T-Schemes for Mathematical Modelling of Vorticity Generation on Smooths Airfoils in Vortex Particle Methods

Abstract

New numerical schemes are proposed for solving the boundary integral equation that arises in CFD vortex particle methods of when simulating a plane flow around smooth airfoils. They are based on considering the 2-nd kind integral equation with respect to vortex sheet intensity with a bounded or absolutely integrable kernel instead of traditionally solved singular integral equations of the 1-st kind with Hilbert-type singularity. To solve it, the Galerkin approach is used. It is shown that when approximating the airfoil boundary with a polygon, it is possible to develop schemes of the 1-st and 2-nd order of accuracy, considering a piecewise-constant or piecewise-linear (discontinuous or continuous) distribution of the solution along the panels. The necessary formulae are presented for calculating the components of the matrix and the right-hand side of the system of linear algebraic equations, that is a discrete analogue of the integral equation. They are suitable for modelling of the vorticity generation when simulating the flow around either single airfoil or system of airfoils, including moving and/or deformable ones. The developed schemes can be used in the framework of the viscous vortex domains method as well as other modifications of vortex particle methods, since they only concern the convective velocities of the flow near the airfoil and are not related to methods for modeling viscous diffusion of vorticity

This research was supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-15-2020-808)

Please cite this article in English as:

Marchevskii I.K., Sokol K.S., Izmailova Yu.A. T-schemes for mathematical modelling of vorticity generation on smooths airfoils in vortex particle methods. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2022, no. 6 (105), pp. 33--59 (in Russ.). DOI: https://doi.org/10.18698/1812-3368-2022-6-33-59

References

[1] Cottet G.-H., Koumoutsakos P.D. Vortex methods: theory and practice. Cambridge Univ. Press, 2000.

[2] Lewis R.I. Vortex element methods for fluid dynamic analysis of engineering systems. Cambridge Univ. Press, 1991.

[3] Branlard E. Wind turbine aerodynamics and vorticity-based methods. Cham, Springer, 2017. DOI: https://doi.org/10.1007/978-3-319-55164-7

[4] Golovkin M.A., Golovkin V.A., Kalyavkin V.M. Voprosy vikhrevoy gidromekhaniki [Issues of vortex hydromechanics]. Moscow, FIZMATLIT Publ., 2009.

[5] Mimeau C., Mortazavi I. A review of vortex methods and their applications: from creation to recent advances. Fluids, 2021, vol. 6, no. 2, art. 68. DOI: https://doi.org/10.3390/fluids6020068

[6] Leonard A. Vortex methods for flow simulation. J. Comput. Phys., 1980, vol. 37, iss. 3, pp. 289--335. DOI: https://doi.org/10.1016/0021-9991(80)90040-6

[7] Sarpkaya T. Computational methods with vortices --- the 1988 Freeman scholar lecture. J. Fluids Eng., 1989, vol. 111, iss. 1, pp. 5--52.DOI: https://doi.org/10.1115/1.3243601

[8] Dynnikova G.Ya. An analog of the Bernoulli and Cauchy --- Lagrange integrals for a time-dependent vortex flow of an ideal incompressible fluid. Fluid Dyn., 2000, vol. 35, no. 1, pp. 24--32. DOI: https://doi.org/10.1007/BF02698782

[9] Dynnikova G.Ya. On the added mass in a viscous incompressible fluid. Doklady Akademii nauk, 2019, vol. 488, no. 5, pp. 493--497 (in Russ.). DOI: https://doi.org/10.31857/S0869-56524885493-497

[10] Lifanov I.K. Metod singulyarnykh integralnykh uravneniy i chislennyy eksperiment [Methods of singular integral equations and numerical experiment]. Moscow, Yanus Publ., 1995.

[11] Dynnikova G.Ya. Lagrange method for Navier --- Stokes equations solving. Doklady Akademii nauk, 2004, vol. 399, no. 1, pp. 42--46 (in Russ.).

[12] Kempka S.N., Glass M.W., Peery J.S., et al. Accuracy considerations for implementing velocity boundary conditions in vorticity formulations. SANDIA report SAND-96-0583, 1996.

