Multiphase Problems Research Articles

Triple junction benchmark for multiphase-field and multi-order parameter models

Due to their ability to encompass complex multi-phase problems, multiphase-field and multi-order parameter methods are increasing in popularity. Multiple model formulations and modifications have been published since their original emergence but a comprehensive comparison has not yet been conducted. We establish the triple junction as a well-defined benchmark for the construction of interfacial energy contributions. Metrics such as the overall free energy of the system and the dihedral angle at the junction are derived from analytic considerations and subsequently evaluated in numerical studies. Various multiphase-field and multi-order parameter models are compared quantitatively over a large range of interfacial energy ratios. The analytic derivations for the two benchmark cases form a strong foundation for identification of model-specific errors or a limited application range. The mutual comparison of combinations of various gradient and potential energy terms yields new insights into the sources of spurious phase generation. The systematic combination of terms covers previously published models as well as new combinations. This work provides guidelines for the choice of a suitable, problem-specific model formulation and the development of new models. Furthermore, our goal is to be as comprehensible as possible to ensure high reproducibility and facilitate the start for beginners in the field.

A density-based WCSPH numerical scheme for investigation of multiphase flows with intricate interfaces at high density ratios

Numerical investigation of multiphase problems having complex interface at high density ratios is one of the numerical challenges associated with particle scattering and divergence. Fewer problems have been performed using density-based smooth particle hydrodynamics (WCSPH) in order to solve complex joint surface currents, and most of the simulations in this field have been performed using incompressible smooth particle hydrodynamics (ISPH). Solving high density flows by smooth particle hydrodynamics is associated with particle dispersion and divergence. Various methods have been used to eliminate the scattering of particles, such as a repulsive force at the interface or the corrected density re-value, but there is a problem of particle disintegration at interface at higher times. Numerical simulations of SPH multiphase problems with high density ratio present divergence and dispersion of soluble particles over time simulation. However, this problem is solved in the current model so that continuity is preserved and provides a more accurate solution over time simulation. In the present work, in order to simulate multiphase flows with complex surfaces and high density ratios fluids investigation, a new density-based smooth particle hydrodynamic approach has been utilized. To prevent the particles scattering at the interface, an incompatible particle removal method is used. In the present study, a particle displacement optimization scheme for regularization at phase interface is created by implementing a two-stage change algorithm to maintain the regular particle distribution continuously and conservatively. In order to investigate the accuracy of the current numerical scheme, it is firstly compared with two-phase Poiseuille flow with three fluids having diverse values of viscosity, Reynolds-Taylor instability, Single Bubble rising in a fully filled container, merging of two bubbles, and Bubble rising in a partially filled container; then it is compared with analytical and numerical solutions.

Enhanced moving finite element method based on error geometric estimation for simultaneous trajectory optimization

The direct method has been widely used in various fields of trajectory optimization due to the need to optimize the performance of a controlled dynamic system. An essential issue in the development of the direct method is the selection of the finite element mesh. Although there have been many pieces of research on adaptive mesh refinement, few works focused on finite element layout optimization before the refinement. This paper proposes an enhanced moving finite element method (EMFE), taking the length of each finite element as a variable, based on the geometric estimation of non-collocation-point error (NCPE) and generalized finite element length (GFEL). The features of EMFE include flexible problem reformulation, stable numerical calculation and the ability to locate breakpoints for singular control problems (the control variables appear linearly in the problems) and finding the local minimum numerical error. The theoretical basis of the ability to locate the breakpoints for singular control problems and the feature of NCPE estimate has also been investigated. Three numerical examples, two of which are singular control problems for lunar landing and the other is a multi-phase nonlinear problem for launch vehicles, demonstrated the performance and features of the proposed EMFE method.

