Stretched Cartesian grids for solution of the incompressible Navier–Stokes equations

Two Cartesian grid stretching functions are investigated for solving the unsteady incompressible Navier–Stokes equations using the pressure–velocity formulation. The first function is developed for the Fourier method and is a generalization of earlier work. This function concentrates more points at the centre of the computational box while allowing the box to remain finite. The second stretching function is for the second‐order central finite difference scheme, which uses a staggered grid in the computational domain. This function is derived to allow a direct discretization of the Laplacian operator in the pressure equation while preserving the consistent behaviour exhibited by the uniform grid scheme. Both functions are analysed for their effects on the matrix of the discretized pressure equation. It is shown that while the second function does not spoil the matrix diagonal dominance, the first one can. Limits to stretching of the first method are derived for the cases of mappings in one and two directions. A limit is also derived for the second function in order to prevent a strong distortion of a sine wave. The performances of the two types of stretching are examined in simulations of periodic co‐flowing jets and a time developing boundary layer. Copyright © 2000 John Wiley & Sons, Ltd.     

Lattice‐Boltzmann and meshless point collocation solvers for fluid flow and conjugate heat transfer

The applicability and performance of the lattice‐Boltzmann (LB) and meshless point collocation methods as CFD solvers in flow and conjugate heat transfer processes are investigated in this work. Lid‐driven cavity flow and flow in a slit with an obstacle including heat transfer are considered as case studies. A comparison of the computational efficiency accuracy of the two methods with that of a finite volume method as implemented in a commercial package (ANSYS CFX, ANSYS Inc., Canonsburg, PA) is made. Utilizing the analogy between heat and mass transfer, an advection–diffusion LB model was adopted to simulate the heat transfer part of the slit flow problem followed by a rigorous mapping of the mass transfer variables to the heat transfer quantities of interest, thus circumventing the need for a thermal LB model. Direct comparison among the results of the three methods revealed excellent agreement over a wide range of Reynolds and Prandtl number values. Furthermore, an integrated computational scheme is proposed, utilizing the rapid convergence of the LB model in the flow part of the conjugate heat transfer problem with that of the meshless collocation method for the heat transfer part. The meshless treatment remains sufficiently rapid even for conduction‐controlled processes in contrast to the LB method, which is very rapid in the convection‐controlled case only. A single, common computational grid, composed of regularly distributed nodes is used, saving significant computational and coding time and ensuring convergence of the discrete Laplacian operator in the heat transfer part of the computations. Copyright © 2012 John Wiley & Sons, Ltd.     

A non‐diffusive,divergence‐free,finite volume‐based double projection method on non‐staggered grids

Second‐order accurate projection methods for simulating time‐dependent incompressible flows on cell‐centred grids substantially belong to the class either of exact or approximate projections. In the exact method, the continuity constraint can be satisfied to machine‐accuracy but the divergence and Laplacian operators show a four‐dimension nullspace therefore spurious oscillating solutions can be introduced. In the approximate method, the continuity constraint is relaxed, the continuity equation being satisfied up to the magnitude of the local truncation error, but the compact Laplacian operator has only the constant mode. An original formulation for allowing the discrete continuity equation to be satisfied to machine‐accuracy, while using a finite volume based projection method, is illustrated. The procedure exploits the Helmholtz–Hodge decomposition theorem for deriving an additional velocity field that enforces the discrete continuity without altering the vorticity field. This is accomplished by solving a second elliptic field for a scalar field obtained by prescribing that its additional discrete gradients ensure discrete continuity based on the previously adopted linear interpolation of the velocity. The resulting numerical scheme is applied to several flow problems and is proved to be accurate, stable and efficient. This paper has to be considered as the companion of: 'F. M. Denaro, A 3D second‐order accurate projection‐based finite volume code on non‐staggered, non‐uniform structured grids with continuity preserving properties: application to buoyancy‐driven flows. IJNMF 2006; 52 (4):393–432. Now, we illustrate the details and the rigorous theoretical framework. Copyright © 2006 John Wiley & Sons, Ltd.     

