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A new approach to optimize the geometry of a turbine airfoil by simultaneously designing multiple 2D sections of the airfoil is presented in this paper. The complexity of 3D geometry modeling is circumvented by generating multiple 2D airfoil sections and constraining their geometry in the radial direction using first- and second-order polynomials that ensure smoothness in the radial direction. The flow fields of candidate geometries obtained during optimization are evaluated using a quasi-3D, inviscid, CFD analysis code. An inviscid flow solver is used to reduce the execution time of the analysis. Multiple evaluation criteria based on the Mach number profile obtained from the analysis of each airfoil cross-section are used for computing a quality metric. A key contribution of the paper is the development of metrics that emulate the perception of the human designer in visually evaluating the Mach Number distribution. A mathematical representation of the evaluation criteria coupled with a parametric geometry generator enables the use of formal optimization techniques in the design. The proposed approach is implemented in the optimal design of a low-pressure turbine nozzle.

Turbine airfoil design has long been a domain of expert designers who use their knowledge and experience along with analysis codes to make design decisions. To increase productivity and performance companies are relying heavily on the vastly improved abilities of computers to use optimization and machine learning approaches. In the past, when the analysis codes took considerable time to execute on slow machines, designers used their domain knowledge and experience to effectively prune the search space and design comparable turbine airfoils efficiently. As the analysis code execution has become faster, the primary bottleneck in the design process has shifted from analyses to manual tasks of designers. Human designers are much better in forming a visual perspective and in decision-making abilities than computers. On the other hand, computers are much quicker in running codes, parsing results and analyzing numerical data. Hence, by capturing the decision-making ability of the designers and coupling them with the computation speed of computers, very efficient design systems can be created. This paper is focused on automation of a part of the process for aerodynamic design of turbines by automating the manual process and using optimization techniques to drive it.

The turbine aerodynamic design is a three-step process that is, pitchline analysis, throughflow analysis, and blade-to-blade analysis (see Figure

The turbine design process.

The objective of airfoil design is to define the airfoil shape so as to ensure structural integrity and minimize losses. The primary sources of losses in an airfoil are profile loss, shock loss, secondary flow loss, tip clearance loss, and end-wall loss. Profile loss is associated with boundary layer growth over the blade profile causing viscous and turbulent dissipation. This also includes loss due to boundary layer separation because of conditions such as extreme angles of incidence and high inlet Mach number. Shock losses arise due to viscous dissipation within the shock wave which results in increase in static pressure and subsequent thickening of the boundary layer, which may lead to flow separation downstream of the shock. End-wall loss is associated with boundary layer growth on the inner and outer walls on the annulus. Secondary flow losses arise from flows, which are present when a wall boundary layer is turned through an angle by an adjacent curved surface. Tip clearance loss is caused by leakage flows in the tip clearance region of the rotor blade, where the leaked flow fails to contribute to the work output and also interacts with the end-wall boundary layer [

To compute all these losses a 3D viscous analysis is required; however, due to the computational load of such a code, a quasi-3D analysis code is often used in the design process. Thus the impact of the blade geometry on 3D losses cannot be determined and only 2D losses can be minimized, that is, profile and shock losses. A viscous quasi-3D analysis though less computationally intense is still too expensive for use in design optimization, and an inviscid quasi-3D code is used instead. Consequently, viscous losses are not computed from the analysis code and airfoil performance is gauged by the characteristics of the Mach number distribution on the blade surface.

Several basic methods for airfoil design have been suggested in the literature, that is, inverse, direct, and adjoint. In inverse methods [

In direct design methods, the airfoil design problem is posed as a constrained optimization problem where parametric geometry represents the design variables, metrics based on the Mach Number distribution comprise the objective function, and mechanical and structural requirements are imposed as constraints. Airfoil design is optimized by incrementally changing the airfoil shape and evaluating the design at each step [

Jameson [

The most practical formulation for low-speed turbine airfoil designs still remains the direct optimization formulation based on 2D inviscid blade-to-blade solvers. This work automates the direct design process as described in the next section.

