High Order Compact (HOC) formulations are becoming more popular in Computational Fluid Dynamics (CFD) field of study. Compact formulations provide more accurate solutions in a compact stencil. High order compact schemes provide fourth order spatial accuracy in a 3×3 stencil. Even though, a high order compact (HOC) formulation have advantages in many ways, one disadvantage may be that there is not many different iterative schemes available for this type of formulation. When a HOC formulation is used the resulting equations are either solved directly or with a point or sometimes with a line iterative solver. A direct solver requires a lot of computer resources (ie. RAM, CPU time) especially when large number of grids are used. Point or line iterations are more preferred.
In the literature, it is possible to find numerous different type of iterative numerical methods for the Navier-Stokes equations. These numerical methods could not be easily used in HOC schemes because of the final form of the HOC formulations found in the literature. It would be very useful if any iterative numerical methods described in books and papers for Navier-Stokes equations could be applied to high order compact (HOC) formulations.
In this study, you will find a new fourth order compact (HOC) formulation. The main difference of this formulation from the ones found in the literature is the way that the final form of the equations are written. The main advantage of this formulation is that, any iterative numerical method used for Navier-Stokes equations, can be applied to this HOC formulation. Moreover if someone already have a second order accurate (O Dx2) code for the solution of steady 2-D incompressible Navier-Stokes equations, they can easily convert their existing code to fourth order accuracy (O Dx4) by just adding some coefficients into their existing code. Using this new compact formulation, we have solved the steady 2-D incompressible driven cavity flow at very high Reynolds numbers using a very fine grid mesh to demonstrate the efficiency of this new formulation.
In the following web pages, you will find detailed information about the fourth order compact formulation Erturk and Gokcol have presented and its application to driven cavity flow, tabulated datas and figures and more.
In a previous study Erturk et. al. have have presented a new, efficient and stable numerical method for the solution of streamfunction and vorticity equations. With this method they have presented steady solutions of driven cavity flow at very high Reynolds numbers (up to Re=21,000) using very fine grid mesh.
In the following study, E. Erturk have analysed the nature of the cavity flow at high Reynolds numbers. In the literature, while some studies claims that the flow in a driven cavity is not steady at a Reynolds number and present unsteady solutions, some other studies present steady solutions at even higher Reynolds numbers. There is a controversy on whether the flow in a cavity is steady or not at high Reynolds numbers. E. Erturk have analysed the cavity flow problem in terms of physical, mathematical and also numerical aspects with a brief literature survey on experimental, analytical and also numerical studies on cavity flow and then presented very fine grid numerical solutions of driven cavity flow at high Reynolds numbers obtained by solving the governing equations simply with Successive Over Relaxation (SOR) method.
In the following study, Erturk and Gokcol have analysed the steady 2-D steady incompressible flow inside a lid driven triangular cavity. The triangular cavity flow is solved using Successive Over Relaxation (SOR) method using a fine grid mesh at high Reynolds numbers and accurate solutions are presented.
In the literature, there are not much benchmark problems with non-orthogonal grids for numerical methods to compare solutions with each other. The skewed cavity problem can be a perfect test case for body fitted non-orthogonal grids and yet it is as simple as the cavity flow in terms of programming point of view. The test case is similar to driven cavity flow but the geometry is a parallelogram rather than a square. In this test case, the skewness of the geometry can be easily changed by changing the skew angle (a). In the following study Erturk and Dursun have presented thenumerical solutions of the driven skewed cavity flow problem for skew angles 15° £ a £ 165°, with body fitted non-orthogonal skewed grid mesh of (513×513). By changing the skew angle to extreme values it is possible to test numerical methods for grid skewness in terms of stability, efficiency and accuracy. The numerical solutions of the flow in a skewed cavity will be presented for Reynolds number of 100 and 1000 for a wide variety of skew angles ranging between a=15° and a=165° with Da=15° increments.