Volume preserving immersed boundary methods for two‐phase fluid flows

In this article, we propose a simple area‐preserving correction scheme for two‐phase immiscible incompressible flows with an immersed boundary method (IBM). The IBM was originally developed to model blood flow in the heart and has been widely applied to biofluid dynamics problems with complex geometries and immersed elastic membranes. The main idea of the IBM is to use a regular Eulerian computational grid for the fluid mechanics along with a Lagrangian representation of the immersed boundary. Using the discrete Dirac delta function and the indicator function, we can include the surface tension force, variable viscosity and mass density, and gravitational force effects. The principal advantage of the IBM for two‐phase fluid flows is its inherent accuracy due in part to its ability to use a large number of interfacial marker points on the interface. However, because the interface between two fluids is moved in a discrete manner, this can result in a lack of volume conservation. The idea of an area preserving correction scheme is to correct the interface location normally to the interface so that the area remains constant. Various numerical experiments are presented to illustrate the efficiency and accuracy of the proposed conservative IBM for two‐phase fluid flows. Copyright © 2011 John Wiley & Sons, Ltd.     

A divergence‐free interpolation scheme for the immersed boundary method

The immersed boundary approach for the modeling of complex geometries in incompressible flows is examined critically from the perspective of satisfying boundary conditions and mass conservation. It is shown that the system of discretized equations for mass and momentum can be inconsistent, if the velocity is used in defining the force density to satisfy the boundary conditions. As a result, the velocity is generally not divergence free and the pressure at locations in the vicinity of the immersed boundary is not physical. However, the use of the pseudo‐velocities in defining the force density, as frequently done when the governing equations are solved using a fractional step or projection method, combined with the use of the specified velocity on the immersed boundary, is shown to result in a consistent set of equations which allows a divergence‐free velocity but, depending on the time step, is shown to have the undesirable effects of inaccurately satisfying the boundary conditions and allowing a significant permeability of the immersed boundary. If the time step is reduced sufficiently, the boundary conditions on the immersed boundary can be satisfied. However, this entails an unacceptable increase in computational expense. Two new methods that satisfy the boundary conditions and allow a divergence‐free velocity while avoiding the increased computational expense are presented and shown to be second‐order accurate in space. The first new method is based on local time step reduction. This method is suitable for problems where the immersed boundary does not move. For these problems, the first new method is shown to be closely related to the second new method. The second new method uses an optimization scheme to minimize the deviation from the interpolation stencil used to represent the immersed boundary while ensuring a divergence‐free velocity. This method performs well for all problems, including those where the immersed boundary moves relative to the grid. Additional results include showing that the force density that is added to satisfy the boundary conditions at the immersed boundary is unbounded as the time step is reduced and that the pressure in the vicinity of the immersed boundary is unphysical, being strongly a function of the time step. A method of computing the total force on an immersed boundary which takes into account the specifics of the numerical solver used in the iterative process and correctly computes the total force irrespective of the residual level is also presented. Copyright © 2007 John Wiley & Sons, Ltd.     

Three‐dimensional immersed boundary conditions for moving solids in the lattice‐Boltzmann method

This paper establishes the range of validity for a previously published three‐dimensional moving solid boundary condition for the lattice‐Boltzmann method. This method was reasonably formulated from a mass and momentum balance perspective, but was only verified for a small range of (primarily two‐dimensional) problems. One of the advantages of this boundary condition is that it offers resolution at the sub‐grid scale, allowing for accurate and stable calculation of the force and torque for solids which are moving through a lattice, even for small solid sizes relative to the computational grid size. We verify the boundary condition for creeping flows by comparison to analytical solutions that include both the force and the torque on fixed and moving spheres, and then follow this with comparisons to experimental and empirical results for both fixed as well moving spheres in inertial flows. Finally, we compare simulation results to numerical results of other investigators for the settling of an offset sphere and the drafting–kissing–tumbling of two sedimenting spheres. We found that an accurate calculation of the collision‐operator weighting used to obtain sub‐grid‐scale resolution was necessary in order to prevent spikes in the velocities, forces, and moments when solid objects cross‐computational cells. The wide range of comparisons collected and presented in this paper can be used to establish the validity of other numerical models, in addition to the one examined here. Published in 2007 by John Wiley & Sons, Ltd.     

