Close Search Advanced
Journals
Resources
About Us
Open Access

Stability of Projection Methods for Incompressible Flows Using High Order Pressure-Velocity Pairs of Same Degree: Continuous and Discontinuous Galerkin Formulations

Stability of Projection Methods for Incompressible Flows Using High Order Pressure-Velocity Pairs of Same Degree: Continuous and Discontinuous Galerkin Formulations
Preview
Add to basket

Year:    2014

Communications in Computational Physics, Vol. 16 (2014), Iss. 3 : pp. 817–840

Abstract

This paper presents limits for stability of projection type schemes when using high order pressure-velocity pairs of same degree. Two high order $h/p$ variational methods encompassing continuous and discontinuous Galerkin formulations are used to explain previously observed lower limits on the time step for projection type schemes to be stable [18], when $h$- or $p$-refinement strategies are considered. In addition, the analysis included in this work shows that these stability limits depend not only on the time step but on the product of the latter and the kinematic viscosity, which is of particular importance in the study of high Reynolds number flows. We show that high order methods prove advantageous in stabilising the simulations when small time steps and low kinematic viscosities are used.
Drawing upon this analysis, we demonstrate how the effects of this instability can be reduced in the discontinuous scheme by introducing a stabilisation term into the global system. Finally, we show that these lower limits are compatible with Courant-Friedrichs-Lewy (CFL) type restrictions, given that a sufficiently high polynomial order or a mall enough mesh spacing is selected.

Submit Article

You do not have full access to this article.

Already a Subscriber? Sign in as an individual or via your institution

Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cicp.290114.170414a

Communications in Computational Physics, Vol. 16 (2014), Iss. 3 : pp. 817–840

Published online:    2014-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    24

Keywords:   

  1. Flow Structures in the Turbulent Wake of a Cross Flow Wind Turbine

    Ramos Casado, Gemma | Beltrán, Víctor | Le Clainche, Soledad | Ferrer, Esteban | Vega, José M.

    2018 Wind Energy Symposium, (2018),

    https://doi.org/10.2514/6.2018-0253 [Citations: 3]

  2. Stability evaluation of high-order splitting method for incompressible flow based on discontinuous velocity and continuous pressure

    Xu, Liyang | Xu, Xinhai | Ren, Xiaoguang | Guo, Yunrui | Feng, Yongquan | Yang, Xuejun

    Advances in Mechanical Engineering, Vol. 11 (2019), Iss. 10

    https://doi.org/10.1177/1687814019855586 [Citations: 1]

  3. On the stability of projection methods for the incompressible Navier–Stokes equations based on high-order discontinuous Galerkin discretizations

    Fehn, Niklas | Wall, Wolfgang A. | Kronbichler, Martin

    Journal of Computational Physics, Vol. 351 (2017), Iss. P.392

    https://doi.org/10.1016/j.jcp.2017.09.031 [Citations: 34]

  4. Robust and efficient discontinuous Galerkin methods for under-resolved turbulent incompressible flows

    Fehn, Niklas | Wall, Wolfgang A. | Kronbichler, Martin

    Journal of Computational Physics, Vol. 372 (2018), Iss. P.667

    https://doi.org/10.1016/j.jcp.2018.06.037 [Citations: 29]

  5. Analysis of a linearization scheme for an interior penalty discontinuous Galerkin method for two‐phase flow in porous media with dynamic capillarity effects

    Karpinski, Stefan | Pop, Iuliu Sorin | Radu, Florin Adrian

    International Journal for Numerical Methods in Engineering, Vol. 112 (2017), Iss. 6 P.553

    https://doi.org/10.1002/nme.5526 [Citations: 17]
  6. Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

    Implicit Large Eddy Simulations for NACA0012 Airfoils Using Compressible and Incompressible Discontinuous Galerkin Solvers

    Ferrer, Esteban | Manzanero, Juan | Rueda-Ramirez, Andres M. | Rubio, Gonzalo | Valero, Eusebio

    2020

    https://doi.org/10.1007/978-3-030-39647-3_38 [Citations: 2]

  7. A high-order semi-explicit discontinuous Galerkin solver for 3D incompressible flow with application to DNS and LES of turbulent channel flow

    Krank, Benjamin | Fehn, Niklas | Wall, Wolfgang A. | Kronbichler, Martin

    Journal of Computational Physics, Vol. 348 (2017), Iss. P.634

    https://doi.org/10.1016/j.jcp.2017.07.039 [Citations: 61]
  8. CFD for Wind and Tidal Offshore Turbines

