A finite element thermally coupled flow formulation for phase‐change problems

A finite element, thermally coupled incompressible flow formulation considering phase‐change effects is presented. This formulation accounts for natural convection, temperature‐dependent material properties and isothermal and non‐isothermal phase‐change models. In this context, the full Navier–Stokes equations are solved using a generalized streamline operator (GSO) technique. The highly non‐linear phase‐change effects are treated with a temperature‐based algorithm, which provides stability and convergence of the numerical solution. The Boussinesq approximation is used in order to consider the temperature‐dependent density variation. Furthermore, the numerical solution of the coupled problem is approached with a staggered incremental‐iterative solution scheme, such that the convergence criteria are written in terms of the residual vectors. Finally, this formulation is used for the solutions of solidification and melting problems validating some numerical results with other existing solutions obtained with different methodologies. Copyright © 2000 John Wiley & Sons, Ltd.     

An iterative implementation of the Uzawa algorithm for 3-D fluid flow problems

A new iterative algorithm for the solution of the three-dimensional Navier–Stokes equations by the finite element method is presented. This algorithm is based on a combination of the Uzawa and the Arrow–Hurwicz algorithms and uses a preconditioning technique to enhance convergence. Numerical tests are presented for the cubic cavity problem with two elements, namely the linear brick Q₁?P₀ and the enriched linear brick Q₁⁺ ? P₁. It is shown that the proposed methodology is optimal with the enriched element and that the CPU time varies as NEQ^1·44, where NEQ is the number of equations.     

Approximation of a Solution to the Euler Equation by Solutions of the Navier–Stokes Equation

We show that a smooth solution u ⁰ of the Euler boundary value problem on a time interval (0, T ₀) can be approximated by a family of solutions of the Navier–Stokes problem in a topology of weak or strong solutions on the same time interval (0, T ₀). The solutions of the Navier–Stokes problem satisfy Navier’s boundary condition, which must be “naturally inhomogeneous” if we deal with the strong solutions. We provide information on the rate of convergence of the solutions of the Navier–Stokes problem to the solution of the Euler problem for ν → 0. We also discuss possibilities when Navier’s boundary condition becomes homogeneous.     

Design and analysis of a Schwarz coupling method for a dimensionally heterogeneous problem

In the present work, we propose and analyse an efficient iterative coupling method for a dimensionally heterogeneous problem. We consider the case of a 2D Laplace equation with non‐symmetric boundary conditions coupled with a corresponding 1D Laplace equation. We first show how to obtain the 1D model from the 2D one by integration along one direction, by analogy with the link between shallow water equations and the Navier–Stokes system. Then we focus on the design of a Schwarz‐like iterative coupling method. We discuss the choice of boundary conditions at coupling interfaces. We prove the convergence of such algorithms and give some theoretical results related to the choice of the location of the coupling interface, and to the control of the difference between a global 2D reference solution and the 2D coupled solution. These theoretical results are illustrated numerically. Copyright © 2014 John Wiley & Sons, Ltd.     

On 3D Lagrangian Navier–Stokes α Model with a Class of Vorticity-Slip Boundary Conditions

This paper concerns the 3-dimensional Lagrangian Navier–Stokes α model and the limiting Navier–Stokes system on smooth bounded domains with a class of vorticity-slip boundary conditions and the Navier-slip boundary conditions. It establishes the spectrum properties and regularity estimates of the associated Stokes operators, the local well-posedness of the strong solution and global existence of weak solutions for initial boundary value problems for such systems. Furthermore, the vanishing α limit to a weak solution of the corresponding initial-boundary value problem of the Navier–Stokes system is proved and a rate of convergence is shown for the strong solution.     

