A high order Discontinuous Galerkin Finite Element solver for the incompressible Navier–Stokes equations

The paper presents an unsteady high order Discontinuous Galerkin (DG) solver that has been developed, verified and validated for the solution of the two-dimensional incompressible Navier–Stokes equations. A second order stiffly stable method is used to discretise the equations in time. Spatial discretisation is accomplished using a modal DG approach, in which the inter-element fluxes are approximated using the Symmetric Interior Penalty Galerkin formulation. The non-linear terms in the Navier–Stokes equations are expressed in the convective form and approximated through the Lesaint–Raviart fluxes modified for DG methods.Verification of the solver is performed for a series of test problems; purely elliptic, unsteady Stokes and full Navier–Stokes. The resulting method leads to a stable scheme for the unsteady Stokes and Navier–Stokes equations when equal order approximation is used for velocity and pressure. For the validation of the full Navier–Stokes solver, we consider unsteady laminar flow past a square cylinder at a Reynolds number of 100 (unsteady wake). The DG solver shows favourably comparisons to experimental data and a continuous Spectral code.     

An efficient solver for the RANS equations and a one-equation turbulence model

A three-stage Runge-Kutta (RK) scheme with multigrid and an implicit preconditioner has been shown to be an effective solver for the fluid dynamic equations. Using the algebraic turbulence model of Baldwin and Lomax, this scheme has been used to solve the compressible Reynolds-averaged Navier–Stokes (RANS) equations for transonic and low-speed flows. In this paper we focus on the convergence of the RK/Implicit scheme when the effects of turbulence are represented by the one-equation model of Spalart and Allmaras. With the present scheme the RANS equations and the partial differential equation of the turbulence model are solved in a loosely coupled manner. This approach allows the convergence behavior of each system to be examined. Point symmetric Gauss-Seidel supplemented with local line relaxation is used to approximate the inverse of the implicit operator of the RANS solver. To solve the turbulence equation we consider three alternative methods: diagonally dominant alternating direction implicit (DDADI), symmetric line Gauss-Seidel (SLGS), and a two-stage RK scheme with implicit preconditioning. Computational results are presented for airfoil flows, and comparisons are made with experimental data. We demonstrate that the two-dimensional RANS equations and a transport-type equation for turbulence modeling can be efficiently solved with an indirectly coupled algorithm that uses RK/Implicit schemes.     

Numerical solution of incompressible Navier–Stokes equations using a fractional-step approach

A fractional step method for the solution of steady and unsteady incompressible Navier–Stokes equations is outlined. The method is based on a finite-volume formulation and uses the pressure in the cell center and the mass fluxes across the faces of each cell as dependent variables. Implicit treatment of convective and viscous terms in the momentum equations enables the numerical stability restrictions to be relaxed. The linearization error in the implicit solution of momentum equations is reduced by using three subiterations in order to achieve second order temporal accuracy for time-accurate calculations. In spatial discretizations of the momentum equations, a high-order (third and fifth) flux-difference splitting for the convective terms and a second-order central difference for the viscous terms are used. The resulting algebraic equations are solved with a line-relaxation scheme which allows the use of large time step. A four color ZEBRA scheme is employed after the line-relaxation procedure in the solution of the Poisson equation for pressure. This procedure is applied to a Couette flow problem using a distorted computational grid to show that the method minimizes grid effects. Additional benchmark cases include the unsteady laminar flow over a circular cylinder for Reynolds numbers of 200, and a 3-D, steady, turbulent wingtip vortex wake propagation study. The solution algorithm does a very good job in resolving the vortex core when fifth-order upwind differencing and a modified production term in the Baldwin–Barth one-equation turbulence model are used with adequate grid resolution.     

Incompressible Navier–Stokes solutions by a triangular spectral/p element projection method

The application of the fractional step projection method recently proposed by Guermond and Quartapelle to the numerical approximation of unsteady Navier–Stokes solutions by means of a spectral/p element method is considered. In particular we illustrate the second-order pressure correction technique and evaluate its accuracy properties in some test cases. Stability with respect to the compatibility condition between the approximation spaces for velocity and pressure is also addressed. The high (spectral) accuracy in space and the second-order accuracy in time are verified by two simple test cases with analytical solution. A more interesting problem is solved showing the ability of the method to produce very accurate results also for problems in complex geometries.     

