Fractional Step Approach Research Articles

Pressure boundary conditions for immersed-boundary methods

Immersed boundary methods have seen an enormous increase in popularity over the past two decades, especially for problems involving complex moving/deforming boundaries. In most cases, the boundary conditions on the immersed body are enforced via forcing functions in the momentum equations, which in the case of fractional step methods may be problematic due to: i) creation of slip-errors resulting from the lack of explicitly enforcing boundary conditions on the (pseudo-)pressure on the immersed body; ii) coupling of the solution in the fluid and solid domains via the Poisson equation. Examples of fractional-step formulations that simultaneously enforce velocity and pressure boundary conditions have also been developed, but in most cases the standard Poisson equation is replaced by a more complex system which requires expensive iterative solvers. In this work we propose a new formulation to enforce appropriate boundary conditions on the pseudo-pressure as part of a fractional-step approach. The overall treatment is inspired by the ghost-fluid method typically utilized in two-phase flows. The main advantage of the algorithm is that a standard Poisson equation is solved, with all the modifications needed to enforce the boundary conditions being incorporated within the right-hand side. As a result, fast solvers based on trigonometric transformations can be utilized. We demonstrate the accuracy and robustness of the formulation for a series of problems with increasing complexity.

Two approaches to compute unsteady compressible two-phase flow models with stiff relaxation terms

The paper deals with the numerical modeling of two-phase flows while using Baer–Nunziato type models. Focus is given here on the numerical treatment of source terms that involve three (or four) relaxation time scales. A new coupled approach relying on the continuous analysis of the system of ODEs is compared with a more widely used strategy grounded on the fractional step approach. Properties of schemes are given in both cases. Several numerical applications show that the coupled approach should be preferred for both stability and accuracy reasons.

A coupled MPM and CBFEM framework for large deformation simulation of porous media interacting with pore and free fluid

The hydromechanical interaction of porous media with pore and free fluid widely exists in geotechnical engineering. In some situations, the solid phase would exhibit large deformation, and the boundaries between the solid and fluid phases might change substantially, making it challenging for computational simulation. A framework coupling the material point method (MPM) with the characteristic-based finite element method (CBFEM) was proposed for simulating the behavior of porous media and fluid systems with large deformation. It employs the MPM to discretize the solid domain and solve corresponding momentum equations in the Lagrangian framework in favor of its advantage for large deformation simulation, and the CBFEM to discretize the fluid domain and solve the Navier-Stokes and free-surface equations of the fluid phase in the Eulerian framework in favor of its maturity and high efficiency. The fractional-step approach was adopted to solve velocity and pressure separately in a time step to make the MPM-CBFEM coupling more flexible. In addition, an implicit drag-force term was proposed to grow the time step in undrained conditions. A series of quasi-static and transient problems were simulated to show the performance and prospects of the proposed framework for the large deformation of the porous media and fluid systems.

A random choice scheme for scalar advection

SummaryThis paper is dedicated to a numerical method based on a random choice as proposed in Glimm's scheme. It is applied to the problem of advection of a scalar quantity. The numerical scheme proposed here relies on a fractional step approach for which: the first step is performed using any classical finite‐volume scheme, and the second step is a cell‐wise update. This second step is a projection based on a random choice. The resulting scheme possesses a very low level of numerical diffusion. In order to assess the capabilities of this approach, several test cases have been investigated including convergence studies with respect to the mesh‐size. The algorithm performs very well on one‐dimensional and multi‐dimensional problems. This algorithm is very easy to implement even for multi‐processor computations.

Fractional-step finite difference schemes for incompressible elasticity on overset grids

Efficient finite-difference schemes for the numerical solution of the time-dependent equations of incompressible linear elasticity on complex geometry are presented. The schemes are second-order accurate in space and time, and solve the equations in displacement-pressure form. The algorithms use a fractional-step approach in which the time-step of the displacements is performed separately from the solution of a Poisson problem to update the pressure. Complex geometry is treated with curvilinear overset grids. Compatibility boundary conditions and an upwind dissipation for wave equations in second-order form are included to ensure stability for overset grids, and for the difficult case of traction (free-surface) boundary conditions. A divergence damping term is added to keep the dilatation small. The stability of the schemes is studied with GKS mode analysis. Exact eigen-mode solutions are obtained for several problems involving rectangular, cylindrical and spherical configurations, and these are valuable as benchmark problems. The accuracy and stability of the approach in two and three space dimensions is illustrated by comparing the numerical solutions with the exact solutions of the benchmark problems.

