Aerospace Technic and Technology

The subject of this work is the development of compact analogs of vortex flows models, which are used in the modeling of vortex structures observed during the flight of an aircraft and the motion of a body in a fluid. In particular, two significant misunderstandings prevailing in this area of science are highlighted. The first misunderstanding is that the stationary motion of fluid parcels in a circle is treated as an inviscid vortex. Therefore, any vortex flow model that does not explicitly contain viscosity is considered to describe inviscid vortex motion. It has been proven that this is not so: the stationary viscous motion of fluid parcels in circular orbits corresponds to the self-balance of one force - the force of viscosity. This conclusion, in an explicit form, was made for the first time. This is very important because it changes our ideas about force balance, where two or more forces of different natures must necessarily be present. Overcoming this misunderstanding opens the way for creating compact analogs of existing models of vortex motions. Along the way, one more - the second general misunderstanding in the field of vortex dynamics was eliminated. Wherever we read it, we can see that the compactness of the vortex flow is associated with the compactness of the vorticity field. This is facilitated by the fact that the equations for vorticity and not for velocity are considered. As a result, except for one or two models of vortices, which correspond to the rotation of the entire space, up to infinity, this violates the fundamental law of physics - the law of conservation and transformation of energy. It is about the fact that, as a second misunderstanding, an error is assumed when calculating the kinetic energy of the vortex flows: the Jacobian in cylindrical (polar) coordinates is not considered. As a result, all the mentioned models of vortex flows, which correspond to the hyperbolic law as their asymptotics in the periphery, have infinite kinetic energy. Certainly, this does not correspond to the formation and evolution of compact vortex structures. Therefore, in this work, based on overcoming the aforementioned misunderstandings, many previously obtained models of compact vortex flows, as well as those obtained for the first time, are presented. In particular, this applies to the turbulent vortex flow during the formation of a vortex sheet, which is a compact analog of the Burgers-Rott vortex - both the classical one corresponding to laminar motion and the one consisting of a laminar flow in the core and a turbulent flow on the periphery of the vortex. Research methods are entirely theoretical. Well-known theorems of theoretical mechanics, mathematical theory of field, and calculus of variations, etc. are used. The obtained solutions are compared with the existing corresponding analogs of non-compact flows. Conclusions. Using the methods of calculus of variation, it was possible to show the possibility of the formation of quasi-solid-like rotational motion in a boundary layer of an incompressible fluid.   The very presence of viscosity, or rather its taking into account (boundary layer), indicates a possible transition of the flow from plane-parallel motion to the just-mentioned rotational one due to the Kelvin-Helmholtz instability. In addition, two new models of the Burgers-Rott vortex flow were obtained in this study. The first model uses the general solution obtained by Burgers, but this model corresponds to a combined vortex: although the velocity field is continuous, the vorticity field has a discontinuity - at the point of maximum velocity. It is proved that such an approach is quite possible: the equation of motion is satisfied everywhere, i.e., at every point in space, and the tangential stresses are continuous functions. Since the periphery of the Burgers-Rott vortex is an unstable flow, another model is proposed - with a laminar core and a turbulent periphery. Certainly, the motion of fluid parcels in the peripheral region is described by a velocity distribution other than that of Burgers. Finally, the possible use of known models of compact vortex flows to simulate the von Karman vortex street is considered, with an indication of the advantages of these models.

aircraft; vortex flows; Burgers-Rott vortex; vortex sheet; Karman vortex street; two misuderstandings in vortex dynamics

Shekhovtcov, A. V. Inertsionno-vikhrevoy printsip generatsii usiliy na kryl'yakh nasekomykh [Inertially-vortex principle of generating forces at insect wings]. Prykladna hidromekhanika – Applied hydromechanics, 2011, vol. 13, no.1, pp. 61-76. Available at: http://dspace.nbuv.gov.ua/handle/123456789/116271. (accessed 10.02.2023) (In Russian).

Lamb, H. Hydrodynamics. Cambridge University Press Publ., 1945. 738 p. Available at: https://archive.org/details/hydrodynamics0000lamb/page/n1/mode/2up. (accessed 10.02.2023).

Lin, M., Cui, G. & Zhang, Zh. A new vortex sheet model for simulating aircraft wake vortex evolution. Chines Journal of Aeronautics, 2017, vol. 30, iss. 4, pp. 1315-1326. DOI: 10.1016/j.cja.2017.04.015.