Internal stabilizability of the Navier–Stokes equations

One shows that the steady-state solutions to Navier–Stokes equations in d-dimensional domains Ω, d=2,3 with Dirichlet and slip curl boundary conditions are exponentially stabilizable by proportional controllers with the support in open subsets ωΩ such that Ωω is sufficiently “small”.     

Development of a parallelized 2D/2D-axisymmetric Navier–Stokes equation solver for all-speed gas flows

A parallelized 2D/2D-axisymmetric pressure-based, extended SIMPLE finite-volume Navier–Stokes equation solver using Cartesians grids has been developed for simulating compressible, viscous, heat conductive and rarefied gas flows at all speeds with conjugate heat transfer. The discretized equations are solved by the parallel Krylov–Schwarz (KS) algorithm, in which the ILU and BiCGStab or GMRES scheme are used as the preconditioner and linear matrix equation solver, respectively. Developed code was validated by comparing previous published simulations wherever available for both low- and high-speed gas flows. Parallel performance for a typical 2D driven cavity problem is tested on the IBM-1350 at NCHC of Taiwan up to 32 processors. Future applications of this code are discussed briefly at the end.     

TURBINS: An immersed boundary, Navier–Stokes code for the simulation of gravity and turbidity currents interacting with complex topographies

An accurate, three-dimensional Navier–Stokes based immersed boundary code called TURBINS has been developed, validated and tested, for the purpose of simulating density-driven gravity and turbidity currents propagating over complex topographies. The code is second order accurate in space and third order in time, uses MPI, and employs a domain decomposition approach. It makes use of multigrid preconditioners and Krylov iterative solvers for the systems of linear equations obtained by the finite difference discretization of the governing equations. TURBINS utilizes the direct forcing variant of the immersed boundary approach and enforces the no-slip boundary condition via the first grid point inside the solid, which yields very accurate wall shear stress results. The results of test simulations are discussed for uniform flow around a circular cylinder, and for two- and three-dimensional lock-exchange gravity currents.     

Textbook multigrid efficiency for the incompressible Navier–Stokes equations: high Reynolds number wakes and boundary layers

Textbook multigrid efficiencies for high Reynolds number simulations based on the incompressible Navier–Stokes equations are attained for a model problem of flow past a finite flat plate. Elements of the full approximation scheme multigrid algorithm, including distributed relaxation, defect correction, and boundary treatment, are presented for the three main physical aspects encountered: entering flow, wake flow, and boundary layer flow. Textbook efficiencies, i.e., reduction of algebraic errors below discretization errors in one full multigrid cycle, are attained for second order accurate simulations at a laminar Reynolds number of 10,000.     

Divide and Congruence Applied to η-Bisimulation

We present congruence formats for η- and rooted η-bisimulation equivalence. These formats are derived using a method for decomposing modal formulas in process algebra. To decide whether a process algebra term satisfies a modal formula, one can check whether its subterms satisfy formulas that are obtained by decomposing the original formula. The decomposition uses the structural operational semantics that underlies the process algebra.     

An explicit Chebyshev pseudospectral multigrid method for incompressible Navier-Stokes equations

The two-dimensional steady incompressible Navier-Stokes equations in the form of primitive variables have been solved by Chebyshev pseudospectral method. The pressure and velocities are coupled by artificial compressibility method and the NS equations are solved by pseudotime method with an explicit four-step Runge-Kutta integrator. In order to reduce the computational time cost, we propose the spectral multigrid algorithm in full approximation storage (FAS) scheme and implement it through V-cycle multigrid and full multigrid (FMG) strategies. Four iterative methods are designed including the single grid method; the full single grid method; the V-cycle multigrid method and the FMG method. The accuracy and efficiency of the numerical methods are validated by three test problems: the modified one-dimensional Burgers equation; the Taylor vortices and the two-dimensional lid driven cavity flow. The computational results fit well with the exact or benchmark solutions. The spectral accuracy can be maintained by the single grid method as well as the multigrid ones, while the time cost is greatly reduced by the latter. For the lid driven cavity flow problem, the FMG is proved to be the most efficient one among the four iterative methods. A speedup of nearly two orders of magnitude can be achieved by the three-level multigrid method and at least one order of magnitude by the two-level multigrid method.     

