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Implicit Methods for Simulating Low Reynolds Number Free Surface Flows: Improvements on MAC-Type Methods

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The Impact of Applications on Mathematics

Part of the book series: Mathematics for Industry ((MFI,volume 1))

Abstract

This paper is concerned with describing the main improvements introduced to the MAC (Marker-And-Cell) method for the numerical simulation of low Reynolds number free surface flows, namely: a stable implicit treatment of the pressure boundary condition for projection methods, a semi-implicit method based on the Crank–Nicolson (C–N) discretization of the momentum equations, a more accurate method for moving the massless particles representing the free surface and a viscoelastic model based on the Pom-Pom constitutive law, are discussed. Low Reynolds number free surface flows appear in a number of important industrial processes in the oil, food, cosmetic and medical industries and their simulation present a challenge for explicit MAC-type methods due to their parabolic time step constraint. The simulation of moving boundary problems presents a number of difficulties for a numerical method. For the semi-implicit (C–N) MAC method the main difficulty appears in applying the projection method to uncouple velocity and pressure, this is in addition to other difficulties of correctly imposing the boundary conditions on the free surface and the free surface representation itself.

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Acknowledgments

The authors would like to acknowledge the financial support of FAPESP (projects nos. 2013/07375-0, 2011/09194-7, 2009/15892-9) and CNPq (projects nos. 305447/2010-6 , 473589/2013-3).

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Authors and Affiliations

  1. Instituto de Ciências Matemáticas e Computação (ICMC), Universidade de São Paulo, Av. do Trabalhador SãoCarlense 400, São Carlos, SP, Brazil

    José A. Cuminato & Rafael A. Figueiredo

  2. Faculdade de Ciência e Tecnologia (FCT), Universidade Estadual Paulista Julio de Mesquita Filho, Roberto Simonsen, 305, Presidente Prudente, SP, Brazil

    Cassio M. Oishi

Authors
  1. José A. Cuminato
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  2. Cassio M. Oishi
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  3. Rafael A. Figueiredo
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Corresponding author

Correspondence to José A. Cuminato .

Editor information

Editors and Affiliations

  1. Institute for Mathematics and Industry, Kyushu University, Fukuoka, Japan

    Masato Wakayama

  2. CSIRO (Commonwealth Scientific and Industrial Research Organisation), Canberra, Aust Capital Terr, Australia

    Robert S. Anderssen

  3. School of Mathematical Sciences, Fudan University, Shanghai, China

    Jin Cheng

  4. Kyushu University, Fukuoka, Japan

    Yasuhide Fukumoto

  5. Institute of Natural and Mathematical Sciences, Massey University, Palmerston North, Auckland, New Zealand

    Robert McKibbin

  6. Institute of Mathematics, Free University of Berlin, Berlin, Germany

    Konrad Polthier

  7. Kyushu University, Fukuoka, Japan

    Tsuyoshi Takagi

  8. Department of Mathematics, National University of Singapore, Singapore, Singapur, Singapore

    Kim-Chuan Toh

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Cuminato, J.A., Oishi, C.M., Figueiredo, R.A. (2014). Implicit Methods for Simulating Low Reynolds Number Free Surface Flows: Improvements on MAC-Type Methods. In: Wakayama, M., et al. The Impact of Applications on Mathematics. Mathematics for Industry, vol 1. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54907-9_9

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  • DOI: https://doi.org/10.1007/978-4-431-54907-9_9

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  • Publisher Name: Springer, Tokyo

  • Print ISBN: 978-4-431-54906-2

  • Online ISBN: 978-4-431-54907-9

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