Phononic materials have rich frequency-dependent characteristics that can be designed into their geometry. These characteristics can open up new avenues for passively controlling flow instabilities with intrinsic frequency content. Recent numerical studies demonstrated the potential of phononic materials to control flow instabilities through the band gap resonance, which induces out-of-phase motion between the fluid and solid surface. Here, we introduce a phononic material design that exhibits tailorable band gap resonances that can passively control flow instabilities with characteristic frequency content. To this end, we first characterized the band structure of an infinitely periodic phononic material using finite element simulations. We next constructed a finite version of the phononic material with an encasing, which enables a smooth solid interface between the phononic material and the moving fluid in aerodynamic applications. Then, we performed harmonic analysis on this finite encased phononic material to numerically investigate its structural response under boundary loading exerted on the interface. The simulation results show that the proposed phononic material can achieve a wide out-of-phase frequency range above the band gap resonance, which could be used to mitigate flow instabilities. Finally, the finite phononic material with encasing was 3D printed and experimentally examined to validate the structural simulation results. These results can be used to inform future designs of phononic materials for passive flow control applications.