Airfoil design typically starts with an initial design that is iteratively changed until a satisfactory geometry is obtained. There are four distinct steps in the process:

The parametric representations of the airfoils used in this work are based on the standard design tools and practices used at General Electric. There are separate models for the high-pressure and low-pressure turbine blades. The high-pressure turbine blades are subject to very high temperatures and need to be cooled. As a result, these airfoils are made thick to accommodate cooling passages inside the blades. For such thick airfoils, suction and pressure surfaces need to be manipulated independently of each other. So the airfoil is represented as a combination of two separate curves, one for the pressure side and the other for the suction side (see Figure

Parametric representation of a high-pressure airfoil section.

Parametric representation of a low-pressure airfoil section.

stagger: angle of line joining leading & trailing edge of the airfoil to axial,

tmaxx: maximum thickness of the airfoil,

c1: point of maximum thickness,

c2: trailing edge included angle,

c3: curvature of meanline,

ratu: curvature near leading edge on upper surface,

ratl: curvature near leading edge on lower surface,

pcttle: incidence angle ,

ti: leading edge bluntness,

e: ellipse ratio of the approximate ellipse fitted in the nose.

A computational fluid dynamics code Ises [

A sample Mach number distribution.

Designers use multiple criteria to evaluate a design and use their judgment in determining the relative weight of the different criteria. Pattern recognition [

Diffusion is defined as the deceleration of the flow along the blade surface. It is measured as the cumulative aggregate of all flow diffusions at each point along the airfoil surface. As the flow diffuses, the boundary layer thickens, and the momentum loss in the boundary layer increases. In this case, the increased drag causes a significant loss of momentum; flow separation may result, causing much larger losses. Thus, the objective of the design is to minimize the diffusion effect. Since the impact of diffusion on the pressure and suction sides is different, separate terms are defined for the suction and pressure sides. A mathematical definition of diffusion follows:

Deviation is a measure of uniformity of flow acceleration along the blade surface. It is defined as the root mean square error between the Mach Number distribution on the airfoil surface and a polynomial fit of the data. Even if the flow is constantly accelerating, the rate of acceleration can have an impact on the thickening of the boundary layer causing a larger momentum loss in the boundary layer. The objective is to obtain as uniform acceleration as possible. Since the effect of nonuniform acceleration can be different on the pressure and suction surfaces, separate deviation terms are defined for the two surfaces. Figure

Hypothetical Mach number distribution and a first-order fit of the data.

Incidence deviation is defined as the root mean square error between the Mach Number distribution at the nose and a polynomial fit of the data. It is a measure of the distortion of the Mach Number at the leading edge caused by a poor incidence angle of the blade. Mach Number gradients are very high at the leading edge of the airfoil resulting in extreme sensitivity to the leading edge shape necessitating a separate metric. Parameters defining the nose geometry primarily drive the incidence deviation metric during optimization. Figure

Mach number distribution at the leading edge of a blade and a 2nd-order fit of the data.

where

Leading edge crossover is a measure of the adverse pressure gradient at the leading edge of the airfoil. Due to the boundary conditions and the shape of the airfoil, the suction side pressure may become higher than that on the pressure side close to the leading edge. This results in an increase in the loading on the rest of the airfoil, an increased risk of flow separation, and additional mechanical stresses in the airfoil. The leading edge crossover metric is computed by finding the points of intersection of the pressure and suction side Mach number distributions of the airfoil and computing the areas of the regions where the pressure side has a lower pressure than the suction side. Figure

Mach number distribution showing the intersection of the suction and pressure side Mach number distributions at the leading edge.

A weighted sum of the above metrics is used to define the objective function. In addition to the above metrics, diffusion on the unguided portion of the suction side is controlled by imposing constraints that are described in the next section.

Constraints are imposed on the airfoil geometry to ensure that the airfoil is manufacturable and structurally feasible as well as for ensuring high aerodynamic efficiency. The structural and manufacturing constraints are based on the airfoil geometry and the aerodynamic constraints are derived from the Mach Number distribution on the airfoil. There aerodynamic constraints are defined, that is, peak-exit-ratio, peak-location, and inlet-valley-ratio. These are listed below and can be interpreted from the Mach Number distribution shown in Figure

Peak-exit-ratio is defined as the ratio of the peak Mach number on the suction surface to the Mach number at the trailing edge of the blade. This is a measure of flow acceleration on the unguided portion of the airfoil (between the throat and the trailing edge). A very high turning on the unguided portion of the airfoil can lead to separation of flow or the formation of a shock on the trailing edge. By putting a constraint on the maximum peak-exit-ratio, chances of separation are minimized.

