Incompressible Two-phase Flow Problem Research Articles

A conservative finite volume method for incompressible two-phase flows on unstructured meshes

A robust finite volume framework for solving incompressible two-phase flow problems based on an explicit dual-time stepping based artificial compressibility formulation has been developed by Parameswaran and Mandal (2019) for structured meshes. This artificial compressibility framework is extended to unstructured meshes in the present paper. In order to further enhance the applicability to real life engineering problems, the surface tension effect is also modeled here. To deal with arbitrary unstructured mesh topology, least square and Green-Gauss integral based methods are used. The efficacy of the proposed method is demonstrated using a set of scalar advection based test problems and a set of standard incompressible two-phase flow test problems involving gravitational, viscous and surface tension forces. The numerical results show improvement in accuracy compared to the results of the structured mesh formulation.

A stable interface-preserving reinitialization equation for conservative level set method

Existing artificial compression based reinitialization approach for conservative level set method has a few drawbacks, like distortion of fluid–fluid interface and formation of unphysical fluid patches away from the interface. In this paper, a novel reinitialization approach has been presented which circumvents these limitations by reformulating the existing reinitialization equation. The new reinitialization equation by design ensures that the interface contour is kept invariant under the action of level set reinitialization. The unphysical patch formation away from the interface is also resolved here by avoiding the use of ill-conditioned contour normal vectors. Owing to the use of much simpler terms, a significant reduction in the numerical computations is achieved with the new reinitialization equation. The new reinitialization equation also enables one to choose a larger time step during the reinitialization iteration, leading to an overall reduction in computational efforts. In order to evaluate the performance of the present formulation, a set of test problems involving reinitialization of stationary level set functions is carried out first. Then, the efficacy of the proposed reinitialization approach is demonstrated using a set of standard two-dimensional scalar advection based test problems and incompressible two-phase flow problems. Finally, the ability to deal with complex mesh types is demonstrated by solving a test problem on an unstructured mesh consisting of finite volume cells having triangular and quadrilateral shapes.

Inferring incompressible two-phase flow fields from the interface motion using physics-informed neural networks

In this work, physics-informed neural networks are applied to incompressible two-phase flow problems. We investigate the forward problem, where the governing equations are solved from initial and boundary conditions, as well as the inverse problem, where continuous velocity and pressure fields are inferred from scattered-time data on the interface position. We employ a volume of fluid approach, i.e. the auxiliary variable here is the volume fraction of the fluids within each phase. For the forward problem, we solve the two-phase Couette and Poiseuille flow. For the inverse problem, three classical test cases for two-phase modeling are investigated: (i) drop in a shear flow, (ii) oscillating drop and (iii) rising bubble. Data of the interface position over time is generated by numerical simulation. An effective way to distribute spatial training points to fit the interface, i.e. the volume fraction field, and the residual points is proposed. Furthermore, we show that appropriate weighting of losses associated with the residual of the partial differential equations is crucial for successful training. The benefit of using adaptive activation functions is evaluated for both the forward and inverse problem.

A multinumerics scheme for incompressible two-phase flow

The incompressible two-phase flow problem is solved by a method that combines cell-centered finite volume with discontinuous Galerkin in non-overlapping subdomains. The primary unknowns are the wetting phase pressure and the capillary pressure. The nonlinear equations are solved fully implicitly at each time step. Fluxes at the interface between subdomains are defined implicitly to allow for seamless propagation of saturation fronts. Numerical results show the robustness and efficiency of the method for homogeneous and heterogeneous porous media.

An HDG formulation for incompressible and immiscible two-phase porous media flow problems

ABSTRACTWe develop a high-order hybridisable discontinuous Galerkin (HDG) formulation to solve the immiscible and incompressible two-phase flow problem in a heterogeneous porous media. The HDG method is locally conservative, has fewer degrees of freedom than other discontinuous Galerkin methods due to the hybridisation procedure, provides built-in stabilisation for arbitrary polynomial degrees and, if the error of the temporal discretisation is low enough, the pressure, the saturation and their fluxes converge with order P+1 in -norm, being P the polynomial degree. In addition, an element-wise post-process can be applied to obtain a convergence rate of P+2 in -norm for the scalar variables. All of these advantages make the HDG method suitable for solving multiphase flow trough porous media. We show numerical evidence of the convergences rates. Finally, to assess the capabilities of the proposed formulation, we apply it to several cases of water-flooding technique for oil recovery.

