A Mach‐uniform preconditioner for incompressible and subsonic flows

In this study, a novel Mach‐uniform preconditioning method is developed for the solution of Euler equations at low subsonic and incompressible flow conditions. In contrast to the methods developed earlier in which the conservation of mass equation is preconditioned, in the present method, the conservation of energy equation is preconditioned, which enforces the divergence free constraint on the velocity field even at the limiting case of incompressible, zero Mach number flows. Despite most preconditioners, the proposed Mach‐uniform preconditioning method does not have a singularity point at zero Mach number. The preconditioned system of equations preserves the strong conservation form of Euler equations for compressible flows and recovers the artificial compressibility equations in the case of zero Mach number. A two‐dimensional Euler solver is developed for validation and performance evaluation of the present formulation for a wide range of Mach number flows. The validation cases studied show the convergence acceleration, stability, and accuracy of the present Mach‐uniform preconditioner in comparison to the non‐preconditioned compressible flow solutions. The convergence acceleration obtained with the present formulation is similar to those of the well‐known preconditioned system of equations for low subsonic flows and to those of the artificial compressibility method for incompressible flows. Copyright © 2013 John Wiley & Sons, Ltd.     

On the Inviscid Limit for the Compressible Navier–Stokes System in an Impermeable Bounded Domain

In this paper we investigate the issue of the inviscid limit for the compressible Navier–Stokes system in an impermeable fixed bounded domain. We consider two kinds of boundary conditions. The first one is the no-slip condition. In this case we extend the famous conditional result (Kato, T.: Remarks on zero viscosity limit for nonstationary Navier–Stokes flows with boundary. In: Seminar on nonlinear partial differential equations, vol. 2, pp. 85–98. Math. Sci. Res. Inst. Publ., Berkeley 1984) obtained by Kato in the homogeneous incompressible case. Kato proved that if the energy dissipation rate of the viscous flow in a boundary layer of width proportional to the viscosity vanishes then the solutions of the incompressible Navier–Stokes equations converge to some solutions of the incompressible Euler equations in the energy space. We provide here a natural extension of this result to the compressible case. The other case is the Navier condition which encodes that the fluid slips with some friction on the boundary. In this case we show that the convergence to the Euler equations holds true in the energy space, as least when the friction is not too large. In both cases we use in a crucial way some relative energy estimates proved recently by Feireisl et al. in J. Math. Fluid Mech. 14:717–730 (2012).     

Analysis of preconditioned iterative solvers for incompressible flow problems

Solving efficiently the incompressible Navier–Stokes equations is a major challenge, especially in the three‐dimensional case. The approach investigated by Elman et al. (Finite Elements and Fast Iterative Solvers. Oxford University Press: Oxford, 2005) consists in applying a preconditioned GMRES method to the linearized problem at each iteration of a nonlinear scheme. The preconditioner is built as an approximation of an ideal block‐preconditioner that guarantees convergence in 2 or 3 iterations. In this paper, we investigate the numerical behavior for the three‐dimensional lid‐driven cavity problem with wedge elements; the ultimate motivation of this analysis is indeed the development of a preconditioned Krylov solver for stratified oceanic flows which can be efficiently tackled using such meshes. Numerical results for steady‐state solutions of both the Stokes and the Navier–Stokes problems are presented. Theoretical bounds on the spectrum and the rate of convergence appear to be in agreement with the numerical experiments. Sensitivity analysis on different aspects of the structure of the preconditioner and the block decomposition strategies are also discussed. Copyright © 2011 John Wiley & Sons, Ltd.     

