Two-phase Navier-Stokes Equations Research Articles

Arbitrary Lagrangian-Eulerian finite element approximations for axisymmetric two-phase flow

We analyze numerical approximations for axisymmetric two-phase flow in the arbitrary Lagrangian-Eulerian (ALE) framework. We consider a parametric formulation for the evolving fluid interface in terms of a one-dimensional generating curve. For the two-phase Navier-Stokes equations, we introduce both conservative and nonconservative ALE weak formulations in the 2d meridian half-plane. Piecewise linear parametric elements are employed for discretizing the moving interface, which is then coupled to a moving finite element approximation of the bulk equations. This leads to a variety of ALE methods, which enjoy either an equidistribution property or unconditional stability. Furthermore, we adapt these introduced methods with the help of suitable time-weighted discrete normals, so that the volume of the two phases is exactly preserved on the discrete level. Numerical results for rising bubbles and oscillating droplets are presented to show the efficiency and accuracy of these introduced methods.

Multiphase flow coupling analysis of shield synchronous grouting filling law with time-dependent viscosity

Owing to the non-visual characteristics of the shield synchronous grouting engineering, the law of grout filling in the shield tail gap remains obscure. There is a current deficiency in effective three-dimensional filling and diffusion models. Building upon a large-diameter, eight-hole shield tunneling grouting project in Hangzhou, this study aims to construct a holistic three-dimensional model of grout filling in the shield tail gap. Employing COMSOL Multiphysics and grounded on the two-phase Navier–Stokes equations, this study simulated the filling of grouting fluid in the shield tail gap. Utilizing the Brookfield DV3T rheometer, the study ascertained the time-dependent expression of grout viscosity and systematically analyzed the impacts of time-dependent grout viscosity, density, and injection pressure on the filling diffusion morphology, pressure field, velocity field, and the buoyancy experienced by the segmental lining. The results indicate that the injection pressure is positively correlated with the circumferential pressure field of the segmental lining, though the influence on the ring's total buoyancy is minimal. The rate of grout viscosity development significantly affects the overall diffusion morphology: conventional grout tends to be underfilled at the top of the shield tail gap while rapid-setting grout is more likely to be underfilled at the bottom. The grout density is positively correlated with the grout displacement speed and the total buoyancy of the segmental lining. In light of these insights, utilizing low-density, rapid-setting grout under high-pressure grouting for shield tail filling can ensure adequate filling rates while mitigating the adverse effects of segment buoyancy.

An unstructured finite-volume level set / front tracking method for two-phase flows with large density-ratios

We extend the unstructured LEvel set / froNT tracking (LENT) method [1,2] for handling two-phase flows with strongly different densities (high-density ratios) by providing the theoretical basis for the numerical consistency between the mass and momentum conservation in the collocated Finite Volume discretization of the single-field two-phase Navier-Stokes equations. Our analysis provides the theoretical basis for the mass conservation equation introduced by Ghods and Herrmann [3] and used in [4–8]. We use a mass flux that is consistent with mass conservation in the implicit Finite Volume discretization of the two-phase momentum convection term, and solve the single-field Navier-Stokes equations with our SAAMPLE segregated solution algorithm [2]. The proposed ρLENT method recovers exact numerical stability for the two-phase momentum advection of a spherical droplet with density ratios ρ−/ρ+∈[1,104]. Numerical stability is demonstrated for in terms of the relative L∞ velocity error norm, for density-ratios in the range of [1,104], dynamic viscosity-ratios in the range of [1,104] and very strong surface tension forces, for challenging mercury/air and water/air fluid pairings. In addition, the solver performs well in cases characterized by strong interaction between two phases, i.e., oscillating droplets and rising bubbles. The proposed ρLENT method1 is applicable to any other two-phase flow simulation method that discretizes the single-field two-phase Navier-Stokes Equations using the collocated unstructured Finite Volume Method but does not solve an advection equation for the phase indicator using a flux-based approach, by adding the proposed geometrical approximation of the mass flux and the auxiliary mass conservation equation to the solution algorithm.