Fitted finite element discretization of two‐phase Stokes flow

We propose a novel fitted finite element method for two‐phase Stokes flow problems that uses piecewise linear finite elements to approximate the moving interface. The method can be shown to be unconditionally stable. Moreover, spherical stationary solutions are captured exactly by the numerical approximation. In addition, the meshes describing the discrete interface in general do not deteriorate in time, which means that in numerical simulations, a smoothing or a remeshing of the interface mesh is not necessary. We present several numerical experiments for our numerical method, which demonstrate the accuracy and robustness of the proposed algorithm. Copyright © 2016 John Wiley & Sons, Ltd.     

Simulation of two‐phase flow–body interaction problems using direct forcing/fictitious domain–level set method

In the present paper, a direct forcing/fictitious domain (DF/FD)–level set method is proposed to simulate the twophase flow–body interaction. The DF/FD does not sacrifice accuracy and robustness by employing a discrete δ (Dirac delta) function to transfer quantities between the Eulerian nodes and Lagrangian points explicitly as the immersed boundary method. The advantages of this approach are the simple concept, the easy implementation and the utilization of original governing equation without modification. The main idea is to combine DF/FD method with the level set method in the Cartesian coordinates. We present the results of a number of test cases to illustrate the effectiveness of the proposed method for single‐phase flow–body interaction problem and the two‐phase flows with a stationary body. Eventually, the simulations of various water entry problems have been conducted to validate the capability and the accuracy of the present method on solving the twophase flow–body interaction. Consequently, the present results are found to be in good agreement with those of previous studies. Copyright © 2013 John Wiley & Sons, Ltd.     

Projection and partitioned solution for two‐phase flow problems

Multiphase flow through porous media is a highly nonlinear process that can be solved numerically with the aid of finite elements (FE) in space and finite differences (FD) in time. For an accurate solution much refined FE grids are generally required with the major computational effort consisting of the resolution to the nonlinearity frequently obtained with the classical Picard linearization approach. The efficiency of the repeated solution to the linear systems within each individual time step represents the key to improve the performance of a multiphase flow simulator. The present paper discusses the performance of the projection solvers (GMRES with restart, TFQMR, and BiCGSTAB) for two global schemes based on a different nodal ordering of the unknowns (ORD1 and ORD2) and a scheme (SPLIT) based on the straightforward inversion of the lumped mass matrix which allows for the preliminary elimination and substitution of the unknown saturations. It is shown that SPLIT is between two and three time faster than ORD1 and ORD2, irrespective of the solver used. Copyright © 2005 John Wiley & Sons, Ltd.     

A relaxation multiresolution scheme for accelerating realistic two‐phase flows calculations in pipelines

We wish to demonstrate that it is judicious to combine various existing computational techniques that appeared for academic cases in seemingly unrelated areas, namely, semi‐implicit relaxation schemes for hyperbolic systems and adaptive multiresolution algorithms, in order to achieve fast and accurate simulations of realistic two‐phase flows problems in oil transportation. By ‘realistic’ we mean problems that are modelled by partial differential equation (PDE) systems closed by sophisticated thermodynamics and hydrodynamics laws, set out over a terrain‐induced geometry and associated with time‐dependent boundary conditions. Although the combination of these techniques is not a straightforward matter, it is made possible via a careful examination of the objectives of the simulation problem and suitable adaptations of which we shall give the details. Significant benchmarks demonstrate the efficiency of the proposed method. Copyright © 2006 John Wiley & Sons, Ltd.     

Interface‐fitted moving mesh method for axisymmetric two‐phase flow in microchannels

A boundary‐fitted moving mesh scheme is presented for the simulation of two‐phase flow in two‐dimensional and axisymmetric geometries. The incompressible Navier‐Stokes equations are solved using the finite element method, and the mini element is used to satisfy the inf‐sup condition. The interface between the phases is represented explicitly by an interface adapted mesh, thus allowing a sharp transition of the fluid properties. Surface tension is modelled as a volume force and is discretized in a consistent manner, thus allowing to obtain exact equilibrium (up to rounding errors) with the pressure gradient. This is demonstrated for a spherical droplet moving in a constant flow field. The curvature of the interface, required for the surface tension term, is efficiently computed with simple but very accurate geometric formulas. An adaptive moving mesh technique, where smoothing mesh velocities and remeshing are used to preserve the mesh quality, is developed and presented. Mesh refinement strategies, allowing tailoring of the refinement of the computational mesh, are also discussed. Accuracy and robustness of the present method are demonstrated on several validation test cases. The method is developed with the prospect of being applied to microfluidic flows and the simulation of microchannel evaporators used for electronics cooling. Therefore, the simulation results for the flow of a bubble in a microchannel are presented and compared to experimental data.     