GMRES physics-based preconditioner for all Reynolds and Mach numbers: numerical examples

This paper presents several numerical results using a vectorized version of a 3D finite element compressible and nearly incompressible Euler and Navier–Stokes code. The assumptions were set on laminar flows and Newtonian fluids. The goal of this research is to show the capabilities of the present code to treat a wide range of problems appearing in laminar fluid dynamics towards the unification from incompressible to compressible and from inviscid to viscous flow codes. Several authors with different approaches have tried to attain this target in CFD with relative success. At the beginning the methods based on operator splitting and perturbation were preferred, but lately, with the wide usage of time-marching algorithms, the preconditioning mass matrix (PMM) has become very popular. With this kind of relaxation scheme it is possible to accelerate the rate of convergence to steady state solutions with the modification of the mass matrix under certain restrictions. The selection of the mass matrix is not an easy task, but we have certain freedom to define it in order to improve the condition number of the system. In this paper we have used a physics-based preconditioner for the GMRES implicit solver developed previously by us and an SUPG formulation for the semidiscretization of the spatial operator. In sections 2 and 3 we present some theoretical aspects related to the physical problem and the mathematical model, showing the inviscid and viscous flow equations to be solved and the variational formulation involved in the finite element analysis. Section 4 deals with the numerical solution of non-linear systems of equations, with some emphasis on the preconditioned matrix-free GMRES solver. Section 5 shows how boundary conditions were treated for both Euler and Navier–Stokes problems. Section 6 contains some aspects about vectorization on the Cray C90. The performance reached by this implementation is close to 1 Gflop using multitasking. Section 7 presents several numerical examples for both models covering a wide range of interesting problems, such as inviscid low subsonic, transonic and supersonic regimes and viscous problems with interaction between boundary layers and shock waves in either attached or separated flows. © 1997 John Wiley & Sons, Ltd.     

Implementing a high‐order accurate implicit operator scheme for solving steady incompressible viscous flows using artificial compressibility method

This paper uses a fourth‐order compact finite‐difference scheme for solving steady incompressible flows. The high‐order compact method applied is an alternating direction implicit operator scheme, which has been used by Ekaterinaris for computing two‐dimensional compressible flows. Herein, this numerical scheme is efficiently implemented to solve the incompressible Navier–Stokes equations in the primitive variables formulation using the artificial compressibility method. For space discretizing the convective fluxes, fourth‐order centered spatial accuracy of the implicit operators is efficiently obtained by performing compact space differentiation in which the method uses block‐tridiagonal matrix inversions. To stabilize the numerical solution, numerical dissipation terms and/or filters are used. In this study, the high‐order compact implicit operator scheme is also extended for computing three‐dimensional incompressible flows. The accuracy and efficiency of this high‐order compact method are demonstrated for different incompressible flow problems. A sensitivity study is also conducted to evaluate the effects of grid resolution and pseudocompressibility parameter on accuracy and convergence rate of the solution. The effects of filtering and numerical dissipation on the solution are also investigated. Test cases considered herein for validating the results are incompressible flows in a 2‐D backward facing step, a 2‐D cavity and a 3‐D cavity at different flow conditions. Results obtained for these cases are in good agreement with the available numerical and experimental results. The study shows that the scheme is robust, efficient and accurate for solving incompressible flow problems. Copyright © 2010 John Wiley & Sons, Ltd.     

Implicit preconditioned high‐order compact scheme for the simulation of the three‐dimensional incompressible Navier–Stokes equations with pseudo‐compressibility method

This paper combines the pseudo‐compressibility procedure, the preconditioning technique for accelerating the time marching for stiff hyperbolic equations, and high‐order accurate central compact scheme to establish the code for efficiently and accurately solving incompressible flows numerically based on the finite difference discretization. The spatial scheme consists of the sixth‐order compact scheme and 10th‐order numerical filter operator for guaranteeing computational stability. The preconditioned pseudo‐compressible Navier–Stokes equations are marched temporally using the implicit lower–upper symmetric Gauss–Seidel time integration method, and the time accuracy is improved by the dual‐time step method for the unsteady problems. The efficiency and reliability of the present procedure are demonstrated by applications to Taylor decaying vortices phenomena, double periodic shear layer rolling‐up problem, laminar flow over a flat plate, low Reynolds number unsteady flow around a circular cylinder at Re = 200, high Reynolds number turbulence flow past the S809 airfoil, and the three‐dimensional flows through two 90°curved ducts of square and circular cross sections, respectively. It is found that the numerical results of the present algorithm are in good agreement with theoretical solutions or experimental data. Copyright © 2011 John Wiley & Sons, Ltd.     

