Regularized Delta Function Research Articles

Benchmarking the immersed boundary method for viscoelastic flows

We present and analyze a series of benchmark tests regarding the application of the immersed boundary (IB) method to viscoelastic flows through and around non-trivial, stationary geometries. The IB method is widely used to simulate biological fluid dynamics and other modeling scenarios in which a structure is immersed in a fluid. Although the IB method has been most commonly used to model systems involving viscous incompressible fluids, it also can be applied to visoelastic fluids and has enabled the study of a wide variety of dynamical problems including the settling of vesicles and the swimming of elastic filaments in fluids modeled by the Oldroyd-B constitutive equation. In the viscoelastic context, however, relatively little work has explored the accuracy or convergence properties of this numerical scheme. Herein, we present benchmarking results for an IB solver applied to viscoelastic flows in and around non-trivial geometries using either the idealized Oldroyd-B constitutive model or the more physically realistic, polymer-entanglement-based Rolie-Poly constitutive equations. We use two-dimensional numerical test cases along with results from rheology experiments to benchmark the IB method and compare it to more complex finite element and finite volume viscoelastic flow solvers. Additionally, we analyze different choices of regularized delta function and relative Lagrangian grid spacings which allow us to identify and recommend the key choices of these numerical parameters depending on the present flow regime.

A general integration kernel formulation for immersed boundary method

This study proposes a new robust and accurate immersed boundary method for the immersion of solid bodies within a fluid with a Cartesian grid. The present method introduces the signed distance fields to recognize the immersed geometry contours, eliminating the need of Lagrangian points. To fully maximize the advantages offered by signed distance fields, a general integration kernel formulation is introduced into the direct forcing method to replace the conventional regularized delta function. With the combination of signed distance fields and kernel function, an interpolation along the radial direction instead of three-dimensional directions is feasible, which further reduces the extra calculation cost involved by immersed boundary method. The numerical results at low Reynolds numbers are compared to experimental and previous numerical results, which shows the efficiency and accuracy of this new method. Upon thorough validation, the proposed method in this paper demonstrates excellent performance across various scenarios, including static and moving cases as well as two- and three-dimensional configurations. And our method greatly reduces the cost of pretreatment of immersed geometry contours and apparently improves the convenience of the method.

Immersed-boundary/soft-sphere method for particle–particle-fluid interaction in a viscous flow: An OpenFOAM solver

We developed a stable OpenFOAM solver for Immersed Boundary Method based on direct forcing and regularized delta function. The soft-sphere model and a lubrication model were implemented to consider particle–particle collision in a viscous flow. We proposed a fluid–structure interaction (FSI) coupling method to accurately calculate the fluid forcing term and particle velocity. Our solver was validated for fixed and moving bodies, including rotation. The accuracy of various FSI schemes was evaluated in predicting the solid and fluid flow behavior in a viscous flow. It was demonstrated that neglecting or simplifying the fluid momentum change affects the accuracy of the solid velocity and fluid flow dynamic; for higher solid-to-fluid density ratios, a larger deviation was predicted. Furthermore, the FSI schemes highly influenced the behavior of the formed vortices.The solver was validated to predict the effective restitution coefficient of particles in a viscous flow as a function of the Stokes number. We also thoroughly analyzed the dynamic flow behavior of colliding particles through the pressure and velocity field and fluid force. This analysis helped us accurately determine the rebound velocity of particles in case of high Stokes numbers when the effect of viscous force is significant.

