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A Structure-Preserving Finite Volume Scheme for a Hyperbolic Reformulation of the Navier–Stokes–Korteweg Equations

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  • Firas Dhaouadi

    (Laboratory of Applied Mathematics, DICAM, University of Trento, Via Mesiano 77, 38123 Trento, Italy)

  • Michael Dumbser

    (Laboratory of Applied Mathematics, DICAM, University of Trento, Via Mesiano 77, 38123 Trento, Italy)

Abstract

In this paper, we present a new explicit second-order accurate structure-preserving finite volume scheme for the first-order hyperbolic reformulation of the Navier–Stokes–Korteweg equations. The model combines the unified Godunov-Peshkov-Romenski model of continuum mechanics with a recently proposed hyperbolic reformulation of the Euler–Korteweg system. The considered PDE system includes an evolution equation for a gradient field that is by construction endowed with a curl-free constraint. The new numerical scheme presented here relies on the use of vertex-based staggered grids and is proven to preserve the curl constraint exactly at the discrete level, up to machine precision. Besides a theoretical proof, we also show evidence of this property via a set of numerical tests, including a stationary droplet, non-condensing bubbles as well as non-stationary Ostwald ripening test cases with several bubbles. We present quantitative and qualitative comparisons of the numerical solution, both, when the new structure-preserving discretization is applied and when it is not. In particular for under-resolved simulations on coarse grids we show that some numerical solutions tend to blow up when the curl-free constraint is not respected.

Suggested Citation

  • Firas Dhaouadi & Michael Dumbser, 2023. "A Structure-Preserving Finite Volume Scheme for a Hyperbolic Reformulation of the Navier–Stokes–Korteweg Equations," Mathematics, MDPI, vol. 11(4), pages 1-25, February.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:4:p:876-:d:1062591
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    References listed on IDEAS

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    1. Dhaouadi, Firas & Gavrilyuk, Sergey & Vila, Jean-Paul, 2022. "Hyperbolic relaxation models for thin films down an inclined plane," Applied Mathematics and Computation, Elsevier, vol. 433(C).
    2. De Lorenzo, M. & Pelanti, M. & Lafon, Ph., 2018. "HLLC-type and path-conservative schemes for a single-velocity six-equation two-phase flow model: A comparative study," Applied Mathematics and Computation, Elsevier, vol. 333(C), pages 95-117.
    3. Abgrall, Rémi & Busto, Saray & Dumbser, Michael, 2023. "A simple and general framework for the construction of thermodynamically compatible schemes for computational fluid and solid mechanics," Applied Mathematics and Computation, Elsevier, vol. 440(C).
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