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8: This figure shows how micro-bubbles are generated as time progresses from top to bottom. The effect of increasing resolution is also demonstrated as the mesh is refined from left to right. All images are from simulations of the nominal problem (impact for U = 1m/s and R = 1.7mm in a water-air system)
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Micro-bubbles have been observed in various contexts such as in raindrops impacting liquid pools, boiling heat transfer, aerosol generation, breaking oceanic waves and ship wakes. Due to their long residence time under the free surface, micro-bubbles impact air-sea mass transport, formation of white caps and the signature of seafaring vessels. As s...
Citations
... Older methods for interface capture include algebraic approximations such as Compressive schemes and THINC (Tangent of hyperbola for interface capturing) schemes. Geometric approaches are more recent, complicated, and precise (Mirjalili, S. 2019). An interface within a computational cell is geometrically reconstructed using a plane in three-dimensional (3D) simulations using methods such as Simple Line Interface Calculation (SLIC) (Noh & Woodward, 1976) or the more recent Piecewise Linear Interface Calculation (PLIC). ...
The present work develops a compressible VOF-LPT coupled solver in OpenFOAM and utilizes it to simulate a LJICF numerically. This methodology helps accurately predict a complex primary breakup in the Eulerian framework and the secondary atomization of spherical droplets using a computationally efficient LPT method. The coupled solver with AMR is rigorously validated for a liquid jet in crossflow at varying operating conditions. We have further carried out a thorough investigation to study the effect of momentum flux ratio and weber number on the various flow features and liquid jet break-up phenomenon in a crossflow while identifying the stream-wise location of the liquid jet breakup region. At low momentum flux ratios in the bag breakup regime, the predictions reveal that the liquid jet breakup occurs due to the growth of similar instability as usually observed in the high-speed liquid sheet atomization. The short wavelength assumption of the inviscid dispersion relation resembles the Kelvin-Helmholtz type instability observed in this case, as opposed to Rayleigh-Taylor instability at high momentum flux ratio in the surface breakup regime. It is also proposed that the shear breakup along the transverse edges of the liquid column occurs due to the shear layer instability of the air passing around the liquid column. The simulation wavelength closely matches the Williamson correlation for shear layer instability around cylinders: a shape similar to the cross-section of the bottom of the liquid column. The results show a distinct streamer or bifurcation phenomenon at low momentum flux ratios and moderate weber numbers. Further investigation suggests that the internal liquid boundary layer and the three-dimensional flow field behind the liquid jet are responsible for streamer formation.
... The balanced-force method is directly extended from the one for the two-phase flows, e.g., in [27,35], because the surface force in Eq.(23) has a form of a scalar times the gradient of another scalar, the same structure as the two-phase surface tension models, e.g., in [11,85,39]. The two-phase balanced-force method has been extensively studied and popularly used, e.g., in [27,35,34,57]. ...
... We perform the steady drop problem to quantify the numerical force balance of the present scheme. Ample studies of the two-phase spurious current have been performed, e.g., in [33,35,56,57,3,40,47,67,11,27,6,94]. Due to the consistency of reduction analyzed in Section 2.5 and Section 3.4.3, the two-phase behavior of the spurious current will be automatically recovered by the multiphase flow model and scheme, and, therefore, it is not to be repeated in the present study. ...
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.
... The gradient term ∇ e F 2 can be absorbed in to the pressure term, and, as a result, the surface force in Eq. [43 ] can be applied to reduce the spurious current caused by the numerical force imbalance. [70 ], [196 ], [197 ]. An alternative choice of modeling the interfacial tensions is called the generalized continuous surface tension force formulation which is a geometry-based model [65 ], [67 ], [69 ]. ...
... This formulation is applied in [196 ] to the two-phase steady drop problem and the results ...
... However, such an exact force balance is seldom achievable on the discrete level, and the force imbalance introduced by the discretization drives the fluid to move, generating the so-called spurious current. Ample studies of the two-phase spurious current have been performed in, e.g., [39 ], [43 ], [44 ], [113 ], [118 ], [196 ], [206 ], [221 ]- [224 ], based on the continuous surface [39 ] and/or the ghost fluid method (GFM) [40 ]. This section investigates the performance of the surface force from the Phase-Field method. ...