Kinetic modeling of immersed boundary layer for accurate evaluation of local surface stresses and hydrodynamic forces with diffuse interface immersed boundary method

The motivation of this paper is to examine the evaluation of local surface stresses and hydrodynamic forces acting on a stationary or moving body using a diffuse interface immersed boundary method (IBM). This task is not trivial for the diffuse IBM because it uses a smoothed regularized delta function in the transfer steps between Lagrangian and Eulerian locations. In our earlier work [D. Xu et al., Phys. Rev. E 105, 035306 (2022)], a particle distribution function (PDF) discontinuity-based kinetic immersed boundary method (KIBM) was proposed based on the Boltzmann equation. This paper is a continuation of our work on the improvement of the KIBM in the framework of the diffuse interface IBM. In the present study, the concept of the immersed boundary layer (IBL) is brought forward, and the dynamic effects of particle advection and collision in the IBL are coupled and evaluated within a numerical time step scale in a kinetic manner. Consequently, the PDFs on both sides of the IBL are reconstructed, and the general immersed boundary force density can be obtained accurately and efficiently. Meantime, the local surface stress distribution acting on the body wall from the actual fluid can be conveniently and accurately calculated by the moment of the PDFs. Finally, some commonly used problems involving incompressible fluid flows in the continuum flow regime with stationary and moving boundaries are simulated by the present KIBM, and the results show that the present KIBM can significantly accelerate the rate of convergence and has a good agreement with other numerical and experimental results.

Particle distribution function discontinuity-based kinetic immersed boundary method for Boltzmann equation and its applications to incompressible viscous flows.

In this paper, a general immersed boundary force density is introduced for the Boltzmann equation and finally expressed as the desired particle distribution function discontinuity across the immersed boundary. Because of its independence of any specific boundary conditions and any specific solvers for the Boltzmann equation, it actually establishes a unified framework to incorporate various types of boundary conditions and several different kinds of solvers for the Boltzmann equation. Hence, a particle distribution function discontinuity-based kinetic immersed boundary method (KIBM) for the Boltzmann equation is proposed based on this general immersed boundary force density. Subsequently, this paper primarily focuses on the isothermal incompressible fluid-solid flows, and uses the discrete unified gas kinetic scheme to solve the Boltzmann Bhatnagar-Gross-Krook model equation. Meanwhile, the regularized delta function and the bounce-back rule combined with an iterative IBM correction procedure are employed in obtaining the general immersed boundary force density to enforce the no-penetration and no-slip boundary conditions on the solid wall. Finally, some numerical experiments for typical incompressible fluid-solid flows show that the present KIBM could provide good agreement with other numerical and experimental results.

Passive concentration dynamics incorporated into the library IB2d, a two-dimensional implementation of the immersed boundary method

In this paper, we present an open-source software library that can be used to numerically simulate the advection and diffusion of a chemical concentration or heat density in a viscous fluid where a moving, elastic boundary drives the fluid and acts as a source or sink. The fully-coupled fluid-structure interaction problem of an elastic boundary in a viscous fluid is solved using Peskin’s immersed boundary method. The addition or removal of the concentration or heat density from the boundary is solved using an immersed boundary-like approach in which the concentration is spread from the immersed boundary to the fluid using a regularized delta function. The concentration or density over time is then described by the advection-diffusion equation and numerically solved. This functionality has been added to our software library, IB2d, which provides an easy-to-use immersed boundary method in two dimensions with full implementations in MATLAB and Python. We provide four examples that illustrate the usefulness of the method. A simple rubber band that resists stretching and absorbs and releases a chemical concentration is simulated as a first example. Complete convergence results are presented for this benchmark case. Three more biological examples are presented: (1) an oscillating row of cylinders, representative of an idealized appendage used for filter-feeding or sniffing, (2) an oscillating plate in a background flow is considered to study the case of heat dissipation in a vibrating leaf, and (3) a simplified model of a pulsing soft coral where carbon dioxide is taken up and oxygen is released as a byproduct from the moving tentacles. This method is applicable to a broad range of problems in the life sciences, including chemical sensing by antennae, heat dissipation in plants and other structures, the advection-diffusion of morphogens during development, filter-feeding by marine organisms, and the release of waste products from organisms in flows.

