Analytical Models Development of Compact Monopole Vortex Flows




Compact vortex, Analytical models, Ideal fluid, Viscous fluid, Laminar flow, Turbulent flow, Vortex diffusion, Vortex generation


Background. Mathematical description of vortex flows as a part of basic fundamental concepts of continuum mechanics, hydro- and gas dynamics, thermal physics, theoretical basics of chemical engineering, geophysics, meteorology.

Objective. Replenishment of existing information in traditional scientific, technical and educational publications about vortices and their analytical models by new data for scientists, teachers, post-graduates and students with the purpose of using in further investigations of fluids and gas vortex motion.

Methods. Analytical review of existing vortex current models, including monopole compact vortices, on a basis of the latest data from scientific articles and specialized monographs and their comparison with traditional data containing in the known textbooks and educational materials.

Results. The lack of modern compact vortices models was revealed in traditional scientific, technical and educational literary sources. They include the compact analogs of the point vortex and Rankine vortex: quasi-point vortex and compact compensated vortex. On basis of these ones and similar to them solutions (circular vortex, vortex with triad constant vorticity zones) the models of compact helical flows and vortices with helical symmetry were elaborated. The solutions of such tasks were analyzed: diffusion of compact vortices (Taylor vortex, Kloosterziel solution), turbulent diffusion of compact vortex, solutions for quasi-compact (laminar and turbulent) vortex-source, vortex-sink and also solution of the problem about compact turbulent vortex generation by rotating cylinder. All considered models are consistent with the energy conservation law and have advantages in their use.

Conclusions. The article contains series of the latest analytical models that describe both laminar and turbulent dynamics of monopole vortex flows which have not been reflected in traditional publications up to the present. The further research must be directed to search of analytical models for the coherent vortical structures in flows of viscous fluids, particularly near curved surfaces, where known in hydromechanics “wall law” is disturbed and heat and mass transfer anomalies take place.


H.J. Pearson and P.F. Linden, “The final stage of decay of turbulence in stably stratified fluid”, J. Fluid. Mech., vol. 134, pp. 195–203, 1983. doi: 10.1017/S0022112083003304

L.G. Loitsyansky, Mechanics of Fluid and Gas. Moscow, SU: Nauka, 1987 (in Russian).

V.I. Putyata and M.M. Sidlyar, Hydromechanics. Kyiv, SU: Kyiv University Publ., 1963 (in Ukrainian).

V.F. Kozlov, “Geophysical hydrodynamics of vortex patches”, Morskoi Gidrofizicheskii Zhurnal, no. 1, pp. 26–35, 1994 (in Russian).

M.E. Stern, “Minimal properties of planetary eddies”, J. Marine Res., vol. 33, no. 1, pp. 1–13, 1975.

P.V. Lukianov, “One-dimensional models of compact vortices”, Naukovi Visti NTUU KPI, no. 4, pp. 145–150, 2010 (in Ukrainian).

P.G. Saffman, Vortex Dynamics. Moscow, Russia: Nauchny Mir, 2000 (in Russian).

P.V. Lukianov, “Model of the quasi-point vortex”, Naukovi Visti NTUU KPI, no. 4, pp. 139–142, 2011 (in Ukrainian).

P.V. Lukianov, “Compact compensated vortex models and their using in fluid and gas mechanics”, Applied Hydromechanics, vol. 13, no. 2, pp. 37–43, 2011 (in Russian).

V.F. Kozlov, “Stationary models of baroclinic compensated vortices”, Izvestiya AN. Fizika Atmosfery i Okeana, vol. 28, no. 6, pp. 615–624, 1992 (in Russian).

S.A. Arseniev et al., “Self-organization of tornados and hurricanes in atmospheric flows with mesoscale vortices”, DAN, vol. 395, no. 6, pp. 1–6, 2004 (in Russian).

V.Ya. Rudyak and S.O. Savchenko, “Modelling of twisted submerged jet inducted by vorticity sink”, Sibirski Zhurnal Industrial’noji Matematiki, no. 4, pp. 139–149, 2002 (in Russian).

S.V. Alekseenko et al., Theory of Concentrated Vortices: An Introduction. Springer-Verlag, Berlin Heidelberg, 2007.

D.G. Dritschel, “Generalized helical Beltrami flows in hydrodynamics and magnetohydrodynamics”, J. Fluid. Mech., vol. 222, pp. 525–541, 1991. doi: 10.1017/S0022112091001209

P.A. Kuibin and V.L. Okulov, “One-dimensional solutions for the flows with helical symmetry”, Teplofizika i Aeromechanika, no. 4, pp. 311–315, 1996.