A novel Roe solver for incompressible two-phase flow problems

In this paper, a novel Roe-type Riemann solver is developed for solving incompressible two-phase flow problems. The two-phase flow is modeled by coupling the artificial compressibility based Navier-Stokes equations and an interface advection equation in a conservative level set framework. The proposed Roe solver does not assume a locally constant density field while evaluating the flux Jacobian. As a result, the solver is capable of capturing the exact jump conditions across the contact and shear waves, which in turn helps in enhancing the accuracy of the interface motion. Moreover, mass conservation characteristics are significantly improved by adopting conservative level set framework together with finite volume discretization of the system of equations. A dual time stepping method with explicit pseudo-time marching scheme is used for modeling the unsteady terms. The efficacy of the proposed solver is demonstrated using various standard incompressible two-phase flow problems, involving low amplitude sloshing, broken dam, Rayleigh-Taylor instability and droplet splash.

Incompressible two-phase flows with an inextensible Newtonian fluid interface

We introduce a diffuse interface approximation for an incompressible two-phase flow problem with an inextensible Newtonian fluid interface. This approach allows to model lipid membranes as viscous fluids. In the present setting the membranes are assumed to be stationary. We validate the model and the numerical approach, which is based on a stream function formulation for the surface flow problem, an operator splitting approach and a semi-implicit adaptive finite element discretization, against observed flow patterns in vesicles, which are adhered to a solid surface and are subjected to shear flow. The influence of the Gaussian curvature on the surface flow pattern is discussed.

TP or not TP, that is the question

We give here a comparative study on the mathematical analysis of two (classes of) discretization schemes for the computation of approximate solutions to incompressible two-phase flow problems in homogeneous porous media. The first scheme is the well-known finite volume scheme with a two-point flux approximation, classically used in industry. The second class contains the so-called approximate gradient schemes, which include finite elements with mass lumping, mixed finite elements, and mimetic finite differences. Both (classes of) schemes are nonconforming and can be expressed using discrete function and gradient reconstructions within a variational formulation. Each class has its specific advantages and drawbacks: monotony properties are natural with the two-point finite volume scheme, but meshes are restricted due to consistency issues; on the contrary, gradient schemes can be used on general meshes, but monotony properties are difficult to obtain.

High Order Schemes on Three-Dimensional General Polyhedral Meshes — Application to the Level Set Method

AbstractIn this article, we detail the methodology developed to construct arbitrarily high order schemes — linear and WENO — on 3D mixed-element unstructured meshes made up of general convex polyhedral elements. The approach is tailored specifically for the solution of scalar level set equations for application to incompressible two-phase flow problems. The construction of WENO schemes on 3D unstructured meshes is notoriously difficult, as it involves a much higher level of complexity than 2D approaches. This due to the multiplicity of geometrical considerations introduced by the extra dimension, especially on mixed-element meshes. Therefore, we have specifically developed a number of algorithms to handle mixed-element meshes composed of convex polyhedra with convex polygonal faces. The contribution of this work concerns several areas of interest: the formulation of an improved methodology in 3D, the minimisation of computational runtime in the implementation through the maximum use of pre-processing operations, the generation of novel methods to handle complex 3D mixed-element meshes and finally the application of the method to the transport of a scalar level set.

The ninth-mode sectorial oscillation of acoustically levitated drops

We report on the ninth-mode sectorial oscillation of acoustically levitated drops excited by actively modulating sound pressure. A numerical computation based on the level set method was performed to model drop shape evolution by solving an incompressible two-phase flow problem. The calculated shapes of the oscillating drop are in good agreement with experimental observations. The relationship between the oscillation frequency and parameters describing the flattened drops is studied both experimentally and numerically. The frequency of the ninth-mode sectorial oscillation decreases with increasing equatorial radius and can be well-described by a modified Rayleigh equation.

