# On a New Approach to 3D Meshless Flows Simulation Around Bluff Bodies by Using Vortex Method

## Main Article Content

## Abstract

The original approach is suggested for vortex sheet intensity computation when 3D flow around bluff bodies is being simulated using meshless Lagrangian vortex methods. Integral equation approximate solution is based on its approximation by the system of linear algebraic equations, which coefficients are expressed through improper integrals. In order to compute these integrals the numerical scheme is developed which allows to exclude the singularities, i.e., to split the integrals into regular and singular parts. Regular parts can be integrated numerically with high precision by using Gaussian quadrature formulae and for singular parts exact analytical expressions are derived. The developed approach allows to raise significantly the accuracy of vortex method. It is shown that the developed semi-analytical procedure of integrals computation is much more accurate in comparison with direct numerical integration. The test problem of flow simulation around the sphere is considered. The exact analytical solution is known for it, and the developed approach provides more accurate results in comparison with `classical' 3D vortex method, especially when non-uniform unstructured triangular meshes are used for bodies surface representation.

## Article Details

How to Cite

Marchevsky, I., & Shcheglov, G.
(2016).
On a New Approach to 3D Meshless Flows Simulation Around Bluff Bodies by Using Vortex Method.

*Proceedings Of The Conference Algoritmy,*, 103-112. Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/399/316
Section

Articles

## References

[1] G.-H. Cottet and P.D. Koumoutsakos, Vortex Methods: Theory and Practice. CUP, 2000.

[2] I.K. Marchevsky and G.A. Shcheglov, 3D vortex structures dynamics simulation using vortex fragmentons, in ECCOMAS 2012, e-Book, Vienna (2012), pp. 5716–5735.

[3] S.N. Kempka, M.W. Glass, J.S. Peery and J.H. Strickland, Accuracy considerations for implementing velocity boundary conditions in vorticity formulations. SANDIA REPORT SAND96-0583, UC-700, 1996.

[4] K.S. Kuzmina and I.K. Marchevsky, On Numerical Schemes in 2D Vortex Element Method for Flow Simulation Around Moving and Deformable Airfoils, in Advanced Problems in Mechanics: Proceedings of the XLII School-Conference, St.Petersburg (2014), pp. 335–344.

[5] J.D. Anderson, A history of aerodynamics. Cambrige University Press, 1997.

[6] I.K. Lifanov, L.N. Poltavskii and G. Vainikko, Hypersingular Integral Equations and Their Applications, Chapman & Hall / CRC Press, Boca Raton, 2004.

[7] V.A. Antonov, I.I. Nikiforov and K.V. Kholshevnikov, Elements of the theory of the gravitational potential, and some instances of its explicit expression. St.-Petersburg University Press, 2008.

[8] I.S. Gradshteyn, I.M. Ryzhik, A. Jeffrey (Ed.) and D. Zwillinger (Ed.), Gradshteyn and Ryzhik’s Table of Integrals, Series, and Products. Elsevier, 2007.

[9] A. Van Oosterom and J. Strackee, The Solid Angle of a Plane Triangle. IEEE Trans. Biom. Eng. BME-30 (1983) 2, pp. 125–126.

[10] O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu, The Finite Element Method: Its Basis and Fundamentals. Elsevier, 2013.

[11] L.M. Milne-Thomson, Theoretical Hydrodynamics. London, Macmillan & Co ltd, 1962.

[2] I.K. Marchevsky and G.A. Shcheglov, 3D vortex structures dynamics simulation using vortex fragmentons, in ECCOMAS 2012, e-Book, Vienna (2012), pp. 5716–5735.

[3] S.N. Kempka, M.W. Glass, J.S. Peery and J.H. Strickland, Accuracy considerations for implementing velocity boundary conditions in vorticity formulations. SANDIA REPORT SAND96-0583, UC-700, 1996.

[4] K.S. Kuzmina and I.K. Marchevsky, On Numerical Schemes in 2D Vortex Element Method for Flow Simulation Around Moving and Deformable Airfoils, in Advanced Problems in Mechanics: Proceedings of the XLII School-Conference, St.Petersburg (2014), pp. 335–344.

[5] J.D. Anderson, A history of aerodynamics. Cambrige University Press, 1997.

[6] I.K. Lifanov, L.N. Poltavskii and G. Vainikko, Hypersingular Integral Equations and Their Applications, Chapman & Hall / CRC Press, Boca Raton, 2004.

[7] V.A. Antonov, I.I. Nikiforov and K.V. Kholshevnikov, Elements of the theory of the gravitational potential, and some instances of its explicit expression. St.-Petersburg University Press, 2008.

[8] I.S. Gradshteyn, I.M. Ryzhik, A. Jeffrey (Ed.) and D. Zwillinger (Ed.), Gradshteyn and Ryzhik’s Table of Integrals, Series, and Products. Elsevier, 2007.

[9] A. Van Oosterom and J. Strackee, The Solid Angle of a Plane Triangle. IEEE Trans. Biom. Eng. BME-30 (1983) 2, pp. 125–126.

[10] O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu, The Finite Element Method: Its Basis and Fundamentals. Elsevier, 2013.

[11] L.M. Milne-Thomson, Theoretical Hydrodynamics. London, Macmillan & Co ltd, 1962.