A robust numerical method for flow through a pipe driven by an oscillating pressure gradient

Abstract

SUMMARYThe 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 . 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 and require a bounded time, independent of the value of , 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.

Effect of an oscillating time-dependent pressure gradient on Dean flow: transient solution

“…A fully developed flow due to an oscillatory pressure gradient with timedependent curvature in a tube was reported by Waters and Pedley [13]. In recent past, Ansari et al [14] solved numerically Navier-Stokes equations responsible for the motion of a viscous and incompressible fluid through a pipe driven by an oscillatory pressure gradient.…”