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.     

An energy stable monolithic Eulerian fluid‐structure finite element method

When written in an Eulerian frame, the conservation laws of continuum mechanics are similar for fluids and solids leading to a single set of variables for a monolithic formulation. Such formulations are well adapted to large displacement fluid‐structure configurations, but stability is a challenging problem because of moving geometries. In this article, the method is presented; time implicit discretizations are proposed with iterative algorithms well posed at each step, at least for small displacements; stability is discussed for an implicit in time finite element method in space by showing that energy decreases with time. The key numerical ingredient is the Characterics‐Galerkin method coupled with a powerful mesh generator. A numerical section discusses implementation issues and presents a few simple tests. It is also shown that contacts are easily handled by extending the method to variational inequalities. This paper deals only with incompressible neo‐Hookean Mooney‐Rivlin hyperelastic material in 2 dimensions in a Newtonian fluid, but the method is not limited to these; compressible and 3D cases will be presented later.     

Nonisothermal two‐ and three‐dimensional flow simulations of inelastic and viscoelastic fluids by a finite‐volume method

This paper applies the finite‐volume method to computations of steady flows of viscous and viscoelastic incompressible fluids in complex two and three‐dimensional geometries. The materials adopted in the study obey different constitutive laws: Newtonian, purely viscous Carreau–Yasuda as also Upper‐Convected Maxwell and Phan‐Thien/Tanner differential models, with a Williams–Landel–Ferry (WLF) equation for temperature dependence. Specific analyses are made depending on the rheological model. A staggered grid is used for discretizing the equations and unknowns. Stockage possibilities allow us to solve problems involving a great number of degrees of freedom, up to 1 500 000 unknowns with a desk computer. In relation to the fluid properties, our numerical simulations provide flow characteristics for various 2D and 3D configurations and demonstrate the possibilities of the code to solve problems involving complex nonlinear constitutive equations with thermal effects. Copyright © 2009 John Wiley & Sons, Ltd.     

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.     

Large‐eddy simulation with complex 2‐D geometries using a parallel finite‐element/spectral algorithm

A parallel stabilized finite‐element/spectral formulation is presented for incompressible large‐eddy simulation with complex 2‐D geometries. A unique discretization scheme is developed consisting of a streamline‐upwind Petrov–Galerkin/Pressure‐Stabilized Petrov–Galerkin (SUPG/PSPG) finite‐element discretization in the 2‐D plane with a collocated spectral/pseudospectral discretization in the out‐of‐plane direction. This formulation provides an efficient approach for solving 3‐D flows over arbitrary 2‐D geometries. Utilizing this discretization and through explicit temporal treatment of the non‐linear terms, the system of equations for each Fourier mode is decoupled within each time step. A novel parallelization approach is then taken, where the computational work is partitioned in Fourier space. A validation of the algorithm is presented via comparison of results for flow past a circular cylinder with published values for Re=195, 300, and 3900. Copyright © 2003 John Wiley & Sons, Ltd.     

Interface‐preserving level‐set reinitialization for DG‐FEM

This paper presents a contribution to level‐set reinitialization in the context of discontinuous Galerkin finite element methods. We focus on high‐order polynomials for the discretization and level set geometries, which are comparable to the element size. In contrast to hyperbolic and geometric reinitialization techniques, our method relies on solving a nonlinear elliptic PDE iteratively. We critically compare two different variants of the algorithm experimentally in numerical studies. The results demonstrate that the method is stable for nontrivial test cases and shows high‐order accuracy. Copyright © 2016 John Wiley & Sons, Ltd.     

A high‐order finite difference method for numerical simulations of supersonic turbulent flows

This paper describes a new class of three‐dimensional finite difference schemes for high‐speed turbulent flows in complex geometries based on the high‐order monotonicity‐preserving (MP) method. Simulations conducted for various 1D, 2D, and 3D problems indicate that the new high‐order MP schemes can preserve sharp changes in the flow variables without spurious oscillations and are able to capture the turbulence at the smallest computed scales. Our results also indicate that the MP method has less numerical dissipation and faster grid convergence than the weighted essentially non‐oscillatory method. However, both of these methods are computationally more demanding than the COMP method and are only used for the inviscid fluxes. To reduce the computational cost for reacting flows, the scalar equations are solved by the COMP method, which is shown to yield similar results to those obtained by the MP in supersonic turbulent flows with strong shock waves. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical simulation of flow‐induced noise using LES/SAS and Lighthill's acoustic analogy

