. Appl Ocean Res 51:14–24CrossRefGoogle Scholar
  9. 9.
    Hess JL, Smith AMO (1962) Calculation of non-lifting potential flow about arbitrary three-dimensional bodies. Report. No. E.S. 40622, Douglas Aircraft CompanyGoogle Scholar
  10. 10.
    Sahin I, Cranet JW, Watsont KP (1997) Application of a panel method to hydrodynamics of underwater vehicles. Ocean Eng 24(6):501–512CrossRefGoogle Scholar
  11. 11.
    Lamantia M, Dabnichki P (2012) Added mass effect on flapping foil. Eng Anal Boundary Elem 36:579–590MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Tyagi A, Sen D (2006) Calculation of transverse hydrodynamic coefficients using computational fluid dynamic approach. Ocean Eng 33:798–809CrossRefGoogle Scholar
  13. 13.
    Pan YC, Zhang HX, Zhou QD (2012) Numerical prediction of submarine hydrodynamic coefficients using CFD simulation. J Hydrodyn Ser. B, Elsevier 24(6): 840–847Google Scholar
  14. 14.
    Malik SA, Guang P (2013) Transient numerical simulation for hydrodynamic derivates predictions of an axisymmetric submersible vehicle. Res J Appl Sci Eng Technol 5(21):5003–5011Google Scholar
  15. 15.
    Leong ZQ, Ranmuthugala D, Penesis I, Nguyen HD (2015) RANS-based CFD prediction of the hydrodynamic coefficients of DARPA SUBOFF geometry in straight-line and rotating arm maneuvers. Trans RINA, Part A1, Int J Marit Eng 157:A41–A52Google Scholar
  16. 16.
    Abkowitz MA (1969) Stability and motion control of ocean vessels. M.I.T. Press, Massachusetts Institute of TechnologyGoogle Scholar
  17. 17.
    Brayshaw I (199) Hydrodynamic coefficients of underwater vehicles. Maritime Platforms Division, Aeronautical and Maritime Research Laboratories. DSTO, MelbourneGoogle Scholar
  18. 18.
    Oller ED (2003) Forces and moments due to unsteady motion of an underwater vehicle. Massachusetts Institute of Technology, MassachusettsCrossRefGoogle Scholar
  19. 19.
    Kim H, Akimoto H, Islam H (2015) Estimation of the hydrodynamic derivatives by RANS simulation of planar motion mechanism test. Ocean Eng 108:129–139CrossRefGoogle Scholar
  20. 20.
    Ferziger JH, Peric M (2002) Computational methods for fluid dynamics, 3rd edn. Springer-Verlag, GermanyCrossRefzbMATHGoogle Scholar
  21. 21.
    Anderson JM, Streitlien K, Barrett DS, Triantafyllou MS (1998) Oscillating foils of high propulsive efficiency. J Fluid Mech 360:41–72MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Menter FR (1994) Two-equation eddy-viscosity turbulence models for engineering applications. AIAA Journal 32(8):1598–1605CrossRefGoogle Scholar
  23. 23.
    Wilcox DC (1998) Turbulence modeling for CFD, 2nd edn. DCW Industries Inc, CaliforniaGoogle Scholar
  24. 24.
    Theodorsen T (1992) General theory of aerodynamic instability and the mechanism of flutter. NACA Report 496, 1935. Also included in ‘A Modern View of Theodore Theodorsen’ published by AIAA, pp 2–21Google Scholar
  25. 25.
    Wright JR, Cooper JE (2015) Introduction to aircraft aeroelasticity and loads, 1st edn. Wiley, New YorkGoogle Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2018

Authors and Affiliations

  • M. H. Shojaeefard
    • 1
  • A. Khorampanahi
    • 1
  • M. Mirzaei
    • 2
  1. 1.School of Mechanical EngineeringIran University of Science and TechnologyTehranIran
  2. 2.Hydro-Aeronautical Research Center, Shiraz UniversityShirazIran

Personalised recommendations