Key Symbols

Indexes and Other Signs

Abbreviations

Review of research on the application of the complex tangential lean and its optimization, as well as conducted computational research has shown that using of complex lean gives the possibility to increase aerodynamic efficiency of turbine cascades. However, as previously noted, research on optimization of complex tangential lean with preserving mass flow rate through the cascade with high precision, currently we do not have. Using developed optimization approach it is possible to preserve in optimal cascade mass flow rate at the level of the initial cascade with a high accuracy.

Complex tangential lean reduces integral losses by reducing secondary losses. It is known, that with increasing *l/b *there is a reducing in the part of the secondary losses in integral losses and, accordingly, the benefit from optimization has to diminish.

Relative height criterion was taken not *l/b, *but the cascade’s characteristic relation *a/l*, by analogy with the flows in the swivel tubes of rectangular cross-section.

Optimization problem is solved using two methods of stacking line parameterization. Research of the efficiency of the algorithm consists in attempts of optimization of turbine cascade at different *a/l =* 0,16; 0,23; 044 by changing of the blade height. It should be noted that for the blades with *a/l* ≤ 0.16 optimization, using both methods of stacking parameterization, no longer led tot he reducing losses compared to the cascade without lean.

The size of the throat varies slightly due to the changing of stagger angle of the profile, which is associated with the preserving of the mass flow rate.

Special attention was given to the FMM accuracy, since it determines the validity of the results obtained with used optimization approach. Criterion of the accuracy is deviation of the values of the target function and the constraint function, which we obtain in FMM and in checking CFD calculation.

##### 6.7.1 Optimization with Various *a/l* Using Method 1

*The results of the optimization for a/l = 0.44*

Taking into account the experience of previous studies, in Table 6.3 the ranges of parameters variation have shown. The correctness of their choice is confirmed by the fact that the optimal combination of varied parameters falls in this range already at the first step of the optimization.

Then, a plan is created in accordance with the algorithm and relevant CFD calculations are produced (Table 6.4). The objective function – integral losses ζ, restriction function – mass flow rate through the calculation channel G

A potential imperfection of the proposed optimization approach can be precision of the obtained FMM. Checking CFD calculation of optimal variant shows that the accuracy of the FMM is high because the objective function optimal values, projected by FMM, and restriction function with high enough precision coincide with their values received as a result of CFD calculation (divergence in ζ is 0.005%, and in G – 0.017%). Differences of the mass flow rate in optimal and initial variant, based on the results of the CFD calculations, is 0.014%.

Turbine cascade, obtained as the result of optimization, is shown at Fig. 6.20. The losses of the optimal cascade were reduced on 0.384% in absolute value or on 6.31% in relative value.

The distribution of the coefficients of losses and the actual outlet angles along the height of the blades for the initial and the optimal variants are represented at Fig. 6.21 and 6.22 respectively. These figures show that increasing of the losses in optimal variant took place in the central part of the cascade along with reduction of the losses at the end areas, while actual outlet angle, on contrary, decreased in the central part and at the end areas.

*The results of the optimization for a/l = 0.23*

Optimization results by method 1 at *a/l* = 0.23 were quite unexpected. Initial ranges of variation of parameters optimization were far from the final result. For this reason, the optimal variant was received only on the fifth stage of the optimization. Ranges of variation of parameters optimization of the fifth step are shown in the Table 6.5. Thus, to obtain the optimal variant in this case, 65 CFD calculations were required plus the calculation of initial variant and checking CFD calculation.

The appropriate calculation’s plan for the fifth step with the results of determining the value of the target function, and restriction function listed in the Table 6.5.

Losses deviation between optimal variants for FMM and CFD made up 0.020% and the corresponding deviation of G in relative terms made up 0.006%. The mass flow rate was preserved with an accuracy of 0.004% in relative terms.

Turbine cascade, obtained as the result of the optimization, is shown at Fig. 6.23. The losses of the optimal cascade were reduced on 0.102% in absolute value or on 1.89% in relative value.

##### 6.7.2 Optimization with Various *a/l* Using Method 2

*The results of the optimization for a/l = 0.44*

In this case were taken the ranges of parameters variation listed in the Table 6.7.

The solution was obtained in the first step of the optimization. The results of calculation using optimization method 2 have shown in the Table 6.8.

In this case, losses deviation between optimal variants for FMM and CFD made up 0.006%, and the corresponding deviation of G in relative terms made up 0.017%. The mass flow rate was preserved with an accuracy of 0.017% in relative terms.