On the Lagrangian-Eulerian coupling in the immersed finite element/difference method

The immersed boundary (IB) method is a non-body conforming approach to fluid-structure interaction (FSI) that uses an Eulerian description of the momentum, viscosity, and incompressibility of a coupled fluid-structure system and a Lagrangian description of the deformations, stresses, and resultant forces of the immersed structure. Integral transforms with Dirac delta function kernels couple the Eulerian and Lagrangian variables, and in practice, discretizations of these integral transforms use regularized delta function kernels. Many different kernel functions have been proposed, but prior numerical work investigating the impact of the choice of kernel function on the accuracy of the methodology has often been limited to simplified test cases or Stokes flow conditions that may not reflect the method's performance in applications, particularly at intermediate-to-high Reynolds numbers, or under different loading conditions. This work systematically studies the effect of the choice of regularized delta function in several fluid-structure interaction benchmark tests using the immersed finite element/difference (IFED) method, which is an extension of the IB method that uses a finite element structural discretization combined with a Cartesian grid finite difference method for the incompressible Navier-Stokes equations. Whereas the conventional IB method spreads forces from the nodes of the structural mesh and interpolates velocities to those nodes, the IFED formulation evaluates the regularized delta function on a collection of interaction points that can be chosen to be denser than the nodes of the Lagrangian mesh. This opens the possibility of using structural discretizations with wide node spacings that would produce gaps in the Eulerian force in nodally coupled schemes (e.g., if the node spacing is comparable to or broader than the support of the regularized delta function). Earlier work with this methodology suggested that such coarse structural meshes can yield improved accuracy for shear-dominated cases and, further, found that accuracy improves when the structural mesh spacing is increased. However, these results were limited to simple test cases that did not include substantial pressure loading on the structure. This study investigates the effect of varying the relative mesh widths of the Lagrangian and Eulerian discretizations in a broader range of tests. Our results indicate that kernels satisfying a commonly imposed even–odd condition require higher resolution to achieve similar accuracy as kernels that do not satisfy this condition. We also find that narrower kernels are more robust, in the sense that they yield results that are less sensitive to relative changes in the Eulerian and Lagrangian mesh spacings, and that structural meshes that are substantially coarser than the Cartesian grid can yield high accuracy for shear-dominated cases but not for cases with large normal forces. We verify our results in a large-scale FSI model of a bovine pericardial bioprosthetic heart valve in a pulse duplicator.

A near-exact analytic solution of progress variable and pdf for single-step Arrhenius chemistry

A new chemical reaction rate source term in the transport equation of a single progress variable is proposed and implications for premixed flame modelling are discussed. This surrogate source term approximates the Arrhenius one over a large range of activation energies and density ratios almost perfectly. An analytic, invertible flame profile and the corresponding 1D laminar flame pdf are derived. The laminar flame Eigenvalue is evaluated analytically for unity and non-unity Lewis number and compared to classical results. A method to evaluate the progress variable pdf from DNS data is introduced. It is based on regularized delta functions and a transformation of the DNS progress variable field with the inverse of the 1D laminar flame profile, which behaves like a signed distance function. The pdf factorizes into the flat flame pdf, a geometrical wrinkling factor and a correction factor accounting for changes in the inner flame structure. The pdfs of RANS-like filter volumes and their ingredients are evaluated from DNS datasets of statistically flat turbulent flames featuring different turbulence intensity levels. For all DNS cases the wrinkling and correction factors vary little with progress variable. The wrinkling factor rises with turbulence intensity. The pdf is similar to that of a 1D laminar flame pdf, scaled by the wrinkling factor. Opportunities for improvements to premixed turbulent flame modelling are discussed.

