High-order compact schemes for incompressible flows: A simple and efficient method with quasi-spectral accuracy (original) (raw)
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High-order finite difference schemes for incompressible flows
International Journal for Numerical Methods in Fluids, 2011
This paper presents a new high-order approach to the numerical solution of the incompressible Stokes and Navier–Stokes equations. The class of schemes developed is based upon a velocity–pressure–pressure gradient formulation, which allows: (i) high-order finite difference stencils to be applied on non-staggered grids; (ii) high-order pressure gradient approximations to be made using standard Padé schemes, and (iii) a variety of boundary conditions to be incorporated in a natural manner. Results are presented in detail for a selection of two-dimensional steady-state test problems, using the fourth-order scheme to demonstrate the accuracy and the robustness of the proposed methods. Furthermore, extensions to higher orders and time-dependent problems are illustrated, whereas the extension to three-dimensional problems is also discussed. Copyright © 2010 John Wiley & Sons, Ltd.
High-Order Methods for Incompressible Fluid Flow
Applied Mechanics Reviews, 2003
In this book, the author has documented his lecture notes on computational fluid dynamics ͑CFD͒ which he has developed over the past 20 years to form this textbook. This textbook is intended for senior undergraduate and first-year graduate students who will be developing or using codes in the numerical simulation of fluid flows or other physical phenomena governed by partial differential equations. The book is organized into 11 chapters. The fundamental numerical methods discussed in this book are based on the finite difference method as a method of discretization on Cartesian mesh systems, in the physical domain, or in the computational domain after coordinate transformation. The author has discussed the finite volume method for discretization for arbitrary mesh systems including unstructured meshes. The fundamental theory and techniques are presented in Chapter 2, which include the Taylor expansion and the complex mode analysis. Further, the accuracy and stability analyses for these methods are also discussed. The ordinary differential equations ͑ODEs͒ and their integration are given in Chapter 3. The general discussions of PDEs are given in Chapter 4, which include the discussions on the type and classifications of PDEs and the concepts of characteristic surfaces, computational relations, and the jump conditions associated with conservation laws. The connection between the physical phenomena of wave propagation,
A high-order finite difference method for incompressible fluid turbulence simulations
International Journal for Numerical Methods in Fluids, 2003
A Hermitian-Fourier numerical method for solving the Navier-Stokes equations with one non-homogeneous direction had been presented by Schiestel and Viazzo (Internat. J. Comput. Fluids 1995; 24(6):739). In the present paper, an extension of the method is devised for solving problems with two non-homogeneous directions. This extension is indeed not trivial since new algorithms will be necessary, in particular for pressure calculation. The method uses Hermitian ÿnite di erences in the non-periodic directions whereas Fourier pseudo-spectral developments are used in the remaining periodic direction. Pressure-velocity coupling is solved by a simpliÿed Poisson equation for the pressure correction using direct method of solution that preserves Hermitian accuracy for pressure. The turbulent ow after a backward facing step has been used as a test case to show the capabilities of the method. The applications in view are mainly concerning the numerical simulation of turbulent and transitional ows.
High order accurate solution of the incompressible Navier–Stokes equations
Journal of Computational Physics, 2005
High order methods are of great interest in the study of turbulent flows in complex geometries by means of direct simulation. With this goal in mind, the incompressible Navier-Stokes equations are discretized in space by a compact fourth order finite difference method on a staggered grid. The equations are integrated in time by a second order semi-implicit method. Stable boundary conditions are implemented and the grid is allowed to be curvilinear in two space dimensions. In every time step, a system of linear equations is solved for the velocity and the pressure by an outer and an inner iteration with preconditioning. The convergence properties of the iterative method are analyzed. The order of accuracy of the method is demonstrated in numerical experiments. The method is used to compute the flow in a channel, the driven cavity and a constricted channel.
Fully Conservative Higher Order Finite Difference Schemes for Incompressible Flow
Journal of Computational Physics, 1998
Conservation properties of the mass, momentum, and kinetic energy equations for incompressible flow are specified as analytical requirements for a proper set of discrete equations. Existing finite difference schemes in regular and staggered grid systems are checked for violations of the conservation requirements and a few important discrepancies are pointed out. In particular, it is found that none of the existing higher order schemes for a staggered mesh system simultaneously conserve mass, momentum, and kinetic energy. This deficiency is corrected through the derivation of a general family of fully conservative higher order accurate finite difference schemes for staggered grid systems. Finite difference schemes in a collocated grid system are also analyzed, and a violation of kinetic energy conservation is revealed. The predicted conservation properties are demonstrated numerically in simulations of inviscid white noise, performed in a two-dimensional periodic domain. The proposed fourth order schemes in a staggered grid system are generalized for the case of a nonuniform mesh, and the resulting scheme is used to perform large eddy simulations of turbulent channel flow.
