ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Sizykh G. B. New Lagrangian view of vorticity evolution in two-dimensional flows of liquid and gas. Izvestiya VUZ. Applied Nonlinear Dynamics, 2022, vol. 30, iss. 1, pp. 30-36. DOI: 10.18500/0869-6632-2022-30-1-30-36

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text:
(downloads: 722)
Article type: 
Short communication

New Lagrangian view of vorticity evolution in two-dimensional flows of liquid and gas

Sizykh Grigory B., Moscow Institute of Physics and Technology

Purpose of the study is to obtain formulas for such a speed of imaginary particles that the circulation of the speed of a (real) fluid along any circuit consisting of these imaginary particles changes (in the process of motion of imaginary particles) according to a given time law. (Until now, only those speeds of imaginary particles were known, at which the mentioned circulation during the motion remained unchanged). Method. Without implementation of asymptotic, numerical and other approximate methods, a rigorous analysis of the dynamic equation of motion (flow) of any continuous fluid medium, from an ideal liquid to a viscous gas, is carried out. Plane-parallel and nonswirling axisymmetric flows are considered. The concept of motion of imaginary particles is used, based on the K. Zoravsky criterion (which is also called A. A. Fridman’s theorem). Results. Formulas for the speed of imaginary particles are proposed. These formulas include the parameters of the (real) flow, their spatial derivatives and the function of time, which determines the law of the change in time of the (real fluid) velocity circulation along the contours moving together with the imaginary particles. In addition, it turned out that for a given function of time (and, as a consequence, for a given law of change in circulation with respect to time), the speed of imaginary particles is determined ambiguously. As a result, a method is proposed to change the speed and direction of motion of imaginary particles in a certain range (while maintaining the selected law of changes in circulation in time). For a viscous incompressible fluid, formulas are proposed that do not include pressure and its derivatives. Conclusion. A new Lagrangian point of view on the vorticity evolution in two-dimensional flows of fluids of all types is proposed. Formulas are obtained for the velocity of such movement of contours, at which the real fluid velocity circulation along any contour changes according to a given time law. This theoretical result can be used in computational fluid dynamics to limit the number of domains when using a gridless method for calculating flows of a viscous incompressible fluid (the method of viscous vortex domains).

  1. Rosenhead L. The formation of vortices from a surface of discontinuity. Proc. R. Soc. Lond. A. 1931;134(823):170–192. DOI: 10.1098/rspa.1931.0189.
  2. Belotserkovskii SM, Nisht MI. Separated and Unseparated Ideal Liquid Flow around Thin Wings. Moscow: Nauka; 1978. 352 p. (in Russian).
  3. Cottet GH, Koumoutsakos PD. Vortex Methods: Theory and Practice. Cambridge: Cambridge University Press; 2000. 320 p. DOI: 10.1017/CBO9780511526442.
  4. Golubkin VN, Sizykh GB. Some general properties of plane-parallel viscous flows. Fluid Dynamics. 1987;22(3):479–481. DOI: 10.1007/BF01051932.
  5. Brutyan MA, Golubkin VN, Krapivskii PL. On the Bernoulli equation for axisymmetric viscous fluid flows. TsAGI Science Journal. 1988;19(2):98–100 (in Russian).
  6. Dynnikova GY. Lagrange method for Navier–Stokes equations solving. Proceedings of the Academy of Sciences. 2004;399(1):42–46 (in Russian).
  7. Markov VV, Sizykh GB. Vorticity evolution in liquids and gases. Fluid Dynamics. 2015;50(2): 186–192. DOI: 10.1134/S0015462815020027.
  8. Dynnikova GY, Dynnikov YA, Guvernyuk SV, Malakhova TV. Stability of a reverse Karman vortex street. Physics of Fluids. 2021;33(2):024102. DOI: 10.1063/5.0035575.
  9. Kuzmina K, Marchevsky I, Soldatova I, Izmailova Y. On the scope of Lagrangian vortex methods for two-dimensional flow simulations and the POD technique application for data storing and analyzing. Entropy. 2021;23(1):118. DOI: 10.3390/e23010118. 
  10. Leonova D, Marchevsky I, Ryatina E. Fast methods for vortex influence computation in meshless Lagrangian vortex methods for 2D incompressible flows simulation. WIT Transactions on Engineering Sciences. 2019;126:255–267. DOI: 10.2495/BE420231.
  11. Sizykh GB. Entropy value on the surface of a non-symmetric convex bow part of a body in the supersonic flow. Fluid Dynamics. 2019;54(7):907–911. DOI: 10.1134/S0015462819070139.
  12. Sizykh GB. Closed vortex lines in fluid and gas. Journal of Samara State Technical University. Ser. Physical and Mathematical Sciences. 2019;23(3):407–416 (in Russian). DOI: 10.14498/vsgtu1723.
  13. Mironyuk IY, Usov LA. The invariant of stagnation streamline for a stationary vortex flow of an ideal incompressible fluid around a body. Journal of Samara State Technical University. Ser. Physical and Mathematical Sciences. 2020;24(4):780–789 (in Russian). DOI: 10.14498/vsgtu1815.
  14. Kotsur OS. On the existence of local formulae of the transfer velocity of local tubes that conserve their strengths. Proceedings of MIPT. 2019;11(1):76–85 (in Russian).
  15. Mironyuk IY, Usov LA. Stagnation points on vortex lines in flows of an ideal gas. Proceedings of MIPT. 2020;12(4):171–176 (in Russian). DOI: 10.53815/20726759_2020_12_4_171.
  16. Sizykh GB. On the collinearity of vortex and the velocity behind a detached bow shock. Proceedings of MIPT. 2021;13(3):144–147 (in Russian). DOI: 10.53815/20726759_2021_13_3_144.
  17. Sizykh GB. Second integral generalization of the Crocco invariant for 3D flows behind detached bow shock wave. Journal of Samara State Technical University. Ser. Physical and Mathematical Sciences. 2021;25(3):588–595 (in Russian). DOI: 10.14498/vsgtu1861.
  18. Prim R, Truesdell C. A derivation of Zorawski’s criterion for permanent vector-lines. Proc. Amer. Math. Soc. 1950;1:32–34.
  19. Truesdell C. The Kinematics of Vorticity. Bloomington: Indiana University Press; 1954. 232 p.
  20. Friedman AA. Experience in the Hydromechanics of Compressible Fluid. Moscow: ONTI; 1934. 370 p. (in Russian).