For certain Reynolds numbers, airfoils are subject to sporadic high-amplitude fluctuations in the aerodynamic forces. These extreme excursions may be seen as prototypical of the kind of unsteady and intermittent dynamics relevant to the flow around airfoils and wings in a variety of real-world applications. Here we investigate the instability mechanisms at the heart of these extreme events, and how they may be harnessed for efficient data-driven forecasting. Through a wavelet and spectral analysis of the pressure and vorticity, we find that the extreme events arise due to the instability of a specific frequency component distinct from the vortex shedding mode. During these events, this extreme event frequency draws energy from the energetically dominant vortex shedding flow and undergoes an abrupt transfer of energy from small to large scales. We propose a preprocessing algorithm to extract this extreme event frequency from the surface pressure data, which in conjunction with an extreme event-tailored loss function, allows us to avoid the commonly used long short-term memory architecture in favor of a simple feed-forward network—a significant reduction in cost over the previous state-of-the-art. Our model requires only three pressure sensors, and it is robust to their location—showing promise for the use of our model in dynamically varying applications. Finally, we show that relying solely on the statistics of the pressure and drag data for optimal sensor placement fails to improve model prediction over uniform or random sensor placement.
The conceptual picture underlying resolvent analysis(RA) is that the nonlinear term in the Navier-Stokes(NS) equations acts as an intrinsic forcing to the linear dynamics, a description inspired by control theory. The inverse of the linear operator, defined as the resolvent, is interpreted as a transfer function between the forcing and the velocity response. From a theoretical point of view this is an attractive approach since it allows for the vast mathematical machinery of control theory to be brought to bear on the problem. However, from a practical point of view, this is not always advantageous. The inversion of the linear operator inherent in the control theoretic definition obscures the physical interpretation of the governing equations and is prohibitive to analytical manipulation, and for large systems leads to significant computational cost and memory requirements. In this work we suggest an alternative, inverse free, definition of the resolvent basis based on an extension of the Courant-Fischer-Weyl min-max principle in which resolvent modes are defined as stationary points of a constrained variational problem. This definition leads to a straightforward approach to approximate the resolvent (response) modes of complex flows as expansions in any arbitrary basis. The proposed method avoids matrix inversions and requires only the spectral decomposition of a matrix of significantly reduced size as compared to the original system. To illustrate this method and the advantages of the variational formulation we present three examples. First, we consider streamwise constant fluctuations in turbulent channel flow where an asymptotic analysis allows us to derive closed form expressions for the optimal resolvent modes. Second, to illustrate the cost saving potential, and investigate the limits, of the proposed method we apply our method to both a 2-dimensional, 3-component equilibrium solution in Couette flow and, finally, to a streamwise developing turbulent boundary layer. For these larger systems we achieve a model reduction of up to two orders of magnitude. Such savings have the potential to open up RA to the investigation of larger domains and more complex flow configurations.
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