This thesis is on weakly-imposed Dirichlet boundary conditions in incompressible and compressible flow. Our target is internal flows. The weakly-imposed conditions are compared with strongly-imposed conditions for mean flow solutions from the incompressible-flow space–time variational multiscale (ST-VMS) method. The numerical tests include ST finite element analysis with linear basis functions in space and time, and ST Isogeometiric Analysis (IGA) with NURBS basis functions in space and linear basis functions in time. The thesis contains three test computations: 1) Incompressible-flow 2D ST finite element computation of laminar flow between parallel plates. 2) Incompressible-flow 3D ST finite element computation of laminar flow in a pipe. 3) Incompressible-flow 3D ST-IGA computation of high-Reynolds number turbulent flow in a pipe. The NURBS mesh generation strategy for the compressible flow in a pipe is also included in the thesis. The weakly-imposed Dirichlet boundary conditions are tested at different spatial and temporal resolutions and penalty parameter values.