Multi-phase and dual aero/propulsive rocket landing guidance using successive convex programming

This paper aims to suggest a new landing guidance algorithm for reusable launch vehicles (RLVs) to enable generation of fuel-efficient trajectories based on successive convex programming. To this end, a dual aero/propulsive landing guidance problem is first formulated to fully exploit the additional moment generated by the aerodynamic control to reduce the propulsion demand required for attitude control. As the result of the aerodynamic landing phase could greatly affect the fuel-optimal trajectory during the vertical landing phase, the formulation is further extended to the multi-phase optimal guidance problem using state-triggered constraints. The proposed guidance strategy is then obtained by solving the formulated optimal control problem based on the successive convex optimization framework using an interior point method The main contribution of this study lies in forming new RLV landing guidance problems to get an optimal trajectory and transforming the corresponding nonconvex problem into a convex optimization problem by introducing appropriate combinations for convexification techniques. Additionally, this paper introduces several practical constraints, such as the maximum slew rate of aerodynamic control fins and nozzle angles, which are not considered in previous works. In this paper, the performance of the proposed method with the potential for online computation is investigated through numerical simulations.

Numerical simulation of drop deformation under simple shear flow of Giesekus fluids by SPH

PurposeThe purpose of this study is to investigate the effects of the shear-thinning viscoelastic behavior of the surrounding matrix on droplet deformation by weakly compressible smoothed particle hydrodynamics (WC-SPH). Also, the effect of the presence of another droplet is examined.Design/methodology/approachA modified consistent weakly compressible SPH method is proposed. After code verification, a complete parameter study is performed for a drop under the simple shear flow of a Giesekus liquid. The investigated parameters are 0.048≤Ca ≤ 14.4, 0.1≤c ≤ 10, 0.04≤De ≤ 10, 0≤α ≤ 1 and 0.12≤Re ≤ 12.FindingsIt is demonstrated that the rheological behavior of the surrounding fluid could dramatically affect the droplet deformation. It is shown that the droplet deformation is increased by increasing Re and Ca. In contrast, the droplet deformation is decreased by increasing a, De and polymer content. Also, it is indicated the presence of another droplet could drastically affect the flow field, and the primary stress difference (N1) is resonated between two droplets.Originality/valueThe main originality of this paper is to introduce a new consistent WC-SPH algorithm. The proposed method is very versatile for tackling the shear-thinning viscoelastic multiphase problems. Furthermore, a complete parameter study is performed for a drop under the simple shear flow of Giesekus liquid. Another novelty of the current paper is studying the effect of the presence of a second droplet. To the best of the authors’ knowledge, this is performed for the first time.

Generalized and efficient wall boundary condition treatment in GPU-accelerated smoothed particle hydrodynamics

This paper presents a generalized and efficient wall boundary treatment in the smoothed particle hydrodynamics (SPH) method for 3-D complex and arbitrary geometries with single- and multi-phase flows to be executed on graphics processing units (GPUs). Using a force balance between the wall and fluid particles with a novel penalty method, a pressure boundary condition is applied on the wall dummy particles which effectively prevents non-physical particle penetration into the wall also in highly violent impacts and multi-phase flows with high density ratios. A new density reinitialization scheme is also presented to enhance the accuracy and robustness. The proposed method is simple in comparison with previous wall boundary formulations on GPUs and enforces no additional memory caching and thus is well suited for massively parallel architecture of GPUs. The method is validated in various test cases involving inviscid violent single- and multi-phase flows, demonstrating very good robustness, accuracy and performance. The new wall boundary treatment is able to improve the high accuracy of its previous version by Adami et al. [1] also in 3-D and multi-phase problems, while it is efficiently executable on GPUs. Therefore, the method can be a reliable solution for the long-lasting wall boundary condition challenge in SPH for a broad range of problems.

