If the computer had been invented and used for computational fluid dynamics at the same time as the first successful flights of the powered aircraft, we would likely have missed out on many fundamental theories that have shaped our understanding of aircraft design. Luckily, early aerodynamicists did not have access to a computer, and therefore had to manipulate the governing equations of fluid motion the Navier-Stokes and Euler Equations to estimate aerodynamic properties such as lift and drag.
Through their manipulations and ingenuity, they discovered key concepts that have shaped our aircraft for the past century. Had we started first with CFD, we may never have discovered these basic relationships that are fundamental to wing and aircraft design.
The predominant theory for wing aerodynamics is called Lifting-Line Theory, first published by Ludwig Prandtl in Many scholars and researchers have since studied the theories, and results of this theory are impressively accurate and simple.
Induced drag is a quadratic function of lift. The elliptic lift distribution is the lift distribution that minimizes induced drag. The elliptic wing is the planform that minimizes induced drag with zero twist. Linearly tapered wings of taper ratio near 0. Lift on a finite wing is very nearly a linear function of angle of attack, even after adding in the complexities of downwash! The optimum amount of twist on a wing is directly proportional to lift coefficient. Washout can reduce the induced drag for a given lift coefficient and wing geometry.
Any wing can be twisted to produce an elliptic lift distribution. The twist distribution needed to produce an elliptic lift distribution is a simple function of span and can be computed for any planform.
Once structural weight for supporting the lift distribution is considered, the elliptic lift distribution is NOT the optimum lift distribution to minimize induced drag for a given aircraft weight Imagine trying to learn all of these things from CFD! Nearly impossible. Imagine what else there is to learn out there!
Is prandtl lifting line theory predicts lift distribution over a three dimensional wing? It is also known as the lanchester-prandtl wing theory. The theory was expressed independently by Frederick w. Is lift over each wing segment is correspond?
Instead, this local amount of lift is strongly affected by the lift generated at neighboring wing section. Is it difficult to predict the amount of lift that a wing geometry will generate? The lifting line theory yields the lift distribution along the span-wise direction, based only on the wing geometry and flow conditions.
Is lifting line theory applies to circulation? Is lift distribution over a wing can be modeled with the concept of circulation? A vortex is shed downstream for every span wise change in lift. Modeling the local lift with local circulation allows us to account for the influence of one section over its neighbors. Is vortex filament cannot begin or terminate in the air?
This shed vortex, hose strength is the derivative of the local wing. Is shed vortex can be modeled at vertical velocity? The up wash and downwash induced by the shed vortex can be computed at each neighbor segment. This sideways influence up wash on the outboard, downwash on the in a board.
Is change in lift distribution is known as lift section? Is local induced change the angle of attack on a given section of a wing? In turn, the integral sum of the lift on each down washed wing section is equal to the total desired amount of lift. Is additional term can be added to make an aircraft wing station?
Is lifting line theory applies to circulation?
12 Things we Learn from Lifting-Line Theory
Is lift distribution over a wing can be modeled with the concept of circulation? A vortex is shed downstream for every span wise change in lift. Modeling the local lift with local circulation allows us to account for the influence of one section over its neighbors. Is vortex filament cannot begin or terminate in the air?
Assumptions of the Lifting-Line Theory
This shed vortex, hose strength is the derivative of the local wing. The vortex strength in the trailing sheet will be a function of the changes in vortex strength along the wing span.
The mathematical function describing the vortex sheet strength is thus obtained by differentiating the bound vortex distribution. This final boundary equation contains all the unknown coefficients of the wing model's vortex distribution, along with the wing's geometry and the stream conditions.
Prandtl Lifting Line Tool
It can be used to find coefficients A1,A2, A3, Assuming that higher order coefficients become increasingly small and make negligible contribution to the result, one method of solution is to truncate the series at term AN. This set can be solved for coefficients A1 to AN. A cosine distribution of span-wise locations should be used for the boundary conditions to match the assumed wing loading distribution.
Clearly the number of coefficients used will determine the accuracy of the solution. If the wing loading is highly non-elliptical then a larger number of coefficients should be included.
The fact that the two-dimensional airfoil section is part of a finite wing is manifested by a modification in the angle of attack at which the section is operating. Finally, experimental data for two-dimensional airfoils may be used to predict the viscous drag of the wing.
If the wing is highly swept or of very low AR, then the spanwise-flow component is large, the assumption is invalid, and the simple lifting-line theory breaks down.
250+ TOP MCQs on Prandtl’s Classical Lifting-Line Theory and Answers
The simple theory has been extended to treat swept wings, but this is not discussed here. The method becomes complicated and the problem is solved more easily numerically. Because the lifting line represents the wing, it is positioned at the wing quarter- chord.