Structure-preserving discretizations of two-phase Navier–Stokes flow using fitted and unfitted approaches

We consider the numerical approximation of a sharp-interface model for two-phase flow, which is given by the incompressible Navier–Stokes equations in the bulk domain together with the classical interface conditions on the interface. We propose structure-preserving finite element methods for the model, meaning in particular that volume preservation and energy decay are satisfied on the discrete level. For the evolving fluid interface, we employ parametric finite element approximations that introduce an implicit tangential velocity to improve the quality of the interface mesh. For the two-phase Navier–Stokes equations, we consider two different approaches: an unfitted and a fitted finite element method, respectively. In the unfitted approach, the constructed method is based on an Eulerian weak formulation, while in the fitted approach a novel arbitrary Lagrangian–Eulerian (ALE) weak formulation is introduced. Using suitable discretizations of these two formulations, we introduce two finite element methods and prove their structure-preserving properties. Numerical results are presented to show the accuracy and efficiency of the introduced methods.

Pulsating pressurization of two-phase fluid in a pipe filled with water and a little gas

Although two-phase flows containing gas and water have received extensive attention, the pulsating pressurization effect of a two-phase fluid in a pipe is unclear and the influence of the gas-phase content has not been revealed. This paper discusses the pulsating pressurization of such a two-phase fluid. First, the two-phase Navier–Stokes equations are derived and an algorithm is developed based on MacCormack's method. The reliability of the algorithm is examined and validated using Poiseuille's theory and existing experimental two-phase flow data. Finally, the influence of several key factors is discussed, including the gas-phase fraction and pipe slenderness. Our results show that a significant pulsating supercharging phenomenon occurs when the gas-phase fraction is less than 10−3. When the gas-phase fraction is greater than this critical value, the pulsating supercharging effect decreases significantly with the increasing gas-phase fraction. The equivalent elastic modulus of the two-phase fluid rapidly decreases as the gas-phase fraction increases, and the pressure disturbance is absorbed by the gas bubbles, causing an apparent weakening of the pulsating supercharging effect. Thus, decreasing the gas-phase content can enhance the pulsating supercharging effect. The pipe slenderness has a very limited influence on the pulsating pressurization process, and the maximum reduction is only 7.3% for slenderness ratios of up to 2000. Moreover, we derive and propose a new mathematical expression for the inlet boundary that is applicable to gas–liquid two-phase flows. To our knowledge, this paper extends the pulsating pressurization range from the single-phase to two-phase fluid for the first time and reports different physical phenomena and regularity. The present research clarifies the pulsating pressurization phenomenon in two-phase flows, providing a valuable reference for pulsating pressurization design.

An adjoint method for control of liquid-gas flows using a sharp interface model

We present a computational, continuous adjoint framework for the control of liquid-gas flows using a sharp interface model. The two-phase Navier–Stokes equations are solved using a mass-conserving geometric Volume-of-Fluid method, while the adjoint equations consider a level set-based representation of the interface. To facilitate the accurate transport of a surface adjoint variable, a geometric surface transport method is formulated and applied. We verify our method by comparing adjoint-calculated gradients against analytical gradients or finite difference approximations. The method is then applied to a variety of benchmark two-phase flow problems, including the multi-dimensional inflow control of droplet position and optimal control of the initial velocity profile in a temporally evolving liquid-gas mixing layer.