D. Lucas and D.G. Dritschel, “A family of helically symmetric vortex equilibria”, J. Fluid Mech., vol. 634, pp. 245–268, 2009. doi: 10.1017/s0022//2009007319

P.V. Lukianov, “The models of compact compensated vortex flows with helical symmetry”, Applied Hydromechanics, vol. 15, no. 3. pp. 37–42, 2013 (in Russian).

E.J. Hopfinger and G.J.F. van Heijst, “Vorticies in rotating fluids”, Ann. Rev. Fluid Mech., vol. 25, ­­pp. 241–289, 1993. doi: 10.1146/annurev.fl.25.010193.001325

G.I. Taylor, “Distribution of velocity and temperature between concentric cylinder”, Proc. Roy. Soc., vol. A 151, pp. 494–512, 1935. doi: 10.1098/rspa.1935.0163

S. Leibovich, “Vortex stability and breakdown – survey and extension”, AIAA J., vol. 22, no. 9, pp. 1192–1206, 1984.

M.P. Escudier et al., “The dynamics of confined vortices”, Proc. Royal Soc. London, vol. A 382, pp. 335–360, 1982. doi: 10.1098/rspa.1982.0105

I.S. Gromeka, Paper Collection. Moscow, SU: Academy of Science of the USSR, 1952 (in Russian).

O.F. Vasiliev, Introduction to Helical and Circulating Flows Mechanics. Moscow, Lieningrad, SU: Gosenergoizdat, 1958 (in Russian).

P.V. Lukianov, “Compact helical vortices”, Applied Hydromechanics, vol. 13, no. 3, pp. 61–68, 2011 (in Russian).

R.C. Kloosterziel, “On the large-time asymptotics of the diffusion equation on infinite domains”, J. Eng. Math., vol. 24, pp. 213–236, 1990. doi: 10.1007/BF00058467

M. Beckers, “Dynamics of pacake-like vortices in stratified fluid: experiments, model and numerical simulations”, J. Fluid Mech., vol. 433. pp. 1–27, 2001. doi: 10.1017/S0022112001003482

P.V. Lukianov, “Vortex diffusion in the layer of viscous stratified fluid”, Applied Hydromechanics, vol. 8, no. 3, pp. 63–77, 2006 (in Russian).

А.M. Gaifullin, “Self-similar non-steady viscous fluid flow”, Miechanika Zhidkosti i Gaza, no. 4, pp. 29–35, 2005 (in Russian). doi: 10.1017/s0022//2009007319

P.V. Lukianov, “Quasi-compact vortex-source and vortex-sink flows”, Applied Hydromechanics, vol. 14, no. 2, pp. 23–29, 2012 (in Ukrainian).

P.V. Lukianov, “Compact turbulent vortex generation: Approximate model for relatively large time moments”, Naukovi Visti NTUU KPI, no. 4, pp. 127–131, 2013 (in Ukrainian).

A.A. Townsend, The Structure of Turbulent Shear Flow. Cambridge, UK: Cambridge University Press, 1956.

V.V. Meleshko and H. Aref, “A bibliography of vortex dynamics 1858–1956”, Adv. Appl. Mech., vol. 41, pp. 197–291, 2007.

G.R. Flierl, “Isolated eddy models in geophysics”, Annu. Rev. Fluid, vol. 19, pp. 493–530 1987. doi: 10.1146/annurev.fl.19.010187.002425

M.P. Satijn et al., “Vortex models based on similarity solutions of the two-dimensional diffusion equation”, Phys. Fluids, vol. 16, no. 11, pp. 3997–4011, 2004. doi: 10.1063/1.1804548

V.N. Turick, “On the mutual susceptibility of vortical structures and their control”, Visnyk NTUU KPI. Ser. Mashinobuduvannja, no. 56, pp. 286–299, 2009 (in Russian).

V.N. Turick, “Coherent vortical structures in the limited twisting flows”, Visnyk Cherkass’kogo Technologichnogo Universitetu, no. 2, pp. 58–67, 2004 (in Russian).

V.A. Kochin and V.N. Turick, “Measurement procedure features of hot-wire anemometer experiment for flow structure study in vortex chamber”, Visnyk NTUU KPI. Ser. Mashinobuduvannja, no. 47, pp. 54–57, 2005 (in Russian).

V.N. Turick, “On hydrodynamic instability of flows in vortex chambers”, Promyslova Gidravlika i Pnevmatyka, no. 3, pp. 32–37, 2006 (in Russian).