Simulation of free and forced convection incompressible flows using an adaptive parallel/vector finite element procedure

The numerical simulation of complex flows demands efficient algorithms and fast computer platforms. The use of adaptive techniques permits adjusting the discretisation according to the analysis requirements, but creates variable computational loads that are difficult to manage in a parallel/vector program. This paper describes the approach adopted to implement an adaptive finite element incompressible Navier–Stokes solver on the Cray J90 machine. Performance measurements for the simulation of free and forced convection incompressible flows indicate that the techniques employed result in a fast parallel/vector code. Copyright © 1999 John Wiley & Sons, Ltd.     

A nodal‐based implementation of a stabilized finite element method for incompressible flow problems

The objective of this paper is twofold. First, a stabilized finite element method (FEM) for the incompressible Navier–Stokes is presented and several numerical experiments are conducted to check its performance. This method is capable of dealing with all the instabilities that the standard Galerkin method presents, namely the pressure instability, the instability arising in convection‐dominated situations and the less popular instabilities found when the Navier–Stokes equations have a dominant Coriolis force or when there is a dominant absorption term arising from the small permeability of the medium where the flow takes place. The second objective is to describe a nodal‐based implementation of the finite element formulation introduced. This implementation is based on an a priori calculation of the integrals appearing in the formulation and then the construction of the matrix and right‐hand side vector of the final algebraic system to be solved. After appropriate approximations, this matrix and this vector can be constructed directly for each nodal point, without the need to loop over the elements, thus making the calculations much faster. In order to be able to do this, all the variables have to be defined at the nodes of the finite element mesh, not on the elements. This is also so for the stabilization parameters of the formulation. However, doing this gives rise to questions regarding the consistency and the conservation properties of the final scheme, which are addressed in this paper. Copyright © 2000 John Wiley & Sons, Ltd.     

Development of a convection–diffusion–reaction model for solving Maxwell's equations in frequency domain

We propose in this study a numerically accurate and computationally efficient convection–diffusion–reaction finite difference scheme to discretize the full‐vector and semi‐vector optical waveguide equations. The scheme formulated in a grid stencil of five nodal points for solving the three‐dimensional waveguide equations employs the locally analytic solution. In this three‐dimensional study, calculations were carried out for the investigation of wave propagation in diffused channel, rectangular and rib types of optical waveguide. Copyright © 2011 John Wiley & Sons, Ltd.     

A substructure‐based iterative inner solver coupled with Uzawa's algorithm for the Stokes problem

A domain decomposition method with Lagrange multipliers for the Stokes problem is developed and analysed. A common approach to solve the Stokes problem, termed the Uzawa algorithm, is to decouple the velocity and the pressure. This approach yields the Schur complement system for the pressure Lagrange multiplier which is solved with an iterative solver. Each outer iteration of the Uzawa procedure involves the inversion of a Laplacian in each spatial direction. The objective of this paper is to effectively solve this inner system (the vector Laplacian system) by applying the finite‐element tearing and interconnecting (FETI) method. Previously calculated search directions for the FETI solver are reused in subsequent outer Uzawa iterations. The advantage of the approach proposed in this paper is that pressure is continuous across the entire computational domain. Numerical tests are performed by solving the driven cavity problem. An analysis of the number of outer Uzawa iterations and inner FETI iterations is reported. Results show that the total number of inner iterations is almost numerically scalable since it grows asymptotically with the mesh size and the number of subdomains. Copyright © 2003 John Wiley & Sons, Ltd.     

A new stable space–time formulation for two‐dimensional and three‐dimensional incompressible viscous flow

A space–time finite element method for the incompressible Navier–Stokes equations in a bounded domain in ?^d (with d=2 or 3) is presented. The method is based on the time‐discontinuous Galerkin method with the use of simplex‐type meshes together with the requirement that the space–time finite element discretization for the velocity and the pressure satisfy the inf–sup stability condition of Brezzi and Babu?ka. The finite element discretization for the pressure consists of piecewise linear functions, while piecewise linear functions enriched with a bubble function are used for the velocity. The stability proof and numerical results for some two‐dimensional problems are presented. Copyright © 2001 John Wiley & Sons, Ltd.     

