A multi‐level adaptive mesh refinement method for level set simulations of multiphase flow on unstructured meshes

An adaptive mesh refinement (AMR) technique is proposed for level set simulations of incompressible multiphase flows. The present AMR technique is implemented for two‐dimensional/three‐dimensional unstructured meshes and extended to multi‐level refinement. Smooth variation of the element size is guaranteed near the interface region with the use of multi‐level refinement. A Courant–Friedrich–Lewy condition for zone adaption frequency is newly introduced to obtain a mass‐conservative solution of incompressible multiphase flows. Finite elements around the interface are dynamically refined using the classical element subdivision method. Accordingly, finite element method is employed to solve the problems governed by the incompressible Navier–Stokes equations, using the level set method for dynamically updated meshes. The accuracy of the adaptive solutions is found to be comparable with that of non‐adaptive solutions only if a similar mesh resolution near the interface is provided. Because of the substantial reduction in the total number of nodes, the adaptive simulations with two‐level refinement used to solve the incompressible Navier–Stokes equations with a free surface are about four times faster than the non‐adaptive ones. Further, the overhead of the present AMR procedure is found to be very small, as compared with the total CPU time for an adaptive simulation. Copyright © 2016 John Wiley & Sons, Ltd.     

Immersed stress method for fluid–structure interaction using anisotropic mesh adaptation

This paper presents advancements toward a monolithic solution procedure and anisotropic mesh adaptation for the numerical solution of fluid–structure interaction with complex geometry. First, a new stabilized three‐field stress, velocity, and pressure finite element formulation is presented for modeling the interaction between the fluid (laminar or turbulent) and the rigid body. The presence of the structure will be taken into account by means of an extra stress in the Navier–Stokes equations. The system is solved using a finite element variational multiscale method. We combine this method with anisotropic mesh adaptation to ensure an accurate capturing of the discontinuities at the fluid–solid interface. We assess the behavior and accuracy of the proposed formulation in the simulation of 2D and 3D time‐dependent numerical examples such as the flow past a circular cylinder and turbulent flows behind an immersed helicopter in a forward flight. Copyright © 2013 John Wiley & Sons, Ltd.     

High‐order finite elements applied to the discrete Boltzmann equation

A discontinuous Galerkin approach for solving the discrete Boltzmann equation is presented, allowing to compute approximate solutions for fluid flow problems. Based on a two‐dimensional high‐order finite element and an explicit Euler time stepping scheme, the D2Q9 model is discretized and the results are compared to the exact solution of the Navier–Stokes equation. Four numerical examples are considered, including stationary and instationary problems with curved boundaries. It is demonstrated that the proposed method allows to obtain the desired, highly efficient exponential convergence. Copyright © 2006 John Wiley & Sons, Ltd.     

ANM for stationary Navier–Stokes equations and with Petrov–Galerkin formulation

This paper deals with the use of the asymptotic numerical method (ANM) for solving non‐linear problems, with particular emphasis on the stationary Navier–Stokes equation and the Petrov–Galerkin formulation. ANM is a combination of a perturbation technique and a finite element method allowing to transform a non‐linear problem into a succession of linear ones that admit the same tangent matrix. This method has been applied with success in non‐linear elasticity and fluid mechanics. In this paper, we apply the same kind of technique for solving Navier–Stokes equation with the so‐called Petrov–Galerkin weighting. The main difficulty comes from the fact that the non‐linearity is no more quadratic and it is not evident, in this case, to be able to compute a large number of terms of the perturbation series. Several examples of fluid mechanic are presented to demonstrate the performance of such a method. Copyright © 2001 John Wiley & Sons, Ltd.     

A three-step Oseen correction method for the steady Navier–Stokes equations

We present and analyze a two-grid scheme based on mixed finite element approximations for the steady incompressible Navier–Stokes equations. This numerical scheme aims at the simulations of high Reynolds number flows and consists of three steps: in the first step, we solve a finite element variational multiscale-stabilized nonlinear Navier–Stokes system on a coarse mesh, and then, in the second and third steps, we solve Oseen-linearized and -stabilized problems which have the same stiffness matrices with only different right-hand sides on a fine mesh. We provide error bounds for the approximate solutions, derive algorithmic parameter scalings from the analysis, and present some numerical results to verify the theoretical predictions and demonstrate the effectiveness of the proposed method.     