Fuel-Optimal Guidance for End-to-End Human-Mars Entry, Powered-Descent, and Landing Mission

This article investigates the fuel-optimal guidance problem of the end-to-end human-Mars entry, powered-descent, and landing (EDL) mission. It applies a unified modeling scheme and develops a computationally efficient new optimization algorithm to solve the multiphase optimal guidance problem. The end-to-end EDL guidance problem is first modeled as a multiphase optimal control problem with different dynamics and constraints at each phase. Via polynomial approximation and discretization techniques, this multiphase optimal control problem is then reformulated as a polynomial programming problem. By introducing intermediate variables and quadratic equality constraints, a polynomial program is equivalently converted into a nonconvex quadratically constrained quadratic program (QCQP). Then, a novel customized alternating direction method of multipliers (ADMM) is proposed to efficiently solve the large-scale QCQP with convergence proof to a local optimum under certain conditions on the algorithmic parameters. The fuel savings under the end-to-end human-Mars EDL guidance are verified by comparing to the fuel consumption using the separate phase guidance approach. Furthermore, the computational efficiency of the customized ADMM algorithm is validated by comparing to the state-of-the-art nonlinear programming method. The robustness of the customized ADMM algorithm is verified via extensive simulation cases with random initial conditions.

Flow instabilities in fluid displacement through enlarged regions in annular ducts

This work presents numerical results of flow displacement and the instabilities that appear when a viscoplastic fluid is displaced by a Newtonian one in a vertical annular duct with an enlarged region. Hydrodynamic instabilities in displacement flows may be due to viscosity and density differences, inertia effects and geometry irregularities like eccentricity and variable cross sections. If one of the fluids is non-Newtonian, the situation can be more challenging due to the dependence of the rheology on the flow kinematics. The geometry analyzed is found in the well cementation process in the oil industry, where the drilling fluid (viscoplastic behavior) is displaced by a spacer fluid (Newtonian). The drilling fluid wash out must be perfect in order to guarantee the success of the cementation operation, thus preventing the collapse of the oil well. Situations with excessive wellbore enlargement can occur, hampering the removal of the drilling fluid and the establishment of a perfect zonal isolation. The numerical solution of the problem is obtained using the finite volume technique to solve the governing conservation equations of mass and momentum. The volume of fluid method is employed to deal with the multiphase problem. The aim of this study is to look at how the governing parameters affect the flow displacement pattern and efficiency inside the enlarged region, with the goal to optimize the displacement process. The results focus on real case scenario for unstable situations, and present new findings that helps understanding the role of viscous, inertia and buoyancy effects. • Instabilities in flow displacement in enlarged annular tubes. • Well cementation process in oil industry. • Viscous, inertia and buoyancy effects on the displacement efficiency. • Displacement of a yield stress fluid by a Newtonian one.

Toward a universal description of multiphase turbulence phenomena based on the vorticity transport equation

Understanding the evolution of turbulence in multiphase flows remains a challenge due to the complex inter-phase interactions at different scales. This paper attempts to enlighten the multiphase turbulence phenomenon from a new perspective by exploiting the classical concept of vorticity and its role in the evolution of the turbulent energy cascade. We start with the vorticity transport equations for two different multiphase flow formulations, which are one-fluid and two-fluid models. By extending the decaying homogeneous isotropic turbulence (HIT) problem to the multiphase flow context, we performed two highly resolved simulations of HIT in the presence of (i) a thin interface layer and (ii) homogeneously distributed solid particle. These two configurations allow for the investigation of interfacial turbulence and particulate turbulence, respectively. In addition to the analysis of the global flow characteristic in both cases, we evaluate the spectral contribution of each production/dissipation mechanism in the vorticity transport equation to the distribution of vortical energy (enstrophy) across the scales. We base our discussion on the role of the main inter-phase interaction mechanisms in vorticity transport (i.e., the surface tension for interfacial turbulence and drag force for particulate turbulence) and unveil a similar contribution from these mechanisms to the multiphase turbulence cascade. The results also explain the deviation of kinetic energy and enstrophy spectra of multiphase HIT problems from their single-phase similitudes, confirming the validity of this approach for establishing a universal description of multiphase turbulence.