Revisiting wind wave growth with fully coupled direct numerical simulations

We investigate wind wave growth by direct numerical simulations solving for the two-phase Navier–Stokes equations. We consider the ratio of the wave speed$c$to the wind friction velocity$u_*$from$c/u_*= 2$to 8, i.e. in the slow to intermediate wave regime; and initial wave steepness$ak$from 0.1 to 0.3; the two being varied independently. The turbulent wind and the travelling, nearly monochromatic waves are fully coupled without any subgrid-scale models. The wall friction Reynolds number is 720. The novel fully coupled approach captures the simultaneous evolution of the wave amplitude and shape, together with the underwater boundary layer (drift current), up to wave breaking. The wave energy growth computed from the time-dependent surface elevation is in quantitative agreement with that computed from the surface pressure distribution, which confirms the leading role of the pressure forcing for finite amplitude gravity waves. The phase shift and the amplitude of the principal mode of surface pressure distribution are systematically reported, to provide direct evidence for possible wind wave growth theories. Intermittent and localised airflow separation is observed for steep waves with small wave age, but its effect on setting the phase-averaged pressure distribution is not drastically different from that of non-separated sheltering. We find that the wave form drag force is not a strong function of wave age but closely related to wave steepness. In addition, the history of wind wave coupling can affect the wave form drag, due to the wave crest shape and other complex coupling effects. The normalised wave growth rate we obtain agrees with previous studies. We make an effort to clarify various commonly adopted underlying assumptions, and to reconcile the scattering of the data between different previous theoretical, numerical and experimental results, as we revisit this longstanding problem with new numerical evidence.

Phase-field Lattice Boltzmann model for liquid bridges and coalescence in wet granular media

In this paper a phase-field based Lattice Boltzmann model is formulated to simulate the formation and coalescence of capillary bridges between spherical solid particles as well as the associated capillary forces. To capture the air–water interface, Allen–Cahn equation is coupled with two-phase Navier–Stokes equation through a surface tension term. The capillary force arising from the water interaction with a curved solid surface is formulated along with a numerical integration scheme. Two benchmark examples are considered to validate the proposed model, namely: the spreading of a drop on a spherical particle and capillary rise in a narrow tube. Then, the capillary forces due to isolated and coalesced liquid in two and three spherical particle configurations are computed to illustrate the pendular regime and its transition to the funicular regime. Numerical simulation results are found to be in good agreement with available experimental and numerical data for wetting in a doublet and a triplet of particles, respectively. The numerical results indicate that the proposed model provides a viable framework within which the complex capillary interface evolution during the wetting of a large assembly of particles could be captured. • The proposed model is used to simulate capillary bridges in an air–water system. • Transition of capillary bridges from pendular to funicular regimes is handled. • A numerical procedure is proposed to compute capillary forces. • Validation on small granular assemblies composed of two and three particles.

Differentiability results and sensitivity calculation for optimal control of incompressible two-phase Navier-Stokes equations with surface tension

We analyze optimal control problems for two-phase Navier-Stokes equations with surface tension. Based on Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_p$$\\end{document}-maximal regularity of the underlying linear problem and recent well-posedness results of the problem for sufficiently small data we show the differentiability of the solution with respect to initial and distributed controls for appropriate spaces resulting from the Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_p$$\\end{document}-maximal regularity setting. We consider first a formulation where the interface is transformed to a hyperplane. Then we deduce differentiability results for the solution in the physical coordinates. Finally, we state an equivalent Volume-of-Fluid type formulation and use the obtained differentiability results to derive rigorosly the corresponding sensitivity equations of the Volume-of-Fluid type formulation. For objective functionals involving the velocity field or the discontinuous pressure or phase indciator field we derive differentiability results with respect to controls and state formulas for the derivative. The results of the paper form an analytical foundation for stating optimality conditions, justifying the application of derivative based optimization methods and for studying the convergence of discrete sensitivity schemes based on Volume-of-Fluid discretizations for optimal control of two-phase Navier-Stokes equations.

On the Rayleigh–Taylor instability for the two-phase Navier–Stokes equations in cylindrical domains

We are considering the two-phase Navier–Stokes equations with surface tension, modelling the dynamic behaviour of two immiscible and incompressible fluids in a cylindrical domain, which are separated by a sharp interface forming a contact angle with the fixed boundary. In the case that the heavy fluid is situated on top of the light fluid, one expects the effect which is known as Rayleigh–Taylor instability . Our main result implies the existence of a critical surface tension with the following property: In the case that the surface tension of the interface separating the two fluids is smaller than the critical surface tension, Rayleigh–Taylor instability occurs. On the contrary, if the surface tension of the interface is larger than the critical value, one has exponential stability of the flat interfaces. The last part of this article is concerned with the bifurcation of nontrivial equilibria in multiple eigenvalues. The invariance of the corresponding bifurcation equation with respect to rotations and reflections yields the existence of bifurcating subcritical equilibria.