This dissertation focuses on a consistent and conservative Phase-Field method for multiphase flow problems, and it includes both model and scheme development. The first general question addressed in the present study is the multiphase volume distribution problem. A consistent and conservative volume distribution algorithm is developed to solve the problem, which eliminates the production of local voids, overfilling, or fictitious phases, but follows the mass conservation of each phase. One of its applications is to determine the Lagrange multipliers that enforce the mass conservation in the Phase-Field equation, and a reduction consistent conservative Allen-Cahn Phase-Field equation is developed. Another application is to remedy the mass change due to implementing the contact angle boundary condition in the Phase-Field equations whose highest spatial derivatives are second-order. As a result, using a 2nd-order Phase-Field equation to study moving contact line problems becomes possible.
The second general question addressed in the present study is the coupling between a given physically admissible Phase-Field equation to the hydrodynamics. To answer this general question, the present study proposes the consistency of mass conservation and the consistency of mass and momentum transport, and they are first implemented to the Phase-Field equation written in a conservative form. The momentum equation resulting from these two consistency conditions is Galilean invariant and compatible with the kinetic energy conservation, regardless of the details of the Phase-Field equation. It is further illustrated that the 2nd law of thermodynamics and consistency of reduction of the entire multiphase system only rely on the properties of the Phase-Field equation. All the consistency conditions are physically supported by the control volume analysis and mixture theory. If the Phase-Field equation has terms that are not in a conservative form, those terms are treated by the proposed consistent formulation. As a result, the proposed consistency conditions can always be implemented. This is critical for large-density-ratio problems.
The consistent and conservative numerical framework is developed to preserve the physical properties of the multiphase model. Several new techniques are developed, including the gradient-based phase selection procedure, the momentum conservative method for the surface force, the boundedness mapping resulting from the volume distribution algorithm, the "DGT" operator for the viscous force, and the correspondences of numerical operators in the discrete Phase-Field and momentum equations. With these novel techniques, numerical analyses ensure that the mass of each phase and momentum of the multiphase mixture are conserved, the order parameters are bounded in their physical interval, the summation of the volume fractions of the phases is unity, and all the consistency conditions are satisfied, on the fully discrete level and for an arbitrary number of phases. Violation of the consistency conditions results in inconsistent errors proportional to the density contrasts of the phases. All the numerical analyses are carefully validated, and various challenging multiphase flows are simulated. The results are in good agreement with the exact/asymptotic solutions and with the existing numerical/experimental data.
The multiphase flow problems are extended to including mass (or heat) transfer in moving phases and solidification/melting driven by inhomogeneous temperature. These are accomplished by implementing an additional consistency condition, i.e., consistency of volume fraction conservation, and the diffuse domain approach. Various problems are solved robustly and accurately despite the wide range of material properties in those problems.
... The balanced-force method is directly extended from the one for the two-phase flows, e.g., in [27,35], because the surface force in Eq.(23) has a form of a scalar times the gradient of another scalar, the same structure as the two-phase surface tension models, e.g., in [11,85,39]. The two-phase balanced-force method has been extensively studied and popularly used, e.g., in [27,35,34,57]. ...
... We perform the steady drop problem to quantify the numerical force balance of the present scheme. Ample studies of the two-phase spurious current have been performed, e.g., in [33,35,56,57,3,40,47,67,11,27,6,94]. Due to the consistency of reduction analyzed in Section 2.5 and Section 3.4.3, the two-phase behavior of the spurious current will be automatically recovered by the multiphase flow model and scheme, and, therefore, it is not to be repeated in the present study. ...
In the present study, we consider incompressible multiphase flows, where there can be an arbitrary number of immiscible phases appearing simultaneously, with each phase having its own density and viscosity, and with each pair of phases having an interfacial tension at their interfaces and a contact angle at a wall. A Phase-Field model for this kind of problems is derived from the consistency conditions, which are the consistency of reduction, the consistency of mass conservation, and the consistency of mass and momentum transport, and the energy law, which requires that the total energy of the multiphase system is not increased by the interfacial tensions. A 2nd-order decoupled semi-implicit scheme is developed and it maintains many physical properties of the Phase-Field model in the discrete level. The proposed scheme is shown to satisfy all the consistency conditions, to conserve mass, to avoid generating any fictitious phases, local voids, or overfilling, and to conserve momentum without interfacial tensions. Two methods, which are the balanced-force method and the conservative method, are proposed to discretize the surface force. We show that using the conservative method results in the momentum conservation even including interfacial tensions, while the balanced-force method leads to an essentially momentum-conserving scheme but has a better balanced-force property. In addition, our numerical tests show that the energy law is preserved by the scheme, that the numerical solution of the Phase-Field model convergences to the sharp-interface solution, and that the proposed model and scheme are robust and effective to study complicated multiphase dynamics, especially for those including large-density ratios.
