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The present study develops a physics-constrained neural network (PCNN) to predict sequential patterns and motions of multiphase flows (MPFs), which includes strong interactions among various fluid phases. To predict the order parameters, which locate individual phases in the future time, a neural network (NN) is applied to quickly infer the dynamics of the phases by encoding observations. The multiphase consistent and conservative boundedness mapping algorithm (MCBOM) is next implemented to correct the predicted order parameters. This enforces the predicted order parameters to strictly satisfy the mass conservation, the summation of the volume fractions of the phases to be unity, the consistency of reduction, and the boundedness of the order parameters. Then, the density of the fluid mixture is updated from the corrected order parameters. Finally, the velocity in the future time is predicted by another NN with the same network structure, but the conservation of momentum is included in the loss function to shrink the parameter space. The proposed PCNN for MPFs sequentially performs (NN)-(MCBOM)-(NN), which avoids nonphysical behaviors of the order parameters, accelerates the convergence, and requires fewer data to make predictions. Numerical experiments demonstrate that the proposed PCNN is capable of predicting MPFs effectively. [ABSTRACT FROM AUTHOR]

In the present study, a general numerical approach, consisting of a consistent limiter and the multiphase reduction-consistent formulation , is developed to solve the multiphase Euler/Phase-Field model for compressible N -phase (N ⩾ 1) flows in a consistent and conservative fashion. The proposed method is general in the sense that it admits N different phases and is not restricted to a specific Phase-Field formulation. Not only is the volume fraction bounded and the mass non-negative, the volume fractions also sum up to unity, preventing the production of fictitious phases, local voids, or overfilling. Velocity, pressure, and temperature equilibria are maintained across material interfaces. Various compressible N -phase problems including shocks and interfaces are performed to verify the analysis and demonstrate the efficacy of the method for N ⩾ 2. Unphysical behavior in N -phase calculations (production of fictitious phases, local voids, and overfilling) due to inappropriate numerical treatment is discussed. Three variants for the Phase-Field mechanism are proposed, one of which is most efficient with increasing number of phases. Including the Phase-Field mechanism produces controllable interface thickness, thus effectively preventing numerical mixing of different phases due to numerical diffusion in shock-capturing schemes. • A general numerical approach is developed for compressible N-phase flows. • A consistent limiter prevents fictitious phases, local voids, or overfilling. • The multiphase reduction-consistent formulation tackles the Phase-Field mechanisms. • The proposed approach maintains equilibrium and preserves physical bounds. • The Phase-Field mechanism prevents the numerical mixing of different phases. [ABSTRACT FROM AUTHOR]

A key challenge with interface-capturing methods for compressible multiphase flows is that the numerical diffusion of upwind schemes causes material interfaces to continuously diffuse over time, possibly in a non-uniform fashion. In the present study, a consistent and conservative Phase-Field model for compressible multiphase flows is derived to control mixing between different fluids at material interfaces due to numerical diffusion. The model is general as it admits an arbitrary number of phases and has no prior assumption of the formulation of the Phase-Field mechanism. Necessary physical constraints to formulate the Phase-Field mechanism are obtained by analyzing physical principles, including the second law of thermodynamics. To numerically solve the model, two consistency requirements and a general numerical strategy, called reduction consistent formulation , are proposed. The proposed approach is consistent, conservative, bound/positivity preserving for the volume fraction/mass, and prevents spurious errors at interfaces. Various two-phase compressible flow problems including shocks and interfaces are performed to verify the analysis and demonstrate the efficacy of the method. Finally, temperature equilibrium and the incompressible limit are also discussed. [ABSTRACT FROM AUTHOR]

