In my talk, I will present ingredients for high-performance implementations of solvers in fluid dynamics and coupled problems, with an emphasis on the application in biomechanical flows. Time integration is done by projection methods based on the BDF family in an implicit/explicit fashion. My focus will be on spatial discretizations based on high-order finite element methods, specifically, the discontinuous Galerkin as well as H(div)-conforming Raviart--Thomas families. These methods have beneficial properties for transport-dominated scenarios at higher Reynolds numbers, because they provide controlled dissipation via upwinding and can enforce the divergence-free condition of incompressible flows. However, the background of these methods is formally in the high-accuracy regime with advantages primarily in asymptotic convergence rates, which has contributed to the opinion that they could be inappropriate for practical applications especially in biomechanics, because the geometries to be represented are complicated, requiring efficient high-order meshing, and because uncertainties in simulation parameters make it impossible to exploit the full accuracy. I will show that matrix-free evaluation of discrete operators can render those advanced numerical schemes attractive nonetheless, with a similar or higher throughput as classical low-order methods for the same number of degrees of freedom. Matrix-free solver design, where finite element integrals are evaluated on the fly to form the matrix-vector product in iterative solvers, necessiate new paradigms in PDE software because the implementation of finite element components must be integrated into the linear solvers, rather than outsourcing all linear algebra to external packages. I will present the design in the generic library deal.II, based on research in the EuroHPC Centre of Excellence dealii-X. Our aim is to separate the optimized backend solver tuning for GPUs and massively parallel architectures from the application codes, which only need to provide an implementation of the operation in quadrature points of cell and face integrals. As a result, the loop over the mesh, the MPI ghost communication and parallelization strategies as well as technologies to employ accelerators can be handled by the library. I will present results in several flow scenarios, including the application for fluid-structure interaction problems with partitioned schemes and for unfitted finite element computations.