A discrete forcing method dedicated to moving bodies in two‐phase flow

The numerical simulation of interaction between structures and two‐phase flows is a major concern for many industrial applications. To address this challenge, the motion of structures has to be tracked accurately. In this work, a discrete forcing method based on a porous medium approach is proposed to follow a nondeformable rigid body with an imposed velocity by using a finite‐volume Navier‐Stokes solver code dedicated to multiphase flows and based on a two‐fluid approach. To deal with the action reaction principle at the solid wall interfaces in a conservative way, a porosity is introduced allowing to locate the solid and insuring no diffusion of the fluid‐structure interface. The volumetric fraction equilibrium is adapted to this novelty. Mass and momentum balance equations are formulated on a fixed Cartesian grid. Interface tracking is addressed in detail going from the definition of the porosity to the changes in the discretization of the momentum balance equation. This so‐called time‐ and space‐dependent porosity method is then validated by using analytical and elementary test cases.     

Simplex space‐time meshes in two‐phase flow simulations

In this paper, we present the numerical solution of 2‐phase flow problems of engineering significance with a space‐time finite element method that allows for local temporal refinement. Arbitrary temporal refinement is applied to preselected regions of the mesh and is governed by a quantity that is part of the solution process, namely, the interface position in 2‐phase flow. Because of local effects such as surface tension, jumps in material properties, etc, the interface can in general be considered a region that requires high flexibility and high resolution, both in space and in time. The new method, which leads to tetrahedral (for 2D problems) and pentatope (for 3D problems) meshes, offers an efficient yet accurate approach to the underlying 2‐phase flow problems.     

Spurious currents in finite element based level set methods for two‐phase flow

A study of spurious currents in continuous finite element based simulations of the incompressible Navier–Stokes equations for two‐phase flows is presented on the basis of computations on a circular drop in equilibrium. The conservative and the standard level set methods are used. It is shown that a sharp surface tension force, expressed as a line integral along the interface, can give rise to large spurious currents and oscillations in the pressure that do not decrease with mesh refinement. If instead a regularized surface tension representation is used, exact force balance at the interface is possible, both for a fully coupled discretization approach and for a fractional step projection method. However, the numerical curvature calculation introduces errors that cause spurious currents. Different ways to extend the curvature from the interface to the whole domain are discussed and investigated. The impact of using different finite element spaces and stabilization methods is also considered. Copyright © 2011 John Wiley & Sons, Ltd.     

A surface remeshing technique for a Lagrangian description of 3D two‐fluid flow problems

Classical Lagrangian schemes applied to update the front position between two immiscible incompressible fluids have been long recognized to provide a sharp representation of the interface. However, the main drawback of these approaches is the progressive distortion in the distribution of the markers used to identify the material front. To avoid this problem, a 3D interface remeshing algorithm is proposed in this work. In addition, the remeshed front is enforced to preserve the global volume. These aspects are incorporated in an existing fluid dynamics formulation for the analysis of two‐fluid flows problems. The resulting formulation, called as the 3D‐moving Lagrangian interface remeshing technique, is applied in the numerical analysis of two‐fluid flow problems. Copyright © 2009 John Wiley & Sons, Ltd.     

A lattice Boltzmann fictitious domain method for modeling red blood cell deformation and multiple‐cell hydrodynamic interactions in flow

To model red blood cell (RBC) deformation and multiple‐cell interactions in flow, the recently developed technique derived from the lattice Boltzmann method and the distributed Lagrange multiplier/fictitious domain method is extended to employ the mesoscopic network model for simulations of RBCs in flow. The flow is simulated by the lattice Boltzmann method with an external force, while the network model is used for modeling RBC deformation. The fluid–RBC interactions are enforced by the Lagrange multiplier. To validate parameters of the RBC network model, stretching tests on both coarse and fine meshes are performed and compared with the corresponding experimental data. Furthermore, RBC deformation in pipe and shear flows is simulated, revealing the capacity of the current method for modeling RBC deformation in various flows. Moreover, hydrodynamic interactions between two RBCs are studied in pipe flow. Numerical results illustrate that the leading cell always has a larger flow velocity and deformation, while the following cells move slower and deform less.Copyright © 2013 John Wiley & Sons, Ltd.     

