Modular algorithms for the numerical solution of the flow rate boundary value problem in fluid-dynamics

In many engineering fluid dynamics problems, the prescription of defective boundary conditions, such as the flow rate, on artificial boundaries can be source of numerical inaccuracies. At the practical level, in the engineering literature this problem has been solved by choosing a velocity profile fitting the given defective datum (practical approach). After a brief description of this method and of its limitations, we review two strategies based on the introduction of Lagrange multipliers. In particular we focus on the derivation and the analysis of the continuous formulations and on the algebraic properties of the related linear systems. We describe two algorithms for their solution which are modular, that is could be solved by using pre-existing fluid solvers. Finally, we present some realistic numerical results showing the effectiveness of the strategies considered in comparison with the practical approach.