An Adjoint-Based Methodology for Sensitivity Analysis of Time-Periodic Flows With Reduced Time Integration

Sensitivity analysis plays a vital role in understanding the impact of control parameter variations on system output, particularly in cases where an objective functional evaluates the output's merit. The adjoint method has gained popularity due to its efficient computation, especially when dealing with a large number of control parameters and a few functionals. While the discrete form of the adjoint method is prevalent, exploring its continuous counterpart can offer valuable insights into the underlying mathematical problem, particularly in characterizing the boundary conditions. This paper presents an investigation into the continuous form of the adjoint method applied to time-dependent viscous flows, where the time dependence is either imposed by boundary conditions or arises from the system dynamics itself. The proposed approach enables the computation of sensitivities with respect to both geometric and operational control parameters using the same adjoint solution. For time-periodic flows, a special formulation is developed to mitigate the computational costs associated with time integration. Results demonstrate that the methodology proposed in a previous work can be successfully extended to time-dependent flows with fixed time spans. In such applications, time-accurate simulations of physics and adjoint fields are sufficient. However, periodic flows necessitate the application of the Leibniz Rule because the period might depend on the control parameters, which introduces additional terms to the adjoint-based sensitivity gradient. In that case, time integration can be limited to a minimum common multiple of all appearing periods in the flow. Although the accurate estimation of such multiple poses a challenge, the approach promises significant benefits for sensitivity analysis of fully established periodic flows. It leads to substantial cuts in computational costs and avoids transient data contamination.