A new high-order method for the simulation of incompressible wall-bounded turbulent flows
Journal of Computational Physics, 2014
A new high-order method for the accurate simulation of incompressible wall-bounded flows is presented. In the stream-and spanwise directions the discretisation is performed by standard Fourier series, while in the wall-normal direction the method combines highorder collocated compact finite differences with the influence matrix method to calculate the pressure boundary conditions that render the velocity field exactly divergence-free. The main advantage over Chebyshev collocation is that in wall-normal direction, the grid can be chosen freely and thus excessive clustering near the wall is avoided. This can be done while maintaining the high-order approximation as offered by compact finite differences. The discrete Poisson equation is solved in a novel way that avoids any full matrices and thus improves numerical efficiency. Both explicit and implicit discretisations of the viscous terms are described, with the implicit method being more complex, but also having a wider range of applications. The method is validated by simulating two-dimensional Tollmien-Schlichting waves, forced transition in turbulent channel flow, and fully turbulent channel flow at friction Reynolds number Re τ = 395, and comparing our data with analytical and existing numerical results. In all cases, the results show excellent agreement showing that the method simulates all physical processes correctly.
Applied Mathematics and Computation, 2010
This article presents a time-accurate numerical method using high-order accurate compact finite difference scheme for the incompressible Navier-Stokes equations. The method relies on the artificial compressibility formulation, which endows the governing equations a hyperbolic-parabolic nature. The convective terms are discretized with a third-order upwind compact scheme based on flux-difference splitting, and the viscous terms are approximated with a fourth-order central compact scheme. Dual-time stepping is implemented for time-accurate calculation in conjunction with Beam-Warming approximate factorization scheme. The present compact scheme is compared with an established non-compact scheme via analysis in a model equation and numerical tests in four benchmark flow problems. Comparisons demonstrate that the present third-order upwind compact scheme is more accurate than the non-compact scheme while having the same computational cost as the latter.
The nature of boundary conditions, and how they are implemented, can have a significant impact on the stability and accuracy of a Computational Fluid Dynamics (CFD) solver. The objective of this paper is to assess how di erent boundary conditions impact the performance of compact discontinuous high-order spectral element methods (such as the discontinuous Galerkin method and the Flux Reconstruction approach), when these schemes are used to solve the Euler and compressible Navier-Stokes equations on unstructured grids. Speci cally, the paper will investigate inflow/outflow and wall boundary conditions. In all studies the boundary conditions were enforced by modifying the boundary flux. For Riemann invariant (characteristic), slip and no-slip conditions we have considered a direct and an indirect enforcement of the boundary conditions, the first obtained by calculating the flux using the known solution at the given boundary while the second achieved by using a ghost state and by solving a Riemann problem. All computations were performed using the open-source software Nektar++ (www.nektar.info).
Higher-Order Compact Schemes for Numerical Simulation of Incompressible Flows, Part II: Applications
Numerical Heat Transfer, Part B: Fundamentals, 2001
A higher order a.(:curat(' mnnerical t)roce(ture has been deveh)l)ed for solving incompressibh' • *The authors were partially supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-19480 while the first two authors were in residence at the Institute for Computer Applications in Science and Engineering (ICASI£,), NASA Langley Resear(:h Center, Hampton, VA 23681-2199. Additional sllpport was provided l)y the NASA Graduate Studenl ll.esearch Program. • _l)epartment of M(,ehanical Engineering, Ohl Dominion l!niversfly, N_)rfolk, Vir_zinia 2352.(t. • {Acouslic and Flow Met hods Branch,
A staggered grid, high-order accurate method for the incompressible Navier–Stokes equations
Journal of Computational Physics, 2006
A high-order accurate, finite-difference method for the numerical solution of the incompressible Navier–Stokes equations is presented. Fourth-order accurate discretizations of the convective and viscous fluxes are obtained on staggered meshes using explicit or compact finite-difference formulas. High-order accuracy in time is obtained by marching the solution with Runge–Kutta methods. The incompressibility constraint is enforced for each Runge–Kutta stage iteratively either