In core-annular flow of two different fluids, for a set of suitable flow conditions, various shapes of saturated waves such as bamboo, snake and corkscrew waves are observed. Some of the dominant parameters such as thickness ratio of the fluid, Reynolds number, viscosity ratio, density ratio, interfacial surface tension, and the direction of gravitational forces determine the final shape of the saturated wave and their ultimate stability in a nonlinear regime. When the flow rate ratio is high, it is sometimes difficult to determine the differences between the final shape of the waves for up-flow and down-flow. For some combinations of thickness ratio, viscosity ratio, density ratio, Reynolds number and surface tension, waves tend to break and bubbles start to form. Interfacial surface tensions between these two fluids play a very important role in stabilizing the waves from breaking. In this study, new sets of waves were discovered for core-annular flow, which modulate at certain flow parameter ranges. The critical parameter ranges are identified where the waves shift from saturated bamboo waves and bifurcate into modulated bamboo waves. A thorough analysis is performed for the first time to depict the windows of these critical parameters at which this transition takes place. A bifurcation diagram is constructed to capture the regime. A detailed wave shape analysis is performed to characterize these wave shapes and their periods of oscillation. Due to challenges associated with large computational domain and enormous computational power requires to resolve the interfacial instability, a three-dimensional true non-axisymmetric model was never studied before. For the first time, effort is being undertaken to construct a viable 3-D Core-annular flow. A general purpose computational fluid dynamics package ANSYS Fluent is used for this analysis. Three dimensional models for both up-flow and down-flow were constructed and a novel explanation is presented to distinguish between the Bamboo waves, Cork-Screw waves, and Snake waves. The sensitivity of down-flow on initial conditions was also verified with 3-D models on some parameter space from selected publication.