Some schwarz methods for integral equations on surfaces-h and p versions

Abstact We present new results from 11, 7, 12 on various Schwarz methods for the h and p versions of the boundary element methods applied to prototype first kind integral equations on surfaces. When those integral equations (weakly/hypersingular) are solved numerically by the Galerkin boundary element method, the resulting matrices become ill-conditioned. Hence, for an efficient solution procedure appropriate preconditioners are necessary to reduce the numbers of CG-iterations. In the p version where accuracy of the Galerkin solution is achieved by increasing the polynomial degree the use of suitable Schwarz preconditioners (presented in the paper) leads to only polylogarithmically growing condition numbers. For the h version where accuracy is achieved by reducing the mesh size we present a multi-level additive Schwarz method which is competitive with the multigrid method. Communicated by: U. Langer Dedicated to George C. Hsiao on the occasion of his 70th birthday.     

Correction to ‘Mixed ℋ︁2/ℋ︁∞ Nonlinear Filtering’

In Section 5 of Aliyu and Boukas (Int. J. Robust Nonlinear Control 2009; 19 :394–417), the authors have presented certainty‐equivalent filters for the mixed ??₂/??_∞ filtering problem for affine nonlinear systems. Sufficient conditions for the solvability of the problem with a finite‐dimensional filter are given in terms of a pair of coupled Hamilton–Jacobi–Isaacs equations (HJIEs). In this note, we supply a correction to these HJIEs. Moreover, for linear systems this correction is not necessary. Copyright © 2011 John Wiley & Sons, Ltd.     

ε‐Optimal Value and Approximate Multidimensional Dual Dynamic Programming

This paper develops an approximate dual dynamic programming for an ε? optimal multidimensional control problem governed by first order hyperbolic equations. The problem considered is of the Dieudonne‐Rashevsky type and contains as a particular case class‐qualified deposit and Capital theory problems. It is proven that each Lipschitz continuous function satisfying the dual Hamilton‐Jacobi inequality can be used to define an ε‐optimal value.     

An efficient spectral method based on an orthogonal decomposition of the velocity for transition analysis in wall bounded flow

As a consequence of the Helmholtz–Hodge theorem, any divergence-free vector field can be decomposed in two L²-orthogonal, solenoidal vector fields expressed in terms of projections of the velocity and vorticity fields, on an arbitrary direction in space. Based on this type of decomposition and the choice of the wall-normal direction, an efficient spectral code is developed for incompressible flows developing between two parallel walls. The method relies on a weak formulation of the Navier–Stokes equations in the two corresponding divergence-free subspaces. The approximation is based on Fourier expansions in two directions and on the Chebyshev basis proposed by Moser et al. in the third direction in order to satisfy the wall boundary conditions. The method accuracy is validated for the plane Poiseuille linear stability problem and compared with the case of a spectral collocation method. Simulations of by-pass transition in boundary layers developing between two parallel walls are then presented. Since, by construction, the two orthogonal vector fields of the decomposition are associated respectively to the Orr–Sommerfeld and the Squire modes of the linear stability theory, the method makes it possible to evaluate kinetic energy transfers due to the coupling between these two scalar modes and their interactions with the base flow. The decomposition is also used to describe the structure of finite-length streaks in the earlier stages of transition.     

Numerical Study of a Modified Time-Stepping θ-Scheme for Incompressible Flow Simulations

In [Turek (1996). Int. J. Numer. Meth. Fluids 22, 987–1011], we had performed numerical comparisons for different time stepping schemes for the incompressible Navier–Stokes equations. In this paper, we present the numerical analysis in the context of the Navier–Stokes equations for a modified time-stepping θ-scheme which has been recently proposed by Glowinski [Glowinski (2003). In: Ciarlet, P. G., and Lions, J. L. (eds.), Handbook of Numerical Analysis, Vol. IX, North-Holland, Amsterdam, pp. 3–1176]. Like the well-known classical Fractional-Step-θ-scheme which had been introduced by Glowinski [Glowinski (1985). In Murman, E. M. and Abarbanel, S. S. (eds.), Progress and Supercomputing in Computational Fluid Dynamics, Birkh?user, Boston MA; Bristeau et al. (1987). Comput. Phys. Rep. 6, 73–187], too, and which is still one of the most popular time stepping schemes, with or without operator splitting techniques, this new scheme consists of 3 substeps with nonequidistant substepping to build one macro time step. However, in contrast to the Fractional-Step-θ-scheme, the second substep can be formulated as an extrapolation step for previously computed data only, and the two remaining substeps look like a Backward Euler step so that no expensive operator evaluations for the right hand side vector with older solutions, as for instance in the Crank–Nicolson scheme, have to be performed. This modified scheme is implicit, strongly A-stable and second order accurate, too, which promises some advantageous behavior, particularly in implicit CFD simulations for the nonstationary Navier–Stokes equations. Representative numerical results, based on the software package FEATFLOW [Turek (2000). FEATFLOW Finite element software for the incompressible Navier–Stokes equations: User Manual, Release 1.2, University of Dortmund] are obtained for typical flow problems with benchmark character which provide a fair rating of the solution schemes, particularly in long time simulations.Dedicated to David Gottlieb on the occasion of his 60th anniversary     

