An energy dissipative semi-discrete finite-difference method on staggered meshes for the 3D compressible isothermal Navier–Stokes–Cahn–Hilliard equations

Vladislav Balashov; Alexander Zlotnik

doi:10.3934/jcd.2020012

Article Contents

2020, Volume 7, Issue 2: 291-312. Doi: 10.3934/jcd.2020012

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An energy dissipative semi-discrete finite-difference method on staggered meshes for the 3D compressible isothermal Navier–Stokes–Cahn–Hilliard equations

Vladislav Balashov^1, and
Alexander Zlotnik^2, ,

1.
Keldysh Institute of Applied Mathematics, Miusskaya sqr., 4, 125047 Moscow, Russia
2.
National Research University Higher School of Economics, Pokrovskii bd. 11, 109028 Moscow, Russia, Keldysh Institute of Applied Mathematics, Miusskaya sqr., 4, 125047 Moscow, Russia

^* Corresponding author
^* Corresponding author

Received: September 2019

Published: July 2020

The study was supported by the Russian Science Foundation, project no. 19-11-00169

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Abstract

We consider the initial-boundary value problem for the 3D regularized compressible isothermal Navier–Stokes–Cahn–Hilliard equations describing flows of a two-component two-phase mixture taking into account capillary effects. We construct a new numerical semi-discrete finite-difference method using staggered meshes for the main unknown functions. The method allows one to improve qualitatively the computational flow dynamics by eliminating the so-called parasitic currents and keeping the component concentration inside the physically reasonable range $ (0,1) $. This is achieved, first, by discretizing the non-divergent potential form of terms responsible for the capillary effects and establishing the dissipativity of the discrete full energy. Second, a logarithmic (or the Flory–Huggins potential) form for the non-convex bulk free energy is used. The regularization of equations is accomplished to increase essentially the time step of the explicit discretization in time. We include 3D numerical results for two typical problems that confirm the theoretical predictions.

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Mathematics Subject Classification: 65M06, 65P99, 76T99.

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Figure 1. $ \tilde\Psi_0(C)\equiv\Psi_0(\tilde\rho, C) $ for $ \omega_2 > \omega_1 $ and some fixed $ \tilde\rho>0 $

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Figure 4. Distributions of $ C $ and $ \rho $ along the segment $ x_1\in[0.5X, X] $ and $ x_2 = x_3 = 0.5X $, at the vicinity of the interface at $ t = 20\cdot10^3\Delta t $

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Figure 2. Location of nodes of $ \omega_{h\bar{k},\bar{l}^*,\bar{m}^*} $ (thick dots) and $ \omega_{h\bar{k}^*,\bar{l},\bar{m}} $ (red crosses), where $ u_k $ and $ \Pi_{lk} $, $ l\neq k $, are respectively defined

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Figure 3. Droplet interface evolution in the case (Ⅰ) for $ R = 0.25X $. Distribution of $ C(x) $ in the section $ x_1,x_2\in[0.5X,0.87X] $, $ x_3 = 0.5X $, is represented

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Figure 5. Evolution of $ \sigma_L(t) $: for $ R = 0.18X $ in the cases (Ⅰ)-(Ⅲ) (left) and for several $ R $ in the case (Ⅲ) (right)

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Figure 6. Observable dependence of $ \Delta p $ on $ 1/R_a $ for different $ R $ in (46)

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Figure 7. Evolution of $ {\bar{E}}_{\text{kin}} $ and $ \mathcal{E}_h-\tilde{E} $ for the droplet with $ R = 0.25X $ and $ \tilde{E} = 9.16034\cdot 10^{-5}\, \text{J} $

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Figure 8. $ {\bar{E}}_{\text{kin}} (t) $ computed by schemes $ A $ (from this paper) and $ B $ (from [6])

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Figure 9. Evolution of $ C_{\min} $ and $ C_{\max} $ (for spinodal decomposition)

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Figure 10. $ \bar{E}_\text{kin}(t) $ for $ \alpha^\ast = 0 $ and $ 0.5 $ and some $ \Delta t $ (the break in the graph line means that computations collapse due to instability)

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Figure 12. $ \bar{E}_{\text{kin}}(t) $ for some $ \alpha^\ast\geq0.5 $ and $ \Delta t $ (the break in the graph line means that computations collapse due to instability)

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Figure 13. Evolution of $ {\bar{E}}_{\text{kin}} $ and $ \mathcal{E}_h $ (for spinodal decomposition)

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Figure 14. Isosurfaces $ C = 0.5 $ at different time moments (for spinodal decomposition)

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