Finite difference methods for incompressible flow problems

For fluid flow problems with simple geometry, finite difference methods provide an effective technique for computing the solution to incompressible fluid motion. My research interest in finite difference methods stems from a desire to use finite difference methods to compute "benchmark" solutions that can be used for comparison with solutions obtained with other solution techniques e.g. vortex methods. I've worked on developing high-order accurate finite difference methods ("A High Order Explicit Method for the Computation of Flow About a Circular Cylinder"), as well as on some of the technical details that one encounters when using finite difference techniques for the vorticity-stream function formulation ("Vorticity Boundary Conditions and Boundary Vorticity Generation for Two-Dimensional Viscous Incompressible Flows" , "Observations on Vorticity Creation Boundary Conditions"), and when one uses a pressure-velocity formulation ("Derivation and Solution of the Discrete Pressure Equations for the Incompressible Navier Stokes Equations").


"A High Order Explicit Method for the Computation of Flow About a Circular Cylinder", Christopher R. Anderson and Marc B. Reider, J. of Comp. Physics, 125, 207-224, (1996)

Abstract : In this paper a difference method for computing the solutions of the incompressible Navier-Stokes equations for flow about a circular cylinder in two dimensions is presented. The stream function/vorticity formulation of the equation is used and the numerical method incorporates recent developments in the computation of vorticity boundary conditions as well as far-field boundary conditions. Three schemes are discussed, one of second order accuracy, one of third order accuracy, and a hybrid method which is second order accurate in the computation of vorticity transport and fourth order accurate in the determination of the stream function. Fully resolved solutions for flow past cylinders have been computed over a range of Reynolds Reynolds numbers from 1000 to 9500. Comparisons are made between the results obtained with methods of different orders of accuracy as well the effectiveness of vorticity and far-field boundary conditions.


"Vorticity Boundary Conditions and Boundary Vorticity Generation for Two-Dimensional Viscous Incompressible Flows", Christopher R. Anderson, J. of Comp. Phys., Vol. 80, No. 1, January 1989. 72-97

Abstract : In this paper we present boundary conditions appropriate for the vorticity formulation of the two-dimensional, incompressible, viscous Navier-Stokes equations. These boundary conditions are incorporated into a finite difference scheme and the resulting method is of the vorticity creation type; i.e., vorticity is generated at the boundary to ensure that the tangential velocity boundary condition is satisfied. The results of computations are presented for flow past a circular cylinder. A difference scheme and computational results for a model problem, the Prandtl boundary layer equations describing flow over a semi-infinite flat plate, are also presented.

"Observations on Vorticity Creation Boundary Conditions", Christopher R. Anderson, Mathematical Aspects of Vortex Dynamics, ed. R. Caflisch, Proceedings of the Workshop on the Mathematical Aspects of Vortex Dynamics, Virginia, 1988. SIAM Publications, 1998. pg. 144-159.

Abstract: In this paper we discuss some issues related to the problem of vorticity boundary conditions. We outline the origins of the problem and briefly discuss three solution procedures which have been used to overcome it. We primarily focus on one method -- that which employs boundary vorticity creation. We give an example which shows the equivalence between creation type boundary conditions and those which exploit the relationship between the vorticity and the stream function on the boundary. We also show how one can obtain greater than first order accuracy in time for methods employing creation type boundary conditions. Lastly, we show by example the problems that arise when the boundary conditions on the vorticity are such that the no-slip condition on the velocity is satisfied only to truncation error rather than to roundoff error.


"Derivation and Solution of the Discrete Pressure Equations for the Incompressible Navier Stokes Equations", C.R. Anderson, Pre-print, Lawrence Berkeley Laboratory report LBL-26353. December 1988.

Abstract : In this paper we derive the equations for the pressure that must be solved in order to advance several commonly used finite difference approximations to the Navier-Stokes equations on rectangular domains. We consider the equations for both staggered and non-staggered grids. We present solution techniques for these equations that require on the order of O(NlogN) operations where N is the number of points in the computational domain. domain.