  In this paper spectral methods are applied to investigate the hydrodynamic
instability of swirling flow with application to Francis hydraulic turbine.
Spectral methods imply representing the problem solution as truncated series of
smooth global functions. An L2 - projection and the collocation methods are
developed assessing both analytically methodology and computational techniques
using symbolic and numerical conversions. Remarks concerning the efficiency and
the accuracy of each method in this case are presented. The model of the
trailing vortex is used to validate the numerical algorithms with existing
results in the literature. All the results are compared to existing ones and
they prove to agree quite well. The advantages of using this methods in flow
control problems are pointed out.