A note on pressure and pressure-correction-based fractional-step approaches for low Reynolds number incompressible flows

A comparative study on pressure and pressure-correction-based fractional-step methods for incompressible flows is presented. Implicit fractional-step methods are shown to give rise to a splitting error due to the implicit temporal discretization of the intermediate momentum equations. Through mathematical reasoning and a numerical example, a relative advantage of pressure-correction-based approaches over pressure-based approach for the computation of low Reynolds number flows using implicit fractional-step methods is demonstrated.

A mass–momentum consistent coupling for mesh-adaptive two-phase flow simulations

We present a novel mass–momentum consistent coupling between a geometric volume-of-fluid scheme and an incompressible flow solver with differing directional-splitting approaches. The advection of the volume fraction is performed using a direction-split algorithm, whereas the momentum advection algorithm uses a traditional unsplit, fractional-step approach. Both algorithms employ finite-volume discretizations based on Cartesian meshes. In solving the mass–momentum consistency problem, momentum fluxes at the cell faces are weighted by the density fluxes based on the already advected volume fraction. The success of our approach lies on introducing a Favre-averaged velocity interpolation at the liquid/gas interface along with a minmod slope limiter. Mesh-convergence studies show that when the minmod slope limiter is used, the two-phase solver retains an accuracy between first and second order, but when a purely upwind scheme is considered, its accuracy drops to first order. Finally, after considering several validation problems, the solver is shown to agree well with reference numerical and experimental data while retaining its robustness and efficiency.

Application of a New Hybrid Immersed Boundary Method for the Simulation of Viscous Compressible Flows

In this paper, we present a hybrid immersed boundary method (HIBM) for the simulation of two- and three-dimensional compressible viscous flows around stationary and moving obstacles. The proposed approach combines the compressible boundary condition-enforced IBM proposed by Qiu et al. and the Brinkman penalization methods. The boundary condition-enforced IBM uses a fractional step approach alongside a Dirac delta function to satisfy the boundary conditions on the body surface. Although this method works properly in subsonic regimes, it cannot correctly simulate the wake region for a supersonic flow. On the other hand, the penalization method considers the body as a porous media with a low permeability that forces the velocity and energy inside the body to converge to the body velocity and energy. However, unlike the boundary condition-enforced IBM, the streamlines penetrate the body. In the present approach, the positive features of the above-mentioned methods were included and their drawbacks were excluded. The proposed approach was applied using the finite volume method and an E-CUSP scheme. The performance of the proposed method was numerically evaluated in simulating compressible fluid flow around both stationary and moving boundaries, showing a close agreement with other numerical and experimental data available in the literature. Further, the effects of geometry, Reynolds, and Mach numbers were investigated on the supersonic flow field around elliptical cylinders of various aspect ratios. The results revealed that increasing the aspect ratio led to an increase in the shock standoff distance, recirculation zone, and drag coefficient.

Various choices of source terms for a class of two-fluid two-velocity models

The source terms of the Baer–Nunziato model involve highly non-linear return to equilibrium terms. In order to perform numerical simulations of realistic situations, accounting for this relaxation effects is mandatory. Unfortunately, with the classical forms retained for these source terms in the literature, building efficient, robust and accurate numerical schemes is a tricky task. In this paper, we propose different non-classical forms for these source terms. As for the classical ones, they all agree with the second law of thermodynamics and they are thus associated with a growth of an entropy. The great advantage of some of these new forms of source terms is that they are more linear with respect to the conservative variables. Consequently, this allows to propose more robust, efficient and accurate numerical schemes, in particular when considering fractional step approaches for which source terms and convection terms are solved separately.

