A projection scheme for incompressible multiphase flow using adaptive Eulerian grid: 3D validation

A three‐dimensional finite element method for incompressible multiphase flows with capillary interfaces is developed based on a (formally) second‐order projection scheme. The discretization is on a fixed (Eulerian) reference grid with an edge‐based local h‐refinement in the neighbourhood of the interfaces. The fluid phases are identified and advected using the level‐set function. The reference grid is then temporarily reconnected around the interface to maintain optimal interpolations accounting for the singularities of the primary variables. Using a time splitting procedure, the convection substep is integrated with an explicit scheme. The remaining generalized Stokes problem is solved by means of a pressure‐stabilized projection. This method is simple and efficient, as demonstrated by a wide range of difficult free‐surface validation problems, considered in the paper. Copyright © 2005 John Wiley & Sons, Ltd.     

An adaptive numerical scheme for solving incompressible 2‐phase and free‐surface flows

In this paper, we present a numerical scheme for solving 2‐phase or free‐surface flows. Here, the interface/free surface is modeled using the level‐set formulation, and the underlying mesh is adapted at each iteration of the flow solver. This adaptation allows us to obtain a precise approximation for the interface/free‐surface location. In addition, it enables us to solve the time‐discretized fluid equation only in the fluid domain in the case of free‐surface problems. Fluids here are considered incompressible. Therefore, their motion is described by the incompressible Navier‐Stokes equation, which is temporally discretized using the method of characteristics and is solved at each time iteration by a first‐order Lagrange‐Galerkin method. The level‐set function representing the interface/free surface satisfies an advection equation that is also solved using the method of characteristics. The algorithm is completed by some intermediate steps like the construction of a convenient initial level‐set function (redistancing) as well as the construction of a convenient flow for the level‐set advection equation. Numerical results are presented for both bifluid and free‐surface problems.     

A curvature‐free multiphase flow solver via surface stress‐based formulation

In this note, we show the link between the classical continuous surface stress and continuous surface force approaches together with special finite element method techniques toward a fully implicit level set method. Based on a modified surface stress formulation, neither normals nor curvature has to be explicitly calculated. The method is space‐dimension independent. Prototypical numerical tests of benchmarking character for a rising 2D bubble are provided for validating the accuracy of this new approach. We show additionally that the explicit redistancing can be avoided using a nonlinear PDE so that a fully implicit and even monolithic formulation of the corresponding multiphase problem gets feasible.     

Momentum‐based approximation of incompressible multiphase fluid flows

We introduce a time stepping technique using the momentum as dependent variable to solve incompressible multiphase problems. The main advantage of this approach is that the mass matrix is time‐independent making this technique suitable for spectral methods. A level set method is applied to reconstruct the fluid properties such as density. We also introduce a stabilization method using an entropy‐viscosity technique and a compression technique to limit the flattening of the level set function. We extend our algorithm to immiscible conducting fluids by coupling the incompressible Navier‐Stokes and the Maxwell equations. We validate the proposed algorithm against analytical and manufactured solutions. Results on test cases such as Newton's bucket problem and a variation thereof are provided. Surface tension effects are tested on benchmark problems involving bubbles. A numerical simulation of a phenomenon related to the industrial production of aluminium is presented at the end of the paper.     

A hybrid phase field multiple relaxation time lattice Boltzmann method for the incompressible multiphase flow with large density contrast

A hybrid phase field multiple relaxation time lattice Boltzmann method (LBM) is presented in this paper for simulation of multiphase flows with large density contrast. In the present method, the flow field is solved by a lattice Boltzmann equation. Concurrently, the interface of two fluids is captured by solving the macroscopic Cahn‐Hilliard equation using the upwind scheme. To be specific, for simulation of the flow field, an lattice Boltzmann equation (LBE) model developed in Shao et al. (Physical Review E, 89 (2014), 033309) for consideration of density contrast in the momentum equation is used. Moreover, in the present work, the multiple relaxation time collision operator is applied to this LBE to enable simulation of problems with large viscosity contrast or high Reynolds number. For the interface capturing, instead of solving another set of LBE as in many phase field LBMs, the macroscopic Cahn‐Hilliard equation is directly solved by using a weighted essentially non‐oscillatory scheme. In this way, the present hybrid phase field LBM shares full advantages of the phase field LBM while enhancing numerical stability. The ability of the present method to simulate multiphase flow problems with large density contrast is demonstrated by several numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.     

