A cell‐based smoothed finite element method with semi‐implicit CBS procedures for incompressible laminar viscous flows

In this paper, the cell‐based smoothed finite element method (CS‐FEM) with the semi‐implicit characteristic‐based split (CBS) scheme (CBS/CS‐FEM) is proposed for computational fluid dynamics. The 3‐node triangular (T3) element and 4‐node quadrilateral (Q4) element are used for present CBS/CS‐FEM for two‐dimensional flows. The 8‐node hexahedral element (H8) is used for three‐dimensional flows. Two types of CS‐FEM are implemented in this paper. One is standard CS‐FEM with quadrilateral gradient smoothing cells for Q4 element and hexahedron cells for H8 element. Another is called as n‐sided CS‐FEM (nCS‐FEM) whose gradient smoothing cells are triangles for Q4 element and pyramids for H8 element. To verify the proposed methods, benchmarking problems are tested for two‐dimensional and three‐dimensional flows. The benchmarks show that CBS/CS‐FEM and CBS/nCS‐FEM are capable to solve incompressible laminar flow and can produce reliable results for both steady and unsteady flows. The proposed CBS/CS‐FEM method has merits on better robustness against distorted mesh with only slight more computation time and without losing accuracy, which is important for problems with heavy mesh distortion. The blood flow in carotid bifurcation is also simulated to show capabilities of proposed methods for realistic and complicated flow problems.     

A discontinuous Galerkin‐based sharp‐interface method to simulate three‐dimensional compressible two‐phase flow

A numerical method for the simulation of compressible two‐phase flows is presented in this paper. The sharp‐interface approach consists of several components: a discontinuous Galerkin solver for compressible fluid flow, a level‐set tracking algorithm to follow the movement of the interface and a coupling of both by a ghost‐fluid approach with use of a local Riemann solver at the interface. There are several novel techniques used: the discontinuous Galerkin scheme allows locally a subcell resolution to enhance the interface resolution and an interior finite volume Total Variation Diminishing (TVD) approximation at the interface. The level‐set equation is solved by the same discontinuous Galerkin scheme. To obtain a very good approximation of the interface curvature, the accuracy of the level‐set field is improved and smoothed by an additional P_NP_M‐reconstruction. The capabilities of the method for the simulation of compressible two‐phase flow are demonstrated for a droplet at equilibrium, an oscillating ellipsoidal droplet, and a shock‐droplet interaction problem at Mach 3. Copyright © 2015 John Wiley & Sons, Ltd.     

An immersed smoothed point interpolation method (IS‐PIM) for fluid‐structure interaction problems

An immersed smoothed point interpolation method using 3‐node triangular background cells is proposed to solve 2D fluid‐structure interaction problems for solids with large deformation/displacement placed in incompressible viscous fluid. In the framework of immersed‐type method, the governing equations can be decomposed into 3 parts on the basis of the fictitious fluid assumption. The incompressible Navier‐Stokes equations are solved using the semi‐implicit characteristic‐based split scheme, and solids are simulated using the newly developed edge‐based smoothed point interpolation method. The fictitious fluid domain can be used to calculate the coupling force. The numerical results show that immersed smoothed point interpolation method can avoid remeshing for moving solid based on immersed operation and simulate the contact phenomenon without an additional treatment between the solid and the fluid boundary. The influence from information transfer between solid domain and fluid domain on fluid‐structure interaction problems has been investigated. The numerical results show that the proposed interpolation schemes will generally improve the accuracy for simulating both fluid flows and solid structures.     

A projection scheme for incompressible multiphase flow using adaptive Eulerian grid

This paper presents a finite element method for incompressible multiphase flows with capillary interfaces based on a (formally) second‐order projection scheme. The discretization is on a fixed Eulerian grid. The fluid phases are identified and advected using a level set function. The grid is temporarily adapted around the interfaces in order to maintain optimal interpolations accounting for the pressure jump and the discontinuity of the normal velocity derivatives. The least‐squares method for computing the curvature is used, combined with piecewise linear approximation to the interface. The time integration is based on a formally second order splitting scheme. The convection substep is integrated over an Eulerian grid using an explicit scheme. The remaining generalized Stokes problem is solved by means of a formally second order pressure‐stabilized projection scheme. The pressure boundary condition on the free interface is imposed in a strong form (pointwise) at the pressure‐computation substep. This allows capturing significant pressure jumps across the interface without creating spurious instabilities. This method is simple and efficient, as demonstrated by the numerical experiments on a wide range of free‐surface problems. Copyright © 2004 John Wiley & Sons, Ltd.     

