The Research Of Numerical Perturbation Algorithm And The Application Of Perturbation Schemes

In the rapid development of computational ?uid dynamics(CFD) over halfa century, the research of algorithm has always been a core subject in CFD.Numerical perturbation algorithm presented by Gao Zhi in recent years, is anew high-accurate and high-resolution algorithm. This algorithm (referred asGao-algorithm) is fundamentally di?erent from other high-accurate and high-resolution algorithms. Gao-algorithm transforms the function perturbation tech-nique in the field of the applied mathematics and mechanics into numerical per-turbation one. By numerical perturbation of convection discrete term in thecondition of maintaining the original form of the scheme, one can obtain thehigh accuracy and high resolution Gao-schemes, including the upwind and cen-tral finite di?erence and finite volume Gao-schemes. Therefore, Gao-schemeshave the properties of high accurate, high resolution, wide stability, simplicity ofprogramming and code maintenance, are of the theoretical significance and wideapplication prospects.Gao-algorithm and corresponding Gao-schemes are still under development,obviously, Gao-schemes will form a scheme family with a large members. Nowa number of important family members using the three-node has been built,but some which may be expected are not presented, for examples, numericalperturbation reconstruction schemes of both the multi-node convection upwindscheme and the schemes with convective conservation ?ux and di?usive naturalvariable. Gao-schemes have been applied to calculations of incompressible andcompressible ?ows, two phase ?ow, etc. but only limited in cavity ?ows.This study focused on the following three aspects:1. To begin with, the discrete scheme of the convective-di?usion equation, inwhich the viscous and convective terms are discrete separately as second-ordercentral and third-order upwind di?erence scheme(3UDS), is reconstructedby using the numerical perturbation algorithm presented by Gao Zhi. Thereconstruct methods have two, one is global reconstruction using all nodesin the discrete grid, the other is upstream and downstream reconstructionusing separately upstream and downstream nodes in the discrete grid. Twokind of higher-order accurate schemes called Gao's upwind di?erence schemes (GUDS, for bravety)are developed by using two reconstruct methods statedabove. GUDS is so simple as 3UDS, but GUDS have more high accurate than3UDS. Both global GUDS and 3UDS are conditional stable schemes, whilesome GUDS of upstream and downstream reconstruction are absolute stableschemes. Excellent properties of GUDS for the convective-di?usion equationare proved by analysis and four numerical tests.The results show that GUDSof upstream and downstream reconstruction provide a new e?ective way for3UDS needing no artificial viscosity and are suitable for solving linear andnonlinear high Reynolds number problems.2. Furthermore, the second-order central finite volume scheme (2CVS) for theconvective-di?usion equation is reconstructed by the dual numerical pertur-bation algorithm presented by the Gao Zhi, some higher-order accurate ofinterpolation approximation, absolute positive central finite volume schemes(GCVS) are obtained. Excellent properties of GCVS for the convective-di?usion equation are proved by analysis and four numerical tests in thispaper.3. Finally, a brief discussion of the basic principle and calculation steps ofthe Semi-Implicit Method for the Pressure Linked Equation called SIMPLE,which is the very popular algorithm for incompressible pressure-based algo-rithm currently. Then try to couple GCVS with the SIMPLE algorithm, andnumerically simulate the the lid-deriven cavity ?ow. The results of numericalsimulations to lid-driven cavity ?ows shows that the GCVS scheme is validand has great application prospect.