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Research Papers

2D Viscous Aerodynamic Shape Design Optimization for Turbine Blades Based on Adjoint Method

[+] Author and Article Information
Haitao Li, Liming Song, Yingchen Li

Institute of Turbomachinery, Xi’an Jiaotong University, Xi’an 710049, P.R.China

Zhenping Feng

Institute of Turbomachinery, Xi’an Jiaotong University, Xi’an 710049, P.R.Chinazpfeng@mail.xjtu.edu.cn

J. Turbomach 133(3), 031014 (Nov 16, 2010) (8 pages) doi:10.1115/1.4001234 History: Received August 21, 2009; Revised November 05, 2009; Published November 16, 2010; Online November 16, 2010

This paper presents an adjoint optimization technique and its application to the design of a transonic turbine cascade. Capable of a quick and exact sensitivity analysis and using little computational resources, the adjoint method has been a focus of research in aerodynamic shape design optimization. The goal of this work is to extend the adjoint method into turbomachinery design applications for viscous and compressible flow, and to further improve the aerodynamic performance. In the work, the minimization of the entropy generation rate with the mass flow rate constraint was considered as the cost function of the optimization, and was applied in the direct design process. The adjoint boundary conditions of the corresponding cost function were derived in detail, using the nonslip boundary condition on the blade wall, while the flow viscous effect on the cascade inlet and outlet was neglected. Numerical techniques used in Computational Fluid Dynamics (CFD) were employed to solve the adjoint linear partial difference equations. With the solved adjoint variables, the final expression of the cost function gradient with respect to the design variables was formulated. Combined with quasi-Newton algorithm, an aerodynamic design approach based on the adjoint method for turbine blades was presented, which was independent of the Navier–Stokes solver being used. Finally, to validate the present optimization algorithm, the aerodynamic design cases of a transonic turbine blade with and without mass flow rate restriction were performed and analyzed.

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Figures

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Figure 1

Gradient of different cost functions with respect to design variables. (a) Gradient of outlet entropy generation rate; (b) gradient of averaged outlet total pressure; and (c) normalized gradient comparison.

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Figure 2

H-O-H grid of computation domain

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Figure 3

Designed blade and control dots

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Figure 4

Gradient components variation

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Figure 5

Blade geometry and control dots variation in the initial and optimal case

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Figure 6

Adjoint fields of four components at the last iterative step (σ=50). (a) distribution of ψ1; (b) distribution of ψ2; (c) distribution of ψ3; (d) distribution of ψ4.

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Figure 7

Evolution of the cost function and constraint

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Figure 8

Isentropic-Mach number distribution

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Figure 9

Mach number distributions for the blade passage (left: initial; middle:σ=0 optimal; right:σ=50 optimal)

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