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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 resolution and computed tomography angiography. However, computational fluid mechanics rely on accurate boundary conditions which is difficult to obtain. In this paper we address the problem of recovering high resolution information from noisy, low-resolution measurements of blood flow using variational data assimilation (also known as 4DVar). We show that accurate flow reconstruction is obtained with proper regularisation even in the presence of significant noise and for a range of regularisation parameters spanning orders of magnitude. Numerical experiments are performed in both 2D and 3D and with pulsatile flow relevant for physiological flow in cerebral aneurysms.