AN ARTIFICIAL BOUNDARY CONDITION FOR TWO-DIMENSIONAL INCOMPRESSIBLE VISCOUS FLOWS USING THE METHOD OF LINES

We design an artificial boundary condition for the steady incompressible Navier–Stokes equations in streamfunction–vorticity formulation in a flat channel with slip boundary conditions on the wall. The new boundary condition is derived from the Oseen equations and the method of lines. A numerical experiment for the non-linear Navier–Stokes equations is presented. The artificial boundary condition is compared with Dirichlet and Neumann boundary conditions for the flow past a rectangular cylinder in a flat channel. The numerical results show that our boundary condition is more accurate.     

A high‐order discontinuous Galerkin method for all‐speed flows

In this article, we present a discontinuous Galerkin (DG) method designed to improve the accuracy and efficiency of steady solutions of the compressible fully coupled Reynolds‐averaged Navier–Stokes and k ? ω turbulence model equations for solving all‐speed flows. The system of equations is iterated to steady state by means of an implicit scheme. The DG solution is extended to the incompressible limit by implementing a low Mach number preconditioning technique. A full preconditioning approach is adopted, which modifies both the unsteady terms of the governing equations and the dissipative term of the numerical flux function by means of a new preconditioner, on the basis of a modified version of Turkel's preconditioning matrix. At sonic speed the preconditioner reduces to the identity matrix thus recovering the non‐preconditioned DG discretization. An artificial viscosity term is added to the DG discretized equations to stabilize the solution in the presence of shocks when piecewise approximations of order of accuracy higher than 1 are used. Moreover, several rescaling techniques are implemented in order to overcome ill‐conditioning problems that, in addition to the low Mach number stiffness, can limit the performance of the flow solver. These approaches, through a proper manipulation of the governing equations, reduce unbalances between residuals as a result of the dependence on the size of elements in the computational mesh and because of the inherent differences between turbulent and mean‐flow variables, influencing both the evolution of the Courant Friedrichs Lewy (CFL) number and the inexact solution of the linear systems. The performance of the method is demonstrated by solving three turbulent aerodynamic test cases: the flat plate, the L1T2 high‐lift configuration and the RAE2822 airfoil (Case 9). The computations are performed at different Mach numbers using various degrees of polynomial approximations to analyze the influence of the proposed numerical strategies on the accuracy, efficiency and robustness of a high‐order DG solver at different flow regimes. Copyright © 2014 John Wiley & Sons, Ltd.     

On simulation of outflow boundary conditions in finite difference calculations for incompressible fluid

For incompressible Navier–Stokes equations in primitive variables, a method of setting absorbing outflow boundary conditions on an artificial boundary is considered. The advection equations used on the outflow boundary are convenient for finite difference (FD) methods, where a weak formulation of a problem is inapplicable. An unsteady viscous incompressible Navier–Stokes flow in a channel with a moving damper is modeled. An accurate comparison and analysis of numerical and mechanical situations are carried out for a variety of boundary conditions and Reynolds numbers. The proposed outflow conditions provide that the problem with Dirichlet boundary conditions should be solved on each time step.     

Fourth‐order compact formulation of Navier–Stokes equations and driven cavity flow at high Reynolds numbers

A new fourth‐order compact formulation for the steady 2‐D incompressible Navier–Stokes equations is presented. The formulation is in the same form of the Navier–Stokes equations such that any numerical method that solve the Navier–Stokes equations can easily be applied to this fourth‐order compact formulation. In particular, in this work the formulation is solved with an efficient numerical method that requires the solution of tridiagonal systems using a fine grid mesh of 601 × 601. Using this formulation, the steady 2‐D incompressible flow in a driven cavity is solved up to Reynolds number with Re = 20 000 fourth‐order spatial accuracy. Detailed solutions are presented. Copyright © 2005 John Wiley & Sons, Ltd.     