    Flow Scales in Cross-Flow Turbines

    Ferrer, Esteban | Le Clainche, Soledad

    2015

    https://doi.org/10.1007/978-3-319-16202-7_1 [Citations: 2]

  9. A spectral deferred correction method for incompressible flow with variable viscosity

    Stiller, Jörg

    Journal of Computational Physics, Vol. 423 (2020), Iss. P.109840

    https://doi.org/10.1016/j.jcp.2020.109840 [Citations: 4]

  10. : A high-order discontinuous Galerkin solver for flow simulations and multi-physics applications

    Ferrer, E. | Rubio, G. | Ntoukas, G. | Laskowski, W. | Mariño, O.A. | Colombo, S. | Mateo-Gabín, A. | Marbona, H. | Manrique de Lara, F. | Huergo, D. | Manzanero, J. | Rueda-Ramírez, A.M. | Kopriva, D.A. | Valero, E.

    Computer Physics Communications, Vol. 287 (2023), Iss. P.108700

    https://doi.org/10.1016/j.cpc.2023.108700 [Citations: 23]

  11. A pressure-based solver for low-Mach number flow using a discontinuous Galerkin method

    Hennink, Aldo | Tiberga, Marco | Lathouwers, Danny

    Journal of Computational Physics, Vol. 425 (2021), Iss. P.109877

    https://doi.org/10.1016/j.jcp.2020.109877 [Citations: 7]

  12. A wavelet‐based variational multiscale method for the LES of incompressible flows in a high‐order DG‐FEM framework

    Pinto, Brijesh | de la Llave Plata, Marta | Lamballais, Eric

    International Journal for Numerical Methods in Fluids, Vol. 92 (2020), Iss. 4 P.285

    https://doi.org/10.1002/fld.4784 [Citations: 1]

  13. Nektar++: An open-source spectral/hp element framework

    Cantwell, C.D. | Moxey, D. | Comerford, A. | Bolis, A. | Rocco, G. | Mengaldo, G. | De Grazia, D. | Yakovlev, S. | Lombard, J.-E. | Ekelschot, D. | Jordi, B. | Xu, H. | Mohamied, Y. | Eskilsson, C. | Nelson, B. | Vos, P. | Biotto, C. | Kirby, R.M. | Sherwin, S.J.

    Computer Physics Communications, Vol. 192 (2015), Iss. P.205

    https://doi.org/10.1016/j.cpc.2015.02.008 [Citations: 419]

  14. Finite surface discretization for incompressible Navier-Stokes equations and coupled conservation laws

    Hokpunna, Arpiruk | Misaka, Takashi | Obayashi, Shigeru | Wongwises, Somchai | Manhart, Michael

    Journal of Computational Physics, Vol. 423 (2020), Iss. P.109790

    https://doi.org/10.1016/j.jcp.2020.109790 [Citations: 4]

  15. Efficiency of high‐performance discontinuous Galerkin spectral element methods for under‐resolved turbulent incompressible flows

    Fehn, Niklas | Wall, Wolfgang A. | Kronbichler, Martin

    International Journal for Numerical Methods in Fluids, Vol. 88 (2018), Iss. 1 P.32

    https://doi.org/10.1002/fld.4511 [Citations: 35]

  16. Implicit formulation of material point method for analysis of incompressible materials

    Kularathna, Shyamini | Soga, Kenichi

    Computer Methods in Applied Mechanics and Engineering, Vol. 313 (2017), Iss. P.673

    https://doi.org/10.1016/j.cma.2016.10.013 [Citations: 57]

  17. Large Eddy Simulation of an Inverted Multi-element Wing in Ground Effect

    Slaughter, James | Moxey, David | Sherwin, Spencer

    Flow, Turbulence and Combustion, Vol. 110 (2023), Iss. 4 P.917

    https://doi.org/10.1007/s10494-023-00404-7 [Citations: 2]
  18. Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2016

    Towards p-Adaptive Spectral/hp Element Methods for Modelling Industrial Flows

    Moxey, D. | Cantwell, C. D. | Mengaldo, G. | Serson, D. | Ekelschot, D. | Peiró, J. | Sherwin, S. J. | Kirby, R. M.

    2017

    https://doi.org/10.1007/978-3-319-65870-4_4 [Citations: 2]

  19. A GPU accelerated discontinuous Galerkin incompressible flow solver

    Karakus, A. | Chalmers, N. | Świrydowicz, K. | Warburton, T.

