## Prandtl Lifting Line Theory (3-D Potential Flow) |

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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.
The span-wise lift distribution is assumed to be elliptical with a small modification due to wing planform geometry. The assumed vortex line strength is thus a Fourier series approximation. The required strength of the distribution coefficients
(A where is the 3-D
wing angle of attack, The vortex strength distribution 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. A solution for the magnitude of the Fourier coefficients
Then find the 2-D section lift coefficient as a function of the local flow incidence and the bound vortex strength at this span location. where a Rearranging and substituting for the local angle of incidence. Substituting for G
and a If a fixed number of coefficients ( A)
is obtained by the reduction of the resulting matrix of equations._{1},A_{2},A_{3},A_{4}...A_{N}It should be noted that in cases were the wing
loading is symmetric then even coefficients ( The lift coefficient for the wing at a given angle of attack will be obtained by integrating the spanwise vortex distribution. so that , where AR is wing aspect ratio. The downwash velocity induced at any span location can be calculated once the strength of the wing loading is known. The variation in local flow angles can then be found. A consequence of this downwash flow is that the direction of action of each section's lift vector is rotated relative to the freestream direction. The local lift vectors are rotated backward and hence give rise to a lift induced drag. By integrating the component of section lift coefficient that acts parallel to the freestream across the span, the induced drag coefficient can be found. so that No real information about pitching moment coefficient can be deduced from lifting line theory since the lift distribution is collapsed to a single line along the 1/4 chord.
## Special Case of Elliptical LoadingIf the wing planform is elliptical, then it can be assumed that the wing load distribution is also a purely elliptical function. In this case a single general boundary condition equation results containing only one unknown, the vortex line strength at the wing root. The exact solution of this equation leads to the following simple answer for lift coefficient and induced drag coefficient. ,
The following computer program allows the user
to define wing planforms (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 and that the loading will be symmetric
about the wing root. 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 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 Executable Program : Lifting-Line (3-D) Theory (249k)
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