Turbine cascade, obtained as the result of the optimization, is shown at Fig. 6.26. The losses of the optimal cascade were reduced on 0.266% in absolute value or on 4.20% in relative value.

Comparing increasing of aerodynamic efficiency using methods 1 and 2, we can conclude that for this turbine cascade with *a/l* = 0.44 method 1 is more preferred because its use has significantly reduced integral losses.

The distribution of the coefficients of losses and the actual outlet angles along the height of the blades for the initial and the optimal variants are represented at Fig. 6.27 and 6.28 respectively. These figures show that increasing of the losses in optimal variant took place in the central part of the cascade along with reduction of the losses at the end areas, while actual outlet angle, on contrary, decreased in the central part and at the end areas.

*The results of the optimization for a/l = 0.23*

Ranges of variation of parameters optimization are shown in the Table 6.5. In this case, the optimal variant was obtained at the first step of optimization.

The plan of calculations with the results of determining the value of the objective function, and the restriction function listed in the Table 6.10.

In this case the losses deviation between optimal variants for FMM and CFD made up 0.032% and the corresponding deviation of G in relative terms made up 0.008%. The mass flow rate was preserved with an accuracy of 0.009% in relative terms.

Turbine cascade, received as the result of the optimization, is shown at Fig. 6.29. Losses in the new cascade made up 5.314% that on 0.145% in absolute value or on 2.69% in relative value is smaller than in the initial variant.

Application of the optimal complex lean by the method 2 when *a/l* = 0.23 resulted in a reduction of losses in end areas, but in the core of the flow losses remained almost unchanged. On contrary, actual outlet angle decreased in the central part of the cascade and increased in the end areas, the same as in the previous cases.

Thus, the optimization method 2 gave the possibility to reduce integral losses greater than method 1.

##### 6.7.3 Reasons of Increasing the Efficiency of Optimized Cascades

Consider the flow in the initial cascade for *a/l*=0.23. Comparing the field of the total pressure at the value of *ī* = 0.96 and *ī* = 0.5 of the initial blade (Fig. 6.32), it can be seen that the thickness of the boundary layer in the similar areas much less at the height *ī* = 0.96 of the blade than in the core flow.

At the same time, the boundary layer thickness on its low-pressure side plays an essential role in the losses formation on the blade. This agrees well with the corresponding graphs of the losses coefficient distribution in height of the blade (see. Fig. 6.30).

Explanation of the reducing of boundary layer thickness in indicated areas at the periphery and the root can be given based on the analysis of the flow on the suction side of the blade (Fig. 6.33). On Fig. 6.33 line S marked the line of detachment of the channel vortex.

It is the line of distinction of the two main border flows on the suction side of the blade:

- flow along the blade from the leading edge to the trailing edge;
- cross-over flow, that comes from the ends of the channel and penetrates, due to its inertia, on the blade (this flow is a part of the peripheral and root channel vortices) (Fig. 6.34).

Thus, on the suction side of the blades, there are areas with variously formed boundary layers. This explains the different thickness of the boundary layer in the appropriate places on the suction side of the blade.

As a result, the important conclusion can be formulated: the channel vortex leads to the formation on the suction side of the blade up to the line S of the boundary layer thickness less than the thickness of the boundary layer of the main flow around the blades. Thus the secondary flows have not only negative, but a positive effect, and properly distribution the stream structure can be used for creating optimal forms of turbine blades.

From the last considerations, it is possible to prove the reduction mechanism of aerodynamic losses for turbine cascade by applying optimal complex lean: the complex lean leads to the movement of the line S in the direction of the flow core and therefore to increasing the areas with smaller losses.

Displacement of the line S, when using complex tangential lean, takes place because of pressure gradient appearance on the suction side of the blade, which occurs when complex tangential lean is used. The result of this aerodynamic effect is certain offset of the saddle point from the pressure side of the blade to the suction side, therefore, to more earlier descent of the channel vortex from the end of the blade to the blade itself (Fig. 6.35). Descended on the blade, vortex, in turn, under the influence of pressure gradient slightly shifted on the blade toward the core of the flow, that leads to an increase of the part of the flow on the suction side of the blade with a smaller boundary layer than in the core of the flow.

As a result the proposed optimization algorithm made possible to find the optimum position of the line channel vortex detachment on the suction side of the blade by complex the lean optimization.