Incompressible Roe‐like scheme for turbulent flows

In the current study, numerical investigation of incompressible turbulent flow is presented. By the artificial compressibility method, momentum and continuity equations are coupled. Considering Reynolds averaged Navier–Stokes equations, the Spalart–Allmaras turbulence model, which has accurate results in two‐dimensional problems, is used to calculate Reynolds stresses. For convective fluxes a Roe‐like scheme is proposed for the steady Reynolds averaged Navier–Stokes equations. Also, Jameson averaging method was implemented. In comparison, the proposed characteristics‐based upwind incompressible turbulent Roe‐like scheme, demonstrated very accurate results, high stability, and fast convergence. The fifth‐order Runge–Kutta scheme is used for time discretization. The local time stepping and implicit residual smoothing were applied as the convergence acceleration techniques. Suitable boundary conditions have been implemented considering flow behavior. The problem has been studied at high Reynolds numbers for cross flow around the horizontal circular cylinder and NACA0012 hydrofoil. Results were compared with those of others and a good agreement has been observed. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

Application of a preconditioned high‐order accurate artificial compressibility‐based incompressible flow solver in wide range of Reynolds numbers

In the present study, the preconditioned incompressible Navier‐Stokes equations with the artificial compressibility method formulated in the generalized curvilinear coordinates are numerically solved by using a high‐order compact finite‐difference scheme for accurately and efficiently computing the incompressible flows in a wide range of Reynolds numbers. A fourth‐order compact finite‐difference scheme is utilized to accurately discretize the spatial derivative terms of the governing equations, and the time integration is carried out based on the dual time‐stepping method. The capability of the proposed solution methodology for the computations of the steady and unsteady incompressible viscous flows from very low to high Reynolds numbers is investigated through the simulation of different 2‐dimensional benchmark problems, and the results obtained are compared with the existing analytical, numerical, and experimental data. A sensitivity analysis is also performed to evaluate the effects of the size of the computational domain and other numerical parameters on the accuracy and performance of the solution algorithm. The present solution procedure is also extended to 3 dimensions and applied for computing the incompressible flow over a sphere. Indications are that the application of the preconditioning in the solution algorithm together with the high‐order discretization method in the generalized curvilinear coordinates provides an accurate and robust solution method for simulating the incompressible flows over practical geometries in a wide range of Reynolds numbers including the creeping flows.     

Flux‐difference splitting‐based upwind compact schemes for the incompressible Navier–Stokes equations

Third‐order and fifth‐order upwind compact finite difference schemes based on flux‐difference splitting are proposed for solving the incompressible Navier–Stokes equations in conjunction with the artificial compressibility (AC) method. Since the governing equations in the AC method are hyperbolic, flux‐difference splitting (FDS) originally developed for the compressible Euler equations can be used. In the present upwind compact schemes, the split derivatives for the convective terms at grid points are linked to the differences of split fluxes between neighboring grid points, and these differences are computed by using FDS. The viscous terms are approximated with a sixth‐order central compact scheme. Comparisons with 2D benchmark solutions demonstrate that the present compact schemes are simple, efficient, and high‐order accurate. Copyright © 2008 John Wiley & Sons, Ltd.     

A momentum coupling method for the unsteady incompressible Navier–Stokes equations on the staggered grid

A new numerical method is developed to efficiently solve the unsteady incompressible Navier–Stokes equations with second-order accuracy in time and space. In contrast to the SIMPLE algorithms, the present formulation directly solves the discrete x- and y-momentum equations in a coupled form. It is found that the present implicit formulation retrieves some cross convection terms overlooked by the conventional iterative methods, which contribute to accuracy and fast convergence. The finite volume method is applied on the fully staggered grid to solve the vector-form momentum equations. The preconditioned conjugate gradient squared method (PCGS) has proved very efficient in solving the associate linearized large, sparse block-matrix system. Comparison with the SIMPLE algorithm has indicated that the present momentum coupling method is fast and robust in solving unsteady as well as steady viscous flow problems. © 1998 John Wiley & Sons, Ltd.     