Numerical simulation of bubble and droplet deformation by a level set approach with surface tension in three dimensions

AbstractIn this paper we present a three‐dimensional Navier–Stokes solver for incompressible two‐phase flow problems with surface tension and apply the proposed scheme to the simulation of bubble and droplet deformation. One of the main concerns of this study is the impact of surface tension and its discretization on the overall convergence behavior and conservation properties.Our approach employs a standard finite difference/finite volume discretization on uniform Cartesian staggered grids and uses Chorin's projection approach. The free surface between the two fluid phases is tracked with a level set (LS) technique. Here, the interface conditions are implicitly incorporated into the momentum equations by the continuum surface force method. Surface tension is evaluated using a smoothed delta function and a third‐order interpolation. The problem of mass conservation for the two phases is treated by a reinitialization of the LS function employing a regularized signum function and a global fixed point iteration. All convective terms are discretized by a WENO scheme of fifth order. Altogether, our approach exhibits a second‐order convergence away from the free surface. The discretization of surface tension requires a smoothing scheme near the free surface, which leads to a first‐order convergence in the smoothing region.We discuss the details of the proposed numerical scheme and present the results of several numerical experiments concerning mass conservation, convergence of curvature, and the application of our solver to the simulation of two rising bubble problems, one with small and one with large jumps in material parameters, and the simulation of a droplet deformation due to a shear flow in three space dimensions. Furthermore, we compare our three‐dimensional results with those of quasi‐two‐dimensional and two‐dimensional simulations. This comparison clearly shows the need for full three‐dimensional simulations of droplet and bubble deformation to capture the correct physical behavior. Copyright © 2009 John Wiley & Sons, Ltd.

Analysis of [formula omitted] discontinuous Galerkin methods for incompressible two-phase flow

In this paper, we prove the convergence of a class of discontinuous Galerkin methods for solving the fully coupled incompressible two-phase flow problem in the non-degenerate case. Estimates in both the mesh size and the polynomial degrees are obtained. Numerical convergence rates confirm the theoretical results.

A level set formulation for the numerical simulation of impact of surge fronts

In this paper we present a level set-based algorithm for the solution of incompressible two-phase flow problems. The technique is applied to the numerical simulation of impact of two surge fronts resulting from the collapse of liquid columns. The incompressible Navier-Stokes equations are solved using a projection method based on forward Euler time-stepping. The Hamilton-Jacobi type equation for the transport of level set function is carried out by a high resolution fifth-order accurate WENO scheme. For efficient implementation of the WENO scheme we have proposed grid staggering for the level set function. The solution of the pressure Poisson equation is obtained using an efficient preconditioned conjugate gradient method. It is shown that the present formulation works very well for large density and viscosity ratios. For the purpose of validation, we have simulated small-amplitude free sloshing of liquid in a container and the well-known two-dimensional broken-dam problem of Martin and Moyce. Simulations of impact of surge fronts have been carried out and the results are discussed.

A stabilized finite element method using a discontinuous level set approach for solving two phase incompressible flows

A numerical method for the simulation of three-dimensional incompressible two-phase flows is presented. The proposed algorithm combines an implicit pressure stabilized finite element method for the solution of incompressible two-phase flow problems with a level set method implemented with a quadrature-free discontinuous Galerkin (DG) method [E. Marchandise, J.-F. Remacle, N. Chevaugeon, A quadrature free discontinuous Galerkin method for the level set equation, Journal of Computational Physics 212 (2006) 338–357]. The use of a fast contouring algorithm [N. Chevaugeon, E. Marchandise, C. Geuzaine, J.-F. Remacle, Efficient visualization of high order finite elements, International Journal for Numerical Methods in Engineering] permits us to localize the interface accurately. By doing so, we can compute the discontinuous integrals without neither introducing an interface thickness nor reinitializing the level set.The capability of the resulting algorithm is demonstrated with “large scale” numerical examples (free surface flows: dam break, sloshing) and “small scale” ones (two phase Poiseuille, Rayleigh–Taylor instability).