A DLM immersed boundary method based wave-structure interaction solver for high density ratio multiphase flows

In this paper we present a robust immersed boundary (IB) method for high density ratio multiphase flows that is capable of modeling complex wave-structure interaction (WSI) problems arising in marine and coastal engineering applications. The IB/WSI methodology is enabled by combining the distributed Lagrange multiplier (DLM) method of Sharma and Patankar (2005) [13] with a robust level set method based multiphase flow solver. The fluid solver integrates the conservative form of the variable-coefficient incompressible Navier-Stokes equations using a hybrid preconditioner and ensures consistent transport of mass and momentum at a discrete level. The consistent transport scheme preserves the numerical stability of the method in the presence of large density ratios found in problems involving air, water, and an immersed structure. The air-water interface is captured by the level set method on an Eulerian grid, whereas the free-surface piercing immersed structure is represented on a Lagrangian mesh. The Lagrangian structure is free to move on the background Cartesian grid without conforming to the grid lines. The fluid-structure interaction (FSI) coupling is mediated via Peskin's regularized delta functions in an implicit manner, which obviates the need to integrate the hydrodynamic stress tensor on the complex surface of the immersed structure. The IB/WSI numerical scheme is implemented within an adaptive mesh refinement (AMR) framework, in which the Lagrangian structure and the air-water interface are embedded on the finest mesh level to capture the thin boundary layers and the vortical structures arising from WSI. We use a well balanced gravitational force discretization that eliminates spurious velocity currents in the hydrostatic limit due to density variation in the three phases (air, water and solid). We also show that using a non-conservative and an inconsistent fluid solver can lead to catastrophic failure of the numerical scheme for large density ratio variations that are prevalent in WSI applications. An effective wave generation and absorption technique for a numerical wave tank is presented and used to simulate a benchmark case of water wave distortion due to a submerged structure. The numerical scheme is tested on several benchmark WSI problems from numerical and experimental literature in both two and three dimensions to demonstrate the applicability of the IB/WSI method to practical marine and coastal engineering problems.

A boundary thickening-based direct forcing immersed boundary method for fully resolved simulation of particle-laden flows

A boundary thickening-based direct forcing (BTDF) immersed boundary (IB) method is proposed for fully resolved simulation of incompressible viscous flows laden with finite size particles. By slightly thickening the boundary thickness, the local communications between the Lagrangian points on the solid boundary and their neighboring fluid Eulerian grids are improved, based on an implicit direct forcing (IDF) approach and a partition-of-unity condition for the regularized delta function. This strategy yields a simple, yet much better imposition of the non-slip and non-penetration boundary conditions than the conventional direct forcing (DF) technique. In particular, the present BTDF method can achieve a numerical accuracy comparable with other representative improved methods, such as multi-direct forcing (MDF), implicit velocity correction (IVC) and the reproducing kernel particle method (RKPM), while its computing cost remains much lower than them and nearly equivalent to the conventional DF method. The dependence of the optimum thickness value of boundary thickening on the form of the different regularized delta functions is also revealed. By coupling the lattice Boltzmann method (LBM) with BTDF-IB, the robustness of the present BTDF IB method is demonstrated using numerical simulations of creeping flow (Re=0.1), steady vortex separating flow (Re=40) and unsteady vortex shedding flow (Re=200) around a circular cylinder. The accuracy and robustness of the present method for moving particle-laden flows are performed on three benchmark cases, such as free sedimentation of single circular cylinder, DKT sedimentation of two particles and Rayleigh-Taylor instability of 504 particles sedimentation in an enclosed cavity. The computational efficiency between the present BTDF method and the MDF method is also compared at last.

Assessment of numerical methods for fully resolved simulations of particle-laden turbulent flows

During the last decade, many approaches for resolved-particle simulation (RPS) have been developed for numerical studies of finite-size particle-laden turbulent flows. In this paper, three RPS approaches are compared for a particle-laden decaying turbulence case. These methods are, the Volume-of-Fluid Lagrangian method, based on the viscosity penalty method (VoF-Lag); a direct forcing Immersed Boundary Method, based on a regularized delta function approach for the fluid/solid coupling (IBM); and the Bounce Back scheme developed for Lattice Boltzmann method (LBM-BB). The physics and the numerical performances of the methods are analyzed. Modulation of turbulence is observed for all the methods, with a faster decay of turbulent kinetic energy compared to the single-phase case. Lagrangian particle statistics, such as the velocity probability density function and the velocity autocorrelation function, show minor differences among the three methods. However, major differences between the codes are observed in the evolution of the particle kinetic energy. These differences are related to the treatment of the initial condition when the particles are inserted in an initially single-phase turbulence. The averaged particle/fluid slip velocity is also analyzed, showing similar behavior as compared to the results referred in the literature. The computational performances of the different methods differ significantly. The VoF-Lag method appears to be computationally most expensive. Indeed, this method is not adapted to turbulent cases. The IBM and LBM-BB implementations show very good scaling.

