Three-Dimensional Aerodynamic Design of Turbine Blades Using the Adjoint Method

2008 ◽  
Author(s):  
Yingchen Li ◽  
Zhenping Feng
Keyword(s):  
Boundary Conditions ◽  
Adjoint Method ◽  
Turbine Blades ◽  
Three Dimensional ◽  
Adjoint Equations ◽  
Aerodynamic Design ◽  
Characteristic Analysis ◽  
Axial Turbine ◽  
B Spline ◽  
Continuous Adjoint

The adjoint method is attractive because of its low computational cost and high efficiency. Although it has been one of the hot issues in aerodynamic design, it is not so widely used in turbomachinery applications as it is in the aeronautical field. The purpose of this work is to apply the adjoint method to three-dimensional (3D) aerodynamic inverse design of axial turbine blades for inviscid compressible flow. The 3D continuous adjoint system of Euler equations is formulated for turbine internal flow. The 3D blade profile is parameterized with Non-uniform B-Spline patch, and the coordinates of the B-Spline control points are selected as the design variables. Characteristic analysis of adjoint equations is taken to set inlet/outlet boundary conditions. To avoid the discontinuity of boundary conditions of adjoint equations in the spanwise direction, a method for solving an ordinary differential equation is developed to smooth the residual distribution of aerodynamic parameter on blade surface. 3D adjoint equations are numerically solved by using time-marching method and finite volume method. Finally, combining the grid perturbation technique, CFD technique and quasi-Newton algorithm, the aerodynamic design approach for 3D axial turbine blades is presented and several numerical examples are demonstrated to validate this approach.

S2 Surface Quasi-3D Aerodynamic Design Using the Continuous Adjoint Method for Multi-Stage Turbine

2014 ◽  
Author(s):  
Lei Chen ◽  
Jiang Chen
Keyword(s):  
Objective Function ◽  
Adjoint Method ◽  
Adjoint Equations ◽  
Design System ◽  
Shape Design ◽  
Aerodynamic Design ◽  
Aerodynamic Shape ◽  
Multi Stage ◽  
Gradient Based ◽  
Continuous Adjoint

The adjoint method eliminates the dependence of the gradient of the objective function with respect to design variables on the flow field making the obtainment of the gradient both accurate and fast. For this reason, the adjoint method has become the focus of attention in recent years. This paper develops a continuous adjoint formulation for through-flow aerodynamic shape design in a multi-stage gas turbine environment based on a S2 surface quasi-3D problem governed by the Euler equations with source terms. Given the general expression of the objective function calculated via a boundary integral, the adjoint equations and their boundary conditions are derived in detail by introducing adjoint variable vectors. As a result, the final expression of the objective function gradient only includes the terms pertinent to those physical shape variations that are calculated by metric variations. The adjoint system is solved numerically by a finite-difference method with explicit Euler time-marching scheme and a Jameson spatial scheme which employs first and third order dissipative flux. Integrating the blade stagger angles and passage perturbation parameterization with the simple steepest decent method, a gradient-based aerodynamic shape design system is constructed. Finally, the application of the adjoint method is validated through a 5-stage turbine blade and passage optimization with an objective function of entropy generation. The result demonstrates that the gradient-based system can be used for turbine aerodynamic design.

Inverse Problem in Aerodynamic Shape Design of Turbomachinery Blades

2006 ◽  
Author(s):  
Yingchen Li ◽  
Dianliang Yang ◽  
Zhenping Feng
Keyword(s):  
Optimization Algorithm ◽  
Adjoint Method ◽  
Adjoint Equations ◽  
Shape Design ◽  
Computational Domain ◽  
Aerodynamic Design ◽  
Aerodynamic Shape ◽  
Design Variables ◽  
Turbomachinery Blades ◽  
Continuous Adjoint

There are various methods for aerodynamic shape design in turbomachinery blades, but at the state of the art the shape design has still been a formidable problem. Optimal shape design based on adjoint method has been developed rapidly in the last decades in aeronautic field with the start of Jameson’s work. As a gradient-based optimization, the adjoint method introduces an adjoint system and the sensitivity derivative is computed by solving a linear adjoint equation, which makes the computational cost almost independent of the number of design variables. Because the adjoint method realizes the quick and exact sensitivity analysis and saves large computational resources, it has been the highlight in aerodynamic shape design of CFD field. Combining the continuous adjoint method with quasi-Newton method, we developed an optimization algorithm for turbomachinery aerodynamic design governed by two-dimensional Euler equations in this paper. The blade shape to be optimized is parameterized by non-uniform B-spline and the computational domain is discreted with H-grids. Then the adjoint equations and their boundary conditions are deduced in detail, both in computational and physical spaces, and are solved numerically by using time-marching finite difference method based on Jameson’s diffusion scheme. With the solved adjoint variables, the final expression of the cost function gradient with respect to the design variables is formulated. Finally, several numerical cases of turbomachinery blade aerodynamic design are presented and analyzed to validate the present optimization algorithm.