ℋ︁∞ and l2–l∞ filtering for two‐dimensional linear parameter‐varying systems

In this paper, the ??_∞ and l₂–l_∞ filtering problem is investigated for two‐dimensional (2‐D) discrete‐time linear parameter‐varying (LPV) systems. Based on the well‐known Fornasini–Marchesini local state‐space (FMLSS) model, the mathematical model of 2‐D systems under consideration is established by incorporating the parameter‐varying phenomenon. The purpose of the problem addressed is to design full‐order ??_∞ and l₂–l_∞ filters such that the filtering error dynamics is asymptotic stable and the prescribed noise attenuation levels in ??_∞ and l₂–l_∞ senses can be achieved, respectively. Sufficient conditions are derived for existence of such filters in terms of parameterized linear matrix inequalities (PLMIs), and the corresponding filter synthesis problem is then transformed into a convex optimization problem that can be efficiently solved by using standard software packages. A simulation example is exploited to demonstrate the usefulness and effectiveness of the proposed design method. Copyright © 2007 John Wiley & Sons, Ltd.     

μ-Bases and singularities of rational planar curves

We provide a technique to detect the singularities of rational planar curves and to compute the correct order of each singularity including the infinitely near singularities without resorting to blow ups. Our approach employs the given parametrization of the curve and uses a μ-basis for the parametrization to construct two planar algebraic curves whose intersection points correspond to the parameters of the singularities including infinitely near singularities with proper multiplicity. This approach extends Abhyankar's method of t-resultants from planar polynomial curves to rational planar curves. We also derive the classical result that for a rational planar curve of degree n the sum of all the singularities with proper multiplicity is (n−1)(n−2)/2. Examples are provided to flesh out our results.     

A finite volume method to solve the 3D Navier–Stokes equations on unstructured collocated meshes

A new method to solve the Navier–Stokes equations for incompressible viscous flows and the transport of a scalar quantity is proposed. This method is based upon a fractional time step scheme and the finite volume method on unstructured meshes. The governing equations are discretized using a collocated, cell-centered arrangement of velocity and pressure. The solution variables are stored at the cell-circumcenters. Theoretical results and numerical properties of the scheme are provided. Predictions of lid-driven cavity flow, flows past a cylinder and heat transport in a cylinder are performed to validate the method.     

Improved parallelized hybrid DSMC–NS method

In this paper, an improved parallelized hybrid DSMC–NS (Navier–Stokes method) algorithm, as compared to the previous work [1], is presented. A detailed kinetic velocity sampling study is conducted with a two-dimensional supersonic flow (M_∞ = 4) past a 25° finite wedge. It shows most of the boundary layer region is in nearly thermal equilibrium, even with very high continuum breakdown parameter based on velocity, velocity gradient and local mean free path. A new continuum breakdown parameter based on pressure is designed to effectively “exclude” the “false” breakdown region such as the boundary layer. An improved hybrid DSMC–NS algorithm is verified using the same wedge flow case. Results show that the improved algorithm can greatly reduce the computational cost while maintaining essentially the same accuracy. A hypersonic flow (M_∞ = 12) past a square cylinder is also employed to exhibit the capability of the improved hybrid DSMC–NS method.     

A Compact Fourth-Order Finite Difference Scheme for Unsteady Viscous Incompressible Flows

In this paper, we extend a previous work on a compact scheme for the steady Navier–Stokes equations [Li, Tang, and Fornberg (1995), Int. J. Numer. Methods Fluids, 20, 1137–1151] to the unsteady case. By exploiting the coupling relation between the streamfunction and vorticity equations, the Navier–Stokes equations are discretized in space within a 3×3 stencil such that a fourth order accuracy is achieved. The time derivatives are discretized in such a way as to maintain the compactness of the stencil. We explore several known time-stepping approaches including second-order BDF method, fourth-order BDF method and the Crank–Nicolson method. Numerical solutions are obtained for the driven cavity problem and are compared with solutions available in the literature. For large values of the Reynolds number, it is found that high-order time discretizations outperform the low-order ones.