Computation of multiphase systems with phase field models

Badalassi, Vittorio; Ceniceros, Héctor D.; Banerjee, Sanjoy

doi:10.1016/s0021-9991(03)00280-8

Cited by 543 publications

(464 citation statements)

References 41 publications

(75 reference statements)

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“…We note that the basic idea of this type of strategy for dealing with a diffusion term with a variable coefficient was discussed in [7] (section 9, page 114), and was also used by other researchers (e.g. [2,24]). The feature of the algorithm that enables a successful treatment of the 4-th spatial order of the Cahn-Hilliard equation with spectral elements is the term S η 2 (φ n+1 −φ * ,n+1 ), which was called a stabilization term in [26].…”

Section: The Algorithmmentioning

confidence: 99%

“…Because of the matched kinematic viscosity requirement for the existence of the exact solution, we have µ 1 = µ 2 = ρ 1 ν = 0.01. We use a fixed interface mobility γ 1 = 10 −5 , and the mixing energy density λ is computed according to equation (2). Let us first look into the convergence of simulation results with respect to spatial and temporal parameters.…”

Section: Capillary Wavementioning

confidence: 99%

“…In the simulations we employ an interface thickness η = 0.01, and the mixing energy density λ is determined based on equation (2). We choose the mobility γ 1 such that λγ 1 has a constant value, λγ 1 = 10 −7 .…”

Section: Rising Air Bubble In Water and Moving Contact Linesmentioning

confidence: 99%

“…[13,2,17,30,29,4,11,12]. The case with different densities for the two fluids is considered in a few studies with the phase field approach [16,5], which, however, all lead to linear algebraic systems with variable (time-dependent) coefficient matrices for the pressure and the velocity after discretization, resulting in a high computational cost.…”

Section: Introductionmentioning

confidence: 99%

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A time-stepping scheme involving constant coefficient matrices for phase-field simulations of two-phase incompressible flows with large density ratios

Dong

Shen

2012

Journal of Computational Physics

160

193

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We present an efficient time-stepping scheme for simulations of the coupled Navier-Stokes CahnHilliard equations for the phase field approach. The scheme has several attractive characteristics: (i) It is suitable for large density ratios, and numerical experiments with density ratios up to 1000 have been presented; (ii) It involves only constant (time-independent) coefficient matrices for all flow variables, which can be pre-computed during pre-processing, so it effectively overcomes the performance bottleneck induced by variable coefficient matrices associated with the variable density and variable viscosity; (iii) It completely de-couples the computations of the velocity, pressure, and the phase field function. Strategy for spectral-element type spatial discretizations to overcome the difficulty associated with the large spatial order of the Cahn-Hilliard equation is also discussed. Ample numerical simulations demonstrate that the current algorithm, together with the Navier-Stokes Cahn-Hilliard phase field approach, is an efficient and effective method for studying two-phase flows involving large density ratios, moving contact lines, and interfacial topology changes.

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Section: The Algorithmmentioning

confidence: 99%

Section: Capillary Wavementioning

confidence: 99%

Section: Rising Air Bubble In Water and Moving Contact Linesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A time-stepping scheme involving constant coefficient matrices for phase-field simulations of two-phase incompressible flows with large density ratios

Dong

Shen

2012

Journal of Computational Physics

160

193

View full text Add to dashboard Cite

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“…This class of linear systems can be formally regarded as a special case of the generalized saddle-point problem [4,5,16] and the skew-Hamiltonian linear system [25,32]. It frequently arises from finite-element discretization and firstorder linearization of the two-phase flow problems based on Cahn-Hilliard equation [3,17,18], order-reduction and sinc discretization of the third-order linear ordinary differential equations [30], finite-element discretizations of elliptic PDE-constrained optimization problems such as the distributed control problems [6,23,24,29], real equivalent formulations of complex symmetric linear systems [2,15,20], linear quadratic control problems [22,25], and H ∞ control problems [28,32,34]. being the symmetric and the skew-symmetric parts and A T being the transpose of the matrix A; see [1,14].…”

Section: Introductionmentioning

confidence: 99%

Additive block diagonal preconditioning for block two-by-two linear systems of skew-Hamiltonian coefficient matrices

2013

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For a class of block two-by-two systems of linear equations with certain skew-Hamiltonian coefficient matrices, we construct additive block diagonal preconditioning matrices and discuss the eigen-properties of the corresponding preconditioned matrices. The additive block diagonal preconditioners can be employed to accelerate the convergence rates of Krylov subspace iteration methods such as MINRES and GMRES. Numerical experiments show that MINRES preconditioned by the exact and the inexact additive block diagonal preconditioners are effective, robust and scalable solvers for the block two-by-two linear systems arising from the Galerkin finite-element discretizations of a class of distributed control problems.

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Numerical study of turbulent liquid‐liquid dispersions

2015

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A numerical approach is developed to gain fundamental insight in liquid‐liquid dispersion formation under well‐controlled turbulent conditions. The approach is based on a free energy lattice Boltzmann equation method, and relies on detailed resolution of the interaction of the dispersed and continuous phase at the microscopic level, including drop breakup and coalescence. The capability of the numerical technique to perform direct numerical simulations of turbulently agitated liquid‐liquid dispersions is assessed. Three‐dimensional simulations are carried out in fully periodic cubic domains with grids of size . The liquids are of equal density. Viscosity ratios (dispersed phase over continuous phase) are in the range 0.3–1.0. The dispersed phase volume fraction varies from 0.001 to 0.2. The process of dispersion formation is followed and visualized. The size of each drop in the dispersion is measured in‐line with no disturbance of the flow. However, the numerical method is plagued by numerical dissolution of drops that are smaller than 10 times the lattice spacing. It is shown that to mitigate this effect it is necessary to increase the resolution of the Kolmogorov scales, such as to have a minimum drop size in the range 20–30 lattice units [lu]. Four levels of Kolmogorov length scale resolution have been considered , 2.5, 5, and 10 [lu]. In addition, the numerical dissolution reduces if the concentration of the dispersed phase is increased. © 2015 American Institute of Chemical Engineers AIChE J, 61: 2618–2633, 2015

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Computation of multiphase systems with phase field models

Cited by 543 publications

References 41 publications

A time-stepping scheme involving constant coefficient matrices for phase-field simulations of two-phase incompressible flows with large density ratios

A time-stepping scheme involving constant coefficient matrices for phase-field simulations of two-phase incompressible flows with large density ratios

Additive block diagonal preconditioning for block two-by-two linear systems of skew-Hamiltonian coefficient matrices

Numerical study of turbulent liquid‐liquid dispersions

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