Phase-contrast magnetic resonance imaging (MRI) provides time-resolved quantification of blood flow dynamics that can aid clinical diagnosis. Long in vivo scan times due to repeated three-dimensional (3D) volume sampling over cardiac phases and breathing cycles necessitate accelerated imaging techniques that leverage data correlations. Standard compressed sensing reconstruction methods require tuning of hyperparameters and are computationally expensive, which diminishes the potential reduction of examination times. We propose an efficient model-based deep neural reconstruction network and evaluate its performance on clinical aortic flow data. The network is shown to reconstruct undersampled 4D flow MRI data in under a minute on standard consumer hardware. Remarkably, the relatively low amounts of tunable parameters allowed the network to be trained on images from 11 reference scans while generalizing well to retrospective and prospective undersampled data for various acceleration factors and anatomies.