Review of design optimization methods for turbomachinery aerodynamics

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. DOI: 10.1115/1.3072498

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

Based on control theory, adjoint system for the general problem of turbomachinery aerodynamic optimization was studied and developed in the present paper by using the variation technique in the grid node coordinates combined with Jacobian Matrics of flow fluxes. Then the adjoint system for aerodynamic design optimization of turbine cascade governed by compressible Navier-Stokes equations was derived in detail. With the purpose of saving computation resources, the mathematic method presented in this paper avoids the coordinate system transforming in the traditional derivation process of the adjoint system and makes the adjoint system much more sententious. Given the general expression of objective functions consisting of both boundary integral and field integral, the adjoint equations and their boundary conditions were derived, and the final expression of the objective function gradient including only boundary integrals was formulated to reduce the CPU cost, especially for the complex 3D configurations. The adjoint system was solved numerically by using the finite volume method with an explicit 5-step Runge-Kutta scheme and Riemann approximate solution of Roe’s scheme combined with multi-grid technique and local time step to accelerate the convergence procedure. Finally, based on the aerodynamic optimization theory in the present work, 2D and 3D inviscid and viscous inverse design programs of axial turbomachinery cascade for both pressure distribution and isentropic Mach number distribution on the blade wall were developed, and several design optimization cases were performed successfully to demonstrate the ability and economy of the present optimization system.

Aerodynamic inverse design optimization for turbine cascades based on control theory

Abstract This paper develops a continuous adjoint formulation for the aerodynamic shape design of a turbine in a multi-stage environment based on S2 surface governed by the Euler equations with source terms. First, given the general expression of the objective function, the adjoint equations and their boundary conditions are derived by introducing the adjoint variable vectors. Then, the final expression of the objective function gradient only includes the terms pertinent to the physical shape variations. The adjoint system is solved numerically by a finite-difference method with the Jameson spatial scheme employing first and third order dissipative flux and the time-marching is conducted by Runge-Kutta time method. Integrating the blade stagger angles, stacking lines and passage perturbation parameterization with the Quasi-Newton method of BFGS, a gradient-based aerodynamic optimization design system is constructed. Finally, the application of the adjoint method is validated through the blade and passage optimization of a 2-stage turbine with an objective function of entropy generation. The efficiency increased by 0.37% with the deviations of the mass flow rate and the pressure ratio within 1% via the optimization, which demonstrates the capability of the gradient-based system for turbine aerodynamic design.

Aerodynamic Optimization Design of Multi-stage Turbine Using the Continuous Adjoint Method

Improvements in aerodynamic turbomachinery design have gained attraction due to the increased demand for a more sustainable future. Several optimization approaches have been presented and employed in the realm of aerodynamic design. However, among all of them, the adjoint approach has emerged as a hot research topic for aerodynamic optimization in the field of turbomachinery. The ability of this method to efficiently compute the derivatives of objective functions for several design variables has made it a promising optimization tool. This study provides a comprehensive review of all significant studies undertaken since the turn of the 21st century when the adjoint method was employed for the aerodynamic optimization of turbomachinery applications. The application of the adjoint approach in that context is extensively discussed under various aspects, including shape optimization in both steady and unsteady flows, varied eddy viscosity, non-ideal compressible fluid-dynamics, multi-objective and multi-point optimizations, multidisciplinary optimization, coupling adjoint method with other approaches, parametrization methods, and uncertainty quantification. Finally, the review concludes by highlighting key points and outlooks on future developments.

A review on aerodynamic optimization of turbomachinery using adjoint method

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.Copyright © 2006 by ASME

Inverse Problem in Aerodynamic Shape Design of Turbomachinery Blades

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Inverse Problem in Aerodynamic Shape Design of Turbomachinery Blades