This is a scientific web page about the two-dimensional steady incompressible flow in a driven cavity. The steady incompressible 2-D Navier-Stokes equations are solved numerically. Driven cavity flow, numerical methods, steady incompressible flow, finite difference, Navier Stokes equations. Driven cavity flow, numerical methods, steady incompressible flow, finite difference, Navier Stokes equations. Driven cavity flow, numerical methods, steady incompressible flow, finite difference, Navier Stokes equations. Driven cavity flow, numerical methods, steady incompressible flow, finite difference, Navier Stokes equations. Driven cavity flow, numerical methods, steady incompressible flow, finite difference, Navier Stokes equations. Driven cavity flow, numerical methods, steady incompressible flow, finite difference, Navier Stokes equations. Driven cavity flow, numerical methods, steady incompressible flow, finite difference, Navier Stokes equations. Driven cavity flow, numerical methods, steady incompressible flow, finite difference, Navier Stokes equations. Driven cavity flow, numerical methods, steady incompressible flow, finite difference, Navier Stokes equations. Driven cavity flow, numerical methods, steady incompressible flow, finite difference, Navier Stokes equations. Driven cavity flow, numerical methods, steady incompressible flow, finite difference, Navier Stokes equations. Driven cavity flow, numerical methods, steady incompressible flow, finite difference, Navier Stokes equations. Driven cavity flow, numerical methods, steady incompressible flow, finite difference, Navier Stokes equations. Driven cavity flow, numerical methods, steady incompressible flow, finite difference, Navier Stokes equations. Driven cavity flow, numerical methods, steady incompressible flow, finite difference, Navier Stokes equations. This is a scientific web page about the two-dimensional steady incompressible flow in a driven cavity. The steady incompressible 2-D Navier-Stokes equations are solved numerically. Driven cavity flow, numerical methods, steady incompressible flow, finite difference, Navier Stokes equations. Driven cavity flow, numerical methods, steady incompressible flow, finite difference, Navier Stokes equations. Driven cavity flow, numerical methods, steady incompressible flow, finite difference, Navier Stokes equations. Driven cavity flow, numerical methods, steady incompressible flow, finite difference, Navier Stokes equations. Driven cavity flow, numerical methods, steady incompressible flow, finite difference, Navier Stokes equations. Driven cavity flow, numerical methods, steady incompressible flow, finite difference, Navier Stokes equations. Driven cavity flow, numerical methods, steady incompressible flow, finite difference, Navier Stokes equations. Driven cavity flow, numerical methods, steady incompressible flow, finite difference, Navier Stokes equations. Driven cavity flow, numerical methods, steady incompressible flow, finite difference, Navier Stokes equations. Driven cavity flow, numerical methods, steady incompressible flow, finite difference, Navier Stokes equations. Driven cavity flow, numerical methods, steady incompressible flow, finite difference, Navier Stokes equations. Driven cavity flow, numerical methods, steady incompressible flow, finite difference, Navier Stokes equations. Driven cavity flow, numerical methods, steady incompressible flow, finite difference, Navier Stokes equations. Driven cavity flow, numerical methods, steady incompressible flow, finite difference, Navier Stokes equations. Driven cavity flow, numerical methods, steady incompressible flow, finite difference, Navier Stokes equations.

www.cavityflow.com Numerical Performance of Fourth
Order Navier-Stokes (FONS) Equations

 

In Computational Fluid Dynamics (CFD) field of study High Order Compact formulations are becoming more popular. Compact formulations provide more accurate solutions in a compact stencil.

In finite difference, in order to achieve fourth order spatial accuracy, standard five point discretization can be used. When a five point discretization is used, the points near the boundaries have to be treated specially. Another way to achieve fourth order spatial accuracy is to use High Order Compact schemes. High Order Compact schemes provide fourth order spatial accuracy in a 3×3 stencil, hence this type of formulations can be used near the boundaries without a complexity.

To the best of the our knowledge, in the literature there is not a study that documents the numerical performance of high order compact formulation the Navier-Stokes equations compared to regular second order formulation of the Navier-Stokes equations in terms of numerical stability and convergence for a chosen iterative method. In this study using the FONS equations introduced by Erturk and Gokcol (Int. J. Numer. Methods Fluids, 50, 421-436), we will numerically solve the Navier-Stokes equations for both fourth order (O (Dx4 )) and second order (O (Dx2 )) spatial accuracy. This way we will be able to compare the convergence and stability characteristics of both formulations. In this study we will also document the extra CPU work that is needed for convergence when a second order accurate code is converted into a fourth order accurate code using the introduced FONS equations by Erturk and Gokcol (Int. J. Numer. Methods Fluids, 50, 421-436). The stability and convergence characteristics of both formulations and also the extra CPU work can show variation depending on the iterative numerical method used for the solution therefore in this study we will use two different line iterative semi-implicit numerical methods. Using these two numerical methods we will solve the benchmark driven cavity flow problem. First we will solve the cavity flow with second order (O (Dx2 )) spatial accuracy then we will solve the same flow with fourth order (O (Dx4 )) spatial accuracy. We will document the stability characteristics, such as the maximum allowable time increment (Dt), convergence characteristics, such as the number of iterations and the CPU time necessary for a chosen convergence criteria and also the extra CPU work that is needed to increase the spatial accuracy of the numerical solution from second order to fourth order using the FONS equations.

In the following web pages, you will find detailed information about the numerical performance of the fourth order compact formulation of the steady 2-D incompressible Navier-Stokes equations.

 

 

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