The theory of Compressive Sensing (CS) enables the efficient acquisition of signals which are sparse or compressible in an appropriate domain. In the sub-field of CS known as model-based CS, prior knowledge of the signal sparsity profile is used to improve compression and sparse signal recovery rates. In this paper, we show that by exploiting the periodic support of Periodic Clustered Sparse (PCS) signals, model-based CS improves upon classical CS. We quantify this improvement in terms of simulations performed with a proposed greedy algorithm for PCS signal recovery and provide sampling bounds for the recovery of PCS signals from compressive measurements.