An improved KGF‐SPH with a novel discrete scheme of Laplacian operator for viscous incompressible fluid flows

The kernel gradient free (KGF) smoothed particle hydrodynamics (SPH) method is a modified finite particle method (FPM) which has higher order accuracy than the conventional SPH method. In KGF‐SPH, no kernel gradient is required in the whole computation, and this leads to good flexibility in the selection of smoothing functions and it is also associated with a symmetric corrective matrix. When modeling viscous incompressible flows with SPH, FPM or KGF‐SPH, it is usual to approximate the Laplacian term with nested approximation on velocity, and this may introduce numerical errors from the nested approximation, and also cause difficulties in dealing with boundary conditions. In this paper, an improved KGF‐SPH method is presented for modeling viscous, incompressible fluid flows with a novel discrete scheme of Laplacian operator. The improved KGF‐SPH method avoids nested approximation of first order derivatives, and keeps the good feature of ‘kernel gradient free’. The two‐dimensional incompressible fluid flow of shear cavity, both in Euler frame and Lagrangian frame, are simulated by SPH, FPM, the original KGF‐SPH and improved KGF‐SPH. The numerical results show that the improved KGF‐SPH with the novel discrete scheme of Laplacian operator are more accurate than SPH, and more stable than FPM and the original KGF‐SPH. 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.     

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.     

Fluid–solid interaction problems with thermal convection using the immersed element‐free Galerkin method

In this work, the immersed element‐free Galerkin method (IEFGM) is proposed for the solution of fluid–structure interaction (FSI) problems. In this technique, the FSI is represented as a volumetric force in the momentum equations. In IEFGM, a Lagrangian solid domain moves on top of an Eulerian fluid domain that spans over the entire computational region. The fluid domain is modeled using the finite element method and the solid domain is modeled using the element‐free Galerkin method. The continuity between the solid and fluid domains is satisfied by means of a local approximation, in the vicinity of the solid domain, of the velocity field and the FSI force. Such an approximation is achieved using the moving least‐squares technique. The method was applied to simulate the motion of a deformable disk moving in a viscous fluid due to the action of the gravitational force and the thermal convection of the fluid. An analysis of the main factors affecting the shape and trajectory of the solid body is presented. The method shows a distinct advantage for simulating FSI problems with highly deformable solids. Copyright © 2009 John Wiley & Sons, Ltd.     

On pressure separation algorithms (PSepA) for improving the accuracy of incompressible flow simulations

We investigate a special technique called ‘pressure separation algorithm’ (PSepA) (see Applied Mathematics and Computation 2005; 165 :275–290 for an introduction) that is able to significantly improve the accuracy of incompressible flow simulations for problems with large pressure gradients. In our numerical studies with the computational fluid dynamics package FEATFLOW ( www.featflow.de ), we mainly focus on low‐order Stokes elements with nonconforming finite element approximations for the velocity and piecewise constant pressure functions. However, preliminary numerical tests show that this advantageous behavior can also be obtained for higher‐order discretizations, for instance, with Q₂/P₁ finite elements. We analyze the application of this simple, but very efficient, algorithm to several stationary and nonstationary benchmark configurations in 2D and 3D (driven cavity and flow around obstacles), and we also demonstrate its effect to spurious velocities in multiphase flow simulations (‘static bubble’ configuration) if combined with edge‐oriented, resp., interior penalty finite element method stabilization techniques. Copyright © 2008 John Wiley & Sons, Ltd.     

