Preliminary Study Of Fourier Analysis And Refinement Of The SIMPLE-serials Algorithms

The SIMPLE algorithm, originally proposed by Patankar in1972, is widely used in numerical simulation of incompressible fluid. Since then several advances have been made in refining the SIMPLE algorithm in terms of computation efficiency, simulating accuracy and grid arrangement, and the SIMPLE algorithm and its variants have evolved into a serials algorithms which dominate in incompressible fluid modeling. At the same time, it is deserved to make some explanations to these remarkable refinements from a better rigorous mathematical view than numerical validation. Although several theoretical researches have been done to study the convergence properties of SIMPLE-serials algorithms so far, most work still focus on the stability analysis of numerical schemes employed for discretizing convection term. It is obvious that the convergence properties are also closely linked to other factors, such as relaxation factors and grid arrangement, which basically maintain blank to researchers. Consequently, the primary object of this paper aims at giving an insight into the convergence properties of SIMPLE-serials from a systematic and comprehensive view, and makes some theoretical enhancements for further improvement of SIMPLE-serials algorithms. The main contents are listed as follows:1. The error amplification matrixes and eigenvalue equations corresponding to six representative SIMPLE-serials algorithms formulated on both collocated and staggered grid are derived for the following convergence property analysis.2. An approaching Fourier method is proposed based on the traditional Fourier analysis which is widely used for convergence properties analysis.3. The approaching Fourier method is employed for checking the effects of relaxation factors, interpolation methods for interfacial velocity, grid arrangement and grid orthogonality on the SIMPLE-serials algorithm and verifying the comparative progresses of SIMPLE-variants with SIMPLE algorithm.4. Based on the above analysis, the improved versions of SIMPLE and SIMPLER algorithms are proposed, and the improvements are verified by three numerical cases in combination with the approaching Fourier analysis.