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Nonlinear Vibration
Published in Haym Benaroya, Mark Nagurka, Seon Han, Mechanical Vibration, 2017
Haym Benaroya, Mark Nagurka, Seon Han
She worked closely with the leading German aerodynamicists of the time, Ludwig Prandlt and Albert Betz, director of the institute. Before Lotz’s arrival, Prandlt had been working on the equation for his lifting line theory for the spanwise lift distribution of an airplane wing. Applying her mathematical skills, Lotz solved the equation, and developed a relatively convenient method for practical use. Lotz was then promoted to head of this dominant group dealing with aerodynamics. In 1938, Irmgard married Dr. Wilhelm Flügge, a civil engineer, and the pair moved first to Berlin and later to the small town of Saulgau. He had accepted a position as a department head at the Deutche Versuchsanstalt fur Luftfahrt (DVL) in Berlin.
Force-System Resultants and Equilibrium
Published in Richard C. Dorf, The Engineering Handbook, 2018
The induced drag of a wing depends on the downwash distribution along its span. It can be shown that the downwash distribution depends on the lift distribution across the wing span. Lift per unit span can vary for several reasons: (1) changes in chord, (2) changes in angle of attack, and (3) variations in airfoil shape. An elliptical distribution of lift across the span, as shown in Figure 193.12, provides a minimum induced drag. Lifting-line theory can be used to predict the downwash for an elliptical lift
Three-dimensional aerodynamics
Published in Martin O. L. Hansen, Aerodynamics of Wind Turbines, 2015
In the lifting line theory it is assumed that the three-dimensionality is limited to the downwash, i.e. the spanwise flow is still small compared to the streamwise velocity and 2-D data can therefore be used locally if the geometric angle of attack is modified by the downwash. This assumption is reasonable for long slender wings such as on a glider plane or a wind turbine. One method to determine the value of the vortices quantitatively, and thus the induced velocities, is Multhopp’s solution of Prandtl’s integral equation. This method is thoroughly described in Schlichting and Truckenbrodt (1959) and will not be shown here, but it is important to understand that the vortex system produced by a three-dimensional wing changes the local inflow conditions seen by the wing, i.e. although the flow is locally 2-D one cannot apply the geometric angle of attack when estimating the forces on the wing. This error was made in early propeller theory and the discrepancy between measured and computed performance was believed to be caused by incorrect 2-D airfoil data. On a rotating blade Coriolis and centrifugal forces play an important role in separated boundary layers, i.e. after stall. In a separated boundary layer the velocity and thus the momentum is relatively small compared to the centrifugal force, which therefore starts to pump fluid in the spanwise direction towards the tip. When the fluid moves radially towards the tip, the Coriolis force points towards the trailing edge and acts as a favourable pressure gradient. The effect of the centrifugal and Coriolis force is to alter the 2-D airfoil data after stall. Whenever such data are needed, for example to compute the performance of a wind turbine at high wind speeds, much engineering skill and experience is needed to construct such post-stall data in order to obtain an acceptable result, see also Snel et al. (1993), and Chaviaropoulos and Hansen (2000).
Vorticity Confinement Applied to Accurate Prediction of Convection of Wing Tip Vortices and Induced Drag
Published in International Journal of Computational Fluid Dynamics, 2021
Alex Povitsky, Kristopher C. Pierson
Table 2 shows that surface integration consistently over-predicted the drag force, sometimes causing the drag coefficient to be over two times the value obtained by lifting line theory. The wake-integral method consistently under-predicted the drag force but with significantly less error in comparison to surface integration. The drag coefficient was typically under-predicted by 25%, with higher accuracy at lower Mach numbers and lower accuracy at higher Mach numbers. Lifting line theory generally does not include effects such as the non-uniform downwash of an elliptically loaded wing and the non-planar character of the wake shed from a curved trailing edge (Sears 1974; Sears 1990). These effects become more pronounced at larger Mach numbers that contribute to the discrepancy between far-field integration results and those obtained by lifting line theory.
Optimal Wing and Horizontal Tail Plane Design for Maximizing the Aircraft Performance in Cruise Flight
Published in Cybernetics and Systems, 2023
Rubén Ferrero-Guillén, Javier Díez-González, José Manuel Alija, Alberto Martínez-Gutiérrez, Paula Verde, Hilde Perez
There are many different methodologies to obtain the lift distribution (Multhopp, 1950; Gabor, Koreanschi, and Botez 2016). Among them, the Prandtl Lifting-Line Theory (Sivells and Neely 1947) is one of the most rounded and expanded techniques for the initial steps of design of the wing and HTP in aircraft projects. Even, despite being a traditional theory, it is still being used and codified in CFD simulations (Phillips and Snyder 2000).