Peak-location is the normalized location of the peak Mach number on the suction side. It is desirable to have an increasing Mach number as far along on the suction side as possible to prevent a thickening of the boundary layer. Imposing a constraint which allows the peak to occur after 65% of the blade width guards against upstream diffusion and helps in achieving a smooth accelerating Mach number on the suction side.

Inlet-valley-ratio is the ratio of the Mach number at the inlet of the airfoil on the pressure side, to the minimum Mach number on the pressure side. This constraint controls the diffusion near the inlet on the pressure side and restricts the thickening of the boundary layer, reducing chances of flow separation. Sample constraints are described below

Constraints are also imposed on

To ensure mechanical and structural feasibility, constraints are imposed on the blade geometry. The primary geometry parameters are cross section area, maximum thickness of airfoil, wedge angle, and nose radius. In cooled airfoils, the constraints on the geometry stem from the necessity to construct cooling channels in the airfoil; these constraints are dictated by manufacturing requirements. In low-pressure airfoils, these constraints are primarily driven by stresses and manufacturing limitations. Most of these constraints have soft limits on them; that is, it is best to have the responses within a given range, beyond a threshold of the range a penalty that increases nonlinearly with increased violation of the constraint is added to the objective function. The geometry-based constraints used in the problem formulation are listed below:

−

−

The design for 3D airfoils was done on multiple 2D planes, which were stacked together and the full 3D geometry was obtained by interpolating between the sections. In order to maintain a smooth geometry across multiple sections, geometry continuity must be maintained from section to section. During the manual design process, designers change multiple sections simultaneously and mentally keep track of the relative shapes of the sections while changing them. Ensuring smoothness is often an iterative process; once the individual airfoil sections are optimized, the sections are assembled into a stack with polynomial fits across the sections. The sections are then tweaked to obtain radial smoothness. The manual design process often alternates between smoothing the airfoil stack and optimizing the airfoil sections. In an automated design system, geometry constraints between adjacent sections can be enforced via curve fitting. Coefficients of the fit can be varied instead of the geometry parameters of individual airfoils.

The starting design for the optimization of the airfoil aerodynamic design problem is usually infeasible and the problem is sometimes over constrained. A lot of the constraints are formulated based on the prior experience of the designers and are fairly conservative. Constraint boundaries, however, are flexible and can be changed at the designer discretion. A constraint is relaxed if it results in a significant reduction in the violation of some other constraint or in the improvement of turbine performance. For some constraints even when the response quantity is within bounds, the designers prefer to keep the value of the response quantity either as low as possible (within bounds), as high as possible (within bounds), or at the midpoint of the constraint bounds to achieve greater reliability, performance, or cost advantage. The design process is thus a complex sequence of tradeoffs with soft constraint boundaries.

To simulate this behavior during optimization a set of Penalty Functions are used in the formulation. These functions vary linearly within the constraint bounds and exponentially outside the bounds. Imposing such penalty functions on the response quantity ensures that the response quantity values remain in close proximity to their allowable limits, while allowing sufficient flexibility in traversal of the search space during optimization. The basic form of the penalty function used for incorporating constraints into the objective function in the optimization formulation is shown below:

The allowable limits and the response quantities are transformed such that they are symmetric about the mid point of the constraint bounds (we will call it the origin). This transformation is shown in (

This penalty function defined in (

To obtain a monotonically increasing penalty function a variation of the penalty function shown in (

Two different penalty functions defined above were used in the turbine design problem formulation. The first function uses (

The objective function includes the performance metrics and constraints where the violations are included via penalty functions. Two different types of penalty terms,

Wsdiff * Suction_Diffusion + Wpdiff * Pressure_Diffusion + Wsdev * Suction_Deviation +Wpdev * Pressure_Deviation + Wledev * Le_Deviation + Wcross * Le_Crossover +

Here W’s are the weighting coefficients associated with the fitness measures. These factors can vary for different problems based on the requirements of the specific problem. The design variables and typical range of variations are listed in Table

Design variables used in the formulation of the airfoil design problem.

Design variable | Lower bound | Upper bound |
---|---|---|

c1 | 0.2 | 0.5 |

c2 | 0.25 | 0.75 |

c3 | 0.25 | 0.75 |

Tmaxx | 0.05 | 0.15 |

Staggr | 8 | 40 |

Pcttle | 0.25 | |

Ratl | 0 | 4 |

Ratu | 0 | 4 |

Ti | 0 | 1 |

E | 1 | 5 |

Constrained variables used in the formulation of the airfoil design problem.