Sohn, S. I. An inviscid model of unsteady separated vortical flow for a moving plate. Theor. Comput. Fluid Dyn., 2020, vol. 34, pp. 187-213. DOI: 10.1007/s00162-020-00524-0.

Solano, D., Xu, L. & Alben, S. Long-time simulations of vortex sheet dynamics. Available at: https://lsa.umich.edu/content/dam/math-assets/math-document/reu-documents/Daniel%20Solano.pdf (accessed 10.02.2023)

Hoepffner, J. & Fontelos, M. A model for global structure of self-similar vortex sheet roll-up. Available at: http://www.lmm.jussieu.fr/~hoepffner/research/spirals.pdf. (accessed 10.02.2023).

Saffman, P. G. Vortex Dynamics. Cambridge University Press Publ., 1993. 311 p. DOI: 10.1017/CBO9780511624063.

Prandtl, L. The generation of vortices in fluid of small viscosity. The Aeronautical Journal, 1927, vol. 31, iss. 200, pp. 718-741. DOI: 10.1017/S0368393100139872.

Batchelor, G. K. An introduction to Fluid Dynamics. Cambridge University Press, 2000. 615 p. DOI: 10.1017/CBO9780511800955.

Sherbokos, O. H. Rozpodil nestatsionarnoho tysku na kryli z heneratoramy vykhriv [Distribution of unsteady wing pressure with vortex generators]. Aviacijno-kosmicna tehnika i tehnologia – Aerospace technic and technology, 2012, no. 2(89), pp. 60-65. Available at: http://195.88.72.95:57772/csp/nauchportal/Arhiv/AKTT/2012/AKTT212/SCHerbon.pdf. (accessed 10.02.2023) (In Ukrainian).

Lukianov, P. V. & Song, L. Optimal character and different nature of flows in laminar boundary layers of incompressible fluid flow. Problems of friction and wear, 2022, no. 4(97), pp. 52-60. DOI: 10.18372/0370-2197.4(97).16959.

Lukianov, P. V. & Song, L. Unsteady incompressible laminar boundary layer: time and space variable molecular viscosity. Aviacijno-kosmicna tehnika i tehnologia – Aerospace technic and technology, 2023, no. 3(187), pp. 50-60. DOI: 10.32620/aktt.2023.3.06.

Meiron, D. I., Baker, G. R. & Orszag, S. A. Analytic structure of vortex sheet dynamics. Part1. Kelvin-Helmholtz instability. J. Fluid Mech., 1982, vol. 114, pp. 283-298. DOI: 10.1017/S0022112082000159.

Baker, G., Caflisch, R. E. & Siegel, M. Singularity formation during Rayleigh-Taylor instability. J. Fluid. Mech., 1993, vol. 252, pp. 51-78. DOI: 10.1017/S0022112093003660.

McNally, C. P., Lyra, W. & Passy, J.-C. A well-posed Kelvin-Helmholtz instability test and comparison. The Astropjysical Journal Supplement Series, 2012, vol. 201, no. 2, article no. 18, pp. 1-17. DOI: 10.1088/0067-0049/201/2/18.

Shadloo, M. S. & Yildiz, M. Numerical modeling of Kelvin-Helmholtz instability using smoothed particle hydrodynamics. International Journal for numerical methods in engineering, 2011, vol. 87, iss. 10, pp. 988-1006. DOI: 10.1002/nme.3149.

Pozrikidis, C. & Higdon, J. J. L Nonlinear Kelvin-Helmholtz instability of a finite vortex layer. J. Fluid Mech., 1985, vol. 157, pp. 225-263. DOI: 10.1017/S0022112085002361.

Kelly, R. E. The Stability of unsteady Kelvin-Helmholtz flow. J. Fluid Mech., 1965, vol. 22, iss. 3, pp. 547-560. DOI: 10.1017/S0022112065000964.

Miura, A. Self-Organization in the Two-dimensional Kelvin-Helmholtz Instability. Physical Review Letters. 1999, vol. 83, iss. 8, pp. 1586-1589. DOI: 10.1103/PhysRevLett.83.1586.

Berlok, T. & Pfrommer, C. On the Kelvin-Helmholtz instability with smooth initial conditions – linear theory and simulations. Monthly Notices of the Royal Astronomical Society, 2019, vol. 485, iss. 1, pp. 908-923. DOI: 10.1093/mnras/stz379.