We present a finite element (FE) formulation of Lighthill's acoustic analogy for the hybrid computation of noise generated by turbulent flows. In the present approach, the flow field is computed using large eddy simulation and scale adaptive simulation turbulence models. The acoustic propagation is obtained by solving the variational formulation of Lighthill's acoustic analogy with the FE method. In order to preserve the acoustic energy, we compute the inhomogeneous part of Lighthill's wave equation by applying the FE formulation on the fine flow grid. The resulting acoustic nodal loads are then conservatively interpolated to the coarser acoustic grid. Subsequently, the radiated acoustic field can be solved in both time and frequency domains. In the latter case, an enhanced perfectly matched layer technique is employed, allowing one to truncate the computational domain in the acoustic near field, without compromising the numerical solution. Our hybrid approach is validated by comparing the numerical results of the acoustic field induced by a corotating vortex pair with the corresponding analytical solution. To demonstrate the applicability of our scheme, we present full 3D numerical results for the computed acoustic field generated by the turbulent flow around square cylinder geometries. The sound pressure levels obtained compare well with measured values. Copyright © 2009 John Wiley & Sons, Ltd.     

The improved space–time conservation element and solution element scheme for two‐dimensional dam‐break flow simulation

A new numerical scheme, namely space–time conservation element and solution element (CE/SE) method, has been used for the solution of the two‐dimensional (2D) dam‐break problem. Distinguishing from the well‐established traditional numerical methods (such as characteristics, finite difference, finite element, and finite‐volume methods), the CE/SE scheme has many non‐traditional features in both concept and methodology: space and time are treated in a unified way, which is the most important characteristic for the CE/SE method; the CEs and SEs are introduced, both local and global flux conservations in space and time rather than space only are enforced; an explicit scheme with a stagger grid is adopted. Furthermore, this scheme is robust and easy to implement. In this paper, an improved CE/SE scheme is extended to solve the 2D shallow water equations with the source terms, which usually plays a critical role in dam‐break flows. To demonstrate the accuracy, robustness and efficiency of the improved CE/SE method, both 1D and 2D dam‐break problems are simulated numerically, and the results are consistent with either the analytical solutions or experimental results. Copyright © 2011 John Wiley & Sons, Ltd.     

Augmented mixed finite element methods for a vorticity‐based velocity–pressure–stress formulation of the Stokes problem in 2D

In this paper, we consider an augmented velocity–pressure–stress formulation of the 2D Stokes problem, in which the stress is defined in terms of the vorticity and the pressure, and then we introduce and analyze stable mixed finite element methods to solve the associated Galerkin scheme. In this way, we further extend similar procedures applied recently to linear elasticity and to other mixed formulations for incompressible fluid flows. Indeed, our approach is based on the introduction of the Galerkin least‐squares‐type terms arising from the corresponding constitutive and equilibrium equations, and from the Dirichlet boundary condition for the velocity, all of them multiplied by stabilization parameters. Then, we show that these parameters can be suitably chosen so that the resulting operator equation induces a strongly coercive bilinear form, whence the associated Galerkin scheme becomes well posed for any choice of finite element subspaces. In particular, we can use continuous piecewise linear velocities, piecewise constant pressures, and rotated Raviart–Thomas elements for the stresses. Next, we derive reliable and efficient residual‐based a posteriori error estimators for the augmented mixed finite element schemes. In addition, several numerical experiments illustrating the performance of the augmented mixed finite element methods, confirming the properties of the a posteriori estimators, and showing the behavior of the associated adaptive algorithms are reported. The present work should be considered as a first step aiming finally to derive augmented mixed finite element methods for vorticity‐based formulations of the 3D Stokes problem. Copyright © 2010 John Wiley & Sons, Ltd.     

Mathematical modelling of two‐phase non‐Newtonian flow in a helical pipe

Governing equations for a two‐phase 3D helical pipe flow of a non‐Newtonian fluid with large particles are derived in an orthogonal helical coordinate system. The Lagrangian approach is utilized to model solid particle trajectories. The interaction between solid particles and the fluid that carries them is accounted for by a source term in the momentum equation for the fluid. The force‐coupling method (FCM), developed by M.R. Maxey and his group, is adopted; in this method the momentum source term is no longer a Dirac delta function but is spread on a numerical mesh by using a finite‐sized envelop with a spherical Gaussian distribution. The influence of inter‐particle and particle–wall collisions is also taken into account. Copyright © 2005 John Wiley & Sons, Ltd.     

