Gradient-index phononic crystals (GRIN-PC), characterized by layers with spatially changing refractive indices, have recently been investigated as part of the effort to realize flat lenses in acoustic and elastic regimes. Such gradient-index lens must be inversely designed from the corresponding refractive indices in order to manipulate the target wave. Unfortunately, estimating the index of this type of lens is not straightforward and requires substantial iterative computation in general, which greatly limits the applicability of GRIN-PC to flat lenses. In this work, we propose a novel design of a GRIN-PC in which neighboring layers are separated by partitions, thus preventing waves in each layer from interacting with other layers. This partitioned GRIN-PC design enables us readily to control the phase gradient accurately at the lens' end, resulting in direct calculation of indices for target wave manipulation. A detailed methodology for partitioned GRIN-PC based collimator and Bessel-beam generator is proposed and experimentally validated to confirm the versatile use of our design in wave engineering applications.