Vortex breakdown in the small Mach number regimes

The effect of compressibility for flow inside the cylinder with the top rotating lid (Vogel-Escudier flow) is examined. Three-dimensional Navier-Stokes equations for compressible flow in Cartesian coordinates are used to simulate the flow using open-source OpenFOAM software. The Mach number (Ma) of the flow is varied from 0.1 to 0.3, and the Reynolds number ( Re ) is varied from 1000 to 5000 for a fixed aspect ratio (Gamma = 2.5) of the cylinder. The flow is found to have a transition from a steady axisymmetric state to a non-axisymmetric state exhibiting multiple azimuthal waves as the Mach number and the Reynolds number are varied. The flow field changes significantly with an increase in Ma for unsteady flow at higher Re. An increase in Ma increases the side wall azimuthal instability, as found in the perturbation contour plots and time series analysis. Further, we reconstruct phase portraits to show the dynamics of the flow finally becoming chaotic as the Reynolds number is increased to 5000. Finally, we support the argument with Lyapunov exponents for the higher Re samples. The Lyapunov Exponent is found to increase with Ma.