Performance of algebraic multi‐grid solvers based on unsmoothed and smoothed aggregation schemes

A comparison is made of the performance of two algebraic multi‐grid (AMG0 and AMG1) solvers for the solution of discrete, coupled, elliptic field problems. In AMG0, the basis functions for each coarse grid/level approximation (CGA) are obtained directly by unsmoothed aggregation, an appropriate scaling being applied to each CGA to improve consistency. In AMG1 they are assembled using a smoothed aggregation with a constrained energy optimization method providing the smoothing. Although more costly, smoothed basis functions provide a better (more consistent) CGA. Thus, AMG1 might be viewed as a benchmark for the assessment of the simpler AMG0. Selected test problems for D'Arcy flow in pipe networks, Fick diffusion, plane strain elasticity and Navier–Stokes flow (in a Stokes approximation) are used in making the comparison. They are discretized on the basis of both structured and unstructured finite element meshes. The range of discrete equation sets covers both symmetric positive definite systems and systems that may be non‐symmetric and/or indefinite. Both global and local mesh refinements to at least one order of resolving power are examined. Some of these include anisotropic refinements involving elements of large aspect ratio; in some hydrodynamics cases, the anisotropy is extreme, with aspect ratios exceeding two orders. As expected, AMG1 delivers typical multi‐grid convergence rates, which for all practical purposes are independent of mesh bandwidth. AMG0 rates are slower. They may also be more discernibly mesh‐dependent. However, for the range of mesh bandwidths examined, the overall cost effectiveness of the two solvers is remarkably similar when a full convergence to machine accuracy is demanded. Thus, the shorter solution times for AMG1 do not necessarily compensate for the extra time required for its costly grid generation. This depends on the severity of the problem and the demanded level of convergence. For problems requiring few iterations, where grid generation costs represent a significant penalty, AMG0 has the advantage. For problems requiring a large investment in iterations, AMG1 has the edge. However, for the toughest problems addressed (vector and coupled vector–scalar fields discretized exclusively using finite elements of extreme aspect ratio) AMG1 is more robust: AMG0 has failed on some of these tests. However, but for this deficiency AMG0 would be the preferred linear approximation solver for Navier–Stokes solution algorithms in view of its much lower grid generation costs. Copyright © 2001 John Wiley & Sons, Ltd.     

Time‐accurate stabilized finite‐element model for weakly nonlinear and weakly dispersive water waves

Introduction of a time‐accurate stabilized finite‐element approximation for the numerical investigation of weakly nonlinear and weakly dispersive water waves is presented in this paper. To make the time approximation match the order of accuracy of the spatial representation of the linear triangular elements by the Galerkin finite‐element method, the fourth‐order time integration of implicit multistage Padé method is used for the development of the numerical scheme. The streamline‐upwind Petrov–Galerkin (SUPG) method with crosswind diffusion is employed to stabilize the scheme and suppress the spurious oscillations, usually common in the numerical computation of convection‐dominated flow problems. The performance of numerical stabilization and accuracy is addressed. Treatments of various boundary conditions, including the open boundary conditions, the perfect reflecting boundary conditions along boundaries with irregular geometry, are also described. Numerical results showing the comparisons with analytical solutions, experimental measurements, and other published numerical results are presented and discussed. Copyright © 2007 John Wiley & Sons, Ltd.     

Improving the CBS‐based partitioned semi‐implicit coupling algorithm for fluid‐structure interaction

We detail in this work 2 simple but effective alternatives to improve the characteristic‐based split–based partitioned semi‐implicit coupling algorithm for fluid‐structure interaction. The basic idea lies in introducing the end‐of‐step velocity into the implicit stages of the 2 algorithms integrating different splits. The algorithm built upon the second‐order pressure split is further stabilized via the pressure gradient projection with particular emphasis on the extremely low mass ratio. The smoothed finite element method is exploited for spatial discretization of fluid and solid equations. Even without any accelerators, both the semi‐implicit solvers incorporating fixed‐point iterations engender visible improvements versus the previously published data for several benchmarks.     

Application of the lattice‐Boltzmann method to study flow and dispersion in channels with and without expansion and contraction geometry

The lattice‐Boltzmann (LB) method, derived from lattice gas automata, is a relatively new technique for studying transport problems. The LB method is investigated for its accuracy to study fluid dynamics and dispersion problems. Two problems of relevance to flow and dispersion in porous media are addressed: (i) Poiseuille flow between parallel plates (which is analogous to flow in pore throats in two‐dimensional porous networks), and (ii) flow through an expansion–contraction geometry (which is analogous to flow in pore bodies in two‐dimensional porous networks). The results obtained from the LB simulations are compared with analytical solutions when available, and with solutions obtained from a finite element code (FIDAP) when analytical results are not available. Excellent agreement is found between the LB results and the analytical/FIDAP solutions in most cases, indicating the utility of the lattice‐Boltzmann method for solving fluid dynamics and dispersion problems. Copyright © 1999 John Wiley & Sons, Ltd.     

