Adjoint Calculation of Sensitivities of Turbomachinery Objective Functions

This paper presents an overview of the steady and harmonic adjoint methods for turbomachinery design using the `discrete ' approach in which the discretized nonlinear Euler/Navier-Stokes equations are linearized and the resulting matrix is then transposed. Steady adjoint solvers give the linear sensitivity of steady-state functionals such as mass and average exit angle to arbitrary changes in the geometry of the blades and this linear sensitivity information can then be used as part of a nonlinear optimization procedure. The harmonic adjoint method is based on a single frequency of unsteadiness and allows one to determine the generalized force acting on the blades due to arbitrary incoming time-periodic gusts. When the forcing is due to the wakes of the upstream blades the adjoint approach can be used to tailor the shape of the incoming wakes to greatly reduce the level of forced vibration they induce. The presented suite of test-cases includes the Inlet Guide Vane and the rotor of a High Pressure Turbine. 1