Modification of the LS-STAG Immersed Boundary Method for Simulating Turbulent Flows

Authors: Marchevsky I.K., Puzikova V.V. Published: 27.09.2017
Published in issue: #5(74)/2017  
DOI: 10.18698/1812-3368-2017-5-19-34

Category: Mathematics | Chapter: Mathematical Physics  
Keywords: immersed boundary method, LS-STAG method, turbulence models, Reynolds-averaged Navier — Stokes equations, large eddy simulation, detached eddy simulation, airfoil

We constructed the LS-STAG discretisation for 2D Reynolds-averaged Navier -- Stokes equations, filtered Navier -- Stokes equations (as used for large eddy simulation and detached eddy simulation) and equations employed in the Smagorinsky, Spalart -- Allmaras, k−ε, k−ω and k−ω Menter's Shear Stress Transport turbulence models. We added a fourth grid to the LS-STAG mesh consisting of three staggered grids. We computed the following parameters at the centres of the additional mesh cells: turbulent shear stress and, depending on the turbulence model used, turbulence kinetic energy, turbulent viscosity, and turbulent kinetic energy dissipation rate. We verified the developed numerical method by solving the problem of flow around a circular airfoil when the flow has a high Reynolds number (102...107). The obtained results are in good agreement with published experimental data and numerical results of other researchers. Our modification of the LS-STAG immersed boundary method made it possible to model the so-called "drag crisis" phenomenon for a circular airfoil when Re = 105...106


[1] Mittal R., Iaccarino G. Immersed boundary methods. Ann. Rev. Fluid Mech., 2005, vol. 37, pp. 239–261. DOI: 10.1146/annurev.fluid.37.061903.175743 Available at: http://www.annualreviews.org/doi/abs/10.1146/annurev.fluid.37.061903.175743

[2] Cheny Y., Botella O. The LS-STAG method: A new immersed boundary/level-set method for the computation of incompressible viscous flows in complex moving geometries with good conservation properties. J. Comput. Phys., 2010, vol. 229, iss. 4, pp. 1043–1076. DOI: 10.1016/j.jcp.2009.10.007

[3] Spalart P.R. Strategies for turbulence modelling and simulations. Int. J. Heat and Fluid Flow, 2000, vol. 21, iss. 3, pp. 252–263. DOI: 10.1016/S0142-727X(00)00007-2

[4] Spalart P.R., Allmaras S.R. A one-equation turbulence model for aerodynamic flows. Recherche Aerospatiale, 1994, no. 1, pp. 5–21.

[5] Jones W.P., Launder B.E. The prediction of laminarization with a two-equation model of turbulence. Int. J. Heat Mass Transfer, 1972, vol. 15, iss. 2, pp. 301–314. DOI: 10.1016/0017-9310(72)90076-2

[6] Wilcox D.C. Reassessment of the scale-determining equation for advanced turbulence models. AIAA Journal, 1988, vol. 26, no. 11, pp. 1299–1310. DOI: 10.2514/3.10041 Available at: https://arc.aiaa.org/doi/abs/10.2514/3.10041?journalCode=aiaaj

[7] Menter F.R. Two-equation eddy-viscosity turbulence models for engineering applications. AIAA Journal, 1994, vol. 32, no. 8, pp. 1598–1605. DOI: 10.2514/3.12149 Available at: https://arc.aiaa.org/doi/abs/10.2514/3.12149?journalCode=aiaaj

[8] Smagorinsky J. General circulation experiments with the primitive equations. I. The basic experiment. Monthly Weather Review, 1963, vol. 91, no. 3, pp. 99–164. DOI: 10.1175/1520-0493(1963)091<0099:GCEWTP>2.3.CO;2

[9] Osher S., Fedkiw R.P. Level set methods and dynamic implicit surfaces. New York, Springer, 2003. 273 p.

[10] Zdravkovich M.M. Flow around circular cylinders. Oxford University Press, 1997. 694 p.

[11] Henderson R.D. Nonlinear dynamics and pattern formation in turbulent wake transition. J. Fluid Mech., 1997, vol. 352, pp. 65–112. DOI: 10.1017/S0022112097007465

[12] He J.W., Glovinski R., Metcalfe R., Nordlander A., Triaux J.P. Active control and drag optimization for flow past a circular cylinder. Part I: Oscillatory cylinder rotation. J. Comput. Phys., 2000, vol. 163, iss. 1, pp. 87–17. DOI: 10.1006/jcph.2000.6556

[13] Zahm A.F. Flow and drag formulae for simple quadrics. NACA Technical Report 253. Washington, US Government Printing Office, 1927. 29 p.

[14] Breuer M. Large eddy simulation of the subcritical flow past a circular cylinder: Numerical and modelling aspects. Int. J. Numer. Meth. Fluids, 1998, vol. 28, iss. 9, pp. 1281–1302. DOI: 10.1002/(SICI)1097-0363(19981215)28:9<1281::AID-FLD759>3.0.CO;2-#

[15] Blackburn H.M., Schmidt S. Large eddy simulation of flow past a circular cylinder. Proc. 14th Australasian Fluid Mechanics Conf., Adelaide University, 2001, pp. 689–692.

[16] Rahman M.M., Karim M.M., Alim M.A. Numerical investigation of unsteady flow past a circular cylinder using 2-D finite volume method. J. Naval Arch. and Marine Eng., 2007, vol. 4, no. 1, pp. 27–42. DOI: 10.3329/jname.v4i1.914

[17] Patel Y. Numerical investigation of flow past a circular cylinder and in a staggered tube bundle using various turbulence models. Masters thesis. Lappeenranta University of Technology, 2010. 87 p.

[18] Wieselsberger von C. Neuere feststellungen uber die gesetze des flussigkeits und luftwiderstandes. Phys. Zeit, 1921, vol. 22, pp. 321–328.

[19] Singh S.P., Mittal S. Flow past a cylinder: Shear layer instability and drag crisis. Int. J. Num. Meth. In Fluids, 2005, vol. 47, iss. 1, pp. 75–98. DOI: 10.1002/fld.807

[20] Henderson R.D. Details of the drag curve near the onset of vortex shedding. Physics of Fluids, 1995, no. 7, pp. 2102–2104. DOI: 10.1063/1.868459 Available at: http://aip.scitation.org/doi/10.1063/1.868459

[21] Marchevsky I.K., Puzikova V.V. Flow-around simulation of circular cylinder performing rotary oscillations by LS-STAG method. Vestn. Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Estestv. Nauki [Herald of the Bauman Moscow State Tech. Univ., Nat. Sci.], 2014, no. 3, pp. 93–107 (in Russ.).