- Chapter 7 and 8 of « Methodes Numeriques, Algorithmes, analyse et applications ». The book is available on my enseirb account in ~mfaverge/public_html/IS104/Methodes-Numeriques-Algorithmes-analyse-et-applications.pdf
- The slides made to help presenting this chapter here. They are also available on my account at enseirb alongside with the book. Slides 1-30 focus on the integration methods, and slides 31-61 focus on the polynomial interpolation.
In order to interpolate the points of the airfoil into a sufficiently smooth curve, cubic splines are used. Points of the curve are joined by a piecewise-polynomial curve, such that each polynomial has a degree at most 3, and such that the joins between two consecutive polynomials preserve the continuity of the curve, its derivative and its second derivative.
Chapter 3.3 of the Numerical Recipes describes the algorithm computing this sequence of polynomials. Here is an example of spline interpolating the points of the function :
Computing the length of plane curves
Note that it is necessary to compute the derivative of the function , which is relatively easy when has been interpolated by a cubic spline first.
Modelling the airflow
Near the airfoil, the air moves along a curve very similar to the airfoil, whereas when the air is situated further, it moves horizontally, without disturbance. The difficulty here consists in determining how the air flows between these two extreme states.
Let (resp. ) be the minimal (resp. maximal) height of the airfoil. In the following, we suppose that the airflow is disturbed by the wing only in a vertical interval . Out of this interval, , the air flows in a rectilinear way.
Let be the curve representing the upper surface of the wing. The family of curves describing the airflow above the wing are given by the following equations :
For a fixed , this equation defines a curve situated between the upper side of the wing and the maximal altitude beyond which the air is not disturbed by the wing.
The pressure on each slice is computed in the following manner : the air is supposed to flow along the curves on each part of the wing. And the air pressure as a moving fluid can be approximated with the Bernoulli law :
For the sake of simplicity, let’s suppose that is constant on the whole area. The variations of pressure are therefore linked to the variations of speed of the air. Since the air flows in a laminar manner, it takes the same time to pass through the zone (from the leading edge to the trailing edge of the wing), independently of the slice along which it flows. As a consequence, it is possible to compute the speed of the air as a simple function of the length of the slice.
The force induced by the air on the wing is merely due to the static pressure : it is normally weaker on the upper part of the wing than on the lower part. This difference induces a force, named the lift, that sustains the plane in the air during its flight. If we are only interested by dynamic pressure, it is possible to compute a map of this pressure around the wing that looks like the following :
Black areas correspond to zones where the air is not disturbed, and lighter areas correspond to zones where the air must travel along a longer path. It can be notices that the air flows faster above than below.
- Write a function performing the computation of a pressure map around the wing.