Using the Matlab program “Lifting_Line_Model_elliptic.m”, you study the effects of the geometrical AoA and aspect ratio AR on the lift and induced drag coefficients and span efficiency of an elliptic wing.
Statement of Work for Final Exam of AAE3610
This is a mini-research project on the application of the lifting-line theory.
- Formulation
Given a wing planform whose chord is given by , geometrical angle of attack (AoA)
, and the zero-lift AoA
,, the fundamental equation of the lifting-line theory is converted to a system of algebraic equations for the coefficients
in the Fourier series for the circulation distribution
, i.e.,
,
where
,
.
and m, n = 1, 2, 3 …. N. Here, the transformation is used to convert
to
(rad).
By selecting N spanwise positions (or N cuts), we calculate the coefficients An (n = 1, 2, 3 … N). Symbolically, the solution is expressed as
,
where is the inverse of the matrix
. After the coefficients An are determined, the circulation distribution
can be calculated. Therefore, the lift coefficient
, the induced drag coefficient
and the span efficiency e can be calculated.
- Programs
There are two Matlab programs in the folder “Matlab_Programs_Aero” on e-learning: “Lifting_Line_Model_elliptic.m” and “Lifting_Line_Model_bird.m”. You will used and adapt these programs to do your project.
The Matlab program “Lifting_Line_Model_elliptic.m” considers an elliptic wing whose chord is given by
where is the wing root chord. The aspect ratio
. For the geometrical AoA
and the zero-lift AoA
, by selecting N spanwise positions
, the coefficients An (n = 1, 2, 3 ….N) are obtained by solving the fundamental equation of lifting line theory for the circulation based on the lifting line theory. The circulation distribution
, chord distribution and planform is plotted. The output data include the lift coefficient, induced drag coefficient, wing aspect ratio AR, and span efficiency.
The Matlab program “Lifting_Line_Model_bird.m” is basically the same as “Lifting_Line_Model_elliptic.m”, except it considers a generic bird wing planform given by
,
where is the wing root chord.
Tasks:
- Using the Matlab program “Lifting_Line_Model_elliptic.m”, you study the effects of the geometrical AoA and aspect ratio AR on the lift and induced drag coefficients and span efficiency of an elliptic wing.
- For a fixed AR, changing the geometrical AoA, you calculate the lift, induced drag coefficients and span efficiency, and plot the results as a function of the geometrical AoA. Discuss the results.
- For a fixed geometrical AoA (for example 3 degrees), changing the AR, you calculate the lift, induced drag coefficients and span efficiency, and plot the results as a function of the AR. Discuss the results.
- You can give a geometrical angle of attack (AoA)
as a function of
that is called the wing twist, and then study of the wing twist. For example, you can choose
(rad),where
is twist at he wing root and
is .is the rate of twist change along the span.
- Using the Matlab program “Lifting_Line_Model_bird.m”, you repeat the above tasks (a), (b) and (c) for a generic bird wing. You discuss the comparisons between the results between the elliptic wing and bird wing.
Delivery::
A technical report including the title, introduction, theory, results (figures and tables) and discussions, and conclusions, references.
The due for the final report will be in the Final Week (The exact date will be set later).
