Numerical assessment of stress concentration in Francis runner blade — crown/band junctions

The paper presents the results of numerical investigation for stress distribution near runner blade-crown/band junctions. Starting with the geometrical investigation of the geometries for a real turbine, different junction geometries were identified. Eight different junction geometries were numerically investigated in order to determine the stress distribution. The applied load for structural analysis was the pressure distribution, obtained from a Computational Fluid Dynamics analysis of the runner blade. The results of the stress distribution will allow to quantify the effect of different junction geometry on stress concentration and integrity of the runner.

Explicit birational geometry of 3-folds and 4-folds of general type, III

Nonsingular projective 3-folds $V$ of general type can be naturally classified into 18 families according to the pluricanonical section index${\it\delta}(V):=\text{min}\{m\mid P_{m}\geqslant 2\}$ since $1\leqslant {\it\delta}(V)\leqslant 18$ due to our previous series (I, II). Based on our further classification to 3-folds with ${\it\delta}(V)\geqslant 13$ and an intensive geometrical investigation to those with ${\it\delta}(V)\leqslant 12$, we prove that $\text{Vol}(V)\geqslant \frac{1}{1680}$ and that the pluricanonical map ${\rm\Phi}_{m}$ is birational for all $m\geqslant 61$, which greatly improves known results. An optimal birationality of ${\rm\Phi}_{m}$ for the case ${\it\delta}(V)=2$ is obtained. As an effective application, we study projective 4-folds of general type with $p_{g}\geqslant 2$ in the last section.