The improvement of numerical modeling in the solution of incompressible viscous flow problems using finite element method based on spherical Hankel shape functions

In this paper, the finite element method with new spherical Hankel shape functions is developed for simulating 2‐dimensional incompressible viscous fluid problems. In order to approximate the hydrodynamic variables, the finite element method based on new shape functions is reformulated. The governing equations are the Navier‐Stokes equations solved by the finite element method with the classic Lagrange and spherical Hankel shape functions. The new shape functions are derived using the first and second kinds of Bessel functions. In addition, these functions have properties such as piecewise continuity. For the enrichment of Hankel radial basis functions, polynomial terms are added to the functional expansion that only employs spherical Hankel radial basis functions in the approximation. In addition, the participation of spherical Bessel function fields has enhanced the robustness and efficiency of the interpolation. To demonstrate the efficiency and accuracy of these shape functions, 4 benchmark tests in fluid mechanics are considered. Then, the present model results are compared with the classic finite element results and available analytical and numerical solutions. The results show that the proposed method, even with less number of elements, is more accurate than the classic finite element method.     

A variable relaxation scheme for multiphase,multicomponent flow

We propose a variable relaxation scheme for the simulation of 1D, two-phase, multicomponent flow in porous media. For these strongly nonlinear systems, traditional high order upwind schemes are impractical: Riemann solutions are not directly available when the phase behavior is complex, and the systems are weakly hyperbolic at isolated points. Relaxation schemes avoid the dependency on the eigenstructure and nonlinear Riemann solvers by approximating the original system with a strongly hyperbolic linear system. We exploit the known information about the eigenvalues to construct first order and second order variable relaxation schemes with much reduced numerical diffusion as compared to the standard relaxation formulations. The proposed second order variable relaxation scheme is competitive in accuracy and efficiency with a third order component-wise ENO reconstruction, and performs at least as well as second order component-wise TVD schemes.     

A volume-of-fluid method for incompressible free surface flows

This paper proposes a hybrid volume-of-fluid (VOF) level-set method for simulating incompressible two-phase flows. Motion of the free surface is represented by a VOF algorithm that uses high resolution differencing schemes to algebraically preserve both the sharpness of interface and the boundedness of volume fraction. The VOF method is specifically based on a simple order high resolution scheme lower than that of a comparable method, but still leading to a nearly equivalent order of accuracy. Retaining the mass conservation property, the hybrid algorithm couples the proposed VOF method with a level-set distancing algorithm in an implicit manner when the normal and the curvature of the interface need to be accurate for consideration of surface tension. For practical purposes, it is developed to be efficiently and easily extensible to three-dimensional applications with a minor implementation complexity. The accuracy and convergence properties of the method are verified through a wide range of tests: advection of rigid interfaces of different shapes, a three-dimensional air bubble's rising in viscous liquids, a two-dimensional dam-break, and a three-dimensional dam-break over an obstacle mounted on the bottom of a tank. The standard advection tests show that the volume advection algorithm is comparable in accuracy with geometric interface reconstruction algorithms of higher accuracy than other interface capturing-based methods found in the literature. The numerical results for the remainder of tests show a good agreement with other numerical solutions or available experimental data. Copyright © 2009 John Wiley & Sons, Ltd.     

A stabilized incremental projection scheme for the incompressible Navier–Stokes equations

It is well known that any spatial discretization of the saddle‐point Stokes problem should satisfy the Ladyzhenskaya–Brezzi–Babuska (LBB) stability condition in order to prevent the appearance of spurious pressure modes. Particularly, if an equal‐order approximation is applied, the Schur complement (or, as called some times, the Uzawa matrix) of the pressure system has a non‐trivial null space that gives rise to such modes. An idea in the past was that all the schemes that solve a Poisson equation for the pressure rather than the Uzawa pressure equation (splitting/projection methods) should overcome this difficulty; this idea was wrong. There is numerical evidence that at least the so‐called incremental projection scheme still suffers from spurious pressure oscillations if an equal‐order approximation is applied. The present paper tries to distinguish which projection requires LBB‐compliant approximation and which does not. Moreover, a stabilized version of the incremental projection scheme is derived. Proper bounds for the stabilization parameter are also given. The numerical results show that the stabilized scheme does indeed achieve second‐order accuracy and does not produce spurious (node to node) pressure oscillations. Copyright © 2001 John Wiley & Sons, Ltd.     

Anisotropic adaptive stabilized finite element solver for RANS models

Aerodynamic characteristics of various geometries are predicted using a finite element formulation coupled with several numerical techniques to ensure stability and accuracy of the method. First, an edge‐based error estimator and anisotropic mesh adaptation are used to detect automatically all flow features under the constraint of a fixed number of elements, thus controlling the computational cost. A variational multiscale‐stabilized finite element method is used to solve the incompressible Navier‐Stokes equations. Finally, the Spalart‐Allmaras turbulence model is solved using the streamline upwind Petrov‐Galerkin method. This paper is meant to show that the combination of anisotropic unsteady mesh adaptation with stabilized finite element methods provides an adequate framework for solving turbulent flows at high Reynolds numbers. The proposed method was validated on several test cases by confrontation with literature of both numerical and experimental results, in terms of accuracy on the prediction of the drag and lift coefficients as well as their evolution in time for unsteady cases.     

