Multigrid solutions of the Euler and Navier–Stokes equations on unstructured grids

A computationally efficient multigrid algorithm for upwind edge‐based finite element schemes is developed for the solution of the two‐dimensional Euler and Navier–Stokes equations on unstructured triangular grids. The basic smoother is based upon a Galerkin approximation employing an edge‐based formulation with the explicit addition of an upwind‐type local extremum diminishing (LED) method. An explicit time stepping method is used to advance the solution towards the steady state. Fully unstructured grids are employed to increase the flexibility of the proposed algorithm. A full approximation storage (FAS) algorithm is used as the basic multigrid acceleration procedure. Copyright © 1999 John Wiley & Sons, Ltd.     

A numerical investigation of a spectral‐type nodal collocation discontinuous Galerkin approximation of the Euler and Navier–Stokes equations

In this paper, we focus on the applicability of spectral‐type collocation discontinuous Galerkin methods to the steady state numerical solution of the inviscid and viscous Navier–Stokes equations on meshes consisting of curved quadrilateral elements. The solution is approximated with piecewise Lagrange polynomials based on both Legendre–Gauss and Legendre–Gauss–Lobatto interpolation nodes. For the sake of computational efficiency, the interpolation nodes can be used also as quadrature points. In this case, however, the effect of the nonlinearities in the equations and/or curved elements leads to aliasing and/or commutation errors that may result in inaccurate or unstable computations. By a thorough numerical testing on a set of well known test cases available in the literature, it is here shown that the two sets of nodes behave very differently, with a clear advantage of the Legendre–Gauss nodes, which always displayed an accurate and robust behaviour in all the test cases considered.Copyright © 2012 John Wiley & Sons, Ltd.     

Fictitious domain approach for numerical modelling of Navier–Stokes equations

This study investigates a fictitious domain model for the numerical solution of various incompressible viscous flows. It is based on the so‐called Navier–Stokes/Brinkman and energy equations with discontinuous coefficients all over an auxiliary embedding domain. The solid obstacles or walls are taken into account by a penalty technique. Some volumic control terms are directly introduced in the governing equations in order to prescribe immersed boundary conditions. The implicit numerical scheme, which uses an upwind finite volume method on staggered Cartesian grids, is of second‐order accuracy in time and space. A multigrid local mesh refinement is also implemented, using the multi‐level Zoom Flux Interface Correction (FIC) method, in order to increase the precision where it is needed in the domain. At each time step, some iterations of the augmented Lagrangian method combined with a preconditioned Krylov algorithm allow the divergence‐free velocity and pressure fields be solved for. The tested cases concern external steady or unsteady flows around a circular cylinder, heated or not, and the channel flow behind a backward‐facing step. The numerical results are shown in good agreement with other published numerical or experimental data. Copyright © 2000 John Wiley & Sons, Ltd.     

An efficient upwind/relaxation algorithm for the Euler and Navier–Stokes equations

An efficient Euler and full Navier–Stokes solver based on a flux splitting scheme is presented. The original Van Leer flux vector splitting form is extended to arbitrary body-fitted co-ordinates in the physical domain so that it can be used with a finite volume scheme. The block matrix is inverted by Gauss–Seidel iteration. It is verified that the often used reflection boundary condition will produce incorrect flux crossing the wall and cause too large numerical dissipation if flux vector splitting is used. To remove such errors, an appropriate treatment of wall boundary conditions is suggested. Inviscid and viscous steady transonic internal flows are analysed, including the case of shock-induced boundary layer separation.     

A Chimera method for the incompressible Navier–Stokes equations

The Chimera method was developed three decades ago as a meshing simplification tool. Different components are meshed independently and then glued together using a domain decomposition technique to couple the equations solved on each component. This coupling is achieved via transmission conditions (in the finite element context) or by imposing the continuity of fluxes (in the finite volume context). Historically, the method has then been used extensively to treat moving objects, as the independent meshes are free to move with respect to the others. At each time step, the main task consists in recomputing the interpolation of the transmission conditions or fluxes. This paper presents a Chimera method applied to the Navier–Stokes equations. After an introduction on the Chimera method, we describe in two different sections the two independent steps of the method: the hole cutting to create the interfaces of the subdomains and the coupling of the subdomains. Then, we present the Navier–Stokes solver considered in this work. Implementation aspects are then detailed in order to apply efficiently the method to this specific parallel Navier–Stokes solver. We conclude with some examples to demonstrate the reliability and application of the proposed method. Copyright © 2014 John Wiley & Sons, Ltd.     