    Journal of Computational Physics, Vol. 390 (2019), Iss. P.380

    https://doi.org/10.1016/j.jcp.2019.04.010 [Citations: 20]

  20. Onset of three-dimensional flow instabilities in lid-driven circular cavities

    González, L. M. | Ferrer, E. | Díaz-Ojeda, H. R.

    Physics of Fluids, Vol. 29 (2017), Iss. 6

    https://doi.org/10.1063/1.4984242 [Citations: 20]
  21. Recent Advances in CFD for Wind and Tidal Offshore Turbines

    Simple Models for Cross Flow Turbines

    Ferrer, Esteban | Le Clainche, Soledad

    2019

    https://doi.org/10.1007/978-3-030-11887-7_1 [Citations: 1]

  22. The Bassi Rebay 1 scheme is a special case of the Symmetric Interior Penalty formulation for discontinuous Galerkin discretisations with Gauss–Lobatto points

    Manzanero, Juan | Rueda-Ramírez, Andrés M. | Rubio, Gonzalo | Ferrer, Esteban

    Journal of Computational Physics, Vol. 363 (2018), Iss. P.1

    https://doi.org/10.1016/j.jcp.2018.02.035 [Citations: 19]

  23. Molecular dynamics simulations in hybrid particle-continuum schemes: Pitfalls and caveats

    Stalter, S. | Yelash, L. | Emamy, N. | Statt, A. | Hanke, M. | Lukáčová-Medvid’ová, M. | Virnau, P.

    Computer Physics Communications, Vol. 224 (2018), Iss. P.198

    https://doi.org/10.1016/j.cpc.2017.10.016 [Citations: 14]

  24. An interior penalty stabilised incompressible discontinuous Galerkin–Fourier solver for implicit large eddy simulations

    Ferrer, Esteban

    Journal of Computational Physics, Vol. 348 (2017), Iss. P.754

    https://doi.org/10.1016/j.jcp.2017.07.049 [Citations: 48]

  25. A high-order generalised differential quadrature element method for simulating 2D and 3D incompressible flows on unstructured meshes

    Liu, Yaguang | Shu, Chang | Yu, Peng | Liu, Yangyang | Zhang, Hua | Lu, Chun

    Computers & Mathematics with Applications, Vol. 174 (2024), Iss. P.230

    https://doi.org/10.1016/j.camwa.2024.08.027 [Citations: 0]

  26. Spectral/hp element simulation of flow past a Formula One front wing: Validation against experiments

    Buscariolo, Filipe F. | Hoessler, Julien | Moxey, David | Jassim, Ayad | Gouder, Kevin | Basley, Jeremy | Murai, Yushi | Assi, Gustavo R.S. | Sherwin, Spencer J.

    Journal of Wind Engineering and Industrial Aerodynamics, Vol. 221 (2022), Iss. P.104832

    https://doi.org/10.1016/j.jweia.2021.104832 [Citations: 7]

  27. Design of a Smagorinsky spectral Vanishing Viscosity turbulence model for discontinuous Galerkin methods

    Manzanero, Juan | Ferrer, Esteban | Rubio, Gonzalo | Valero, Eusebio

    Computers & Fluids, Vol. 200 (2020), Iss. P.104440

    https://doi.org/10.1016/j.compfluid.2020.104440 [Citations: 31]

  28. A high-order nodal discontinuous Galerkin method for simulation of three-dimensional non-cavitating/cavitating flows

    Hajihassanpour, Mahya | Hejranfar, Kazem

    Finite Elements in Analysis and Design, Vol. 200 (2022), Iss. P.103681

    https://doi.org/10.1016/j.finel.2021.103681 [Citations: 0]

  29. Multimaterial Eulerian finite element formulation for pressure‐sensitive adhesives

    Nishiguchi, Koji | Okazawa, Shigenobu | Tsubokura, Makoto

    International Journal for Numerical Methods in Engineering, Vol. 114 (2018), Iss. 13 P.1368

    https://doi.org/10.1002/nme.5790 [Citations: 9]

  30. Adaptation strategies for high order discontinuous Galerkin methods based on Tau-estimation

    Kompenhans, Moritz | Rubio, Gonzalo | Ferrer, Esteban | Valero, Eusebio

    Journal of Computational Physics, Vol. 306 (2016), Iss. P.216

    https://doi.org/10.1016/j.jcp.2015.11.032 [Citations: 36]

  31. Blade–wake interactions in cross-flow turbines

    Ferrer, Esteban | Willden, Richard H.J.