Vapor-liquid phase transition in fluctuating hydrodynamics: The most probable transition path and its computation

In this paper, we consider transition events between metastable states in fluid systems modeled by two-phase Navier-Stokes equations with thermal noise. The vapor-liquid phase transition which occurs during the vapor condensation is such an example. We propose a numerical method to compute the transition pathway. We first approximate the continuum hydrodynamics model by the smoothed particle hydrodynamics (SPH). In the SPH formulation, the dynamics of the system take the form of Langevin equations. This allows us to derive conditions that characterize the most probable transition pathway. Then we compute the most probable transition pathway by solving these conditions in the space of mass and momentum density fields using the string method. The force, which is needed to evolve the string in the space of the density fields, is computed by discretizing the deterministic model using the finite difference method. The numerical method is applied to study the condensation of vapor in a periodic cell and the Cassie-Wenzel transition on pillared solid substrates.

High-resolution direct simulation of deep water breaking waves: transition to turbulence, bubbles and droplets production

We present high-resolution three-dimensional (3-D) direct numerical simulations of breaking waves solving for the two-phase Navier–Stokes equations. We investigate the role of the Reynolds number (Re, wave inertia relative to viscous effects) and Bond number (Bo, wave scale over the capillary length) on the energy, bubble and droplet statistics of strong plunging breakers. We explore the asymptotic regimes at highReandBo, and compare with laboratory breaking waves. Energetically, the breaking wave transitions from laminar to 3-D turbulent flow on a time scale that depends on the turbulentReup to a limiting value$Re_\lambda \sim 100$, consistent with the mixing transition in other canonical turbulent flows. We characterize the role of capillary effects on the impacting jet and ingested main cavity shape and subsequent fragmentation process, and extend the buoyant-energetic scaling from Deikeet al.(J. Fluid Mech., vol. 801, 2016, pp. 91–129) to account for the cavity shape and its scale separation from the Hinze scale,$r_H$. We confirm two regimes in the bubble size distribution,$N(r/r_H)\propto (r/r_H)^{-10/3}$for$r>r_H$, and$\propto (r/r_H)^{-3/2}$for$r<r_H$. Bubbles are resolved up to one order of magnitude below$r_H$, and we observe a good collapse of the numerical data compared to laboratory breaking waves (Deane & Stokes,Nature, vol. 418 (6900), 2002, pp. 839–844). We resolve droplet statistics at highBoin good agreement with recent experiments (Erininet al.,Geophys. Res. Lett., vol. 46 (14), 2019, pp. 8244–8251), with a distribution shape close to$N_d(r_d)\propto r_d^{-2}$. The evolution of the droplet statistics appears controlled by the details of the impact process and subsequent splash-up. We discuss velocity distributions for the droplets, finding ejection velocities up to four times the phase speed of the wave, which are produced during the most intense splashing events of the breaking process.

Wind wave growth in the viscous regime

How water surface waves grow under wind forcing has long been an interesting and challenging question. For short gravity-capillary waves, the viscous effects are important but have not been well studied. In this paper, we simulate the wind-wave growth by directly solving the two-phase Navier-Stokes equations. The numerical method features a momentum conserving scheme, interface reconstruction using Volume of Fluid, and adaptive mesh refinement (AMR). As a result, we observe concurrent growth of the irrotational traveling wave and the rotational drift layer (current). The growth rate of the wave and the evolution of the drift layer under different forcing parameters are discussed respectively.