Aerodynamic Design of Turbine Blades by Using Adjoint-Based Method and N-S Equation

2007 ◽  
Author(s):  
Yingchen Li ◽  
Zhenping Feng
Keyword(s):  
Adjoint Method ◽  
Adjoint Equation ◽  
Turbine Blades ◽  
Computational Cost ◽  
Internal Flow ◽  
Aerodynamic Design ◽  
Continuous Adjoint Method ◽  
Continuous Adjoint ◽  
Quasi Newton ◽  
Low Computational Cost

Because of the low computational cost, adjoint method has been a highlight issue in aerodynamic design since being firstly introduced in aeronautical field. The purpose of this work is to expand the adjoint method into turbomachinery applications for viscous and compressible flow. The continuous adjoint method is formulated in two-dimensional blade design by using N-S equation. Combining Thompson’s theory of time-related boundary condition, we present the boundary conditions of the adjoint equation for internal flow and discuss the restrictions of cost function in the case of given surface temperature and adiabatic conditions on blade walls. Numerical techniques used in CFD are employed here to solve the adjoint linear PDE. In conjunction with quasi-Newton algorithm, the aerodynamic design approach for turbine blades is presented, which is independent of the N-S solver being used. Several numerical examples are implemented to validate this approach and then the pressure inverse design of a cascade blade is taken.

Three-Dimensional Aerodynamic Design Optimization of a Turbine Blade by Using an Adjoint Method

2009 ◽  
Author(s):  
Jiaqi Luo ◽  
Juntao Xiong ◽  
Feng Liu ◽  
Ivan McBean
Keyword(s):  
Design Optimization ◽  
Turbine Blade ◽  
Adjoint Method ◽  
Euler Method ◽  
Adjoint Equations ◽  
Design Parameters ◽  
Aerodynamic Design ◽  
Blade Profile ◽  
Profile Shape ◽  
Aerodynamic Design Optimization

This paper presents the application of an adjoint method to the aerodynamic design optimization of a turbine blade. With the adjoint method, the complete gradient information needed for optimization can be obtained by solving the governing flow equations and their corresponding adjoint equations only once, regardless of the number of design parameters. The formulations including imposition of appropriate boundary conditions for the adjoint equations of the Euler equations for turbomachinery problems are presented. Two design cases are demonstrated for a turbine cascade that involves a high tip flare, characteristic of steam turbine blading in low pressure turbines. The results demonstrate that the design optimization method is effective and the redesigned blade yielded weaker shock and compression waves in the supersonic region of the flow while satisfying the specified constraint. The relative effects of changing blade profile stagger, modifying the blade profile shape, and changing both stagger and profile shape at the same time are examined and compared. Navier-Stokes calculations are performed to confirm the performance at both the design and off-design conditions of the designed blade profile by the Euler method.

Time Accurate Euler Simulation of Interaction of Nozzle Wake and Secondary Flow With Rotor Blade in an Axial Turbine Stage Using Nonreflecting Boundary Conditions

1995 ◽  
Author(s):  
S. Fan ◽  
B. Lakshminarayana
Keyword(s):  
Flow Field ◽  
Boundary Conditions ◽  
Unsteady Flow ◽  
Secondary Flow ◽  
Rotor Blade ◽  
Three Dimensional ◽  
Computational Domain ◽  
Turbine Stage ◽  
Axial Turbine ◽  
Unsteady Pressure

The objective of this paper is to investigate the three dimensional unsteady flow interactions in a turbomachine stage. A three-dimensional time accurate Euler code has been developed using an explicit four-stage Runge-Kutta scheme. Three-dimensional unsteady non-reflecting boundary conditions are formulated at the inlet and at the outlet of the computational domain to remove the spurious numerical reflections. The three-dimensional code is first validated for 2-D and 3-D cascades with harmonic vortical inlet distortions. The effectiveness of non reflecting boundary conditions is demonstrated. The unsteady Euler solver is then used to simulate the propagation of nozzle wake and secondary flow through rotor and the resulting unsteady pressure field in an axial turbine stage. The three dimensional and time dependent propagation of nozzle wakes in the rotor blade row and the effects of nozzle secondary flow on the rotor unsteady surface pressure and passage flow field are studied. It was found that the unsteady flow field in the rotor is highly three-dimensional and the nozzle secondary flow has significant contribution to the unsteady pressure on the blade surfaces. Even though the steady flow at the midspan is nearly two-dimensional, the unsteady flow is 3-D and the unsteady pressure distribution can not by predicted by a 2-D analysis.

