Fourth‒order method for solving the Navier–Stokes equations in a constricting channel

✍ Scribed by P. F. de A. Mancera; R. Hunt

• Publisher: John Wiley and Sons
• Year: 1997
• Tongue: English
• Weight: 285 KB
• Volume: 25
• Category: Article
• ISSN: 0271-2091
• DOI: 10.1002/(sici)1097-0363(19971130)25:10<1119::aid-fld610>3.0.co;2-4

✦ Synopsis

A fourth-order numerical method for solving the Navier±Stokes equations in streamfunctionavorticity formulation on a two-dimensional non-uniform orthogonal grid has been tested on the ¯uid ¯ow in a constricted symmetric channel. The family of grids is generated algebraically using a conformal transformation followed by a non-uniform stretching of the mesh cells in which the shape of the channel boundary can vary from a smooth constriction to one which one possesses a very sharp but smooth corner. The generality of the grids allows the use of long channels upstream and downstream as well as having a re®ned grid near the sharp corner. Derivatives in the governing equations are replaced by fourth-order central differences and the vorticity is eliminated, either before or after the discretization, to form a wide difference molecule for the streamfunction. Extra boundary conditions, necessary for wide-molecule methods, are supplied by a procedure proposed by Henshaw et al. The ensuing set of non-linear equations is solved using Newton iteration. Results have been obtained for Reynolds numbers up to 250 for three constrictions, the ®rst being smooth, the second having a moderately sharp corner and the third with a very sharp corner. Estimates of the error incurred show that the results are very accurate and substantially better than those of the corresponding second-order method. The observed order of the method has been shown to be close to four, demonstrating that the method is genuinely fourth-order.