The inverse problem for incompressible internal flows

Abstract

A duct design algorithm is presented by which duct wall shapes are produced directly from prescribed wall friction velocity distributions. The method is based on matched axisymmetric core flow and integral boundary layer solutions in which the flow is assumed to be in viscid in the core, turbulent in the boundary layer and incompressible throughout. An inverse boundary layer analysis is based on Coles profiles coupled with momentum and entrainment equations. Wall friction velocity is used as the major boundary condition which leads to the calculation of the required core velocity at the boundary layer edge. This velocity distribution is simultaneously employed as the major boundary condition in an inverse core flow calculation which is based on an inverted form of the Stokes-Beltrami equation. The displacement surface method is used to match boundary layer and core solutions which together lead to a predicted duct shape. Numerical results are presented, some of which include separated boundary layer flows. An assessment of the influence of wall friction velocity distribution on duct shape and wall pressure field is included.