# Fractional derivatives for vortex simulations

## Main Article Content

## Abstract

Two modifications of the incompressible Navier--Stokes equations are investigated. The first modification is based on assuming hyperviscosity such that the Laplacian operator is replaced with a fractional Laplacian. The second modification consists of using fractional time derivatives. Both models are tested on the classical Backward Facing Step benchmark problem with different expansion ratios. The simulation results are in a good accordance with real measurements.

## Article Details

How to Cite

SZEKERES, Béla J.; IZSÁK, Ferenc.
Fractional derivatives for vortex simulations.

**Proceedings of the Conference Algoritmy**, [S.l.], p. 175-182, feb. 2016. Available at: <http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/406>. Date accessed: 22 sep. 2017.
Section

Articles

## References

[1] K. Spyksma, M. Magcalas, N. Campbell, Quantifying effects of hyperviscosity on isotropic turbulence, Physics of Fluids, 24, 12 (2012)

[2] Harlow, F. H., Welch, E. J.: Numerical Calculation of Time-Dependent Viscous Incompressible Flow with Free Surface, The Physics of Fluids 8,12, 2182–2189 (1965)

[3] F. Hirsch, G. Lacombe, Elements of Functional Analysis, Graduate Texts in Mathematics, Springer-Verlag, New York, 192 (1993)

[4] I. Podlubny, Fractional Differential Equations, Academic Press Inc., San Diego, CA (1999)

[5] S. V. Patankar, D. B. Spalding, A calculation procedure for heat, mass and momentum transfer in three–dimensional parabolic flows, Int. J. Heat and Mass Transfer 15, 1787– 1806 (1972)

[6] M. Ilic, F. Liu, I. Turner, V. Anh, Numerical approximation of a fractional-in-space diffusion equation. I, Fract. Calc. Appl. Anal., 8, 323–341 (2005)

[7] B. J. Szekeres, F. Izsa ́ k, Finite element approximation of fractional order elliptic boundary value problems, J. Comput. Appl. Math., 292, 553–561 (2016)

[8] B. F. Armaly, F. Durst, J. C. F. Pereira, B. Schnung, Experimental and theoretical investigation of backward-facing step flow, 127, 473–496 (1983)

[9] Anwar-ul-Haque, F. Ahmad, S. Yamada, S. R. Chaudhry, Assessment of Turbulence Models for Turbulent Flow over Backward Facing Step, Proceedings of the World Congress on Engineering 2007 Vol II, (2007)

[10] H. Le, P. Moin, J. Kim, Direct numerical simulation of turbulent flow over a backward-facing step, J. Fluid Mech., 330, 349–374 (1997)

[11] L. Jongebloed, Numerical Study using FLUENT of the Separation and Reattachment Points for Backwards-Facing Step Flow, Masters Project, (2008)

[12] S. Thangam, N. Hur, A highly-resolved numerical study of turbulent separated flow past a backward-facing step, International Journal of Engineering Science, 29, 5, 607–615 (1991)

[13] S. Jovic, D. M. Driver, Backward-facing step measurements at low Reynolds number, Reh = 5000, NASA STI/Recon Technical Report N, 94, 33290 (1994)

[14] J. Kim, S. J. Kline, J. P. Johnston, Investigation of a Reattaching Turbulent Shear Layer: Flow Over a Backward-Facing Step, J. Fluids Eng. 102, 3, 302–308 (1980)

[15] M.Ilic ́, F. Liu, I. Turner, V. Anh, Numerical approximation of a fractional-in-space diffusion equation. II. With nonhomogeneous boundary conditions, Fract. Calc. Appl. Anal., 9, 4, 333-349 (2006)

[16] F. Liu, P. Zhuang, V. Anh, I. Turner, A fractional-order implicit difference approximation for the space-time fractional diffusion equation, ANZIAM J., 47, C48–C68 (2005)

[17] M. Ilic ́, I. W. Turner, D. P. Simpson, A restarted Lanczos approximation to functions of a symmetric matrix, IMA J. Numer. Anal., 30, 4, 1044–1061 (2010)

[18] N. J. Higham, L. Lin, An improved Schur–Pad ́ e algorithm for fractional powers of a matrix and their Fr ́ echet derivatives, SIAM J. Matrix Anal. Appl., 34, 3, 1341–1360 (2013)

[2] Harlow, F. H., Welch, E. J.: Numerical Calculation of Time-Dependent Viscous Incompressible Flow with Free Surface, The Physics of Fluids 8,12, 2182–2189 (1965)

[3] F. Hirsch, G. Lacombe, Elements of Functional Analysis, Graduate Texts in Mathematics, Springer-Verlag, New York, 192 (1993)

[4] I. Podlubny, Fractional Differential Equations, Academic Press Inc., San Diego, CA (1999)

[5] S. V. Patankar, D. B. Spalding, A calculation procedure for heat, mass and momentum transfer in three–dimensional parabolic flows, Int. J. Heat and Mass Transfer 15, 1787– 1806 (1972)

[6] M. Ilic, F. Liu, I. Turner, V. Anh, Numerical approximation of a fractional-in-space diffusion equation. I, Fract. Calc. Appl. Anal., 8, 323–341 (2005)

[7] B. J. Szekeres, F. Izsa ́ k, Finite element approximation of fractional order elliptic boundary value problems, J. Comput. Appl. Math., 292, 553–561 (2016)

[8] B. F. Armaly, F. Durst, J. C. F. Pereira, B. Schnung, Experimental and theoretical investigation of backward-facing step flow, 127, 473–496 (1983)

[9] Anwar-ul-Haque, F. Ahmad, S. Yamada, S. R. Chaudhry, Assessment of Turbulence Models for Turbulent Flow over Backward Facing Step, Proceedings of the World Congress on Engineering 2007 Vol II, (2007)

[10] H. Le, P. Moin, J. Kim, Direct numerical simulation of turbulent flow over a backward-facing step, J. Fluid Mech., 330, 349–374 (1997)

[11] L. Jongebloed, Numerical Study using FLUENT of the Separation and Reattachment Points for Backwards-Facing Step Flow, Masters Project, (2008)

[12] S. Thangam, N. Hur, A highly-resolved numerical study of turbulent separated flow past a backward-facing step, International Journal of Engineering Science, 29, 5, 607–615 (1991)

[13] S. Jovic, D. M. Driver, Backward-facing step measurements at low Reynolds number, Reh = 5000, NASA STI/Recon Technical Report N, 94, 33290 (1994)

[14] J. Kim, S. J. Kline, J. P. Johnston, Investigation of a Reattaching Turbulent Shear Layer: Flow Over a Backward-Facing Step, J. Fluids Eng. 102, 3, 302–308 (1980)

[15] M.Ilic ́, F. Liu, I. Turner, V. Anh, Numerical approximation of a fractional-in-space diffusion equation. II. With nonhomogeneous boundary conditions, Fract. Calc. Appl. Anal., 9, 4, 333-349 (2006)

[16] F. Liu, P. Zhuang, V. Anh, I. Turner, A fractional-order implicit difference approximation for the space-time fractional diffusion equation, ANZIAM J., 47, C48–C68 (2005)

[17] M. Ilic ́, I. W. Turner, D. P. Simpson, A restarted Lanczos approximation to functions of a symmetric matrix, IMA J. Numer. Anal., 30, 4, 1044–1061 (2010)

[18] N. J. Higham, L. Lin, An improved Schur–Pad ́ e algorithm for fractional powers of a matrix and their Fr ́ echet derivatives, SIAM J. Matrix Anal. Appl., 34, 3, 1341–1360 (2013)