An Adaptive Time Filter Based Finite Element Method for the Velocity-Vorticity-Temperature Model of the Incompressible Non-Isothermal Fluid Flows

Year 2020, Volume: 33 Issue: 3, 696 - 713, 01.09.2020

Mine Akbaş

https://doi.org/10.35378/gujs.681778

Cited By: 2

Abstract

This paper studies a velocity-vorticity-temperature (VVT) model of the Boussinesq equations and introduces a numerical method for solving that. The proposed numerical method adds separate three minimally intrusive steps, one for each fluid variable, except pressure, to the standard semi-implicit backward-Euler (BE) approximation of VVT-model. The key idea in these intrusive steps is to post-process the BE approximate solutions with 2-step, second order, linear time filters. The paper provides full mathematical analysis of the proposed numerical method, and two numerical experiments for that. The first numerical experiment verifies the predicted convergence rates while the second one shows the effectiveness of the method on a benchmark problem. 

Keywords

Finite element method, The Boussinesq equations, Time filter

References

• Referans1: Adams, R.,“Sobolev Spaces”. 2nd ed. Newyork: Academic Press, 1975, pp 30-73. Referans2: Akbas, M., Rebholz, L. G., Zerfas C., “Optimal Vorticity Accuracy in an Efficient Velocity-Vorticity Method for the Navier-Stokes Equations”, Calcolo, 55-3: 1-29, (2018). Referans3: Cıbık, A., Kaya, S., “A Projection Based Stabilized Finite Element Method for Natural Convection Problem”, J. Math. Anal. Appl., 381-2: 469-484, (2011). Referans4: Cıbık, A., Kaya S., “A Projection Based Stabilized Finite Element Method for Natural Convection Problem”, J. Math. Anal. Appl., 381-2: 469-484, (2013). Referans5: Cengel, Y. A. and Ghajar, A. J.,“ Heat and Mass Transfer: Fundamentals and Applications”, 5th ed. New York: McGraw-Hill Education, (2015). Referans6: Frutos, J., Garcia-Archilla, B., John,V., Novo, J., “Grad-Div Stabilization for the Evolutionary Oseen Problem with Inf-Sup Stable Finite Elements”, J. Sci. Comput., 66-3 : 991-1024, (2016). Referans7: Decaria, V., Layton, W., Zhao, H., “A Time-Accurate, Adaptive Discretization for Fluid Flow Problems”, Inter. J. Numer. Anal. Mod., 17-2: 254-280, (2020). Referans8: Gresho, P. M., Sani, R. L., “Incompressible Flow and the Finite Element Method, Volume 2, Isothermal Laminar Flow”, Wiley, (1998). Referans9: Guzel, A. and Layton, W., “Time Filters Increase Accuracy of the Fully Implicit Method”, BIT Numer. Math., 58-2: 301-315, (2018). Referans10: John, V., Matthies, G. and Rang, J. A., “Comparison of Time-Discretization/Linearization Approaches for the Incompressible Navier-Stokes Equations”, Comput. Methods Appl. Mech. Eng., 195: 5995-6010, (2006). Referans11: Frutos, J., Garcia-Archilla , B., Novo, J., “The Post-Processed Mixed Finite-Element Method for the Navier-Stokes Equations: Refined Error Bounds”, SIAM J. Numer. Anal., 46-1: 201-230, (2008). Referans12: Heister, T., Olshanskii, M.A., Rebholz, L. G., “Unconditional Long-Time Stability of a Velocity-Vorticity Method for 2D Navier-Stokes Equations”, Numer. Math., 135(1): 143-167, (2017). Referans13: Heywood, J. G., Rannacher, R., “Finite Element Approximation of the Nonstationary Navier-Stokes Problem, Part II: Error Analysis for the Second-Order Time Discretization”, SIAM J. Numer. Anal., 27(2): 353-384, (1990). Referans14: Franca, L. P., John, V., Matthies, G., and Tobiska, L., “An Inf-Sup Stable and Residual-Free Bubble Element for the Oseen Equations”, SIAM J. Numer. Anal., 45(6): 2392-2407, (2007). Referans15: Layton, W., “Introduction to Finite Element Methods”, SIAM, Philadelphia, (2008). Referans16: Lee, H. L., Olshanskii, M. A., Rebholz, L. G., “On Error Analysis for the 3D Navier-Stokes Equations in Velocity-Vorticity-Helicity Form”, SIAM J. Numer. Anal., 49(2): 711-732, (2011). Referans17: Olshanskii, M. A., “A Fluid Solver Based on Vorticity-Helical Density Equations with Application to a Natural Convection in a Cubic Cavity”, Int. J. Numer. Methods Fluids., 69(5): 983-994, (2012). Referans18: Wang, C., Liu, J. -G. “Analysis of Finite Difference Schemes for Unsteady Navier-Stokes Equations in Vorticity Formulation”, Numer. Math., 91(3): 543-576, (2002). Referans19: Wang, C., Liu, J. -G., Johnston, H., “Analysis of a Fourth Order Finite Difference Method for the Incompressible Boussinesq Equations”, Numer. Math., 97(3) : 555-594, (2004). Referans20: Wong, K. L., Baker, A. J., “A 3D incompressible Navier-Stokes Velocity-Vorticity Weak Form Finite Element Algorithm”, Int. J. Numer. Methods Fluids, 38(2), 99-123, (2002).