Abstract
In this work we present a high-order discontinuous Galerkin approach for the simulation of variable density incompressible (VDI) flows. Here, the density is treated as a purely advected property tracking possibly multiple (more than two) components, while the fluids interface is captured in a diffuse fashion by the high-degree polynomial solution thus not requiring any geometrical reconstruction. Specific care is devoted to deal with density over/undershoots, spurious oscillations at flows interfaces and Godunov numerical fluxes at inter-element boundaries. Time integration is performed with high-order implicit schemes thus preventing any time step restriction condition. Promising results with high-degree polynomial representation and relatively coarse meshes are achieved on numerical experiments involving high-density ratios (water–air) and the possible interaction of more than two components.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods. Springer, Berlin (2008)
Bassi, F., Botti, L., Colombo, A., Ghidoni, A., Massa, F.: Linearly implicit Rosenbrock-type Runge-Kutta schemes for the Discontinuous Galerkin solution of compressible and incompressible unsteady flows. Comput. Fluids (2015). https://doi.org/10.1016/j.compfluid.2015.06.007
Elsworth, D.T., Toro, E.F.: Riemann solvers for solving the incompressible Navier-Stokes equations using the artificial compressibility method. College of Aeronautics, Cranfield Institute of Technology, 9208 (1992)
Bassi, F., Crivellini, A., Di Pietro, D.A., Rebay, S.: An artificial compressibility flux for the discontinuous Galerkin solution of the incompressible Navier-Stokes equations. J. Comput. Phys. (2006). https://doi.org/10.1016/j.jcp.2006.03.006
Bassi, F., Massa, F., Botti, L., Colombo, A.: Artificial compressibility Godunov fluxes for variable density incompressible flows. Comput. Fluids (2018). https://doi.org/10.1016/j.compfluid.2017.09.010
Bassi, F., Rebay, S., Mariotti, G., Pedinotti, S., Savini, M.: A high-order accurate discontinuous finite element method for inviscid and viscous turbomachinery flows. In: Proceedings of 2nd European Conference on Turbomachinery Fluid Dynamics and Thermodynamics, pp. 99–108 (1997)
Brezzi, F., Manzini, G., Marini, D., Pietra, P., Russo, A.: Discontinuous Galerkin approximations for elliptic problems. Numer. Meth. Part. D. E. 16, 365–378 (2000)
Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2002)
Persson, P.-O., Peraire, J.: Sub-cell shock capturing for discontinuous Galerkin Methods. In: 44th AIAA Aerospace Sciences Meeting and Exhibit, Nevada (2006)
Klöckner, A., Warburton, T., Hesthaven, J.S.: Viscous shock capturing in a time-explicit discontinuous Galerkin Method. Math. Model. Nat. Phenom. (2011). https://doi.org/10.1051/mmnp/20116303
Jaffre, J., Johnson, C., Szepessy, A.: Convergence of the discontinuous Galerkin finite element method for hyperbolic conservation laws. Math. Models Methods Appl. Sci. 5(3), 286–367 (1995)
Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Springer, Berlin (2010)
Söderlind, G.: Digital filters in adaptive time-stepping. ACM Trans. Math. Softw. V 1–24 (2005)
Saad, Y., Shults, M.H.: A generalised minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986)
Smith, B., Bjørstad, P., Gropp, W.: Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, Cambridge (1996)
Ritter, A.: Die Fortpflanzung der Wasserwellen. Z. Ver. Deut. Ing 36, 947–954 (1892)
Dressler, F.: Comparison of theories and experiments for the hydraulic dam-break wave. Proc. Int. Assoc. of Sci. Hydrol. Assemblée Générale 3(38), 319–328 (1954)
Acknowledgements
F. Massa is supported by the Supporting Talented Researchers (STaRS) programm of the University of Bergamo.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Massa, F.C., Bassi, F., Botti, L., Colombo, A. (2020). An Implicit High-Order Discontinuous Galerkin Approach for Variable Density Incompressible Flows. In: Lamanna, G., Tonini, S., Cossali, G., Weigand, B. (eds) Droplet Interactions and Spray Processes. Fluid Mechanics and Its Applications, vol 121. Springer, Cham. https://doi.org/10.1007/978-3-030-33338-6_15
Download citation
DOI: https://doi.org/10.1007/978-3-030-33338-6_15
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-33337-9
Online ISBN: 978-3-030-33338-6
eBook Packages: EngineeringEngineering (R0)