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1. Effect of Viscous Unsteady Aerodynamics on Flutter Calculation

   https://doi.org/10.2514/6.2019-2036  

   Taha, Haitham E.; Rezaei, Amir S. (January 2019, AIAA SciTech)

   Many studies over the 1960’s reported failure in predicting accurate flutter boundaries using the classical theory of unsteady aerodynamics even at zero angle of attack and/or lift conditions. Since the flutter phenomenon lies in the intersection between unsteady aerodynamics and structural dynamics, and because the structural dynamics of slender beams can be fairly predicted, it was inferred that the problem stems from the classical theory of unsteady aerodynamics. As a result, a research flurry occurred over the 1970’s and 1980’s investigating such a theory, with particular emphasis on the applicability of the Kutta condition to unsteady flows. There was almost a consensus that the Kutta condition must to be relaxed at high frequencies and low Reynolds numbers, which was also concluded from several recent studies of the unsteady aerodynamics of bio-inspired flight. Realizing that vorticity generation and lift development are essentially viscous processes, we develop a viscous extension of the classical theory of unsteady aerodynamics, equivalently an unsteady extension of the boundary layer theory. We rely on a special boundary layer theory that pays close attention to the details in the vicinity of the trailing edge: the triple deck theory. We use such a theory to relax the Kutta condition and determine a viscous correction to the inviscid unsteady lift. Using the developed viscous unsteady model, we develop a Reynolds-number-dependent lift frequency response (i.e., a viscous extension of Theodorsen’s). It is found that viscosity induces a significant phase lag to the lift development beyond Theodorsen’s inviscid solution, particularly at high frequencies and low Reynolds numbers. Since flutter, similar to any typical hopf bifurcation, is mainly dictated by the phase difference between the applied loads and the motion, it is expected that the viscosity-induced lag will affect the flutter boundary. To assess such an effect, we couple the developed unsteady viscous aerodynamic theory with a structural dynamic model of a typical section to perform aeroelastic simulation and analysis. We compare the flutter boundary determined using the developed viscous unsteady model to that of Theodorsen’s. 

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2. Unsteady Viscous Lift Frequency Response Using The Triple Deck Theory

   https://doi.org/10.2514/6.2018-0038  

   Taha, Haitham E.; Rezaei, Amir S. (January 2018, AIAA Aerospace Sciences Meeting)

   Many reports, studies, and evidences invoke an alternative auxiliary condition to the Kutta condition for the analysis of unsteady flows around wings with sharp edges. Some of these reports, because of the observed discrepancies at zero lift conditions, suggested a fundamental revision to the classical theory of unsteady aerodynamics. That is, since the vorticity generation and lift development are essentially viscous processes, a purely inviscid theory of unsteady aerodynamics might be fundamentally flawed. In this work, an unsteady boundary layer approach (triple deck theory) is adopted to develop a viscous correction to the unsteady lift frequency response; i.e., a Reynolds-number dependent ex- tension of Theodorsen function. The developed model is weakly nonlinear, but so compact and efficient that it may be used in flight dynamics, control, and aeroelastic analyses. It is found that the viscous correction induces a signi cant phase shift to the circulatory lift component, particularly at low Reynolds numbers and high-frequencies, that matches expectations based on previous experimental results. This enhancement in the predicted phase of the unsteady loads is envisaged to boost the current predictability of aeroelastic flutter boundaries. High- fidelity computational fluid dynamic simulations are also carried out to validate the results of the developed theoretical model. 

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3. Extension of the Unsteady Vortex Lattice Method: viscous Oseen vortices and thickness effects

   https://doi.org/10.2514/6.2019-1852  

   Mesalles Ripoll, Pol; Rezaei, Amir S.; Taha, Haitham E. (January 2019, AIAA SciTech)

   In this study, the standard unsteady vortex lattice method (UVLM) is modified by different approaches to accommodate the viscous effects on the flow field and lift force history of a moving airfoil. Firstly, the conventional inviscid vortices are replaced with ones that satisfy Oseen’s approximation of the Navier–Stokes equations. These are known as viscous vortices and are a function of the Reynolds number. Compared to the standard UVLM, the results show a different behavior for the wake deformation at lower Reynolds numbers where the viscous effects are more dominant. The second modification is implemented by an elegant alteration in the conservation of circulation condition based on the fact that the whole domain composed of the fluid and solid can be treated as a single kinematic entity. The no-slip boundary condition enables considering the moving solid vorticity in the Kelvin condition, where the results for the frequency response indicate more phase lag compared to the standard UVLM or potential flow theories (i.e., Theodorsen), which increases at higher reduced frequencies. Furthermore, the effect of the position of the shed vortices is analyzed and seen to affect the amplitude of the resulting lift history. Finally, the no-slip boundary condition is also explicitly considered by implementing source singularities along the surface of a thick airfoil, resulting in a better prediction of the lift magnitude compared to CFD results. The combination of all the aforementioned modifications has the potential to enhance the performance of the UVLM in case of viscous flows at relatively high reduced frequencies. 

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4. State Space Modeling of Viscous Unsteady Aerodynamic Loads

   https://doi.org/10.2514/6.2021-1830  

   Taha, Haitham E.; Rezaei, Amir S. (January 2021, AIAA SciTech)

   null (Ed.)

   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. 

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5. Describing function of the nonlinear dynamics of viscous unsteady lift response for a pitching airfoil

   https://doi.org/10.1063/5.0173643  

   Selim, Y M; Taha, H E; El_Bayoumi, G M (November 2023, Physics of Fluids)

   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. 

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