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A new finite‐volume approach with adaptive upwind convection
Author(s) -
Kinney Robert B.,
Mahdi Hashim S.
Publication year - 1988
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1620260608
Subject(s) - finite volume method , mathematics , convection–diffusion equation , péclet number , weighting , convection , upwind scheme , mechanics , mathematical analysis , physics , discretization , acoustics
A new finite‐volume approach is developed and applied to the two‐dimensional continuity and convective‐diffusive energy equations. The variation of the field variables is approximated by bi‐quadratic interpolation formulae over the space occupied by the finite volume and the region surrounding it. These are used in the integral conservation laws for energy and mass. The convective transport is modelled using a new upstream‐weighting approach which uses volume averages for the energy transported across the boundaries of the finite volume. The weighting is dependent on the skewness of the velocity field to the surfaces of the finite volume as well as its strength. It is adaptive to local flow conditions. Two test cases are treated which have exact solutions. The first is not new and involves a rotating shaft. The errors are less than 0.06 per cent for this case. The second case is new and involves convection past a source and sink. In contrast to the first case, the global Peclet number is a strong parameter, and cell Peclet numbers ( Pe h ) range from 0 to 20. The maximum error is 2.3 per cent for Pe h = 4, and there is no evidence of numerical diffusion for even the largest value of Pe h . For both test cases, the maximum error occurs at moderate values of Pe h and diminishes at the extreme low and high values.