Black hole systems (X- ray binaries) exhibit a range of disk behaviors, including state transitions and variability, radio jets, and quasi-periodic oscillations (QPOs). A striking timing feature is the occurrence of QPOs over a wide frequency range. This thesis presents a numerical study of transonic accretion flows to understand the physical mechanisms responsible for these time-dependent observational behavior of black hole systems. 

The evolution of viscous transonic accretion around a non-rotating as well as rotating black hole is examined using a total variation diminishing (TVD) numerical scheme that accurately models angular momentum transport due to viscosity. The thermodynamic properties of the flow are described by a relativistic variable adiabatic index equation of state. When the viscosity parameter exceeds a critical value, the initially steady shock becomes unstable and begins to oscillate. Estimated radiative output indicates that it also exhibits periodic variability associated with these oscillations. For a $10\,M_{\odot}$ black hole, the corresponding power density spectra show a dominant peak. The results show that the frequency range of quasi-periodic oscillations strongly depends on the spin of the black hole, with rapidly rotating systems producing a broader range of variability. Axisymmetric hydrodynamic simulations are carried out using a finite-volume code based on the HLL scheme, incorporating both viscous effects and radiative cooling. The interplay between viscous heating, radiative cooling, and gravitational attraction determines the position and stability of the shock. Variations in angular momentum transport across different layers generate turbulence eddies that enhance the oscillatory behavior and outflow dynamics. Finally, this thesis provides a detailed understanding of the dynamics of the viscous transonic accretion disks around black holes to explain important timing properties, such as quasi-periodic oscillations (QPOs), as well as spectral characteristics, including the correlation between QPO frequency and photon index.