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

Abstract This paper develops the unsteady continuous adjoint method for aeroacoustic optimization problems governed by time-dependent turbulent flows. To predict the 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 distribution 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. 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. The time window during which the simulation is performed and the one during which the objective function is evaluated do not coincide. This is reflected on the adjoint boundary conditions along the body and the time integration of the sensitivity derivatives. 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 residuals 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 in 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.

[1]  F. Farassat,et al.  An Analytical Comparison of the Acoustic Analogy and Kirchhoff Formulation for Moving Surfaces , 1997 .

[2]  Joaquim R. R. A. Martins,et al.  Aerodynamic shape optimization of wind turbine blades using a Reynolds‐averaged Navier–Stokes model and an adjoint method , 2017 .

[3]  Qiqi Wang,et al.  Minimal Repetition Dynamic Checkpointing Algorithm for Unsteady Adjoint Calculation , 2009, SIAM J. Sci. Comput..

[4]  A. Roshko.  On the development of turbulent wakes from vortex streets , 1953 .

[5]  Niklas Kühl,et al.  Adjoint volume-of-fluid approaches for the hydrodynamic optimisation of ships , 2018 .

[6]  A. Lyrintzis.  Surface Integral Methods in Computational Aeroacoustics—From the (CFD) Near-Field to the (Acoustic) Far-Field , 2003 .

[7]  P. Spalart,et al.  A hybrid RANS-LES approach with delayed-DES and wall-modelled LES capabilities , 2008 .

[8]  Lars Davidson,et al.  The Sub-Critical Flow pas a Generic Side Mirror and its Impact on Sound Generation and Propagation , 2006 .

[9]  K. Giannakoglou,et al.  Continuous Adjoint Methods for Turbulent Flows, Applied to Shape and Topology Optimization: Industrial Applications , 2016 .

[10]  W. K. Anderson,et al.  Aerodynamic design optimization on unstructured grids with a continuous adjoint formulation , 1997 .

[11]  D. L. Hawkings,et al.  Sound generation by turbulence and surfaces in arbitrary motion , 1969, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[12]  Kristine R. Meadows,et al.  TOWARDS A HIGHLY ACCURATE IMPLEMENTATION OF THE KIRCHHOFF APPROACH FOR COMPUTATIONAL AEROACOUSTICS , 1996 .

[13]  Kyriakos C. Giannakoglou,et al.  Noise reduction in car aerodynamics using a surrogate objective function and the continuous adjoint method with wall functions , 2015 .

[14]  Niles A. Pierce,et al.  An Introduction to the Adjoint Approach to Design , 2000 .

[15]  P. Spalart,et al.  Physical and Numerical Upgrades in the Detached-Eddy Simulation of Complex Turbulent Flows , 2002 .

[16]  Peter Unterlechner,et al.  Wind Noise Caused by the Side-Mirror and A-Pillar of a Generic Vehicle Model , 2012 .

[17]  ’ GEORGES.TRIANTAFYLLOU,et al.  Three-dimensional dynamics and transition to turbulence in the wake of bluff objects , 2005 .

[18]  Juan J. Alonso,et al.  An Efficient Unsteady Aerodynamic and Aeroacoustic Design Framework Using Discrete Adjoint , 2016 .

[19]  Wolfgang Schröder,et al.  A Discrete Adjoint Approach for Trailing-Edge Noise Minimization Using Porous Material , 2015 .

[20]  Kyriakos C. Giannakoglou,et al.  On the proper treatment of grid sensitivities in continuous adjoint methods for shape optimization , 2015, J. Comput. Phys..

[21]  Carsten Othmer,et al.  Adjoint methods for car aerodynamics , 2014, Journal of Mathematics in Industry.

[22]  F. Farassat,et al.  Modeling aerodynamically generated sound of helicopter rotors , 2003 .

[23]  Kyriakos C. Giannakoglou,et al.  Continuous adjoint approach to the Spalart–Allmaras turbulence model for incompressible flows , 2009 .

[24]  P. Spalart.  A One-Equation Turbulence Model for Aerodynamic Flows , 1992 .

[25]  Antony Jameson,et al.  Aerodynamic design via control theory , 1988, J. Sci. Comput..

[26]  Frank Thiele,et al.  Optimal Control of Unsteady Flows Using a Discrete and a Continuous Adjoint Approach , 2011, System Modelling and Optimization.

[27]  M. Giles,et al.  Adjoint equations in CFD: duality, boundary conditions and solution behaviour , 1997 .

[28]  F. Farassat,et al.  Extension of Kirchhoff's formula to radiation from moving surfaces , 1988 .

[29]  J. Sesterhenn,et al.  Prediction of the Interior Noise Level for Automotive Applications Based on Time-Domain Methods , 2016 .