Low‐Reynolds‐number flow around an oscillating circular cylinder using a cell viscousboundary element method

Flow fields from transversely oscillating circular cylinders in water at rest are studied by numerical solutions of the two‐dimensional unsteady incompressible Navier–Stokes equations adopting a primitive‐variable formulation. These findings are successfully compared with experimental observations. The cell viscous boundary element scheme developed is first validated to examine convergence of solution and the influence of discretization within the numerical scheme of study before the comparisons are undertaken. A hybrid approach utilising boundary element and finite element methods is adopted in the cell viscous boundary element method. That is, cell equations are generated using the principles of a boundary element method with global equations derived following the procedures of finite element methods. The influence of key parameters, i.e. Reynolds number Re, Keulegan–Carpenter number KC and Stokes' number β, on overall flow characteristics and vortex shedding mechanisms are investigated through comparisons with experimental findings and theoretical predictions. The latter extends the study into assessment of the values of the drag coefficient, added mass or inertia coefficient with key parameters and the variation of lift and in‐line force results with time derived from the Morison's equation. The cell viscous boundary element method as described herein is shown to produce solutions which agree very favourably with experimental observations, measurements and other theoretical findings. Copyright © 2001 John Wiley & Sons, Ltd.     

Finite element solution of free‐surface ship‐wave problems

An unstructured finite element solver to evaluate the ship‐wave problem is presented. The scheme uses a non‐structured finite element algorithm for the Euler or Navier–Stokes flow as for the free‐surface boundary problem. The incompressible flow equations are solved via a fractional step method whereas the non‐linear free‐surface equation is solved via a reference surface which allows fixed and moving meshes. A new non‐structured stabilized approximation is used to eliminate spurious numerical oscillations of the free surface. Copyright © 1999 John Wiley & Sons, Ltd.     

Application of the level set method to the finite element solution of two‐phase flows

This paper presents a method to solve two‐phase flows using the finite element method. On one hand, the algorithm used to solve the Navier–Stokes equations provides the neccessary stabilization for using the efficient and accurate three‐node triangles for both the velocity and pressure fields. On the other hand, the interface position is described by the zero‐level set of an indicator function. To maintain accuracy, even for large‐density ratios, the pseudoconcentration function is corrected at the end of each time step using an algorithm successfully used in the finite difference context. Coupling of both problems is solved in a staggered way. As demonstrated by the solution of a number of numerical tests, the procedure allows dealing with problems involving two interacting fluids with a large‐density ratio. Copyright © 2001 John Wiley & Sons, Ltd.     

A combined BEM–FEM model for the velocity–vorticity formulation of the Navier–Stokes equations in three dimensions

This paper describes a combined boundary element and finite element model for the solution of velocity–vorticity formulation of the Navier–Stokes equations in three dimensions. In the velocity–vorticity formulation of the Navier–Stokes equations, the Poisson type velocity equations are solved using the boundary element method (BEM) and the vorticity transport equations are solved using the finite element method (FEM) and both are combined to form an iterative scheme. The vorticity boundary conditions for the solution of vorticity transport equations are exactly obtained directly from the BEM solution of the velocity Poisson equations. Here the results of medium Reynolds number of up to 1000, in a typical cubic cavity flow are presented and compared with other numerical models. The combined BEM–FEM model are generally in fairly close agreement with the results of other numerical models, even for a coarse mesh.     

A discrete splitting finite element method for numerical simulations of incompressible Navier–Stokes flows

The presence of the pressure and the convection terms in incompressible Navier–Stokes equations makes their numerical simulation a challenging task. The indefinite system as a consequence of the absence of the pressure in continuity equation is ill‐conditioned. This difficulty has been overcome by various splitting techniques, but these techniques incur the ambiguity of numerical boundary conditions for the pressure as well as for the intermediate velocity (whenever introduced). We present a new and straightforward discrete splitting technique which never resorts to numerical boundary conditions. The non‐linear convection term can be treated by four different approaches, and here we present a new linear implicit time scheme. These two new techniques are implemented with a finite element method and numerical verifications are made. Copyright © 2005 John Wiley & Sons, Ltd.     

Non‐weak/strong solutions in gas dynamics: a C11 p‐version STLSFEF in Eulerian frame of reference using ρ, u,p primitive variables

This paper presents a space–time least squares finite element formulation of one‐dimensional transient Navier–Stokes equations (governing differential equations: GDE) for compressible flow in Eulerian frame of reference using ρ, u, p as primitive variables with C¹¹ type p‐version hierarchical interpolations in space and time. Time marching procedure is utilized to compute time evolutions for all values of time. For high speed gas dynamics the C¹¹ type interpolations in space and time possess the same orders of continuity in space and time as the GDE. It is demonstrated that with this approach accurate numerical solutions of Navier–Stokes equations are possible without any assumptions or approximations. In the approach presented here SUPG, SUPG/DC, SUPG/DC/LS operators are neither used nor needed. Time accurate numerical simulations show resolution of shock structure (i.e. shock speed, shock relations and shock width) to be in excellent agreement with the analytical solutions. The role of diffusion i.e. viscosity (physical or artificial) and thermal conductivity on shock structure is demonstrated. Riemann shock tube is used as a model problem. True time evolutions are reported beginning with the first time step until steady shock conditions are achieved. In this approach, when the computed error functionals become zero (computationally), the computed non‐weak solutions have characteristics as those of the strong solutions of the gas dynamics equations. Copyright © 2001 John Wiley & Sons, Ltd.     

