The method of fundamental solutions for the Oseen steady‐state viscous flow past obstacles of known or unknown shapes

Abstract

In this paper, the steady-state Oseen viscous flow equations past a known or unknown obstacle are solved numerically using the method of fundamental solutions (MFS), which is free of meshes, singularities, and numerical integrations. The direct problem is linear and well-posed, whereas the inverse problem is nonlinear and ill-posed. For the direct problem, the MFS computations of the fluid flow characteristics (velocity, pressure, drag, and lift coefficients) are in very good agreement with the previously published results obtained using other methods for the Oseen flow past circular and elliptic cylinders, as well as past two circular cylinders. In the inverse obstacle problem the boundary data and the internal measurement of the fluid velocity are minimized using the MATLAB © optimization toolbox lsqnonlin routine. Regularization was found necessary in the case the measured data are contaminated with noise. Numerical results show accurate and stable reconstructions of various star-shaped obstacles of circular, bean, or peanut cross-section.

Identification of obstacles immersed in a stationary Oseen fluid via boundary measurements

“…For simplicity, we only consider the two-dimensional case. The MFS for the exterior stationary Oseen fluid flow past an arbitrary obstacle has been recently introduced in [12]. For our interior problem in the annular domain Ω\D, it approximates the fluid velocity u = (u 1 , u 2 ) and pressure p as…”

“…Example 3: Bean-shaped obstacle. We attempt to reconstruct a more irregular shape than the previous example, given by a bean-shaped obstacle D with the parametrisation [3,12,14] r(ϑ) = 1 + 0.9 cos(ϑ) + 0.1 sin(ϑ) 1 + 0.75 cos(ϑ) , ϑ ∈ [0, 2π).…”

“…However, the case of Oseen low Reynolds number flow has been somehow overlooked and it is the purpose of this paper to consider it in some detail. In a recent paper [12] we have developed the method of fundamental solutions (MFS) [13] with Tikhonov's regularization for solving both direct and inverse problems for the Oseen steady-state fluid flow past arbitrary obstacles of known or unknown shapes. In that exterior inverse problem, the extra data needed to replace the obstacle's shape missing information was taken as the internal fluid velocity on a curve surrounding the obstacle at one [12] or multiple incoming flow directions [14].…”

“…In a recent paper [12] we have developed the method of fundamental solutions (MFS) [13] with Tikhonov's regularization for solving both direct and inverse problems for the Oseen steady-state fluid flow past arbitrary obstacles of known or unknown shapes. In that exterior inverse problem, the extra data needed to replace the obstacle's shape missing information was taken as the internal fluid velocity on a curve surrounding the obstacle at one [12] or multiple incoming flow directions [14]. In contrast to the previous exterior problem formulated in an unbounded domain, the present paper deals with the interior obstacle situation in a bounded domain in which the Oseen equations hold in the annular domain formed within the region between the unknown obstacle, on which the no-slip fluid velocity condition holds, and an exterior known boundary on which both the fluid velocity and traction are prescribed.…”

The method of fundamental solutions for Brinkman flows. Part I. Exterior domains

“…In terms of the perturbations u and p, the inverse problem requires solving for the unknown triplet (u, p, Ω) satisfying (1.7)-(1.11) and (4.2). This problem and its MFS combined nonlinear minimization is similar to that previously treated in [7] for Oseen flow. We assume that Ω is starshaped with respect to the origin, parametrised by the radial polar coordinate r(ϑ) ∈ (0, r max ] for ϑ ∈ [0, 2π), where r max > 0 is an a priori known upper bound of the size of the obstacle, namely,…”