One key factor for improving the accuracy and efficiency of numerical simulations is the use of time error indicators. A posterior time error indicators specifically evaluate errors arising from time discretization. A time integration scheme capable of operating variable time step sizes is required to utilize such indicators effectively. This work focuses on Backward Difference Formulas (BDF) due to their inherent stability originating from their implicit nature. Nevertheless, the proposed time error indicator is versatile and can be extended to more complex time integration schemes. This study integrates a novel time error indicator into a stabilized finite element flow solver within the Variational Multi-Scale (VMS) framework [1]. The approach explicitly quantifies errors from time discretization in monolithic algorithms and introduces an additional term to account for segregation errors in fractional step methods. Since spatial and temporal errors are interdependent, developing a time-adaptive algorithm necessitates an understanding of spatial errors as well. To address this, the Zienkiewicz-Zhu [2,3] spatial error indicator is employed for the velocity field. The proposed time-adaptive algorithm incorporates a user-defined factor that establishes the ratio between spatial and temporal error indicators. This factor can either be fixed a priori or dynamically adjusted during the simulation based on parameters such as the Courant–Friedrichs–Lewy (CFL) number, nonlinearities, and solver residual norms. Numerical experiments conducted on benchmark flow problems, including incompressible flow over a cylinder, isentropic compressible flow cases including the Aeolian tones of a low Mach viscous flow [4], and aerodynamic sound generation by flow past a NACA 0012 airfoil at 5° angle of attack [5], demonstrate the effectiveness of the proposed adaptive time-stepping strategy. This approach enhances accuracy and efficiency in the fractional step method by balancing spatial and temporal errors, improving solution fidelity and resource utilization for complex flow problems.