Numerical modeling of liquid sloshing in flexible tank with FSI approach

PurposeThe paper aims to present a numerical modeling procedure for the analysis of liquid sloshing in a flexible tank subjected to an external excitation, with taking into account the effects of fluid–structure interaction (FSI).Design/methodology/approachA numerical model based on coupling a two-phase flow solver and an elastic solid solver is developed in OpenFOAM code. The Arbitrary Lagrangian–Eulerian formulation is adopted for the two-phase Navier–Stokes equations in a moving domain. The volume of fluid (VOF) method is applied for the air–liquid interface tracking. The finite volume method is used for the spatial discretization of both the fluid and the structure dynamics equations. The FSI coupling problem is solved by an explicit coupling scheme. The model is validated for linear and nonlinear sloshing cases. Then, it is used to analyze the effects of the liquid sloshing on the dynamic response of the tank and the effects of the tank flexibility on the liquid sloshing.FindingsThe obtained results show that the flexibility of the tank walls amplifies the amplitude of the sloshing and increases the fluctuation period of the air–liquid interface. Furthermore, it is found that the bending moment acting on the tank walls may be underestimated when rigid walls assumption is adopted as usually done in sloshing tank modeling. Also, tank walls flexibility causes a phase shift in the free surface dynamic response.Originality/valueA review of previous studies on liquid sloshing in flexible tanks revealed that FSI effects have not been clearly and comprehensively analyzed for large-amplitude liquid sloshing. Many physical and numerical aspects of this problem still require clarifications and enhancements. The added value of the present work and its originality lie in the investigation of large-amplitude liquid sloshing in flexible tanks by using a staggered coupling approach. This approach is carried out by an original combination of a linear solid solver with a two phase fluid solver in OpenFOAM code. In addition, FSI effects on some response quantities, identified and analyzed herein, have not been found in the previous works.

Time-Decay Estimates for Linearized Two-Phase Navier–Stokes Equations with Surface Tension and Gravity

The aim of this paper is to show time-decay estimates of solutions to linearized two-phase Navier-Stokes equations with surface tension and gravity. The original two-phase Navier-Stokes equations describe the two-phase incompressible viscous flow with a sharp interface that is close to the hyperplane xN=0 in the N-dimensional Euclidean space, N≥2. It is well-known that the Rayleigh–Taylor instability occurs when the upper fluid is heavier than the lower one, while this paper assumes that the lower fluid is heavier than the upper one and proves time-decay estimates of Lp-Lq type for the linearized equations. Our approach is based on solution formulas for a resolvent problem associated with the linearized equations.

A novel diffuse-interface model and a fully-discrete maximum-principle-preserving energy-stable method for two-phase flow with surface tension and non-matching densities

Two well-established classes of the interface capturing models are the level-set and phase-field models. Level-set formulations satisfy the maximum principle for the density but are not energy-stable. On the other hand, phase-field models do satisfy the second law of thermodynamics but lack the maximum principle for the density. In this paper we derive a novel model for incompressible immiscible two-phase flow with non-matching densities and surface tension that is both energetically-stable and satisfies the maximum principle for the density. The model finds its place at the intersection of level-set and phase-field models. Its derivation is based on a diffusification of the incompressible two-phase Navier–Stokes equations with non-matching densities and surface tension and involves functional entropy variables. Additionally, we present an associated fully-discrete energy-stable method. Isogeometric analysis is used for the spatial discretization and the temporal-integration is performed with a new time-integration scheme that is a perturbation of the second-order midpoint scheme. The fully-discrete scheme is unconditionally energy-dissipative, pointwise divergence-free and satisfies the maximum principle for the density. Numerical examples in two and three dimensions verify the energetic-stability of the methodology.

Unique solvability of elliptic problems associated with two-phase incompressible flows in unbounded domains

This paper shows the unique solvability of elliptic problems associated with two-phase incompressible flows, which are governed by the two-phase Navier-Stokes equations with a sharp moving interface, in unbounded domains such as the whole space separated by a compact interface and the whole space separated by a non-compact interface. As a by-product, we obtain the Helmholtz-Weyl decomposition for two-phase incompressible flows.

