Grid induced errors in stream function solutions of the Navier-Stokes equations

Abstract

This paper examines the effects of the flow Rcynolds numbcr, grid stretching, and velocity gradients on the of computed solutions to the Navier-Stokes stream function cquations. It is shown that second order truncation crrors become small only as the flow Reynolds number approaches zero. For sufficiently high values of Rcynolds number, the highly nonlinear flow solutions that result from thc large velocity gradients can result in large truncation error, cvcn for the case of non-uniform grids. Thcse errors bchavc as momentum sources and sinks within the cornputcd flow domain, causing significant crror in the prcdictions of boundary laycr separation and rcattachmcnt. The streamrunction-vorticity mcthod is shown to bc unsuitablc to the modcling of viscid-inviscid interactive flows. Limitations on the use of highcr order finite diffcrcncc approximations are discussed. The present work is motivated by a nccd to compute the high Reynolds number, interactive flow about axisymmetric propcller-hull configurations whcre significant streamline curvature cffccts are expected to be encountered. In the region on the body and near the propcllcr, the intcractions bctwcen a rclativcly thick, turbulent boundary laycr and the inviscid outer flow are important. Accurate computational modcling requires a complcte solution of the Navicr-Stokes equations. Incompressible flow is assumed. Thc assumption of axial symmetry allows the simplification of thc Navicr-Stokes equation into a system of two equations that can be solved sequentially: the Poisson cauation for stream function and the vorticitv transuort I differcncc approximation to the elliptic Poisson equation for the strcam function was made by Thom and Apelt[l]. The method has subsequcntly been used in many investigations[2-5]. The appropriate finite difference approximation for a derivative can be obtained from the Taylor series expansion[6] by truncating the highcr order terms of the Taylor series. The difference between the derivative and its finitc difference approximation is known as the truncation error. The magnitude of the truncation error is characterized by its order, which is the limiting behavior of the truncation error as the finite diffcrence step size approaches zero. In the computational fluid dynamics literature, thc conccpt of truncation error is mcntioncd with little or no discussion about the crror magnitudes relativc to thc tcrms being approximated. For example, in the text by Andcrson, et a1.[6], the brief discussion of truncation crrors concludes with the statcment that ...Naturally, we solve only the finite difference equations and hope that thc truncation crror is small. Of the many published cornputations using tlic vorticity-stream function approach[2-5], the magnitude of errors in their computations is not always discussed. Those who do, cvaluate the accuracy of the mcthod by noting the changes in thc computcd drag cocfficicnts due to grid refinements. Such tests revcal only the crros duc to the use of inadequatcly rcfincd grids. Briley[Z] for cxamplc, reporled a 1.3% change in computed skin friction cocfficicnt when the number of grid points in the radial dircction was increased from 31 to 41. Such a modcst rcfincmcnt of grid step size docs not change significantly thc truncation crrors. Thus the truncation errors tended to cancel cach other out when the change in drag is computed, masking the true magnitude of the truncation errors. cquation. Most of the computations mentioned abovc wcrc madc at very low Reynolds numbcrs. Brileyl21 computcd flow Reynolds number based on momentum thickncss of between 15 and 30. Masliyah and EpsteinL31 investigated flows about sphcriods with Reynolds numbers up to 100. In these studies the second order finite diffcrcnce appearcd IC provide sufficient accuracy. In fact, Jcnson[4], in computing the flow over suheres at Reynolds number of 5. Finite differcnce approximations are commonly used to rcducc the two partial differential equations to algebraic relations that can be easily solved by a computer. One of thc carliest efforts to obtain a numcrical solution by finite _. _--__ dernonstratcd no improvement in Glution can bc difference approximation. Thus the mcthod using second order accurate finite differcncing appears to work wcll for low Reynolds numbers (of the ordcr of 50). * derived from lhe added complication of a fourth ordcr finite Lccturcr, Nanyang Technological Univcrsity, Singapore Professor, Dcpt. of Aeronautics & Astronautics, Univcrsity of Washington, Seattle, WA 98195 f * RescarchScientist, Advanced Propulsion, United Tcchnologics Rcscarch Ccntcr, Fast Hartford, (3 06108 The same method when applicd to high Reynolds numbcr flows gives solutions of limitcd duc LO significant Copyright