Meshfree Collocation Framework for Multi-Phase Coupling Nonlinear Dynamic System in a Porous Enclosure

This study is aimed at establishing an effective incremental-iterative framework, for the first time, using reproducing kernel collocation method for solving multi-phase dynamic coupling problems in a porous enclosure. The central difference method is utilized in the temporal discretization while direct collocation with reproducing kernel approximation is adopted in the spatial discretization, in particular, with Newton–Raphson method for nonlinear iteration. Under the proposed numerical framework, a determined and well-conditioned nonlinear system is ensured during nonlinear iteration. In addition to the traditional mesh-based weak-form methods, it is shown that the present study provides an alternative effective approach to solve unsteady-state double-diffusive convection problems.

Diffuse interface relaxation model for two-phase compressible flows with diffusion processes

The paper addresses a two-temperature model for simulating compressible two-phase flow taking into account diffusion processes related to the heat conduction and viscosity of the phases. This model is reduced from the two-phase Baer-Nunziato model in the limit of complete velocity relaxation and consists of the phase mass and energy balance equations, the mixture momentum equation, and a transport equation for the volume fraction. Terms describing effects of mechanical relaxation, temperature relaxation, and thermal conduction on volume fraction evolution are derived and demonstrated to be significant for heat conduction problems. The thermal conduction leads to instantaneous thermal relaxation so that the temperature equilibrium is always maintained in the interface region with meeting the entropy relations. A numerical method is developed to solve the model governing equations that ensures the pressure-velocity-temperature (PVT) equilibrium condition in its high-order extension. We solve the hyperbolic part of the governing equations with the Godunov method with the HLLC approximate Riemann solver. The non-linear parabolic part is solved with an efficient Chebyshev explicit iterative method without dealing with large sparse matrices. To verify the model and numerical methods proposed, we demonstrate numerical results of several numerical tests such as the multiphase shock tube problem, the multiphase impact problem, and the planar ablative Rayleigh–Taylor instability problem.

Large-eddy simulation of particle-laden isotropic turbulence using machine-learned subgrid-scale model

We apply a machine-learned subgrid-scale model to large-eddy simulations (LES) of heavy particles in isotropic turbulence with different Stokes numbers. The data-driven model, originally developed for high Reynolds number isotropic turbulent flows based on the gene expression programming (GEP) method, has explicit model equations and is for the first time tested in multiphase problems. The performance of the GEP model has been investigated in detail, focusing on the particle statistics including particle acceleration, velocity, and clustering. Compared with the commonly used dynamic Smagorinsky model, the GEP model provides significantly improved predictions on the particle statistics with Stokes numbers varying from 0.01 to 20, showing satisfactory agreement with the results from direct numerical simulations. The reasons for the enhanced predictions of the GEP model are further discussed. As the GEP model is less dissipative and it introduces high-order terms closely related to vorticity distribution, the fine-scale structures usually missing in LES simulations can be better recovered, which are believed to be closely related to the intermittency of particle motion and also particle clustering.

Interactions Between Shock Waves and Liquid Droplet Clusters: Interfacial Physics

Abstract This study quantitatively investigates the behaviors of single and multiple liquid cylinders placed in the path of a traveling normal shock wave using high-fidelity numerical simulations. The research is motivated by next-generation liquid-fueled scramjet and rotating detonation engines (RDE) where the liquid fuel interacts with shock waves and undergoes deformation, fragmentation, atomization, and vaporization before it mixes with the air and subsequently burns—the focus of this study is on the deformation and interfacial physics. The mathematical formulation to investigate this multiphase problem is based on a modified five-equation Kapila model that incorporates pressure-relaxation, viscous, and surface tension effects. A diffuse interface method is used to capture the liquid–gas interface. Two configurations are studied in this effort: (1) a single column of diameter 22 mm exposed to a shock wave traveling at Mach 2.4 and (2) a two identical cylinder system with diameters of 4.8 mm and 30 mm apart, and exposed to a shock wave moving a Mach number of 1.47. The computational results show excellent agreement with high-speed images and droplet deformation measured in the experiments. For both cases, it is found that the shock and the flow field in its wake leads to the flattening of the cylinder, followed by the formation of instability waves that are amplified by the baroclinic torque and the continuous reflections of the waves transmitted inside the liquid interior, eventually leading to ligament stripping. Based on the spatiotemporal evolution of the liquid and gaseous flowfields, time evolution of shock strength and parent droplet's mass and translation distance are also discussed.