An aspect ratio agglomeration multigrid for unstructured grids

A robust aspect ratio‐based agglomeration algorithm to generate high quality of coarse grids for unstructured and hybrid grids is proposed in this paper. The algorithm focuses on multigrid techniques for the numerical solution of Euler and Navier–Stokes equations, which conform to cell‐centered finite volume special discretization scheme, combines vertex‐based isotropic agglomeration and cell‐based directional agglomeration to yield large increases in convergence rates. Aspect ratio is used as fusing weight to capture the degree of cell convexity and give an indication of cell stretching. Agglomeration front queue is established to propagate inward from the boundaries, which stores isotropic vertex and also high‐stretched cell marked with different flag according to aspect ratio. We conduct the present method to solve Euler and Navier–Stokes equations on unstructured and hybrid grids and compare the results with single grid as well as MGridGen, which shows that the present method is efficient in reducing computational time for large‐scale system equations. Copyright © 2013 John Wiley & Sons, Ltd.     

Finite element and implicit Runge–Kutta implementation of an acoustics–convection upstream resolution algorithm for the time‐dependent two‐dimensional Euler equations

The second of a two‐paper series, this paper details a solver for the characteristics‐bias system from the acoustics–convection upstream resolution algorithm for the Euler and Navier–Stokes equations. An integral formulation leads to several surface integrals that allow effective enforcement of boundary conditions. Also presented is a new multi‐dimensional procedure to enforce a pressure boundary condition at a subsonic outlet, a procedure that remains accurate and stable. A classical finite element Galerkin discretization of the integral formulation on any prescribed grid directly yields an optimal discretely conservative upstream approximation for the Euler and Navier–Stokes equations, an approximation that remains multi‐dimensional independently of the orientation of the reference axes and computational cells. The time‐dependent discrete equations are then integrated in time via an implicit Runge–Kutta procedure that in this paper is proven to remain absolutely non‐linearly stable for the spatially‐discrete Euler and Navier–Stokes equations and shown to converge rapidly to steady states, with maximum Courant number exceeding 100 for the linearized version. Even on relatively coarse grids, the acoustics–convection upstream resolution algorithm generates essentially non‐oscillatory solutions for subsonic, transonic and supersonic flows, encompassing oblique‐ and interacting‐shock fields that converge within 40 time steps and reflect reference exact solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

Global Solutions for a Coupled Compressible Navier–Stokes/Allen–Cahn System in 1D

In this paper, we investigate a coupled compressible Navier–Stokes/Allen–Cahn system which describes the motion of a mixture of two viscous compressible fluids. We prove the existence and uniqueness of global classical solution, the existence of weak solutions and the existence of unique strong solution of the Navier–Stokes/Allen–Cahn system in 1D for initial data ρ ₀ without vacuum states.     

Adjoint‐based error estimation and mesh adaptation for hybridized discontinuous Galerkin methods

We present a robust and efficient target‐based mesh adaptation methodology, building on hybridized discontinuous Galerkin schemes for (nonlinear) convection–diffusion problems, including the compressible Euler and Navier–Stokes equations. The hybridization of finite element discretizations has the main advantage that the resulting set of algebraic equations has globally coupled degrees of freedom (DOFs) only on the skeleton of the computational mesh. Consequently, solving for these DOFs involves the solution of a potentially much smaller system. This not only reduces storage requirements but also allows for a faster solution with iterative solvers. The mesh adaptation is driven by an error estimate obtained via a discrete adjoint approach. Furthermore, the computed target functional can be corrected with this error estimate to obtain an even more accurate value. The aim of this paper is twofold: Firstly, to show the superiority of adjoint‐based mesh adaptation over uniform and residual‐based mesh refinement and secondly, to investigate the efficiency of the global error estimate. Copyright © 2014 John Wiley & Sons, Ltd.     

A preconditioned semi‐staggered dilation‐free finite volume method for the incompressible Navier–Stokes equations on all‐hexahedral elements

A new semi‐staggered finite volume method is presented for the solution of the incompressible Navier–Stokes equations on all‐quadrilateral (2D)/hexahedral (3D) meshes. The velocity components are defined at element node points while the pressure term is defined at element centroids. The continuity equation is satisfied exactly within each elements. The checkerboard pressure oscillations are prevented using a special filtering matrix as a preconditioner for the saddle‐point problem resulting from second‐order discretization of the incompressible Navier–Stokes equations. The preconditioned saddle‐point problem is solved using block preconditioners with GMRES solver. In order to achieve higher performance FORTRAN source code is based on highly efficient PETSc and HYPRE libraries. As test cases the 2D/3D lid‐driven cavity flow problem and the 3D flow past array of circular cylinders are solved in order to verify the accuracy of the proposed method. Copyright © 2005 John Wiley & Sons, Ltd.     

