Search for: All records

In recent years, there has been a growing interest in low-Reynolds-number, unsteady flight applications, leading to renewed scrutiny of the Kutta condition. As an alternative, various methods have been proposed, including the combination of potential flow with the triple-deck boundary layer theory to introduce a viscous correction for Theodorsen's unsteady lift. In this research article, we present a dynamical system approach for the total circulatory unsteady lift generation of a pitching airfoil. The system's input is the pitching angle, and the output is the total circulatory lift. By employing the triple-deck boundary layer theory instead of the Kutta condition, a new nonlinearity in the system emerges, necessitating further investigation to understand its impact on the unsteady lift model. To achieve this, we utilize the describing function method to represent the frequency response of the total circulatory lift. Our analysis focuses on a pitching flat plate about the mid-chord point. The results demonstrate that there is an additional phase lag due to viscous effects, which increase as the reduced frequency increases, the Reynolds number decreases, and/or the pitching amplitude increases. 

In this paper we revive a special, less-common, variational principle in analytical mechanics (Hertz’ principle of least curvature) to develop a novel variational analogue of Euler's equations for the dynamics of an ideal fluid. The new variational formulation is fundamentally different from those formulations based on Hamilton's principle of least action. Using this new variational formulation, we generalize the century-old problem of the flow over a two-dimensional body; we developed a variational closure condition that is, unlike the Kutta condition, derived from first principles. The developed variational principle reduces to the classical Kutta–Zhukovsky condition in the special case of a sharp-edged airfoil, which challenges the accepted wisdom about the Kutta condition being a manifestation of viscous effects. Rather, we found that it represents conservation of momentum. Moreover, the developed variational principle provides, for the first time, a theoretical model for lift over smooth shapes without sharp edges where the Kutta condition is not applicable. We discuss how this fundamental divergence from current theory can explain discrepancies in computational studies and experiments with superfluids. 

In this paper, we started by summarizing our recently developed viscous unsteady theory based on coupling potential flow with the triple deck boundary layer theory. This approach provides a viscous extension of potential flow unsteady aerodynamics. As such, a Reynolds- number-dependence could be determined. We then developed a finite-state approximation of such a theory, presenting it in a state space model. This novel nonlinear state space model of the viscous unsteady aerodynamic loads is expected to serve aerodynamicists better than the classical Theodorsen’s model, as it captures viscous effects (i.e., Reynolds number dependence) as well as nonlinearity and additional lag in the lift dynamics; and allows simulation of arbitrary time-varying airfoil motions (not necessarily harmonic). Moreover, being in a state space form makes it quite convenient for simulation and coupling with structural dynamics to perform aeroelasticity, flight dynamics analysis, and control design. We then proceeded to develop a linearization of such a model, which enables analytical results. So, we derived an analytical representation of the viscous lift frequency response function, which is an explicit function of, not only frequency, but also Reynolds number. We also developed a state space model of the linearized response. We finally simulated the nonlinear and linear models to a non- harmonic, small-amplitude pitching maneuver at 100 , 000 Reynolds number and compared the resulting lift and pitching moment with potential flow, in reference to relatively higher fidelity computations of the Unsteady Reynolds-Averaged Navier-Stokes equations.