Analysis of the immersed boundary method for a finite element Stokes problem

Convergence results are presented for the immersed boundary (IB) method applied to a model Stokes problem. As a discretization method, we use the finite element method. First, the immersed force field is approximated using a regularized delta function. Its error in the W−1, p norm is examined for 1 ≤ p < n/(n − 1), with n representing the space dimension. Subsequently, we consider IB discretization of the Stokes problem and examine the regularization and discretization errors separately. Consequently, error estimate of order h1 − α in the W1, 1 × L1 norm for the velocity and pressure is derived, where α is an arbitrary small positive number. The validity of those theoretical results is confirmed from numerical examples.

An Immersed Boundary method with divergence-free velocity interpolation and force spreading

The Immersed Boundary (IB) method is a mathematical framework for constructing robust numerical methods to study fluid–structure interaction in problems involving an elastic structure immersed in a viscous fluid. The IB formulation uses an Eulerian representation of the fluid and a Lagrangian representation of the structure. The Lagrangian and Eulerian frames are coupled by integral transforms with delta function kernels. The discretized IB equations use approximations to these transforms with regularized delta function kernels to interpolate the fluid velocity to the structure, and to spread structural forces to the fluid. It is well-known that the conventional IB method can suffer from poor volume conservation since the interpolated Lagrangian velocity field is not generally divergence-free, and so this can cause spurious volume changes. In practice, the lack of volume conservation is especially pronounced for cases where there are large pressure differences across thin structural boundaries. The aim of this paper is to greatly reduce the volume error of the IB method by introducing velocity-interpolation and force-spreading schemes with the properties that the interpolated velocity field in which the structure moves is at least C1 and satisfies a continuous divergence-free condition, and that the force-spreading operator is the adjoint of the velocity-interpolation operator. We confirm through numerical experiments in two and three spatial dimensions that this new IB method is able to achieve substantial improvement in volume conservation compared to other existing IB methods, at the expense of a modest increase in the computational cost. Further, the new method provides smoother Lagrangian forces (tractions) than traditional IB methods. The method presented here is restricted to periodic computational domains. Its generalization to non-periodic domains is important future work.

A free energy-based surface tension force model for simulation of multiphase flows by level-set method

In this paper, a free energy-based surface tension force (FESF) model is presented for accurately resolving the surface tension force in numerical simulation of multiphase flows by the level set method. By using the analytical form of order parameter along the normal direction to the interface in the phase-field method and the free energy principle, FESF model offers an explicit and analytical formulation for the surface tension force. The only variable in this formulation is the normal distance to the interface, which can be substituted by the distance function solved by the level set method. On one hand, as compared to conventional continuum surface force (CSF) model in the level set method, FESF model introduces no regularized delta function, due to which it suffers less from numerical diffusions and performs better in mass conservation. On the other hand, as compared to the phase field surface tension force (PFSF) model, the evaluation of surface tension force in FESF model is based on an analytical approach rather than numerical approximations of spatial derivatives. Therefore, better numerical stability and higher accuracy can be expected. Various numerical examples are tested to validate the robustness of the proposed FESF model. It turns out that FESF model performs better than CSF model and PFSF model in terms of accuracy, stability, convergence speed and mass conservation. It is also shown in numerical tests that FESF model can effectively simulate problems with high density/viscosity ratio, high Reynolds number and severe topological interfacial changes.

