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Research Papers

Adjoint Aerodynamic Design Optimization for Blades in Multistage Turbomachines—Part I: Methodology and Verification

[+] Author and Article Information
D. X. Wang

School of Engineering, Durham University, Durham DH1 3LE, UK

L. He

Department of Engineering Science, Oxford University, Parks Road, Oxford OX1 3PJ, UK

J. Turbomach 132(2), 021011 (Jan 13, 2010) (14 pages) doi:10.1115/1.3072498 History: Received August 31, 2008; Revised October 07, 2008; Published January 13, 2010; Online January 13, 2010

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 multistage 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. Attention is paid to the exactly opposite propagation of the adjoint characteristics against the physical flow characteristics, providing a simple and consistent guidance in the adjoint method development and applications. The adjoint mixing-plane treatment is formulated to have the two fundamental features of its counterpart in the physical flow domain: conservation and nonreflectiveness across the interface. 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.

Copyright © 2010 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

An interface between a rotor and a stator configuration

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Figure 2

Overview (upper) and close-up (lower) of the mesh around a NACA0012 aerofoil

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Figure 3

Mach number contours around NACA0012 at zero angle of attack

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Figure 4

Adjoint field (λ1) around NACA0012 at zero angle of attack

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Figure 5

Lift coefficient/gradient versus angle of attack for NACA0012 (FDM: finite difference method, ADJ: adjoint method, and CL: calculated lift coefficient)

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Figure 6

Pressure coefficient distributions over the initial, target, and designed aerofoils’ surface for the inverse design

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Figure 7

Comparison between initial, target, and designed aerofoil profiles for the inverse design

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Figure 8

History of the objective function for the inverse design

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Figure 9

Computational mesh of the 2D compressor stage

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Figure 10

Relative Mach number contours (2D configuration of the compressor stage)

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Figure 11

Adjoint field (λ1) with the tangential force on the stator blade as the objective function

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Figure 12

Adjoint field (λ3) with mass flow rate as the objective function

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Figure 13

Gradients of the tangential force on the stator blade (FDM: finite difference method and ADJ: adjoint method)

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Figure 14

Gradients of stage mass flow rate (FDM: finite difference method and ADJ: adjoint method)

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Figure 15

Comparison of the original rotor blade and the optimized rotor blade

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Figure 16

Relative Mach contours of the original blade passage and the optimized blade passage

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Figure 17

Comparison of the original stator blade and the optimized stator blade

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