A semi-implicit direct forcing immersed boundary method for periodically moving immersed bodies: A Schur complement approach

An extended immersed boundary methodology utilizing a semi-implicit direct forcing approach was formulated for the simulation of incompressible flows in the presence of periodically moving immersed bodies. The methodology utilizes a Schur complement approach to enforce no-slip kinematic constraints for immersed surfaces. The methodology is split into an “embarrassingly” parallel pre-computing stage and a time integration stage, both of which take advantage of the general parallel file system (GPFS) for efficient writing and reading of large amounts of data. The methodology can be embedded straight forwardly into the whole family of pressure–velocity segregated solvers of incompressible Navier–Stokes equations based on projection or fractional step approaches. The methodology accurately meets the no-slip kinematic constraints on the surfaces of immersed oscillating bodies. In this study, it was extensively verified by applying it for the simulation of a number of representative flows developing in the presence of an oscillating sphere. The capabilities of the methodology for the simulation of incompressible flow generated by a number of bodies whose motion is governed by general periodic kinematics were demonstrated by simulation of the flow developing in the presence of two out-of-phase oscillating spheres. The physical characteristics of the generated flows in terms of the time evolutions of the total drag coefficients were presented as a function of Reynolds values. The vortical structures inherent in the generated flows were visualized by presenting the isosurfaces of the λ2 criterion.

On the use of Glimm‐like schemes for transport equations on multidimensional domain

AbstractThis article is dedicated to a numerical method based on Glimm's idea and applied to the specific problem of advection of an indicator function. The numerical scheme relies on a fractional step approach for which: the first step is performed using any classical finite‐volume scheme, and the second step sharpens the approximated solution issued from the first step. This second step is based on a random choice as proposed in the original idea of Glimm. In order to assess the capabilities of this approach, several test cases have been investigated including convergence studies with respect to the mesh‐size. The method proposed in this article is very easy to implement, even for multiprocessor computations, and it naturally extends to multidimensional domains. Several numerical test cases, including both one‐ and two‐dimensional domains, have been considered in order to examine its behavior in terms of robustness, accuracy, and convergence rate.

Parallel and fully implicit simulations of the thermal convection in the Earth’s outer core

A parallel and fully implicit method is developed for numerical simulations of the thermal convection of an incompressible fluid in a rotating spherical shell. The method is based on a pseudo-compressibility approach with dual time stepping in order to tackle with the numerical difficulty of the solenoidal condition in the incompressible Navier-Stokes equations. The numerical solution of the nonlinear governing equations is based on the method of lines, whereby the partial differential equations are transformed into ordinary differential equations by a suitable spatial discretization. A second-order cell-centered finite volume discretization based on a collocated cubed-sphere grid is used here. In order to relax the time step limit and accurately integrate the semi-discrete nonlinear ordinary differential equations in time, we employ a second-order explicit-first-step, single-diagonal-coefficient, diagonally implicit Runge–Kutta (ESDIRK) method with adaptive time stepping. The nonlinear algebraic system arising at each pseudo-time step is solved by a Newton–Krylov–Schwarz algorithm to achieve good load balance on modern supercomputers. The numerical results are in good agreement with the benchmark solutions and more accurate than those obtained with a fractional step approach in our previous research. Large-scale tests on the Tianhe-2 supercomputer indicate that the fully implicit solver can achieve good parallel scalabilities in both strong and weak senses.

Unsteady flow simulation using the curvilinear multiple-relaxation-time lattice Boltzmann method: Danube River case study

In order to introduce the lattice Boltzmann method as a competitive approach compared to traditionally based mathematical models when river and coastal hydraulics are considered, the innovative curvilinear multi-relaxation-time lattice Boltzmann method for solving 2D flow equations transformed in curvilinear coordinate system is utilized. The curvilinear approach implemented in the multi-relaxation-time lattice Boltzmann method is tested on a geometrically simple non-prismatic channel and on a section of the Danube River. Steady flow tangential velocity measurements are used in case of the channel model verification, while in case of the Danube unsteady flow is examined. Corresponding velocity and water level measurements acquired during a period of five days are used for calibration and verification of the Danube model, while an alternative mathematical model based on the traditional fractional-step approach has been exploited for comparison. The presented results confirm the aptness of the proposed model to simulate flow in natural water bodies effectively and accurately, therefore it is fully competitive with the classic computational fluid dynamics models.