On the efficiency and quality of numerical solutions in CFD problems using the interface strip preconditioner for domain decomposition methods

In this paper, some pathologies found for simple tests solved by means of preconditioned full iterative schemes are presented. According to these results (Sections 4 and 5), the accuracy deterioration observed should be considered as a warning for the final application given to these solutions. Even though it is well known that full iterative solvers are not the best selection for comparison, they were chosen because they are widely used by the computational fluid dynamic (CFD) community for a diversity of complex fluid dynamics applications. FEM simulated solutions are compared with analytical solutions or measured data for problems that have been considered as ‘benchmarks’ in the CFD literature. For this purpose, the study of the solution obtained via parallelized iterative methods that have been extensively used (e.g. conjugate gradients (CG), GMRes global iteration and its variants, ‘overlapping’ and ‘non‐overlapping’ additive Schwarz domain decomposition schemes) in CFD computations and those obtained with the new interface strip preconditioner (J. Comput. Meth. Sci. Engng 2003; Int. J. Numer. Meth. Engng 2005; 62 (13):1873–1894) for the Schur complement method is carried out. The idea is to present the new solver as an alternative to obtain more accurate and faster solutions in the context of monolithic and non‐monolithic schemes applied to a internal/external viscous compressible/incompressible flows around bodies of complex shapes. Therefore, the target of this work is to show how the reliability of CFD codes is affected by the solver selection and why domain decomposition methods should be viewed not only as a more efficient strategy, but also to guarantee the solution quality. Copyright © 2006 John Wiley & Sons, Ltd.     

Monotone nonlinear finite‐volume method for nonisothermal two‐phase two‐component flow in porous media

This article presents a new nonlinear finite‐volume scheme for the nonisothermal two‐phase two‐component flow equations in porous media. The face fluxes are approximated by a nonlinear two‐point flux approximation, where transmissibilities nonlinearly depend on primary variables. Thereby, we mainly follow the ideas proposed by Le Potier combined with a harmonic averaging point interpolation strategy for the approximation of arbitrary heterogeneous permeability fields on polygonal grids. The behavior of this interpolation strategy is analyzed, and its limitation for highly anisotropic permeability tensors is demonstrated. Moreover, the condition numbers of occurring matrices are compared with linear finite‐volume schemes. Additionally, the convergence behavior of iterative solvers is investigated. Finally, it is shown that the nonlinear scheme is more efficient than its linear counterpart. Copyright © 2016 John Wiley & Sons, Ltd.     

Resolution sensitivity of momentum‐exchange and immersed boundary methods for solid–fluid interaction in the lattice Boltzmann method

In the lattice Boltzmann method (LBM), the mechanism of fluid–solid interaction can be effectively captured by appropriately enforcing the no‐slip conditions in shear direction, and bounce‐back of the non‐equilibrium distribution portion in the normal direction at fluid–solid interfaces. Among various solid–fluid interaction schemes being proposed for LBM in recent decades, two simple fluid–solid interaction methods—the momentum exchange algorithm (MEA) and the immersed boundary scheme (IBS)—were developed based on the above concept. In this paper, MEA and IBS are implemented in a D2Q9 LBGK system and applied to measure the wall correction factors of drag force upon a stationary circular particle midway in the Poiseuille channel flow at very low Reynolds number and drag coefficients at low to moderate Reynolds numbers. MEA and IBS are also employed to compare the fluid‐induced torque over the cylinder in the Taylor–Couette flow, and the steady velocity of a particle settling under the influence of gravity inside a tube. The above experiments show that IBS seems to be more accurate and less demanding on lattice resolution. Copyright © 2010 John Wiley & Sons, Ltd.     