A least-squares finite element method for time-dependent incompressible flows with thermal convection

The time-dependent Navier–Stokes equations and the energy balance equation for an incompressible, constant property fluid in the Boussinesq approximation are solved by a least-squares finite element method based on a velocity–pressure–vorticity–temperature–heat-flux ( u –P–ω–T– q ) formulation discretized by backward finite differencing in time. The discretization scheme leads to the minimization of the residual in the l²-norm for each time step. Isoparametric bilinear quadrilateral elements and reduced integration are employed. Three examples, thermally driven cavity flow at Rayleigh numbers up to 10⁶, lid-driven cavity flow at Reynolds numbers up to 10⁴ and flow over a square obstacle at Reynolds number 200, are presented to validate the method.     

Error estimate and adaptive refinement for incompressible Navier‐Stokes equations using the discrete least squares meshless method

In this paper, an adaptive refinement strategy based on a node‐moving technique is proposed and used for the efficient solution of the steady‐state incompressible Navier–Stokes equations. The value of a least squares functional of the residual of the governing differential equation and its boundary conditions at nodal points is regarded as a measure of error and used to predict the areas of poor solutions. A node‐moving technique is then used to move the nodal points to the zones of higher numerical errors. The problem is then resolved on the refined distribution of nodes for higher accuracy. A spring analogy is used for the node‐moving methodology in which nodal points are connected to their neighbors by virtual springs. The stiffness of each spring is assumed to be proportional to the errors of its two end points and its initial length. The new positions of the nodal points are found such that the spring system attains its equilibrium state. Some numerical examples are used to illustrate the ability of the proposed scheme for the adaptive solution of the steady‐state incompressible Navier–Stokes equations. The results demonstrate a considerable improvement of the results with a reasonable computational effort by using the proposed adaptive strategy. Copyright © 2011 John Wiley & Sons, Ltd.     

New stabilized finite element method for time‐dependent incompressible flow problems

A new stabilized finite element method is considered for the time‐dependent Stokes problem, based on the lowest‐order P₁?P₀ and Q₁?P₀ elements that do not satisfy the discrete inf–sup condition. The new stabilized method is characterized by the features that it does not require approximation of the pressure derivatives, specification of mesh‐dependent parameters and edge‐based data structures, always leads to symmetric linear systems and hence can be applied to existing codes with a little additional effort. The stability of the method is derived under some regularity assumptions. Error estimates for the approximate velocity and pressure are obtained by applying the technique of the Galerkin finite element method. Some numerical results are also given, which show that the new stabilized method is highly efficient for the time‐dependent Stokes problem. Copyright © 2009 John Wiley & Sons, Ltd.     

An improved projection scheme applied to pseudospectral methods for the incompressible Navier–Stokes equations

An improved projection scheme is proposed and applied to pseudospectral collocation-Chebyshev approximation for the incompressible Navier–Stokes equations. It consists of introducing a correct predictor for the pressure, one which is consistent with a divergence-free velocity field at each time step. The main objective is to allow a time variation of the pressure gradient at boundaries. From different test problems, it is shown that this method, associated with a multistep second-order time scheme, provides a time accuracy of the same order as the temporal scheme used for the pressure, and also improves the prediction of the velocity slip. Moreover, it does not exhibit any numerical boundary layer mentioned as a drawback of fractional steps algorithm, and does not require the use of staggered grids for the velocity and the pressure. Its effectiveness is validated by comparison with a previous time-splitting algorithm proposed by Goda (K. Goda, J. Comput. Phys., 30 , 76–95 (1979)) and implemented by Gresho (P. Gresho, Int. j. numer. methods fluids, 11 , 587–620 (1990)) to finite element approximations. Steady and unsteady solutions for the regularized driven cavity and the rotating cavity submitted to throughflow are also used to assess the efficiency of this algorithm. © 1998 John Wiley & Sons, Ltd.     