An immersed‐boundary finite‐volume method for direct simulation of flows with suspended paramagnetic particles

In the present study, we have proposed an immersed‐boundary finite‐volume method for the direct numerical simulation of flows with inertialess paramagnetic particles suspended in a nonmagnetic fluid under an external magnetic field without the need for any model such as the dipole–dipole interaction. In the proposed method, the magnetic field (or force) is described by the numerical solution of the Maxwell equation without current, where the smoothed representation technique is employed to tackle the discontinuity of magnetic permeability across the particle–fluid interface. The flow field, on the other hand, is described by the solution of the continuity and momentum equations, where the discrete‐forcing‐based immersed‐boundary method is employed to satisfy the no‐slip condition at the interface. To validate the method, we performed numerical simulations on the two‐dimensional motion of two and three paramagnetic particles in a nonmagnetic fluid subjected to an external uniform magnetic field and then compared the results with the existing finite‐element and semi‐analytical solutions. Comparison shows that the proposed method is robust in the direct simulation of such magnetic particulate flows. This method can be extended to more general flows without difficulty: three‐dimensional particulate flows, flows with a great number of particles, or flows under an arbitrary external magnetic field. Copyright © 2010 John Wiley & Sons, Ltd.     

A local domain‐free discretization method for simulation of incompressible flows over moving bodies

This paper presents a local domain‐free discretization (DFD) method for the simulation of unsteady flows over moving bodies governed by the incompressible Navier–Stokes equations. The discretization strategy of DFD is that the discrete form of partial differential equations at an interior point may involve some points outside the solution domain. All the mesh points are classified as interior points, exterior dependent points and exterior independent points. The functional values at the exterior dependent points are updated at each time step by the approximate form of solution near the boundary. When the body is moving, only the status of points is changed and the mesh can stay fixed. The issue of ‘freshly cleared nodes/cells’ encountered in usual sharp interface methods does not pose any particular difficulty in the presented method. The Galerkin finite‐element approximation is used for spatial discretization, and the discrete equations are integrated in time via a dual‐time‐stepping scheme based on artificial compressibility. In order to validate the present method for moving‐boundary flow problems, two groups of flow phenomena have been simulated: (1) flows over a fixed circular cylinder, a harmonic in‐line oscillating cylinder in fluid at rest and a transversely oscillating cylinder in uniform flow; (2) flows over a pure pitching airfoil, a heaving–pitching airfoil and a deforming airfoil. The predictions show good agreement with the published numerical results or experimental data. Copyright © 2010 John Wiley & Sons, Ltd.     

Simulation of flow‐flexible body interactions with large deformation

A modified front‐tracking method was proposed for the simulation of fluid‐flexible body interactions with large deformations. A large deformable body was modeled by restructuring the body using a grid adaptation. Discontinuities in the viscosity at the fluid‐structure interface were incorporated by distributing the viscosity across the interface using an indicator function. A viscosity gradient field was created near the interface, and a smooth transition occurred between the structure and the fluid. The fluid motion was defined on the Eulerian domain and was solved using the fractional step method on a staggered Cartesian grid system. The solid motion was described by Lagrangian variables and was solved by the finite element method on an unstructured triangular mesh. The fluid motion and the structure motion were independently solved, and their interaction force was calculated using a feedback law. The interaction force was the restoring force of a stiff spring with damping, and spread from the Lagrangian coordinates to the Eulerian grid by a smoothed approximation of the Dirac delta function. In the numerical simulations, we validated the effect of the grid adaptation on the solid solver using a vibrating circular ring. The effects of the viscosity gradient field were verified by solving the deformation of a circular disk in a linear shear flow, including an elastic ring moving through a channel with constriction, deformation of a suspended catenary, and a swimming jellyfish. A comparison of the numerical results with the theoretical solutions was presented. Copyright © 2011 John Wiley & Sons, Ltd.     

