Calculation of the flow around a parabola skeleton with an angle of attack alpha:
Changes of a lift distribution, caused by a deflection of the red marked flap (with its trailing edge) around 10° downwards
The methods considered here belong to the group of singularity methods. By these an inviscid, irrotational and steady flow around a wing is modeled by the distribution of a system of singularities (sources, drains, dipoles and vortices) over the wing and the superposition of this system and the undisturbed incoming flow.
Due to the linearity of the method's underlying Laplace-equation, the problem of the flow around a wing having an angle of attack can be split up into that of the flow around a skeleton-area having an angle of attack (lift-problem) and that of the flow around a displacement-body having no angle of attack (thickness-problem). If further on only thin profiles, whose expansion in x-direction is much smaller than that in z-direction, are considered, the thickness-problem can be neglected. The theory for the solution of the lift-problem is called skeleton-theory.
Skeleton-theory (see picture)
In skeleton-theory, initially only the two-dimensional airfoil (in the picture shown in black) is considered. Additionally, due to the simplification to thin airfoils, the airfoil can be represented by its skeleton-line (in the picture shown in red) and its chord (in the picture shown in blue). Along the chord a distribution of potential vortices is arranged. The integration of the vortex-densities of these vortices over the chord results in the total circulation from which, according to Joukowsky, the lift of the airfoil can be calculated. If the vortex-density is chosen in accordance with the approach of Ackermann and Birnbaum, it can be seen that the flow around a parabola skeleton having an angle of attack alpha can be replaced by the superposition of the flow around a flat plate having the angle of attack alpha (1st Birnbaum distribution) and the flow around a skeleton in a chord-parallel flow (2nd Birnbaum distribution). Introducing a zero-lift-angle (an angle at which the airfoil produces no lift) of the skeleton alpha_0, the same flow can be expressed by the superposition of the flow around a flat plate having the angle of attack (alpha - alpha_0) and the flow around a skeleton having the angle of attack alpha_0. This second variant has the advantage that only the plate produces lift and the skeleton is only needed to calculate the zero-lift-angle. So, using that variant, it is possible (in the scope of the previously made simplifications) to represent an airfoil by a flat plate having an angle of attack.
Extension to the three-dimensional case
For the extension to the three-dimensional-case, the vortices along the chord are extended perpendicular to their plane of rotation i.e. in spanwise-direction (the former two-dimensional vortex can be seen as a cross section of the emerging vortex-tube). If the wing has a limited span, the vortex has to begin and to end somwhere in spanwise direction. In these points - or more general: at each position where the circulation of the vortex changes - this change has to be transported in x-direction into infinity. So each of the potential vortices, which are arranged along the chord, has to be extended to a vortex-system which is - in the most simple case - similar to a horseshoe. For that reason it is called a horseshoe-vortex. The vortex running along the spanwidth is called a lifting vortex (it produces the lift in superposition with the flow), while the vortices running along the x-direction are producing no lift and so are called non-lifting vortices.
Vortex-lattice and lifting-line methods
Since the distribution of the circulation is not constant in spanwise (y-) direction, several vortices with different circulation have to be set next to each other along the wing span. Thus the wing is finally replaced by a system of discrete vortices (or more pecisely: Horseshoe-vortex-systems) of a constant circulation in x- and y-direction. Due to their appearance, the methods using this arrangement are called vortex-lattice methods.
A fundamental disadvantage of vortex-lattice methods is that they need a relatively large number of vortices along the wing span for obtaining a sufficiently exact value for the wing lift. And even then, the value for the induced drag is determined quite inaccurately. In this matter, an advancement relative to the vortex-lattice methods are the multi-lifting-line methods. Within these, the circulation along each vortex in spanwise direction is not constant, but varies on the basis of a predetermined function. Furthermore, the multi-lifting-line methods allow an arrangement of multiple vortices along the x-direction one after another. In this way, the number of necessary vortices along the wing span can be reduced significantly and the induced drag can be determined much more accurately. On the other hand, the procedure becomes more complex, for two single non-lifting vortices are replaced by a vortex-sheet along the vortex-span, going along the x-axis to infinity.
Singularity arrangement within LIFTING_LINE (see picture)
LIFTING_LINE is such a kind of method, in which the spanwise distribution of circulation of each vortex is described by a quadratic function. The wing is modelled by segmenting its chord-plane into several panels in spanwise and chord direction (upper part of the picture). On each of these panels one of the previously mentioned vortex-systems is arranged in a way, that the lifting vortex lies along the t/4-line of the panel (middle part of the picture). For the determination of the quadratic distribution of the circulation (lower part of the picture), each panel has to fulfill three boundary conditions: One transitional condition at each spanwise border of the panel and the fulfilment of the kinematic flow condition in the (spanwise) middle of the panel at 3/4 of its chord. The transitional conditions typically are that the circulation and its derivative on two neighboring panels shall be equal at the lateral edges or that the circulation shall be zero at the outer edge of the outermost panel. The kinematic flow condition defines the angle with which the flow goes through the panel - quasi its angle of attack as predetermined by the skeleton-theory (see chapter Skeleton-theory above).
Computation of the aerodynamic coefficients within LIFTING_LINE
The main task of LIFTING_LINE is the computation of the aerodynamic coefficients for lift, pitching moment and induced drag. The lift is determined - as already stated in the section on skeleton-theory - by the integration of the circulation over the entire wing; the pitching moment follows directly from this (the lift force acts along the t/4 line of each panel). The induced drag is determined by projecting the circulation into the Trefftz-layer, far behind the wing.
LIFTING_LINE was written in FORTRAN 77 and FORTRAN 90 and can be used basically on all operating systems, as long as the appropriate compilers are available. For the use within automatic parametric variation loops the method was designed in such a way, that allows to control it entirely via its command line parameters and its input file. The input file is based on the ASCII-format and contains the essential geometric data, the transitional conditions and other necessary parameters.
Basically, the geometry of the calculable configurations is almost arbitrary; the maximum number of lifting surfaces is only limited by the array-size, which is predefined at the time of compilation. Problems can arise from an unfavourable coinciding of vortex-singularities and points, where the kinematic flow condition is considered. Other problems can occur dur to numeric difficulties. Furthermore, the general simplifications (thin airfoils, small angles) are to be considered. Beyond that, LIFING_LINE assumes an incompressible flow - if necessary, a compressibility correction (Göthert) with the appropriate geometric modifications may be used.
The results (in particular the integrated aerodynamic coefficients as well as the coefficient-distributions over the wing span) are presented in various output files. In most of the files the input-format of the visualization program TECPLOT of the company AMTEC (in a directly readable ASCII-format) is used.
Example of a lift-distribution, computed with LIFTING_LINE (see picture)
The presented example shows a TECPLOT-visualization of a lift-distribution over the appropriate wing, which was computed using LIFTING_LINE. In the lower picture, the red marked flap is quasi deflected by 10° (with its trailing edge) downwards, as the kinematic flow condition on the appropriate panels was modified by 10°. The black line shows the resulting total lift-distribution over the wing span (along the line where the lift forces act).
 K. H. Horstmann : "Ein Mehrfach-Traglinienverfahren und seine Verwendung für Entwurf und Nachrechnung nichtplanarer Flügelanordnungen"DFVLR-FB 87-51, 1987
With friendly permission of Prof. Dr. Horstmann, the program LIFTING_LINE is made available by the Institute of Aerodynamics and Flow Technology to interested persons. Please adress requests to: Carsten Liersch, DLR AS/KE