Optimization of Endwall Contours of a Turbine Blade Row Using an Adjoint Method

Abstract

This paper presents the application of a viscous adjoint method in the optimization of a low-aspect-ratio turbine blade through spanwise restaggering and endwall contouring. A generalized wall-function method is implemented in a Navier-Stokes flow solver coupled with Menter's SST k-ω Coordinates in the computational domain INTRODUCTIONAt present, it is difficult to further improve the performance of turbomachinery through traditional design procedures because significant efficiency gains have already been obtained. However, with the rapidly increased computing capacity and advances in numerical methods, Computational Fluid Dynamics (CFD) coupled with advanced optimization algorithms provides a costeffective way to improve the design of turbomachinery as compared to classical methods based on manual iteration. Many design optimization approaches such as response surface methodology [1-4], genetic algorithms [5][6][7] and finite difference methods [8] were applied in design development and most of them are widely used nowadays.In the optimization of designs based on CFD analysis, the flow solver which plays an important role in the optimization system is required to provide physically accurate flow solutions. However, not all turbulence models can sufficiently model complex flows. Wilcox gave an overview of the turbulence models [9] and demonstrated that models based on the ω-equation can support satisfactory solutions for most of the flows. In the present study, the flow solutions come from a turbomachinery flow code Turbo90, in which Menter's SST k-ω turbulence model [10] is adopted coupled with a third-order Roe scheme for the convective terms.For the designs of complex geometries, meshes with millions of cells are required to resolve the flow. The problem of reducing the computer time without loss in numerical accuracy is of particular importance for the optimization of designs. In the past several decades, research has been done to improve the efficiency of flow solvers. An effective method for reducing computational effort is to reduce the computational grid without loss of accuracy. Hereby for the improved modeling of boundary layer regions with limited grid resolution, the wall function methods were developed originally through flat-plate flows with zero streamwise pressure gradient. Most of the earliest wall functions were developed from the "law of the wall" [11,12], which accounts for the flow structure in the logarithmic layer. However, the entire boundary region in a fully-turbulent flow can be subdivided into three parts, a viscous sublayer, a buffer layer and a logarithmic layer. By using the earliest wall function methods, the first grid point away from the solid wall should locate in the logarithmic layer and obviously it is a severe constraint, which is inevitably to be violated in complex flows. In order to overcome such drawbacks, wall functions that account for the entire boundary layer are needed. Kalitzin [13] proposed a general-

A review on aerodynamic optimization of turbomachinery using adjoint method

“…Therefore, Luo et al proposed an improved optimization method relying on the continuous adjoint method to tackle the mentioned problem in inviscid 79,80 and viscous flows. 139–141 First, the inviscid continuous adjoint method was derived to investigate restaggering blade profiles. 79,80 A blade profile restaggering method combined with blade profile change was extended and applied to direct and inverse optimization cases.…”

Multi-Objective Design Optimization of a Transonic Compressor Rotor Using an Adjoint Equation Method

“…Liu et al 7,8 successfully applied this method to the optimization of cascade and turbine blade. Luo et al [9][10][11] performed design optimization of turbine blades to reduce shock loss, profile loss and secondary flow loss through modifying blade profile, restaggering and endwall contouring by using a continuous adjoint method; subsequently, they successfully performed multi-point optimization of a transonic compressor rotor to improve the aerodynamic performance over the whole operating range. 12 Wang et al 13,14 and Nadarajah et al 15 presented the applications of the adjoint method to the design optimizations of multi-stages.…”

Multi-Point Design Optimization of a Transonic Compressor Blade by Using an Adjoint Method

“…Liu et al [14,15] introduced this method to the design optimization of cascade and turbine blade. Luo et al [16][17][18] performed design optimizations of turbine blades to reduce shock loss, profile loss, and secondary flow loss through blade profiling, restaggering and endwall contouring by using a continuous adjoint method. Wang et al [19,20] presented the applications of the adjoint method to the design optimizations of multistages.…”

“…In the present study, the continuous adjoint method developed for single-point design optimization of turbine nozzle vanes [16][17][18] is extended to study multipoint design optimization of transonic compressor rotor blades. Multipoint optimization is achieved by using a combined cost function consisting of weighted performance measures at the operating conditions of interest.…”