Data Compression Algorithms for Adjoint Based Sensitivity Studies of Unsteady Flows

Adjoint based sensitivity studies are effective means to understand how flow quantities of interest and grid quality could affect the flow simulations. However, applying an unsteady adjoint method to three-dimensional complex flows remains a challenging topic. One of the challenges is that the flow variables at all previous time steps will be needed when solving the unsteady adjoint equation backwards in time. The straightforward treatment of storing all the previous flow solutions could be prohibitive for simulations with a large number of grid points and time steps. To avoid storing the full trajectory, the checkpointing method only stores the flow solutions at some carefully selected time steps called checkpoints, and re-computes the flow solutions between checkpoints when they are needed by the adjoint solver. However, the re-computation increases the computational cost by multiple times of the cost of solving the flow equations, which may be unacceptable for some applications. Alternatively, in this study, several data compression algorithms with much less extra cost were considered for alleviating the storage problem. In these data compression algorithms, the full flow solutions were projected onto a small set of bases which were either generated by the proper orthogonal decomposition (POD) method or by the Gram-Schmidt orthogonalization. Only a small set of bases and corresponding expansion coefficients need to be stored and they could recover the flow solutions at every time step with reasonable accuracy. The data compression algorithms were implemented in the numerical test cases, and the computed adjoint solutions were compared with that obtained by using full flow solutions. The comparisons demonstrated that the data compression algorithms were able to greatly reduce the storage requirements while maintaining sufficient accuracy.

Unsteady Adjoint of Pressure Loss for a Fundamental Transonic Turbine Vane

High-fidelity simulations, e.g., large eddy simulation (LES), are often needed for accurately predicting pressure losses due to wake mixing and boundary layer development in turbomachinery applications. An unsteady adjoint of high-fidelity simulations is useful for design optimization in such aerodynamic applications. In this paper, we present unsteady adjoint solutions using a large eddy simulation model for an inlet guide vane from von Karman Institute (VKI) using aerothermal objectives. The unsteady adjoint method is effective in capturing the gradient for a short time interval aerothermal objective, whereas the method provides diverging gradients for long time-averaged thermal objectives. As the boundary layer on the suction side near the trailing edge of the vane is turbulent, it poses a challenge for the adjoint solver. The chaotic dynamics cause the adjoint solution to diverge exponentially from the trailing edge region when solved backward in time. This results in the corruption of the sensitivities obtained from the adjoint solutions. An energy analysis of the unsteady compressible Navier–Stokes adjoint equations indicates that adding artificial viscosity to the adjoint equations can dissipate the adjoint energy while potentially maintaining the accuracy of the adjoint sensitivities. Analyzing the growth term of the adjoint energy provides a metric for identifying the regions in the flow where the adjoint term is diverging. Results for the vane obtained from simulations performed on the Titan supercomputer are demonstrated.