Implementation of a stabilized finite element formulation for the incompressible Navier–Stokes equations based on a pressure gradient projection

We discuss in this paper some implementation aspects of a finite element formulation for the incompressible Navier–Stokes equations which allows the use of equal order velocity–pressure interpolations. The method consists in introducing the projection of the pressure gradient and adding the difference between the pressure Laplacian and the divergence of this new field to the incompressibility equation, both multiplied by suitable algorithmic parameters. The main purpose of this paper is to discuss how to deal with the new variable in the implementation of the algorithm. Obviously, it could be treated as one extra unknown, either explicitly or as a condensed variable. However, we take for granted that the only way for the algorithm to be efficient is to uncouple it from the velocity–pressure calculation in one way or another. Here we discuss some iterative schemes to perform this uncoupling of the pressure gradient projection (PGP) from the calculation of the velocity and the pressure, both for the stationary and the transient Navier–Stokes equations. In the first case, the strategies analyzed refer to the interaction of the linearization loop and the iterative segregation of the PGP, whereas in the second the main dilemma concerns the explicit or implicit treatment of the PGP. Copyright © 2001 John Wiley & Sons, Ltd.     

The two‐dimensional streamline upwind scheme for the convection–reaction equation

This paper is concerned with the development of the finite element method in simulating scalar transport, governed by the convection–reaction (CR) equation. A feature of the proposed finite element model is its ability to provide nodally exact solutions in the one‐dimensional case. Details of the derivation of the upwind scheme on quadratic elements are given. Extension of the one‐dimensional nodally exact scheme to the two‐dimensional model equation involves the use of a streamline upwind operator. As the modified equations show in the four types of element, physically relevant discretization error terms are added to the flow direction and help stabilize the discrete system. The proposed method is referred to as the streamline upwind Petrov–Galerkin finite element model. This model has been validated against test problems that are amenable to analytical solutions. In addition to a fundamental study of the scheme, numerical results that demonstrate the validity of the method are presented. Copyright © 2001 John Wiley & Sons, Ltd.     

Analysis of the local truncation error in the pressure‐free projection method for incompressible flows: a new accurate expression of the intermediate boundary conditions

The numerical integration of the Navier–Stokes equations for incompressible flows demands efficient and accurate solution algorithms for pressure–velocity splitting. Such decoupling was traditionally performed by adopting the Fractional Time‐Step Method that is based on a formal separation between convective–diffusive momentum terms from the pressure gradient term. This idea is strictly related to the fundamental theorem on the Helmholtz–Hodge orthogonal decomposition of a vector field in a finite domain, from which the name projection methods originates. The aim of this paper is to provide an original evaluation of the local truncation error (LTE) for analysing the actual accuracy achieved by solving the de‐coupled system. The LTE sources are formally subdivided in two categories: errors intrinsically due to the splitting of the original system and errors due to the assignment of the boundary conditions. The main goal of the present paper consists in both providing the LTE analysis and proposing a remedy for the inaccuracy of some types of intermediate boundary conditions associated with the prediction equation. Such evaluations will be directly performed in the physical space for both the time continuous formulation and the finite volume discretization along with the discrete Adams–Bashforth/Crank–Nicolson time integration. A new proposal for a boundary condition expression, congruent with the discrete prediction equation is herein derived, fulfilling the goal of accomplishing the closure of the problem with fully second order accuracy. In our knowledge, this procedure is new in the literature and can be easily implemented for confined flows. The LTE is clearly highlighted and many computations demonstrate that our proposal is efficient and accurate and the goal of adopting the pressure‐free method in a finite domain with fully second order accuracy is reached. Copyright © 2003 John Wiley & Sons, Ltd.     

