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A Locally-Refined Locally-Conservative Quadtree Finite-Volume Staggered-Grid Scheme

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Droplet Interactions and Spray Processes

Part of the book series: Fluid Mechanics and Its Applications ((FMIA,volume 121))

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Abstract

Simulating flow between porous media and adjacent free-flow regions in sufficient detail becomes computationally expensive when complex flow profiles develop. This is e.g. the case if a variety of strongly coupled physical processes is involved or if the surface between the two flow domains is rough. However, it is often sufficient to only use a fine grid resolution in regions of interest. Here, we present a locally-refined quadtree finite-volume staggered-grid scheme for the two-dimensional Navier-Stokes equations. Local mass and momentum conservation at lines, at which the sides of two finer control volumes touch the side of one coarser control volume, is ensured by defining the fluxes at the sides of coarser control volumes to be equal to the negative sum of fluxes at the two sides of finer control volumes. The method has been successfully applied to locally resolve the flow details in the vicinity of dividing streamlines in a fluid-flow test case. Being developed for the fully-coupled fully-implicit solution of the Navier-Stokes equations, this locally-refined grid scheme is a good basis for increasing the efficiency of simulations of free flow, which is strongly coupled to flow in adjacent domains.

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References

  1. Alkämper, M., Dedner, A., Klöfkorn, R., Nolte, M.: The dune-alugrid module. arXiv preprint arXiv:1407.6954 (2014)

  2. Anderson, R.W., Pember, R.B., Elliott, N.S.: A dynamically adaptive arbitrary Lagrangian-Eulerian method for hydrodynamics. In: Computational Fluid and Solid Mechanics, pp. 1232–1236. Elsevier (2003)

    Google Scholar 

  3. Angelidis, D., Chawdhary, S., Sotiropoulos, F.: Unstructured cartesian refinement with sharp interface immersed boundary method for 3D unsteady incompressible flows. J. Comput. Phys. 325, 272–300 (2016)

    Article  MathSciNet  Google Scholar 

  4. Beltman, R., Anthonissen, M.J.H., Koren, B.: Mimetic staggered discretization of incompressible Navier-Stokes for barycentric dual mesh. In: International Conference on Finite Volumes for Complex Applications, pp. 467–475. Springer, Heidelberg (2017)

    Google Scholar 

  5. Bennett, B.A.V., Smooke, M.D.: Local rectangular refinement with application to axisymmetric laminar flames. Combust. Theor. Model. 2(3), 221–258 (1998)

    Article  Google Scholar 

  6. Chauhan, V.P., Stylianopoulos, T., Boucher, Y., Jain, R.K.: Delivery of molecular and nanoscale medicine to tumors: transport barriers and strategies. Annu. Rev. Chem. Biomol. Eng. 2, 281–298 (2011)

    Article  Google Scholar 

  7. Chénier, E., Eymard, R., Herbin, R.: An extension of the mac scheme to some unstructured meshes. In: Fořt, J., Fürst, J., Halama, J., Herbin, R., Hubert, F. (eds.) Finite Volumes for Complex Applications VI Problems & Perspectives, pp. 253–261. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  8. Davis, T.A.: Algorithm 832: Umfpack v4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004)

    Google Scholar 

  9. Defraeye, T.: Advanced computational modelling for drying processes—a review. Appl. Energy 131, 323–344 (2014)

    Article  Google Scholar 

  10. Gerya, T.V., May, D.A., Duretz, T.: An adaptive staggered grid finite difference method for modeling geodynamic stokes flows with strongly variable viscosity. Geochem. Geophys. Geosyst. 14(4), 1200–1225 (2013)

    Article  Google Scholar 

  11. Harlow, F.H., Welch, J.E.: Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids 8(12), 2182–2189 (1965)

    Article  MathSciNet  Google Scholar 

  12. Huang, Y.L., Liu, J.G., Wang, W.C.: A generalized mac scheme on curvilinear domains. SIAM J. Sci. Comput. 35(5), B953–B986 (2013)

    Article  MathSciNet  Google Scholar 

  13. Jambhekar, V.A., Helmig, R., Schröder, N., Shokri, N.: Free-flow-porous-media coupling for evaporation-driven transport and precipitation of salt in soil. Transp. Porous Media 110(2), 251–280 (2015)

    Article  Google Scholar 

  14. Koch, T., Gläser, D., Weishaupt, K., Ackermann, S., Beck, M., Becker, B., Burbulla, S., Class, H., Coltman, E., Emmert, S., Fetzer, T., Grüninger, C., Heck, K., Hommel, J., Kurz, T., Lipp, M., Mohammadi, F., Scherrer, S., Schneider, M., Seitz, G., Stadler, L., Utz, M., Weinhardt, F., Flemisch, B.: Dumu\(^\text{x}\)3—An Open-source Simulator for Solving Flow and Transport Problems in Porous Media with a Focus on Model Coupling. arXiv:1909.05052 (2019)