Unified lattice Boltzmann method with improved schemes for multiphase flow simulation: Application to droplet dynamics under realistic conditions

As a powerful mesoscale approach, the lattice Boltzmann method (LBM) has been widely used for the numerical study of complex multiphase flows. Recently, Luo etal. [Philos. Trans. R. Soc. A: Math. Phys. Eng. Sci. 379, 20200397 (2021)10.1098/rsta.2020.0397] proposed a unified lattice Boltzmann method (ULBM) to integrate the widely used lattice Boltzmann collision operators into a unified framework. In this study, we incorporate additional features into this ULBM in order to simulate multiphase flow under realistic conditions. A nonorthogonal moment set [Fei etal., Phys. Rev. E 97, 053309 (2018)10.1103/PhysRevE.97.053309] and the entropic-multi-relaxation-time (KBC) lattice Boltzmann model are used to construct the collision operator. An extended combined pseudopotential model is proposed to realize multiphase flow simulation at high-density ratio with tunable surface tension over a wide range. The numerical results indicate that the improved ULBM can significantly decrease the spurious velocities and adjust the surface tension without appreciably changing the density ratio. The ULBM is validated through reproducing various droplet dynamics experiments, such as binary droplet collision and droplet impingement on superhydrophobic surfaces. Finally, the extended ULBM is applied to complex droplet dynamics, including droplet pancake bouncing and droplet splashing. The maximum Weber number and Reynolds number in the simulation reach 800 and 7200, respectively, at a density ratio of 1000. The study demonstrates the generality and versatility of ULBM for incorporating schemes to tackle challenging multiphase problems.

Integrated topology and packaging optimization for multi-phase multi-component problems

Integrated topology and packaging optimization for multi-phase multi-component problems

U-FNO—An enhanced Fourier neural operator-based deep-learning model for multiphase flow

Numerical simulation of multiphase flow in porous media is essential for many geoscience applications. Machine learning models trained with numerical simulation data can provide a faster alternative to traditional simulators. Here we present U-FNO, a novel neural network architecture for solving multiphase flow problems with superior accuracy, speed, and data efficiency. U-FNO is designed based on the newly proposed Fourier neural operator (FNO), which has shown excellent performance in single-phase flows. We extend the FNO-based architecture to a highly complex CO2-water multiphase problem with wide ranges of permeability and porosity heterogeneity, anisotropy, reservoir conditions, injection configurations, flow rates, and multiphase flow properties. The U-FNO architecture is more accurate in gas saturation and pressure buildup predictions than the original FNO and a state-of-the-art convolutional neural network (CNN) benchmark. Meanwhile, it has superior data utilization efficiency, requiring only a third of the training data to achieve the equivalent accuracy as CNN. U-FNO provides superior performance in highly heterogeneous geological formations and critically important applications such as gas saturation and pressure buildup “fronts” determination. The trained model can serve as a general-purpose alternative to routine numerical simulations of 2D-radial CO2 injection problems with significant speed-ups than traditional simulators.

A three-phase coupled numerical model for the hydraulic fracturing of rock

In this work, a three-phase coupled numerical model is developed for rock fracturing by integrating the lattice Boltzmann method (LBM) and the distinct lattice spring model (DLSM). The multiphase flow is handled by using the LBM based on the Shan–Chen potential, whereas the rock fracturing is solved by using the DLSM with a cohesive-like model. During the calculation, the solid and fluid models are executed in their own timeline independently, and the coupling is realized from boundary exchange within each step. The influence of fracturing on the multiphase distribution is considered by using a phase update operation. The three-phase model is verified against a few multiphase benchmark problems. Then, its ability to model the fracturing of rock is demonstrated through a comparison with the hydraulic fracturing test. Following this, the influence phase distribution along cavities on the hydraulic fracturing path was investigated by using the coupled model. Our results show that the three-phase coupled model can be a useful numerical tool for handling rock fracturing problems involving multiphase fluids.

