Journal of Engineering Mathematics

, Volume 55, Issue 1–4, pp 313–338 | Cite as

Basic properties of free stratified flows

  • Yuli D. Chashechkin
  • Vasiliy G. Baydulov
  • Anatoliy V. Kistovich


Results of analytical studies of the governing equations of stratified rotating fluids based on the unification of theories of continuous and discrete groups, perturbations and modern numerical visualizations are described. Symmetries of basic systems and their simplified versions, different approximations and constitutive turbulent models are compared. A new method to calculate discrete groups analytically, which does not need a preliminary search for continuous groups, is developed. As an example of the practical use of the developed algorithm, a complete classification of cellular and roll structures of Bénard convection is presented. A complete classification of 3D periodic motions in compressible viscous stratified and rotating fluids, including regular (wave) and singular elements, is performed by perturbation methods. In all cases, in a viscous fluid, besides waves there are two different periodic boundary layers. In a homogeneous fluid the split boundary layers are merged, thus forming a joint doubly-degenerate structure. Stratification and rotation reduce the degeneration of the 3D periodic boundary layers. Calculations of a 3D periodic wave beam emitted by an oscillating part of a sloping plane are visualized by a computer-graphics method. The existence of thin extended components on the edges of the 3D wave cone is demonstrated.


boundary layers complete equations continuous and discrete groups exact solutions internal waves 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    L. D. Landau and E. M. Lifshitz, Course of theoretical physics, In: Vol. 6: Fluid Mechanics. Moscow: (1986) 136 pp; New York: Pergamon (1987).Google Scholar
  2. 2.
    Pedlosky J. (1987). Geophysical Fluid Dynamics 2nd edition. Springer-Verlag, New York, 710 pp.MATHGoogle Scholar
  3. 3.
    Lighthill J. (1978). Waves in Fluids. Cambridge University Press, CambridgeMATHGoogle Scholar
  4. 4.
    Ovsiannikov L.V. (1982). Group Analysis of Differential Equations. Academic Press, New YorkMATHGoogle Scholar
  5. 5.
    Ovsiannikov L.V. (1994). Programme SUBMODELS. Gas dynamics. J. Appl. Math. Mech. 58(4):30–55Google Scholar
  6. 6.
    V. G. Baidulov and Yu. D. Chashechkin, Invariant properties of the equations of motion of stratified fluids. Dok. Phys. 47(12) (2002) 888–891. (Translated from Dokl. Akad. Nauk 387(6) 760–763.)Google Scholar
  7. 7.
    E. V. Bruyatskii, Turbulentnye Stratifitsirovannye Struinye Techeniya. (in Russian, Turbulent Stratified Jet Flows). Kiev: Naukova Dumka (1986) 296 pp.Google Scholar
  8. 8.
    Rodi W. (1980). Turbulence Models and Their Application in Hydraulics – A State of the Art Review. SBF Report 80/T/125. University of Karlsruhe, KarlsruheGoogle Scholar
  9. 9.
    Kistovich A.V., Chashechkin Yu.D. (2001). A Regular method of searching for discrete symmetries in models of physical processes. Dokl. Phys. 46(10):718–721MathSciNetGoogle Scholar
  10. 10.
    Kistovich A.V., Chashechkin Yu.D. (2001). Regular method for searching of differential equations discrete symmetries. Regular and Chaotic Dynamics 6(3):327–336MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    A. V. Kistovich and Yu. D. Chashechkin, Types of discrete symmetries of convection in a plane fluid layer. Dok. Phys. 47(6) (2002) 458–460. (Translated from Dokl. Akad. Nauk 384(5), 630–633.)Google Scholar
  12. 12.
    White D.B. (1988). The planforms and onset of convection with a temperature-dependent viscosity. J. Fluid Mech. 191:247–286CrossRefADSGoogle Scholar
  13. 13.
    Palm E., Ellingsen T., Gjevik B. (1967). On the occurrence of cellular motion in Benard convection. J. Fluid Mech. 30:651–661MATHCrossRefADSGoogle Scholar
  14. 14.
    Chashechkin Yu.D., Kistovich A.V. (2004). Classification of three-dimensional periodic fluid flows. Dokl. Phys. 49(3):183–186CrossRefMathSciNetGoogle Scholar
  15. 15.
    Chashechkin Yu.D., Mitkin V.V. (2001). Experimental study of a fine structure of 2D wakes and mixing past an obstacle in a continuously stratified fluid. Dyn. Atmosph. Oceans 34:165–187CrossRefADSGoogle Scholar
  16. 16.
    Chashechkin Yu.D., Kistovich A.V. (2003). Calculation of the structure of periodic flows in a continuously stratified fluid with allowance for diffusion. Dokl. Phys. 48(12):710–714CrossRefMathSciNetGoogle Scholar
  17. 17.
    S. A. Gabov and A. G. Sveshnikov, Zadachi Dinamiki Stratifitsirovannykh Zhidkostei. (in Russian, Problems of Stratified Fluids Dynamics). Moskva: Nauka (1986) 288 pp.Google Scholar
  18. 18.
    Kistovich Yu.V., Chashechkin Yu.D. (1999). Non-linear generation of periodic internal waves by a boundary-layer flow around a rotating axially symmetric body. Dokl. Phys. 44(8):573–576ADSMathSciNetGoogle Scholar
  19. 19.
    Il’inykh Yu.S., Chashechkin Yu.D. (2004). Generation of periodic motions by a disk performing torsional oscillations in a viscous continuously stratified fluid. Fluid Dyn. 39(1):148–161MATHCrossRefGoogle Scholar
  20. 20.
    Yu. V. Kistovich and Yu. D. Chashechkin, A new mechanism of the non-linear generation of internal waves. Dokl. Phys. 47(2) (2002) 163–167. (Translated from Dokl. Akad. Nauk 382(6) 772–776).Google Scholar
  21. 21.
    Kistovich Yu.V., Chashechkin Yu.D. (2001). Some exactly solvable problems of the radiation of three-dimensional periodic internal waves. J. Appl. Mech. Tech. Phys. 42(2):228–236CrossRefMathSciNetGoogle Scholar
  22. 22.
    Holm D.D., Kimura Y. (1991). Zero-helicity Lagrangian kinematics of three-dimensional advection. Phys. Fluids A3:1033–1038ADSMathSciNetGoogle Scholar
  23. 23.
    Vasiliev A.Yu., Chashechkin Yu.D. (2003). The generation of beams of three–dimensional periodic internal waves in an exponentially stratified fluid. J. Appl. Math. Mech. 67(3):397–405CrossRefMathSciNetGoogle Scholar
  24. 24.
    McEwan A.D., Plumb P.A. (1977). Off-resonant amplification of finite internal wave packets. Dyn. Atmosph. Oceans 2:83–105CrossRefGoogle Scholar
  25. 25.
    Teoh S.G., Ivey G.N., Imberger J. (1997). Laboratory study of the interaction between two internal wave rays. J. Fluid Mech. 336: 91–121CrossRefADSGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  • Yuli D. Chashechkin
    • 1
  • Vasiliy G. Baydulov
    • 1
  • Anatoliy V. Kistovich
    • 1
  1. 1.Laboratory of Fluid MechanicsInstitute for Problems in Mechanics of the RAS 101/1 prospect VernadskogoMoscowRussia

Personalised recommendations