  15. Koch, T., Gläser, D., Weishaupt, K., Ackermann, S., Beck, M., Becker, B., Burbulla, S., Class, H., Coltman, E., Fetzer, T., Flemisch, B., Grüninger, C., Heck, K., Hommel, J., Kurz, T., Lipp, M., Mohammadi, F., Schneider, M., Seitz, G., Scholz, S., Weinhardt, F.: Dumux 3.0.0 (2018). https://doi.org/10.5281/zenodo.2479595

  16. Kovasznay, L.I.G.: Laminar flow behind a two-dimensional grid. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 44, pp. 58–62. Cambridge University Press (1948)

    Google Scholar 

  17. Kramer, R.M.J., Pantano, C., Pullin, D.I.: Nondissipative and energy-stable high-order finite-difference interface schemes for 2-d patch-refined grids. J. Comput. Phys. 228(14), 5280–5297 (2009)

    Article  MathSciNet  Google Scholar 

  18. Löhner, R.: Applied Computational Fluid Dynamics Techniques: An Introduction Based on Finite Element Methods. Wiley (2008)

    Google Scholar 

  19. Manhart, M.: A zonal grid algorithm for dns of turbulent boundary layers. Comput. Fluids 33(3), 435–461 (2004)

    Article  Google Scholar 

  20. Minion, M.L.: A projection method for locally refined grids. J. Comput. Phys. 127(1), 158–178 (1996)

    Article  MathSciNet  Google Scholar 

  21. Nicolaides, R.A.: Flow discretization by complementary volume techniques. In: 9th AIAA CFD Meeting, pp. 464–470. Buffalo, New York (1989)

    Google Scholar 

  22. Olshanskii, M.A., Terekhov, K.M., Vassilevski, Y.V.: An octree-based solver for the incompressible Navier-Stokes equations with enhanced stability and low dissipation. Comput. Fluids 84, 231–246 (2013)

    Article  MathSciNet  Google Scholar 

  23. Papadakis, G., Bergeles, G.: A local grid refinement method for three-dimensional turbulent recirculating flows. Int. J. Numer. Methods Fluids 31(7), 1157–1172 (1999)

    Article  Google Scholar 

  24. Vittoz, L., Oger, G., Li, Z., De Leffe, M., Le Touzé, D.: A high-order finite volume solver on locally refined Cartesian meshes–benchmark session. In: International Conference on Finite Volumes for Complex Applications, pp. 73–89. Springer, Heidelberg (2017)

    Google Scholar 

  25. Weishaupt, K., Joekar-Niasar, V., Helmig, R.: An efficient coupling of free flow and porous media flow using the pore-network modeling approach. J. Comput. Phys. X 1, 100011 (2019)

    Google Scholar 

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Acknowledgements

Melanie Lipp is supported by a scholarship of the Landesgraduiertenförderung Baden-Württemberg at the University of Stuttgart. We thank the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) for supporting this work by funding SFB 1313, Project Number 327154368. The code to produce the results of this chapter can be downloaded from https://git.iws.uni-stuttgart.de/dumux-pub/Lipp2019a.git.

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Authors and Affiliations

  1. Institute for Modelling Hydraulic and Environmental Systems, University of Stuttgart, Pfaffenwaldring 61, 70569, Stuttgart, Germany

    Melanie Lipp & Rainer Helmig

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  1. Melanie Lipp
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  2. Rainer Helmig
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Correspondence to Melanie Lipp .

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Editors and Affiliations

  1. Institute of Aerospace Thermodynamics, University of Stuttgart, Stuttgart, Baden-Württemberg, Germany

    Grazia Lamanna

  2. Dipartimento di Ingegneria e Scienze Applicate, Università degli Studi di Bergamo, Dalmine, Bergamo, Italy

    Simona Tonini

  3. Dipartimento di Ingegneria e Scienze Applicate, Università degli Studi di Bergamo, Dalmine, Bergamo, Italy

    Gianpietro Elvio Cossali

  4. Institute of Aerospace Thermodynamics, University of Stuttgart, Stuttgart, Baden-Württemberg, Germany

    Bernhard Weigand

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Lipp, M., Helmig, R. (2020). A Locally-Refined Locally-Conservative Quadtree Finite-Volume Staggered-Grid Scheme. 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_12

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  • DOI: https://doi.org/10.1007/978-3-030-33338-6_12

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-33337-9

  • Online ISBN: 978-3-030-33338-6

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