A comparative analysis of binding in ultralong-range Rydberg molecules

We perform a comparative analysis of different computational approaches employed to explore the electronic structure of ultralong-range Rydberg molecules. Employing the Fermi pseudopotential approach, where the interaction is approximated by an s-wave bare delta function potential, one encounters a non-convergent behavior in basis set diagonalization. Nevertheless, the energy shifts within the first order perturbation theory coincide with those obtained by an alternative approach relying on Green’s function calculation with the quantum defect theory. A pseudopotential that yields exactly the results obtained with the quantum defect theory, i.e. beyond first order perturbation theory, is the regularized delta function potential. The origin of the discrepancies between the different approaches is analytically explained.

Accuracy of three-dimensional analysis of regularized singularities

SUMMARY In computational mechanics, the quadrature of discontinuous and singular functions is often required. To avoid specialized quadrature procedures, discontinuous and singular fields can be regularized. However, regularization changes the algebraic structure of the solving equations, and this can lead to high errors. We show how to acquire accurate and consistent results when regularization is carried out. A three-dimensional analysis of a tensile butt joint is performed through a regularized extended finite element method. The accuracy obtained via Gaussian quadrature is compared with that obtained by means of CUBPACK adaptive quadrature FORTRAN tool. The use of regularized functions with non-compact and compact support is investigated through an error evaluation procedure based on the use of their Fourier transform. The proposed procedure leads to the remarkable conclusion that regularized delta functions with non-compact support exhibit superior performance. Copyright © 2014 John Wiley & Sons, Ltd.

Numerical Study of Flow Over a Cylinder Using an Immersed Boundary Finite Volume Method

solver. Based on how the mo mentum forcing is emp loyed in the Navier-Stokes equation, IB method is classified into two: continuous forcing approach and discrete forcing approach. The continuous forcing approach of IB method utilizes Eu lerian variables for the fluid region and Lagrangian variables for the solid region. The interaction between these two variables is lin ked by the Dirac delta function. The discrete version of IB method employs interpolation techniques to obtain the desired no-slip conditions at the boundary and the mo mentum fo rcing is obtained directly fro m the discretized Nav ier-Stokes equation. The continuous forcing approach is used for the simulation of flu id-flexible body interaction. For the simu lation of elastic bodies interacting with the fluid, the Lagrangian force is the elastic force which can be obtained by applying Hooke’s law. But when dealing with rigid bodies, this law is not well posed. Hence in such cases methods like the virtual boundary method proposed by Go ldstein et al. [2] are emp loyed. This method is based on the feedback forcing scheme which can enforce the no slip boundary condition on the rigid boundary immersed in the flu id. The basic difference between the virtual boundary method and that of Lai and Peskin [3] is that, the boundary points are exactly prescribed in the former method, but allowed to move slightly fro m their equilibriu m positions in the latter. Recently, Shin et al. [4] proposed a new version of the IB method which combines the feedback forcing scheme of virtual boundary method along with Peskin’s regularized delta function. Fro m the literature rev iew on IB method, it is inves tigated that discrete forcing approach of IB method is suited for simulating rig id body interaction with the fluid whereas continuous forcing approach is suited for both rigid and elastic bodies. Also, for moving body problems, the continuous forcing approach of IB method is far superior because of the simplicity in computing the mo mentu m fo rcing function. The present work is based on the continuous forcing approach of IB method. The two– dimensional flow past a stationary circular cylinder is a bench mark problem in fluid dynamics and has been widely investigated experimentally and nu merically. A lso, this problem has been used as the validation problem for most of the IB methods. With this perspective, a numerical model based on the Peskin’s IB method [5] is developed to simulate the two – dimensional flow over a stationary cylinder. Here, the cylinder is modeled by discrete number of IB points. The Navier -Stokes equations governing the fluid flow over and surrounding the cylinder are solved on a staggered Cartesian grid system using the fractional-step based finite-volu me method. We develop a FORTRAN code to perform numerical simulat ions to demonstrate the fluid dynamics of flow over a stationary circular cylinder. Nu merical simulations are carried out for d ifferent values of Reynolds numbers to capture the flow behaviour over and surrounding the cylinder. The variation of mean drag coefficient with Reynolds number is also investigated.