A coupled continuous and discontinuous finite element method for the incompressible flows

In this paper, we develop a coupled continuous Galerkin and discontinuous Galerkin finite element method based on a split scheme to solve the incompressible Navier–Stokes equations. In order to use the equal order interpolation functions for velocity and pressure, we decouple the original Navier–Stokes equations and obtain three distinct equations through the split method, which are nonlinear hyperbolic, elliptic, and Helmholtz equations, respectively. The hybrid method combines the merits of discontinuous Galerkin (DG) and finite element method (FEM). Therefore, DG is concerned to accomplish the spatial discretization of the nonlinear hyperbolic equation to avoid using the stabilization approaches that appeared in FEM. Moreover, FEM is utilized to deal with the Poisson and Helmholtz equations to reduce the computational cost compared with DG. As for the temporal discretization, a second‐order stiffly stable approach is employed. Several typical benchmarks, namely, the Poiseuille flow, the backward‐facing step flow, and the flow around the cylinder with a wide range of Reynolds numbers, are considered to demonstrate and validate the feasibility, accuracy, and efficiency of this coupled method. Copyright © 2016 John Wiley & Sons, Ltd.     

A stabilized finite element method for the incompressible Navier–Stokes equations using a hierarchical basis

Stabilized finite element methods have been shown to yield robust, accurate numerical solutions to both the compressible and incompressible Navier–Stokes equations for laminar and turbulent flows. The present work focuses on the application of higher‐order, hierarchical basis functions to the incompressible Navier–Stokes equations using a stabilized finite element method. It is shown on a variety of problems that the most cost‐effective simulations (in terms of CPU time, memory, and disk storage) can be obtained using higher‐order basis functions when compared with the traditional linear basis. In addition, algorithms will be presented for the efficient implementation of these methods within the traditional finite element data structures. Copyright © 2001 John Wiley & Sons, Ltd.     

A Hermite finite element method for incompressible fluid flow

We describe some Hermite stream function and velocity finite elements and a divergence‐free finite element method for the computation of incompressible flow. Divergence‐free velocity bases defined on (but not limited to) rectangles are presented, which produce pointwise divergence‐free flow fields (∇· u _h≡0). The discrete velocity satisfies a flow equation that does not involve pressure. The pressure can be recovered as a function of the velocity if needed. The method is formulated in primitive variables and applied to the stationary lid‐driven cavity and backward‐facing step test problems. Copyright © 2009 John Wiley & Sons, Ltd.     

A locally DIV‐free projection scheme for incompressible flows based on non‐conforming finite elements

We present a projection scheme whose end‐of‐step velocity is locally pointwise divergence free, using a continuous ?₁ approximation for the velocity in the momentum equation, a first‐order Crouzeix–Raviart approximation at the projection step, and a ?₀ approximation for the pressure in both steps. The analysis of the scheme is done only for grids that guarantee the existence of a divergence free conforming ?₁ interpolant for the velocity. Optimal estimates for the velocity error in L²‐ and H¹‐norms are deduced. The numerical results demonstrate that these estimates should also hold on grids on which the continuous ?₁ approximation for the velocity locks. Since the end‐of‐step velocity is locally solenoidal, the scheme is recommendable for problems requiring good mass conservation. Copyright © 2005 John Wiley & Sons, Ltd.     

A Fully Coupled Solver for Incompressible Navier–Stokes Equations using Operator Splitting

The steady incompressible Navier–Stokes equations are coupled by a Poisson equation for the pressure from which the continuity equation is subtracted. The equivalence to the original N–S problem is proved. Fictitious time is added and vectorial operator-splitting is employed leaving the system coupled at each fractional-time step which allows satisfaction of the boundary conditions without introducing artificial conditions for the pressure. Conservative second-order approximations for the convective terms are employed on a staggered grid. The splitting algorithm for the 3D case is verified through an analytic solution test. The stability of the method at high values of Reynolds number is illustrated by accurate numerical solutions for the flow in a lid-driven rectangular cavity with aspect ratio 1 and 2, as well as for the flow after a back-facing step.     

Time‐related element‐free Taylor–Galerkin method with non‐splitting decoupling process for incompressible steady flow

The time‐related element‐free Taylor–Galerkin method with non‐splitting decoupling process (EFTG‐NSD) is proposed for the simulation of steady flows. The goal of the present paper is twofold. One is to raise the efficiency of the time‐related methods for solving steady flow problems, and the other is to obtain a good stability. The EFTG‐NSD method, which uses the time‐related Navier–Stokes equations to describe steady flows, does not care about the intermediate process and obtains solution of steady flows through time marching. Different from the classical time‐related fractional step methods, the EFTG‐NSD method decouples the Navier–Stokes equations without any operator‐splitting and correction. Because the elimination of correction at each iteration step reduces the computation cost, the EFTG‐NSD method possesses higher computation efficiency. In addition, the EFTG‐NSD method has a good stability due to the use of the Taylor–Galerkin formula in time and space discretization. Furthermore, the method combining element‐free Galerkin method with Taylor–Galerkin method is an important supplement of the element‐free Galerkin method for solving flow problems. Copyright © 2011 John Wiley & Sons, Ltd.