The unsteady continuous adjoint method for minimizing flow-induced sound radiation

This paper develops the unsteady continuous adjoint method for aeroacoustic
problems governed by time-dependent turbulent flows. To predict flow-induced
sound radiation from a body in free-stream, an incompressible Improved Delayed
Detached Eddy Simulation is firstly performed. The generated noise is then
propagated with the Kirchhoff Integral method, which uses the pressure distri-
bution on the body surface to compute the sound pressure at selected receivers.
The continuous adjoint method for the aforementioned process is developed and
presented for the first time. In the adjoint process, the differentiated Kirchhoff
Integral is used to compute the boundary condition of the adjoint velocity on the
body surface and, then, the unsteady adjoint equations are solved backwards
in time. It should be noted that the time window where the simulation is per-
formed and the time window over which the objective function is evaluated do
not coincide. This is reflected on the adjoint boundary condition along the body
and the time integration of the sensitivity derivatives. Furthermore, to ensure
the consistency of the continuous adjoint-based gradients, grid sensitivities are
taken into account. By incorporating the grid displacement equations during
the mathematical development of the adjoint method, the surface integral of
the residual of the flow PDEs, which is commonly omitted, is replaced with a volume integral of the grid sensitivities and gives rise to the adjoint grid displacement equations along with an accurate sensitivity derivatives expression.
The proposed method is verified against Finite Differences on a 3D turbulent
flow around a cylinder and, then, applied to a real-world test case, concerned
with the flow-induced sound radiation of the side mirror of the generic SAE
vehicle.