Optimal control finite element approximation for penalty variational formulation of three-dimensional Navier–Stokes problem

An optimal control finite element approximation of the penalization type for the solution of incompressible viscous Navier–Stokes equations is presented. This paper has proved the convergence of its minimizing sequence and the existence of the solution for the optimal control problem corresponding to the penalty variational problem of the Navier–Stokes equations. Based on this method and the conjugate gradient algorithm, the program for solving the three-dimensional Navier–Stokes problem has been developed, and some numerical results have also been provided.     

An essentially non‐oscillatory semi‐Lagrangian method for tidal flow simulations

We develop an essentially non‐oscillatory semi‐Lagrangian method for solving two‐dimensional tidal flows. The governing equations are derived from the incompressible Navier–Stokes equations with assumptions of shallow water flows including bed frictions, eddy viscosity, wind shear stresses and Coriolis forces. The method employs the modified method of characteristics to discretize the convective term in a finite element framework. Limiters are incorporated in the method to reconstruct an essentially non‐oscillatory algorithm at minor additional cost. The central idea consists in combining linear and quadratic interpolation procedures using nodes of the finite element where departure points are localized. The resulting semi‐discretized system is then solved by an explicit Runge–Kutta Chebyshev scheme with extended stages. This scheme adds in a natural way a stabilizing stage to the conventional Runge–Kutta method using the Chebyshev polynomials. The proposed method is verified for the recirculation tidal flow in a channel with forward‐facing step. We also apply the method for simulation of tidal flows in the Strait of Gibraltar. In both test problems, the proposed method demonstrates its ability to handle the interaction between water free‐surface and bed frictions. Copyright © 2009 John Wiley & Sons, Ltd.     

An alternative approach for numerical solutions of the Navier–Stokes equations

Conventional approaches for solving the Navier–Stokes equations of incompressible fluid dynamics are the primitive‐variable approach and the vorticity–velocity approach. In this paper, an alternative approach is presented. In this approach, pressure and one of the velocity components are eliminated from the governing equations. The result is one higher‐order partial differential equation with one unknown for two‐dimensional problems or two higher‐order partial differential equations with two unknowns for three‐dimensional problems. A meshless collocation method based on radial basis functions for solving the Navier–Stokes equations using this approach is presented. The proposed method is used to solve a two‐ and a three‐dimensional test problem of which exact solutions are known. It is found that, with appropriate values of the method parameters, solutions of satisfactory accuracy can be obtained. Copyright © 2006 John Wiley & Sons, Ltd.     

Output error estimation strategies for discontinuous Galerkin discretizations of unsteady convection‐dominated flows

We study practical strategies for estimating numerical errors in scalar outputs calculated from unsteady simulations of convection‐dominated flows, including those governed by the compressible Navier–Stokes equations. The discretization is a discontinuous Galerkin finite element method in space and time on static spatial meshes. Time‐integral quantities are considered for scalar outputs and these are shown to superconverge with temporal refinement. Output error estimates are calculated using the adjoint‐weighted residual method, where the unsteady adjoint solution is obtained using a discrete approach with an iterative solver. We investigate the accuracy versus computational cost trade‐off for various approximations of the fine‐space adjoint and find that exact adjoint solutions are accurate but expensive. To reduce the cost, we propose a local temporal reconstruction that takes advantage of superconvergence properties at Radau points, and a spatial reconstruction based on nearest‐neighbor elements. This inexact adjoint yields output error estimates at a computational cost of less than 2.5 times that of the forward problem for the cases tested. The calculated error estimates account for numerical error arising from both the spatial and temporal discretizations, and we present a method for identifying the percentage contributions of each discretization to the output error. Copyright © 2011 John Wiley & Sons, Ltd.     

A new least-squares finite element model for combined Navier–Stokes and Darcy flows in geometrically complicated domains with solid and porous boundaries

In this paper we present a novel method for linking Navier–Stokes and Darcy equations along a porous inner boundary in a flow regime which is governed by both types of these equations. The method is based on a least-squares finite element technique and uses isoparametric C¹ continuous Hermite elements for domain discretization. We show that our technique is superior to previously developed models for the combined Navier–Stokes/Darcy flows. The previous works use weighted residual finite element procedures in conjunction with C⁰ elements which are inherently incapable of linking Navier–Stokes and Darcy equations. The paper includes the application of our model to a geometrically complicated axisymmetric slurry filtration system.     