On coupling the Reynolds‐averaged Navier–Stokes equations with two‐equation turbulence model equations

Two methods for coupling the Reynolds‐averaged Navier–Stokes equations with the q–ω turbulence model equations on structured grid systems have been studied; namely a loosely coupled method and a strongly coupled method. The loosely coupled method first solves the Navier–Stokes equations with the turbulent viscosity fixed. In a subsequent step, the turbulence model equations are solved with all flow quantities fixed. On the other hand, the strongly coupled method solves the Reynolds‐averaged Navier–Stokes equations and the turbulence model equations simultaneously. In this paper, numerical stabilities of both methods in conjunction with the approximated factorization‐alternative direction implicit method are analysed. The effect of the turbulent kinetic energy terms in the governing equations on the convergence characteristics is also studied. The performance of the two methods is compared for several two‐ and three‐dimensional problems. Copyright © 2005 John Wiley & Sons, Ltd.     

A NON–LINEAR ADAPTED TRI–TREE MULTIGRID SOLVER FOR FINITE ELEMENT FORMULATION OF THE NAVIER–STOKES EQUATIONS

An iterative adaptive equation multigrid solver for solving the implicit Navier–Stokes equations simultaneously with tri-tree grid generation is developed. The tri-tree grid generator builds a hierarchical grid structur e which is mapped to a finite element grid at each hierarchical level. For each hierarchical finite element multigrid the Navier–Stokes equations are solved approximately. The solution at each level is projected onto the next finer grid and used as a start vector for the iterative equation solver at the finer level. When the finest grid is reached, the equation solver is iterated until a tolerated solution is reached. The iterative multigrid equation solver is preconditioned by incomplete LU factorization with coupled node fill-in. The non-linear Navier–Stokes equations are linearized by both the Newton method and grid adaption. The efficiency and behaviour of the present adaptive method are compared with those of the previously developed iterative equation solver which is preconditioned by incomplete LU factorization with coupled node fill-in.     

Flow simulation on moving boundary‐fitted grids and application to fluid–structure interaction problems

We present a method for the parallel numerical simulation of transient three‐dimensional fluid–structure interaction problems. Here, we consider the interaction of incompressible flow in the fluid domain and linear elastic deformation in the solid domain. The coupled problem is tackled by an approach based on the classical alternating Schwarz method with non‐overlapping subdomains, the subproblems are solved alternatingly and the coupling conditions are realized via the exchange of boundary conditions. The elasticity problem is solved by a standard linear finite element method. A main issue is that the flow solver has to be able to handle time‐dependent domains. To this end, we present a technique to solve the incompressible Navier–Stokes equation in three‐dimensional domains with moving boundaries. This numerical method is a generalization of a finite volume discretization using curvilinear coordinates to time‐dependent coordinate transformations. It corresponds to a discretization of the arbitrary Lagrangian–Eulerian formulation of the Navier–Stokes equations. Here the grid velocity is treated in such a way that the so‐called Geometric Conservation Law is implicitly satisfied. Altogether, our approach results in a scheme which is an extension of the well‐known MAC‐method to a staggered mesh in moving boundary‐fitted coordinates which uses grid‐dependent velocity components as the primary variables. To validate our method, we present some numerical results which show that second‐order convergence in space is obtained on moving grids. Finally, we give the results of a fully coupled fluid–structure interaction problem. It turns out that already a simple explicit coupling with one iteration of the Schwarz method, i.e. one solution of the fluid problem and one solution of the elasticity problem per time step, yields a convergent, simple, yet efficient overall method for fluid–structure interaction problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Parallel‐in‐time simulation of the unsteady Navier–Stokes equations for incompressible flow