    International Journal of Marine Energy, Vol. 11 (2015), Iss. P.71

    https://doi.org/10.1016/j.ijome.2015.06.001 [Citations: 17]

  32. On the performance of the DG method with a directional do-nothing boundary condition

    Garcia, Aureo Quintas | Gomes, Francisco Augusto Aparecido | Hecke, Mildred Ballin

    Journal of the Brazilian Society of Mechanical Sciences and Engineering, Vol. 39 (2017), Iss. 10 P.3919

    https://doi.org/10.1007/s40430-017-0868-8 [Citations: 1]

  33. A Reduced Order Model to Predict Transient Flows around Straight Bladed Vertical Axis Wind Turbines

    Le Clainche, Soledad | Ferrer, Esteban

    Energies, Vol. 11 (2018), Iss. 3 P.566

    https://doi.org/10.3390/en11030566 [Citations: 43]

  34. Comparisons of p-adaptation strategies based on truncation- and discretisation-errors for high order discontinuous Galerkin methods

    Kompenhans, Moritz | Rubio, Gonzalo | Ferrer, Esteban | Valero, Eusebio

    Computers & Fluids, Vol. 139 (2016), Iss. P.36

    https://doi.org/10.1016/j.compfluid.2016.03.026 [Citations: 41]

  35. Dispersion-Dissipation Analysis for Advection Problems with Nonconstant Coefficients: Applications to Discontinuous Galerkin Formulations

    Manzanero, Juan | Rubio, Gonzalo | Ferrer, Esteban | Valero, Eusebio

    SIAM Journal on Scientific Computing, Vol. 40 (2018), Iss. 2 P.A747

    https://doi.org/10.1137/16M1101143 [Citations: 25]

  36. Implicit-explicit and explicit projection schemes for the unsteady incompressible Navier–Stokes equations using a high-order dG method

    Emamy, Nehzat | Kummer, Florian | Mrosek, Markus | Karcher, Martin | Oberlack, Martin

    Computers & Fluids, Vol. 154 (2017), Iss. P.285

    https://doi.org/10.1016/j.compfluid.2017.06.003 [Citations: 4]

  37. A pressure correction scheme for generalized form of energy-stable open boundary conditions for incompressible flows

    Dong, S. | Shen, J.

    Journal of Computational Physics, Vol. 291 (2015), Iss. P.254

    https://doi.org/10.1016/j.jcp.2015.03.012 [Citations: 28]

  38. High-order arbitrary Lagrangian–Eulerian discontinuous Galerkin methods for the incompressible Navier–Stokes equations

    Fehn, Niklas | Heinz, Johannes | Wall, Wolfgang A. | Kronbichler, Martin

    Journal of Computational Physics, Vol. 430 (2021), Iss. P.110040

    https://doi.org/10.1016/j.jcp.2020.110040 [Citations: 8]

  39. An entropy–stable discontinuous Galerkin approximation for the incompressible Navier–Stokes equations with variable density and artificial compressibility

    Manzanero, Juan | Rubio, Gonzalo | Kopriva, David A. | Ferrer, Esteban | Valero, Eusebio

    Journal of Computational Physics, Vol. 408 (2020), Iss. P.109241

    https://doi.org/10.1016/j.jcp.2020.109241 [Citations: 13]

  40. A high-order Discontinuous Galerkin solver for unsteady incompressible turbulent flows

    Noventa, G. | Massa, F. | Bassi, F. | Colombo, A. | Franchina, N. | Ghidoni, A.

    Computers & Fluids, Vol. 139 (2016), Iss. P.248

    https://doi.org/10.1016/j.compfluid.2016.03.007 [Citations: 25]

  41. An implicit dual-time stepping high-order nodal discontinuous Galerkin method for solving incompressible flows on triangle elements

    Hajihassanpour, M. | Hejranfar, K.

    Mathematics and Computers in Simulation, Vol. 168 (2020), Iss. P.173

    https://doi.org/10.1016/j.matcom.2019.08.011 [Citations: 5]

  42. A stable discontinuous Galerkin method based on high‐order dual splitting scheme without additional stabilization term for incompressible flows

    Ma, Mengxia | Ouyang, Jie | Wang, Xiaodong | Zhang, Chenhui

    International Journal for Numerical Methods in Fluids, Vol. 93 (2021), Iss. 8 P.2660

    https://doi.org/10.1002/fld.4992 [Citations: 2]