An unfitted interior penalty discontinuous Galerkin method for incompressible Navier–Stokes two‐phase flow

A discontinuous Galerkin method for the solution of the immiscible and incompressible two‐phase flow problem based on the nonsymmetric interior penalty method is presented. Therefore, the incompressible Navier–Stokes equation is solved for a domain decomposed into two subdomains with different values of viscosity and density as well as a singular surface tension force. On the basis of a piecewise linear approximation of the interface, meshes for both phases are cut out of a structured mesh. The discontinuous finite elements are defined on the resulting Cartesian cut‐cell mesh and may therefore approximate the discontinuities of the pressure and the velocity derivatives across the interface with high accuracy. As the mesh resolves the interface, regularization of the density and viscosity jumps across the interface is not required. This preserves the local conservation property of the velocity field even in the vicinity of the interface and constitutes a significant advantage compared with standard methods that require regularization of these discontinuities and cannot represent the jumps and kinks in pressure and velocity. A powerful subtessellation algorithm is incorporated to allow the usage of standard time integrators (such as Crank–Nicholson) on the time‐dependent mesh. The presented discretization is applicable to both the two‐dimensional and three‐dimensional cases. The performance of our approach is demonstrated by application to a two‐dimensional benchmark problem, allowing for a thorough comparison with other numerical methods. Copyright © 2012 John Wiley & Sons, Ltd.     

A computational study of gas flow in a De‐Laval micronozzle at different throat diameters

A numerical study has been carried out to investigate the gas flows in a micronozzle using a continuum model under both slip and no‐slip boundary conditions. The governing equations were solved with a finite volume method. The numerical model was validated with available experimental data. Numerical results of exit thrust showed good agreement with experimental data except at very low Reynolds numbers. For parametric studies on the effect of geometric scaling, the nozzle throat diameter was varied from 10 to 0.1 mm, whereas throat Reynolds number was varied from 5 to 2000. A correlation has also been developed to calculate the specific impulse at specified throat diameter and Reynolds number. The effect of different gases on the specific impulse of the nozzle, such as helium, nitrogen, argon and carbon dioxide, was also examined. Copyright © 2008 John Wiley & Sons, Ltd.     

Enablers for high‐order level set methods in fluid mechanics

In the context of numerical simulations of multiphysics flows, accurate tracking of an interface and consistent computation of its geometric properties are crucial. In this paper, we investigate a level set technique that satisfies these requirements and ensures local third‐order accuracy on the level set function (near the interface) and first‐order accuracy on the curvature, even for long‐time computations. The method is developed in a finite differences framework on Cartesian grids. As in usual level set strategies, reinitialization steps are involved. Several reinitialization algorithms are reviewed and mixed to design an accurate and fast reinitialization procedure. When coupled with a time evolution of the interface, the reinitialization procedure is performed only when there are too large deformations of the isocontours. This strategy limits the number of reinitialization steps and shows a good balance between accuracy and computational cost. Numerical results compare well with usual level set strategies and confirm the necessity of the reinitialization procedure, together with a limited number of reinitialization steps. Copyright © 2015 John Wiley & Sons, Ltd.     

Development of temporal and spatial high‐order schemes for two‐fluid seven‐equation two‐pressure model and its applications in two‐phase flow benchmark problems

Current existing main nuclear thermal‐hydraulics (T‐H) system analysis codes, such as RALAP5, TRACE, and CATHARE, play a crucial role in the nuclear engineering field for the design and safety analysis of nuclear reactor systems. However, two‐fluid model used in these T‐H system analysis codes is ill posed, easily leading to numerical oscillations, and the classical first‐order methods for temporal and special discretization are widely employed for numerical simulations, yielding excessive numerical diffusion. Two‐fluid seven‐equation two‐pressure model is of particular interest due to the inherent well‐posed advantage. Moreover, high‐order accuracy schemes have also attracted great attention to overcome the challenge of serious numerical diffusion induced by low‐order time and space schemes for accurately simulating nuclear T‐H problems. In this paper, the semi‐implicit solution algorithm with high‐order accuracy in space and time is developed for this well‐posed two‐fluid model and the robustness and accuracy are verified and assessed against several important two‐phase flow benchmark tests in the nuclear engineering T‐H field, which include two linear advection problems, the oscillation problem of the liquid column, the Ransom water faucet problem, the reversed water faucet problem, and the two‐phase shock tube problem. The following conclusions are achieved. (1) The proposed semi‐implicit solution algorithm is robust in solving two‐phase flows, even for fast transients and discontinuous solutions. (2) High‐order schemes in both time and space could prevent excessive numerical diffusion effectively and the numerical simulation results are more accurate than those of first‐order time and space schemes, which demonstrates the advantage of using high‐order schemes.