A 3D incompressible Navier–Stokes velocity–vorticity weak form finite element algorithm

The velocity–vorticity formulation is selected to develop a time‐accurate CFD finite element algorithm for the incompressible Navier–Stokes equations in three dimensions.The finite element implementation uses equal order trilinear finite elements on a non‐staggered hexahedral mesh. A second order vorticity kinematic boundary condition is derived for the no slip wall boundary condition which also enforces the incompressibility constraint. A biconjugate gradient stabilized (BiCGSTAB) sparse iterative solver is utilized to solve the fully coupled system of equations as a Newton algorithm. The solver yields an efficient parallel solution algorithm on distributed‐memory machines, such as the IBM SP2. Three dimensional laminar flow solutions for a square channel, a lid‐driven cavity, and a thermal cavity are established and compared with available benchmark solutions. Copyright © 2002 John Wiley & Sons, Ltd.     

Enhancement of the accuracy of the finite volume particle method for the simulation of incompressible flows

A finite volume particle (FVP) method for simulation of incompressible flows that provides enhanced accuracy is proposed. In this enhanced FVP method, a dummy neighbor particle is introduced for each particle in the calculation and used for the discretization of the gradient model and Laplacian model. The error‐compensating term produced by introducing the dummy neighbor particle enables higher order terms to be calculated. The proposed gradient model and Laplacian model are applied in both pressure and pressure gradient calculations. This enhanced FVP scheme provides more accurate simulations of incompressible flows. Several 2‐dimensional numerical simulations are given to confirm its enhanced performance.     

AN ADAPTIVE FINITE ELEMENT METHOD FOR A TWO-EQUATION TURBULENCE MODEL IN WALL-BOUNDED FLOWS

This paper presents an adaptive finite element method for solving incompressible turbulent flows using a k–ϵ model of turbulence. Solutions are obtained in primitive variables using a highly accurate quadratic finite element on unstructured grids. A projection error estimator is presented that takes into account the relative importance of the errors in velocity, pressure and turbulence variables. The efficiency and convergence rate of the methodology are evaluated by solving problems with known analytical solutions. The method is then applied to turbulent flow over a backward-facing step and predictions are compared with experimental measurements. © 1997 John Wiley & Sons, Ltd.     

A multigrid method for steady incompressible navier-stokes equations based on flux difference splitting

The steady Navier–Stokes equations in primitive variables are discretized in conservative form by a vertex-centred finite volume method Flux difference splitting is applied to the convective part to obtain an upwind discretization. The diffusive part is discretized in the central way. In its first-order formulation, flux difference splitting leads to a discretization of so-called vector positive type. This allows the use of classical relaxation methods in collective form. An alternating line Gauss–Seidel relaxation method is chosen here. This relaxation method is used as a smoother in a multigrid method. The components of this multigrid method are: full approximation scheme with F-cycles, bilinear prolongation, full weighting for residual restriction and injection of grid functions. Higher-order accuracy is achieved by the flux extrapolation method. In this approach the first-order convective fluxes are modified by adding second-order corrections involving flux limiting. Here the simple MinMod limiter is chosen. In the multigrid formulation the second-order discrete system is solved by defect correction. Computational results are shown for the well known GAMM backward-facing step problem and for a channel with a half-circular obstruction.     

A stabilized finite element predictor–corrector scheme for the incompressible Navier–Stokes equations using a nodal‐based implementation

A finite element model to solve the incompressible Navier–Stokes equations based on the stabilization with orthogonal subscales, a predictor–corrector scheme to segregate the pressure and a nodal based implementation is presented in this paper. The stabilization consists of adding a least‐squares form of the component orthogonal to the finite element space of the convective and pressure gradient terms, which allows to deal with convection‐dominated flows and to use equal velocity–pressure interpolation. The pressure segregation is inspired in fractional step schemes, although the converged solution corresponds to that of a monolithic time integration. Finally, the nodal‐based implementation is based on an a priori calculation of the integrals appearing in the formulation and then the construction of the matrix and right‐hand side vector of the final algebraic system to be solved. After appropriate approximations, this matrix and this vector can be constructed directly for each nodal point, without the need to loop over the elements and thus making the calculations much faster. Some issues related to this implementation for fractional step and our predictor–corrector scheme, which is the main contribution of this paper, are discussed. Copyright © 2004 John Wiley & Sons, Ltd.     