A discontinuous Galerkin method for the Navier–Stokes equations

The foundations of a new discontinuous Galerkin method for simulating compressible viscous flows with shocks on standard unstructured grids are presented in this paper. The new method is based on a discontinuous Galerkin formulation both for the advective and the diffusive contributions. High‐order accuracy is achieved by using a recently developed hierarchical spectral basis. This basis is formed by combining Jacobi polynomials of high‐order weights written in a new co‐ordinate system. It retains a tensor‐product property, and provides accurate numerical quadrature. The formulation is conservative, and monotonicity is enforced by appropriately lowering the basis order and performing h‐refinement around discontinuities. Convergence results are shown for analytical two‐ and three‐dimensional solutions of diffusion and Navier–Stokes equations that demonstrate exponential convergence of the new method, even for highly distorted elements. Flow simulations for subsonic, transonic and supersonic flows are also presented that demonstrate discretization flexibility using hp‐type refinement. Unlike other high‐order methods, the new method uses standard finite volume grids consisting of arbitrary triangulizations and tetrahedrizations. Copyright © 1999 John Wiley & Sons, Ltd.     

Treatment of non-linearities in the numerical solution of the incompressible Navier–Stokes equations

The solution of the full non-linear set of discrete fluid flow equations is usually obtained by solving a sequence of linear equations. The type of linearization used can significantly affect the rate of convergence of the sequence to the final solution. The first objective of the present study was to determine the extent to which a full Newton–Raphson linearization of all non-linear terms enhances convergence relative to that obtained using the ‘standard’ incompressible flow linearization. A direct solution procedure was employed in this evaluation. It was found that the full linearization enhances convergence, especially when grid curvature effects are important. The direct solution of the linear set is uneconomical. The second objective of the paper was to show how the equations can be effectively solved by an iterative scheme, based on a coupled-equation line solver, which implicitly retains all the inter-equation couplings. This solution method was found to be competitive with the highly refined segregated solution methods that represent the current state-of-the-art.     

A semi‐implicit numerical method for the free‐surface Navier–Stokes equations

Semi‐implicit methods are known for being the basis of simple, efficient, accurate, and stable numerical algorithms for simulating a large variety of geophysical free‐surface flows. Geophysical flows are typically characterized by having a small vertical scale as compared with their horizontal extents. Hence, the hydrostatic approximation often applies, and the free surface can be conveniently represented by a single‐valued function of the horizontal coordinates. In the present investigation, semi‐implicit methods are extended to complex free‐surface flows that are governed by the full incompressible Navier–Stokes equations and are delimited by solid boundaries and arbitrarily shaped free‐surfaces. The primary dependent variables are the velocity components and the pressure. Finite difference equations for momentum, and a finite volume discretization for continuity, are derived in such a fashion that, after simple manipulation, the resulting pressure equation yields a well‐posed piecewise linear system from which both the pressure and the fluid volume within each computational cell are naturally derived. This system is efficiently solved by a nested Newton type iterative scheme, and the resulting fluid volumes are assured to be nonnegative and bounded from above by the available cell volumes. The time step size is not restricted by stability conditions dictated by surface wave speed, but can be freely chosen just to achieve the desired accuracy. Several examples illustrate the model applicability to a large range of complex free‐surface flows and demonstrate the effectiveness of the proposed algorithm. Copyright © 2013 John Wiley & Sons, Ltd.     

Adaptive mesh refinement‐enhanced local DFD method and its application to solve Navier–Stokes equations

Recently, the domain‐free discretization (DFD) method was presented to efficiently solve problems with complex geometries without introducing the coordinate transformation. In order to exploit the high performance of the DFD method, in this paper, the local DFD method with the use of Cartesian mesh is presented, where the physical domain is covered by a Cartesian mesh and the local DFD method is applied for numerical discretization. In order to further improve the efficiency of the solver, the newly developed solution‐based adaptive mesh refinement (AMR) technique is also introduced. The proposed methods are then applied to the simulation of natural convection in concentric annuli between a square outer cylinder and a circular inner cylinder. Numerical experiments show that the present numerical results agree very well with available data in the literature, and AMR‐enhanced local DFD method is an effective tool for the computation of flow problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Newton linearization of the incompressible Navier–Stokes equations