Constrained variable | Lower bound | Upper bound |
---|---|---|

peak_exit | 0 | 1.3 |

peak_exit_loc | 0.6 | 1 |

peak_imin | 0 | 1.5 |

Carea | 0 | 10 |

rad_le | 0.035 | 5 |

Wedge | 2 | 6.6 |

Te | 0 | 30 |

Ovt | −1 | 4 |

delta1 | −30 | 30 |

tmax_c | 0.04 | 0.2 |

Pctz | 0 | 2 |

During the design of an airfoil, multiple sections are designed concurrently, and the objective function is a sum of the objective functions of all the cross sections being designed. Constraints for all the sections are also included in the problem formulation. Polynomial fits are used to represent the radial variation of the design variables; thus the objective function becomes a combination of the coefficients of the fits across multiple sections rather than individual parameters for each section. A second-order polynomial fit is used in the formulation; so corresponding to each metric we have three coefficients.

The solution to the problem can be attempted using a variety of optimization techniques including numerical optimization, genetic algorithms, simulated annealing, and heuristic search. In the current investigation, the BFGS variable metric method implemented in an optimization code ADS was used. A one-dimensional search technique was used in which the search was bounded followed by use of polynomial interpolation.

In the test case presented here, a low-pressure turbine nozzle is optimized. The flowpath of the low-pressure turbine used in the investigation is shown in Figure

Flowpath of the turbine used in the current investigation.

In the current investigation, stage 5 nozzle was designed using sections from five streamlines equally spaced along the blade span (hub to tip). Figure

Schematic representation of an airfoil showing streamlines along the radial direction.

To ensure slope and curvature smoothness of the geometry, second-order polynomials were used to represent the radial distribution of geometry parameters. Thus there are three design variables for each geometry parameter, that is, C_{0}, C_{1}, and C_{2}. These are the coefficients of the second-order polynomial representing the geometry parameter. The coefficients of the fit match well with the starting design since the design is based on a previously designed airfoil. Subsequently the smoothness is maintained since the parameters are not changed directly but rather the coefficients of the polynomials are varied. The geometry parameters which describe the low-pressure turbine airfoil geometry are

Performance constraints based on design practice guidelines are shown in the list below. These are hard constraints; one or more of these are usually violated in the starting solution. Even within bounds, constraints like Peak-exit-ratio and Inlet-valley-ratio are included in the objective function and minimized. Detailed definition of each of these is provided earlier in Section

0.0 < Peak-exit-ratio <1.2,

0.65 < Peak-location <1.0,

0 < Inlet-valley-ratio <2.0.

The geometry constraints for manufacturing requirements and mechanical stress limits are

The results of the optimization for a low-pressure turbine nozzle are contained in Tables

Design variables.

Name | Lower bound | Upper bound | Initial value | Final value |
---|---|---|---|---|

peak_exit | 0 | 1.3 | 1.15239 | 1.09774 |

peak_exit_loc | 0.6 | 1 | 0.655226 | 0.702005 |

peak_imin | 0 | 1.5 | 3.84903 | 1.63175 |

rad_le | 0.035 | 5 | 0.0177245 | 0.0177331 |

Wedge | 2 | 6.6 | 2.3837 | 2.87872 |

Ugt | 5 | 18 | 16.0004 | 12.2213 |

Te | 0 | 30 | 0.0299102 | 0.0299102 |

Ovt | 4 | 3.22223 | 1.77036 | |

Stagger | 0 | 50 | 30.9603 | 39.1863 |

Deltal | 30 | |||

Tmax | 0 | 2 | 0.141241 | 0.138897 |

tmax_c | 0.04 | 0.2 | 0.094285 | 0.0837007 |

Pctz | 0 | 2 | 0.389531 | 0.40175 |

Carea | 0 | 10 | 0.165028 | 0.174538 |

Objective function components.