Effect of regularized delta function on accuracy of immersed boundary method

The immersed boundary method is an effective technique for modeling and simulating fluid-structure interactions especially in the area of biomechanics. The effect of the regularized delta function on the accuracy is an important subject in the property study. A method of manufactured solutions is used in the research. The computational code is first verified to be mistake-free by using smooth manufactured solutions. Then, a jump in the manufactured solution for pressure is introduced to study the accuracy of the immersed boundary method. Four kinds of regularized delta functions are used to test the effect on the accuracy analysis. By analyzing the discretization errors, the accuracy of the immersed boundary method is proved to be first-order. The results show that the regularized delta function cannot improve the accuracy, but it can change the discretization errors in the entire computational domain.

A second-order accurate immersed boundary method for fully resolved simulations of particle-laden flows

An immersed boundary method (IBM) with second-order spatial accuracy is presented for fully resolved simulations of incompressible viscous flows laden with rigid particles. The method is based on the computationally efficient direct-forcing method of Uhlmann [M. Uhlmann, An immersed boundary method with direct forcing for simulation of particulate flows, J. Comput. Phys. 209 (2005) 448–476] that is embedded in a finite-volume/pressure-correction method. The IBM consists of two grids: a fixed uniform Eulerian grid for the fluid phase and a uniform Lagrangian grid attached to and moving with the particles. A regularized delta function is used to communicate between the two grids and proved to be effective in suppressing grid locking. Without significant loss of efficiency, the original method is improved by: (1) a better approximation of the no-slip/no-penetration (ns/np) condition on the surface of the particles by a multidirect forcing scheme, (2) a correction for the excess in the effective particle diameter by a slight retraction of the Lagrangian grid from the surface towards the interior of the particles with a fraction of the Eulerian grid spacing, and (3) an enhancement of the numerical stability for particle–fluid mass density ratios near unity by a direct account of the inertia of the fluid contained within the particles. The new IBM contains two new parameters: the number of iterations Ns of the multidirect forcing scheme and the retraction distance rd. The effect of Ns and rd on the accuracy is investigated for five different flows. The results show that rd has a strong influence on the effective particle diameter and little influence on the error in the ns/np condition, while exactly the opposite holds for Ns. A novel finding of this study is the demonstration that rd has a strong influence on the order of grid convergence. It is found that for spheres the choice of rd=0.3Δx yields second-order accuracy compared to first-order accuracy of the original method that corresponds to rd=0. Finally, Ns=2 appears optimal for reducing the error in the ns/np condition and maintaining the computational efficiency of the method.

A numerical method for two phase flows with insoluble surfactants

In many practical multiphase flow problems, i.e. treatment of gas emboli and various microfluidic applications, the effect of interfacial surfactants, or surface reacting agents, on the surface tension between the fluids is important. The surfactant concentration on an interface separating the fluids can be modeled with a time dependent differential equation defined on the moving and deforming interface. The equations for the location of the interface and the surfactant concentration on the interface are coupled with the Navier–Stokes equations. These equations include the singular surface tension forces from the interface on the fluid, which depend on the interfacial surfactant concentration. A new accurate and inexpensive numerical method for simulating the evolution of insoluble surfactants is presented in this paper. It is based on an explicit yet Eulerian discretization of the interface, which for two dimensional flows allows for the use of uniform one dimensional grids to discretize the equation for the interfacial surfactant concentration. A finite difference method is used to solve the Navier–Stokes equations on a regular grid with the forces from the interface spread to this grid using a regularized delta function. The timestepping is based on a Strang splitting approach. Drop deformation in shear flows in two dimensions is considered. Specifically, the effect of surfactant concentration on the deformation of the drops is studied for different sets of flow parameters.