Applications of the artificial compressibility method for turbulent open channel flows

A three‐dimensional (3‐D) numerical method for solving the Navier–Stokes equations with a standard k–ε turbulence model is presented. In order to couple pressure with velocity directly, the pressure is divided into hydrostatic and hydrodynamic parts and the artificial compressibility method (ACM) is employed for the hydrodynamic pressure. By introducing a pseudo‐time derivative of the hydrodynamic pressure into the continuity equation, the incompressible Navier–Stokes equations are changed from elliptic‐parabolic to hyperbolic‐parabolic equations. In this paper, a third‐order monotone upstream‐centred scheme for conservation laws (MUSCL) method is used for the hyperbolic equations. A system of discrete equations is solved implicitly using the lower–upper symmetric Gauss–Seidel (LU‐SGS) method. This newly developed numerical method is validated against experimental data with good agreement. Copyright © 2005 John Wiley & Sons, Ltd.     

An adaptive multigrid finite-volume scheme for incompressible Navier–Stokes equations

An algorithm for the solutions of the two-dimensional incompressible Navier–Stokes equations is presented. The algorithm can be used to compute both steady-state and time-dependent flow problems. It is based on an artificial compressibility method and uses higher-order upwind finite-volume techniques for the convective terms and a second-order finite-volume technique for the viscous terms. Three upwind schemes for discretizing convective terms are proposed here. An interesting result is that the solutions computed by one of them is not sensitive to the value of the artificial compressibility parameter. A second-order, two-step Runge–Kutta integration coupling with an implicit residual smoothing and with a multigrid method is used for achieving fast convergence for both steady- and unsteady-state problems. The numerical results agree well with experimental and other numerical data. A comparison with an analytically exact solution is performed to verify the space and time accuracy of the algorithm.     

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.     

Continuous and discrete adjoint approaches for aerodynamic shape optimization with low Mach number preconditioning

Discrete and continuous adjoint approaches for use in aerodynamic shape optimization problems at all flow speeds are developed and assessed. They are based on the Navier–Stokes equations with low Mach number preconditioning. By alleviating the large disparity between acoustic waves and fluid speeds, the preconditioned flow and adjoint equations are numerically solved with affordable CPU cost, even at the so‐called incompressible flow conditions. Either by employing the adjoint to the preconditioned flow equations or by preconditioning the adjoint to the ‘standard’ flow equations (under certain conditions the two formulations become equivalent, as proved in this paper), efficient optimization methods with reasonable cost per optimization cycle, even at very low Mach numbers, are derived. During the mathematical development, a couple of assumptions are made which are proved to be harmless to the accuracy in the computed gradients and the effectiveness of the optimization method. The proposed approaches are validated in inviscid and viscous flows in external aerodynamics and turbomachinery flows at various Mach numbers. Copyright © 2007 John Wiley & Sons, Ltd.     

Analysis of iterative methods for the viscous/inviscid coupled problem via a spectral element approximation

Based on a new global variational formulation, a spectral element approximation of the incompressible Navier–Stokes/Euler coupled problem gives rise to a global discrete saddle problem. The classical Uzawa algorithm decouples the original saddle problem into two positive definite symmetric systems. Iterative solutions of such systems are feasible and attractive for large problems. It is shown that, provided an appropriate pre‐conditioner is chosen for the pressure system, the nested conjugate gradient methods can be applied to obtain rapid convergence rates. Detailed numerical examples are given to prove the quality of the pre‐conditioner. Thanks to the rapid iterative convergence, the global Uzawa algorithm takes advantage of this as compared with the classical iteration by sub‐domain procedures. Furthermore, a generalization of the pre‐conditioned iterative algorithm to flow simulation is carried out. Comparisons of computational complexity between the Navier–Stokes/Euler coupled solution and the full Navier–Stokes solution are made. It is shown that the gain obtained by using the Navier–Stokes/Euler coupled solution is generally considerable. Copyright © 2000 John Wiley & Sons, Ltd.