In the present work, a general formulation is proposed to implement the contact angle boundary conditions for the second-order Phase-Field models, which is applicable to N -phase (N ⩾ 2) moving contact line problems. To remedy the issue of mass change due to the contact angle boundary condition, a source term or Lagrange multiplier is added to the original second-order Phase-Field models, which is determined by the consistent and conservative volume distribution algorithm so that the summation of the order parameters and the consistency of reduction are not influenced. To physically couple the proposed formulation to the hydrodynamics, especially for large-density-ratio problems, the consistent formulation is employed. The reduction-consistent conservative Allen-Cahn models are chosen as examples to illustrate the application of the proposed formulation. The numerical scheme that preserves the consistency and conservation of the proposed formulation is employed to demonstrate its effectiveness. Results produced by the proposed formulation are in good agreement with the exact and/or asymptotic solutions. The proposed method captures complex dynamics of moving contact line problems having large density ratios. • A general formulation is proposed to implement contact angle BCs in 2nd-order Phase-Field models. • The novel formulation applies to both two-phase and N -phase (N ≥ 2) problems. • The novel formulation prevents producing fictitious phase, local voids, or overfilling. • The novel formulation is robust for large-density-ratio problems. • The novel formulation is effective for moving contact line problems. [ABSTRACT FROM AUTHOR]

In the present study, a consistent and conservative Phase-Field method, including both the model and scheme, is developed for multiphase flows with an arbitrary number of immiscible and incompressible fluid phases. The consistency of mass conservation and the consistency of mass and momentum transport are implemented to address the issue of physically coupling the Phase-Field equation, which locates different phases, to the hydrodynamics. These two consistency conditions, as illustrated, provide the "optimal" coupling because (i) the new momentum equation resulting from them is Galilean invariant and implies the kinetic energy conservation, regardless of the details of the Phase-Field equation, and (ii) failures of satisfying the second law of thermodynamics or the consistency of reduction of the multiphase flow model only result from the same failures of the Phase-Field equation but are not due to the new momentum equation. Physical interpretation of the consistency conditions and their formulations are first provided, and general formulations that are obtained from the consistency conditions and independent of the interpretation of the velocity are summarized. Then, the present consistent and conservative multiphase flow model is completed by selecting a reduction consistent Phase-Field equation. Several novel techniques are developed to inherit the physical properties of the multiphase flows after discretization, including the gradient-based phase selection procedure, the momentum conservative method for the surface force, and the general theorems to preserve the consistency conditions on the discrete level. Equipped with those novel techniques, a consistent and conservative scheme for the present multiphase flow model is developed and analyzed. The scheme satisfies the consistency conditions, conserves the mass and momentum, and assures the summation of the volume fractions to be unity, on the fully discrete level and for an arbitrary number of phases. All those properties are numerically validated. Numerical applications demonstrate that the present model and scheme are robust and effective in studying complicated multiphase dynamics, especially for those with large-density ratios. [ABSTRACT FROM AUTHOR]

In the present study, a consistent and conservative Phase-Field model is developed to study thermo-gas-liquid-solid flows with liquid-solid phase change. The proposed model is derived with the help of the consistency conditions and exactly reduces to the consistent and conservative Phase-Field method for incompressible two-phase flows, the fictitious domain Brinkman penalization (FD/BP) method for fluid-structure interactions, and the Phase-Field model of solidification of pure material. It honors the mass conservation, defines the volume fractions of individual phases unambiguously, and therefore captures the volume change due to phase change. The momentum is conserved when the solid phase is absent, but it changes when the solid phase appears due to the no-slip condition at the solid boundary. The proposed model also conserves the energy, preserves the temperature equilibrium, and is Galilean invariant. A novel continuous surface tension force to confine its contribution at the gas-liquid interface and a drag force modified from the Carman-Kozeny equation to reduce solid velocity to zero are proposed. The issue of initiating phase change in the original Phase-Field model of solidification is addressed by physically modifying the interpolation function. The corresponding consistent scheme is developed to solve the model, and the numerical results agree well with the analytical solutions and the existing experimental and numerical data. Two challenging problems having a wide range of material properties and complex dynamics are conducted to demonstrate the capability of the proposed model. • A Phase-Field model for multiphase flows with solidification/melting is developed. • Different parts of the model are physically coupled with the consistency conditions. • The model satisfies the mass, momentum, and energy conservation, and Galilean invariance. • Novel surface tension and drag forces are proposed. • A consistent and mass conserving scheme for the model is developed. [ABSTRACT FROM AUTHOR]