The present study aims to accelerate the convergence to incompressible Navier–Stokes solution. For the sake of computational efficiency, Newton linearization of equations is invoked on non‐staggered grids to shorten the sequence to the final solution of the non‐linear differential system of equations. For the sake of accuracy, the resulting convection–diffusion–reaction finite‐difference equation is solved line‐by‐line using the proposed nodally exact one‐dimensional scheme. The matrix size is reduced and, at the same time, the CPU time is considerably saved due to the decrease of stencil points. The effectiveness of the implemented Newton linearization is demonstrated through computational exercises. Copyright © 2004 John Wiley & Sons, Ltd.     

Upwind discretization of the steady Navier–Stokes equations

A discretization method is presented for the full, steady, compressible Navier–Stokes equations. The method makes use of quadrilateral finite volumes and consists of an upwind discretization of the convective part and a central discretization of the diffusive part. In the present paper the emphasis lies on the discretization of the convective part. The solution method applied solves the steady equations directly by means of a non-linear relaxation method accelerated by multigrid. The solution method requires the discretization to be continuously differentiable. For two upwind schemes which satisfy this requirement (Osher's and van Leer's scheme), results of a quantitative error analysis are presented. Osher's scheme appears to be increasingly more accurate than van Leer's scheme with increasing Reynolds number. A suitable higher-order accurate discretization of the convection terms is derived. On the basis of this higher-order scheme, to preserve monotonicity, a new limiter is constructed. Numerical results are presented for a subsonic flat plate flow and a supersonic flat plate flow with oblique shock wave–boundary layer interaction. The results obtained agree with the predictions made. Useful properties of the discretization method are that it allows an easy check of false diffusion and that it needs no tuning of parameters.     

On the discontinuous Galerkin method for the numerical solution of the Navier–Stokes equations

The paper deals with the use of the discontinuous Galerkin finite element method (DGFEM) for the numerical solution of viscous compressible flows. We start with a scalar convection–diffusion equation and present a discretization with the aid of the non‐symmetric variant of DGFEM with interior and boundary penalty terms. We also mention some theoretical results. Then we extend the scheme to the system of the Navier–Stokes equations and discuss the treatment of stabilization terms. Several numerical examples are presented. Copyright © 2004 John Wiley & Sons, Ltd.     

Vectorized poisson solvers for the Navier–Stokes equations

The advent of vector and massively parallel computers offers researchers the possibility of enormous gains in execution time for scientific and engineering programs. From the numerical point of view, such programs are frequently based on the inversion of sparse, diagonally banded matrices. Conventional scalar solvers often perform poorly on vector machines due to short effective vector lengths, and thus appropriate methods must be chosen for use with vector machines. In this paper a number of commonly used solvers are tested for the Navier–Stokes equations, in both scalar and vector form, on two vector architecture machines. A new method is presented which performs well in both vector and scalar form on a range of vector architectures.     

A kinetic theory solution method for the Navier–Stokes equations

The kinetic-theory-based solution methods for the Euler equations proposed by Pullin and Reitz are here extended to provide new finite volume numerical methods for the solution of the unsteady Navier–Stokes equations. Two approaches have been taken. In the first, the equilibrium interface method (EIM), the forward- and backward-flowing molecular fluxes between two cells are assumed to come into kinetic equilibrium at the interface between the cells. Once the resulting equilibrium states at all cell interfaces are known, the evaluation of the Navier–Stokes fluxes is straightforward. In the second method, standard kinetic theory is used to evaluate the artificial dissipation terms which appear in Pullin's Euler solver. These terms are subtracted from the fluxes and the Navier–Stokes dissipative fluxes are added in. The new methods have been tested in a 1D steady flow to yield a solution for the interior structure of a shock wave and in a 2D unsteady boundary layer flow. The 1D solutions are shown to be remarkably accurate for cell sizes large compared to the length scale of the gradients in the flow and to converge to the exact solutions as the cell size is decreased. The steady-state solutions obtained with EIM agree with those of other methods, yet require a considerably reduced computational effort.     

