Computational and Theoretical Analyses of the Precessing Vortex Rope in a Simplified Draft Tube of a Scaled Model of a Francis Turbine

Results on flows in a draft tube of a constant-head, constant-specific speed, model Francis turbine are presented based on computational fluid dynamics (CFD) simulations and theoretical analysis. A three-dimensional, unsteady, Navier-Stokes solver with the detached-eddy simulation (DES) model and the realizable k-? (RKE) model is used to analyze the vortex rope formed at different discharge coefficients. The dominant amplitude of the pressure fluctuations at a fixed point in the draft tube increases by 13 times, and the length of the rope increases by 3.4 times when the operating point of the turbine shifts from a discharge coefficient of 0.37 to 0.34. A perturbation analysis based on a steady, axisymmetric, inviscid, incompressible model for the mean flow is performed to obtain a Sturm-Liouville (SL) system, the solutions of which are oscillatory if the discharge coefficient is greater than 0.3635, and nonoscillatory otherwise.