Fluidic oscillators (FOs) are used in a variety of applications, including process control and process intensification. Despite the simple design and operation of FOs, the fluid dynamics of FOs exhibit rich complexities. The inherently unstable flow, jet oscillations, and resulting vortices influence mixing and other transport processes. In this work, we computationally investigated the fluid dynamics of a new design of a planar FO with backflow limbs. The design comprised of two symmetric backflow limbs leading to bistable flow. The unsteady flow dynamics, internal recirculation, jet oscillations, secondary flow vortices were computationally studied over a range of inlet Reynolds numbers (2400-12,000). The nature and frequency of the jet oscillations were quantified. The computed jet oscillation frequency was compared with the experimentally measured (using imaging techniques) jet oscillation frequency. The flow model was then used to quantitatively understand mixing, heat transfer, and residence time distribution. The approach and the results presented in this work will provide a basis for designing FO's with desired flow and transport characteristics for various engineering applications.