An Integral Equation Method for the Flow in an Infinite Tunnel

Abstract

The steady plane irrotational incompressible parallel flow in an infinite tunnel with an obstacle (e.g. an airfoil) is studied. The boundary value problem (BVP) admits the treatment by the boundary integral equation method (BIEM) that transforms the BVP to an integral equation with boundary integral operators. Since these operators are neither compact nor of Fredholm. type, existence and uniqueness results cannot be derived from Fredholm's theory and numerical problems occur because the domain is unbounded. To overcome these difficulties the BVP is transformed to a finite ring-domain. The parallel flow in the tunnel is transformed to the flow of a source/sink system in the ring-domain. Using BIEM for the transformed problem we obtain an integral equation with a compact linear operator. A procedure for computation of the numerical solution is derived by a Nystrom-like method with simple quadrature rules. Engineers are interested in the tangential component of the flor around the obstacle. We do this job in the ring-domain and re-transform the results for the original obstacle.