Residual‐based artificial viscosity for simulation of turbulent compressible flow using adaptive finite element methods

In this paper, we present a finite element method with a residual‐based artificial viscosity for simulation of turbulent compressible flow, with adaptive mesh refinement based on a posteriori error estimation with sensitivity information from an associated dual problem. The artificial viscosity acts as a numerical stabilization, as shock capturing, and as turbulence capturing for large eddy simulation of turbulent flow. The adaptive method resolves parts of the flow indicated by the a posteriori error estimates but leaves shocks and turbulence under‐resolved in a large eddy simulation. The method is tested for examples in 2D and 3D and is validated against experimental data. Copyright © 2012 John Wiley & Sons, Ltd.     

Bubble‐based stabilized finite element methods for time‐dependent convection–diffusion–reaction problems

In this paper, we propose a numerical algorithm for time‐dependent convection–diffusion–reaction problems and compare its performance with the well‐known numerical methods in the literature. Time discretization is performed by using fractional‐step θ‐scheme, while an economical form of the residual‐free bubble method is used for the space discretization. We compare the proposed algorithm with the classical stabilized finite element methods over several benchmark problems for a wide range of problem configurations. The effect of the order in the sequence of discretization (in time and in space) to the quality of the approximation is also investigated. Numerical experiments show the improvement through the proposed algorithm over the classical methods in either cases. Copyright © 2016 John Wiley & Sons, Ltd.     

A least‐square weighted residual method for level set formulation

In this paper, a least‐square weighted residual method (LSWRM) for level set (LS) formulation is introduced to achieve interface capturing in two‐dimensional (2D) and three‐dimensional (3D) problems. An LSWRM was adopted for two semi‐discretized advection and reinitialization equations of the LS formulation. The present LSWRM provided good mathematical properties such as natural numerical diffusion and the symmetry of the resulting algebraic systems for the advection and reinitialization equations. The proposed method was validated by solving some 2D and 3D benchmark problems such as those involving a rotating slotted disk, the rotation of a slotted sphere, and a time‐reversed single‐vortex flow and a deformation problem of a spherical fluid. The numerical results were compared with those obtained from essentially non‐oscillatory type formulations and particle LS methods. Further, the proposed LSWRM for the LS formulation was coupled with a splitting finite element method code to solve the incompressible Navier–Stokes equations, and then, the collapse of a 3D broken dam flow was well simulated; in the simulation, the entrapping of air and the splashing of the surge front of water were reproduced. The mass conservation of the present method was found to be satisfactory during the entire simulation. Copyright © 2011 John Wiley & Sons, Ltd.     

Smoothed‐profile method for momentum and heat transfer in particulate flows

The smoothed‐profile method for the motion of solid bodies suspended in a fluid phase is investigated when combined with a high‐order spatial discretization. The performance of the combined method is tested for a wide range of flow and geometry parameters as well as for static and for moving particles. Moreover, a sensitivity analysis is conducted with respect to the smoothed‐profile function. The algorithm is extended to include thermal effects in Boussinesq approximation. Several benchmark problems are considered to demonstrate the potential of the technique. The implementation of the energy equation is verified by dedicated tests. All simulations are compared with either theoretical, numerical, or experimental data. The results demonstrate the accuracy and efficiency of the smoothed‐profile method for non‐isothermal problems in combination with a discontinuous finite‐element solver for the fluid flow, which allows for a flexible handling of the grid and the order of spectral approximation in each element. Copyright © 2016 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.     

Anisotropic adaptive stabilized finite element solver for RANS models

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

Three‐dimensional simulation of planar contraction viscoelastic flow by penalty finite element method

The planar contraction flow is a benchmark problem for the numerical investigation of viscoelastic flow. The mathematical model of three‐dimensional viscoelastic fluids flow is established and the numerical simulation of its planar contraction flow is conducted by using the penalty finite element method with a differential Phan‐Thien–Tanner constitutive model. The discrete elastic viscous split stress formulation in cooperating with the inconsistent streamline upwind scheme is employed to improve the computation stability. The distributions of velocity and stress obtained by simulation are compared with that of Quinzani's experimental results detected by laser–doppler velocimetry and flow‐induced birefringence technologies. It shows that the numerical results agree well with the experimental results. The numerical methods proposed in the study can be well used to predict complex flow patterns of viscoelastic fluids. Copyright © 2009 John Wiley & Sons, Ltd.     

An efficient and energy stable scheme for a phase‐field model for the moving contact line problem

In this paper, we propose for the first time a linearly coupled, energy stable scheme for the Navier–Stokes–Cahn–Hilliard system with generalized Navier boundary condition. We rigorously prove the unconditional energy stability for the proposed time discretization as well as for a fully discrete finite element scheme. Using numerical tests, we verify the accuracy, confirm the decreasing property of the discrete energy, and demonstrate the effectiveness of our method through numerical simulations in both 2‐D and 3‐D. Copyright © 2015 John Wiley & Sons, Ltd.