Transient oscillation response characteristics of an electrohydrodynamic settling drop subjected to a uniform electric field

The transient oscillation response of an electrohydrodynamic settling drop under a uniform electric field is numerically investigated. The governing equations are solved in the lattice Boltzmann framework through the application of the leaky dielectric model and the pseudopotential model for the multi-phase electrohydrodynamic problem. A viscous drop with inertia is considered for non-density matched settling systems. Numerical simulations are performed over a range of electric capillary numbers CaE, Eotvos numbers Eo, and Ohnesorge numbers Oh. The results indicate that three typical development stages, namely, the electric stress-dominated stage, the force competition stage, and the inertia-dominated stage, are identified in terms of the deformation evolution characteristics. Our study also demonstrates the role of the three dimensionless numbers in the deformation response at each stage. It is found that, at the earlier stage of settling, the maximum achievable deformation is sensitive to CaE and Oh, while the influence of Eo on the first oscillatory peak at the deformation-time curve is approximately neglectable. Moreover, the deformation response time is determined by the interaction of the electric field, the gravitational field, and viscosity. Specifically, the corresponding oscillatory peak time correlates positively with Eo and Oh numbers and exponentially grows with CaE.

An open source code for two-phase rarefied flows: rarefiedMultiphaseFoam

This paper presents the development and benchmarking tests of an open source code for providing solutions of two-phase rarefied flows. The solver is named in rarefiedMultiphaseFoam and it is developed in OpenFOAM, based on dsmcFoamPlus, which is used for the rarefied gas phase. The solver can produce both steady and transient results for arbitrary 2D/3D two-phase rarefied flows and includes a phase change model for materials that experience melting and solidification processes. The benchmarking tests include momentum and heat exchange between gas phase and solid phases, particle phase change, a converging nozzle that creates a solid particle beam, and transport of solid particles. The tests yield good agreement with analytical and experimental data where available, and are compared to previous numerical results from the literature. Program summaryProgram Title: rarefiedMulitphaseFoamCPC Library link to program files:https://doi.org/10.17632/8vxrc5gsb8.1Licensing provisions: GNU General Public License 3Programming language: C++Nature of problem: rarefiedMultiphaseFoam has been developed to help investigate multiphase problems with rarefied gas and solid phases. The rarefied gas phase is simulated with the direct simulation Monte Carlo (DSMC) method, with both one and two way coupling models available to simulate the momentum and heat transfer between the phases. It provides an easily extended, parallelised, environment.Solution method: rarefiedMultiphaseFoam implements an explicit time-stepping solver with stochastic molecular collisions appropriate for studying rarefied gas flow problems and uses a Green's function model to calculate the energy transfers between phases.

Onboard fuel-optimal guidance for human-Mars entry, powered-descent, and landing mission based on feature learning

This paper investigates the end-to-end human-Mars entry, powered-descent, and landing (EDL) guidance problem by developing an on-board learning-based optimal control method (L-OCM) to achieve the goal of precise and fuel-efficient planetary landing. First, the end-to-end EDL guidance problem is formulated as a multi-phase optimal control problem with hybrid dynamics and constraints. Then a customized alternating direction method of multipliers is applied to solve the end-to-end EDL guidance problem with varying initial states off-line. After that, the L-OCM is developed to generate real-time optimal guidance commands. To be specific, supported by the optimal control theory, the necessary conditions of optimality for optimal control of the entry phase and powered-descent phase are derived, respectively, which leads to two two-point-boundary-value-problems (TPBVPs). Then, critical parameters are identified to approximate the complete solutions of the TPBVPs. To find the implicit relationship between the initial states and these critical parameters, deep neural networks are constructed to learn the values of these critical parameters in real-time with training data obtained from the off-line solutions. Furthermore, when random disturbances are considered during the EDL process, the single stage L-OCM is extended to the multi-stage L-OCM to regenerate real-time optimal guidance commands. Finally, the proposed L-OCM is implemented in extensive simulation cases to verify the effectiveness and efficiency of the new method.