Nayfen, A. H. & Saric, W. S. Non-linear Kelvin-Helmholtz instability. J. Fluid Mech., 1971, vol. 46, iss. 2, pp. 209-231. DOI: 10.1017/S0022112071000491.

Belotserkovsky, S. M. & Nisht, M. I. Otryvnoye i bezotryvnoye obtekaniye tonkikh kryl'yev ideal'noy zhidkost'yu [Separated and continuous flow of an ideal fluid around thin wings]. Moscow, Nauka Publ., 1978. 352 p. Available at: https://libarch.nmu.org.ua/handle/GenofondUA/50182 (accessed 10.02.2023) (In Russian).

Golovkin, M. A., Golovkin, V. A. & Kalyavin, V. M. Voprosy vikhrevoy gidrodinamiki [Problems of vortex hydrodynamics]. Moscow, Fizmatlit Publ., 2009. 263 p. Available at: https://biblioclub.ru/index.php?page=book&id=76681 (accessed 10.02.2023) (In Russian).

Bulanchuk, H. H., Bulanchuk, O. N. & Dovhyy, S. A. Metod diskretnykh vikhrevykh ramok so vstavkoy promezhutochnykh tochek na vikhrevoy pelene [Method of discrete vortex frames with insertion of intermediate points on the vortex sheet]. Visnyk Kharʹkovskoho natsionalʹnoho universytetu – Bulletin of Kharkiv National University, 2009, no. 853, pp. 37-46. (In Russian).

Chorin, A. J. Vortex models and boundary layer instability. SIAM J. Sci. Stat. Comput., 1980, vol. 1, no.1, pp. 1-21. Available at: https://math.berkeley.edu/~chorin/C80.pdf. (accessed 10.02.2023).

Ovcharov, M. M. Metodika rascheta nelineynykh aerodinamicheskikh kharakteristik letatel'nogo apparata pri nesimmetrichnom obtekanii [Method of calculation of nonlinear aerodynamic characteristics of aircraft at asymmetrical flowing around]. Aviacijno-kosmicna tehnika i tehnologia – Aerospace technic and technology, 2012, no. 6(93), pp. 48-52. Available at: http://195.88.72.95:57772/csp/nauchportal/Arhiv/AKTT/2012/AKTT612/Ovcharov.pdf. (accessed 10.02.2023). (In Russian).

Gómes-Serrano, J., Park, J., Shi, J. & Yao, Y. Remarks on Stationary Uniformly-rotating Vortex Sheets: Rigidity Results. Commun. Math. Phys., 2021, vol. 386, pp. 1845-1879. DOI: 10.1007/s00220-021-04146-3.

Lukyanov, P. V. Diffuziya izolirovannogo kvazidvumernogo vikhrya v sloye stratifitsirovannoy zhidkosti [Vortex diffusion in a layer of viscous stratified fluid]. Prykladna hidromekhanika – Applied Hydromechanics, 2006, vol. 8(80), no. 3, pp. 63-77. Available at: http://hydromech.org.ua/content/pdf/ph/ph-08-3(63-77).pdf. (accessed 10.02.2023). (In Russian).

Luk‘yanov, P. V. Modelʹ kvazitochkovoho vykhoru [Model of Quasi-Point Vortex]. Naukovi visti NTUU KPI – Research Bulletin of National Technical University of Ukraine “Kyiv Polytechnic Institute”, 2011, no. 4 (78), pp. 139-142. Available at: https://ela.kpi.ua/handle/123456789/36756. (accessed 10.02.2023). (In Ukrainian).

Blyuss, B. O., Luk‘yanov, P. V. & Dzyuba, S. V. Modelyuvannya kvazitochkovoho turbulentnoho vykhoru v zakruchenykh techiyakh ridyny v zbahachuvalʹnomu ustatkuvanni [Simulation of quasi-point turbulent vortex in the swirling flows of fluids in the preparatory equipment]. Heotekhnichna mekhanika. Mizhvidomchiy Zbirnyk Naukovykh Pratsʹ – Geotechnical mechanics. Interdepartmental Collection of Scientific Works, Dnipro, 2018, no. 143, pp. 19-25. DOI: 10.15407/geotm2018.143.049. (In Ukrainian).

