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A robust numerical method for flow through a pipe driven by an oscillating pressure gradient
Author(s) -
Ansari Ali R.,
Miller John J. H.,
Shishkin Grigori I.
Publication year - 2006
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.1290
Subject(s) - mathematics , discretization , mathematical analysis , numerical analysis , reynolds number , pressure gradient , boundary value problem , boundary layer , turbulence , mechanics , physics
Abstract The problem of periodic flow of an incompressible fluid through a pipe, which is driven by an oscillating pressure gradient (e.g. a reciprocating piston), is investigated in the case of a large Reynolds number. This process is described by a singularly perturbed parabolic equation with a periodic right‐hand side, where the singular perturbation parameter is the viscosity ν. The periodic solution of this problem is a solution of the Navier–Stokes equations with cylindrical symmetry. We are interested in constructing a parameter‐robust numerical method for this problem, i.e. a numerical method generating numerical approximations that converge uniformly with respect to the parameter ν and require a bounded time, independent of the value of ν, for their computation. Our method comprises a standard monotone discretization of the problem on non‐standard piecewise uniform meshes condensing in a neighbourhood of the boundary layer. The transition point between segments of the mesh with different step sizes is chosen in accordance with the behaviour of the analytic solution in the boundary layer region. In this paper we construct the numerical method and discuss the results of extensive numerical experiments, which show experimentally that the method is parameter‐robust. Copyright © 2006 John Wiley & Sons, Ltd.