In this paper, we present an application of a parallel‐in‐time algorithm for the solution of the unsteady Navier–Stokes model equations that are of parabolic–elliptic type. This method is based on the alternated use of a coarse global sequential solver and a fine local parallel one. A standard finite volume/finite differences first‐order approach is used for discretization of the unsteady two‐dimensional Navier–Stokes equations. The Taylor vortex decay problem and the confined flow around a square cylinder were selected as unsteady flow examples to illustrate and analyse the properties of the parallel‐in‐time method through numerical experiments. The influence of several parameters on the computing time required to perform a parallel‐in‐time calculation on a PC cluster was verified. Among them we have analysed the influence of the number of processors, the number of iterations for convergence, the resolution of the spatial domain and the influence of the time‐step sizes ratio between the coarse and fine grids. Significant computer time saving was achieved when compared with the single processor computing time, particularly when the spatial dimension of the problem is low and the temporal scale is large. Copyright © 2004 John Wiley & Sons, Ltd.     

A discussion and comparison of numerical techniques used to solve the Navier–Stokes and Euler equations

This paper is intended to provide some background to a number of widely used methods for solving the Navier–Stokes and Euler equations. The difference between coupled and uncoupled iterative schemes is discussed together with methods for solving the equations. Methods covered include time marching (both explicit and implicit), pressure correction and a Newton–Raphson technique. The relationship between the methods is illustrated.     

Spectral p‐multigrid discontinuous Galerkin solution of the Navier–Stokes equations

Discontinuous Galerkin (DG) methods are very well suited for the construction of very high‐order approximations of the Euler and Navier–Stokes equations on unstructured and possibly nonconforming grids, but are rather demanding in terms of computational resources. In order to improve the computational efficiency of this class of methods, a high‐order spectral element DG approximation of the Navier–Stokes equations coupled with a p‐multigrid solution strategy based on a semi‐implicit Runge–Kutta smoother is considered here. The effectiveness of the proposed approach in the solution of compressible shockless flow problems is demonstrated on 2D inviscid and viscous test cases by comparison with both a p‐multigrid scheme with non‐spectral elements and a spectral element DG approach with an implicit time integration scheme. Copyright © 2010 John Wiley & Sons, Ltd.     

Stabilisation of AMG solvers for saddle‐point stokes problems

When a block factorisation is used to precondition the saddle‐point equations of the discrete Stokes problem, the stability that this gives for the relaxation of residual errors may not be conserved in the coarse‐grid approximations (CGA) of algebraic multi‐grid (AMG) solvers. If the same first‐order interpolation is used in the inter‐grid transfer operators for the scalar and the vector fields, the conditioning degrades with each coarsening step until eventually a critical coarsening is reached beyond which residual errors are no longer damped and will become divergent with any further coarsening. It is shown that by introducing the same block pre‐conditioner as an integral part of the coarsening algorithm, stable smoothing can be maintained at all levels of the CGA. The pre‐conditioning need only be applied at preselected grid levels, one immediately before the critical threshold and others beyond that level if required. Excessive complexity in the CGA is thereby avoided. The method is purely algebraic and may be used for both classical AMG solvers and for smoothed‐aggregation AMG solvers. It should be applicable to other coupled vector and scalar fields in science and engineering that involve second‐order (block‐diagonal) and first‐order (block‐off‐diagonal) discrete difference operators. Copyright © 2015 John Wiley & Sons, Ltd.     

