This mini-symposium aims to bring together new trends in the development of innovative numerical methods (finite elements, fictitious domain methods, moving interfaces, multi-scale methods, inverse problems, domain decomposition and HPC, etc.) for fluid flow simulation. Machine learning based techniques are also welcome. The focus of this mini-symposium also includes the challenges and contributions to the mathematical analysis and computational implementation of the methods, as well as their applications in the computational simulation of realistic fluid problems.