ADVECTION–DIFFUSION PROCESSES AND RESIDENCE TIMES IN SEMIENCLOSED MARINE BASINS

This paper addresses the problem of estimating the residence times in a marine basin of a passive constituent released in the sea. The dispersion process is described by an advection–diffusion model and the hydrodynamics is assumed to be known. We have performed the analysis of two different scenarios: (i) basins with unidirectional flows, in three space dimensions and under the rigid lid approximation, and (ii) basins with flows forced by the tide, under the shallow water approximation. Let the random variable τ be defined as the time spent in the basin by a particle released at a given point. The probability distribution of τ is obtained from the solution of the advection–diffusion problem and the residence time of a particle is defined as the mean value of τ. Two different numerical approximations have been used to solve the continuous problem: the finite volume and Monte Carlo methods. For both continuous and discrete formulations it is proved that if all the particles eventually leave the basin, then the residence time has a finite value. We present here the results obtained for two study cases: a two- dimensional basin with a steady flow and a one-dimensional channel with flow induced by the tide. The results obtained by the finite volume and Monte Carlo methods are in very good agreement for both scenarios.     

Chemically Reacting Flows and Continuous Flow Stirred Tank Reactors: The Large Diffusivity Limit

In this paper we consider the relationship between chemically reacting flows and continuous flow stirred tank reactors (CSTR). In particular, we show that in the limit as the chemical and thermal diffusivities go to infinity, the solutions of the reacting flow PDE approach the solutions to the CSTR ODE. We further show that the global attractors for the reacting flow come arbitrarily close top the CSTR global attractor as the diffusivities go to infinity. An important feature of the reacting flow model we use is Robin–Neumann boundary conditions for the chemistry and temperature, which we use to mimic the CSTR inflow and outflow terms. The key in our analysis is an examination of how the Laplacian with Robin–Neumann boundary conditions converges to the Laplacian with Neumann boundary conditions as the diffusivity goes to infinity.     

Numerical implementation of the Crank–Nicolson/Adams–Bashforth scheme for the time‐dependent Navier–Stokes equations

This article considers numerical implementation of the Crank–Nicolson/Adams–Bashforth scheme for the two‐dimensional non‐stationary Navier–Stokes equations. A finite element method is applied for the spatial approximation of the velocity and pressure. The time discretization is based on the Crank–Nicolson scheme for the linear term and the explicit Adams–Bashforth scheme for the nonlinear term. Comparison with other methods, through a series of numerical experiments, shows that this method is almost unconditionally stable and convergent, i.e. stable and convergent when the time step is smaller than a given constant. Copyright © 2009 John Wiley & Sons, Ltd.     

Kernel‐based vector field reconstruction in computational fluid dynamic models

Kernel‐based reconstruction methods are applied to obtain highly accurate approximations of local vector fields from normal components assigned at the edges of a computational mesh. The theoretical background of kernel‐based reconstructions for vector‐valued functions is first reviewed, before the reconstruction method is adapted to the specific requirements of relevant applications in computational fluid dynamics. To this end, important computational aspects concerning the design of the reconstruction scheme like the selection of suitable stencils are explained in detail. Extensive numerical examples and comparisons concerning hydrodynamic models show that the proposed kernel‐based reconstruction improves the accuracy of standard finite element discretizations, including Raviart–Thomas (RT) elements, quite significantly, while retaining discrete conservation properties of important physical quantities, such as mass, vorticity, or potential enstrophy. Copyright © 2010 John Wiley & Sons, Ltd.