A Jacobian‐free‐based IIM for incompressible flows involving moving interfaces with Dirichlet boundary conditions

In this paper, a finite difference marker‐and‐cell (MAC) scheme is presented for the steady Stokes equations with moving interfaces and Dirichlet boundary condition. The moving interfaces are represented by Lagrangian control points and their position is updated implicitly using a Jacobian‐free approach within each time step. The forces at the moving interfaces are calculated from the position of the interfaces and interpolated using cubic splines and then applied to the fluid through the related jump conditions. The proposed Jacobian‐free Newton–generalized minimum residual (GMRES) method avoids the need to form and store the matrix explicitly in the computation of the inverse of the Jacobian and betters numerical stability. The Stokes equations are discretized on a MAC grid via a second‐order finite difference scheme with the incorporation of jump contributions and the resulting saddle point system is solved by the conjugate gradient Uzawa‐type method. Numerical results demonstrate very well the accuracy and effectiveness of the proposed method. The present algorithm has been applied to solve incompressible Navier–Stokes flows with moving interfaces. Copyright © 2010 John Wiley & Sons, Ltd.     

BEM simulations of potential flow with viscous effects as applied to a rising bubble

A model for the unsteady rise and deformation of non-oscillating bubbles under buoyancy force at high Reynolds numbers has been implemented using a boundary element method. Results such as the evolution of the bubble shape, variations of the transient velocity with rise height and the terminal velocity for different size bubbles have been compared to recent experimental data in clean water and to numerical solutions of the unsteady Navier–Stokes equation. The aim is to capture the essential physical ingredients that couple bubble deformation and the transient approach towards terminal velocity. This model requires very modest computational resources and yet has the flexibility to be extended to more general applications.     

A posteriori finite element bounds to linear functional outputs of the three‐dimensional Navier–Stokes equations

An implicit a posteriori finite element error estimation method is presented to inexpensively calculate lower and upper bounds for a linear functional output of the numerical solutions to the three‐dimensional Navier–Stokes (N–S) equations. The novelty of this research is to utilize an augmented Lagrangian based on a coarse mesh linearization of the N–S equations and the finite element tearing and interconnecting (FETI) procedure. The latter approach extends the a posteriori bound method to the three‐dimensional Crouzeix–Raviart space for N–S problems. The computational advantage of the bound procedure is that a single coupled non‐symmetric large problem can be decomposed into several uncoupled symmetric small problems. A simple model problem, which is selected here to illustrate the procedure, is to find the lower and upper bounds of the average velocity of a pressure driven, incompressible, steady Newtonian fluid flow moving at low Reynolds numbers through an endless square channel which has an array of rectangular obstacles. Numerical results show that the bounds for this output are rigorous, i.e. always in the asymptotic certainty regime, that they are sharp and that the required computational resources decrease significantly. Parallel implementation on a Beowulf cluster is also reported. Copyright © 2004 John Wiley & Sons, Ltd.     

The enriched space–time finite element method (EST) for simultaneous solution of fluid–structure interaction

The paper introduces a weighted residual‐based approach for the numerical investigation of the interaction of fluid flow and thin flexible structures. The presented method enables one to treat strongly coupled systems involving large structural motion and deformation of multiple‐flow‐immersed solid objects. The fluid flow is described by the incompressible Navier–Stokes equations. The current configuration of the thin structure of linear elastic material with non‐linear kinematics is mapped to the flow using the zero iso‐contour of an updated level set function. The formulation of fluid, structure and coupling conditions uniformly uses velocities as unknowns. The integration of the weak form is performed on a space–time finite element discretization of the domain. Interfacial constraints of the multi‐field problem are ensured by distributed Lagrange multipliers. The proposed formulation and discretization techniques lead to a monolithic algebraic system, well suited for strongly coupled fluid–structure systems. Embedding a thin structure into a flow results in non‐smooth fields for the fluid. Based on the concept of the extended finite element method, the space–time approximations of fluid pressure and velocity are properly enriched to capture weakly and strongly discontinuous solutions. This leads to the present enriched space–time (EST) method. Numerical examples of fluid–structure interaction show the eligibility of the developed numerical approach in order to describe the behavior of such coupled systems. The test cases demonstrate the application of the proposed technique to problems where mesh moving strategies often fail. Copyright © 2007 John Wiley & Sons, Ltd.