Constrained Aerodynamic Shape Research Articles

Optimum Shape Design for Multirow Turbomachinery Configurations Using a Discrete Adjoint Approach and an Efficient Radial Basis Function Deformation Scheme for Complex Multiblock Grids

This paper proposes a framework for fully automatic gradient-based constrained aerodynamic shape optimization in a multirow turbomachinery environment. The concept of adjoint-based gradient calculation is discussed and the development of the discrete adjoint equations for a turbomachinery Reynolds-averaged Navier–Stokes (RANS) solver, particularly the derivation of flow-consistent adjoint boundary conditions as well as the implementation of a discrete adjoint mixing-plane formulation, are described in detail. A parallelized, automatic grid perturbation scheme utilizing radial basis functions (RBFs), which is accurate and robust as well as able to handle highly resolved complex multiblock turbomachinery grid configurations, is developed and employed to calculate the gradient from the adjoint solution. The adjoint solver is validated by comparing its sensitivities with finite-difference gradients obtained from the flow solver. A sequential quadratic programming (SQP) algorithm is then utilized to determine an improved blade shape based on the gradient information from the objective functional and the constraints. The developed optimization method is used to redesign a single-stage transonic flow compressor in both inviscid and viscous flow. The design objective is to maximize the isentropic efficiency while constraining the mass flow rate and the total pressure ratio.

Adjoint-Based Constrained Aerodynamic Shape Optimization for Multistage Turbomachines

This paper develops the discrete adjoint equations for a turbomachinery Reynolds-averaged Navier–Stokes solver and proposes a framework for fully automatic gradient-based constrained aerodynamic shape optimization in a multistage turbomachinery environment. The systematic approach for the development of the discrete adjoint solver is discussed. Special emphasis is put on the development of the turbomachinery-specific features of the adjoint solver (that is, on the derivation of flow-consistent adjoint inlet and outlet boundary conditions) and, to allow for a concurrent rotor–stator optimization and stage coupling, on the development of an exact adjoint counterpart to the nonreflective, conservative mixing-plane formulation used in the flow solver. The adjoint solver is validated by comparing its sensitivities with finite difference gradients obtained from the flow solver. A sequential-quadratic programming algorithm is used to determine an improved blade shape based on the gradient information provided by the adjoint solution. Optimization within a sequential-quadratic programming framework avoids the time-consuming task to determine the weights of the individual constraints by including the constraints directly into the design problem. The functionality of the proposed optimization method is demonstrated by the redesign of two two-dimensional transonic compressor configurations. The design objective is to maximize the isentropic efficiency while constraining the mass flow rate and the total pressure ratio.