A fourth-order accurate compact scheme for the solution of steady Navier-Stokes equations on non-uniform grids (original) (raw)
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International Journal for Numerical Methods in Fluids, 2006
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 of 20,000 with fourth order spatial accuracy. Detailed solutions are presented.
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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
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Computers & Mathematics with Applications, 2017
We present a high-order finite difference scheme for Navier-Stokes equations in irregular domains. The scheme is an extension of a fourth-order scheme for Navier-Stokes equations in streamfunction formulation on a rectangular domain (Ben-Artzi et al., 2010). The discretization offered here contains two types of interior points. The first is regular interior points, where all eight neighboring points of a grid point are inside the domain and not too close to the boundary. The second is interior points where at least one of the closest eight neighbors is outside the computational domain or too close to the boundary. In the second case we design discrete operators which approximate spatial derivatives of the streamfunction on irregular meshes, using discretizations of pure derivatives in the x, y and along the diagonals of the element.
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.
A Fourth-Order Compact Scheme for the Navier-Stokes Equations in Irregular Domains
2016
We present a high-order finite difference scheme for Navier-Stokes equations in irregular domains. The discretization offered here contains two types of interior points. The first is regular interior points, where all eight neighboring points of a grid point are inside the domain and not too close to the boundary. The second is interior points where at least one of the closest eight neighbors is outside the computational domain or too close to the boundary. In the second case we design discrete operators which approximate spatial derivatives of the streamfunction on irregular meshes, using discretizations of pure derivatives in the x, y and along the diagonals of the element.
Fourth order convergence of compact finite difference solver for 2D incompressible flow
We study a fourth order finite difference method for the unsteady incompressible Navier-Stokes equations in vorticity formulation. The scheme is essentially compact and can be implemented very efficiently. Either Briley's formula, or a new higher order formula, which will be derived in this paper, can be chosen as the vorticity boundary condition. By formal Taylor expansion, the new formula for the vorticity on the boundary gives 4th order accuracy; while Briley's formula provides only 3rd order accuracy. However, the use of either formula results in a stable method and achieves full 4th order accuracy. The convergence analysis of the scheme with our new formula will be given in this paper, while that with Briley's formula has been established in earlier literature. The consistency analysis is easier than that of Briley's formula, no Strang type analysis is needed. In the stability analysis part, we adopt the technique of controlling some local terms by the diffusion term via discrete elliptic regularity. Physical no-slip boundary conditions are used throughout.
An efficient transient Navier-Stokes solver on compact nonuniform space grids
Journal of Computational and Applied …, 2008
In this paper, we propose an implicit higher-order compact (HOC) finite difference scheme for solving the two-dimensional (2D) unsteady Navier-Stokes (N-S) equations on nonuniform space grids. This temporally second-order accurate scheme which requires no transformation from the physical to the computational plane is at least third-order accurate in space, which has been demonstrated with numerical experiments. It efficiently captures both transient and steady-state solutions of the N-S equations with Dirichlet as well as Neumann boundary conditions. The proposed scheme is likely to be very useful for the computation of transient viscous flows involving free and wall bounded shear layers which invariably contain spatial scale variation. Numerical results are presented and compared with analytical as well as established numerical data. Excellent comparison is obtained in all the cases.
Exponential Compact Higher Order Scheme for Steady Incompressible Navier-Stokes Equations
Exponential Compact Higher Order Scheme for Steady Incompressible Navier-Stokes Equations, 2012
An Exponential Compact Higher Order (ECHO) scheme is developed for the stream function vorticity form of steady viscous incompressible Navier-Stokes equations. The coupled equations are solved using an iterative procedure. The developed scheme has been validated initially using two model problems with known analytic solutions and demonstrated the fourth order rate of convergence and about fifth order for highly convection dominated problems. Thus the developed scheme requires smaller number of grid points for the computation. Finally the tested code is used to simulate the lid-driven square cavity flow up to Reynolds number 10,000. Keywords: Navier-Stokes equations, ECHO, stream function, vorticity, high Reynolds number