Coupled Navier–Stokes—Molecular dynamics simulations using a multi‐physics flow simulation framework

Simulation of nano‐scale channel flows using a coupled Navier–Stokes/Molecular Dynamics (MD) method is presented. The flow cases serve as examples of the application of a multi‐physics computational framework put forward in this work. The framework employs a set of (partially) overlapping sub‐domains in which different levels of physical modelling are used to describe the flow. This way, numerical simulations based on the Navier–Stokes equations can be extended to flows in which the continuum and/or Newtonian flow assumptions break down in regions of the domain, by locally increasing the level of detail in the model. Then, the use of multiple levels of physical modelling can reduce the overall computational cost for a given level of fidelity. The present work describes the structure of a parallel computational framework for such simulations, including details of a Navier–Stokes/MD coupling, the convergence behaviour of coupled simulations as well as the parallel implementation. For the cases considered here, micro‐scale MD problems are constructed to provide viscous stresses for the Navier–Stokes equations. The first problem is the planar Poiseuille flow, for which the viscous fluxes on each cell face in the finite‐volume discretization are evaluated using MD. The second example deals with fully developed three‐dimensional channel flow, with molecular level modelling of the shear stresses in a group of cells in the domain corners. An important aspect in using shear stresses evaluated with MD in Navier–Stokes simulations is the scatter in the data due to the sampling of a finite ensemble over a limited interval. In the coupled simulations, this prevents the convergence of the system in terms of the reduction of the norm of the residual vector of the finite‐volume discretization of the macro‐domain. Solutions to this problem are discussed in the present work, along with an analysis of the effect of number of realizations and sample duration. The averaging of the apparent viscosity for each cell face, i.e. the ratio of the shear stress predicted from MD and the imposed velocity gradient, over a number of macro‐scale time steps is shown to be a simple but effective method to reach a good level of convergence of the coupled system. Finally, the parallel efficiency of the developed method is demonstrated. Copyright © 2009 John Wiley & Sons, Ltd.     

ℋ︁‐LU factorization in preconditioners for augmented Lagrangian and grad‐div stabilized saddle point systems

The (mixed finite element) discretization of the linearized Navier–Stokes equations leads to a linear system of equations of saddle point type. The iterative solution of this linear system requires the construction of suitable preconditioners, especially in the case of high Reynolds numbers. In the past, a stabilizing approach has been suggested which does not change the exact solution but influences the accuracy of the discrete solution as well as the effectiveness of iterative solvers. This stabilization technique can be performed on the continuous side before the discretization, where it is known as ‘grad‐div’ (GD) stabilization, as well as on the discrete side where it is known as an ‘augmented Lagrangian’ (AL) technique (and does not change the discrete solution). In this paper, we study the applicability of ??‐LU factorizations to solve the arising subproblems in the different variants of stabilized saddle point systems. We consider both the saddle point systems that arise from the stabilization in the continuous as well as on the discrete setting. Recently, a modified AL preconditioner has been proposed for the system resulting from the discrete stabilization. We provide a straightforward generalization of this approach to the GD stabilization. We conclude the paper with numerical tests for a variety of problems to illustrate the behavior of the considered preconditioners as well as the suitability of ??‐LU factorization in the preconditioners. Copyright © 2010 John Wiley & Sons, Ltd.     

An iterative stabilized fractional step algorithm for numerical solution of incompressible N–S equations

Stabilized fractional step algorithm has been widely employed for numerical solution of incompressible Navier–Stokes equations. However, smaller time step sizes are required to use for existing explicit and semi‐implicit versions of the algorithm due to their fully or partially explicit nature particularly for highly viscous flow problems. The purpose of this paper is to present two modified versions of the fractional step algorithm using characteristic based split and Taylor–Galerkin like based split. The proposed modified versions of the algorithm are based on introducing an iterative procedure into the algorithm and allow much larger time step sizes than those required to the preceding ones. A numerical study of stability at acceptable convergence rate and accuracy as well as capability in circumventing the restriction imposed by the LBB condition for the proposed iterative versions of the algorithm is carried out with the plane Poisseuille flow problem under different Reynolds numbers ranging from low to high viscosities. Numerical experiments in the plane Poisseuille flow and the lid‐driven cavity flow problems demonstrate the improved performance of the proposed versions of the algorithm, which are further applied to numerical simulation of the polymer injection moulding process. Copyright © 2005 John Wiley & Sons, Ltd.