Numerical appraisal of three low Mach number algorithms for radiative–convective flows in enclosures

The present study investigates three different algorithms for the numerical simulation of non-Boussinesq convection with thermal radiative heat transfer based on a low-Mach number formulation. The solution methodology employs a fractional step approach based on the finite-volume method on arbitrary polyhedral meshes. The three algorithms compute the coupled governing equations in a segregated manner using the conservative form of momentum equations in conjunction with a variable coefficient pressure Poisson equation. The first algorithm (Algorithm A) uses conservation of mass and energy equation to compute density and temperature. The other two algorithms (Algorithm B) and (Algorithm C) calculates temperature and density from the equation of state respectively and solves a conservative form of the continuity and energy equation to obtain density and temperature respectively. The energy and mass conservation errors arising due to the use of Algorithms B and C are derived concerning various non-dimensional parameters governing the flow and heat transfer. The significance of these errors is highlighted by performing investigations over a range of Rayleigh, Prandtl, and Planck numbers for various two and three-dimensional natural convection problems with radiative heat transfer. Finally, the role of balancing of the pressure and buoyancy terms is emphasized for robust calculations of large temperature difference thermo-buoyant convection with radiative heat transfer.

A Mach-sensitive implicit–explicit scheme adapted to compressible multi-scale flows

The method presented below focuses on the numerical approximation of the Euler compressible system. It pursues a two-fold objective: being able to accurately follow slow material waves as well as strong shock waves in the context of low Mach number flows. The resulting implicit–explicit(IMEX) fractional step approach leans on a dynamic splitting designed to react to the time fluctuations of the maximal flow Mach number. When the latter rises suddenly, the IMEX scheme, so far driven by a material-wave Courant number, turn into a time-explicit approximate Riemann solver constrained by an acoustic-wave Courant number. It is also possible to enrich the dynamic splitting in order to capture high pressure jumps even when the flow Mach number is low. One-dimensional low Mach number test cases involving single or multiple waves confirm that the present approach is as accurate and efficient as an IMEX Lagrange-Projection method. Besides, numerical results suggest that the stability of the present method holds for any Mach number if the Courant number related to the convective subsystem arising from the splitting is of order unity.

A Mach-sensitive splitting approach for Euler-like systems

Herein, a Mach-sensitive fractional step approach is proposed for Euler-like systems. The key idea is to introduce a time-dependent splitting which dynamically decouples convection from acoustic phenomenon following the fluctuations of the flow Mach number. By doing so, one seeks to maintain the accuracy of the computed solution for all Mach number regimes. Indeed, when the Mach number takes high values, a time-explicit resolution of the overall Euler-like system is entirely performed in one of the present splitting step. On the contrary, in the low-Mach number case, convection is totally separated from the acoustic waves production. Then, by performing an appropriate correction on the acoustic step of the splitting, the numerical diffusion can be significantly reduced. A study made on both convective and acoustic subsystems of the present approach has revealed some key properties as hyperbolicity and positivity of the density and internal energy in the case of an ideal gas thermodynamics. The one-dimensional results made on a wide range of Mach numbers using an ideal and a stiffened gas thermodynamics show that the present approach is as accurate and CPU-consuming as a state of the art Lagrange-Projection-type method.

Unified one‐fluid formulation for incompressible flexible solids and multiphase flows: Application to hydrodynamics using the immersed structural potential method (ISPM)

SummaryIn this paper, we present a two‐dimensional computational framework for the simulation of fluid‐structure interaction problems involving incompressible flexible solids and multiphase flows, further extending the application range of classical immersed computational approaches to the context of hydrodynamics. The proposed method aims to overcome shortcomings such as the restriction of having to deal with similar density ratios among different phases or the restriction to solve single‐phase flows. First, a variation of classical immersed techniques, pioneered with the immersed boundary method (IBM), is presented by rearranging the governing equations, which define the behaviour of the multiple physics involved. The formulation is compatible with the “one‐fluid” formulation for two‐phase flows and can deal with large density ratios with the help of an anisotropic Poisson solver. Second, immersed deformable structures and fluid phases are modelled in an identical manner except for the computation of the deviatoric stresses. The numerical technique followed in this paper builds upon the immersed structural potential method developed by the authors, by adding a level set–based method for the capturing of the fluid‐fluid interfaces and an interface Lagrangian‐based meshless technique for the tracking of the fluid‐structure interface. The spatial discretisation is based on the standard marker‐and‐cell method used in conjunction with a fractional step approach for the pressure/velocity decoupling, a second‐order time integrator, and a fixed‐point iterative scheme. The paper presents a wide d range of two‐dimensional applications involving multiphase flows interacting with immersed deformable solids, including benchmarking against both experimental and alternative numerical schemes.

