Unsteady continuous adjoint to URANS coupled with FW-H analogy for aeroacoustic shape optimization

Abstract

In this paper, an unsteady aeroacoustic shape optimization tool is presented based on the continuous adjoint method. Aeroacoustic noise radiated by bodies in unsteady flows can be computed using a hybrid acoustic prediction tool, where the near-field flow and acoustics result from an unsteady CFD simulation while the acoustic propagation to far-field relies upon an acoustic analogy. The CFD solution is performed using the in-house GPU-enabled unsteady RANS solver for which a continuous adjoint solver is also available. Next to this, the noise prediction tool is developed based on the permeable version of the Ffowcs Williams and Hawkings analogy in the frequency domain and its implementation is verified by comparison to a well-known analytical solution of the sound field from a monopole source in uniform flow as well as comparing the results computed by the FW-H analogy with that of a time-accurate CFD run. The full mathematical formulation of the continuous adjoint method for the aforementioned acoustic analogy, coupled with the adjoint to the mean-flow and turbulence model equations, is presented. The accuracy of the gradients computed by the hybrid adjoint method is verified through comparisons with finite differences in two cases, both governed mainly by the tonal component in noise generation; these include the vortex shedding cylinder in a laminar flow and the rod-airfoil canonical benchmark in a turbulent flow. Finally, the programmed software is used to optimize the shape of the airfoil in the last test case, aiming at min. noise for a receiver located at far-field.