We develop an on-the-fly reduced-order model (ROM) integrated with a flow simulation, gradually replacing a corresponding full-order model (FOM) of a physics solver. Unlike offline methods requiring a separate FOM--only simulation prior to model reduction, our approach constructs a ROM dynamically during the simulation, replacing the FOM when deemed credible. Dynamic mode decomposition (DMD) is employed for online ROM construction, with a single snapshot vector used for rank-1 updates in each iteration. Demonstrated on a flow over a cylinder with Re=100, our hybrid FOM/ROM simulation is verified in terms of the Strouhal number, resulting in a 1.6-times speedup compared to the FOM solver.