Multi-fidelity modeling framework for nonlinear unsteady aerodynamics of airfoils

Aerodynamic data can be obtained from different sources, which vary in fidelity, availability and cost. As the fidelity of data increases, the cost of data acquisition usually becomes higher. Therefore, to obtain accurate unsteady aerodynamic model with very low cost and the desired level of accuracy, this paper proposes an unsteady multi-fidelity aerodynamic modeling framework. The approach integrates ideas from data fusion, multi-fidelity modeling, nonlinear system identification and machine learning. Data fusion reduces the total cost of data generation for model construction, while multi-fidelity modeling with a nonlinear autoregressive with exogenous input (NARX) description provides a general framework for unsteady aerodynamics. The correction term from the low-fidelity model to the high-fidelity result is then identified by a machine learning approach, i.e., a multi-kernel neural network. To validate the proposed method, unsteady aerodynamics of a NACA0012 airfoil pitching at Mach number 0.8 is modeled. The high-fidelity data is obtained from a Navier–Stokes-equation-based solver, while the low-fidelity solution is taken from an Euler-equation-based flow solver. The main difference between two types of data is that the high-fidelity solution takes into account the viscous effect, while the low-fidelity solution is based the invisicid flow assumption. Besides, to mimic the practical situation where high-fidelity data are limited in amount and diversity due to high cost (e.g., the experimental condition), only three high-fidelity unsteady aerodynamic solutions from harmonic motion are available. After performing a multi-fidelity analysis on a typical harmonic motion, the model is applied to the prediction of aerodynamic loads from either new harmonic motions or random motions. The multi-fidelity model shows a good agreement with the high-fidelity solution, indicating that by using only a few high-fidelity data and a low-fidelity model, high-fidelity results can be accurately reproduced. Furthermore, the model convergence with respect to increasing training data, and the comparison with a single high-fidelity reduced-order model (ROM) are also studied. The proposed approach becomes more accurate as the number of high-fidelity samples increases, and outperforms a single aerodynamic ROM in most of test cases. Compared with ROM method, additional computational cost for the proposed approach is small, therefore the total time cost of model training is still low.