Dynamic wetting failure in curtain coating by the Volume-of-Fluid method

In this paper, we investigate dynamic wetting in the curtain coating configuration. The two-phase Navier–Stokes equations are solved by a Volume-of-Fluid method on an adaptive Cartesian mesh. We introduce the Navier boundary condition to regularize the solution at the triple point and remove the implicit numerical slip induced by the cell-centered interface advection. We use a constant contact angle to describe the dynamic contact line. The resolution of the governing equations allows us to predict the substrate velocity at which wetting failure occurs. The model predictions are compared with prior computations of Liu et al. [C.Y. Liu, E. Vandre, M. Carvalho, S. Kumar, J. Fluid Mech. 808, 290 (2016); C.Y. Liu, M. Carvalho, S. Kumar, Chem. Eng. Sci. 195, 74 (2019)] and experimental observations of Blake et al. [T. Blake, M. Bracke, Y. Shikhmurzaev, Phys. Fluids 11, 1995 (1999)] and Marston et al. [J. Marston, V. Hawkins, S. Decent, M. Simmons, Exp. Fluids 46, 549 (2009)].

Boundary conditions for dynamic wetting - A mathematical analysis

The moving contact line paradox discussed in the famous paper by Huh and Scriven has lead to an extensive scientific discussion about singularities in continuum mechanical models of dynamic wetting in the framework of the two-phase Navier–Stokes equations. Since the no-slip condition introduces a non-integrable and therefore unphysical singularity into the model, various models to relax the singularity have been proposed. Many of the relaxation mechanisms still retain a weak (integrable) singularity, while other approaches look for completely regular solutions with finite curvature and pressure at the moving contact line. In particular, the model introduced recently in [A.V. Lukyanov, T. Pryer, Langmuir 33, 8582 (2017)] aims for regular solutions through modified boundary conditions. The present work applies the mathematical tool of compatibility analysis to continuum models of dynamic wetting. The basic idea is that the boundary conditions have to be compatible at the contact line in order to allow for regular solutions. Remarkably, the method allows to compute explicit expressions for the pressure and the curvature locally at the moving contact line for regular solutions to the model of Lukyanov and Pryer. It is found that solutions may still be singular for the latter model.

Hysteretic heat transfer study of liquid–liquid two-phase flow in a T-junction microchannel

Liquid–liquid two-phase flow in microchannels is capable of boosting the heat removal rate in cooling processes. Formation of different two-phase flow patterns which affect the heat transfer rate is numerically investigated here in a T-junction containing water-oil flow. For this purpose, the finite element method (FEM) is applied to solve the unsteady two-phase Navier–Stokes equations along with the level set (LS) equation in order to capture the interface between phases. It is shown that the two-phase flow pattern in microchannels depends on the flow initial condition which causes hysteresis effect in two-phase flow. In this study, the hysteresis is observed in flow pattern and consequently in the heat transfer rate. The effect of wall contact angle on the hydrodynamics and heat transfer in the microchannel is investigated to gain useful insight into the hysteresis phenomenon. It is observed that the hysteresis is significant in super-hydrophilic microchannels, while it disappears at the contact angle of 75°. The effect of water to oil flow rate ratio (Qwat/Qoil) on the heat transfer is also studied. The flow rate ratio has a negligible effect on the Nusselt number (Nu) in the dripping regime, while the Nu decreases with an increase of Qwat/Qoil in the co-flow regime. The thickness of the oil film, velocity, and temperature distribution are studied in the co-flow regime. It is revealed that the normalized slip velocity reduces at higher values of Qwat/Qoil, which causes a reduction in the averaged Nu. In dripping regimes, higher flow rate ratios lead to a more frequent generation of droplet/slugs at a smaller size. The passage of the slugs or droplets increases the local Nu. Larger droplets generated at lower flow rate ratios cause a larger increase in the local Nu than smaller droplets. The temperature and velocity field around the droplets are also illustrated to investigate the heat transfer improvement. The generated vortex at the tip of the oil jet causes an increase in the velocity and Nu on the water side.