Several cardiovascular diseases are caused from localised abnormal blood flow such as in the case of stenosis or aneurysms. Prevailing theories propose that the development is caused by abnormal wall shear stress in focused areas. Computational fluid mechanics have arisen as a promising tool for a more precise and quantitative analysis, in particular because the anatomy is often readily available even by standard imaging techniques such as magnetic resonance and computed tomography angiography. However, computational fluid mechanics rely on accurate initial and boundary conditions, which are difficult to obtain. In this paper, we address the problem of recovering high‐resolution information from noisy and low‐resolution physical measurements of blood flow (for example, from phase‐contrast magnetic resonance imaging [PC‐MRI]) using variational data assimilation based on a transient Navier‐Stokes model. Numerical experiments are performed in both 3D (2D space and time) and 4D (3D space and time) and with pulsatile flow relevant for physiological flow in cerebral aneurysms. The results demonstrate that, with suitable regularisation, the model accurately reconstructs flow, even in the presence of significant noise.