On a new Lagrangian view on the evolution of vorticity in~spatial flows

I. A. Maksimenko ; V. V. Markov

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 26 (2022) no. 1, pp. 179-189.

Résumé

The purpose of the study is to extend to the spatial case proposed by G. B. Sizykh approach to a two-dimensional vorticity evolution, which is based on the idea of considering a vorticity evolution in the form of such a motion of vortex lines and tubes that the intensity of these tubes changes over time according to a predefined law. Method. Thorough analysis is determined by describing the flow velocity field of an ideal incompressible fluid and a viscous gas in the general case, using the idea of the movement of imaginary particles. Results. For any given time law of change of velocity circulation (i. e. for an exponential decay) of a real fluid along the contours the method of evaluating the field of velocity of such contours and vortex tubes is proposed (e. g. getting a field of imaginary particles, which transfer them). It is established that for a given time law the velocity of imaginary particles is determined ambiguously, and the method of how to adjust their motion preserving defined law of circulation change is proposed. Conclusion. A new Lagrangian approach to the evolution of vorticity in three-dimensional flows is derived, as well as the expressions for the contours' velocity, which imply stated changing over the time of the velocity circulation of a real fluid along any contour. This theoretical result can be utilized in spatial modifications of the viscous vortex domain method to limit the number of vector tubes used in calculations.

Détail

Keywords: contour velocity, contour intensity, imaginary fluid motion, Zoravski's criterion, Friedmann's theorem
Mots-clés : viscous vortex domain method.

Bibliographie

[1] Rosenhead L., “The formation of vortices from a surface of discontinuity”, P. Roy. Soc. Lond., 1931, 170–192 | DOI

[2] Belotserkovskii S. M., Nisht M. I., Otryvnoe i bezotryvnoe obtekanie tonkikh kryl'ev ideal'noi zhidkost'iu [Separated and Unseparated Ideal Liquid Flow around thin Wings], Nauka, Moscow, 1978, 352 pp. (In Russian)

[3] Cottet G.-H., Koumoutsakos P., Vortex Methods. Theory and Practice, Cambridge Univ. Press, 2000, xiv+313 pp. | DOI

[4] Aparinov A. A., Setukha A. V., Zhelannikov A. I., “Numerical simulation of separated flow over three-dimensional complex shape bodies with some vortex method”, AIP Conference Proceedings, 1629:1 (2014), 69 | DOI

[5] Aparinov A. A., Kritskii B. S., Setukha A. V., “Numerical modeling of helicopter main rotor behavior near a small-scale helideck by the vortex method”, Russ. Aeronaut., 60:4 (2017), 500–507 | DOI

[6] Aparinov A. A., Aparinov V. A., Setukha A. V, “Supercomputer modeling of parachute flight dynamics”, Supercomputing Frontiers and Innovations, 5:3 (2018), 121–125 | DOI

[7] Golubkin V. N., Sizykh G. B., “Some general properties of plane-parallel viscous flows”, Fluid Dyn., 22:3 (1987), 479–481

[8] Brutyan M. A., Golubkin V. N., Krapivskii P. L., “On the Bernoulli equation for axisymmetric viscous fluid flows”, Uch. zap. TsAGI [TsAGI Science Journal], 19:2 (1988), 98–100 (In Russian)

[9] Dynnikova G. Ya., “The Lagrangian approach to the solution of non-stationary Navier–Stokes equations”, Dokl. Math., 49:11 (2004), 648–652 | MR

[10] Andronov P. R., Guvernyuk S. V., Dynnikova G. Ya., Vikhrevye metody rascheta nestatsionarnykh gidrodinamicheskikh nagruzok [Vortex Methods for Calculating Non-Stationary Hydrodynamic Loads], Moscow Univ., Moscow, 2006, 184 pp. (In Russian)

[11] Markov V. V., Sizykh G. B., “Vorticity evolution in liquids and gases”, Fluid Dyn., 50:2 (2015), 186–192 | DOI | Zbl

