zbMATH — the first resource for mathematics

Slope limiting the velocity field in a discontinuous Galerkin divergence-free two-phase flow solver. (English) Zbl 07132269
Summary: Solving the Navier-Stokes equations when the density field contains a large and sharp discontinuity – such as a water/air free surface – is numerically challenging. Convective instabilities cause Gibbs oscillations which quickly destroy the solution. We investigate the use of slope limiters for the velocity field to overcome this problem in a way that does not compromise on the mass-conservation properties. The equations are discretised using a symmetric interior-penalty discontinuous Galerkin finite element method that is divergence-free to machine precision.
A slope limiter made specifically for exactly divergence-free (solenoidal) fields is presented and used to illustrate the difficulties in obtaining convectively stable fields that are also exactly solenoidal. The lessons learned from this are applied in constructing a simpler method based on the use of an existing scalar slope limiter applied to each velocity component.
We show by numerical examples how both presented slope limiting methods are vastly superior to the naive non-limited method. The methods can solve difficult two-phase problems with high density-ratios and high Reynolds numbers – typical for marine and offshore water/air simulations – in a way that conserves mass and stops unbounded energy growth caused by the Gibbs phenomenon.
76 Fluid mechanics
FEniCS; SciPy; SyFi
Full Text: DOI
[1] Liu, C.; Walkington, N. J., Convergence of numerical approximations of the incompressible Navier-Stokes equations with variable density and viscosity, SIAM J Numer Anal, 45, 3, 1287-1304 (2007) · Zbl 1138.76048
[2] Guermond, J.-L.; Quartapelle, L., A projection FEM for variable density incompressible flows, J Comput Phys, 165, 1, 167-188 (2000) · Zbl 0994.76051
[3] Pyo, J.-H.; Shen, J., Gauge-Uzawa methods for incompressible flows with variable density, J Comput Phys, 221, 1, 181-197 (2007) · Zbl 1109.76037
[4] Guermond, J.-L.; Salgado, A., A splitting method for incompressible flows with variable density based on a pressure Poisson equation, J Comput Phys, 228, 8, 2834-2846 (2009) · Zbl 1159.76028
[5] Hirt, C. W.; Nichols, B. D., Volume of fluid (VOF) method for the dynamics of free boundaries, J Comput Phys, 39, 1, 201-225 (1981) · Zbl 0462.76020
[6] Ubbink, O., Numerical prediction of two fluid systems with sharp interfaces (1997), Imperial College, University of London, Ph.D. thesis
[7] Muzaferija, S.; Peric, M.; Sames, P. C.; Schellin, T. E., A two-fluid Navier-Stokes solver to simulate water entry, Proceedings of the twenty-second symposium on naval hydrodynamics, 638-651 (1999), The National Academies Press: The National Academies Press Washington, DC
[8] Osher, S.; Sethian, J. A., Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J Comput Phys, 79, 1, 12-49 (1988) · Zbl 0659.65132
[9] Olsson, E.; Kreiss, G., A conservative level set method for two phase flow, J Comput Phys, 210, 1, 225-246 (2005) · Zbl 1154.76368
[10] Touré, M. K.; Fahsi, A.; Soulaïmani, A., Stabilised finite-element methods for solving the level set equation with mass conservation, Int J Comput Fluid Dyn, 30, 1, 38-55 (2016)
[11] Sweby, P. K., High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J Numer Anal, 21, 5, 995-1011 (1984) · Zbl 0565.65048
[12] Leonard, B. P., Simple high-accuracy resolution program for convective modelling of discontinuities, Int J Numer Methods Fluids, 8, 10, 1291-1318 (1988) · Zbl 0667.76125
[13] Peskin, C. S., The immersed boundary method, Acta Numer, 11, 479-517 (2002) · Zbl 1123.74309
[14] Cockburn, B.; Kanschat, G.; Schötzau, D., The local discontinuous Galerkin method for the Oseen equations, Math Comput, 73, 246, 569-594 (2004) · Zbl 1066.76036
[15] Cockburn, B.; Kanschat, G.; Schötzau, D., A locally conservative LDG method for the incompressible Navier-Stokes equations, Math Comput, 74, 251, 1067-1096 (2005) · Zbl 1069.76029
[16] Arnold, D. N., An interior penalty finite element method with discontinuous elements, SIAM J Numer Anal, 19, 4, 742-760 (1982) · Zbl 0482.65060