• A consistent and conservative multiphase volume distribution algorithm is proposed. • The general and physical Lagrange multipliers that enforce mass conservation is determined. • A reduction-consistent multiphase conservative Allen-Cahn model is developed. • The Phase-Field models are consistently coupled with the momentum equation. • Consistent, conservative, and bounded schemes are developed for multiphase flows. In the present study, the multiphase volume distribution problem, where there can be an arbitrary number of phases, is addressed using a consistent and conservative volume distribution algorithm. The proposed algorithm satisfies the summation constraint, the conservation constraint, and the consistency of reduction. The first application of the volume distribution algorithm is to determine the Lagrange multipliers in multiphase Phase-Field models that enforce the mass conservation, and a multiphase conservative Allen-Cahn model that satisfies the consistency of reduction is developed. A corresponding consistent and conservative numerical scheme is developed for the model. The multiphase conservative Allen-Cahn model has a better ability than the multiphase Cahn-Hilliard model to preserve under-resolved structures. The second application is to develop a numerical procedure, called the boundedness mapping, to map the order parameters, obtained numerically from a multiphase model, into their physical interval, and at the same time to preserve the physical properties of the order parameters. Along with the consistent and conservative schemes for the multiphase Phase-Field models, the numerical solutions of the order parameters are reduction consistent, conservative, and bounded, which are theoretically analyzed and numerically validated. Then, the multiphase Phase-Field models are coupled with the momentum equation by satisfying the consistency of mass conservation and the consistency of mass and momentum transport , thanks to the consistent formulation. It is demonstrated that the proposed model and scheme converge to the sharp-interface solution and are capable of capturing the complicated multiphase dynamics even when there is a large density and/or viscosity ratio. [ABSTRACT FROM AUTHOR]

• A general model for multiphase and multicomponent incompressible flows is developed. • Consistency conditions for multiphase and multicomponent flows are provided. • The model satisfies the conservation of mass and momentum. • The model satisfies a physical energy law and is Galilean invariance. • A consistent and conservative scheme for the model is developed. In the present work, we propose a consistent and conservative model for multiphase and multicomponent incompressible flows, where there can be arbitrary numbers of phases and components. Each phase has a background fluid called the pure phase, each pair of phases is immiscible, and components are dissolvable in some specific phases. The model is developed based on the multiphase Phase-Field model including the contact angle boundary condition, the diffuse domain approach, and the analyses on the proposed consistency conditions for multiphase and multicomponent flows. The model conserves the mass of individual pure phases, the amount of each component in its dissolvable region, and thus the mass of the fluid mixture, and the momentum of the flow. It ensures that no fictitious phases or components can be generated and that the summation of the volume fractions from the Phase-Field model is unity everywhere so that there is no local void or overfilling. It satisfies a physical energy law and it is Galilean invariant. A corresponding numerical scheme is developed for the proposed model, whose formal accuracy is 2nd-order in both time and space. It is shown to be consistent and conservative and its solution is demonstrated to preserve the Galilean invariance and energy law. Numerical tests indicate that the proposed model and scheme are effective and robust to study various challenging multiphase and multicomponent flows. [ABSTRACT FROM AUTHOR]

• A conservative, consistent two-phase phase-field model is developed. • The two-phase model satisfies the conservation of mass and momentum and energy law. • A consistent and conservative numerical scheme is developed. • A boundedness mapping algorithm is developed to preserve the maximum principle. In the present work, we consider the conservative Allen-Cahn model and applied it to two-phase flows in a consistent and conservative manner. The consistent formulation is proposed, where the conservative Allen-Cahn equation is reformulated in a conservative form using an auxiliary variable. As a result, the consistency analysis is performed and the resulting two-phase model honors the consistency of reduction , the consistency of mass conservation and the consistency of mass and momentum transport , which are important to reproduce the physical momentum and kinetic energy transport, to achieve mass and momentum conservation, and to satisfy the energy law of the two-phase system. A consistent and conservative scheme is developed, and its properties are carefully analyzed and validated. In order to honor the maximum principle of the conservative Allen-Cahn model, we proposed a boundedness mapping algorithm, which preserves the properties of consistency and conservation of the scheme. The applications of the consistent formulation and the proposed scheme to realistic two-phase flows show that they are accurate, robust and effective for complicated two-phase problems. The applicability of the consistent formulation and consistency analysis to multiphase flows and to the improved Cahn-Hilliard model is discussed. [ABSTRACT FROM AUTHOR]