Pseudostress–velocity formulation for incompressible Navier–Stokes equations

This paper presents a numerical algorithm using the pseudostress–velocity formulation to solve incompressible Newtonian flows. The pseudostress–velocity formulation is a variation of the stress–velocity formulation, which does not require symmetric tensor spaces in the finite element discretization. Hence its discretization is greatly simplified. The discrete system is further decoupled into an H ( div ) problem for the pseudostress and a post‐process resolving the velocity. This can be done conveniently by using the penalty method for steady‐state flows or by using the time discretization for nonsteady‐state flows. We apply this formulation to the 2D lid‐driven cavity problem and study its grid convergence rate. Also, computational results of the time‐dependent‐driven cavity problem and the flow past rectangular problem are reported. Copyright © 2009 John Wiley & Sons, Ltd.     

A comparison of preconditioners for incompressible Navier–Stokes solvers

We consider solution methods for large systems of linear equations that arise from the finite element discretization of the incompressible Navier–Stokes equations. These systems are of the so‐called saddle point type, which means that there is a large block of zeros on the main diagonal. To solve these types of systems efficiently, several block preconditioners have been published. These types of preconditioners require adaptation of standard finite element packages. The alternative is to apply a standard ILU preconditioner in combination with a suitable renumbering of unknowns. We introduce a reordering technique for the degrees of freedom that makes the application of ILU relatively fast. We compare the performance of this technique with some block preconditioners. The performance appears to depend on grid size, Reynolds number and quality of the mesh. For medium‐sized problems, which are of practical interest, we show that the reordering technique is competitive with the block preconditioners. Its simple implementation makes it worthwhile to implement it in the standard finite element method software. Copyright © 2007 John Wiley & Sons, Ltd.     

A parallel adaptive numerical method with generalized curvilinear coordinate transformation for compressible Navier–Stokes equations

A fourth‐order finite‐volume method for solving the Navier–Stokes equations on a mapped grid with adaptive mesh refinement is proposed, implemented, and demonstrated for the prediction of unsteady compressible viscous flows. The method employs fourth‐order quadrature rules for evaluating face‐averaged fluxes. Our approach is freestream preserving, guaranteed by the way of computing the averages of the metric terms on the faces of cells. The standard Runge–Kutta marching method is used for time discretization. Solutions of a smooth flow are obtained in order to verify that the method is formally fourth‐order accurate when applying the nonlinear viscous operators on mapped grids. Solutions of a shock tube problem are obtained to demonstrate the effectiveness of adaptive mesh refinement in resolving discontinuities. A Mach reflection problem is solved to demonstrate the mapped algorithm on a non‐rectangular physical domain. The simulation is compared against experimental results. Future work will consider mapped multiblock grids for practical engineering geometries. Copyright © 2016 John Wiley & Sons, Ltd.     

A stabilized incremental projection scheme for the incompressible Navier–Stokes equations

It is well known that any spatial discretization of the saddle‐point Stokes problem should satisfy the Ladyzhenskaya–Brezzi–Babuska (LBB) stability condition in order to prevent the appearance of spurious pressure modes. Particularly, if an equal‐order approximation is applied, the Schur complement (or, as called some times, the Uzawa matrix) of the pressure system has a non‐trivial null space that gives rise to such modes. An idea in the past was that all the schemes that solve a Poisson equation for the pressure rather than the Uzawa pressure equation (splitting/projection methods) should overcome this difficulty; this idea was wrong. There is numerical evidence that at least the so‐called incremental projection scheme still suffers from spurious pressure oscillations if an equal‐order approximation is applied. The present paper tries to distinguish which projection requires LBB‐compliant approximation and which does not. Moreover, a stabilized version of the incremental projection scheme is derived. Proper bounds for the stabilization parameter are also given. The numerical results show that the stabilized scheme does indeed achieve second‐order accuracy and does not produce spurious (node to node) pressure oscillations. Copyright © 2001 John Wiley & Sons, Ltd.     

A complete boundary integral formulation for compressible Navier–Stokes equations

A complete boundary integral formulation for compressible Navier–Stokes equations with time discretization by operator splitting is developed using the fundamental solutions of the Helmholtz operator equation with different order. The numerical results for wall pressure and wall skin friction of two‐dimensional compressible laminar viscous flow around airfoils are in good agreement with field numerical methods. Copyright © 2004 John Wiley & Sons, Ltd.