In this work we propose a variational reconstruction algorithm for enhancement and denoising of flow fields that is reminiscent of total-variation (TV) regularization used in image processing, but which also takes into account physical properties of flow such as curl and divergence. We point out the invariance properties of the scheme with respect to transformations of the coordinate system such as shifts, rotations, and changes of scale. To demonstrate the utility of the reconstruction method, we use it first to denoise a simulated phantom where the scheme is found to be superior to its quadratic (L-2) variant both in terms of SNR and in preservation of discontinuities. We then use the scheme to enhance the quality of pathline visualizations in an application to 4D (3D+ time) flow-sensitive magnetic resonance imaging of blood flow in the aorta.