Optimization of a Centrifugal Impeller Using Evolutionary Algorithms

2008 ◽  
Author(s):  
Andreas Bartold ◽  
Franz Joos
Keyword(s):  
Evolutionary Algorithms ◽  
Three Dimensional ◽  
Optimization Method ◽  
Navier Stokes ◽  
Centrifugal Impeller ◽  
Aerodynamic Design ◽  
B Spline ◽  
Spline Curves ◽  
Automated Optimization ◽  
Isentropic Efficiency

This paper presents the development and application of an automated optimization method for aerodynamic design of centrifugal impellers. The algorithm used for the optimization is an evolutionary algorithm. Within this method the shape of the centrifugal impeller is described using B-Spline curves. The method introduced is used for redesigning an existing impeller with regard to maximization of the isentropic efficiency at a fixed operating point. Here the isentropic efficiency is calculated using the solution of a compressible three-dimensional Reynolds-averaged Navier-Stokes solver. The presentation will show that the method presented provides a new design that outperforms the original impeller with respect to the particular objective and demonstrates its usefulness.

Turbine Blade Aerodynamic Optimization on Unstructured Grids Using a Continuous Adjoint Method

2012 ◽  
Author(s):  
M. Zeinalpour ◽  
K. Mazaheri ◽  
A. Irannejad
Keyword(s):  
Turbine Blade ◽  
Adjoint Method ◽  
Suction Side ◽  
Adjoint Equations ◽  
Inverse Design ◽  
Aerodynamic Optimization ◽  
Design Variables ◽  
Continuous Adjoint Method ◽  
Flow Variables ◽  
Continuous Adjoint

A gradient based optimization using the continuous adjoint method for inverse design of a turbine blade cascade is presented. The advantage of the adjoint method is that the objective function gradients can be evaluated by solving the adjoint equations with coefficients depending on the flow variables. This method is particularly suitable for aerodynamic design optimization for which the number of design variables is large. Bezier polynomials are used to parameterize suction side of the turbine blade. The numerical convective fluxes of both flow and adjoint equations are computed by using a Roe-type approximate Riemann solver. An approximate linearization is applied to simplify the calculation of the numerical flux of adjoint variables on the faces of computational cell. The problem examined is that of the inverse design of NASA C3X blade that reproduces a given pressure distributions over its surfaces. Adjoint results show a good agreement with those obtained by finite-difference method.

Adjoint Aerodynamic Design Optimization for Blades in Multi-Stage Turbomachines: Part I—Methodology and Verification

2008 ◽  
Author(s):  
D. X. Wang ◽  
L. He
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Design Optimization ◽  
Difference Method ◽  
Adjoint Method ◽  
Adjoint Equations ◽  
Steepest Descent Method ◽  
Compressor Stage ◽  
Multi Stage ◽  
Continuous Adjoint

The adjoint method for blade design optimization will be described in this two-part paper. The main objective is to develop the capability of carrying out aerodynamic blading shape design optimization in a multi-stage turbomachinery environment. To this end, an adjoint mixing-plane treatment has been proposed. In the first part, the numerical elements pertinent to the present approach will be described. The gradients of a single objective function of a weighted sum of objectives and constraints with respect to detailed blade shape perturbations are obtained very efficiently by the continuous adjoint method. The steepest descent method is used to drive the design to an optimum. The adjoint mixing-plane treatment enables the adjoint equations to be solved in a multi-stage environment. The adjoint solver is verified by comparing gradient results with a direct finite difference method and through a 2D inverse design. The adjoint mixing-plane treatment is verified by comparing gradient results against those by the finite difference method for a 2D compressor stage. The redesign of the 2D compressor stage further demonstrates the validity of the adjoint mixing-plane treatment and the benefit of using it in a multi-bladerow environment.

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