This study presents a new 2D boundary element method (BEM) flow simulation code. The purpose of the BEM code is to investigate fluidic control using injection and suction of the ambient fluid on foil profiles. In the BEM, a wall-normal transpiration velocity boundary condition can be imposed directly, and the simulated results show good agreement with pressure distributions with wall-normal jet control in the literature, from RANS FEM simulations — validating the BEM for high Re flows over foil profiles without flow separation, with wall-normal control. Use of the BEM allowed for a novel approach to the inverse problem; by assuming that the magnitude of the transpiration velocity remained small, the BEM system was able to be inverted, with additional required information provided by heuristics. To incorporate multiple constraints and objectives, the target pressure profiles were created by modifying the no-flux pressure profile through a series of optimisations using the SLSQP method. The BEM inversion was then able to generate transpiration velocity distributions that produced pressure profiles which deviated from the target profiles by a mean absolute error of less than 10% for the cases tested. This process constitutes a viable way to generate near-optimal solutions to the inverse problem.
Nankivell et al. (2026) studied this question.