[17] Cockburn, B.; Shu, C.-W., The Runge-Kutta discontinuous Galerkin method for conservation laws v, J Comput Phys, 141, 2, 199-224 (1998) · Zbl 0920.65059
[18] Cockburn, B.; Shu, C.-W., Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J Sci Comput, 16, 3, 173-261 (2001) · Zbl 1065.76135
[19] Kuzmin, D., A vertex-based hierarchical slope limiter for p-adaptive discontinuous Galerkin methods, J Comput Appl Math, 233, 12, 3077-3085 (2010) · Zbl 1252.76045
[20] Michoski, C.; Dawson, C.; Kubatko, E. J.; Wirasaet, D.; Brus, S.; Westerink, J. J., A comparison of artificial viscosity, limiters, and filters, for high order discontinuous Galerkin solutions in nonlinear settings, J Sci Comput, 66, 1, 406-434 (2016) · Zbl 1338.65228
[21] Zingan, V.; Guermond, J.-L.; Morel, J.; Popov, B., Implementation of the entropy viscosity method with the discontinuous Galerkin method, Comput Methods Appl Mech Eng, 253, 479-490 (2013) · Zbl 1297.76109
[22] Kuzmin, D., Slope limiting for discontinuous Galerkin approximations with a possibly non-orthogonal Taylor basis, Int J Numer Methods Fluids, 71, 9, 1178-1190 (2013)
[23] Unverdi, S. O.; Tryggvason, G., A front-tracking method for viscous, incompressible, multi-fluid flows, J Comput Phys, 100, 1, 25-37 (1992) · Zbl 0758.76047
[24] Sussman, M.; Smereka, P.; Osher, S., A level set approach for computing solutions to incompressible two-phase flow, J Comput Phys, 114, 1, 146-159 (1994) · Zbl 0808.76077
[25] Hundsdorfer, W.; Ruuth, S. J.; Spiteri, R. J., Monotonicity-preserving linear multistep methods, SIAM J Numer Anal, 41, 2, 605-623 (2003) · Zbl 1050.65070
[26] Nitsche, J. A., Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 36, 1, 9-15 (1971) · Zbl 0229.65079
[27] Epshteyn, Y.; Riviére, B., Estimation of penalty parameters for symmetric interior penalty Galerkin methods, J Comput Appl Math, 206, 2, 843-872 (2007) · Zbl 1141.65078
[28] Shahbazi, K.; Fischer, P. F.; Ethier, C. R., A high-order discontinuous Galerkin method for the unsteady incompressible Navier-Stokes equations, J Comput Phys, 222, 1, 391-407 (2007) · Zbl 1216.76034
[29] Cockburn, B.; Kanschat, G.; Schötzau, D., A note on discontinuous Galerkin divergence-free solutions of the Navier-Stokes equations, J Sci Comput, 31, 1-2, 61-73 (2007) · Zbl 1151.76527
[30] Kirby, R. C.; Logg, A.; Rognes, M. E.; Terrel, A. R., Common and unusual finite elements, Automated solution of differential equations by the finite element method. Automated solution of differential equations by the finite element method, Lecture Notes in Computational Science and Engineering, 95-119 (2012), Springer, Berlin, Heidelberg
[31] Nédélec, J.-C., A new family of mixed finite elements in R3, Numerische Mathematik, 50, 1, 57-81 (1986) · Zbl 0625.65107
[32] Brezzi, F.; Fortin, M., Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, 15 (1991), Springer-Verlag, New York · Zbl 0788.73002
[33] Baker, G. A.; Jureidini, W. N.; Karakashian, O. A., Piecewise solenoidal vector fields and the Stokes problem, SIAM J Numer Anal, 27, 6, 1466-1485 (1990) · Zbl 0719.76047
[34] Jones E., Oliphant T., Peterson P., others. SciPy: Open source scientific tools for Python. 2001. http://www.scipy.org/.
[35] Landet, T., Ocellaris: a DG FEM solver for free-surface flows, J Open Source Softw, 4, 35, 1239 (2019)
[36] Logg, A.; Mardal, K.-A.; Wells, G., Automated solution of differential equations by the finite element method: the FEniCS book (2012), Springer Science & Business Media · Zbl 1247.65105
[37] T. Landet, Ocellaris DG FEM software and input files to reproduce results, Zenodo, 2017, 10.5281/zenodo.845352.
[38] Martin, J. C.; Moyce, W. J., Part IV. An experimental study of the collapse of liquid columns on a rigid horizontal plane, Philos Trans R Soc Lond A Math Phys Eng Sci, 244, 882, 312-324 (1952)
[39] Guermond, J.-L.; de Luna, M. Q.; Thompson, T., An conservative anti-diffusion technique for the level set method, J Comput Appl Math, 321, 448-468 (2017) · Zbl 1457.76126
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.