[12] Dynnikova G. Ya., “Calculation of flow around a circular cylinder on the basis of two-dimensional Navier–Stokes equations at large Reynolds numbers with high resolution in a boundary layer”, Dokl. Phys., 53:10 (2008), 544–547 | DOI | Zbl

[13] Dynnikova G. Ya., Dynnikov Ya. A., Guvernyuk S. V., Malakhova T. V., “Stability of a reverse Karman vortex street”, Physics of Fluids, 33:2 (2021), 024102 | DOI

[14] 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, 23:1 (2021), 118 | DOI

[15] 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, 126 (2019), 255–267 | DOI

[16] Sizykh G. B., “New Lagrangian view of vorticity evolution in two-dimensional flows of liquid and gas”, Izv. VUZ. Applied Nonlinear Dynamics, 30:1 (2022), 30–36 (In Russian) | DOI

[17] Sizykh G. B., “Evolution of vorticity in swirling axisymmetric flows of a viscous incompressible fluid”, TsAGI Science Journal, 46:3 (2015), 209–217 | DOI

[18] Prosviryakov E. Yu., “Recovery of radial-axial velocity in axisymmetric swirling flows of a viscous incompressible fluid in the Lagrangian consideration of vorticity evolution”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 31:3 (2021), 505–516 (In Russian) | DOI | Zbl

[19] Grant J. R., Marshall J. S., “Diffusion velocity for a three-dimensional vorticity field”, Theor. Comput. Fluid Dyn., 19:6 (2005), 377–390 | DOI | Zbl

[20] Kotsur O. S., “Mathematical modelling of the elliptical vortex ring in a viscous fluid with the vortex filament method”, Mathematics and Mathematical Modeling, 2021, no. 3, 46–61 | DOI

[21] Sizykh G. B., “Entropy value on the surface of a non-symmetric convex bow part of a body in the supersonic flow”, Fluid Dyn., 54:7 (2019), 907–911 | DOI | DOI | Zbl

[22] Sizykh G. B., “Closed vortex lines in fluid and gas”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 23:3 (2019), 407–416 | DOI | Zbl

[23] Mironyuk I. Yu., Usov L. A., “The invariant of stagnation streamline for a stationary vortex flow of an ideal incompressible fluid around a body”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 24:4 (2020), 780–789 (In Russian) | DOI | Zbl

[24] Kotsur O. S., “On the existence of local formulae of the transfer velocity of local tubes that conserve their strengths”, Proceedings of MIPT, 11:1 (2019), 76–85 (In Russian)

[25] Mironyuk I. Yu., Usov L. A., “Stagnation points on vortex lines in flows of an ideal gas”, Proceedings of MIPT, 12:4 (2020), 171–176 (In Russian) | DOI

[26] Sizykh G. B., “On the collinearity of vortex and the velocity behind a detached bow shock”, Proceedings of MIPT, 13:3 (2021), 144–147 (In Russian) | DOI

[27] Sizykh G. B., “Second integral generalization of the Crocco invariant for 3D flows behind detached bow shock wave”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 25:3 (2021), 588–595 (In Russian) | DOI

[28] Sizykh G. B., “Integral invariant of ideal gas flows behind a detached bow shock”, Fluid Dyn., 56:8 (2021), 1027–1030 | DOI | DOI

[29] Prim R., Truesdell C., “A derivation of Zorawski's criterion for permanent vector-lines”, Proc. Am. Math. Soc., 1 (1950), 32–34 | Zbl

[30] Truesdell C., The Kinematics of Vorticity, Indiana Univ. Press, Bloomington, 1954, xx+232 pp. | Zbl

[31] Friedmann A. A., Opyt gidromekhaniki szhimaemoi zhidkosti [Experience in the Hydromechanics of Compressible Fluid], ONTI, Moscow, 1934, 368 pp. (In Russian)