Least‐squares meshfree method for incompressible Navier–Stokes problems

A least‐squares meshfree method based on the first‐order velocity–pressure–vorticity formulation for two‐dimensional incompressible Navier–Stokes problem is presented. The convective term is linearized by successive substitution or Newton's method. The discretization of all governing equations is implemented by the least‐squares method. Equal‐order moving least‐squares approximation is employed with Gauss quadrature in the background cells. The boundary conditions are enforced by the penalty method. The matrix‐free element‐by‐element Jacobi preconditioned conjugate method is applied to solve the discretized linear systems. Cavity flow for steady Navier–Stokes problem and the flow over a square obstacle for time‐dependent Navier–Stokes problem are investigated for the presented least‐squares meshfree method. The effects of inaccurate integration on the accuracy of the solution are investigated. Copyright © 2004 John Wiley & Sons, Ltd.     

Die swell measurements of second‐order fluids: numerical experiments

An analysis of the flow of a second‐order fluid is presented. Reference values for some variables are defined, and with these a non‐dimensional formulation of the governing equations. From this formulation, three dimensionless numbers appear; one is the Reynolds number, and two numbers that are called the first‐ and second‐dimensionless normal stress (NSD) coefficients. The equations of motion are solved by a finite element method using a commercially available program (Fidap), and the steady state converged solution was used to measure the die swell. The factors that influence die swell and that are studied in this work include: the die geometry for circular cross sectional dies, including tubular, converging, diverging, half‐converging/half‐tubular shapes; fluid characteristics such as Reynolds number and first‐ and second‐DNS coefficients (both positive and negative values); and flow rates, as determined by the maximum velocity in a parabolic velocity profile at the entrance to the die. The results suggest that shear and deformation histories of the fluid directly influence not only swell characteristics, but also convergence characteristics of the numerical simulation. © 1999 John Wiley & Sons, Ltd.     

A new VOF‐based numerical scheme for the simulation of fluid flow with free surface. Part II: application to the cavity filling and sloshing problems

Finite element analysis of fluid flow with moving free surface has been performed in 2‐D and 3‐D. The new VOF‐based numerical algorithm that has been proposed by the present authors (Int. J. Numer. Meth. Fluids, submitted) was applied to several 2‐D and 3‐D free surface flow problems. The proposed free surface tracking scheme is based on two numerical tools; the orientation vector to represent the free surface orientation in each cell and the baby‐cell to determine the fluid volume flux at each cell boundary. The proposed numerical algorithm has been applied to 2‐D and 3‐D cavity filling and sloshing problems in order to demonstrate the versatility and effectiveness of the scheme. The proposed numerical algorithm resolved successfully the free surfaces interacting with each other. The simulated results demonstrated applicability of the proposed numerical algorithm to the practical problems of large free surface motion. It has been also demonstrated that the proposed free surface tracking scheme can be easily implemented in any irregular non‐uniform grid systems and can be extended to 3‐D free surface flow problems without additional efforts. Copyright © 2003 John Wiley & Sons, Ltd.     

A coupled finite volume method for the solution of flow processes on complex geometries

CFD modelling of ‘real‐life’ thermo‐fluid processes often requires solutions in complex three‐dimensional geometries, which can result in meshes containing aspects that are badly distorted. Cell‐centred finite volume methods (CC‐FV), typical of most commercial CFD tools, are computationally efficient, but can lead to convergence problems on meshes that feature cells with highly non‐orthogonal shapes. The control volume‐finite element method (CVFE) uses a vertex‐based approach and handles distorted meshes with relative ease, but is computationally expensive. A combined vertex‐based—cell‐centre technique (CFVM), detailed in this paper, allows solutions on distorted meshes where purely cell‐centred solutions procedures fail. The method utilizes the ability of the vertex‐based approach to resolve the flow field on a distorted mesh, enabling well established cell‐centred physical models to be employed in the solution of other transported quantities. The vertex‐based flow code is verified against a manufactured 3D solution and error norms are compared on meshes with various degrees of distortion. The CFVM method is validated with benchmark solutions for thermally driven flow and turbulent flow. Finally, the method is illustrated on three‐dimensional turbulent flow over an aircraft wing on a distorted mesh where purely cell‐centred techniques fail. The CFVM is relatively straightforward to embed within generic CC based CFD tools allowing it to be employed in a wide variety of processing applications. Copyright © 2006 John Wiley & Sons, Ltd.