Luk‘yanov, P. V. Heneratsiya kompaktnoho turbulentnoho vykhora: nablyzhena modelʹ dlya vidnosno velykykh momentiv chasu [Compact Turbulent Vortex Generation Approximate Model for Relatevely large Time Moments]. Naukovi visti NTUU KPI – Research Bulletin of National Technical University of Ukraine “Kyiv Polytechnic Institute”, 2013, no. 4 (90), pp. 127-131. Available at: https://ela.kpi.ua/jspui/handle/123456789/7334. (accessed 10.02.2023). (In Ukrainian).

Luk'yanov, P. V. Universal'naya model' kompaktnogo vikhrya i yeyo primeneniye dlya kompaktnykh kvazivintovykh potokov [Universal model of compact vortex and the using of the model for compact quasi-helical fluxes]. Materialy XIII mizhnarodnoyi naukovo-tekhnichnoyi konferentsiyi AVIA-2017, 19-21 kvitnya 2017 r. – Materials of the XIII international scientific and technical conference AVIA-2017, April 19-21, 2017, Kyiv, 2017, vol. 1, pp. 7.16-7.21. Available at: http://avia.nau.edu.ua/doc/avia-2017/AVIA_2017.pdf. (accessed 10.02.2023). (In Russian).

Luk'yanov, P. V. Zastosuvannya teorem Helʹmholʹtsa ta Stoksa pro vykhorovu trubku dlya obhruntuvannya yiyi kompaktnosti [The using of Helmholz and Stokes theorems for vortex tube for it compactnees graunding]. Materialy XXII mizhnarodnoyi naukovo-tekhnichnoyi konferentsiyi FS PHP «Promyslova hidravlika i pnevmatyka», Kyyiv, 17-18 lystopada, Vinnytsya - Materials of the XXII international scientific and technical conference of FS PGP "Industrial hydraulics and pneumatics". Kyiv, November 17-18. Vinnytsia, «Hlobus-Pres» Publ., 2021, pp. 30-32. (In Ukrainian).

Luk'yanov, P. V. Inertsiyna stiykistʹ yak rezulʹtat spivvidnoshennya perenosnoho i vidnosnoho obertanʹ nestyslyvoyi ridyny [Inertial Stability as a Result of the Relation between Transport and Relative Fluid Rotation]. Naukovi visti NTTU «KPI» – Research Bulletin of National Technical University of Ukraine “Kyiv Polytechnic Institute”, 2014, no. 4(96), pp. 133-138. Available at: https://ela.kpi.ua/handle/123456789/9935. (accessed 10.02.2023). (In Ukrainian).

Burgers, J. M. A mathematical model illustrating the theory of turbulence. Advances in applied mechanics. 1948, vol. 1, pp. 171-199. DOI: 10.1016/S0065-2156(08)70100-5.

Rott, N. On the viscous core of a line vortex. Journal of Applied Mathematics and Physics (ZAMP), 1958, vol. 9, pp. 543-553. DOI: 10.1007/BF02424773.

Betyayev, S. K. Gidrodinamika: problemy i paradoksy [Hydrodynamics: problems and paradoxes]. Uspekhi fizicheskikh nauky – Advances in Physical Sciences, 1995, vol. 165, no. 3, pp. 299-330. (In Russian).

Gorshkov, K. A., Soustova, I. A., Sergeyev, D. A. Ob ustoychivosti vikhrevykh dorozhek v stratifitsirovannoy zhidkosti [On the stability of vortex streets in a stratified fluid]. Izvestiya RAN. Fizika atmosfery i okeana – News of the Russian Academy of Sciences. Atmospheric and Ocean Physics, 2007, vol. 43, no. 6, pp. 851-860. Available at: https://naukarus.com/ob-ustoychivosti-vihrevyh-dorozhek-v-stratifitsirovannoy-zhidkosti. (accessed 10.02.2023). (In Russian).

Loytsyanskiy, L. G. & Lur'ye, A. I. Kurs teoreticheskoy mekhaniki. T. 2. Dinamika [Theoretical Mechanics course. Vol. 2. Dynamics]. Moscow, Drofa Publ., 2006. 719 p. Available at: http://pdf.lib.vntu.edu.ua/books/2019/Loytsyanskiy_P_2_2006_719.pdf. (accessed 10.02.2023). (In Russian).

Loytsyanskiy, L. G. Mekhanika zhidkosti i gaza [Mechanics of liquid and gas]. 6-ye perer. i dop. Moscow, Nauka, Gl. red. fiz.-mat. lit. Publ., 1987. 840 p Available at: http://www.knigograd.com.ua/index.php?dispatch=products.view&product_id=70436. (accessed 10.02.2023). (In Russian).