Parameter | Weight | Initial value | Final value |
---|---|---|---|

dev_suction | 10 | 0.177097 | 0.0644436 |

dev_pressure | 2.5 | 0.0450l37 | 0.0112053 |

diff_suction | 5 | 0.186932 | 0.128518 |

diff_pressure | 1.5 | 0.308536 | 0.0656558 |

le_crossover | 50 | 0 | 0 |

le_smoothness | 2.5 | 0.0035042 | 0.00875813 |

peak_exit | l.9005 | l.80434 | |

peak_exit_loc | 0 | 0 | |

peak_imin | 12.287 | 2.43818 | |

Carea | 0.212921 | 0.225093 | |

tmax/c | 0 | 0 | |

Wedge | 0 | 0 | |

Ugt | 0 | 0 | |

Ovt | 0 | 0 | |

Objective function value | 18.4653 | 6.53994 |

Constraint variables.

Name | Lower bound | Upper bound | Initial value | Final value |
---|---|---|---|---|

c1 | 0.2 | 0.5 | 0.35 | 0.35 |

c2 | 0.25 | 0.75 | 0.5 | 0.632109 |

c3 | 0.25 | 0.75 | 0.5 | 0.569811 |

Tmaxx | 0.05 | 0.15 | 0.139516 | 0.139516 |

Stagger | 8 | 40 | 31.643 | 39.4644 |

Pcttle | 0.25 | 0 | 0 | |

Ratl | 0 | 4 | 1.25 | 2.7032 |

Ratu | 0 | 4 | 2.59442 | 2.72716 |

Ti | 0 | 1 | 0.5 | 1 |

E | 1 | 5 | 3 | 2.04457 |

Initial and final airfoil sections and Mach Number distributions for streamline at the nozzle tip [SL 3].

Initial and final airfoil sections and Mach number distributions for streamline between the nozzle tip and pitch [SL 5].

Initial and final airfoil sections and Mach number distributions for streamline at the nozzle pitch [SL 7].

Initial and final airfoil sections and Mach Number distributions for streamline between the nozzle pitch and root [SL 9].

Initial and final airfoil sections and Mach Number distributions for streamline at the nozzle root [SL 11].

Change in objective function for low-pressure turbine nozzle optimization using gradient-based search (ADS: ISTRAT = 3, IOPT = 3, IONED = 3).

Fletcher-Reeves algorithm for unconstrained minimization with Golden Section search was also used for optimization of the nozzle. Airfoil sections very similar to those obtained in the previous example were obtained. The iteration history for the optimization is shown in Figure

Change in objective function for low-pressure turbine nozzle optimization using gradient-based search (ADS

The optimized airfoils are more front loaded compared to the initial design which may or may not be desirable based on the designer preference. The objective function weights could be changed such that the loading on the leading edge is reduced. During the optimization, radial smoothness of the airfoil geometry is ensured by maintaining a 1st- or 2nd-order polynomial smoothness in the distribution of each design variable in the radial direction. Figure

A 3D model of an airfoil showing the passage between adjacent airfoils.

The paper presents a mathematical formulation for design of turbine airfoils using 2D geometry models and 2D inviscid analysis codes. The reduced computational complexity of the new formulation compared to 3D viscous analysis makes the airfoil design problem amenable to the use of formal optimization methods. The paper presents results from design of a low-pressure turbine nozzle. There are three primary contributions of this work:

Airfoil design is a labor intensive, repetitive, and cumbersome task for the designers and is a bottleneck for both the design cycle and rapid generation of inputs for complex multistage analyses. Automating the design process significantly cuts down the design cycle time and facilitates the task of running multistage analysis by rapidly generating airfoil geometries.

While designing an airfoil, it is hard to establish the existence of a unique optimum. Multiple evaluation criteria which are weighted together to define the objective function and the relative importance of these are determined based on designer experience. Furthermore, the analysis codes are not exact, and even with precisely defined quality metrics, a significant margin of error remains. In manual design the evaluation criteria are implicitly considered by the designer, with weighting factors based on past experience and individual biases. Subjectivity is introduced into the design process since the evaluation criteria for the design are partially based on heuristics abstracted from designer experiences. Thus in order to completely understand the results of airfoil optimization, an evaluation of the qualitative changes to the design is essential after the optimization is completed. Over time as the metrics to evaluate airfoil design become more acceptable, a standard metric will emerge, till such time designers will need to tinker with the weights to suit their own preferences.

This appendix presents the initial and final design configuration for the low-pressure turbine airfoil during optimization. The values are presented at the pitchline (approx. axial mid-point of the airfoil) of the turbine.

See Tables

The support of General Electric Corporate Research and Development is acknowledged in this work.