### 3D PRANDTL LIFTING LINE THEORY

### Application to high aspect ratio, unswept wings.

A simple solution for unswept three-dimensional wings can be obtained by using Prandtl's lifting line model. For incompressible, inviscid flow, the wing is modelled as a single bound vortex line located at the 1/4 chord position and an associated shed vortex sheet.**Γ**is the circulation (vortex strength), defined a function across the wing span,

**V**is free-stream velocity,

_{∞}**b**is wing span,

**s**is wing semi-span,

**c**is wing chord and

**y**is the distance across the span measured from the wing root. The mapping of angle (

**θ**) to semi-span (

**s**) position is done using Fourier series and allows variation of the model to suit different geometries.

**A**) for a given geometry and set of free-stream conditions can be calculated by applying a surface flow boundary condition. The equation used is based on the usual condition of zero flow normal to the surface.

_{n}**α**is the 3-D wing angle of attack,

**θ**is the wing twist angle and

_{t}**w**is the velocity induced by trailing vortex sheet. The downwash velocity is caused by the shed vorticies trailing behind the 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.

_{i}**r**is the distance across the span between the vortex element and the point at which downwash is being calculated.

**r = ( y – y**). The full downwash will be,

_{i}**a**is the section lift slope (

_{0}**dC**),

_{L}/dα_{2D}**α**is the section zero lift angle and

_{0}**c**is the chord length. This can be rearranged in terms of vortex strength,

**A**,

_{1}**A**,

_{2}**A**, ... 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

_{3}**A**. By applying the boundary condition at

_{N}**N**span locations a set of simultaneous linear equations can be constructed. This set can be solved for coefficients

**A**to

_{1}*. 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. This occurs when analysing wings with part span flaps. This type of geometry causes a discontinuity in the spanwise loading and hence requires a much larger number of coefficients to accurately describe the distribution. Where the the wing loading is symmetric about the wing root, the contribution of even functions will become zero. Coefficients*

**A**_{N}**A**,

_{2}**A**,

_{4}**A**, ... are all zero and can be dropped from the analysis. Once the coefficients of the load distribution are known the total lift of the wing can be found by integration.

_{6}**AR**is the aspect ratio of the wing

### Special Case of Purely Elliptical Wing Loading

If the wing planform is elliptical,**c=c**then it can be shown that the wing load distribution is also a purely elliptical function and all coefficients except

_{0}cos(θ)**A**will vanish.

_{1}**A**and the loading will be approximately elliptical. In this case, a single general boundary condition equation results containing only one unknown, the vortex line strength at the wing root.

_{1}### Reference

*Houghton & Carruthers "Aerodynamics for Engineering Students", Arnold Ed 3 1982.*

### Software : Prandtl Lifting Line Program

The following computer program allows the user to define wing plan-forms (without sweep) and to define wing root and wing tip section properties. The program assumes a linear variation of section properties between wing root and tip. For the MS Windows downloadable version, and that the wing loading is assumed to be symmetric about the wing root.**a**, ie.

_{0}, α_{0}**C**are required for the wing root and the wing tip sections. This information can be obtained from published 2-D experimental data or theoretical techniques such as thin aerofoil theory. The program uses the above lifting line equations to get solutions for lift coefficient versus angle of attack and induced drag coefficient versus lift coefficient

_{L}*(2D)*= a_{0}(α_{2D}- α_{0})^{2}. For a given angle of attack the program will display the resulting distribution of section lift coefficient across the span, the distribution of downwash at the wing and a listing of the solution Fourier coefficients for this angle. A flapped section can also be input. The percentage of wing span with flap must be input to create a flapped wing section. The flap section properties are assumed to be those entered for the wing root section. These section properties will be kept constant across the flapped portion of the span. The section properties used outboard of the flap will also be constant and assumed to be equal to those of the wing tip.

### Software

- Prandtl Lifting Line Solver for 3D unswept wings (MS Windows downloadable)