Recovery of radial-axial velocity in axisymmetric swirling flows of a viscous incompressible fluid in the Lagrangian consideration of vorticity evolution

E. Yu. Prosviryakov

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 31 (2021) no. 3, pp. 505-516

Résumé

Swirling laminar axisymmetric flows of viscous incompressible fluids in a potential field of body forces are considered. The study of flows is carried out in a cylindrical coordinate system. In the flows, the regions in which the axial derivative of the circumferential velocity cannot take on zero value in some open neighborhood (essentially swirling flows) and the regions in which this derivative is equal to zero (the region with layered swirl) are considered separately. It is shown that a well-known method (the method of viscous vortex domains) developed for non-swirling flows can be used for regions with layered swirling. For substantially swirling flows, a formula is obtained for calculating the radial-axial velocity of an imaginary fluid through the circumferential vorticity component, the circumferential circulation of a real fluid, and the partial derivatives of these functions. The particles of this imaginary fluid “transfer” vortex tubes of the radial-axial vorticity component while maintaining the intensity of these tubes, and also “transfer” the circumferential circulation and the product of the circular vorticity component by some function of the distance to the axis of symmetry. A non-integral method for reconstructing the velocity field from the vorticity field is proposed. It is reduced to solving a system of linear algebraic equations in two variables. The obtained result is proposed to be used to extend the method of viscous vortex domains to swirling axisymmetric flows.

Détail

Keywords: Navier–Stokes equations, swirling flow, the discrete vortex method, the Helmholtz vortex theorem
Mots-clés : method of viscous vortex domains.

Bibliographie

[1] Rosenhead L., “The formation of vortices from a surface of discontinuity”, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 134:823 (1931), 170–192 | DOI

[2] Belotserkovskii S. M., Nisht M. I., Separated and unseparated ideal liquid flow around thin wings, Nauka, M., 1978

[3] Cottet G.-H., Koumoutsakos P. D., Vortex methods: theory and practice, Cambridge University Press, Cambridge, 2000 | DOI

[4] Gutnikov V. A., Lifanov I. K., Setukha A. V., “On modelling the aerodynamics of buildings and structures using the closed vortex method”, Izvestiya Rossiiskoi Akademii Nauk. Mekhanika Zhikosti i Gaza, 2006, no. 4, 78–92 | Zbl

[5] Kochin N. E., Kibel' I. A., Roze N. V., Theoretical fluid mechanics, v. 1, Fizmatgiz, M., 1963

[6] Golubkin V. N., Sizykh G. B., “On some general properties of plane-parallel viscous fluid flows”, Izvestiya Akademii Nauk SSSR. Mekhanika Zhikosti i Gaza, 1987, no. 3, 176–178

[7] Brutyan M. A., Golubkin V. N., Krapivskii P. L., “On the Bernoulli equation for axisymmetric viscous fluid flows”, Uchenie zapiski Tsentral'nogo Aerogidrodinamicheskogo Instituta, 19:2 (1988), 98–100

[8] Ogami Y., Akamatsu T., “Viscous flow simulation using the discrete vortex model — the diffusion velocity method”, Computers and Fluids, 19:3–4 (1991), 433–441 | DOI | Zbl

[9] Dynnikova G. Ya., “Lagrange method for Navier–Stokes equations solving”, Doklady Akademii Nauk, 399:1 (2004), 42–46

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

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

[12] Golubkin V. N., Markov V. V., Sizykh G. B., “The integral invariant of the equations of motion of a viscous gas”, Journal of Applied Mathematics and Mechanics, 79:6 (2015), 566–571 | DOI | Zbl

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

[14] Golubkin V. N., Sizykh G. B., “On the vorticity behind 3-D detached bow shock wave”, Advances in Aerodynamics, 1:1 (2019) | DOI

[15] Sizykh G. B., “Closed vortex lines in fluid and gas”, Vestnik Samarskogo Gosudarstvennogo Tekhnicheskogo Universiteta. Ser. Fiziko-Matematicheskie Nauki, 23:3 (2019), 407–416 | DOI | Zbl

[16] Golubkin V. N., Sizykh G. B., “Generalization of the Crocco invariant for 3D gas flows behind detached bow shock wave”, Russian Mathematics, 63:12 (2019), 45–48 | DOI | DOI | Zbl

[17] Mironyuk I. Yu., Usov L. A., “The invariant of stagnation streamline for a stationary vortex flow of an ideal incompressible fluid around a body”, Vestnik Samarskogo Gosudarstvennogo Tekhnicheskogo Universiteta. Ser. Fiziko-Matematicheskie Nauki, 24:4 (2020), 780–789 (in Russian) | DOI | Zbl

[18] Mironyuk I. Yu., Usov L. A., “Stagnation points on vortex lines in flows of an ideal gas”, Trydy Moskovskogo Fiziko-Tekhnicheskogo Instituta, 12:4(48) (2020), 171–176 (in Russian)