Tyvand, P. A. Viscous Rankine vortices. Physics of Fluids, 2022, vol. 34, iss. 7, article no. 073603. DOI: 10.1063/5.0090143.

Horiuti, K. A classification method for vortex sheet and tube structures in turbulent flows. Physics of Fluids, 2001, no. 13, iss. 12, pp. 3756-3774. DOI: 10.1063/1.1410981.

Horiuti, K. & Takagi, Y. Identification method for vortex sheet structures in turbulent flows. Physics of Fluids, 2005, no. 17, iss. 12, article no. 121703. DOI: 10.1063/1.2147610.

Millionshchikov, M. D. Turbulentnye techeniya v pogranichnom sloe i v trubakh [Turbulent flows in boundary layer and in pipes]. Moscow, Nauka Publ., 1969. 52 p. (In Russian.)

Pellone, S., Di Stefano, C. A., Rasmus, A. M., Kuranz, C. C. & Johnsen, E. Vortex-sheet modeling of hydrodynamic instabilities produced by an oblique shock interacting with a perturbed interface in the HED regime. Phys. Plasmas, 2021, vol. 28, iss. 2, article no. 022303. DOI: 10.1063/5.0029247.

Bernard, P. S. A Vortex Method for Wall Bounded Turbulent Flows. Esaim: Proceedings, 1996, vol. 1, pp. 15-31. DOI: 10.1051/proc:1996002.

Lin, C. L., Kaminski, A. K. & Smyth, W. D. The effect of boundary proximity on Kelvin-Helmholtz instability and turbulence. J. Fluid Mech., 2023, vol. 966, pp. A2-1 – A2-27. DOI: 10.1017/jfm.2023.412.

Eldredge, J. D. & Jones, A. R. Leading-Edge Vortices: Mechanics and Modelling. Ann. Rev. Fluid Mech., 2019, vol. 51, pp. 75-104. DOI: 10.1146/annurev-fluid-010518-040334.

Rayleigh Lord. On the Dynamics of Revolving Fluids. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 1917, vol. 93, no. 648, pp. 148-154. Available at: http://www.jstor.org/stable/93794. (accessed 10.02.2023).

Kloosterziel, R. C. & van Heijst, G. J. F. An experimental study of unstable barotropic vortices in a rotating fluid. Journal of Fluid Mechanics, 1991, vol. 223, pp. 1-24. DOI: 10.1017/S0022112091001301.

Reynolds, O. IV. On the Dynamical Theory of Incompressible Viscous Fluids and the Determination of the Criterion. Philosophical Transactions of the Royal Society of London (A), 1895, vol. 186, pp. 123-164. DOI: 10.1098/rsta.1895.0004.

Boussinesq, J. Essai sur la theorie des eaux courantes. Nabu Press, 2011. 778 p. Available at: https://www.amazon.co.jp/-/en/Joseph-Boussinesq/dp/1271798727. (accessed 10.02.2023) (in French).

Kiya, M. & Arie, M. Formation of Vortex Street in Laminar Boundary Layer. Journal of Applied Mechanics, 1980, vol. 47, iss. 2, pp. 227-233. DOI: 10.1115/1.3153647.

Aref, H. & van der Giessen, E. A bibliography of Vortex Dynamics 1858-1956. Advances in Applied Mechanics, 2007, vol. 41, pp. 197-292. DOI: 10.1016/S0065-2156(07)41003-1.

Beckers, M., Verzicco, R., Clercx, H. J. H. & Van Heijst, G. J. F. Dynamics of pancake-like vortices in a stratified fluid: experiments, model and numerical simulations. Journal of Fluid Mechanics, 2001, vol. 433, pp. 1-27. DOI: 10.1017/S0022112001003482.

Richardson, L. F. Atmospheric diffusion shown on a distance-neighbor graph. Proc. R. Soc. London. Ser. A, 1926, no. 110, iss. 756, pp. 709-737. DOI: 10.1098/rspa.1926.0043.

Garcia, C. Karman Vortex Street in incompressible fluid models. Nonlinearity, 2020, vol. 33, no. 4, article no. 1625. DOI: 10.1088/1361-6544/ab6309.

Stern, M. E. Horizontal Entraiment and Detraiment in Large-Scale Eddies. J. Phys. Ocean, 1987, vol. 17, no. 10, pp. 1688-1695. DOI: 10.1175/1520-0485(1987)017<1688:HEADIL>2.0.CO;2.