Abstract
This study presents a subgrid artificial viscosity method for approximating solutions to the Boussinesq equations. The stability is obtained by adding a term via an artificial viscosity and then removing it only on the coarse mesh scale. The method includes both vorticity in the viscous term and a grad-div stabilization. We analyze the method from both analytical and computational point of view and show that it is unconditionally stable and optimally convergent. Several numerical experiments are provided that support the derived theoretical results and demonstrate the efficiency and accuracy of the method.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bejan, A. & Krauss, A. D.: Heat Transfer Handbook. John Wiley & Sons (2003)
Galvin, K. J.: New subgrid artificial viscosity Galerkin methods for the Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 200, 242–250 (2011)
Layton, W.: A connection between subgrid scale eddy viscosity and mixed methods. Appl. Math. Comput. 133, 147–157 (2002)
Borggaard, J., Illiescu, T., Lee, H. & Roop, J. P.: A two level Smagorinsky model. Multiscale Model. Simul. Mech. Engrg. 7, 599–621 (2008)
Çıbık, A. & Kaya, S.: A projection-based stabilized finite element method for steady-state natural convection problem. Int. J. Math. Anal. 381, 469–484 (2011)
Hecht, F.: New development in FreeFem++.J. Numer. Math. 20, 251–265 (2012)
Belenli, M. A., Kaya, S. & Rebholz, L.G.: An explicitly decoupled variational multiscale method for incompressible, non-isothermal flows. Comput. Meth.Appl. Math. 15, 1–20 (2015)
Liu, J. G., Wang, C. & Johnston, H.: A fourth order scheme for incompressible Boussinesq equations. J. Sci. Comput. 18, 253–285 (2003)
Wan, D. C., Patnaik, B. S. V. & Wei, G. W.: A new benchmark quality solution for the buoyancy driven- cavity by discrete singular convolution. Numerical Heat Transfer, Part B 40, 199–228 (2001)
Younes, A., Makrad, A., Zidane, A., Shao, Q. & Bouhala, L.: A combination of Crouzeix-Raviart, Discontinuous Galerkin and MPFA methods for buoyancy-driven flows. Int. J. Numer. Meth. Heat Fluid Flow. 24(3), 735–759 (2014)
Kosec, G. & S̆arler, B.: Solution of a low Prandtl number natural convection benchmark by a local meshless method. Int. J. Numer. Meth. Heat Fluid Flow.23(1), 189–204 (2013)
Girault, V. & Raviart, P. A.: Finite element method for Navier-Stokes equations. Springer, Berlin (1986)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Demir, M., Kaya, S. (2021). Numerical Investigation of the Boussinesq Equations Through a Subgrid Artificial Viscosity Method. In: Vermolen, F.J., Vuik, C. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2019. Lecture Notes in Computational Science and Engineering, vol 139. Springer, Cham. https://doi.org/10.1007/978-3-030-55874-1_28
Download citation
DOI: https://doi.org/10.1007/978-3-030-55874-1_28
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-55873-4
Online ISBN: 978-3-030-55874-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)