#### 6.1 Problem Statement

The rapid development of computational aerodynamics methods not only puts on the agenda introduction of the spatial calculations into the turbines design practice, but also raises the need to develop the blades shape and other turbine flow path elements optimization methods taking into account 3D flow [24].

Formulations of the blades spatial optimization problems, which essentially cannot be solved by using one-dimensional and two-dimensional models, for minimization of the secondary flows loses, arising at the tip and the hub of the blades, are of the greatest interest [35].

Analyzing the results of the research, three main reasons for formation of the secondary flows in the turbine cascades could be singled out:

1. Turning of the flow. In channels with flow turning (including the turbine cascades) the transverse gradient of pressure arises, under influence of which whirlwind is forming at the ends of the channel.
2. Interaction of the boundary layer, accumulated on the end wall in front of the blade with the leading edge of the blade. For this reason, a horseshoe-shaped whirlwind is formed, which is then divided into two parts on both sides of the blade.
3. Vortex wedge. In almost every corner areas, which are generated between vortical structures and walls of turbine channel cascade, the forming or dissipating of corner vortexes may take place. Some from them are there constantly, some are dissipating depending rom the flow parameters and the type of the wedge.

To the most important causes inducing the secondary losses, relate the following factors:

1. Natural accumulation of the boundary layer on end wall at the entrance till the lines of detachment;
2. Braked the detachable bubble in the field of entrance edge between two lines of the detachment;
3. Increasing the new boundary layer after the line of detachment;
4. Losses in the corners between the high and low pressure sides and end wall (the most significant losses are situated in the corner between the low pressure side and end wall);
5. Influence of tangential stresses along the 3D lines of detachment;
6. Loss caused by tangential stresses between the channel vortex and the low pressure side of the blade, as well as the process of mixing of the crosscut flow with the flow in the channel along with a three-dimensional line of detachment;
7. Dissipation of all vortexes and complete mixing of heterogeneous flow field at the cascade exit.

Such a complex character of the influence of secondary flows on cascade losses requires investigation of means of their minimization through appropriate selection of the blades shape in the end zones. Management of the end phenomena may be implemented by changing the shape of the profile along the height of the blades, using the complex tangential lean, profiling of the form of the flow path, utilization of the additional aerodynamic elements in channels between the blades.

The most evident way to control the flow nearby the ends of the blades is a lean. The simple problem statement of lean optimization involves parameterization of the blades shape by means of deformation of the stacking line in accordance with the chosen law. Selection of the deformation parameters, using one of the methods of direct search, leads to the definition of a profile shape that ensures minimum integral losses in the cascade. Despite its apparent simplicity, this approach requires overcoming several problems associated with the efficiency of the solving of the putted task, in particular: rapid and reliable ways of building a parameterized blades and corresponding with them calculation grids in the blade passages; development of the mechanisms of data exchange with CFD-solvers; elaboration of recommendations for solver settings, providing sufficient accuracy and speed of calculations; the choice of optimization method, usable for solving the problems with difficult computable objective functions and with various kinds of restrictions.