Unsteady optimal control of shear flows based on the discrete and continuous adjoint Navier-Stokes equations.

• Applicants: Professor Dr.-Ing. Holger Foysi; Professor Dr. Nicolas R. Gauger
• Subject Area: Fluid Mechanics
• Term: from 2013 to 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 235772517

• Final Report Year: 2018

Final Report Abstract

The developed hand-discrete and continuous adjoint solvers provided a good accuracy of sensitivities for all test cases. The performed optimization studies using these adjoint solvers showed good success in reducing the jet noise for relatively small control horizons. For large control horizons, however, the sensitivities grew too much causing the failure of gradient-based optimization algorithms. To investigate this issue further, AD based tangent-linear solver, which realizes an exact linearization of the solver, has been generated. The behavior of the sensitivities has been investigated on a LES test case with a large control horizon. The simulation results showed an overflow of sensitivities after a certain time iteration. The sensitivity values in the initial time iterations matched perfectly with the finite difference approximations. This behavior indicates that the physical instability, which is inherently present the in the turbulent flows, is the reason why the sensitivities overflow and the system becomes uncontrollable. Regularization methods such as Least Square Shadowing is proven to solve this problem, but it comes at the expense of the increased run-time and memory requirements. To overcome these issues, a new approach that couples the one-shot method with Least Square Shadowing has been suggested during this project. The new approach has been applied to a one dimensional convection diffusion equation driven by an oscillating upstream boundary condition. Solving the corresponding least squares problem provides a uniform reduction of the residuals over the entire time domain, independent of the time domain length. Its integration into a simultaneous one-shot method has been demonstrated for an inverse design optimization problem. By relaxing the initial condition, changes to the trajectory at early time chunks and changes to the design parameters can be accommodated by a change in the initial condition of the trajectory. Transporting information is thereby no longer limited to forward-in-time propagation as it is the case for unsteady PDEs with fixed initial conditions. The formulation is therefore especially beneficial for simultaneous optimization of statistical quantities subject to chaotic and turbulent flow constraints.

Publications

• “Spanwise reflection symmetry breaking and turbulence control: Plane Couette flow”, Journal of Fluid Mechanics, Vol. 745, pp. 300-320, 2014
  G. Chagelishvili, G. Khujadze, H. Foysi, M. Oberlack
  
• “Numerical optimization of the pseudopotential-based Lattice Boltzmann method”, J. Comput. Science, Vol. 17, Part 2, pp. 384–393, 2016
  K. Küllmer, A. Krämer, D. Reith , W. Joppich, H. Foysi
  
• ”Simultaneous one-shot optimization with unsteady PDEs”, Journal of Computational and Applied Mathematics, Vol. 294, pp. 12-22, 2016
  S. Günther, N.R. Gauger, Q. Wang
  
• A non-intrusive parallel-in-time adjoint solver with the XBraid library
  S. Günther, N.R. Gauger and J.B. Schroder
  
• ”A framework for simultaneous aerodynamic design optimization in the presence of chaos”, Journal of Computational Physics, Vol. 328, pp. 387-398, 2017
  S. Günther, N.R. Gauger, Q. Wang
  
• ”On the stability of gradient based turbulent flow control without regularization”, submitted to Journal of Computational and Applied Mathematics, 2018
  E. Özkaya, N.R. Gauger, H. Foysi, D. Marinc