[13] Bykhovskiy E.B., Smirnov N.V. Orthogonal decomposition of the space of vector functions square-summable on a given domain, and the operators of vector analysis. Trudy MIAN SSSR, 1960, vol. 59, pp. 5--36 (in Russ.).

[14] Morino L. Helmholtz decomposition revisited: vorticity generation and trailing edge condition. Comput. Mech., 1986, vol. 1, no. 1, pp. 65--90. DOI: https://doi.org/10.1007/BF00298638

[15] Wu J.C., Thompson J.F. Numerical solutions of time-dependent incompressible Navier --- Stokes equations using an integro-differential formulation. Comput. Fluids, 1973, vol. 1, iss. 2, pp. 197--215. DOI: https://doi.org/10.1016/0045-7930(73)90018-2

[16] Markov V.V., Sizykh G.B. Vorticity evolution in liquids and gases. Fluid Dyn., 2015, vol. 50, no. 2, pp. 186--192. DOI: https://doi.org/10.1134/S0015462815020027

[17] Ogami Y., Akamatsu T. Viscous flow simulation using the discrete vortex model --- the diffusion velocity method. Comput. Fluids, 1991, vol. 19, no. 3-4, pp. 433--441. DOI: https://doi.org/10.1016/0045-7930(91)90068-S

[18] Batchelor G.K. An introduction to fluid dynamics. Cambridge Univ. Press, 1973.

[19] Marchevskii I.K. Razrabotka i realizatsiya T-skhem chislennogo resheniya granichnykh integralnykh uravneniy v matematicheskikh modelyakh vikhrevykh metodov vychislitelnoy gidrodinamiki. Dis. d-ra fiz.-mat. nauk [Development and implementation of T-schemes for numerical solution of boundary integral equations in mathematical models of vortex methods of computational fluid dynamics. Dr. Phys.-Math. Sc. Diss.]. Moscow, BMSTU, 2021 (in Russ.).

[20] Kuzmina K.S., Marchevskii I.K., Moreva V.S. Vortex sheet intensity computation in incompressible flow simulation around an airfoil by using vortex methods. Math. Models Comput. Simul., 2018, vol. 10, no. 3, pp. 276--287. DOI: https://doi.org/10.1134/S2070048218030092

[21] Kuzmina K.S., Marchevskii I.K. On the calculation of the vortex sheet and point vortices effects at approximate solution of the boundary integral equation in 2D vortex methods of computational hydrodynamics. Fluid Dyn., 2019, vol. 54, no. 7, pp. 991--1001. DOI: https://doi.org/10.1134/S0015462819070103

[22] Morgenthal G. Aerodynamic analysis of structures using high-resolution vortex particle methods. Ph.D. thesis. Univ. of Cambridge, 2002.

[23] Milani G. Concepts of adaptivity for vortex particle methods and applications to bluff body aerodynamics. Ph.D. thesis. Bauhaus-Universitat Weimar, 2018.

[24] Lin H., Vezza M. Implementation of a vortex method for the prediction of separated incompressible flows. Aero Report no. 9425. Univ. of Glasgow, 1994.

[25] Taylor I., Vezza M. Prediction of unsteady flow around square and rectangular section cylinders using a discrete vortex method. J. Wind. Eng. Ind. Aerodyn., 1999, vol. 82, iss. 1-3, pp. 247--269. DOI: https://doi.org/10.1016/S0167-6105(99)00038-0

[26] Kuzmina K., Marchevsky I., Soldatova I., et al. On the scope of Lagrangian vortex methods for two-dimensional flow simulations and the POD technique application for data storing and analyzing. Entropy, 2021, vol. 23, no. 1, art. 118. DOI: https://doi.org/10.3390/e23010118

[27] Kuzmina K., Marchevsky I., Ryatina E. Exact solutions of boundary integral equation arising in vortex methods for incompressible flow simulation around elliptical and Zhukovsky airfoils. J. Phys.: Conf. Ser., 2019, vol. 1348, art. 012099. DOI: https://doi.org/10.1088/1742-6596/1348/1/012099