Relation Between Convective Instability and Global Instability on a Rotating Disk

Background: The velocity fluctuations grow dominantly by convective instability form 32 spiral vortices which are stationary with respect to the disk. However, recent researches suggest that the global instability plays a role in the boundary layer transition. Objective: The study looks into the relation between convective instability and global instability. Method: A finite difference method is used to carry out numerical simulation. The full Navier-Stokes perturbation equations and the continuity equation solved by simulation code. Results: A disturbance is continuatively introduced to excite the convectively unstable mode, which successfully generates a flow field with 32 spiral and stationary vortices. Next, a short-duration artificial disturbance with an azimuthal wavenumber of 64 is introduced at Reynolds number of 530 in order to introduce a velocity fluctuation of the traveling mode, which is globally unstable. It is shown that the source of vibration for the globally unstable mode exists between Reynolds number of 560 and 670. Finally, the global and traveling wavenumber 64 component is excited in a flow field which is dominated by the convective and stationary wavenumber 32 component. It is shown that the wavenumber 64 component grows by the global instability even when the excitation is very weak. Conclusion: The results suggest that the reason why the globally unstable mode has not been observed in experiments is because the boundary layer transition caused by the convective instability takes place before the globally unstable mode can start to grow by itself.