An adaptive Fixed-Mesh ALE method for free surface flows

In this work we present a Fixed-Mesh ALE method for the numerical simulation of free surface flows capable of using an adaptive finite element mesh covering a background domain. This mesh is successively refined and unrefined at each time step in order to focus the computational effort on the spatial regions where it is required. Some of the main ingredients of the formulation are the use of an Arbitrary-Lagrangian–Eulerian formulation for computing temporal derivatives, the use of stabilization terms for stabilizing convection, stabilizing the lack of compatibility between velocity and pressure interpolation spaces, and stabilizing the ill-conditioning introduced by the cuts on the background finite element mesh, and the coupling of the algorithm with an adaptive mesh refinement procedure suitable for running on distributed memory environments. Algorithmic steps for the projection between meshes are presented together with the algebraic fractional step approach used for improving the condition number of the linear systems to be solved. The method is tested in several numerical examples. The expected convergence rates both in space and time are observed. Smooth solution fields for both velocity and pressure are obtained (as a result of the contribution of the stabilization terms). Finally, a good agreement between the numerical results and the reference experimental data is obtained.

Lagrangian finite element model for the 3D simulation of glass forming processes

We propose here a numerical model for a three-dimensional simulation of glass forming processes. Using the basic philosophy of the Particle Finite Element method (PFEM), we introduce several new features adapting the strategy to suit the problem of interest. A modified fractional step method for the solution of the flow equations is applied. This approach, on the one hand, inherits the computational efficiency of the original fractional step approach, and on the other hand shows better mass conservation features. These features are particularly attractive taking into account the importance of the correct prediction of the glass product’s wall thickness. A smart mesh update strategy and a simple mechanical contact scheme are introduced. In order to account for temperature-dependent viscosity, the heat equation is coupled to the mechanical model. Viscosity is obtained from the temperature field via an empirical law. The model is validated and an example modeling the processes in the final blow mold of the bottle manufacturing process is proposed.

Computational study of no-slip and rarefied slip flows in infinite structured porous media

No-slip and rarefied slip flows in infinite structured porous media are numerically investigated in this study using hybrid FE/FV methods. The hybrid FE/FV flow solver methodology is based on the pressure correction method which involves a fractional step approach to first obtain an intermediate velocity by solving the momentum equations with the matrix-free, implicit, cell-centered finite volume method. The poisson equation resulting from the fractional step approach is then solved by node based Galerkin finite element method for an auxiliary variable which is closely related to pressure and is used to update the velocity and pressure fields.Three dimensional numerical simulations are conducted using our hybrid FE/FV flow solver for analyzing fluid flow in infinite structured porous media. In this study, structured porous media is represented by packed beds of spherical particles in different packing configurations including Simple (SC), Body Centered Cubic (BCC), and FCC (Face Centered Cubic) arrangements. Finite contact spacing's (2% of particle diameter) are maintained between the spherical particles to avoid highly skewed elements during the grid generation process. Our results are compared with available experimental and numerical data for both continuum no-slip and also rarefied slip flows. For continuum no-slip flows, packing form of structured beds had a considerable influence on the pressure drop, particularly at higher Reynolds numbers. However, for rarefied flows, the influence of packing form seemed to be negligible. It was also observed that infinite random porous media correlations are unsuitable for predicting the friction factors and/or pressure drop in structured porous media for continuum no-slip flows. Additionally rarefaction reduces the pressure drop and impacts the friction factor correlations. Due to lack of consistent generalized correlations in structured porous media and also since continuum correlations are not applicable for rarefied flows, new improved correlations for predicting friction factors in infinite structured microporous media are proposed.